ORCID Profile
0000-0001-5061-9563
Current Organisation
University of Adelaide
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Biological Mathematics | Theoretical and applied mechanics | Theoretical and Applied Mechanics | Applied Mathematics | Biological mathematics | Cell development proliferation and death | Applied mathematics |
Expanding Knowledge in the Biological Sciences | Expanding Knowledge in the Mathematical Sciences
Publisher: Cold Spring Harbor Laboratory
Date: 10-02-2021
DOI: 10.1101/2021.02.10.430170
Abstract: Multi-cellular modelling, where tissues and organs are represented by a collection of in idual interacting agents, is a well established field, encapsulating several different approaches. In particular, off-lattice models, which represent cells using points that are free to move in space, have been applied to numerous biological problems in both two and three dimensions. The fact that a cell can be represented by point objects is useful in a wide range of settings, particularly when large populations are involved. However, a purely point-based representation is not naturally equipped to deal with objects that inherently have length like cell boundaries or external membranes. In this paper we introduce a novel off-lattice modelling framework that exploits rigid body mechanics to represent cells using a collection of one-dimensional edges (rather than zero-dimensional points) in a viscosity-dominated system. The rigid body framework can be used, among other things, to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, and to model rod shaped cells like bacteria. We demonstrate the value of our new framework by using it in these three applications, showing that this approach can replicate established results, as well as offer solutions to problems that limit the scope of current off-lattice multi-cellular models.
Publisher: American Physical Society (APS)
Date: 12-01-2022
Publisher: Oxford University Press (OUP)
Date: 16-05-2018
Abstract: We develop a continuum model for the aggregation of cells cultured in a nutrient-rich medium in a culture well. We consider a 2D geometry, representing a vertical slice through the culture well, and assume that the cell layer depth is small compared with the typical lengthscale of the culture well. We adopt a continuum mechanics approach, treating the cells and culture medium as a two-phase mixture. Specifically, the cells and culture medium are treated as fluids. Additionally, the cell phase can generate forces in response to environmental cues, which include the concentration of a chemoattractant that is produced by the cells within the culture medium. The model leads to a system of coupled nonlinear partial differential equations for the volume fraction and velocity of the cell phase, the culture medium pressure and the chemoattractant concentration, which must be solved subject to appropriate boundary and initial conditions. To gain insight into the system, we consider two model reductions, appropriate when the cell layer depth is thin compared to the typical length scale of the culture well: a (simple) 1D and a (more involved) thin-film extensional flow reduction. By investigating the resulting systems of equations analytically and numerically, we identify conditions under which small litude perturbations to a homogeneous steady state (corresponding to a spatially uniform cell distribution) can lead to a spatially varying steady state (pattern formation). Our analysis reveals that the simpler 1D reduction has the same qualitative features as the thin-film extensional flow reduction in the linear and weakly nonlinear regimes, motivating the use of the simpler 1D modelling approach when a qualitative understanding of the system is required. However, the thin-film extensional flow reduction may be more appropriate when detailed quantitative agreement between modelling predictions and experimental data is desired. Furthermore, full numerical simulations of the two model reductions in regions of parameter space when the system is not close to marginal stability reveal significant differences in the evolution of the volume fraction and velocity of the cell phase, and chemoattractant concentration.
Publisher: Elsevier BV
Date: 02-2017
DOI: 10.1016/J.JTBI.2017.11.014
Abstract: Understanding the underlying mechanisms that produce the huge variety of swarming and aggregation patterns in animals and cells is fundamental in ecology, developmental biology, and regenerative medicine, to name but a few ex les. Depending upon the nature of the interactions between in iduals (cells or animals), a variety of different large-scale spatial patterns can be observed in their distribution ex les include cell aggregates, stripes of different coloured skin cells, etc. For the case where all in iduals are of the same type (i.e., all interactions are alike), a considerable literature already exists on how the collective organisation depends on the inter-in idual interactions. Here, we focus on the less studied case where there are two different types of in iduals present. Whilst a number of continuum models of this scenario exist, it can be difficult to compare these models to experimental data, since real cells and animals are discrete. In order to overcome this problem, we develop an agent-based model to simulate some archetypal mechanisms involving attraction and repulsion. However, with this approach (as with experiments), each realisation of the model is different, due to stochastic effects. In order to make useful comparisons between simulations and experimental data, we need to identify the robust features of the spatial distributions of the two species which persist over many realisations of the model (for ex le, the size of aggregates, degree of segregation or intermixing of the two species). In some cases, it is possible to do this by simple visual inspection. In others, the features of the pattern are not so clear to the unaided eye. In this paper, we introduce a pair correlation function (PCF), which allows us to analyse multi-species spatial distributions quantitatively. We show how the differing strengths of inter-in idual attraction and repulsion between species give rise to different spatial patterns, and how the PCF can be used to quantify these differences, even when it might be impossible to recognise them visually.
