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

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 DNA polymerase

cytosol mitochondria dT dC dG dA dTMP dCMP dGMP dAMP dTTP dCTP dGTP dATP

5NT NT2

cytosol nucleus dUDP dUTP U-DNA dN dN

flux activation inhibition

ATP

RNR

R1 R2 R2 R1 R1 R1 R1 R1 R1 R1 R1 R2 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 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

ATP activates at hexamerization site??

RNR Literature

R2 R2

Michaelis-Menten Model

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. ) ( ) ( ] [ ] [ ] [ ] [ ] [ ] [ 1 / ] [ 1 1 / ] [ / ] [ ] [ ] [

max

E P E ES P E k E ES E E E ES ES E k K S E K S K S E k K S S V

cat cat m m m cat m

+ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + E + S ES

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= R22 G= GDP t= dTTP

) 2 ( ] ][ [ ] [ ] [ ) 1 ( ] ][ [ ] [ ] [

_ _ _ _ S E d T S E d T

K S E S S K S E E E − − = − − = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = ⇒ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + =

S E d T S E d T T S E d S E d T S E d T S E d T

K S K S E ES versus K S K S E ES K S E E K S E E

_ _ _ _ _ _ _ _ _ _ _ _

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

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)

( ) ( )

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

_ _ 2 _ _ _ _ T S E d T T S E d T T S E d T

S K E S K E S K S ES + + − ± + − + =

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

_ _ _ _

= = − − = − − = S E K S E S S d S d K S E E E d E d

S E d T S E d T

τ τ

Michaelis-Menten Model [S] vs. [ST]

(3)

I ES d S E d I E d T I ES d S E d S E d T I ES d S E d I E d S E d T

K K I S E K I E I I K K I S E K S E S S K K I S E K I E K S E E E

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] ][ [ ] [ ] [ − − − = − − − = − − − − =

E ES EI ESI E ES EI ESI E ES EI ESI

S EI d I E d I E d T S EI d I E d S E d T S EI d I E d I E d S E d T

K K S I E K I E I I K K S I E K S E S S K K S I E K I E K S E E E

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] ][ [ ] [ ] [ − − − = − − − = − − − − =

E ES EI

EI T ES T EI ES T

K I E I I K S E S S K I E K S E E E ] ][ [ ] [ ] [ ] ][ [ ] [ ] [ ] ][ [ ] ][ [ ] [ ] [ − − = − − = − − − =

ESI EI T ESI ES T ESI EI ES T

K I S E K I E I I K I S E K S E S S K I S E K I E K S E E E ] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] [ ] [ ] ][ ][ [ ] ][ [ ] ][ [ ] [ ] [ − − − = − − − = − − − − =

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 Kd= Kd’ Model

= =

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 ] [ ] [ ] ][ [ ] [ ] [ ] ][ [ ] [ ] [ I I K S E S S K S E E E

T ES T ES T

− = − − = − − = ] [ and ] [ ] [ ] [ ], [ ] [ else , ] [ and ] [ ] [ ] [ ], [ ] [ ) ] [ ] [ ( if = − = = = − = = > E E S S E ES S S E E S ES S E

T T T T T T T T

as KES approaches 0

Enzyme, Substrate and I nhibitor

Total number of spur graph models is 16+ 4= 20

Radivoyevitch, (2008) BMC Systems Biology 2:15

Rt Spur Graph Models

[ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

RRtt RRt Rt T RRtt RRt RR Rt T

K t R K t R K t R t t = K t R K t R K R K t R R R p =

2 2 2 2 2 2 2

2 2 2 2 − − − − − − − − −

[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ] [ ] [ ]

. ) ( ; ) ( 2 2 2 2

2 2 2 2 2 2 2

= = − − − − − − − − − t R K t R K t R K t R t t = d t d K t R K t R K R K t R R R p = d R d

RRtt RRt Rt T RRtt RRt RR Rt T

τ τ

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 3D 3E 3F 3G 3H 3A

R RRtt RRt R RR RRtt R RR RRt R Rt R RRtt R RRt

3J 3I 3K

R RR R

3L 3M 3N 3O 3P

R Rt R RRtt R RRt R RR

3Q 3R 3S 3T

R R t t RR t t Rt R t RRt t Rt Rt RRtt Kd_R_R Kd_Rt_R Kd_Rt_Rt Kd_R_t Kd_R_t Kd_RRt_t Kd_RR_t

= = = =

2A R R t t RR t t Rt R t RRt t Rt Rt RRtt Kd_R_R Kd_Rt_R Kd_Rt_Rt Kd_R_t Kd_R_t Kd_RRt_t Kd_RR_t

= = =

2B R R t t RR t t Rt R t RRt t Rt Rt RRtt Kd_R_R Kd_Rt_R Kd_Rt_Rt Kd_R_t Kd_R_t Kd_RRt_t Kd_RR_t 2C

| | | | |

= = =

| |

R R t t RR t t Rt R t RRt t Rt Rt RRtt Kd_R_R Kd_Rt_R Kd_Rt_Rt Kd_R_t Kd_R_t Kd_RRt_t Kd_RR_t 2D

| |

R R t t RR t t Rt R t RRt t Rt Rt RRtt Kd_R_R Kd_Rt_R Kd_Rt_Rt Kd_R_t Kd_R_t Kd_RRt_t Kd_RR_t 2E

= =

2A 2B 2C 2D 2E 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.

2F1 2F2 2F3 2F4 2F5 2F6 2F7 2F8

Standardize: take E-shapes and sub E-shapes as defaults Use n-shapes if

• needed. Other

shapes are possible

3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 3A

Rt Grid Graph Models

AICc = 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

] [ ] [ 2 ] [ 2 ] [ 2 180 ] [ ) 1 ]( [ ] [ 90

T T T a

R RRtt RRt RR R p R R M + + + − + =

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 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.

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

Model Space Predictions

Best next 10 measurements if 3Rp is rejected

• 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. This reduces Rt model space fitting times by a factor of 5.

• The best of a best-guess lot of ~10 models may be adequate in many cases

Final Remarks

Acknowledgements

Case Comprehensive Cancer Center NIH (K25 CA104791) Anders Hofer (Umea) Thank you