Effective properties and collective dynamics in bacterial suspensions

Open Access
Author:
Ryan, Shawn David
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
May 06, 2014
Committee Members:
  • Leonid Berlyand, Dissertation Advisor
  • Leonid Berlyand, Committee Chair
  • Anna L Mazzucato, Committee Member
  • Xiantao Li, Committee Member
  • Ralph H Colby, Committee Member
Keywords:
  • mathematical biology
  • active biosystems
  • collective motion
  • effective viscosity
  • coupled PDE/ODE system
  • kinetic theory
Abstract:
This dissertation introduces novel computationally efficient PDE models, which are used to investigate the origin of self-organization in bacterial suspensions. The key feature of these models is the incorporation of interbacterial interactions motivated by recent experimental observations suggesting their importance in the emergence of collective swimming. Results on well-posedness, effective properties and the onset of the collective state are established through rigorous asymptotic and numerical analysis. Each problem considered is highly multiscale in that microscopic interactions result in changes in the macroscopic state. This work provides a better understanding of the physical mechanisms governing the transition to collective motion. Throughout this dissertation, novel models are employed where a bacterium is represented as a point force dipole subject to two types of interactions: hydrodynamic interactions and excluded volume type interactions introduced through the use of a short-range Lennard-Jones type repelling potential. The point dipole model accounts for the particle size through this potential and shape via Jeffery’s equations modeling how an ellipsoid interacts with the surrounding fluid. Confirming experimental observation, the mathematical analysis reveals that the alignment of asymmetrical particles and the presence of self-propulsion change the effective rheological properties of the suspension such as a drastic reduction in the effective viscosity. By providing explicit formulas for the effective viscosity as well as the effective normal stress differences, the theory presented herein can describe the complete rheological behavior of an active suspension undergoing planar shear in terms of known physical parameters. The first few chapters (1-4) of this dissertation are concerned with introducing the PDE/ODE model for the suspension allowing for the investigation of this decrease in the effective viscosity. The main challenge is added complexity due to the incorporation of interbacterial interactions, in contrast to previous models valid only in the dilute regime. Rigorous mathematical analysis is then performed on the associated nonlinear non-local kinetic equation governing the evolution of the particle distribution function for bacterial positions and orientations. Using this approach, an explicit asymptotic formula for the effective viscosity in terms of known physical parameters is derived. This formula reveals the physical mech- anisms responsible for the striking decrease in the effective viscosity observed in experiment; namely, the combination of self-propulsion, a non-uniform spatial distribution of bacteria due to interactions, and a non-spherical shape of bacteria. The model developed in this dissertation also allows for computationally efficient GPU numerical simulations containing a large number of particles, which are in agreement with the analytical results and experiment. This work is the first to capture the qualitative behavior of the effective viscosity observed in active suspensions for all experimental concentrations. In addition to the effective viscosity, the effective normal stress coefficients are also computed. The main mathematical result, which is presented in Chapter 4, is that the model proposed is well-posed and provides the existence of unique particle trajectories for all time. The later chapters (5-6) of this dissertation explore more recent work, which involves understanding the onset of collective motion by investigating the spatiotemporal correlations associated with bacterial velocities. Numerical analysis of a thin film PDE model provides novel understanding of which physical mechanisms govern the onset, size, and duration of the collective state. Using the proposed model, this work confirms the recent experimental observation that particle size and shape rather than the concentration of bacteria or swimming speed governs the size and duration of the collective state in bacterial suspensions. In addition to verifying experimental observation, this work studies the effects of system pa- rameters that are difficult to control in experiment such as the particle aspect ratio as well as the decoupling of swimming speed and the tumbling rate of bacteria. The current state of experiments does not allow for such an investigation, but our theory provides testable predictions for future work. The results of the analysis in this dissertation exemplify the delicate balance between hydrodynamic interactions and collisions governing collective motion in bacterial suspensions and provide important insights into its mesoscopic nature.