Biomedical subjects

J A Sherratt

Publications and source records attributed to J A Sherratt.

At least 19 recordsLinked to original sources

PubMed2003

The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations.

Recent field data indicate that in a number of cyclic populations, the cycles are organized spatially with the form of a periodic traveling wave. One way in which this type of wave is generated is when dispersing individuals encounter landscape features that impede movement in certain directions. In this article, we investigate the dependence of such periodic waves on ecological parameters and on the form of the landscape feature. Using a standard predator-prey model as a prototype for a cyclic population, we calculate the speed and amplitude of waves generated by a large landscape feature. This enables us to determine parameters for which the waves are stable; in other cases, they evolve into irregular oscillations. We then undertake for the first time a detailed study of the effects of the size and shape of a landscape feature on the waves that it generates. We show that size rather than shape is the key wave-forming property, with smaller obstacles generating waves with longer wavelength and waves from larger landscape features dominating those from smaller ones. Our results suggest that periodic traveling waves may be much more common than has previously been assumed in real ecological systems, and they enable quantitative predictions on the properties of these waves for particular cases.

Computer Simulation↗

PubMed2002

Generation of periodic waves by landscape features in cyclic predator-prey systems.

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator-prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular 'target patterns' or spirals. This is, to our knowledge, a new mechanism for periodic-wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator-prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland-England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large-scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real-world environments, our results suggest that complex spatio-temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.

Animals↗

PubMed2002

Lipoprotein oxidation and its significance for atherosclerosis: a mathematical approach.

Atherosclerosis is a chronic disease which involves the build up of cholesterol and fatty deposits within the arterial wall. This results in the narrowing of the vessel lumen, which eventually restricts blood flow to vital organs such as the heart and lungs. These events may culminate in a heart attack or stroke, the commonest causes of death in the U.K. population. In this paper we study the early stages of atherosclerosis which include the build up of cholesterol within subendothelial cells to form what is known as a fatty streak, the earliest identifiable evidence of atherosclerosis. The deposition of cholesterol is believed to be a consequence of oxidation of circulating cholesterol-rich lipoproteins, in particular low density lipoproteins (LDLs). Via a mathematical model we investigate this process of oxidation within the context of an in vitro framework. We first recreate existing experimental results and then extend the model to investigate phenomenon not studied by current experimental protocols. We find that the model displays hysteresis which reveals some interesting insights into possible in vivo events. Mathematical analysis of this behaviour predicts that vitamin E supplementation is not as beneficial as high density lipoproteins (HDLs) and vitamin C. Furthermore, the scavenging of oxidants by HDL can provide an important first line of defence against LDL oxidation.

Arteriosclerosis↗

PubMed2002

Mathematical modelling of flow through vascular networks: implications for tumour-induced angiogenesis and chemotherapy strategies.

Angiogenesis, the formation of blood vessels from a pre-existing vasculature, is a process whereby capillary sprouts are formed in response to externally supplied chemical stimuli. The sprouts then grow and develop, driven initially by endothelial cell migration, and organize themselves into a branched, connected network structure. Subsequent cell proliferation near the sprout-tip permits further extension of the capillary and ultimately completes the process. Angiogenesis occurs during embryogenesis, wound healing, arthritis and during the growth of solid tumours. In this paper we initially generate theoretical capillary networks (which are morphologically similar to those networks observed in vivo) using the discrete mathematical model of Anderson and Chaplain. This discrete model describes the formation of a capillary sprout network via endothelial cell migratory and proliferative responses to external chemical stimuli (tumour angiogenic factors, TAF) supplied by a nearby solid tumour, and also the endothelial cell interactions with the extracellular matrix. The main aim of this paper is to extend this work to examine fluid flow through these theoretical network structures. In order to achieve this we make use of flow modelling tools and techniques (specifically, flow through interconnected networks) from the field of petroleum engineering. Having modelled the flow of a basic fluid through our network, we then examine the effects of fluid viscosity, blood vessel size (i.e., diameter of the capillaries), and network structure/geometry, upon: (i) the rate of flow through the network; (ii) the amount of fluid present in the complete network at any one time; and (iii) the amount of fluid reaching the tumour. The incorporation of fluid flow through the generated vascular networks has highlighted issues that may have major implications for the study of nutrient supply to the tumour (blood/oxygen supply) and, more importantly, for the delivery of chemotherapeutic drugs to the tumour. Indeed, there are also implications for the delivery of anti-angiogenesis drugs to the network itself. Results clearly highlight the important roles played by the structure and morphology of the network, which is, in turn, linked to the size and geometry of the nearby tumour. The connectedness of the network, as measured by the number of loops formed in the network (the anastomosis density), is also found to be of primary significance. Moreover, under certain conditions, the results of our flow simulations show that an injected chemotherapy drug may bypass the tumour altogether.

