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 Reaction networks Classical scaling and the law of mass action Multiple scales Example: Michaelis-Menten equation Example: Model of a viral infection References Abstract Collaboration with David Anderson, Karen Ball, George Craciun, Hye- Won Kang, Lea Popovic, Greg Rempala

2 First Prev Next Go To Go Back Full Screen Close Quit 2 Bilingual dictionary Chemistry propensity master equation nonlinear diffusion approximation Langevin 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 Reaction networks Standard notation for chemical reactions A + B κ 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.

4 First Prev Next Go To Go Back Full Screen Close Quit 4 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.

5 First Prev Next Go To Go Back Full Screen Close Quit 5 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).

6 First Prev Next Go To Go Back Full Screen Close Quit 6 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 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.

7 First Prev Next Go To Go Back Full Screen Close Quit 7 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). For example, for 2A 1 A 2 and z 1 = N 1 x 1, 1 N κ kx 1 (x 1 1) = Nκ k z 1 (z 1 1 N ) Nκ kz 2 1.

8 First Prev Next Go To Go Back Full Screen Close Quit 8 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

9 First Prev Next Go To Go Back Full Screen Close Quit 9 Multiple scales Let N >> 1. For each species i, define the normalized abundances (or simply, the abundances) by Z i (t) = N α i X i (t), where α i should be selected so that Z i = O(1). Note that the abundance may be the species number (α i = ) or the species concentration or something else. The rate constants may also vary over several orders of magnitude κ k = κ kn β k, so for a binary reaction κ kx i x j = N β k+α i +α j κ k z i z j

10 First Prev Next Go To Go Back Full Screen Close Quit 1 A parameterized family of models Let Z N i (t) = Z i () + k N α i Y k ( N β k+ν k α λ k (Z N (s))ds)(ν ik ν ik ). Then the true model is Z = Z N.

11 First Prev Next Go To Go Back Full Screen Close Quit 11 Example: 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 1X A (s)x E (s)ds) + Y 2 ( κ 1X A (s)x E (s)ds) + Y 2 ( κ 3X AE (s)ds) X B (t) = Y 3 ( κ 3X AE (s)ds) κ 2, κ 3 >> κ 1 κ 2X AE (s)ds) κ 2X AE (s)ds)

12 First Prev Next Go To Go Back Full Screen Close Quit 12 Scaling Note that M = X AE (t) + X E (t) is constant. Let N = O(X A ) >> M. Setting β 2 = β 3 = 1, α A = 1, α E = α AE =, κ 1 = κ 1, κ 2 = κ 2N 1, κ 3 = κ 3N 1 V E (t) = M 1 X E (s)ds, Z A (t) = N 1 X A (t) Z A (t) = Z A () N 1 Y 1 (N = Z A () N 1 Y 1 (N κ 1 MZ A (s)m 1 X E (s)ds) + N 1 Y 2 (Nκ 2 X AE (s)ds) κ 1 MZ A (s)dv E (s)) + N 1 Y 2 (Nκ 2 M(t V E (t)))

13 First Prev Next Go To Go Back Full Screen Close Quit 13 Analysis Similarly, X E (t) = X E () Y 1 (N +Y 3 (Nκ 3 M(t V E (t))) and dividing by N and letting N, ( (κ 2 + κ 3 )M(t V E (t))) Also lim N lim N κ 1 MZ A (s)dv E (s)) + Y 2 (Nκ 2 M(t V E (t))) ) κ 1 MZ A (s)dv E (s) =. ( ) Z A (t) Z A () + κ 1 MZ A (s)dv E (s) κ 2 M(t V E (t)) =

14 First Prev Next Go To Go Back Full Screen Close Quit 14 Derivation of Michaelis-Menten equation Theorem 1 (Darden [2, 3]) Assume that N and ZA N() = X A()/N x A (). Then (ZA N, V E N) converges to (x A(t), v E (t)) satisfying x A (t) = x A () = and hence v E (s) = κ 1 Mx A (s) v E (s)ds + κ 1 x A (s) v E (s)ds + κ 2 +κ 3 κ 2 +κ 3 +κ 1 x A (s) and ẋ A (t) = Mκ 1κ 3 x A (t) κ 2 + κ 3 + κ 1 x A (t). κ 2 M(1 v E (s))ds (κ 2 + κ 3 )(1 v E (s))ds,

