Combinatorially Complex Combinatorially Complex Equilibrium Model Equilibrium Model

SelectionSelection

Tom RadivoyevitchAssistant ProfessorEpidemiology and BiostatisticsCase Western Reserve University

Email: txr24@case.eduWebsite: http://epbi-radivot.cwru.edu/

Ultimate GoalUltimate Goal

EXPERIMENTALBIOLOGY

COMPUTERMODELING

CONTROLTHEORY

models

control lawsdata

hypotheses

proposed clinical trial

validated process model development

control system design methods development

Present Future

• Safer flying airplanes with autopilotsSafer flying airplanes with autopilots• Ultimate Goal: individualized, state feedback based Ultimate Goal: individualized, state feedback based

clinical trialsclinical trials

• Better understanding => better controlBetter understanding => better control• Conceptual models help trial designs today Conceptual models help trial designs today • Computer models of airplanes help train pilots and Computer models of airplanes help train pilots and

autopilotsautopilots

Radivoyevitch et al. (2006) BMC Cancer 6:104

Systems Cancer BiologySystems Cancer Biology

dNTP Supply System

Figure 1. dNTP supply. Many anticancer agents act on or through this system to kill cells. The most central enzyme of this system is RNR.

UDP

CDP

GDP

ADP

dTTP

dCTP

dGTP

dATP

dT

dC

dG

dA

DNA

dUMP

dU

TS

CDA

dCK

DN

A p

olym

eras

eTK1

cytosol

mitochondria

dT

dC

dG

dA

TK

2dG

K

dTMP

dCMP

dGMP

dAMP

dTTP

dCTP

dGTP

dATP

5NT

NT2

cytosol

nucleus

dUDP

dUTP

U-DNA

dUTPasedN

dN

dCK

flux activation inhibition

ATP RN

R

dCK

R1

R2 R2

R1 R1

R1 R1

R1 R1

R1

R1

R1

R1

R1 R1

R1

R1

R1

R1

R2 R2

UDP, CDP, GDP, ADP bind to catalytic site

dTTP, dGTP, dATP, ATP bind to selectivity site

dATP inhibits at activity site, ATP activates at activity site?

Selectivity site binding promotes R1 dimers. R2 is always a dimer.

ATP drives hexamer. Controversy: dATP drives inactive tetramer vs. inactive hexamer

Controversy: Hexamer binds one R22 vs. three R22

Total concentrations of R1, R22, dTTP, dGTP, dATP, ATP and NDPscontrol the distribution of R1-R2 complexes and this changes in S, G1-G2 and G0

ATP activates at hexamerization site??RNR LiteratureRNR Literature

R2 R2

Michaelis-Menten ModelMichaelis-Menten Model

With RNR: no NDP and no R2 dimer => kcat of complex is zero.Otherwise, many different R1-R2-NDP complexes can have many different kcat values.

)(0)(

][][

][0

][][

][

1/][

10

1/][

/][

][

][

00

00

00max

EPEESPEk

EES

EE

EES

ESEk

KSE

KS

KSEk

KS

SV

cat

cat

mm

mcat

m

E + S ES

0.005 0.010 0.020 0.050 0.100 0.200 0.500

02

04

06

08

01

00

Total [r] (uM)

Pe

rce

nt A

ctiv

ity

solid line = Eqs. (1-2) dotted = Eq. (3)

Data from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry 40(6), 1651-166

Model Parameter Initial Value Optimal Value Confidence Interval

RRGGttr1.1.0 RRGGtt_r 0.020 0.012 (0.007, 0.024)

SSE 1070.252 823.793

AIC 45.006 42.650

MM Kd 0.020 0.033 (0.022, 0.049)

SSE 2016.335 1143.682

AIC 50.706 45.603

R=R1 r=R22

G=GDP t=dTTP

)2(]][[

][][0

)1(]][[

][][0

__

__

SEdT

SEdT

K

SESS

K

SEEE

SEd

T

SEd

T

T

SEd

SEdT

SEd

TSEd

T

KS

KS

EESversus

KS

KS

EES

KS

EEK

SEE

__

__

__

__

__

__ ][1

][

][][][

1

][

][][][

1

1][][

][1][][

Substitute this in here to get a quadratic in [S] which solves as

Bigger systems of higher polynomials cannot be solved algebraically => use ODEs (above)

