 Methodology article
 Open Access
 Published:
Constructing stochastic models from deterministic process equations by propensity adjustment
BMC Systems Biology volume 5, Article number: 187 (2011)
Abstract
Background
Gillespie's stochastic simulation algorithm (SSA) for chemical reactions admits three kinds of elementary processes, namely, mass action reactions of 0^{th}, 1^{st} or 2^{nd} order. All other types of reaction processes, for instance those containing noninteger kinetic orders or following other types of kinetic laws, are assumed to be convertible to one of the three elementary kinds, so that SSA can validly be applied. However, the conversion to elementary reactions is often difficult, if not impossible. Within deterministic contexts, a strategy of model reduction is often used. Such a reduction simplifies the actual system of reactions by merging or approximating intermediate steps and omitting reactants such as transient complexes. It would be valuable to adopt a similar reduction strategy to stochastic modelling. Indeed, efforts have been devoted to manipulating the chemical master equation (CME) in order to achieve a proper propensity function for a reduced stochastic system. However, manipulations of CME are almost always complicated, and successes have been limited to relative simple cases.
Results
We propose a rather general strategy for converting a deterministic process model into a corresponding stochastic model and characterize the mathematical connections between the two. The deterministic framework is assumed to be a generalized mass action system and the stochastic analogue is in the format of the chemical master equation. The analysis identifies situations: where a direct conversion is valid; where internal noise affecting the system needs to be taken into account; and where the propensity function must be mathematically adjusted. The conversion from deterministic to stochastic models is illustrated with several representative examples, including reversible reactions with feedback controls, MichaelisMenten enzyme kinetics, a genetic regulatory motif, and stochastic focusing.
Conclusions
The construction of a stochastic model for a biochemical network requires the utilization of information associated with an equationbased model. The conversion strategy proposed here guides a model design process that ensures a valid transition between deterministic and stochastic models.
Background
Most stochastic models of biochemical reactions are based on the fundamental assumption that no more than one reaction can occur at the exact same time. A consequence of this assumption is that only elementary chemical reactions can be converted directly into stochastic analogues [1]. These include: 1) zeroorder reactions, such as the generation of molecules at a constant rate; 2) firstorder reactions, with examples including elemental chemical reactions as well as transport and decay processes; and 3) secondorder reactions, which include heterogeneous and homogeneous bimolecular reactions (dimerization). Reactions with integer kinetic orders other than 0, 1 and 2 are to be treated as combinations of sequential elementary reactions. The advantage of the premise of nonsimultaneous reaction steps is that the stochastic reaction rate can be calculated from a deterministic, equationbased model with some degree of rigor, even though the derivation is usually not based on first physical principles but instead depends on other assumptions and on macroscopic information, such as a fixed rate constant in the equationbased model. The severe disadvantage is that this rigorous treatment is not practical for modelling larger biochemical reaction systems. The reasons include the following. First, in many cases, elementary reaction rates are not available. Secondly, even in the case that all reaction parameters are available, the computational expense is very significant when the system involves many species and reactions, and this fact ultimately leads to a combinatorial explosion of required computations. Within a deterministic modelling framework, the common practice in this situation is to fit the transient and steadystate experimental data with a phenomenological, (differential) equationbased model, which explicitly or implicitly eliminates or merges some intermediate species and reactions. The bestknown examples are probably MichaelisMenten and Hill rate laws, which are ultimately explicit, but in truth approximate a multivariate system of underlying chemical processes.
Similar model reduction efforts have been carried out for stochastic modelling. For instance, the use of a complexorder function (which corresponds to a reduced equationbased model) was shown to be justified for some types of stochastic simulations. A prominent example is again the MichaelisMenten rate law, which can be reduced from a system of elementary reactions to an explicit function by means of the quasisteadystate assumption (see Result section and [2, 3]). However, model reduction within the stochastic framework has proven to be far more difficult than in the deterministic counterpart. The difficulties are mainly due to the fact that the reduction must be carried out on the chemical master equation (CME). This process is nontrivial and has succeeded only in simple cases.
In general, the construction of a stochastic model for a large biochemical network requires the use of information available from an equationbased model. In the past, several strategies have been proposed for this purpose and within the context of Gillespie's exact stochastic simulation algorithm (SSA; [1]) and its variants [4]. For example, Tian and Burrage [5] proposed that a stochastic model could be directly formulated from the deterministic model through a Poisson leaping procedure. However, a rigorous mathematical justification for such a conversion is lacking. Typical momentbased approaches [6–8] derive ODEs for the statistical moments of the stochastic model from an equationbased model where the 0^{th}, 1^{st} and 2^{nd} order reactions follow mass action rate laws. More recently the moment method was extended to cover models consisting of rational rate laws [9]. Moreover, it was realized that the moment method is complementary to, but cannot fully replace, stochastic simulations, because it does not cover situations like genetic switches [6, 10].
In this article, we explore the mathematical connection between deterministic and stochastic frameworks for the pertinent case of Generalized Mass Action (GMA) systems, which are frequently used in Biochemical Systems Theory (BST; [11–13]). Specifically, we address two questions: First, under what conditions can a deterministic, equationbased model be converted directly into a stochastic simulation model? And second, what is a proper way of implementing this conversion? We will develop a method to answer these questions and demonstrate it for functions in the canonical powerlaw format of GMA systems. However, the results are applicable to other functions and formats as well, as we will demonstrate with several examples.
Representations of systems of biochemical reactions
Consider a wellstirred biochemical reaction system with constant volume and temperature, where N_{ s } different chemical species {\left\{{S}_{s}\right\}}_{s=1}^{{N}_{s}}, interact through N_{ r } unidirectional reaction channels {\left\{{R}_{r}\right\}}_{r=1}^{{N}_{r}}. Each reaction channel can be characterized as
where {\underset{}{v}}_{rs} and {\stackrel{\u0304}{v}}_{rs} are the counts of molecular species S_{ s } consumed and produced due to reaction R_{ r }, respectively, and k_{ r } is the rate constant. The changed amount of {S}_{s}{v}_{rs}={\stackrel{\u0304}{v}}_{rs}{\underset{}{v}}_{rs}, which is due to the firing of reaction R_{ r }, defines the stoichiometric coefficient of S_{ s } in R_{ r }. The stoichiometric coefficients of all species can be summarized according to each reaction R_{ r } in the stoichiometric vector
The stoichiometric vectors of all reactions can further be arranged as the stoichiometric matrix of the system
The size of the system is defined as Φ = AU, where A is the Avogadro number and U is the reaction volume.
The modelling of biochemical reaction networks typically uses one of two conceptual frameworks: deterministic or stochastic. In a deterministic framework, the state of the system is given by the a nonnegative vector \left[\mathbf{X}\left(t\right)\right]={\left[\left[{X}_{1}\left(t\right)\right],\dots ,\left[{X}_{{N}_{s}}\left(t\right)\right]\right]}^{T}\in {\mathbb{R}}^{{N}_{s}}, where component [X_{ s }(t)] represents the concentration of species S_{ s }, measured in moles per unit volume. The temporal evolution of the state of the system is modelled by a set of ordinary differential equations, which in our case are assumed to follow a generalized mass action (GMA) kinetic law. By contrast, in a stochastic framework, the state of the systems is characterized by a vector \mathbf{x}\left(t\right)={\left[{x}_{1}\left(t\right),\dots ,{x}_{{N}_{s}}\left(t\right)\right]}^{T}\in {\mathbb{Z}}^{{N}_{s}}, whose values are nonnegative integers. Specifically, x_{ s }(t) = Φ [X_{ s } (t)] is the count of S_{ s } molecules, which is a sample value of the random variable X_{ s }(t). The system dynamics of this process is typically described with the chemical master equation (CME). Both GMA and CME will be discussed in detail in the following sections.
