Landscape and flux theory of non-equilibrium dynamical systems with application to biology

We present a review of the recently developed landscape and flux theory for non-equilibrium dynamical systems. We point out that the global natures of the associated dynamics for non-equilibrium system are determined by two key factors: the underlying landscape and, importantly, a curl probability flux. The landscape (U) reflects the probability of states (P) () and provides a global characterization and a stability measure of the system. The curl flux term measures how much detailed balance is broken and is one of the two main driving forces for the non-equilibrium dynamics in addition to the landscape gradient. Equilibrium dynamics resembles electron motion in an electric field, while non-equilibrium dynamics resembles electron motion in both electric and magnetic fields. The landscape and flux theory has many interesting consequences including (1) the fact that irreversible kinetic paths do not necessarily pass through the landscape saddles; (2) non-equilibrium transition state theory at the new saddle on the optimal paths for small but finite fluctuations; (3) a generalized fluctuation–dissipation relationship for non-equilibrium dynamical systems where the response function is not just equal to the fluctuations at the steady state alone as in the equilibrium case but there is an additional contribution from the curl flux in maintaining the steady state; (4) non-equilibrium thermodynamics where the free energy change is not just equal to the entropy production alone, as in the equilibrium case, but also there is an additional house-keeping contribution from the non-zero curl flux in maintaining the steady state; (5) gauge theory and a geometrical connection where the flux is found to be the origin of the gauge field curvature and the topological phase in analogy to the Berry phase in quantum mechanics; (6) coupled landscapes where non-adiabaticity of multiple landscapes in non-equilibrium dynamics can be analyzed using the landscape and flux theory and an eddy current emerges from the non-zero curl flux; (7) stochastic spatial dynamics where landscape and flux theory can be generalized for non-equilibrium field theory. We provide concrete examples of biological systems to demonstrate the new insights from the landscape and flux theory. These include models of (1) the cell cycle where the landscape attracts the system down to an oscillation attractor while the flux drives the coherent motion on the oscillation ring, the different phases of the cell cycle are identified as local basins on the cycle path and biological checkpoints are identified as local barriers or transition states between the local basins on the cell-cycle path; (2) stem cell differentiation where the Waddington landscape for development as well as the differentiation and reprogramming paths can be quantified; (3) cancer biology where cancer can be described as a disease of having multiple cellular states and the cancer state as well as the normal state can be quantified as basins of attractions on the underlying landscape while the transitions between normal and cancer states can be quantified as the transitions between the two attractors; (4) evolution where more general evolution dynamics beyond Wright and Fisher can be quantified using the specific example of allele frequency-dependent selection; (5) ecology where the landscape and flux as well as the global stability of predator–prey, cooperation and competition are quantified; (6) neural networks where general asymmetrical connections are considered for learning and memory, gene self-regulators where non-adiabatic dynamics of gene expression can be described with the landscape and flux in expanded dimensions and analytically treated; (7) chaotic strange attractor where the flux is crucial for the chaotic dynamics; (8) development in space where spatial landscape can be used to describe the process and pattern formation. We also give the philosophical implications of the theory and the outlook for future studies.