Potential Energy Landscape and Robustness of A Gene Regulatory Network: Toggle Switch

Studying the gene regulatory networks is crucial for understanding the biological function of the genes. The underlying nature of gene regulatory networks has been explored by many experimental techniques. It has often been found that cellular networks are in general quite robust and perform their biological functions in the midst of environmental perturbations. There have recently been an increasing number of studies on the global topological structures of the networks. However, there are so far very few studies of why the network should be robust and perform the biological functions from the physical point of view. To probe the global physical properties, one often has to explore different parameters. Since the parameter space is huge, the issue of global robustness is hard to address directly from these local approaches. Here we will explore the nature of the network from another angle: we formulate the problem in terms of the potential energy function or potential energy landscape. If the potential landscape of the cellular network is known, the global properties can be explored. This is analogous to the fact that the global thermodynamic properties can be explored when knowing the inherent interaction potentials in a system. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This is realized by explicitly constructing the steady state probability of the gene switch in the protein concentration space in the presence of the intrinsic statistical fluctuations due to the small number of proteins in the cell. We have explored the global phase space for the system. We found the protein synthesis rate and unbinding rate of proteins to the gene are small relative to the protein degradation rate, the gene switch is mono-stable with only one stable basin of attraction. When both the protein synthesis rate and unbinding rate of proteins to the gene are large compared with the protein degradation rate, two global basins of attraction emerge for a toggle switch. These basins correspond to the stable biologically functional states. The energy barrier between the two basins determines the time scale of conversion from one to the other. We found as the protein synthesis rate and protein unbinding rate to the gene relative to the protein degradation rate become larger, the energy barrier becomes larger. This leads to the robustness of the biological basins of the gene switches. The technique used here is general and can be applied to explore the potential energy landscape of the gene networks. The potential energy landscape provides a framework to study the global nature of the gene regulatory network. The basins of attractions in the underlying landscape represent the biological functional states. The robustness of the network can be explained as the high barriers among the basins of attractions. By further exploring the underlying potential energy landscapes, more detail features of the network such as which genes and which regulatory links are crucial in function can be uncovered. This is important for identifying the key elements in regulating the diseases such as cancers.