Research ArticleBifurcation Analysis for Phage Lambda withBinding Energy Uncertainty

Ning Xu12 Xue Lei3 Ping Ao3 and Jun Zhang12

1 Joint Institute of UMich-SJTU Shanghai Jiao Tong University Shanghai 200240 China2 Key Laboratory of System Control and Information Processing Ministry of Education Shanghai 200240 China3 Shanghai Center for Systems Biomedicine Shanghai Jiao Tong University Shanghai 200240 China

Correspondence should be addressed to Jun Zhang zhangjun12sjtueducn

Received 30 October 2013 Accepted 25 December 2013 Published 3 February 2014

Academic Editor Jose Nacher

Copyright copy 2014 Ning Xu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In a phage 120582 genetic switch model bistable dynamical behavior can be destroyed due to the bifurcation caused by inappropriatelychosen model parameters Since the values of many parameters with biological significance often cannot be accurately acquired itis thus of fundamental importance to analyze how and to which extent the system dynamics is influenced by model parametersespecially those parameters pertaining to binding energies In this paper we apply a Jacobian method to investigate the relationbetween bifurcation and parameter uncertainties on a phage 120582 OR model By introducing bistable range as a measure of systemrobustness we find that RNApolymerase binding energies have theminimumbistable ranges among all the binding energies whichimplies that the uncertainties on these parameters tend to demolish the bistability more easily Moreover parameters describingmutual prohibition between proteins Cro and CI have finite bistable ranges whereas those describing self-prohibition have infinitybistable ranges Hence the former are more sensitive to parameter uncertainties than the latter These results help to understandthe influence of parameter uncertainties on the bifurcation and thus bistability

1 Introduction

Bistability is a salient feature of phage 120582 genetic switchMathematical models with a number of parameters havebeen established to describe its bistable behavior Howeverparameter variation in phage 120582 model may considerablychange the system dynamics Therefore in the case whenthe model parameters are not precisely known it is usefulto study the influence of parameter uncertainty on systemdynamics

Since 1980s researchers have developed several modelsfor phage 120582 genetic switch Reference [1] proposed a quan-titative model for phage 120582 according to the principles ofstatistical thermodynamics The model by [2] included twovariables that is concentrations of proteins CI and CroReference [3] extended this model and pointed out thediscrepancy between experimental and theoretical resultsMoreover models with four variables were built to includemore ingredients such as mRNA concentration in phage 120582[4] Other useful models include Markov chain model [5]

delayed reaction stochastic model [6] fuzzy logic model [7]decision making [8] and qualitative model [9] Based onthesemodels significant research progresses on phage 120582 havebeen made [10ndash15]

Most of these models contain parameters that cannot beprecisely determined For example themodel by [2] hasmorethan 10 binding energy related parameters each of whichmayvary up to plusmn006 kcalmol and thus affects system dynamicsNonspecific binding also introduces additional factors thatinfluence the system models [16] Reference [17] developeda model that has 13 binding energy related parameterswith uncertainty ranges around 05 kcalmol Moreover [18]pointed out that some parameters such as protein synthesisrates can be changed as surrounding environment changesReference [19] reported that single point mutation of DNAleads to plusmn2 kcalmol change for binding free energy Due tothese parameter uncertainties it is often desired to analyzethe effect of these parameters on system dynamics

Researchers have been using sensitivity and robustnessto study the influence of parameter uncertainty Sensitivity

Hindawi Publishing CorporationComputational Biology JournalVolume 2014 Article ID 465216 11 pageshttpdxdoiorg1011552014465216

2 Computational Biology Journal

and robustness are closely related because a robust systemis not sensitive to parameter variations [20] By using lysisfrequency as a sensitivity measure [21] simulated a mutationmodel introduced by [22] and concluded that spontaneousswitching from lysogen to lysis depends sensitively on modelparameters By systematically perturbing binding energieswith plusmn1 kcalmol [23] concluded that lysogenic activity atpromoter 119875

119877119872remains stable under this perturbation

In this paper we investigate bifurcation by using bistablerange as a system robustness measure In a typical case phage120582 genetic switch model has three equilibria Two of them arestable corresponding to the lysogenic and lytic states andthe third one is a saddle point When some parameters areperturbed it is possible that one stable equilibrium and thesaddle point coalesce then both of them disappear resultingin the loss of bistable behavior Hence bistable region can bedefined as the parameter region in which bistable behaviorexists [18] The bistable region for a parameter is large if thesystem dynamics is not sensitive to that parameter

We are interested in analyzing bistable ranges for bindingenergy related parameters Some other parameters such asprotein degradation rate and protein synthesis rate withrespect to gene transcription have already been proved thattheir variations may lead to the loss of bistability [3 24]Our calculations are based on a phage 120582 OR model [3]Although DNA looping formed by cooperations betweenthe left operators and the right operators affects the stabilityof the genetic switch [25] the OR model only includesthe effects of the right operators which are believed toplay a critical role for lysogeny maintenance [26] We willcalculate bistable ranges for all binding energies which arecrucial in determining the affinity that proteins bind tophage 120582 genes Furthermore we define Z-shaped bifurcationand calculate Jacobian matrix to study robustness We findout that different binding energy affects lysogeny stabilitydifferently some of them may destroy bistability whereasothers have little influence on system dynamics Thereforewe need to precisely determine the values of those sensitiveparameters in the model because otherwise we may not getan effective bistable switch

2 Phage 120582 Modeling

We first present phage 120582 gene regulatory networks and thenintroduce their mathematical model Lysogeny and lysis aretwo possible states for phage 120582 It is believed that wild-typephage 120582 has quite a stable lysogenic state since lysogeny lossrate is even lower than 10minus8 per cell generation [27] Howevercertain conditions such as the exposure to UV light [28 29]the presence of mitomycin C [30] or the starvation of thehost cells [31] can induce phage 120582 from lysogeny to lysisMathematical models have been built to successfully explainstability and high efficiency of genetic switch

The concentrations of two proteins CI and Cro can serveas an indication of which state phage 120582 is in lysogeny orlysis In lysogeny the concentration of protein CI in the cell issignificantly higher than that of protein Cro whereas in lysis

Gene for CI expression Gene for Cro expression

Cro dimerCI dimer

OR1OR3 OR2middot middot middot

PRM PR

Figure 1 Three binding sites on a DNA segment The sites 1198741198771

1198741198772 and 119874

1198773 can be bound by CI dimer Cro dimer or RNA

polymeraseThis DNA segment consists of promoters119875119877119872

and119875119877 If

the RNA polymerase binds to 1198741198773 cI gene is expressed if the RNA

polymerase binds to1198741198771 and119874

1198772 cro gene is expressedThe amount

of Cro and CI in the cell directly determines which state phage 120582 isin

Table 1 Parameter values for our model

Δ1198661119888= minus120 kcalmoldagger Δ119866

2119888= minus108 kcalmoldagger

Δ1198663119888= minus134 kcalmoldagger Δ119866

1119903= minus125 kcalmolDagger

Δ1198662119903= minus105 kcalmolDagger Δ119866

3119903= minus95 kcalmolDagger

Δ11986612119903

= minus27 kcalmolDagger Δ11986623119903

= minus29 kcalmolDagger

Δ11986611987712

= minus125 kcalmoldagger Δ1198661198773= minus115 kcalmoldagger

119870119903= 556 times 10

minus9Mdagger 119870119888= 326 times 10

minus7Mdagger

119873RNAP = 30 times 10minus8Msect

1205831= 20 times 10

minus2minminus1dagger

1205832= 36 times 10

minus2minminus1dagger 119877 = 199 cal(molsdotK)119879 = 310K 119896 = 111

para

daggerdenotes the parameter values from [14] Daggerfrom [32] sectfrom [21] and parafrom[3]

protein Cro dominates the cell The amount of CI and Crothus implies which state phage 120582 is in

Figure 1 illustrates the gene segment regulating cro andcI gene expression Three binding sites 119874

1198771 1198741198772 and 119874

1198773

control the expression of two kinds of regulatory proteinsThe 119875

119877119872promoter covers the whole binding site 119874

1198773 and

part of 1198741198772 controlling cI gene expression When RNA

polymerase (RNAP) binds to the site1198741198773 the gene expression

is on resulting in the production of protein CI The 119875119877

promoter covers all of 1198741198771 and part of 119874

1198772 controlling

cro expression When RNAP binds to sites 1198741198771 and 119874

1198772

simultaneously the 119888119903119900 gene expression is on The bindingsites may be occupied by either Cro or CI protein UsuallyCro and CI first form their dimers so that they are able tobind to 119874