Publisher: AIP Publishing
Date: 2017
DOI: 10.1063/1.4973670
Abstract: We consider the steady, gravity-driven flow of a thin film of viscous fluid down a helically wound shallow channel of arbitrary cross-sectional shape with arbitrary torsion and curvature. This extends our previous work [D. J. Arnold et al., “Thin-film flow in helically-wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] on channels of rectangular cross section. The Navier-Stokes equations are expressed in a novel, non-orthogonal coordinate system fitted to the channel bottom. By assuming that the channel depth is small compared to its width and that the fluid depth in the vertical direction is also small compared to its typical horizontal extent, we are able to solve for the velocity components and pressure analytically. Using these results, a differential equation for the free surface shape is obtained, which must in general be solved numerically. Motivated by the aim of understanding flows in static spiral particle separators used in mineral processing, we investigate the effect of cross-sectional shape on the secondary flow in the channel cross section. We show that the competition between gravity and inertia in non-rectangular channels is qualitatively similar to that in rectangular channels, but that the cross-sectional shape has a strong influence on the breakup of the secondary flow into multiple clockwise-rotating cells. This may be triggered by small changes to the channel geometry, such as one or more bumps in the channel bottom that are small relative to the fluid depth. In contrast to the secondary flow which is quite sensitive to small bumps in the channel bottom, the free-surface profile is relatively insensitive to these. The sensitivity of the flow to the channel geometry may have important implications for the design of efficient spiral particle separators.
Publisher: Cold Spring Harbor Laboratory
Date: 10-12-2021
DOI: 10.1101/2021.12.08.471846
Abstract: Biological tissues are composed of cells surrounded by the extracellular matrix (ECM). The ECM can be thought of as a fibrous polymer network, acting as a natural scaffolding to provide mechanical support to the cells. Reciprocal mechanical and chemical interactions between the cells and the ECM are crucial in regulating the development of tissues and maintaining their functionality. Hence, to maintain in vivo -like behaviour when cells are cultured in vitro , they are often seeded in a gel, which aims to mimic the ECM. In this paper, we present a mathematical model that incorporate cell-gel interactions together with osmotic pressure to study the mechanical behaviour of biological gels. In particular, we consider an experiment where cells are seeded within a gel, which gradually compacts due to forces exerted on it by the cells. Adopting a one-dimensional Cartesian geometry for simplicity, we use a combination of analytical techniques and numerical simulations to investigate how cell traction forces interact with osmotic effects (which can lead to either gel swelling or contraction depending on the gel’s composition). Our results show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e. the chemical potentials) and the strength of the cell traction forces. We observe an unusual case where the gel oscillates between swelling and contraction. We also consider on how physical parameters like drag and viscosity affect the manner in which the gel evolves.
Publisher: American Society of Mechanical Engineers
Date: 17-06-2020
Abstract: Collagen is an important structural protein in the human body, and its molecules form structural aggregates at multiple length scales (i.e., microfibrils, fibrils, fibers, and bundles of different sizes) in biological tissues and organs [1]. The mechanical properties of most tissues are dependent on the underlying network of collagen fibers, proteoglycans, and other extracellular matrix components [2]. Similarly, the properties of in vitro tissue analogs, often created from collagen or fibrin gels, are also dependent on the organization of the biopolymers [3]. The overall mechanical response is intrinsically multi-scale and dynamic in both materials. As a result, a satisfactory description of the microstructure is important for exploring the essential physics of the tissue.
Publisher: Cambridge University Press (CUP)
Date: 06-2008
DOI: 10.1017/S0956792508007377
Abstract: Motivated by the aim of modelling the mechanical behaviour of biological gels (such as collagen gels) which have a fibrous microstructure, we consider the extensional flow of a thin two-dimensional film of incompressible, transversely isotropic viscous fluid. Neglecting inertia, and the effects of gravity and surface tension, leading-order equations are derived from a perturbation expansion of the full flow problem in powers of the (small) inverse aspect ratio. The existence and uniqueness of the solution of the reduced system of equations for small times is then proven. Special cases, in which the solution may be determined explicitly, are considered and we discuss the physical interpretation of the results.
