Stochastic models for chemical reactions

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1 First Prev Next Go To Go Back Full Screen Close Quit 1 Stochastic models for chemical reactions Formulating Markov models Two stochastic equations Simulation schemes Reaction Networks Scaling limit Central limit theorem General approaches to averaging Well-mixed reactions Michaelis-Menten equation Model of viral infection References Software Abstract

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Bilingual dictionary Chemistry propensity master equation nonlinear diffusion approximation Van Kampen approximation quasi steady state/partial equilibrium Probability intensity forward equation diffusion approximation central limit theorem averaging

3 First Prev Next Go To Go Back Full Screen Close Quit 3 Intensities for continuous-time Markov chains Assume X is a continuous time Markov chain in E Z d. The Q-matrix, Q = {q kl }, for the chain gives and hence P {X(t + t) = l X(t) = k} q kl t, k l E, E[f(X(t+ t)) f(x(t)) F X t ] l q X(t),l (f(l) f(x(t)) t Af(X(t)) t Alternative notation: Define β l (k) = q k,k+l. Then Af(k) = l β l (k)(f(k + l) f(k))

4 First Prev Next Go To Go Back Full Screen Close Quit 4 Martingale problems is made precise by the requirement that f(x(t)) f(x()) Af(X(s))ds be a {Ft X }-martingale for f in an appropriate domain D(A). X is called a solution of the martingale problem for A.

5 First Prev Next Go To Go Back Full Screen Close Quit 5 Backward and forward equations Defining u(x, t) = E[f(X(t)) X() = x], one can derive the backward equation t u(t, x) = Au(t, x) and setting ν t (G) = P {X(t) G} and ν t f = E fdν t, the martingale property gives the forward equation (in weak form) ν t f = ν f + ν s Afds, f D(A).

6 First Prev Next Go To Go Back Full Screen Close Quit 6 The forward/master equation Taking f = 1 {k} and setting p k (t) = ν t ({k}), ṗ k (t) = l p k l (t)β l (k l) p k (t) l β l (k) giving the usual form of the forward equation (the master equation in the chemical literature).

7 First Prev Next Go To Go Back Full Screen Close Quit 7 Time change equation X(t) = X() + l ln l (t) where N l (t) is the number of jumps of l at or before time t. N l is a counting process with intensity (propensity in the chemical literature) β l (X(t)), that is, N l (t) β l (X(s))ds is a martingale. Consequently, we can write N l (t) = Y l ( β l (X(s))ds), where the Y l are independent, unit Poisson processes, and X(t) = X() + l ly l ( β l (X(s))ds).

8 First Prev Next Go To Go Back Full Screen Close Quit 8 Random jump equation Alternatively, setting β(k) = l β l(k), and N(t) = Y ( X(t) = X() + β(x(s))ds) F (X(s ), ξ N(s ) )dn(s) where Y is a unit Poisson process, {ξ i } are iid uniform [, 1], and P {F (k, ξ) = l} = β l(k) β(k).

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Connections to simulation schemes Simulating the random-jump equation gives Gillespie s [7, 8] direct method (the stochastic simulation algorithm SSA). Simulating the time-change equation gives the next jump) method as defined by Gibson and Bruck [6]. reaction (next For = τ (x) < τ 1 (x) <, satisfying τ k (x) = τ k (x τk ), where { x τk x(s) s < τ (s) = k (x) x(τ k (x) ) s τ k (x) (typically, τ k+1 (x) = τ k (x) + g k+1 (x(τ k ))), simulation of ˆX(t) = X() + ( ) ly l β l ( ˆX(τ k ))(τ k+1 t τ k t) l k gives Gillespie s [9] τ-leap method

10 First Prev Next Go To Go Back Full Screen Close Quit 1 Reaction networks Standard notation for chemical reactions A + B k C is interpreted as a molecule of A combines with a molecule of B to give a molecule of C. A + B C means that the reaction can go in either direction, that is, a molecule of C can dissociate into a molecule of A and a molecule of B We consider a network of reactions involving m chemical species, A 1,..., A m. m m ν ik A i i=1 i=1 ν ika i where the ν ik and ν ik are nonnegative integers

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Markov chain models X(t) number of molecules of each species in the system at time t. ν k number of molecules of each chemical species consumed in the kth reaction. ν k number of molecules of each species created by the kth reaction. λ k (x) rate at which the kth reaction occurs. (The propensity/intensity.) If the kth reaction occurs at time t, the new state becomes X(t) = X(t ) + ν k ν k. The number of times that the kth reaction occurs by time t is given by the counting process satisfying R k (t) = Y k ( λ k (X(s))ds), where the Y k are independent unit Poisson processes.

