On the dynamics of a self-gravitating medium with random and non-random initial conditions

The dynamics of a one-dimensional self-gravitating medium, with initial density almost uniform is studied. Numerical experiments are performed with ordered and with Gaussian random initial conditions. The phase space portraits are shown to be qualitatively similar to shock waves, in particular with initial conditions of Brownian type. The PDF of the mass distribution is investigated.Comment: Latex, figures in eps, 23 pages, 11 figures. Revised versio

Lattice Kinetic Monte Carlo Simulations of Platelet Aggregation and Deposition

Platelet aggregation is an essential process in forming a stable clot to prevent blood loss. The response of platelets to a complex signal of pro-clotting agonists determines the stability and size of the resulting clot. An underdeveloped clot represents a bleeding risk, while an overdeveloped clot can cause vessel occlusion, which can lead to heart attack or stroke. A multiscale model was developed to study the integration of platelet signaling within the complex phenomena driven by flow. The model is built upon a lattice kinetic Monte Carlo algorithm (LKMC) to track platelet motion and binding. First, a new method for including flow-driven particle motion in LKMC was derived from a timescale analysis of particle motion. Simple methods for simulating flow-driven motion were found to exhibit concentration dependent velocities violating the assumptions in the model. The nature of the error was analyzed mathematically and resolved by considering the chain length distribution on the lattice. The accuracy of the method was found to scale linearly with the lattice spacing. Second, the LKMC method was extended to study particle aggregation in complex flows. The LKMC results for simple flows were compared directly to a continuum population balance equation (PBE) approach. A contact time model was introduced to capture nonideal collisions in the LKMC model and a connection to the continuum collision efficiency was derived. The particle size distribution for a baffled geometry with regions of standing vortices and squeezing flows was determined using the LKMC method for varying baffle heights. Finally, the LKMC method was incorporated within a multiscale model to simulate platelet aggregation including platelet signaling (neural network model), blood flow (lattice Boltzmann method), and the release of soluble platelet agonists (finite element method). The neural network model for platelet signaling was trained on patient-specific, experimental measurements of intracellular calcium enabling patient-specific predictions of platelet function in flow. The model accurately predicted the order of potency for three antiplatelet therapies, donor-specific aggregate size, and donor-specific response to antiplatelet therapy as compared to microfluidic experiments of platelet aggregation

Taxis Equations for Amoeboid Cells

The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally-measured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.Comment: 35 pages, submitted to the Journal of Mathematical Biolog

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

Active Brownian Particles. From Individual to Collective Stochastic Dynamics

We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of such self-propelled entities in the framework of statistical mechanics. Examples of such active units in complex physico-chemical and biological systems are chemically powered nano-rods, localized patterns in reaction-diffusion system, motile cells or macroscopic animals. Based on the description of individual motion of point-like active particles by stochastic differential equations, we discuss different velocity-dependent friction functions, the impact of various types of fluctuations and calculate characteristic observables such as stationary velocity distributions or diffusion coefficients. Finally, we consider not only the free and confined individual active dynamics but also different types of interaction between active particles. The resulting collective dynamical behavior of large assemblies and aggregates of active units is discussed and an overview over some recent results on spatiotemporal pattern formation in such systems is given.Comment: 161 pages, Review, Eur Phys J Special-Topics, accepte

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

Crystal morphology is known to be of great importance to the end-use properties of crystal products, and to affect down-stream processing such as filtration and drying. However, it has been previously regarded as too challenging to achieve automatic closed-loop control. Previous work has focused on controlling the crystal size distribution, where the size of a crystal is often defined as the diameter of a sphere that has the same volume as the crystal. This paper reviews the new advances in morphological population balance models for modelling and simulating the crystal shape distribution (CShD), measuring and estimating crystal facet growth kinetics, and two- and three-dimensional imaging for on-line characterisation of the crystal morphology and CShD. A framework is presented that integrates the various components to achieve the ultimate objective of model-based closed-loop control of the CShD. The knowledge gaps and challenges that require further research are also identified