Journal
JOURNAL OF THE ROYAL SOCIETY INTERFACE
Volume 9, Issue 77, Pages 3539-3553Publisher
ROYAL SOC
DOI: 10.1098/rsif.2012.0434
Keywords
multi-stable dynamical system; non-equilibrium dynamics; quasi-potential; state transition; epigenetic landscape; Freidlin-Wentzell theory
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Funding
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- Canadian Institutes of Health Research (CIHR)
- Alberta Innovates
- Institute for Systems Biology (ISB), Seattle
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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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