BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES

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1 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES EDUARDO S. ZERON 1. Historical Introduction Since the first observation of current oscillations during electrodissolution of an iron wire in nitric acid by Fechner [2] in 1828, the experimental evidence of electrochemical instabilities resulting in nonlinear behaviour such as oscillations, multistability, and chaos have been constantly accumulating. A simple oscillating or chaotic electrochemical cell can be built by placing an aluminium anode in a solution of sodium chloride in tap water. The oscillating current is produced in this case by the competing forces of corrosion by the chloride ions and protection by the carbonate ions from the tap water. It is quite easy to understand why the first oscillating chemical reactions were discovered around 1930 experimenting with an electrochemical cell, for the galvanometer was invented in 1820 and that it was for 90 years one of the few instruments sensible enough to do scientific measurements. Consider that Edwin H. Armstrong [1] published until 1914 an explanation of the Audion s operation. However Fechner s discovery was met with skepticism, because the idea of an oscillating chemical reaction seems to contradict the laws of thermodynamic and it is difficult to understand why two competing forces (dissolving and passivation of iron in nitric acid) should produce an oscillating chemical reaction instead of simply converging to a state of equilibrium. Date: May

2 2 EDUARDO S. ZERON This theoretical atmosphere changed in 1910, when Alfred J. Lotka published his famous paper entitled Contribution to the Theory of Periodic Reactions [4]. This work contains a detailed analysis of the following hypothetical autocatalytic model that produces damped chemical oscillations, (1) E dx a X, X + Y b 2Y, = a[e] bxy, and dy Y c, = bxy cy. Lotka explained in detail why the non-linear (autocatalytic) term bxy is absolutely necessary for producing (damped) oscillations. Even so it is explicitly stated in [4] that: No reaction is known which follows the above law, and as a matter of fact the case here considered was suggested by the consideration of matters lying outside the field of physical chemistry, A simple modification of system (1) is able to produce sustained oscillations. One only needs to change the constant term a[e] by ax in (1) in order to obtain sustained oscillations instead of damped ones. The new model is better know as the Lotka-Volterra equations and gives a simple description of the dynamical behaviour of biological systems in which two species interact, one as a predator and the other one as its prey, (2) dx = ax bxy and dy = bxy cy. Even when no chemical clocks (homogeneous oscillating chemical reaction) was known at the time Lotka analysed system (1), several chemical clocks were discovered quite soon. The fist one is the Bray- Liebhafsky reaction discovered by W.C. Bray in 1921, when he investigated the role of the iodate IO3 ions in the catalytic degradation of hydrogen peroxide to oxygen and water. The oscillations are produced

3 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 3 by the following complementary reactions. 5H 2 O 2 + I 2 2IO 3 + 2H + + 4H 2 O, 5H 2 O 2 + 2IO 3 + 2H + I 2 + 5O 2 + 6H 2 O. More popular chemical clocks are the Belousov-Zhabotinsky and the Briggs-Rauscher reactions. Our interest in Lotka system (1) comes from the fact that these differential equations are involved in the analysis of the Circadian Rhythm, but the oscillations are produced by stochastic resonances; i.e. the stochastic simulations of system (1) oscillate when the respective deterministic simulations do not. Thus we begin analysing when the deterministic systems (1) and (2) oscillate, and then we introduce the stochastic analysis. 2. The deterministic Lotka-Volterra system We began analysing the classical Lotka-Volterra system (2) dx = ax bxy and dy = bxy cy, where the constants a, b, and c are all strictly positive. There are two stationary states, the trivial one (0, 0) that is a saddle point (it is associated to a positive eingenvalue and a negative one); and the second steady state ( c, a ) that is associated to a pair of pure imaginary b b eigenvalues. It is easy to calculate that any initial condition different from the steady states yields to an oscillatory solution. second equation in (2) by the first one to obtain dy dx = bxy cy ax bxy and Integrating with respect to x yields ( a y b ) dy dx = b c x. (3) a ln y by + c ln x bx = r. Divide the Homework 1. Let r be a given real constant. Prove that the locus of all points (x, y) that satisfies (3) is either the empty set or a closed curve in the plane. Hint: Prove first that a ln y by is a concave function