1198771 1198741198772 and 119874

1198773 prohibiting the gene expressions

on these sitesSeveral phage 120582 models have been established based on

this regulating mechanism In this paper we adopt the well-known two-dimensionalORmodel stated in [3]with updatedparameter values given in Table 1 This model involves three119874119877gene binding sites which are believed to be the most

critical factors for the stability of genetic switch [33] Thistwo-dimensional model is in line with the general biologicalswitch model for two-gene networks [34] To keep it simplewe eliminate the effects from other activities in the cell suchas nonspecific binding [16 17] stochastic gene expression

Computational Biology Journal 3

[35] and intrinsic and extrinsic noises [36] The model isdescribed by the following system equations

119889119873119903

119889119905= 1198601119891119903minus 1205831119873119903

119889119873119888

119889119905= 1198602119891119888minus 1205832119873119888

(1)

where

119891119903= 11987528+ 11987529+ 11987530+ 11987540

119891119888=

33

sum

119894=31

119875119894+ 119896(

40

sum

119895=34

119875119895)

(2)

119875119904=

exp (minusΔ119866119904119877119879)119873

119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

sum40

119898=1exp (minusΔ119866

119898119877119879)119873

119894(119898)

1199032119873119895(119898)

1198882119873119896(119898)

RNAP

(119904 = 1 2 40)

(3)

The subscript 119904 corresponds to 40 binding configurationsshown in Table 2 The conventions between CI dimer (119873

1199032)

Cro dimer (1198731198882) and their monomers (119873

119903and119873

119888) are

1198731199032=1

2119873119903+1

8119870119903minus1

8radic1198702119903+ 8119873119903119870119903

1198731198882=1

2119873119888+1

8119870119888minus1

8radic1198702119888+ 8119873119888119870119888

(4)

From (1)ndash(4)119873119903119873119888represent concentrations of proteins

CI and Cro The parameters 1198601and 119860

2are synthesis rates of

CI and Cro respectively 1205831and 120583

2are the overall rates that

decrease protein concentration which are caused by normalprotein degradation and cell growth [37]The function 119891

119903(or

119891119888) represents the overall probability that the 119875

119877(or 119875119877119872

)promoter is occupied by RNAP 119875

119904is the probability that

119904th configuration occurs and Δ119866119904is the total binding energy

needed for this configuration The exponents 119894(119904) 119895(119904) and119896(119904) represent the numbers of CI dimers Cro dimers andRNAP at 119904th configuration respectively Δ119866

119904is calculated

by summing up the needed single binding energies listed inTable 1 including cooperation energy For example Δ119866

5=

Δ11986612119903

+Δ1198661119903+Δ1198662119903 The left right arrows in Table 2 indicate

the cooperations 119896 is the coefficient for positive feedbackWhen CI dimer binds to 119874

1198772 positive feedback makes the

transcription of cI gene more efficient hence promotes itsexpression 119870

119903 119870119888are dissociation constants for CI and Cro

proteins respectively 119877 represents the universal gas constantand 119879 the absolute temperature

3 Bifurcation in Phage 120582

The model used in this paper is two-dimensional nonlinearordinary differential equations To get all the equilibria we setthe left hand side of (1) to zero and solve the transcendentalequations It can be found that in general the model has threeequilibria However the number of equilibria may vary dueto bifurcation under parameter variation In phage 120582 model

Table 2 40 Binding configurations for the three sites

119904 1198741198773 119874

1198772 119874

1198771

12 CI23 CI24 CI25 CI2harr CI26 CI2 CI27 CI2harr CI28 CI2harr CI2harr CI29 Cro210 Cro211 Cro212 Cro2 Cro213 Cro2 Cro214 Cro2 Cro215 Cro2 Cro2 Cro216 Cro2 CI217 Cro2 CI218 Cro2 Cro2 CI219 Cro2 CI220 CI2 Cro221 Cro2 CI2 Cro222 CI2 Cro223 CI2 Cro224 CI2 Cro2 Cro225 Cro2 CI2harr CI226 CI2 Cro2 CI227 CI2harr CI2 Cro228 RNAP29 CI2 RNAP30 Cro2 RNAP31 RNAP CI232 RNAP CI2harr CI233 RNAP CI2 Cro234 RNAP35 RNAP CI236 RNAP Cro237 RNAP Cro238 RNAP Cro2 CI239 RNAP Cro2 Cro240 RNAP RNAP

frequently occurring is the saddle node bifurcation whoseprecise mathematical definition is given as follows [38]

Definition 1 (saddle node bifurcation) A saddle node bifur-cation in dynamical system takes place when two equilibriumpoints coalesce and disappear

In (1) two parameters 1198601and 119860

2are not specified

in Table 1 They represent proportional constants relatingthe transcription initiation rate to the absolute rate of CI and

4 Computational Biology Journal

CA CBAB

Nr at equilibrium Nc at equilibrium

A2 A2

Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860

2 This is a schematic plot and not drawn to scale

Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860

2 Set 119860

1= 8 times

10minus9Mmin and divide the whole plane into three regions

as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860

2is small in region A for example 119860

2asymp

5times10minus9Mmin there exists only one stable point and thus one

single curve segment for both 119873119903and 119873

119888 At the boundary

of regions A and B saddle node bifurcation occurs 1198602

increases to region B for example a typical value 1198602asymp

5 times 10minus8Mmin and two new equilibria (one saddle and

one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1

At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860

2asymp 5 times 10

minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860

2increases the saddle pointmoves toward the

lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2

We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]

Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the

phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909

0)119911 near 119909

0if none of the eigenvalues

of 119869(1199090) are on the 119895120596 axis

For a given equilibrium 1199090 in most cases we can

determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909

0) For a two-dimensional system its

Jacobian matrix can be calculated as

119869 =

[[[[

[

1205971198921

120597119873119903

1205971198921

120597119873119888

1205971198922

120597119873119903

1205971198922

120597119873119888

]]]]

]

(5)

The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]

Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911

dagger 120583 = 120583dagger only

if the following relations are satisfied

119892 (119911dagger 120583dagger) = 0 (6)

10038161003816100381610038161003816100381610038161003816

120597119892

120597119911(119911dagger 120583dagger)

10038161003816100381610038161003816100381610038161003816= 0 (7)

From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860

2 the system in (1)

can be rewritten as

119889119873119903

119889119905= 1198921(119873119888 119873119903 1198602)

119889119873119888

119889119905= 1198922(119873119888 119873119903 1198602)

(8)

From Theorems 2 and 3 for a given 1198602 the Jacobian matrix

is evaluated at equilibrium points 119873dagger119888and 119873

dagger

119903 These two

equilibrium points can be represented as a function of 1198602as

shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen

Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin

To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860

2 The Jacobian matrix

can be calculated analytically (see the Appendix for details)By solving the value of 119860

2that makes the determinant of

the Jacobian matrix vanish we get the critical value of 1198602

at which bifurcation occurs When 1198602is small (less than

56 times 10minus9Mmin in Figure 3) there are only two negative

eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]

As 1198602exceeds 56 times 10

minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860

2continues

increasing only one equilibrium point is left (1198602greater than

34 times 10minus7Mmin in this case) and both of its eigenvalues

are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation

Computational Biology Journal 5

0

002

004

Eige

nval

ues o

f Jac

obia

n

At equilibrium for lysogenAt equilibrium for lysisAt saddle point

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

minus012

minus01

minus008

minus006

minus004

minus002

A2 (Mmin)

Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860

1fixed at

8 times 10minus9Mmin and 119860

2varying The solid line (black) shows

the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle

4 Bifurcation Analysis of BindingEnergy Uncertainty

In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866

1119903and Δ119866

2119888represents Gibbs free

energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860

1and 119860

2at appropriate

values 1198601= 8 times 10

minus9Mmin and 1198602= 5 times 10

minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873

119903and119873

119888as functions of Δ119866rsquos

119873lowast

119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

119873lowast

119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

(9)

where 119873lowast119903and 119873lowast

119888are the concentrations of proteins CI and

Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by

119878119873119903

(Δ119866119904) =

1205971198911

120597Δ119866119904

119878119873119888

(Δ119866119904) =

1205971198912

120597Δ119866119904

(10)

where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the

numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation

Since there are ten differentΔ119866s we divide them into fourgroups

(1) Δ1198661119903 Δ1198662119903 and Δ119866

3119903 binding free energies of CI

(2) Δ1198661119888 Δ1198662119888 and Δ119866

3119888 binding free energies of Cro

(3) Δ11986612119903

and Δ11986623119903

the cooperation between proteinsCIrsquos if both 119874

1198771 and 119874

1198772 or 119874

1198772 and 119874

1198773 are bound

by CI dimers(4) Δ119866

11987712and Δ119866

1198773 RNAP binding energies

We then study each of these four groups

41 Δ1198661119903 Δ1198662119903 and Δ119866

3119903 We systematically perturb the

values of Δ1198661119903 Δ1198662119903 and Δ119866

3119903to investigate their impact

on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes

Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and

Δ1198663119903

variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866

119904 We find that

the energy Δ1198663119903does not show Z-shaped bifurcation while

the other two parameters Δ1198661119903and Δ119866

2119903show quite similar

behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866

1119903and Δ119866

2119903

play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874

1198771

and 1198741198772 When either of the two sites is occupied by protein

dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866

3119903indicates the

likelihood of CI dimer binding to the site1198741198773 If the absolute

value of Δ1198663119903is too large it is easy for CI dimer to bind to

1198741198773 thus cI gene expression is prohibited resulting in lysis

If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation

vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866

3119903is less than its nominal value in

Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866

3119903 Comparedwith other parameters the

system dynamics is more robust to Δ1198663119903variation

Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866

1119903is less than the nominal value

6 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1r

(a) Z-shaped bifurcation for Δ1198661119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2r

(b) Z-shaped bifurcation for Δ1198662119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3r

(c) Bifurcation for Δ1198663119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1c

(d) Bifurcation for Δ1198661119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2c

(e) Bifurcation for Δ1198662119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3c

(f) Z-shaped bifurcation for Δ1198663119888

Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866

3119888

All vertical axes are plotted on a log scale

at minus1133 kcalmol or Δ1198662119903

is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866

3119903 however lysogenic state always exists if it is

greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874

1198771 and119874

1198772 helps to keep the DNA

of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874

1198771 and 119874

1198772

become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

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BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Computational Biology Journal 3

[35] and intrinsic and extrinsic noises [36] The model isdescribed by the following system equations

119889119873119903

119889119905= 1198601119891119903minus 1205831119873119903

119889119873119888

119889119905= 1198602119891119888minus 1205832119873119888

(1)

where

119891119903= 11987528+ 11987529+ 11987530+ 11987540

119891119888=

33

sum

119894=31

119875119894+ 119896(

40

sum

119895=34

119875119895)

(2)

119875119904=

exp (minusΔ119866119904119877119879)119873

119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

sum40

119898=1exp (minusΔ119866

119898119877119879)119873

119894(119898)

1199032119873119895(119898)

1198882119873119896(119898)

RNAP

(119904 = 1 2 40)

(3)

The subscript 119904 corresponds to 40 binding configurationsshown in Table 2 The conventions between CI dimer (119873

1199032)

Cro dimer (1198731198882) and their monomers (119873

119903and119873

119888) are

1198731199032=1

2119873119903+1

8119870119903minus1

8radic1198702119903+ 8119873119903119870119903

1198731198882=1

2119873119888+1

8119870119888minus1

8radic1198702119888+ 8119873119888119870119888

(4)

From (1)ndash(4)119873119903119873119888represent concentrations of proteins

CI and Cro The parameters 1198601and 119860

2are synthesis rates of

CI and Cro respectively 1205831and 120583

2are the overall rates that

decrease protein concentration which are caused by normalprotein degradation and cell growth [37]The function 119891

119903(or

119891119888) represents the overall probability that the 119875

119877(or 119875119877119872

)promoter is occupied by RNAP 119875

119904is the probability that

119904th configuration occurs and Δ119866119904is the total binding energy

needed for this configuration The exponents 119894(119904) 119895(119904) and119896(119904) represent the numbers of CI dimers Cro dimers andRNAP at 119904th configuration respectively Δ119866

119904is calculated

by summing up the needed single binding energies listed inTable 1 including cooperation energy For example Δ119866

5=

Δ11986612119903

+Δ1198661119903+Δ1198662119903 The left right arrows in Table 2 indicate

the cooperations 119896 is the coefficient for positive feedbackWhen CI dimer binds to 119874

1198772 positive feedback makes the

transcription of cI gene more efficient hence promotes itsexpression 119870

119903 119870119888are dissociation constants for CI and Cro

proteins respectively 119877 represents the universal gas constantand 119879 the absolute temperature

3 Bifurcation in Phage 120582

The model used in this paper is two-dimensional nonlinearordinary differential equations To get all the equilibria we setthe left hand side of (1) to zero and solve the transcendentalequations It can be found that in general the model has threeequilibria However the number of equilibria may vary dueto bifurcation under parameter variation In phage 120582 model

Table 2 40 Binding configurations for the three sites

119904 1198741198773 119874

1198772 119874

1198771

12 CI23 CI24 CI25 CI2harr CI26 CI2 CI27 CI2harr CI28 CI2harr CI2harr CI29 Cro210 Cro211 Cro212 Cro2 Cro213 Cro2 Cro214 Cro2 Cro215 Cro2 Cro2 Cro216 Cro2 CI217 Cro2 CI218 Cro2 Cro2 CI219 Cro2 CI220 CI2 Cro221 Cro2 CI2 Cro222 CI2 Cro223 CI2 Cro224 CI2 Cro2 Cro225 Cro2 CI2harr CI226 CI2 Cro2 CI227 CI2harr CI2 Cro228 RNAP29 CI2 RNAP30 Cro2 RNAP31 RNAP CI232 RNAP CI2harr CI233 RNAP CI2 Cro234 RNAP35 RNAP CI236 RNAP Cro237 RNAP Cro238 RNAP Cro2 CI239 RNAP Cro2 Cro240 RNAP RNAP

frequently occurring is the saddle node bifurcation whoseprecise mathematical definition is given as follows [38]

Definition 1 (saddle node bifurcation) A saddle node bifur-cation in dynamical system takes place when two equilibriumpoints coalesce and disappear

In (1) two parameters 1198601and 119860

2are not specified

in Table 1 They represent proportional constants relatingthe transcription initiation rate to the absolute rate of CI and

4 Computational Biology Journal

CA CBAB

Nr at equilibrium Nc at equilibrium

A2 A2

Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860

2 This is a schematic plot and not drawn to scale

Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860

2 Set 119860

1= 8 times

10minus9Mmin and divide the whole plane into three regions

as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860

2is small in region A for example 119860

2asymp

5times10minus9Mmin there exists only one stable point and thus one

single curve segment for both 119873119903and 119873

119888 At the boundary

of regions A and B saddle node bifurcation occurs 1198602

increases to region B for example a typical value 1198602asymp

5 times 10minus8Mmin and two new equilibria (one saddle and

one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1

At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860

2asymp 5 times 10

minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860

2increases the saddle pointmoves toward the

lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2

We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]

Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the

phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909

0)119911 near 119909

0if none of the eigenvalues

of 119869(1199090) are on the 119895120596 axis

For a given equilibrium 1199090 in most cases we can

determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909

0) For a two-dimensional system its

Jacobian matrix can be calculated as

119869 =

[[[[

[

1205971198921

120597119873119903

1205971198921

120597119873119888

1205971198922

120597119873119903

1205971198922

120597119873119888

]]]]

]

(5)

The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]

Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911

dagger 120583 = 120583dagger only

if the following relations are satisfied

119892 (119911dagger 120583dagger) = 0 (6)

10038161003816100381610038161003816100381610038161003816

120597119892

120597119911(119911dagger 120583dagger)

10038161003816100381610038161003816100381610038161003816= 0 (7)

From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860

2 the system in (1)

can be rewritten as

119889119873119903

119889119905= 1198921(119873119888 119873119903 1198602)

119889119873119888

119889119905= 1198922(119873119888 119873119903 1198602)

(8)

From Theorems 2 and 3 for a given 1198602 the Jacobian matrix

is evaluated at equilibrium points 119873dagger119888and 119873

dagger

119903 These two

equilibrium points can be represented as a function of 1198602as

shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen

Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin

To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860

2 The Jacobian matrix

can be calculated analytically (see the Appendix for details)By solving the value of 119860

2that makes the determinant of

the Jacobian matrix vanish we get the critical value of 1198602

at which bifurcation occurs When 1198602is small (less than

56 times 10minus9Mmin in Figure 3) there are only two negative

eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]

As 1198602exceeds 56 times 10

minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860

2continues

increasing only one equilibrium point is left (1198602greater than

34 times 10minus7Mmin in this case) and both of its eigenvalues

are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation

Computational Biology Journal 5

0

002

004

Eige

nval

ues o

f Jac

obia

n

At equilibrium for lysogenAt equilibrium for lysisAt saddle point

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

minus012

minus01

minus008

minus006

minus004

minus002

A2 (Mmin)

Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860

1fixed at

8 times 10minus9Mmin and 119860

2varying The solid line (black) shows

the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle

4 Bifurcation Analysis of BindingEnergy Uncertainty

In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866

1119903and Δ119866

2119888represents Gibbs free

energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860

1and 119860

2at appropriate

values 1198601= 8 times 10

minus9Mmin and 1198602= 5 times 10

minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873

119903and119873

119888as functions of Δ119866rsquos

119873lowast

119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

119873lowast

119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

(9)

where 119873lowast119903and 119873lowast

119888are the concentrations of proteins CI and

Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by

119878119873119903

(Δ119866119904) =

1205971198911

120597Δ119866119904

119878119873119888

(Δ119866119904) =

1205971198912

120597Δ119866119904

(10)

where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the

numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation

Since there are ten differentΔ119866s we divide them into fourgroups

(1) Δ1198661119903 Δ1198662119903 and Δ119866

3119903 binding free energies of CI

(2) Δ1198661119888 Δ1198662119888 and Δ119866

3119888 binding free energies of Cro

(3) Δ11986612119903

and Δ11986623119903

the cooperation between proteinsCIrsquos if both 119874

1198771 and 119874

1198772 or 119874

1198772 and 119874

1198773 are bound

by CI dimers(4) Δ119866

11987712and Δ119866

1198773 RNAP binding energies

We then study each of these four groups

41 Δ1198661119903 Δ1198662119903 and Δ119866

3119903 We systematically perturb the

values of Δ1198661119903 Δ1198662119903 and Δ119866

3119903to investigate their impact

on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes

Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and

Δ1198663119903

variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866

119904 We find that

the energy Δ1198663119903does not show Z-shaped bifurcation while

the other two parameters Δ1198661119903and Δ119866

2119903show quite similar

behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866

1119903and Δ119866

2119903

play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874

1198771

and 1198741198772 When either of the two sites is occupied by protein

dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866

3119903indicates the

likelihood of CI dimer binding to the site1198741198773 If the absolute

value of Δ1198663119903is too large it is easy for CI dimer to bind to

1198741198773 thus cI gene expression is prohibited resulting in lysis

If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation

vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866

3119903is less than its nominal value in

Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866

3119903 Comparedwith other parameters the

system dynamics is more robust to Δ1198663119903variation

Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866

1119903is less than the nominal value

6 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1r

(a) Z-shaped bifurcation for Δ1198661119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2r

(b) Z-shaped bifurcation for Δ1198662119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3r

(c) Bifurcation for Δ1198663119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1c

(d) Bifurcation for Δ1198661119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2c

(e) Bifurcation for Δ1198662119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3c

(f) Z-shaped bifurcation for Δ1198663119888

Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866

3119888

All vertical axes are plotted on a log scale

at minus1133 kcalmol or Δ1198662119903

is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866

3119903 however lysogenic state always exists if it is

greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874

1198771 and119874

1198772 helps to keep the DNA

of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874

1198771 and 119874

1198772

become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

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Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

4 Computational Biology Journal

CA CBAB

Nr at equilibrium Nc at equilibrium

A2 A2

Figure 2 Saddle node bifurcation with the variation of Crosynthesis rate 119860

2 This is a schematic plot and not drawn to scale

Cro protein synthesis [3] We use Figure 2 to illustrate Z-shaped bifurcation with the variation of 119860

2 Set 119860

1= 8 times

10minus9Mmin and divide the whole plane into three regions

as shown in Figure 2 labeled by A B and C Initially whenthe value of 119860

2is small in region A for example 119860

2asymp

5times10minus9Mmin there exists only one stable point and thus one

single curve segment for both 119873119903and 119873

119888 At the boundary

of regions A and B saddle node bifurcation occurs 1198602

increases to region B for example a typical value 1198602asymp

5 times 10minus8Mmin and two new equilibria (one saddle and

one stable equilibrium) emerge resulting in three equilibriaNote that this is a saddle node bifurcation with the reverseddirection in comparison to Definition 1

At the boundary of regions B and C saddle nodebifurcation takes place again leading to the vanishing oftwo equilibria (one saddle point and one stable equilibrium)In region C say 119860

2asymp 5 times 10

minus7Mmin only one stableequilibrium remains We notice the motion of the saddlepoint When the saddle point shows up at the boundary ofregions A and B the lysis stable equilibrium appears as wellAs parameter119860

2increases the saddle pointmoves toward the

lysogen stable point and finallymergeswith it at the boundaryof regions B and C Two successive bifurcations form a Z-shaped curve are shown in Figure 2

We have not yet specified the boundaries between theseregions quantitatively Next we solve the values of the bound-aries by using the Jacobian method Let us first introduce thefollowing two theorems [38]

Theorem 2 (Hartman-Grobman Theorem) If the Jacobianof the nonlinear system = 119892(119909) at equilibrium namely119869(1199090) has no zero or purely imaginary eigenvalues then the

phase trajectory of the system = 119892(119909) resembles that of thelinearized system = 119869(119909

0)119911 near 119909

0if none of the eigenvalues

of 119869(1199090) are on the 119895120596 axis

For a given equilibrium 1199090 in most cases we can

determine its dynamical behavior qualitatively by calculatingits Jacobian matrix 119869(119909

0) For a two-dimensional system its

Jacobian matrix can be calculated as

119869 =

[[[[

[

1205971198921

120597119873119903

1205971198921

120597119873119888

1205971198922

120597119873119903

1205971198922

120597119873119888

]]]]

]

(5)

The following theorem gives the necessary conditions forthe critical point that bifurcation occurs [38]

Theorem 3 For any nonlinear system = 119892(119911 120583) the systemgoes through a saddle node bifurcation at 119911 = 119911

dagger 120583 = 120583dagger only

if the following relations are satisfied

119892 (119911dagger 120583dagger) = 0 (6)

10038161003816100381610038161003816100381610038161003816

120597119892

120597119911(119911dagger 120583dagger)

10038161003816100381610038161003816100381610038161003816= 0 (7)

From (6) 119911dagger is an equilibrium point In a planar system(7) is the determinant of system Jacobian When the deter-minant of Jacobian matrix vanishes at least one eigenvalueis zero Jacobian is a characterization of planar systemdynamics For variation of parameter 119860

2 the system in (1)

can be rewritten as

119889119873119903

119889119905= 1198921(119873119888 119873119903 1198602)

119889119873119888

119889119905= 1198922(119873119888 119873119903 1198602)

(8)

From Theorems 2 and 3 for a given 1198602 the Jacobian matrix

is evaluated at equilibrium points 119873dagger119888and 119873

dagger

119903 These two

equilibrium points can be represented as a function of 1198602as

shown in Figure 2ByTheorem 3we know that the bifurcation appearswhen

Jacobian has a zero eigenvalue From Figure 3 we can obtainthe approximate values of critical points by checking thepoints of the eigenvalue curves crossing zero line Thesevalues are 56 times 10minus9Mmin and 34 times 10minus7Mmin

To determine the stability of an equilibrium point wecalculate the eigenvalue of the Jacobian matrix Figure 3 plotsthe eigenvalues as a function of 119860

2 The Jacobian matrix

can be calculated analytically (see the Appendix for details)By solving the value of 119860

2that makes the determinant of

the Jacobian matrix vanish we get the critical value of 1198602

at which bifurcation occurs When 1198602is small (less than

56 times 10minus9Mmin in Figure 3) there are only two negative

eigenvalues corresponding to one equilibrium point Thisequilibrium is thus stable [38]

As 1198602exceeds 56 times 10

minus9Mmin we have the casewhen there are three equilibria and correspondingly sixeigenvalues Among these three equilibria two are stableand the third is unstable (saddle point) This is becausepositive eigenvalue appears at saddle point (see dash-dotline in Figure 3) For a two-dimensional linear system if theJacobian has one positive and one negative eigenvalues at theequilibrium the equilibrium is a saddle point If119860

2continues

increasing only one equilibrium point is left (1198602greater than

34 times 10minus7Mmin in this case) and both of its eigenvalues

are negative The curves in Figure 3 have two intersectionswith zero eigenvalue which indicates that the eigenvalues ofJacobian vanish twice forming a Z-shaped bifurcation

Computational Biology Journal 5

0

002

004

Eige

nval

ues o

f Jac

obia

n

At equilibrium for lysogenAt equilibrium for lysisAt saddle point

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

minus012

minus01

minus008

minus006

minus004

minus002

A2 (Mmin)

Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860

1fixed at

8 times 10minus9Mmin and 119860

2varying The solid line (black) shows

the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle

4 Bifurcation Analysis of BindingEnergy Uncertainty

In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866

1119903and Δ119866

2119888represents Gibbs free

energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860

1and 119860

2at appropriate

values 1198601= 8 times 10

minus9Mmin and 1198602= 5 times 10

minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873

119903and119873

119888as functions of Δ119866rsquos

119873lowast

119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

119873lowast

119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

(9)

where 119873lowast119903and 119873lowast

119888are the concentrations of proteins CI and

Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by

119878119873119903

(Δ119866119904) =

1205971198911

120597Δ119866119904

119878119873119888

(Δ119866119904) =

1205971198912

120597Δ119866119904

(10)

where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the

numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation

Since there are ten differentΔ119866s we divide them into fourgroups

(1) Δ1198661119903 Δ1198662119903 and Δ119866

3119903 binding free energies of CI

(2) Δ1198661119888 Δ1198662119888 and Δ119866

3119888 binding free energies of Cro

(3) Δ11986612119903

and Δ11986623119903

the cooperation between proteinsCIrsquos if both 119874

1198771 and 119874

1198772 or 119874

1198772 and 119874

1198773 are bound

by CI dimers(4) Δ119866

11987712and Δ119866

1198773 RNAP binding energies

We then study each of these four groups

41 Δ1198661119903 Δ1198662119903 and Δ119866

3119903 We systematically perturb the

values of Δ1198661119903 Δ1198662119903 and Δ119866

3119903to investigate their impact

on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes

Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and

Δ1198663119903

variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866

119904 We find that

the energy Δ1198663119903does not show Z-shaped bifurcation while

the other two parameters Δ1198661119903and Δ119866

2119903show quite similar

behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866

1119903and Δ119866

2119903

play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874

1198771

and 1198741198772 When either of the two sites is occupied by protein

dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866

3119903indicates the

likelihood of CI dimer binding to the site1198741198773 If the absolute

value of Δ1198663119903is too large it is easy for CI dimer to bind to

1198741198773 thus cI gene expression is prohibited resulting in lysis

If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation

vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866

3119903is less than its nominal value in

Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866

3119903 Comparedwith other parameters the

system dynamics is more robust to Δ1198663119903variation

Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866

1119903is less than the nominal value

6 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1r

(a) Z-shaped bifurcation for Δ1198661119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2r

(b) Z-shaped bifurcation for Δ1198662119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3r

(c) Bifurcation for Δ1198663119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1c

(d) Bifurcation for Δ1198661119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2c

(e) Bifurcation for Δ1198662119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3c

(f) Z-shaped bifurcation for Δ1198663119888

Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866

3119888

All vertical axes are plotted on a log scale

at minus1133 kcalmol or Δ1198662119903

is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866

3119903 however lysogenic state always exists if it is

greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874

1198771 and119874

1198772 helps to keep the DNA

of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874

1198771 and 119874

1198772

become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

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BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Computational Biology Journal 5

0

002

004

Eige

nval

ues o

f Jac

obia

n

At equilibrium for lysogenAt equilibrium for lysisAt saddle point

10minus9

10minus8

10minus7

10minus6

10minus5

10minus4

minus012

minus01

minus008

minus006

minus004

minus002

A2 (Mmin)

Figure 3 Jacobian eigenvalues at equilibria The figure shows howthe eigenvalues of Jacobian at equilibria change with 119860

1fixed at

8 times 10minus9Mmin and 119860

2varying The solid line (black) shows

the eigenvalues of lysis equilibrium The dashed line (blue) is forlysogeny equilibrium The dot-dashed line (red) is for saddle

4 Bifurcation Analysis of BindingEnergy Uncertainty

In this section we will analyze bifurcation for Δ119866rsquos byusing the Jacobian method In Table 1 the symbol Δ119866 withsubscripts such as Δ119866

1119903and Δ119866

2119888represents Gibbs free

energy required for CI or Cro dimers to bind to the ORsites Because of the difficulty in determining the precisevalues of the binding energy it is useful to study how theequilibria change as these Gibbs free binding energies vary Inthis section we set synthesis rates 119860

1and 119860

2at appropriate

values 1198601= 8 times 10

minus9Mmin and 1198602= 5 times 10

minus8Mmin Tocalculate the equilibria we set the left hand side of (1) to zeroand obtain the expression of119873

119903and119873

119888as functions of Δ119866rsquos

119873lowast

119903= 1198911(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

119873lowast

119888= 1198912(Δ1198661119888 Δ1198662119888 Δ119866

1198773)

(9)

where 119873lowast119903and 119873lowast

119888are the concentrations of proteins CI and

Cro at the equilibrium point Once (9) is obtained we candetermine the quantitative behavior around equilibria whenchanging Δ119866rsquos The relative sensitivities are given by

119878119873119903

(Δ119866119904) =

1205971198911

120597Δ119866119904

119878119873119888

(Δ119866119904) =

1205971198912

120597Δ119866119904

(10)

where the subscript 119904 isin 1119903 2119903 11987712 1198773 Since the systemequilibria cannot be analytically solved we may apply the

numerical procedure to calculate (10) as a measure of param-eter sensitivity As to the study on system robustness theJacobianmethod can be used to study bifurcation introducedby parameter variation

Since there are ten differentΔ119866s we divide them into fourgroups

(1) Δ1198661119903 Δ1198662119903 and Δ119866

3119903 binding free energies of CI

(2) Δ1198661119888 Δ1198662119888 and Δ119866

3119888 binding free energies of Cro

(3) Δ11986612119903

and Δ11986623119903

the cooperation between proteinsCIrsquos if both 119874

1198771 and 119874

1198772 or 119874

1198772 and 119874

1198773 are bound

by CI dimers(4) Δ119866

11987712and Δ119866

1198773 RNAP binding energies

We then study each of these four groups

41 Δ1198661119903 Δ1198662119903 and Δ119866

3119903 We systematically perturb the

values of Δ1198661119903 Δ1198662119903 and Δ119866

3119903to investigate their impact

on the system dynamics Two proper robustness measuresare the range of parameters in which lysogeny exists (stablerange) and inwhich bistable dynamics exists (bistable range)In stable range it is possible to keep phage 120582 dormant andintegrate into the gene of the host On the other hand bistablerange defines the parameter range to keep bistability and themaximal uncertainty tolerance In this case it implies thatthe system favors dynamics with two stable equilibria and asaddle Further if the stochastic force is strong enough phage120582 can switch from the lysogenic stable point to the lytic stablepoint even without the parameter changes

Figures 4(a)ndash4(c) illustrate the impact of Δ1198661119903 Δ1198662119903 and

Δ1198663119903

variations We set parameter values as in Table 1 andfind out all the equilibria when varying oneΔ119866

119904 We find that

the energy Δ1198663119903does not show Z-shaped bifurcation while

the other two parameters Δ1198661119903and Δ119866

2119903show quite similar

behaviors since Figures 4(a) and 4(b) are almost identicallythe sameThe result is not surprising because Δ119866

1119903and Δ119866

2119903

play similar roles in system dynamics These two parametersdetermine the likelihood that CI dimer binds to sites 119874

1198771

and 1198741198772 When either of the two sites is occupied by protein

dimers the expression of 119888119903119900 gene is prohibited resulting inlysogenic stateNonetheless the parameterΔ119866

3119903indicates the

likelihood of CI dimer binding to the site1198741198773 If the absolute

value of Δ1198663119903is too large it is easy for CI dimer to bind to

1198741198773 thus cI gene expression is prohibited resulting in lysis

If the absolute value ofΔ1198663119903is small the Z-shaped bifurcation

vanishes In this case cI gene expression is not prohibited butit has no influence on cro gene expression Bistable behavioralways exists when Δ119866

3119903is less than its nominal value in

Table 1 Without stopping or weakening the expression of crogene single lysogen stable point layout cannot be achievedjust by increasingΔ119866

3119903 Comparedwith other parameters the

system dynamics is more robust to Δ1198663119903variation

Table 3 lists the stable range bistable range and the lengthof bistable range for all tenΔ119866s It can be shownbyTheorem 2that one of the three equilibria is a saddle and the other twoare stable points Jacobian matrix is also used to determinethe boundary of stable and bistable ranges according toTheorem 3 As long as Δ119866

1119903is less than the nominal value

6 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1r

(a) Z-shaped bifurcation for Δ1198661119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2r

(b) Z-shaped bifurcation for Δ1198662119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3r

(c) Bifurcation for Δ1198663119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1c

(d) Bifurcation for Δ1198661119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2c

(e) Bifurcation for Δ1198662119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3c

