ORCID Profile
0000-0002-6304-9333
Current Organisation
University of Oxford
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Biological Mathematics | Applied Mathematics | Biological mathematics | Applied mathematics | Systems biology | Microbiology not elsewhere classified | Biological physics
Skin and Related Disorders | Cancer and Related Disorders | Expanding Knowledge in the Mathematical Sciences |
Publisher: AIP Publishing
Date: 04-09-2015
DOI: 10.1063/1.4929993
Abstract: Unlike standard applications of transport theory, the transport of molecules and cells during embryonic development often takes place within growing multidimensional tissues. In this work, we consider a model of diffusion on uniformly growing lines, disks, and spheres. An exact solution of the partial differential equation governing the diffusion of a population of in iduals on the growing domain is derived. Using this solution, we study the survival probability, S(t). For the standard non-growing case with an absorbing boundary, we observe that S(t) decays to zero in the long time limit. In contrast, when the domain grows linearly or exponentially with time, we show that S(t) decays to a constant, positive value, indicating that a proportion of the diffusing substance remains on the growing domain indefinitely. Comparing S(t) for diffusion on lines, disks, and spheres indicates that there are minimal differences in S(t) in the limit of zero growth and minimal differences in S(t) in the limit of fast growth. In contrast, for intermediate growth rates, we observe modest differences in S(t) between different geometries. These differences can be quantified by evaluating the exact expressions derived and presented here.
Publisher: American Physical Society (APS)
Date: 15-11-2013
Publisher: The Royal Society
Date: 06-2016
DOI: 10.1098/RSOB.160056
Abstract: Mathematical models are becoming increasingly integrated with experimental efforts in the study of biological systems. Collective cell migration in developmental biology is a particularly fruitful application area for the development of theoretical models to predict the behaviour of complex multicellular systems with many interacting parts. In this context, mathematical models provide a tool to assess the consistency of experimental observations with testable mechanistic hypotheses. In this review, we showcase ex les from recent years of multidisciplinary investigations of neural crest cell migration. The neural crest model system has been used to study how collective migration of cell populations is shaped by cell–cell interactions, cell–environmental interactions and heterogeneity between cells. The wide range of emergent behaviours exhibited by neural crest cells in different embryonal locations and in different organisms helps us chart out the spectrum of collective cell migration. At the same time, this ersity in migratory characteristics highlights the need to reconcile or unify the array of currently hypothesized mechanisms through the next generation of experimental data and generalized theoretical descriptions.
Publisher: Elsevier BV
Date: 08-2018
Publisher: The Company of Biologists
Date: 06-2015
DOI: 10.1242/DEV.117507
Abstract: Neural crest (NC) cell migration is crucial to the formation of peripheral tissues during vertebrate development. However, how NC cells respond to different microenvironments to maintain persistence of direction and cohesion in multicellular streams remains unclear. To address this, we profiled eight subregions of a typical cranial NC cell migratory stream. Hierarchical clustering showed significant differences in the expression profiles of the lead three subregions compared with newly emerged cells. Multiplexed imaging of mRNA expression using fluorescent hybridization chain reaction (HCR) quantitatively confirmed the expression profiles of lead cells. Computational modeling predicted that a small fraction of lead cells that detect directional information is optimal for successful stream migration. Single-cell profiling then revealed a unique molecular signature that is consistent and stable over time in a subset of lead cells within the most advanced portion of the migratory front, which we term trailblazers. Model simulations that forced a lead cell behavior in the trailing subpopulation predicted cell bunching near the migratory domain entrance. Misexpression of the trailblazer molecular signature by perturbation of two upstream transcription factors agreed with the in silico prediction and showed alterations to NC cell migration distance and stream shape. These data are the first to characterize the molecular ersity within an NC cell migratory stream and offer insights into how molecular patterns are transduced into cell behaviors.
Publisher: The Open Journal
Date: 13-03-2020
DOI: 10.21105/JOSS.01848
Publisher: Springer Science and Business Media LLC
Date: 26-09-2020
Publisher: IOP Publishing
Date: 12-05-2021
Abstract: The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial–mesenchymal transition (EMT). Chemical signals, such as TGF- β , produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.
