Modeling stochasticity and variability in gene regulatory networks

Variability at the molecular level, defined as the phenotypic differences within a genetically identical population of cells exposed to the same environmental conditions, has been observed experimentally [1-4]. Understanding mechanisms that drive variability in molecular networks is an important goal of molecular systems biology, for which mathematical modeling can be very helpful. Different modeling strategies have been used for this purpose and, depending on the level of abstraction of the mathematical models, there are several ways to introduce stochasticity. Dynamic mathematical models can be broadly divided into two classes: continuous, such as systems of differential equations (and their stochastic variants) and discrete, such as Boolean networks and their generalizations (and their stochastic variants). This article will focus on stochasticity and discrete models.Discrete models do not require detailed information about kinetic rate constants and they tend to be more intuitive. In turn, they only provide qualitative information about the system. The most general setting is as follows. Network nodes represent genes, proteins, and other molecular components of gene regulation, while network edges describe biological interactions among network nodes that are given as logical rules representing their interactions. Time in this framework is implicit and progresses in discrete steps. More formally, let x1, ..., xn be variables, which can take values in finite sets X1, ..., Xn, respectively. Let X = X1 × ？ × Xn be the Cartesian product. A discrete dynamical system (DDS) in the variables x1, ..., xn is a functionwhere each coordinate function fi: X → Xi is a function in a subset of {x1, ..., xn}. Dynamics is generated by iteration of f, and different update schemes can be used for this purpose. As an example, if Xi = {0, 1} for all i, then each fi is a Boolean rule and f is a Boolean network where all the variables are updated simultaneously. We will assume that each