4 4 EDUARDO S. ZERON that attains its maximum at a/b and that converges to when y > 0 goes to 0+ or Simple oscillating systems. It is quite easy to build systems that have a stable limit cycle, and so they are able to produce sustained oscillations. Consider the following system, (4) dr = a r and dθ = b. We obviously have that θ = c 0 +bt and that r converges to the steady state a. If we use the change of variables: r = x 2 + y 2, θ = arctan(y/x); we can rewrite (4) as follows and 2xẋ + 2yẏ = a x 2 y 2 and x = r cos(θ), y = r sin(θ); xẏ yẋ x 2 + y 2 = b. Solving for ẋ and ẏ yields a system that obviously has only one stable limit circle x 2 +y 2 = r, 4 dx 4 dy = = ax 2by x, x 2 + y2 ay + 2bx y. x 2 + y2 Homework 2. Use the change of variables below with d > 0 in order to build another system that has only one stable limit circle, r = 2d x 2 + y 2, θ = arctan(y/x); and x = r d cos(θ), y = r d sin(θ) The original Lotka system. We analyse now the original Lotka system (1) dx dy = ac bxy and = bxy cy, where [E]=c and the constants a, b, and c are all strictly positive. There is only one steady state ( c, a) associated to the eigenvalues b ab ± a 2 b 2 4abc 2

5 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 5 Both eigenvalues have strictly negative real part, so that the steady state ( c, a) is locally stable. Moreover the solutions to (1) are damped b oscillations when 4c > ab. Nevertheless the stochastic simulations of Lotka system (1) produce sustained oscillations. We have to explain first what oscillations do mean (and how to identify them) in a stochastic process. 3. Probability Space A probability space (X, Ω, p) is composed from a fixed set X called the base space, a collection Ω of measurable subsets of X called the space of events, and a probability measure p 0 that quantifies the size of every element or event B Ω. Notice that each B is at the same time an element of Ω and a subset of X. The collection Ω of (events) subsets of X must be a σ-algebra; i.e., it must satisfy the following properties: The empty set and the total set X are in Ω, If B Ω, the complement X \ B also lies in Ω, Given a countable collection {B k } k=1 of elements in Ω, their union k B k and intersection k B k and both in Ω. The probability measure p is a function p : Ω R that satisfies the following properties: p( ) = 0 and p(b) 0 for every B Ω, p(x) = 1 and p(a) p(b) for all A B in Ω, Given a countable collection {B k } k=1 of elements disjoint by pairs in Ω, the measure of the union p ( k B k) is equal to the sum k p(b k). Example 3. Maybe one of the simplest examples is obtained when X is a countable set, Ω = 2 X is the collection of all subsets of X, and each element x X is endowed with a fix real value µ(x) = c x 0 in such a way that µ(x) x X is equal to one.

6 6 EDUARDO S. ZERON It is easy to verify that Ω is a σ-algebra; and that µ is a probability measure when every subset B X is quantified by the sum µ(b) := µ(x). x B The space of probability (X, 2 X, µ) is known as a discrete probability space. One can take for example the natural numbers X = N and the Poisson distribution µ(x) = λx e λ x! for the parameter λ > 0 real. It is not possible to use the previous structure when X is not countable, but even so, we can construct several structures as follows. Example 4. Let X be an open or closed subset of the real space R n. Define Ω as the Borel σ-algebra; i.e., it is generated by all the relative open and closed subsets of X under countable unions, intersections, unions of intersections, etcetera. Given an integrable function f : X R such that f(x) 0 for every x X and f(x)dx equal to one, x X we can define the probability measure p(b) := f(x)dx for all B Ω, x B in such a way that (X, Ω, p) is a well defined space of probability. The function f is known as a probability density function. Notice that the probability of a point p(x) = 0 is zero for every x X. We can take for example the normal (Gaussian) probability density function f(x) = e x2 /2 2π defined on the real line X = R. 4. Random Variables Let (X, Ω, p) be a probability space (either discrete or not). A random variable is any (measurable) function g : X R; i.e., for every open set in the real line U R, the inverse image g 1 (U) X is an element in the σ-algebra Ω, so that the probability p ( g 1 (U) ) is well defined.

