A Monte Carlo method for in silico modeling and visualization of Waddington’s epigenetic landscape with intermediate details

Abstract

Waddington’s epigenetic landscape is a classic metaphor for describing the cellular dynamics during the development modulated by gene regulation. Quantifying Waddington’s epigenetic landscape by mathematical modeling would be useful for understanding the mechanisms of cell fate determination. A few computational methods have been proposed for quantitative modeling of landscape; however, to model and visualize the landscape of a high dimensional gene regulatory system with realistic details is still challenging. Here, we propose a Monte Carlo method for modeling the Waddington’s epigenetic landscape of a gene regulatory network (GRN). The method estimates the probability distribution of cellular states by collecting a large number of time-course simulations with random initial conditions. By projecting all the trajectories into a2-dimensional plane of dimensions𝑖and𝑗, we can approximately calculate the quasi-potential𝑈(𝑥𝑖,𝑥𝑗,∗) = −ln𝑃(𝑥𝑖,𝑥𝑗,∗), where𝑃(𝑥𝑖,𝑥𝑗,∗)is the estimated probability of an equilibrium steady state or a non-equilibrium state. Compared to the state-of-the-art methods, our Monte Carlo method can quantify the global potential landscape (or emergence behavior) of GRN for a high dimensional system. The potential landscapes show that not only attractors represent stability, but the paths between attractors are also part of the stability or robustness of biological systems. We demonstrate the novelty and reliability of our method by plotting the potential landscapes of a few published models of GRN.