][4][][(][][(5.][][ __2

____ TSEdTTSEdTTSEdT SKESKESKSES

0)0]([,0)0]([

]][[][][

][

]][[][][

][

__

__

SE

K

SESS

d

Sd

K

SEEE

d

Ed

SEdT

SEdT

Michaelis-Menten ModelMichaelis-Menten Model [S] vs. [S[S] vs. [STT] ]

(3)

IESdSEdIEdT

IESdSEdSEdT

IESdSEdIEdSEdT

KK

ISE

K

IEII

KK

ISE

K

SESS

KK

ISE

K

IE

K

SEEE

______

______

________

]][][[]][[][][0

]][][[]][[][][0

]][][[]][[]][[][][0

E ES

EI ESI

E ES

EI ESI

E ES

EI ESI

SEIdIEdIEdT

SEIdIEdSEdT

SEIdIEdIEdSEdT

KK

SIE

K

IEII

KK

SIE

K

SESS

KK

SIE

K

IE

K

SEEE

______

______

________

]][][[]][[][][0

]][][[]][[][][0

]][][[]][[]][[][][0

E ES

EI

EIT

EST

EIEST

K

IEII

K

SESS

K

IE

K

SEEE

]][[][][0

]][[][][0

]][[]][[][][0

ESIEIT

ESIEST

ESIEIEST

K

ISE

K

IEII

K

ISE

K

SESS

K

ISE

K

IE

K

SEEE

]][][[]][[][][0

]][][[]][[][][0

]][][[]][[]][[][][0

E ES

EI ESI

E

EI ESI

E ES

ESI

E

EI ESI

E ES

ESI

=

=E

EI

E

ESI

E ES E

Competitive inhibition

uncompetitive inhibition if kcat_ESI=0

E | ES

EI | ESI

noncompetitive inhibition Example of Kd=Kd’ Model

==

Let p be the probability that an E molecule is undamaged.Then in each model [ET] can be replace with p[ET] to double the number of models to 2*(23+3+1)=24.

E

EI

E

ESI

E ESKj=0 Models

][][0

]][[][][0

]][[][][0

II

K

SESS

K

SEEE

T

EST

EST

0][ and ][][][ ],[][ else ,0][ and ][][][ ],[][ )][][( if EESSEESSSEESESSE TTTTTTTT

as KES approaches 0

Enzyme, Substrate and InhibitorEnzyme, Substrate and Inhibitor

Total number of spur graph models is 16+4=20 Radivoyevitch, (2008) BMC Systems Biology 2:15

Rt Spur Graph ModelsRt Spur Graph Models

RRttRRtRt

T

RRttRRtRRRtT

K

tR

K

tR

K

tRtt=

K

tR

K

tR

K

R

K

tRRRp=

222

2222

20

2220

.0)0(;0)0(

2

222

222

2222

tR

K

tR

K

tR

K

tRtt=

d

td

K

tR

K

tR

K

R

K

tRRRp=

d

Rd

RRttRRtRtT

RRttRRtRRRtT

R RR

RRtt

RRt Rt

R

RRtt

RRt Rt

R RR

RRtt

Rt

R RR

RRt Rt

R RR

RRtt

RRt

R

RRtt

Rt

R

RRt Rt

R RR

Rt

3B 3C 3D 3E

3F 3G 3H

3A

R

RRtt

RRt

R RR

RRtt

R RR

RRt

R

Rt

R

RRtt

R

RRt

3J3I 3K

R RR R

3L 3M 3N 3O 3P

R

Rt

R

RRtt

R

RRt

R RR

3Q 3R 3S 3T

R = R1 t = dTTP

for dTTP induced R1 dimerization

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

Kd_R_R

Kd_Rt_R

Kd_Rt_Rt

Kd_R_t

Kd_R_tKd_RRt_t

Kd_RR_t=

=

=

=

2A

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

Kd_R_R

Kd_Rt_R

Kd_Rt_Rt

Kd_R_t

Kd_R_tKd_RRt_t

Kd_RR_t==

=

2B

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

Kd_R_R

Kd_Rt_R

Kd_Rt_Rt

Kd_R_t

Kd_R_tKd_RRt_t

Kd_RR_t

2C

|

|

|

|

|

=

=

=

|

|

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

Kd_R_R

Kd_Rt_R

Kd_Rt_Rt

Kd_R_t

Kd_R_tKd_RRt_t

Kd_RR_t

2D

|

|

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

Kd_R_R

Kd_Rt_R

Kd_Rt_Rt

Kd_R_t

Kd_R_tKd_RRt_t

Kd_RR_t

2E

=

=

2A 2B 2C 2D 2E 2G 2I 2K

|

|

|

|

|

|

| |

|

|

|

|

|

|

|

=

= =

=

2F 2H 2J 2L 2M 2N

Figure 3. Spur graph models. The following models are equivalent: 3A=2F, 3B=2H, 3C=2J, 3D=2L, 3E=2N