7 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 7 In general, one can think of (X, Ω, p) as the set of all possible experiments that can be done in a physical system, while x g(x) is a specific physical measure that we do in the experiment x X. For example X can be the space of all human beings that live or lived on earth, while g(x) is the weight of a given human being x X in particular. Any random variable g : X R automatically induces a structure of probability space on its image g(x). For example, if (X, Ω, µ) is a discrete space of probability, then X and its image g(x) are both countable. Hence we can consider the σ-algebra 2 g(x) composed from all subsets of g(x), and the new probability measure ρ g defined on g(x) by the formula: ρ g (E) := µ ( g 1 (E) ) = µ(x) = µ ( g 1 (y) ) x g 1 (E) y E for every subset E g(x). Likewise, given an arbitrary space of probability (X, Ω, p), consider a random variable h : X R. One can construct a second space of probability (h(x), S, ρ h ), where S is the Borel σ-algebra over h(x) and ρ h is the probability measure defined by: ρ h (U) := p ( h 1 (U) ) for every U S. Moreover, if we suppose that ρ h (U) = 0 for every set U S whose Lebesgue measure λ(u) = 0, there exists a probability density function f h : h(x) R such that f h (y) 0 and ρ h (U) := p ( h 1 (U) ) = dp(x) = f h (y)dy x h 1 (U) y U for every U S. A quite important point with the random variables is that random variables can always be added, multiplied, etcetera, because its image lies in the real numbers. In particular we can define

8 8 EDUARDO S. ZERON la mean and variance of the random variables introduced above. E(g) = x X E(h) = x X g(x)µ(x) = y g(x) h(x)dp(x) = V (g) = E(g 2 ) E(g) 2. yµ ( g 1 (y) ) = y h(x) yf h (y)dy, y g(x) yρ g (y), Now then what is the probability density function association to the sum of two random variables? Consider a pair of random variables g and h defined on a discrete probability space (X, Ω, µ) and with image in the integer numbers Z. event for z Z? What is the probability of the following D z = {(w, x) X 2 g(w) + h(x) = z}. The first problem that we have to solve is to define the probability in the product X 2. A simple solution is to define the product measure µ 2 (A B) = µ(a)µ(b) for all subsets A and B of X. Whence for all points y, z Z: µ 2( g 1 (y) h 1 (z) ) = µ ( g 1 (y) ) µ ( h 1 (z) ) = ρ g (y)ρ h (z). In general, given any probability measure p defined on the product space X 2, we say that the random variables g and h are independent with respect con to the probability p if and only if p ( g 1 (y) h 1 (z) ) = p ( g 1 (y) Y ) p ( X h 1 (z) ). We can now calculate the probability of the event D z as a convolution of the associated measures ρ g and ρ h, µ 2 (D z ) = = y Z w,x X, g(w)+h(x)=z µ 2 (w, x) = y Z µ 2( g 1 (y) h 1 (z y) ) µ ( g 1 (y) ) µ ( h 1 (z y) ) = y Z ρ g (y)ρ h (z y).

