Quasi-potential landscape in complex multi-stable systems

TL;DR

This paper reviews methods for constructing a quasi-potential landscape in multi-stable systems, introduces a new vector field decomposition technique, and discusses its implications for understanding cell differentiation dynamics.

Contribution

It proposes a novel vector field decomposition method to compute a global quasi-potential function applicable to multiple attractors in complex systems.

Findings

The new method computes a quasi-potential equivalent to Freidlin-Wentzell potential for multiple attractors.

Existing methods are limited to pairwise attractor transitions, which this approach overcomes.

The approach enhances understanding of stability and transition pathways in developmental systems.

Abstract

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.