An accelerated algorithm for discrete stochastic simulation of reaction–diffusion systems using gradient-based diffusion and tau-leaping

Abstract

Stochastic simulation of reaction–diffusion systems enables the investigation of stochastic events arising from the small numbers and heterogeneous distribution of molecular species in biological cells. Stochastic variations in intracellular microdomains and in diffusional gradients play a significant part in the spatiotemporal activity and behavior of cells. Although an exact stochastic simulation that simulates every individual reaction and diffusion event gives a most accurate trajectory of the system's state over time, it can be too slow for many practical applications. We present an accelerated algorithm for discrete stochastic simulation of reaction–diffusion systems designed to improve the speed of simulation by reducing the number of time-steps required to complete a simulation run. This method is unique in that it employs two strategies that have not been incorporated in existing spatial stochastic simulation algorithms. First, diffusive transfers between neighboring subvolumes are based on concentration gradients. This treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method that was originally developed for speeding up nonspatial or homogeneous stochastic simulation algorithms. This method calculates each leap time in a unified step for both reaction and diffusion processes while satisfying the leap condition that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Numerical results are presented that illustrate the improvement in simulation speed achieved by incorporating these two new strategies.

INTRODUCTION

Reaction–diffusion systems, involving a set of molecular species within a given space domain, describe changes in the molecular populations produced over time by chemical reactions and diffusion. Chemical reactions coupled with molecular transport have been proposed to play an important role in many cellular processes^{1, 2} that include, but are not limited to morphogenesis,³ cell polarity,⁴ self-organization,^{5, 6} self-regulation,⁴ and cell division.⁷ In addition, cell signaling is critically influenced by both reaction and diffusion processes because reactants involved in a cell's signaling pathways are heterogeneously distributed throughout the cell.⁸ For example, depolarization of a neuronal dendrite or spine may cause calcium influx into that dendrite or spine, but not other dendrites or spines;^{9, 10, 11, 12} this heterogeneous distribution produces diffusional gradients that influence the degree of localization (e.g., Ras diffusion¹³ and synapse specificity¹⁴).

A classical or deterministic formulation of such reaction–diffusion systems represents the molecular populations by continuously varying and spatially dependent molecular concentrations, and describes the time evolution of those concentrations by a set of partial differential equations. Although this traditional formulation is valid in the thermodynamic limit of large numbers of molecules, many processes occurring in biological cells contain only a small and discrete number of molecular species. When the molecular population numbers are small, stochastic effects can have a large impact on the behavior of the biological system such as in gene expression^{15, 16, 17, 18} and transcriptional noise.^{19, 20, 21} To account for the stochastic effects resulting from the small and discrete number of molecular species in small systems, McQuarrie²² suggested formulating chemical kinetics in a probabilistic or stochastic framework.

The stochastic simulation algorithm (SSA), formulated by Gillespie,^{23, 24} numerically simulates the time evolution of a well-stirred chemically reacting system in a thermal equilibrium by defining the state of the system to be the integer numbers of molecular populations, which is changed by discrete amounts upon the occurrence of a chemical reaction. The SSA is exact in that it generates a trajectory of the corresponding chemical master equation (CME), which describes the time evolution of the probability density functions of molecular populations but cannot be solved analytically in most cases due to the high dimensionality of the state space.^{22, 25}

Methods for stochastic simulation of reaction–diffusion systems include Brownian dynamics formulation and spatial extension of the SSA. In the Brownian dynamics formulation of stochastic chemical kinetics, Brownian random walks of individual molecules are tracked and reactions are simulated when reactant molecules collide.^{26, 27, 28, 29, 30} The Brownian dynamics approach of tracking individual molecules may be the most accurate way to depict a system of chemically reacting and diffusing molecules at a microscopic scale, but it may become computationally intractable when simulating a system with more than 10⁶ molecules.^{27, 31, 32} This difficulty may be mitigated by the first-passage kinetic Monte Carlo method³³ that uses exact solutions for the first-passage statistics of Brownian random walks to bring colliding molecules together more efficiently; however, its numerical efficiency decreases when the density of walkers increases.^{33, 34} The SSA has been extended to incorporate diffusion by dividing the spatial domain into a collection of well-stirred subvolumes and treating diffusion jumps between neighboring subvolumes as a set of first-order reactions.^{35, 36, 37} Analogous to the CME, the reaction–diffusion master equation (RDME)^{38, 39, 40} then describes the time evolution of the probability of the system being in a given state.^{35, 36, 37} The size of the subvolumes should be small enough to capture the spatial variation due to diffusion with a desired resolution and, at the same time, large enough to satisfy the assumption that each subvolume is well-stirred.^{41, 42, 43, 44} Isaacson et al.^{42, 43} have shown that when these lower and upper bounds on the subvolume size are satisfied, the RDME may be interpreted as an asymptotic approximation to Brownian dynamics type models. Alternatively, to circumvent the error introduced by the subvolume size being too small, modified formulas of reaction rates that depend on the subvolume size have been proposed to yield more accurate results.^{41, 44}

Although faster than Brownian dynamics simulation algorithms, the SSA for reaction–diffusion systems can be slow because it simulates every single reaction or diffusion event. Different methods have been proposed to speed up the SSA for well-stirred chemically reacting systems that do not include diffusion, and these fall into two categories. The first category of methods reduces the time that the SSA spends in each time-step of a simulation.^{45, 46, 47, 48, 49} The second category of methods reduces the number of time-steps needed to complete a simulation by employing approximate simulation strategies that sacrifice some of the exactness of the SSA.^{50, 51, 52, 53, 54, 55, 56, 57, 58, 59} The methods in the first category are different but equivalent formulations of the SSA and therefore exact, whereas those in the second category mix approximate and exact strategies in an attempt to optimize both accuracy and speed. Previous efforts to speed up the SSA for reaction–diffusion systems include deterministic–stochastic hybrid methods^{60, 61, 62} in which diffusion is treated deterministically everywhere and reactions are treated stochastically, and stochastic–stochastic hybrid methods^{62, 63, 64, 65} in which reaction and diffusion processes are executed independently of each other by alternating the execution of each process at predetermined intervals of time.

In this paper, we present a new simulation algorithm that improves the speed of the SSA for reaction–diffusion systems by reducing the number of time-steps needed to complete a simulation run. Our method is unique in that it employs two strategies that have not been incorporated in existing SSAs for reaction–diffusion systems. First, we treat diffusive transfers between neighboring subvolumes based on concentration gradients. Our treatment necessitates sampling of only the net or observed diffusion events from higher to lower concentration gradients rather than sampling all diffusion events regardless of local concentration gradients. Second, we extend the non-negative Poisson tau-leaping method⁶⁶ that was developed to accelerate nonspatial or homogeneous SSAs. As in homogeneous tau-leaping,⁶⁶ the difficulty in spatial tau-leaping is in choosing a time-step that satisfies the leap condition⁵³ that the propensities do not change appreciably during the leap and ensuring that leaping does not cause molecular populations to become negative. Our method calculates each leap time in a unified step for both reaction and diffusion processes that underlie the corresponding RDME, and therefore is distinct from previous efforts to speed up spatial SSAs that treat two processes separately.^{60, 61, 62, 63, 64, 65}

In the subsequent sections, we first explain relevant previous work that includes Gillespie's direct method to stochastically simulate reactions in a well-stirred volume in thermal equilibrium²³ and its extension to simulate reaction–diffusion systems,^{35, 37, 67, 68} and the non-negative Poisson tau-leaping strategy for well-stirred, chemically reacting systems.⁵¹ We then present our gradient-based diffusion and spatial tau-leaping strategies, and describe how to incorporate them into spatial SSAs. We show numerical results from several test cases that illustrate improved performance of our algorithm in reducing the number of time-steps required to complete a simulation run while maintaining accuracy. We conclude with discussion and future work.

