School of Mathematics and Statistics - Research Publications
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ItemWhy we do not expect dispersal probability density functions based on a single mechanism to fit real seed shadows(WILEY, 2018-05)Bullock et al. (Journal of Ecology 105:6‐19, 2017) have suggested that the theory behind the Wald Analytical Long Distance (WALD) model for wind dispersal from a point source needs to be re‐examined. This is on the basis that an inverse Gaussian probability density function (pdf) does not provide the best fit to seed shadows around individual source plants known to be dispersed by wind. We present two reasons why we would not necessarily expect any of the standard mechanistically derived pdfs to fit real seed shadows any better than empirical functions. Firstly, the derivation of “off‐the‐shelf” pdfs such as the Gaussian, exponential and inverse Gaussian involves only one of the processes and factors that together generate a real seed shadow. It is implausible to expect that a single‐process model, no matter how sophisticated in detail, will capture the behaviour of an entire, complex system, which may involve a number of sequential random processes, or a superposition of parallel random processes, or both. Secondly, even if there is only one process involved and we have a perfect model for that process, the basic parameters of the model would be difficult to pin down precisely. Moreover, these parameters are unlikely to remain constant over a dispersal season, so that effectively we observe the outcome of a linear combination of dispersal events with different parameter values, constituting a form of averaging over the parameters of the distribution. Simple examples show that averaging a pdf over its parameters can lead to a pdf from an entirely different class. Synthesis. The failure of the inverse Gaussian model to fit seed shadow data is not in itself a reason to doubt the validity of the Wald Analytical Long Distance model for movement of particles through the air under specified environmental conditions. A greater awareness is needed of the differences between the Wald Analytical Long Distance and the inverse Gaussian (or Wald) and the purposes for which they are used. The complexity of dispersing populations of seeds means that any of the standard mechanistically derived pdfs will actually be merely empirical in this context. Shape and flexibility of a pdf is far more important for adequately describing data than some perceived higher status.
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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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ItemTheoretical size distribution of fossil taxa: analysis of a null model(BIOMED CENTRAL LTD, 2007)BACKGROUND: This article deals with the theoretical size distribution (of number of sub-taxa) of a fossil taxon arising from a simple null model of macroevolution. MODEL: New species arise through speciations occurring independently and at random at a fixed probability rate, while extinctions either occur independently and at random (background extinctions) or cataclysmically. In addition new genera are assumed to arise through speciations of a very radical nature, again assumed to occur independently and at random at a fixed probability rate. CONCLUSION: The size distributions of the pioneering genus (following a cataclysm) and of derived genera are determined. Also the distribution of the number of genera is considered along with a comparison of the probability of a monospecific genus with that of a monogeneric family.
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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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Item3D hybrid modelling of vascular network formation(ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD, 2017-02-07)We develop an off-lattice, agent-based model to describe vasculogenesis, the de novo formation of blood vessels from endothelial progenitor cells during development. The endothelial cells that comprise our vessel network are viewed as linearly elastic spheres that move in response to the forces they experience. We distinguish two types of endothelial cells: vessel elements are contained within the network and tip cells are located at the ends of vessels. Tip cells move in response to mechanical forces caused by interactions with neighbouring vessel elements and the local tissue environment, chemotactic forces and a persistence force which accounts for their tendency to continue moving in the same direction. Vessel elements are subject to similar mechanical forces but are insensitive to chemotaxis. An angular persistence force representing interactions with the local tissue is introduced to stabilise buckling instabilities caused by cell proliferation. Only vessel elements proliferate, at rates which depend on their degree of stretch: elongated elements have increased rates of proliferation, and compressed elements have reduced rates. Following division, the fate of the new cell depends on the local mechanical environment: the probability of forming a new sprout is increased if the parent vessel is highly compressed and the probability of being incorporated into the parent vessel increased if the parent is stretched. Simulation results reveal that our hybrid model can reproduce the key qualitative features of vasculogenesis. Extensive parameter sensitivity analyses show that significant changes in network size and morphology are induced by varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and the sprouting probability to mechanical stretch. Varying the chemotactic sensitivity directly influences the directionality of the networks. The degree of branching, and thereby the density of the networks, is influenced by the sprouting probability. Glyphs that simultaneously depict several network properties are introduced to show how these and other network quantities change over time and also as model parameters vary. We also show how equivalent glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, in the longer term, for model validation. We conclude that our biomechanical hybrid model can generate vascular networks that are qualitatively similar to those generated from in vitro and in vivo experiments.
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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.