Publisher: Cambridge University Press (CUP)
Date: 23-12-2014
DOI: 10.1017/JFM.2014.703
Abstract: Laminar helically-symmetric gravity-driven thin-film flow down a helically-wound channel of rectangular cross-section is considered. We extend the work of Stokes et al. ( Phys. Fluids , vol. 25 (8), 2013, 083103) and Lee et al. ( Phys. Fluids , vol. 26 (4), 2014, 043302) to channels with arbitrary curvature and torsion or, equivalently, arbitrary curvature and slope. We use a non-orthogonal coordinate system and, remarkably, find an exact steady-state solution. We find that the free-surface shape and flow have a complicated dependence on the curvature, slope and flux down the channel. Moderate to large channel slope has a significant effect on the flow in the region of the channel near the inside wall, particularly when the curvature of the channel is large. This work has application to flow in static spiral particle separators used in mineral processing.
Publisher: Proceedings of the National Academy of Sciences
Date: 21-07-2009
Abstract: Reconstructive microsurgery is a clinical technique used to transfer large amounts of a patient's tissue from one location used to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. It would thus be helpful to surgeons to understand the relationship between the tissue volume and blood vessel diameter to ensure success in these operations. As a first step towards addressing this question, we present a simple mathematical model that might be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery.
Publisher: Royal Society of Chemistry (RSC)
Date: 2016
DOI: 10.1039/C6RA11699J
Abstract: Thermo-reversible microgels to culture and harvest uniform-sized tumour spheroids with a narrow size-distribution.
Publisher: Cambridge University Press (CUP)
Date: 04-2018
Publisher: Elsevier BV
Date: 08-2021
Publisher: Springer Science and Business Media LLC
Date: 05-01-2022
Publisher: Elsevier BV
Date: 07-2014
DOI: 10.1016/J.JTBI.2014.02.033
Abstract: Many cell types form clumps or aggregates when cultured in vitro through a variety of mechanisms including rapid cell proliferation, chemotaxis, or direct cell-to-cell contact. In this paper we develop an agent-based model to explore the formation of aggregates in cultures where cells are initially distributed uniformly, at random, on a two-dimensional substrate. Our model includes unbiased random cell motion, together with two mechanisms which can produce cell aggregates: (i) rapid cell proliferation and (ii) a biased cell motility mechanism where cells can sense other cells within a finite range, and will tend to move towards areas with higher numbers of cells. We then introduce a pair-correlation function which allows us to quantify aspects of the spatial patterns produced by our agent-based model. In particular, these pair-correlation functions are able to detect differences between domains populated uniformly at random (i.e. at the exclusion complete spatial randomness (ECSR) state) and those where the proliferation and biased motion rules have been employed - even when such differences are not obvious to the naked eye. The pair-correlation function can also detect the emergence of a characteristic inter-aggregate distance which occurs when the biased motion mechanism is dominant, and is not observed when cell proliferation is the main mechanism of aggregate formation. This suggests that applying the pair-correlation function to experimental images of cell aggregates may provide information about the mechanism associated with observed aggregates. As a proof of concept, we perform such analysis for images of cancer cell aggregates, which are known to be associated with rapid proliferation. The results of our analysis are consistent with the predictions of the proliferation-based simulations, which supports the potential usefulness of pair correlation functions for providing insight into the mechanisms of aggregate formation.
Publisher: Springer Science and Business Media LLC
Date: 28-03-2009
Publisher: AIP Publishing
Date: 07-2019
DOI: 10.1063/1.5092814
Abstract: Particle-laden flows in helical channels are of interest for their applications in spiral particle separators used in the mining and mineral processing industries. In this paper, we extend the previous work of Lee, Stokes, and Bertozzi [“Behaviour of a particle-laden flow in a spiral channel,” Phys. Fluids 26, 043302 (2014)] by studying thin-film flows of monodisperse particle-laden fluid in helically wound channels of arbitrary centerline curvature and torsion and arbitrary cross-sectional shape. In the case where the particles are uniformly distributed through the depth of the film, significant analytic progress can be made yielding insight into the influence of channel geometry on particle distribution across the channel cross section: the governing equations reduce to a single nonlinear ordinary differential equation, which is readily integrated numerically to obtain the solution subject to appropriate boundary conditions. Motivated by possible application to the design of spiral separators, we consider the effects of changing the channel centerline geometry, the cross-sectional shape and the particle density on the resulting flows, and the radial distribution of particles. Our results support the findings in the work of Arnold, Stokes, and Green [“Thin-film flow in helically wound rectangular channels of arbitrary torsion and curvature,” J. Fluid Mech. 764, 76–94 (2015)] regarding the effect of channel centerline geometry and cross-sectional shape on flows in particle-free regions. In particle-rich regions, similar effects are seen although the primary velocity is lower due to increased effective mixture viscosity. Of key interest is the effect of channel geometry on the focusing of the particles for given fluxes of fluid and particles. We find that introducing a trench into the channel cross section, a feature often used in commercial spiral particle separators, leads to a smaller radial width of the particle-rich region, i.e., sharper focusing of the particles, which is consistent with experiments showing that channel geometry influences particle separation in a spiral separator.
