Mathematical modeling of gene expression: a guide for the perplexed biologist

Abstract

The detailed analysis of transcriptional networks holds a key for understanding central biological processes, and interest in this field has exploded due to new large-scale data acquisition techniques. Mathematical modeling can provide essential insights, but the diversity of modeling approaches can be a daunting prospect to investigators new to this area. For those interested in beginning a transcriptional mathematical modeling project, we provide here an overview of major types of models and their applications to transcriptional networks. In this discussion of recent literature on thermodynamic, Boolean, and differential equation models, we focus on considerations critical for choosing and validating a modeling approach that will be useful for quantitative understanding of biological systems.

Figure 1. Analytical modeling approaches used in gene regulation studies

A) Thermodynamic or fractional occupancy model of gene expression. The first column shows a simplified enhancer region with two binding sites for a repressor (R) and an activator (A). The mathematical formulation below represents binding efficiency for the repressor site. In the second column, all four possible states of this enhancer region are shown. The third column represents the probability of this state occurring, which is not simply ¼ but rather a function of the protein concentration and quality of the binding site(s). The fourth column indicates the efficiencies with which a particular state drives gene expression. This maybe a simple additive expression of activators minus repressors, or a more complex expression. The last column represents the total expressions coming from each state (the probability that a state will occur multiplied by the potential of this configuration of proteins) and their summation, which provides a measure of the total output of the cis-element. B) Differential equation model of gene expression. In this case regulatory relationship between two genes is depicted. Synthesis of gene 1 (G ₁) involves expression of mRNA (M ₁) and translation of protein (P₁), which regulates gene 2 (G ₂). Both mRNA and protein are subjects to turnover and protein is subject to diffusion. mRNA and protein synthesis, degradation and diffusion events are shown at left. This process can be modeled with reaction-diffusion equations as shown at right. Each molecular constituent is assigned such an equation. C) Boolean model of gene expression. The network describing the regulatory relationships among four proteins is shown; the directed arrows show activation, and the blunt arrows show repression. Starting from initial state, three temporal steps are demonstrated. In this model, protein turnover occurs in one time interval and repression is assumed to be dominant over activation. Here, superscripts [1] and [0] indicate active or inactive states respectively. (A color version of the figure is available online.)