1. Combinatorially Complex Equilibrium Model Selection Tom Radivoyevitch Assistant Professor Epidemiology and Biostatistics Case Western Reserve University Email: txr24@case.edutxr24@case.edu Website: http://epbi-radivot.cwru.edu/http://epbi-radivot.cwru.edu/

2. Ultimate Goal Present Future Safer flying airplanes with autopilots Safer flying airplanes with autopilots Ultimate Goal: individualized, state feedback based clinical trials Ultimate Goal: individualized, state feedback based clinical trials Better understanding => better control Better understanding => better control Conceptual models help trial designs today Conceptual models help trial designs today Computer models of airplanes help train pilots and autopilots Computer models of airplanes help train pilots and autopilots Radivoyevitch et al. (2006) BMC Cancer 6:104 Systems Cancer Biology

3. 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 DNA polymerase TK1 cytosol mitochondria dT dC dG dA TK2 dGK dTMP dCMP dGMP dAMP dTTP dCTP dGTP dATP 5NT NT2 cytosol nucleus dUDP dUTP U-DNA dUTPase dN dCK flux activation inhibition ATP RNR dCK

4. R1 R2 R1 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 R2 2 vs. three R2 2 Total concentrations of R1, R2 2, dTTP, dGTP, dATP, ATP and NDPs control the distribution of R1-R2 complexes and this changes in S, G 1 -G 2 and G 0 ATP activates at hexamerization site?? RNR Literature R2

5. Michaelis-Menten Model With RNR: no NDP and no R2 dimer => k cat of complex is zero. Otherwise, many different R1-R2-NDP complexes can have many different k cat values. E + S ES

6. 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 R=R1 r=R2 2 G=GDP t=dTTP 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) Michaelis-Menten Model [S] vs. [S T ] [S] vs. [S T ] (3)

7. E ES EI ESI E ES EI ESI E ES EI ESI E ES EI 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 k cat_ESI =0 E | ES EI | ESI noncompetitive inhibition Example of K d =K d ’ Model = = Let p be the probability that an E molecule is undamaged. Then in each model [E T ] can be replace with p[E T ] to double the number of models to 2*(2 3 +3+1)=24. E EI E ESI E ES K j =0 Models as K ES approaches 0 Enzyme, Substrate and Inhibitor

8. Total number of spur graph models is 16+4=20 Radivoyevitch, (2008) BMC Systems Biology 2:15 Rt Spur Graph Models 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 3D3E 3F3G3H 3A R RRtt RRt R RR RRtt R RR RRt R Rt R RRtt R RRt 3J 3I 3K R RR R 3L 3M3N 3O 3P R Rt R RRtt R RRt R RR 3Q 3R 3S3T R = R1 t = dTTP for dTTP induced R1 dimerization

9. R t RR t Rt R t RRt t Rt RRtt K d_R_R K d_Rt_R K d_Rt_Rt K d_R_t K d_RRt_t K d_RR_t = = = = 2A R t RR t Rt R t RRt t Rt RRtt K d_R_R K d_Rt_R K d_Rt_Rt K d_R_t K d_RRt_t K d_RR_t = = = 2B R t RR t Rt R t RRt t Rt RRtt K d_R_R K d_Rt_R K d_Rt_Rt K d_R_t K d_RRt_t K d_RR_t 2C | | | | | = = = | | R t RR t Rt R t RRt t Rt RRtt K d_R_R K d_Rt_R K d_Rt_Rt K d_R_t K d_RRt_t K d_RR_t 2D | | R t RR t Rt R t RRt t Rt RRtt K d_R_R K d_Rt_R K d_Rt_Rt K d_R_t K d_RRt_t K d_RR_t 2E = = Rt Grid Graph Models R t RR t Rt R t RRt t RRtt K R_R K R_t K RRt_t K RR_t = = = = = = = = = = = = = = = = = = = R t RR t Rt R t RRt t Rt RRtt K R_R K Rt_R K Rt_Rt K R_t | | | | | | | | | | | | | | | ? ?

10. AIC c = 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 Data 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 µ M, the plateau point where R exists solely as RRtt. Experiment becomes a titration scan of [t T ] to estimate [R T ], but [R T ]=7.6 µ M was already known.

11. Model Space Fit with New Data Radivoyevitch, (2008) BMC Systems Biology 2:15

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

13. 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

14. Model Space Predictions Best next 10 measurements if 3Rp is rejected Show new data + R Code

15. Comments on Methods

16. 2+5+9+13 = 28 parameters => 2 28 =2.5x10 8 spur graph models via K j =∞ hypotheses 28 models with 1 parameter, 428 models with 2, 3278 models with 3, 20475 with 4 [dATP] uM X = ATP Kashlan et al. Biochemistry 2002 41:462

18. Kolomogorov-Smirnov Test p < 10 -16 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 included R6X8 or R6X9, save one, single edge model R6X7 This suggests that less than 3 a-sites are occupied in hexamer. 2081 Models with SSE < 2 min (SSE)