(f) Z-shaped bifurcation for Δ1198663119888

Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866

3119888

All vertical axes are plotted on a log scale

at minus1133 kcalmol or Δ1198662119903

is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866

3119903 however lysogenic state always exists if it is

greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874

1198771 and119874

1198772 helps to keep the DNA

of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874

1198771 and 119874

1198772

become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

6 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1r

(a) Z-shaped bifurcation for Δ1198661119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2r

(b) Z-shaped bifurcation for Δ1198662119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3r

(c) Bifurcation for Δ1198663119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG1c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG1c

(d) Bifurcation for Δ1198661119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG2c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG2c

(e) Bifurcation for Δ1198662119888

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG3c

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG3c

(f) Z-shaped bifurcation for Δ1198663119888

Figure 4 Movement of equilibria under parameter variation Z-shaped bifurcation can be generated by variation of Δ1198661119903 Δ1198662119903 and Δ119866

3119888

All vertical axes are plotted on a log scale

at minus1133 kcalmol or Δ1198662119903

is less than minus934 kcalmol thesystem always has lysogenic stateThey are critical bifurcationpoints For Δ119866

3119903 however lysogenic state always exists if it is

greater than minus1233 kcalmol Consequently for CI protein alarger binding energy to119874

1198771 and119874

1198772 helps to keep the DNA

of phage 120582 silent in the host The system tends to be unstableif the values of binding energies for CI to 119874

1198771 and 119874

1198772

become small thus forming Z-shaped bifurcation Parametervariations may lead to Z-shaped bifurcation if and only if thebistable range is bounded

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Computational Biology Journal 7

Table 3 Robustness measures of binding energies

Variable Stable Range(kcalmol)

Bistable Range(kcalmol)

Length(kcalmol)

Δ1198661119903 (minusinfin minus1133) (minus1549 minus1133) 416

Δ1198662119903 (minusinfin minus934) (minus1304 minus934) 370

Δ1198663119903 (minus1233 +infin) (minus1233 +infin) infin

Δ1198661119888 (minusinfin +infin) (minus1514 +infin) infin

Δ1198662119888 (minusinfin +infin) (minus1512 +infin) infin

Δ1198663119888 (minus1567 +infin) (minus1567 minus1224) 343

Δ11986612119903 (minusinfin minus155) (minus603 minus155) 448

Δ11986623119903 (minus771infin) (minus771 +infin) infin

Δ11986611987712 (minus1367infin) (minus1367 minus1086) 281

Δ1198661198773 (minusinfin minus1050) (minus1266 minus1050) 216

When the stochastic noise is strong enough the noisemayinduce the system to lytic stateMoreover if free energy variesdue to gene mutation bistable behavior might be destroyedand only one stable equilibrium is left This is what themathematical model implies about the parameter influenceon the system dynamics

42 Δ1198661119888 Δ1198662119888 and Δ119866

3119888 The bifurcation results of Δ119866

1119888

Δ1198662119888 andΔ119866

3119888are shown in Figures 4(d)ndash4(f)The equilibria

behaviors with respect toΔ1198661119888andΔ119866

2119888variation are similar

to Δ1198663119903 Δ1198661119888represents the likelihood that Cro dimer binds

to site 1198741198771 If Cro dimer binds to site 119874

1198771 the expression

of cro is prohibited resulting in only one single lysogenicstable point When Δ119866

1119888or Δ119866

2119888is small the prohibition

effect is weak but this is not enough to drive the systemdynamics to only one single lysis stable point Hence bistablelayout always exists if these two parameters are small Theinfluence ofΔ119866

3119888is similar to the influence ofΔ119866

1119903andΔ119866

2119903

The bistable range of Δ1198663119888

is about 343 kcalmol which iscomparable to those of Δ119866

1119903and Δ119866

2119903

The six parameters we have shown so far are the basicbinding energies The values of Δ119866

3119903 Δ1198661119888 and Δ119866

2119888decide

the likelihood that negative feedback (ie self-prohibition)happens It is known that the sites 119874

1198771 and 119874

1198772 are used

to transcript and then translate protein Cro The site 1198741198773

contributes to the production of protein CI If protein CIbinds to 119874

1198773 or Cro binds to 119874

1198771 and 119874

1198772 they can stop

their own transcription Thus we call the phenomenon neg-ative feedback Figure 4 shows that such negative feedbackcannot take the system to Z-shaped bifurcationThe negativefeedback occurs when the amount of Cro or CI is too largeFor instance the binding between CI dimer and 119874

1198773 readily

occurs when the amount of protein CI is too large On theother hand Δ119866

1119903 Δ1198662119903 and Δ119866

3119888represent the likelihood

of positive feedback (ie mutual prohibition) For instancewith largeΔ119866

3119888 the binding between Cro and119874

1198773 is favored

stopping the expression of cI gene If the value of Δ1198663119903

issmall and the negative feedback is weakened but it is notenough to make the system dynamics change from singlelysis equilibrium to single lysogenic equilibrium Parametersassociated with positive feedback can show Z-shaped bifur-cation whereas parameters associatedwith negative feedback

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG12r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG12r

(a) Z-shaped bifurcation for Δ11986612119903

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔG23r

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔG23r

(b) Bifurcation for Δ11986623119903

Figure 5 Z-shaped bifurcation generated by variation of 11986612119903

and11986623119903The variation ofΔ119866

12119903also leads to Z-shaped bifurcation while

Δ11986623119903

is not sensitive enough to create Z-shaped bifurcation

cannotHence the system ismore robust to negative feedbackparameters than to positive feedback parameters

43 Δ11986612119903

and Δ11986623119903

The parameters Δ11986612119903

and Δ11986623119903

determine the energy difference caused by the cooperationof adjacent binding CI dimers Although [32] discovered thatadjacent Cro dimers also have cooperations we ignore thiscooperation effect since it has a small cooperation energyand focus on CI cooperation in our analysis Figure 5 showsthe bifurcation results for the parameters Δ119866

12119903and Δ119866

23119903

The nominal values of Δ11986612119903

and Δ11986623119903

are close to zeroSince protein cooperation process is an exothermic reactionΔ11986612119903

and Δ11986623119903

must be less than zero We find out thatthe variation of Δ119866

12119903leads to Z-shaped bifurcation whereas

Δ11986623119903

does not The value of Δ11986623119903

affects the expression ofboth cro and cI genes whereas Δ119866

12119903only promotes mutual

prohibition from CI to Cro Δ11986623119903

with large value prohibitsthe binding of RNAP to both sites119874

1198772 and119874

1198773 thus both 119888119868

and 119888119903119900 gene expressions are closed

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

8 Computational Biology Journal

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR12

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR12

(a) Z-shaped bifurcation for Δ11986611987712

10minus6

10minus8

10minus10

10minus12

Nr

Nc

minus20 minus15 minus10 minus5 0

ΔGR3

10minus6

10minus8

10minus10

10minus12

minus20 minus15 minus10 minus5 0

ΔGR3

(b) Z-shaped bifurcation for Δ1198661198773

Figure 6 Z-shaped bifurcation generated by variation of 11986611987712

and1198661198773 Both of them are able to generate Z-shaped bifurcation Their

bistable ranges are obviously smaller than any other parametersTheappearance of ldquoZrdquo shows steep slope

Moreover we know the length of bifurcation range forΔ11986612119903

is 448 kcalmol which is the maximal value among allthe finite values in Table 3 Comparedwith the influence fromother parameters the system is relatively more robust to thecooperation energy variations The bistable range for Δ119866

12119903

is from minus603 kcalmol to minus155 kcalmol (Table 3) In ourmodel if Δ119866

12119903is equal to the center value at minus379 kcalmol

the system is most robust to parameter uncertainty becauseof maximum tolerance of uncertainty This result is con-sistent with the calculation by [39] where they calculatedbiological behaviors in phage 120582 and found that cooperationenergy is minus37 kcalmol in comparison to the in vitro valueminus27 kcalmol used in this work The 10 kcalmol differencebetween these two values might be caused by the loopingeffect [40] Hence the looping effect makes the system morerobust to Δ119866

12119903from this point of view

44 Δ11986611987712

and Δ1198661198773 Figure 6 illustrates the influence of

Δ1198661198773

and Δ11986611987712

variations From Figure 6 we find that Z-shaped bifurcation appears for both parameters Note that the

appearance of Z is different from the previous parameters asdiscussed above As in Table 3 the length of bistable rangeforΔ119866

