Quasi-potential landscape in complex multi-stable systems

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.

Figure 1.

Schematic quasi-potential U and the transitions among stable steady states (attractors) in a one-dimensional multi-stable dynamical system. The transition rate P_A→B is not determined by ΔU_AB but by . Similarly, the transition rate P_A→C is not determined by ΔU_AC but by .

Figure 2.

Curl is not necessarily a driving force for a limit circle. The vector field of a divergence-free dynamical system has open trajectories. The governing equations of the dynamical system are as follows: dx/dt =−y and dy/dt =−x. (Online version in colour.)

Figure 3.

(a) Vector field of the driving forces of the dynamic system in example 1. Note how the non-vanishing remainder components U_⊥ cause the vector field to be non-symmetric even if the underlying quasi-potential U^norm is symmetric. (b) The quasi-potential U^norm reconstructed from the normal decomposition and the positions of nine points for comparison in (c). (c) Different quasi-potentials constructed from the Helmholtz decomposition, the normal decomposition, −lnP decomposition and the Freidlin–Wentzell action function for comparison of the nine positions (states) in phase space. (Online version in colour.)

Figure 4.

(a) Quasi-potential function U^norm derived from the normal decomposition. The white circles represent attractors A, B, C, D; the grey circles denote the saddle points. (b) Vector field plot of the gradient component of the normal decomposition super-positioned on the contour plot of U^norm. (c) Vector field plot of the F_⊥, the remainder component of the normal decomposition super-positioned on the contour plot of U^norm. (d) Quasi-potential U^norm calculated from the normal decomposition and the U^prob calculated from the −ln P decomposition with various noise levels (D = 10, 15 and 20). (Online version in colour.)

Figure 5.

(a) The LAP for the attractor transition from x_A to x_D. The white dot denotes the starting point; the black dot is the end point. The attractors A, B, C, D are noted as in figure 4. V(t) is the Wentzell action function at every time step, and V is the total Wentzell action function at each time t. (b) The LAP from attractor from x_D to x_A is different from the LAP from x_A to x_D. Here the Wentzell action function is smaller than for the transition from x_A to x_D because the attractor state x_D has a ‘higher elevation’ than x_A in the quasi-potential ‘landscape’ U^norm. (c) The LAP for the attractor transition from x_A to x_C. It is clear that the Wentzell action function applies during the ‘uphill’ process, while it is zero during the ‘downhill’ process. (d) The LAP for the attractor transition from x_C to x_A. The Wentzell action function is smaller than for the transition from x_A to x_C because attractor x_C is ‘higher’ than x_A in the quasi-potential U^norm. (Online version in colour.)

Figure 6.

(a) Quasi-potential U^prob at the attractors and ‘saddle’ points derived from the –lnP decomposition with noise level D = 20. (b) Quasi-potential U^norm at the same points derived from the normal decomposition. Attractors A, B, C, D are denoted as in figure 4. (c) Potential barriers ΔU that need to be overcome for the transition from each attractor to every other one. ΔU were calculated using the quasi-potential function from the normal decomposition U^norm and –lnP decomposition U^prob based on the topology of attractors and ‘saddle’ points in between. Then ΔU were compared with the Freidlin–Wentzell action functions V calculated from numeric minimization. (Online version in colour.)