Robert Dillon

Pattern formation in generalized Turing systems

Turings model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.

Simulation of swimming organisms: coupling internal mechanics with external fluid dynamics

Problems in biological fluid dynamics typically involve the interaction of an elastic structure with its surrounding fluid. A unified computational approach, based on an immersed boundary framework, couples the internal force-generating mechanisms of organisms and cells with an external, viscous, incompressible fluid. Computational simulation, in conjunction with laboratory experiment, can provide valuable insight into complex biological systems that involve the interaction of an elastic structure with a viscous, incompressible fluid. This biological fluid-dynamics setting presents several more challenges than those traditionally faced in computational fluid dynamics - specifically, dynamic flow situations dominate, and capturing time-dependent geometries with large structural deformations is necessary. In addition, the shape of the elastic structures is not preset: fluid dynamics determines it. This article presents our recent progress on coupling the internal molecular motor mechanisms of beating cilia and flagella with an external fluid, as well as the three-dimensional (3D) undulatory swimming of nematodes and leeches. We expect these computational models to provide a testbed for examining different theories of internal force-generation mechanisms.

Fluid Dynamic Models of Flagellar and Ciliary Beating

Abstract:  We have developed a fluid–mechanical model of a eucaryotic axoneme that couples the internal force generation of dynein molecular motors, the passive elastic mechanics of microtubules, and forces due to nexin links with a surrounding incompressible fluid. This model has been used to examine both ciliary beating and flagellar motility. In this article, we show preliminary simulation results for sperm motility in both viscous and viscoelastic fluids, as well as multiciliary interaction with a mucus layer.

Short- and long-range effects of Sonic hedgehog in limb development

The secreted protein Sonic hedgehog (Shh) and its transmembrane receptor Patched (Ptc) control a major signal transduction pathway in early vertebrate limb development. Ligand-free Ptc interacts with the transmembrane protein Smoothened (Smo) and blocks expression of Smo-controlled genes including ptc. Ligand-bound Ptc removes the block and leads to further expression of ptc, which in turn restricts the range of Shh transport. Currently it is not certain that Shh functions as a morphogen on the 300-μm scale of early chick limb development, because it has been difficult to determine how far different forms of Shh are transported. We develop a model to study the effects of two forms of Shh used experimentally and propose a mechanism for Shh signal transduction based on a two-state model for the Ptc-Smo interaction. Recent bead- and tissue-implant experiments can be explained by using this model without postulating different diffusivities for the two forms of Shh; a difference in other parameters such as the rate of release of Shh from the bead or transplant can explain the results equally well. The model also predicts that lower concentrations of Shh in a bead will produce a response similar to that after a tissue transplant. Our results provide an explanation for the counterintuitive experimental results and show that the same signal transduction mechanism can explain both short- and long-range Shh signaling. We conclude that Shh can function as a long-range morphogen.

Modeling and simulation of dielectrophoretic particle–particle interactions and assembly

Electric field induced particle-particle interactions and assembly are of great interest due to their useful applications in micro devices. The behavior of particles becomes more complex if multiple particles interact with each other at the same time. In this paper, we present a numerical study of two dimensional DC dielectrophoresis based particle-particle interactions and assembly for multiple particles using a hybrid immersed interface-immersed boundary method. The immersed interface method is employed to capture the physics of electrostatics in a fluid media with suspended particles. Particle interaction based dielectrophoretic forces are obtained using Maxwells stress tensor without any boundary or volume integration. This electrostatic force distribution mimics the actual physics of the immersed particles in a fluid media. The corresponding particle response and hydrodynamic interactions are captured through the immersed boundary method by solving the transient Navier-Stokes equations. The interaction and assembly of multiple electrically similar and dissimilar particles are studied for various initial positions and orientations. Numerical results show that in a fluid media, similar particles form a chain parallel to the applied electric field, whereas dissimilar particles form a chain perpendicular to the applied electric field. Irrespective of initial position and orientation, particles first align themselves parallel or perpendicular to the electric field depending on the similarity or dissimilarity of particles. The acceleration and deceleration of particles are also observed and analyzed at different phases of the assembly process. This comprehensive study can be used to explain the multiple particle interaction and assembly phenomena observed in experiments.

A microscale model of bacterial and biofilm dynamics in porous media.

A microscale model for the transport and coupled reaction of microbes and chemicals in an idealized two‐dimensional porous media has been developed. This model includes the flow, transport, and bioreaction of nutrients, electron acceptors, and microbial cells in a saturated granular porous media. The fluid and chemicals are represented as a continuum, but the bacterial cells and solid granular particles are represented discretely. Bacterial cells can attach to the particle surfaces or be advected in the bulk fluid. The bacterial cells can also be motile and move preferentially via a run and tumble mechanism toward a chemoattractant. The bacteria consume oxygen and nutrients and alter the profiles of these chemicals. Attachment of bacterial cells to the soil matrix and growth of bacteria can change the local permeability. The coupling of mass transport and bioreaction can produce spatial gradients of nutrients and electron acceptor concentrations. We describe a numerical method for the microscale model, show results of a convergence study, and present example simulations of the model system.