Motivation for the powerlaw formalism: reactions in crowded media
Powerlaw functions with noninteger kinetics have proven very useful in biochemical systems analysis, and forty years of research have demonstrated their wide applicability (e.g., see [11–13]). Generically, this type of description of a biochemical reaction can be seen either as a Taylor approximation in logarithmic space or as a heuristic or phenomenological model that has been applied successfully hundreds of times and in different contexts, even though it is difficult or impossible in many situations to trace it back to first mechanistic principles. A particularly interesting line of support for the powerlaw format can be seen in the example of a bimolecular reaction occurring in a spatially restricted environment. Savageau demonstrated that the kinetics of such a reaction can be validly formulated as a generalization of the law of mass action, where noninteger kinetic orders are allowed [14, 15]. Neff and colleagues [16–18] showed with careful experiments that this formulation is actually more accurate than alternative approaches.
Within the conceptual framework of powerlaw representations, the rate of the association reaction between molecules of species S_{1} and S_{2} is given as k{\left[{X}_{1}\left(t\right)\right]}^{{f}_{1}}{\left[{X}_{2}\left(t\right)\right]}^{{f}_{2}}. Here, k is the rate constant and f_{1} and f_{2} are realvalued kinetic orders, which are no longer necessarily positive integers as it is assumed in a mass action law. As an example, consider the reversible bimolecular reaction {S}_{1}+{S}_{2}\underset{{k}_{b}}{\overset{{k}_{f}}{\rightleftharpoons}}{S}_{3}. Like Neff and colleagues [17], we begin by formulating a discrete update function for the population of S_{3} molecules as
The first term on the righthand side of this equation, f ([X_{1}], [X_{2}])Δt x_{1}x_{2}, describes the production of S_{3}: it depends on the totality of possible collisions x_{1} x_{2} and also on some fraction f ([X_{1}], [X_{2}])Δt that actually reacts and forms the product. In a dilute environment, f ([X_{1}], [X_{2}]) equals a traditional rate constant, and the reaction obeys the law of mass action, while in a spatially restricted environment, such as the cytoplasm, one needs to take crowding effects into account. As shown in Savageau [14, 15], the desired fraction of a reaction in a crowded environment becomes a rate function that depends on the current concentrations of S_{1} and S_{2}. The second term, g ([X_{3}]) Δtx_{3}, describes the fraction g ([X_{3}]) Δt of species S_{3} that dissociates back into S_{1} and S_{2}. This fraction may depend on some functional form of [X_{3}] because in a crowded environment the complex may not be able to dissociate effectively. Thus, rate constants in the generalized mass action setting become rate functions (cf. [17]).
By taking the limit Δt → 0, one obtains the differential equation
Savageau used Taylor series expansion to approximate the functions f and g in the logarithmic space (log [X_{1}], log[X_{2}]) around some operating point (a, b). The result for f is
where k_{ f }, α, and β are constants related to the chosen operating point (a, b). The final step is achieved by ignoring all higher order terms (HOT) beyond the constant and linear terms. Transformation back to the Cartesian space yields
The same procedure leads to the powerlaw expression for the degradation term: g ([X_{3}]) ≈ k_{ d } [X_{3}]^{γ}. By combining constants we arrive at a powerlaw representation for the dynamics of species S_{3} as
where a = α + 1, b = β + 1, and c = γ + 1. As long as k_{ f }, k_{ d }, a, b and c remain more or less constant throughout a relevant range, the powerlaw model is mathematically well justified. In actual applications, the values of rate constants and kinetic orders can be estimated from experimental data [19]. When the functions f and g are originally not in powerlaw format, they can be locally approximated by powerlaw functions with a procedure similar to the one shown above (Equations (3) to (5)). An illustration will be given in the example section.
The Generalized Mass Action (GMA) format
In the GMA format within Biochemical Systems Theory, each process is represented as a univariate or multivariate powerlaw function. GMA models may be developed de novo or as an approximation of some other nonlinear rate laws. GMA models characterize the time evolution of the system state given that the system was in the state X (t_{0}) at some initial time t_{0}. Generically, the state of the system is changed within a sufficiently small time interval by one out of the N_{ r } possible reactions that can occur in the system. The reaction velocity through reaction channel R_{ r } is:
for those {v}_{rs}={\stackrel{\u0304}{v}}_{rs}{\underset{}{v}}_{rs}\ne 0, s = 1, ..., N_{ s }. As shown in the example of a bimolecular reaction, the kinetic order f_{ rs } associated with species S_{ s } captures the effects of both reactant properties (such as the stoichiometric coefficient v_{ rs }) and environmental influences (such as temperature, pressure, molecular crowding effects, etc.). Therefore f_{ rs } does not necessary equal an integer v_{ rs }, which is assumed to be the case in mass action kinetics, but is possibly realvalued and may be negative. Summing up the contributions of all reactions, one obtains a GMA model describing the dynamics of S_{ s } as
for every s = 1, ..., N_{ s }. Each reaction contributes either a production flux or a degradation flux to the dynamics of a certain species. Positive terms (v_{ rs } > 0) represent the production of S_{ s }, while negative terms (v_{ rs } < 0) describe degradation. If f_{ rs } is positive, then S_{ s } accelerates the reaction R_{ r }; a negative value represents that S_{ s } inhibits the reaction, and f_{ rs } = 0 implies that S_{ s } has no influence on the reaction. The rate constant k_{ r } for reaction R_{ r }, is either positive or zero. Both, the rate constant and the kinetic order, are to be estimated from data.
Proper use of equationbased functions for stochastic simulations
The fundamental concept of a stochastic simulation is the propensity function α(X), and α(X)dt describes the probability that a reaction will change the value of a system variable within the next (infinitesimal) time interval (t, t +dt). While a formal definition will be given later (Equation 18), it is easy to intuit that the propensity function is in some sense analogous to the rate in the corresponding deterministic model. In fact, the propensity function is traditionally assumed to be α(X) = f_{ s }(X), if the deterministic model is X_{ s }' = f_{ s }(X, t), s = 1, ..., N_{ s }. However, a proper justification for this common practice is by and large missing. Indeed, we will show that the direct use of a rate function as the propensity function in a stochastic simulation algorithm requires that at least one of the following assumptions be true:

1)
f is a linear function;

2)
the reaction is monomolecular;

3)
all X_{ i } in the system are noisefree variables, i.e., without (or with ignorable) fluctuations, which implies that the covariance of any two participating reactants is zero (or close to zero).
Each of these assumptions constitutes a sufficient condition for the direct use of a rate function as the propensity function and applies, in principle, to GMA as well as other systems. The validity of these conditions will be discussed later. Specifically, the first condition will be addressed in the Results section under the headings "0^{th}order reaction kinetics" and "1^{st}order reaction kinetics, " while the second condition will be discussed under the heading "Realvalued order monomolecular reaction kinetics." The third condition will be the focus of Equations (2936) and their associated explanations.
