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AuthorGoriely, Alain (19)Byrne, Helen M. (18)Erban, Radek (17)Goriely, A. (17)Vella, Dominic (14)View MoreJournalPhysical Review E (18)Bulletin of Mathematical Biology (15)SIAM Journal on Applied Mathematics (14)Journal of Fluid Mechanics (12)Journal of Theoretical Biology (12)View MoreKAUST Grant Number

KUK-C1-013-04 (327)

PublisherElsevier BV (50)Springer Nature (49)Society for Industrial & Applied Mathematics (SIAM) (36)American Physical Society (APS) (27)AIP Publishing (19)View MoreSubjectAsymptotic analysis (10)Preconditioning (7)lubrication theory (6)capillary flows (5)Elasticity (5)View MoreTypeArticle (319)Book Chapter (5)Conference Paper (3)Year (Issue Date)2017 (3)2016 (2)2015 (21)2014 (54)2013 (83)View MoreItem AvailabilityMetadata Only (314)Open Access (13)

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Turing’s Theory of Morphogenesis: Where We Started, Where We Are and Where We Want to Go

Woolley, Thomas E.; Baker, Ruth E.; Maini, Philip K. (Theory and Applications of Computability, Springer International Publishing, 2017-05-05) [Book Chapter]

Over 60 years have passed since Alan Turing first postulated a mechanism for biological pattern formation. Although Turing did not have the chance to extend his theories before his unfortunate death two years later, his work has not gone unnoticed. Indeed, many researchers have since taken up the gauntlet and extended his revolutionary and counter-intuitive ideas. Here, we reproduce the basics of his theory as well as review some of the recent generalisations and applications that have led our mathematical models to be closer representations of the biology than ever before. Finally, we take a look to the future and discuss open questions that not only show that there is still much life in the theory, but also that the best may be yet to come.

The role of the Hes1 crosstalk hub in Notch-Wnt interactions of the intestinal crypt

Kay, Sophie K.; Harrington, Heather A.; Shepherd, Sarah; Brennan, Keith; Dale, Trevor; Osborne, James M.; Gavaghan, David J.; Byrne, Helen M. (PLOS Computational Biology, Public Library of Science (PLoS), 2017-03-01) [Article]

The Notch pathway plays a vital role in determining whether cells in the intestinal epithelium adopt a secretory or an absorptive phenotype. Cell fate specification is coordinated via Notch’s interaction with the canonical Wnt pathway. Here, we propose a new mathematical model of the Notch and Wnt pathways, in which the Hes1 promoter acts as a hub for pathway crosstalk. Computational simulations of the model can assist in understanding how healthy intestinal tissue is maintained, and predict the likely consequences of biochemical knockouts upon cell fate selection processes. Chemical reaction network theory (CRNT) is a powerful, generalised framework which assesses the capacity of our model for monostability or multistability, by analysing properties of the underlying network structure without recourse to specific parameter values or functional forms for reaction rates. CRNT highlights the role of β-catenin in stabilising the Notch pathway and damping oscillations, demonstrating that Wnt-mediated actions on the Hes1 promoter can induce dynamic transitions in the Notch system, from multistability to monostability. Time-dependent model simulations of cell pairs reveal the stabilising influence of Wnt upon the Notch pathway, in which β-catenin- and Dsh-mediated action on the Hes1 promoter are key in shaping the subcellular dynamics. Where Notch-mediated transcription of Hes1 dominates, there is Notch oscillation and maintenance of fate flexibility; Wnt-mediated transcription of Hes1 favours bistability akin to cell fate selection. Cells could therefore regulate the proportion of Wnt- and Notch-mediated control of the Hes1 promoter to coordinate the timing of cell fate selection as they migrate through the intestinal epithelium and are subject to reduced Wnt stimuli. Furthermore, mutant cells characterised by hyperstimulation of the Wnt pathway may, through coupling with Notch, invert cell fate in neighbouring healthy cells, enabling an aberrant cell to maintain its neighbours in mitotically active states.