15 First Prev Next Go To Go Back Full Screen Close Quit 15 Example: Model of a viral infection Srivastava, You, Summers, and Yin [5], Haseltine and Rawlings [4], 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 κ 1 T + G G κ 2 T T + stuff κ 3 T + S T κ 4 S κ 5 G + S κ 6 V

16 First Prev Next Go To Go Back Full Screen Close Quit 16 Stochastic system X 1 (t) = X 1 () + Y b ( X 2 (t) = X 2 () + Y a ( X 3 (t) = X 3 () + Y c ( κ 2X 2 (s)ds) Y d ( κ 1X 1 (s)ds) Y b ( Y f ( κ 6X 2 (s)x 3 (s)ds) κ 3X 1 (s)ds) Y e ( Y f ( κ 6X 2 (s)x 3 (s)ds) κ 4X 1 (s)ds) κ 2X 2 (s)ds) κ 5X 3 (s)ds)

17 Figure 1: Simulation (Haseltine and Rawlings 22) First Prev Next Go To Go Back Full Screen Close Quit 17

18 First Prev Next Go To Go Back Full Screen Close Quit 18 Scaling parameters 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 κ κ N 2/3 κ 3 1 N κ κ κ N 5/3

19 First Prev Next Go To Go Back Full Screen Close Quit 19 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)

20 First Prev Next Go To Go Back Full Screen Close Quit 2 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)

21 First Prev Next Go To Go Back Full Screen Close Quit 21 Fast time scale Define V N i (t) = Z i (N 2/3 t). V N 1 (t) = V N 1 () + Y b ( V N 2 (t) = V N 2 () + N 2/3 Y a ( V3 N (t) = V3 N () + N 1 Y c ( 2.5N 2/3 V2 N (s)ds) Y d ( N 2/3 V1 N (s)ds) N 2/3 Y b ( N 2/3 Y f (N 2/3 2.5N 2/3 V N 2 (s)ds) N 5/3 V1 N (s)ds) N 1 Y e ( N 1 Y f (.25N 2/3 V N 1 (s)ds).75v2 N (s)v3 N (s)ds).75n 2/3 V N 2 (s)v N 3 (s)ds) 2N 5/3 V N 3 (s)ds)

22 First Prev Next Go To Go Back Full Screen Close Quit 22 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 (1).

23 First Prev Next Go To Go Back Full Screen Close Quit 23 Law of large numbers Theorem 2 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) = V 2 () + 7.5V 2 (s)ds) 3.75V 2 (s) 2 ds. (1)

24 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ= 3 X1:Viral Template X1:Viral Template 2.5 The Number of X The Number of X Time Time The Whole System The Reduced System when γ= 12 X3:Viral Structural Protein 12 X3:Viral Structural Protein The Number of X The Number of X Time Time

25 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ= 3 X2:Viral Genome X2:Viral Genome 25 The Number of X The Number of X Time Time

26 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ=2/3 25 X2:Viral Genome X2:Viral Genome The Number of X The Number of X Time Time

27 First Prev Next Go To Go Back Full Screen Close Quit The Whole System The Reduced System when γ=2/3 35 X1:Viral Template X1:Viral Template 3 The Number of X The Number of X Time Time 16 The Whole System X3:Viral Structural Protein 16 The Reduced System when γ=2/3 X3:Viral Structural Protein The Number of X The Number of X Time Time

28 First Prev Next Go To Go Back Full Screen Close Quit 28 References [1] Karen Ball, Thomas G. Kurtz, Lea Popovic, and Greg Rempala. Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Probab., 16(4): , 26. [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] 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. [5] 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.

29 First Prev Next Go To Go Back Full Screen Close Quit 29 Abstract Stochastic models for chemical reactions Attempts to model chemical reactions within biological cells have 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. The methods will be illustrated with derivation of the Michaelis-Menten model for enzyme reactions and a simple model of viral infection of a cell.