In reality, the rates of reactions in biochemical systems are commonly nonlinear functions of the reactant species, and fluctuations within each species are not necessarily ignorable. Therefore, to the valid use of an equationbased model in a stochastic simulation mandates that we know how to define a proper propensity function. The following section addresses this issue. It uses statistical techniques to characterize estimates for both the mean and variance of the propensity function, and these features will allow an assessment of the validity of the assumption α(X) = f_{ s }(X) and prescribe adjustments if the assumption is not valid.
Methods
Deriving the mean and variance of a powerlaw function of random variables
Consider a generic powerlaw function of random variables X_{ s } with the format PL\left(\mathbf{X}\right)=k\prod _{s=1}^{{N}_{s}}{X}_{s}^{{f}_{s}}. Estimates of its mean μ_{ PL } and variance σ_{ PL } are given as
(for details, see Additional file 1). Here,
and μ_{ s } = E[X_{ s }] and {\sigma}_{s}^{2}=\mathsf{\text{E}}\left[{\left({X}_{s}{\mu}_{s}\right)}^{2}\right] are the mean and variance of random variable X_{ s }, respectively. If we choose to express cov [logX_{ i }, logX_{ j }] as a function of μ_{ s }, σ_{ s }^{2} and covariance σ_{ ij } = cov [X_{ i }, X_{ j }], using a Taylor approximation, we obtain
where
Since many biochemical variables approximately follow a lognormal distribution [20–22], it is valuable to consider the special situation where (X_{1}, ..., X_{ s })is lognormally distributed (i.e., (logX_{1}, ..., logX_{ s }) is normally distributed). In such a case, a simpler alternative way to calculate cov [logX_{ i }, logX_{ j }] is
[23]. By substituting this result into (8)(10), one obtains
where
The approximation formulae for μ_{ PL } and σ_{ PL }^{2} in eqns. (8)(10) provide an easy numerical implementation if observation data are available to estimate cov [logX_{ i }, logX_{ j }]. Furthermore, Equations (11)(13) demonstrate how μ_{ PL } and σ_{ PL }^{2} are related to μ_{ s }, σ_{ s }^{2} and σ_{ ij }; however, the price of this insight is paid by the possible inaccuracy introduced through the Taylor approximation. Equations (15)(17) also provide a functional dependence of μ_{ PL } and σ_{ PL }^{2} on (μ_{ s }, σ_{ s }^{2}, σ_{ ij }), but it is only valid if the additional assumption of lognormality is acceptable.
Deriving proper propensity functions for stochastic simulations from differential equationbased models
Assuming that the GMA model faithfully captures the average behaviour of a biochemical reaction system and recalling \left[\mathbf{X}\left(t\right)\right]={\left(\left[{X}_{1}\left(t\right)\right],\dots ,\left[{X}_{{N}_{s}}\left(t\right)\right]\right)}^{T}, the expected metabolite numbers are defined as the expectation
where Φ is the system size as defined above.
To describe the reaction channel R_{ r } stochastically, one needs the state update vector v_{ r }and must characterize the quantity of molecules flowing through of reaction channel R_{ r } during a small time interval. The key concept of this type of description is the propensity function α_{ r }(x), which is defined as
[1]. Because of the probabilistic nature of the propensity function, X(t) is no longer deterministic, and the result is instead stochastic and based on the transition probability
which follows the chemical master equation (CME)
Updating CME requires knowledge of every possible combination of all species counts within the population, which immediately implies that it can be solved analytically for only a few very simple systems and that numerical solutions are usually prohibitively expensive [24]. To address the inherent intractability of CME, Gillespie developed an algorithm, called the Stochastic Simulation Algorithm (SSA), to simulate CME models [1]. SSA is an exact procedure for numerically simulating the time evolution of a wellstirred reaction system. It is rigorously based on the same microphysical premise that underlies CME and gives a more realistic representation of a system's evolution than a deterministic reaction rate equation represented by ODEs. SSA requires knowledge of the propensity function, which however is truly available only for elementary reactions. These reactions include: 1) 0^{th} order reactions, exemplified with the generation of a molecule at a constant rate; 2) 1^{st} order monomolecular reactions, such as an elemental chemical conversion or decay of a single molecule; 3) 2^{nd} order bimolecular reactions, including reactive collisions between two molecules of the same or different species. The reactive collision of more than two molecules at exactly the same time is considered highly unlikely and modelled as two or more sequential bimolecular reactions.
For elementary reactions, the propensity function of reaction R_{ r } is computed as the product of a stochastic rate constant c_{ r } and the number h_{ r } of distinct combinations of reactant molecules, i.e.
Here {h}_{r}\left(\mathbf{x}\right)=\left\{\begin{array}{c}\hfill \prod _{s=1}^{{N}_{s}}\left(\begin{array}{c}\hfill {x}_{s}\hfill \\ \hfill {\underset{}{v}}_{rs}\hfill \end{array}\right)\approx \frac{\prod _{s=1}^{{N}_{s}}{x}_{s}^{{\underset{}{v}}_{rs}}}{\prod _{s=1}^{{N}_{s}}{\underset{}{v}}_{rs}!},\mathsf{\text{for}}\phantom{\rule{2.77695pt}{0ex}}{x}_{s}\ge {\underset{}{v}}_{rs}>0\hfill \\ \hfill 0,\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{otherwise}}\hfill \end{array}\right., where x_{ s } is the sample value of random variable X_{ s }. The approximation is invoked when x_{ s } is large and (x_{ s }  1), ..., (x_{ s }  v_{ rs } + 1) are approximately equal to x_{ s }.
In Gillespie's original formulation [1] c_{ r } is a constant that only depends on the physical properties of the reactant molecules and the temperature of the system, and c_{ r }dt is the probability that a particular combination of reactant molecules will react within the next infinitesimally small time interval (t, t + dt). The constant c_{ r } can be calculated from the corresponding deterministic rate constants, if they are known.
Since the assumption of mass action kinetics is not valid generally, especially in spatially restricted environments and in situations dominated by macromolecular crowding, we address the broader scenario where c_{ r }is not a constant but a function of the reactant concentrations. Thus, we denote c_{ r }as a stochastic rate function, while retaining the definition of h_{ r } as above. Knowing that any positivevalued differentiable function can be approximated locally by a powerlaw function, we assume the functional form of the stochastic rate function as
Here, κ_{ r } and ε_{ rs } are constants that will be specified in the next section, and r = 1, ..., N_{ r }. Note that ε_{ rs } are now realvalued. Once the stochastic rate function is determined (see below), the propensity function can be calculated as
In order to identify the functional expression for a stochastic rate function, and thus the propensity function, we consider the connection between the stochastic and the deterministic equation models. By multiplying CME with x and summing over all x, we obtain
Similarly, the expectation for any species X_{ s }(t) is given as
The details of these derivations are shown in Additional file 1.
