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

Finding a multidimensional potential landscape is the key for addressing important global issues, such as the robustness of cellular networks. We have uncovered the underlying potential energy landscape of a simple gene regulatory network: a toggle switch. This was 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 explored the global phase space for the system. We found that the protein synthesis rate and the unbinding rate of proteins to the gene were small relative to the protein degradation rate; the gene switch is monostable with only one stable basin of attraction. When both the protein synthesis rate and the 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 biologically stable functional states. The potential 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 became larger, the potential energy barrier became larger. This also corresponded to systems with less noise or the fluctuations on the protein numbers. It 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.: Cellular networks are at the heart of systems biology at present. To understand how cellular networks function in these highly fluctuating environments, a global approach is needed. Here we provide a global framework, in terms of potential landscapes, for studying the gene regulatory networks in the presence of the intrinsic statistical fluctuations. We uncovered the underlying landscape for the network. We identified the basins of attraction of the landscape as the biological functional states. The potential barrier between the two basins determines the time scale of conversion from one to the other. The robustness of the biological functional states of the network, the gene switches in this case, can be guaranteed if the conversions among the basins of attraction are not frequent, or, in other words, the barriers among the basins are relatively large. More detailed features of the network, such as the key genes or regulating links relevant to diseases (i.e., cancers), can be uncovered from the underlying landscape. Our technique is general and can be applied to explore the potential landscape of more realistic gene networks. Furthermore, our approach can also be helpful in guiding the network optimal design for synthetic biology.