PREVIOUS WORK

We briefly describe Gillespie's direct method²³ and its extension to simulate reaction–diffusion systems,^{35, 37, 67, 68} and the non-negative Poisson tau-leaping strategy by Cao et al.^{51, 66} on which our algorithm in Sec. 3 is based.

Gillespie's direct method to simulate reactions in a well-stirred volume in thermal equilibrium

Gillespie's direct method²³ assumes a well-stirred volume Ω in thermal equilibrium that contains N molecular species {S₁, …, S_N}. The state vector X(t) = (X₁(t), …, X_N(t)) denotes the number of molecules of S_i (i = 1, 2, …, N) in Ω at time t that are involved in M chemical reaction channels {R₁, …, R_M}. The dynamics of each reaction channel R_j (j = 1, 2, …, M) is characterized by a propensity function a_j and the state change vector v_j. Given x = X(t), a_j(x)dt = c_jh_j(t)dt gives the probability that an R_j reaction will occur in Ω in the next time interval [t, t + dt) where c_jdt denotes the average probability that one particular combination of R_j reactant molecules will react in the next time interval [t, t + dt), and h_j(t) is the number of distinct combinations of R_j reactant molecules found to be present in Ω at time t. The state change vector v_j = (v_1j, …, v_Nj) gives the change in the molecular populations induced by one R_j reaction.

The joint probability density function p(τ, μ|x, t) gives the probability that the next reaction occurs in the infinitesimal time interval [t+τ, t+τ+dτ) and is a R_μ reaction

$$p(\tau ,\mu |x,t)=a_{\mu }\left(x\right)\exp (-a_0\left(x\right)\tau ),$$

where

$$a_0\left(x\right)=\sum _{j=1}^M{a}_j\left(x\right).$$

At each time-step, two random numbers r₁ and r₂ from the uniform distribution in the unit interval are generated. The next reaction occurs at t+τ where

$$\tau =\frac{1}{a_0\left(x\right)}\ln \frac{1}{r_1}$$ (1)

and the index μ of the next reaction is the smallest integer that satisfies

$$\sum _{j=1}^{\mu }{a}_j\left(x\right)>r_2{a}_0\left(x\right).$$ (2)

The state vector is then updated by X(t + τ) = x + v_μ, and the process repeats until the end of the desired simulation time. Each simulation run generates a trajectory of the following CME that describes the time evolution of P(x, t|x₀, t₀), the probability that X(t) = x given X(t₀) = x₀,

$$\begin{array}{ccc}\hfill \frac{\partial P(x,t|x_0,t_0)}{\partial t}& =& \sum _{j=1}^M[a_j(x-v_j)P(x-v_j,t|x_0,t_0)\hfill \\ & & -a_j\left(x\right)P(x,t|x_0,t_0\left)\right].\hfill \end{array}$$ (3)

Gillespie's direct method extended to simulate reaction–diffusion systems

Diffusion is introduced into the above SSA by partitioning the system volume Ω into N_V cubical subvolumes {V₁, …, _VNV}, each of which is assumed to be well-stirred or homogeneous. N molecular species are still involved in M reactions in each subvolume, and diffusive transfers of a molecular species S_i from a subvolume V_f to a neighboring subvolume V_g are treated as unimolecular reactions, $S_i^f\to S_i^g$.^{35, 36, 37, 67} In each subvolume, these unimolecular reactions denoting the diffusive transfers to neighboring subvolumes are appended to the list of M reactions taking place in the subvolume. Gillespie's direct method is then applied to the entire system that contains N·N_V molecular species involved in the reaction channels comprising N_V appended lists of reactions.

Non-negative Poisson tau-leaping for Gillespie's SSA

The non-negative Poisson tau-leaping strategy by Cao et al.^{51, 66} tries to advance the state of the system by the largest value of the leap time τ while simultaneously allowing multiple reaction events to occur and satisfying the leap condition,⁵³ which states that no propensity function is likely to change its value by a significant amount during [t, t + τ). Under this condition that bounds the relative change in each propensity function a_j(x) during the leap,^{53, 54} the state vector is updated as

$$X(t+\tau )\dot{=}x+\sum _{j=1}^M{P}_j\left(a_j\left(x\right)\tau \right)v_j,$$ (4)

where P_j(a_j(x)τ) is a statistically independent Poisson random variable. To avoid any of the population of reactant species becoming negative due to the unboundedness of the Poisson random numbers, their strategy introduces a control parameter n_c, a small positive integer (e.g., 10), such that any reaction channel with a positive propensity function that is currently within n_c firings of exhausting one of its reactants is classified as a critical reaction. No more than one firing of all the critical reactions can occur during the leap time τ while multiple firings of noncritical reactions are allowed. The algorithm becomes identical to the SSA if n_c is chosen so large that every reaction is critical.^{51, 53, 54, 66}

Cao et al.⁵¹ choose a leap time τ by bounding the estimated relative changes in the molecular populations by ɛ_i = ɛ∕g_i, which also bounds the estimated change in each reaction propensity function during τ:

$$\left|〈X_i(t+\tau )-X_i\left(t\right)〉\right|\le \underset{i\in I_{\mathrm{rs}}}{max}\{{\varepsilon }_i·X_i\left(t\right),1\},$$ (5)

where ɛ is a user-specified accuracy-control parameter such that 0 < ɛ ≪ 1, g_i depends on the highest order reaction in which X_i is involved as well as its stoichiometry and is between 1 and 5.5 for first, second, and third order reactions (as defined by Cao et al.⁵¹), and I_rs denotes the set of indices of all reactant species. The bound enforced in Eq. 5 in addition to requiring the changes in molecular populations to be bounded by a small fraction ɛ_i of those populations, takes into account the discreteness of the populations by ensuring that the amount of the changes is never less than 1.

Because the statistically independent Poisson random variables P_j(a_j(x)τ) in Eq. 4 have means and variances equal to a_j(x)τ, the means and variances of X_i(t + τ) − X_i(t) are ${\sum }_{j=1}^M\left[a_j\left(x\right)\tau \right]v_{ji}$ and ${\sum }_{j=1}^M\left[a_j\left(x\right)\tau \right]v_{ji}^2$ for every i ∈ I_rs, respectively. Thus, based on the propensities, stoichiometries, and current state of the system, the set of inequalities,

$$\begin{array}{c}\hfill |⟨X_i(t+\tau )-X_i\left(t\right)⟩|\le \underset{i\in I_{rs}}{max}\{{\varepsilon }_i·X_i\left(t\right),1\},\end{array}$$

and

$$\begin{array}{c}\hfill \sqrt{\mathrm{var}\{X_i(t+\tau )-X_i\left(t\right)\}}\le \underset{i\in I_{\mathrm{rs}}}{max}\{{\varepsilon }_i·X_i\left(t\right),1\},\end{array}$$

yields the largest permissible value of τ.