We can use these results directly to compute the propensity function for a stochastic GMA model, assuming that its deterministic counterpart is well defined. Specifically, we start with the deterministic GMA equation for X_{ s },
where v_{ rs }, k_{ r } and f_{rs' }are again the stoichiometric coefficients, rate constants, and kinetic orders, respectively. By substituting \left[{X}_{s}\right]=\frac{E\left[{X}_{s}\right]}{\mathrm{\Phi}} from Equation (18) into this GMA model, we obtain a "particlebased" equation of the format
Elementary operations allow us to rewrite this equation as
where {F}_{r}=\sum _{{s}^{\prime}=1}^{{N}_{s}}{f}_{r{s}^{\prime}}. In this formulation, the differential operator is justified only when large numbers of molecules are involved. The assumption that the deterministic equations precisely capture the average behaviour of the biochemical reaction system directly equates the stochastic CME (25) to the deterministic equation based model (28)
Now we have two choices for approximating the expectation of the propensity function on lefthand side of equation (29):

1)
adopt a zerocovariance assumption as was done in [25], which implies ignoring random fluctuations within every species as well as their correlations. This assumption is only justified for some special cases such as monomolecular and bimolecular reactions under the thermodynamic limit (cf. [4, 6]), but is not necessary valid in generality. Here the thermodynamic limit is defined as a finite concentration limit which the system reaches when both population and volume approach infinity. Under this assumption, the left hand side of (29) becomes
\begin{array}{cc}\hfill E\left[{\alpha}_{r}\left(\mathbf{x}\right)\right]& =E\left[\frac{{\kappa}_{r}}{\prod _{s=1}^{{N}_{s}}{\underset{}{v}}_{rs}!}\prod _{s=1}^{{N}_{s}}{x}_{s}^{{\underset{}{v}}_{rs}+{\epsilon}_{rs}}\right]\hfill \\ =\frac{{\kappa}_{r}}{\prod _{s=1}^{{N}_{s}}{\underset{}{v}}_{rs}!}\prod _{s=1}^{{N}_{s}}E{\left[{X}_{s}\right]}^{{\underset{}{v}}_{rs}+{\epsilon}_{rs}}\hfill \end{array}(30)
for every r = 1, ..., N_{ r }, and Equation (24) yields
and
Here, the index r_0 is used to distinguish this 0covariance propensity function from a second type of propensity in the next section.
With the zerocovariance assumption, one can substitute (32) back into the equation for the expectation for each species, which yields
for every s = 1, ..., N_{ s }.. Note that this result is exactly equivalent to the equationbased model (27).
Equation (33) is based on assumption that both the fluctuations within species and their correlations are ignorable, which is not necessarily true in reality. If one uses it in simulations where the assumptions are not satisfied, it is possible that the means for the molecular species are significantly different from the corresponding equationbased model values. This discrepancy arises because the evolution of each species in the stochastic simulation is in truth affected by the covariance which is not necessarily zero, as it was assumed. This phenomenon was observed by Paulsson and collaborators [26] and further discussed in different momentbased approaches [6, 7]. To assess the applicability limit of the propensity defined by (32), we can apply approximation techniques as shown in eqns. (8)(10) on the functional expression of α_{r_0 }and obtain mean and variance as
where
for every s = 1, ..., N_{ s }. These expressions demonstrate that even with large numbers of molecules the mean of CME does not always converge to the GMA model. Indeed, the convergence is only guaranteed in one of the following special situations: 1) the reaction is of 0^{th} order; 2) the reaction is a real valueorder monomolecular reaction, with 1^{st} order reaction as a special case; 3) the covariance contribution in (34) is sufficiently small to be ignored for all participating reactant species of a particular reaction channel. Except for these three special situations, the covariance as shown in (34) significantly affects the mean dynamics. Therefore, stochastic simulations using zerocovariance propensity functions will in general yield means different from what the deterministic GMA model produces. How large these differences are cannot be said in generality. Under the assumption that the GMA model correctly captures the mean dynamics of every species, this conclusion means that α_{r_0 }is not necessarily an accurate propensity function for stochastic simulations, and the direct conversion of the equationbased model into a propensity function must be considered with caution.
Moreover, there is no theoretical basis to assume that there are no fluctuations in the molecular species or that these are independent. Therefore, we need to consider the second treatment of the expectation of the propensity function and study the possible effects of a nonzero covariance.

2)
We again assume that the GMA model is well defined, which implies that information regarding the species correlations and fluctuations has been captured in the parameters of the GMA model on the left hand size of Equations (7) and (28). To gain information regarding correlations, we use Taylor expansion to approximate the propensity function (see Additional file 1 for details):
\begin{array}{c}E\left[{\alpha}_{r}(X(t))\right]=E\left[\frac{{\kappa}_{r}}{{\displaystyle \prod _{s=1}^{{N}_{s}}{\underset{\xaf}{v}}_{rs}}!}{\displaystyle \prod _{s=1}^{{N}_{s}}{X}_{s}^{{\underset{\xaf}{v}}_{rs}}{}^{+{\epsilon}_{rs}}}\right]\\ \approx \frac{{\kappa}_{r}}{{\displaystyle \prod _{s=1}^{{N}_{s}}{\underset{\xaf}{v}}_{rs}!}}{\displaystyle \prod _{s=1}^{{N}_{s}}E{\left[{X}_{s}\right]}^{{\underset{\xaf}{v}}_{rs}+{\epsilon}_{rs}}}\\ \times \mathrm{exp}\left({\displaystyle \sum _{i<j}^{{N}_{s}}\left({\underset{\xaf}{v}}_{ri}+{\epsilon}_{ri}\right)\left({\underset{\xaf}{v}}_{rj}+{\epsilon}_{rj}\right)}\text{}\text{cov}\left[1\text{og}{X}_{i},\mathrm{log}{X}_{j}\right]\right)\end{array}(37)
After substitution of (37) in (29), one obtains
Given the state x of the system at time t, the stochastic rate function of reaction R_{ r } is
Here it is important to understand that although the random variables {X_{ s }}_{s∈S}appear in the expression c_{ r }(x), c_{ r }(x) is not a function of random variables but a deterministic function. The reason is that the cov [logX_{ i }(t), logX_{ j }(t)] in the composition of c_{ r }(x), which as the numerical characteristic of the random variables {X_{ s }}_{s∈S}, is deterministic. Therefore, the stochastic rate function c_{ r }(x) is a welljustified deterministic function that is affected by both the state of the system \left[{x}_{1},\dots ,{x}_{{N}_{s}}\right] and cov [logX_{ i }(t), logX_{ j }(t)], the numerical characteristic of fluctuations in the random variables {X_{ s }}_{s∈S}.
Given the expression c_{ r }(x), the propensity function is
These results are based on the assumption that there are large numbers of molecules for all reactant species participating in reaction R_{ r }. For simplicity of discussion, we define the propensity adjustment factor (paf) of reaction R_{ r } as
paf is a function of time t and represents the contribution of the reactants to correlations among species in the calculation of the propensity function for reaction R_{ r }. We denote the propensity function in (39), which accounts for the contribution of the covariance, as α_{r_cov}, in order to distinguish it from the propensity function α_{r_0 }(32), which is based on the assumption of zerocovariance, i.e.,
Remembering that cov [logX_{ i }(t), logX_{ j }(t)], which is a component in both the stochastic rate function c_{ r }(x) and now in the function paf(t), is a deterministic function rather than a function of random variables, paf(t) is a deterministic correction to the kinetic constant k_{ r } in the construction of α_{r_cov }in (41), which corrects the stochastic simulation toward the correct average.
