Mathematical models for chemotaxis and their applications in self-organisation phenomena

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog

Development and applications of a model for cellular response to multiple chemotactic cues

The chemotactic response of a cell population to a single chemical species has been characterized experimentally for many cell types and has been extensively studied from a theoretical standpoint. However, cells frequently have multiple receptor types and can detect and respond chemotactically to more than one chemical. How these signals are integrated within the cell is not known, and we therefore adopt a macroscopic phenomenological approach to this problem. In this paper we derive and analyze chemotactic models based on partial differential (chemotaxis) equations for cell movement in response to multiple chemotactic cues. Our derivation generalizes the approach of Othmer and Stevens [29], who have recently developed a modeling framework for studying different chemotactic responses to a single chemical species. The importance of such a generalization is illustrated by the effect of multiple chemical cues on the chemotactic sensitivity and the spatial pattern of cell densities in several examples. We demonstrate that the model can generate the complex patterns observed on the skin of certain animal species and we indicate how the chemotactic response can be viewed as a form of positional indicator

Travelling waves in hyperbolic chemotaxis equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s [Keller and Segel, J. Theor. Biol., 1971]. The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically

Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics

The ability to achieve near lossless coupling between a waveguide and a resonator is fundamental to many quantum-optical studies as well as to practical applications of such structures. The nature of loss at the junction is described by a figure of merit called ideality. It is shown here that under appropriate conditions ideality in excess of 99.97% is possible using fiber-taper coupling to high-Q silica microspheres. To verify this level of coupling, a technique is introduced that can both measure ideality over a range of coupling strengths and provide a practical diagnostic of parasitic coupling within the fiber-taper-waveguide junction

Measurement of spontaneous emission from a two-dimensional photonic band gap defined microcavity at near-infrared wavelengths

An active, photonic band gap-based microcavity emitter in the near infrared is demonstrated. We present direct measurement of the spontaneous emission power and spectrum from a microcavity formed using a two-dimensional photonic band gap structure in a half wavelength thick slab waveguide. The appearance of cavity resonance peaks in the spectrum correspond to the photonic band gap energy. For detuned band gaps, no resonances are observed. For devices with correctly tuned band gaps, a two-time enhancement of the extraction efficiency was demonstrated compared to detuned band gaps and unpatterned material

Photonic bandgap disk laser

A two-dimensional photonic crystal defined hexagonal disk laser which relies on Bragg reflection rather than the total internal reflection as in traditional microdisk lasers is described. The devices are fabricated using a selective etch to form free standing membranes suspended in air. Room temperature lasing at 1650nm for a 150nm thick, ~15μm wide cavity fabricated in InP/GaAsP is demonstrated with pulsed optical pumping

Modelling the movement of interacting cell populations

Mathematical modelling of cell movement has traditionally focussed on a single population of cells, often moving in response to various chemical and environmental cues. In this paper, we consider models for movement in two or more interacting cell populations. We begin by discussing intuitive ideas underlying the extension of models for a single-cell population to two interacting populations. We then consider more formal model development using transition probability methods, and we discuss how the same equations can be obtained as the limiting form of a velocity-jump process. We illustrate the models we have developed via two examples. The first of these is a generic model for competing cell populations, and the second concerns aggregation in cell populations moving in response to chemical gradients. © 2003 Elsevier Ltd. All rights reserved

A user's guide to PDE models for chemotaxis

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399-415, 1970; 30:225- 234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display "auto-aggregation", has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller-Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work. © Springer-Verlag 2008

Transport and anisotropic diffusion models for movement in oriented habitats

A common feature of many living organisms is the ability to move and navigate in heterogeneous environments. While models for spatial spread of populations are often based on the diffusion equation, here we aim to advertise the use of transport models; in particular in cases where data from individual tracking are available. Rather than developing a full general theory of transport models, we focus on the specific case of animal movement in oriented habitats. The orientations can be given by magnetic cues, elevation profiles, food sources, or disturbances such as seismic lines or roads. In this case we are able to present and contrast the three most common scaling limits, (i) the parabolic scaling, (ii) the hyperbolic scaling, and (iii) the moment closure method. We clearly state the underlying assumptions and guide the reader to an understanding of which scaling method is used in what kind of situations. One interesting result is that the macroscopic drift velocity is given by the mean direction of the underlying linear features, and the diffusion is given by the variance-covariance matrix of the underlying oriented habitat. We illustrate our findings with specific applications to wolf movement in habitats with seismic lines. © 2013 Springer-Verlag Berlin Heidelberg

Mathematical modelling of glioma growth: The use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion

The nonuniform growth of certain forms of cancer can present significant complications for their treatment, a particularly acute problem in gliomas. A number of experimental results have suggested that invasion is facilitated by the directed movement of cells along the aligned neural fibre tracts that form a large component of the white matter. Diffusion tensor imaging (DTI) provides a window for visualising this anisotropy and gaining insight on the potential invasive pathways. In this paper we develop a mesoscopic model for glioma invasion based on the individual migration pathways of invading cells along the fibre tracts. Via scaling we obtain a macroscopic model that allows us to explore the overall growth of a tumour. To connect DTI data to parameters in the macroscopic model we assume that directional guidance along fibre tracts is described by a bimodal von Mises-Fisher distribution (a normal distribution on a unit sphere) and parametrised according to the directionality and degree of anisotropy in the diffusion tensors. We demonstrate the results in a simple model for glioma growth, exploiting both synthetic and genuine DTI datasets to reveal the potentially crucial role of anisotropic structure on invasion. © 2013 Elsevier Ltd