Quasi-potential landscape in complex multi-stable systems

Developmental dynamics of multicellular organism is a process that takes place in a multi-stable system in which each attractor state represents a cell type and attractor transitions correspond to cell differentiation paths. This new understanding has revived the idea of a quasi-potential landscape, first proposed by Waddington as a metaphor. To describe development one is interested in the “relative stabilities” of N attractors (N>2). Existing theories of state transition between local minima on some potential landscape deal with the exit in the transition between a pair attractor but do not offer the notion of a global potential function that relate more than two attractors to each other. Several ad hoc methods have been used in systems biology to compute a landscape in non-gradient systems, such as gene regulatory networks. Here we present an overview of the currently available methods, discuss their limitations and propose a new decomposition of vector fields that permit the computation of a quasi-potential function that is equivalent to the Freidlin-Wentzell potential but is not limited to two attractors. Several examples of decomposition are given and the significance of such a quasi-potential function is discussed.

💡 Research Summary

The paper addresses a fundamental challenge in modeling developmental dynamics of multicellular organisms: how to quantify the relative stability of many cell‑type attractors (N > 2) in a non‑gradient gene‑regulatory network. Classical potential‑landscape concepts, dating back to Waddington, work well for gradient systems where a scalar potential directly governs the flow. However, most realistic biological networks contain non‑conservative feedback loops, making the vector field non‑gradient and preventing the definition of a global potential that simultaneously relates more than two attractors. Existing approaches either treat each pair of attractors separately (using Kramers‑type escape rates) or employ ad‑hoc constructions that lack a rigorous theoretical foundation.

The authors propose a systematic decomposition of an arbitrary vector field f(x) into a gradient (conservative) part and a solenoidal (rotational) part: f(x) = −∇U(x) + J(x), with ∇·J = 0.
Here U(x) is the quasi‑potential they seek, while J(x) captures the non‑conservative circulation that does not affect escape probabilities in the small‑noise limit. By imposing a variational principle that minimizes the Freidlin‑Wentzell action functional, they show that the resulting U(x) coincides with the Freidlin‑Wentzell potential, which is known to govern the exponential scaling of transition rates in stochastic dynamical systems. Crucially, the formulation does not rely on pairwise comparisons; a single scalar field U(x) simultaneously encodes the depths of all N attractors, allowing direct ranking of their relative stabilities.

To compute U(x) in practice, the authors derive a stationary Fokker‑Planck‑type equation: ∇·