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.

中文翻译：

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非平衡动力系统的景观和通量理论在生物学中的应用

我们回顾了最近开发的非平衡动力系统的景观和通量理论。我们指出，非平衡系统相关动力学的全局性质由两个关键因素决定：潜在的景观，重要的是，卷曲概率通量。景观 (U) 反映了状态 (P) () 的概率，并提供了系统的全局特征和稳定性度量。卷曲通量项衡量详细平衡被打破的程度，它是除景观梯度之外的非平衡动力学的两个主要驱动力之一。平衡动力学类似于电场中的电子运动，而非平衡动力学类似于电场和磁场中的电子运动。景观和通量理论有许多有趣的结果，包括（1）不可逆的动力学路径不一定穿过景观鞍点；(2) 新马鞍上的非平衡过渡态理论在小但有限涨落的最优路径上；(3) 非平衡动力系统的广义涨落-耗散关系，其中响应函数不仅等于平衡情况下单独的稳态涨落，而且旋度通量在维持稳态方面有额外贡献状态; (4) 非平衡热力学，其中自由能变化不仅仅等于熵产生，就像在平衡情况下一样，而且非零卷曲通量在维持稳态方面也有额外的管家贡献; (5) 规范理论和几何联系，其中通量被发现是规范场曲和拓扑相的起源，类似于量子力学中的贝里相；(6) 耦合景观，其中可以使用景观和通量理论分析非平衡动力学中多个景观的非绝热性，并且从非零旋度通量中出现涡流；(7) 随机空间动力学，其中景观和通量理论可以推广到非平衡场理论。我们提供了生物系统的具体例子来展示景观和通量理论的新见解。其中包括 (1) 细胞周期模型，其中景观将系统吸引到振荡吸引子，而通量驱动振荡环上的相干运动，细胞周期的不同阶段被确定为循环路径上的局部盆地，生物检查点被确定为细胞周期路径上局部盆地之间的局部屏障或过渡状态；(2) 干细胞分化，其中 Waddington 发展景观以及分化和重编程路径可以量化；(3) 癌症生物学，其中癌症可以被描述为具有多种细胞状态的疾病，癌症状态和正常状态可以量化为基础景观上的吸引力盆地，而正常状态和癌症状态之间的转变可以量化作为两个吸引子之间的过渡；(4) 进化，其中可以使用等位基因频率相关选择的具体例子量化除 Wright 和 Fisher 之外的更一般的进化动力学；(5) 生态，其中捕食者-猎物、合作和竞争的景观和通量以及全球稳定性被量化；(6) 神经网络，其中一般不对称连接被考虑用于学习和记忆，基因自我调节器，其中基因表达的非绝热动态可以用扩展维度的景观和通量来描述并进行分析处理；(7) 混沌奇异吸引子，其中通量对混沌动力学至关重要；(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(5) 生态，其中捕食者-猎物、合作和竞争的景观和通量以及全球稳定性被量化；(6) 神经网络，其中一般不对称连接被考虑用于学习和记忆，基因自我调节器，其中基因表达的非绝热动态可以用扩展维度的景观和通量来描述并进行分析处理；(7) 混沌奇异吸引子，其中通量对混沌动力学至关重要；(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(5) 生态，其中捕食者-猎物、合作和竞争的景观和通量以及全球稳定性被量化；(6) 神经网络，其中一般不对称连接被考虑用于学习和记忆，基因自我调节器，其中基因表达的非绝热动态可以用扩展维度的景观和通量来描述并进行分析处理；(7) 混沌奇异吸引子，其中通量对混沌动力学至关重要；(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(6) 神经网络，其中一般不对称连接被考虑用于学习和记忆，基因自我调节器，其中基因表达的非绝热动态可以用扩展维度的景观和通量来描述并进行分析处理；(7) 混沌奇异吸引子，其中通量对混沌动力学至关重要；(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(6) 神经网络，其中一般不对称连接被考虑用于学习和记忆，基因自我调节器，其中基因表达的非绝热动态可以用扩展维度的景观和通量来描述并进行分析处理；(7) 混沌奇异吸引子，其中通量对混沌动力学至关重要；(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。(8)空间发展，可以用空间景观来描述其形成过程和格局。我们还给出了该理论的哲学含义和未来研究的展望。