153 research outputs found
A Mathematical Model for Lymphangiogenesis in Normal and Diabetic Wounds
Several studies suggest that one possible cause of impaired wound healing is
failed or insufficient lymphangiogenesis, that is the formation of new
lymphatic capillaries. Although many mathematical models have been developed to
describe the formation of blood capillaries (angiogenesis) very few have been
proposed for the regeneration of the lymphatic network. Moreover,
lymphangiogenesis is markedly distinct from angiogenesis, occurring at
different times and in a different manner. Here a model of five ordinary
differential equations is presented to describe the formation of lymphatic
capillaries following a skin wound. The variables represent different cell
densities and growth factor concentrations, and where possible the parameters
are estimated from experimental and clinical data. The system is then solved
numerically and the results are compared with the available biological
literature. Finally, a parameter sensitivity analysis of the model is taken as
a starting point for suggesting new therapeutic approaches targeting the
enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper
understanding of the phenomenon in question, clarifying the main factors
involved. In particular, the balance between TGF- and VEGF levels,
rather than their absolute values, is identified as crucial to effective
lymphangiogenesis. In addition, the results indicate lowering the
macrophage-mediated activation of TGF- and increasing the basal
lymphatic endothelial cell growth rate, \emph{inter alia}, as potential
treatments. It is hoped the findings of this paper may be considered in the
development of future experiments investigating novel lymphangiogenic
therapies
Fractional Patlak-Keller-Segel equations for chemotactic superdiffusion
The long range movement of certain organisms in the presence of a
chemoattractant can be governed by long distance runs, according to an
approximate Levy distribution. This article clarifies the form of biologically
relevant model equations: We derive Patlak-Keller-Segel-like equations
involving nonlocal, fractional Laplacians from a microscopic model for cell
movement. Starting from a power-law distribution of run times, we derive a
kinetic equation in which the collision term takes into account the long range
behaviour of the individuals. A fractional chemotactic equation is obtained in
a biologically relevant regime. Apart from chemotaxis, our work has
implications for biological diffusion in numerous processes.Comment: 20 pages, 4 figures, to appear in SIAM Journal on Applied Mathematic
Navigating the flow:individual and continuum models for homing in flowing environments
Navigation for aquatic and airborne species often takes place in the face of complicated flows, from persistent currents to highly unpredictable storms. Hydrodynamic models are capable of simulating flow dynamics and provide the impetus for much individual-based modelling, in which particle-sized individuals are immersed into a flowing medium. These models yield insights on the impact of currents on population distributions from fish eggs to large organisms, yet their computational demands and intractability reduce their capacity to generate the broader, less parameter-specific, insights allowed by traditional continuous approaches. In this paper, we formulate an individual-based model for navigation within a flowing field and apply scaling to derive its corresponding macroscopic and continuous model. We apply it to various movement classes, from drifters that simply go with the flow to navigators that respond to environmental orienteering cues. The utility of the model is demonstrated via its application to ‘homing’ problems and, in particular, the navigation of the marine green turtle Chelonia mydas to Ascension Island
Phenotype switching in chemotaxis aggregation models controls the spontaneous emergence of large densities
We consider a phenotype-switching chemotaxis model for aggregation, in which
a chemotactic population is capable of switching back and forth between a
chemotaxing state (performing chemotactic movement) and a secreting state
(producing the attractant). We show that the switching rate provides a powerful
mechanism for controlling the densities of spontaneously emerging aggregates.
