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An introduction to moment closure techniques

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An internal seminar introducing the moment closure technique for stochastic kinetic models

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An introduction to moment closure techniques

  1. 1. An introduction to moment closure techniques Colin Gillespie School of Mathematics & Statistics Newcastle University July 30, 2008 Colin Gillespie An introduction to moment closure techniques
  2. 2. Modelling Let’s start with a simple birth-death model. Birth-death model X −→ X − 1 and X −→ X + 1 which has the propensity functions µX and λX . 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]. Colin Gillespie An introduction to moment closure techniques
  3. 3. Deterministic Solution: λ < µ 50 40 Population 30 20 10 0 0 1 2 3 4 Time Colin Gillespie An introduction to moment closure techniques
  4. 4. Stochastic Simulation It’s very easy to simulate the birth-death process using Gillespie’s method: 1 Update reaction clock; 2 Choose a reaction to occur; 3 Repeat. Colin Gillespie An introduction to moment closure techniques
  5. 5. Four Stochastic Simulations 4 Simulations of a birth-death process 0 1 2 3 4 Simulation 3 Simulation 4 50 40 30 20 10 Population 0 Simulation 1 Simulation 2 50 40 30 20 10 0 0 1 2 3 4 Time Colin Gillespie An introduction to moment closure techniques
  6. 6. Stochastic Mean and Variance If we simulated the process a large number of times (say 109 ), then we could calculate the population mean and variance. We could construct an approximate 95% tolerance interval √ Mean ± 2 Variance Colin Gillespie An introduction to moment closure techniques
  7. 7. Four Stochastic SimulationsMean (Green), tolerance interval (red), simulation(blue) 0 1 2 3 4 Simulation 3 Simulation 4 50 40 30 20 10 Population 0 Simulation 1 Simulation 2 50 40 30 20 10 0 0 1 2 3 4 Colin Gillespie An introduction to moment closure techniques
  8. 8. Mean and Variance In this talk we will look at a quick method for estimating the mean and variance, without using stochastic simulation Colin Gillespie An introduction to moment closure techniques
  9. 9. Moment generating function Let pn (t) be the probability that the population is of size n at time t. The moment generating function is defined as ∞ M(θ; t) ≡ pn (t)enθ . n=0 If we differentiate M(θ; t) w.r.t θ and set θ = 0, we get E[N(t)], i.e. the mean. If we differentiate M(θ; t) w.r.t θ twice, and set θ = 0, we get E[N(t)2 ] and hence Var[N(t)] = E[N(t)2 ] − E[N(t)]2 . Colin Gillespie An introduction to moment closure techniques
  10. 10. General idea The birth-death process has the following CME: dpn = λ(n − 1)pn−1 + µ(n + 1)pn+1 − (λ + µ)npn dt After multiplying the CME by enθ and summing over n, we obtain ∂M ∂M = [λ(eθ − 1) + µ(e−θ − 1)] ∂t ∂θ Colin Gillespie An introduction to moment closure techniques
  11. 11. Moment Equations 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. This is a single ODE that we can solve to obtain a value for the mean. 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 So instead of simulating the process 109 to estimate the mean and variance, we can simply solve two ODEs. Colin Gillespie An introduction to moment closure techniques
  12. 12. Moment Equations 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. This is a single ODE that we can solve to obtain a value for the mean. 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 So instead of simulating the process 109 to estimate the mean and variance, we can simply solve two ODEs. Colin Gillespie An introduction to moment closure techniques
  13. 13. Part I ExamplesColin Gillespie An introduction to moment closure techniques
  14. 14. Simple Dimerisation model The dimerisation model has the following biochemical reactions: Dimerisation 2X1 −→ X2 and X2 −→ 2X1 We can formulate the dimer model in terms of moment equations, namely, 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 2 variance of X1 . Colin Gillespie An introduction to moment closure techniques
  15. 15. Simple Dimerisation model The dimerisation model has the following biochemical reactions: Dimerisation 2X1 −→ X2 and X2 −→ 2X1 We can formulate the dimer model in terms of moment equations, namely, 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 2 variance of X1 . The i th moment equation depends on the (i + 1)th equation. Colin Gillespie An introduction to moment closure techniques
