Delay dynamics: biochemical importance, large deviations, and statistical coherence

合成生物学的一个主要目标是为医学和工业应用创造实用的、工程化的基因电路。数学在这种电路的设计中起着至关重要的作用，它为应用科学家和工程师提供了描述如何实现预期设计目标的预测模型。这些预测模型必须考虑到细胞内功能性蛋白质合成的“延迟”。最近的高分辨率实验室数据和理论研究表明，这种类型的延迟有助于自然发生的生物体获得生物学上理想的结果：它可以加速信号传导，稳定生化网络，并产生可调谐的振荡。首席研究员和他的同事们发展了数学理论，以帮助解释延迟如何塑造遗传电路的动力学，并研究某些相关的无限维模型。合成生物学家利用这一理论来设计和实现用于医学和工业的合成遗传电路。“延迟”现在被认为是遗传调控网络和生物化学网络的一个重要组成部分。考虑到合成生物学技术创新的步伐和高分辨率生物数据的可用性，需要一种适用于延迟系统动力学的数学理论。首席研究员（PI）和他的同事利用并扩展了概率论和抽象遍历理论的技术，在多个层面上研究随机性和延迟之间的相互作用。首先，PI和他的同事研究了延迟随机过程的罕见事件问题。具体的主题包括大偏差率、最佳过渡路径、平均首次通过时间、数值模拟和重要抽样。此外，PI和他的同事开发了一种方法，该方法可以使用时间序列数据来估计生化延迟的分布。这种方法是有价值的，因为延迟分布是难以测量的实验。其次，PI和他的合作者研究了在取延迟随机模型的热力学极限后可能出现的随机性。这种随机性可称为“统计一致性”。它源于确定性极限中的动态不稳定性。当取时滞随机微分方程的热力学极限时，得到时滞微分方程（DDE）。从遍历论的观点来看，DDE可以看作是在合适的函数巴拿赫空间上生成解半群的泛函微分方程（FDE）。类似地，某些演化偏微分方程在合适的Banach空间上生成解半群。这些解半群是无限维动力系统；PI和他的同事们研究了FDE和PDE背景下的统计一致性。他们通过分离产生非均匀双曲动力学的特定几何机制来做到这一点。在这方面，必须发展关于Sinai-Ruelle-Bowen （SRB）措施的新概念。SRB测量形成了统计相干性的基石，因为它们控制着函数空间中大型轨道集的渐近分布。至关重要的是，可检查的条件，意味着存在SRB措施的具体fde和pde被开发。统计性质，如中心极限定理的行为，相关性的衰减，不变性原理，极值统计，和随机稳定性的研究。这项工作为生化系统的建模、实验研究和设计提供了信息，并向抽象的动力系统社区介绍了大量新的物理相关示例。

A major goal of synthetic biology is the creation of practical, engineered genetic circuits for medical and industrial applications. Mathematics plays a vital role in the design of such circuits by providing applied scientists and engineers with predictive models that describe how to achieve desired design goals. These predictive models must take into account the 'delay' that results from the synthesis of functional protein inside cells. Recent high-resolution laboratory data and theoretical investigations have shown that this type of delay helps naturally-occurring organisms achieve biologically desirable results: it can accelerate signaling, stabilize biochemical networks, and create tunable oscillations. The principal investigator and his colleagues develop the mathematical theory needed to help explain how delay shapes the dynamics of genetic circuits as well as study certain associated infinite-dimensional models. Synthetic biologists use this theory to inform the design and implementation of synthetic genetic circuits for use in medicine and industry.'Delay' is now recognized as an important component of genetic regulatory networks and biochemical networks more generally. Given the pace of technological innovation in synthetic biology and the availability of high-resolution biological data, there is need for a mathematical theory applicable to the dynamics of delay systems. The principal investigator (PI) and his colleagues leverage and extend techniques from both probability theory and abstract ergodic theory to study the interplay between stochasticity and delay on multiple levels. First, the PI and his colleagues study the rare events problem for delay stochastic processes. Specific topics include large deviations rates, optimal transition paths, mean first passage times, numerical simulation, and importance sampling. Further, the PI and his colleagues develop a method by which time series data may be used to estimate the distribution of biochemical delay. Such a method is valuable because delay distributions are difficult to measure experimentally. Second, the PI and his collaborators study the stochasticity that can arise after one takes thermodynamic limits of delay stochastic models. Such stochasticity may be called 'statistical coherence.' it arises from dynamical instabilities in the deterministic limits. When one takes the thermodynamic limit of a delay stochastic differential equation, one obtains a delay differential equation (DDE). From the ergodic-theoretic point of view, a DDE may be viewed as a functional differential equation (FDE) that generates a solution semigroup on a suitable Banach space of functions. Similarly, certain evolution partial differential equations (PDEs) generate solution semigroups on suitable Banach spaces. These solution semigroups are infinite-dimensional dynamical systems; the PI and his colleagues study statistical coherence in the FDE and PDE contexts. They do this by isolating specific geometric mechanisms that produce nonuniformly hyperbolic dynamics. In this context new notions of Sinai-Ruelle-Bowen (SRB) measure must be developed. SRB measures form a cornerstone of statistical coherence because they govern the asymptotic distribution of large sets of orbits in the function space. Crucially, checkable conditions that imply the existence of SRB measures for concrete FDEs and PDEs are developed. Statistical properties such as central limit theorem behavior, decay of correlations, invariance principles, extreme value statistics, and stochastic stability are investigated. This work informs the modeling, experimental study, and design of biochemical systems and introduces a large new class of physically relevant examples to the abstract dynamical systems community.