Generalized Chemical Langevin Equation for Chemical Reaction Systems

Transcription

1 ATCH Communications in athematical and in Computer Chemistry ATCH Commun. ath. Comput. Chem. 78 (207) ISSN Generalized Chemical Langevin Equation for Chemical Reaction Systems Tao Li a, Jie Li b a School of Environment and Architecture, University of Shanghai for Science and Technology, Shanghai 20093, China litao@usst.edu.cn b School of Civil Engineering, Tongi University, Shanghai , China liie@tongi.edu.cn (Received November 28, 206) Abstract The stochastic dynamical behaviors of chemical reaction systems driven by white Gaussian noise can be investigated by the chemical Langevin equation (CLE). For chemical systems driven by colored Gaussian noise, it is not always suitable to probe their stochastic dynamic responses by the CLE. In this paper, a stochastic kinetic model of chemical reaction systems called the generalized chemical Langevin equation (GCLE) is proposed to simulate the stochastic response of chemical reaction systems with intrinsic noise. The solution procedure based on Karhunen-Leove (K-L) decomposition and onte Carlo simulation is also presented. The numerical examples of the Brusselator and Oregonator are used to illustrate the effectiveness of the GCLE. The results show that the GCLE is applicable to stochastic chemical reaction systems driven by either colored or white noise. The evolution process of the concentration of the chemical reactants can be comprehensively described by the GCLE. The results also indicate that the stochastic dynamic response of the chemical reaction systems is significantly affected by the correlation time of the intrinsic noise and relative size of the system. Introduction In recent years, stochastic simulations of chemical kinetics have received increasing attention [-3]. any approaches have been proposed to describe the stochastic dynamical behaviors of chemical reaction systems, such as the chemical master equation (CE) [4], chemical Fokker-Plank equation (CFPE) [5], and chemical Langevin equation (CLE) [6]. When reaction species are in low concentrations, the internal noise in the stochastic chemical reaction systems described by the CE is approximately equivalent to the inverse square root

2 -282- of the mean number of molecules. Owing to consideration of internal noise, the dynamical behavior of chemical systems described by the CE is more accurate than that described by deterministic rate equations [7]. Because an exact solution of the CE is not possible in most complex cases, many studies have focused on investigating chemical systems using the approximative schemes [8-0]. Although onte Carlo simulations using the stochastic simulation algorithm (SSA) have been comprehensively developed [-7], they are timeconsuming for most chemical reaction systems. The CFPE, which was proposed by Kramers and oyal in the 940s [8, 9], is a secondorder partial differential equation obtained by Taylor series expansion of the CE. The mean and variance of molecule numbers obtained using the CFPE are exactly consistent with those obtained by the CE when the chemical systems are composed of purely unimolecular reactions. However, when at least one reaction in the chemical system is bimolecular, the mean and variance of molecular numbers predicted by the CFPE are not exact [20-22]. Because of the difficulties in solving the CFPEs of complex chemical systems in which more than three types of molecular species are involved, the accuracy of CFPE predictions for such cases has not been well studied. Generally, the difference between the CFPE and the CE is quite small [23, 24]. The third simulation approach is the CLE. The CLE is used to describe the concentration evolution of each molecular species. The solution of the CLE can be obtained by the standard stochastic integration method. Hence, the calculation speed of the CLE is naturally faster than that of the CE. The internal noise of chemical reaction systems described by the CLE is presumed to be white Gaussian noise. A white noise process is defined to mean a process whose power spectral density is a constant over the whole spectrum. Accordingly, a colored noise process is defined to mean a process whose power spectral density changes over the spectrum [25]. In fact, the existence of the white noise is impossible. It s only a kind of assumption of the practical random process with short relative time [26]. The number of reactions that occur in a certain time interval is considered to be independent Gaussian random variable. However, the number of times that a specific reaction occurs in a certain time interval is generally affected by other reactions in the same chemical reaction system. Namely, these random variables are not independent. Therefore, the internal noise in stochastic chemical reaction systems should not be considered to be white Gaussian noise. These internal fluctuations could be either white or colored Gaussian noise. For chemical systems driven by colored Gaussian noise, the CLE is not a suitable mathematical model to simulate their stochastic dynamical behavior. In

