Multi-scale modeling of a wound-healing cell migration assay

Quantitative image analysis of the shape and size of circular wound sites generated by vertically stamped scratches

A protocol for quantitative image analysis of wound generation is important to better understand the integrative process of wound healing and the closure mechanism. Here, we present a method for quantitative analysis of microscopic images of circular wound sites generated by vertically stamped scratches. To demonstrate proof-of-concept validation, we used two types of mechanical stamping tools, a mechanical pencil lead (type 1; brittle) and polydimethylsiloxane (PDMS) pillars (type 2; ductile), to create circular wound sites. We also present a method for analysis of microscopic images of the generated wound sites by suggesting new parameters, such as controlled area transfer ratio, modified shape factor, and roundness index, specifically to investigate the shape and size of wounds via house-coded image processing. We believe that this approach can be potentially useful by providing a better way of studying vertical wound generation for future skin wound generation and care applications compared with its counterpart, conventional horizontal wound generation.

Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model

This paper studies the global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with appropriate initial and mixed boundary conditions. Under some practicable regularity criteria on diffusion item and nonlinearity, we establish the local existence and uniqueness of classical solutions based on a contraction mapping. This local solution can be continued for all positive time by employing the methods of energy estimates, Lp-theory, and Schauder estimate of linear parabolic equations. A straightforward application of global existence result of classical solutions to a density-dependent diffusion model of in vitro glioblastoma growth is also presented.

Logistic Proliferation of Cells in Scratch Assays is Delayed

Scratch assays are used to study how a population of cells re-colonises a vacant region on a two-dimensional substrate after a cell monolayer is scratched. These experiments are used in many applications including drug design for the treatment of cancer and chronic wounds. To provide insights into the mechanisms that drive scratch assays, solutions of continuum reaction-diffusion models have been calibrated to data from scratch assays. These models typically include a logistic source term to describe carrying capacity-limited proliferation; however, the choice of using a logistic source term is often made without examining whether it is valid. Here we study the proliferation of PC-3 prostate cancer cells in a scratch assay. All experimental results for the scratch assay are compared with equivalent results from a proliferation assay where the cell monolayer is not scratched. Visual inspection of the time evolution of the cell density away from the location of the scratch reveals a series of sigmoid curves that could be naively calibrated to the solution of the logistic growth model. However, careful analysis of the per capita growth rate as a function of density reveals several key differences between the proliferation of cells in scratch and proliferation assays. Our findings suggest that the logistic growth model is valid for the entire duration of the proliferation assay. On the other hand, guided by data, we suggest that there are two phases of proliferation in a scratch assay; at short time, we have a disturbance phase where proliferation is not logistic, and this is followed by a growth phase where proliferation appears to be logistic. These two phases are observed across a large number of experiments performed at different initial cell densities. Overall our study shows that simply calibrating the solution of a continuum model to a scratch assay might produce misleading parameter estimates, and this issue can be resolved by making a distinction between the disturbance and growth phases. Repeating our procedure for other scratch assays will provide insight into the roles of the disturbance and growth phases for different cell lines and scratch assays performed on different substrates.

Quantifying the roles of random motility and directed motility using advection-diffusion theory for a 3T3 fibroblast cell migration assay stimulated with an electric field

Background

Directed cell migration can be driven by a range of external stimuli, such as spatial gradients of: chemical signals (chemotaxis); adhesion sites (haptotaxis); or temperature (thermotaxis). Continuum models of cell migration typically include a diffusion term to capture the undirected component of cell motility and an advection term to capture the directed component of cell motility. However, there is no consensus in the literature about the form that the advection term takes. Some theoretical studies suggest that the advection term ought to include receptor saturation effects. However, others adopt a much simpler constant coefficient. One of the limitations of including receptor saturation effects is that it introduces several additional unknown parameters into the model. Therefore, a relevant research question is to investigate whether directed cell migration is best described by a simple constant tactic coefficient or a more complicated model incorporating saturation effects.

Results

We study directed cell migration using an experimental device in which the directed component of the cell motility is driven by a spatial gradient of electric potential, which is known as electrotaxis. The electric field (EF) is proportional to the spatial gradient of the electric potential. The spatial variation of electric potential across the experimental device varies in such a way that there are several subregions on the device in which the EF takes on different values that are approximately constant within those subregions. We use cell trajectory data to quantify the motion of 3T3 fibroblast cells at different locations on the device to examine how different values of the EF influences cell motility. The undirected (random) motility of the cells is quantified in terms of the cell diffusivity, D, and the directed motility is quantified in terms of a cell drift velocity, v. Estimates D and v are obtained under a range of four different EF conditions, which correspond to normal physiological conditions. Our results suggest that there is no anisotropy in D, and that D appears to be approximately independent of the EF and the electric potential. The drift velocity increases approximately linearly with the EF, suggesting that the simplest linear advection term, with no additional saturation parameters, provides a good explanation of these physiologically relevant data.

