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AuthorGaffney, E. A. (5)Breward, C. J. W. (3)Erban, Radek (3)Baker, Ruth E. (2)Cummings, L. J. (2)View MoreJournal

Bulletin of Mathematical Biology (15)

KAUST Grant Number
KUK-C1-013-04 (15)

PublisherSpringer Nature (15)SubjectBlinking (2)Gene expression (2)Time delays (2)Turing pattern formation (2)Biomechanics (1)View MoreTypeArticle (15)Year (Issue Date)2014 (2)2013 (4)2012 (2)2011 (1)2010 (4)View MoreItem AvailabilityMetadata Only (14)Open Access (1)

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Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

Seirin Lee, S.; Gaffney, E. A. (Bulletin of Mathematical Biology, Springer Nature, 2010-03-23) [Article]

Turing's pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental self-organisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing's model has received limited attention. Here, we novelly focus on the Gierer-Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing's mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99-130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing's model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing's mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing's mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning. © 2010 Society for Mathematical Biology.

Diffusion of Finite-Size Particles in Confined Geometries

Bruna, Maria; Chapman, S. Jonathan (Bulletin of Mathematical Biology, Springer Nature, 2013-05-10) [Article]

The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle's dimensions. The result is a nonlinear diffusion equation for the one-particle probability density function, with an overall collective diffusion that depends on both the excluded-volume and the narrow confinement. By including both these effects, the equation is able to interpolate between severe confinement (for example, single-file diffusion) and unconfined diffusion. Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared. As an application, the case of diffusion under a ratchet potential is considered, and the change in transport properties due to excluded-volume and confinement effects is examined. © 2013 Society for Mathematical Biology.

A Fibrocontractive Mechanochemical Model of Dermal Wound Closure Incorporating Realistic Growth Factor Kinetics

Murphy, Kelly E.; Hall, Cameron L.; Maini, Philip K.; McCue, Scott W.; McElwain, D. L. Sean (Bulletin of Mathematical Biology, Springer Nature, 2012-01-13) [Article]

Fibroblasts and their activated phenotype, myofibroblasts, are the primary cell types involved in the contraction associated with dermal wound healing. Recent experimental evidence indicates that the transformation from fibroblasts to myofibroblasts involves two distinct processes: The cells are stimulated to change phenotype by the combined actions of transforming growth factor β (TGFβ) and mechanical tension. This observation indicates a need for a detailed exploration of the effect of the strong interactions between the mechanical changes and growth factors in dermal wound healing. We review the experimental findings in detail and develop a model of dermal wound healing that incorporates these phenomena. Our model includes the interactions between TGFβ and collagenase, providing a more biologically realistic form for the growth factor kinetics than those included in previous mechanochemical descriptions. A comparison is made between the model predictions and experimental data on human dermal wound healing and all the essential features are well matched. © 2012 Society for Mathematical Biology.

Cyclic Loading of Growing Tissue in a Bioreactor: Mathematical Model and Asymptotic Analysis

Pohlmeyer, J. V.; Cummings, L. J. (Bulletin of Mathematical Biology, Springer Nature, 2013-10-24) [Article]

A simplified 2D mathematical model for tissue growth within a cyclically-loaded tissue engineering scaffold is presented and analyzed. Such cyclic loading has the potential to improve yield and functionality of tissue such as bone and cartilage when grown on a scaffold within a perfusion bioreactor. The cyclic compression affects the flow of the perfused nutrient, leading to flow properties that are inherently unsteady, though periodic, on a timescale short compared with that of tissue proliferation. A two-timescale analysis based on these well-separated timescales is exploited to derive a closed model for the tissue growth on the long timescale of proliferation. Some sample numerical results are given for the final model, and discussed. © 2013 Society for Mathematical Biology.

Coupling Fluid and Solute Dynamics Within the Ocular Surface Tear Film: A Modelling Study of Black Line Osmolarity

Zubkov, V. S.; Breward, C. J. W.; Gaffney, E. A. (Bulletin of Mathematical Biology, Springer Nature, 2012-07-06) [Article]

We present a mathematical model describing the spatial distribution of tear film osmolarity across the ocular surface of a human eye during one blink cycle, incorporating detailed fluid and solute dynamics. Based on the lubrication approximation, our model comprises three coupled equations tracking the depth of the aqueous layer of the tear film, the concentration of the polar lipid, and the concentration of physiological salts contained in the aqueous layer. Diffusive boundary layers in the salt concentration occur at the thinnest regions of the tear film, the black lines. Thus, despite large Peclet numbers, diffusion ameliorates osmolarity around the black lines, but nonetheless is insufficient to eliminate the build-up of solute in these regions. More generally, a heterogeneous distribution of solute concentration is predicted across the ocular surface, indicating that measurements of lower meniscus osmolarity are not globally representative, especially in the presence of dry eye. Vertical saccadic eyelid motion can reduce osmolarity at the lower black line, raising the prospect that select eyeball motions more generally can assist in alleviating tear film hyperosmolarity. Finally, our results indicate that measured evaporative rates will induce excessive hyperosmolarity at the black lines, even for the healthy eye. This suggests that further evaporative retardation at the black lines, for instance due to the cellular glycocalyx at the ocular surface or increasing concentrations of mucus, will be important for controlling hyperosmolarity as the black line thins. © 2012 Society for Mathematical Biology.

