Reaction time distributions in chemical kinetics: Oscillations and other weird behaviors

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1 Introduction The algorithm Results Summary Reaction time distributions in chemical kinetics: Oscillations and other weird behaviors Ramon Xulvi-Brunet Escuela Politécnica Nacional

2 Outline Introduction The algorithm Results Summary 1 Introduction Deterministic approach to Chemical Kinetics Stochastic approach to Chemical Kinetics 2 The algorithm The Gillespie Algorithm Our algorithm 3 Results Exponential time distribution Gaussian time distribution Gaussian time distribution of the passage time among network configurations

3 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Reaction rates aa + bb cc + dd reaction rate = 1 d[a] = 1 d[b] = 1 d[c] = 1 d[d] = k[a] x [B] y a dt b dt c dt d dt The reaction rate law relates the rate of a reaction to the concentrations of the reactants. x and y are reactant orders (ultimately determined by the experiment). k is the reaction rate constant. According to Arrhenius law, k depends on temperature: k(t) exp ( E a /RT).

4 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Limitations of the deterministic approach Reactions for which this approach is not adequate Processes where a few molecules activate an avalanche type reaction (such as for certain visual and blood clotting mechanisms). Diffussion controled chemical reactions. Denaturation of proteins or polypeptides. etc. Modeling limitations Concentration is a continous variable, but molecules are discrete entities. Chemical reactions are inherently stochastic. Fluctuations and correlations cannot be studied.

5 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Advantages and disadvantages of the deterministic approach. Advantages Mathematically, the approach is based on ordinary differencial equations, which are much easier to solve than their stochastic counterpart. "Straightforward" chemical reactions are easy to model using this approach. Disadvantages The previuosly exposed limitations. The approach only works for large amounts of molecules of each of the species involved in the chemical process. Its underlying physical assumptions are unrealistic and its consequent predictions may be unreliable.

6 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Fundamental hypothesis The reaction parameter c µ which characterizes reaction R µ is defined as follows: c µ δt = probability, to first order in δt, that a particular combination of R µ reactant molecules will react accordingly in the next time interval δt. Notice the following: 1 The stochastic approach will consequently work with a single master equation, which is a differential-difference equation. 2 The function satisfying the master equation measures the probability of finding various molecule populations at each instant of time. 3 Reaction "rates" vs reactions "probatilities per unit time".

7 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Limitations of the stochastic approach Reactions for which this approach is not adequate All the chemical reactions that are spatially inhomogeneous. (This is also a limitation of the deterministic approach). Practical limitations The master equation is very difficult of solve, even numerically, for systems which are not quite simple. (This is the basic reason why the stochastic approach, which has clear advantages with respect the deterministic approach, is mainly used for simulating chemical reactions).

8 Introduction The algorithm Results Summary Deterministic approach to Chemical Kinetics Stochastic approach to Ch Advantages and disadvantages of the stochastic approach. Advantages The stochastic approach is always valid whenever the deterministic approach is valid, and is sometimes valid when the deterministic approach is not. Its underlying physical assumptions are more realistic and its consequent predictions are more reliable. It works for whatever amount of molecules. It allows the study of fluctuations and correlations. Disadvantages Mathematically, it is based on a differential-difference equation, which is more difficult to solve than a set of ordinary differential equations.

9 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm The goal of the Gillespie algorithm. Goal: To create a feasible, computational method for numerically calculating the stochastic time evolution of any spatially homegeneous chemical system. Remarks: The goal of this numerical method is not to try to solve the master equation for a given system. The goal is to simulate the very Markov process that the master equation describes analytically. The simulation algorithm will be fully equivalent to the master equation, even though the master equation itself will never be explicitly used.

10 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm The framework of the Gillespie algorithm. Given: A volume V which contains N chemically active species S i, i = 1, 2,..., N (possibly, together with molecules of several inert species as well), that can participate in M chemical reactions R µ (µ = 1,..., M), each characterized by a numerical reaction parameter c µ. Then: Let X i be the current number of molecules of chemical species S i in V. The method will simulate the time evolution of the N quantities X i, knowing only their initial values X (o) i, the forms of the M reactions R µ, and the values associated to the reaction parameters c µ.