Publisher: IOP Publishing
Date: 27-05-2019
Abstract: Moving fronts of cells are essential for development, repair and disease progression. Therefore, understanding and quantifying the details of the mechanisms that drive the movement of cell fronts is of wide interest. Quantitatively identifying the role of intercellular interactions, and in particular the role of cell pushing, remains an open question. In this work, we report a combined experimental-modelling approach showing that intercellular interactions contribute significantly to the spatial spreading of a population of cells. We use a novel experimental data set with PC-3 prostate cancer cells that have been pretreated with Mitomycin-C to suppress proliferation. This allows us to experimentally separate the effects of cell migration from cell proliferation, thereby enabling us to focus on the migration process in detail as the population of cells recolonizes an initially-vacant region in a series of two-dimensional experiments. We quantitatively model the experiments using a stochastic modelling framework, based on Langevin dynamics, which explicitly incorporates random motility and various intercellular forces including: (i) long range attraction (adhesion) and (ii) finite size effects that drive short range repulsion (pushing). Quantitatively comparing the ability of this model to describe the experimentally observed population-level behaviour provides us with quantitative insight into the roles of random motility and intercellular interactions. To quantify the mechanisms at play, we calibrate the stochastic model to match experimental cell density profiles to obtain estimates of cell diffusivity, D , and the litude of intercellular forces, f 0 . Our analysis shows that taking a standard modelling approach which ignores intercellular forces provides a poor match to the experimental data whereas incorporating intercellular forces, including short-range pushing and longer range attraction, leads to a faithful representation of the experimental observations. These results demonstrate a significant role of cell pushing during cell front movement and invasion.
Publisher: Cambridge University Press (CUP)
Date: 2013
DOI: 10.1017/S1446181113000059
Abstract: In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and co-workers [A. M. Berezhkovskii, C. S le and S. Y. Shvartsman, “How long does it take to establish a morphogen gradient?” Biophys. J. 99 (2010) L59–L61] rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single-species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalize the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications, for ex le the analysis of models describing coupled chemical decay and cell differentiation processes.
Publisher: Cold Spring Harbor Laboratory
Date: 10-12-2020
DOI: 10.1101/2020.12.09.418434
Abstract: The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF- β , produced by surrounding tissue can be up-taken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.
Publisher: Springer Science and Business Media LLC
Date: 27-02-2019
DOI: 10.1007/S11538-019-00589-X
Abstract: Reaction-diffusion models describing the movement, reproduction and death of in iduals within a population are key mathematical modelling tools with widespread applications in mathematical biology. A erse range of such continuum models have been applied in various biological contexts by choosing different flux and source terms in the reaction-diffusion framework. For ex le, to describe the collective spreading of cell populations, the flux term may be chosen to reflect various movement mechanisms, such as random motion (diffusion), adhesion, haptotaxis, chemokinesis and chemotaxis. The choice of flux terms in specific applications, such as wound healing, is usually made heuristically, and rarely it is tested quantitatively against detailed cell density data. More generally, in mathematical biology, the questions of model validation and model selection have not received the same attention as the questions of model development and model analysis. Many studies do not consider model validation or model selection, and those that do often base the selection of the model on residual error criteria after model calibration is performed using nonlinear regression techniques. In this work, we present a model selection case study, in the context of cell invasion, with a very detailed experimental data set. Using Bayesian analysis and information criteria, we demonstrate that model selection and model validation should account for both residual errors and model complexity. These considerations are often overlooked in the mathematical biology literature. The results we present here provide a straightforward methodology that can be used to guide model selection across a range of applications. Furthermore, the case study we present provides a clear ex le where neglecting the role of model complexity can give rise to misleading outcomes.
Publisher: Elsevier BV
Date: 11-2019
Publisher: The Royal Society
Date: 10-2021
Abstract: Equation learning aims to infer differential equation models from data. While a number of studies have shown that differential equation models can be successfully identified when the data are sufficiently detailed and corrupted with relatively small amounts of noise, the relationship between observation noise and uncertainty in the learned differential equation models remains unexplored. We demonstrate that for noisy datasets there exists great variation in both the structure of the learned differential equation models and their parameter values. We explore how to exploit multiple datasets to quantify uncertainty in the learned models, and at the same time draw mechanistic conclusions about the target differential equations. We showcase our results using simulation data from a relatively straightforward agent-based model (ABM) which has a well-characterized partial differential equation description that provides highly accurate predictions of averaged ABM behaviours in relevant regions of parameter space. Our approach combines equation learning methods with Bayesian inference approaches so that a quantification of uncertainty can be given by the posterior parameter distribution of the learned model.
Publisher: The Royal Society
Date: 23-10-2019
Abstract: Understanding how cells proliferate, migrate and die in various environments is essential in determining how organisms develop and repair themselves. Continuum mathematical models, such as the logistic equation and the Fisher–Kolmogorov equation, can describe the global characteristics observed in commonly used cell biology assays, such as proliferation and scratch assays. However, these continuum models do not account for single-cell-level mechanics observed in high-throughput experiments. Mathematical modelling frameworks that represent in idual cells, often called agent-based models, can successfully describe key single-cell-level features of these assays but are computationally infeasible when dealing with large populations. In this work, we propose an agent-based model with crowding effects that is computationally efficient and matches the logistic and Fisher–Kolmogorov equations in parameter regimes relevant to proliferation and scratch assays, respectively. This stochastic agent-based model allows multiple agents to be contained within compartments on an underlying lattice, thereby reducing the computational storage compared to existing agent-based models that allow one agent per site only. We propose a systematic method to determine a suitable compartment size. Implementing this compartment-based model with this compartment size provides a balance between computational storage, local resolution of agent behaviour and agreement with classical continuum descriptions.