Angiogenesis Inducing Agents↗

PubMed2001

Modelling tumour acidity and invasion.

The intracellular pH (pHi) of mammalian cells is tightly regulated by the concerted action of a number of different pumps in the plasma membrane. Despite the acidic extracellular environment (pHe 6.8-7.0) of some tumours, the pHi of solid tumours is neutral or slightly alkaline compared to normal tissues (pHi 7.0-7.4). This gives rise to a reversed pH gradient across the cell membrane between tumours and normal tissue, which has been implicated in many aspects of tumour progression. One such area is tumour invasion: the incubation of tumour cells at low pH has been shown to induce more aggressive invasive behaviour in vitro. In this paper we use mathematical models to investigate whether altered proteolytic activity at low pH is responsible for the stimulation of a more metastatic phenotype. We examined the effect of culture pH on the secretion and activity of two different classes of proteinases: the metalloproteinases (MMPs), and the cysteine proteinases (such as cathepsin B). The modelling predicts that, in addition to metalloproteinase activity, the pH-induced peripheral redistribution of cathepsin B could be a major factor involved in the acquisition of a more metastatic phenotype in malignant cells at low pHe.

Animals↗

PubMed2001

A new mathematical model for avascular tumour growth.

The early development of solid tumours has been extensively studied, both experimentally via the multicellular spheroid assay, and theoretically using mathematical modelling. The vast majority of previous models apply specifically to multicell spheroids, which have a characteristic structure of a proliferating rim and a necrotic core, separated by a band of quiescent cells. Many previous models represent these as discrete layers, separated by moving boundaries. Here, the authors develop a new model, formulated in terms of continuum densities of proliferating, quiescent and necrotic cells, together with a generic nutrient/growth factor. The model is oriented towards an in vivo rather than in vitro setting, and crucially allows for nutrient supply from underlying tissue, which will arise in the two-dimensional setting of a tumour growing within an epithelium. In addition, the model involves a new representation of cell movement, which reflects contact inhibition of migration. Model solutions are able to reproduce the classic three layer structure familiar from multicellular spheroids, but also show that new behaviour can occur as a result of the nutrient supply from underlying tissue. The authors analyse these different solution types by approximate solution of the travelling wave equations, enabling a detailed classification of wave front solutions.

Cell Division↗

PubMed2001

Healing by numbers: mathematics meets medicine.

The molecular biology revolution is providing vast quantities of information about the cell biology and biochemistry underlying clinical medicine. How can the pieces of this jigsaw puzzle be put together? This is currently a major biological challenge, and help is at hand from an unlikely source - mathematics.

Humans↗

PubMed2000

Mathematical modelling of nitric oxide activity in wound healing can explain keloid and hypertrophic scarring.

Keloid and hypertrophic lesions are both types of scarring pathologies which arise as a consequence of excess collagen deposition during the wound healing process. The exact mechanism by which this occurs is not understood and currently no effective treatment exists. In this paper, we study the possible role of nitric oxide in excess scar formation. In recent years, the physiological role of this free radical in mammalian tissue has been extensively studied; in particular numerous groups have studied its role in wound healing. We describe a mathematical model which offers a possible explanation for keloid scarring in terms of the presence of higher than normal nitric oxide concentrations related to the fact that nitric oxide stimulates synthesis of collagen by fibroblasts. As a consequence of this, we put forward a suggestion for a treatment strategy involving the surgical excision of the keloid lesion combined with the application of a low-dose nitric oxide inhibitor. In addition, we show that a quasi-steady-state analysis of our model reveals a possible approach to distinguishing between hypertrophic and keloid lesions, a task which has to date proven to be clinically difficult. We also present an extended model which confirms these results in the context of a more complicated and biologically more realistic system. The fuller model demonstrates additional features of keloid and hypertrophic scarring which we were not able to consider in the basic model, and as a consequence further supports our hypothesis that nitric oxide activity could play a key role in keloid scarring.

