Multi-scale stochastic systems motivated by biological models

系统生物学的挑战是确定蛋白质组和基因组的相互作用如何产生细胞的功能和行为。细胞是固有的“嘈杂”系统，复杂的生化途径如何受到内在和外在随机性的影响尚不清楚。细胞网络可以用化学动力学原理建立数学模型。在细胞水平上，化学动力学通常由每个细胞只有几个拷贝的调节分子的作用所控制。由于这些成分的随机波动而产生的固有噪声可能会产生重大后果，为了充分描述可能的结果集，随机建模是必要的。在许多情况下，反应速率也可以变化几个数量级，这意味着丰度的有效变化会在多个时间尺度上发生波动。许多重要的细胞内过程，从生化信号到基因调控，也受到相关分子类型的位置和运动的极大影响。***这些系统的随机模型使用密度相关的连续时间马尔可夫链来表示由它们之间规定的相互作用集连接的不同类型物种的进化。为了预测和模拟的目的，减少问题的建模和计算复杂性是有用的，同时仍然捕获系统的基本随机行为。可以利用模型的多尺度特性来降低一些复杂性。严谨的方法提出了许多有趣和具有挑战性的数学问题。我的研究计划的主要目标是开发这种随机模型的模型简化技术，并确定由于噪声可能产生的影响。这涉及到将非线性动力学与随机平均、扩散近似和多时间尺度上的密度相关马尔可夫链的大偏差相结合的新结果。具体的短期和长期目标将使我们更好地理解：(i)随机反应和分子运动在结果空间传播中的综合效应；以及（ii）罕见事件在多尺度反应动力学中的作用，以及它们在增强和空间扩展双稳态动力学和特定初始结果中的作用。从长远来看，它们将帮助我们确定一组在多时间尺度的非均相反应系统中产生的空间现象，以及它们产生的条件。在此过程中开发的数学工具将使我们能够对非均相反应扩散系统的长期行为进行概率分析。这是一个正在进行的研究项目，为此我已经为多尺度密度相关的连续时间马尔可夫链开发了一个合适的数学框架，并得出了一些相关的结果

The challenge of systems biology is identifying how interactions of the proteome and genome produce the function and behaviour of a cell. Cells are inherently "noisy" systems, and how complex biochemical pathways are affected by intrinsic and extrinsic stochasticity is not well understood. A cellular network can be modelled mathematically using the principles of chemical kinetics. At the cellular level, chemical dynamics are often dominated by the action of regulatory molecules present at levels of only a few copies per cell. Intrinsic noise due to random fluctuations of these components can have significant consequences, and stochastic modelling is necessary in order to fully describe the set of possible outcomes. In many cases reaction rates can also vary over several orders of magnitude, which implies that fluctuations of effective changes in abundances occur on multiple time-scales. Many important intracellular processes, from biochemical signalling to gene regulation, are also greatly affected by the location and motility of the molecular types involved. ***Stochastic models of these systems use density dependent continuous time Markov chains to represent the evolution of different types of species connected by the prescribed set of interactions between them. For prediction and simulation purposes it is useful to reduce the modelling and computational complexity of the problem, while still capturing the essential stochastic behaviour of the system. The multiscale nature of the models may be exploited in order to reduce some of the complexity. A rigorous approach poses a number of interesting and challenging mathematical questions. The primary goal of my research program is to develop model reduction techniques for such stochastic models and identify the possible effects due to noise. This involves deriving novel results combining nonlinear dynamics with stochastic averaging, diffusion approximations and large deviations for density dependent Markov chains on multiple time scales. Specific short and long term goals will allow us to better understand: (i) the combined effect of stochastic reactions and molecular movement in spatial propagation of outcomes; as well as (ii) the effect of rare events in multi-scale reaction dynamics, and their role in enhancing and spatially spreading bistable dynamics and specific initial outcomes. In the long run, they will help us identify a set of spatial phenomena arising in heterogeneous reaction systems with multiple time-scales, and conditions under which they arise. The mathematical tools developed along the way will allow us to give a probabilistic analysis of the long term behaviour of heterogeneous reaction diffusion systems. This is an ongoing research program, for which I have already developed an appropriate mathematical framework for multi-scale density dependent continuous time Markov chains and derived a number of relevant results.**