ODEs Based Signaling Pathways Dynamics P.S. Thiagarajan School of - PowerPoint PPT Presentation

Probabilistic Approximations of ODEs Based Signaling Pathways Dynamics P.S. Thiagarajan School of Computing, National University of Singapore

Biopathways  Biopathways:  Metabolic Pathways  Signaling Pathways  Gene Regulatory Networks

Signaling Pathways • Chemical reactions in response to external signals (ligands) • Signals pass into the nucleus through a series of protein modifications „Data transfer‟ mechanism of the cell

A Common Modeling Approach • View a pathway as a network of bio-chemical reactions • Model the network as a system of ODEs  One for each molecular species  Reaction kinetics: Mass action law, Michelis-Menten, Hill, etc. • Study the ODE system dynamics.

The ODEs model k 1 k 3 S  E  E ES P k 2 Assume mass law. ฀  ฀  dS dt   k 1  S  E  k 2  ES ฀  dE dt   k 1  S  E  ( k 2  k 3 )  ES dES dt  k 1  S  E  ( k 2  k 3 )  ES dP dt  k 3  ES ฀ 

 Alternative approach:  Keep track of exact number of molecules of each type. Simulate the dynamics by executing one reaction at a time stochastically (CTMCs)  Stochastic simulations (Gillespie‟s algorithm)  Kappa , BioNetGen, PRISM, Bio-Pepa, ..

ODEs: Major Hurdles • Many unknown rate constants. • Must be estimated using limited data:  Low precision , population-based, noisy

Major Hurdles • High dimensional non-linear system  no closed-form solutions  must resort to numerical simulations  point values of initial states/data will not be available  a large number of numerical simulations needed for answering each analysis question

“Polling” based approximation • Start with an ODEs system. • Discretize the time and value domains. • Assume a (uniform) distribution of initial states • Generate a “sufficiently” large number of trajectories by  Sampling the initial states and numerical simulations.

The “exit poll” Idea • Encode this collection of discretized trajectories as a dynamic Bayesian network. • ODEs  DBN • Pay the one-time cost of constructing the DBN approximation. • Do analysis using Bayesian inferencing on the DBN.

Time Discretization • Observe the system only at a finite number of time points. x(t) x(0) = 2 2 x(t) = t 3 + 4t + 2 t max t 0 t 1 t 2 ... ... ... ...

Value Discretization • Observe only with bounded precision x(t) E D C B A t 0 t 1 t 2 t max ... ... ... ...

Symbolic trajectories • A trajectory is recorded as a finite sequence of discrete values . (C,0) (D,1) (D,2) (E,3) (D,4) (C,5) (B,6) x(t) E D C B A t 0 t 1 t 2 t max ... ... ... ...

Collection of Trajectories • Assume a prior distribution of the initial states. • Uncountably many trajectories. Represented as a set of (timed) finite sequences. (C,0) (D,2) (D,1) (E,3) (D,4) (C,5) (B,6) ... ... (D,3) x(t) E D C B A t t 0 t 1 t 2 t max ... ... ... ...

Piecing trajectories together.. • In fact, a probabilistic transition system. • Pr( (D, 2) (E, 3) ) is 0.8 (C,0) (D,2) (D,1) (E,3) (D,4) (C,5) (B,6) the “fraction” of the 0.2 ... ... (D,3) trajectories residing in D at t = 2 that land in E at E D t = 3. C B A t t 0 t 1 t 2 t max ... ... ... ...

The Justification • The value space of the variables is assumed to be a compact subset C of • In Z’ = F(Z), F is assumed to be continuously differentiable in C.  Mass-law, Michaelis- Menton,… • Then the solution  t : C  C (for each t) exists, is unique, a bijection, continuous and hence measurable. • But the transition probabilities can’t be computed.

A computational approximation (s, i) – States; (s, i)  (s‟, i+1) -- Transitions 1000 800 0.8 Sample, say, 1000 times the (C,0) (D,1) (D,2) (E,3) (D,4) (C,5) (B,6) initial states. 0.2 ... ... (D,3) Through numerical simulation, generate 1000 E trajectories. D Pr((s, i)  (s‟ i+1)) is the C fraction of the trajectories that are in s at t i which land in s‟ B at t i+1 . A t t 0 t 1 t 2 t max ... ... ... ...

Infeasible Size! • But the transition system will be huge.  O(T . k n )  k  2 and n (  50-100).

Compact Representation • Exploit the network structure (additional independence assumptions) to construct a DBN instead. • The DBN is a factored form of the probabilistic transition system.