Publisher: Public Library of Science (PLoS)
Date: 07-07-2021
DOI: 10.1371/JOURNAL.PCBI.1008353
Abstract: Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are: solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the aggregation of juvenile wingless locust nymphs. In this paper we attempt to understand how the distribution of food resources affect the group formation process. We do this by introducing a multi-population partial differential equation model that includes non-local locust interactions, local locust and food interactions, and gregarisation. Our results suggest that, food acts to increase the maximum density of locust groups, lowers the percentage of the population that needs to be gregarious for group formation, and decreases both the required density of locusts and time for group formation around an optimal food width. Finally, by looking at foraging efficiency within the numerical experiments we find that there exists a foraging advantage to being gregarious.
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 27-07-2016
Publisher: Springer Science and Business Media LLC
Date: 02-09-2016
DOI: 10.1007/S00285-015-0927-7
Abstract: 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.
Publisher: The Royal Society
Date: 08-2018
DOI: 10.1098/RSOS.180456
Abstract: Suspensions of self-motile, elongated particles are a topic of significant current interest, exemplifying a form of ‘active matter’. Ex les include self-propelling bacteria, algae and sperm, and artificial swimmers. Ericksen's model of a transversely isotropic fluid (Ericksen 1960 Colloid Polym. Sci. 173 , 117–122 ( doi:10.1007/bf01502416 )) treats suspensions of non-motile particles as a continuum with an evolving preferred direction this model describes fibrous materials as erse as extracellular matrix, textile tufts and plant cell walls. Director-dependent effects are incorporated through a modified stress tensor with four viscosity-like parameters. By making fundamental connections with recent models for active suspensions, we propose a modification to Ericksen's model, mainly the inclusion of self-motility this can be considered the simplest description of an oriented suspension including transversely isotropic effects. Motivated by the fact that transversely isotropic fluids exhibit modified flow stability, we conduct a linear stability analysis of two distinct cases, aligned and isotropic suspensions of elongated active particles. Novel aspects include the anisotropic rheology and translational diffusion. In general, anisotropic effects increase the instability of small perturbations, while translational diffusion stabilizes a range of wave-directions and, in some cases, a finite range of wavenumbers, thus emphasizing that both anisotropy and translational diffusion can have important effects in these systems.
Publisher: Elsevier BV
Date: 07-2018
DOI: 10.1016/J.JTBI.2018.04.004
Abstract: Previous experiments have shown that mature yeast mat biofilms develop a floral morphology, characterised by the formation of petal-like structures. In this work, we investigate the hypothesis that nutrient-limited growth is the mechanism by which these floral patterns form. To do this, we use a combination of experiments and mathematical analysis. In mat formation experiments of the yeast species Saccharomyces cerevisiae, we observe that mats expand radially at a roughly constant speed, and eventually undergo a transition from circular to floral morphology. To determine the extent to which nutrient-limited growth can explain these features, we adopt a previously proposed mathematical model for yeast growth. The model consists of a coupled system of reaction-diffusion equations for the yeast cell density and nutrient concentration, with a non-linear, degenerate diffusion term for cell spread. Using geometric singular perturbation theory and numerics, we show that the model admits travelling wave solutions in one dimension, which enables us to infer the diffusion ratio from experimental data. We then use a linear stability analysis to show that two-dimensional planar travelling wave solutions for feasible experimental parameters are linearly unstable to non-planar perturbations. This provides a potential mechanism by which petals can form, and allows us to predict the characteristic petal width. There is good agreement between these predictions, numerical solutions to the model, and experimental data. We therefore conclude that the non-linear cell diffusion mechanism provides a possible explanation for pattern formation in yeast mat biofilms, without the need to invoke other mechanisms such as flow of extracellular fluid, cell adhesion, or changes to cellular shape or behaviour.
