Moment Closure Based Parameter Inference of Stochastic Kinetic Models

1. Moment Closure Based Parameter Inference of Stochastic Kinetic Models Colin Gillespie School of Mathematics & Statistics

2. Overview Talk outline An introduction to moment closure Parameter inference Conclusion 2/25

3. Birth-death process Birth-death model X −→ 2X and 2X −→ X which has the propensity functions λX and µX . Deterministic representation The deterministic model is dX (t ) = ( λ − µ )X (t ) , dt which can be solved to give X (t ) = X (0) exp[(λ − µ)t ]. 3/25

4. Birth-death process Birth-death model X −→ 2X and 2X −→ X which has the propensity functions λX and µX . Deterministic representation The deterministic model is dX (t ) = ( λ − µ )X (t ) , dt which can be solved to give X (t ) = X (0) exp[(λ − µ)t ]. 3/25

5. Stochastic representation In the stochastic framework, each reaction has a probability of occurring 50 The analogous version of the 40 birth-death process is the difference Population equation 30 20 dpn = λ(n − 1)pn−1 + µ(n + 1)pn+1 10 dt − (λ + µ)npn 0 0 1 2 3 4 Time Usually called the forward Kolmogorov equation or chemical master equation 4/25

6. Moment equations Multiply the CME by enθ and sum over n, to obtain ∂M ∂M = [λ(eθ − 1) + µ(e−θ − 1)] ∂t ∂θ where ∞ M (θ; t ) = ∑ e n θ pn ( t ) n =0 If we differentiate this p.d.e. w.r.t θ and set θ = 0, we get dE[N (t )] = (λ − µ)E[N (t )] dt where E[N (t )] is the mean 5/25

7. The mean equation dE[N (t )] = (λ − µ)E[N (t )] dt This ODE is solvable - the associated forward Kolmogorov equation is also solvable The equation for the mean and deterministic ODE are identical When the rate laws are linear, the stochastic mean and deterministic solution always correspond 6/25

8. The variance equation If we differentiate the p.d.e. w.r.t θ twice and set θ = 0, we get: dE[N (t )2 ] = (λ − µ)E[N (t )] + 2(λ − µ)E[N (t )2 ] dt and hence the variance Var[N (t )] = E[N (t )2 ] − E[N (t )]2 . Differentiating three times gives an expression for the skewness, etc 7/25

9. Simple dimerisation model Dimerisation 2X1 −→ X2 and X2 −→ 2X1 with propensities 0.5k1 X1 (X1 − 1) and k2 X2 . 8/25

10. Dimerisation moment equations We formulate the dimer model in terms of moment equations dE[X1 ] 2 = 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ] dt 2 dE[X1 ] 2 2 = k1 (E[X1 X2 ] − E[X1 X2 ]) + 0.5k1 (E[X1 ] − E[X1 ]) dt 2 + k2 (E[X1 ] − 2E[X1 ]) where E[X1 ] is the mean of X1 and E[X1 ] − E[X1 ]2 is the variance 2 The i th moment equation depends on the (i + 1)th equation 9/25

11. Deterministic approximates stochastic Rewriting dE[X1 ] 2 = 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ] dt in terms of its variance, i.e. E[X1 ] = Var[X1 ] + E[X1 ]2 , we get 2 dE[X1 ] = 0.5k1 E [X1 ](E[X1 ] − 1) + 0.5k1 Var[X1 ] − k2 E[X1 ] (1) dt Setting Var[X1 ] = 0 in (1), recovers the deterministic equation So we can consider the deterministic model as an approximation to the stochastic When we have polynomial rate laws, setting the variance to zero results in the deterministic equation 10/25

12. Deterministic approximates stochastic Rewriting dE[X1 ] 2 = 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ] dt in terms of its variance, i.e. E[X1 ] = Var[X1 ] + E[X1 ]2 , we get 2 dE[X1 ] = 0.5k1 E [X1 ](E[X1 ] − 1) + 0.5k1 Var[X1 ] − k2 E[X1 ] (1) dt Setting Var[X1 ] = 0 in (1), recovers the deterministic equation So we can consider the deterministic model as an approximation to the stochastic When we have polynomial rate laws, setting the variance to zero results in the deterministic equation 10/25

