Dynamics of defect formation

A dynamic symmetry-breaking transition with noise and inertia is analyzed. Exact solution of the linearized equation that describes the critical region allows precise calculation (exponent and prefactor) of the number of defects produced as a function of the rate of increase of the critical parameter. The procedure is valid in both the overdamped and underdamped limits. In one space dimension, we perform quantitative comparison with numerical simulations of the nonlinear nonautonomous stochastic partial differential equation and report on signatures of underdamped dynamics.Comment: 4 pages, LaTeX, 4 figures. Submitted to Physical Revie

Urbanisation factors impacting on ant (Hymenoptera: Formicidae) biodiversity in the Perth metropolitan area, Western Australia: Two case studies

Two synchronous projects undertaken in 2011 examined the likely impact of increasing urban densification on invertebrate populations within urban settlement in Perth, Western Australia. One project analysed the ant fauna found in 20 gardens and lawns in small to very small properties (these having a bungalow or duplex (semi-detached) as the main residential building, and a lawn or garden area of 43 m2–332 m2) east, south, north and west of the Central Business District (CBD). The other project examined the ant fauna at 14 sites, principally in native regrowth along the Kwinana Freeway, a major artery that runs north to south through Perth’s suburbs. The gardens and lawns produced a very depauperate fauna of 26 ant species, of which a maximum of 20 were native and at least six species were exotic. The ant fauna from regrowth adjacent to the Kwinana Freeway and at two additional sites (one a bush control) was more than twice as rich, the 56 species collected including only two exotics. In the garden project, ant richness, evenness and abundance were not significantly correlated with size of the garden area. The same applied even when the exotic Pheidole megacephala-dominated gardens were removed from the analysis. Ordination analysis combining the two sets of data revealed a distinct clustering of most of the regrowth sites, whereas the bush control stood alone and garden or lawn sites exhibited a much looser pattern of association. We suggest that increasing the density of Perth suburbs is resulting in drastic loss of native invertebrate fauna, of which ants are a useful bioindicator. However, native vegetation regrowth along major arterial roads could act as a reservoir for invertebrate species that might otherwise disappear entirely from the Perth metropolitan area

A Novel Stochastic Multi-Scale Model of Francisella tularensis Infection to Predict Risk of Infection in a Laboratory

We present a multi-scale model of the within-phagocyte, within-host and population-level infection dynamics of Francisella tularensis, which extends the mechanistic one proposed by Wood et al. (2014). Our multi-scale model incorporates key aspects of the interaction between host phagocytes and extracellular bacteria, accounts for inter-phagocyte variability in the number of bacteria released upon phagocyte rupture, and allows one to compute the probability of response, and mean time until response, of an infected individual as a function of the initial infection dose. A Bayesian approach is applied to parameterize both the within-phagocyte and within-host models using infection data. Finally, we show how dose response probabilities at the individual level can be used to estimate the airborne propagation of Francisella tularensis in indoor settings (such as a microbiology laboratory) at the population level, by means of a deterministic zonal ventilation model

Diffusion in a Disk with a Circular Inclusion

We consider diffusion in a disk, representing a cell with a circular interior compartment. Using bipolar coordinates, we perform exact calculations, not restricted by the size or location of the intracellular compartment. We find Green functions, hitting densities and mean times to move from the compartment to the cellular surface and vice versa. For molecules with diffusivity $D$, mean times are proportional to $R^2/D$, where $R$ is the radius of the cell. We find explicit expressions for the dependence on $a^2$ (the fraction of the cell occupied by the intracellular compartment) and on the displacement of the compartment from the center of the cell. We consider distributions of initial conditions that are (i) uniform on the nuclear surface, (ii) uniform on the cellular surface, or (iii) given by the hitting density of particles diffusing from the nuclear to the cellular surface

Sampling from T cell receptor repertoires

Modern single-cell sequencing techniques allow the unique TCR signature of each of a sample of hundreds of T cells to be read. The mathematical challenge is to extrapolate from the properties of a sample to those of the whole repertoire of an individual, made up of many millions of T cells. We consider the distribution of the number of repeats of any TCR in a sample, the mean number of samples needed to find a repeat with probability one half, and the relationship between the true distribution of clonal sizes and that experimentally observed in the sample. We consider two special cases, where the distribution of clonal sizes is geometric, and where a subset of clones in the repertoire is expanded

A new mechanism shapes the naïve CD8+ T cell repertoire: the selection for full diversity

During thymic T cell differentiation, TCR repertoires are shaped by negative, positive and agonist selection. In the thymus and in the periphery, repertoires are also shaped by strong inter-clonal and intra-clonal competition to survive death by neglect. Understanding the impact of these events on the T cell repertoire requires direct evaluation of TCR expression in peripheral naïve T cells. Several studies have evaluated TCR diversity, with contradictory results. Some of these studies had intrinsic technical limitations since they used material obtained from T cell pools, preventing the direct evaluation of clone sizes. Indeed with these approaches, identical TCRs may correspond to different cells expressing the same receptor, or to several amplicons from the same T cell. We here overcame this limitation by evaluating TCRB expression in individual naïve CD8+ T cells. Of the 2269 Tcrb sequences we obtained from 13 mice, 99% were unique. Mathematical analysis of this data showed that the average number of naïve peripheral CD8+ T cells expressing the same TCRB is 1.1 cell. Since TCRA co-expression studies could only increase repertoire diversity, these results reveal that the number of naïve T cells with unique TCRs approaches the number of naïve cells. Since thymocytes undergo multiple rounds of divisions after TCRB rearrangement; and 3–5% of thymocytes survive thymic selection events; the number of cells expressing the same TCRB was expected to be much higher. Thus, these results suggest a new repertoire selection mechanism, which strongly selects for full TCRB diversity

Stochastic dynamics of Francisella tularensis infection and replication

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo

Mathematics in Modern Immunology

Mathematical and statistical methods enable multi-disciplinary approaches that catalyse discovery. Together with experimental methods, they identify key hypotheses, de ne measurable observables and reconcile disparate results. We collect a representative sample of studies in cell biology that illustrate the benefits of modelling-experimental collaborations and that have proven valuable or even ground-breaking. We conclude that it is possible to nd excellent examples of synergy between mathematical modelling and experiment in Immunology, which have brought signi cant insight that would not be available without these collaborations, but that much remains to be discovered

Coexisting periodic attractors in injection locked diode lasers

We present experimental evidence for coexisting periodic attractors in a semiconductor laser subject to external optical injection. The coexisting attractors appear after the semiconductor laser has undergone a Hopf bifurcation from the locked steady state. We consider the single mode rate equations and derive a third order differential equation for the phase of the laser field. We then analyze the bifurcation diagram of the time periodic states in terms of the frequency detuning and the injection rate and show the existence of multiple periodic attractors.Comment: LaTex, 14 pages, 6 postscript figures include