3 -283- this paper, we propose an effective mathematical model to simulate the stochastic dynamical response of chemical reaction systems driven by colored Gaussian noise. The rest of this paper is organized as follows. In Sec. 2, we briefly review the CLE, discussing its dynamical conditions in more detail. In Sec. 3, the GCLE is derived from an original equation describing the numbers of the types of molecules in the system at time t+ τ. Randomness originating from both internal noise and system parameters is involved in the GCLE. The internal noise is not limited to statistically independent white Gaussian noise, but it could also be colored Gaussian noise with a specific covariance function. The solution procedure of the GCLE, which is based on Karhunen-Leove decomposition and onte Carlo simulation, is also presented in Sec. 3. In Sec. 4, a numerical example of the Brusselator is used to demonstrate the practicality and effectiveness of the GCLE. Finally, we conclude the work in Sec Chemical Langevin Equation Here, we investigate the chemical reaction system consisting of N >> molecular specific { S S S } that chemically interact through >> reaction channel { },,..., N 2 C, C2,..., C. These chemical reactions occur in the time interval [ t, t+τ ) at constant temperature and inside a fixed volume Ω. The dynamical state of the chemical system is denoted as ( ) Y ( t) = Y ( t), Y ( t),..., Y ( t), where Y( t ) is the number of Si molecules in the system at time 2 N i t. The evolution of Y ( t) can be described by the following chemical kinetic equation [6]: dyi( t) = + Γ /2 υ ia ( Y( t)) υ ia ( Y( t)) ( t), = =,..., N) () where a ( Y ( t)) is the propensity function of the reaction channel C (=,2,,), a ( Y ( t)) is denoted as the probability that C reaction will occur somewhere inside Ω in the next infinitesimal time interval [t, t+) for given Y( t ) = y (in this paper all boldface vectors consist of N components, and each component represents a type of molecular species), υ i is the change of the number of Si 2 molecules produced by the C reaction, and Γ ( t), Γ ( t),..., Γ ( t) are uncorrelated independent white Gaussian noise. Eq. () is known as the CLE. The CLE is obtained based on the following two dynamical conditions:

4 -284- Condition : the time interval τ is small enough so that none of the propensity functions change their values in the state Y ( t) during [ t, t+ τ). The propensity functions then satisfy Y y [ + τ) [ ] (2) a ( ( t')) a ( ), t' t, t,,. t This condition means that the number of occurrences of each reaction step C in the time interval [ t, t+ τ) will be a statistically independent random variable that follows the Poisson distribution. Condition 2: the time interval τ is large enough so that the number of occurrences of each reaction step C in the time interval τ will be much larger than. This premise leads to the conclusion that a Poisson random variable can be substituted by a normal random variable with the same mean and variance. See ref. [6] for more detailed explanations of the above conditions. If we analyze the propensity function of the reaction channel C from its original expression, something is different from Condition. The propensity function a ( Y ( t)) of Eq. () has the following form [27]: a ( Y( t )) s h ( Y=( t )), (3) where s is the specific probability rate constant for reaction channel C which is associated with the reaction rate k, and h ( Y ( t )) is the number of distinct combinations of C reactant molecules available in the state Y ( t ). If the reaction rate for reaction channel C is considered to be a constant, s is also a constant. Because h( Y ( t)) is a deterministic function that is connected with υ i, a ( Y ( t )) naturally becomes a constant when t= t. This result directly leads to the conclusion of Condition, i.e., the number of occurrences of each reaction step C in the time interval [ t, t+ τ) will be a statistically independent random variable. However, because of the existence of randomness in chemical reaction systems, the reaction rates for some chemical reactions are not constant. In a real chemical reaction system, the reaction rates are always random variables following specific probability distributions [28]. As shown in ref. [27], for some simple reaction systems, the probability rate s for reaction channel C is associated with the reaction rate. Therefore, it can be considered that the probability rate s for reaction channel C is a random variable that satisfies a certain probability distribution. Accordingly, a ( Y ( t )) is a random variable following the same probability distribution. This random variable evolves over time t,

5 -285- becoming a random process. Therefore, a more reasonable description for stochastic chemical reaction systems is that the number of occurrences of each reaction step C in the time interval τ will be a random process rather than a random variable. Furthermore, it is not necessary to satisfy Condition if the values of the propensity functions change over time t. 3 Generalized chemical Langevin equation 3. Kinetic equation for chemical reaction systems Owing to the physical interpretation of υ i, the number of S i molecules in a chemical system at time ( t+ τ) can be expressed as Y( t+ τ) = y + A ( y, τ) υ, (4) i ti t i = where the system s state Y( t) at the current time t is y t, y ti is the ith component of vector y t and represents the number of S i molecules at the current time t, is the number of reaction channels, and A ( y, τ) is the number of the C reaction that occurs in the time interval [ t, t+ τ)for any τ > 0. t As shown in the ref. [6], because the values of the propensity functions for the reactions that occur in the time interval τ do not change, each A ( y, τ) will be a statistically independent Poisson random variable. This conclusion is mainly based on the premise of Condition, i.e., the propensity function for reaction channel C in time period τ remains constant at the value a ( y ). t However, according to our analysis of the stochastic chemical reaction system in Sec. II, the number of times that the reaction channel would occur in the time interval [ t, t+ τ) is a random process rather than a random variable. The propensity function for each reaction channel C at time t ( t t < t+ τ) can be expressed as Y = y [ + τ) [ ] (5) a ( ( t')) a ( ), t' t, t,,. t The number of occurrences of each reaction channel C at time t ( t t < t+ τ) is still a random variable that can be considered to be a Poisson random variable. Each Poisson random variable P( a ( y ), τ) can be approximated by a normal random variable with the t same mean and variance η( a ( yt' ) τ, a ( y t' ) τ) [6]. Therefore, the random process A( y t, τ) t