The leap time τ thus chosen is compared with the estimated time to the next critical reaction τ_c, which is calculated by applying the SSA to the set of critical reactions and their reactant species. If τ is smaller than τ_c, all noncritical reactions j fire P_j(a_j(x)τ) times and no critical reaction fires in the next time interval τ. Otherwise, all noncritical reactions j fire P_j(a_j(x)τ_c) times and only one critical reaction chosen for the next time-step fires in the next time interval τ_c. When τ is not only small enough to satisfy the leap condition, but also large enough that the expected number of firings of each reaction channel R_j during τ is ≫ 1, it has been shown that such leaping strategy is equivalent to the chemical Langevin equation.⁴⁷

A NEW ALGORITHM BASED ON GRADIENT-BASED DIFFUSION AND TAU-LEAPING

Our objective is to develop a new simulation algorithm that improves the speed of the SSA for reaction–diffusion systems by reducing the number of time-steps needed to complete a simulation run while maintaining accuracy. Our goal is accomplished in two ways: gradient-based diffusion and unified spatial tau-leaping.

Gradient-based diffusion

Simulating a given reaction–diffusion system for a simulation period becomes more efficient when the number of time-steps to complete the simulation becomes less numerous. The number of time-steps to complete a simulation is reduced when the duration of each time-step can be increased without sacrificing accuracy. Equation 1 shows that the length of the next time-step τ is inversely proportional to a₀(x), which is the sum of all values of current propensity functions. The value of each propensity function is proportional to the number of distinct combinations of reactant molecules for the reaction. The value of a₀(x) is then reduced by decreasing the number of distinct molecular reactant combinations for those reactions that represent diffusive transfers. Consequently, τ has a larger value, which leads to reducing the number of total time-steps needed to complete a simulation run. Therefore, we have developed a method to reduce the distinct molecular reactant combinations for reactions denoting diffusive transfers between neighboring subvolumes.

We first state the assumptions and definitions to be used throughout the paper. We next derive, from the diffusion equation, a diffusion propensity function that gives the probability that a diffusive transfer of a molecular species will occur from a subvolume V_g to its neighboring subvolume V_f in the next time interval [t, t+dt). We then present our algorithm based on the newly derived diffusion propensity function for reaction–diffusion systems.

Assumptions and definitions

Assume a closed volume Ω with von Neumann boundary conditions on dΩ that contains N molecular species {S₁, …, S_N} involved in M chemical reaction channels {R₁, …, R_M} and is divided into N_V cubical subvolumes {V₁, …, _VNV}. Assume that each subvolume is small enough to capture the length scale of interest and large enough to be well-stirred. For the well-stirred assumption to be physically valid, it was shown that the side length of the cubic subvolume, u, must satisfy τ_R ≫ τ_D, where τ_R and τ_D denote the typical time interval between reaction events and the typical time interval between diffusion events, respectively.⁴² Further assume that n_d molecular species (1 ≤ n_d ≤ N) diffuse with diffusion coefficients greater than zero. Let the state vector X^g(t) = ($X_1^g\left(t\right)$, …, $X_N^g\left(t\right)$) denote the number of molecules of S_i (i = 1, 2, …, N) in subvolume V_g (g = 1, 2, …, N_V) at time t. Then X(t) = (X¹(t), …,^XN_V(t)) denotes the N_V by N state matrix of the system at time t. N molecular species in each subvolume V_g are involved in M chemical reaction channels {R₁, …, R_M} and n_dn_g diffusion channels {F_i^gf} where n_g is the number of V_g's neighboring subvolumes; i ∈ I_d where I_d is the set of indices of n_d molecular species with diffusion coefficients greater than zero; f denotes the index of each neighboring subvolume of V_g, f ∈{1, …, N_V} (f ≠ g). The dynamics of each diffusion channel F_i^gf is then characterized by a propensity function b_i^gf(x)dt (defined in Sec. 3A2), and $E_i^f$–$E_i^g$ gives the change in the molecular populations induced by one F_i^gf diffusion where $E_i^g$ denotes the N_V by N matrix whose entries are all zero except for the entry (g, i).

Diffusion propensity function

Before we derive our diffusion propensity function, we briefly review the analogous propensity functions for diffusive transfers between neighboring subvolumes in previous spatial SSAs for reaction–diffusion systems^{35, 36, 37, 67} to show the equivalent theoretical basis while, at the same time, highlighting the advantage of our method. In these previous SSAs extended to reaction–diffusion systems, as the diffusive transfers are treated as first-order reactions, the reaction propensity for a reaction $S_i^f\to S_i^g$ has been given by $d_i^{gf}$$h_{i}^{\it gf} (t)$. In this formulation, $d_i^{gf}$ is related to the diffusion rate constant D_i by $d_i^{gf}$ = D_i∕u² , where u is the side length of the subvolumes, which are assumed to be cubes; $h_i^{gf}\left(t\right)$, the number of distinct molecular reactant combinations for this reaction found to be present in V_g at time t, is the number of S_i molecules$X_i^g\left(t\right)$. This formulation treats diffusive transfers between subvolumes as two independent birth or death processes regardless of a concentration gradient of S_i that may exist between two subvolumes.

Our diffusion propensity function takes into account the fact that the direction of movement of molecules by diffusion occurs along the concentration gradient from a higher concentration to a lower concentration as described by the diffusion equation

$$\frac{\partial S_i}{\partial t}=-\nabla J=D_i{\nabla }^2{S}_i,$$

where J = −D_i∇S_i by Fick's law. Thus by definition, a net movement of S_i molecules between two subvolumes occurs when the concentration of S_i is higher in one subvolume than in the other. When the concentrations of S_i are equal in two neighboring subvolumes, no net accumulation or decrement of molecules by diffusion occurs between the subvolumes although a continuous exchange of molecules may occur. Accordingly, for our diffusion propensity function, we use $d_i^{gf}$ = D_i∕u² as in previous algorithms,^{35, 36, 37, 67} but redefine $h_i^{gf}\left(t\right)$ in order to incorporate the concentration gradient that directs the net movement of molecules described in the diffusion equation.

For diffusive transfers, we redefine $h_i^{gf}\left(t\right)$ to be the number of S_i molecules available for this net movement from V_g to V_f by diffusion at time t:

$$h_i^{gf}\left(t\right)=\left\{\begin{array}{c}{X}_i^g\left(t\right)-X_i^f\left(t\right),\hfill \\ 0,\hfill \end{array}\right.\begin{array}{cc}& \mathrm{if}{X}_i^g\left(t\right)>X_i^f\left(t\right)\\ & \mathrm{otherwise}.\end{array}$$ (6)

Our diffusion propensity b_i^gf(x)dt = $d_i^{gf}$$ h_{i}^{\it gf} (t)$dt then gives the probability that a net movement of a S_i molecule by diffusion will occur from V_g to V_f in the next time interval [t, t+dt). The advantage of this newly defined diffusion propensity function comes from the fact that this $h_i^{gf}\left(t\right)$ is always less than or equal to$X_i^g\left(t\right)$; consequently, the diffusion propensity $d_i^{gf}$$h_{i}^{\it gf} (t)$ is always less than or equal to $d_i^{gf}$$X_{i} ^{g} (t)$. This decrease in the value of each diffusion propensity leads to a decrease in a₀(x) and ultimately to a greater step size τ, which is illustrated in Sec. 4.