19. ModelParameterInitial ValueOptimal ValueConfidence Interval IIIIIIIIIIIJIIIIIIIIIIIIIIIII.20R6X8100.000^1363.101^13(59.878^13, 66.175^13) SSE7.5030.069 AIC37.176-33.163 cpu0.0001.792fit succeeded IIIIIJIIIIIJIIIIIIIIIIIIIIIII.108R2X4100.000^5432.997^5(311.064^5, 601.845^5) R6X8100.000^1362.796^13(59.878^13, 66.175^13) SSE6.9500.061 AIC39.211-31.737 cpu0.0005.230fit succeeded IIIIIIIIIIIIJIIIIIIIIIIIIIIII.21R6X9100.000^1470.367^14(66.686^14, 74.228^14) SSE7.5100.077 AIC37.191-31.514 cpu0.0001.733fit succeeded 1.8 Days on 16 cores [1] 1000000.00000 -33.16309 -31.73658 -29.99075 Time difference of 2623.881 mins Fitted = 3406, out of a total of 3410 (2623*16)/3410 = 12 min per model

20. Human Thymidine Kinase 1 Max(E 0 )=200 pM Min(S T )=50 nM => [S] ≈ [S T ] These K are complete dissociation constants

21. = = = = = = _ = = = = = = = = = = = = _ E ES ES 2 ES 3 ES 4 E ES ES 2 ES 3 ES 4 ES ES 2 ES 3 ES 4 Complete vs. Binary K

22. library(ccems) # The TK1 data is included in ccems. topology = list( heads=c("E1S0"), # E1S0 = substrate free E sites=list( c=list( # c for catalytic site t=c("E1S1","E1S2","E1S3","E1S4") ) # t for tetramer ) ) # TK1 is 25kDa = 25mg/umole, so 1 mg =.04 umoles g = mkg(topology, activity=T,TCC=F) dd=subset(TK1,(year==2000),select=c(E,S,v)) names(dd)[1:2]= c("ET","ST")#v was in µmoles/min/mg dd=transform(dd, ET=ET/4,v=ET*v/(.04*60))# now uM/sec tops=ems(dd,g,maxTotalPs=8,kIC=10,topN=96)#9 min on 1 cpu K = e ∆G/RT => K = (0, ∞) maps to ∆G = (-∞, ∞) Model weights e ∆AIC /∑e ∆AIC ModelParameterInitialFinal95% Wald CI DDDM.DDDDE1S0_S1.0000.773(0.647, 0.925) E1S3_S1.0000.189(0.116, 0.307) kE1S11.0007.003(6.821, 7.171) SSE14.0020.018 AIC53.536-46.085 cpu0.0000.029fit succeeded DDLL.DDDDE1S0_S1.0000.922(0.715, 1.191) E1S2_S1.0000.380(0.298, 0.485) kE1S11.0007.078(6.890, 7.243) SSE14.0020.021 AIC53.536-44.322 cpu0.0000.029fit succeeded JJII.DD00E1S11.000^10.556^1(0.372^1, 0.831^1) E1S21.000^20.585^2(0.525^2, 0.653^2) kE1S11.00014.123(13.736, 14.440) SSE16.0980.022 AIC55.628-43.500 cpu0.0000.031fit succeeded

24. $MunchPetersen1993$ma E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 [1,] 1 1 1 0.32 3.9 3.9 3.9 3.9 $Birringer2006$ma E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 [1,] 0.77 0.74 0.77 0.77 0.29 0.29 0.29 0.29 $Frederiksen2004$ma E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 [1,] 0.45 0.36 0.36 0.36 1.1 2 2.2 3.9 $Berenstein2000$ma E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 [1,] 0.78 0.78 0.68 0.21 6.8 6.8 7 7 $Li2004$ma E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 [1,] 0.56 0.56 0.56 0.56 4.9 4.9 4.7 4.7 $Frederiksen2004 $Frederiksen2004$df aic sse wgts E1S0_S E1S1_S E1S2_S E1S3_S kE1S1 kE1S2 kE1S3 kE1S4 DDDD.DFFF -15 0.026 0.31 0.52 0.52 0.52 0.52 0.55 3.91 3.91 3.9 DDDD.DDDM -15 0.027 0.26 0.13 0.13 0.13 0.13 0.44 0.44 0.44 3.9 DFFF.DDDD -15 0.027 0.21 1.40 0.53 0.53 0.53 3.91 3.91 3.91 3.9 DDDD.DDDD -14 0.053 0.11 0.73 0.73 0.73 0.73 4.14 4.14 4.14 4.1 DDDD.DDLL -14 0.029 0.11 0.35 0.35 0.35 0.35 1.23 1.23 3.81 3.8

25. Acknowledgements Case Comprehensive Cancer Center Case Comprehensive Cancer Center NIH (K25 CA104791) NIH (K25 CA104791) Chris Dealwis (CWRU Pharmacology) Chris Dealwis (CWRU Pharmacology) Anders Hofer (Umea) Anders Hofer (Umea) Thank you Thank you