Conclusions

We find that the simplest linear advection term in a continuum model of directed cell motility is sufficient to describe a range of different electrotaxis experiments for 3T3 fibroblast cells subject to normal physiological values of the electric field. This is useful information because alternative models that include saturation effects involve additional parameters that need to be estimated before a partial differential equation model can be applied to interpret or predict a cell migration experiment.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-017-0413-5) contains supplementary material, which is available to authorized users.

Using approximate Bayesian computation to quantify cell-cell adhesion parameters in a cell migratory process

In this work, we implement approximate Bayesian computational methods to improve the design of a wound-healing assay used to quantify cell–cell interactions. This is important as cell–cell interactions, such as adhesion and repulsion, have been shown to play a role in cell migration. Initially, we demonstrate with a model of an unrealistic experiment that we are able to identify model parameters that describe agent motility and adhesion, given we choose appropriate summary statistics for our model data. Following this, we replace our model of an unrealistic experiment with a model representative of a practically realisable experiment. We demonstrate that, given the current (and commonly used) experimental set-up, our model parameters cannot be accurately identified using approximate Bayesian computation methods. We compare new experimental designs through simulation, and show more accurate identification of model parameters is possible by expanding the size of the domain upon which the experiment is performed, as opposed to increasing the number of experimental replicates. The results presented in this work, therefore, describe time and cost-saving alterations for a commonly performed experiment for identifying cell motility parameters. Moreover, this work will be of interest to those concerned with performing experiments that allow for the accurate identification of parameters governing cell migratory processes, especially cell migratory processes in which cell–cell adhesion or repulsion are known to play a significant role.

Cell motility is a central process in wound healing and relies on complex cell-cell interactions. A team of mathematicians led by Ruth Baker and Kit Yates at the University of Oxford utilised computer simulations to re-design wound-healing assays that efficiently identify cell motility parameters. New experimental designs through computer simulation can more accurately identify cell motility parameters by expanding the size of the domain upon which the experiment is performed, as opposed to increasing the number of experimental replicates. The results describe time and cost-saving alterations for an experimental method for evaluate complex cell-cell interactions.

A novel approach to quantify the wound closure dynamic

The Wound Healing (WH) assay is widely used to investigate cell migration in vitro, in order to reach a better understanding of many physiological and pathological phenomena. Several experimental factors, such as uneven cell density among different samples, can affect the reproducibility and reliability of this assay, leading to a discrepancy in the wound closure kinetics among data sets corresponding to the same cell sample. We observed a linear relationship between the wound closure velocity and cell density, and suggested a novel methodological approach, based on transport phenomena concepts, to overcome this source of error on the analysis of the Wound Healing assay. In particular, we propose a simple scaling of the experimental data, based on the interpretation of the wound closure as a diffusion-reaction process. We applied our methodology to the MDA-MB-231 breast cancer cells, whose motility was perturbed by silencing or over-expressing genes involved in the control of cell migration. Our methodological approach leads to a significant improvement in the reproducibility and reliability in the in vitro WH assay.

Novel human bioactive peptides identified in Apolipoprotein B: Evaluation of their therapeutic potential

Host defence peptides (HDPs) are short, cationic amphipathic peptides that play a key role in the response to infection and inflammation in all complex life forms. It is increasingly emerging that HDPs generally have a modest direct activity against a broad range of microorganisms, and that their anti-infective properties are mainly due to their ability to modulate the immune response. Here, we report the recombinant production and characterization of two novel HDPs identified in human Apolipoprotein B (residues 887-922) by using a bioinformatics method recently developed by our group. We focused our attention on two variants of the identified HDP, here named r(P)ApoB_L and r(P)ApoB_S, 38- and 26-residue long, respectively. Both HDPs were found to be endowed with a broad-spectrum antimicrobial activity while they show neither toxic nor haemolytic effects towards eukaryotic cells. Interestingly, both HDPs were found to display a significant anti-biofilm activity, and to act in synergy with either commonly used antibiotics or EDTA. The latter was selected for its ability to affect bacterial outer membrane permeability, and to sensitize bacteria to several antibiotics. Circular dichroism analyses showed that SDS, TFE, and LPS significantly alter r(P)ApoB_L conformation, whereas slighter or no significant effects were detected in the case of r(P)ApoB_S peptide. Interestingly, both ApoB derived peptides were found to elicit anti-inflammatory effects, being able to mitigate the production of pro-inflammatory interleukin-6 and nitric oxide in LPS induced murine macrophages. It should also be emphasized that r(P)ApoB_L peptide was found to play a role in human keratinocytes wound closure in vitro. Altogether, these findings open interesting perspectives on the therapeutic use of the herein identified HDPs.

Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions

Two-dimensional collective cell migration assays are used to study cancer and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding. Previous discrete models of collective cell migration assays involve a nearest-neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site; and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but some experimental evidence suggests that cells do not always proliferate logistically. Motivated by these observations, we introduce a generalised proliferation mechanism which allows non-nearest neighbour proliferation events to take place over a template of [Formula: see text] concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a crowding function, f(C), which accounts for the density of agents within a group of sites rather than dealing with the occupancy of a single randomly chosen site. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, [Formula: see text], where λ is the proliferation rate, is generalised to a universal growth function, [Formula: see text]. Comparing the solution of the continuum description with averaged simulation data indicates that the continuum model performs well for many choices of f(C) and r. For nonlinear f(C), the quality of the continuum-discrete match increases with r.

Comparison between fibroblast wound healing and cell random migration assays in vitro

Cell migration plays a key role in many biological processes, including cancer growth and invasion, embryogenesis, angiogenesis, inflammatory response, and tissue repair. In this work, we compare two well-established experimental approaches for the investigation of cell motility in vitro: the cell random migration (CRM) and the wound healing (WH) assay. In the former, extensive tracking of individual live cells trajectories by time-lapse microscopy and elaborate data processing are used to calculate two intrinsic motility parameters of the cell population under investigation, i.e. the diffusion coefficient and the persistence time. In the WH assay, a scratch is made in a confluent cell monolayer and the closure time of the exposed area is taken as an easy-to-measure, empirical estimate of cell migration. To compare WH and CRM we applied the two assays to investigate the motility of skin fibroblasts isolated from wild type and transgenic mice (TgPED) overexpressing the protein PED/PEA-15, which is highly expressed in patients with type 2 diabetes. Our main result is that the cell motility parameters derived from CRM can be also estimated from a time-resolved analysis of the WH assay, thus showing that the latter is also amenable to a quantitative analysis for the characterization of cell migration. To our knowledge this is the first quantitative comparison of these two widely used techniques.

Modeling keratinocyte wound healing dynamics: Cell-cell adhesion promotes sustained collective migration

The in vitro migration of keratinocyte cell sheets displays behavioral and biochemical similarities to the in vivo wound healing response of keratinocytes in animal model systems. In both cases, ligand-dependent Epidermal Growth Factor Receptor (EGFR) activation is sufficient to elicit collective cell migration into the wound. Previous mathematical modeling studies of in vitro wound healing assays assume that physical connections between cells have a hindering effect on cell migration, but biological literature suggests a more complicated story. By combining mathematical modeling and experimental observations of collectively migrating sheets of keratinocytes, we investigate the role of cell-cell adhesion during in vitro keratinocyte wound healing assays. We develop and compare two nonlinear diffusion models of the wound healing process in which cell-cell adhesion either hinders or promotes migration. Both models can accurately fit the leading edge propagation of cell sheets during wound healing when using a time-dependent rate of cell-cell adhesion strength. The model that assumes a positive role of cell-cell adhesion on migration, however, is robust to changes in the leading edge definition and yields a qualitatively accurate density profile. Using RNAi for the critical adherens junction protein, α-catenin, we demonstrate that cell sheets with wild type cell-cell adhesion expression maintain migration into the wound longer than cell sheets with decreased cell-cell adhesion expression, which fails to exhibit collective migration. Our modeling and experimental data thus suggest that cell-cell adhesion promotes sustained migration as cells pull neighboring cells into the wound during wound healing.