A SSA based on gradient-based diffusion propensity functions for reaction–diffusion systems

The time evolution of the state matrix x = X(t) (as defined in Sec. 3A1 above) is simulated by generating two random numbers r₁ and r₂ from the uniform distribution in the unit interval at each time-step. At time t, the next reaction occurs at t+τ, where

$$\tau =\frac{1}{a_0\left(x\right)}\ln \frac{1}{r_1}$$ (7)

and

$$a_0\left(x\right)=\sum _{g=1}^{N_V}\sum _{j=1}^M{a}_j^g\left(x\right)+\sum _{g=1}^{N_V}\sum _{f=1}^{N_V}\sum _{i=1}^N{b}_i^{gf}\left(x\right).$$ (8)

Note that b_i^gf(x) = 0 in Eq. 8 when i∉I_d, and when V_f and V_g are not neighboring subvolumes. The next reaction is R_μ in V_κ if there is the first integer pair (μ, κ) that satisfies

$$\sum _{g=1}^{\kappa }\sum _{j=1}^{\mu }{a}_j^g\left(x\right)>r_2{a}_0\left(x\right).$$

Otherwise, the next reaction is$F_{\mu }^{\kappa \rho }$ with the first integer triple (μ, κ, ρ) that satisfies

$$\sum _{g=1}^{N_V}\sum _{j=1}^M{a}_j^g\left(x\right)+\sum _{g=1}^{\kappa }\sum _{f=1}^{\rho }\sum _{i=1}^{\mu }{b}_i^{gf}\left(x\right)>r_2{a}_0\left(x\right).$$

In the former case with the integer pair (μ, κ), the state matrix is updated by X(t+τ) = x + e_κ^Tv_μ where e_κ^T is the column vector of length N_V whose entries are all zero except for the κ-th entry, which is one, such that e_κ^Tv_μ denotes the N_V by N matrix formed by the product of the column vector e_κ^T with the row vector v_μ. In the latter case with the integer triple (μ, κ, ρ), the state matrix is updated by X(t+τ) = x − $E_{\mu }^{\kappa }$ + $E_{\mu }^{\rho }$. The process repeats until the end of the desired simulation time. This simulation method follows the probability distribution underlying the RDME given by

$$\begin{array}{ccc}\hfill & & \frac{\partial P(x,t|x_0,t_0)}{\partial t}\hfill \\ \hfill & & =\sum _{g=1}^{N_V}\sum _{f=1}^{N_V}\sum _{i=1}^N[b_i^{gf}\left(x+E_i^g-E_i^f\right)P\left(x+E_i^g-E_i^f,t|x_0,t_0\right)\hfill \\ \hfill & & -b_i^{fg}\left(x\right)P(x,t|x_0,t_0\left)\right]\hfill \\ \hfill & & +\sum _{g=1}^{N_V}\sum _{j=1}^M[a_j^g\left(x-e_g^T{v}_j\right)P\left(x-e_g^T{v}_j,t|x_0,t_0\right)\hfill \\ \hfill & & -a_j^g\left(x\right)P(x,t|x_0,t_0\left)\right],\hfill \end{array}$$

where P(x, t|x₀, t₀) is the probability that X(t) = x given X(t₀) = x₀. As has been noted previously, the above RDME is separated into two sums.³⁷ The first term corresponds to diffusion processes in the system, whereas the second term corresponds to the reaction processes and is the sum of the CMEs for individual subvolumes in Ω.

Unified tau-leaping for reaction–diffusion systems

Our second approach to reduce the number of time-steps required to simulate a reaction–diffusion system in order to make such simulations more efficient employs the tau-leaping strategy,^{51, 53, 54, 66} which has been developed and refined to accelerate SSAs to simulate reaction systems in a well-stirred domain. Our extension of the tau-leaping strategy to reaction–diffusion systems is distinct from previous approaches^{62, 63, 64} that treat reaction and diffusion processes independently of each other by alternating the execution of each process at predetermined intervals of time, an approximation strategy developed to speed up simulation of diffusion processes in reaction–diffusion systems. Our method calculates each leap time after considering both reaction and diffusion processes. In combination with gradient-based diffusion, the next leap time is calculated after considering reactions and diffusions in a unified way in order to better satisfy the leap condition.⁵³ We first show how to estimate τ that satisfies the leap condition on both reaction and diffusion propensity functions. We then extend our gradient-based diffusion method to incorporate the new tau-selection strategy.

Given the system in state x at time t, we estimate the largest value of τ that satisfies the leap condition that during [t, t + τ), no propensity function is likely to change its value by a significant amount. Similar to Eq. 4 to the degree that the leap condition is satisfied, the system can be advanced by the leap time τ as

$$\begin{array}{ccc}\hfill X(t+\tau )& \dot{=}& x+\sum _{g=1}^{N_V}\sum _{j=1}^M{P}_j\left(a_j^g\left(x\right)\tau \right)e_g^T{v}_j\hfill \\ & & +\sum _{g=1}^{N_V}\sum _{f=1}^{N_V}\sum _{i=1}^N{P}_i\left(b_i^{gf}\left(x\right)\tau \right)\left(E_i^f-E_i^g\right).\hfill \end{array}$$

To avoid any of the reactant species populations becoming negative, our unified non-negative Poisson tau-leaping strategy maintains a control parameter n_c, which is a small positive integer (e.g., 10), similar to the non-negative Poisson tau-leaping algorithm for well-stirred chemically reacting systems.⁶⁶ Any reaction channel with a positive propensity function that is currently within n_c firings of exhausting one of its reactants is classified as a critical reaction, and any diffusion channel with a positive propensity function whose $h_i^{gf}\left(t\right)$ value is less than n_c is classified as a critical diffusion. No more than one firing of all critical reactions and diffusions can occur during the leap. The algorithm becomes exact if n_c is chosen so large that every reaction and diffusion is critical.

The largest value of the tentative leap time τ is chosen so that the estimated change in each reaction propensity function during τ is bounded by the estimated change $X_i^g(t+\tau )-X_i^g\left(t\right)$ as described in Sec. 2. In addition, the estimated change in each gradient-based diffusion propensity function during τ is bounded by requiring the estimated change $h_i^{gf}(t+\tau )-h_i^{gf}\left(t\right)$ be less than the amount max{$\varepsilon ·h_i^{gf}\left(t\right)$, 1}, in order to prevent negative populations. Therefore, τ is selected by enforcing the bounds

$$\left|〈X_i^g(t+\tau )-X_i^g\left(t\right)〉\right|\le \max \left\{{\varepsilon }_i·X_i^g\left(t\right),1\right\},$$

$$\sqrt{\mathrm{var}\left\{X_i^g(t+\tau )-X_i^g\left(t\right)\right\}}\le \max \left\{{\varepsilon }_i·X_i^g\left(t\right),1\right\},$$

$$\left|〈h_i^{gf}(t+\tau )-h_i^{gf}\left(t\right)〉\right|\le \max \left\{\varepsilon ·h_i^{gf}\left(t\right),1\right\},\mathrm{and}$$

$$\sqrt{\mathrm{var}\left\{h_i^{gf}(t+\tau )-h_i^{gf}\left(t\right)\right\}}\le \max \left\{\varepsilon ·h_i^{gf}\left(t\right),1\right\},$$ (9)

where ɛ_i and ɛ have been described in Sec. 2.

The unified non-negative Poisson tau-leaping procedure, also based on gradient-based diffusion, can be summarized as follows:

1. At t = t₀, initialize the system's state to x = X(t₀). Calculate all $a_j^g\left(x\right)$ and $b_i^{gf}\left(x\right)$ and their sum a₀(x).