Nematic Equilibria on a Two-Dimensional Annulus

Lewis, A. H.; Aarts, D. G. A. L.; Howell, P. D.; Majumdar, A. (Studies in Applied Mathematics, Wiley-Blackwell, 2017-01-16) [Article]

We study planar nematic equilibria on a two-dimensional annulus with strong and weak tangent anchoring, in the Oseen–Frank theoretical framework. We analyze a radially invariant defect-free state and compute analytic stability criteria for this state in terms of the elastic anisotropy, annular aspect ratio, and anchoring strength. In the strong anchoring case, we define and characterize a new spiral-like equilibrium which emerges as the defect-free state loses stability. In the weak anchoring case, we compute stability diagrams that quantify the response of the defect-free state to radial and azimuthal perturbations. We study sector equilibria on sectors of an annulus, including the effects of weak anchoring and elastic anisotropy, giving novel insights into the correlation between preferred numbers of boundary defects and the geometry. We numerically demonstrate that these sector configurations can approximate experimentally observed equilibria with boundary defects.

Numerical algebraic geometry for model selection and its application to the life sciences

Gross, Elizabeth; Davis, Brent; Ho, Kenneth L.; Bates, Daniel J.; Harrington, Heather A. (Journal of The Royal Society Interface, The Royal Society, 2016-10-12) [Article]

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.

Anisotropy in wavelet-based phase field models

Korzec, Maciek; Münch, Andreas; Süli, Endre; Wagner, Barbara (Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences (AIMS), 2016-04-01) [Article]

When describing the anisotropic evolution of microstructures in solids using phase-field models, the anisotropy of the crystalline phases is usually introduced into the interfacial energy by directional dependencies of the gradient energy coefficients. We consider an alternative approach based on a wavelet analogue of the Laplace operator that is intrinsically anisotropic and linear. The paper focuses on the classical coupled temperature/Ginzburg--Landau type phase-field model for dendritic growth. For the model based on the wavelet analogue, existence, uniqueness and continuous dependence on initial data are proved for weak solutions. Numerical studies of the wavelet based phase-field model show dendritic growth similar to the results obtained for classical phase-field models.

Mathematical and Statistical Techniques for Systems Medicine: The Wnt Signaling Pathway as a Case Study

MacLean, Adam L.; Harrington, Heather A.; Stumpf, Michael P. H.; Byrne, Helen M. (Methods in Molecular Biology, Springer Nature, 2015-12-16) [Book Chapter]

The last decade has seen an explosion in models that describe phenomena in systems medicine. Such models are especially useful for studying signaling pathways, such as the Wnt pathway. In this chapter we use the Wnt pathway to showcase current mathematical and statistical techniques that enable modelers to gain insight into (models of) gene regulation and generate testable predictions. We introduce a range of modeling frameworks, but focus on ordinary differential equation (ODE) models since they remain the most widely used approach in systems biology and medicine and continue to offer great potential. We present methods for the analysis of a single model, comprising applications of standard dynamical systems approaches such as nondimensionalization, steady state, asymptotic and sensitivity analysis, and more recent statistical and algebraic approaches to compare models with data. We present parameter estimation and model comparison techniques, focusing on Bayesian analysis and coplanarity via algebraic geometry. Our intention is that this (non-exhaustive) review may serve as a useful starting point for the analysis of models in systems medicine.

Channelization of plumes beneath ice shelves

Dallaston, M. C.; Hewitt, I. J.; Wells, A. J. (Journal of Fluid Mechanics, Cambridge University Press (CUP), 2015-11-11) [Article]

© 2015 Cambridge University Press. We study a simplified model of ice-ocean interaction beneath a floating ice shelf, and investigate the possibility for channels to form in the ice shelf base due to spatial variations in conditions at the grounding line. The model combines an extensional thin-film description of viscous ice flow in the shelf, with melting at its base driven by a turbulent ocean plume. Small transverse perturbations to the one-dimensional steady state are considered, driven either by ice thickness or subglacial discharge variations across the grounding line. Either forcing leads to the growth of channels downstream, with melting driven by locally enhanced ocean velocities, and thus heat transfer. Narrow channels are smoothed out due to turbulent mixing in the ocean plume, leading to a preferred wavelength for channel growth. In the absence of perturbations at the grounding line, linear stability analysis suggests that the one-dimensional state is stable to initial perturbations, chiefly due to the background ice advection.

Hybrid approaches for multiple-species stochastic reaction-diffusion models.