9 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 9 A similar result can be calculated when (X, Ω, p) is a non-discrete probability space. Consider now a countable number of random variables {g k } k=1 defined over a discrete probability space (X, Ω, µ). Endow every product X n with the standard product measure µ n, so that the random variables g 1, g 2,..., g n are all independent by pairs with respect to the probability measure µ n. Finally assume that the means and variances are all equal and finite, i.e., E(g k ) = c < and V (g k ) = σ 2 < k = 1, 2, 3,... The Central Limit Theorem yields that the following sequence of random variables converge in distribution to a Gaussian distribution, n [ g1 + g g ] n c ; σ n i.e., for every real number t R, we have that [ n [ g1 + g g ] ] t n e s2 /2 lim n µn c < t = ds. σ n 2π A similar result can be obtained when (x, Ω, p) is a non discrete Markov Chain. 5. Markov Chains Markov Chains allows us to analyse the dynamic of those systems for which there exists a fixed sequence of finite times t 0 < t 1 < t 2 < < t k < < with lim k t k =, such that the dynamic of the system is static inside each temporal interval [t k, t k+1 ). The state of the system can then be represented by a sequence of probability measures p k, one for every interval [t k, t k+1 ), where the index k 0 runs over all the natural numbers. In particular p k (x) is the probability that the system is at the state x in the time interval [t k, t k+1 ). The state of the system and the probability measures p k can only change as a jump at the times t k for every k 1. In a formal sense, a Markov Chain is a probability space (X, Ω, p k ) with a countable number of probability measures p k : Ω R indexed

10 10 EDUARDO S. ZERON by the integer k 0 and that can only change in discrete steps. Moreover the exact values of the measure p k+1 is exclusively given by the values of the previous measure p k, and the relation is linear. Hence, if we assume that the probability space (X, Ω, p k ) is discrete, there exits a collection of real numbers q k (x, y) in such a way that: p k+1 (y) = p k (x) q k (x, y) for y X, k N. x X The coefficients q k (x, y) cannot be arbitrary, every q k must lie in the real interval [0, 1] and for all fixed elements x X and k N, q k (x, y) = 1. y X Both conditions are necessary to guarantee that each p k+1 is indeed a probability measure. Suppose for example that p k is a probability measure of mass one at the fix element w X; i.e., { 1 if x = w, p k (x) = 0 otherwise. Then p k+1 (y) = q k (w, y) for every y X, so that q k (w, y) must lie in the closed interval [0, 1]. The second condition is necessary to guarantee that p k+1 is indeed a probability measure with total sum equal to one; i.e., y X p k+1 (y) = x X [ p k (x) ] q k (x, y) y X = x X p k (x) = 1. Notice in particular that the probability measure p 0 is indeed the initial state of the system; and that we can calculate the other probability measures p k in an inductive form. Moreover, if we suppose that the elements of the countable set X are indexed by the positive integers, we can talk about the first, second, third, etcetera, element in X. This indexation allow us to see the probability measures p k (x) (resp. coefficients q k (x, y)) as maybe infinite vectors (resp. matrixes), P k := ( p k (x) ) x X and Q k := [ q k (x, y) ] (x,y) X 2.

11 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 11 Previous notation allows us to write k P k+1 = P k Q k = P 0 Q 0 Q 1 Q k = P 0 Q j, for every k = 0, 1, 2,... Matrixes Q k are known as the transfer matrixes of the system. Notice that the number one is always an eigenvalue to every Q k, actually (1, 1,...) is the associated eigenvector; i.e., 1 = Q k j=0 Those matrices whose entries lie in the real interval [0, 1] and that satisfy the above identity are known as stochastic matrixes. Suppose for example that we have a random walk in a circle with (let us say) six fixed positions. Enumerate the positions clockwise from one to six. Assuming that we are at any given position: we move to a new position after throwing a dice, so we move to the adjacent position at the left side when we get a one or a two, otherwise we move to the adjacent right position. Given any initial position, it is trivial to simulate millions of random walks around the circle, and so to calculate the probability of being at any position at a given time. However it is really easy to calculate the probability of being at any position in the circle without having to do any simulation. Suppose that the time is indexed by integer numbers k 0. The probability that we are at any of the six positions in the circle can be expressed as an horizontal vector P k = (a 1, a 2,..., a 6 ) R 6, where each 0 a j 1 is indeed the probability that we are standing on the j-th position at the time k 0. Now then, if we assume that the dice is fear, we can accept that the probability of moving to the position at the left side (resp. right side) is equal to 1/3 (resp. 2/3), so

