Nested Stochastic Simulation Algorithm for KMC with Multiple Time-Scales

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1 Nested Stochastic Simulation Algorithm for KMC with Multiple Time-Scales Eric Vanden-Eijnden Courant Institute Join work with Weinan E and Di Liu

2 Chemical Systems: Evolution of an isothermal, spatially homogeneous mixture of chemically reacting molecules contained in a fixed volume V. N S species of molecules S i, i = 1,..., N S involved in M R reactions R j, j = 1,..., M R. Let x i be the number of molecules of species S i. Each reaction R j is characterized by a rate function a j (x) and a state change (or stochiometric) vector ν j : R j = (a j, ν j ), R = {R 1,..., R MR }.

3 Continuous-time Markov Chain: Given state x = (x 1,..., x NS ), the occurrences of the reactions on an infinitesimal time interval dt are independent of each other and the probability for reaction R j to happen during this time interval is given by a j (x)dt. The state of the system after reaction R j is x + ν j. Equivalently: Given that the state of the system is X t = x at time t; 1. The probability that the next reaction happens after time t+s is e a(x)s where a(x) = M R j=1 a j(x). 2. Given that a reaction happens at time t + s, the probability that it be reaction j is a j (x)/a(x).

4 Gillespie s Stochastic Simulation Algorithm (SSA, aka BKL): D. T. Gillespie, J. Comp. Phys. 22, 403 (1976) A. B. Bortz, M. H. Kalos and J. L. Lebowitz, J. Comp. Phys. 17, 10 (1975) Assume that the system is at state X n at time t n, then: 1. Generate two independent random numbers r 1 and r 2 with uniform distribution on the unit interval (0, 1]. Let δt n+1 = ln r 1 a(x n ), and k n+1 be the natural number such that (a 0 = 0) 1 a(x n ) k n+1 1 j=0 a j (X n ) < r 2 1 a(x n ) k n+1 j=0 a j (X n ), 2. Update the time and the state of the system by Then repeat. t n+1 = t n + δt n+1, X n+1 = X n + ν kn+1. Exact realization of the process X t

5 Systems with two disparate time scales.: Assume that the reactions can be grouped as R s = {(a s (x), ν s )}, (Slow reactions) R f = {(ε 1 a f (x), ν f )} (Fast reactions) where ε 1 represents the ratio of time scales of the system. Then: The time-step between reactions is O(ε) and with probability 1 O(ε) a fast reaction happens. Difficult to simulate the evolution up to the O(1) time-scale of the slow reactions! Several recent works on this topic e.g. by E. L. Haseltine and J. B. Rawlings, J. of Chem. Phys., (2002); C. V. Rao and A. P. Akin J. of Chem. Phy. 118, (2003); Y. Cao, D. Gillespie, and L. Petzold, J. Chem. Phys. 122, (2005); A. Samant and D. G. Vlachos J. Chem. Phys. 123, (2005).

6 Simple example: with a S 1 1 S 2 a2 }{{} fast a, S 3 2 S 3 a4 }{{} slow a 1 = 10 5 x 1, ν 1 = ( 1, +1, 0, 0), a 2 = 10 5 x 2, ν 2 = (+1, 1, 0, 0), a 3 = x 2, ν 3 = ( 0, 1, +1, 0), a 4 = x 3, ν 4 = ( 0, +1, 1, 0), a 5 = 10 5 x 3, ν 5 = ( 0, 0, 1, +1), a 6 = 10 5 x 4, ν 6 = ( 0, 0, +1, 1). a S 5 3 S 4 a6 }{{} fast i.e. the first and third reactions are faster than the second one.. Every species is involved in at least one fast reaction so there is no slow species. But the variables y 1 = x 1 + x 2 and y 2 = x 3 + x 4 are conserved during the fast reactions and only evolve during the slow reaction.

7 x 1 +x 2 x 3 12 molecules time x 10!3 Fig. 1. Evolution of slow variable y 1 = x 1 + x 2 and fast variable x 3 on the intermediate time scale. Evolution of slow variable y 1 = x 1 + x 2 and 2.2 Effective fast dynamics variable onxthe 3 on slowthe time intermediate scale time scale. For the kind of systems discussed above, very often we are interested mostly in

8 Solution: Nested Stochastic Simulation Algorithm: W. E, D. Liu, and E. V.-E., J. Chem Phys. in press; J. Comp Phys. submitted Inner SSA: Run N independent replicas of SSA with the fast reactions R f = {(ε 1 a f, ν f })} only, for a time interval of T 0 + T f. During this calculation, compute the modified slow rates from ã s j = 1 N N k=1 1 T f Tf +T 0 T 0 a s j (Xk τ )dτ, j = 1,, M RS where X k τ is the result of the k-th replica of this auxiliary virtual fast process at virtual time τ whose initial value is X k t=0 = X n. Outer SSA: Run one step of SSA for the modified slow reactions R s = (ã s, ν s ) to generate (t n+1, X n+1 ) from (t n, X n ). Then repeat. Totally seamless!