13. Simple dimerisation model To close the equations, we assume an underlying distribution The easiest option is to assume an underlying Normal distribution, i.e. E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3 3 2 But we could also use, the Poisson 3 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3 or the Log normal 2 3 3 E [ X1 ] E [ X1 ] = E [ X1 ] 11/25

14. Simple dimerisation model To close the equations, we assume an underlying distribution The easiest option is to assume an underlying Normal distribution, i.e. E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3 3 2 But we could also use, the Poisson 3 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3 or the Log normal 2 3 3 E [ X1 ] E [ X1 ] = E [ X1 ] 11/25

15. Heat shock model Proctor et al, 2005. Stochastic kinetic model of the heat shock system twenty-three reactions seventeen chemical species A single stochastic simulation up to t = 2000 takes about 35 minutes. If we convert the model to moment equations, we get 139 equations ADP Native Protein 1200 6000000 5950000 1000 5900000 800 Population 5850000 600 5800000 400 5750000 200 5700000 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Time Gillespie, CS, 2009 12/25

16. Density plots: heat shock model Time t=200 Time t=2000 0.006 Density 0.004 0.002 0.000 600 800 1000 1200 1400 600 800 1000 1200 1400 ADP population 13/25

17. P53-Mdm2 oscillation model Proctor and Grey, 2008 300 16 chemical species Around a dozen reactions 250 The model contains an event 200 Population At t = 1, set X = 0 150 If we convert the model to moment 100 equations, we get 139 equations. 50 However, in this case the moment 0 closure approximation doesn’t do to 0 5 10 15 20 25 30 well! Time 14/25

20. What went wrong? The moment closure (tends) to fail when there is a large difference between the deterministic and stochastic formulations In this particular case, strongly correlated species Typically when the MC approximation fails, it gives a negative variance The MC approximation does work well for other parameter values for the p53 model 15/25

21. Parameter inference 4 3 Population 2 Simple immigration-death 1 process k1 0 R1 : ∅ − X → 0 10 20 30 40 50 k2 Time R2 : X − ∅ → The CME can be solved Discrete time course data The likelihood can be very ﬂat 16/25

22. Parameter inference 4 3 q Population 2 q Simple immigration-death 1 q q q q q process k1 0 q q q q R1 : ∅ − X → 0 10 20 30 40 50 k2 Time R2 : X − ∅ → The CME can be solved Discrete time course data The likelihood can be very ﬂat 16/25

23. Parameter inference 4 3 q Population 2 q Simple immigration-death 1 q q q q q process k1 0 q q q q R1 : ∅ − X → 0 10 20 30 40 50 k2 Time R2 : X − ∅ → 10 8 The CME can be solved 6 Discrete time course data k2 4 The likelihood can be very ﬂat 2 0 0 2 4 6 8 10 k1 16/25

24. Lotka-Volterra model Species Predator Prey The Lotka-Volterra predator prey system, describes the time evolution of two 400 species, Y1 and Y2 Prey birth: Y1 → 2Y1 300 Population Interaction: Y1 + Y2 → 2Y2 200 Predator death: Y2 → ∅ Since the Lotka-Volterra model 100 contains a non-linear rate law, the i th moment equation depends on the 0 (i + 1)th moment. 0 10 20 30 40 Time 17/25

28. Parameter estimation Let Y(tu ) = (Y1 (tu ), Y2 (tu )) be the vector of the observed predator and prey To infer c1 , c2 and c3 , we need to estimate Pr[Y(tu )| Y(tu −1 ), c] i.e. the solution of the forward Kolmogorov equation We will use moment closure to estimate this distribution: Y(tu ) | Y(tu −1 ), c ∼ N (ψu −1 , Σu −1 ) where ψu −1 and Σu −1 are calculated using the moment closure approximation 18/25

