the limiting dynamics of a bistable molecular switch with ......the limiting dynamics of a bistable...

Journal of Mathematical Biology manuscript No.(will be inserted by the editor)

The Limiting Dynamics of a Bistable Molecular SwitchWith and Without Noise

Michael C. Mackey · Marta Tyran-Kaminska

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Abstract We consider the dynamics of a population of organisms containing twomutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in theabsence of any noise, and then go on to consider the effects of both intrinsic noise(due to bursting transcription or translation) and extrinsic noise when there isa dominant slow variable in the system. We show analytically how the steadystate distribution in the population can range from a single unimodal distributionthrough a bimodal distribution and give the explicit analytic form for the invariantstationary density which can be shown to be globally asymptotically stable.

Keywords Stochastic modelling, bistable switch, mutual repression

1 Introduction

In electrical circuits there are only two elementary ways to produce bistable be-havior. Either with positive feedback (e.g. A stimulates B and B stimulates A)or with double negative feedback (A inhibits B and B inhibits A). This elemen-tary fact, known to all electrical engineering students, has, in recent years, cometo the attention of molecular biologists who have rushed to implicate one or theother mechanism as the source of putative or real bistable behavior in a varietyof biological systems. Indeed, some laboratories have used this insight to engineerin vitro systems to have bistable behavior and one of the first was Gardner et al(2000) who engineered switch like behavior of the type we study in this paper.Some especially well written, if somewhat dated, surveys are to be found in Ferrell(2002), Tyson et al (2003), and Angeli et al (2004).

Michael C. MackeyDepartments of Physiology, Physics & Mathematics and Centre for Applied Mathematics inBioscience and Medicine (CAMBAM), McGill University, 3655 Promenade Sir William Osler,Montreal, QC, CANADA, H3G 1Y6E-mail: [email protected]

Marta Tyran-KaminskaInstitute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, POLANDE-mail: [email protected]

2 Michael C. Mackey, Marta Tyran-Kaminska

Gene regulatory networks are, however, noisy affairs either because of the exis-tence of outside (extrinsic) noise due to small numbers of participating molecules orinternal (intrinsic) noise due, e.g. to the intermittent production of mRNA and/orprotein (bursting). In such noisy dynamical systems experimentalists will oftentake a populational level approach and infer the existence of underlying bistablebehavior based on the existence of bimodal densities of some molecular constituentover some range of experimental parameter values.

There have been few theoretical studies of the effects of intrinsic or extrinsicnoise in bistable dynamical systems in a molecular biology context. Mackey et al(2011) did carry out such a study that included the positive-positive feedbacksituation alluded to above. This paper extends these results to the situation ofnegative-negative feedback. The paper is organized as follows. Section 2 lays thegroundwork by developing the deterministic model that we wish to consider. Thisis followed in Section 3 with an analysis of the possible steady states of the sys-tem, conditions for the coexistence of multiple steady states, and considerations oftheir global stability. Section 4 briefly considers how the existence of fast and slowvariables enables the simplification of the dynamics, while the following Section 5introduces bursting transcriptional or translational noise and derives the station-ary population density in a variety of situations. Section 6 considers the countersource of noise as coming from extrinsic fluctuations in either degradation ratesor production rates. The paper concludes with a short discussion.

2 The bistable genetic switch

2.1 Biological background

The paradigmatic molecular biology example of a bistable switch due to reciprocalnegative feedback is the bacteriophage (or phage) λ, which is a virus capable ofinfecting Escherichia coli bacteria. Originally described in Jacob and Monod (1961)and very nicely treated in Ptashne (1986), it is but one of scores of mutuallyinhibitory bistable switchs that have been found since.

2.2 Model development

Figure 1 gives a cartoon representation of the situation we are modeling here.The reader may find reference to that figure helpful while following the modeldevelopment below.

Polynikis et al (2009) offers a nice survey of techniques applicable to the ap-proach we take in this section. We consider two operons X and Y such that theeffector ofX, denoted by Ex, inhibits the transcriptional production of mRNA fromoperon Y and vice versa. We take the approach of Goodwin (1965) as extended anddeveloped in (Griffith 1968a,b; Othmer 1976; Selgrade 1979) and consider initiallya single operon a where a ∈ {x, y}. If we let the mRNA and effector concentrationsbe denoted by (Ma, Ea) then we assume that the dynamics for operon a are given

The Limiting Dynamics of a Bistable Molecular Switch 3

Regx Ox SGxOperon X

Rx + Ey RxEy

SGy Oy RegyOperon Y

Mx

Ry + ExRyExMy

Fig. 1 A schematic depiction of the elements of a bistable genetic switch. There are twooperons (X and Y ). For each, the regulatory region (Regx or Regy) produces a repressormolecule (Rx or Ry) that is inactive unless it is combined with the effector produced bythe opposing operon (Ey or Ex respectively). In the combined form (RxEy or RyEx) therepressor-effector complex binds to the operator region (Ox or Oy respectively) and blockstranscription of the corresponding structural gene (SGx or SGy). When the operator region isnot complexed with the active form of the repressor, transcription of the structural gene cantake place and mRNA (Mx or My) is produced. Translation of the mRNA then produces aneffector molecule (Ex or Ey). These effector molecules then are capable of interacting with therepressor molecule of the opposing gene.

