Power-law scaling of the depth of potential wells in multistable switches from feedback-regulated networks

The depth of a potential well plays a critical role in noise-assisted rate processes. Prevailing qualitative understanding suggests that the sizes of the fluctuations relative to the size of the basin of attraction in the bifurcation diagram dictate the possibility of noise-driven cellular fate transitions regulated by multistable switches. However, the quantitative relation between the size of basins of attraction and the depth of the wells in the pseudopotential energy of the dynamical systems is unknown. We show that, in multistable switches due to saddle-node bifurcations, the depth of the wells follows power-law scaling with the size of the basins of attraction, with the scaling exponent, α, ranging between 2.5 and 3.0 across various models and parameter combinations. Power-law scaling also holds for the well depth with the distance from the bifurcation point, with the scaling exponent, β, ranging between 1.4 and 1.8. By investigating various models of bi- and tristability with random parameter sampling, we report median scaling exponents of α[over ¯]=2.85±0.12 and β[over ¯]=1.5±0.08. Scaling laws provide a route to determine the well depth, in relative scale, from the bifurcation diagram, bypassing the challenging task of direct calculation of pseudopotential energy in multidimensional dynamical systems.