Quantifying the effect of experimental design choices for in vitro scratch assays

Scratch assays are often used to investigate potential drug treatments for chronic wounds and cancer. Interpreting these experiments with a mathematical model allows us to estimate the cell diffusivity, D, and the cell proliferation rate, λ. However, the influence of the experimental design on the estimates of D and λ is unclear. Here we apply an approximate Bayesian computation (ABC) parameter inference method, which produces a posterior distribution of D and λ, to new sets of synthetic data, generated from an idealised mathematical model, and experimental data for a non-adhesive mesenchymal population of fibroblast cells. The posterior distribution allows us to quantify the amount of information obtained about D and λ. We investigate two types of scratch assay, as well as varying the number and timing of the experimental observations captured. Our results show that a scrape assay, involving one cell front, provides more precise estimates of D and λ, and is more computationally efficient to interpret than a wound assay, with two opposingly directed cell fronts. We find that recording two observations, after making the initial observation, is sufficient to estimate D and λ, and that the final observation time should correspond to the time taken for the cell front to move across the field of view. These results provide guidance for estimating D and λ, while simultaneously minimising the time and cost associated with performing and interpreting the experiment.

Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias

The ability of cells to undergo collective movement plays a fundamental role in tissue repair, development and cancer. Interactions occurring at the level of individual cells may lead to the development of spatial structure which will affect the dynamics of migrating cells at a population level. Models that try to predict population-level behaviour often take a mean-field approach, which assumes that individuals interact with one another in proportion to their average density and ignores the presence of any small-scale spatial structure. In this work, we develop a lattice-free individual-based model (IBM) that uses random walk theory to model the stochastic interactions occurring at the scale of individual migrating cells. We incorporate a mechanism for local directional bias such that an individual's direction of movement is dependent on the degree of cell crowding in its neighbourhood. As an alternative to the mean-field approach, we also employ spatial moment theory to develop a population-level model which accounts for spatial structure and predicts how these individual-level interactions propagate to the scale of the whole population. The IBM is used to derive an equation for dynamics of the second spatial moment (the average density of pairs of cells) which incorporates the neighbour-dependent directional bias, and we solve this numerically for a spatially homogeneous case.

One-dimensional collective migration of a proliferating cell monolayer

The importance of collective cellular migration during embryogenesis and tissue repair asks for a sound understanding of underlying principles and mechanisms. Here, we address recent in vitro experiments on cell monolayers, which show that the advancement of the leading edge relies on cell proliferation and protrusive activity at the tissue margin. Within a simple viscoelastic mechanical model amenable to detailed analysis, we identify a key parameter responsible for tissue expansion, and we determine the dependence of the monolayer velocity as a function of measurable rheological parameters. Our results allow us to discuss the effects of pharmacological perturbations on the observed tissue dynamics.

A data-motivated density-dependent diffusion model of in vitro glioblastoma growth

Glioblastoma multiforme is an aggressive brain cancer that is extremely fatal. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. We propose a single equation which uses density-dependent diffusion to capture the behavior of both proliferation and migration. We analyze the model to determine the existence of traveling wave solutions. To prove the viability of the density-dependent diffusion function chosen, we compare our model with well-known in vitro experimental data.

Calculating the Fickian diffusivity for a lattice-based random walk with agents and obstacles of different shapes and sizes

Random walk models are often used to interpret experimental observations of the motion of biological cells and molecules. A key aim in applying a random walk model to mimic an in vitro experiment is to estimate the Fickian diffusivity (or Fickian diffusion coefficient), D. However, many in vivo experiments are complicated by the fact that the motion of cells and molecules is hindered by the presence of obstacles. Crowded transport processes have been modeled using repeated stochastic simulations in which a motile agent undergoes a random walk on a lattice that is populated by immobile obstacles. Early studies considered the most straightforward case in which the motile agent and the obstacles are the same size. More recent studies considered stochastic random walk simulations describing the motion of an agent through an environment populated by obstacles of different shapes and sizes. Here, we build on previous simulation studies by analyzing a general class of lattice-based random walk models with agents and obstacles of various shapes and sizes. Our analysis provides exact calculations of the Fickian diffusivity, allowing us to draw conclusions about the role of the size, shape and density of the obstacles, as well as examining the role of the size and shape of the motile agent. Since our analysis is exact, we calculate D directly without the need for random walk simulations. In summary, we find that the shape, size and density of obstacles has a major influence on the exact Fickian diffusivity. Furthermore, our results indicate that the difference in diffusivity for symmetric and asymmetric obstacles is significant.

