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

合成生物学的一个主要目标是为医疗和工业应用创造实用的工程遗传电路。 数学在此类电路的设计中发挥着至关重要的作用，它为应用科学家和工程师提供了描述如何实现所需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实现生物学上期望的结果：它可以加速信号传导、稳定生化网络并产生可调振荡。 首席研究员和他的同事们开发了所需的数学理论，以帮助解释延迟如何塑造遗传电路的动力学，以及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基因电路。“延迟”现在被认为是基因调控网络和生物化学网络的重要组成部分。 鉴于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，需要一个适用于延迟系统动力学的数学理论。 首席研究员（PI）和他的同事们利用和扩展了概率论和抽象遍历理论的技术，以研究多个层面上的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，数值模拟和重要性抽样。 此外，PI和他的同事开发了一种方法，通过该方法，时间序列数据可以用于估计生化延迟的分布。 这种方法是有价值的，因为延迟分布是很难测量实验。 第二，PI和他的合作者研究的随机性，可以出现在一个热力学限制延迟随机模型。 这种随机性可称为“统计相干性”。它是由确定性极限中的动力学不稳定性引起的。 当人们取延迟随机微分方程的热力学极限时，就会得到延迟微分方程（DDE）。 从遍历理论的角度来看，一个DDE可以被看作是一个泛函微分方程（FDE），它在一个合适的Banach函数空间上生成一个解半群。 类似地，某些发展偏微分方程（PDE）在适当的Banach空间上生成解半群。 这些解半群是无限维动力系统; PI和他的同事研究FDE和PDE背景下的统计相干性。他们通过隔离产生非均匀双曲动力学的特定几何机制来做到这一点。 在这种情况下，新概念的Sinai-Ruelle-Bowen（SRB）措施必须制定。 SRB措施形成了统计一致性的基石，因为它们管理的函数空间中的大型轨道集的渐近分布。 至关重要的是，可检查的条件，意味着存在的SRB措施的具体FDES和PDE的发展。 统计性质，如中心极限定理的行为，衰减的相关性，不变性原则，极值统计，和随机稳定性进行了研究。 这项工作通知建模，实验研究和设计的生化系统，并介绍了一个大的新类的物理相关的例子，抽象的动力系统社区。

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.