Trajectory-based energy landscapes of gene regulatory networks.

Multistability and natural biological variability can result in significant heterogeneity within a cell population, leading to challenges in understanding and modulating cell behavior. Energy landscapes can offer qualitatively intuitive visualizations of cell phenotype and facilitate a more quantitative understanding of cellular dynamics, but current methods for landscape generation are mathematically involved and often require specific system properties (e.g., ergodicity or independent gene/protein probability distributions) that do not always hold. Here, we present a simple kinetic Monte Carlo-based method for landscape generation from a system of ordinary differential equations using only simulation trajectories initialized throughout the phase space of interest. The resulting landscape produces three quantitative features relevant to understanding cell behavior: stability (reflected by the depth or potential of landscape valleys), velocity (representing average directional movement on the landscape), and variance in velocity (indicative of landscape positions with heterogeneous movements). We applied this method to a genetic toggle switch, a core decision-making network in binary cellular responses, to elucidate effects of biologically relevant intrinsic and extrinsic cues. Intrinsic noise, such as stochasticity in transcription-translation and differences in cell cycle position, manifests through changes in valley width and position, reflecting increased population heterogeneity and more probabilistic cell fate transitions. The landscapes also capture the effect of an external inducer, revealing a quantitative correlation between the rate of cell fate transition and the energy barrier above a threshold inducer concentration determined by the permissivity of the valley. Further, in tracking dynamically changing landscapes under time-varying external cues, we unexpectedly found that an oscillatory inducer input can modulate cell fate heterogeneity and lead to periodic cell fate transitions entrained to the input frequency, depending on the intrinsic degradation rate of the switch. The landscape generation approach outlined herein is generalizable to other network topologies and may provide new quantitative insights into their dynamics.