Quasi-potential landscape in complex multi-stable systems |
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Authors: | Joseph Xu Zhou M. D. S. Aliyu Erik Aurell Sui Huang |
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Affiliation: | 1.Institute for Systems Biology, Seattle, WA, USA;2.Institute for Biocomplexity and Informatics, University of Calgary, Calgary, Alberta, Canada;3.Mathematics and Statistics Department, University of Calgary, Calgary, Alberta, Canada;4.Department of Computational Biology, KTH, Stockholm, Sweden;5.Kavli Institute for Theoretical Physics China, CAS, Beijing, People''s Republic of China |
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Abstract: | The developmental dynamics of multicellular organisms 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 part in the transition between two attractors in pair-attractor systems but do not offer the notion of a global potential function that relates 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 currently available methods, discuss their limitations and propose a new decomposition of vector fields that permits 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. |
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Keywords: | multi-stable dynamical system, non-equilibrium dynamics, quasi-potential, state transition, epigenetic landscape, Freidlin– Wentzell theory |
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