In contrast to the propensity function α_{r_0}, α_{r_cov }leads to accurate stochastic simulations. To illustrate this difference, we analyze \frac{d}{dt}E\left[{X}_{s}\left(t\right)\right] as follows: We apply the approximation techniques in eqns. (9)(11) in order to obtain the mean and variance of the propensity function α_{r_cov}:
Here
By substituting (42) back into the derivation of CME (26), one obtains
for every s = 1, ..., N_{ s }, which is equivalent in approximation to the GMA model (28). In the other words, the mean of every molecular species obtained by using α_{r_cov }in the CME derived equation (27) is approximately identical to the corresponding macroscopic variable in the GMA model.
Calculation of cov [logX_{ i }(t), logX_{ j }(t)]
When data in the form of multiple time series for all the reactants are available, it is possible to compute cov [logX_{ i }(t), logX_{ j }(t)] directly from these data. Once this covariance is known, the function paf, α_{r_cov }and the mean dynamics can all be assessed. Alas, the availability of several time series data for all reactants under comparable conditions is rare, so that cov [logX_{ i }(t), logX_{ j }(t)] must be estimated in a different manner.
If one can validly assume that the covariance based on α_{r_0 }does not differ significantly from the covariance based on α_{r_cov}, one may calculate cov [logX_{ i }(t), logX_{ j }(t)] by one of following methods.
Method 1:
One uses α_{r_0 }to generate multiple sets of time series data of all reactants and then computes cov [logX_{ i }(t), logX_{ j }(t)].
Method 2:
First, cov [logX_{ i }(t), logX_{ j }(t)] is expressed as a function of mean and covariance in one of the following ways; either as
or as Equation (14):
The first functional expression of cov [logX_{ i }(t), logX_{ j }(t)] is achieved by Taylor approximation, whereas the second expression is obtained by the additional assumption that the concentrations (X_{1}, ..., X_{ s }) are lognormally distributed [8, 23]. The consideration of a lognormal distribution is often justified by the fact that many biochemical data have indeed been observed to be lognormally distributed (e.g., [20–22]).
Second, one uses α_{r_0 }to approximate the mean and covariance either by direct simulation, as shown in method 1, or by a momentbased approach, which is explained in Additional file 2, and which yields the differential equations
For convenience of computational implementation, the above equations can be written in matrix format
Here for r = 1, ..., N_{ r }, and s, m, n = 1, ..., N_{ s }, \mu ={\left({\mu}_{1},\dots ,{\mu}_{{N}_{s}}\right)}^{T}, (V)_{ rs }= v_{ rs }, \alpha ={\left({\alpha}_{1},...,{\alpha}_{{N}_{r}}\right)}^{T}, {\alpha}^{\u2033}={\left({\alpha}_{r}^{{}^{\u2033}},.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}}.\phantom{\rule{2.77695pt}{0ex}},{\alpha}_{{N}_{r}}^{{}^{\u2033}}\right)}^{T}, {\left({\alpha}_{r}^{{}^{\u2033}}\right)}_{mn}=\frac{{\partial}^{2}{\alpha}_{r}\left(\mathbf{X}\right)}{\partial {X}_{m}\partial {X}_{n}}, {\alpha}_{r}^{{}^{\u2033}}\odot \sigma =\sum _{m,n=1}^{{N}_{s}}\frac{{\partial}^{2}{\alpha}_{r}\left(\mathbf{X}\right)}{\partial {X}_{m}\partial {X}_{n}}{}_{\mathbf{X}=\mu}{\sigma}_{mn}, {\alpha}^{{}^{\u2033}}\odot \sigma \triangleq {\left({\alpha}_{1}^{{}^{\u2033}}\odot \sigma ,\dots ,{\alpha}_{{N}_{r}}^{{}^{\u2033}}\odot \sigma \right)}^{T}, {\alpha}^{\prime}=\left({{\alpha}_{1}}^{\prime},\dots ,{{\alpha}_{{N}_{r}}}^{\prime}\right), {{\alpha}_{r}}^{\prime}={\left(\frac{\partial {\alpha}_{r}\left(\mathbf{\mu}\right)}{\partial {X}_{1}},\dots ,\frac{\partial {\alpha}_{r}\left(\mathbf{\mu}\right)}{\partial {X}_{{N}_{s}}}\right)}^{T}, and Λ is a diagonal matrix with {\left(\mathrm{\Lambda}\right)}_{rr}={\alpha}_{r}\left(\mu \right)+\frac{1}{2}\sum _{m,n=1}^{{N}_{s}}\frac{{\partial}^{2}{\alpha}_{r}\left(\mu \right)}{\partial {X}_{m}\partial {X}_{n}}{\sigma}_{mn}.
Statistical criteria for propensity adjustment
Suppose an equationbased model captures the average behavior of a stochastic system and one intends to find the propensity function for a stochastic simulation that will reproduce that means. One can use the 95% confidence interval to evaluate the need for a propensity adjustment. Specifically, for stable systems that will reach a steady state, we use the reversible reaction model as an example. If the steady state of the ODE x_{ st } is within the 95% confidence interval of n runs of stochastic simulations, i.e. {x}_{st}\in \left[{\mu}_{st}1.96\frac{{\delta}_{st}}{\sqrt{n}},{\mu}_{st}+1.96\frac{{\delta}_{st}}{\sqrt{n}}\right], then the rate function in the original ODEs can be used as the propensity without adjustment; otherwise propensity adjustment is needed. Here μ_{ st } and δ_{ st } can be attained from either a momentbase method or from n independent runs of stochastic simulations using propensity without adjustment. An example discussing a reversible reaction with feedback controls can be found in the results section.
For other systems that do not reach a steady state, but where instead transient characteristics are of the highest interest, one can judge the need of propensity adjustment by whether the pertinent characteristics of the ODEs are within the 95% confidence interval of the corresponding characteristic, which is given by a prediction from the momentbased method or from n runs of stochastic simulations. The Repressilator example in the result section will serve as a demonstration.
Results
Generic special cases
It is generally not valid to translate a rate from a deterministic biochemical model into a propensity function of the corresponding stochastic simulation without adjustment (see Equations. (34)(36)). However, in some situations, the propensity adjustment (e.g., Equations (40)(44)) is not needed, and in some other cases it becomes relatively simple.

1)
0^{th}order reaction kinetics
Consider a very simple equationbased model of the type
for all s = 1, ..., N_{ s }, f_{ rs } = 0. According to Equations (40)(44), one obtains
Thus, for a 0^{th}order reaction, its rate equation can be taken directly as the propensity function in stochastic simulations.

2)
1^{st}order reaction kinetics
Direct application of Equations (40)(44) yields
f_{ rs } = δ_{ sj }, i, j = 1, ..., N_{ s }. Therefore, according to Equations (40)(44)
Thus, for 1^{st}order reactions, the rate equation can again be taken directly as the propensity function in stochastic simulations.

3)
Realvalued order monomolecular reaction kinetics
Consider a reaction with kinetics of the type
f_{ rj } ≠ 0, f_{ rs } = 0, for any s ≠ j, s = 1, ..., N_{ s }. Equations (40)(44) lead to
Thus, for reaction kinetics involving a single variable and a realvalued order, the rate equation can again be taken as the propensity function in stochastic simulations.