RNAi-based therapeutic nanostrategy: IL-8 gene silencing in pancreatic cancer cells using gold nanorods delivery vehicles

RNA interference (RNAi)-based gene silencing possesses great ability for therapeutic intervention in pancreatic cancer. Among various oncogene mutations, Interleukin-8 (IL-8) gene mutations are found to be overexpressed in many pancreatic cell lines. In this work, we demonstrate IL-8 gene silencing by employing an RNAi-based gene therapy approach and this is achieved by using gold nanorods (AuNRs) for efficient delivery of IL-8 small interfering RNA (siRNA) to the pancreatic cell lines of MiaPaCa-2 and Panc-1. Upon comparing to Panc-1 cells, we found that the dominant expression of the IL-8 gene in MiaPaCa-2 cells resulted in an aggressive behavior towards the processes of cell invasion and metastasis. We have hence investigated the suitability of using AuNRs as novel non-viral nanocarriers for the efficient uptake and delivery of IL-8 siRNA in realizing gene knockdown of both MiaPaCa-2 and Panc-1 cells. Flow cytometry and fluorescence imaging techniques have been applied to confirm transfection and release of IL-8 siRNA. The ratio of AuNRs and siRNA has been optimized and transfection efficiencies as high as 88.40 ± 2.14% have been achieved. Upon successful delivery of IL-8 siRNA into cancer cells, the effects of IL-8 gene knockdown are quantified in terms of gene expression, cell invasion, cell migration and cell apoptosis assays. Statistical comparative studies for both MiaPaCa-2 and Panc-1 cells are presented in this work. IL-8 gene silencing has been demonstrated with knockdown efficiencies of 81.02 ± 10.14% and 75.73 ± 6.41% in MiaPaCa-2 and Panc-1 cells, respectively. Our results are then compared with a commercial transfection reagent, Oligofectamine, serving as positive control. The gene knockdown results illustrate the potential role of AuNRs as non-viral gene delivery vehicles for RNAi-based targeted cancer therapy applications.

Estimating cell diffusivity and cell proliferation rate by interpreting IncuCyte ZOOM™ assay data using the Fisher-Kolmogorov model

Background

Standard methods for quantifying IncuCyte ZOOM™ assays involve measurements that quantify how rapidly the initially-vacant area becomes re-colonised with cells as a function of time. Unfortunately, these measurements give no insight into the details of the cellular-level mechanisms acting to close the initially-vacant area. We provide an alternative method enabling us to quantify the role of cell motility and cell proliferation separately. To achieve this we calibrate standard data available from IncuCyte ZOOM™ images to the solution of the Fisher-Kolmogorov model.

Results

The Fisher-Kolmogorov model is a reaction-diffusion equation that has been used to describe collective cell spreading driven by cell migration, characterised by a cell diffusivity, D, and carrying capacity limited proliferation with proliferation rate, λ, and carrying capacity density, K. By analysing temporal changes in cell density in several subregions located well-behind the initial position of the leading edge we estimate λ and K. Given these estimates, we then apply automatic leading edge detection algorithms to the images produced by the IncuCyte ZOOM™ assay and match this data with a numerical solution of the Fisher-Kolmogorov equation to provide an estimate of D. We demonstrate this method by applying it to interpret a suite of IncuCyte ZOOM™ assays using PC-3 prostate cancer cells and obtain estimates of D, λ and K. Comparing estimates of D, λ and K for a control assay with estimates of D, λ and K for assays where epidermal growth factor (EGF) is applied in varying concentrations confirms that EGF enhances the rate of scratch closure and that this stimulation is driven by an increase in D and λ, whereas K is relatively unaffected by EGF.

Conclusions

Our approach for estimating D, λ and K from an IncuCyte ZOOM™ assay provides more detail about cellular-level behaviour than standard methods for analysing these assays. In particular, our approach can be used to quantify the balance of cell migration and cell proliferation and, as we demonstrate, allow us to quantify how the addition of growth factors affects these processes individually.

Interpreting scratch assays using pair density dynamics and approximate Bayesian computation

Quantifying the impact of biochemical compounds on collective cell spreading is an essential element of drug design, with various applications including developing treatments for chronic wounds and cancer. Scratch assays are a technically simple and inexpensive method used to study collective cell spreading; however, most previous interpretations of scratch assays are qualitative and do not provide estimates of the cell diffusivity, D, or the cell proliferation rate, λ. Estimating D and λ is important for investigating the efficacy of a potential treatment and provides insight into the mechanism through which the potential treatment acts. While a few methods for estimating D and λ have been proposed, these previous methods lead to point estimates of D and λ, and provide no insight into the uncertainty in these estimates. Here, we compare various types of information that can be extracted from images of a scratch assay, and quantify D and λ using discrete computational simulations and approximate Bayesian computation. We show that it is possible to robustly recover estimates of D and λ from synthetic data, as well as a new set of experimental data. For the first time, our approach also provides a method to estimate the uncertainty in our estimates of D and λ. We anticipate that our approach can be generalized to deal with more realistic experimental scenarios in which we are interested in estimating D and λ, as well as additional relevant parameters such as the strength of cell-to-cell adhesion or the strength of cell-to-substrate adhesion.

