Potential and flux field landscape theory. I. Global stability and dynamics of spatially dependent non-equilibrium systems

Abstract

We established a potential and flux field landscape theory to quantify the global stability and dynamics of general spatially dependent non-equilibrium deterministic and stochastic systems. We extended our potential and flux landscape theory for spatially independent non-equilibrium stochastic systems described by Fokker-Planck equations to spatially dependent stochastic systems governed by general functional Fokker-Planck equations as well as functional Kramers-Moyal equations derived from master equations. Our general theory is applied to reaction-diffusion systems. For equilibrium spatially dependent systems with detailed balance, the potential field landscape alone, defined in terms of the steady state probability distribution functional, determines the global stability and dynamics of the system. The global stability of the system is closely related to the topography of the potential field landscape in terms of the basins of attraction and barrier heights in the field configuration state space. The effective driving force of the system is generated by the functional gradient of the potential field alone. For non-equilibrium spatially dependent systems, the curl probability flux field is indispensable in breaking detailed balance and creating non-equilibrium condition for the system. A complete characterization of the non-equilibrium dynamics of the spatially dependent system requires both the potential field and the curl probability flux field. While the non-equilibrium potential field landscape attracts the system down along the functional gradient similar to an electron moving in an electric field, the non-equilibrium flux field drives the system in a curly way similar to an electron moving in a magnetic field. In the small fluctuation limit, the intrinsic potential field as the small fluctuation limit of the potential field for spatially dependent non-equilibrium systems, which is closely related to the steady state probability distribution functional, is found to be a Lyapunov functional of the deterministic spatially dependent system. Therefore, the intrinsic potential landscape can characterize the global stability of the deterministic system. The relative entropy functional of the stochastic spatially dependent non-equilibrium system is found to be the Lyapunov functional of the stochastic dynamics of the system. Therefore, the relative entropy functional quantifies the global stability of the stochastic system with finite fluctuations. Our theory offers an alternative general approach to other field-theoretic techniques, to study the global stability and dynamics of spatially dependent non-equilibrium field systems. It can be applied to many physical, chemical, and biological spatially dependent non-equilibrium systems.

Parameter asymmetry and time‐scale separation in core genetic commitment circuits

“…Therefore, it is not only continuous nonlinear dynamics that is relevant, but it is also stochasticity induced by the paucity of some key molecular regulator(s) in the system. Thus, the notion of stochastic potential is of paramount importance (see Wang et al [13][14][15][16][17][18][19][20][21][22][23]). The role of energy in the establishment and maintenance of living systems at steady-states far from equilibrium is related, but is not addressed here (see Qian et al [24][25][26][27][28][29]).…”

Potential landscape and flux field theory for turbulence and nonequilibrium fluid systems

“…It is particularly suited for the study of the global dynamics and nonequilibrium thermodynamics of stochastic field systems governed by the Langevin and Fokker-Planck field equations [35,36,37]. The potential landscape and flux framework, which has its historical origin in the energy landscape theory in protein folding dynamics, was initially developed for nonequilibrium biological systems and has been applied extensively in that area and beyond [38,39,40,41].…”

Stability analysis of an autocatalytic protein model

“…[12][13][14][15] The genetic regulatory system can be modeled in terms of the probability distribution of molecule numbers, which follows a master equation. [2][3][4][5][7][8][9][19][20][21][22][23][24][25][26][27][28][29][30][31] At the limit where the number of molecules is large and the fluctuation of molecule numbers becomes negligible, the system can be described in terms of the molecule density, which follows a system of first-order differential equations. 2,[10][11][12][13][14][15][16]32,[32][33][34] This set of differential equations describes the time-dependent production and decay of various RNA and protein molecules, as well as binding of transcription factors to DNA that stimulate or inhibit gene transcription.…”

“…Depending on the network structure, the gene network may exhibit multistability, that is, there may be more than one steady state. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] Multistability forms the basis for explaining the response of the cell to the environment, and it is also important for understanding stem cell differentiation into different tissues. 17,18 The existence of a positive feedback loop in the regulatory network is necessary for multistability.…”