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首页> 外文期刊>The Journal of Chemical Physics >Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond
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Construction of the landscape for multi-stable systems: Potential landscape, quasi-potential, A-type integral and beyond

机译：多稳态系统的景观建设：潜在的景观，准势，A型积分和超越

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摘要

Motivated by the famous Waddington's epigenetic landscape metaphor in developmental biology, biophysicists and applied mathematicians made different proposals to construct the landscape for multi-stable complex systems. We aim to summarize and elucidate the relationships among these theories from a mathematical point of view. We systematically investigate and compare three different but closely related realizations in the recent literature: the Wang's potential landscape theory from steady state distribution of stochastic differential equations (SDEs), the Freidlin-Wentzell quasi-potential from the large deviation theory, and the construction through SDE decomposition and A-type integral. We revisit that the quasi-potential is the zero noise limit of the potential landscape, and the potential function in the third proposal coincides with the quasi-potential. We compare the difference between local and global quasi-potential through the viewpoint of exchange of limit order for time and noise amplitude. We argue that local quasi-potentials are responsible for getting transition rates between neighboring stable states, while the global quasi-potential mainly characterizes the residence time of the states as the system reaches stationarity. The difference between these two is prominent when the transitivity property is broken. The most probable transition path by minimizing the Onsager-Machlup or Freidlin-Wentzell action functional is also discussed. As a consequence of the established connections among different proposals, we arrive at the novel result which guarantees the existence of SDE decomposition while denies its uniqueness in general cases. It is, therefore, clarified that the A-type integral is more appropriate to be applied to the decomposed SDEs rather than its primitive form as believed by previous researchers. Our results contribute to a deeper understanding of landscape theories for biological systems. (C) 2016 AIP Publishing LLC.
机译：由着名的Waddington的表观遗工隐喻在发育生物学中，生物物理学家和应用数学家制作了不同的建议，构建了多稳定复杂系统的景观。我们的目标是总结和阐明这些理论之间的关系，从数学的角度来看。我们在最近的文献中系统地调查并比较三种不同但密切相关的实现：王的潜在景观理论从随机微分方程（SDES）的稳态分布，来自大偏差理论的弗赖林 - Wentzell准势，以及通过SDE分解和A型积分。我们重新审视了准潜力是潜在景观的零噪声限制，第三提案中的潜在功能与准电位一致。通过对时间和噪声幅度的限制顺序交换的观点来看，我们将本地和全球准潜力之间的差异进行比较。我们认为，当地的准潜力负责在邻国稳定状态之间获得转换率，而全球准势主要是当系统达到平稳性时州的停留时间。当传递性财产破坏时，这两个之间的差异是突出的。还讨论了最小化onSager-Machlup或Freidlin-Wenzell动作功能来最可能的过渡路径。由于不同建议之间的建立联系，我们到达了新颖的结果，保证了SDE分解的存在，同时否定了其唯一性。因此，阐明了α型积分更适合于应用于分解的SDES而不是其原始形式，以便以前的研究人员所捕获。我们的结果有助于更深入地了解生物系统的景观理论。 （c）2016 AIP发布LLC。

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