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Practical stochastic model for chemical kinetics |

Leonardo Silva-Dias; Alejandro López-Castillo Departamento de Química, Universidade Federal de São Carlos, 13560-970 São Carlos - SP, Brasil Recebido em 06/04/2015 Endereço para correspondência
*e-mail: alcastil@ufscar.br RESUMO
The Practical Stochastic Model is a simple and robust method to describe coupled chemical reactions. The connection between this stochastic method and a deterministic method was initially established to understand how the parameters and variables that describe the concentration in both methods were related. It was necessary to define two main concepts to make this connection: the filling of compartments or dilutions and the rate of reaction enhancement. The parameters, variables, and the time of the stochastic methods were scaled with the size of the compartment and were compared with a deterministic method. The deterministic approach was employed as an initial reference to achieve a consistent stochastic result. Finally, an independent robust stochastic method was obtained. This method could be compared with the Stochastic Simulation Algorithm developed by Gillespie, 1977. The Practical Stochastic Model produced absolute values that were essential to describe non-linear chemical reactions with a simple structure, and allowed for a correct description of the chemical kinetics. Palavras-chave: Stochastic Method; Ehrenfest Urn Model; Monte Carlo Method; non-linear chemical reaction.
The Ehrenfest Urn Model (EUM), or as it is also named, the Stochastic Monte Carlo Method, The Practical Stochastic Model (PSM) can be compared with the Stochastic Simulation Algorithm (SSA) as developed by Gillespie The development of the PSM was based on the earlier works for simple first and second order reactions Some definitions concerning the theory employed in this study are to be considered as follows: 1) Stochastic - a Numerical Method that considers a time evolution of a discrete chemically reacting system; The initial Monte Carlo studies, The PSM method can accurately describe the linear and non-linear chemical kinetic reactions, where the fluctuations are naturally obtained, simulating the finiteness of the systems. This method is consistent, robust, simple, practical, and independent of a deterministic Ordinary Differential Equation (ODE). The PSM is a simple model of a "particle" exchange between containers, where the state of the system is defined in terms of the container occupancy. This method has already been used for many years in the classroom to simulate the kinetic behavior of first and second order reactions. To introduce the method, a vector with only 10 positions can be used with a manual random selection. 10 is a good number to perform a manual selection, as it is small enough to manually realize first order kinetics in approximately 15 minutes (with second order reactions, the time increases as N This method is a very easy and efficient statistical integration tool and avoids the use of more advanced mathematical techniques, but still allows for accurate descriptions of the various chemical aspects related to reactions, including the relaxation time (the necessary time for a reaction to reach equilibrium); thermodynamic equilibrium and fluctuations; and a description of the steady states. This method is efficient for teaching chemical kinetics and becomes a very powerful tool in a simple way for simulating chemical reactions.
The PSM was built step-by-step with simple (first and second-order) reactions, and it was completed considering non-linear reactions as do the Lotka The PSM is based on Monte Carlo steps and could be compared to the Stochastic Simulation Algorithm (SSA) as developed by Gillespie The PSM approach was developed for comparing its solution to the deterministic methodology. This stochastic simulation assumes that the compartments can have any occupation from 0 to The necessary steps to simulate a typical reaction are (as in the explicit example that is given in Supplementary Material (SM) - Section A.3):
In order to simplify the discussions, it is assumed that In the PSM approach, the step size time is effectively proportional to 1/
In the first step, it was necessary to establish connections between the stochastic method and the deterministic method. For that, several kinds of chemical reactions were used, in order to obtain the main stochastic rules. Those rules are simple and intuitive and contribute to an understanding of the method and its generalizations. Finally, the developed stochastic method is very accurate and can potentially describe any kind of chemical kinetic reaction. The main steps of those connections are given below:
The stochastic and deterministic time flow connections were obtained from several simulations (see SM - Section B). The flow of time obtained from the deterministic solution is numerically equivalent to the number corresponding to the Monte Carlo step divided by Each step of the first order reaction occurs with a probability ( The first and second-order reactions of the stochastic approach can be described by (scaled) relative values of the concentrations (parameters and variables). However, a set of coupled non-linear reactions should be described by absolute values. For example, the stability matrix trace ( _{4})^{2} or tr = B - 1 -A^{2} (with unitary probabilities) for the Brusselator system, where A and B are the concentration parameters (SM - Section A.2). For example, if one can correctly describe the concentration evolution in time for the Brusselator system with tr>1, it is necessary to require that B>1.
