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.

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.

Multiple types of data are required to identify the mechanisms influencing the spatial expansion of melanoma cell colonies.

BACKGROUND: The expansion of cell colonies is driven by a delicate balance of several mechanisms including cell motility, cell-to-cell adhesion and cell proliferation. New approaches that can be used to independently identify and quantify the role of each mechanism will help us understand how each mechanism contributes to the expansion process. Standard mathematical modelling approaches to describe such cell colony expansion typically neglect cell-to-cell adhesion, despite the fact that cell-to-cell adhesion is thought to play an important role. RESULTS: We use a combined experimental and mathematical modelling approach to determine the cell diffusivity, D, cell-to-cell adhesion strength, q, and cell proliferation rate, λ, in an expanding colony of MM127 melanoma cells. Using a circular barrier assay, we extract several types of experimental data and use a mathematical model to independently estimate D, q and λ. In our first set of experiments, we suppress cell proliferation and analyse three different types of data to estimate D and q. We find that standard types of data, such as the area enclosed by the leading edge of the expanding colony and more detailed cell density profiles throughout the expanding colony, does not provide sufficient information to uniquely identify D and q. We find that additional data relating to the degree of cell-to-cell clustering is required to provide independent estimates of q, and in turn D. In our second set of experiments, where proliferation is not suppressed, we use data describing temporal changes in cell density to determine the cell proliferation rate. In summary, we find that our experiments are best described using the range D=161-243µm2 hour-1, q=0.3-0.5 (low to moderate strength) and λ=0.0305-0.0398 hour-1, and with these parameters we can accurately predict the temporal variations in the spatial extent and cell density profile throughout the expanding melanoma cell colony. CONCLUSIONS: Our systematic approach to identify the cell diffusivity, cell-to-cell adhesion strength and cell proliferation rate highlights the importance of integrating multiple types of data to accurately quantify the factors influencing the spatial expansion of melanoma cell colonies.

Sensitivity of edge detection methods for quantifying cell migration assays.

Quantitative imaging methods to analyze cell migration assays are not standardized. Here we present a suite of two-dimensional barrier assays describing the collective spreading of an initially-confined population of 3T3 fibroblast cells. To quantify the motility rate we apply two different automatic image detection methods to locate the position of the leading edge of the spreading population after 24, 48 and 72 hours. These results are compared with a manual edge detection method where we systematically vary the detection threshold. Our results indicate that the observed spreading rates are very sensitive to the choice of image analysis tools and we show that a standard measure of cell migration can vary by as much as 25% for the same experimental images depending on the details of the image analysis tools. Our results imply that it is very difficult, if not impossible, to meaningfully compare previously published measures of cell migration since previous results have been obtained using different image analysis techniques and the details of these techniques are not always reported. Using a mathematical model, we provide a physical interpretation of our edge detection results. The physical interpretation is important since edge detection algorithms alone do not specify any physical measure, or physical definition, of the leading edge of the spreading population. Our modeling indicates that variations in the image threshold parameter correspond to a consistent variation in the local cell density. This means that varying the threshold parameter is equivalent to varying the location of the leading edge in the range of approximately 1-5% of the maximum cell density.

Experimental and modelling investigation of monolayer development with clustering.

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 individuals 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 individuals, pairs of individuals, triplets of individuals, 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.

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.

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.