An agent-based model approach to multi-phase life-cycle for contact inhibited, anchorage dependent cells

Cellular agent-based models are a technique that can be easily adapted to describe nuances of a particular cell type. Within we have concentrated on the cellular particularities of the human Endothelial Cell, explicitly the effects both of anchorage dependency and of heightened scaffold binding on the total confluence time of a system. By expansion of a discrete, homogeneous, asynchronous cellular model to account for several states per cell (phases within a cell's life); we accommodate and track dependencies of confluence time and population dynamics on these factors. Increasing the total motility time, analogous to weakening the binding between lattice and cell, affects the system in unique ways from increasing the average cellular velocity; each degree of freedom allows for control over the time length the system achieves logistic growth and confluence. These additional factors may allow for greater control over behaviors of the system. Examinations of system's dependence on both seed state velocity and binding are also enclosed.

Spatial and temporal dynamics of cell generations within an invasion wave: a link to cell lineage tracing

Mathematical models of a cell invasion wave have included both continuum partial differential equation (PDE) approaches and discrete agent-based cellular automata (CA) approaches. Here we are interested in modelling the spatial and temporal dynamics of the number of divisions (generation number) that cells have undergone by any time point within an invasion wave. In the CA framework this is performed from agent lineage tracings, while in the PDE approach a multi-species generalized Fisher equation is derived for the cell density within each generation. Both paradigms exhibit qualitatively similar cell generation densities that are spatially organized, with agents of low generation number rapidly attaining a steady state (with average generation number increasing linearly with distance) behind the moving wave and with evolving high generation number at the wavefront. This regularity in the generation spatial distributions is in contrast to the highly stochastic nature of the underlying lineage dynamics of the population. In addition, we construct a method for determining the lineage tracings of all agents without labelling and tracking the agents, but through either a knowledge of the spatial distribution of the generations or the number of agents in each generation. This involves determining generation-dependent proliferation probabilities and using these to define a generation-dependent Galton-Watson (GDGW) process. Monte-Carlo simulations of the GDGW process are used to determine the individual lineage tracings. The lineages of the GDGW process are analyzed using Lorenz curves and found to be similar to outcomes generated by direct lineage tracing in CA realizations. This analysis provides the basis for a potentially useful technique for deducing cell lineage data when imaging every cell is not feasible.

Assessing the role of spatial correlations during collective cell spreading

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.

Influence of individual cell motility on the 2D front roughness dynamics of tumour cell colonies

The dynamics of in situ 2D HeLa cell quasi-linear and quasi-radial colony fronts in a standard culture medium is investigated. For quasi-radial colonies, as the cell population increased, a kinetic transition from an exponential to a constant front average velocity regime was observed. Special attention was paid to individual cell motility evolution under constant average colony front velocity looking for its impact on the dynamics of the 2D colony front roughness. From the directionalities and velocity components of cell trajectories in colonies with different cell populations, the influence of both local cell density and cell crowding effects on individual cell motility was determined. The average dynamic behaviour of individual cells in the colony and its dependence on both local spatio-temporal heterogeneities and growth geometry suggested that cell motion undergoes under a concerted cell migration mechanism, in which both a limiting random walk-like and a limiting ballistic-like contribution were involved. These results were interesting to infer how biased cell trajectories influenced both the 2D colony spreading dynamics and the front roughness characteristics by local biased contributions to individual cell motion. These data are consistent with previous experimental and theoretical cell colony spreading data and provide additional evidence of the validity of the Kardar-Parisi-Zhang equation, within a certain range of time and colony front size, for describing the dynamics of 2D colony front roughness.

Are in vitro estimates of cell diffusivity and cell proliferation rate sensitive to assay geometry?

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.

Do pioneer cells exist?

