Mathematical Models of dNTP Supply

Tom Radivoyevitch, PhDAssistant Professor

Epidemiology and Biostatistics

CCCC Developmental Therapeutics Program

Single-Loop Temperature Control

5

0

10

+-

setpointKp

Ki∫ Σ hot plate water temperature

-5 0 5 10 15 20

2535

45

minutes

tem

pera

ture

(C)

-5 0 5 10 15 20

0246

8

minutes

cont

rol e

ffort

-5 0 5 10 15 20

2535

45

minutes

tem

pera

ture

(C)

-5 0 5 10 15 20

0123

45

minutes

cont

rol e

ffort

-5 0 5 10 15 20

2535

45

minutes

tem

pera

ture

(C)

-5 0 5 10 15 20

0123

45minutes

cont

rol e

ffort

-5 0 5 10 15 20

2535

45

minutes

tem

pera

ture

(C)

-5 0 5 10 15 20

02

46

8

minutes

cont

rol e

ffort

controllerprocess

process output = controller input; process input = controller output (= control effort)

Power Plant Process & Controls

air/o2

stack

turbine

turbine

condenser

gas flow

recirc

T

T

fuel

P

θ

PI+

-Megawatts Demanded

Megawatts Supplied

PI Reheat TempSetpoint (1050F: variations break expensive thick pipes)

PI Setpoint (3500 psi)

+

-+

-

Boiler pressure ~ plasma [dN]Fuel input ~ de novo dNTP supply Cold Water bypass ~ cN-II/dNK cycle ~ police car in idle to catch speedersTwo temp control ~ drugs kill bad and good, rescue saves good and bad

Ultimate Goal

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

• Safer flying airplanes with autopilots• Individualized, feedback-based therapies

FuturePresent

EXPERIMENTALBIOLOGY

MODELS CONTROLLER-MODEL PAIRSdata

simulations

proposed controller

process model development control system design

or

dNTP Supply

Many anticancer agents target or traverse this system.

UDP

CDP

GDP

ADP

dTTP

dCTP

dGTP

dATP

dT

dC

dG

dADNA

dUMP

dU

TS

DCTD

dCK

DN

A po

lym

eras

eTK1

cytosol

mitochondria

dT

dC

dG

dA

TK2

dGK

dTMPdCMP

dGMP

dAMP

dTTPdCTP

dGTP

dATP

5NT

NT2

cytosol

nucleus

dUDP

dUTPdUTPase dN

dN

dCK

flux activation inhibition

ATPordATP

RN

R

dCK

Metabolism of Metabolism of dNTPsdNTPs + Analogs+ Analogs

Metabolism of Metabolism of DNA + DrugDNA + Drug--DNADNA

Damage DrivenDamage Drivenor or

SS--phase Drivenphase Driven

Nucleoside Nucleoside demand is eitherdemand is either

Focus on cancers Focus on cancers caused by DNA repair caused by DNA repair system failuressystem failures

DNA repairDNA repair

Problem StatementProblem Statement

salvage

De novo

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

y

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x

y2 -

y

MMR+

MMR-

Best time to hit with IRIU-DNA levels

Time

MMR- Treatment Hypothesis

dNTPs + Analogs DNA + Drug-DNA

Damage Drivenor

S-phase Driven

dNTP demand is either

DNA repair

Salvage

De novo

IUdR

p53- Treatment Hypothesis

• Residual DSBs at 24 hours kill cells• dNTP supply inhibition retards DSB repair• p53- cells are slower at DSB repair• Best dNTP supply inhibition timing post IR

is just after p53+ cells complete DSB repair

• Questions: Prolonged RNR↓ => plasma [dN]↓? Compensation by RNR

overexpression? Is dCK expression also increased?

