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# Moment Closure Based Parameter Inference of Stochastic Kinetic Models

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Talk on moment closure parameter which I gave at the SIAM conference on life sciences 2012,

http://www.siam.org/meetings/ls12/

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### Moment Closure Based Parameter Inference of Stochastic Kinetic Models

1. 1. Moment Closure Based Parameter Inference of Stochastic Kinetic Models Colin GillespieSchool of Mathematics & Statistics
2. 2. OverviewTalk outline An introduction to moment closure Parameter inference Conclusion 2/25
3. 3. Birth-death processBirth-death model X −→ 2X and 2X −→ Xwhich has the propensity functions λX and µX .Deterministic representationThe deterministic model is dX (t ) = ( λ − µ )X (t ) , dtwhich can be solved to give X (t ) = X (0) exp[(λ − µ)t ]. 3/25
4. 4. Birth-death processBirth-death model X −→ 2X and 2X −→ Xwhich has the propensity functions λX and µX .Deterministic representationThe deterministic model is dX (t ) = ( λ − µ )X (t ) , dtwhich can be solved to give X (t ) = X (0) exp[(λ − µ)t ]. 3/25
5. 5. Stochastic representationIn the stochastic framework, eachreaction has a probability of occurring 50The analogous version of the 40birth-death process is the difference Populationequation 30 20dpn = λ(n − 1)pn−1 + µ(n + 1)pn+1 10 dt − (λ + µ)npn 0 0 1 2 3 4 TimeUsually called the forward Kolmogorovequation or chemical master equation 4/25
6. 6. Moment equationsMultiply the CME by enθ and sum over n, to obtain ∂M ∂M = [λ(eθ − 1) + µ(e−θ − 1)] ∂t ∂θwhere ∞ M (θ; t ) = ∑ e n θ pn ( t ) n =0If we differentiate this p.d.e. w.r.t θ and set θ = 0, we get dE[N (t )] = (λ − µ)E[N (t )] dtwhere E[N (t )] is the mean 5/25
7. 7. The mean equation dE[N (t )] = (λ − µ)E[N (t )] dtThis ODE is solvable - the associated forward Kolmogorov equation isalso solvableThe equation for the mean and deterministic ODE are identicalWhen the rate laws are linear, the stochastic mean and deterministicsolution always correspond 6/25
8. 8. The variance equationIf 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 ] dtand 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. 9. Simple dimerisation modelDimerisation 2X1 −→ X2 and X2 −→ 2X1with propensities 0.5k1 X1 (X1 − 1) and k2 X2 . 8/25
10. 10. Dimerisation moment equationsWe 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 2The i th moment equation depends on the (i + 1)th equation 9/25
11. 11. Deterministic approximates stochasticRewriting dE[X1 ] 2 = 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ] dtin 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. 12. Deterministic approximates stochasticRewriting dE[X1 ] 2 = 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ] dtin 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. 13. Simple dimerisation modelTo close the equations, we assume an underlying distributionThe easiest option is to assume an underlying Normal distribution, i.e. E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3 3 2But we could also use, the Poisson 3 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3or the Log normal 2 3 3 E [ X1 ] E [ X1 ] = E [ X1 ] 11/25
14. 14. Simple dimerisation modelTo close the equations, we assume an underlying distributionThe easiest option is to assume an underlying Normal distribution, i.e. E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3 3 2But we could also use, the Poisson 3 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3or the Log normal 2 3 3 E [ X1 ] E [ X1 ] = E [ X1 ] 11/25
15. 15. Heat shock modelProctor et al, 2005. Stochastic kinetic model of the heat shock system twenty-three reactions seventeen chemical speciesA 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. 16. Density plots: heat shock model Time t=200 Time t=2000 0.006Density 0.004 0.002 0.000 600 800 1000 1200 1400 600 800 1000 1200 1400 ADP population 13/25
17. 17. P53-Mdm2 oscillation modelProctor and Grey, 2008 300 16 chemical species Around a dozen reactions 250The model contains an event 200 Population At t = 1, set X = 0 150If we convert the model to moment 100equations, we get 139 equations. 50However, in this case the moment 0closure approximation doesn’t do to 0 5 10 15 20 25 30well! Time 14/25
18. 18. P53-Mdm2 oscillation modelProctor and Grey, 2008 300 16 chemical species Around a dozen reactions 250The model contains an event 200 Population At t = 1, set X = 0 150If we convert the model to moment 100equations, we get 139 equations. 50However, in this case the moment 0closure approximation doesn’t do to 0 5 10 15 20 25 30well! Time 14/25
19. 19. P53-Mdm2 oscillation modelProctor and Grey, 2008 300 16 chemical species Around a dozen reactions 250The model contains an event 200 Population At t = 1, set X = 0 150If we convert the model to moment 100equations, we get 139 equations. 50However, in this case the moment 0closure approximation doesn’t do to 0 5 10 15 20 25 30well! Time 14/25
