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This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the τ-leap and midpoint τ-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Stochastic differential equations (SDE) are a powerful tool to model biological regulatory processes with intrinsic and extrinsic noise. However, numerical simulations of SDE models may be problematic if the values of noise terms are negative and large, which is not realistic for biological systems since the molecular copy numbers or protein concentrations should be non-negative. To address this issue, we propose the composite Patankar-Euler methods to obtain positive simulations of SDE models. A SDE model is separated into three parts, namely, the positive-valued drift terms, negative-valued drift terms, and diffusion terms. We first propose the deterministic Patankar-Euler method to avoid negative solutions generated from the negative-valued drift terms. The stochastic Patankar-Euler method is designed to avoid negative solutions generated from both the negative-valued drift terms and diffusion terms. These Patankar-Euler methods have the strong convergence order of a half. The composite Patankar-Euler methods are the combinations of the explicit Euler method, deterministic Patankar-Euler method, and stochastic Patankar-Euler method. Three SDE system models are used to examine the effectiveness, accuracy, and convergence properties of the composite Patankar-Euler methods. Numerical results suggest that the composite Patankar-Euler methods are effective methods to ensure positive simulations when any appropriate stepsize is used. [ABSTRACT FROM AUTHOR]

Recent research indicates that quiescent tumor cells are significantly less radiosensitive with a greater repair capacity than proliferative cells. In order to better predict patient-specific responses to radiotherapy, we develop a mathematical model with treatment terms to describe dynamical behaviors of tumor growth. The global stabilities of the tumor-free equilibrium, the tumor-present equilibrium and the corresponding sufficient criteria are obtained. In addition, we simulate volumetric imaging data from 12 head-and-neck cancer patients and estimate the patient-specific responses to radiotherapy. Results indicate that radiosensitivity of proliferative cells is a critical factor that determines a successful radiotherapy. By comparison with previous simulation results, we find that the model presented in this paper is more suitable to describe the radiotherapy procedure of head-and-neck cancer. Finally, we discuss the influences of different radiotherapy strategies on therapeutic effect. The results show that treatment strategies with large dose or short treatment cycle can obtain better treatment effect. [ABSTRACT FROM AUTHOR]

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein. [ABSTRACT FROM AUTHOR]