The DBN representation k 1 k 3 S  E  E ES P k 2 Assume mass law. ฀  ฀  dS dt   k 1  S  E  k 2  ES ฀  dE dt   k 1  S  E  ( k 2  k 3 )  ES dES dt  k 1  S  E  ( k 2  k 3 )  ES dP dt  k 3  ES ฀ 

S k 1 k 3 S  E  E ES P k 2 ES P ฀  ฀  ฀  E dS dt   k 1  S  E  k 2  ES dE dt   k 1  S  E  ( k 2  k 3 )  ES Dependency diagram dES dt  k 1  S  E  ( k 2  k 3 )  ES dP dt  k 3  ES ฀ 

S k 1 k 3 S  E  E ES P k 2 ES P ฀  ฀  ฀  dS E dt   k 1  S  E  k 2  ES dE dt   k 1  S  E  ( k 2  k 3 )  ES Dependency diagram dES dt  k 1  S  E  ( k 2  k 3 )  ES dP dt  k 3  ES ฀ 

The DBN Representation k 1 k 3 S  E  E ES P k 2 ... ... S 0 S 1 S 2 S 3 dS dt   k 1  S  E  k 2  ES ฀  ฀  ฀  dE ... ... E 0 E 1 E 2 E 3 dt   k 1  S  E  ( k 2  k 3 )  ES dES ... ... dt  k 1  S  E  ( k 2  k 3 )  ES ES 0 ES 1 ES 2 ES 3 dP dt  k 3  ES ... ... P 0 P 1 P 2 P 3 ฀ 

P(S 2 = C |S 1 = B ,E 1 = C ,ES 1 = B )= 0.2 P(S 2 = C |S 1 = B ,E 1 = C ,ES 1 = C )= 0.1 P(S 2 = A |S 1 = A ,E 1 = A ,ES 1 = C )= 0.05 . . . • Each node has a CPT ... ... S 0 S 1 S 2 S 3 associated with it. • This specifies the local ... ... E 0 E 1 E 2 E 3 (probabilistic) dynamics. ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3 A B C

P(S 2 = C |S 1 = B ,E 1 = C ,ES 1 = B )= 0.2 P(S 2 = C |S 1 = B ,E 1 = C ,ES 1 = C )= 0.1 P(S 2 = A |S 1 = A ,E 1 = A ,ES 1 = C )= 0.05 . . . • A Fill up the entries in B ... ... S 0 S 1 S 2 S 3 the CPTs by C sampling, simulations and counting ... ... E 0 E 1 E 2 E 3 ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3

Computational Approximation 500 100 • A Fill up the entries in B ... ... S 0 S 1 S 2 S 3 the CPTs by C sampling, simulations and counting ... ... E 0 E 1 E 2 E 3 1000 ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3

The Technique P(S 2 = C |S 1 = B ,E 1 = C ,ES 1 = B )= 100/500= 0.2 500 100 • A Fill up the entries in B ... ... S 0 S 1 S 2 S 3 the CPTs by C sampling, simulations and counting ... ... E 0 E 1 E 2 E 3 ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3

The Technique The size of the DBN is: ... ... S 0 S 1 S 2 S 3 O(T . n . k d ) ... ... E 0 E 1 E 2 E 3 ... ... ES 0 ES 1 ES 2 ES 3 d will be usually much smaller than n. ... ... P 0 P 1 P 2 P 3

Unknown rate constants k 1  0.1 k 3 S  E  E ES P k 2  0.2 ... ... S 0 S 1 S 2 S 3 ฀  ฀  ฀  dS ... ... E 0 E 1 E 2 E 3 dt   0.1  S  E  0.2  ES dE dt   0.1  S  E  (0.2  k 3 )  ES ... ... ES 0 ES 1 ES 2 ES 3 dES dt  0.1  S  E  (0.2  k 3 )  ES ... ... P 0 P 1 P 2 P 3 dP dt  k 3  ES 0 k 3 dk 3 = 0 dt ฀ 

Unknown rate constants During the numerical generation of a ... ... S 0 S 1 S 2 S 3 trajectory, the value ... ... E 0 E 1 E 2 E 3 of k 3 does not change after sampling. ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3 ... ... 3 0 1 2 k 3 k 3 k 3 k 3 1 0 P(k 3 = A |k 3 = A )= 1

Unknown rate constants P(ES 2 = A |S 1 = C ,E 1 = B ,ES 1 = A ,k 3 = A )= 0.4 1 During the numerical generation of a ... ... S 0 S 1 S 2 S 3 trajectory, the value ... ... E 0 E 1 E 2 E 3 of k 3 does not change after sampling. ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3 ... ... 3 0 1 2 k 3 k 3 k 2 k 3 1 0 P(k 3 = A |k 3 = A )= 1

Unknown rate constants ... ... S 0 S 1 S 2 S 3 ... ... E 0 E 1 E 2 E 3 ... ... ES 0 ES 1 ES 2 ES 3 ... ... P 0 P 1 P 2 P 3 Sample uniformly ... ... 3 0 1 2 across all the k 3 k 3 k 3 k 3 Intervals.

DBN based Analysis • Use Bayesian inferencing to do parameter estimation, sensitivity analysis, probabilistic model checking … • Exact inferencing is not feasible for large models. • We do approximate inferencing. • Factored Frontier algorithm .

Parameter Estimation 1. For each choice of (interval) values for unknown parameters, ... ... run FF, compare with S 0 S 1 S 2 S 3 experimental data and assign a score using FF. ... ... E 0 E 1 E 2 E 3 2. Return parameter estimates as ... ... maximal likelihoods. ES 0 ES 1 ES 2 ES 3 3. FF can be then used on the ... ... P 0 P 1 P 2 P 3 calibrated model to do sensitivity analysis, probabilistic verification etc. ... ... 3 2 0 1 k 3 k 3 k 3 k 3

DBN based Analysis • Our experiments with signaling pathways models (taken from the BioModels data base ) show:  The one-time cost of constructing the DBN can be easily amortized by using it to do parameter estimation and sensitivity analysis.  Good compromise between efficiency and accuracy.