Abstract:
The bending and twisting deformations of bio-filaments on intermediate length scales
of the order of 5-100 nm play a key role in many biological processes. For instance, the looping behavior of DNA molecules plays a pivotal role in gene expression. Therefore, modeling these bending and twisting deformations in bio-filaments is useful in many applications. While traditionally all-atom molecular dynamics approach, semiempirical approaches and continuum approach have been successfully applied to model such deformations, we focus on the continuum approach using Kirchhoff elastic rod model equations in this thesis. While continuum modeling has shown promise, one key ingredient that remains a challenge to accurately estimate is the respective constitutive law that describes the bending and twisting stiffnesses based on the atomic composition and arrangement. Further, these bio-filaments are typically present in viscous fluid media and their deformations are significantly affected by the forces resulting from interactions with randomly moving solvent particles. Since these thermal fluctuations are arising from stochastic forces, if these forces can be modeled, qualitative observations and predictions about the bio-filament deformations could be made that are useful in characterizing various properties such as entropic stiffness. Thus the objective of this thesis is to develop a continuum model for bio-filaments that takes into account thermal fluctuations. Such a model will serve two purposes; of helping make qualitative predictions and observations, and help estimate constitutive law for various bio-filaments from experimental data. Such a model can potentially be applied to model deformations of actin fibers, microtubules, DNA, flagella of bacteria, cilia of eukaryotic cells, etc.
There are two widely followed approaches in general to model thermal fluctuations on particles, Langevin dynamics, in which viscous drag force and random forces due to motion of solvent particles are accounted for in Newton’s second law, and Brownian motion in which inertia forces are neglected. In this dissertation, we explore a simpler third alternative that is to model them using a quasi-static approach. In a quasi-static approach, we neglect both inertia forces and viscous forces and consider the stochastic
forces due to solvent medium. So, our model simulates deformation patterns (both planar and 3-dimensional) of filaments and therefore, can be used to infer the mechanical behavior and entropic properties. This approach will not however be able to capture dynamic changes in these deformation patterns with time. We also consider various options on how the stochastic forces can be discretized in relation to the continuum filament. In this approach, filament is treated as a continuous elastic filament with discrete stochastic forces acting on it. We show that the simulated ensemble of filaments generated through quasi-static dynamics significantly resembles the experimental AFM image of DNA. At last, the generated model is validated by comparisons with published results from equilibrium rod theories and laboratory-scale experiments.