Robust dynamics in minimal hybrid models of genetic networks

Abstract

Many gene-regulatory networks necessarily display robust dynamics that are insensitive to noise and stable under evolution. We propose that a class of hybrid systems can be used to relate the structure of these networks to their dynamics and provide insight into the origin of robustness. In these systems, the genes are represented by logical functions, and the controlling transcription factor protein molecules are real variables, which are produced and destroyed. As the transcription factor concentrations cross thresholds, they control the production of other transcription factors. We discuss mathematical analysis of these systems and show how the concepts of robustness and minimality can be used to generate putative logical organizations based on observed symbolic sequences. We apply the methods to control of the cell cycle in yeast.

Figure 1.

Analysis of the three-gene repressilator system described in table 1 and shown in panel (a). The state transition diagram in (b) has a one-to-one correspondence between edge directions and truth-table values as indicated by the edge labellings. The dynamics in equation (3.1) with (c)  and (d) n = 10. x₁, solid line; x₂, dashed line; x₃, dashed-dotted line.

Figure 2.

Integrated dynamics of the robust, minimal network identified from the on/off state sequence in table 2 with all thresholds (θ_i) fixed at 0.5 and all degradation rates (γ_i) set at 1. The explicit regulatory structure of the minimal network is shown in table 4.