Cells rely on a mechanical structure called the cortex to maintain their shape and to respond to chemical and mechanical cues in the environment. The cortex, which lies just below the outer membrane of the cell, consists primarily of polymerized filaments of the protein actin. These actin filaments are cross-linked by a variety of proteins that can reorient and even move the filaments, making the cortex a highly dynamic structure. Despite the importance of the actin cortex for critical cell functions, it is not clear how small scale interactions give rise to large scale patterns and functions associated with the cortex. This project will use mathematical modeling to bridge disparate time and space scales associated with cortical actin dynamics, directly linking molecular level interactions to the emergence of cell level functional structures. For this project, novel models will be constructed that draw on a variety of mathematical approaches, including stochastic models which take into account infrequent interactions, and continuum models which can track the moving boundary of the cortex as well movement of proteins in time and space. The development and analysis of these biologically based models will result in an improved understanding of actin and its regulation in a cellular context. This research will also be leveraged to recruit and retain women in Science, Technology, Engineering, and Mathematics fields through the development of an integrated undergraduate course in mathematical modeling, with recruitment of students through a number of existing programs at Ohio State University, including Women in Mathematics and Science and Women in Engineering. Students involved in this project, and in the undergraduate course, will be exposed to real world mathematics applications and will interact with peers from other disciplines in a supportive interdisciplinary environment.
Mathematical modeling is an ideal tool for the investigation of complex biological systems, where experimental techniques are not available or are not feasible. At the macro-scale level, systems of partial differential equations will be used to investigate the interplay between biochemical and mechanical actin dynamics, and the consequence of these dynamics on cell shape. Asymptotic analysis will be used to explicitly study the role of curvature in actin dynamics, and moving boundary simulations will be used to study changes in cell shape. At the mesoscale level, integro-differential equations, which use integral kernels to describe actin filament dynamics, will be used to study the formation of large scale actin filament patterns such as asters, vortices and aggregates. At the micro-scale level, stochastic individual-based models will be used to determine physical properties of the local actin meshwork and to determine under what conditions the actin meshwork behaves as a viscoelastic material. By enforcing consistency across multiple time and space scales in the proposed models, mechanisms that give rise to cellular structures will be revealed. Analysis and simulations of these models will present unique challenges leading to improvement of techniques in several areas of applied mathematics, including asymptotic analysis, moving boundary simulations, nonlocal model analysis, and stochastic simulations. The proposed models will be motivated and validated in a single experimental organism, avoiding complications from combining in vitro data with in vivo data from multiple organisms and cell types. The integrated computational and analytical models resulting from this research will increase our understanding of a fundamental protein that is critical for proper cell function.