Most mathematical models of collective cell spreading make the standard assumption that the cell diffusivity and cell proliferation rate are constants that do not vary across the cell population. Here we present a combined experimental and mathematical modeling study which aims to investigate how differences in the cell diffusivity and cell proliferation rate amongst a population of cells can impact the collective behavior of the population. We present data from a three-dimensional transwell migration assay that suggests that the cell diffusivity of some groups of cells within the population can be as much as three times higher than the cell diffusivity of other groups of cells within the population. Using this information, we explore the consequences of explicitly representing this variability in a mathematical model of a scratch assay where we treat the total population of cells as two, possibly distinct, subpopulations. Our results show that when we make the standard assumption that all cells within the population behave identically we observe the formation of moving fronts of cells where both subpopulations are well-mixed and indistinguishable. In contrast, when we consider the same system where the two subpopulations are distinct, we observe a very different outcome where the spreading population becomes spatially organized with the more motile subpopulation dominating at the leading edge while the less motile subpopulation is practically absent from the leading edge. These modeling predictions are consistent with previous experimental observations and suggest that standard mathematical approaches, where we treat the cell diffusivity and cell proliferation rate as constants, might not be appropriate.

Cell migration

Cell migration is fundamental to establishing and maintaining the proper organization of multicellular organisms. Morphogenesis can be viewed as a consequence, in part, of cell locomotion, from large-scale migrations of epithelial sheets during gastrulation, to the movement of individual cells during development of the nervous system. In an adult organism, cell migration is essential for proper immune response, wound repair, and tissue homeostasis, while aberrant cell migration is found in various pathologies. Indeed, as our knowledge of migration increases, we can look forward to, for example, abating the spread of highly malignant cancer cells, retarding the invasion of white cells in the inflammatory process, or enhancing the healing of wounds. This article is organized in two main sections. The first section is devoted to the single-cell migrating in isolation such as occurs when leukocytes migrate during the immune response or when fibroblasts squeeze through connective tissue. The second section is devoted to cells collectively migrating as part of multicellular clusters or sheets. This second type of migration is prevalent in development, wound healing, and in some forms of cancer metastasis.

Bioinspired conical copper wire with gradient wettability for continuous and efficient fog collection

Inspired by the efficient fog collection on cactus spines, conical copper wires with gradient wettability are fabricated through gradient electrochemical corrosion and subsequent gradient chemical modification. These dual-gradient copper wires' fog-collection ability is demonstrated to be higher than that of conical copper wires with pure hydrophobic surfaces or pure hydrophilic surfaces, and the underlying mechanism is also analyzed.

Travelling waves for a velocity-jump model of cell migration and proliferation

Cell invasion, characterised by moving fronts of cells, is an essential aspect of development, repair and disease. Typically, mathematical models of cell invasion are based on the Fisher-Kolmogorov equation. These traditional parabolic models cannot be used to represent experimental measurements of individual cell velocities within the invading population since they imply that information propagates with infinite speed. To overcome this limitation we study combined cell motility and proliferation based on a velocity-jump process where information propagates with finite speed. The model treats the total population of cells as two interacting subpopulations: a subpopulation of left-moving cells, L(x,t), and a subpopulation of right-moving cells, R(x,t). This leads to a system of hyperbolic partial differential equations that includes a turning rate, Λ⩾0, describing the rate at which individuals in the population change direction of movement. We present exact travelling wave solutions of the system of partial differential equations for the special case where Λ=0 and in the limit that Λ→∞. For intermediate turning rates, 0<Λ<∞, we analyse the travelling waves using the phase plane and we demonstrate a transition from smooth monotone travelling waves to smooth nonmonotone travelling waves as Λ decreases through a critical value Λcrit. We conclude by providing a qualitative comparison between the travelling wave solutions of our model and experimental observations of cell invasion. This comparison indicates that the small Λ limit produces results that are consistent with experimental observations.

Systems-based approaches toward wound healing

Wound healing in the pediatric patient is of utmost clinical and social importance because hypertrophic scarring can have aesthetic and psychological sequelae, from early childhood to late adolescence. Wound healing is a well-orchestrated reparative response affecting the damaged tissue at the cellular, tissue, organ, and system scales. Although tremendous progress has been made toward understanding wound healing at the individual temporal and spatial scales, its effects across the scales remain severely understudied and poorly understood. Here, we discuss the critical need for systems-based computational modeling of wound healing across the scales, from short-term to long-term and from small to large. We illustrate the state of the art in systems modeling by means of three key signaling mechanisms: oxygen tension-regulating angiogenesis and revascularization; transforming growth factor-β (TGF-β) kinetics controlling collagen deposition; and mechanical stretch stimulating cellular mitosis and extracellular matrix (ECM) remodeling. The complex network of biochemical and biomechanical signaling mechanisms and the multiscale character of the healing process make systems modeling an integral tool in exploring personalized strategies for wound repair. A better mechanistic understanding of wound healing in the pediatric patient could open new avenues in treating children with skin disorders such as birth defects, skin cancer, wounds, and burn injuries.