by

dMa

dt= bd,aϕm,afa(Ea)− γMa

Ma, a ∈ {y, x} (1)

dEadt

= βEaMa − γEaEa. (2)

It is assumed that the rate of mRNA production is proportional to the fraction oftime the operator region is active and that the maximum level of transcription isbd,a, and that the enzyme production rate is proportional to the amount of mRNA.Note in particular that the production of Mx is regulated by Ey and vice versa,and that the components (Ma, Ea) are subject to random loss. The function f iscalculated next.

To compute f we temporarily suppress the subscript a and then restore it atthe end. For the mutually repressible systems we consider here, in the presence

of the effector molecule E the repressor R is active (able to bind to the operatorregion), and thus block DNA transcription. The effector binds with the inactiveform R of the repressor, and when bound to the effector the repressor becomesactive. We take this reaction to be of the form

R+ nEK1

REn K1 =REnR · En , (3)

Here, n is the number of effector molecules that inactivate the repressor R. Thereis an interaction between the operator O and repressor R described by

O +R · EnK2

OREn K2 =OREn

O ·R · En .


The total operator is given by

Otot = O +OREn = O +K2O ·R · En = O(1 +K2R · En),

while the total repressor Rtot is

Rtot = R+K1R · En +K2O ·R,

so the fraction of operators not bound by repressor is given by

f(E) =O

Otot=

1

1 +K2R · En.

If the amount operator O is large compared with the amount of repressor R boundto it

f(E) =1 +K1E

n

1 + (K1 +K2Rtot)En=

1 +K1En

1 +KEn,

where K = K1 + K2Rtot. When E is large there will be maximal repression, buteven then there will still be a basal level of mRNA production proportional toK1K

−1 < 1 (this is known as leakage). The variation of the DNA transcriptionrate with effector level is given by ϕ = ϕmf or

ϕ(E) = ϕm1 +K1E

n

1 +KEn= ϕmf(E), (4)

where ϕm is the maximal DNA transcription rate (in units of inverse time).Now explicitly including the proper subscripts we have

ϕa(Ea) = ϕm,a1 +K1,aE

naa

1 +KaEnaa

= ϕm,afa(Ea), (5)

where Ka = K1,a +K2,aRtot,a.We next rewrite Equations 1-2 by defining dimensionless concentrations. Equa-

tion 4 becomesϕa(ea) = ϕm,afa(ea), (6)

where the dimensionless rate ϕm,a is defined by

ϕm,a =ϕm,aβE,aγM,aγE,a

and fa(ea) =1 + enaa

1 +∆aenaa, (7)

∆a = KaK−11,a, and the dimensionless effector concentration (ea) is defined by

Ea = ηaea with ηa =1

na√K1,a

.

Similarly using a dimensionless intermediate mRNA concentration (ma) given by

Ma = maηaγEaβEa

,

Equations 1-2 take the form

dma

dt= γMa

[κd,afa(ea)−ma],

deadt

= γEa(ma − ea),


with

κd,a = bd,aϕm,a, and bd,a =bd,aηa

which are both dimensionless.Thus the equations governing the dynamics of this system are given by the

four differential equations

dmx

dt= γMx

[κd,xfx(ey)−mx],

dexdt

= γEx(mx − ex),

dmy

dt= γMy

[κd,yfy(ex)−my],

deydt

= γEy (my − ey)

where

fx(ey) =1 + enxy

1 +∆xenxy

and fy(ex) =1 + e

nyx

1 +∆yenyx.

To make the model equations somewhat more straightforward, denote dimen-sionless concentrations by (mx, ex,my, ey) = (x1, x2, y1, y2) (with obvious changesin the other subscripts) to obtain

dx1

dt= γx1 [κd,xfx(y2)− x1], (8)

dx2

dt= γx2(x1 − x2), (9)

dy1

dt= γy1 [κd,yfy(x2)− y1], (10)

dy2

dt= γy2(y1 − y2), (11)

Throughout, γi is a random loss rate (time−1), and so Equations 8-11 are notdimensionless. In addition to the loss rates explicitly appearing, we have the pa-rameters κd,x, κd,y. Since

fx(y2) =1 + ynx2

1 +∆xynx2

and fy(x2) =1 + x

ny2

1 +∆yxny2

. (12)

we have as well the four parameters ∆x,∆y, nx, ny to consider. Note that

fx(0) = 1, limy2→∞

fx(y2) = ∆−1x < 1, fy(0) = 1, lim

x2→∞fy(x2) = ∆−1

y < 1.