  16. 16. Simple Dimerisation model The dimerisation model has the following biochemical reactions: Dimerisation 2X1 −→ X2 and X2 −→ 2X1 We can formulate the dimer model in terms of moment equations, namely, dE[X1 ] = 0.5k1 E[X1 ](E[X1 ] − 1) + 0.5k1 Var[X1 ] − k2 E[X1 ] dt 2 where E[X1 ] is the mean of X1 and E[X1 ] − E[X1 ]2 is the variance of X1 . The deterministic equation is an approximation to the stochastic mean. Colin Gillespie An introduction to moment closure techniques
  17. 17. Simple Dimerisation model To close the equations, we usually assume that the underlying distribution is Normal or Lognormal. 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 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3 3 or the Lognormal 2 3 3 E[X1 ] E[X1 ] = E[X1 ] Colin Gillespie An introduction to moment closure techniques
  18. 18. Simple Dimerisation model To close the equations, we usually assume that the underlying distribution is Normal or Lognormal. 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 E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3 3 or the Lognormal 2 3 3 E[X1 ] E[X1 ] = E[X1 ] Colin Gillespie An introduction to moment closure techniques
  19. 19. Simple Dimerisation model 300 Protein Population 250 200 150 0 5 10 15 20 25 Time Colin Gillespie An introduction to moment closure techniques
  20. 20. Heat Shock Model Proctor et al, 2005 - 23 reactions, 17 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. A python script automatically generates the ODEs from an SBML file These can be solved in less than 5 minutes using Maple Hopefully I’ll start outputting in sundials, so this should be even quicker Colin Gillespie An introduction to moment closure techniques
  21. 21. Heat Shock Model 1200 600 Native Protein (10,000’s) 1000 590 800 ADP 600 580 400 570 200 560 0 500 1000 1500 2000 0 500 1000 1500 2000 Time Time Colin Gillespie An introduction to moment closure techniques
  22. 22. Univariate Distributions 600 800 1000 1200 1400 Time t=200 Time t=2000 0.006 Density 0.004 0.002 0.000 600 800 1000 1200 1400 ADP Colin Gillespie An introduction to moment closure techniques
  23. 23. Bivariate Distributions at time t = 2000 7e+06 6e+06 NatP 5e+06 4e+06 800 900 1000 1100 1200 ADP Colin Gillespie An introduction to moment closure techniques
  24. 24. P53-Mdm2 Oscillations model Proctor and Grey, 2008 - 16 chemical species and about a dozen reactions. The model contains two events. If we convert the model to moment equations, we get 139 equations. However, in this case the moment closure approximation doesn’t do to well! Colin Gillespie An introduction to moment closure techniques
  25. 25. P53-Mdm2 Oscillations model Proctor and Grey, 2008 - 16 chemical species and about a dozen reactions. The model contains two events. If we convert the model to moment equations, we get 139 equations. However, in this case the moment closure approximation doesn’t do to well! Colin Gillespie An introduction to moment closure techniques
  26. 26. P53 MeanMC(black), True (red) 300 250 200 P53 Population 150 100 50 0 0 5 10 15 20 25 30 Time Colin Gillespie An introduction to moment closure techniques
  27. 27. P53 MeanMC(black), True (red), Deterministic(green) 300 250 200 P53 Population 150 100 50 0 0 5 10 15 20 25 30 Time Colin Gillespie An introduction to moment closure techniques
  28. 28. What went wrong? The Moment closure (tends) to fail when there is a large difference between the deterministic and stochastic formulations. I believe it failed because of 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. Colin Gillespie An introduction to moment closure techniques
  29. 29. Software Python script that takes in a SBML file and outputs the moment equations. Currently outputs as a Maple file (University has a site licence) Hopefully it will soon output as a sundials/GSL C file (Sort of) supports events. Currently only handles polynomial rate laws, but could be upgrade to handle more complicated rate laws. Colin Gillespie An introduction to moment closure techniques
  30. 30. References For an introduction to Moment closure see papers by Matis et al over the last 20 years. Gillespie, C.S. Moment closure approximations for mass-action models. IET Systems Biology, in press Colin Gillespie An introduction to moment closure techniques

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