From Microscopic to Macroscopic Descriptions of Cell Migration on Growing Domains

Baker, Ruth E.; Yates, Christian A.; Erban, Radek (Bulletin of Mathematical Biology, Springer Nature, 2009-10-28) [Article]

Cell migration and growth are essential components of the development of multicellular organisms. The role of various cues in directing cell migration is widespread, in particular, the role of signals in the environment in the control of cell motility and directional guidance. In many cases, especially in developmental biology, growth of the domain also plays a large role in the distribution of cells and, in some cases, cell or signal distribution may actually drive domain growth. There is an almost ubiquitous use of partial differential equations (PDEs) for modelling the time evolution of cellular density and environmental cues. In the last 20 years, a lot of attention has been devoted to connecting macroscopic PDEs with more detailed microscopic models of cellular motility, including models of directional sensing and signal transduction pathways. However, domain growth is largely omitted in the literature. In this paper, individual-based models describing cell movement and domain growth are studied, and correspondence with a macroscopic-level PDE describing the evolution of cell density is demonstrated. The individual-based models are formulated in terms of random walkers on a lattice. Domain growth provides an extra mathematical challenge by making the lattice size variable over time. A reaction-diffusion master equation formalism is generalised to the case of growing lattices and used in the derivation of the macroscopic PDEs. © 2009 Society for Mathematical Biology.

Travelling Waves in Hyperbolic Chemotaxis Equations

Xue, Chuan; Hwang, Hyung Ju; Painter, Kevin J.; Erban, Radek (Bulletin of Mathematical Biology, Springer Nature, 2010-10-16) [Article]

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235-248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically. © 2010 Society for Mathematical Biology.

Stochastic simulations of normal aging and Werner's syndrome.

Qi, Qi; Wattis, Jonathan A D; Byrne, Helen M. (Bulletin of Mathematical Biology, Springer Nature, 2014-04-26) [Article]

Human cells typically consist of 23 pairs of chromosomes. Telomeres are repetitive sequences of DNA located at the ends of chromosomes. During cell replication, a number of basepairs are lost from the end of the chromosome and this shortening restricts the number of divisions that a cell can complete before it becomes senescent, or non-replicative. In this paper, we use Monte Carlo simulations to form a stochastic model of telomere shortening to investigate how telomere shortening affects normal aging. Using this model, we study various hypotheses for the way in which shortening occurs by comparing their impact on aging at the chromosome and cell levels. We consider different types of length-dependent loss and replication probabilities to describe these processes. After analyzing a simple model for a population of independent chromosomes, we simulate a population of cells in which each cell has 46 chromosomes and the shortest telomere governs the replicative potential of the cell. We generalize these simulations to Werner's syndrome, a condition in which large sections of DNA are removed during cell division and, amongst other conditions, results in rapid aging. Since the mechanisms governing the loss of additional basepairs are not known, we use our model to simulate a variety of possible forms for the rate at which additional telomeres are lost per replication and several expressions for how the probability of cell division depends on telomere length. As well as the evolution of the mean telomere length, we consider the standard deviation and the shape of the distribution. We compare our results with a variety of data from the literature, covering both experimental data and previous models. We find good agreement for the evolution of telomere length when plotted against population doubling.

Stochastic Turing Patterns: Analysis of Compartment-Based Approaches

Cao, Yang; Erban, Radek (Bulletin of Mathematical Biology, Springer Nature, 2014-11-25) [Article]

© 2014, Society for Mathematical Biology. Turing patterns can be observed in reaction-diffusion systems where chemical species have different diffusion constants. In recent years, several studies investigated the effects of noise on Turing patterns and showed that the parameter regimes, for which stochastic Turing patterns are observed, can be larger than the parameter regimes predicted by deterministic models, which are written in terms of partial differential equations (PDEs) for species concentrations. A common stochastic reaction-diffusion approach is written in terms of compartment-based (lattice-based) models, where the domain of interest is divided into artificial compartments and the number of molecules in each compartment is simulated. In this paper, the dependence of stochastic Turing patterns on the compartment size is investigated. It has previously been shown (for relatively simpler systems) that a modeler should not choose compartment sizes which are too small or too large, and that the optimal compartment size depends on the diffusion constant. Taking these results into account, we propose and study a compartment-based model of Turing patterns where each chemical species is described using a different set of compartments. It is shown that the parameter regions where spatial patterns form are different from the regions obtained by classical deterministic PDE-based models, but they are also different from the results obtained for the stochastic reaction-diffusion models which use a single set of compartments for all chemical species. In particular, it is argued that some previously reported results on the effect of noise on Turing patterns in biological systems need to be reinterpreted.

The Effect of Polar Lipids on Tear Film Dynamics

Aydemir, E.; Breward, C. J. W.; Witelski, T. P. (Bulletin of Mathematical Biology, Springer Nature, 2010-06-17) [Article]

In this paper, we present a mathematical model describing the effect of polar lipids, excreted by glands in the eyelid and present on the surface of the tear film, on the evolution of a pre-corneal tear film. We aim to explain the interesting experimentally observed phenomenon that the tear film continues to move upward even after the upper eyelid has become stationary. The polar lipid is an insoluble surface species that locally alters the surface tension of the tear film. In the lubrication limit, the model reduces to two coupled non-linear partial differential equations for the film thickness and the concentration of lipid. We solve the system numerically and observe that increasing the concentration of the lipid increases the flow of liquid up the eye. We further exploit the size of the parameters in the problem to explain the initial evolution of the system. © 2010 Society for Mathematical Biology.

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