Publisher: Cold Spring Harbor Laboratory
Date: 21-09-2020
DOI: 10.1101/2020.09.21.305896
Abstract: Locust swarms are a major threat to agriculture, affecting every continent except Antarctica and impacting the lives of 1 in 10 people. Locusts are short horned grasshoppers that exhibit two behaviour types depending on their local population density. These are solitarious, where they will actively avoid other locusts, and gregarious where they will seek them out. It is in this gregarious state that locusts can form massive and destructive flying swarms or plagues. However, these swarms are usually preceded by the formation of hopper bands by the juvenile wingless locust nymphs. It is thus important to understand the hopper band formation process to control locust outbreaks. On longer time-scales, environmental conditions such as rain events synchronize locust lifecycles and can lead to repeated outbreaks. On shorter time-scales, changes in resource distributions at both small and large spatial scales have an effect on locust gregarisation. It is these short time-scale locust-resource relationships and their effect on hopper band formation that are of interest. In this paper we investigate not only the effect of food on both the formation and characteristics of locust hopper bands but also a possible evolutionary explanation for gregarisation in this context. We do this by deriving a multi-population aggregation equation that includes non-local inter-in idual interactions and local inter-in idual and food interactions. By performing a series of numerical experiments we find that there exists an optimal food width for locust hopper band formation, and by looking at foraging efficiency within the model framework we uncover a possible evolutionary reason for gregarisation. Locusts are short horned grass hoppers that live in two diametrically opposed behavioural states. In the first, solitarious, they will actively avoid other locusts, whereas the second, gregarious, they will actively seek them out. It is in this gregarious state that locusts form the recognisable and destructive flying adult swarms. However, prior to swarm formation juvenile flightless locusts will form marching hopper bands and make their way from food source to food source. Predicting where these hopper bands might form is key to controlling locust outbreaks. Research has shown that changes in food distributions can affect the transition from solitarious to gregarious. In this paper we construct a mathematical model of locust-locust and locust-food interactions to investigate how and why isolated food distributions affect hopper band formation. Our findings suggest that there is an optimal food width for hopper band formation and that being gregarious increases a locusts ability to forage when food width decreases.
Publisher: The Royal Society
Date: 09-2019
Abstract: In the presence of glycoproteins, bacterial and yeast biofilms are hypothesized to expand by sliding motility. This involves a sheet of cells spreading as a unit, facilitated by cell proliferation and weak adhesion to the substratum. In this paper, we derive an extensional flow model for biofilm expansion by sliding motility to test this hypothesis. We model the biofilm as a two-phase (living cells and an extracellular matrix) viscous fluid mixture, and model nutrient depletion and uptake from the substratum. Applying the thin-film approximation simplifies the model, and reduces it to one-dimensional axisymmetric form. Comparison with Saccharomyces cerevisiae mat formation experiments reveals good agreement between experimental expansion speed and numerical solutions to the model with O ( 1 ) parameters estimated from experiments. This confirms that sliding motility is a possible mechanism for yeast biofilm expansion. Having established the biological relevance of the model, we then demonstrate how the model parameters affect expansion speed, enabling us to predict biofilm expansion for different experimental conditions. Finally, we show that our model can explain the ridge formation observed in some biofilms. This is especially true if surface tension is low, as hypothesized for sliding motility.
Publisher: The Royal Society
Date: 10-2016
Abstract: Automatic identification of the necrotic zone boundary is important in the assessment of treatments on in vitro tumour spheroids. This has been difficult especially when the difference in cell density between the necrotic and viable zones of a tumour spheroid is small. To help overcome this problem, we developed novel one-dimensional pair-correlation functions (PCFs) to provide quantitative estimates of the radial distance of the necrotic zone boundary from the centre of a tumour spheroid. We validate our approach on synthetic tumour spheroids in which the position of the necrotic zone boundary is known a priori . It is then applied to nine real tumour spheroids imaged with light sheet-based fluorescence microscopy. PCF estimates of the necrotic zone boundary are compared with those of a human expert and an existing standard computational method.
Publisher: American Physiological Society
Date: 10-2016
DOI: 10.1152/JAPPLPHYSIOL.00435.2016
Abstract: This study presents a structure-function analysis of the mammalian left ventricle and examines the performance of the cardiac capillary network, mitochondria, and myofibrils at rest and during simulated heavy exercise. Left ventricular external mechanical work rate was calculated from cardiac output and systemic mean arterial blood pressure in resting sheep ( Ovis aries n = 4) and goats ( Capra hircus n = 4) under mild sedation, followed by perfusion-fixation of the left ventricle and quantification of the cardiac capillary-tissue geometry and cardiomyocyte ultrastructure. The investigation was then extended to heavy exercise by increasing cardiac work according to published hemodynamics of sheep and goats performing sustained treadmill exercise. Left ventricular work rate averaged 0.017 W/cm 3 of tissue at rest and was estimated to increase to ∼0.060 W/cm 3 during heavy exercise. According to an oxygen transport model we applied to the left ventricular tissue, we predicted that oxygen consumption increases from 195 nmol O 2 ·s −1 ·cm −3 of tissue at rest to ∼600 nmol O 2 ·s −1 ·cm −3 during heavy exercise, which is within 90% of the oxygen demand rate and consistent with work remaining predominantly aerobic. Mitochondria represent 21-22% of cardiomyocyte volume and consume oxygen at a rate of 1,150 nmol O 2 ·s −1 ·cm −3 of mitochondria at rest and ∼3,600 nmol O 2 ·s −1 ·cm −3 during heavy exercise, which is within 80% of maximum in vitro rates and consistent with mitochondria operating near their functional limits. Myofibrils represent 65–66% of cardiomyocyte volume, and according to a Laplacian model of the left ventricular chamber, generate peak fiber tensions in the range of 50 to 70 kPa at rest and during heavy exercise, which is less than maximum tension of isolated cardiac tissue (120–140 kPa) and is explained by an apparent reserve capacity for tension development built into the left ventricle.