6 -286- that consists of these normal random variables is a Gaussian random process ω. Its covariance function is D( y, y 2). Because this study focuses on evolution of the dynamical state Y ( t) of the chemical reaction system in a microscopic infinitesimal interval, could be substituted for τ. Recalling that y t represents Y t, Eq. (3) becomes Y( t+ ) = Y( t) + υ ω. (6) i i i = The linear combination theorem for the Gaussian random process is involved [29]: ω = ω + n, (7) where ω is the mean vector of the random process ω, and n is a Gaussian random process with zero mean, and the covariance function D( y, y 2). ω can be expressed as where a ( ( t)) E η ( y ) t E η ( 2) t ω E ω y = =... E η ( tn) y a ( yt) a ( yt) a ( yt2) a ( yt2) = a ( ( t)),... =... = Y a ( ytn) a ( ytn) Y is a vector consisting of ( a ( t), a ( t2),..., a ( tn) ) y y y. (8) Substituting Eqs. (7) and (8) into Eq. (6) gives,..., N) Substituting for τ in Eq. (9) gives Y( t+ τ) = Y( t) + υ a ( Y( t)) + υ n. i i i i = = Y( t+ ) = Y( t) + υ a ( Y( t)) + υ n. i i i i = =,..., N) The kinetic equation of chemical reaction systems is given by (9) (0)

7 -287- dyi( t) = υ a ( Y( t)) + υ Γ ( t), i i = =,..., N) () where Γ ( t) = n /. Considering that the derivative of a Gaussian random process { X( t), t T} Gaussian process if this Gaussian random process { X( t), t T} will still be a is mean-square-differentiable during time interval T, Γ ( t) is a Gaussian random process with zero mean and covariance function D( y, y 2) in Eq. (),. In Eq. (7), when n( t ) is Gaussian white noise, n( t ) satisfies the following equation: n( t) σ ( t) n(0,; t) σ ( t) n(0,; t) n( t2) σ ( t2) n(0,; t2) 0 σ ( t2)... 0 n(0,; t2) n( t) = = n( tn) σ ( tn) n(0,; tn) σ ( tn) n(0,; tn) a ( yt ) n(0,; t) = 0 a ( (0,; 2)... 0 n t2) yt = a ( ) n(0,; tn) tn y a ( y ) t n(0,; t) 0 a ( y )... 0 n(0,; t2) a ( ) n(0,; y tn tn ) t2 =. The following approximation is used a( y ) t 0 a( yt) a( t) y This means that is a ( Y( t)). (2) (3)

8 -288- a ( y ) t n (0,; t) 0 a ( (0,; 2) ) y n t t n ( t) = a ( ) n (0,; tn) ytn = a ( Y( t)) κ, (4) where κ is a Gaussian random process consisting of a series of standard normal random variables. Substituting Eqs. (4), (7) and (8) into Eq. (6) gives Y( t+ τ) = Y( t) + υ a ( Y( t)) i i i = + =,..., N) υ a ( Y ( t) κ ( ). (5) /2 /2 i After rearranging Eq. (5), the following form is obtained Eq. (6) can also be expressed as Yi( t+ τ) Yi( t) = υ i a ( Y( t)) + =,..., N) = υ a ( Y ( t) κ /( ). (6) /2 /2 i dyi( t) = + Γ /2 υ i a ( Y( t)) υ ia ( Y( t) ( t), = =,..., N) /2 where Γ ( t) = κ /( ) is a white Gaussian process with zero mean. It is worth noting that Eq. (7) is the CLE. Eq. () can be deduced from Eq. (4) if Condition 2 is satisfied. The internal noise Γ ( t) of Eq. () can be either statistically independent Gaussian white noise or Gaussian color noise with a certain covariance function. If Γ ( t) is white Gaussian noise, Eq. () is equivalent to the CLE (i.e., Eq. (7)). If Γ ( t) is colored Gaussian noise, we can investigate the dynamical behavior of the stochastic chemical reaction systems by solving Eq. (). (7)