20. 20. What went wrong?The moment closure (tends) to fail when there is a large differencebetween the deterministic and stochastic formulationsIn this particular case, strongly correlated speciesTypically when the MC approximation fails, it gives a negativevarianceThe MC approximation does work well for other parameter values forthe p53 model 15/25
21. 21. Parameter inference 4 3Population 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. 22. Parameter inference 4 3 qPopulation 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. 23. Parameter inference 4 3 qPopulation 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 datak2 4 The likelihood can be very ﬂat 2 0 0 2 4 6 8 10 k1 16/25
24. 24. Lotka-Volterra model Species Predator PreyThe Lotka-Volterra predator prey system,describes the time evolution of two 400species, 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
25. 25. Lotka-Volterra model Species Predator PreyThe Lotka-Volterra predator prey system,describes the time evolution of two 400species, 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
26. 26. Lotka-Volterra model Species Predator PreyThe Lotka-Volterra predator prey system,describes the time evolution of two 400species, 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
27. 27. Lotka-Volterra model Species Predator PreyThe Lotka-Volterra predator prey system,describes the time evolution of two 400species, 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. 28. Parameter estimationLet Y(tu ) = (Y1 (tu ), Y2 (tu )) be the vector of the observed predatorand preyTo infer c1 , c2 and c3 , we need to estimate Pr[Y(tu )| Y(tu −1 ), c]i.e. the solution of the forward Kolmogorov equationWe 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 closureapproximation 18/25
29. 29. Parameter estimationLet Y(tu ) = (Y1 (tu ), Y2 (tu )) be the vector of the observed predatorand preyTo infer c1 , c2 and c3 , we need to estimate Pr[Y(tu )| Y(tu −1 ), c]i.e. the solution of the forward Kolmogorov equationWe 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 closureapproximation 18/25
30. 30. Bayesian parameter inferenceSummarising our beliefs about c and the unobserved predatorpopulation Y2 (0) via uninformative priorsThe joint posterior for parameters and unobserved states (for a singledata set) is 40 p (y2 , c | y1 ) ∝ p (c) p (y2 (0)) ∏ p (y(tu ) | y(tu−1 ), c) u =1For the results shown, we used a vanilla Metropolis-Hasting step toexplore the parameter and state spacesFor more complicated models, we can use a Durham & Gallant stylebridge (Milner, G & Wilkinson, 2012) 19/25
31. 31. Bayesian parameter inferenceSummarising our beliefs about c and the unobserved predatorpopulation Y2 (0) via uninformative priorsThe joint posterior for parameters and unobserved states (for a singledata set) is 40 p (y2 , c | y1 ) ∝ p (c) p (y2 (0)) ∏ p (y(tu ) | y(tu−1 ), c) u =1For the results shown, we used a vanilla Metropolis-Hasting step toexplore the parameter and state spacesFor more complicated models, we can use a Durham & Gallant stylebridge (Milner, G & Wilkinson, 2012) 19/25
32. 32. Results c1 c2 c3 Exact q q q Fully Obs. Diffusion q q qMom. Clos. q q q Exact q q q Partially Obs. Diffusion q q qMom. 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. 33. Auto regulation systemThis system contains twelve reactions and six speciesThe species populations ranges from zero (for species i) to around65,000 for species GThe moment closure approximation yields a closed set oftwenty-seven ODEs Six ODEs for the means Six ODEs for the variances Fifteen ODEs for the covariance terms 21/25
34. 34. Stochastic realisation 30 Species 25 gPopulation 20 i 15 10 r_g 5 r_i 0 0 10 20 30 40 50 Time 15 65100 10 65050 GI 5 65000 0 0 10 20 30 40 50 0 10 20 30 40 50 Time Time 22/25
35. 35. Stochastic realisation 30 Species 25 gPopulation 20 i 15 10 r_g 5 r_i 0 0 10 20 30 40 50 Time 15 65100 10 65050 GI 5 65000 0 0 10 20 30 40 50 0 10 20 30 40 50 Time Time 22/25
36. 36. Stochastic realisation 30 Species 25 gPopulation 20 i 15 10 r_g 5 r_i 0 0 10 20 30 40 50 Time 15 65100 10 65050 GI 5 65000 0 0 10 20 30 40 50 0 10 20 30 40 50 Time Time 22/25
37. 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 allParameter 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
38. 38. 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 allParameter 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
39. 39. 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 allParameter 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
40. 40. 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 allParameter 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. 41. Future workTechniques for assessing the moment closure approximationBetter closure techniques Computer emulation for momentsUsing the moment closure approximation as a proposal distribution inan MCMC algorithm The proposal can be (almost) anything we want The likelihood can be calculated using anything we want 24/25
42. 42. Acknowledgements Peter Milner Darren WilkinsonReferences 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