2. Iterate while (t < t_end)

   1. Identify the currently critical reactions and diffusions.
   2. Compute the candidate time-step τ^′ for all noncritical reactions and diffusions that satisfies the bounds in Eq. 9.
   3. If τ^′ is less than some small multiple (e.g., 10) of 1∕a₀(x)

      • Execute some modest number (e.g., 100) of SSA steps.
        Else
      • Compute the second candidate time-step τ^″ as the time that the next critical reaction or diffusion occurs.
      • If (τ^′ < τ^″)

        • Take τ = τ^′. No critical reaction or diffusion fires during this leap time τ. All noncritical reactions and diffusions fire $P_j\left(a_j^g\left(x\right)\tau \right)$ and $P_i\left(b_i^{gf}\left(x\right)\tau \right)$ times, respectively.
          Else
        • Take τ = τ^″. Obtain the index of the next critical reaction or diffusion, which is the only critical reaction or diffusion that fires once in this step. All noncritical reactions and diffusions fire $P_j\left(a_j^g\left(x\right)\tau \right)$ and $P_i\left(b_i^{gf}\left(x\right)\tau \right)$ times, respectively.
      • Update t and x.
   4. Recalculate those $a_j^g\left(x\right)$ and $b_i^{gf}\left(x\right)$ whose reactant species’ populations changed and update a₀(x).

NUMERICAL RESULTS

We have applied our algorithm to three different test cases: 1D reaction–diffusion, 2D diffusion, and 2D reaction–diffusion. The first two problems, the A+B annihilation problem⁶³ and the 2D diffusion problem,⁶⁹ have been used in previous studies as test cases for reaction–diffusion in one dimension and diffusion in two dimensions, respectively. The third problem, cyclic adenosine monophosphate (cAMP) activation of protein kinase A that describes part of a ubiquitous mammalian signaling pathway,^{70, 71} is chosen to further confirm and illustrate the accuracy and improved performance of our algorithm in simulating a larger set of reactions and diffusion.

We present numerical simulation results of these three test cases in Secs. 4A, 4B, 4C and, in Sec. 4D, show the improved performance of our algorithm in terms of reducing the number of time-steps needed to complete a simulation run and the corresponding reduction in simulation time. To better understand how each of two strategies, gradient-based diffusion (Gradient) and unified tau-leaping (Leap), contributes to improving the efficiency of the overall algorithm (Gradient–Leap), we use two strategies individually and in combination in simulations.

Simulation results from the Gradient, Leap, and Gradient–Leap methods are compared with simulation results obtained by using Gillespie's direct (Direct) method extended to reaction and diffusion to validate the accuracy and demonstrate the improved efficiency of our individual and combined methods. Thus, for each of three test cases, four sets of simulations—each set consisting of 1000 simulations—were run. To further validate accuracy, numerical simulation results are compared with a deterministic solution for the 2D diffusion problem.

To quantify accuracy, we use three measures. The first measure is the space-averaged Kolmogorov distance to quantify the distance between two cumulative distributions, F₁(x) and F₂(x), which is defined as

$${〈K(F_1\left(x\right),F_2\left(x\right))〉}_{N_V}=\frac{1}{N_V}\sum _{i=1}^{N_V}K(F_1\left(x_i\right),F_2\left(x_i\right)),$$ (10)

where K(F₁(x_i), F₂(x_i)) = $\underset{\mathrm{all}{x}_i}{max}|F_1\left(x_i\right)-F_2\left(x_i\right)|$.^{63, 72} We estimate each distribution function F(x) by an empirical distribution function of a sample of size 1000. The Kolmogorov self-distance⁷² is defined as the Kolmogorov distance between two independent samples obtained from the exact simulation algorithm and is observed to be not arbitrarily small even when the sample size is as large as 10 000. It represents the amount of noise due to the natural fluctuations in the system.⁷² This inherent noise makes the Kolmogorov–Smirnov (K–S) test reject the null hypothesis that two samples obtained from the Direct method are drawn from the same distribution. Therefore, in lieu of the K–S test, we compare the Kolmogorov distance with the self-distance. The second measure computes the space-averaged mean number of species S:

$$S_{\mathrm{mean}}=\frac{1}{N_v}{\sum }_{g=1}^{N_V}〈X_i^g\left(t\right)〉,$$ (11)

where $〈X_i^g\left(t\right)〉$ denotes the number of S_i molecules in subvolume V_g at time t, averaged over 1000 simulations. The third measure computes the space-averaged variance of the S population:

$$S_{\mathrm{var}}=\frac{1}{N_v}{\sum }_{g=1}^{N_V}{\sigma }^2\left(\left\{X_i^g\left(t\right)\right\}\right),$$ (12)

where $\left\{X_i^g\left(t\right)\right\}$ denotes the set containing 1000 $X_i^g\left(t\right)$’s. We compare the values of S_mean and S_var obtained from the Leap, Gradient, and Gradient–Leap methods with those from the Direct method.

Reaction–diffusion in 1D: A+B annihilation problem

The 1D A+B annihilation problem⁶³ is defined by the reaction

$$\mathrm{A}+\mathrm{B}\stackrel{k_1}{\to }Ø$$

on a 40 × 0.4 × 0.4 μm volume. The simulation volume is divided into 100 × 1 × 1 subvolumes. The reaction rate constant k₁ is 10⁻¹ nM⁻¹  s⁻¹. The diffusion coefficient D for both A and B is 5 μm² s⁻¹. Initially, 1000 molecules of A are evenly distributed across the subvolumes, and 1000 molecules of B are all placed in one of the outermost subvolumes. Starting from this initial condition, the problem is numerically simulated for 100 s.

Figure 1 shows the ensemble average numbers of A and B molecules over time and space obtained from four sets of simulations by using four different methods: Direct, Leap, Gradient, and Gradient–Leap. Initially, 1000 molecules of B are placed in the rightmost subvolume in the figure. As these B molecules diffuse and react with A molecules, both reactant molecules are destroyed. Figure 1 shows an excellent qualitative agreement between all of the ensemble means, which is further demonstrated with the space-averaged means in Fig. 2 (middle row).

Figure 1.

Ensemble average numbers of A and B molecules over time and space for the 1D A+B annihilation problem. The simulation methods are Direct, Leap, Gradient, and Gradient—Leap (from top to bottom).

Figure 2.

Quantification of accuracy for the A+B annihilation problem with both species diffusing. Space-averaged Kolmogorov distances (top row), space-averaged means (middle row), and space-averaged variances (bottom row) over time for species A are shown on the left, and for species B are shown on the right. The Kolmogorov distances are calculated for four pairs of samples: (1) Direct and Direct (self-distance), (2) Direct and Leap, (3) Direct and Gradient, and (4) Direct and Gradient–Leap. The means completely overlap for all four methods for both species, thus not all traces are visible. Note that the y-axes have different ranges.

Figure 2 shows space-averaged Kolmogorov distances, space-averaged means, and space-averaged variances of species A and B. The top row shows the Kolmogorov distances over time of four pairs of samples: (1) Direct and Direct (self-distance), (2) Direct and Leap, (3) Direct and Gradient, and (4) Direct and Gradient–Leap. All paired-distance values are comparable to the values previously reported.⁶³ The distances of pairs (1) and (2) nearly overlap each other, which is expected as the tau-leaping method is an approximation of the exact method that converges to the exact method as the step-size becomes smaller.⁵³ The distances of pairs (3) and (4) are larger than the other two paired-distances because both gradient-based methods approximate frequent diffusion events by the less-frequent net diffusion event, which has a smoothing effect on fluctuations due to all diffusion events in the system trajectory. Space-averaged variances (bottom row) show that the smaller fluctuations result in sample distributions with reduced variances, which explains the larger Kolmogorov distances. For the Leap method, the space-averaged variance values closely match those from the Direct method.