Cicatrix, Hypertrophic↗

PubMed2000

Lateral induction by juxtacrine signaling is a new mechanism for pattern formation.

Many signaling molecules in epithelia are now known to function in a membrane-bound form, binding to receptors on immediately neighbouring cells. This "juxtacrine" mode of communication has been well studied in the case of lateral inhibition, where ligand binding at the cell surface downregulates ligand and receptor expression, and is known to generate spatial patterns with a wavelength of exactly two cells. However, recent evidence shows that a number of juxtacrine signals can lead to the opposite phenomenon of lateral induction. Here, we use mathematical modeling to show that such positive feedback, in combination with juxtacrine communication, provides a novel mechanism for the generation of spatial patterns, with wavelengths that vary with parameters and can be many cell lengths.

Body Patterning↗

PubMed2000

Mathematical modelling of juxtacrine patterning.

Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell-cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429-446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that juxtacrine signalling provides a novel and robust mechanism for the generation of spatial patterns.

Animals↗

PubMed2000

Keratinocyte growth factor signalling: a mathematical model of dermal-epidermal interaction in epidermal wound healing.

A wealth of growth factors are known to regulate the various cell functions involved in the repair process. An understanding of their therapeutic value is essential to achieve improved wound healing. Keratinocyte growth factor (KGF) seems to have a unique role as a mediator of mesenchymal-epithelial interactions: it originates from mesenchymal cells, yet acts exclusively on epithelial cells. In this paper, we study KGF's role in epidermal wound healing, since its production is substantially up-regulated after injury. We begin by modelling the dermal-epidermal signalling mechanism of KGF to investigate how this extra production affects the signal range. We then incorporate the effect of KGF on cell proliferation, and using travelling wave analysis we obtain an approximation for the rate of healing. Our modelling shows that the large up-regulation of KGF post-wounding extends the KGF signal range but is above optimal for the rate of wound closure. We predict that other functions of KGF may be more important than its role as a mitogen for the healing process.

Cell Communication↗

PubMed1999

Mathematical modelling of extracellular matrix dynamics using discrete cells: fiber orientation and tissue regeneration.

Matrix orientation plays a crucial role in determining the severity of scar tissue after dermal wounding. We present a model framework which allows us to examine the interaction of many of the factors involved in orientation and alignment. Within this framework, cells are considered as discrete objects, while the matrix is modelled as a continuum. Using numerical simulations, we investigate the effect on alignment of changing cell properties and of varying cell interactions with collagen and fibrin.

Animals↗

PubMed1999

How far can a juxtacrine signal travel?

Juxtacrine signalling is the process of cell communication in which ligand and receptors are both anchored in the cell membrane. We develop three mathematical models for this process, involving different mathematical representations of the dynamics of membrane-bound ligand and free and bound receptors, within an epithelial sheet. We consider the dynamics of this system following a localized disturbance, such as would be provided by a source of ligand or by the generation of a free edge via wounding. We study the ability of the juxtacrine mechanism to transmit a signal away from this disturbance, and show analytically that the spatial half-life of the signal can in fact be arbitrarily large. This result is quite general, since we use a generic reaction kinetic scheme; the key assumption is that ligand and receptor production are both upregulated by binding. Moreover, the result applies to all three of our model formulations. We conclude by discussing applications of the result to the particular case of the transforming growth factor alpha binding to epidermal growth factor receptor in epidermal wound healing.

Animals↗

PubMed1999

The mathematical modelling of cell kinetics in corneal epithelial wound healing.

This paper considers the comparison of experimental spatial and temporal data of mitotic rates measured during corneal epithelial wound healing (CEWH) of a rat model with the predictions of a computer modelling framework. We begin by briefly showing that previous models, used in the study of corneal epithelial wound healing speeds, are inadequate for the study of cell kinetics. We proceed to formulate a new modelling framework more suited to such a study. This framework is simulated in its simplest form, and the results from this motivate a new realisation of the modelling framework, including a caricature of age structuring. Finally, a model with a simple representation of juxtacrine signalling is considered. The final model captures many, though not all, of the trends of the experimental data. This paper thus lays a foundation for the modelling of the cell kinetics of corneal epithelial wound healing, and yields valuable insight regarding the important mechanisms a model should consider in order to reproduce the observed experimental trends.

Animals↗

PubMed1999

Mathematical modelling of tumour acidity: regulation of intracellular pH.