3 Steady states and dynamics

The dynamics of this model for a bistable switch can be analyzed as follows. Thissection is an elaboration of aspects of the work presented in Cherry and Adler(2000). Set W = (x1, x2, y1, y2) so the system (8)-(11) generates a flow St(W ). Theflow St(W

0) ∈ R+4 for all initial conditions W 0 = (x0

1, x02, , y

01 , y

02) ∈ R+

4 and t > 0.


The steady states of the system (8)-(11) are given by x∗1 = x∗2 = x∗, y∗1 = y∗2 =y∗ where (x∗, y∗) is the solution of

x1 = x2 = κd,xfx(y2) (13)

y1 = y2 = κd,yfy(x2). (14)

For each solution (x∗, y∗) of (13)-(14) there is a steady state W ∗ of the model, andthe parameters (κd,x, κd,y,∆x,∆y, nx, ny) will determine whether W ∗ is unique orhas multiple values.

3.1 Graphical investigation of the steady states

Figure 2 gives a graphical picture of the five qualitative possibilities for steadystate solutions of the pair of equations (8)-(11).

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

ABCDE

x2

y2

Fig. 2 A graphical representation of the possible steady state solutions of Equations 13 and 14.We have plotted the y1 and x1 isoclines (y2 = κd,yfy(x2) and x2 = κd,xfx(y2) respectively),and assumed that the y1 isocline (the graph of y2 = κd,yfy(x2)) is not changed but that x1

isocline (the graph of x2 = κd,xfx(y2)) is varied as indicated by the labels A to E, e.g. bydecreasing κd,x. (A) There is a single steady state at a large value of x2 and a correspondinglysmall value of y2. In this case operon X of the bistable switch is in the “ON” state whileoperon Y is in the “OFF” state. This steady state is globally stable. (B) A decrease in κd,xnow leads to a situation in which there are two steady states, the largest (locally stable one)corresponding to the intersection of the two graphs, and the second smaller (half stable) onewhere the two graphs are tangent. (C) Further decreases in κd,x now result in three steadystates. For the largest (locally stable) one the operon X is in the on state while Y is in the offstate. The smallest one (also locally stable) corresponds to operon Y in the ON state and Xis in the OFF state. The intermediate steady state is unstable. (D) This case is like B in thatthere are two steady states, one (locally stable) defined by the intersection of the two graphsin which Y is ON and the second at the tangency of the two graphs is again half stable. (E)Finally, for sufficiently small κd,x there is a single globally stable steady state in which Y isON and X is OFF.


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

x

ABCDE

Fig. 3 A graphical representation of the possible steady state solutions of the equationx/κd,x = fx(κd,yfy(x)) := Fx(x). The smooth monotone increasing graph is that of Fx(x) asgiven in Equation 16, while the straight line is that of x/κd,x for different values of κd,x. Thefive straight lines correspond to the five possibilities (A through E) in Figure 2.

An alternative, but equivalent, way of examining the steady state of this modelis by examining the solution of either one of the pair of equations

x

κd,x= fx(κd,yfy(x)) := Fx(x),

y

κd,y= fy(κd,xfx(y)) := Fy(y). (15)

We choose to deal with the first. Note that since both fx and fy are monotonedecreasing functions of their arguments, the composition of the two

Fx(x) =1 + (κd,yfy(x))nx

1 +∆x(κd,yfy(x))nx=

1 +

(κd,y

1 + xny

1 +∆yxny

)nx1 +∆x

(κd,y

1 + xny

1 +∆yxny

)nx (16)

is a monotone increasing function of x with

Fx(0) =1 + κnxd,y

1 +∆xκnxd,y

:= Fx,0

and

limx→∞

Fx(x) =1 + (κd,y∆

−1y )nx

1 +∆x(κd,y∆−1y )nx

:= Fx,∞ > Fx,0.

In Figure 3 we have shown graphically the same sequence of steady states as weillustrated in Figure 2


0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

κd,x

x

Fig. 4 The plot of κd,x versus x obtained from Equation 17. The figure was constructed forthe following parameters: nx ∈ {1, 2, 3}, ny ∈ {1, 2, 3, 4, 5}, κy = 2, ∆x = 12, ∆y = 10. Thesolid lines correspond to nx = 1 and we increase ny from 1 (the lowest line) to 5 (the top one).The dashed lines correspond to nx = 2 and the dotted to nx = 3.

3.2 Analytic investigation of the steady states

Single versus multiple steady states. This model for a bistable genetic switchmay have one [W ∗1 ( E of Figure 2 or Figure 3) or W ∗3 (A)], two [W ∗1 ,W

∗2 = W ∗3 (D)

or W ∗1 = W ∗2 ,W∗3 (B)], or three [W ∗1 ,W

∗2 ,W

∗3 (C)] steady states, with the ordering

0 � W ∗1 � W ∗2 � W ∗3 , indicating that W ∗1 corresponds to operon X in the OFFstate and operon Y in the ON state while at W ∗3 X is ON and Y is OFF.

