Many biochemical processes at the sub-cellular level involve a small number of molecules. The local numbers of these molecules vary in space and time, and exhibit random fluctuations that can only be captured with stochastic simulations. We present a novel stochastic operator-splitting algorithm to model such reaction-diffusion phenomena. The reaction and diffusion steps employ stochastic simulation algorithms and Brownian dynamics, respectively. Through theoretical analysis, we have developed an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We have used three examples to demonstrate the accuracy and robustness of the proposed algorithm. The first example deals with diffusion of two chemical species undergoing an irreversible bimolecular reaction. It is used to validate our algorithm by comparing its results with the solution obtained from a corresponding deterministic partial differential equation at low and high number of molecules. In this example, we also compare the results from our method to those obtained using a Gillespie multi-particle (GMP) method. The second example, which models simplified RNA synthesis, is used to study the performance of our algorithm in reaction-and diffusion-controlled regimes and to investigate the effects of local inhomogeneity. The third example models reaction-diffusion of CheY molecules through the cytoplasm of Escherichia coli during chemotaxis. It is used to compare the algorithm's performance against the GMP method. Our analysis demonstrates that the proposed algorithm enables accurate simulation of the kinetics of complex and spatially heterogeneous systems. It is also computationally more efficient than commonly used alternatives, such as the GMP method.

Mouse oocytes matured in vitro in chemically defined medium were not penetrated by spermatozoa. The time required for dissolution of the zona pellucida of such oocytes by alpha-chymotrypsin was much longer than that for ovulated oocytes. Addition of fetal calf serum to the medium for maturation of oocytes improved the incidence of sperm penetration and shortened the time of enzymic dissolution of the zona pellucida. These results suggest that the low rate of fertilization of oocytes matured in vitro is mainly due to qualitative changes of the zona pellucida, which could be overcome by a factor or factors in fetal calf serum.

Deterministic models of biochemical processes at the subcellular level might become inadequate when a cascade of chemical reactions is induced by a few molecules. Inherent randomness of such phenomena calls for the use of stochastic simulations. However, being computationally intensive, such simulations become infeasible for large and complex reaction networks. To improve their computational efficiency in handling these networks, we present a hybrid approach, in which slow reactions and fluxes are handled through exact stochastic simulation and their fast counterparts are treated partially deterministically through chemical Langevin equation. The classification of reactions as fast or slow is accompanied by the assumption that in the time-scale of fast reactions, slow reactions do not occur and hence do not affect the probability of the state. Our new approach also handles reactions with complex rate expressions such as Michaelis-Menten kinetics. Fluxes which cannot be modeled explicitly through reactions, such as flux of Ca 2+ from endoplasmic reticulum to the cytosol through inositol 1,4,5-trisphosphate receptor channels, are handled deterministically. The proposed hybrid algorithm is used to model the regulation of the dynamics of cytosolic calcium ions in mouse macrophage RAW 264.7 cells. At relatively large number of molecules, the response characteristics obtained with the stochastic and deterministic simulations coincide, which validates our approach in the limit of large numbers. At low doses, the response characteristics of some key chemical species, such as levels of cytosolic calcium, predicted with stochastic simulations, differ quantitatively from their deterministic counterparts. These observations are ubiquitous throughout dose response, sensitivity, and gene-knockdown response analyses. While the relative differences between the peak-heights of the cytosolic ͓Ca 2+ ͔ time-courses obtained from stochastic ͑mean of 16 realizations͒ and deterministic simulations are merely 1%-4% for most perturbations, it is specially sensitive to levels of G ␤␥ ͑relative difference as large as 90% at very low G ␤␥ ͒.

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