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Equations for the system state The state of the system satisfies X(t) = X() + k = X() + k R k (t)(ν k ν k ) Y k ( λ k (X(s))ds)(ν k ν k ) = (ν ν)r(t) ν is the matrix with columns given by the ν k. ν is the matrix with columns given by the ν k. R(t) is the vector with components R k (t). Basic assumption: The system is well-mixed.

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Rates for the law of mass action For a binary reaction A 1 + A 2 A 3 or A 1 + A 2 A 3 + A 4 λ k (x) = κ k x 1 x 2 For A 1 A 2 or A 1 A 2 + A 3, λ k (x) = κ k x 1. For 2A 1 A 2, λ k (x) = κ k x 1 (x 1 1). For a binary reaction A 1 +A 2 A 3, the rate should vary inversely with volume, so it would be better to write λ N k (x) = κ k N 1 x 1 x 2 = Nκ k z 1 z 2, where classically, N is a scaling parameter taken to be the volume of the system times Avogadro s number and z i = N 1 x i is the concentration in moles per unit volume. Note that unary reaction rates also satisfy λ k (x) = κ k x i = Nκ k z i.

14 First Prev Next Go To Go Back Full Screen Close Quit 14 General form for classical scaling All the rates naturally satisfy λ N k (x) Nκ k i z ν ik i N λ k (z).

15 First Prev Next Go To Go Back Full Screen Close Quit 15 First scaling limit Setting C N (t) = N 1 X(t) C N (t) = C N () + k C N () + k N 1 Y k ( N 1 Y k (N λ N k (X(s))ds)(ν k ν k ) λ k (C N (s))ds)(ν k ν k ) The law of large numbers for the Poisson process implies N 1 Y (Nu) u, C N (t) C N () + κ k Ci N (s) ν ik (ν k ν k )ds, k i which in the large volume limit gives the classical deterministic law of mass action Ċ(t) = κ k C i (t) ν ik (ν k ν k ) F (C(t)). k i skip

16 First Prev Next Go To Go Back Full Screen Close Quit 16 Central limit theorem/van Kampen approximation V N (t) N(C N (t) C(t)) V N () + N( 1 N Y k(n k F (C(s))ds) λ N k (C N (s))ds)(ν k ν k ) = V N () + 1 Ỹ k (N λ N k (C N (s))ds)(ν k ν k ) k N + N(F N (C N (s)) F (C(s)))ds V N () + W k ( λ k (C(s))ds)(ν k ν k ) k + F (C(s)))V N (s)ds

17 First Prev Next Go To Go Back Full Screen Close Quit 17 Gaussian limit V N converges to the solution of V (t) = V () + k W k ( λ k (C(s))ds)(ν k ν k ) + F (C(s)))V (s)ds C N (t) C(t) + 1 N V (t)

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Diffusion approximation where C N (t) = C N () + k N 1 Y k ( C N () + N 1/2 W k ( k + F (C N (s))ds, F (c) = k λ k (X N (s))ds)(ν k ν k ) λ k (c)(ν k ν k ). λ k (C N (s))ds)(ν k ν k ) The diffusion approximation is given by the equation C N (t) = C N ()+ N 1/2 W k ( k λ k ( C N (s))ds)(ν k ν k )+ F ( C N (s))ds.

19 First Prev Next Go To Go Back Full Screen Close Quit 19 Itô formulation The time-change formulation is equivalent to the Itô equation C N (t) = N 1/2 λ k ( C N (s))d W k (s)(ν k ν k ) C N () + k + F ( C N (s))ds = C N () + N 1/2 σ( C N (s))d W (s) + k where σ(c) is the matrix with columns λk (c)(ν k ν k). F ( C N (s))ds, See Kurtz [11], Ethier and Kurtz [4], Chapter 1, Gardiner [5], Chapter 7, and Van Kampen, [14].

