A Model for Collective Dynamics in Ant Raids

Ant raiding, the process of identifying and returning food to the nest or bivouac, is a fascinating example of collective motion in nature. During such raids ants lay pheromones to form trails for others to find a food source. In this work a coupled PDE/ODE model is introduced to study ant dynamics and pheromone concentration. The key idea is the introduction of two forms of ant dynamics: foraging and returning, each governed by different environmental and social cues. The model accounts for all aspects of the raiding cycle including local collisional interactions, the laying of pheromone along a trail, and the transition from one class of ants to another. Through analysis of an order parameter measuring the orientational order in the system, the model shows self-organization into a collective state consisting of lanes of ants moving in opposite directions as well as the transition back to the individual state once the food source is depleted matching prior experimental results. This indicates that in the absence of direct communication ants naturally form an efficient method for transporting food to the nest/bivouac. The model exhibits a continuous kinetic phase transition in the order parameter as a function of certain system parameters. The associated critical exponents are found, shedding light on the behavior of the system near the transition.Comment: Preprint Version, 30 pgs., 18 figures, complete version with supplementary movies to appear in Journal of Mathematical Biology (Springer

Role of Hydrodynamic Interactions in Chemotaxis of Bacterial Populations

How bacteria sense local chemical gradients and decide to move has been a fascinating area of recent study. Chemotaxis of bacterial populations has been traditionally modeled using either individual-based models describing the motion of a single bacterium as a velocity jump process, or macroscopic PDE models that describe the evolution of the bacterial density. In these models, the hydrodynamic interaction between the bacteria is usually ignored. However, hydrodynamic interaction has been shown to induce collective bacterial motion and self-organization resulting in larger mesoscale structures. In this paper, the role of hydrodynamic interactions in bacterial chemotaxis is investigated by extending a hybrid computational model that incorporates hydrodynamic interactions and adding components from a classical velocity jump model. It is shown that by including hydrodynamic interactions, a suspension with a low initial volume fraction can exhibit locally high concentrations in bacterial aggregates. Also, it is shown that hydrodynamic interactions enhance the merging of the small aggregates into larger ones and lead to qualitatively different aggregate behavior than possible with pure chemotaxis models. Namely, differences in the shape, number, and dynamics of these emergent clusters

An Elastica Model of the Buckling of a Nanoscale Sheet Perpendicular to a Rigid Substrate

We study a variation on the classical problem of the buckling of an elastica. The elastica models a nanoscale sheet that interacts with a rigid substrate by intermolecular forces. We formulate a buckling problem in which the sheet is perpendicular to the substrate and a load is applied to the edge of the sheet further from the substrate. Our study is motivated by problems in nanomechanics such as the bending of a graphene sheet interacting with a rigid substrate by van der Waals forces. After identifying a trivial branch, we combine computation and analysis to determine the stability and bifurcations of solutions along this branch. We also study the boundary-layer problem that arises if the length of the sheet is large compared to the characteristic length over which the van der Waals interaction is significant

Pulsatile Flow Through Idealized Renal Tubules: Fluid-structure Interaction and Dynamic Pathologies

Kidney tubules are lined with flow-sensing structures, yet information about the flow itself is not easily obtained. We aim to generate a multiscale biomechanical model for analyzing fluid flow and fluid-structure interactions within an elastic kidney tubule when the driving pressure is pulsatile. We developed a two-dimensional macroscopic mathematical model of a single fluid-filled tubule corresponding to a distal nephron segment and determined both flow dynamics and wall strains over a range of driving frequencies and wall compliances using finite-element analysis. The results presented here demonstrate good agreement with available analytical solutions and form a foundation for future inclusion of elastohydrodynamic coupling by neighboring tubules. Overall, we are interested in exploring the idea of dynamic pathology to better understand the progression of chronic kidney diseases such as Polycystic Kidney Disease

Fluid-Structure Interaction Modelling of Neighboring Tubes with Primary Cilium Analysis