In order to develop a concise stochastic theory, it is necessary to understand the relationship between the stochastic and deterministic methods for the absolute values of the concentrations (parameters and variables). It is easier to start the analysis firstly by considering the absolute values of concentration parameters (e.g., A) bigger than unity with concentration variables less than unity. The parameters are ever constants for deterministic and stochastic methods. They can be understood as reagents or products, which are consumed or produced, but are kept as constants, on average, for a stochastic method. If [ N. In other words, it is necessary to increase the probability, by adding more random selection steps to the reaction, especially where the A parameter appears, in order to overcome that limitation.That step must be Several combinations were performed to show the correctness of the relation [ N^{1/2 18} (see SM - Section C), the relation of [A] = mN is trivially obtained. That relation means that the time flow decreases m-times, while the velocity and the concentration become _{A}/Nm-times larger.Similarly, one can also study the absolute values of the dependent variables (e.g., X] N, and the excess is given by Ex = N - _{X}N = ([X] - 1) N, where N is the maximum number of occupations of the X compartment, as defined above. A relative excess can be defined as e = Ex/N.The real number N = [X] is taken into account when [X] > 1. A natural number m can be defined by the application of a truncation operation over r, i.e., m = trunk (r). The usual procedure does not change if r< 1, i.e., just choosing the random number (m=1) once is enough. However, if m= trunk (r) > 2, it is necessary to add (m - 1) random choices.If [ The general PSM scheme is shown in SM - Section A.1. The complete flowchart for the
The Lotka
The Lotka system is described by three reactants: one parameter ( q, respectively. These reactions are described by the following ODE's: _{3}d[X]/dt=q and _{2}AX-q_{1}XYd[Y]/dt=q. This system implicitly considers a constant flux, since A and P are substances with a constant concentration._{1} XY-q_{3}Y
Figure 1. Flowchart of the stochastic method for an A+X→Y reaction
The deterministic solutions q = 0.6, and _{2}q = 0.7, _{3}A=4, X_{0}=1, and Y_{0}=1.5, as obtained by the Runge-Kutta method, are shown in Figure 2, where X_{0} and Y_{0} are the initial conditions. The stochastic solutions, considering the procedures of Section 3.1 and Section 3.2, are also shown in Figure 2 for A=10000/10000=1 with m=4 (_{A}A=4), X_{0}=10000/10000=1, and Y_{0}=10000/10000=1 with acc= 5000/10000=0.5 (_{Y} Y_{0}=1.5) and with the same q's as above.
Figure 2. Concentrations X(t) and Y(t) as a function of time for the Lotka system using q _{1} = 0.5, q_{2} = 0.6, and q_{3} = 0.7, from a deterministic method (gray lines) with A=4, X_{0}=1, and Y_{0}=1.5, and from the stochastic solution (black lines), with N_{A}=10000 (m_{A}=4), N_{X}=10000, N_{Y}=15000, and N=10000
The occurrence of a limit cycle in the space of configuration (
Figure 3. Limit Cycle for the Lotka system obtained from both methods. Similar comments are as in Figure 2
This simple stochastic method describes the Lotka system very well when compared to a deterministic one. The absolute values of the parameters and variables and the oscillations are correctly described. The "dissipative" aspect as shown in the limit cycle is due to the finiteness of the system. That aspect could be interpreted mathematically as a numerical error for a deterministic map approximation.
