Realization of Waddington's Metaphor: Potential Landscape, Quasi-potential, A-type Integral and Beyond

TL;DR

This paper systematically investigates and connects different mathematical realizations of Waddington's epigenetic landscape, clarifying their relationships and implications for biological systems.

Contribution

It establishes connections among potential landscape, quasi-potential, and SDE decomposition theories, clarifies their relationships, and discusses their implications in biological modeling.

Findings

Quasi-potential is the zero noise limit of the potential landscape.

Potential function in SDE decomposition coincides with the quasi-potential.

The A-type integral is more suitable for decomposed SDEs.

Abstract

Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to realize this metaphor in a rationalized way. We adopt comprehensive perspectives to systematically investigate three different but closely related realizations in recent literature: namely the potential landscape theory from the steady state distribution of stochastic differential equations (SDEs), the quasi-potential from the large deviation theory, and the construction through SDE decomposition and A-type integral.The connections among these theories are established in this paper. We demonstrate that the quasi-potential is the zero noise limit of the potential landscape. We also show that the potential function in the third proposal coincides with the quasi-potential. The most probable transition path by minimizing the Onsager-Machlup or Freidlin-Wentzell action functional is discussed as well. Furthermore, we compare the difference between local and global quasi-potential through the exchange of limit order for time and noise amplitude. As a consequence of such explorations, we arrive at the existence result for the SDE decomposition while deny its uniqueness in general cases. It is also clarified that the A-type integral is more appropriate to be applied to the decomposed SDEs rather than the original one. Our results contribute to a better understanding of existing landscape theories for biological systems.