We have developed a numerical model of two osculating cylindrical elastic renal tubules to investigate the impact of neighboring tubules on the stress applied to a primary cilium. We hypothesize that the stress at the base of the primary cilium will depend on the mechanical coupling of the tubules due to local constrained motion of the tubule wall. The objective of this work was to determine the in-plane stresses of a primary cilium attached to the inner wall of one renal tubule subject to the applied pulsatile flow, with a neighboring renal tube filled with stagnant fluid in close proximity to the primary tubule. We used the commercial software COMSOL (R) to model the fluid-structure interaction of the applied flow and tubule wall, and we applied a boundary load to the face of the primary cilium during this simulation to produces a stress at its base. We confirm our hypothesis by observing that on average the in-plane stresses are greater at the base of the cilium when there is a neighboring renal tube versus if there is no neighboring tube at all. In combination with the hypothesized function of a cilium as a biological fluid flow sensor, these results indicate that flow signaling may also depend on how the tubule wall is constrained by neighboring tubules. Our results may be limited in their interpretation due to the simplified nature of our model geometry, and further improvements to the model may potentially lead to the design of future experiments

Curvature-driven Foam Coarsening on a Sphere: A Computer Simulation

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state

Curvature-driven Foam Coarsening on a Sphere: A Computer Simulation

The von Neumann-Mullins law for the area evolution of a cell in the plane describes how a dry foam coarsens in time. Recent theory and experiment suggest that the dynamics are different on the surface of a three-dimensional object such as a sphere. This work considers the dynamics of dry foams on the surface of a sphere. Starting from first principles, we use computer simulation to show that curvature-driven motion of the cell boundaries leads to exponential growth and decay of the areas of cells, in contrast to the planar case where the growth is linear. We describe the evolution and distribution of cells to the final stationary state

A Kinetic Model for Semidilute Bacterial Suspensions

Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated with the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal spheroids representing self-propelled particles) with Lennard-Jones--type repulsion. A planar shear background flow is imposed on the suspension through the novel use of Lees--Edwards quasi-periodic boundary conditions applied to a representative volume. We show the existence and uniqueness of solutions for all time to the equations of motion for particle configurations---dipole orientations and relative positions. This result follows from first establishing the regularity of the solution to the fluid equations. The existence and uniqueness result allows us to define the Liouville equation for the probability density of configurations. We show that this probability density defines the average bulk stress in the suspension underlying the definition of many macroscopic quantities of interest, in particular the effective viscosity. These effective properties are determined by microscopic interactions highlighting the multiscale nature of this work

A Kinetic Model for Semidilute Bacterial Suspensions

Suspensions of self-propelled microscopic particles, such as swimming bacteria, exhibit collective motion leading to remarkable experimentally observable macroscopic properties. Rigorous mathematical analysis of this emergent behavior can provide significant insight into the mechanisms behind these experimental observations; however, there are many theoretical questions remaining unanswered. In this paper, we study a coupled PDE/ODE system first introduced in the physics literature and used to investigate numerically the effective viscosity of a bacterial suspension. We then examine the kinetic theory associated with the coupled system, which is designed to capture the long-time behavior of a Stokesian suspension of point force dipoles (infinitesimal spheroids representing self-propelled particles) with Lennard-Jones--type repulsion. A planar shear background flow is imposed on the suspension through the novel use of Lees--Edwards quasi-periodic boundary conditions applied to a representative volume. We show the existence and uniqueness of solutions for all time to the equations of motion for particle configurations---dipole orientations and relative positions. This result follows from first establishing the regularity of the solution to the fluid equations. The existence and uniqueness result allows us to define the Liouville equation for the probability density of configurations. We show that this probability density defines the average bulk stress in the suspension underlying the definition of many macroscopic quantities of interest, in particular the effective viscosity. These effective properties are determined by microscopic interactions highlighting the multiscale nature of this work

Rayleigh Approximation to Ground State of the Bose and Coulomb Glasses

Glasses are rigid systems in which competing interactions prevent simultaneous minimization of local energies. This leads to frustration and highly degenerate ground states the nature and properties of which are still far from being thoroughly understood. We report an analytical approach based on the method of functional equations that allows us to construct the Rayleigh approximation to the ground state of a two-dimensional (2D) random Coulomb system with logarithmic interactions. We realize a model for 2D Coulomb glass as a cylindrical type II superconductor containing randomly located columnar defects (CD) which trap superconducting vortices induced by applied magnetic field. Our findings break ground for analytical studies of glassy systems, marking an important step towards understanding their properties