4,011 research outputs found
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
Experimental demonstration of evanescent coupling from optical fibre tapers to photonic crystal waveguides
Experimental results demonstrating nearly complete mode-selective evanescent coupling to a photonic crystal waveguide from an optical fibre taper are presented. Codirectional coupling with 98% maximum power transfer to a photonic crystal waveguide of length 65 μm and with a coupling bandwidth of 20 nm is realised
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
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
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
Wavelength- and material-dependent absorption in GaAs and AlGaAs microcavities
The quality factors of modes in nearly identical GaAs and
Al_{0.18}Ga_{0.82}As microdisks are tracked over three wavelength ranges
centered at 980 nm, 1460 nm, and 1600 nm, with quality factors measured as high
as 6.62x10^5 in the 1600-nm band. After accounting for surface scattering, the
remaining loss is due to sub-bandgap absorption in the bulk and on the
surfaces. We observe the absorption is, on average, 80 percent greater in
AlGaAs than in GaAs and in both materials is 540 percent higher at 980 nm than
at 1600nm.Comment: 4 pages, 2 figures, 1 table, minor changes to disucssion of Qrad and
Urbach tai
Position-squared coupling in a tunable photonic crystal optomechanical cavity
We present the design, fabrication, and characterization of a planar silicon
photonic crystal cavity in which large position-squared optomechanical coupling
is realized. The device consists of a double-slotted photonic crystal structure
in which motion of a central beam mode couples to two high-Q optical modes
localized around each slot. Electrostatic tuning of the structure is used to
controllably hybridize the optical modes into supermodes which couple in a
quadratic fashion to the motion of the beam. From independent measurements of
the anti-crossing of the optical modes and of the optical spring effect, the
position-squared vacuum coupling rate is measured to be as large as 245 Hz to
the fundamental in-plane mechanical resonance of the structure at 8.7MHz, which
in displacement units corresponds to a coupling coefficient of 1 THz/nm.
This level of position-squared coupling is approximately five orders of
magnitude larger than in conventional Fabry-Perot cavity systems.Comment: 11 pages, 6 figure
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
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