4)
2^{nd}order reaction kinetics
This type of reaction can be expressed as
i, j ∈ {1, ..., N_{ s }}, i ≠ j, f_{ ri } = f_{ rj } = 1, and f_{ rs } = 0, for all s ≠ i, j. Therefore, according to Equations (40)(44)
Thus, the proper propensity function for 2^{nd}order reactions is different from the rate equation. The difference can be ignored only if the contribution from the covariance is insignificant. In general, the rate equation yields only an approximate propensity function for stochastic simulations, and the approximation quality must be assessed on a casebycase basis.

5)
Bimolecular reaction with realvalued order kinetics
This type of reaction can be formulated as
i, j ∈ {1, ..., N_{ s }}, i ≠ j, f_{ ri }, f_{ rj } ≠ 0, and f_{ rs } = 0, for all s ≠ i, j. According to Equations (40)(44) we obtain
For bimolecular reactions of complex order, the propensity function is different from the rate equation. The difference can be ignored only if the contribution from the covariance is insignificant.
Powerlaw representation of a reversible reaction with feedback controls
We consider a reversible reaction with feedback controls (see Figure 1) whose average behaviour is accurately described by the following GMA model
Here S_{3} feeds back to inhibit the forward reaction and S_{1} feeds back on the reverse reaction and accelerates it. The task is to develop a stochastic model whose performance converges to that of the deterministic GMA model. We can see from equations (52) that three variables x_{1}, x_{2} and x_{3} contribute to the forward flux {k}_{f}{\mathrm{\Phi}}^{1{f}_{1}{f}_{2}{f}_{3}}{x}_{1}^{{f}_{1}}{x}_{2}^{{f}_{2}}{x}_{3}^{{f}_{3}} and two variables x_{1} and x_{3} contribute to the backward flux {k}_{b}{\mathrm{\Phi}}^{1{g}_{1}{g}_{3}}{x}_{1}^{{g}_{1}}{x}_{3}^{{g}_{3}}. Because several variables are involved, their covariance has the potential of affecting the forward and the backward propensity functions in a stochastic simulation. To obtain the covariance information, we formulate the moment equations (53) from the ODE model (52).
To simplify the calculation, as explained in detail in Additional file 2, we set the third central moment to zero and obtain a closedform set of ODEs. Expressed differently, the rate of change in mean and covariance depends only on the functions of mean and covariance themselves, but not on higherorder moments. Thus,
Here μ = (μ_{1}, μ_{2}, μ_{3})^{T}, V=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right], \alpha ={\left({\alpha}_{1},{\alpha}_{2}\right)}^{T}=\left[\begin{array}{c}\hfill {k}_{f}{\mathrm{\Phi}}^{1{f}_{1}{f}_{2}{f}_{3}}{x}_{1}^{{f}_{1}}{x}_{2}^{{f}_{2}}{x}_{3}^{{f}_{3}}\hfill \\ \hfill {k}_{b}{\mathrm{\Phi}}^{1{g}_{1}{g}_{3}}{x}_{1}^{{g}_{1}}{x}_{3}^{{g}_{3}}\hfill \end{array}\right].
Moreover, for r = 1, 2 and m, n = 1, 2, 3, {\left({\alpha}_{r}^{{}^{\u2033}}\right)}_{mn}=\frac{{\partial}^{2}{\alpha}_{r}\left(\mathbf{X}\right)}{\partial {x}_{m}\partial {x}_{n}}, α" = (α_{1}", α_{2}")^{T}, \sigma =\left[\begin{array}{ccc}\hfill {\sigma}_{11}\hfill & \hfill {\sigma}_{12}\hfill & \hfill {\sigma}_{13}\hfill \\ \hfill {\sigma}_{21}\hfill & \hfill {\sigma}_{22}\hfill & \hfill {\sigma}_{23}\hfill \\ \hfill {\sigma}_{31}\hfill & \hfill {\sigma}_{32}\hfill & \hfill {\sigma}_{33}\hfill \end{array}\right], {\alpha}_{r}^{{}^{\u2033}}\odot \sigma \triangleq \sum _{m,n=1}^{3}\frac{{\partial}^{2}{\alpha}_{r}\left(\mathbf{X}\right)}{\partial {x}_{m}\partial {x}_{n}}{}_{\mathbf{X}=\mu}{\sigma}_{mn}, α"⊙σ ≜ (α_{1}"⊙ σ, α_{2}"⊙ σ)^{T}, α' = (α_{1}', α_{2}'), {{\alpha}_{r}}^{\prime}={\left(\frac{\partial {\alpha}_{r}\left(\mu \right)}{\partial {x}_{1}},\frac{\partial {\alpha}_{r}\left(\mu \right)}{\partial {x}_{2}},\frac{\partial {\alpha}_{r}\left(\mu \right)}{\partial {x}_{3}}\right)}^{T}, and \mathrm{\Lambda}=\left[\begin{array}{cc}\hfill {\alpha}_{1}\left(\mu \right)+\frac{1}{2}\sum _{m,n=1}^{3}\frac{{\partial}^{2}{\alpha}_{1}\left(\mu \right)}{\partial {x}_{m}\partial {x}_{n}}{\sigma}_{mn}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\alpha}_{2}\left(\mu \right)+\frac{1}{2}\sum _{m,n=1}^{3}\frac{{\partial}^{2}{\alpha}_{2}\left(\mu \right)}{\partial {x}_{m}\partial {x}_{n}}{\sigma}_{mn}\hfill \end{array}\right].
Two initial conditions are chosen for representative simulations; they differ by a factor of 20 in species populations and reaction volume between the upper and lower panels of Figure 2. The purpose is to observe the thermodynamic limit of the systems: both scenarios have the same initial concentrations, but the system in the lower panel case has a larger species populations and reaction volume and can thus be regarded as the thermodynamic limit sample of system in the upper panel. As demonstrated by the figures in the first column, the moment approach predicts that for both population sizes the average trajectories of the stochastic model (without propensity adjustment) dynamics is lower than that of the equationbased model: the differences are about 10% of the steadystate value of the equationbased model in the upper figure and 1% in the lower figure; for 100 runs of the stochastic simulation, the steadystate value of the equationbased model lies outside the 95% confidence interval in the upper figure, while it is inside the interval in the lower figure. Therefore, we can expect that the propensity adjustment will significantly contribute to the stochastic simulation for the upper case while not for the lower case. This expectation is confirmed by the simulation results in the third and fourth columns. With the common assumption that the deterministic equations precisely capture the system's average behaviour, the case in the upper panel represents the situation where propensity adjustment is needed, while the lower panel represents the situation that a propensity without adjustment is sufficient when the system approaches its thermodynamics limit. This example furthermore demonstrates that either the moment approach or the stochastic simulations without propensity adjustment can be used to estimate whether there is a need to construct a propensity adjustment function for stochastic simulations.