Other non-linear coupled chemical reactions were studied and were contemplated by using the Brusselator system, which has a trimolecular reaction (or bimolecular autocatalysis). Its chemical reactions are: q = _{2}q= q_{3}_{4} = 1. This system is described by four reactants: two parameters (A and B) with a constant concentration and two variables (X and Y) as chemical intermediaries. These chemical equations are described by the following ODEs: d[X]/dt= A-(B+1) X+X and ^{2}Yd[Y]/dt=BX-X. This system has important differences on stability behavior when in comparison to the Lotka one.^{2}Y^{12}The deterministic solutions B=1.4 and X_{0}=Y_{0}=10000/10000=1 are also shown in Figure 4.
Figure 4. Concentrations X(t) and Y(t) as a function of time for the Brusselator system using q _{1} = q_{2} = q_{3}= q_{4}= 1 from the deterministic method (gray lines) with A=0.5, B=1.4, X_{0}=1, and Y_{0}=1 - and from the stochastic one (black lines) with N_{A}=5000, N_{B}=7000 (mB=2), N_{X}=10000, N_{Y}=10000, and N=10000
The occurrence of a limit cycle in the space of configuration (
Figure 5. Limit Cycle for the Brusselator system from both methods. Similar comments as in Figure 4
The PSM describes the Brusselator system very well when compared to the deterministic method. The absolute values of the parameters and variables and the oscillations are correctly described by the Lotka and Brusselator systems. However, the response of the stochastic method is slow when in comparison to the deterministic one, if the derivatives B=2.4 and X_{0}=Y_{0}=10000/10000=1 for the stochastic method when in comparison to A=0.5, B=2.4, X_{0}=1, Y_{0}=1, and tr=1.15 for the deterministic one. These deterministic and stochastic solutions X(t) and Y(t) of the Brusselator system are shown in Figure 6.
Figure 6. Concentrations X(t) and Y(t) as a function of time for the Brusselator system using q _{1} = q_{2} = q_{3}= q_{4}= 1 from the deterministic method (gray lines) with A=0.5, B=2.4, X_{0}=1, and Y_{0}=1 - and from the stochastic one (black lines), with N_{A}=5000, N_{B}=8000 (m_{B}=3), N_{X}=10000, N_{Y}=10000, and N=10000
A zoom of the concentrations
Figure 7. [X] deterministic and stochastic (red solid lines) and d[X]/dt deterministic (red dashed line); [Y] deterministic and stochastic (blue solid lines) and d[Y]/dt deterministic (dashed blue line) for the Brusselator system. Similar comments as in Figure 6
The stochastic method can show a "slow" response to adjust to the sudden changes of variables, perhaps due to the discreteness of Generic chemical reactions can be described by different techniques such as the ODE approach (an infinitesimal description), or by a stochastic simulation (a discreteness description), which are the first principle procedures using a different nature. One can consider that the "slow" response of the stochastic method (see above), in comparison to the deterministic one, could be a best answer of the real chemical reactions, since the ODE approach is an infinitesimal description of chemical reactions, which essentially have a discrete nature. These two non-linear systems are described correctly by the Practical Stochastic Method, which is simple, robust, auto-sufficient, and independent of a deterministic procedure. This stochastic model can achieve absolute values, which is very important, in order to describe non-linear reactions.
The PSM is an improvement over the simple Ehrenfest Urn model, which is based on the random selection of compartment positions, in order to describe the path of equilibrium in simple chemical reactions. The connection between the stochastic and deterministic methods was established in order to achieve a consistent stochastic one. For that, it was necessary to define two main concepts: a) the filling of the compartment, or by dilution; b) the rate of the reaction enhancement. This simple stochastic method is an alternative to describe finite chemical kinetic reaction systems, where fluctuations are naturally obtained. This stochastic method is robust and is a consistent approach in possible applications for linear and non-linear chemical systems. The "Practical Stochastic Model for Chemical Kinetics" can manage absolute values, which is very important, when describing non-linear reactions.
The financial support given by the Sao Paulo Research Foundation (FAPESP, Fundaçao de Amparo a Pesquisa do Estado de Sao Paulo) Grant 2010/11385-2 and by the CNPq (Brazilian Science Funding Agencies) is fully acknowledged with special thanks.
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