Specifically, in two- and three-dimensional settings it is shown that when both
switching rates coincide and are suitably large, then the densities of both the
chemotaxing and the secreting population will exceed any prescribed level at
some points in the considered domain. This is complemented by two results
asserting the absence of such aggregation phenomena in corresponding scenarios
in which one of the switching rates remains within some bounded interval
Consistent Robustness Analysis (CRA) Identifies Biologically Relevant Properties of Regulatory Network Models
A number of studies have previously demonstrated that "goodness of fit" is insufficient in reliably classifying the credibility of a biological model. Robustness and/or sensitivity analysis is commonly employed as a secondary method for evaluating the suitability of a particular model. The results of such analyses invariably depend on the particular parameter set tested, yet many parameter values for biological models are uncertain.Here, we propose a novel robustness analysis that aims to determine the "common robustness" of the model with multiple, biologically plausible parameter sets, rather than the local robustness for a particular parameter set. Our method is applied to two published models of the Arabidopsis circadian clock (the one-loop [1] and two-loop [2] models). The results reinforce current findings suggesting the greater reliability of the two-loop model and pinpoint the crucial role of TOC1 in the circadian network.Consistent Robustness Analysis can indicate both the relative plausibility of different models and also the critical components and processes controlling each model
A stochastic cellular automaton model to describe the evolution of the snow-covered area across a high-elevation mountain catchment
Variations in the extent and duration of snow cover impinge on surface albedo and snowmelt rate, influencing the energy and water budgets. Monitoring snow coverage is therefore crucial for both optimising the supply of snowpack-derived water and understanding how climate change could impact on this source, vital for sustaining human activities and the natural environment during the dry season. Mountainous sites can be characterised by complex morphologies, cloud cover and forests that can introduce errors into the estimates of snow cover obtained from remote sensing. Consequently, there is a need to develop simulation models capable of predicting how snow coverage evolves across a season. Cellular Automata models have previously been used to simulate snowmelt dynamics, but at a coarser scale that limits insight into the precise factors driving snowmelt at different stages. To address this information gap, we formulate a novel, fine-scale stochastic Cellular Automaton model that describes snow coverage across a high-elevation catchment. Exploiting its refinement, the model is used to explore the interplay between three factors proposed to play a critical role: terrain elevation, sun incidence angle, and the extent of nearby snow. We calibrate the model via a randomised parameter search, fitting simulation data against snow cover masks estimated from Sentinel-2 satellite images. Our analysis shows that
Aggregation of biological particles under radial directional guidance
AbstractMany biological environments display an almost radially-symmetric structure, allowing proteins, cells or animals to move in an oriented fashion. Motivated by specific examples of cell movement in tissues, pigment protein movement in pigment cells and animal movement near watering holes, we consider a class of radially-symmetric anisotropic diffusion problems, which we call thestar problem. The corresponding diffusion tensorD(x) is radially symmetric with isotropic diffusion at the origin. We show that the anisotropic geometry of the environment can lead to strong aggregations and blow-up at the origin. We classify the nature of aggregation and blow-up solutions and provide corresponding numerical simulations. A surprising element of this strong aggregation mechanism is that it is entirely based on geometry and does not derive from chemotaxis, adhesion or other well known aggregating mechanisms. We use these aggregate solutions to discuss the process of pigmentation changes in animals, cancer invasion in an oriented fibrous habitat (such as collagen fibres), and sheep distributions around watering holes.JTB classification:21.050, 21.160, 52.250, 71.060</jats:sec
Adding Adhesion to a Chemical Signaling Model for Somite Formation
Somites are condensations of mesodermal cells that form along the two sides of the neural tube during early vertebrate development. They are one of the first instances of a periodic pattern, and give rise to repeated structures such as the vertebrae. A number of theories for the mechanisms underpinning somite formation have been proposed. For example, in the “clock and wavefront” model (Cooke and Zeeman in J. Theor. Biol. 58:455– 476, 1976), a cellular oscillator coupled to a determination wave progressing along the anterior-posterior axis serves to group cells into a presumptive somite. More recently, a chemical signaling model has been developed and analyzed by Maini and coworkers (Collier et al. in J. Theor. Biol. 207:305–316, 2000; Schnell et al. in C. R. Biol. 325:179– 189, 2002; McInerney et al. in Math. Med. Biol. 21:85–113, 2004), with equations for two chemical regulators with entrained dynamics. One of the chemicals is identified as a somitic factor, which is assumed to translate into a pattern of cellular aggregations via its effect on cell–cell adhesion. Here, the authors propose an extension to this model that includes an explicit equation for an adhesive cell population. They represent cell adhesion via an integral over the sensing region of the cell, based on a model developed previousl
The impact of phenotypic heterogeneity on chemotactic self-organisation
The capacity to aggregate through chemosensitive movement forms a paradigm of
self-organisation, with examples spanning cellular and animal systems. A basic
mechanism assumes a phenotypically homogeneous population that secretes its own
attractant, with the well known system introduced more than five decades ago by
Keller and Segel proving resolutely popular in modelling studies. The typical
assumption of population phenotypic homogeneity, however, often lies at odds
with the heterogeneity of natural systems, where populations may comprise
distinct phenotypes that vary according to their chemotactic ability,
attractant secretion, {\it etc}. To initiate an understanding into how this
diversity can impact on autoaggregation, we propose a simple extension to the
classical Keller and Segel model, in which the population is divided into two
distinct phenotypes: those performing chemotaxis and those producing
attractant. Using a combination of linear stability analysis and numerical
simulations, we demonstrate that switching between these phenotypic states
alters the capacity of a population to self-aggregate. Further, we show that
switching based on the local environment (population density or chemoattractant
level) leads to diverse patterning and provides a route through which a
population can effectively curb the size and density of an aggregate. We
discuss the results in the context of real world examples of chemotactic
aggregation, as well as theoretical aspects of the model such as global
existence and blow-up of solutions
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