Analytic conditions for the existence of one or more steady states can be ob-tained by first noting that we must have

x

κd,x= fx(κd,yfy(x)) := Fx(x) (17)

satisfied. In Figure 4 we have illustrated Equation 17 for various values of param-eters.

In addition to this criteria, we have a second relation at our disposal at thedelineation points between the existence of two and three steady state. Thesepoints are also determined by a second relation since x/κd is tangent to Fx(x) (seeFigure 3 B,D). Thus we must also have

1

κd,x=dFx(x)

dx. (18)

Now the problem is to derive values for x± at which a tangency occurs, as well as tofigure out some way to make a parametric plot of a combination of κd,x, κd,y,∆x,∆yfor given values of nx, ny.

Indeed, from Equations 13 and 14 we have

x = κd,xfx(y) and y = κd,yfy(x). (19)


Additionally at a tangency between fx(y) and fy(x) we must have

x = κd,xfx(κd,yfy(x)),

so1 = κd,xκd,yf

′xf′y.

However,

κd,xκd,y =xy

fx(y)fy(x)

so we have an implicit relationship between x and y given by

fx(y)

yf ′x(y)=xf ′y(x)

fy(x)

that, when written explicitly becomes

L(y) := − (1 + ynx)(1 +∆xynx)

nx(∆x − 1)ynx= − ny(∆y − 1)xny

(1 + xny )(1 +∆yxny ):= R(x). (20)

Now L(y) has a maximum at ymax = ∆−1/2nxx and

L(ymax) := Lmax = − (1 +√∆x)2

nx(∆x − 1),

while R(x) has a minimum at xmin = ∆−1/2nyy given by

R(xmin) := Rmin = − ny(∆y − 1)

(1 +√∆y)2

.

A necessary condition for there to be a solution to Equation 20, and thus a neces-sary condition for bistability, is that Lmax ≥ Rmin or

nxny ≥(1 +

√∆x)2(1 +

√∆y)2

(∆x − 1)(∆y − 1)≥ 1. (21)

This is interesting in the sense that if either nx OR ny is one but the other islarger than one then the possibility of bistability behavior still persists, while inthe situation of Mackey et al (2011) this is impossible. However, note from Figure4 that this necessary condition is far from what is sufficient since it would appearfrom Equation 17 that a necessary and sufficient condition is more like nxny ' 4.

Going back to Equation 20, we can write

∆xy2nx + [nx(∆x − 1)R(x) + (∆x + 1)]ynx + 1 = 0, (22)

which has two positive solutions y± given by

y± = nx

√√√√∆x − 1

2∆x

{−nxR(x)− ∆x + 1

∆x − 1±√

[nxR(x)]2 + 2nxR(x)∆x + 1

∆x − 1+ 1

},

(23)provided that

[nxR(x)]2 + 2nxR(x)∆x + 1

∆x − 1+ 1 ≥ 0


0 1 2 3 4 5 60

1

2

3

4

5

6

7

κd,x

κd,y

Fig. 5 The parametric plot of κd,x versus κd,y obtained from Equation 24 where we used thefollowing parameters: nx = 2, ny = 3, ∆x = 12, ∆y = 10. The blue line is for y(x) = y− andthe green for y(x) = y+.

and

−nxR(x)− ∆x + 1

∆x − 1−√

[nxR(x)]2 + 2nxR(x)∆x + 1

∆x − 1+ 1 ≥ 0.

Substitution of the result into Equations 19 gives explicitly

κd,x(x) =x

fx(y(x))and κd,y(x) =

y(x)

fy(x), (24)

where y(x) is either y+ or y− as given by (23). In Figure 5 we have plotted κd,x(x)versus κd,y(x) with x as the parametric variable. Inside the region bounded bythe blue line (below) and green line (above) we are assured of the existence ofbistable behaviour while outside this region there will be only a single globallystable steady state. Thus, for example, for a constant value of κd,y such thatbistability is possible, then increasing κd,x from 0 there will be a minimal valueκd,x− at which bistability is first seen and this will persist as κd,x is increased untila second value κd,x− < κd,x+ is reached where the bistable behaviour once againdisappears.

