Stochastic Pattern Formation in Mathematical Biology

数学生物学中的随机模式形成

• 批准号: 2441793
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2441793
• 关键词: Stochastic; Pattern; Formation; Mathematical; Biology

Many varied biological systems exhibit visually striking patterns both static and dynamic, from the pigmentations of animal skins to the spatial distribution of vegetation growth. Deterministic continuum (partial differential equation) models have been widely used to analyse and successfully predict various characteristics of these patterns via the diffusion-driven `Turing instability' mechanism. Initially proposed as an explanatory model for embryogenesis, the Turing instability describes the spontaneous emergence of spatially inhomogeneous structure from simple homogeneous initial conditions through the coupled dynamics of localised `reactions' and spatial diffusion.Such PDE models often suffer from issues of robustness and parameter tuning. Sometimes a continuum of pattern instabilities is possible, and the model cannot predict which pattern will persist over time. Further, for realistic reaction rates, the predicted instability thresholds typically require an order of magnitude (or greater) disparity in species diffusivities, which is rarely observed in real-world systems.Stochastic models can avoid these problems while simultaneously allowing analysis of the impacts of noise and (on microscopic scales) discreteness on the pattern formation. Such models have also expanded the Turing mechanism to include random fluctuation-driven `stochastic Turing patterns'. Compartment-based (mesoscopic) stochastic models offer an analytically tractable approach to analysing the spatiotemporal characteristics and statistics of patterning instabilities, both diffusion-driven and fluctuation-driven. The aims of this project are first and foremost to develop a cohesive analytical framework for the identification, classification, and prediction of spontaneous patterning instabilities; starting with compartment-based models, but eventually spanning a range of discrete stochastic reaction-diffusion set-ups, encompassing both deterministic and stochastic Turing patterns. Secondly, this project will apply this framework to analysing patterning instabilities in more recently developed stochastic reaction-diffusion models, such as volume exclusion processes. With this, we aim to further the analysis of patterning instabilities by enabling comparison between different models and types of instability, to better understand the underlying physical mechanisms driving pattern formation.

许多不同的生物系统显示出视觉上引人注目的静态和动态模式，从动物皮肤的色素到植物生长的空间分布。确定性连续介质(偏微分方程)模型已被广泛应用于通过扩散驱动的“图灵不稳定性”机制来分析和成功预测这些模式的各种特征。图灵不稳定性最初是作为胚胎发生的解释模型提出的，它描述了从简单的均匀初始条件通过局域反应和空间扩散的耦合动力学自发出现的空间不均匀结构。这类PDE模型经常遇到稳健性和参数调整的问题。有时，模式不稳定的连续体是可能的，并且模型无法预测哪种模式将随着时间的推移而持续存在。此外，对于真实的反应速率，预测的不稳定性阈值通常需要物种扩散系数的数量级(或更大)的差异，这在现实系统中很少观察到。随机模型可以避免这些问题，同时允许分析噪声和(在微观尺度上)离散性对图案形成的影响。这些模型还扩展了图灵机制，将随机波动驱动的“随机图灵模式”包括在内。基于隔室的(介观)随机模型提供了一种分析容易处理的方法来分析扩散驱动和波动驱动的图案化不稳定性的时空特征和统计数据。这个项目的目标首先是开发一个连贯的分析框架，用于自发模式不稳定性的识别、分类和预测；从基于隔室的模型开始，但最终跨越一系列离散的随机反应-扩散设置，包括确定性和随机图灵模式。其次，这个项目将应用这个框架来分析最近发展起来的随机反应扩散模型中的图案不稳定性，例如体积排除过程。在此基础上，我们的目标是通过比较不同模型和不同类型的不稳定性来进一步分析图案化不稳定性，以更好地理解驱动图案形成的潜在物理机制。