Thomas H. Solomon

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.

We present experiments on chaotic motion of self-propelled (active) particles in a time-independent, two-dimensional vortex chain flow. We track Tetraselmis microbes and calculate the variance of a spreading distribution of these microbes in the flow. For small non-dimensional swimming speed v0, we find subdiffusion with variance ⟨x2⟩∼tγ with γ<1; transport is diffusive (γ=1) for larger v0. Subdiffusion for small v0 is due to dynamic trapping of microbes to islands of ordered trajectories surrounded by a sea of chaotic motion; these islands disappear for larger v0. We calculate Lagrangian-averaged trajectories (LATs) from the experimental data and use the LATs to measure trapping time probability distributions P(t). We find regimes with P(t)∼t−ν with ν<2 for small v0, consistent with the measured subdiffusion.

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 present experiments on one-way barriers that block reaction fronts in a fully three-dimensional (3D) fluid flow. Fluorescent Belousov-Zhabotinsky reaction fronts are imaged with laser-scanning in a laminar, overlapping vortex flow. The barriers are analyzed with a 3D extension to burning invariant manifold (BIM) theory that was previously applied to two-dimensional advection-reaction-diffusion processes. We discover tube and sheet barriers that guide the front evolution. The experimentally determined barriers are explained by BIMs calculated from a model of the flow.

We present experiments on reaction fronts and self-sustaining trigger wave patterns in an advection-reaction-diffusion system with chaotic mixing. The flow is a two-dimensional array of oscillating vortices, and the reaction is the excitable Belousov-Zhabotinsky chemical reaction. Reaction fronts are found to mode-lock for a wide range of frequencies, and the mode-locking results in "faceted" fronts that line up along the directions of the underlying vortex array. Self-sustaining trigger wave patterns are also found composed of large scale spiral and spiral-pair patterns, similar to patterns in the reaction-diffusion (no flow) limit but with a significantly larger typical length scale and with clear anisotropy that reflects the vortex array.