-+

24 h

R Packages

Combinatorially Complex Equilibrium Model Selection

(ccems, CRAN 2009)

Systems Biology Markup Language interface to R

(SBMLR, BIOC 2004)

Model networks of enzymes Model enzymes

R1

R2 R2

R1 R1

R1 R1

R1 R1

R1 R1

R1

R1

R1 R1

R1 R1

R1

R1

R2 R2

R2 R2

Enzyme Modeling Overview

• Model enzymes as quasi-equilibria (e.g. E ES) • Combinatorially Complex Equilibria:

• few reactants => many possible complexes• R package: Combinatorially Complex Equilibrium Model Selection

(ccems) implements methods for activity and mass data• Hypotheses: complete K = ∞ [Complex] = 0 vs binary K1 = K2

• Generate a set of possible models, fit them, and select the best • Model Selection: Akaike Information Criterion (AIC)

• AIC decreases with P and then increases• Billions of models, but only thousands near AIC upturn

• Generate 1P, 2P, 3P model space chunks sequentially• Use structures to constrain complexity and simplicity of models

Ribonucleotide Reductase

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

ATP, dATP, dTTP, dGTP bind to selectivity site

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

5 catalytic site states x 5 s-site states x 3 a-site states x 2 h-site states = 150 statesÞ (150)6 different hexamer complexes => 2^(150)6 models 2^(150)6 = ~1 followed by a trillion zeros1 trillion complexes => 1 trillion (1 followed by only 12 zeros) 1-parameter models

ATP activates at hexamerization site??

R2 R2

RNR is Combinatorially Complex

Michaelis-Menten Model

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

E + S ES

mm

m

m

mT

Tmm

mT

T

TT

TT

KSSV

KSKSV

KSKSkE

kEKSKS

KSkEv

EESEESEESkE

kEEES

EEES

ESkE

EPEESPEkEESkv

][][

1/][/][

1/][/][][

][1/][

101/][

/][][

1]/[][10

1]/[][]/[][][

][][][

][0][][

][][

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

maxmax1

1

1

1

1

1

mm K

SEES

ESESK ][

][][

][]][[

Þbut so

Key perspective

Free Concentrations Versus Totals

0.005 0.010 0.020 0.050 0.100 0.200 0.500

020

4060

8010

0

Total [r] (uM)

Per

cent

Act

ivity 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

_

_

SET

SET

KSESS

KSEEE

Þ

Þ

SE

T

SE

T

T

SE

SET

SE

TSE

T

KS

KS

EESversus

KS

KS

EES

KS

EEKSEE

_

_

_

_

_

_ ][1

][

][][][1

][

][][][1

1][][][1][][

Substitute this in here to get a quadratic in [S] whose solution is

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

][4][][(][][(5.][][ _2

__ TSETTSETTSET SKESKESKSES

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

]][[][][][

]][[][][][

_

_

SE

KSESS

dSd

KSEEE

dEd

SET

SET

[S] vs. [ST]

(3)

Enzyme, Substrate and InhibitorE ES

EI ESI

E ES

EI ESI

SEIIEIET

SEIIESET

SEIIEIESET

KKSIE

KIEII

KKSIE

KSESS

KKSIE

KIE

KSEEE

___

___

____

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

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

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

E ES

EI

EIT

EST

EIEST

KIEII

KSESS

KIE

KSEEE

]][[][][0

]][[][][0

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

ESIEIT

ESIEST

ESIEIEST

KISE

KIEII

KISE

KSESS

KISE

KIE

KSEEE

]][][[]][[][][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 K=K’ Model

==

dTTP induced R1 dimerizationComplete K Hypotheses

Radivoyevitch, (2008) BMC Systems Biology 2:15

RRttRRtRt

T

RRttRRtRRRtT

KtR

KtR

KtRtt=

KtR

KtR

KR

KtRRRp=

222

2222

20

2220

.0)0(;0)0(

2

222

222

2222

tRKtR

KtR

KtRtt=

dtd

KtR

KtR

KR

KtRRRp=

dRd

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

IJJJJJIJ JJJI JIJJ

IJIJ IJJI JJII

JJJJ

R

RRtt

RRt

R RR

RRtt

R RR

RRt

R

Rt

R

RRtt

R

RRt

JIIJIIJJ JIJI

R RR RIJII IIIJ IIJI JIII IIII

R

Rt

R

RRtt

R

RRt

R RRI0II III0 II0I 0III

R = R1 t = dTTP

(RR, Rt, RRt, RRtt)