Publisher: Elsevier BV
Date: 02-2014
Publisher: Elsevier BV
Date: 07-2020
Publisher: Elsevier BV
Date: 09-2014
DOI: 10.1016/J.JTBI.2014.04.026
Abstract: Cells respond to various biochemical and physical cues during wound-healing and tumour progression. in vitro assays used to study these processes are typically conducted in one particular geometry and it is unclear how the assay geometry affects the capacity of cell populations to spread, or whether the relevant mechanisms, such as cell motility and cell proliferation, are somehow sensitive to the geometry of the assay. In this work we use a circular barrier assay to characterise the spreading of cell populations in two different geometries. Assay 1 describes a tumour-like geometry where a cell population spreads outwards into an open space. Assay 2 describes a wound-like geometry where a cell population spreads inwards to close a void. We use a combination of discrete and continuum mathematical models and automated image processing methods to obtain independent estimates of the effective cell diffusivity, D, and the effective cell proliferation rate, λ. Using our parameterised mathematical model we confirm that our estimates of D and λ accurately predict the time-evolution of the location of the leading edge and the cell density profiles for both assay 1 and assay 2. Our work suggests that the effective cell diffusivity is up to 50% lower for assay 2 compared to assay 1, whereas the effective cell proliferation rate is up to 30% lower for assay 2 compared to assay 1.
Publisher: The Company of Biologists
Date: 15-08-2012
DOI: 10.1242/DEV.081471
Abstract: Long-distance cell migration is an important feature of embryonic development, adult morphogenesis and cancer, yet the mechanisms that drive subpopulations of cells to distinct targets are poorly understood. Here, we use the embryonic neural crest (NC) in tandem with theoretical studies to evaluate model mechanisms of long-distance cell migration. We find that a simple chemotaxis model is insufficient to explain our experimental data. Instead, model simulations predict that NC cell migration requires leading cells to respond to long-range guidance signals and trailing cells to short-range cues in order to maintain a directed, multicellular stream. Experiments confirm differences in leading versus trailing NC cell subpopulations, manifested in unique cell orientation and gene expression patterns that respond to non-linear tissue growth of the migratory domain. Ablation experiments that delete the trailing NC cell subpopulation reveal that leading NC cells distribute all along the migratory pathway and develop a leading/trailing cellular orientation and gene expression profile that is predicted by model simulations. Transplantation experiments and model predictions that move trailing NC cells to the migratory front, or vice versa, reveal that cells adopt a gene expression profile and cell behaviors corresponding to the new position within the migratory stream. These results offer a mechanistic model in which leading cells create and respond to a cell-induced chemotactic gradient and transmit guidance information to trailing cells that use short-range signals to move in a directional manner.
Publisher: Elsevier BV
Date: 11-2017
Publisher: Elsevier BV
Date: 12-2019
DOI: 10.1016/J.JTBI.2019.109997
Abstract: Variability in cell populations is frequently observed in both in vitro and in vivo settings. Intrinsic differences within populations of cells, such as differences in cell sizes or differences in rates of cell motility, can be present even within a population of cells from the same cell line. We refer to this variability as cell heterogeneity. Mathematical models of cell migration, for ex le, in the context of tumour growth and metastatic invasion, often account for both undirected (random) migration and directed migration that is mediated by cell-to-cell contacts and cell-to-cell adhesion. A key feature of standard models is that they often assume that the population is composed of identical cells with constant properties. This leads to relatively simple single-species homogeneous models that neglect the role of heterogeneity. In this work, we use a continuum modelling approach to explore the role of heterogeneity in spatial spreading of cell populations. We employ a three-species heterogeneous model of cell motility that explicitly incorporates different types of experimentally-motivated heterogeneity in cell sizes: (i) monotonically decreasing (ii) uniform (iii) non-monotonic and (iv) monotonically increasing distributions of cell size. Comparing the density profiles generated by the three-species heterogeneous model with density profiles predicted by a more standard single-species homogeneous model reveals that when we are dealing with monotonically decreasing and uniform distributions a simple and computationally efficient single-species homogeneous model can be remarkably accurate in describing the evolution of a heterogeneous cell population. In contrast, we find that the simpler single-species homogeneous model performs relatively poorly when applied to non-monotonic and monotonically increasing distributions of cell sizes. Additional results for heterogeneity in parameters describing both undirected and directed cell migration are also considered, and we find that similar results apply.
