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Foundations of Chemical Kinetics Lecture 22: The chemical master equation Marc R. Roussel Department of Chemistry and Biochemistry The chemical master equation I So far, we have looked at transitions between states of a single molecule using the master equation. I If we want to go back to describing reactions at the level of elementary reactions, i.e. neglecting internal degrees of freedom, we can use the chemical master equation (CME), which is a master equation for the probabilities that the system has any given composition as a function of time. Derivation of the chemical master equation Preliminaries I I I I I Let N(t) = (N1 (t), N2 (t), . . . , Nn (t)) be the composition vector of the system, where Ni is the number of molecules of type i and n is the number of distinct chemical species. Let N be the space of all possible vectors N. Let P(N , t) be the probability distribution over the space N , i.e. the vector containing the probabilities of all possible states, P(N, t), at time t. Assume the system is well-mixed, i.e. that the probability of finding a particular molecule inside a subvolume ∆V is ∆V /V . Reactions are random events: I I First-order reactions occur when some essentially random condition is met within a molecule (e.g. IVR putting enough energy in a reactive mode). In a well-mixed system, the collisions necessary for a reaction to occur are random events. Derivation of the chemical master equation Preliminaries Boltzmann’s Stosszahlansatz (assumption of molecular chaos): Collisions cause a rapid loss of memory, i.e. particle trajectories can be treated as essentially random. Consequence: Chemical reactions can be treated as Markov (memoryless) processes, provided the Stosszahlansatz is satisfied. This in turn means that we can write a master equation for P(N , t). Limitation: A model based on this ansatz will not be valid for time scales shorter than the mean collision time (gas phase). Derivation of the chemical master equation I I I I I The transitions here are chemical reactions that change the composition of the system. The transition rates appearing in the master equation ought to depend on the chemical composition. Let R be the set of reactions occurring in a chemical system. In this theory, the transition rates are called reaction propensities, denoted ar (N), for r ∈ R. The (chemical) master equation is X dP(N, t) X = ar (N − ν r )P(N − ν r , t) − ar (N)P(N, t) dt r ∈R r ∈R where ν r is the stoichiometric vector of reaction r , i.e. the vector that gives the change in the numbers of each species as a result of reaction r . The reaction propensities I ar (N) is the probability per unit time that reaction r occurs given that the composition of the system is N. First-order reactions: Xi → I Suppose that the probability per unit time that any given molecule of Xi reacts is κr . I If ∆t is sufficiently small, then the probability that a molecule of Xi reacts given that there are Ni molecules of this type is κr Ni ∆t. I Therefore a = κ N . r r i The reaction propensities (continued) Second-order reactions: Xi + Xj → with j 6= i I We start by figuring out the probability that a particular pair of molecules of types i and j meet and react in time ∆t. I From collision theory, the reaction volume explored per unit time in the gas phase is vij σij . I This represents the fraction (v σ /V )∆t of the ij ij total volume V in time ∆t. I This is the probability of a collision between any particular pair of molecules in time ∆t. I If a fraction η of the collisions lead to reaction, r then the reaction probability per unit time is (ηr vij σij /V )∆t = κr ∆t. The reaction propensities (continued) Second-order reactions: Xi + Xj → (continued) I Reaction probability per unit time for a particular pair of molecules of types i and j: κr ∆t I There are N N pairs of molecules, so if we take i j ∆t sufficiently small, the probability that one reaction of type r occurs per unit time is ar = κr Ni Nj . The reaction propensities (continued) Second-order reactions: Xi + Xi → I Everything is as above, except I I I The collision rate is 12 vii σii , leading to κr = ( 21 ηr vii σii /V ). The number of different pairs of two molecules both of type i is Ni (Ni − 1)/2. The propensity is therefore ar = κr Ni (Ni − 1)/2. The reaction propensities (continued) In general: The reaction propensity can always be written as the product of a stochastic rate constant with units of inverse time, and of a combinatorial factor which gives the number of different combinations of reactant molecules present prior to the reaction: ar = κr hr (N) κ1 −− * − −C Example 1: A + B ) κ−1 I The composition vector is (NA , NB , NC ). I The stoichiometry vector for the forward reaction is ν 1 = (−1, −1, 1). I The stoichiometry vector for the reverse reaction is I The propensity of the forward reaction is a1 = κ1 NA NB . I The propensity of the reverse reaction is I Suppose that we start out with four molecules of A, three of B and none of C. Then the space of all possible compositions is N = {(4, 3, 0), (3, 2, 1), (2, 1, 2), (1, 0, 3)} I The probability space is P(N ) = {P(NA , NB , NC )}. κ1 −− * − −C Example: A + B ) κ−1 (continued) I The chemical master equation is dP(NA , NB , NC ) = dt κ1 (NA + 1)(NB + 1)P(NA + 1, NB + 1, NC − 1) + κ−1 (NC + 1)P(NA − 1, NB − 1, NC + 1) − (κ1 NA NB + κ−1 NC ) P(NA , NB , NC ) κ1 −− * − −C Example 1: A + B ) κ−1 (continued) I The CME is actually the four equations dP(4, 3, 0) = κ−1 P(3, 2, 1) − 12κ1 P(4, 3, 0) dt dP(3, 2, 1) = 12κ1 P(4, 3, 0) + 2κ−1 P(2, 1, 2) dt − (6κ1 + κ−1 ) P(3, 2, 1) dP(2, 1, 2) = 6κ1 P(3, 2, 1) + 3κ−1 P(1, 0, 3) dt − (2κ1 + 2κ−1 ) P(2, 1, 2) dP(1, 0, 3) = 2κ1 P(2, 1, 2) − 3κ−1 P(1, 0, 3) dt Properties of the CME I At fixed concentrations, the number of molecules of each kind is proportional to V . I Consider a chain of isomerizations X1 X2 . . . Xn and suppose we start out with N molecules of X1 . I The number of different compositions is (N + n − 1)! ≈ N!(n − 1)! eN n−1 n−1 , which grows as N n−1 ∝ V ρ , where ρ is the number of reactions. Curse of dimensionality