Steady-state patterns in a reaction diffusion system with mixed boundary conditions

A number of models for pattern formation and regulation are based on the hypothesis that a diffusible morphogen supplies positional information that can interpreted by cells. Such models fall into two main classes:- those in which pattern arises from distributed sources and/or sinks of the morphogens, and those which can spontaneously produce pattern via the interaction of reaction and transport. In source-sink models, specialized cells maintain the concentration of the morphogen at fixed levels, and given a suitable distribution of sources and sinks, a tissue can be proportioned into any number of cell types with a threshold interpretation mechanism. However, the spatial pattern established is strongly dependent on the distances between the sources and sinks, and additional hyoptheses must be invoked to ensure that the pattern is invariant under changes in the scale of the system

Spatial variation in boundary conditions can govern selection and location of eyespots in butterfly wings

Despite being the subject of widespread study, many aspects of the development of eyespot patterns in butterfly wings remain poorly understood. In this work, we examine, through numerical simulations, a mathematical model for eyespot focus point formation in which a reaction-diffusion system is assumed to play the role of the patterning mechanism. In the model, changes in the boundary conditions at the veins at the proximal boundary alone are capable of determining whether or not an eyespot focus forms in a given wing cell and the eventual position of focus points within the wing cell. Furthermore, an auxiliary surface reaction diffusion system posed along the entire proximal boundary of the wing cells is proposed as the mechanism that generates the necessary changes in the proximal boundary profiles. In order to illustrate the robustness of the model, we perform simulations on a curved wing geometry that is somewhat closer to a biological realistic domain than the rectangular wing cells previously considered, and we also illustrate the ability of the model to reproduce experimental results on artificial selection of eyespots.Publisher PD

Pattern formation in heterogeneous domains

Development of spatial pattern in the early embryo results from the interaction of several processes in a complex hierarchy of mechanisms. Most models for morphogenesis to date have, however, focussed on a particular mechanism. Although such models are capable of capturing some aspects of development they are inconsistent with key experimental observations. Here we consider a two-step hierarchy of patterning mechanisms in which the spatial pattern of a control chemical regulates morphogen diffusivity in an overlaying reaction diffusion system

Cell_motility: a cross-platform, open source application for the study of cell motion paths

BACKGROUND: Migration is an important aspect of cellular behaviour and is therefore widely studied in cell biology. Numerous components are known to participate in this process in a highly dynamic manner. In order to obtain a better insight in cell migration, mutants or drugs are used and their motive phenotype is then linked with the disturbing factors. One of the typical approaches to study motion paths of individual cells relies on fitting mean square displacements to a persistent random walk function. Since the numerous calculations involved often rely on diverse commercial software packages, the analysis can be expensive, labour-intensive and error-prone work. Additionally, due to the nature of algorithms employed the calculations involved are not readily reproducible without access to the exact software package(s) used. RESULTS: We here present the cell_motility software, an open source Java application under the GNU-GPL license that provides a clear and concise analysis workbench for large amounts of cell motion data. Apart from performing the necessary calculations, the software also visualizes the original motion paths as well as the results of the calculations to help the user interpret the data. The application features an intuitive graphical user interface as well as full user and developer documentation and both source and binary files can be freely downloaded from the project website at . CONCLUSION: In providing a free, open source software solution for the automated processing of cell motion data, we aim to achieve two important goals: labs can greatly simplify their data analysis pipeline as switching between different computational software packages becomes obsolete (thus reducing the chances for human error during data manipulation and transfer) and secondly, to provide scientists in the field with a freely available common platform to perform their analyses, enabling more efficient data quality control through peer reviewing

Multiscale models for movement in oriented environments and their application to hilltopping in butterflies

Hilltopping butterflies direct their movement in response to topography, facilitating mating encounters via accumulation at summits. In this paper, we take hilltopping as a case study to explore the impact of complex orienteering cues on population dynamics. The modelling employs a standard multiscale framework, in which an individual's movement path is described as a stochastic 'velocity-jump' process and scaling applied to generate a macroscopic model capable of simulating large populations in landscapes. In this manner, the terms and parameters of the macroscopic model directly relate to statistical inputs of the individual-level model (mean speeds, turning rates and turning distributions). Applied to hilltopping in butterflies, we demonstrate how hilltopping acts to aggregate populations at summits, optimising mating for low-density species. However, for abundant populations, hilltopping is not only less effective but also possibly disadvantageous, with hilltopping males recording a lower mating rate than their non-hilltopping competitors. © 2013 Springer Science+Business Media Dordrecht