Publisher: Springer Science and Business Media LLC
Date: 16-04-2018
DOI: 10.1038/S41598-018-23649-Z
Abstract: The emergence of diffusion-limited growth (DLG) within a microbial colony on a solid substrate is studied using a combination of mathematical modelling and experiments. Using an agent-based model of the interaction between microbial cells and a diffusing nutrient, it is shown that growth directed towards a nutrient source may be used as an indicator that DLG is influencing the colony morphology. A continuous reaction–diffusion model for microbial growth is employed to identify the parameter regime in which DLG is expected to arise. Comparisons between the model and experimental data are used to argue that the bacterium Bacillus subtilis can undergo DLG, while the yeast Saccharomyces cerevisiae cannot, and thus the non-uniform growth exhibited by this yeast must be caused by the pseudohyphal growth mode rather than limited nutrient availability. Experiments testing directly for DLG features in yeast colonies are used to confirm this hypothesis.
Publisher: Elsevier BV
Date: 11-2023
Publisher: World Scientific Pub Co Pte Lt
Date: 19-11-2013
DOI: 10.1142/S0218202512500479
Abstract: We present a mathematical model for cell-induced gel contraction in vitro. The core of the model consists of conservation equations for the mass of the gel and the number of cells, coupled to a force balance on the gel. On the basis of previously reported experimental findings for collagen gels, which are frequently used experimentally, the gel is treated as a compressible viscous fluid while inertial effects are neglected. The flow is assumed to be isothermal, and a linear pressure–density relation is adopted. The force exerted on the gel by cells is assumed to depend upon the local environment surrounding the cell: influences can include the cell and extracellular matrix density, and the concentration of a diffusible chemical produced by the cells. We consider the simple, but experimentally relevant, case of spherically symmetric gels. For cell-free gels, we show how simple experiments might be used to determine the parameters in the model. When the cell-derived forces are given by a prescribed function of position, we are able to obtain the early time and steady-state behavior of the solution analytically. We perform numerical simulations which generate predictions of how the gel density evolves during compaction under differing assumptions concerning the factors influencing the force exerted by the cells. These results are compared with some previous observations reported in the literature.
Publisher: Elsevier BV
Date: 11-2022
DOI: 10.1016/J.JINSPHYS.2022.104443
Abstract: Density dependent phase polyphenism is the exhibiting of two or more distinct phenotypes from a single genotype depending on local population density. The most well known insect to exhibit this phenomenon is the locust, with whom the profound effect on behaviour leads to the classification of the two phases solitarious, where locusts actively avoid other locusts, and gregarious, where locusts are strongly attracted to other locusts. It has been shown that food distributions at both small and large scales have an effect on the process of gregarisation. While gregarisation offers advantages, such as greater predator avoidance, the relationship between phase polyphenism and potential foraging benefits is still not fully understood. In this paper, we explore the effect of gregarisation on foraging within increasingly heterogeneous environments using a partial differential equation model. We first consider a single two dimensional simulation of a spatially heterogeneous environment to understand the mechanics of gregarious/solitarious foraging. We then look at the steady state foraging advantage (measured as the ratio of per-capita contact with food) in environments ranging from homogeneous to very spatially heterogeneous. Finally, we perform a parameter sensitivity analysis to find which model parameters have the greatest effect on foraging advantage. We find that during the aggregation stage, prior to the onset of marching (which we do not model here), in increasingly heterogeneous food environments it is better to be gregarious than solitarious. In addition, we find that this is intrinsic to the gregarious/solitarious behavioural dynamic as it occurs almost regardless of the model parameters. That is to say, it doesn't matter how fast the organisms disperse or how strong their long range interactions as long as there is the solitarious/gregarious behaviour the gregarious foraging advantage will exist.