9 Generalized chemical Langevin equation The randomness of stochastic dynamical systems may originate from external excitations, internal fluctuations, or the internal parameters. For a chemical reaction system, its elementary reactions are described as [30] N k N < > υ i i υ i i i= i= Y Y, i=,..., N; =,...,, (8) where υ < i is the number of consumed S i molecules in the C reaction step, υ > i is the number of generated S i molecules in the C reaction step, and k is the reaction rate of the C reaction. In the C reaction step, the change of the number of S i molecules is denoted by υ = υ υ. The propensity functions of the C reaction that occur in unit time are [30] > < i i i N Y! a ( Y ( t)) = Vk, (9) i < i= ( Yi υ i) i! V υ < where V is the volume of the chemical reaction system, and C reaction. i V υ< is the reduced volume of the Generally, the reaction rates k ( =, 2,..., ) in Eqs. (8) and (9) are considered to be constants. However, the reaction rates of some reaction steps always have randomness, namely, the parameters of the chemical systems have randomness. Accordingly, because of the randomness of the reaction rates, the propensity functions of the C reaction that occur in the time interval [ t, t+ τ) also have randomness. Based on Eq. (9), considering the randomness of the reaction rates k, the propensity functions can be expressed as a ( X ( t), ξk), where ξ k is a random variable representing the reaction rate of the reaction C. Substituting a ( X ( t), ξ ) into Eq. (), the kinetic equation of the stochastic chemical systems is k dyi( t) = υ a ( Y( t), ξ ) + υ Γ ( t). i k i = =,..., N) In Eq. (20), the randomness originates from the parameters of the stochastic chemical reaction systems and internal noise, which are not limited to white Gaussian noise. Because of the similar form to the CLE, we call it the GCLE. (20)

10 -290- It is seen from the above derivation that the GCLE is equivalent to the CLE when its covariance function of the internal fluctuations is equal to zero and the reaction rates are considered to be constant, namely, the CLE is a special form of expressions of the GCLE. 3.3 Solution of the GCLE 3.3. White Gaussian noise If Γ ( t) is white Gaussian noise, Eq. (20) has the following form ( t) dyi( t) = + Γ /2 υ i a ( Y( t), ξk) υ ia ( Y( t) ( t). = =,..., N) Γ can be replaced by the set of random variables ( )(,2,.., ) becomes dyi( t) = + /2 υ i a ( Y( t), ξk) υ ia ( Y( t) ζ ( t). = =,..., N, =,2,..., ) (2) ζ t =. Eq. (2) then The total number of the random variables in Eq. (22) is equal to the sum of the number of ξ k plus the number of ζ. Thus, Eq. (22) becomes a general stochastic kinetic equation with 2 random variables. It can be solved by the stochastic differential method, onte Carlo method, and so forth. (22) Colored Gaussian noise From Sec. 2, υ i is the change of the number of Si molecules in the C reaction step. Γ ( t) will be the additive noise with zero mean if υ i is considered to be a constant. Assuming that Γ ( t) have the same covariance function D( y, y 2), each Γ ( t) can be approximated by a same random process Γ ( t), i.e., Γ ( t) =Γ 2( t) =... =Γ ( t) =Γ ( t). Then Eq. (20) then becomes dyi( t) = υ a ( Y( t), ξ ) +Γ( t) υ. i k i = =,..., N) (23) Letting ϒ ( t) =Γ( t) υ i, where ϒ ( t) is also a Gaussian process because υ i is assumed to = be constant, Eq. (23) can then be rewritten as

11 -29- dyi( t) = υ i a ( Y( t), ξk) +ϒ( t), =,..., N) (24) where ϒ( t) is colored Gaussian noise with zero mean and covariance function D( y, y 2). According to random process theory, colored Gaussian noise ϒ ( t) can be expanded to the sum of a series of random variables [29]. The expansion of ϒ ( t) can be achieved by the K-L decomposition method. Eq. (24) then becomes dyi( t) = υ a ( Y( t), ξ ) + ζ ( t) λ ϕ, i k h h h = h=,..., N) (25) where ζ ( t h )( h=,2,..., ) are a set of independent random variables, and λ h and ϕ h are the hth eigenvalue and eigenvector of the covariance function D( y, y2) respectively. The eigenvalue λ h and eigenvector ϕ h are the solution of the following Fredholm equation: t D ( y, y2) f h( y) d ( y) =λ f h h( y 2). (26) 0 The first S sets of eigenvalues and eigenvectors are obtained according to the 97% energy principle [29, 3]. The GCLE for stochastic chemical systems subected to disturbance by colored Gaussian noise is S dyi( t) = υ a ( Y( t), ξ ) + ζ ( t) λ ϕ. i k h h h = h=,..., N) The numbers of random variables ξ k( k =, 2,..., ) and ζ ( t h )( h =,2,..., S ) are and S, (27) respectively. The total number of random variables in Eq. (27) is equal to plus S. The randomness considered in Eq. (27) originates from internal fluctuations and the parameters of the chemical system. It can be solved by the general stochastic differential method or the onte Carlo simulation method. In this paper, the onte Carlo simulation method is used to solve the GCLE. Here, we give the solving procedure of the GCLE using the onte Carlo simulation method. Step : Decompose the Gaussian process ϒ ( t) by the K-L decomposition method to give Eq. (27) consisting of a certain number of random variables.

