Nonlocal interactions by repulsive-attractive potentials: radial ins/stability

In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins/stability are sharp.Comment: 42 pages, 7 figure

On the structural response of eukaryotic cells

textThe actin, microtubule and intermediate filament cytoskeletal polymer assemblies, along with their accessory proteins, govern the mechanical or structural response of an eukaryotic cell to an external stress. Using statistical mechanics tools, this dissertation investigates the molecular properties, such as mesh size, persistence length and filament length, that determine the structural strength of the in vivo polymer networks, with an emphasis on the actin network or the cortex of cells. Our study of actin shows how the wide range of shear moduli from 1 Pa to 1 kPa that spans the viscous sol-like state to the elastic gel-like state witnessed in eukaryotic cells can be achieved through transient crosslinking and the spatial distribution of actin and actin crosslinking proteins alone. Thus, this gel-sol transition is achieved without the action of any severing or capping proteins that depolymerize the actin network. In order to understand how the microscopic quantities controlling the structural properties of these in vivo polymers are related to the deformation of a cell observed experimentally, a cell model is created by us. It starts with modeling the actin cortex as a thick shell and increases in complexity to include microtubules and the nucleus. Our cell model predicts that the structural response of the cell amplifies changes in molecular properties such as the in vivo actin concentration. Hence, the sensitivity of the structural response to cytoskeletal changes can be used to distinguish between different cells such as normal and cancer cells and can serve as an indicator of disease.Physic

Biomolecular design elements : cortical microtubes and DNA-coated colloids

This thesis deals with the self-organizing properties of systems of biomolecules

Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems

From physics to pharmacology?

Over the last fifty years there has been an explosion of biological data, leading to the realization that to fully explain biological mechanisms it is necessary to interpret them as complex dynamical systems. The first stage of this interpretation is to determine which components (proteins, genes or metabolites) of the system interact. This is usually represented by a graph, or network. The behavior of this network can then be investigated using mathematical modeling. In vivo these biological networks show several remarkable (and seemingly paradoxical) properties including robustness, plasticity and sensitivity. Erroneous behavior of these networks is often associated with disease. Hence understanding the system-level properties can have important implications for the treatment of disease. Systems biology is an organized approach to quantitatively describe and elucidate the behavior of these complex networks. This review focuses on the progress and future challenges of a systems approach to biology

Modelling the endothelial cell response to fluid flow

In vitro endothelial cells respond to fluid flow by elongating in the direction of flow. How the mechanical signal is transformed into an organised and directed response is poorly understood. The most studied and crucial aspects to this response are; actin filament alignment, mechano-transduction, signal transduction, Rho GTPase localised activation and lamellipodium formation. The goal of this project is to understand how these separate facets interact and lead to a coordinated response. The flow is modelled over a 3D virtual cell, which naturally gives the force the flow exerts on the cell surface via a boundary integral representation. This force is coupled to a Kelvin-body model of mechano-transduction which links, via a focal adhesion associated protein, Src, to a partial differential equation model (PDE) of the Rho GTPases Rac and Rho. The PDEs are integrated over a 2D projection of the 3D cell giving a time course for protein concentration at any point in the cell. It is demonstrated that a mechano-transducer that can respond to the normal component of the force is likely to be a necessary (though perhaps not sufficient) component of the signalling network. In some processes cross talk between the GTPases is thought to be important in forming spatially segregated zones of activation, for example in the front and back of migratory cells. This research shows that local signalling in endothelial cells could be initiated by the force normal to the surface of the cell and maintained by limited diffusion. Modelling indicates the EC signalling response to fluid flow may be attenuated by a change in morphology. Rac and Rho activation and deactivation are validated against experimentally reported time courses that have been taken for whole cell averages. However it will be demonstrated that these time courses do not characterise the process and therefore there is a need for more quantitative local measure of protein activation