An Integrative Computational Model of Multiciliary Beating

The coordinated beating of motile cilia is responsible for ovum transport in the oviduct, transport of mucus in the respiratory tract, and is the basis of motility in many single-celled organisms. The beating of a single motile cilium is achieved by the ATP-driven activation cycles of thousands of dynein molecular motors that cause neighboring microtubule doublets within the ciliary axoneme to slide relative to each other. The precise nature of the spatial and temporal coordination of these individual motors is still not completely understood. The emergent geometry and dynamics of ciliary beating is a consequence of the coupling of these internal force-generating motors, the passive elastic properties of the axonemal structure, and the external viscous, incompressible fluid. Here, we extend our integrative model of a single cilium that couples internal force generation with the surrounding fluid to the investigation of multiciliary interaction. This computational model allows us to predict the geometry of beating, along with the detailed description of the time-dependent flow field both near and away from the cilia. We show that synchrony and metachrony can, indeed, arise from hydrodynamic coupling. We also investigate the effects of viscosity and neighboring cilia on ciliary beat frequency. Moreover, since we have precise flow information, we also measure the dependence of the total flow pumped per cilium per beat upon parameters such as viscosity and ciliary spacing.

A single cell-based model of the ductal tumour microarchitecture.

The preinvasive intraductal tumours, such as the breast or prostate carcinomas, develop in many different architectural forms. There are, however, no experimental models explaining why cancer cells grow in these various configurations. We use a mathematical model to compare different proliferative conditions that can lead to such distinct microarchitectures. In order to simulate different scenarios of tumour growth, we employed a single cell-based technique that allows us to model development of the whole tumour tissue by focusing on biomechanical processes of individual cells and on communication between cells and their microenvironment. Formation of four specific intraductal tumour patterns, micropapillary, cribriform, tufting and solid, are presented in this paper together with a discussion on gradual dedifferentiation of ductal epithelial cells that gives rise to these distinct carcinomas. We introduce two versions of our cell-based model to show that the obtained results do not depend on a particularly chosen cell structure.

A 3D Motile Rod-Shaped Monotrichous Bacterial Model

We introduce a 3D model for a motile rod-shaped bacterial cell with a single polar flagellum which is based on the configuration of a monotrichous type of bacteria such as Pseudomonas aeruginosa. The structure of the model bacterial cell consists of a cylindrical body together with the flagellar forces produced by the rotation of a helical flagellum. The rod-shaped cell body is composed of a set of immersed boundary points and elastic links. The helical flagellum is assumed to be rigid and modeled as a set of discrete points along the helical flagellum and flagellar hook. A set of flagellar forces are applied along this helical curve as the flagellum rotates. An additional set of torque balance forces are applied on the cell body to induce counter-rotation of the body and provide torque balance. The three-dimensional Navier–Stokes equations for incompressible fluid are used to describe the fluid dynamics of the coupled fluid–microorganism system using Peskin’s immersed boundary method. A study of numerical convergence is presented along with simulations of a single swimming cell, the hydrodynamic interaction of two cells, and the interaction of a small cluster of cells.

Hybrid immersed interface-immersed boundary methods for AC dielectrophoresis

Abstract Dielectrophoresis, a nonlinear electrokinetic transport mechanism, has become popular in many engineering applications including manipulation, characterization and actuation of biomaterials, particles and biological cells. In this paper, we present a hybrid immersed interface–immersed boundary method to study AC dielectrophoresis where an algorithm is developed to solve the complex Poisson equation using a real variable formulation. An immersed interface method is employed to obtain the AC electric field in a fluid media with suspended particles and an immersed boundary method is used for the fluid equations and particle transport. The convergence of the proposed algorithm as well as validation of the hybrid scheme with experimental results is presented. In this paper, the Maxwell stress tensor is used to calculate the dielectrophoretic force acting on particles by considering the physical effect of particles in the computational domain. Thus, this study eliminates the approximations used in point dipole methods for calculating dielectrophoretic force. A comparative study between Maxwell stress tensor and point dipole methods for computing dielectrophoretic forces are presented. The hybrid method is used to investigate the physics of dielectrophoresis in microfluidic devices using an AC electric field. The numerical results show that with proper design and appropriate selection of applied potential and frequency, global electric field minima can be obtained to facilitate multiple particle trapping by exploiting the mechanism of negative dielectrophoresis. Our numerical results also show that electrically neutral particles form a chain parallel to the applied electric field irrespective of their initial orientation when an AC electric field is applied. This proposed hybrid numerical scheme will help to better understand dielectrophoresis and to design and optimize microfluidic devices.