Mathematical Modelling of Cellular and Physiological Processes

我的科学研究计划的主题是理解和解释复杂的生物过程，特别是细胞和生理过程，通过数学模型的开发和分析。 我将专注于描述时空现象的模型。 研究的一个领域是了解在马达蛋白存在下微管所表现出的空间模式。 微管是细胞质中的蛋白质聚合物，其在为细胞提供结构、细胞分裂、细胞运动和细胞运输中起关键作用。 在体内和体外，微管被观察到形成各种图案，如星形和漩涡，以及平行和反平行束。 空间模式化是与马达蛋白相互作用的结果，马达蛋白可以沿着微管沿着行走并交联相邻的微管，从而影响微管相对于彼此的取向。 微管和马达蛋白相互作用的详细数学建模将导致深入了解分子组分自组织成各种空间结构的机制。该模型由非线性偏积分微分方程组成，该研究也将推进此类方程的分析技术。 另一个研究领域涉及到一个相对较新的模型，即出生跳跃模型。跳跃模型适用于描述种群的动态（从细胞种群到动物种群的所有水平），其中生长和空间扩散过程不能解耦，特别是在新生成的细胞或个体立即被运送到遥远位置的情况下。 出生跳跃模型已经研究了特定的限制情况下，但它仍然建立一般情况下的理论结果。 这些工作将推进非线性积分微分方程理论的发展，并为出生跳跃模型在生物学中的应用提供坚实的基础。

The main theme of my scientific research program is to understand and explain complex biological processes, in particular cellular and physiological processes, through the development and analysis of mathematical models. I will focus on models that describe spatio-temporal phenomena. One area of investigation is to understand the spatial patterns exhibited by microtubules in the presence of motor proteins. Microtubules are protein polymers in the cytoplasm of cells that play a crucial role in providing structure to cells, cell division, cell motility, and cellular transport. Microtubules are observed to form a variety of patterns, such as asters and vortices, and parallel and anti-parallel bundles, both in vivo and in vitro. The spatial patterning is the result of interactions with motor proteins, which can walk along microtubules and crosslink adjacent microtubules, affecting the orientation of microtubules with respect to each other. Detailed mathematical modelling of the interaction of microtubules and motor proteins will lead to insights into the mechanisms underlying self-organization of molecular components into a variety of spatial structures. The model consists of nonlinear partial integro-differential equations, and this research also will advance techniques for the analysis of such equations. Another area of investigation pertains to a relatively new class of model, namely the birth-jump model. Birth-jump models are appropriate for describing the dynamics of populations (at all levels, from populations of cells to populations of animals) where the growth and spatial spread processes cannot be decoupled, in particular in situations where newly generated cells or individuals are transported immediately to distant locations. The birth-jump model has been studied for specific limiting cases, but it remains to establish theoretical results for the general case. This work will advance the theory of nonlinear integro-differential equations, and provide a solid foundation for the use of birth-jump models in biological applications.