12 12 EDUARDO S. ZERON that we can construct the following stochastic matrix of transference, Q 0 = If we never change the dice, we shall have that the matrixes of transference are all equal; i.e., Q k = Q 0 for k = 1, 2, 3,... Hence, assuming that we stand on position one as the initial condition, we have that the initial probability P 0 has mass one at the first position, and so we can calculate the evolution of the probabilities P k following an inductive process: P 0 = ( 1, 0, 0, 0, 0, 0 ), ( P 1 = P 0 Q 0 = 0, 2 3, 0, 0, 0, 1 ), 3 ( 4 P 2 = P 0 Q 2 0 = 9, 0, 4 9, 0, 1 ) 9, 0, ( P 3 = P 0 Q 3 0 = 0, 4 9, 0, 1 3, 0, 2 ), 9 ( 8 P 4 = P 0 Q =, 0, 27 27, 0, 8 ) 27, 0, P 5 = P 0 Q 5 0 =. = lim P 2k = k lim P 2k+1 = k. ( 1 3, 0, 1 3, 0, 1 ) 3, 0, ( 0, 1 3, 0, 1 3, 0, 1 ). 3 ( 0, 1 10, 0, 3 27, 0, 8 27 ),

13 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 13 Homework 5. Verify the previous calculations and prove that the identities below holds for the initial probability P 0 = (0, 1, 0,...) with mass one at the second position, lim P 2k = k lim P 2k+1 = k ( 0, 1 3, 0, 1 3, 0, 1 ), 3 ( 1 3, 0, 1 3, 0, 1 ) 3, 0. Moreover, given the initial probability P 0 ( 1, 1, 0...), prove that 2 2 ( 1 lim P k = k 6, 1 6, 1 6, 1 6, 1 6, 1 6) 5.1. Simulations. Consider again the general case of a (discrete) Markov Chain (X, Ω, p k ), so that it is defined on a discrete probability space X and which dynamics on time is given by p k+1 (y) = p k (x) q k (x, y) for y X, k N. x X The example analysed in the previous paragraphs indicated us how to simulate any trajectory in a Markov Chain. We obviously use an inductive process. Assume that x(k) X is the state of the system in the temporal interval [t k, t k+1 ), so that x(0) X is the initial state of the system. We only need to choose the new state of the system in X as a random variable x(k+1) = y with the following distribution function: y q k (x(k), y) for y X. Recall that each q k (x(k), y) lies in [0, 1] and the sum y q k(x(k), y) is equal to one. It is very important to remember that Markov Chains do not have memory, in the sense that the new state x(k+1) is only determined by the exact value of the actual state x(k). It is not important what was the value of the previous states x(k 1), x(k 2), x(k 3), etcetera, when we calculate the new state x(k+1), we only consider the value of the actual state x(k).

14 14 EDUARDO S. ZERON 5.2. Transfer matrixes that are invariant on time. A very special case happens when the coefficients q k (x, y) and transfer matrixes Q k are all invariant with respect to the time t k, so that Q k = Q 0 y P k = P 0 Q k 0, for every k = 0, 1, 2,... The importance of this case lies in the facts that the probability distribution (measure) P k = P 0 Q k 0 can be calculated in a simple way and that we can prove the existence of a stationary distribution P satisfying so that P = P Q 0, P k = P for every k 0, when P 0 = P. The distribution P can be calculated as follows, P = lim k P 0 Q k 0, when the limit exists for some initial probability distribution P 0. If the above limit does not exist, we can always use the following formula that always converges to a stationary distribution, p 0 + P 0 Q 0 + P 0 Q P 0 Q k 0 P = lim. k k Example, a system of chemical reactions. Markov Chains can be used to model a chemical reactor (a system of chemical reactions) under the assumptions that all the chemical reactions happen (one at a time) in a fixed sequence of finite times: t 0 < t 1 < t 2 < < t k < < with lim k t k =. Consider a chemical system in which n different chemical species are involved into m different chemical reactions. The state of this system can be represented by a vector of natural numbers x N n, such that the instantaneous molecule count of the l-th chemical species is represented