20 First Prev Next Go To Go Back Full Screen Close Quit 2 General approaches to averaging Models with two time scales: (X, Y ), Y is fast Occupation measure: Γ Y (C [, t]) = 1 C(Y (s))ds Replace integrals involving Y by integrals against Γ Y f(x(s), Y (s))ds = f(x(s), y)γ Y (dy ds) How do we identify η s? E Y [,t] E Y f(x(s), y)η s (dy)ds

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Generator approach Suppose B r f(x, y) = rcf(x, y) + Df(x, y) where C operates on f as a function of y alone. f(x r (t), Y r (t)) r Cf(X r (s), y)γ Y r (dy ds) E Y [,t] Df(X r (s), y)γ Y r (dy ds) E Y [,t] Assuming (X r, Γ Y r ) (X, Γ Y ), dividing by r, we should Cf(X(s), y)γ Y (dy ds) = Cf(X(s), y)η s (dy)ds = E Y [,t] E Y [,t] Suppose that for each x, there exists a unique µ x satisfying E Y Cf(x, y)µ x (dy) =, f D, then η s (dy) = µ X(s) (dy)

22 Well-mixed reactions Consider A + B κ C. The generator for the Markov chain model is Af(m, n) = κmn(f(m 1, n 1) f(m, n)) Spatial model U i V j state (location and configuration) of ith molecule of A state of jth molecule of B Bf(u, v) = m n rc A u i f(u, v) + rc B v j f(u, v) i=1 j=1 + ρ(u i, v j )(f(θ i u, θ j v) f(u, v)) i,j where rc A is a generator modeling the evolution of a molecule of A and rc B models the evolution of a molecule of B. First Prev Next Go To Go Back Full Screen Close Quit 22

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Independent evolution of molecules If there was no reaction rcf(u, v) = m rc A u i f(u, v) + i=1 n rc B v j f(u, v) would model the independent evolution of m molecules of A and r molecules of B. j=1

24 First Prev Next Go To Go Back Full Screen Close Quit 24 Averaging: Markov chain model Assume that the state spaces E A, E B for molecules of A and B are compact and let E = m,n E m A En B. Let Γ r be the occupation measure so f(u r (t), V r (t)) Γ r (C [, t]) = E [,t] 1 C (U r (s), V r (s))ds, (rcf(u, v) + Df(u, v))γ r (du dv ds) is a martingale. Then {(Γ r, XA r, Xr B )} is relatively compact, and assuming all functions are continuous, any limit point (Γ, X A, X B ) of Γ r as r satisfies Cf(u, v)γ(du, dv, ds) =. E [,t] cf. Kurtz [12]

25 First Prev Next Go To Go Back Full Screen Close Quit 25 Averaged generator If f depends only on the numbers of molecules the martingale becomes f(x A (t), X B (t)) E [,t] i,j ρ(u i, v j )(f(x A (s) 1, X B (s) 1) f(x A (s), X B (s)))γ(du, dv, ds). If C A and C B have unique stationary distributions µ A, µ B, then for f(u, v) = m i=1 g(u i) n j=1 h(u j), f(u, v)γ(du, dv, t) = g, µ A XA(s) h, µ B XB(s) ds and setting κ = ρ(u, v )µ A (du )µ B (dv ), f(x A (t), X B (t)) is a martingale. κx A (s)x B (s)f(x A (s) 1, X B (s) 1) f(x A (s), X B (s)))ds

26 First Prev Next Go To Go Back Full Screen Close Quit 26 Averaging: Michaelis-Menten kinetics Consider the reaction system A + E AE B + E modeled as a continuous time Markov chain satisfying X A (t) = X A () Y 1 ( X E (t) = X E () Y 1 ( +Y 3 ( κ 1 X A (s)x E (s)ds) + Y 2 ( κ 1 X A (s)x E (s)ds) + Y 2 ( κ 3 X AE (s)ds) X B (t) = Y 3 ( κ 3 X AE (s)ds) κ 2 X AE (s)ds) κ 2 X AE (s)ds)

27 First Prev Next Go To Go Back Full Screen Close Quit 27 Scaling Note that M = X AE (t) + X E (t) is constant. Let N = O(X A ) define V E (t) = M 1 X E (s)ds, Z A (t) = N 1 X A (t) κ 1 = M 1 γ 1, κ 2 = NM 1 γ 2, κ 3 = NM 1 γ 3 Z A (t) = Z A () N 1 Y 1 (N X E (t) = X E () Y 1 (N +Y 3 (Nγ 3 (t V E (t))) Z B (t) = N 1 Y 3 (Nγ 3 (t V E (t))) γ 1 Z A (s)dv E (s)) + N 1 Y 2 (Nγ 2 (t V E (t))) γ 1 Z A (s)dv E (s)) + Y 2 (Nγ 2 (t V E (t))) Along a subsequence (Z A, Z B, V E ) (x A, x B, v E ) and Z A (s)dv E (s) x A (s)dv E (s) = x A (s) v E (s)ds