3.3 Local and global stability.

Whether or not a steady state W ∗ is locally stable is completely determined bythe eigenvalues that solve the equation

2∏i=1

(λ+ γxi)(λ+ γyi)−2∏i=1

γxiγyiκd,xκd,yf′x∗f′y∗ = 0, (25)


where f ′x∗ = f ′x(x∗), f ′y∗ = f ′y(y∗). Equation 25 can be rewritten in the form

4∑i=1

aiλi + a0 = 0 (26)

where the ai, i > 0 are positive and a0 = (1 − κd,xκd,yf′x∗f′y∗)∏2i=1 γxiγyi . By

Descartes’s rule of signs, (26) has no positive roots for f ′x∗f′y∗ ∈ [0, (κd,xκd,y)−1)

or one positive root otherwise. Denote a locally stable steady state by S, a half orneutrally stable steady state by HS, and unstable steady state by US. Then weknow that there will be:

– A single steady state W ∗1 (S), for κd,x ∈ [0, κd,x−)– Two steady states W ∗1 (S) and W ∗2 = W ∗3 (HS) for κd,x = κd,x−– Three steady states W ∗1 (S), W ∗2 (US), W ∗3 (S) for κd,x ∈ (κd,x−, κd,x+)– Two steady states W ∗1 = W ∗2 (HS) and W ∗3 (S) for κd,x = κd,x+

– One steady state W ∗3 (S) for κd,x+ < κd,x.

Global stability results of others complement this classification.

Theorem 1 (Othmer 1976; Smith 1995, Proposition 2.1, Chapter 4) For the bistable

switch given by Equations 8-12, define Ix = [κd,x∆−1x , 1] and Iy = [κd,y∆

−1y , 1]. There

is an attracting box B ⊂ R+4 , where

B = {(xi, yi) : x1,2 ∈ Ix, y1,2 ∈ Iy},

for which the flow St is directed inward on the surface of B. All W ∗ ∈ B and

1. If there is a single steady state, then it is globally stable.

2. If there are two locally stable steady states, then all flows St(W0) are attracted to

one of them.

4 Fast and slow variables

Identification of fast and slow variables in systems can often be used to achievesimplifications that allow quantitative examination of the relevant dynamics, andparticularly to examine the approach to a steady state and the nature of thatsteady state. A fast variable is one that relaxes much more rapidly to an equilib-rium than does a slow variable (Haken 1983). In chemical systems this separationis often a consequence of differences in degradation rates, and the fastest variableis the one that with the largest degradation rate. We will use this technique inexamining the effects of noise.

If it is the case that there is a single dominant slow variable in the system (8)-(11) relative to all of the other three (and here we assume with loss of generalitythat it is in the X gene) then the four variable system describing the full switchreduces to a single equation

dx

dt= γ[κd,xF(x)− x], (27)

and γ is the dominant (smallest) degradation rate. (Here, and subsequently, wewill drop the subscript x whenever there will not be an confusion when treatingthe situation with a single dominant slow variable to simplify the notation.)


5 Transcriptional and translational bursting as a noise source

It has been quite clearly shown (Cai et al 2006; Chubb et al 2006; Golding et al2005; Raj et al 2006; Sigal et al 2006; Yu et al 2006) that in a number of ex-perimental situations some organisms transcribe mRNA discontinuously and as aconsequence there is a discontinuous production of the corresponding effector pro-teins (i.e. protein is produced in bursts). Experimentally, the amplitude of proteinproduction through bursting translation of mRNA is exponentially distributed atthe single cell level with density

h(y) =1

be−y/b, (28)

where b is the average burst size, and the frequency of bursting ϕ is dependenton the level of the effector. Writing Equation 28 in terms of our dimensionlessvariables we have

h(x) =1

be−x/b. (29)

When bursting is present, the analog of the deterministic single slow variabledynamics discussed above is

dx

dt= −γx+ Ξ(h, ϕ(x)), with ϕ(x) = γϕmF(x), (30)

where Ξ(h, ϕ) denotes a jump Markov process, occurring at a rate ϕ, whose ampli-tude is distributed with density h as given in (29). Set κb = ϕm, so F has the sameform as (16) but with κd,y replaced by κb,y. When we have bursting dynamicsdescribed by the stochastic differential equation (30), it has been shown (Mackeyand Tyran-Kaminska 2008) that the evolution of the density u(t, x) is governed bythe integro-differential equation

∂u(t, x)

∂t− γ ∂(xu(t, x))

∂x= −γκb,xF(x)u(t, x)

+ γκb,x

∫ x

0

F(z)u(t, z)h(x− z)dz.(31)

In a steady state the (stationary) solution u∗(x) of (31) is found by solving

−d(xu∗(x))

dx= −κb,xF(x)u∗(x) + κb,x

∫ x

0

F(z)u∗(z)hx(x− z)dz. (32)

If u∗(x) is unique, then the solution u(t, x) of Equation 31 is said to be asymptot-ically stable (Lasota and Mackey 1994) in that

limt→∞

∫ ∞0

|u(t, x)− u∗(x)|dx = 0

for all initial densities u(0, x). Somewhat surprisingly, it is possible to actuallyobtain a closed form solution for u∗(x) as given in the following


Theorem 1 (Mackey and Tyran-Kaminska 2008, Theorem 7). The unique stationary

density of Equation 31, with F given by Equation 16 and h given by (29), is

u∗(x) =Cxe−x/b exp

[κb,x

∫ x F(z)

zdz

], (33)

C is a normalization constant such that∫∞0u∗(x)dx = 1, and u(t, x) is asymptotically

stable.

