A Unified Term for Directed and Undirected Motility in Collective Cell Invasion

In this paper we develop mathematical models for collective cell motility. Initially we develop a model using a linear diffusion-advection type equation and fit the parameters to data from cell motility assays. This approach is helpful in classifying the results of cell motility assay experiments. In particular, this model can determine degrees of directed versus undirected collective cell motility. Next we develop a model using a nonlinear diffusion term that is able capture in a unified way directed and undirected collective cell motility. Finally we apply the nonlinear diffusion approach to a problem in tumor cell invasion, noting that neither chemotaxis or haptotaxis are present in the system under consideration in this article

Problems in the analysis of the carbohydrate moiety of glycoproteins

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A conjugate gradient minimisation approach to generating holographic traps for ultracold atoms

Direct minimisation of a cost function can in principle provide a versatile and highly controllable route to computational hologram generation. However, to date iterative Fourier transform algorithms have been predominantly used. Here we show that the careful design of cost functions, combined with numerically efficient conjugate gradient minimisation, establishes a practical method for the generation of holograms for a wide range of target light distributions. This results in a guided optimisation process, with a crucial advantage illustrated by the ability to circumvent optical vortex formation during hologram calculation. We demonstrate the implementation of the conjugate gradient method for both discrete and continuous intensity distributions and discuss its applicability to optical trapping of ultracold atoms.Comment: 11 pages, 4 figure

Multi-wavelength holography with a single spatial light modulator for ultracold atom experiments

The authors acknowledge funding from the Leverhulme Trust Research Project Grant RPG-2013-074 and from the EPSRC grant GR/T08272/01.We demonstrate a method to independently and arbitrarily tailor the spatial profile of light of multiple wavelengths and we show possible applications to ultracold atoms experiments. A single spatial light modulator is programmed to create a pattern containing multiple spatially separated structures in the Fourier plane when illuminated with a single wavelength. When the modulator is illuminated with overlapped laser beams of different wavelengths, the position of the structures is wavelength-dependent. Hence, by designing their separations appropriately, a desired overlap of different structures at different wavelengths is obtained. We employ regional phase calculation algorithms and demonstrate several possible experimental scenarios by generating light patterns with 670 nm, 780 nm and 1064 nm laser light which are accurate to the level of a few percent. This technique is easily integrated into cold atom experiments, requiring little optical access.PostprintPeer reviewe

Invariant tensors and cellular categories

Let U be the quantised enveloping algebra associated to a Cartan matrix of finite type. Let W be the tensor product of a finite list of highest weight representations of U. Then the centraliser algebra of W has a basis called the dual canonical basis which gives an integral form. We show that this integral form is cellular by using results due to Lusztig.Comment: 6 pages; to appear in Journal of Algebr

Molecular Dynamics Simulation in Arbitrary Geometries for Nanoscale Fluid Mechanics

Simulations of nanoscale systems where fluid mechanics plays an important role are required to help design and understand nano-devices and biological systems. A simulation method which hybridises molecular dynamics (MD) and continuum computational fluid dynamics (CFD) is demonstrated to be able to accurately represent the relevant physical phenomena and be computationally tractable. An MD code has been written to perform MD simulations in systems where the geometry is described by a mesh of unstructured arbitrary polyhedral cells that have been spatially decomposed into irregular portions for parallel processing. The MD code that has been developed may be used for simulations on its own, or may serve as the MD component of a hybrid method. The code has been implemented using OpenFOAM, an open source C++ CFD toolbox (www.openfoam.org). Two key enabling components are described in detail. 1) Parallel generation of initial configurations of molecules in arbitrary geometries. 2) Calculation of intermolecular pair forces, including between molecules that lie on mesh portions assigned to different, and possibly non-neighbouring processors. To calculate intermolecular forces, the spatial relationship of mesh cells is calculated once at the start of the simulation and only the molecules contained in cells that have part of their surface closer than a cut-off distance are required to interact. Interprocessor force calculations are carried out by creating local copies of molecules from other processors in a layer around the processor in question. The process of creating these copied molecules is described in detail. A case study of flow in a realistic nanoscale mixing channel, where the geometry is drawn and meshed using engineering CAD tools, is simulated to demonstrate the capabilities of the code for complex simulations

Light-induced atomic desorption in a compact system for ultracold atoms

In recent years, light-induced atomic desorption (LIAD) of alkali atoms from the inner surface of a vacuum chamber has been employed in cold atom experiments for the purpose of modulating the alkali background vapour. This is beneficial because larger trapped atom samples can be loaded from vapour at higher pressure, after which the pressure is reduced to increase the lifetime of the sample. We present an analysis, based on the case of rubidium atoms adsorbed on pyrex, of various aspects of LIAD that are useful for this application. Firstly, we study the intensity dependence of LIAD by fitting the experimental data with a rate-equation model, from which we extract a correct prediction for the increase in trapped atom number. Following this, we quantify a figure of merit for the utility of LIAD in cold atom experiments and we show how it can be optimised for realistic experimental parameters.Comment: 10 pages, 8 figure

The Role of Osteocytes in Targeted Bone Remodeling: A Mathematical Model

Until recently many studies of bone remodeling at the cellular level have focused on the behavior of mature osteoblasts and osteoclasts, and their respective precursor cells, with the role of osteocytes and bone lining cells left largely unexplored. This is particularly true with respect to the mathematical modeling of bone remodeling. However, there is increasing evidence that osteocytes play important roles in the cycle of targeted bone remodeling, in serving as a significant source of RANKL to support osteoclastogenesis, and in secreting the bone formation inhibitor sclerostin. Moreover, there is also increasing interest in sclerostin, an osteocyte-secreted bone formation inhibitor, and its role in regulating local response to changes in the bone microenvironment. Here we develop a cell population model of bone remodeling that includes the role of osteocytes, sclerostin, and allows for the possibility of RANKL expression by osteocyte cell populations. This model extends and complements many of the existing mathematical models for bone remodeling but can be used to explore aspects of the process of bone remodeling that were previously beyond the scope of prior modeling work. Through numerical simulations we demonstrate that our model can be used to theoretically explore many of the most recent experimental results for bone remodeling, and can be utilized to assess the effects of novel bone-targeting agents on the bone remodeling process

Towards a New Spatial Representation of Bone Remodeling

Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur \emph{in vitro} and \emph{in vivo}. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as \emph{in vitro} and \emph{in vivo}. This makes it possible to perform \emph{in silica} experiments more closely resembling experimental observations.Comment: Math. Biosci. Eng., 9(2), 201