12 -292- Step 2: Randomly generate a set of vectors whose dimensions are equal to the number of random variables, letting the number of random vectors be equal to the number of onte Carlo simulations. Step 3: Solve the deterministic chemical reaction equation that is formed by Eq. (27), in which the random variables are substituted by each vector randomly generated by a computer. Step 4: Obtain the evolution curves of the mean and variance of the reactant concentrations by solving the above deterministic ordinary differential equations. 4 Numerical example In this section, the dynamic responses of the Brusselator and Oregonator driven by colored Gaussian noise are investigated by the GCLE. To illustrate the effectiveness and feasibility of the GCLE, in this chemical reaction system, randomness from internal noise is involved. 4. GCLE for the Brusselator The elementary reactions of the Brusselator are [30] k k2 k3 k4 X, X Y, 2X + Y 3 X, X, (28) where X and Y represent two types of reactant, and k, k2, k3 and k4 are the reaction rates of different elementary reactions. The corresponding deterministic kinetic equation is dx 2 = k k2x+ k3x y k4x, (29) dy 2 = k2x k3x y, where x and y are the concentrations of reactants X and Y, respectively. For a chemical reaction system with white Gaussian noise, ignoring the randomness from the parameters of the system, the GCLE is equivalent to the CLE. Generally, the CLE is simplified to Eq. (22). Dividing both sides of Eq. (22) by V (here, V is the volume of the chemical reaction system) gives d( Y( )/ ) ( ( )) i t V a Y t = υ i = V /2 a ( Y( t)) + υ i ζ ( t), V = V,..., N) (30)

13 -293- where Y( t)/ V is the concentration of Si molecules. i According to Eq. (30), the chemical kinetic equation of the Brusselator is [30] dx 2 = ( k k2x+ k3x y k4x), 2 kζ ( t) k 2xζ 2( t) k3x yζ3( t) k4xζ 4( t) + +, V dy 2 2 = ( k2x k3x y) + k2xζ 2( t) k3x yζ3( t) / V, a( X( t)) a2( X( t)) a3( X( t)) 2 a4( X( t)) where = k, = k2x, = k3x y, = k4x; V V V V a( Y( t)) a2( Y( t)) a3( Y( t)) 2 a4( Y( t)) = 0, = k2x, = k3x y, = 0; V V V V υ =, υ =, υ =, υ = ; υ y =, υ2 y =, υ3y =, υ4y =. x 2x 3x 4x (3) This work focuses on stochastic chemical reaction systems driven by colored Gaussian noise. The dynamical response of a stochastic chemical reaction system with colored Gaussian noise can be described by the GCLE (i.e. Eq. (24)). Dividing both sides of Eq. (24) by V, Eq. (24) gives d( Y( )/ ) ( ( )) i t V a Y t = υ i + ϒ ( t ). = V V,..., N) Substituting ϒ '( t) = ϒ ( t) for ϒ ( t) gives V d( Y( )/ ) ( ( )) i t V a Y t = υ i + ϒ'( t). = V V,..., N) The volume of the chemical reaction system is constant. The properties of the random process will not change if it is multiplied by a constant coefficient. Thus, ϒ'( t) is still Gaussian color noise with zero mean. Imposing K-L decomposition on ϒ '( t), Eq. (33) then becomes (32) (33) d( Y( )/ ) ( ( )) i t V a Y t = υ i V = S + ζ h( t) λhϕh. V h=,..., N) (34)

14 -294- Combining Eq. (34) with Eqs. (28) and (29), and using the same expressions of the parameters in Eq. (3), the GCLE for the Brusselator is then dx 2 = ( k k2x+ k3x y k4x) S + ζ h( t) λhϕh, V h= dy = + S 2 ( k2x k3x y) ζ h( t) λhϕh. V h= (35) 4.2 Stochastic dynamical behaviors of the Brusselator dx dy Letting = 0, = 0, the stable solution of Eq. (29) can be attained [30]: k xs =, k4 k2k4 ys. = kk3 (36) From ref. [30], the values of k, k2, k3 and k4 are suggested to be k=k4=.0, k2=.8 and k3= Determination of the values of λ h and ϕ h From Sections 2 and 3, we know that the GCLE is equivalent to the CLE when its covariance function is equal to zero. But what we concern is the case of the covariance function D( y, y ) 0. Here, in order to indicate the solving procedure and effectiveness of the GCLE, 2 a simple form of the covariance function is used, although it is known that the exact form of internal fluctuations should be further studied. To readily achieve K-L decomposition on the colored Gaussian noise of the GCLE, we assume that the covariance function of the colored t s Gaussian noise in the GCLE is D( y, y 2) = ϒ ( t) ϒ ( s) = exp( ), where l is the l correlation time of ϒ '( t), and t and s are the different reaction times.

