Copyright: 2012 Thomas et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The work was funded by SULSA (Scottish Universities Life Sciences Alliance) by SynthSys Edinburgh The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Current software implementations offer a broad range of stochastic modeling methods. Available packages can be divided into particle based descriptions and population based descriptions. Particle based methods adopt a microscopic approach that describes the movement of each individual reactant (non-solvent) molecule in space and time by means of Brownian dynamics. Popular software packages include the Greens Function Reaction-Diffusion algorithm [4], Smoldyn [5] and MCell [6]. Population based methods adopt a mesoscopic approach that retains the discreteness of reactants but does not need to simulate individual particle trajectory explicitly. This methodology, used by packages such as Smartcell [7] and MesoRD [8], is based on the reaction diffusion master equation [9], [10]. The basic idea is to divide the reaction volume into smaller subvolumes, with reactions proceeding in each subvolume and molecules entering adjacent subvolumes by diffusion. Next one applies the well-mixed assumption to each subvolume (but not to the whole system) which implies that we can ignore the positions and velocities of individual molecules inside each subvolume. The state of the system is then described by the number of molecules of each species in each subvolume, a description which is considerably reduced compared to that offered by particle based methods. This methodology relies on the knowledge of length scales over which the system is said to be spatially homogeneous [2].
A further reduced population description can be achieved by specifying to the situation in which the concentrations of interacting molecules are approximately spatially homogeneous over the entire reaction volume. Reaction kinetics is governed by two timescales: (i) the diffusion timescale, i.e., the time it takes for two molecules to meet each other and (ii) the reaction timescale, i.e., the time it takes for two molecules to react when they are in close proximity to each other. Concentration homogeneity over the entire compartment in which reactions occur, ensues when the reaction timescale is much larger than the diffusion timescale [2]. The large majority of available software packages, deterministic or stochastic, model this situation. Under such well-mixed conditions the Stochastic Simulation Algorithm (SSA) provides an accurate mesoscopic description of stochastic chemical dynamics. The SSA is a Monte Carlo technique by which one can simulate exact sample paths of the stochastic dynamics. The latter has been rigorously derived from microscopic physics by Gillespie for dilute well-mixed gases and solutions [11], [12]. Over the past two decades, the popularization of the algorithm has led to its broad availability in many software packages (see Table 1). However in many situations of practical interest, the application of the SSA is computationally expensive mainly due to the two reasons: (i) whenever the fluctuations are large, e.g., the case of low copy numbers of molecules, a considerably large amount of ensemble averaging of the stochastic trajectories is needed to obtain statistically accurate results. (ii) the SSA simulates each reaction event explicitly which becomes computationally expensive whenever the copy number of at least one molecular species is large [1].
The chemical master equation (CME) is a mathematically equivalent and hence complementary description to the SSA [9]. The CME is a system of linear ordinary differential equations with an unbounded or a typically very large finite state space given by all combinations of copy numbers of the reactant molecules. Hence the advantage of the CME over the SSA is that it does not require any ensemble averaging and is not based on time-consuming simulation of individual reactions. However the CME does not lend itself easily to numerical or analytical computation, the reason being the large dimensionality of its state space. Hence to-date, software packages exploiting the utility of the CME have been scarce (see Table 1). Direct numerical integration of the CME is possible through the finite state projection method [13] which is implemented in the python package CmePy [14]. However, the state space grows exponentially with the number of species and hence these methods have limited applicability in biologically relevant situations. A different type of approach involves the calculation of the moments of the probability distribution solution of the CME by approximate means. Generally there exists an infinite hierarchy of coupled moment equations for reaction networks with bimolecular interactions. In order to make progress, a common method involves the truncation of the hierarchy by means of a moment-closure scheme. The software MomentClosure [15] implements the normal moment-closure approximation for mass action networks by setting all cumulants higher than a desired order to zero. A variety of alternative closures schemes are implemented in the package StochDynTools [16]. The advantage of these approaches is that they generally present quick ways to investigate the effects of noise without the need for averaging over many realizations of the stochastic process. However, these methods are based on ad hoc assumptions for the choice of the closure scheme and hence their accuracy and range of validity is often unknown.
