STOCHASTIC CHEMICAL KINETICS

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1 STOCHASTIC CHEICAL KINETICS Dan Gillespie Current Support: Caltech (NIGS) Caltech (NIH) University of California at Santa Barbara (NIH) Past Support: Caltech (DARPA/AFOSR, Beckman/BNC)) University of California at Santa Barbara (DOE) olecular Sciences Institute (Sandia/DOE) Office of Naval Research ain Collaborators: Linda Petzold (UCSB) John Doyle (Caltech) A Chemically Reacting System consists of olecules of N chemical species S, S., N - Inside a volume Ω, at some temperature T. elemental reaction channels R, R., - We assume R to be a single instantaneous physical event that changes the population of at least one species. - In practice, elemental means that R must be one of two types: Unimolecular: Si something else, or Bimolecular: S + S something else. i i - All other types of reaction (trimolecular, reversible, etc.) are made up of a series of two or more elemental reactions. E.g.: S+ S2 S2 S+ S2+ S3 S4+ S5 is typically S + S S + S

2 Question: How does a spatially homogeneous (or well-stirred) chemically reacting system evolve in time? The Trad Answer: According to the reaction rate equation (RRE). A set of coupled, first-order ODEs. Derived using ad hoc, phenomenological reasoning. - Is more than the mass action equations of thermodynamics, which apply only to systems in complete equilibrium. Implies the system evolves continuously and deterministically. - Yet molecules come in integer numbers and react stochastically. Is empirically accurate for large (test tube size) systems But is sometimes not adequate for very small (cell-size) systems. * * * Doing it right : olecular Dynamics The most exact way of describing the system s evolution. The motion picture approach: Tracks the position and velocity of every molecule in the system. Simulates every collision, non-reactive as well as reactive. Shows changes in species populations and their spatial distributions. But... it s unfeasibly slow for nearly all realistic systems. 2

3 A great simplification occurs if successive reactive collisions tend to be separated in time by very many non-reactive collisions. The overall effect of the non-reactive collisions is to randomize - the velocities of the molecules (axwell-boltzmann distribution). - the positions of the molecules (spatially uniform or well-stirred), Then, instead of having to describe the system s state as the position, velocity and species of each molecule, we need only give ( ) X( t) X ( t),, X ( t), X i ( t) the number of S i molecules at time t. N But this well-stirred simplification, which... ignores the non-reactive collisions, drastically truncates the definition of the system s state,... comes at a price: X ( t) must now be viewed as a stochastic process. But in fact, the system was never deterministic to begin with. Even if molecules moved according to classical mechanics... - Unimolecular reactions always involve randomness (Q). - Bimolecular reactions usually do too. - A system of many colliding molecules is so sensitive to initial conditions that, for all practical purposes, it evolves randomly. - The system is not isolated. It s in a heat bath, which keeps it at temperature T via essentially random interactions. 3

4 For well-stirred systems, each R is completely characterized by a propensity function a ( x ) : Given the system is in state x, a ( x) dt probability that one R event will occur in the next dt. - The existence and form of a ( x ) follow from molecular physics. a state change vector ( ν,, vn ) - R induces Examples: ν i the change in ν : x x+ ν. { i } X i caused by one R event. ν the stoichiometric matrix. ( ) x ( ) c 2 : ν =,,,, ; ( ) = ( x) = S S a dt c dt x a c x ( ) c : same ν ; ( x) = 2 ( x) = 2 S S S a dt c dt x x a c x x 2 ( r2 )( v2 dt) Prob{ collision in dt} = π. Prob{ R collision } p. Ω 2 2 ( π r2 )( v2dt ) π r2 v2 p a ( x) dt p xx2 = xx2 dt= c xx2 dt Ω Ω a ( ) Prob that a typical S -S x 2 c pair reacts in next dt E R iff " collisional K. E." > E p = exp... Arrhenius! kbt Diffusional motion (well-stirred): π r v is replaced by 4 π r2 ( D + D2 ) 4