28 First Prev Next Go To Go Back Full Screen Close Quit 28 Theorem 1 (Darden [2, 3]) Assume that N, M/N, Mκ 1 γ 1, Mκ 2 /N γ 2, Mκ 3 /N γ 3, and X A ()/N x A (), and Then (N 1 X A, V E ) converges to (x A (t), v E (t)) satisfying x A (t) = x A () = and hence v E (s) = γ 1 x A (s) v E (s)ds + γ 1 x A (s) v E (s)ds + γ 2 +γ 3 γ 2 +γ 3 +γ 1 x A (s) and γ 1 γ 3 x A (t) ẋ A (t) = γ 2 + γ 3 + γ 1 x A (s). γ 2 (1 v E (s))ds (1) (γ 2 + γ 3 )(1 v E (s))ds,

29 First Prev Next Go To Go Back Full Screen Close Quit 29 Quasi-steady state Assume M is constant κ 2 = γ 2 N/M, κ 3 = γ 3 N/M. Then f(x E (t)) f(x E ()) Nγ 1 Z A (s)m 1 X E (s)(f(x E (s) 1) f(x E (s)))ds N(γ 2 + γ 3 )(1 M 1 X E (s))(f(x E (s) + 1) f(x E (s)))ds Since Z A (s) x A (s), Cf(x A (s), k)η s (dk) = becomes M η s (k)[(γ 1 x A (s)m 1 k(f(k 1) f(k) k= so η s is binomial(m, p s ), where p s = +(γ 2 + γ 3 )(1 M 1 k)(f(k + 1) f(k))] = γ 2 +γ 3 γ 2 +γ 3 +γ 1 x A (s).

30 A model of intracellular viral infection Srivastava, You, Summers, and Yin [13], Haseltine and Rawlings [1], Ball, Kurtz, Popovic, and Rampala [1] Three time-varying species, the viral template, the viral genome, and the viral structural protein (indexed, 1, 2, 3 respectively). The model involves six reactions, T + stuff k 1 T + G G k 2 T T + stuff k 3 T + S T k 4 S k 5 G + S k 6 V First Prev Next Go To Go Back Full Screen Close Quit 3

31 First Prev Next Go To Go Back Full Screen Close Quit 31 Stochastic system X 1 (t) = X 1 () + Y b ( X 2 (t) = X 2 () + Y a ( X 3 (t) = X 3 () + Y c ( k 2 X 2 (s)ds) Y d ( k 1 X 1 (s)ds) Y b ( Y f ( k 6 X 2 (s)x 3 (s)ds) k 3 X 1 (s)ds) Y e ( Y f ( k 6 X 2 (s)x 3 (s)ds) k 4 X 1 (s)ds) k 2 X 2 (s)ds) k 5 X 3 (s)ds)

32 Figure 1: Simulation (Haseltine and Rawlings 22) First Prev Next Go To Go Back Full Screen Close Quit 32

33 First Prev Next Go To Go Back Full Screen Close Quit 33 Scaling parameters N scaling parameter Each X i is scaled according to its abundance in the system. For N = 1, X 1 = O(N ), X 2 = O(N 2/3 ), and X 3 = O(N) and we take Z 1 = X 1, Z 2 = X 2 N 2/3, and Z 3 = X 3 N 1. Expressing the rate constants in terms of N = 1 k k N 2/3 k 3 1 N k k k N 5/3

34 First Prev Next Go To Go Back Full Screen Close Quit 34 Normalized system With the scaled rate constants, we have Z N 1 (t) = Z N 1 () + Y b ( Z N 2 (t) = Z N 2 () + N 2/3 Y a ( 2.5Z N 2 (s)ds) Y d ( N 2/3 Y f ( Z3 N (t) = Z3 N () + N 1 Y c ( N 1 Y f ( Z N 1 (s)ds) N 2/3 Y b (.25Z N 1 (s)ds).75z N 2 (s)z N 3 (s)ds) NZ N 1 (s)ds) N 1 Y e (.75Z N 2 (s)z N 3 (s)ds), 2.5Z N 2 (s)ds) 2NZ N 3 (s)ds)

35 First Prev Next Go To Go Back Full Screen Close Quit 35 Limiting system With the scaled rate constants, we have Z 1 (t) = Z 1 () + Y b ( Z 2 (t) = Z 2 () Z 3 (t) = Z 3 () + 2.5Z 2 (s)ds) Y d ( Z 1 (s)ds 2Z 3 (s)ds.25z 1 (s)ds)