15 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 15 by the l-th entry (x l ) of the vector x. Each one of the chemical reactions taking place in this reactor can be represented as usual: (5) x a(j,x) x + v[j], j = 1, 2... m, where a(j, x) 0 (resp. v[j] Z n ) denote the propensity (resp. stoichiometric vector) of the j-th reaction. Therefore each entry of the vector v[j] Z n determines the change produced in the molecule count of the corresponding species by the occurrence of the j-th chemical reaction. For example, the following two representations of a chemical reactions are completely equivalent A + B 1 3A, (x A, x B,...) x Ax B (xa +2, x B 1,...). Here and thereafter we assume that every propensity a(j, x) does not depend on time explicitly. In accordance to the fact that negative molecule counts are impossible, a(j, x) vanishes whenever any entry of x or x + v[j] is negative. We also introduce a countable number of (discrete) probability measures p k in the base space N n, so that p k (x) is the probability that the system is at the state x N n in the time interval [t k, t k+1 ) for k 0. Now then, suppose for some moments that the system is at the state x(k) N n in the temporal interval [t k, t k+1 ), how do we calculate which one of the m-possible chemical reactions happens now? The more natural option is to chose the following chemical reaction in a random way, but taking in consideration that the probability of choosing the j-th reaction should be proportional to the value of the respective propensity a(j, x(k)) 0. Whence the next chemical reaction is chosen as a random variable with the following distribution function j a(j, x(k)) m l=1 a(l, x(k)) for j = 1, 2,..., m. Previous considerations yield a natural way to calculate the respective transfer matrixes associated to the chemical system, and so to determine the temporal evolution of the probability measures. Thus

16 16 EDUARDO S. ZERON for every y X and k N, m p x (y v[j]) a(j, y v[j]) p k+1 (y) = m = l=1 a(l, y v[j]) where j=1 q k (x, y) = a(j, x) n l=1 x N n p k (x) q k (x, y), if y = x + v[j], a(l, x) 0 otherwise. Notice that the sums above are calculated over all possible j-th chemical reactions that begin in the state x = y v[j] and finish in y. We can simulate this model (Markov Chain) according to the procedure described in subsection 5.1. Moreover this model describes very well the random behaviour of a chemical reactor, but it requires the assumption that all the chemical reactions happen (one at a time) in a fixed sequence of finite times: t 0 < t 1 < t 2 < < t k < < with lim k t k =. This assumption does not hold in real life, because the chemical reactions happen at random times as well, so that the values of t k must be chosen in a random form Non-discrete probability spaces. One can easily analyse Markov Chains defined on a non-discrete probability space, but we must proceed with care, because it is not possible to use sums when the number of elements is non-countable. The natural solution is to use integrals instead of sums. In any case we only need to keep in mind that a Markov Chain is a probability space (X, Ω, p k ) with a countable number of probability measures p k : Ω R indexed by natural numbers k 0. What it is really important is that the probability measure p k+1 can be calculated as a linear combination of the values of the previous measure p k (x). In other words Markov Chains do not have memory, in the sense that every state x k+1 only depends on the value of the previous state x k.

17 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES Stochastic processes with continuous time Previous section was devoted to analyse discrete probability spaces (X, Ω, p k ) that are endowed with a countable number of probability measures p k : Ω R indexed by an integer number k 0, and such that the measure p k+1 is calculated as a linear combination of the values of the previous measure p k (x). Following the same tonic we analyse now those stochastic processes that can be expressed as a discrete probability space (X, Ω, p t ) with a non-countable number of probability measures p t : Ω R indexed by a real number t [t 0, ) that it is called time. Moreover the dynamic of p t is governed by a differential equation, where dp t / is a linear combination of the values of the measure p t (x). Hence there exists real numbers q t (x, y) such that dp t (y) = x X p t (x) q t (x, y) for y X, t t 0. A necessary condition is that the sum q t (x, y) = 0 for all x X and t t 0. y X This condition is necessary to guarantee that y p t(y) is constant with respect to the time t; i.e., d p t (y) = [ p t (x) ] q t (x, y) y X x X y X = 0. Thus, if p 0 (t) is the initial probability measure with y p t 0 (y) equal to, then p t is also a probability measure with y p t(y) equal to one for every t t 0. Notice in particular that unlike the analysis done in the previous Chapter 5, the new function y q t (x, y) is not a distribution function, because at least one of the coefficients q t (x, y) must be negative, in order to guarantee that sum of all of them with respect to y X is equal to zero.