9 Justification: Averaging theorem for Markov chains. Consider the u(x, t) = E x f(x t ), where X t is the state variable at time t, and E x denotes expectation conditional on X t=0 = x. Then sup 0 t T u(x, t) ū(x, t) = O(ε) Here ū(x, t) = E x f( X t ), where X t is the process generated by: R = {( ā j (x) = )} a s j (x )µ y(x) (x ), νj s x j=1,...,m RS where µ y(x) (x ) is the equilibrium distribution of the virtual fast process, i.e. the original process where only the fast reactions are kept and X t=0 = x.

10 u(x, t) = E x f(x t ) satisfies the backward Kolomogorov equation: u(x, t) t = M R j=1 a j (x) ( u(x + ν j, t) u(x, t) ) (Lu)(x, t) = ε 1 (L 1 u)(x, t) + (L 0 u)(x, t) Singular perturbation analysis of this equation (Khasminskii,...). Key observations: (i) the slow variables coincide with the maximum ergodic components of the process associated with L 1 ; (ii) they are given by y k = b k x for k = 1,..., K, where {b 1,..., b K } form a basis of the linear subspace such that b ν f j = 0 for all νf j ; and (iii) if x and x belong to the same ergodic component, then x + ν s j and x + ν s j also belong to the same ergodic component for all ν s j.

11 Simple example revisited a S 1 1 S 2 a2 }{{} fast a, S 3 2 S 3 a4 }{{} slow a S 5 3 S 4 a6 }{{} fast. Slow variables: y 1 = x 1 + x 2, y 2 = x 3 + x 4 Equilibrium distribution of the virtual fast process: µ y1,y 2 (x 1, x 2, x 3, x 4 ) = y 1! y 2! x 1! x 2! x 3! x 4! (1/2)y 1(1/2) y 2δ x1 +x 2 =y 1 δ x3 +x 4 =y 2. Effective dynamics: ā s 3 = P x 2 = x 1 + x 2 2 ā s 4 = P x 3 = x 3 + x 4 2 = y 1 2, νs 3 = y 2 2, νs 4 = ( 1, +1), = (+1, 1).

12 x 1 +x 2 x 3 +x x 1 +x 2 x 3 Multiscale Methods for Stochastic Dynamical S molecules time x 10!3

13 Error estimate: For any T > 0, there exist constants C and α independent of (N, T 0, T f ) such that, sup 0 t T E v(x, t) u(x, t) C ε + e αt 0/ε 1 + T f /ε + 1 N(1 + Tf /ε). Efficiency: Given an error tolerance λ: cost = O(N(1 + T 0 /ε + T f /ε)) = O ( ) 1 λ 2 (nested SSA) cost = O ( ) 1 ε (direct SSA)

14 Example: Heat shock response of E. Coli Mechanism of protection that the E. Coli bacteria uses to fight against denaturation (unfolding) of its constituent proteins induced by the increase of temperature. Stochastic petri net for heat shock response (from Srivastava et al.) R. Srivastava, M. Peterson and W. Bently, Biotech. Bioeng. 75, 120 (2001) 14 species, 17 reactions

15 Species Initial value DNA.σ 32 1 mrna.σ σ RNAPσ DNA.DnaJ 1 DNA.FtsH 0 DNA.GroEL 1 DnaJ 4640 FtsH 200 GroEL 4314 DnaJ.UnfoldedProtein Protein σ 32.DnaJ 2959 UnfoldedProtein es and their initial value (in number E. Coli proposed in [8]. Figure 3. SPN of the!32 genetic regulatory circuit. All simulations were based on the SPN shown. The arcs made up of dotted lines were for recombinant protein simulations. of the molecules) in the Petri net model of tion, the generic binding constant between the recombinant protein and the J-complex was varied. In this way, predictions of a!32-mediated stress response could be generated. software package for modeling stochastic activi (Sanders, 1995). The ULTRASAN software was vided by Dr. Sanders (Center for Reliable