Binary K Hypotheses

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=

=

=

=

|

|

|

R Rt t

RRt t

Rt R t

RRt t

RRtt

KR_R

KR_t

KRRt_t

KRR_t=

=

=

==

===

==

=

==

===

==

=

R Rt t

RRt t

Rt R t

RRt t

Rt Rt RRtt

KR_R

KRt_R

KRt_Rt

KR_t

KR_t

|

|

|

|

|

|

| |

|

|

|

|

| |

|

?

?

HIFF

HDFFHDDD

=

5 10 15

100

120

140

160

180

Total [dTTP] (uM)

Ave

rage

Mas

s (k

Da)

III0mIIIJHDFF

Fits to Data

][][2][2][22

][)1]([][)(

)(

11TT

T

RRRttRRtRRM

RpRRMyE

yEy

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

Data from Scott, C. P., et al. (2001) Biochemistry 40(6), 1651-166

Radivoyevitch, (2008) BMC Systems Biology 2:15

HDFF

==

R

RRtt

IIIJIII0m

Model Parameter Initial Value

Optimal Value

Confidence Interval

1 III0m m1 90.000 82.368 (79.838, 84.775)

  SSE 4397.550 525.178

  AIC 71.965 57.090

2 IIIJ R2t2 1.000^3 2.725^3 (2.014^3, 3.682^3)

  SSE 2290.516 557.797

  AIC 67.399 57.512

27 HDFF R2t0 1.000 12369.79 (0, 1308627507869)

  R1t0_t 1.000 1.744 (0.003, 1187.969)

  R2t0_t 1.000 0.010 (0.000, 403.429)

  SSE 25768.23 477.484

  AIC 105.342 77.423

RRttRRtRt

T

RRttRRtRRRtT

KtR

KtR

KtRtt=

KtR

KtR

KR

KtRRRp=

222

2222

20

2220

.0)0(;0)0(

2

222

222

2222

tRKtR

KtR

KtRtt=

dtd

KtR

KtR

KR

KtRRRp=

dRd

RRttRRtRtT

RRttRRtRRRtT

jitR

ji

KtR=tR

ji

ATP-induced R1 Hexamerization

2+5+9+13 = 28 parameters => 228=2.5x108 spur graph models via Kj=∞ hypotheses28 models with 1 parameter, 428 models with 2, 3278 models with 3, 20475 with 4

R = R1X = ATP

18

6

612

4

46

2

22

1

18

6

612

4

46

2

22

1

642

642

0

6420

i XR

i

i XR

i

i XR

i

i RX

i

T

i XR

i

i XR

i

i XR

i

i RX

i

T

iiii

iiii

KXRi

KXRi

KXRi

KXRiXX=

KXR

KXR

KXR

KXRRR=

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

Data of Kashlan et al. Biochemistry 2002 41:462

Fits to R1 Mass Data

28 of top 30 did not include an h-site term; 28/30 ≠ 503/2081 with p < 10-16. This suggests no h-site. Top 13 all include R6X8 or R6X9, save one, single edge model R6X7 This suggests less than 3 a-sites are occupied in hexamer. Radivoyevitch, T. , Biology Direct 4, 50 (2009).

2088 Models with SSE < 2 min (SSE)

142 144 146 148 150 152 154

0.00

0.05

0.10

0.15

0.20

0.25

AIC

dens

ity

no h site (508)h site (1580)

A

142 144 146 148 150

0.05

0.15

0.25

AIC

dens

ity

no h site (77)h site (78)

B

146 148 150 152 154

0.00

0.05

0.10

0.15

0.20

0.25

0.30

AIC

dens

ity

no h site (287)h site (1174)

C

148 150 152 154

0.0

0.1

0.2

0.3

0.4

0.5

AIC

dens

ity

no h site (146)h site (329)

D

50 100 200 500 1000 200010

020

030

040

050

0

[ATP] (uM)

