A new mathematical model is developed to understand the swimming mechanism of bacteria such as Escherichia coli. The bacterium E. coli is a single-celled organism which swims in a viscous fluid by rotating its helical flagellar filaments. Two successive motions are involved in the cell motility: Runs (straight swimming propelled by flagellar bundling) and tumbles (random reorientation by interspersing flagella). This project will focus on the study of the hydrodynamic interaction among flagella and flagellar filament shapes that arise through polymorphic transformations?local changes in helical wavelength, helical diameter, and handedness?during swimming. A generalized version of the immersed boundary method combined with the unconstrained Kirchhoff rod theory is used to study biological fluid mechanics in the bacterium. A new feature of this method is that the interaction of the immersed boundary with the fluid now involves not only translation of the immersed boundary points at the local fluid velocity, but also rotation of the associated triads at the local fluid angular velocity.
This method will find numerous applications in biological fluid dynamics, where filamentous structures interact with a viscous fluid. Examples include the supercoiling of DNA during transcription and replication and protein folding. In addition, one of the challenges in nanotechnology is to develop machines at the nanoscale which can be used in the treatment of disease. Understanding swimming mechanism by means of rotary motors will help to create a nanomachine, operated by self-propelled biomolecular nano motors, that could be used for drug delivery inside the body. Furthermore, such interdisciplinary projects give rise to problems and activities that can be used to attract students to science through the investigator's work with the Women in Science and Engineering program (WISE) at her university.
Some bacteria such as E. coli swim through the fluid by rotating its helical flagella, which is known as a very efficient method of motility. The main goal of this research project is to understand the swimming mechanism of such bacteria using mathematical modeling and scientific computations. During NSF award period, we developed a mathematical model that could analyze the hydrodynamic interaction between flagella as well as the dynamics of an elastic filament in a viscous fluid. In particular, flagellar bundling that occurs when all rotary motors turn counter-clock wise when viewed from outside the bacterium, is an important aspect of bacterial locomotion. Our method was able to capture the complicated dynamics of flagella interacting with fluid. This new method is based on the immersed boundary (IB) method combined with the unstrained Kirchhoff rod theory. The IB method is one of the schemes that solve prevalent fluid-structure interaction problems in nature. Mathematical tools developed for this project can also have many applications. One of the applications is to study the supercoiling dynamics of a double stranded DNA which takes the form of helices. DNA supercoiling dynamics takes place during DNA replication, DNA transcription and protein-DNA interactions, and it depends on the amount of twist and bend along the DNA backbone as well as the ionic strength in solvent. DNA can be considered as an electrostatically charged rod in a fluid. Another application is to study the motility of a helical shaped bacterium, Spiroplasma, which moves around the fluid by propagating a pair of kinks with time delay. In fact, any types of elastic rods may be simulated by the developed method. New findings may contribute to a modeling of helical flagella based on its actual geometry which could not be done by experiments due to the size of a flagellum. This will help to simulate a swimming bacterium at micron scale and to understand the swimming mechanism and hence help to create the nanomachine to deliver the drug inside the body to cure some disease. Animations are made from the computational results in this project and are posted on the PI’s website so that everyone can have an access to them. Results from the projects have been delivered to scientists and students through the classes, seminars and colloquia, and helped them to broaden their scientific views in mathematical biology. This work has made a significant impact on graduate and undergraduate education, and enhanced the mathematical biology program within the Department of Mathematical Sciences at University of Cincinnati.