To demonstrate that the gradient-based methods correctly approximate the variance of nondiffusing species, the 1D A+B annihilation problem was repeated with a single change: the diffusion coefficient of A is zero. Figure 3 shows that for species B, the space-averaged Kolmogorov distances of pairs (1) Direct and Direct (self-distance) and (2) Direct and Leap overlap whereas the distances of pairs (3) Direct and Gradient and (4) Direct and Gradient–Leap are larger than the self-distance, similar to the results presented in Fig. 2. In contrast, for species A, which reacts with B but does not diffuse, the distances of pairs (2), (3), and (4) are all close to the self-distance of pair (1) throughout the simulation interval. As before, the space-averaged means from all four methods are identical for both species throughout the simulation interval. For the diffusing species B, the space-averaged variance values from the Leap method closely match those from the Direct method while the space-averaged variance values from the Gradient and Gradient–Leap methods are smaller than those from the Direct method, similar to the results shown in Fig. 2. The space-averaged variance values of the nondiffusing species A from the Direct and Leap methods are markedly reduced when compared to the respective values shown in Fig. 2 and are similar to the space-averaged variance values from the Gradient and Gradient–Leap method. In this example, the nondiffusing species is present in numbers between 0 and 10 in each subvolume on average. In this simulation, approximating frequent diffusion events by the less-frequent net diffusion event has minimal effect on the sample distribution of a nondiffusing reaction partner.

Figure 3.

Quantification of accuracy for the modified A+B annihilation problem where the diffusion coefficient of A is set to zero. Space-averaged Kolmogorov distances (top row), space-averaged means (middle row), and space-averaged variances (bottom row) for species A are shown on the left, and for species B are shown on the right. The Kolmogorov distances are calculated for four pairs of samples: (1) Direct and Direct (self-distance), (2) Direct and Leap, (3) Direct and Gradient, and (4) Direct and Gradient–Leap. The means completely overlap for all four methods for both species, thus not all traces are visible. Note that the y-axes have different ranges.

Diffusion in 2D

The 2D diffusion problem⁶⁹ is defined by a species A whose diffusion coefficient is 300 μm²s⁻¹ on a 10 × 10.5 × 0.5 μm volume. The simulation volume is divided into 20 × 21 × 1 subvolumes. Initially, 1000 molecules of A are placed in one of the four corner subvolumes. Starting from this initial condition, the problem is numerically simulated for 50 ms.

Figure 4 shows the ensemble average numbers of A molecules over space at the end of the simulation interval. Initially, 1000 molecules of A are placed in the lower-right corner subvolume in the figure. Figure 4 shows an excellent qualitative agreement between all of the ensemble means.

Figure 4.

Ensemble average numbers of A molecules over space for the 2D diffusion problem at the end of simulation, t = 50 ms. The simulation methods are Direct, Leap, Gradient, and Gradient—Leap (from left to right).

Figure 5 shows space-averaged Kolmogorov distances (top row) and space-averaged variances (bottom row) of species A. The space-averaged Kolmogorov distances of pairs (1) Direct and Direct (self-distance) and (2) Direct and Leap nearly overlap whereas the distances of pairs (3) Direct and Gradient and (4) Direct and Gradient–Leap are larger than the self-distance. Similar to the results for the diffusing species in the previous examples, the space-averaged variance values from the Leap method closely match those from the Direct method, whereas the space-averaged variance values from the Gradient and Gradient–Leap methods are smaller than those from the Direct method. This reduced variance results from the smoothing effect that the gradient-based propensity function has on fluctuations caused by diffusion events and explains the larger differences in the Kolmogorov distances of pairs (3) and (4). Although not plotted, the results show that the space-averaged means are 2.381 for all four methods throughout the simulation interval.

Figure 5.

Space-averaged Kolmogorov distances (top row) and space-averaged variances (bottom row) over time for A molecules for the 2D diffusion problem. The Kolmogorov distances are calculated for four pairs of samples: (1) Direct and Direct (self-distance), (2) Direct and Leap, (3) Direct and Gradient, and (4) Direct and Gradient–Leap. Note that the y-axes have different ranges.

Figure 6 shows the ensemble means and standard deviation of A molecules in each of 20 diagonal subvolumes at three different times—1, 30, and 50 ms—overlaid with the solution of the discretized diffusion equation. Ensemble means from all four methods closely match the deterministic solution. As diffusing molecules spread to more subvolumes, which were initially empty, diffusive fluctuations propagate to those subvolumes. The fluctuation in molecule numbers is smaller for the gradient–based methods, which is reflected in the smaller standard deviations, and explains the smaller spaced-averaged variance values and larger Kolmogorov distances shown in Fig. 5. The underestimate of the variance of diffusing species is smaller when the adjacent subvolumes have fewer particles, as occurs initially and when the ensemble mean is quite small.

Figure 6.

Ensemble mean numbers of A molecules over 20 subvolumes that lie diagonally on the simulation volume. They are arranged in increasing distance from the lower-right subvolume for the 2D diffusion problem. The means are overlaid with the deterministic solution at three different times in simulation: t = 1, 30, and 50 ms. The one-sided error bars indicate the standard deviation. The legend is the same for all plots. Note that the y-axes have different ranges.

Reaction–diffusion in 2D: cAMP activation of protein kinase A

The 2D reaction–diffusion problem chosen as the third test case describes cAMP activation of protein kinase A (PKA). PKA is a tetramer that consists of two regulatory subunits (PKAr) and two catalytic subunits (PKAc). Binding of two cAMP molecules to each regulatory subunit of PKA leads to the dissociation of the catalytic subunits, which subsequently phosphorylate various substrate proteins. cAMP activation of PKA is part of a ubiquitous mammalian signaling pathway that translates an extracellular signal into an intracellular biological response.^{70, 71}

The cAMP activation of PKA problem is defined by

$$\mathrm{PKA}+2\mathrm{cAMP}\stackrel{k_1}{\to }\mathrm{PKA}\text{-}{\mathrm{cAMP}}_2$$

$$\mathrm{PKA}\text{-}{\mathrm{cAMP}}_2\stackrel{k_2}{\to }\mathrm{PKA}+2\mathrm{cAMP}$$

$$\mathrm{PKA}\text{-}{\mathrm{cAMP}}_2+2\mathrm{cAMP}\stackrel{k_3}{\to }\mathrm{PKA}\text{-}{\mathrm{cAMP}}_4$$

$$\mathrm{PKA}\text{-}{\mathrm{cAMP}}_4\stackrel{k_4}{\to }\mathrm{PKA}\text{-}{\mathrm{cAMP}}_2+2\mathrm{cAMP}$$

$$\mathrm{PKA}\text{-}{\mathrm{cAMP}}_4\stackrel{k_5}{\to }\mathrm{PKAr}+2\mathrm{PKAc}$$

$$\mathrm{PKAr}+2\mathrm{PKAc}\stackrel{k_6}{\to }\mathrm{PKA}\text{-}{\mathrm{cAMP}}_4$$

on a 10 × 10.5 × 0.5 μm volume. The simulation volume is divided into 20 × 21 × 1 subvolumes. The reaction rate constants are defined as follows: k₁ = 8.696 × 10⁻⁵ nM⁻¹s⁻¹, k₂ = 0.02 nM⁻¹s⁻¹, k₃ = 1.154 × 10⁻⁴ nM⁻¹s⁻¹, k₄ = 0.02 nM⁻¹s⁻¹, k₅ = 0.016 nM⁻¹s⁻¹, and k₆ = 0.0017 nM⁻¹s⁻¹. The diffusion coefficient for cAMP is 300 μm²s⁻¹; the diffusion coefficient for all other species is zero. Initially, 30240 molecules of PKA are evenly distributed across the subvolumes, and 80 000 molecules of cAMP are placed in one of four corner subvolumes. Starting from this initial condition, the problem is numerically simulated for 500 ms.

