Equilibrium Model Selection Tom Radivoyevitch Assistant Professor - PowerPoint PPT Presentation

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

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

RNR Literature ATP activates at hexamerization site?? dATP inhibits at activity site, ATP activates at activity site? R1 R1 R1 R1 dTTP , dGTP , dATP , ATP bind to selectivity site R1 UDP , CDP , GDP , ADP bind to catalytic site R2 R2 R1 R1 R1 R1 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 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

Michaelis-Menten Model E + S ES ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ [ S ] [ S ] / K 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = + m V k E 0 E ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ + + + max cat 0 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ [ S ] K [ S ] / K 1 [ S ] / K 1 m m m ⎛ ⎞ ⎛ ⎞ [ ES ] [ E ] = ⎜ ⎟ + ⎜ ⎟ 0 k E ⎜ ⎟ E ⎜ ⎟ + + 0 0 cat ⎝ ⎠ ⎝ ⎠ [ ES ] [ E ] [ ES ] [ E ] = + k E P ( ES ) 0 E P ( E ) cat 0 0 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.

d [ E ] [ E ][ S ] Michaelis-Menten Model = − − [ E ][ S ] [ E ] [ E ] = − − τ T 0 [ E ] [ E ] ( 1 ) d K T d _ E _ S K [S] vs. [S T ] d _ E _ S [ ] [ ][ ] d S E S = − − [ S ] [ S ] [ E ][ S ] τ T = − − d K 0 [ S ] [ S ] ( 2 ) d _ E _ S T K = = d _ E _ S [ ]( 0 ) 0 , [ ]( 0 ) 0 E S Substitute this in here to get a quadratic in [S] which solves as ( ) ( ) = + − + ± − + + 2 [ ] [ ] . 5 ( [ ] [ ] ( [ ] [ ] 4 [ ] ES S K S E K S E K S T d _ E _ S T T d _ E _ S T T d _ E _ S T Bigger systems of higher polynomials cannot be solved algebraically => use ODEs (above) [ S ] [ S ] T ⎛ ⎞ K K [ S ] 1 ⎜ ⎟ = + ⇒ = ⇒ = = d _ E _ S d _ E _ S [ ] [ ] 1 [ ] [ ] [ ] [ ] [ ] [ ] E E E E ES E versus ES E (3) ⎜ ⎟ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ T T T T K ⎝ ⎠ [ S ] [ S ] [ S ] ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ d _ E _ S + + + T 1 1 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ K K K ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ d _ E _ S d _ E _ S d _ E _ S R= R1 r= R2 2 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) Biochemistry 40 (6), 1651-166

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

Rt Spur Graph Models 3A 3C 3D 3E 3B R R RR R RR R RR R RR 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 RR R R R R R R R Rt RRt Rt RRt RRtt RRtt [ ] ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] 2 2 2 2 [ d R R t R R t R t ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] − − − − − = p R R 2 2 2 2 2 2 2 [ R t R R t R t τ T − − − − − d K K K K 0 = p R R 2 2 2 Rt RR RRt RRtt T K K K K [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] 2 2 2 Rt RR RRt RRtt d t R t R t R t [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] − − − − = t t 2 2 2 2 R t R t R t τ T − − − − d K K K 0 = t t 2 Rt RRt RRtt T [ ] [ ] K K K = = Rt RRt RRtt R ( 0 ) 0 ; t ( 0 ) 0 . Total number of spur graph models is 16+ 4= 20 Radivoyevitch, (2008) BMC Systems Biology 2:15

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

Data and fit from Scott, C. P ., Kashlan, O. Application to Data B., Lear, J. D., and Cooperman, B. S. (2001) Biochemistry 40 (6), 1651-166 3Rp R titration model has lowest AIC RRtt 2E ~10-fold deviations from = Scott et al. (initial values) = AIC c = 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 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. + − + + [ ] [ ]( 1 ) 2 [ ] 2 [ ] 2 [ ] R R p RR RRt RRtt = + T M 90 180 a [ ] [ ] R R T T Radivoyevitch, (2008) BMC Systems Biology 2:15

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

One additional data point here would reject 3Rp If so, new data here would be logical next No need to constrain data collection to such profiles

Best next 10 measurements if 3Rp is rejected Model Space Predictions