Mas

s (k

Da) R6X8 141.23

R6X9 142.88R6X7 143.12R6X10 146.12R6X6 148.92R6X11 149.60R6X12 152.82R6X13 155.68R6X14 158.17R6X15 160.33R6X16 162.20R6X17 163.84R6X18 165.26

Data from Kashlan et al. Biochemistry 2002 41:462

~1/2 a-sites not occupied by ATP? R

1 R

1

R1 R1

R1 R1

aa

[ATP]=~1000[dATP]So system prefers to have 3 a-sites empty and ready for dATPInhibition versus activation is partly due to differences in pockets

a

a

a

a

UDP

CDP

GDP

ADP

dTTP

dCTP

dGTP

dATP

dT

dC

dG

dADNA

dUMP

dU

TS

DCTD

dCK

DN

A po

lym

eras

e

TK1

cytosol

mitochondria

dT

dC

dG

dA

TK2

dGK

dTMPdCMP

dGMP

dAMP

dTTPdCTP

dGTP

dATP

5NT

NT2

cytosol

nucleus

dUDP

dUTPdUTPase dN

dN

dCK

flux activation inhibition

ATPordATP

RN

R

dCK

Fits to RNR Activity Data

36

3

26

2

12

1

2

6622

. 66220iii XR

i

XR

i

XR

i

RT K

XRKXR

KXR

KRRR=

ij XR

ijij

KXR=XR

][][][][ 321 63

62

21

20

iii XRkXRkXRkRkk

[ATP] (M)

CD

P R

educ

tase

Act

ivity

(1/s

ec)

0.1

0.15

0.2

0.25

0.3

0 1000 2000 3000 10000

4.13.18 (AIC=-65, SSE=0.00257) 2.7.12 (AIC=-58.9, SSE=0.00386)i1 4 i2 13 i3 18 k0 0.31 K0 3.66 K1 25.86 K2 21.26 K3 56.23i1 2 i2 7 i3 12 k0 0.31 K0 3.86 K1 14.29 K2 12.48 K3 54.67

Distribution of Model Space SSEs

Models with occupied h-sites are in red, those without are in black. Sizes of spheres are proportional to 1/SSE.

0.002 0.004 0.006 0.008 0.010 0.012

050

100

150

200

250

300

SSE

Pro

babi

lity

Den

sity

occupied h-sites (171 models)no occupied h-sites (54 models)

Microfluidics

Figure 8. T. Thorsen et al. (S. R. Quake Lab) Science 2002

Figure 9. J. Melin and S. R. Quake Annu. Rev. Biophys. Biomol. Struct. 2007. 36:213–31

Figure 9 shows how a peristaltic pump is implemented by three valves that cycle through the control codes 101, 100, 110, 010, 011, 001, where 0 and 1 represent open and closed valves; note that the 0 in this sequence is forced to the right as the sequence progresses.

Adaptive Experimental DesignsFind best next 10 measurement conditions given models of data collected.Need automated analyses in feedback loop of automatic controls of microfluidic chips

µFluidic M-inputCMPM

C1

Mixing Control bits

C2C3

CM

…

TPC1 …C2 C2 C3 buff buff buff

N-plug stream (C1:2C2:C3:C4)/N

Output

Output

C4

Streams of pulses

Filtered output

(a)

(b)

Output

Output Mixer

Dye-3 (C3)Dye-2 (C2)

Solvent Dye-1 (C1)

0b

2b

1b

3b

Dye-3

C3

Dye-1

Dye-2

C2

Water

C1

Mixing Channel

Output

Control Lines

3 mm

C1

C2

C3

Flow velocity = 2 cm/sTP=100 ms, M=4N=20, Levels = 64

Why Systems Biology

Emphasis is on the stochastic component of the model.

Is there something in the black box or are the input wires disconnected from the output wires such that only thermal noise is being measured? Do we have enough data?

Model components: (Deterministic = signal) + (Stochastic = noise)

Statistics Engineering

Emphasis is on the deterministic component of the model

We already know what is in the box, since we built it. The goal is to understand it well enough to be able to control it.