The general formulation of biochemical kinetics considers a number N of distinct chemical species confined in a mesoscopic volume of size under well-mixed conditions. Species interact via R chemical reactions of the type(1)
where j is the reaction index running from 1 to R, Xi denotes chemical species i, kj is the reaction rate of the jth reaction and sij and rij are the stoichiometric coefficients. We associate with each reaction a propensity function such that the probability for the jth reaction to occur in the time interval is given by . The vector denotes a mesoscopic state where ni is the number of molecules of the ith species. Note that our general formulation does not require all reactions to be necessarily elementary, i.e., unimolecular and bimolecular chemical reactions, but can also describe effective reactions. If the jth reaction is elementary then its reaction rate kj is a constant while if it is non-elementary the reaction rate is a function of the instantaneous concentrations, i.e., the elements of the vector .
Over the past two decades the SSA has enjoyed widespread popularity mainly because of the ease by which one can simulate stochastic reaction networks [23], [24]. Given that the system is in state at time t, Gillespie proved using the laws of probability [25] that the probability per unit time for the jth reaction to occur at time is(2)
The SSA generates a stochastic trajectory of the kinetics by sampling a reaction index j and a reaction time according to Eq. (2), followed by an update of the population size for every species i. Note that the net change of the molecule number of species i by reaction j is given by the stoichiometric matrix . Despite its popularity, stochastic simulation has two major shortcomings. Firstly, simulations have to be carried out a significantly large number of times because of the considerable amount of independent realizations needed to obtain accurate statistical averages. Secondly, simulations can become quite slow when the population number of any molecular species is large [1]. Stochastic simulation is a basic component of the software iNA with support for simultaneous simulation of independent realizations using shared memory parallelism of the OpenMP standard [26]. The software features two implementations of the SSA via the direct and the optimized direct method [25], [27]. The output data is presented in terms of mean concentrations, variances and correlations as a function of time which allow for direct statistical interpretation.
An equivalent formulation for the stochastic reaction network described by Eq. (1) is the CME which can be derived from combinatorial arguments [9], [10], [28] and or from microphysics [11], [12]. The CME gives the time-evolution equation for the probability that the system is in a particular mesoscopic state ,(3)
which are exactly the same as those used in deterministic models of biochemical kinetics. Note that is the vector of macroscopic concentrations and is the macroscopic rate function vector, see Methods section. Note that matrices are underlined throughout the article.
The leading order term of the SSE is given by the LNA which has been the key tool in analytical studies of noise [9], [29], [30]. The merit of the method is that it provides a simple means of calculating the fluctuations about the concentration solution of the REs. In particular one is typically interested in the covariance of the time-dependent concentrations(5)where the angled brackets denote the statistical average. Within the LNA the elements of the covariance matrix are determined by the time-dependent equation
(6)Note that the matrix is the Jacobian which gives the extent by which small perturbations of the REs, Eq. (4), decay. The matrix is the diffusion matrix which quantifies the size of the perturbation due to intrinsic noise. Both matrices can be constructed from the stoichiometric coefficients and the macroscopic rate function vector . The diagonal elements of are the variances and hence determine the standard deviation of concentration fluctuations by(7)
Considering higher terms of the expansion one can obtain corrections to the REs which stems from a coupling of the mean concentrations to the higher order moments of the concentration fluctuations. These corrections have been calculated for networks composed of elementary reactions by Grima [31]. The new time-evolution equations which are obtained from this analysis are called effective mesoscopic rate equations (EMREs) and they are here extended for general reaction networks composed of elementary and non-elementary reaction steps (see Methods section). The EMREs are given by(9) 99f5d2560f betlata