Note that u∗ can be written as

u∗(x) = C exp

∫ x(κb,xF(z)

z− 1

b− 1

z

)dz. (34)

Thus from (34) we can write

u′∗(x) = u∗(x)

(κb,xF(x)

x− 1

b− 1

x

), (35)

so for x > 0 we have u′∗(x) = 0 if and only if

1

κb,x

(x

b+ 1)

= F(x). (36)

An easy graphical argument shows there may be zero to three positive roots ofEquation 36, and if there are three roots we denote them by x1 < x2 < x3. Thegraphical arguments in conjunction with (35) show that two general cases mustbe distinguished, exactly as was found in Mackey et al (2011). (In what follows,κb,x, κb,x−, and κb,x+ play exactly the same role as do κd,x, κd,x−, and κd,x+ inthe discussion around Figure 5.)

Case 1. 0 < κb,x < F−10 . In this case, u∗(0) = ∞. If κb,x < κb,x−, there are no

positive solutions, and u∗ will be a monotone decreasing function of x. If κb,x >κb,x−, there are two positive solutions (x2 and x3), and a maximum in u∗ at x3

with a minimum in u∗ at x2.

Case 2. 0 < F−10 < κb,x. Now, u∗(0) = 0 and either there are one, two, or three

positive roots of Equation 36. When there are three, x1, x3 will correspond to thelocation of maxima in u∗ and x2 will be the location of the minimum betweenthem. The condition for the existence of three roots is κb,x− < κb,x < κb,x+.

Thus we can classify the stationary density u∗ for a bistable switch as:

1. Unimodal type 1: u∗(0) = ∞ and u∗ is monotone decreasing for 0 < κb,x <

κb,x− and 0 < κb,x < F−10

2. Unimodal type 2: u∗(0) = 0 and u∗ has a single maximum at(a) x1 > 0 for F−1

0 < κb,x < κb,x− or(b) x3 > 0 for κb,x+ < κb,x and F−1

0 < κb,x3. Bimodal type 1: u∗(0) = ∞ and u∗ has a single maximum at x3 > 0 for

κb,x− < κb,x < F−10

4. Bimodal type 2: u∗(0) = 0 and u∗ has two maxima at x1, x3, 0 < x1 < x3 forκb,x− < κb,x < κb,x+ and F−1

0 < κb,x


The exact determination of these three roots is difficult in general because ofthe complexity of F , but we can derive implicit criteria for when there are exactlytwo roots (x1 and x3) by determining when the graph of the left hand side of (36)is tangent to F . Using this tangency condition, differentiation of (36) yields

1

κb,xb= F ′(x). (37)

Although Equations 36 and 37 offer conceptually simple conditions for delin-eating when there are exactly two roots (and thus to find boundaries betweenmonostable and bistable stationary densities u∗), a moments reflection after look-ing at (16) for F reveals that it is algebraically quite difficult to obtain quantitativeconditions in general. However, (36) and (37) are easily used in determining nu-merically boundaries between monostable and bistable stationary densities.

5.1 Monomeric repression of one of the genes with bursting (nx = 1)

Evaluation of the integral appearing in Equation 33 can be carried out for all(positive) integer values of (nx, ny) in theory, but the calculations become alge-braically complicated. However, if we consider the situation when a single moleculeof the protein from the y gene is capable of repressing the x gene, so nx = 1, thenthe results become more tractable and allow us to examine the role of differentparameters in determining the nature of u∗.

Thus, for nx = 1, F takes the simpler form

F(x) =(1 + κb,y) + (∆y + κb,y)xny

Λ+ Γxny, (38)

where

Λ = 1 +∆xκb,y > 0, Γ = ∆y +∆xκb,y > 0.

Evaluating (33) we have the explicit representation

u∗(x) = Ce−x/bxA−1[Λ+ Γxny ]θ (39)

with

A =κb,x(1 + κb,y)

Λ> 0, θ =

κb,xκb,y(∆x − 1)(∆y − 1))

nyΛΓ> 0.

In Figure 6 we have illustrated the form of u∗(x) in four different situations.Figures 6 A,B show a smooth variation in a Unimodal Type 2 density as κb,x ∈[25, 37] is varied by steps of 2 for nx = 2 and nx = 3 respectively. The behavior isquite different in Figures 6 C,D however for there, with nx = 4 and nx = 6, theform of u∗(x) varies from a Unimodal Type 2 to a Bimodal Type 2 and back againas κb,x is varied.


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

(A)

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

(B)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

(C)

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

(D)

Fig. 6 In this figure we illustrate stationary densities given by Equation 39 where the param-eter values in each panel are taken to be κb,y = 1, ∆x = 12, ∆y = 10, κb,x ∈ [25, 37] changesby 2, where the graph with highest maximum corresponds to κb,x = 25 and the maxima aredecreasing when κb,x is increased. The parameter ny is taken in an increasing order to be2, 3, 4, 6, so that we start with ny = 2 in panel (A) and have ny = 6 in panel (D).

