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Motivated by the famous Waddington's epigenetic landscape metaphor indevelopmental biology, biophysicists and applied mathematicians made differentproposals to realize this metaphor in a rationalized way. We adoptcomprehensive perspectives to systematically investigate three different butclosely related realizations in recent literature: namely the potentiallandscape theory from the steady state distribution of stochastic differentialequations (SDEs), the quasi-potential from the large deviation theory, and theconstruction through SDE decomposition and A-type integral.The connectionsamong these theories are established in this paper. We demonstrate that thequasi-potential is the zero noise limit of the potential landscape. We alsoshow that the potential function in the third proposal coincides with thequasi-potential. The most probable transition path by minimizing theOnsager-Machlup or Freidlin-Wentzell action functional is discussed as well.Furthermore, we compare the difference between local and global quasi-potentialthrough the exchange of limit order for time and noise amplitude. As aconsequence of such explorations, we arrive at the existence result for the SDEdecomposition while deny its uniqueness in general cases. It is also clarifiedthat the A-type integral is more appropriate to be applied to the decomposedSDEs rather than the original one. Our results contribute to a betterunderstanding of existing landscape theories for biological systems.

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