15 -295- Eigenvalues Orders of eigenvalues Figure. First 2-order eigenvalues The analytic solution of the Fredholm equation of the K-L decomposition method, in which the above covariance function D( y, y 2) is used as its kernel function, can be conveniently obtained by solving this integral equation. The first 2-order eigenvalues are shown in Fig.. From Fig., the higher order eigenvalues are close to zero. The first 0-order eigenvalues and eigenvectors are determined to expand the random process ϒ '( t) (i.e. S = 0). Thus Eq. (35) becomes dx 2 = ( k k2x+ k3x y k4x) 0 + ζ h( t) λhϕh, V h= dy = ( k2x k3x y) ζ h( t) λhϕh. V h= There are 0 random variables in Eq. (37). These random variables expand a 0-dimension random-variate space. Eq. (37) can be solved by the onte Carlo simulation method following the three steps mentioned in Sec. IV. (37) Samples of the random process Eq. (37) contains 0 random variables that expand to a 0-dimension random-variate space. There are 0,000 sets of random numbers generated by the computer. Each set of random numbers consisting of 0 components can be considered to be a point in the random-variate space. Thus, we can obtain the reactant concentration evolution of the Brusselator by performing 0,000 onte Carlo simulations. Inserting these 0,000 points into the expansion of the random process in Eq. (37), we can obtain 0,000 samples of the random process of the

16 -296- GCLE. The values of the samples of the random process will change with different correlation times, as shown in Figs Figure 2. Samples of the random process with correlation time l = s. Points 55, 524, 3357, and 8736 represent four random process samples determined by these points. Figure 3. Samples of the random process with correlation time l = 0.5s. Points 55, 524, 3357, and 8736 represent four random process samples determined by these points. Value of samples Value of samples Value of samples Value of samples Time Time Time Figure 4. Samples of the random process with correlation time l = 0.02s. Points 55, 524, 3357, and 8736 represent four random process samples determined by these points. Time Figure 5. Samples of the random process with correlation time l = 0.002s. Points 55, 524, 3357, and 8736 represent four random process samples determined by these points. From Figs. 2-5, the oscillation frequency of the random process sample increases with decreasing correlation time, but the amplitude decreases with decreasing correlation time.

17 -297- Namely, the correlation time has a clear effect on the random process. Treating the internal fluctuations as white noise is not reasonable. In the next section, the dynamic response of the Brusselator subected to colored Gaussian excitation with different correlation times will be investigated. 4.3 ean and standard deviation of the reactant concentration of the Brusselator The GCLE for the Brusselator was solved by 0,000 onte Carlo simulations for correlation time l = 0.02s, relative volume of chemical reaction system V = 00, and reaction timet = 20s. Evolution of the mean and standard deviation of the reactant concentrations is compared with the exact solution of the deterministic chemical rate equation, as shown in Fig. 6. DRE C ean Time (s) C Standard deviation Time (s) Figure 6. ean and standard deviation of the concentration of reactant X of the Brusselator. DRE is the result of the deterministic chemical rate equation and C is the result of the GCLE solved by the onte Carlo method. The gap between the solutions of the deterministic chemical rate equation and that of the GCLE solved by the onte Carlo method is presented as Fig. 7. From Figs. 6 and 7, evolution of the mean and standard deviation of the concentration of reactant X in the Brusselator driven by colored Gaussian excitation can be attained by solving

18 -298- the GCLE for this chemical system. At the beginning of the chemical reaction, the mean of reactant concentration is very close to the solution of the deterministic rate equation. However, the gap between the two gradually increases with increasing reaction time. This means that the internal fluctuation effect of the stochastic chemical system progressively increases with the reaction time. Furthermore, from Fig. 6, the standard deviation of the reactant concentration is much smaller than mean, i.e., the variance coefficients at different times are small. This is mainly because the internal fluctuation effect is relatively weak. 0. Deviation Time (s) Figure 7. Gap between the solutions of the deterministic chemical rate equation and that of the GCLE solved by the onte Carlo method. 4.4 Reactant concentration evolution of the Brusselator driven by colored Gaussian noise with different correlation times The mean and standard deviation of the change of the concentration of reactant X for different correlation times were calculated. The changes of the mean and standard deviation with the different correlation times are shown in Fig. 8. As shown in Fig. 8, the correlation time of colored Gaussian noise affects the mean and standard deviation of the reactant concentration of the Brusselator. The peak value of the standard deviation also changes with the correlation time. The standard deviation increases with increasing correlation time. From Fig. 8, it can be concluded that the effect of internal fluctuations in the stochastic chemical reaction systems is enhanced with increasing correlation time. The influence of the correlation time cannot be ignored. Therefore, a more reasonable description of the internal fluctuations in stochastic chemical reaction systems is that the internal fluctuating forces should be considered to be colored Gaussian noise rather than white Gaussian noise.

19 -299- Figure 8. ean and standard deviation of the concentration of reactant X of the Brusselator driven by colored Gaussian noise with different correlation times. l=2sec, l=sec, l=0.2sec, l=0.02sec and l=0.002sec represent the correlation times of the Brusselator of 2s, s, 0.2s, 0.02s and 0.002s respectively. 4.5 Reactant concentration evolution of the Brusselator with different volumes Figure 9. ean and standard deviation of the concentration of X of the Brusselator with different system volumes. V=4, V=0 and V=00 represent the relative volumes of the Brussselator of 4, 0 and 00 respectively.