29. Parameter estimation Let Y(tu ) = (Y1 (tu ), Y2 (tu )) be the vector of the observed predator and prey To infer c1 , c2 and c3 , we need to estimate Pr[Y(tu )| Y(tu −1 ), c] i.e. the solution of the forward Kolmogorov equation We will use moment closure to estimate this distribution: Y(tu ) | Y(tu −1 ), c ∼ N (ψu −1 , Σu −1 ) where ψu −1 and Σu −1 are calculated using the moment closure approximation 18/25

30. Bayesian parameter inference Summarising our beliefs about c and the unobserved predator population Y2 (0) via uninformative priors The joint posterior for parameters and unobserved states (for a single data set) is 40 p (y2 , c | y1 ) ∝ p (c) p (y2 (0)) ∏ p (y(tu ) | y(tu−1 ), c) u =1 For the results shown, we used a vanilla Metropolis-Hasting step to explore the parameter and state spaces For more complicated models, we can use a Durham & Gallant style bridge (Milner, G & Wilkinson, 2012) 19/25

31. Bayesian parameter inference Summarising our beliefs about c and the unobserved predator population Y2 (0) via uninformative priors The joint posterior for parameters and unobserved states (for a single data set) is 40 p (y2 , c | y1 ) ∝ p (c) p (y2 (0)) ∏ p (y(tu ) | y(tu−1 ), c) u =1 For the results shown, we used a vanilla Metropolis-Hasting step to explore the parameter and state spaces For more complicated models, we can use a Durham & Gallant style bridge (Milner, G & Wilkinson, 2012) 19/25

32. Results c1 c2 c3 Exact q q q Fully Obs. Diffusion q q q Mom. Clos. q q q Exact q q q Partially Obs. Diffusion q q q Mom. Clos. q q q 0.3 0.4 0.5 0.6 0.7 0.8 0.0015 0.0020 0.0025 0.0030 0.0035 0.2 0.3 0.4 Parameter value 20/25

33. Auto regulation system This system contains twelve reactions and six species The species populations ranges from zero (for species i) to around 65,000 for species G The moment closure approximation yields a closed set of twenty-seven ODEs Six ODEs for the means Six ODEs for the variances Fifteen ODEs for the covariance terms 21/25

34. Stochastic realisation 30 Species 25 g Population 20 i 15 10 r_g 5 r_i 0 0 10 20 30 40 50 Time 15 65100 10 65050 G I 5 65000 0 0 10 20 30 40 50 0 10 20 30 40 50 Time Time 22/25

37. Parameter inference Fully Obs. Partially Obs. c1 c2 Posterior distributions for c1 to c8 : mean ± 2 sd. True values in c3 red Given information on all Parameter c4 species, inference is reasonable c5 For most of the parameters, c6 fewer data points results in c7 larger credible regions c8 But not in all cases! 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 Parameter value 23/25

41. Future work Techniques for assessing the moment closure approximation Better closure techniques Computer emulation for moments Using the moment closure approximation as a proposal distribution in an MCMC algorithm The proposal can be (almost) anything we want The likelihood can be calculated using anything we want 24/25

42. Acknowledgements Peter Milner Darren Wilkinson References Gillespie, CS Moment closure approximations for mass-action models. IET Systems Biology 2009. Gillespie, CS, Golightly, A Bayesian inference for generalized stochastic population growth models with application to aphids. Journal of the Royal Statistical Society, Series C 2010. Milner, P, Gillespie, CS, Wilkinson, DJ Moment closure approximations for stochastic kinetic models with rational rate laws. Mathematical Biosciences 2011. Milner, P, Gillespie, CS and Wilkinson, DJ Moment closure based parameter inference of stochastic kinetic models. Statistics and Computing 2012. 25/25

Moment Closure Based Parameter Inference of Stochastic Kinetic Models

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