18 18 EDUARDO S. ZERON Since there are too many different kinds of discrete stochastic processes with continuous time, we shall devote ourselves to analyse a particular kind called birth and death models. These models are really similar to the Markov Chains, in the sense that the state of the system changes as a jump, but it is now supposed that these jumps happen at a random time as well. Systems of chemical reactions are the ideal examples of birth and death models, because the chemical reactions can be seen as processes where molecules are destroyed, transformed, and generated at random times Modelling and realistic simulation of a system of chemical reactions. Consider a chemical system in which n different chemical species are involved into m different chemical reactions. The state of this system can be represented by a vector of natural numbers x N n, such that the instantaneous molecule count of the l-th chemical species is represented by the l-th entry (x l ) of the vector x. Each one of the chemical reactions taking place in this reactor can be represented as usual: (6) x a(j,x) x + v[j], j = 1, 2... m, where a(j, x) 0 (resp. v[j] Z n ) denote the propensity (resp. stoichiometric vector) of the j-th reaction. Therefore each entry of the vector v[j] Z n determines the change produced in the molecule count of the corresponding species by the occurrence of the j-th chemical reaction. In accordance to the fact that negative molecule counts are impossible, a(j, x) vanishes whenever any entry of x or x + v[j] is negative. Nevertheless we introduce know a non-countable quantity of probability measures p t defined in the base space N n and indexed by a real time t t 0, in such a way that p t0 is the initial probability measure. Simulations are calculated with the Gillespie algorithm: Suppose that x(t) N n is the state of the system at a given time t t 0, so

19 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 19 that x(t 0 ) is the initial state of the system. We firstly calculate when the next chemical reaction is happening, and to so we calculate a temporal increment τ in the real interval [0, ) as a random variable with exponential distribution function τ λ exp( λτ), where λ = m a(l, x(t)). l=1 Hence the next chemical reaction happens at the time t+τ. decide now which one of the m-possible chemical reactions is next, and we proceed here as in the case of modelling a chemical reactor as a Markov Chain: The next reaction is chosen as a random variable with the following distribution function j a(j, x(t)) m l=1 a(l, x(t)) for j = 1, 2,..., m. We Therefore, if the j-th chemical reaction is chosen, the new state of the system is given by x(t+τ) = x(t)+v[j]. Moreover in the temporal interval [t, t+τ) the state of the system is considered constant and equal to x(t). It is important to indicate that calculating the time increment τ 0 as a random variable with exponential distribution function is a necessary condition for asserting that the continuous time process has no memory (like in the case of a Markov Chain) Chemical master equation. The differential equation that governs the temporal dynamic of the probability measures p t can be deduced from the following considerations. Supposing that x(t) N n is the state of the system at the time t t 0, the value of each propensity a(j, x(t)) 0 can be interpreted as follow: For every positive and small enough temporal increment δt 0 a(j, x(t))δt is the probability that the j-th chemical reaction happens in the time interval [t, t+δt).