11 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Example of c µ for a simple bimolecular reaction. where: c µ δt = (probability of collision) (probability of reaction) c µ δt = δv coll /V exp ( ɛ /KT) δv coll = πd 2 12 v 12 δt d 12 = center-to-center distance between an S 1 molecule and S 2 molecule. v 12 = average speed of an arbitrary S 1 molecule relative to an arbitrary S 2 molecule. If the velocities of molecules are distributed according to Maxell-Boltzmann, then: c µ = V 1 πd 2 12 ( ) 8kT 1/2 exp ( ɛ /KT) πµ

12 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Key assumptions of the method. The reactant molecules are always randomly distributed uniformly throughout V. That is what allows the introduction of the collision probability δv coll /V. For this, it is sufficient to require that the non-reactive (elastic) molecular encounters occur much more frequently than the reactive (inelastic) molecular encounters. This will allow a uniform redistribution of the molecules inside V prior to each reactive collision, and will also allow the restoration of the Maxell-Boltzmann velocity distributions. If non-reactive collisions do not occur much more frequently than reactive collisions, then this stochastic method for simulating chemically reactive systems is not strictly valid.

13 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Probability that a reaction R µ occurs inside V in δt. probability = h µ c µ δt where h µ = number of molecular reactant combinations for reaction R µ. For: unimolecular reactions: h µ = X 1. biomolecular reactions: h µ = X 1 X 2 (or X 1 (X 1 1)/2). etc. Consequently, the average rate at which R µ occurs: unimolecular reactions: X 1 c µ. biomolecular reactions: X 1 X 2 c µ (or X 1 (X 1 1)/2 X 2 1 /2 ). etc.

14 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Reaction probability density function To carry out the simulation, Gillespie introduced a new perspective of the stochastic approach: P(τ, µ)δτ probability at time t that the next reaction in V will occur in the differential time interval (t + τ, t + τ + δt), and will be an R µ reaction. P(τ, µ) is a joint probability density function on the space of the continuous variable τ (0 τ < ) and the discrete variable µ (µ = 1, 2,..., M). If P 0 (τ) = probability at time t that no reaction occurs in the time interval (t, t + τ), then Theorem P(τ, µ)δτ = P 0 (τ) h µ c µ δτ

15 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Calculation of P 0 (τ). Imagine the interval (t, t + τ) divided into K subintervals of equal length ɛ = τ/k. The probability that no reaction occurs in the first ɛ subinterval (t, t + ɛ) is M [1 h ν c ν ɛ + O(ɛ)] = 1 ν=1 M h ν c ν ɛ + O(ɛ) ν=1 For the next subsequent subinvervals, that probability is clearly the same. Therefore: [ K ( ) M M P 0 (τ) = lim 1 h ν c ν τ/k] = exp h ν c ν τ K ν=1 ν=1

16 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Rewriting P(τ, µ)... M M P(τ, µ) = P(τ, µ) P(τ, µ)/ P(τ, µ) = P 1 (τ) P 2 (µ τ) µ=1 µ=1 where: P 1 (τ) gives the probability that the next reaction will occur between times t + τ and t + τ + δτ, irrespective of which reaction it might be. P 2 (µ τ) gives the probability that the next reaction will be an R µ reaction, given that the next reaction occurs at time t + τ.

17 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Rewriting P(τ, µ)... M P 1 (τ) = P(τ, µ) = P 2 (µ τ) = µ=1 ( ) M M h µ c µ exp h ν c ν τ a exp( aτ) µ=1 P(τ, µ) h M = µ=1 P(τ, µ) ν=1 ( µ c µ exp ) M ν=1 h νc ν τ M µ=1 h µc µ exp( M ν=1 h νc ν τ ) = a µ a where a µ = h µ c µ and a = M µ=1 h µc µ. Note that P 2 (µ τ) is independent of τ and that both density functions are properly normalized.

18 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Iterative steps of the algorithm. At each step: 1 Draw two (pseudo)random numbers r 1 and r 2 from a uniform distribution. 2 Calculate a value τ as τ = (1/a) ln(1/r 1 ). That generates a value τ according to P 1 (τ). 3 Calculate a value µ such that µ 1 ν=1 a ν < r 2 a µ ν=1 a ν. That generates a random integer µ according to P 2 (µ τ). 4 Update time t with t + τ and X i (t + τ ) (i = 1, 2,..., N) according to the reaction R µ selected.

