Angiogenesis: A Model of Cell Differentiation

Angiogenesis is the formation of blood vessels, and is of great importance in the growth of tumours. Attempts have been made to desgin experiments in petri-dishes that mimic the 'Conditions of tumour growth. The first of the experiments is the 'matrigel' assay. Matrigel provides a matrix for the endothelial cells to grow on, and contains all the nutrients that the cells need. It is found that in the matrigel assay blood vessels didn't form, although some transient strucutres formed at early times in the experiment. The second experiment is the 'biocure' assay. In this experiment the petri dish is filled with both endothelial and fibroblast cells. The fibroblasts form a strucutal supporting network for the endothelial cells. Tubules resembling blood-vessels formed after about ten days in the biocure asssay. The process of cell differentiation is thought to be important in the growth of blood vessels. Cells can sense that they are part of a blood vessel, and change their shape to form tubules. Also it is likely that they change their chemical messaging properties, and their abilities to bind to other endothelial cells. A model is developed that describes cell differentiation, and separates cells into different classes. For simplicity the spatial distribution of cells in different classes is ignored. Using simple population dynamics, a set of coupled non-linear ODEs is developed to describe the dynamics of the system. The system is found to have two different long-time states, one corresponding to the formation of blood vessels and one where vessels did not form. The ratio of the cell proliferation rate to the cell maturity rate (the time it takes to realise that it is part of a blood vessel) is critical in determining which is the final state of the system

Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations

Wrinkling of microcapsules in shear flow

Elastic capsules can exhibit short wavelength wrinkling in external shear flow. We analyse this instability of the capsule shape and use the length scale separation between the capsule radius and the wrinkling wavelength to derive analytical results both for the threshold value of the shear rate and for the critical wave-length of the wrinkling. These results can be used to deduce elastic parameters from experiments.Comment: 4 pages, 2 figures, submitted to PR

Modelling an isolated dust grain in a plasma using matched asymptotic expansions

The study of dusty plasmas is of significant practical use and scientific interest. A characteristic feature of dust grains in a plasma is that they are typically smaller than the electron Debye distance, a property which we exploit using the technique of matched asymptotic expansions. We first consider the case of a spherical dust particle in a stationary plasma, employing the Allen–Boyd–Reynolds theory, which assumes cold, collisionless ions. We derive analytical expressions for the electric potential, the ion number density and ion velocity. This requires only one computation that is not specific to a single set of dust–plasma parameters, and sheds new light on the shielding distance of a dust grain. The extension of this calculation to the case of uniform ion streaming past the dust grain, a scenario of interest in many dusty plasmas, is less straightforward. For streaming below a certain threshold we again establish asymptotic solutions but above the streaming threshold there appears to be a fundamental change in the behaviour of the system

Ink Drying in Inkjet Printers

The first problem put to the Study Group for Maths in Industry by Domino UK Ltd concerns ink drying and blocking nozzles in a printer. The goals were as follows: 1. To propose mechanisms for the growth of a plug of dried ink in the open end of a Drop-on-Demand drop generator, 2. To suggest cures to this problem, 3. To consider why oscillating the meniscus appears to alleviate the problem

Elastic instability in stratified core annular flow

We study experimentally the interfacial instability between a layer of dilute polymer solution and water flowing in a thin capillary. The use of microfluidic devices allows us to observe and quantify in great detail the features of the flow. At low velocities, the flow takes the form of a straight jet, while at high velocities, steady or advected wavy jets are produced. We demonstrate that the transition between these flow regimes is purely elastic -- it is caused by viscoelasticity of the polymer solution only. The linear stability analysis of the flow in the short-wave approximation captures quantitatively the flow diagram. Surprisingly, unstable flows are observed for strong velocities, whereas convected flows are observed for low velocities. We demonstrate that this instability can be used to measure rheological properties of dilute polymer solutions that are difficult to assess otherwise.Comment: 4 pages, 4 figure

Additive Equivalence in Turbulent Drag Reduction by Flexible and Rodlike Polymers

We address the "Additive Equivalence" discovered by Virk and coworkers: drag reduction affected by flexible and rigid rodlike polymers added to turbulent wall-bounded flows is limited from above by a very similar Maximum Drag Reduction (MDR) asymptote. Considering the equations of motion of rodlike polymers in wall-bounded turbulent ensembles, we show that although the microscopic mechanism of attaining the MDR is very different, the macroscopic theory is isomorphic, rationalizing the interesting experimental observations.Comment: 8 pages, PRE, submitte

The role of inertia for the rotation of a nearly spherical particle in a general linear flow

We analyse the angular dynamics of a neutrally buoyant nearly spherical particle immersed in a steady general linear flow. The hydrodynamic torque acting on the particle is obtained by means of a reciprocal theorem, regular perturbation theory exploiting the small eccentricity of the nearly spherical particle, and assuming that inertial effects are small, but finite.Comment: 7 pages, 1 figur