Publisher: American Physical Society (APS)
Date: 02-02-2011
Publisher: Springer Science and Business Media LLC
Date: 13-04-2013
DOI: 10.1007/S11538-013-9839-0
Abstract: Standard differential equation-based models of collective cell behaviour, such as the logistic growth model, invoke a mean-field assumption which is equivalent to assuming that in iduals within the population interact with each other in proportion to the average population density. Implementing such assumptions implies that the dynamics of the system are unaffected by spatial structure, such as the formation of patches or clusters within the population. Recent theoretical developments have introduced a class of models, known as moment dynamics models, which aim to account for the dynamics of in iduals, pairs of in iduals, triplets of in iduals, and so on. Such models enable us to describe the dynamics of populations with clustering, however, little progress has been made with regard to applying moment dynamics models to experimental data. Here, we report new experimental results describing the formation of a monolayer of cells using two different cell types: 3T3 fibroblast cells and MDA MB 231 breast cancer cells. Our analysis indicates that the 3T3 fibroblast cells are relatively motile and we observe that the 3T3 fibroblast monolayer forms without clustering. Alternatively, the MDA MB 231 cells are less motile and we observe that the MDA MB 231 monolayer formation is associated with significant clustering. We calibrate a moment dynamics model and a standard mean-field model to both data sets. Our results indicate that the mean-field and moment dynamics models provide similar descriptions of the 3T3 fibroblast monolayer formation whereas these two models give very different predictions for the MDA MD 231 monolayer formation. These outcomes indicate that standard mean-field models of collective cell behaviour are not always appropriate and that care ought to be exercised when implementing such a model.
Publisher: AIP Publishing
Date: 05-05-2016
DOI: 10.1063/1.4948782
Abstract: The motion of cells and molecules through biological environments is often hindered by the presence of other cells and molecules. A common approach to modeling this kind of hindered transport is to examine the mean squared displacement (MSD) of a motile tracer particle in a lattice-based stochastic random walk in which some lattice sites are occupied by obstacles. Unfortunately, stochastic models can be computationally expensive to analyze because we must average over a large ensemble of identically prepared realizations to obtain meaningful results. To overcome this limitation we describe an exact method for analyzing a lattice-based model of the motion of an agent moving through a crowded environment. Using our approach we calculate the exact MSD of the motile agent. Our analysis confirms the existence of a transition period where, at first, the MSD does not follow a power law with time. However, after a sufficiently long period of time, the MSD increases in proportion to time. This latter phase corresponds to Fickian diffusion with a reduced diffusivity owing to the presence of the obstacles. Our main result is to provide a mathematically motivated, reproducible, and objective estimate of the amount of time required for the transport to become Fickian. Our new method to calculate this crossover time does not rely on stochastic simulations.
Publisher: Cold Spring Harbor Laboratory
Date: 11-08-2020
DOI: 10.1101/2020.08.10.245233
Abstract: Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly-adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability , we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically-motivated synthetic data and Markov-chain Monte Carlo (MCMC) methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.
Publisher: AIP Publishing
Date: 04-02-2014
DOI: 10.1063/1.4864000
Abstract: Transport through crowded environments is often classified as anomalous, rather than classical, Fickian diffusion. Several studies have sought to describe such transport processes using either a continuous time random walk or fractional order differential equation. For both these models the transport is characterized by a parameter α, where α = 1 is associated with Fickian diffusion and α & 1 is associated with anomalous subdiffusion. Here, we simulate a single agent migrating through a crowded environment populated by impenetrable, immobile obstacles and estimate α from mean squared displacement data. We also simulate the transport of a population of such agents through a similar crowded environment and match averaged agent density profiles to the solution of a related fractional order differential equation to obtain an alternative estimate of α. We examine the relationship between our estimate of α and the properties of the obstacle field for both a single agent and a population of agents we show that in both cases, α decreases as the obstacle density increases, and that the rate of decrease is greater for smaller obstacles. Our work suggests that it may be inappropriate to model transport through a crowded environment using widely reported approaches including power laws to describe the mean squared displacement and fractional order differential equations to represent the averaged agent density profiles.
Publisher: American Physical Society (APS)
Date: 25-09-2012
Publisher: The Royal Society
Date: 06-2021
Abstract: We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded. To explore whether these data can be used to identify the hopping rates in each layer, we compute various profile likelihoods using two methods: first, an exact likelihood is evaluated using a relatively expensive Markov chain approach and, second, we form an approximate likelihood by assuming the distribution of exit times is given by a Gamma distribution whose first two moments match the moments from the continuum limit description of the stochastic model. Using the exact and approximate likelihoods, we construct various profile likelihoods for a range of problems. In cases where parameter values are not identifiable, we make progress by re-interpreting those data with a reduced model with a smaller number of layers.
Publisher: The Royal Society
Date: 12-2020
Abstract: Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability , we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github .
Publisher: Public Library of Science (PLoS)
Date: 12-2020
DOI: 10.1371/JOURNAL.PCBI.1008462
Abstract: Biologically-informed neural networks (BINNs), an extension of physics-informed neural networks [1], are introduced and used to discover the underlying dynamics of biological systems from sparse experimental data. In the present work, BINNs are trained in a supervised learning framework to approximate in vitro cell biology assay experiments while respecting a generalized form of the governing reaction-diffusion partial differential equation (PDE). By allowing the diffusion and reaction terms to be multilayer perceptrons (MLPs), the nonlinear forms of these terms can be learned while simultaneously converging to the solution of the governing PDE. Further, the trained MLPs are used to guide the selection of biologically interpretable mechanistic forms of the PDE terms which provides new insights into the biological and physical mechanisms that govern the dynamics of the observed system. The method is evaluated on sparse real-world data from wound healing assays with varying initial cell densities [2].
