Quasi-potential landscape in complex multi-stable systems

Abstract

The developmental dynamics of multicellular organisms is a process that takes place in a multistable 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 quasipotential 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.

Construction of an Effective Landscape for Multistate Genetic Switches

“…The cellular decisions are typically regulated by coupled genetic circuits that operate as multistate switches, i.e., circuits whose dynamics permits the coexistence of multiple states [2,3], each of which corresponds to a distinct cell fate or phenotype.…”

Balls, cups, and quasi-potentials: quantifying stability in stochastic systems

“…Knowledge about the potential function thus provides information about the steady-state probability distribution, mean fi rst passage times, and transition frequencies, motivating its use as a stability metric (Wang et al 2011, Zhou et al 2012. The potential function is especially useful because it does not depend on the noise intensity σ (in contrast to the steady-state probability distribution and mean fi rst passage times; see Appendix S1: Section S4 ).…”

“…In subsequent decades, the fl urry of research on alternative stable states largely overlooked this insight. Recently, the quasi-potential has been embraced by researchers analyzing models in other areas of biology, although it often appears under other names, and is disconnected from the Freidlin-Wentzell formulation (but see Zhou et al 2012 ). These applications include gene regulatory networks (Zhou et al 2012, Lv et al 2014, neural networks (Yan et al 2013 ), and evolution (Wang et al 2011, Zhang et al 2012.…”

“…Recently, the quasi-potential has been embraced by researchers analyzing models in other areas of biology, although it often appears under other names, and is disconnected from the Freidlin-Wentzell formulation (but see Zhou et al 2012 ). These applications include gene regulatory networks (Zhou et al 2012, Lv et al 2014, neural networks (Yan et al 2013 ), and evolution (Wang et al 2011, Zhang et al 2012. Very recently, it has been applied to a predator-prey system (Xu et al 2014 ), and with countless other possibilities for application, we argue that the quasi-potential is poised to become a major quantitative tool in ecology.…”

Criticality in Cell Differentiation

“…In the case of a one-variable system described by the deterministic rate equation, ௗ௫ ௗ௧ = ‫,)ݔ(݂‬ the potential function U(x) is given by ‫)ݔ(݂‬ = − ௗ ௗ௫ , i.e., U(x) can be computed by integrating f(x) over x in the appropriate range of x values. This method of computing the potential function cannot, however, be generalized to the case of a multi-variable system [20]. In the stochastic description, the valleys of the potential landscape are the regions in state space in which the steady state probability distribution is maximal.…”