Publisher: Springer Science and Business Media LLC
Date: 25-11-2021
Publisher: Springer Science and Business Media LLC
Date: 16-03-2022
DOI: 10.1007/S00285-022-01730-6
Abstract: Biological tissues are composed of cells surrounded by the extracellular matrix (ECM). The ECM can be thought of as a fibrous polymer network, acting as a natural scaffolding to provide mechanical support to the cells. Reciprocal mechanical and chemical interactions between the cells and the ECM are crucial in regulating the development of tissues and maintaining their functionality. Hence, to maintain in vivo-like behaviour when cells are cultured in vitro, they are often seeded in a gel, which aims to mimic the ECM. In this paper, we present a mathematical model that incorporates cell-gel interactions together with osmotic pressure to study the mechanical behaviour of biological gels. In particular, we consider an experiment where cells are seeded within a gel, which gradually compacts due to forces exerted on it by the cells. Adopting a one-dimensional Cartesian geometry for simplicity, we use a combination of analytical techniques and numerical simulations to investigate how cell traction forces interact with osmotic effects (which can lead to either gel swelling or contraction depending on the gel’s composition). Our results show that a number of qualitatively different behaviours are possible, depending on the composition of the gel (i.e. its chemical potentials) and the strength of the cell traction forces. A novel prediction of our model is that there are cases where the gel oscillates between swelling and contraction to our knowledge, this behaviour has not been reported in experiments. We also consider how physical parameters like drag and viscosity affect the manner in which the gel evolves.
Publisher: Elsevier BV
Date: 07-2014
DOI: 10.1016/J.MBS.2014.04.004
Abstract: Cellular automata are discrete agent-based models, generally used in cell-based applications. There is much interest in obtaining continuum models that describe the mean behaviour of the agents in these models. Previously, continuum models have been derived for agents undergoing motility and proliferation processes, however, these models only hold under restricted conditions. In order to narrow down the reason for these restrictions, we explore three possible sources of error in deriving the model. These sources are the choice of limiting arguments, the use of a discrete-time model as opposed to a continuous-time model and the assumption of independence between the state of sites. We present a rigorous analysis in order to gain a greater understanding of the significance of these three issues. By finding a limiting regime that accurately approximates the conservation equation for the cellular automata, we are able to conclude that the inaccuracy between our approximation and the cellular automata is completely based on the assumption of independence.
Publisher: Public Library of Science (PLoS)
Date: 20-12-2021
Publisher: Elsevier BV
Date: 11-2010
DOI: 10.1016/J.JTBI.2010.08.013
Abstract: Liver cell aggregates may be grown in vitro by co-culturing hepatocytes with stellate cells. This method results in more rapid aggregation than hepatocyte-only culture, and appears to enhance cell viability and the expression of markers of liver-specific functions. We consider the early stages of aggregate formation, and develop a new mathematical model to investigate two alternative hypotheses (based on evidence in the experimental literature) for the role of stellate cells in promoting aggregate formation. Under Hypothesis 1, each population produces a chemical signal which affects the other, and enhanced aggregation is due to chemotaxis. Hypothesis 2 asserts that the interaction between the two cell types is by direct physical contact: the stellates extend long cellular processes which pull the hepatocytes into the aggregates. Under both hypotheses, hepatocytes are attracted to a chemical they themselves produce, and the cells can experience repulsive forces due to overcrowding. We formulate non-local (integro-partial differential) equations to describe the densities of cells, which are coupled to reaction-diffusion equations for the chemical concentrations. The behaviour of the model under each hypothesis is studied using a combination of linear stability analysis and numerical simulations. Our results show how the initial rate of aggregation depends upon the cell seeding ratio, and how the distribution of cells within aggregates depends on the relative strengths of attraction and repulsion between the cell types. Guided by our results, we suggest experiments which could be performed to distinguish between the two hypotheses.
Publisher: Springer Science and Business Media LLC
Date: 18-12-2009
DOI: 10.1007/S11538-008-9387-1
Abstract: The behavior of mammalian cells within three-dimensional structures is an area of intense biological research and underpins the efforts of tissue engineers to regenerate human tissues for clinical applications. In the particular case of hepatocytes (liver cells), the formation of spheroidal multicellular aggregates has been shown to improve cell viability and functionality compared to traditional monolayer culture techniques. We propose a simple mathematical model for the early stages of this aggregation process, when cell clusters form on the surface of the extracellular matrix (ECM) layer on which they are seeded. We focus on interactions between the cells and the viscoelastic ECM substrate. Governing equations for the cells, culture medium, and ECM are derived using the principles of mass and momentum balance. The model is then reduced to a system of four partial differential equations, which are investigated analytically and numerically. The model predicts that provided cells are seeded at a suitable density, aggregates with clearly defined boundaries and a spatially uniform cell density on the interior will form. While the mechanical properties of the ECM do not appear to have a significant effect, strong cell-ECM interactions can inhibit, or possibly prevent, the formation of aggregates. The paper concludes with a discussion of our key findings and suggestions for future work.