Publisher: The Royal Society
Date: 03-2020
Abstract: We examine the practical identifiability of parameters in a spatio-temporal reaction–diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction–diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.
Publisher: Elsevier BV
Date: 05-2016
Publisher: Springer Science and Business Media LLC
Date: 25-01-2018
DOI: 10.1007/S11538-018-0398-2
Abstract: Scratch assays are routinely used to study the collective spreading of cell populations. In general, the rate at which a population of cells spreads is driven by the combined effects of cell migration and proliferation. To examine the effects of cell migration separately from the effects of cell proliferation, scratch assays are often performed after treating the cells with a drug that inhibits proliferation. Mitomycin-C is a drug that is commonly used to suppress cell proliferation in this context. However, in addition to suppressing cell proliferation, mitomycin-C also causes cells to change size during the experiment, as each cell in the population approximately doubles in size as a result of treatment. Therefore, to describe a scratch assay that incorporates the effects of cell-to-cell crowding, cell-to-cell adhesion, and dynamic changes in cell size, we present a new stochastic model that incorporates these mechanisms. Our agent-based stochastic model takes the form of a system of Langevin equations that is the system of stochastic differential equations governing the evolution of the population of agents. We incorporate a time-dependent interaction force that is used to mimic the dynamic increase in size of the agents. To provide a mathematical description of the average behaviour of the stochastic model we present continuum limit descriptions using both a standard mean-field approximation and a more sophisticated moment dynamics approximation that accounts for the density of agents and density of pairs of agents in the stochastic model. Comparing the accuracy of the two continuum descriptions for a typical scratch assay geometry shows that the incorporation of agent growth in the system is associated with a decrease in accuracy of the standard mean-field description. In contrast, the moment dynamics description provides a more accurate prediction of the evolution of the scratch assay when the increase in size of in idual agents is included in the model.
Publisher: American Physical Society (APS)
Date: 10-06-2013
Publisher: The Royal Society
Date: 07-2019
Abstract: Mechanical heterogeneity in biological tissues, in particular stiffness, can be used to distinguish between healthy and diseased states. However, it is often difficult to explore relationships between cellular-level properties and tissue-level outcomes when biological experiments are performed at a single scale only. To overcome this difficulty, we develop a multi-scale mathematical model which provides a clear framework to explore these connections across biological scales. Starting with an in idual-based mechanical model of cell movement, we subsequently derive a novel coarse-grained system of partial differential equations governing the evolution of the cell density due to heterogeneous cellular properties. We demonstrate that solutions of the in idual-based model converge to numerical solutions of the coarse-grained model, for both slowly-varying-in-space and rapidly-varying-in-space cellular properties. We discuss applications of the model, such as determining relative cellular-level properties and an interpretation of data from a breast cancer detection experiment.
Publisher: Cold Spring Harbor Laboratory
Date: 16-08-2019
DOI: 10.1101/738542
Abstract: Vertebrate head morphogenesis involves orchestrated cell growth and tissue movements of the mesoderm and neural crest to form the distinct craniofacial pattern. To better understand structural birth defects, it is important that we learn how these processes are controlled. Here, we examine this question during chick head morphogenesis using time-lapse imaging, computational modeling, and experiment. We find that head mesodermal cells are inherently dynamic in culture and alter cell behaviors in the presence of either ectoderm or neural crest cells. Mesodermal cells in vivo display large-scale whirling motions that rapidly transition to lateral, directed movements after neural crest cells emerge. Computer model simulations predict distinct changes in neural crest migration as the spatio-temporal growth profile of the mesoderm is varied. BrdU-labeling and photoconversion combined with cell density measurements then reveal non-uniform mesoderm growth in space and time. Chemical inhibition of head mesoderm proliferation or ablation of premigratory neural crest alters mesoderm growth and neural crest migration, implying a dynamic feedback between tissue growth and neural crest cell signaling to confer robustness to the system. Dynamic feedback between tissue growth and neural crest cell migration ensures robust neural crest stream formation and head morphogenesis shown by time-lapse microscopy, mathematical modeling and embryo perturbations.
Publisher: American Physical Society (APS)
Date: 29-05-2012
Publisher: American Physical Society (APS)
Date: 26-05-2011
Publisher: The Royal Society
Date: 06-11-2013
Abstract: Cell trajectory data are often reported in the experimental cell biology literature to distinguish between different types of cell migration. Unfortunately, there is no accepted protocol for designing or interpreting such experiments and this makes it difficult to quantitatively compare different published datasets and to understand how changes in experimental design influence our ability to interpret different experiments. Here, we use an in idual-based mathematical model to simulate the key features of a cell trajectory experiment. This shows that our ability to correctly interpret trajectory data is extremely sensitive to the geometry and timing of the experiment, the degree of motility bias and the number of experimental replicates. We show that cell trajectory experiments produce data that are most reliable when the experiment is performed in a quasi-one-dimensional geometry with a large number of identically prepared experiments conducted over a relatively short time-interval rather than a few trajectories recorded over particularly long time-intervals.
