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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/

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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

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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

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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

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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

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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)

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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

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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

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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 = = 2A 2B 2C 2D2E 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. 2F12F22F32F42F52F62F72F8 Standardize: take E-shapes and sub E-shapes as defaults Use n-shapes if needed. Other shapes are possible 3B 3C 3D3E3F3G3H3I 3J3K3L3M 3N 3O3P 3Q 3R 3S 3T 3A Rt Grid Graph Models

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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.

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

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

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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

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

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Comments on Methods

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2+5+9+13 = 28 parameters => 2 28 =2.5x10 8 spur graph models via K j =∞ 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

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Human 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(E 0 )=200 pM Min(S T )=50 nM => OK OK Questionable (but I need to start simple)

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

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Eq. (25): k 1 = k 2 = 0, k 3 > 0 and K 1 = K 2 = K 3 Fits to human thymidine kinase 1 data of Birringer et al. Protein Expr Purif 2006, 47(2):506-515 Eq. (7): k 1 = k 3 and K 2 = ∞ and K 1 = K 3 EQSSEAICc α 1 V∞V∞ V1V1 α2α2 n eq250.00026-57.673.7650.151 eq280.00028-56.932.7560.145 eq70.00029-56.861.6580.145 eq260.00034-55.352.3160.156 eq80.00039-54.11.370.158 eq220.00026-50.480.8330.1486.732 eq200.00027-50.231.2930.1484.635 eq90.00061-49.993.6450.132 eq10.00028-49.881.9410.1441.297 eq60.00028-49.712.7470.1450 eq40.00029-49.60.8150.1462.704 eq240.00065-49.50.2720.149 eq213.00E-04-49.331.4590.1493.996 eq190.00032-48.790.9460.1446.156 eq230.00033-48.492.1450.1553.006 eq180.00037-47.410.5760.20.022 eq50.00037-47.250.960.1570.207 eq270.00105-45.190.3030.127 eq30.00026-38.572.52870.14960.020510.1832 Same top 4 models found by fits to Frederiksen H, Berenstein D, Munch-Petersen B: Eur J Biochem 2004, 271(11):2248-2256. Hill rises from 5 th, 6 th and worse to 2 nd 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

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Acknowledgements Case Comprehensive Cancer Center Case 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

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