Efficient and flexible implementation of Langevin simulation for gene burst production

Gene expression involves bursts of production of both mRNA and protein, and the fluctuations in their number are increased due to such bursts. The Langevin equation is an efficient and versatile means to simulate such number fluctuation. However, how to include these mRNA and protein bursts in the Langevin equation is not intuitively clear. In this work, we estimated the variance in burst production from a general gene expression model and introduced such variation in the Langevin equation. Our approach offers different Langevin expressions for either or both transcriptional and translational bursts considered and saves computer time by including many production events at once in a short burst time. The errors can be controlled to be rather precise (<2%) for the mean and <10% for the standard deviation of the steady-state distribution. Our scheme allows for high-quality stochastic simulations with the Langevin equation for gene expression, which is useful in analysis of biological networks.