Directed cell migration in the presence of obstacles

BACKGROUND: Chemotactic movement is a common feature of many cells and microscopic organisms. In vivo, chemotactic cells have to follow a chemotactic gradient and simultaneously avoid the numerous obstacles present in their migratory path towards the chemotactic source. It is not clear how cells detect and avoid obstacles, in particular whether they need a specialized biological mechanism to do so. RESULTS: We propose that cells can sense the presence of obstacles and avoid them because obstacles interfere with the chemical field. We build a model to test this hypothesis and find that this naturally enables efficient at-a-distance sensing to be achieved with no need for a specific and active obstacle-sensing mechanism. We find that (i) the efficiency of obstacle avoidance depends strongly on whether the chemotactic chemical reacts or remains unabsorbed at the obstacle surface. In particular, it is found that chemotactic cells generally avoid absorbing barriers much more easily than non-absorbing ones. (ii) The typically low noise in a cell's motion hinders the ability to avoid obstacles. We also derive an expression estimating the typical distance traveled by chemotactic cells in a 3D random distribution of obstacles before capture; this is a measure of the distance over which chemotaxis is viable as a means of directing cells from one point to another in vivo. CONCLUSION: Chemotactic cells, in many cases, can avoid obstacles by simply following the spatially perturbed chemical gradients around obstacles. It is thus unlikely that they have developed specialized mechanisms to cope with environments having low to moderate concentrations of obstacles

Trail formation based on directed pheromone deposition

We propose an Individual-Based Model of ant-trail formation. The ants are modeled as self-propelled particles which deposit directed pheromones and interact with them through alignment interaction. The directed pheromones intend to model pieces of trails, while the alignment interaction translates the tendency for an ant to follow a trail when it meets it. Thanks to adequate quantitative descriptors of the trail patterns, the existence of a phase transition as the ant-pheromone interaction frequency is increased can be evidenced. Finally, we propose both kinetic and fluid descriptions of this model and analyze the capabilities of the fluid model to develop trail patterns. We observe that the development of patterns by fluid models require extra trail amplification mechanisms that are not needed at the Individual-Based Model level

Turing patterns on networks

Turing patterns formed by activator-inhibitor systems on networks are considered. The linear stability analysis shows that the Turing instability generally occurs when the inhibitor diffuses sufficiently faster than the activator. Numerical simulations, using a prey-predator model on a scale-free random network, demonstrate that the final, asymptotically reached Turing patterns can be largely different from the critical modes at the onset of instability, and multistability and hysteresis are typically observed. An approximate mean-field theory of nonlinear Turing patterns on the networks is constructed.Comment: 4 pages, 4 figure

Novel Methods for Analysing Bacterial Tracks Reveal Persistence in Rhodobacter sphaeroides

Tracking bacteria using video microscopy is a powerful experimental approach to probe their motile behaviour. The trajectories obtained contain much information relating to the complex patterns of bacterial motility. However, methods for the quantitative analysis of such data are limited. Most swimming bacteria move in approximately straight lines, interspersed with random reorientation phases. It is therefore necessary to segment observed tracks into swimming and reorientation phases to extract useful statistics. We present novel robust analysis tools to discern these two phases in tracks. Our methods comprise a simple and effective protocol for removing spurious tracks from tracking datasets, followed by analysis based on a two-state hidden Markov model, taking advantage of the availability of mutant strains that exhibit swimming-only or reorientating-only motion to generate an empirical prior distribution. Using simulated tracks with varying levels of added noise, we validate our methods and compare them with an existing heuristic method. To our knowledge this is the first example of a systematic assessment of analysis methods in this field. The new methods are substantially more robust to noise and introduce less systematic bias than the heuristic method. We apply our methods to tracks obtained from the bacterial species Rhodobacter sphaeroides and Escherichia coli. Our results demonstrate that R. sphaeroides exhibits persistence over the course of a tumbling event, which is a novel result with important implications in the study of this and similar species

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with two examples arising from cancer invasion