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01/16/2026Copyright: 2012 Thomas et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The work was funded by SULSA (Scottish Universities Life Sciences Alliance) by SynthSys Edinburgh The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Current software implementations offer a broad range of stochastic modeling methods. Available packages can be divided into particle based descriptions and population based descriptions. Particle based methods adopt a microscopic approach that describes the movement of each individual reactant (non-solvent) molecule in space and time by means of Brownian dynamics. Popular software packages include the Greens Function Reaction-Diffusion algorithm [4], Smoldyn [5] and MCell [6]. Population based methods adopt a mesoscopic approach that retains the discreteness of reactants but does not need to simulate individual particle trajectory explicitly. This methodology, used by packages such as Smartcell [7] and MesoRD [8], is based on the reaction diffusion master equation [9], [10]. The basic idea is to divide the reaction volume into smaller subvolumes, with reactions proceeding in each subvolume and molecules entering adjacent subvolumes by diffusion. Next one applies the well-mixed assumption to each subvolume (but not to the whole system) which implies that we can ignore the positions and velocities of individual molecules inside each subvolume. The state of the system is then described by the number of molecules of each species in each subvolume, a description which is considerably reduced compared to that offered by particle based methods. This methodology relies on the knowledge of length scales over which the system is said to be spatially homogeneous [2].
A further reduced population description can be achieved by specifying to the situation in which the concentrations of interacting molecules are approximately spatially homogeneous over the entire reaction volume. Reaction kinetics is governed by two timescales: (i) the diffusion timescale, i.e., the time it takes for two molecules to meet each other and (ii) the reaction timescale, i.e., the time it takes for two molecules to react when they are in close proximity to each other. Concentration homogeneity over the entire compartment in which reactions occur, ensues when the reaction timescale is much larger than the diffusion timescale [2]. The large majority of available software packages, deterministic or stochastic, model this situation. Under such well-mixed conditions the Stochastic Simulation Algorithm (SSA) provides an accurate mesoscopic description of stochastic chemical dynamics. The SSA is a Monte Carlo technique by which one can simulate exact sample paths of the stochastic dynamics. The latter has been rigorously derived from microscopic physics by Gillespie for dilute well-mixed gases and solutions [11], [12]. Over the past two decades, the popularization of the algorithm has led to its broad availability in many software packages (see Table 1). However in many situations of practical interest, the application of the SSA is computationally expensive mainly due to the two reasons: (i) whenever the fluctuations are large, e.g., the case of low copy numbers of molecules, a considerably large amount of ensemble averaging of the stochastic trajectories is needed to obtain statistically accurate results. (ii) the SSA simulates each reaction event explicitly which becomes computationally expensive whenever the copy number of at least one molecular species is large [1].
The chemical master equation (CME) is a mathematically equivalent and hence complementary description to the SSA [9]. The CME is a system of linear ordinary differential equations with an unbounded or a typically very large finite state space given by all combinations of copy numbers of the reactant molecules. Hence the advantage of the CME over the SSA is that it does not require any ensemble averaging and is not based on time-consuming simulation of individual reactions. However the CME does not lend itself easily to numerical or analytical computation, the reason being the large dimensionality of its state space. Hence to-date, software packages exploiting the utility of the CME have been scarce (see Table 1). Direct numerical integration of the CME is possible through the finite state projection method [13] which is implemented in the python package CmePy [14]. However, the state space grows exponentially with the number of species and hence these methods have limited applicability in biologically relevant situations. A different type of approach involves the calculation of the moments of the probability distribution solution of the CME by approximate means. Generally there exists an infinite hierarchy of coupled moment equations for reaction networks with bimolecular interactions. In order to make progress, a common method involves the truncation of the hierarchy by means of a moment-closure scheme. The software MomentClosure [15] implements the normal moment-closure approximation for mass action networks by setting all cumulants higher than a desired order to zero. A variety of alternative closures schemes are implemented in the package StochDynTools [16]. The advantage of these approaches is that they generally present quick ways to investigate the effects of noise without the need for averaging over many realizations of the stochastic process. However, these methods are based on ad hoc assumptions for the choice of the closure scheme and hence their accuracy and range of validity is often unknown.