Publisher: Australian Mathematical Publishing Association, Inc.
Date: 07-03-2022
DOI: 10.21914/ANZIAMJ.V62.16056
Abstract: Aggregations abound in nature, from cell formations to locust swarms. One method of modelling these aggregations is the non-local aggregation equation with the addition of degenerate diffusion. In this article we develop a finite volume based numerical scheme for this style of equation and perform an error, computation time, and convergence analysis. In addition we investigate two methods for approximating the non-local component. References A. J. Bernoff and C. M. Topaz. Nonlocal aggregation models: A primer of swarm equilibria. SIAM Rev. 55.4 (2013), pp. 709–747. doi: 10.1137/130925669 R. Bürger, D. Inzunza, P. Mulet, and L. M. Villada. Implicit-explicit methods for a class of nonlinear nonlocal gradient flow equations modelling collective behaviour. Appl. Numer. Math. 144 (2019), pp. 234–252. doi: 10.1016/j.apnum.2019.04.018 J. A. Carrillo, A. Chertock, and Y. Huang. A finite-volume method for nonlinear nonlocal equations with a gradient flow structure. In: Commun. Comput. Phys. 17.1 (2015), pp. 233–258. doi: 10.4208/cicp.160214.010814a J. R. Dormand and P. J. Prince. A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6.1 (1980), pp. 19–26. doi: 10.1016/0771-050X(80)90013-3 J. von zur Gathen and J. Gerhard. Modern computer algebra. 3rd ed. Cambridge University Press, 2013. doi: 10.1017/CBO9781139856065 F. Georgiou, J. Buhl, J. E. F. Green, B. Lamichhane, and N. Thamwattana. Modelling locust foraging: How and why food affects group formation. PLOS Comput. Biol. 17.7 (2021), e1008353. doi: 10.1371/journal.pcbi.1008353 F. Georgiou, B. P. Lamichhane, and N. Thamwattana. An adaptive numerical scheme for a partial integro-differential equation. Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2018. Ed. by B. Lamichhane, T. Tran, and J. Bunder. Vol. 60. ANZIAM J. 2019, pp. C187–C200. doi: 10.21914/anziamj.v60i0.14066 F. Georgiou, N. Thamwattana, and B. P. Lamichhane. Modelling cell aggregation using a modified swarm model. Proceedings of the 23rd International Congress on Modelling and Simulation, MODSIM2019. Vol. 6. 2019, pp. 22–27. doi: 10.36334/modsim.2019.a1.georgiou J. E. F. Green, S. L. Waters, J. P. Whiteley, L. Edelstein-Keshet, K. M. Shakesheff, and H. M. Byrne. Non-local models for the formation of hepatocyte–stellate cell aggregates. J. Theor. Bio. 267.1 (2010), pp. 106–120. doi: 10.1016/j.jtbi.2010.08.013 R. J. LeVeque. Finite-volume methods for hyperbolic Pproblems. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253 C. F. Van Loan. Introduction to Scientific Computing: A Matrix Vector Approach Using MATLAB. 1997. url: s/higher-education rogram/Van- Loan-Introduction-to-Scientific-Computing-A-Matrix-Vector- Approach-Using-MATLAB-2nd-Edition/PGM215520.html A. Mogilner and L. Edelstein-Keshet. A non-local model for a swarm. J. Math. Bio. 38.6 (1999), pp. 534–570. doi: 10.1007/s002850050158 C. M. Topaz, A. L. Bertozzi, and M. A. Lewis. A nonlocal continuum model for biological aggregation. Bull. Math. Biol. 68 (2006), p. 1601. doi: 10.1007/s11538-006-9088-6 C. M. Topaz, M. R. D’Orsogna, L. Edelstein-Keshet, and A. J. Bernoff. Locust dynamics: Behavioral phase change and swarming. PLOS Comput. Bio. 8.8 (2012), e1002642. doi: 10.1371/journal.pcbi.1002642
Start Date: 06-2013
End Date: 11-2018
Amount: $333,684.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2023
End Date: 12-2025
Amount: $441,000.00
Funder: Australian Research Council
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