5 Two exact, rigorously derivable consequences.... The chemical master equation (CE): P( x, t x, t ) = a ( ) P(, t, t) a ( ) P(, t, t) t x x x x x x = ν ν. P( x, t x, t ) Prob { X( t) = x, given X( t ) = x } for t t. Follows from the probability statement P( x, t+ dt x, t) = P( x, t x, t) ( a ( x) dt) = = But the CE is usually too hard to solve. ( ν ) + P( x ν, t x, t ) a ( x ) dt. Averages: f ( X t ) f x P x t x t x ( ) ( ) (,, ). If we multiply the CE through by x and then sum over x, we find d X( t) = ν a ( X( t) ). dt = If there were no fluctuations, ( ( )) = ( ( ) ) = ( ( )) a X t a X t a X t, and the above would reduce to: dx( t) dt = = ( X( t) ) ν a. - This is the reaction-rate equation (RRE). - It s usually written in terms of the concentration Z( t) X( t) Ω. But as yet, we have no ustification for ignoring fluctuations. 5

6 2. The stochastic simulation algorithm (SSA): A procedure for constructing sample paths or realizations of X ( t). Idea: Generate properly distributed random numbers for - the time τ to the next reaction, - the index of that reaction. p( τ, x, t) dτ probability, given X( t ) = x, that the next reaction will occur in [ t+ τ, t+ τ+ dτ ), and will be R. = P ( τ ) a ( x ) dτ, P ( τ ) Pr( no reactions in time τ ). ( ) P ( τ + dτ ) = P ( τ ) a( x ) dτ, where a ( x) a ( ) x. dp ( τ ) a Implies ( x ) τ = a( x ) P ( τ ). Solution: P ( τ ) = e. dτ a ( x) τ a ( x) τ a ( x) p( τ, x, t) = e a ( x) = a ( x)e a( x ). Thus, - τ is an exponential random variable with mean a( x ), - is an integer random variable with probabilities a ( x) a( x ). The Direct Version of the SSA. In state x at time t, evaluate a ( x ),, a ( ) x, and a ( x) a ( x ). = 2. Draw two unit-interval uniform random numbers r and r 2, and compute τ and according to τ = ln, a( x) r = the smallest integer satisfying ak ( x) > r2 a( x ). 3. Replace t t+ τ and x x+ ν. k= 4. Record ( x, t). Return to Step, or else end the simulation. 6

7 c A Simple Example: S. a ( x ) = c x, ν =. Take X () x dx( t) RRE: = c X( t). Solution is X( t) = x e c t. dt CE: =. P( x, t x,) = c ( x + ) P( x+, t x,) xp( x, t x,) t. x! c x t c t x x = = x!( x x )! Solution: ( ) P( x, t x,) e e ( x,,, x ) which implies X ( t) x e c t ct ct =, sdev { X( t) } x e ( e ) =. SSA: Given X( t) = x, generate τ = ln, then update: c x r t t+ τ, x x. 8 S c =, X () =

8 a ( x) dt Prob that R will fire in next dt CE SSA The SSA... Is exact. It does not entail approximating dt by t. Is procedurally simple, even when the CE is intractable. Comes in a variety of implementations - Direct ethod (Gillespie, 976) - First Reaction ethod (Gillespie, 976) - Next Reaction ethod (Gibson & Bruck, 2) - First Family ethod (Lok, 23) - odified Direct ethod (Cao, Li & Petzold, 24) - Sorting Direct ethod (ccollum, et al. 26) Remains too slow for most practical problems: Simulating every reaction event one at a time ust takes too much time if any reactants are present in very large numbers. 8

9 We would be willing to sacrifice a little exactness if that would buy us a faster simulation. Tau-Leaping Approximately advances the process by a pre-selected time τ, which may encompass more than one reaction event. Key: The Poisson random variable with mean aτ can be defined: P ( aτ ) the number of events that will occur in a time τ, when the probability of an event in any dt is adt, provided a is a positive constant. With X( t ) = x, let us choose τ small enough to satisfy the Leap Condition: Each a ( x ) constant in [ t, t+ τ ]. Then the number of R firings in [ t, t τ ] = + will be P ( a ( x) τ) ( a τ) X( t τ ) x P ( x) + + ν. So = ( a τ) X( t τ ) x P ( x) + + ν - Practical Considerations for Implementing Tau-Leaping - Finding the largest τ that satisfies the Leap Condition. - Accomplished via an accuracy control parameter ε. - We estimate the largest τ for which a a ε,. Avoiding negative populations, and segueing to the SSA. - Accomplished via a second control parameter n c. - We call any reaction that is within c n firings of exhausting any reactant a critical reaction. Then we tau-leap no farther than the next firing of a critical reaction. - Becomes the SSA when all reactions are classified critical. τ 9

