Asymptotic stability of contraction-driven cell motion.

We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a two- dimensional free-boundary model that generalizes a previous one-dimensional model [P. Recho, T. Putelat, and L. Truskinovsky, Phys. Rev. Lett. 111, 108102 (2013)10.1103/PhysRevLett.111.108102] by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We find a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is nonlinear asymptotic stability of traveling solutions that model observable steady cell motion. We derive an explicit asymptotic formula for the stability-determining eigenvalue via asymptotic expansions in small speed. This formula greatly simplifies computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability.

Minimal model of directed cell motility on patterned substrates.

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern geometry and the strength of actin polymerization on the directionality of the crawling cell. These results reveal that high strength of actin polymerization results in motion perpendicular to substrate stripes only when the substrate is relatively nonadhesive; in particular, this suggests potential applications in motile cell sorting and guiding on engineered substrates.

Effective Rheological Properties in Semi-dilute Bacterial Suspensions.

Interactions between swimming bacteria have led to remarkable experimentally observable macroscopic properties such as the reduction in the effective viscosity, enhanced mixing, and diffusion. In this work, we study an individual-based model for a suspension of interacting point dipoles representing bacteria in order to gain greater insight into the physical mechanisms responsible for the drastic reduction in the effective viscosity. In particular, asymptotic analysis is carried out on the corresponding kinetic equation governing the distribution of bacteria orientations. This allows one to derive an explicit asymptotic formula for the effective viscosity of the bacterial suspension in the limit of bacterium non-sphericity. The results show good qualitative agreement with numerical simulations and previous experimental observations. Finally, we justify our approach by proving existence, uniqueness, and regularity properties for this kinetic PDE model.

Effective viscosity of puller-like microswimmers: a renormalization approach.

Effective viscosity (EV) of suspensions of puller-like microswimmers (pullers), for example Chlamydamonas algae, is difficult to measure or simulate for all swimmer concentrations. Although there are good reasons to expect that the EV of pullers is similar to that of passive suspensions, analytical determination of the passive EV for all concentrations remains unsatisfactory. At the same time, the EV of bacterial suspensions is closely linked to collective motion in these systems and is biologically significant. We develop an approach for determining analytical EV estimates at all concentrations for suspensions of pullers as well as for passive suspensions. The proposed methods are based on the ideas of renormalization group (RG) theory and construct the EV formula based on the known asymptotics for small concentrations and near the critical point (i.e. approaching dense packing). For passive suspensions, the method is verified by comparison against known theoretical results. We find that the method performs much better than an earlier RG-based technique. For pullers, the validation is done by comparing them to experiments conducted on Chlamydamonas suspensions.

Correlation properties of collective motion in bacterial suspensions.

The study of collective motion in bacterial suspensions has been of significant recent interest. To better understand the non-trivial spatio-temporal correlations emerging in the course of collective swimming in suspensions of motile bacteria, a simple model is employed: a bacterium is represented as a force dipole with size, through the use of a short-range repelling potential, and shape. The model emphasizes two fundamental mechanisms: dipolar hydrodynamic interactions and short-range bacterial collisions. Using direct particle simulations validated by a dedicated experiment, we show that changing the swimming speed or concentration alters the time scale of sustained collective motion, consistent with experiment. Also, the correlation length in the collective state is almost constant as concentration and swimming speed change even though increasing each greatly increases the input of energy to the system. We demonstrate that the particle shape is critical for the onset of collective effects. In addition, new experimental results are presented illustrating the onset of collective motion with an ultrasound technique. This work exemplifies the delicate balance between various physical mechanisms governing collective motion in bacterial suspensions and provides important insights into its mesoscopic nature.

Viscosity of bacterial suspensions: hydrodynamic interactions and self-induced noise.

The viscosity of a suspension of swimming bacteria is investigated analytically and numerically. We propose a simple model that allows for efficient computation for a large number of bacteria. Our calculations show that long-range hydrodynamic interactions, intrinsic to self-locomoting objects in a viscous fluid, result in a dramatic reduction of the effective viscosity. In agreement with experiments on suspensions of Bacillus subtilis, we show that the viscosity reduction is related to the onset of large-scale collective motion due to interactions between the swimmers. The simulations reveal that the viscosity reduction occurs only for relatively low concentrations of swimmers: Further increases of the concentration yield an increase of the viscosity. We derive an explicit asymptotic formula for the effective viscosity in terms of known physical parameters and show that hydrodynamic interactions are manifested as self-induced noise in the absence of any explicit stochasticity in the system.

A model of hydrodynamic interaction between swimming bacteria.

We study the dynamics and interaction of two swimming bacteria, modeled by self-propelled dumbbell-type structures. We focus on alignment dynamics of a coplanar pair of elongated swimmers, which propel themselves either by "pushing" or "pulling" both in three- and quasi-two-dimensional geometries of space. We derive asymptotic expressions for the dynamics of the pair, which complemented by numerical experiments, indicate that the tendency of bacteria to swim in or swim off depends strongly on the position of the propulsion force. In particular, we observe that positioning of the effective propulsion force inside the dumbbell results in qualitative agreement with the dynamics observed in experiments, such as mutual alignment of converging bacteria.

Three-dimensional model for the effective viscosity of bacterial suspensions.

We derive the effective viscosity of dilute suspensions of swimming bacteria from the microscopic details of the interaction of an elongated body with the background flow. An individual bacterium propels itself forward by rotating its flagella and reorients itself randomly by tumbling. Due to the bacterium's asymmetric shape, interactions with a prescribed generic (such as planar shear or straining) background flow cause the bacteria to preferentially align in directions in which self-propulsion produces a significant reduction in the effective viscosity, in agreement with recent experiments on suspensions of Bacillus subtilis.

Effective viscosity of dilute bacterial suspensions: a two-dimensional model.

Suspensions of self-propelled particles are studied in the framework of two-dimensional (2D) Stokesean hydrodynamics. A formula is obtained for the effective viscosity of such suspensions in the limit of small concentrations. This formula includes the two terms that are found in the 2D version of Einstein's classical result for passive suspensions. To this, the main result of the paper is added, an additional term due to self-propulsion which depends on the physical and geometric properties of the active suspension. This term explains the experimental observation of a decrease in effective viscosity in active suspensions.