Publisher: Elsevier BV
Date: 04-2015
DOI: 10.1016/J.JTBI.2015.01.025
Abstract: Mathematical models describing the movement of multiple interacting subpopulations are relevant to many biological and ecological processes. Standard mean-field partial differential equation descriptions of these processes suffer from the limitation that they implicitly neglect to incorporate the impact of spatial correlations and clustering. To overcome this, we derive a moment dynamics description of a discrete stochastic process which describes the spreading of distinct interacting subpopulations. In particular, we motivate our model by mimicking the geometry of two typical cell biology experiments. Comparing the performance of the moment dynamics model with a traditional mean-field model confirms that the moment dynamics approach always outperforms the traditional mean-field approach. To provide more general insight we summarise the performance of the moment dynamics model and the traditional mean-field model over a wide range of parameter regimes. These results help distinguish between those situations where spatial correlation effects are sufficiently strong, such that a moment dynamics model is required, from other situations where spatial correlation effects are sufficiently weak, such that a traditional mean-field model is adequate.
Publisher: Cold Spring Harbor Laboratory
Date: 09-12-2019
DOI: 10.1101/869495
Abstract: Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than by having some intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.
Publisher: Rockefeller University Press
Date: 15-08-2017
Abstract: Neural crest cells are both highly migratory and significant to vertebrate organogenesis. However, the signals that regulate neural crest cell migration remain unclear. In this study, we test the function of differential screening-selected gene aberrant in neuroblastoma (DAN), a bone morphogenetic protein (BMP) antagonist we detected by analysis of the chick cranial mesoderm. Our analysis shows that, before neural crest cell exit from the hindbrain, DAN is expressed in the mesoderm, and then it becomes absent along cell migratory pathways. Cranial neural crest and metastatic melanoma cells avoid DAN protein stripes in vitro. Addition of DAN reduces the speed of migrating cells in vivo and in vitro, respectively. In vivo loss of function of DAN results in enhanced neural crest cell migration by increasing speed and directionality. Computer model simulations support the hypothesis that DAN restrains cell migration by regulating cell speed. Collectively, our results identify DAN as a novel factor that inhibits uncontrolled neural crest and metastatic melanoma invasion and promotes collective migration in a manner consistent with the inhibition of BMP signaling.
Publisher: The Royal Society
Date: 02-2019
Abstract: Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab ® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.
Publisher: Elsevier BV
Date: 07-2012
Publisher: The Royal Society
Date: 03-2021
Abstract: Agent-based models provide a flexible framework that is frequently used for modelling many biological systems, including cell migration, molecular dynamics, ecology and epidemiology. Analysis of the model dynamics can be challenging due to their inherent stochasticity and heavy computational requirements. Common approaches to the analysis of agent-based models include extensive Monte Carlo simulation of the model or the derivation of coarse-grained differential equation models to predict the expected or averaged output from the agent-based model. Both of these approaches have limitations, however, as extensive computation of complex agent-based models may be infeasible, and coarse-grained differential equation models can fail to accurately describe model dynamics in certain parameter regimes. We propose that methods from the equation learning field provide a promising, novel and unifying approach for agent-based model analysis. Equation learning is a recent field of research from data science that aims to infer differential equation models directly from data. We use this tutorial to review how methods from equation learning can be used to learn differential equation models from agent-based model simulations. We demonstrate that this framework is easy to use, requires few model simulations, and accurately predicts model dynamics in parameter regions where coarse-grained differential equation models fail to do so. We highlight these advantages through several case studies involving two agent-based models that are broadly applicable to biological phenomena: a birth–death–migration model commonly used to explore cell biology experiments and a susceptible–infected–recovered model of infectious disease spread.
Publisher: Elsevier BV
Date: 06-2017
DOI: 10.1016/J.JTBI.2017.04.009
Abstract: Cell migration within tissues involves the interaction of many cells from distinct subpopulations. In this work, we present a discrete model of collective cell migration where the motion of in idual cells is driven by random forces, short range repulsion forces to mimic crowding, and longer range attraction forces to mimic adhesion. This discrete model can be used to simulate a population of cells that is composed of K ≥ 1 distinct subpopulations. To analyse the discrete model we formulate a hierarchy of moment equations that describe the spatial evolution of the density of agents, pairs of agents, triplets of agents, and so forth. To solve the hierarchy of moment equations we introduce two forms of closure: (i) the mean field approximation, which effectively assumes that the distributions of in idual agents are independent and (ii) a moment dynamics description that is based on the Kirkwood superposition approximation. The moment dynamics description provides an approximate way of incorporating spatial patterns, such as agent clustering, into the continuum description. Comparing the performance of the two continuum descriptions confirms that both perform well when adhesive forces are sufficiently weak. In contrast, the moment dynamics description outperforms the mean field model when adhesive forces are sufficiently large. This is a first attempt to provide an accurate continuum description of a lattice-free, multi-species model of collective cell migration.