Quantifying the roles of cell motility and cell proliferation in a circular barrier assay

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.

Cell migration

Cell migration is fundamental to establishing and maintaining the proper organization of multicellular organisms. Morphogenesis can be viewed as a consequence, in part, of cell locomotion, from large-scale migrations of epithelial sheets during gastrulation, to the movement of individual cells during development of the nervous system. In an adult organism, cell migration is essential for proper immune response, wound repair, and tissue homeostasis, while aberrant cell migration is found in various pathologies. Indeed, as our knowledge of migration increases, we can look forward to, for example, abating the spread of highly malignant cancer cells, retarding the invasion of white cells in the inflammatory process, or enhancing the healing of wounds. This article is organized in two main sections. The first section is devoted to the single-cell migrating in isolation such as occurs when leukocytes migrate during the immune response or when fibroblasts squeeze through connective tissue. The second section is devoted to cells collectively migrating as part of multicellular clusters or sheets. This second type of migration is prevalent in development, wound healing, and in some forms of cancer metastasis.

Models of collective cell behaviour with crowding effects: comparing lattice-based and lattice-free approaches

Individual-based models describing the migration and proliferation of a population of cells frequently restrict the cells to a predefined lattice. An implicit assumption of this type of lattice-based model is that a proliferative population will always eventually fill the lattice. Here, we develop a new lattice-free individual-based model that incorporates cell-to-cell crowding effects. We also derive approximate mean-field descriptions for the lattice-free model in two special cases motivated by commonly used experimental set-ups. Lattice-free simulation results are compared with these mean-field descriptions and with a corresponding lattice-based model. Data from a proliferation experiment are used to estimate the parameters for the new model, including the cell proliferation rate, showing that the model fits the data well. An important aspect of the lattice-free model is that the confluent cell density is not predefined, as with lattice-based models, but an emergent model property. As a consequence of the more realistic, irregular configuration of cells in the lattice-free model, the population growth rate is much slower at high cell densities and the population cannot reach the same confluent density as an equivalent lattice-based model.

On the evolution of morphogenetic models: mechano-chemical interactions and an integrated view of cell differentiation, growth, pattern formation and morphogenesis

In the 1950s, embryology was conceptualized as four relatively independent problems: cell differentiation, growth, pattern formation and morphogenesis. The mechanisms underlying the first three traditionally have been viewed as being chemical in nature, whereas those underlying morphogenesis have usually been discussed in terms of mechanics. Often, morphogenesis and its mechanical processes have been regarded as subordinate to chemical ones. However, a growing body of evidence indicates that the biomechanics of cells and tissues affect in striking ways those phenomena often thought of as mainly under the control of cell-cell signalling. This accumulation of data has led to a revival of the mechano-transduction concept in particular, and of complexity in general, causing us now to consider whether we should retain the traditional conceptualization of development. The researchers' semantic preferences for the terms 'patterning', 'pattern formation' or 'morphogenesis' can be used to describe three main 'schools of thought' which emerged in the late 1970s. In the 'molecular school', the term patterning is deeply tied to the positional information concept. In the 'chemical school', the term 'pattern formation' regularly implies reaction-diffusion models. In the 'mechanical school', the term 'morphogenesis' is more frequently used in relation to mechanical instabilities. Major differences among these three schools pertain to the concept of self-organization, and models can be classified as morphostatic or morphodynamic. Various examples illustrate the distorted picture that arises from the distinction among differentiation, growth, pattern formation and morphogenesis, based on the idea that the underlying mechanisms are respectively chemical or mechanical. Emerging quantitative approaches integrate the concepts and methods of complex sciences and emphasize the interplay between hierarchical levels of organization via mechano-chemical interactions. They draw upon recent improvements in mathematical and numerical morphogenetic models and upon considerable progress in collecting new quantitative data. This review highlights a variety of such models, which exhibit important advances, such as hybrid, stochastic and multiscale simulations.

Velocity-jump models with crowding effects

Velocity-jump processes are discrete random-walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity-jump models are often used to represent a type of persistent motion, known as a run and tumble, that is exhibited by some isolated bacteria cells. All previous velocity-jump processes are noninteracting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity-jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion, and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high-cell-density problems using a velocity-jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity-jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.