In this research project a general mathematical framework and computational method are developed which extend the Immersed Boundary Method to account for thermal fluctuations. The thermal fluctuations are taking into account by including appropriate stochastic forcing terms in the fluid equations. These methods are then applied to study the dynamic membrane rearrangements involved in the functions of two cell organelles, the Golgi Apparatus and Mitochondria. The main biological questions that are addressed concern the role that membrane geometry and spatial distribution of biochemical species play in the functions of these organelles, as induced by the membrane biochemistry, osmotic stresses, fluid flow, and thermal fluctuations. With advances in biochemical assays and electron tomography data is now becoming available which hint at how these cell organelles function both in healthy and in diseased cells. While a comprehensive understanding remains elusive, mathematical modeling may help clarify our present understanding and aid in postulating basic mechanisms by which these cell organelles function. The detailed, large-scale mathematical modeling of this project is intended to shed light on some of these mechanisms, by generating and refining experimentally testable hypothesis about cell organelles' dynamical structures. This work may also advance our understanding of cell organelle processes in general, and possibly give insight into the cellular mechanisms which break down during diseased states suggesting new medical intervention strategies. In addition to contributing to basic science, the knowledge gained in this research will be used in training graduate students and postdoctoral researchers, and in the design of research-influenced mathematical biology courses for which materials will be posted on the web. Software packages will also be made available for the general computational methods developed.
With advances in cell biology compelling information has been obtained about many cellular processes from mathematical models which consider primarily the interactions between a collection of biochemical species. An even deeper understanding may become possible if in addition the spatial organization of these components and their interactions with cellular structures are modeled. In this project the role cellular structures play through spatial distribution of biochemical species is studied, with a specific emphasis on modeling at a coarse-level the Golgi Apparatus and Mitochondria cell organelles. The mechanics of many cellular structures can be regarded at a coarse-level as flexible structures which interact with a fluid. This common mechanical feature of biological systems presents many challenges in formulating models which are amenable to mathematical analysis and computational simulation while being realistic enough to capture relevant features of the biological phenomena being studied. The Immersed Boundary Method, a computational method simultaneously accounting for flexible structures and fluid, has been used to perform simulations of these mechanical features in the study of a variety of biological systems, including blood flow around valves in the beating heart, wave propagation in the cochlea, and lift generation in insect flight. Simulating cellular processes at microscopic scales presents further challenges requiring that thermal fluctuations be taking into account for the fluid and immersed structures.