Acyclic spanning subgraphs are reparameterizations of equilvalent models

2F0

Figure 2. Grid graph models.

2F1 2F2 2F3 2F4 2F5 2F6 2F7 2F8Standardize: take E-shapes and sub E-shapes as defaults

Use n-shapes if needed. Other shapes are possible

3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T3A

Rt Grid Graph ModelsRt Grid Graph Models

5 10 15

10

01

20

14

01

60

18

0

Total [dTTP] (uM)

Ave

rag

e M

ass

(kD

a)

AICc = 2P+N*log(SSE/N)+2P(P+1)/(N-P-1)

Data and fit from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry 40(6), 1651-166

~10-fold deviations from Scott et al. (initial values)

titration model has lowest AIC

Radivoyevitch, (2008) BMC Systems Biology 2:15

Application to DataApplication to Data

][

][2][2][2180

][

)1]([][90

TT

Ta R

RRttRRtRR

R

pRRM

2E

=

=

R

RRtt

3Rp

Infinitely tight binding situation wherein free molecule annihilation (the initial linear ramp) continues in a one-to-one fashion with increasing [dTTP]T until [dTTP]T equals [R1]T

=7.6 µM, the plateau point where R exists solely as RRtt.

Experiment becomes a titration scan of [tT] to estimate [RT],

but [RT]=7.6 µM was already known.

Model Space Fit with New DataModel Space Fit with New Data

Radivoyevitch, (2008) BMC Systems Biology 2:15

Yeast R1 structure. Chris Dealwis’ Lab, PNAS 102, 4022-4027, 2006

One additional data point here would reject 3Rp

If so, new data here would be logical next

No need to constrain data collection to orthogonal profiles

Model Space PredictionsModel Space Predictions

Best next 10 measurements if 3Rp is Best next 10 measurements if 3Rp is rejected rejected

Show new Show new datadata+ R Code+ R Code

Fast Total Concentration Constraint (TCC; i.e. g=0) solvers are critical to model

estimation/selection. TCC ODEs (#ODEs = #reactants) solve TCCs faster than kon =1 and koff = Kd systems (#ODEs = #species = high # in combinatorially complex situations)

Semi-exhaustive approach = fit all models with same number of parameters as parallel batch, then fit next batch only if current shows AIC improvement over previous batch. This reduces Rt model space fitting times by a factor of 5.

Comments on Methods

12

1

68

1

44

1

22

1

12

0

68

0

44

0

22

1

642

642

0

6420

i aR

i

i aR

i

i aR

i

i Ra

i

T

i aR

i

i aR

i

i aR

i

i Ra

i

T

iiii

iiii

K

aRi

K

aRi

K

aRi

K

aRiaa=

K

aR

K

aR

K

aR

K

aRRRp=

2+5+9+13 = 28 parameters => 228=2.5x108 spur graph models via Kj=∞ hypotheses

(if average fit = 60 sec, then need 60 cores running for 8 straight years)

28 models with 1 parameter, 428 models with 2, 3278 models with 3, 20475 with 4

[dATP] uM

a = dATP

Kashlan et al. Biochemistry 2002 41:462

)2(][

1

2

0

31

2

21

31

2

32

21

1

E

KK

S

K

S

K

S

KK

Sk

K

Sk

K

Sk

v

Human Thymidine Kinase 1Human Thymidine Kinase 1

Assumptions

(1)binding of enzyme monomers to form enzyme dimers is approximately infinitely tight across the enzyme concentrations of interest

(2)behavior of any higher order enzyme structures (if they exist) is dominated by the behavior of dimers within such structures

(3)enzyme concentrations are low enough that free substrate concentrations approximately equal total substrate concentrations.