Publisher: Springer Science and Business Media LLC
Date: 16-07-2014
DOI: 10.1038/SREP05713
Abstract: Spreading cell fronts are essential features of development, repair and disease processes. Many mathematical models used to describe the motion of cell fronts, such as Fisher's equation, invoke a mean–field assumption which implies that there is no spatial structure, such as cell clustering, present. Here, we examine the presence of spatial structure using a combination of in vitro circular barrier assays, discrete random walk simulations and pair correlation functions. In particular, we analyse discrete simulation data using pair correlation functions to show that spatial structure can form in a spreading population of cells either through sufficiently strong cell–to–cell adhesion or sufficiently rapid cell proliferation. We analyse images from a circular barrier assay describing the spreading of a population of MM127 melanoma cells using the same pair correlation functions. Our results indicate that the spreading melanoma cell populations remain very close to spatially uniform, suggesting that the strength of cell–to–cell adhesion and the rate of cell proliferation are both sufficiently small so as not to induce any spatial patterning in the spreading populations.
Publisher: American Physical Society (APS)
Date: 23-04-2012
Publisher: Cold Spring Harbor Laboratory
Date: 12-11-2019
DOI: 10.1101/839282
Abstract: We examine the practical identifiability of parameters in a spatiotemporal reaction-diffusion model of a scratch assay. Experimental data involves fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction–diffusion model by treating the total population as two interacting subpopulations. Practical identifiability is examined using a Bayesian Markov chain Monte Carlo (MCMC) framework, confirming that the parameters are identifiable when we assume the diffusivities of the subpopulations are identical, but that the parameters are practically non-identifiable when we allow the diffusivities to be distinct. We also assess practical identifiability using a profile likelihood approach, providing similar results to MCMC with the advantage of being an order of magnitude faster to compute. Therefore, we suggest that the profile likelihood ought to be adopted as a screening tool to assess practical identifiability before MCMC computations are performed.
Publisher: Elsevier BV
Date: 11-2015
DOI: 10.1016/J.YDBIO.2015.08.011
Abstract: Embryonic neural crest cells travel in discrete streams to precise locations throughout the head and body. We previously showed that cranial neural crest cells respond chemotactically to vascular endothelial growth factor (VEGF) and that cells within the migratory front have distinct behaviors and gene expression. We proposed a cell-induced gradient model in which lead neural crest cells read out directional information from a chemoattractant profile and instruct trailers to follow. In this study, we show that migrating chick neural crest cells do not display distinct lead and trailer gene expression profiles in culture. However, exposure to VEGF in vitro results in the upregulation of a small subset of genes associated with an in vivo lead cell signature. Timed addition and removal of VEGF in culture reveals the changes in neural crest cell gene expression are rapid. A computational model incorporating an integrate-and-switch mechanism between cellular phenotypes predicts migration efficiency is influenced by the timescale of cell behavior switching. To test the model hypothesis that neural crest cellular phenotypes respond to changes in the VEGF chemoattractant profile, we presented ectopic sources of VEGF to the trailer neural crest cell subpopulation and show erted cell trajectories and stream alterations consistent with model predictions. Gene profiling of trailer cells that erted and encountered VEGF revealed upregulation of a subset of 'lead' genes. Injection of neuropilin1 (Np1)-Fc into the trailer subpopulation or electroporation of VEGF morpholino to reduce VEGF signaling failed to alter trailer neural crest cell trajectories, suggesting trailers do not require VEGF to maintain coordinated migration. These results indicate that VEGF is one of the signals that establishes lead cell identity and its chemoattractant profile is critical to neural crest cell migration.
Publisher: The Royal Society
Date: 08-2018
DOI: 10.1098/RSOS.180384
Abstract: To better understand development, repair and disease progression, it is useful to quantify the behaviour of proliferative and motile cell populations as they grow and expand to fill their local environment. Inferring parameters associated with mechanistic models of cell colony growth using quantitative data collected from carefully designed experiments provides a natural means to elucidate the relative contributions of various processes to the growth of the colony. In this work, we explore how experimental design impacts our ability to infer parameters for simple models of the growth of proliferative and motile cell populations. We adopt a Bayesian approach, which allows us to characterize the uncertainty associated with estimates of the model parameters. Our results suggest that experimental designs that incorporate initial spatial heterogeneities in cell positions facilitate parameter inference without the requirement of cell tracking, while designs that involve uniform initial placement of cells require cell tracking for accurate parameter inference. As cell tracking is an experimental bottleneck in many studies of this type, our recommendations for experimental design provide for significant potential time and cost savings in the analysis of cell colony growth.