5.2 ‘Bang-bang’ repression with bursting

We can partially circumvent the algebraic difficulties of the previous sections byconsidering a limiting case. Consider the situation in which ny becomes large sofy(x) approaches the simpler form

fy(x)→{

1, 0 ≤ x < θ,

∆−1y , θ ≤ x, (40)

where

θ ' 1ny√∆y

ny

√ny − 1

ny + 1→ 1

ny√∆y→ 1,

so we have

F(x)→

F0 =

1 + κnyb,y

1 +∆xκnxb,y

, 0 ≤ x < 1,

F∞ =1 + (κb,y/∆y)ny

1 +∆x(κb,y/∆y)nx, 1 ≤ x.

(41)


The evaluation of (33) is simple and yields a stationary density which is (piecewise)that of the gamma distribution:

u∗(x) = Ce−x/bxA(x)−1 (42)

where

A(x) =

{A0 ≡ κb,xF0, 0 ≤ x < 1,A∞ ≡ κb,xF∞, 1 ≤ x.

Note that u∗(x) is continuous but not differentiable at x = 1, and C is givenexplicitly by

C =1

bA0 [Γ (A0)− Γ (A0, 1/b)] + bA∞Γ (A∞),

where Γ (α) is the gamma function and Γ (α, β) is the incomplete gamma function.In this limiting case the stationary density may display one of three general

forms as we have classified the densities earlier. Namely:

1. If κb,x < F−10 and κb,x < F−1

∞ then u∗(x) will be of Unimodal type 1;2. If κb,x < F−1

0 and κb,x > F−1∞ then u∗(x) will be Bimodal type 1;

3. If κb,x > F−10 (which implies κb,x > F−1

∞ ) then u∗(x) will be Bimodal type 2.

6 The effect of Gaussian distributed noise in molecular degradation or

production rates

For a generic one dimensional stochastic differential equation of the form

dx(t) = α(x)dt+ σ(x)dw(t), (43)

where w is a standard Brownian motion, the corresponding Fokker Planck equation

∂u

∂t= −∂(αu)

∂x+

1

2

∂2(σ2u)

∂x2(44)

can be written in the form of a conservation equation

∂u

∂t+∂J

∂x= 0,

where

J = αu− 1

2

∂(σ2u)

∂x

is the probability current. In a steady state when ∂tu ≡ 0, the current must satisfyJ = constant throughout the domain of the problem. In the particular case whenJ = 0 at one of the boundaries (a reflecting boundary) then J = 0 for all x in thedomain and the steady state solution u∗ of Equation 44 is easily obtained with asingle quadrature as

u∗(x) =C

σ2(x)exp

{2

∫ x α(y)

σ2(y)dy

},

where C is a normalizing constant as before.


6.1 Noise in the degradation rate

In our considerations of the effects of continuous fluctuations, we examine thesituation in which Gaussian fluctuations appear in the degradation rate γx of thegeneric equation (27). Oppenheim et al (1969) have shown that if the fluctuationsare Gaussian distributed the mean numbers of molecules decaying in a time dt issimply γxdt and the standard deviation of these numbers is proportional to

√x.

Thus we take the decay to be given by the sum of a deterministic component γxdtand a stochastic component σγ

√xdw(t), and write Equation 27 as a stochastic

differential equation in the form

dx = γ[κd,xF(x)− x]dt+ σγ√xdw.

(In the situation we consider here, α(x) = γ[κd,xF(x) − x] and σ(x) = σγ√x.)

Within the Ito interpretation of stochastic integration, this equation has a corre-sponding Fokker Planck equation for the evolution of the ensemble density u(t, x)given by (Lasota and Mackey 1994)

∂u

∂t= −

∂[γ(κd,xF(x)− x)u

]∂x

+σ2γ

2

∂2(xu)

∂x2. (45)

Since concentrations of molecules cannot become negative the boundary at x = 0is reflecting and the stationary solution of Equation 45 is given by

u∗(x) =Cxe−x/bw,γ exp

[κd,xbw,γ

∫ x F(z)

zdz

], with bw,γ =

σ2γ

2γ. (46)

We have also the following result.

Theorem 2 (Pichor and Rudnicki 2000, Theorem 2). The unique stationary density

of Equation 45 is given by Equation 46. Further u(t, x) is asymptotically stable.

Remark 1 Note that the stationary solution for the density u∗(x) given by Equation

46 in the presence of noise in the protein degradation rate is identical to the solution in

Equation 33, when transcriptional and/or translational noise in present in the system,

as long as we make the identification of κb,x with κd,x/bw,γ and b with bw,γ . As a

consequence, all of the results of the analysis in Section 5 are applicable in this section.