Max(E0)=200 pMMin(ST)=50 nM => OK

OK

Questionable (but I need to start simple)

row Eq k1 k2 k3 K1 K2 K3 Numerator Denominator

1s 3 k1 k1 k3 K1 K1 K3 31

2

31

1 22KK

Sk

K

Sk

31

2

1

21KK

S

K

S

2 3 k3 k2 k3 K1 K1 K3 31

2

31

23 2KK

Sk

K

Skk

31

2

1

21KK

S

K

S

3 3 k1 k3 k3 K1 K1 K3 31

2

31

31 2KK

Sk

K

Skk

31

2

1

21KK

S

K

S

4 3 k1 k1 k3 K1 K2 K1 21

2

321

1 211

K

SkS

KKk

21

2

21

111

K

SS

KK

5 3 k3 k2 k3 K1 K2 K1 21

2

32

21

3 211

K

SkS

Kk

Kk

21

2

21

111

K

SS

KK

6 3 k1 k3 k3 K1 K2 K1 21

2

32

31

1 211

K

SkS

Kk

Kk

21

2

21

111

K

SS

KK

7 3 k1 k1 k3 K1 K2 K2 21

23

211

211

KK

SkS

KKk

21

2

21

111

KK

SS

KK

8 3 k3 k2 k3 K1 K2 K2 21

23

2

2

1

3 2

KK

SkS

K

k

K

k

21

2

21

111

KK

SS

KK

9 3 k1 k3 k3 K1 K2 K2 21

23

2

3

1

1 2

KK

SkS

K

k

K

k

21

2

21

111

KK

SS

KK

.10 4 k1+k2=2k3 K1 K1 K3

31

2

132

KK

S

K

Sk

31

2

1

21KK

S

K

S

11 11b k1+k2=2k3 K1 K2 K1 2

1

2

212

2

1

1

K

SkkS

K

k

K

k

21

2

21

111

K

SS

KK

12 11b k1+k2=2k3 K1 K2 K2 21

2

212

2

1

1

KK

SkkS

K

k

K

k

21

2

21

111

KK

SS

KK

13 4 k3 k3 k3 K1 K2 K3

31

2

213

112

KK

SS

KKk

31

2

21

111

KK

SS

KK

14s 4 k3 k3 k3 K1 K1 K3

31

2

132

KK

S

K

Sk

31

2

1

21KK

S

K

S

15 4 k3 k3 k3 K1 K2 K1

21

2

213

115.2

K

SS

KKk

21

2

21

111

K

SS

KK

16 4 k3 k3 k3 K1 K2 K2

21

2

213

115.2

KK

SS

KKk

21

2

21

111

KK

SS

KK

17 5 k1 k2 k3 K1 K1 K1 2

21

3

1

21 2S

K

kS

K

kk

2

1

1

K

S

18s 5 k1 k1 k3 K1 K1 K1 2

21

3

1

1 22S

K

kS

K

k

2

1

1

K

S

19 5 k3 k2 k3 K1 K1 K1 2

21

3

1

23 2S

K

kS

K

kk

2

1

1

K

S

20 5 k1 k3 k3 K1 K1 K1 2

21

3

1

31 2S

K

kS

K

kk

2

1

1

K

S

21 8 k3 k3 k3 K1 K1 K1 1

32K

Sk

1

1K

S

22 3 X k2 k3 ∞ K2 K3 2

31

3

2

2 2S

KK

kS

K

k

31

2

2

11

KK

SS

K

23 4 X k3 k3 ∞ K2 K3

2

3123

15.2 S

KKS

Kk

31

2

2

11

KK

SS

K

24 3 X k2 k3 ∞ K2 K2 2

21

3

2

2 2S

KK

kS

K

k

21

2

2

11

KK

SS

K

25 4 X k3 k3 ∞ K2 K2

2

2123

15.2 S

KKS

Kk

21

2

2

11

KK

SS

K

26 3 k1 X k3 K1 ∞ K3 2

31

3

1

1 2S

KK

kS

K

k

31

2

1

11

KK

SS

K

27 4 k3 X k3 K1 ∞ K3

2

3113

15.2 S

KKS

Kk

31

2

1

11

KK

SS

K

28 6 k1 X k3 K1 ∞ K1 2

21

3

1

1 2S

K

kS

K

k

21

2

1

11

K

SS

K

29 7 k3 X k3 K1 ∞ K1

2

211

3

15.2 S

KS

Kk

21

2

1

11

K

SS

K

30 8 k1 k2 X K1 K2 ∞ SK

k

K

k

2

2

1

1 SKK

21

111

31 8 k1 k1 X K1 K2 ∞ SKK

k

211

11 SKK

21

111

32 8 k1 k2 X K1 K1 ∞ SK

kk

1

21 SK1

21

