School of Mathematics and Statistics - Research Publications
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ItemA Multicellular Model of Intestinal Crypt Buckling and Fission(SPRINGER, 2018-02)Crypt fission is an in vivo tissue deformation process that is involved in both intestinal homeostasis and colorectal tumourigenesis. Despite its importance, the mechanics underlying crypt fission are currently poorly understood. Recent experimental development of organoids, organ-like buds cultured from crypt stem cells in vitro, has shown promise in shedding light on crypt fission. Drawing inspiration from observations of organoid growth and fission in vivo, we develop a computational model of a deformable epithelial tissue layer. Results from in silico experiments show the stiffness of cells and the proportions of cell subpopulations affect the nature of deformation in the epithelial layer. In particular, we find that increasing the proportion of stiffer cells in the layer increases the likelihood of crypt fission occurring. This is in agreement with and helps explain recent experimental work.
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ItemA Computational Model for Collective Cellular Motion in Three Dimensions: General Framework and Case Study for Cell Pair Dynamics(PUBLIC LIBRARY SCIENCE, 2013-03-19)Cell migration in healthy and diseased systems is a combination of single and collective cell motion. While single cell motion has received considerable attention, our understanding of collective cell motion remains elusive. A new computational framework for the migration of groups of cells in three dimensions is presented, which focuses on the forces acting at the microscopic scale and the interactions between cells and their extracellular matrix (ECM) environment. Cell-cell adhesion, resistance due to the ECM and the factors regulating the propulsion of each cell through the matrix are considered. In particular, our approach emphasizes the role of receptors that mediate cell-cell and cell-matrix interactions, and examines how variation in their properties induces changes in cellular motion. As an important case study, we analyze two interacting cells. Our results show that the dynamics of cell pairs depends on the magnitude and the stochastic nature of the forces. Stronger intercellular stability is generally promoted by surface receptors that move. We also demonstrate that matrix resistance, cellular stiffness and intensity of adhesion contribute to migration behaviors in different ways, with memory effects present that can alter pair motility. If adhesion weakens with time, our findings show that cell pair break-up depends strongly on the way cells interact with the matrix. Finally, the motility for cells in a larger cluster (size 50 cells) is examined to illustrate the full capabilities of the model and to stress the role of cellular pairs in complex cellular structures. Overall, our framework shows how properties of cells and their environment influence the stability and motility of cellular assemblies. This is an important step in the advancement of the understanding of collective motility, and can contribute to knowledge of complex biological processes involving migration, aggregation and detachment of cells in healthy and diseased systems.
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ItemBuilding a Morphogen Gradient without Diffusion in a Growing Tissue(PUBLIC LIBRARY SCIENCE, 2010-09-30)In many developmental systems, spatial pattern arises from morphogen gradients, which provide positional information for cells to determine their fate. Typically, diffusion is thought to be the mechanism responsible for building a morphogen gradient. An alternative mechanism is investigated here. Using mathematical modeling, we demonstrate how a non-diffusive morphogen concentration gradient can develop in axially growing tissue systems, where growth is due to cell proliferation only. Two distinct cases are considered: in the first, all cell proliferation occurs in a localized zone where active transcription of a morphogen-producing gene occurs, and in the second, cell proliferation is uniformly distributed throughout the tissue, occurring in both the active transcription zone and beyond. A cell containing morphogen mRNA produces the morphogen protein, hence any gradient in mRNA transcripts translates into a corresponding morphogen protein gradient. Proliferation-driven growth gives rise to both advection (the transport term) and dilution (a reaction term). These two key mechanisms determine the resultant mRNA transcript distribution. Using the full range of uniform initial conditions, we show that advection and dilution due to cell proliferation are, in general, sufficient for morphogen gradient formation for both types of axially growing systems. In particular, mRNA transcript degradation is not necessary for gradient formation; it is only necessary with localized proliferation for one special value of the initial concentration. Furthermore, the morphogen concentration decreases with distance away from the transcription zone, except in the case of localized proliferation with the initial concentration sufficiently large, when the concentration can either increase with distance from the transcription zone or sustain a local minimum. In both localized and uniformly distributed proliferation, in order for a concentration gradient to form across the whole domain, transcription must occur in a zone equal to the initial domain size; otherwise, it will only form across part of the tissue.
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ItemIs cell migration or proliferation dominant in the formation of linear arrays of oligodendrocytes?(Elsevier, 2016-10-07)Oligodendrocytes are the myelin-producing cells of the central nervous system that are responsible for electrically insulating axons to speed the propagation of electrical impulses. A striking feature of oligodendrocyte development within white matter is that the cell bodies of many oligodendrocyte progenitor cells become organised into discrete linear arrays of three or more cells before they differentiate into myelin-producing oligodendrocytes. These linear arrays align parallel to the direction of the axons within white matter tracts and are believed to play an important role in the co-ordination of myelination. Guided by experimental data on the abundance and composition of linear arrays in the corpus callosum of the postnatal mouse brain, we construct discrete and continuous models of linear array generation to specifically investigate the relative influence of cell migration, proliferation, differentiation and death of oligodendroglia upon the genesis of linear arrays during early postnatal development. We demonstrate that only models that incorporate significant cell migration can replicate all of the experimental observations on number of arrays, number of cells in arrays and total cell count of oligodendroglia within a given area of the corpus callosum. These models are also necessary to accurately reflect experimental data on the abundance of linear arrays composed of oligodendrocytes that derive from progenitors of different clonal origins.