Figure 7 shows the ensemble average numbers of PKA, cAMP, and the dissociated catalytic subunit of PKA (PKAc) over space at the end of the simulation interval. The 80 000 molecules of cAMP are initialized in the lower-right corner subvolume in the figure. Figure 7 shows an excellent qualitative agreement between all of the ensemble means.

Figure 7.

Ensemble average numbers of PKA, cAMP, and PKAc molecules over space for the cAMP activation of PKA problem at the end of simulation, t = 500 ms. The simulation methods are Direct, Leap, Gradient, and Gradient–Leap (from top to bottom).

Figure 8 shows space-averaged Kolmogorov distances (top row), space-averaged means (middle row), and space-averaged variances (bottom row) of species of PKA, cAMP, and PKAc. The distances of pairs (1) Direct and Direct (self-distance) and (2) Direct and Leap nearly overlap for all species. The distances of pairs (3) Direct and Gradient and (4) Direct and Gradient–Leap are slightly larger than the self-distance for PKA, which reacts with cAMP but does not diffuse, similar to the results for A molecules in Fig. 3. As expected, the distances of pairs (3) and (4) are larger than the self-distance for cAMP since cAMP has a nonzero diffusion coefficient. More importantly, the distances of pairs (3) and (4) are almost identical to the self-distance for PKAc, a downstream molecule that does not diffuse but is the crucial species for cell function. As observed in the previous test cases, the space-averaged means of all methods are almost identical for all molecules. The space-averaged variances exhibit the same pattern elucidated with the A+B annihilation problem, namely, the variance is much smaller when using the gradient-based methods for diffusing molecules such as cAMP. For nondiffusing molecules, such as PKA and PKAc, the space-averaged variance values from the gradient-based methods are similar to those from the Direct method. Figure 8 illustrates that the gradient-based methods can generate a distribution of a nondiffusing species that closely approximates the distribution generated from the Direct method despite their underestimate of variance in the distribution of the diffusing species. The gradient-based methods generate the correct distribution of the crucial molecule PKAc, whose presence or absence leads to phosphorylation of a substrate and no phosphorylation, respectively.

Figure 8.

Quantification of accuracy for the cAMP activation of PKA problem. Space-averaged Kolmogorov distances (top row), space-averaged means (middle row), and space-averaged variances (bottom row) of PKA, cAMP, and PKAc molecules over time for the cAMP activation of PKA problem are shown. The Kolmogorov distances are calculated for four pairs of samples: (1) Direct and Direct (self-distance), (2) Direct and Leap, (3) Direct and Gradient, and (4) Direct and Gradient–Leap. The means completely overlap for all four methods for PKA and cAMP, thus not all traces are visible. For PKAc, the space-averaged means from the Direct method lie below those from the Gradient and Gradient–Leap methods, and thus cannot be seen. Note that the y-axes have different ranges.

Improved performance

Table 1 shows the mean number of time-steps taken for the given simulation interval for three test problems by our three methods (Leap, Gradient, and Gradient–Leap) as well as the Direct method. For the A+B annihilation problem in 1D, the average number of time-steps taken to simulate the system for the given interval by the Leap, Gradient, and Gradient–Leap methods decreased by 42%, 94%, and 94%, respectively, compared with the average number of time-steps taken by the Direct method. For the 2D diffusion problem, the average number of time-steps taken by our methods decreased by 30%, 88%, and 90%, respectively, compared with the average number taken by the Direct method. For the 2D reaction–diffusion problem involving cAMP activation of PKA, compared with the average number of time-steps taken by the Direct method, the average number of time-steps taken by the Leap, Gradient, and Gradient–Leap methods decreased by 87%, 93%, and 98%, respectively. These results show that the gradient-based diffusion strategy, by sampling only the net diffusion events based on concentration gradients rather than sampling all diffusion events, significantly reduces the number of time-steps taken for the given simulation interval for all the three test problems. As anticipated, the Leap method, whether applied to the Direct or Gradient methods, also reduces the number of time-steps, albeit constrained by the control parameter n_c and the leap condition in that as more reactions and diffusions in the system become critical, the tau-leap algorithm more closely resembles an exact algorithm. The reduction in the number of time-steps is most dramatic for the cAMP activation of PKA problem, which has a large number of molecules, suggesting that this algorithm may be especially useful for simulating large signaling pathways.

Table 1.

Mean number of time-steps (and standard deviation) to complete a simulation run.

	1D reaction–diffusion		2D reaction–diffusion
	(A+B annihilation)	2D diffusion	(cAMP activation of PKA)
Direct	25.804 × 105 (61514)	22.246 × 104 (564)	212.50 × 105 (71 780)
Leap	14.955 × 105 (32 499)	15.485 × 104 (2247)	29.489 × 105 (17 015)
Gradient	1.5682 × 105 (1372)	2.6841 × 104 (205)	14.320 × 105 (1037)
Gradient-Leap	1.5587 × 105 (1311)	2.3233 × 104 (214)	4.2223 × 105 (1128)

Next, we examined how reducing the number of time-steps taken to complete a simulation run translates to improving the speed of an exact algorithm for simulating reaction–diffusion systems. All four methods were implemented in JAVA, and simulations were run on the following systems: a 4-processor Xeon 3.2 GHz PC, a 4-processor Xeon 3.0 GHz Linux cluster, and a 8-processor Xeon 2.27 GHz Linux cluster. The Direct method was implemented with the optimizations suggested by Gillespie,²³ which do not recalculate propensity functions whose reactant populations did not change in the previous time-step. The same optimizations were applied to the other three methods. All simulations were initialized with the same set of problem description files.

Table 2 shows the mean simulation time taken by our three methods (Leap, Gradient, and Gradient–Leap) as well as the Direct method for the three test problems. The Gradient–Leap method that combines both the gradient-based diffusion and unified tau-leaping strategies uniformly reduces the simulation time for all the three test cases. For the reaction–diffusion problems in both 1D and 2D, the reduction in the number of time-steps taken to complete a simulation run achieved by the Gradient method alone corresponds to a significant—more than an order of magnitude—reduction in simulation times. A less dramatic yet similar correspondence between the reduced numbers of times steps and the reduced simulation time is also seen for the 2D diffusion problem. The reduced number of time-steps taken by the Leap method alone (whether applied to the Direct or Gradient method) does not always translate to a corresponding reduction in simulation time. This result is in line with the observation made in previous studies^{32, 51} that the larger time-step of the tau-leap algorithm comes at the expense of larger computational requirements per step compared to the exact algorithm. The slight increase in the central processing unit (CPU) time due to the overhead of selecting τ accumulates especially when τ is rejected and the system must evolve following the exact algorithm. This phenomenon is especially pronounced in the case of the 2D diffusion problem where the system has 1000 molecules in total spread out in 420 subvolumes by diffusion, thereby making more and more reactions in the system critical.