Repressilator
Interestingly, a propensity function may even be obtained through powerlaw approximation of some function that describes complex transient behaviours of a reaction network. As an example, consider the socalled Repressilator [27], which is a threecomponent genetic circuit where each component represses its downstream neighbour. More specifically (as shown in Figure 3), gene G_{1} codes for protein x_{1}, whose dimer y_{1} subsequently represses the transcription of the gene G_{2}. Similarly, y_{2}, the dimer of gene G_{2}'s protein product x_{2}, represses the transcription of gene G_{3}, and y_{3}, the dimer of gene G_{3}'s protein product x_{3}, represses the transcription of gene G_{1}. The corresponding differential equation model following mass action kinetics is given by [28]
where i = 1, 2, 3; j = 2, 3, 1; k = 3, 1, 2; the rate constants are explained in the diagram below
Assuming that the reversible dimerization and the dissociation/association of a protein dimer from/to the promoter are much faster than other processes, the full systems can be reduced to
[28]. Here Φ = 1, p\left({x}_{i}\right)=1+4{c}_{p}{x}_{i}+\frac{4{c}_{d}{c}_{p}d{x}_{i}}{{\left(1+{c}_{d}{c}_{p}{x}_{i}^{2}\right)}^{2}}, c_{ p } = κ_{+}/κ_{}, c_{ d } = k_{+}/k_{} and d = d_{0, i}+ d_{ r, i } for i = 1, 2, 3. It has been shown that the simplified ODEs rather accurately approximate the transient dynamics of the full system by retaining the original oscillation period and amplitude.
In [28], the system (55) is further rescaled by setting \stackrel{\u0303}{t}={\gamma}_{m}t,\phantom{\rule{2.77695pt}{0ex}}{\stackrel{\u0303}{x}}_{i}=\sqrt{{c}_{d}{c}_{p}}{x}_{i} and \stackrel{\u0303}{{m}_{i}}=\left(\sigma \sqrt{{c}_{d}{c}_{p}}{m}_{i}\right)/\left({\gamma}_{m}\beta \right), which yields
Intriguingly, one makes the following observation. The scaled ODE system (56) is consistent with the original system (55) in oscillation amplitude and period. However, its corresponding stochastic model produces results that deviate substantially from the average responses. To see the effects of the transition from a deterministic to a stochastic model, we apply SSA to the scaled system (56). The main result is that the oscillation periods of both x_{ i } and m_{ i } are reduced to half (Figure 4). The reason is that, in the stochastic simulation, the oscillation period is very sensitive to the ratio of x_{ i } and m_{ i }, which has been altered by the scaling operation. Therefore, in general one needs to pay attention to how scaling may affect the stochastic performance when the model is generated through the conversion of an ODE model.
We can see from equations (55) that two variables x_{ i } and m_{ i } contribute to the production of x_{ i }; hence, their covariance could affect the propensity function of x_{ i } in the production reaction of a stochastic simulation. Similar to the example of a reversible reaction (Equation 52), it is therefore necessary to evaluate covariance effects and to judge whether the propensity function needs adjusting. Thus, we need to compare the difference between the dynamics of the phenomenological model (55) and the dynamics under the influence of covariance, which can be produced by either stochastic simulation or the moment approach.
The influence of the covariance on the dynamics of the stochastic simulation is relatively easy to assess: we simply use the terms on the righthand side of the differential equations (54) as the propensity functions in SSA and obtain simulation results shown in the 2^{nd} and the 4^{th} panels of Figure 5. Obtaining the covarianceinfluenced dynamics with the momentbased approach is more complicated, and we need to discuss some implementation issues.
First, the momentbased approach requires information regarding the first and the second derivatives of p(x_{ i })^{1}, which have rather complicated functional forms. To simplify the calculation, we replace the function p(x_{ i })^{1} with an approximating powerlaw function. Specifically, suppose the original parameter values are κ_{+} = k_{+} = 5, κ_{} = k_{} = 100 and d = 20. Plotting the data (x_{ i }, p(x_{ i })^{1})in loglog space (Figure 5) indicates that the original function is represented well by a straight line:
for x_{ i } ∈ [30, 300]. In Cartesian space, this line corresponds to the powerlaw function
which models the original function very well (see Figure 5). For x_{ i } ∈ [1, 30], this powerlaw function does not fit the original function precisely; the effect of this imprecision can be evaluated later at after we use this powerlaw function in the momentbased method. Moreover, using the truncated moment equations to estimate the mean and variance involves multiple approximations: First, the function p (x_{ i })^{1} on the righthand side of (55) is replaced by a powerlaw function (see Figure 5). Second, the result is approximated by Taylor expansion to the second order. Third, similar to the example of a reversible reaction, the central moment of the third degree is assumed to be zero, which leads to a closedform ODE for the first two moments.
Solving the technical issues as described, one obtains the corresponding momentbased model of (55) (not shown) with results shown in Figure 6. Suppose one is particularly interested in the period and amplitude of the oscillation within a time interval between 0 and 400 seconds. As shown in Figure 6, the GMA approximation (black dashed line) fits the original ODEs (55) (bold black solid line) very well at the beginning, but as time goes on, the approximation error accumulates. As seen in the time interval [350, 400], the GMA approximation deviates from the original ODEs significantly. However, this does not mean that the GMA approximation cannot be used as a propensity function for stochastic simulations; the momentbased method with the GMA approximation shows that, when the GMA approximation is used as propensity function (without adjustment) for stochastic simulations, the resulting mean (red solid line) consistently fits the trajectory of the original ODEs (bold solid black line) very well up to about t = 400 seconds. The oscillation period and amplitude in the stochastic simulation based on the GMA approximation (without adjustment) are almost identical to those of the original ODEs. Therefore, a propensity adjustment for the GMA approximation is not needed, and the GMA approximation can be used as a propensity function for stochastic simulations. In other words, a stochastic model for the Repressilator system can be generated by using the scheme in (32) without propensity adjustment. Moreover, the imprecision caused by the powerlaw approximation can be tolerated when its corresponding momentbased mean matches the original ODEs sufficiently well with respect to the features of highest interest.
Enzymatic reaction using a quasisteady state assumption (QSSA)
We consider an enzymatic reaction following the MichaelisMenten mechanism:
Here enzyme E reacts with substrate S through a reversible reaction to form complex ES, which can proceed to yield product P and to release the enzyme E. By assuming the law of mass action for the reaction kinetics we obtain a set of differential equations for the system dynamics:
where the total amount of enzyme in the form of free enzyme and complex [E]_{0} ≜ [E] + [ES] is assumed to be constant. In addiction, by making the socalled quasi steady state assumption (QSSA) [29, 30], assuming that the complex ES is essentially in steady state, we can assert \frac{d\left[ES\right]}{dt}\approx 0. As it has been discussed many times in the literature, QSSA reduces the system and leads to the approximate form
which is known as MichaelisMenten kinetics [30]. The characterizing parameters are V_{max} = k_{2}[E]_{0} and K_{ m } = (k_{1} + k_{2})/k_{1}.
Applying QSSA, Rao and Arkin [2] were able to reduce the CME of S and ES to a CME only containing S. For the reduced CME, the propensity function for the overall reaction S → P is
where the volume was scaled so that Φ = 1 and the lowercase letter s denotes the molecule count of species S. Instead of reviewing the relatively complicated manipulations with CME, we show in the following that the techniques described above lead directly from the equationbased model to the propensity function for the reduced system.