Measurements of extracellular pH (pHe) in vivo have shown that the microenvironment in tumours is more acidic than in normal tissue. However, both human and animal tumour cells have been shown to have an intracellular pH (pHi) on the alkaline side of neutrality (pH 7.1-7.2). This gives rise to a reversed pH gradient between tumours and normal tissue which implies that cells within solid tumours are capable of maintaining their level of pHi at physiological levels, despite lower than normal levels of pHe. In this paper the authors describe a mathematical model that provides a possible explanation for the altered pH gradient observed in tumours. The authors examine the influence of changes in the microenvironment on the activity of several membrane based ion transport systems. Using qualitative analysis the authors show that the pHi of tumour cells is less sensitive to external pH than for normal cells, because of their increased reliance on the inefficient glycolytic pathway for energy production. It is shown that under aerobic conditions the lactate-/H+ symporter could be the most active exchanger in the regulation of pHi in tumour cells. However, under more hypoxic conditions lactate extrusion is reduced, and so this exchanger has little effect on resting pHi in these regions. The authors also consider an extended model which incorporates the transfer of acids from the cytosol into acidic organelles. The model demonstrates that one of the major factors involved in the maintenance of cytosolic pH to physiological levels, despite an acidic extracellular pH in hypoxic areas of tumour tissue (median, 6.9-7.0), is enhanced sequestration of cytosolic protons into acidic cellular vesicles such as endoplasmic reticulum, golgi, endosomes, and lysosomes.

Animals↗

PubMed1999

Mathematical modelling of anisotropy in fibrous connective tissue.

We present two modelling frameworks for studying dynamic anistropy in connective tissue, motivated by the problem of fibre alignment in wound healing. The first model is a system of partial differential equations operating on a macroscopic scale. We show that a model consisting of a single extracellular matrix material aligned by fibroblasts via flux and stress exhibits behaviour that is incompatible with experimental observations. We extend the model to two matrix types and show that the results of this extended model are robust and consistent with experiment. The second model represents cells as discrete objects in a continuum of ECM. We show that this model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population.

Anisotropy↗

PubMed1999

Alterations in proteolytic activity at low pH and its association with invasion: a theoretical model.

The extracellular pH (pHe) of solid tumours is often lower than in normal tissues, with median pH values of about 7.0 in tumours and 7.5 in normal tissue. Despite this more acidic tumour microenvironment, non-invasive measurements of intracellular pH (pHi) have shown that the pHi of solid tumours is neutral or slightly alkaline compared to normal tissue (pHi 7.0-7.4). This gives rise to a reversed cellular pH gradient between tumours and normal tissue, which has been implicated in many aspects of tumour progression. One such area is tumour invasion: the incubation of tumour cells at low pH has been shown to induce more aggressive invasive behaviour in vitro. In this paper the authors use mathematical models to investigate whether altered proteolytic activity at low pH is responsible for the stimulation of a more metastatic phenotype. The authors examined the effect of culture pH on the secretion and activity of two different classes of proteinases: the metalloproteinases (MMPs), and the cysteine proteinases (such as cathepsin B). The modelling suggests that changes in MMP activity at low pH do not have significant effects on invasive behaviour. However, the model predicts that the levels of active-cathepsin B are significantly altered by acidic pH. This result suggests a critical role for the cysteine proteinases in tumour progression.

Cell Movement↗

PubMed1998

Spatially varying equilibria of mechanical models: application to dermal wound contraction.

Mechanochemical models based on the Oster-Murray continuum framework have been applied to a variety of biological settings to obtain an understanding of the morphogenesis of living tissues. Wound-healing in mammalian skin is an important example, because a complex sequence of biochemical and biomechanical responses are orchestrated to close a wound by a combination of new tissue formation and wound contraction. Mechanical interactions between dermal fibroblastic cells and the collagen-rich extracellular matrix are crucial in the development of a contracted wound state. We and others have previously proposed mechanochemical models for wound repair to gain a greater understanding of both normal and abnormal healing. In the present work, the existence of spatially varying equilibria of these models is investigated by using a small-stain approximation and phase-plane techniques, with numerical simulations to confirm the analytical predictions. These results are sources of novel insight into the roles of key biological parameters in determining the mechanical properties of a contracted wound. These methods may also be relevant to other morphogenetic scenarios for which similar mechanochemical models have been proposed.

Animals↗