Potential landscape and flux framework of nonequilibrium networks: robustness, dissipation, and coherence of biochemical oscillations

Abstract

We established a theoretical framework for studying nonequilibrium networks with two distinct natures essential for characterizing the global probabilistic dynamics: the underlying potential landscape and the corresponding curl flux. We applied the idea to a biochemical oscillation network and found that the underlying potential landscape for the oscillation limit cycle has a distinct closed ring valley (Mexican hat-like) shape when the fluctuations are small. This global landscape structure leads to attractions of the system to the ring valley. On the ring, we found that the nonequilibrium flux is the driving force for oscillations. Therefore, both structured landscape and flux are needed to guarantee a robust oscillating network. The barrier height separating the oscillation ring and other areas derived from the landscape topography is shown to be correlated with the escaping time from the limit cycle attractor and provides a quantitative measure of the robustness for the network. The landscape becomes shallower and the closed ring valley shape structure becomes weaker (lower barrier height) with larger fluctuations. We observe that the period and the amplitude of the oscillations are more dispersed and oscillations become less coherent when the fluctuations increase. We also found that the entropy production of the whole network, characterizing the dissipation costs from the combined effects of both landscapes and fluxes, decreases when the fluctuations decrease. Therefore, less dissipation leads to more robust networks. Our approach is quite general and applicable to other networks, dynamical systems, and biological evolution. It can help in designing robust networks.

Fig. 1.

Wiring diagram. (A) Cyclin fluctuations during the cell cycle in budding yeast. (B) The gene network. Gene X (representing CLN/CDC28) is self-activating and inhibits gene Y (representing CLB/CDC28) degradation. Y inhibits the synthesis of X.

Fig. 2.

Phase diagram for the network.

Fig. 3.

Potential energy landscape with parameter b = 0.1, c = 100 with Mexican hat like closed ring valley shape (diffusion coefficient D = 0.001) (A) and with shallow shape (D = 1.0) (B). The blue arrows represent the flux, and the white arrows represent the force from negative gradient of the energy landscape.

Fig. 4.

Flux and gradient potential force. Shown are the vector graphs of the flux (A), the residue force (C), the force from negative gradient of the energy landscape (E), and the direction of those forces with diffusion coefficient D = 0.001 (B, D, and F).

Fig. 5.

Barrier heights and escape time. (A) Barrier Heights (U_fix − U_min) and (U_fix − U_max) versus diffusion coefficient for b = 0.1, c = 100. (B) Escape time versus barrier heights for different diffusion coefficients.

Fig. 6.

Entropy production. (A) Entropy production rate versus diffusion coefficient for b = 0.1, c = 100. (B) Entropy production rate versus barrier heights.

Fig. 7.

Period distribution against fluctuations. (A and B) Period distribution (A) and variance (B) with different diffusion coefficients D for b = 0.1, c = 100. (C) Diffusion D and entropy production rate EPR versus standard deviation σ. (D) Barrier heights versus σ.

Fig. 8.

Amplitude distribution against fluctuations. (A) Amplitude distribution of different D for x₂. (B) Standard deviation σ of amplitude versus diffusion D.

Fig. 9.

Coherence against fluctuations. (A) Diffusion coefficient D and entropy production rate versus phase coherence. (B) Barrier heights versus phase coherence.