20 -300- For correlation time l= s, the mean and standard deviation were calculated with the relative volumes of the Brusselator of 4, 0 and 00 (here, the relative volume is the ratio between the system volume and the reactant volume). The results are shown in Fig. 9. From Fig. 9, changing the system volume leads to fluctuations of the mean and standard deviation of the reactant concentration. The mean curve of the reactant concentration becomes smoother with increasing volume. The standard deviation of the reactant concentration clearly decreases with increasing relative volume. This means that the effect of the internal fluctuations will decrease with increasing system volume. 4.6 ean and standard deviation of the reactant concentration of the Oregonator To illustrate the effectiveness of the GCLE, the dynamic responses of the Oregonator, driven by colored Gaussian noise is investigated. The Oregonator consists of five reactions involving three intermediates. This kinetic model is proposed to explain the oscillations observed in the Belousov- Zhabotinskii reaction. The elementary reactions of the Oregonator are [32] k k k k k Y X, X + Y, X 2 X + Z, 2 X, Z Y (38) where X, Y and Z represent three types of reactant, and k, k2, k3, k4 and k5 are the reaction rates of different elementary reactions. The corresponding deterministic kinetic equation is [32] dx = k y k2xy+ k3x 2k4x dy = k y k2xy+ k5z dz = k3x k5z where x, y and z are the concentrations of reactants X, Y and Z, respectively. The GCLE for the Oregonator is then 2, (39)

21 -30- dx 2 = ( k y k2xy+ k3x 2 k4x ) S + ζ h( t) λhϕh, V h= S dy = ( k y k2xy+ k5z) + ζ h( t) λhϕh. V h= S dz = ( k3x k5z) + ζ h( t) λhϕh, V h= dx dy dz Letting = 0, = 0, = 0, the stable solution of Eq. (39) can be attained [32]: k xs = k2 k3 ys = k2 k k zs = k k The values of k, k2, k3, k4 and k5 are suggested to be k =, k2=0.9, k 3 =.8, k 4 =.6 and k 5 = 0.6. The GCLE for the Oregonator was solved by 0,000 onte Carlo simulations for correlation time l = 0.02s, relative volume of chemical reaction system V= 00, and reaction time t = 20s. Evolutions of the mean and standard deviation of concentration of reactant X, Y and Z are compared with the solutions of the deterministic chemical rate equation of the Oregonator. From Fig. 0, evolution of the mean and standard deviation of the concentration of the reactants of the Oregonator driven by colored Gaussian excitation can also be conveniently attained by solving the GCLE. Although the influence of the intrinsic noise makes the dynamic response of the stochastic chemical reaction system different from that of the deterministic chemical system, as shown in Figs. 6 and 0, the existence of the randomness of the chemical reaction system does not change the essential laws of the physical phenomenon in the chemical reaction systems.. (40) (4)

22 -302- ean C-z C-x C-y DRE-z DRE-x DRE-y Time (s) Standard deviation C-z C-x C-y Time (s) Figure 0. ean and standard deviation of the concentration of reactant X, Y and Z of the Oregonator. DRE-x, DRE-y and DRE-z are the results of the deterministic chemical rate equation. C-x, C-y and C-z are the results of the GCLE solved by the onte Carlo method. Concentration of reactant Y DRE CLE GCLE Time (s) Figure. Concentration of reactant Y of the Oregonator obtained by solving the DRE, CLE and GCLE respectively, where DRE is denoted as the solution of the deterministic chemical rate equation, CLE is the solution of the CLE, GCLE represents the result of the GCLE. If the intrinsic noise is considered to be white Gaussian noise, the dynamic response of the stochastic chemical reaction system can be obtained by the CLE. In this case, as presented in Sec. 3., the GCLE is exactly equivalent to the CLE. The numerical results obtained by the CLE for the Oregonator driven by the white Gaussian noise and those obtained by the GCLE for this system driven by the colored Gaussian noise are compared with the solution of the