Publisher: Springer Science and Business Media LLC
Date: 12-2013
Publisher: Public Library of Science (PLoS)
Date: 25-09-2015
Publisher: IOP Publishing
Date: 06-03-2017
Publisher: The Royal Society
Date: 06-05-2013
Abstract: Moving fronts of cells are essential features of embryonic development, wound repair and cancer metastasis. This paper describes a set of experiments to investigate the roles of random motility and proliferation in driving the spread of an initially confined cell population. The experiments include an analysis of cell spreading when proliferation was inhibited. Our data have been analysed using two mathematical models: a lattice-based discrete model and a related continuum partial differential equation model. We obtain independent estimates of the random motility parameter, D , and the intrinsic proliferation rate, λ , and we confirm that these estimates lead to accurate modelling predictions of the position of the leading edge of the moving front as well as the evolution of the cell density profiles. Previous work suggests that systems with a high λ / D ratio will be characterized by steep fronts, whereas systems with a low λ / D ratio will lead to shallow diffuse fronts and this is confirmed in the present study. Our results provide evidence that continuum models, based on the Fisher–Kolmogorov equation, are a reliable platform upon which we can interpret and predict such experimental observations.
Publisher: American Physical Society (APS)
Date: 20-11-2013
Publisher: Elsevier BV
Date: 07-2014
DOI: 10.1016/J.JTBI.2014.02.023
Abstract: Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of in idual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and the asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performance of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.
Publisher: American Physical Society (APS)
Date: 05-10-2010
Publisher: Cold Spring Harbor Laboratory
Date: 30-07-2019
DOI: 10.1101/719666
Abstract: Neural crest migration requires cells to move through an environment filled with dense extracellular matrix and mesoderm to reach targets throughout the vertebrate embryo. Here, we use high-resolution microscopy, computational modeling, and in vitro and in vivo cell invasion assays to investigate the function of Aquaporin-1 (AQP-1) signaling. We find that migrating lead cranial neural crest cells express AQP-1 mRNA and protein, implicating a biological role for water channel protein function during invasion. Differential AQP-1 levels affect neural crest cell speed, direction, and the length and stability of cell filopodia. Further, AQP-1 enhances matrix metalloprotease (MMP) activity and colocalizes with phosphorylated focal adhesion kinases (pFAK). Co-localization of AQP-1 expression with EphB guidance receptors in the same migrating neural crest cells raises novel implications for the concept of guided bulldozing by lead cells during migration.
Publisher: AIP Publishing
Date: 27-06-2016
DOI: 10.1063/1.4953913
Abstract: Diffusive processes are often represented using stochastic random walk frameworks. The amount of time taken for an in idual in a random walk to intersect with an absorbing boundary is a fundamental property that is often referred to as the particle lifetime, or the first passage time. The mean lifetime of particles in a random walk model of diffusion is related to the amount of time required for the diffusive process to reach a steady state. Mathematical analysis describing the mean lifetime of particles in a standard model of diffusion without crowding is well known. However, the lifetime of agents in a random walk with crowding has received much less attention. Since many applications of diffusion in biology and biophysics include crowding effects, here we study a discrete model of diffusion that incorporates crowding. Using simulations, we show that crowding has a dramatic effect on agent lifetimes, and we derive an approximate expression for the mean agent lifetime that includes crowding effects. Our expression matches simulation results very well, and highlights the importance of crowding effects that are sometimes overlooked.
Publisher: Cold Spring Harbor Laboratory
Date: 02-08-2017
DOI: 10.1101/171710
Abstract: To better understand development, repair and disease progression it is useful to quantify the behaviour of proliferative and motile cell populations as they grow and expand to fill their local environment. Inferring parameters associated with mechanistic models of cell colony growth using quantitative data collected from carefully designed experiments provides a natural means to elucidate the relative contributions of various processes to the growth of the colony. In this work we explore how experimental design impacts our ability to infer parameters for simple models of the growth of proliferative and motile cell populations. We adopt a Bayesian approach, which allows us to characterise the uncertainty associated with estimates of the model parameters. Our results suggest that experimental designs that incorporate initial spatial heterogeneities in cell positions facilitate parameter inference without the requirement of cell tracking, whilst designs that involve uniform initial placement of cells require cell tracking for accurate parameter inference. As cell tracking is an experimental bottleneck in many studies of this type, our recommendations for experimental design provide for significant potential time and cost savings in the analysis of cell colony growth.
Location: United Kingdom of Great Britain and Northern Ireland
Start Date: 02-2018
End Date: 08-2022
Amount: $392,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2024
End Date: 12-2026
Amount: $586,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2023
End Date: 12-2029
Amount: $35,000,000.00
Funder: Australian Research Council
View Funded ActivityStart Date: 2012
End Date: 08-2017
Amount: $330,000.00
Funder: Australian Research Council
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