Mathematical Modelling of Cellular and Physiological Processes

我的科研计划的主要主题是通过数学模型的发展和分析来理解和解释复杂的生物过程，特别是细胞和生理过程。我将着重于描述时空现象的模型。******研究的一个领域是了解微管在运动蛋白存在时所表现出的空间模式。微管是细胞细胞质中的蛋白质聚合物，在提供细胞结构、细胞分裂、细胞运动和细胞运输方面起着至关重要的作用。在体内和体外都观察到微管形成各种各样的模式，如紫菀形和旋涡形，以及平行和反平行束。这种空间模式是与运动蛋白相互作用的结果，运动蛋白可以沿着微管行走，并将相邻的微管交联，从而影响微管相对于彼此的方向。微管和运动蛋白相互作用的详细数学建模将导致深入了解分子成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，本研究也将推动此类方程分析技术的发展。******另一个研究领域涉及一类相对较新的模型，即出生-跳跃模型。出生跳跃模型适用于描述生长和空间传播过程不能解耦的种群动态（从细胞种群到动物种群），特别是在新生成的细胞或个体立即被运送到遥远地点的情况下。对于特定的极限情况，出生-跳跃模型已经有了研究，但对于一般情况，还有待建立理论结果。这一工作将促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应用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.