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ItemEvidence for Cooperative Selection of Axons for Myelination by Adjacent Oligodendrocytes in the Optic Nerve(PUBLIC LIBRARY SCIENCE, 2016-11-09)The cellular mechanisms that regulate the topographic arrangement of myelin internodes along axons remain largely uncharacterized. Recent clonal analysis of oligodendrocyte morphologies in the mouse optic nerve revealed that adjacent oligodendrocytes frequently formed adjacent internodes on one or more axons in common, whereas oligodendrocytes in the optic nerve were never observed to myelinate the same axon more than once. By modelling the process of axonal selection at the single cell level, we demonstrate that internode length and primary process length constrain the capacity of oligodendrocytes to myelinate the same axon more than once. On the other hand, probabilistic analysis reveals that the observed juxtaposition of myelin internodes among common sets of axons by adjacent oligodendrocytes is highly unlikely to occur by chance. Our analysis may reveal a hitherto unknown level of communication between adjacent oligodendrocytes in the selection of axons for myelination. Together, our analyses provide novel insights into the mechanisms that define the spatial organization of myelin internodes within white matter at the single cell level.
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ItemWhat is the optimal distribution of myelin along a single axon?(Elsevier, 2017-09-29)The myelin sheath that insulates some axons in the central nervous system allows for faster signal conduction. Previously, axons were thought to be either unmyelinated or fully myelinated. Recent experimental work has discovered a new pattern of myelination (intermittent myelination) along axons in the mouse brain, in which long unmyelinated axon segments are followed by myelinated segments of comparable length. We use a computational model to explore how myelin distribution (in particular intermittent myelination) affects conduction velocity. We find that although fully myelinated axons minimize conduction velocity, varying the spatial distribution of a fixed amount of myelin along a partially myelinated axon leads to considerable variation in the conduction velocity for action potentials. Whether sodium ion channel number or sodium ion channel density is held constant as the area of the unmyelinated segments increases has a strong influence on the optimal pattern of myelin and the conduction velocity.
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ItemDistinguishing cell shoving mechanisms(PUBLIC LIBRARY SCIENCE, 2018-03-12)Motivated by in vitro time-lapse images of ovarian cancer spheroids inducing mesothelial cell clearance, the traditional agent-based model of cell migration, based on simple volume exclusion, was extended to include the possibility that a cell seeking to move into an occupied location may push the resident cell, and any cells neighbouring it, out of the way to occupy that location. In traditional discrete models of motile cells with volume exclusion such a move would be aborted. We introduce a new shoving mechanism which allows cells to choose the direction to shove cells that expends the least amount of shoving effort (to account for the likely resistance of cells to being pushed). We call this motility rule 'smart shoving'. We examine whether agent-based simulations of different shoving mechanisms can be distinguished on the basis of single realisations and averages over many realisations. We emphasise the difficulty in distinguishing cell mechanisms from cellular automata simulations based on snap-shots of cell distributions, site-occupancy averages and the evolution of the number of cells of each species averaged over many realisations. This difficulty suggests the need for higher resolution cell tracking.
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ItemA dynamical model of tumour immunotherapy(ELSEVIER SCIENCE INC, 2014-07)A coupled ordinary differential equation model of tumour-immune dynamics is presented and analysed. The model accounts for biological and clinical factors which regulate the interaction rates of cytotoxic T lymphocytes on the surface of the tumour mass. A phase plane analysis demonstrates that competition between tumour cells and lymphocytes can result in tumour eradication, perpetual oscillations, or unbounded solutions. To investigate the dependence of the dynamic behaviour on model parameters, the equations are solved analytically and conditions for unbounded versus bounded solutions are discussed. An analytic characterisation of the basin of attraction for oscillatory orbits is given. It is also shown that the tumour shape, characterised by a surface area to volume scaling factor, influences the size of the basin, with significant consequences for therapy design. The findings reveal that the tumour volume must surpass a threshold size that depends on lymphocyte parameters for the cancer to be completely eliminated. A semi-analytic procedure to calculate oscillation periods and determine their sensitivity to model parameters is also presented. Numerical results show that the period of oscillations exhibits notable nonlinear dependence on biologically relevant conditions.
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ItemWhen push comes to shove: Exclusion processes with nonlocal consequences(ELSEVIER, 2015-11-01)
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ItemLength-based connectivity metrics and their ecological interpretation(ELSEVIER SCIENCE BV, 2015-11)