Combinatorially Complex Equilibrium Model Selection

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

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

3. DNA polymerase TK1 dNTP Supply System flux activation inhibition nucleus ADP dATP U-DNA dA GDP dGTP DNA dCK dG dCTP CDP dCK dCK RNR mitochondria dC ATP dTTP UDP CDA cytosol cytosol 5NT TS dA dAMP dATP dUMP dUDP dGK dT dG dGMP dGTP dU dUTP dUTPase dC dN dCMP dCTP TK2 dT dTMP dTTP NT2 dN 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.

4. RNR Literature ATP activates at hexamerization site?? dATP inhibits at activity site, ATP activates at activity site? R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 R1 dTTP, dGTP, dATP, ATP bind to selectivity site UDP, CDP, GDP, ADP bind to catalytic site R1 R1 R2 R2 R2 R2 R2 R2 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 NDPs control the distribution of R1-R2 complexes and this changes in S, G1-G2 and G0

5. Michaelis-Menten Model E + S ES 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.

6. Michaelis-Menten Model [S] vs. [ST] 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) (3) R=R1 r=R22 G=GDP t=dTTP solid line = Eqs. (1-2) dotted = Eq. (3) Data from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry40(6), 1651-166

7. Enzyme, Substrate and Inhibitor E ES EI ESI E ES EI ESI E ES EI ESI E ES EI ESI Competitive inhibition E EI ESI E EI ESI E ES EI noncompetitive inhibition Example of Kd=Kd’ Model = E | ES EI | ESI = = uncompetitive inhibition if kcat_ESI=0 E ES ESI E ES ESI = E EI E ESI E ES E 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 ES Kj=0 Models as KES approaches 0

8. Rt Spur Graph Models for dTTP induced R1 dimerization 3A 3C 3D 3E 3B R R RR R RR R RR R RR R = R1 t = dTTP Rt RRt Rt Rt RRt RRt Rt RRt RRtt RRtt RRtt RRtt 3J 3K 3I 3F 3G 3H R RR R RR R R R RR R RRt Rt RRt RRt Rt Rt RRtt RRtt RRtt 3R 3P 3S 3T 3Q 3O 3L 3M 3N R RR R R R R R RR R R R Rt RRt Rt RRt RRtt RRtt Total number of spur graph models is 16+4=20 Radivoyevitch, (2008) BMC Systems Biology 2:15

9. Rt Grid Graph Models 2A 2B 2C 2D 2E Kd_R_R Kd_R_R Kd_R_R Kd_R_R Kd_R_R RR t t R R t t R R t t RR t t RR t t R R t t RR t t RR t t R R t t R R t t | | | = Kd_RR_t = Kd_R_t = Kd_RR_t = Kd_R_t Kd_RR_t = Kd_R_t Kd_RR_t Kd_RR_t = Kd_R_t Kd_R_t Kd_Rt_R Kd_Rt_R Kd_Rt_R Kd_Rt_R Kd_Rt_R RRt t Rt R t RRt t Rt R t Rt R t RRt t RRt t Rt R t RRt t Rt R t | | | Kd_RRt_t Kd_RRt_t Kd_RRt_t Kd_RRt_t Kd_R_t Kd_RRt_t Kd_R_t = Kd_R_t = Kd_R_t Kd_R_t = = = = Kd_Rt_Rt Kd_Rt_Rt Kd_Rt_Rt Kd_Rt_Rt Kd_Rt_Rt Rt Rt RRtt RRtt Rt Rt RRtt Rt Rt | RRtt Rt Rt RRtt Rt Rt | | 2K 2A 2B 2H 2C 2G 2L 2D 2E 2F 2M 2N 2I 2J | | | | | = = | | | | | = = | | | | | 2F0 2F1 2F2 2F3 2F4 2F5 2F6 2F7 2F8 Acyclic spanning subgraphs are reparameterizations of equilvalent models Standardize: take E-shapes and sub E-shapes as defaults Use n-shapes if needed. Other shapes are possible Figure 2. Grid graph models. 3C 3J 3K 3L 3M 3N 3A 3D 3E 3F 3G 3H 3I 3S 3R 3B 3O 3P 3Q 3T Figure 3. Spur graph models. The following models are equivalent: 3A=2F, 3B=2H, 3C=2J, 3D=2L, 3E=2N

10. Data and fit from Scott, C. P., Kashlan, O. B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry40(6), 1651-166 Application to Data 3Rp R titration model has lowest AIC RRtt 2E ~10-fold deviations from Scott et al. (initial values) = = AICc = 2P+N*log(SSE/N)+2P(P+1)/(N-P-1) 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. Radivoyevitch, (2008) BMC Systems Biology 2:15

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. Kashlan et al. Biochemistry 2002 41:462 [dATP] uM a = dATP 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

17. Human ThymidineKinase 1 Assumptions • binding of enzyme monomers to form enzyme dimers is approximately infinitely tight across the enzyme concentrations of interest • behavior of any higher order enzyme structures (if they exist) is dominated by the behavior of dimers within such structures • enzyme concentrations are low enough that free substrate concentrations approximately equal total substrate concentrations. OK Questionable (but I need to start simple) Max(E0)=200 pM Min(ST)=50 nM => OK

19. . 4-parameter model 3-parameter models 2-parameter models

20. Fits to human thymidine kinase 1 data of Birringer et al. Protein Expr Purif 2006, 47(2):506-515 Same top 4 models found by fits to Frederiksen H, Berenstein D, Munch-Petersen B: Eur J Biochem 2004, 271(11):2248-2256. Eq. (25): k1 = k2 = 0, k3 > 0 and K1 = K2 = K3 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 Eq. (7): k1 = k3 and K2 = ∞ and K1 = K3

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