First, we recast the equationbased model into the GMA format [31], by introducing an auxiliary variable [T] ≜ K_{ m } + [S]. The result,
is exactly equivalent to the reduced system in (58) with the initial condition [S]_{0} and [T]_{0} = K_{ m } + [S]_{0}. The corresponding stochastic model has only one reaction channel and the propensity function is
The propensity adjustment factor can be set to 1 because T is a function of s and its covariance with s is therefore 1. By applying t = K_{ m } + s, the propensity function can be simplified as
Thus, we arrive at the propensity function for the reduced system, which is identical to the result of Rao and Arkin obtained through manipulations of CME.
In the above derivation, we used the simplest type of recasting, where a new, auxiliary variable simply consists of an old variable plus a constant. This reformulation of the MichaelisMenten process as a pair of GMA equations is a special case of a much more general recasting technique that permits the equivalent conversion of any system of ordinary differential equations into a powerlaw format [31]. However, this equivalence transformation imposes constraints on the variables of the GMA equations, and it is at this point unclear whether there are mathematical warranties ensuring that the proposed transition from differential to stochastic equations in general preserves these constraints in all cases. This question will require further investigation.
Stochastic Focusing
Stochastic focusing [26] describes the phenomenon that the fluctuations of a chemical species can drive the system to reach a different steady state than what a deterministic ODE model predicts. To demonstrate the utility of propensity adjustment, we derive a stochastic model which produces consistent results with those of the deterministic model.
Following [32], we consider the following reactions system
This system can be interpreted as follows: the intermediate species I is produced at constant rate k_{1} from some source Φ and degrades with rate k_{4} through the catalysis with signalling molecule S; the end product P is converted from species I at rate k_{2} and degrades at rate k_{3}; the signalling molecule S is produced and degrades at rates k_{5} and k_{6}, respectively. Moreover, the value of k_{5} is reduced to half at a certain time point to achieve a significant divergence effect. In order to capture the average dynamics of the system accurately, we use a powerlaw model in GMA format instead of the mass action rate law in [32].
The system size is set to 1. We can see from equations (64) that two variables i and s contribute to the degradation of I and that their covariance could therefore affect the propensity function of I in the degradation reaction of a stochastic simulation. To calculate the propensity adjustment function paf_{4}(t) = exp (f_{ I } f_{ S } cov [log I(t), log S(t)]) for reaction {R}_{4}:I+S\underset{}{\overset{{k}_{4}}{\to}}S, we formulate equations (cf. (60)) for the moments as
Here μ = (μ_{ I }, μ_{ P }, μ_{ S })^{T}, V=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right], \alpha =\left[\begin{array}{c}\hfill {\alpha}_{1}\hfill \\ \hfill {\alpha}_{2}\hfill \\ \hfill {\alpha}_{3}\hfill \\ \hfill {\alpha}_{4}\hfill \\ \hfill {\alpha}_{5}\hfill \\ \hfill {\alpha}_{6}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {k}_{1}\hfill \\ \hfill {k}_{2}i\hfill \\ \hfill {k}_{3}p\hfill \\ \hfill {k}_{4}{i}^{{f}_{I}}{s}^{{f}_{S}}\hfill \\ \hfill {k}_{5}\hfill \\ \hfill {k}_{6}s\hfill \end{array}\right].
Moreover, for r = 1, ..., 6 and m, n = i, p, s, {\left({\alpha}_{r}^{{}^{\u2033}}\right)}_{mn}=\frac{{\partial}^{2}{\alpha}_{r}}{\partial m\partial n}, α" = (α_{1}", ..., α_{6}")^{T}, \sigma =\left[\begin{array}{ccc}\hfill {\sigma}_{11}\hfill & \hfill {\sigma}_{12}\hfill & \hfill {\sigma}_{13}\hfill \\ \hfill {\sigma}_{21}\hfill & \hfill {\sigma}_{22}\hfill & \hfill {\sigma}_{23}\hfill \\ \hfill {\sigma}_{31}\hfill & \hfill {\sigma}_{32}\hfill & \hfill {\sigma}_{33}\hfill \end{array}\right], {\alpha}_{r}^{{}^{\u2033}}\odot \sigma =\sum _{m,n=i,p,s}\frac{{\partial}^{2}{\alpha}_{r}\left(\mathbf{\mu}\right)}{\partial m\partial n}{\sigma}_{mn}, α"⊙ σ ≜ (α_{1} "⊙ σ, ..., α_{6}"⊙ σ)^{T}, α' = (α_{1}', ..., α_{6}'), {{\alpha}_{r}}^{\prime}={\left(\frac{\partial {\alpha}_{r}}{\partial i},\frac{\partial {\alpha}_{r}}{\partial p},\frac{\partial {\alpha}_{r}}{\partial s}\right)}^{T}, and the diagonal matrix Λ is defined by {\left(\mathrm{\Lambda}\right)}_{rr}={\alpha}_{r}+\frac{1}{2}\sum _{m,n=i,p,s}\frac{{\partial}^{2}{\alpha}_{r}}{\partial m\partial n}{\sigma}_{mn}.
The stochastic focusing model without propensity adjustment yields results quite different from those of the deterministic model, as is illustrated in Figure 7. In this figure, the blue lines in the 1^{st} panel are predicted from the moment equations (65) and the blue error bars for μ_{ P } in the 2^{nd} panel are obtained from ten independent stochastic simulations. Both diverge systematically from the black line predicted by ODE model (64). By contrast, the stochastic model with propensity adjustment produces results consistent with the deterministic model, as shown by the 4^{th} panel.
Discussion
It is often implicitly assumed that the rate of a dynamic process can be directly taken as the propensity for a corresponding stochastic process. We have shown here that this is sometimes, but not always, true. Our results fall into three categories. The first develops conditions for a valid conversion of a rate to a propensity, the second presents a general conversion procedure, and the third discusses computational issues of propensity adjustment.
Conditions for the direct use of a rate constant (function) as propensity function
We have shown that the direct use of a rate constant or a rate function f as the propensity function in a stochastic simulation algorithm requires that at least one of the following assumptions be true:

1)
f is a linear function; this assumption has been validated in the Results sections addressing 0^{th}order and 1^{st}order reaction kinetics.

2)
the reaction is monomolecular; this assumptions was evaluated in the Results section describing realvalued order monomolecular reaction kinetics.

3)
all X_{ i } in the system are noisefree variables, i.e., without (or with ignorable) fluctuations; this assumption implies that the covariance of any two participating reactants is zero (or close to zero). This assumption is assessed in equations (29  36).
Each of these three conditions is a sufficient condition for the direct use of a rate function f as the propensity function. Moreover, these statements are valid for functions of a general format, not just for GMA. This is so because the functional formats in cases 1 and 2 above are special cases of the GMA format. For the third case, a formal proof is only given for functions in GMA format, because this structured format allows us to give an explicit estimation on how the covariance can affect the average behavior of a stochastic simulation through equation (34). For functions not in GMA format, the conclusion is still holds, although an analogous explicit estimation is lacking. The argument is as follows. The bimolecular reaction E[α_{ r }(X(t))] contains at least one quadratic moment of the form E[X_{ i }(t)X_{ j }(t)] (cf. [4] and page 38). Therefore, by definition of the covariance, E[X_{ i }(t)X_{ j }(t)] = E[X_{ i }(t)]E[X_{ i }(t)] + cov (X_{ i }(t), X_{ j }(t)), we obtain