Table 2.

Mean simulation time (and standard deviation) in seconds

	1D reaction-diffusion (A+B annihilation)	2D diffusion	2D reaction–diffusion (cAMP activation of PKA)
Direct	1677.8 (292.11)	60.0 (11.97)	18 599 (6134)
Leap	463.2 (70.64)	453.6 (33.96)	32 662 (8322)
Gradient	64.7 (6.08)	13.1 (0.332)	2209 (1345)
Gradient–Leap	81.1 (2.92)	54.4 (2.578)	4795 (1618)

DISCUSSION AND CONCLUSION

We have presented a new algorithm to improve the speed of the SSA for reaction–diffusion systems by reducing the number of time-steps taken for a given simulation interval. Our contribution lies in incorporating two new strategies into the SSA to achieve this goal. First, our method samples only the net or observed diffusion events from higher to lower concentration gradients by treating diffusive transfers between neighboring subvolumes based on concentration gradients, thereby reducing the number of diffusion events to be simulated. Second, our method extends the non-negative Poisson tau-leaping method⁶⁶ that was developed to accelerate the SSA for well-stirred (homogeneous) reaction systems to spatial (nonhomogeneous) reaction–diffusion systems by calculating the leap time for both reaction and diffusion processes in a unified step.

These strategies have been applied individually (Leap and Gradient) and in combination (Gradient–Leap) to three representative problems: 1D reaction–diffusion, 2D diffusion, and 2D reaction–diffusion systems. The numerical simulation results have been compared with the results from an exact algorithm (Direct) and, for the 2D diffusion problem, with the deterministic solution. The accuracy of the results has been quantified by using three metrics. Our results show that the Leap method produces distributions that are nearly indistinguishable from those produced by the Direct method. Our results further show that, compared with the distributions produced by the Direct method, the Gradient and Gradient–Leap methods produce distributions with a correct mean but reduced variance for those molecular species with a nonzero diffusion coefficient, while they produce distributions that are almost indistinguishable for the nondiffusing molecular species. The reduced variance for diffusing species when using the gradient-based methods is expected because sampling the net diffusion events avoids the high frequency components of the fluctuations.

Most chemical systems involve several widely varying time scales and are considered nearly always stiff.⁷³ Previous approaches to reduce the expensive computational requirement in modeling such systems in a homogeneous spatial domain include developing efficient numerical methods to solve the chemical Langevin equation,^{74, 75} and partitioning chemical reactions into fast and slow reactions and then approximating some parts of the CME although it is often not obvious how to partition a system when at least one reactant in the system is involved in both fast and slow reactions.^{55, 76, 77, 78} Analogous to the smoothing effect that our gradient–diffusion based methods have on fluctuations due to diffusion events, taking large time-steps for some fast reactions in a homogeneous spatial domain has a dampening effect on fluctuations due to those reactions.⁷³ Consequently, an accurate reconstruction of statistics, including capturing the correct variance, for reactant species affected by these fast reactions is sacrificed.⁷⁷ Nevertheless, it has been suggested that the less frequent slow reactions tend to have a greater impact on the behavior of the system,⁷⁸ and since it may not even be possible to observe some physical systems at very fine time scales, some sacrifices in capturing the correct statistics of a few reactant species may be acceptable.⁷⁷ For a spatially inhomogeneous domain, the assumption that each subvolume is well-stirred requires that a typical time interval between reaction events is much longer than a typical time interval between diffusion events.⁴² Under this assumption, our results show that in systems that involve both reaction and diffusion, underestimating the variance of the diffusing species does not have adverse effects on capturing the behavior of the system characterized by the statistics of nondiffusing species. Furthermore, the underestimate of the variance of diffusing species decreases as the ensemble mean of the species decreases. Thus, the gradient-based methods are reminiscent of deterministic–stochastic hybrid methods in that diffusing species with large numbers will have a small variance, closer to the deterministic solution. However, as the numbers of molecules decrease, the variance becomes closer to the exact stochastic method resulting in a smooth transition toward the exact stochastic solution. These properties make our new algorithm extremely useful in simulating biological systems in which many of the interactions are between diffusible and nondiffusible molecules and the high frequency fluctuations due to diffusion may be of less interest than the fluctuations attributed to the slower down-stream reactions. On the other hand, when the assumption that each subvolume is well-stirred is not satisfied and the behavior of a system is characterized by one or more slowly diffusing species involved in reactions that occur in much faster time scales, capturing the correct statistics of those diffusing species becomes important. For such systems where some diffusion propensity values are relatively smaller, using the gradient-based diffusion strategy would not be suitable.

The simulation results show that all three methods (Leap, Gradient, and Gradient–Leap) markedly reduce the number of time-steps taken for the given simulation interval for all the three test problems. The Gradient–Leap method that combines both the Gradient and Leap methods uniformly reduces the simulation time for all the three test cases. The reduction in the number of time-steps taken to complete a simulation run achieved by the Gradient method directly translates to an often-significant reduction in the simulation time, although that achieved by the Leap method does not always have the same effect.

Previous efforts^{79, 80, 81} to alleviate the increase in the CPU time due to the computationally expensive τ-selection mechanism within a homogeneous spatial domain have been constrained by the difficulty in having to both calculate tentative leap times for each reaction process and satisfy the leap condition. This computational overhead for τ-selection grows in an inhomogeneous spatial domain as the number of reaction and diffusion processes in the system increases with the number of subvolumes. To compensate for this difficulty, hybrid methods^{60, 61, 62, 63, 64, 65, 82} that treat reaction and diffusion processes separately in determining the leap time have been developed. Although these hybrid methods tend to make simulations run faster, a recent study⁸³ has shown that it is crucial to consider reaction and diffusion events together during τ selection in order to properly account for the effects of incoming diffusion events to achieve an accurate spatial leaping implementation. We are investigating further optimization strategies to improve the τ-selection procedure and potential strategies to combine the τ-selection procedure with the hybrid approaches while maintaining the accuracy provided by the unified tau-leaping strategy.

Understanding the information processing performed by cell signaling pathways requires investigating the mechanisms that produce spatially limited activation of signaling molecules. Live cell imaging⁸⁴ is one technique that has revealed essential information regarding the kinetics (rate constants) and spatial organization of signaling pathways. Stochastic simulation of reaction–diffusion systems is an alternative approach to investigating mechanisms underlying the temporal and spatial aspects of cell signaling pathways, and our method facilitates this approach by making stochastic simulation more efficient while maintaining accuracy. Recent advancements in computing power allow for an increase in the number of signaling pathways implemented in single models, and an increase in the temporal and spatial scale of simulations. Computational simulations that are constrained by experimental data are invaluable for explicating hidden assumptions in a conceptual model, or evaluating the robustness of results to kinetic parameters. Spatial aspects of signaling are particularly important in neurons whose shape is characterized by numerous long thin processes (dendrites) in which restricted diffusion creates microdomains of signaling molecules. Thus, we plan to apply our method to investigate problems such as spatial specificity and temporal sensitivity of plasticity in the irregular spatial domain of neuronal dendrites.