16 Reaction Rate constant Rates magnitude DNA.σ 32 mrna.σ mrna.σ 32 σ 32 + mrna.σ mrna.σ 32 degradation σ 32 RNAPσ RNAPσ 32 σ σ 32 + DnaJ σ 32.DnaJ ( ) DnaJ degradation ( ) σ 32.DnaJ σ 32 + DnaJ DNA.DnaJ + RNAPσ 32 DnaJ + DNA.DnaJ + σ DNA.FtsH + RNAP.σ 32 FtsH + DNA.FtsH + σ FtsH degradation GroEL degradation σ 32.DnaJ + FtsH DnaJ + FtsH DNA.GroEL + RNAPσ 32 GroEL + DNA.GroEL +σ Protein UnfoldedProtein ( ) DnaJ+ UnfoldedProtein DnaJ.UnfoldedProtein ( ) DnaJ.UnfoldedProtein DnaJ+ UnfoldedProtein ( )

17 Nested SSA allows a speed-up by a factor of 100 in this example without any loss of accuracy %)!! %(&! %(!! +,-./ " 5"!!' 4 " 5"!!% %'&! %'!! %&&! %&!! %%&! %%!! %$&! %$!!! "!! #!! $!! %!! &!! '!! (!! )!! *!! "!!! Shown: Growth rate of GroEL (a protein measuring stress response)

18 (N, T f /10 6 ) (1, 1) (1, 4) (1, 16) (1, 64) (1, 256) (1, 1024) CPU σ var(σ 32 ) TABLE III: Efficiency of nested SSA when N = 1. Since we used N 0 = 1000 realizations of the Outer SSA to compute σ 32 and var(σ 32 ), the statistical errors on these quantities is about 0.1. (N, T f /10 6 ) (1, 1) (2, 2) (4, 4) (8, 8) (16, 16) (32, 32) CPU σ var(σ 32 ) TABLE IV: Efficiency of nested SSA with multiple replicas in the Inner SSA. Again the statistical errors on σ 32 and var(σ 32 ) is about 0.1. T f / CPU

19 or C ε T f /ε + 1 N f (1 + T f /ε) N uf (1 + T uf /ε ) Generalization: 1 Nested + T uf /ε 2 SSA with more than two levels: Let us take a look at an example. Consider the (104) following system + 1 Nf (1 + T f /ε) + 1 Nuf (1 + T uf /ε 2 ) a 1 S 1 S 2, S 2 a2 a 3 S 3, S 3 a4 with the reaction rates and state change vectors. e a look at an example. Consider the following system S 1 a 1 S 2, S 2 a2 a 3 S 3, S 3 a4 eaction rates and state change vectors a 5 S 4. (105) a6 a 1 = x 1, ν 1 = 1 ( 1, +1, Nested 0, SSA0), a 2 = x 2, ν 2 = (+1, 1, Direct SSA 0, 0), a 3 = x 2, ν 3 = ( 0, 1, +1, 0), a 4 = x 3, ν 4 = ( 0, +1, 1, 0), (106) a 5 = x 3, ν = ( 0, 0, 1, +1), a 6 = x 4, ν 6 = ( 0, 0, +1, 1). a 1 = x 1, ν 1 = ( 1, +1, 0, 0), a 2 = x 2, ν 2 = (+1, 1, 0, 0), a 3 = 10 5 x 2, ν 3 = ( 0, 1, +1, 0), a 4 = x 3, ν 4 = ( 0, +1, 1, 0), a 5 = x 3, ν 5 = ( 0, 0, 1, +1), a 6 = x 4, ν 6 = ( 0, 0, +1, 1). Empirical distribution 0.2 a 5 a6 S 4. (105) (106) x1 x2 x3 x4 Fig. 6. Empirical distributions obtained by nested SSA and direct SSA cost. The distribution produced by nested SSA is nearly exact, wher produced by direct SSA is totally inaccurate. results:

20 SSA to pick up the set of fast reactions dynamically. Consider the following system: Generalization: Adaptive nested SSA: Fig. 3. Time evolutiona 1 of the reaction rates a S a 32 and a 5 + a 6 S 2, S 2 a2 S 3, 2S 2 + S 3 a4 eaction rates and the state change vectors are Fig. 3. Time evolution of the reaction rates a 1 + a 2 and a 5 + a 6 The a reaction 1 = x 1, rates and the state change ν 1 = vectors ( 1, +1, are 0, 0), a 2 = x 2, ν 2 = (+1, 1, 0, 0), 23 a 3 = 10 4 ax 12 =, x 1, ν 3 = (0, ν 1, 1 = +1, ( 1, 0), +1, 0, 0), a 4 = 10 4 (96) ax 23 =, x 2, ν 4 = (0, ν+1, 2 = 1, (+1, 0), 1, 0, 0), a 5 = 2x 2 a(x 3 = )x 4 x 23, ν 5 = ( 0, ν 2, 3 = 1, (0, +3), 1, +1, 0), a 6 = 2x 4 a(x 4 = )(x 4 x 3, 4 2), ν 6 = ( 0, ν+2, 4 = +1, (0, 3). +1, 1, 0), a 5 = 2x 2 (x 2 1)x 3, ν 5 = ( 0, 2, 1, +3), se that we start with the following initial condition: a 6 = 2x 4 (x 4 1)(x 4 2), ν 6 = ( 0, +2, +1, 3). (x 1, x 2, x 3, x 3 ) = (100, 3, 3, 3). (97) Suppose that we start with the following initial condition: (96) e beginning, when the concentration of S 2 is low, only the transition en S 2 Initial and S 3 data: (x is fast. As the 1, x number 2, x 3, x of 3 ) = (100, 3, 3, 3). (97) S 2 grows, the last two reactions e faster and faster. Figure 5 shows the evolution of the sum of the At the beginning, when the concentration of S on rates a 1 + a 2 and a 5 + a 6. It can be seen that the 2 is low, only the transition reaction rate a 5 + a between S 6 from O(1) to 2 and S O( is fast. As the number of S ) on a time scale of O(1). 2 grows, the last two reactions become faster and faster. Figure 5 shows the evolution of the sum of the t reaction the nested rates SSAa 1 and + a 2 compare and a 5 + itawith 6. It the can direct be seen SSA, thatwe the use reaction the mean rate a 5 + a 6 egrows variance from at time O(1) T to= O(10 4 of x 5 ) 1, on thea number time scale of species of O(1). S 1, as a benchmark. putation of the direct SSA with N 0 = gives To test the nested SSA and compare it with the direct SSA, we use the mean and the variance x 1 = at time ± 0.2, T = 4 var(x of x 1, 1 the ) = number ± of 0.2. species S 1, as a(98) benchmark. a 5 a6 3S 4. (95)

21 A computation A computation of theof direct the direct SSA SSANwith 0 = N 0 = gives gives the ensemble and simulation time of the inner SSA in the nested SSA scheme are chosen to be Direct SSA: x 1 = x 1 = ± , ± 0.2, var(x 1 ) var(x = ) = ± ± 0.2. (98) (98) (N, T f ) = (1, 2 k 10 5 ), (100) The calculation The calculation for different took took values seconds of k = seconds 0, of 1, CPU 2, of.. time.. CPU The ontime error ouron machine. should our machine. be In the Innested the nested SSA, SSA, we dynamically we dynamically change change the set λ = O(2 the of k/2 fast ). set of reactions fast reactions by con-by con-(101) The following table gives 24 the CPU time and the values of the mean and variance of x 1 using N 0 = The 24 sum of the relative errors of the mean and the variance is shown in figure 5. Adaptive Nested SSA: T f / CPU x var(x 1 ) Table 4 Efficiency and accuracy of the adaptive nested SSA. The strategy for adaptively determining the set of fast and slow reactions may not be the best one. Further work in this direction is clearly needed. 25

22 Generalization to other types of KMC Generator (Lf)(x) = j J λ j (x)(f(x + z j (x)) f(x)) Ergodic components of fast process: for any test function f : X R, we have if x X y then lim T 1 T T 0 E xf(x f t )dt = x X µ y (x)f(x), where E x denotes the expectation conditional on X f t=0 = x Assumption: Slow reactions induce an injection on the ergodic components X y, i.e. for each e j, there exists a unique ē j such that for all x X, if x X y then x + e j X yj with y j = y + ē j.

Nested Stochastic Simulation Algorithm for Chemical Kinetic Systems with Disparate Rates. Abstract

Nested Stochastic Simulation Algorithm for Chemical Kinetic Systems with Disparate Rates Weinan E Department of Mathematics and PACM, Princeton University, Princeton, NJ 08544, USA Di Liu and Eric Vanden-Eijnden