Spill, Fabian; Guerrero, Pilar; Alarcon, Tomas; Maini, Philip K; Byrne, Helen M. (Journal of Computational Physics, Elsevier BV, 2015-10-01) [Article]

Reaction-diffusion models are used to describe systems in fields as diverse as physics, chemistry, ecology and biology. The fundamental quantities in such models are individual entities such as atoms and molecules, bacteria, cells or animals, which move and/or react in a stochastic manner. If the number of entities is large, accounting for each individual is inefficient, and often partial differential equation (PDE) models are used in which the stochastic behaviour of individuals is replaced by a description of the averaged, or mean behaviour of the system. In some situations the number of individuals is large in certain regions and small in others. In such cases, a stochastic model may be inefficient in one region, and a PDE model inaccurate in another. To overcome this problem, we develop a scheme which couples a stochastic reaction-diffusion system in one part of the domain with its mean field analogue, i.e. a discretised PDE model, in the other part of the domain. The interface in between the two domains occupies exactly one lattice site and is chosen such that the mean field description is still accurate there. In this way errors due to the flux between the domains are small. Our scheme can account for multiple dynamic interfaces separating multiple stochastic and deterministic domains, and the coupling between the domains conserves the total number of particles. The method preserves stochastic features such as extinction not observable in the mean field description, and is significantly faster to simulate on a computer than the pure stochastic model.

An investigation of the influence of extracellular matrix anisotropy and cell–matrix interactions on tissue architecture

Dyson, R. J.; Green, J. E. F.; Whiteley, J. P.; Byrne, H. M. (Journal of Mathematical Biology, Springer Science + Business Media, 2015-09-02) [Article]

© 2015 Springer-Verlag Berlin Heidelberg Mechanical interactions between cells and the fibrous extracellular matrix (ECM) in which they reside play a key role in tissue development. Mechanical cues from the environment (such as stress, strain and fibre orientation) regulate a range of cell behaviours, including proliferation, differentiation and motility. In turn, the ECM structure is affected by cells exerting forces on the matrix which result in deformation and fibre realignment. In this paper we develop a mathematical model to investigate this mechanical feedback between cells and the ECM. We consider a three-phase mixture of collagen, culture medium and cells, and formulate a system of partial differential equations which represents conservation of mass and momentum for each phase. This modelling framework takes into account the anisotropic mechanical properties of the collagen gel arising from its fibrous microstructure. We also propose a cell–collagen interaction force which depends upon fibre orientation and collagen density. We use a combination of numerical and analytical techniques to study the influence of cell–ECM interactions on pattern formation in tissues. Our results illustrate the wide range of structures which may be formed, and how those that emerge depend upon the importance of cell–ECM interactions.

A spatially-averaged mathematical model of kidney branching morphogenesis

Zubkov, V.S.; Combes, A.N.; Short, K.M.; Lefevre, J.; Hamilton, N.A.; Smyth, I.M.; Little, M.H.; Byrne, H.M. (Journal of Theoretical Biology, Elsevier BV, 2015-08) [Article]

© 2015 Published by Elsevier Ltd. Kidney development is initiated by the outgrowth of an epithelial ureteric bud into a population of mesenchymal cells. Reciprocal morphogenetic responses between these two populations generate a highly branched epithelial ureteric tree with the mesenchyme differentiating into nephrons, the functional units of the kidney. While we understand some of the mechanisms involved, current knowledge fails to explain the variability of organ sizes and nephron endowment in mice and humans. Here we present a spatially-averaged mathematical model of kidney morphogenesis in which the growth of the two key populations is described by a system of time-dependant ordinary differential equations. We assume that branching is symmetric and is invoked when the number of epithelial cells per tip reaches a threshold value. This process continues until the number of mesenchymal cells falls below a critical value that triggers cessation of branching. The mathematical model and its predictions are validated against experimentally quantified C57Bl6 mouse embryonic kidneys. Numerical simulations are performed to determine how the final number of branches changes as key system parameters are varied (such as the growth rate of tip cells, mesenchyme cells, or component cell population exit rate). Our results predict that the developing kidney responds differently to loss of cap and tip cells. They also indicate that the final number of kidney branches is less sensitive to changes in the growth rate of the ureteric tip cells than to changes in the growth rate of the mesenchymal cells. By inference, increasing the growth rate of mesenchymal cells should maximise branch number. Our model also provides a framework for predicting the branching outcome when ureteric tip or mesenchyme cells change behaviour in response to different genetic or environmental developmental stresses.

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