11987712is 281 kcalmol and forΔ119866

1198773is only 216 kcalmol

which is the minimum among all ten parametersNotice that Δ119866

1198773and Δ119866

11987712are the binding free energies

of RNAP to 119875119877119872

promoter (1198741198773) and to 119875

119877promoter (119874

1198771

and 1198741198772) Because RNAP is necessary for gene expression

the result from RNAP could be the most influential Table 3summarizes all the sensitivity measures for the above tenparameters We observe that the last two parameters have theshortest length of bistable ranges which implies that bistablebehavior of phage 120582 is more sensitive to Δ119866

11987712and Δ119866

1198773

than any other parameters For example the bistable ranges ofΔ1198661119903and Δ119866

2119903are around 4 kcalmol while those of Δ119866

11987712

and Δ1198661198773

are less than 3 kcalmol Parameter uncertaintiesof Δ119866

11987712and Δ119866

1198773most easily destroy bistable behaviorThe

concentrations ofCI andCro at equilibriumpoint also changesignificantly under Δ119866

11987712or Δ119866

1198773uncertainty

45 Discussion From the above analysis we find that thebinding free energies to sites119874

1198771 and119874

1198772 have similar effects

on system dynamics As we have shown in Table 3 the resultsof Δ119866

1119903 Δ1198662119903 Δ1198661119888 and Δ119866

2119888are similar This is mainly

because of the similar status for binding sites 1198741198771 and 119874

1198772

both of which control the expression of Cro proteinMoreover the bistable ranges have infinite length for

Δ1198663119903Δ1198661119888Δ1198662119888 andΔ119866

23119903 All of the above four parameters

do not have Z-shaped bifurcation When the values of thoseparameters become close to zero bistable behavior alwaysexistsTheparametersΔ119866

1119888andΔ119866

2119888can be used tomaintain

lysogenic state when they are less than a critical value becausetheir stable ranges do not have any limits For exampleif we introduce gene mutation to make Δ119866

1119888lower than

minus1514 kcalmol bistable behavior disappears and only onelysogenic state is left Consequently the state of phage 120582

stabilizes at lysogenyAs to the other six parameters with limited bistable ranges

in Table 3 their variations lead to Z-shaped bifurcationsWe are interested in why the original system seems moresensitive to these six parameters The free energies Δ119866

1119903

Δ1198662119903 andΔ119866

12119903indicate the likelihood of protein CI binding

to sites 1198741198771 and 119874

1198772 or the likelihood of mutual prohi-

bition occurring Similarly Δ1198663119888indicates the likelihood of

protein Cro binding to site 1198741198773 and also results in mutual

prohibition Our result has shown that compared with self-prohibition the mutual prohibition is more influential onsystem dynamics The binding energy uncertainty related tomutual prohibition may lead to the loss of bistable behavior

Note that the system dynamics of this model is evenmore sensitive to the parameters Δ119866

11987712and Δ119866

1198773 Since

Δ11986611987712

and Δ1198661198773

have the minimum bistable ranges theuncertainties of these two parameters are more likely tolead to the loss of bistable behavior This is consistent withthe conclusion by [23] that perturbation of RNAP bindingenergy significantly changes promoter activity thus affectinggene expression The binding free energies of RNAP to croand cI promoters play a key role for the generation ofproteins

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Computational Biology Journal 9

In building mathematical models for phage 120582 geneticswitch it is crucial to determine the values for those param-eters that are more sensitive to parameter uncertainties suchas Δ119866

11987712and Δ119866

1198773 because otherwise the system dynamics

may deviate from normal bistable behavior Furthermoreour results provide guidelines for the experimentalists inlaboratory on how to choose those parameters that can altermutations more easily

5 Conclusion

In this paper we studied the robustness of a phage 120582model byinvestigating its bifurcation behavior induced by parameteruncertainties We introduced bistable range as a measure ofrobustness and then applied the Jacobianmethod to calculateit It can be concluded that binding energy uncertaintiesmay destroy bistable behavior especially for those sensitiveparameters such as Δ119866

11987712and Δ119866

1198773 Moreover parameters

describingmutual prohibition can formZ-shaped bifurcation(ie Δ119866

1119903 Δ1198662119903 Δ1198663119888 and Δ119866

12119903) whereas parameters

describing self-prohibition (ie Δ1198663119903 Δ1198661119888 and Δ119866

2119888) can-

not Hence system dynamics is more sensitive to mutualprohibition parameters

There also exist other factors affecting the stability androbustness of a phage 120582 model For example DNA loopingcan promote CI expression in lysogenic state [14] and thushelp phage 120582 keep stable and robust [25] Our future workincludes exploring the bifurcation and system robustnesswith respect to these new factors

Appendix

Calculation of Jacobian

From (1)ndash(5) we may represent system Jacobian by

119869 =[[[

[

1198601

120597119875119903

120597119873119903

minus 1205831

1198601

120597119875119903

120597119873119888

1198602

120597119875119888

120597119873119903

1198602

120597119875119888

120597119873119888

minus 1205832

]]]

]

(A1)

Since 119875119903and 119875

119888are sums of 119875

119904 119904 = 1 2 40 as (2) shows

the key part to evaluate their partial derivatives in (A1) is120597119875119904120597119873119903and 120597119875

119904120597119873119888 Since 119875

119904is a function with respect to

Cro and CI dimers we evaluate the partial derivatives byChain Rule

120597119875119904

120597119873119903

=120597119875119904

1205971198731199032

1198891198731199032

119889119873119903

120597119875119904

120597119873119888

=120597119875119904

1205971198731198882

1198891198731198882

119889119873119888

(A2)

Table 4 Exponents for 30 binding configurations in Table 2

119904 119894(119904) 119895(119904) 119896(119904)

1 0 0 02 1 0 03 1 0 04 1 0 05 2 0 06 2 0 07 2 0 08 3 0 09 0 1 010 0 1 011 0 1 012 0 2 013 0 2 014 0 2 015 0 3 016 1 1 017 1 1 018 1 2 019 1 1 020 1 1 021 1 2 022 1 1 023 1 1 024 1 2 025 2 1 026 2 1 027 1 2 028 0 0 129 1 0 130 0 1 131 1 0 132 2 0 133 1 1 134 0 0 135 1 0 136 0 1 137 0 1 138 1 1 139 0 2 140 0 0 2

where

1198891198731199032

119889119873119903

=1

2(1 minus

119870119903

radic1198702119903+ 8119873119903119870119903

)

1198891198731198882

119889119873119888

=1

2(1 minus

119870119888

radic1198702119888+ 8119873119888119870119888

)

(A3)

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

10 Computational Biology Journal

Thenwe have to deal with other terms in (A2) Note that119875119904is

a fractional function in (3) so we write its numerator119860119904and

denominator 119861 as

119872119904= exp(minus

Δ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)

1198882119873119896(119904)

RNAP (A4)

119863 =

40

sum

119904=1

119872119904 (A5)

The partial derivatives for 119875119904with respect to119873

1198882and119873

1199032are

given by

120597119875119904

1205971198731199032

=120597119872119904

1205971198731199032

1

119863minus

120597119863

1205971198731199032

119872119904

1198632

120597119875119904

1205971198731198882

=120597119872119904

1205971198731198882

1

119863minus

120597119863

1205971198731198882

119872119904

1198632

(A6)

The partial derivative of119872119904can be obtained as

120597119872119904

1205971198731199032

= 119894 (119904) exp(minusΔ119866119904

119877119879)119873119894(119904)minus1

1199032119873119895(119904)

1198882119873119896(119904)

RNAP

120597119872119904

1205971198731198882

= 119895 (119904) exp (minusΔ119866119904

119877119879)119873119894(119904)

1199032119873119895(119904)minus1

1198882119873119896(119904)

RNAP

(A7)

The subscript 119904 corresponds to the 40 binding configurationsThe partial derivative of119863 is exactly the sum of 40119872

119904partial

derivatives From the above derivation we may obtain theJacobian matrix analytically This is quite helpful when wecalculate the eigenvalues of Jacobian matrix at equilibriumpoints By referencing Table 2 we may count the exponents(ie 119894(119904) 119895(119904) and 119896(119904)) in (A4) and list them in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research has been supported by the China National 973Project under Grant no 2010CB529200 the Innovation Pro-gram of Shanghai Municipal Education Commission underGrant no 11ZZ20 the Shanghai Pujiang Program underGrant no 11PJ1405800 NSFC under Grant no 61174086 andProject-sponsored by SRF for ROCS SEM

References

[1] G K Ackers A D Johnson and M A Shea ldquoQuantitativemodel for gene regulation by 120582 phage repressorrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 79 no 4 I pp 1129ndash1133 1982

[2] M A Shea and G K Ackers ldquoThe OR control system ofbacteriophage lambda A physical-chemical model for generegulationrdquo Journal of Molecular Biology vol 181 no 2 pp 211ndash230 1985