Predict the best multi-agent drug dose time course schedules

Increasing amounts of data/knowledge

Indirect Approach pro-B Cell Childhood ALL

• T: TEL-AML1 with HR • t : TEL-AML1 with CCR• t : other outcome

• B: BCR-ABL with CCR• b: BCR-ABL with HR• b: censored, missing, or

other outcome

B

b

b

b

b

b

b

b

bb

bb

b

b

b

b

tt t

tt

t

t t

ttt

tt

t

t

t

t

t

t

t

t

t

tt

t

t

t

t

t

t

tt

t

t

t

t

t

t

t

t

t

tt

t

t

tt

t

t

t

t

t

tt

tt

t

T

T

T

t

t

t

t

t

tt

t

t t

tt

t

tt

t

t

t

t

0 2 4 6 8 10 12 140

200

400

600

800

1000

1200

DNTS Flux (uM/hr)

DN

PS

Flu

x (u

M/h

r)

Ross et al: Blood 2003, 102:2951-2959 Yeoh et al: Cancer Cell 2002, 1:133-143

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

Folate Cycle (dTTP Supply)

THF

CH2THF

CH3THF

CHOTHF

DHF

CHODHFHCHO

GAR

FGAR

AICAR

FAICAR

dUMP dTMP

NADP+ NADPHNADP+ NADPH

NADP+ NADPH

MetHcys

Ser

Gly

GART

ATICATIC

TS

ATP

ADP

11R

2R 2

3

4

10

9 8

5 6

7

12 11

13

HCOOH

MTHFDMTHFR

MTR

DHFR

SHMT

FTS

FDS

Morrison PF, Allegra CJ: Folate cycle kinetics in human breast cancer cells. JBiolChem 1989, 264:10552-10566.

Conclusions

• For systems biology to succeed:– move biological research toward systems which

are best understood– specialize modelers to become experts in

biological literatures (e.g. dNTP Supply) • Systems biology is not a service

Acknowledgements

• Case Comprehensive Cancer Center• NIH (K25 CA104791)• Charles Kunos (CWRU)• John Pink (CWRU)• James Jacobberger (CWRU)• Anders Hofer (Umea) • Yun Yen (COH)• And thank you for listening!

Comments on Methods

• 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.

Conjecture

• Greater X/R ratios dominate at high Ligand concentrations. In this limit the system wants to partition as much ATP into a bound form as possible

1e+02 1e+03 1e+04 1e+05 1e+06

010

020

030

040

050

0

[ATP] (uM)

Mas

s (k

Da)

R2X4.R6X8R2X3.R6X9R2X3.R6X12

ccems Sample Code

library(ccems) # Ribonucleotide Reductase Exampletopology <- list( heads=c("R1X0","R2X2","R4X4","R6X6"), sites=list( # s-sites are already filled only in (j>1)-mers a=list( #a-site thread m=c("R1X1"), # monomer 1 d=c("R2X3","R2X4"), # dimer 2 t=c("R4X5","R4X6","R4X7","R4X8"), # tetramer 3 h=c("R6X7","R6X8","R6X9","R6X10", "R6X11", "R6X12") # hexamer 4 ), # tails of a-site threads are heads of h-site threads h=list( # h-site m=c("R1X2"), # monomer 5 d=c("R2X5", "R2X6"), # dimer 6 t=c("R4X9", "R4X10","R4X11", "R4X12"), # tetramer 7 h=c("R6X13", "R6X14", "R6X15","R6X16", "R6X17", "R6X18")# hexamer 8 ) ))g=mkg(topology,TCC=TRUE) dd=subset(RNR,(year==2002)&(fg==1)&(X>0),select=c(R,X,m,year))cpusPerHost=c("localhost" = 4,"compute-0-0"=4,"compute-0-1"=4,"compute-0-2"=4)top10=ems(dd,g,cpusPerHost=cpusPerHost, maxTotalPs=3, ptype="SOCK",KIC=100)