36 First Prev Next Go To Go Back Full Screen Close Quit 36 Fast time scale When τɛ N On the event τɛ N <, = inf{t : Z N 2 (t) ɛ} <, define V N i (t) = Z i (τ N ɛ + N 2/3 t). V N 1 (t) = Z 1 (τ N ɛ ) + Y b ( V2 N (t) = ɛn 2/3 + N 2/3 Y N 2/3 a ( V3 N (t) = Z 3 (τɛ N ) + N 1 Yc ( 2.5N 2/3 V N 2 (s)ds) Y N 2/3 V1 N (s)ds) N 2/3 Y b ( N 2/3 Y f (N 2/3 d ( 2.5N 2/3 V N 2 (s)ds).25n 2/3 V N 1 (s)ds).75v N 2 (s)v N N 5/3 V N 1 (s)ds) N 1 Y N 1 Y f ( e ( 3 (s)ds).75n 2/3 V N 2 (s)v N 3 (s)ds) 2N 5/3 V N 3 (s)ds)

37 First Prev Next Go To Go Back Full Screen Close Quit 37 Averaging As N, dividing the equations for V1 N and V3 N by N 2/3 shows that V N 1 (s)ds 1 V N 3 (s)ds 5 V N 2 (s)ds V N 2 (s)ds. The assertion for V3 N and the fact that V2 N is asymptotically regular imply V2 N (s)v3 N (s)ds 5 V2 N (s) 2 ds. It follows that V2 N converges to the solution of (2).

38 First Prev Next Go To Go Back Full Screen Close Quit 38 Law of large numbers Theorem 2 Conditioning on τ N ɛ <, for each δ > and t >, where V 2 is the solution of lim P { sup V2 N (s) V 2 (s) δ} =, N s t V 2 (t) = ɛ + 7.5V 2 (s)ds) 3.75V 2 (s) 2 ds. (2)

39 First Prev Next Go To Go Back Full Screen Close Quit 39 References [1] Karen Ball, Thomas G. Kurtz, Lea Popovic, and Grzegorz A. Rempala. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab., 26. to appear. [2] Thomas Darden. A pseudo-steady state approximation for stochastic chemical kinetics. Rocky Mountain J. Math., 9(1):51 71, Conference on Deterministic Differential Equations and Stochastic Processes Models for Biological Systems (San Cristobal, N.M., 1977). [3] Thomas A. Darden. Enzyme kinetics: stochastic vs. deterministic models. In Instabilities, bifurcations, and fluctuations in chemical systems (Austin, Tex., 198), pages Univ. Texas Press, Austin, TX, [4] Stewart N. Ethier and Thomas G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, Characterization and convergence. [5] C. W. Gardiner. Handbook of stochastic methods for physics, chemistry and the natural sciences, volume 13 of Springer Series in Synergetics. Springer-Verlag, Berlin, third edition, 24. [6] M. A. Gibson and Bruck J. Efficient exact simulation of chemical systems with many species and many channels. J. Phys. Chem. A, 14(9): , 2. [7] Daniel T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Computational Phys., 22(4):43 434, [8] Daniel T. Gillespie. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81:234 61, [9] Daniel T. Gillespie. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys., 115(4): , 21. [1] Eric L. Haseltine and James B. Rawlings. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys., 117(15): , 22.

40 First Prev Next Go To Go Back Full Screen Close Quit 4 [11] Thomas G. Kurtz. Strong approximation theorems for density dependent Markov chains. Stochastic Processes Appl., 6(3):223 24, 1977/78. [12] Thomas G. Kurtz. Averaging for martingale problems and stochastic approximation. In Applied stochastic analysis (New Brunswick, NJ, 1991), volume 177 of Lecture Notes in Control and Inform. Sci., pages Springer, Berlin, [13] R. Srivastava, L. You, J. Summers, and J. Yin. Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theoret. Biol., 218(3):39 321, 22. [14] N. G. van Kampen. Stochastic processes in physics and chemistry. North-Holland Publishing Co., Amsterdam, Lecture Notes in Mathematics, 888.

41 First Prev Next Go To Go Back Full Screen Close Quit 41 Software StochKit Petzold group, UC Santa Barbara Hy3S Hybrid Stochastic Simulation for Supercomputers Kaznessis Group, University of Minnesota StochSim Computational Cell Biology Group, Cambridge Stochastirator Molecular Sciences Institute, Berkeley

42 First Prev Next Go To Go Back Full Screen Close Quit 42 Abstract Attempts to model chemical reactions within biological cells has led to renewed interest in stochastic models for these systems. The classical stochastic models for chemical reaction networks will be reviewed, and multiscale methods for model reduction will be described. New models motivated by the particular nature of biochemical processes will also be discussed.