Abstract
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.
Acknowledgments
I would like to thank my students, postdoctoral researchers, and collaborators for their efforts and contributions in establishing the landscape–flux theory and its applications for the non-equilibrium dynamical systems. They are Dr Li Xu, Mr. Kun Zhang, Dr Chunhe Li, Dr Haidong Feng, Dr Bo Han, Dr Dave Lepzelter, Dr Liufang Xu, Dr Feng Zhang, Dr Wei Wu, Mr. Han Yan, Mr. Zhedong Zhang, Mr. Lei Zhao, Mr. Bo Huang, Dr Xuefeng Xia, Dr Xidi Wang, Prof. Masaki Sasai, Prof. Zhirong Sun, Prof. Hualin Shi, Prof. Keun-Young Kim, and Prof. Erkang Wang. I also thank Dr Li Xu for the help in preparing references and figures for this review.
I would like to thank Prof. Peter G. Wolynes for his constant encouragement and insights. I thank Prof. Jose N. Onuchic for his supports. I also thank Prof. Hong Qian for useful discussions.
Notation
| = | system variables (concentrations, densities, etc.) |
| = | driving force |
| = | stochastic driving force |
| D | = | a scale factor representing the magnitude of the fluctuations |
| = | diffusion tensor or matrix |
| = | probability |
| = | probability flux |
| = | probability flux of steady state |
| = | the population potential landscape |
| = | the intrinsic potential landscape |
| = | the intrinsic flux velocity |
| = | Lagrangian |
| = | Hamiltonian |
| = | effective energies |
| k | = | kinetic rate constant |
| = | the chemical potential difference in terms of Gibbs free energy |
| = | the action |
| = | Lagrangian for each path |
| dl | = | infinitesimal displacement along 1D path |
| = | the discreterized target function |
| = | the probability distribution functional |
| = | probability current functional |
| = | the internal energy functional |
| = | steady-state probability velocity functional |
| = | functional variables the entropy |
| = | entropy change for the system |
| = | the heat dissipation |
| = | the change of the total entropy |
| = | the transition matrix in population space |
| = | the transition matrix in coherence space |
| = | the coupling transition matrix between population and coherence space |
| = | the reduced density matrix for non-equilibrium quantum system |
| = | the transfer matrix |
| = | the variance |
| = | Allele |
| = | genotype |
| = | the fitness of genotype |
| = | the mean fitness of population |
| = | the additive genetic variance |
| = | the effective input action potential of the neuron i |
| = | the capacitance |
| = | the resistance of the neuron i |
| = | the probability of the protein concentration when the gene is on or off |
| = | the production or synthesis rate when a regulation protein is bound to the gene |
| = | the production or synthesis rate when the regulation protein is not bound to the gene |
| ω | = | the adiabaticity parameter measuring the unbinding rate relative to the degradation rate |
Disclosure statement
No potential conflict of interest was reported by the authors.
Funding
The research studies presented in this review article were supported in part by National Science Foundation [USA: NSF-0447533, NSF-0926287, and NSF-0947767] and National Natural Science Foundation of China [NSFC-21190040, NSFC-11174105, NSFC-91225114, and NSFC-91430217].
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