Simulations of stochastic biological phenomena

This Teaching Resource provides lecture notes, slides, and a student assignment for a two-part lecture that introduces stochastic modeling of biological systems. The first lecture uses biological examples to present the concept of cell-to-cell variability and makes the connection between the variability of single-cell measurements and concepts from statistical mechanics and probability theory. This section makes the point that for low copy number of a species, the usual differential equation formalism is no longer applicable and needs to be replaced by a probabilistic approach based on the so-called Master Equation. As an example, a simple model of gene transcription is discussed in detail, the different contributions to the relevant Master Equation are highlighted, and the equation itself is derived. The second lecture describes how, for more complex and biologically interesting applications, direct solution of the Master Equation becomes difficult. Gillespie's algorithm, which is used in most cases of biological interest, is then introduced as a practical alternative. The lecture delves into the crux of Gillespie's algorithm, which entails the drawing of two random numbers at each time step. It establishes the corresponding formalism, details the connection between chemical rate constants and Gillespie rates, and culminates in a description and explanation of a core MATLAB program for the transcriptional model considered in the first lecture. This core program, written for a single cell, is expanded by the students in the homework assignment to consider both transcription and translation.