23 -303- DRE for the deterministic system, as shown in Fig.. The result of the CLE is also attained by the onte Carlo method. Seen from Fig., the dynamic response of the stochastic chemical reaction system driven by the white Gaussian noise is very close to that of the deterministic system. By comparison, the GCLE may be more suitable for investigating the effect of the internal fluctuation of a stochastic chemical reaction system. 5 Conclusions The GCLE is proposed for dynamic response analysis of the stochastic chemical reaction systems. The limitations of the CLE are discussed. The GCLE is then derived from the initial state equation for changes of molecules in the stochastic chemical reaction systems. The internal noise of chemical systems is considered to be colored Gaussian noise with a certain covariance function rather than white Gaussian noise. Two numerical examples are used to demonstrate the feasibility and effectiveness of the GCLE. The results show that the dynamic response of the stochastic chemical reaction system can be conveniently investigated by the proposed method. The reactant concentration evolution is significantly affected by the correlation time and relative volume of the chemical reaction system. In this paper, for convenience, a simple covariance function of the colored Gaussian noise is used. In order to clarify the mechanism of the stochastic chemical reaction systems, the further work should focus on the exact form of the internal fluctuations. Thus, we can effortlessly obtain the evolution law of the dynamic response of the stochastic chemical reaction systems by the GCLE. Acknowledgements: Supports from the Natural Science Foundation for aor International Cooperation Proect of China (Grant No ) and the Young Teachers Program of Shanghai are gratefully acknowledged. References [] Y. D. Huang, S. Rudiger, J. W. Shuai, Accurate Langevin approaches to simulate arkovian channel dynamics, Phys. Biol. 2 (205) -22. [2] G. H. Huang, W. F. Zeng, A didscrete hidden arkov model for detecting Histone Crotonyllysine sites, ATCH Commun. ath. Comput. Chem. 75 (206) [3] P. Kim, H. C. Lee, Fast probability generating function method for stochastic chemical reaction networks, ATCH Commun. ath. Comput. Chem. 7 (204)

24 -304- [4] D. A. cquarrie, Stochastic approach to chemical kinetics, J. Appl. Probab. 4 (967) [5] C. W.Gardiner, Handbook of Stochastic ethods for Physics, Chemistry and the Natural Sciences, Springer, New York, [6] D. T. Gillespie, The chemical Langevin equation, J. Chem. Phys. 3 (2000) [7] S. Smith, C. Cianci, R. Grima, odel reduction for stochastic chemical systems with abundant species, J. Chem. Phys. 43 (205) #2405. [8] T.. Liggett, Interacting Particle Systems, Springer-Verlag, Berlin, 985. [9] L. Frachebourg, P. L. Krapivsky, Exact results for kinetics of catalytic reactions, Phys. Rev. E 53 (996) R3009-R302. [0] T. Brett, T. Galla, Stochastic processes with distributed delays: Chemical Langevin equation and linear-noise approximation, Phys. Rev. Lett. 0 (203) [] D. T. Gillespie, Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem. 8 (977) [2] D.T. Gillespie, Stochastic simulation of chemical kinetics, Ann. Rev. Phys. Chem. 58 (2007) [3] K. Takahashi, H. Kaizu, B. Hu,. Tomita, A multi-algorithm, multi-timescale method for cell simulation, Bioinformatics 20 (2004) [4] S. Lampoudi, D. T. Gillespie, L. R. Petzold, Effect of excluded volume on 2D discrete stochastic chemical kinetics, J. Comput. Phys. 228 (2009) [5] A. S. Ribeiro, Stochastic and delayed stochastic models of gene expression and regulation, ath. Biosci. 223 (200) -. [6] A. Celani,. Vergassola, Nonlinearity, fluctuations, and response in sensory systems, Phys. Rev. Lett. 08 (202) # [7] F. Antoneli, R. C. Ferreira,. R. S. Briones, A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks, ath. Biosci. 276 (206) [8] H. Kramers, Brownian motion in a field of force and the diffusion model of chemical reactions, Physica 7 (940) [9] J. E. oyal, Stochastic processes and statistical physics, J. R. Stat. Soc. B (949) [20] D. Schnoerr, G. Sanguinetti, R. Grima, The complex chemical Langevin equation, J. Chem. Phys. 4 (204) [2] R. Erban, S. J. Chapman, I. G. Kevrekidis, T. Vechodsk y, Analysis of a stochastic chemical system close to a sniper bifurcation of its mean-field model, SIA J. Appl. ath. 70 (2009) [22] P. Thomas, A. V. Straube, J. Timmer, C. Fleck, R. Grima, Signatures of nonlinearity in single cell noise-induced oscillations, J. Theor. Biol. 335 (203)

25 [23] S. N. Ethier, T. G. Kurtz, Convergence to fleming-viot processes in the weak atomic topology, Stoch. Proc. Appl. 54 (994) -27. [24] R. Grima, P. Thomas, A. V. Straube, How accurate are the nonlinear chemical Fokker- Planck and chemical Langevin equations, J. Chem. Phys. 35 (20) [25] T. T. Soong,. Grigoriu, Radom Vibration of echanical and Structural Systems, PTR, Prentice Hall, 993. [26] W. Q. Zhu, Random Vibration, Science Press, Beiing, 992. [27] D. T. Gillespie, Statistical mechanics and its application, Physica A 87 (992) [28] H. Berthoumieux, Fluctuations in reactive networks subect to extrinsic noise studied in the framework of the chemical Langevin equation, Phys. Rev. E 94 (206) [29] R. G. Ghanem, P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, Berlin, 99. [30] H. W. Xin, Z. H. Hou, Nonlinear Chemistry, Press Univ. Sci. Techn. China, Hefei, [3] Z. J. Liu, Research on the Orthogonal Expansion Theory and Application, Tongi University, Shanghai, 2007, (PhD Thesis). [32] R. J. Field, R.. Noyes, Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys. 60 (974)