33s 8 k1 k1 X K1 K1 ∞ SK

k1

1

2 SK1

21

34s 9 X X k3 ∞ ∞ K3 31

2

32KK

Sk

31

2

1KK

S

)2(][

1

2

0

31

2

21

31

2

32

21

1

E

KK

S

K

S

K

S

KK

Sk

K

Sk

K

Sk

v

)3(

21 221

221

SS

SVSVv

)1(

150

50max

n

n

S

S

S

SV

v

)3(21 2

21

221

SS

SVSVv

)4(

21 221

221

SS

SSVv

)5(

1 21

2211

S

SVSVv

)6(1 22

11

2211

SS

SVSVv

)7(

1

5.22

11

2211

SS

SSVv

)8(1 1

1

S

SVv

)9(1 2

1

21

S

SVv

3-parameter models

2-parameter models

4-parameter model

)18(21 2

21

1

SS

SVv

)19(21 2

21

22

SS

SVv

)20(

21

5.2

21

221

SS

SSVv

.

)21(21

)5(.2

21

222

SS

SSVv

)22()(1

)5(.2

221

221

SS

SSVv

)23()(1

)5(.2

2121

2211

SS

SSVv

)24(

1 21

1

S

SVv

)25(

1 21

221

S

SVv

)26(

1

5.2

1

2211

S

SVSVv

)27(1 22

11

1

SS

SVv

)28(1 22

11

221

SS

SVv

Eq. (25): k1 = k2 = 0, k3 > 0 and K1 = K2 = K3

)7(

1

5.22

11

2211

SS

SSVv

)25(

1 21

221

S

SVv

Fits to human thymidine kinase 1 data of Birringer et al. Protein Expr Purif 2006, 47(2):506-515

Eq. (7): k1 = k3 and K2 = ∞ and K1 = K3

EQ SSE AICc α1 V∞ V1 α2 neq25 0.00026 -57.67 3.765 0.151eq28 0.00028 -56.93 2.756 0.145eq7 0.00029 -56.86 1.658 0.145eq26 0.00034 -55.35 2.316 0.156eq8 0.00039 -54.1 1.37 0.158eq22 0.00026 -50.48 0.833 0.148 6.732eq20 0.00027 -50.23 1.293 0.148 4.635eq9 0.00061 -49.99 3.645 0.132eq1 0.00028 -49.88 1.941 0.144 1.297eq6 0.00028 -49.71 2.747 0.145 0eq4 0.00029 -49.6 0.815 0.146 2.704eq24 0.00065 -49.5 0.272 0.149eq21 3.00E-04 -49.33 1.459 0.149 3.996eq19 0.00032 -48.79 0.946 0.144 6.156eq23 0.00033 -48.49 2.145 0.155 3.006eq18 0.00037 -47.41 0.576 0.2 0.022eq5 0.00037 -47.25 0.96 0.157 0.207eq27 0.00105 -45.19 0.303 0.127eq3 0.00026 -38.57 2.5287 0.1496 0.0205 10.1832

Same top 4 models found by fits to

Frederiksen H, Berenstein D, Munch-Petersen B: Eur J Biochem 2004, 271(11):2248-2256.

1e-02 1e-01 1e+00 1e+01 1e+02

0.0

0.2

0.4

0.6

0.8

1.0

dT (uM)

no

rma

lize

d fl

ux

Eq. 25Eq. 7

Hill rises from 5th, 6th and worse to 2nd if older data is merged in

Conclusion: across the board Hill fits should be avoided since it is not biologically plausible at non-integer n and since it needs competition to guide subsequent experiments

AcknowledgementsAcknowledgements

Case Comprehensive Cancer CenterCase Comprehensive Cancer Center NIH (K25 CA104791)NIH (K25 CA104791) Chris Dealwis (CWRU Pharmacology)Chris Dealwis (CWRU Pharmacology) Sanath Wijerathna (CWRU Pharmacology)Sanath Wijerathna (CWRU Pharmacology) Anders Hofer (Umea) Anders Hofer (Umea) Thank you Thank you