The energy landscape theory has been widely applied to study the stochastic dynamics of biological systems. Different methods have been developed to quantify the energy landscape for gene networks, e.g., using Gaussian approximation (GA) approach to calculate the landscape by solving the diffusion equation approximately from the first two moments. However, how high-order moments influence the landscape construction remains to be elucidated. Also, multistability exists extensively in biological networks. So, how to quantify the landscape for a multistable dynamical system accurately, is a paramount problem. In this work, we prove that the weighted summation from GA (WSGA), provides an effective way to calculate the landscape for multistable systems and limit cycle systems. Meanwhile, we proposed an extended Gaussian approximation (EGA) approach by considering the effects of the third moments, which provides a more accurate way to obtain probability distribution and corresponding landscape. By applying our generalized EGA approach to two specific biological systems: multistable genetic circuit and synthetic oscillatory network, we compared EGA with WSGA by calculating the KL divergence of the probability distribution between these two approaches and simulations, which demonstrated that the EGA provides a more accurate approach to calculate the energy landscape.

Abstract 2

High-dimensional networks producing oscillatory dynamics are ubiquitous in biological systems. Unravelling the mechanism of oscillatory dynamics in biological networks with stochastic perturbations becomes paramountly significant. Although the classical energy landscape theory provides a tool to study this problem in multistable systems and explain cellular functions, it remains challenging to quantify the landscape for periodic oscillatory systems accurately. To tackle this challenge, we propose an approach called the diffusion decomposition of the Gaussian approximation (DDGA). We demonstrate the efficacy of the DDGA in quantifying the energy landscape of oscillatory systems and corresponding stochastic dynamics, in comparison with existing approaches. By further applying the DDGA to high-dimensional biological networks, we are able to uncover more intricate biological mechanisms efficiently, which deepens the understanding of cellular functions.