10 5 45 monomer X, unstable dimer X2, stable dimer X X2 X3 S--> c = S + S --> S2 c2 =.2 S2 --> S + S c3 =.5 S2 --> S3 c4 =.4 - Exact SSA Run. - Intially: X=,; X2=X3=. - 5 reactions per plotted dot - 57,67 reactions total. 5 X Time 5 45 monomer X, unstable dimer X2, stable dimer X X2 X3 S--> c = S + S --> S2 c2 =.2 S2 --> S + S c3 =.5 S2 --> S3 c4 =.4 - Explicit Tau Leaping Run - ε =.4; n c =. - leap per plotted dot leaps total. - Run time speedup over SSA > X. 5 X Time

11 a ( x) dt Prob that R will fire in next dt { a ( x ) const over τ, } CE SSA Tau-Leaping Discrete & Stochastic Speeding up Tau-Leaping: The Langevin Equation Two math facts: - If m, then P( m) N ( m, m). - 2 N ( m, σ ) = m+ σn (,). So, with X( t ) = x, suppose we can choose τ small enough to satisfy the Leap Condition, yet also large enough that a ( x) τ,. Then... X( t+ τ ) x+ P ( a ( x) τ) = + = ν ( a ( ) τ, a ( ) τ) x N x x = ν x+ a ( ) τ + a ( ) τ (,) x x N ν t+ + ν a ( ) + ν a ( ) X ( τ ) x x τ x N (,) τ. = =

12 ( ) ν ( ) X ( t+ τ ) x + ν a x τ + a x N (,) τ = = This is the Langevin leaping formula. It s faster than the ordinary tau-leaping formula, because - a ( ) τ x means lots of reaction events get leapt over in τ ; - normal random numbers can be generated faster than Poissons. It directly implies, and is entirely equivalent to, a SDE called the chemical Langevin equation (CLE): dx( t) ν a ( X( t) ) + ν a ( X( t) ) Γ ( t). dt = = N (,) - Gaussian white noise: Γ ( t) lim lim N, + +. dt dt dt dt - Satisfies Γ ( t) Γ ( t ) = δ δ ( t t ). Our discrete stochastic process X ( t) has now been approximated as a continuous stochastic process. a ( x) dt Prob that R will fire in next dt { a ( x ) const over τ, } CE SSA Tau-Leaping Discrete & Stochastic { a ( x ) τ, } CFPE * CLE Continuous & Stochastic * - J. Chem. Phys. 3:297 (2) - Am. J. Phys. 64:246 (996) - J. Phys. Chem. A 6:563 (22) 2

13 Def: All The Thermodynamic Limit X i, and Ω, with X i Ω constants. a = c x x In the thermodynamic limit, a 's grow like (system size). c a x x2 Ω = xx2 x2 all So in the thermodynamic limit, we see that in the CLE dx( t) ν a ( X( t) ) + ν a ( X( t) ) Γ ( t), dt = = - the deterministic term grows like (system size), - the stochastic term grows like (system size) /2. Rule of Thumb: Relative fluctuations die off as (system size) /2. At the thermodynamic limit the stochastic term disappears, leaving dx( t) ν a ( X( t) ) the RRE derived! dt = X ( t) has now become a continuous deterministic process. a ( x) dt Prob that R will fire in next dt { a ( x ) const over τ, } CE SSA Tau-Leaping Discrete & Stochastic { a ( x ) τ, } CFPE CLE Continuous & Stochastic X i, Ω X i Ω = const i, i RRE Continuous & Deterministic 3

14 A Spectrum of Descriptive athematical odes (for well-stirred systems) a ( x) const a ( x) τ X, Ω in [ t, t+ τ ], X i Ω const i i CE/SSA Tau- Leaping CLE / CFPE RRE exact discrete approximate approximate approximate discrete continuous stochastic stochastic stochastic continuous deterministic increasing numbers of molecules Another ulti-scale Problem Some reactions/species may be very fast, others very slow. Fast and slow are interconnected not easy to separate. Often manifests as dynamical stiffness, a known ODE problem. SSA still works, and is exact. But it s agonizingly slow. Tau-leaping remains accurate, but the Leap Condition restricts τ to the shortest (fastest) time scale of the system. So even it s too slow. One approach: Implicit Tau-Leaping - A stochastic adaptation of the implicit Euler method for ODEs. Another approach: The Slow-Scale Stochastic Simulation Algorithm - Skips over the fast reactions and simulates only the slow reactions, using specially modified propensity functions. An adaptation of the rapid equilibrium / quasi steady-state methods for RREs. 4