Catherine J. Penington

Reproducibility of scratch assays is affected by the initial degree of confluence: Experiments, modelling and model selection.

Scratch assays are difficult to reproduce. Here we identify a previously overlooked source of variability which could partially explain this difficulty. We analyse a suite of scratch assays in which we vary the initial degree of confluence (initial cell density). Our results indicate that the rate of re-colonisation is very sensitive to the initial density. To quantify the relative roles of cell migration and proliferation, we calibrate the solution of the Fisher-Kolmogorov model to cell density profiles to provide estimates of the cell diffusivity, D, and the cell proliferation rate, λ. This procedure indicates that the estimates of D and λ are very sensitive to the initial density. This dependence suggests that the Fisher-Kolmogorov model does not accurately represent the details of the collective cell spreading process, since this model assumes that D and λ are constants that ought to be independent of the initial density. Since higher initial cell density leads to enhanced spreading, we also calibrate the solution of the Porous-Fisher model to the data as this model assumes that the cell flux is an increasing function of the cell density. Estimates of D and λ associated with the Porous-Fisher model are less sensitive to the initial density, suggesting that the Porous-Fisher model provides a better description of the experiments.

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 rates of cell migration and cell proliferation in co-culture barrier assays reveals how skin and melanoma cells interact during melanoma spreading and invasion

Malignant spreading involves the migration of cancer cells amongst other native cell types. For example, in vivo melanoma invasion involves individual melanoma cells migrating through native skin, which is composed of several distinct subpopulations of cells. Here, we aim to quantify how interactions between melanoma and fibroblast cells affect the collective spreading of a heterogeneous population of these cells in vitro. We perform a suite of circular barrier assays that includes: (i) monoculture assays with fibroblast cells; (ii) monoculture assays with SK-MEL-28 melanoma cells; and (iii) a series of co-culture assays initiated with three different ratios of SK-MEL-28 melanoma cells and fibroblast cells. Using immunostaining, detailed cell density histograms are constructed to illustrate how the two subpopulations of cells are spatially arranged within the spreading heterogeneous population. Calibrating the solution of a continuum partial differential equation to the experimental results from the monoculture assays allows us to estimate the cell diffusivity and the cell proliferation rate for the melanoma and the fibroblast cells, separately. Using the parameter estimates from the monoculture assays, we then make a prediction of the spatial spreading in the co-culture assays. Results show that the parameter estimates obtained from the monoculture assays lead to a reasonably accurate prediction of the spatial arrangement of the two subpopulations in the co-culture assays. Overall, the spatial pattern of spreading of the melanoma cells and the fibroblast cells is very similar in monoculture and co-culture conditions. Therefore, we find no clear evidence of any interactions other than cell-to-cell contact and crowding effects.

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.

A computational modelling framework to quantify the effects of passaging cell lines

In vitro cell culture is routinely used to grow and supply a sufficiently large number of cells for various types of cell biology experiments. Previous experimental studies report that cell characteristics evolve as the passage number increases, and various cell lines can behave differently at high passage numbers. To provide insight into the putative mechanisms that might give rise to these differences, we perform in silico experiments using a random walk model to mimic the in vitro cell culture process. Our results show that it is possible for the average proliferation rate to either increase or decrease as the passaging process takes place, and this is due to a competition between the initial heterogeneity and the degree to which passaging damages the cells. We also simulate a suite of scratch assays with cells from near–homogeneous and heterogeneous cell lines, at both high and low passage numbers. Although it is common in the literature to report experimental results without disclosing the passage number, our results show that we obtain significantly different closure rates when performing in silico scratch assays using cells with different passage numbers. Therefore, we suggest that the passage number should always be reported to ensure that the experiment is as reproducible as possible. Furthermore, our modelling also suggests some avenues for further experimental examination that could be used to validate or refine our simulation results.

Continuum approximations for lattice-free multi-species models of collective cell migration

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 individual 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 individual 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.

Exits in order: How crowding affects particle lifetimes

Diffusive processes are often represented using stochastic random walk frameworks. The amount of time taken for an individual 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.

Scaling laws for localised states in a nonlocal amplitude equation

It is well known that, although a uniform magnetic field inhibits the onset of small amplitude thermal convection in a layer of fluid heated from below, isolated convection cells may persist if the fluid motion within them is sufficiently vigorous to expel magnetic flux. Such fully nonlinear (“convecton”) solutions for magnetoconvection have been investigated by several authors. Here we explore a model amplitude equation describing this separation of a fluid layer into a vigorously convecting part and a magnetically-dominated part at rest. Our analysis elucidates the origin of the scaling laws observed numerically to form the boundaries in parameter space of the region of existence of these localised states, and importantly, for the lowest thermal forcing required to sustain them.