20 20 EDUARDO S. ZERON This consideration dictates how the probability measure p t evolves [ m ] m p t+δt (y) = 1 a(j, y)δt p t (y) p t (y v[j])a(j, y v[j]) δt j=1 for every y N n. The term between the square brackets in the above equation is the probability that no chemical reactions happen in the time interval [t, t+δt), while the sum at the right is the probability that some j-th reaction happen (beginning at the state y v[j] and finishing at y). We now proceed to move some terms around, divide by δt, and calculate the limit when δt, at the end we obtain the differential j=1 equation that governs the temporal dynamic of p t, dp t (y) p t+δt (y) p t (y) = lim δt 0 δt m [ ] = a(j, y v[j]) p t (y v[j]) a(j, y) p t (y). j=1 for every y N n. Previous result is known as the chemical master equation, and it yields a way to calculate the exact value of the probability measure p t (y) at every given time t t 0. We only need to solve the chemical master equation with an initial probability measure p t0. On the other hand notice that the identities below hold for every index j = 1, 2,..., m, y N n a(j, y v[j]) p t (y v[j]) = and so d y N n p t (y) = 0. y N n a(j, y) p t (y), 6.3. Langevin equation. Every term a(j, y v[j])p t (y v[j]) can be approximated using a truncated Taylor series, n a(j, y v[j])p t (y v[j]) a(j, y)p t (y) v k [j] da(j, y)p t(y) dy k + n n k=1 l=1 k=1 v k [j]v l [j] d2 a(j, y)p t (y) 2 dy k dy l.

21 BIOCHEMICAL OSCILLATIONS VIA STOCHASTIC PROCESSES 21 These approximations are used to deduce the Fokker-Planck equation, a partial differential equation that also governs the temporal dynamic of the probability measure p t, dp t (y) n [ d = p t (y) dy k + k=1 n n k=1 l=1 d 2 dy k dy l m j=1 ] v k [j]a(j, y) [ p t (y) m j=1 ] v k [j]v l [j]a(j, y). 2 The probability measure p t (y) can be calculated by solving or simulating the chemical master equation or the Fokker-Plank equation for a given time t t 0. Moreover the Fokker-Planck equation can be translated into the associated Langevin equation, this is a classical procedure, dx(t) [ m = j=1 ] [ m v[j]a(j, x(t)) + j=1 v[j] a(j, x(t)) dw ] j(t). The derivatives dw j (t)/ in the equation above are non-correlated white noise terms (derivatives of Brownian movements W j ). Notice that there is one white noise term for each of the m chemical reactions. Moreover the Langevin equation gets reduced to the deterministic chemical equation when the white noise terms dw j (t)/ are removed. Observe that the differential equation dx(t) = m v[j]a(j, x(t)) j=1 represents the deterministic dynamical behaviour of the following system of chemical reactions x a(j,x) x + v[j], for j = 1, 2,..., m. 7. Stochastic Lotka system We come back to the original Lotka chemical model [4], but we introduce a new term to denote the direct degradation of the chemical

22 22 EDUARDO S. ZERON specie X, a b X, X + Y c 2Y, Y e. The associated chemical master equation is given by dp (x, y; t) Hence the Langevin equation is dx (7) dy (8) where = ap (x 1, y; t) ap (x, y, t) + b(x+1)p (x+1, y; t) bxp (x, y; t) + c(x+1)(y 1)P (x+1, y 1; t) cxyp (x, y; t) + e(y+1)p (x, y+1; t) eyp (x, y; t). = a bx cxy + Ẇ1 a Ẇ 2 bx Ẇ 3 cxy, = cxy ey + Ẇ3 cxy Ẇ 4 ey, Ẇk are four non-correlated white noise terms. It is important to recall that the variables x and y denote the number of molecules of the respective chemical species X and Y, so that the dynamic of the concentrations [X] and [Y ] can be calculated by dividing equations (7) and (8) by the volume of the chemical reactor. The quite interesting property is that system (7) (8) cannot produce sustained oscillations when the noise terms Ẇk are removed, but sustained oscillations are indeed induced for some particular sets of parameters (a, b, c, d) when the noise terms Ẇk are included. The real problem is to characterise for which sets of parameters (a, b, c, d) sustained oscillations are produced. References [1] E.H. Armstorng. Operating Features of the Audion. Annals of the New York Academy of Sciences, 27 ( ), issue 1, pp [2] G.T. Fechner. Schweigg J. Chem. Phys. 53 (1828), p [3] J.L. Hudson and T.T. Tsotsis. Electrochemical reaction dynamics: a review. Chem. Eng. Sci. 49 (1994), issue 10, pp [4] A.J. Lotka. Contribution to the Theory of Periodic Reactions. J. Phys. Chem. 14 (1910), issue 3, pp