Extension of Leap Condition in Approximate Stochastic Simulation Algorithms of Biological Networks

Abstract

In the biological systems, Monte Carlo approaches are used to provide the stochastic simulation of the chemical reactions. The major stochastic simulation algorithms (SSAs) are the direct method, also known as the Gillespie algorithm, the first reaction method and the next reaction method. While these methods give accurate generation of the results, they are computationally demanding for large complex systems. To increase the computational efficiency of SSAs, approximate SSAs can be option. The approximate methods rely on the leap condition. This condition means that the propensity function during the time interval $ t $ to $[ t+\tau ]$ should not be altered for the chosen time step $\tau$. Here, to proceed with the system's history axis from one time step to the next, we compute how many times each reaction can be realized in each small time interval $\tau$ so that we can observe plausible simultaneous reactions. Hence, this study aims to generate a realistic and close confidence interval for the parameter which denotes the underlying numbers of simultaneous reactions in the system by satifying the leap condition. For this purpose, the poisson $\tau$-leap algorithm and the approximate Gillespie algorithm, as the extension of the Gillespie algorithm, are handled. In the estimation for the associated parameters in both algorithms, we derive their maximum likelihood estimators, moment estimatora and bayesian estimators. From the derivations, we theoretically show that our novel confidence intervals are narrower than the current confidence intervals under the leap condition.