Potential energy landscape and robustness of a gene regulatory network: toggle switch

Abstract

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.

Figure 1. Toggle Switch: Protein A Represses Gene B (Dotted Line), and Protein B Represses Gene A (Solid Line)

Figure 2. Probability C That Genes Are in the Active State as a Function of X_ad = (g ₁ + g ₀)/2k_A for a Symmetric Switch Showing the Bifurcation

Exact moment equation solutions are compared with Poisson ansatz solutions 0 < X_ad(= g ₁/2) < 100 , k_A = k_B = 1, f_A = f_B = 0.5, h_A = h_B = f_A/500, and g _A0 = g _B0 = 0.

Figure 3. Steady State Probability of Symmetric Toggle Switch (Long Time Limit) as a Function of the Number of Protein A and the Number of Protein B for Different X_ad = g _A1/2k_A

The other parameters are the same as Figure 2.

Figure 4. Potential Energy of Symmetric Toggle Switch as a Function of the Number of Protein A and the Number of Protein B for Different X_ad = g _A1/2k_A

The other parameters are the same as Figure 2.

Figure 5. The Time Scale of the Transition between the Two Stable Minimum Basins of Attractions as a Function of ω = f_A/k_A and X_ad = g _A1/2k_A

The other parameters are the same as Figure 2.

Figure 6. The Time Evolution of the Probability of Symmetric Toggle Switch as a Function of the Numbers of Protein A and the Numbers of Protein B

X_ad = 60, and the other parameters are the same as Figure 2. t = 0, 10, 55, 60, 70, 100.

Figure 7. The Time Evolution of the Potential of Symmetric Toggle Switch as a Function of the Numbers of Protein A and the Numbers of Protein B

X_ad = 60, and the other parameters are the same as Figure 2. t = 10, 55, 60, 70, 100.