[3] J Reinitz and J R Vaisnys ldquoTheoretical and experimentalanalysis of the phage lambda genetic Switch implies missinglevels of co-operativityrdquo Journal of Theoretical Biology vol 145no 3 pp 295ndash318 1990

[4] M Santillan and M C Mackey ldquoWhy the lysogenic stateof phage 120582 is so stable a mathematical modeling approachrdquoBiophysical Journal vol 86 no 1 I pp 75ndash84 2004

[5] S Wang Y Zhang and Q Ouyang ldquoStochastic model of col-iphage lambda regulatory networkrdquo Physical Review E vol 73no 4 Article ID 041922 2006

[6] R Zhu A S Ribeiro D Salahub and S A Kauffman ldquoStudyinggenetic regulatory networks at the molecular level delayedreaction stochastic modelsrdquo Journal of Theoretical Biology vol246 no 4 pp 725ndash745 2007

[7] D Laschov and M Margaliot ldquoMathematical modeling of thelambda switch a fuzzy logic approachrdquo Journal of TheoreticalBiology vol 260 no 4 pp 475ndash489 2009

[8] L Zeng S O Skinner C Zong J Sippy M Feiss and IGolding ldquoDecisionmaking at a subcellular level determines theoutcome of bacteriophage infectionrdquo Cell vol 141 no 4 pp682ndash691 2010

[9] M Chaves and J Gouze ldquoExact control of genetic networks ina qualitative framework the bistable switch examplerdquoAutomat-ica vol 47 no 6 pp 1105ndash1112 2011

[10] E Aurell and K Sneppen ldquoEpigenetics as a first exit problemrdquoPhysical Review Letters vol 88 no 4 Article ID 048101 4 pages2002

[11] X-M Zhu L Yin L Hood and P Ao ldquoRobustness stabilityand efficiency of phage 120582 genetic switch dynamical structureanalysisrdquo Journal of Bioinformatics and Computational Biologyvol 2 no 4 pp 785ndash818 2004

[12] T Tian and K Burrage ldquoBistability and switching in thelysislysogeny genetic regulatory network of bacteriophage 120582rdquoJournal of Theoretical Biology vol 227 no 2 pp 229ndash237 2004

[13] C Lou X Yang X Liu B He and Q Ouyang ldquoA quantitativestudy of 120582-phage SWITCH and its componentsrdquo BiophysicalJournal vol 92 no 8 pp 2685ndash2693 2007

[14] L M Anderson and H Yang ldquoDNA looping can enhancelysogenic CI transcription in phage lambdardquo Proceedings of theNational Academy of Sciences of the United States of Americavol 105 no 15 pp 5827ndash5832 2008

[15] H Zhu T Chen X LeiW Tian Y Cao and P Ao ldquoUnderstandthe noise of CI expression in phage lysogenrdquo in Proceedings ofthe 31st Chinese Control Conference (CCC rsquo12) pp 7432ndash74362012

[16] A Bakk and R Metzler ldquoIn vivo non-specific binding of 120582 CIand Cro repressors is significantrdquo FEBS Letters vol 563 no 1ndash3pp 66ndash68 2004

[17] A Bakk and R Metzler ldquoNonspecific binding of the ORrepressors CI andCro of bacteriophage 120582rdquo Journal ofTheoreticalBiology vol 231 no 4 pp 525ndash533 2004

[18] T S Gardner C R Cantor and J J Collins ldquoConstruction ofa genetic toggle switch in Escherichia colirdquo Nature vol 403 no6767 pp 339ndash342 2000

[19] D S Burz andGK Ackers ldquoCooperativitymutants of bacterio-phage 120582 cI repressor temperature dependence of self-assemblyrdquoBiochemistry vol 35 no 10 pp 3341ndash3350 1996

[20] W S Hlavecek and M A Savageau ldquoSubunit structure of regu-lator proteins influences the design of gene circuitry analysis ofperfectly coupled and completely uncoupled circuitsrdquo Journal ofMolecular Biology vol 248 no 4 pp 739ndash755 1995

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Computational Biology Journal 11

[21] E Aurell S Brown J Johanson and K Sneppen ldquoStabilitypuzzles in phage 120582rdquo Physical Review E vol 65 no 5 Article ID051914 9 pages 2002

[22] J W Little D P Shepley and DWWert ldquoRobustness of a generegulatory circuitrdquoThe EMBO Journal vol 18 no 15 pp 4299ndash4307 1999

[23] A Bakk RMetzler andK Sneppen ldquoSensitivity ofOR in phage120582rdquo Biophysical Journal vol 86 no 1 I pp 58ndash66 2004

[24] T Gedeon KMischaikow K Patterson and E Traldi ldquoBindingcooperativity in phage 120582 is not sufficient to produce an effectiveswitchrdquo Biophysical Journal vol 94 no 9 pp 3384ndash3392 2008

[25] M J Morelli P R Wolde and R J Allen ldquoDNA looping pro-vides stability and robustness to the bacteriophage 120582 switchrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 106 no 20 pp 8101ndash8106 2009

[26] M Ptashne A Genetic Switch Phage Lambda and HigherOrganisms Cell and Blackwell Scientific Cambridge MassUSA 1992

[27] C Zong L So L A Sepulveda S O Skinner and I GoldingldquoLysogen stability is determined by the frequency of activitybursts from the fate-determining generdquo Molecular SystemsBiology vol 6 no 1 article 440 2010

[28] R T Sauer M J Ross andM Ptashne ldquoCleavage of the lambdaand P22 repressors by recA proteinrdquo The Journal of BiologicalChemistry vol 257 no 8 pp 4458ndash4462 1982

[29] M Ptashne A Genetic Switch Phage Lambda Revisited ColdSpring Harbor Laboratory Press 2004

[30] J W Roberts and C W Roberts ldquoProteolytic cleavage ofbacteriophage lambda repressor in inductionrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 72 no 1 pp 147ndash151 1975

[31] N Chia I Golding and N Goldenfeld ldquo120582-prophage inductionmodeled as a cooperative failure mode of lytic repressionrdquoPhysical Review E vol 80 no 3 Article ID 030901 2009

[32] P J Darling J M Holt and G K Ackers ldquoCoupled energeticsof 120582 cro repressor self-assembly and site-specific DNA operatorbinding II cooperative interactions of cro dimersrdquo Journal ofMolecular Biology vol 302 no 3 pp 625ndash638 2000

[33] K Brooks and A J Clark ldquoBehavior of lambda bacteriophagein a recombination deficienct strain of Escherichia colirdquo Journalof Virology vol 1 no 2 pp 283ndash293 1967

[34] J L Cherry and F R Adler ldquoHow to make a biological switchrdquoJournal of Theoretical Biology vol 203 no 2 pp 117ndash133 2000

[35] A Arkin J Ross and H H McAdams ldquoStochastic kineticanalysis of developmental pathway bifurcation in phage 120582-infected Escherichia coli cellsrdquoGenetics vol 149 no 4 pp 1633ndash1648 1998

[36] P S Swain M B Elowitz and E D Siggia ldquoIntrinsic andextrinsic contributions to stochasticity in gene expressionrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 20 pp 12795ndash12800 2002

[37] G Nilsson J G Belasco S N Cohen and A von GabainldquoGrowth-rate dependent regulation of mRNA stability inEscherichia colirdquo Nature vol 312 no 5989 pp 75ndash77 1984

[38] S Sastry Nonlinear Systems Analysis Stability and ControlInterdisciplinary Applied Mathematics Systems and ControlSpringer 1999

[39] X-M Zhu L Yin L Hood and P Ao ldquoCalculating biologicalbehaviors of epigenetic states in the phage 120574 life cyclerdquo Func-tional and Integrative Genomics vol 4 no 3 pp 188ndash195 2004

[40] I B Dodd A J Perkins D Tsemitsidis and J B EganldquoOctamerization of 120582 CI repressor is needed for effectiverepression of PRMand efficient switching from lysogenyrdquoGenesand Development vol 15 no 22 pp 3013ndash3022 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Anatomy Research International

PeptidesInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

International Journal of

Volume 2014

Zoology

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Molecular Biology International

Hindawi Publishing Corporationhttpwwwhindawicom

GenomicsInternational Journal of

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioinformaticsAdvances in

Marine BiologyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Signal TransductionJournal of

BioMed Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Evolutionary BiologyInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Biochemistry Research International

ArchaeaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Genetics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

Virolog y

Hindawi Publishing Corporationhttpwwwhindawicom

Nucleic AcidsJournal of

Volume 2014

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Enzyme Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Microbiology