19 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Our algorithm For all elementary reactions yielding one intermediate, our algorithm works exactly like the Gillespie algorithm. 1 We draw two (pseudo)random numbers r 1 and r 2 from a uniform distribution. 2 We calculate τ as τ = (1/a) ln(1/r 1 ). Here, a = M µ=1 h µc µ, where M is the number of elementary reactions that yield an intermediate. (Not all reactions are considered). 3 We calculate a value µ such that µ 1 ν=1 a ν < r 2 a µ ν=1 a ν. This selects the reaction, out of the reactions that yield an intermediate, that may take place at time t + τ.

20 Introduction The algorithm Results Summary The Gillespie Algorithm Our algorithm Our algorithm For all reactions such as intermediate products or intermediate reactants, the intermediate decays according to a previously given time probability distribution. 1 Each time an intermediate is produced, we draw from the given time probability distribution the time interval τ that the intermediate will live. 2 At any time t, we compute τ min, such that t + τ min is the time that has to pass until one of the existing intermediates dies. 3 At any time t: if τ < τ min, then t is updated as t = t + τ, and, depending on r 2, reaction ν is executed. if τ τ min, then t is updated as t = t + τ min, and the intermediate that was "programmed" to die at t = t + τ min, decays.

21 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim Exponential decay Let s assume that intermediates decay exponentially, and that p 1 is the probability that they decay into products and p 2 is the probability that they decay into the initial reactants. Exponential decay is equivalent to a constant probability per unit time, P, of decaying. If we consider a number of intermediates (not just one), exponential decay means that N(t) = N(0) exp ( Pt). This result can be derived from dn(t) dt = PN(t) which is nothing else than the classical deterministic approach. In terms of the reaction constants, k 1 = Pp 1 and k 2 = Pp 2

22 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim A + B products Exponential decay 1e+06 A + B --> P deterministic approach A B I P 1e+06 A + B --> P stochastic approach exponential A B I P number of molecules number of molecules time time

23 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim S + E E + products Exponential decay S + E --> E + P deterministic approach S E I P S + E --> E + P stochastic approach exponential S E I P number of molecules number of molecules time time

24 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim A + B products Gaussian decay A + B --> P A B I P 1e+06 A + B --> P stochastic approach delayed exponential A B I P molecules number of molecules time time

25 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim S + E E + products Gaussian decay S + E --> E + P stochastic approach delayed exponential A B I P S + E --> E + P stochastic approach delayed exponential A B I P number of molecules number of molecules time time

26 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim An example of network

27 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim A + B products Gaussian decay among configurations (forward) time distribution of network 5 based on gaussian time displacements kinematics with network 5 based on gaussian time displacements (direct) A B I P likelihood molecules time time

28 Introduction The algorithm Results Summary Exponential time distribution Gaussian time distribution Gaussian tim A + B products Gaussian decay among configurations (forward and backward) 1 time distribution of network 7 based on gaussian time displacements (back and forth) kinematics with network 7 based on gaussian time displacements (back and forth) A B I P likelihood probability e time time

29 Introduction The algorithm Results Summary Summary We modified the Gillespie algorithm to be able to consider different time probability distribution laws for the decay of intermediates. We studied the effects of different time distribution laws on the chemical kinetics of a couple of simple reactions. We observed oscillations and other weird behaviors if the time distribution is Gaussian-like. We observed behaviors similar to the classical ones if the time distribution has an exponetial tail. Molecular dynamics simulations of biomolecules may help us to determine realistic time probability distribution laws. Experiments with sufficient accuracy will be necessary for establishing whether time distributions other than the exponential one are real and frequent.

30 Introduction The algorithm Results Summary Acknowledgment PROYECTO PROMETEO Secretaría de Educación Superior Ciencia, Tecnología, e Innovación de la República del Ecuador Escuela Politécnica Nacional Proyecto Prometeo

Principles of Chemical Kinetics

Kinetic Theory of Gases Rates of Chemical Reactions Theories of Chemical Reactions Summary Principles of Chemical Kinetics Ramon Xulvi-Brunet Escuela Politécnica Nacional Kinetic Theory of Gases Rates