6.2 Gaussian fluctuations in production rate

We next examine the situation where there are fluctuations in the productionrate κd,x of equation (27). We take the production as the sum of a deterministiccomponent γκd,xF(x)dt and a stochastic component γσκ

√xF(x)dw(t), and write

Equation 27 as a stochastic differential equation in the form

dx = γ[κd,xF(x)− x]dt+ γσκ√xF(x)dw. (47)

Now, α(x) = γ[κd,xF(x)−x] and σ(x) = γσκ√xF(x). Using Ito stochastic integra-

tion, the Fokker Planck equation for the ensemble density u(t, x) is

∂u

∂t= −

∂[γ(κd,xF(x)− x)u

]∂x

+ γbw,κ∂2[F2(x)xu]

∂x2with bw,κ =

γσ2κ

2. (48)


Again invoking the existence of a reflecting boundary at x = 0, u∗ is the solutionof

bw,κ∂[F2(x)xu∗]

∂x=

(κd,xF(x)

x− 1

)xu∗,

so

u∗(x) =C

xF2(x)exp

[1

bw,κ

∫ x( κd,xzF(z)

− 1

F2(z)

)dz

]. (49)

Furthermore,

Theorem 3 (Pichor and Rudnicki 2000, Theorem 2). The unique stationary density

of Equation 48 is given by Equation 49, and u(t, x) is asymptotically stable.

6.2.1 Production rate noise and monomeric repression in one gene

We now assume, again, that we have monomeric repression such that nx = 1, andwish to evaluate the precise form for the stationary density in (49). We proceedin two steps evaluating the first integral in the exponential to obtain

u∗(x) = CxA/bw,κ−1 [Λ+ Γxny ]2+θ/bw,κ

[(1 + κb,y) + (∆y + κb,y)xny ]2exp

[− 1

bw,κ

∫ xdz

F2(z)

], (50)

where Γ , Λ, A and θ are the same as in (39). The exponential appearing in (50)can be evaluated with some difficulty, and yields

exp

[− 1

bw,κ

∫ xdz

F2(z)

]=

exp

{− x

bw,κ

[α− β2F1

(1,

1

ny; 1 +

1

ny;−∆y + κy

1 + κyxny)

+δ

(1 + κy) + (∆y + κy)xny

]},

(51)

where 2F1 is Gauss’ hypergeometric series, and the constants α, β, δ are given by

α =(∆y +∆xκy)2

(∆y + κy)2> 0,

β = ξ[(ny−1)(∆y+κy)(1+∆xκy)+(ny+1)(1+κy)(∆y+∆xκy)] > 0 for ny > 1

with

ξ =κxκy(∆x − 1)(∆y − 1)

ny(1 + κy)2(∆y + κy)2> 0,

and

δ = (1 + κy)ξ > 0.

From the above we know that for small x, u∗(x) ≈ CxA/bw,κ−1e−constx so whenA/bw,κ < 1, u∗(0) ≈ ∞ while for A/bw,κ > 1, u∗(0) = 0. Furthermore, for large xwe have u∗(x) ≈ Cx(A+nY θ)/bw,κ−1e−constx.


6.2.2 Bang bang regulation with production rate noise

If we assume that the control is of the ‘bang-bang’ type introduced in Section 5.2,and take F to be of the form in (41), then the evalutaion of the final form foru∗(x) given in (49) is straightforward and yields

u∗(x) = Ce−x/b(x)xA(x)−1 (52)

where

(b(x), A(x)) =

b0 = bw,κF2

0 , A0 =κd,xbw,κF0

, 0 ≤ x < 1,

b∞ = bw,κF2∞, A∞ =

κd,xbw,κF∞

, 1 ≤ x,

b0 < b∞, A∞ < A0, and C is given by

C =1

F−20 bA0 [Γ (A0)− Γ (A0, 1/b)] + F−2

∞ bA∞Γ (A∞).

In contrast to the case of bang-bang control with bursting and Gaussian noise inthe degradration rate, u∗(x) is neither continuous nor differentiable at x = 1.

Remark 2 Thus, just as in the case of bang-bang regulation with bursting noise and

Gaussian noise on the degradation rate, we recover a stationary density that is piecewise

that of the gamma distribution.

7 Discussion and conclusions

Here we have considered the behavior of a bistable molecular switch in both itsdeterministic version as well as what happens in the presence of two different kindsof noise. The results that we have obtained in the presence of noise are, unfortu-nately, only partial due to the analytic difficulties in solving for the stationarydensity u∗(x). Two other situations that we have been unable to treat analyticallyare that of two slow variables in the same gene or two slow variables in differentgenes. In both situations we would be led to solve for a two-dimensional stationarydensity (u∗(x1, x2) or u∗(x, y) respectively), but we leave these for a further paper.

Acknowledgements This work was supported by the Natural Sciences and Engineering Re-search Council (NSERC, Canada) and the State Committee for Scientific Research (Poland)Grant N N201 608240. We are grateful to Romain Yvinec (Tours) and Changjing Zhuge (Ts-inghua University, Beijing) for helpful comments on this problem.

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