The general formulation of biochemical kinetics considers a number N of distinct chemical species confined in a mesoscopic volume of size under well-mixed conditions. Species interact via R chemical reactions of the type(1)
where j is the reaction index running from 1 to R, Xi denotes chemical species i, kj is the reaction rate of the jth reaction and sij and rij are the stoichiometric coefficients. We associate with each reaction a propensity function such that the probability for the jth reaction to occur in the time interval is given by . The vector denotes a mesoscopic state where ni is the number of molecules of the ith species. Note that our general formulation does not require all reactions to be necessarily elementary, i.e., unimolecular and bimolecular chemical reactions, but can also describe effective reactions. If the jth reaction is elementary then its reaction rate kj is a constant while if it is non-elementary the reaction rate is a function of the instantaneous concentrations, i.e., the elements of the vector .
Over the past two decades the SSA has enjoyed widespread popularity mainly because of the ease by which one can simulate stochastic reaction networks [23], [24]. Given that the system is in state at time t, Gillespie proved using the laws of probability [25] that the probability per unit time for the jth reaction to occur at time is(2)
The SSA generates a stochastic trajectory of the kinetics by sampling a reaction index j and a reaction time according to Eq. (2), followed by an update of the population size for every species i. Note that the net change of the molecule number of species i by reaction j is given by the stoichiometric matrix . Despite its popularity, stochastic simulation has two major shortcomings. Firstly, simulations have to be carried out a significantly large number of times because of the considerable amount of independent realizations needed to obtain accurate statistical averages. Secondly, simulations can become quite slow when the population number of any molecular species is large [1]. Stochastic simulation is a basic component of the software iNA with support for simultaneous simulation of independent realizations using shared memory parallelism of the OpenMP standard [26]. The software features two implementations of the SSA via the direct and the optimized direct method [25], [27]. The output data is presented in terms of mean concentrations, variances and correlations as a function of time which allow for direct statistical interpretation.
An equivalent formulation for the stochastic reaction network described by Eq. (1) is the CME which can be derived from combinatorial arguments [9], [10], [28] and or from microphysics [11], [12]. The CME gives the time-evolution equation for the probability that the system is in a particular mesoscopic state ,(3)
which are exactly the same as those used in deterministic models of biochemical kinetics. Note that is the vector of macroscopic concentrations and is the macroscopic rate function vector, see Methods section. Note that matrices are underlined throughout the article.
The leading order term of the SSE is given by the LNA which has been the key tool in analytical studies of noise [9], [29], [30]. The merit of the method is that it provides a simple means of calculating the fluctuations about the concentration solution of the REs. In particular one is typically interested in the covariance of the time-dependent concentrations(5)where the angled brackets denote the statistical average. Within the LNA the elements of the covariance matrix are determined by the time-dependent equation
(6)Note that the matrix is the Jacobian which gives the extent by which small perturbations of the REs, Eq. (4), decay. The matrix is the diffusion matrix which quantifies the size of the perturbation due to intrinsic noise. Both matrices can be constructed from the stoichiometric coefficients and the macroscopic rate function vector . The diagonal elements of are the variances and hence determine the standard deviation of concentration fluctuations by(7)
Considering higher terms of the expansion one can obtain corrections to the REs which stems from a coupling of the mean concentrations to the higher order moments of the concentration fluctuations. These corrections have been calculated for networks composed of elementary reactions by Grima [31]. The new time-evolution equations which are obtained from this analysis are called effective mesoscopic rate equations (EMREs) and they are here extended for general reaction networks composed of elementary and non-elementary reaction steps (see Methods section). The EMREs are given by(9) 99f5d2560f betlata