Taylor J. Whitney

Microbial motion is typically analyzed by simplified models in which trajectories exhibit straight runs (perhaps with added Gaussian noise) followed by random, discrete tumbling events. We present the results of a statistical analysis of the angular dynamics for four different swimming microbes: tumbling and smooth-swimming strains of Bacillus subtilis and two Eukaryotic algae, Tetraselmis suecica and Euglena gracilis. We show that the angular statistics closely resemble a Voigt profile, the convolution of a Gaussian (Lévy index α=2) and Lorentzian (Lévy index α=1) distribution. This distribution is ubiquitous for all four microbes. Rather than modeling tumbling as a discrete process, we model tumbling dynamics as a continuous process: Lévy flights in the orientational dynamics using a Lorentzian noise model. This model is analytically solvable. Each individual microbe trajectory has both stochastic behavior (noise) and varying drift behavior, such as helices of different sizes and frequencies and circular arcs with different radii. We model the distribution of different drift behavior via an ensemble theory. The drift behavior (e.g., circular arcs) comes from physical observations of the swimming behavior and explains many of the qualitative features of the experimental data that cannot be explained by a pure noise model. From this theory, we estimate the strength of Lorentzian noise, the physical rotational diffusion constant, and some relevant parameters relating to the distributions of deterministic drift behavior. This analysis shows that in some cases Gaussian noise is not the dominant process responsible for the angular statistics following a Voigt profile.

Microbial motion is typically analyzed by simplified models in which trajectories exhibit straight runs (perhaps with added Gaussian noise) followed by random, discrete tumbling events. We present the results of a statistical analysis of the angular dynamics for four different swimming microbes: tumbling and smooth-swimming strains of \textit{Bacillus subtilis} and two Eukaryotic algae, \textit{Tetraselmis suecica} and \textit{Euglena gracilis}. We show that the angular statistics closely resemble a Voigt profile, the convolution of a Gaussian (Lévy index $α=2$) and Lorentzian (Lévy index $α=1$) distribution. This distribution is ubiquitous for all four microbes. Rather than modeling tumbling as a discrete process, we model tumbling dynamics as a continuous process: Lévy flights in the orientational dynamics using a Lorentzian noise model. This model is analytically solvable. Each individual microbe trajectory has both stochastic behavior (noise) and varying deterministic behavior, such as helices of different sizes and frequencies and circular arcs with different radii. We model the distribution of different deterministic behavior via an ensemble theory. The deterministic behavior (e.g., circular arcs) comes from physical observations of the swimming behavior and explains many of the qualitative features in the data that cannot be explained by a pure noise model. From this theory, we estimate the strength of Lorentzian noise, the physical rotational diffusion constant, and some relevant parameters relating to the distributions of deterministic behavior. This analysis shows that in some cases Gaussian noise is not the dominant process responsible for the angular statistics following a Voigt profile.

We study the transport of rigid ellipsoidal swimmers in a periodic vortex array via numerical simulation and dynamical systems analysis. Via ensemble simulations, we show the counterintuitive result that slower swimming speeds can generate fast ballistic transport, while faster swimming speeds generate chaotic and diffusive transport, which is inherently slower in the long run. To explain this, we use the symmetry of the flow to construct a time-reversible Poincaré return map on a two-dimensional surface of section in phase space. For sufficiently small swimming speeds, we find stable periodic orbits on the surface of section surrounded by invariant tori, similar to Kolmogorov-Arnold-Moser curves. Trajectories within these tori are ballistic. As the swimming speed is increased, the periodic orbits undergo a sequence of period-doubling bifurcations that destroys the ballistic tori. These bifurcations exactly match the ballistic to diffusive transition from the ensemble simulations. Additional ensemble simulations are used to test the robustness of these results to noise. The ballistic behavior is destroyed as the strength of rotational diffusion increases. However, we estimate that the ballistic tori might still be seen in experiments.