Vahid Nourian | Applied Mathematics, University of Waterloo
Modeling and simulation of uni- and multi-flagellar bacterial locomotion in a viscous fluid
Flagellated bacteria categorized as microorganisms, play vital roles in human life such that their beneficial and detrimental effects on different aspects of the ecosystem are indisputable. Flagellated bacteria propel themselves in fluids by rotating one or more flexible filaments, known as flagella, driven by independent flagellar motors. Depending on the rotation direction of the motors and handedness of the helical filaments, the flagella either pull or push the cell body. Entrapment of swimming bacteria near surfaces, observed in some species, may lead to biological processes such as biofilm formation and wound infection. Previous experimental and numerical studies of bacterial locomotion have illustrated that several behaviors exhibited by the bacteria have roots in hydrodynamic interactions between the bacteria components and the surrounding fluid.
In this thesis, we numerically study flagellated bacterial locomotion in bounded and unbounded spaces. The physical properties of the model bacteria in this study are described based on experimental data available for various species of uni-, bi-, and multiflagellated bacteria. Specifically, we choose Vibrio alginolyticus, Magnetococcus marinus and Escherichia coli to focus on their motility to shed light on some of the unique behavior observed in each one. Depending on the species, the model bacteria have either a spherical or a spherocylindrical cell body and the flexible flagellar filaments are connected to the cell body membrane directly or via very flexible straight hooks. The flagella are independently driven by either constant or variable torque motors. Despite a similar flagellar structure, uni- and multiflagellated bacteria employ different mechanisms to swim on straight trajectories or reorient.
Here, we use the boundary element method (BEM) and the Kirchhoff rod model to develop a comprehensive elastohydrodynamic framework in order to model the motility of uni- and multiflagellated bacteria in a Newtonian viscous fluid. For this purpose, the boundary integral equations (BIE) are numerically evaluated over the cell body surface and along the flagella which are described by distributions of regularized Stokeslets and Rotlets. By assuming that the flagella are inextensible and unshearable, the linear theory of elasticity is used to estimate the internal moments along the flagella. Adding the hydrodynamics and elasticity equations to the total force/torque balance and kinematic equations leads to a system of linear equations which are solved to find the velocities and update the swimmer configuration accordingly.
Motivated by experimental observations of Vibrio alginolyticus locomotion in which it is shown that there is an interesting correlation between the near-surface entrapment of bacteria and the concentration of certain ions in the swimming medium, we numerically investigate its motility in different concentrations of NaCl. Our simulations demonstrate that changing the concentration of NaCl in the swimming fluid affects the tendency of pusher-mode bacteria to remain near the surfaces by altering the averaged swimming speed and inducing the different degrees of deformations along the flagellum. In addition to the ion concentration, our results indicate the flagellum/hook stiffness, the flagellar motor torque, and the cell body aspect ratio may affect whether the uniflagellated model bacterium escapes from the surface or becomes trapped in circular orbits.
By simulating the locomotion of a bi-flagellated model bacterium with a spherical cell body, one puller, and one pusher flagellum, we show that the bacteria with such configuration mainly swim along double helical trajectories. Comparing the properties of the obtained trajectories with the Magnetococcus marinus trajectories measured experimentally, indicate that this species has likely puller-pusher configuration. Varying the stiffness, orientations, or positions of the flagella significantly changes the swimming characteristics. Notably, when either the applied torque to the pusher flagellum is higher than a critical value and/or its stiffness is lower than a critical stiffness, the pusher flagellum exhibits overwhirling motion, resulting in a more complicated swimming style and a lower swimming speed. For a moderate flagellum stiffness, the swimming speed is insensitive to the rest orientation of the flagella over a wide range of orientation angles because the flagella deform to maintain alignment with the swimming direction.
Numerical investigation of multiflagellated bacteria locomotion in unbounded fluid indicates that the arrangement of the flagella on the cell body provides no advantages in the average swimming speeds of bacteria. However, the trajectory of the bacteria could be either relatively straight or double helical trajectory depending on the degree of asymmetry that exists in the distribution of the flagella. Our results indicate that the multiflagellated bacteria may have several stable swimming modes in which the swimming properties such as speeds and trajectories could be different. The tumbling event, pause in the flagellar motor, and interaction with other bacteria are likely some reasons which cause the bacteria to switch between the different modes. High viscous torque due to the presence of a no-slip boundary slightly changes the swimming properties of the multiflagellated bacteria such as bundling time, the translational and angular speeds. Remarkably, the flagella arrangement is one of the key factors determining how the swimming properties vary in response to the presence of a surface.