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

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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.