Parvathi Haridas

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.

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.

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.

Spatial structure arising from neighbour-dependent bias in collective cell movement

Mathematical models of collective cell movement often neglect the effects of spatial structure, such as clustering, on the population dynamics. Typically, they assume that individuals interact with one another in proportion to their average density (the mean-field assumption) which means that cell–cell interactions occurring over short spatial ranges are not accounted for. However, in vitro cell culture studies have shown that spatial correlations can play an important role in determining collective behaviour. Here, we take a combined experimental and modelling approach to explore how individual-level interactions give rise to spatial structure in a moving cell population. Using imaging data from in vitro experiments, we quantify the extent of spatial structure in a population of 3T3 fibroblast cells. To understand how this spatial structure arises, we develop a lattice-free individual-based model (IBM) and simulate cell movement in two spatial dimensions. Our model allows an individual’s direction of movement to be affected by interactions with other cells in its neighbourhood, providing insights into how directional bias generates spatial structure. We consider how this behaviour scales up to the population level by using the IBM to derive a continuum description in terms of the dynamics of spatial moments. In particular, we account for spatial correlations between cells by considering dynamics of the second spatial moment (the average density of pairs of cells). Our numerical results suggest that the moment dynamics description can provide a good approximation to averaged simulation results from the underlying IBM. Using our in vitro data, we estimate parameters for the model and show that it can generate similar spatial structure to that observed in a 3T3 fibroblast cell population.

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.

Quantitative comparison of the spreading and invasion of radial growth phase and metastatic melanoma cells in a three-dimensional human skin equivalent model

Background Standard two-dimensional (2D) cell migration assays do not provide information about vertical invasion processes, which are critical for melanoma progression. We provide information about three-dimensional (3D) melanoma cell migration, proliferation and invasion in a 3D melanoma skin equivalent (MSE) model. In particular, we pay careful attention to compare the structure of the tissues in the MSE with similarly-prepared 3D human skin equivalent (HSE) models. The HSE model is identically prepared to the MSE model except that melanoma cells are omitted. Using the MSE model, we examine melanoma migration, proliferation and invasion from two different human melanoma cell lines. One cell line, WM35, is associated with the early phase of the disease where spreading is thought to be confined to the epidermis. The other cell line, SK-MEL-28, is associated with the later phase of the disease where spreading into the dermis is expected. Methods 3D MSE and HSE models are constructed using human de-epidermised dermis (DED) prepared from skin tissue. Primary fibroblasts and primary keratinocytes are used in the MSE and HSE models to ensure the formation of a stratified epidermis, with a well-defined basement membrane. Radial spreading of cells across the surface of the HSE and MSE models is observed. Vertical invasion of melanoma cells downward through the skin is observed and measured using immunohistochemistry. All measurements of invasion are made at day 0, 9, 15 and 20, providing detailed time course data. Results Both HSE and MSE models are similar to native skin in vivo, with a well-defined stratification of the epidermis that is separated from the dermis by a basement membrane. In the HSE and MSE we find fibroblast cells confined to the dermis, and differentiated keratinocytes in the epidermis. In the MSE, melanoma cells form colonies in the epidermis during the early part of the experiment. In the later stage of the experiment, the melanoma cells in the MSE invade deeper into the tissues. Interestingly, both the WM35 and SK-MEL-28 melanoma cells lead to a breakdown of the basement membrane and eventually enter the dermis. However, these two cell lines invade at different rates, with the SK-MEL-28 melanoma cells invading faster than the WM35 cells. Discussion The MSE and HSE models are a reliable platform for studying melanoma invasion in a 3D tissue that is similar to native human skin. Interestingly, we find that the WM35 cell line, that is thought to be associated with radial spreading only, is able to invade into the dermis. The vertical invasion of melanoma cells into the dermal region appears to be associated with a localised disruption of the basement membrane. Presenting our results in terms of time course data, along with images and quantitative measurements of the depth of invasion extends previous 3D work that has often been reported without these details.

A Bayesian sequential learning framework to parameterise continuum models of melanoma invasion into human skin

We present a novel framework to parameterise a mathematical model of cell invasion that describes how a population of melanoma cells invades into human skin tissue. Using simple experimental data extracted from complex experimental images, we estimate three model parameters:(i) the melanoma cell proliferation rate, λ; (ii) the melanoma cell diffusivity, D; and (iii) δ, a constant that determines the rate that melanoma cells degrade the skin tissue. The Bayesian sequential learning frame-work involves a sequence of increasingly-sophisticated experimental data from:(i) a spatially uniform cell proliferation assay; (ii) a two-dimensional circular barrier assay; and, (iii) a three-dimensional invasion assay. The Bayesian sequential learning approach leads to well-defined parameter estimates. In contrast, taking a naive approach that attempts to estimate all parameters from a single set of images from the same experiment fails to produce meaningful results. Overall our approach to inference is simple-to-implement, computationally efficient, and well-suited for many cell biology phenomena that can be described by low dimensional continuum models using ordinary differential equations and partial differential equations. We anticipate that this Bayesian sequential learning framework will be relevant in other biological contexts where it is challenging to extract detailed, quantitative biological measurements from experimental images and so we must rely on using relatively simple measurements from complex images.

Three-dimensional experiments and individual based simulations show that cell proliferation drives melanoma nest formation in human skin tissue

BackgroundMelanoma can be diagnosed by identifying nests of cells on the skin surface. Understanding the processes that drive nest formation is important as these processes could be potential targets for new cancer drugs. Cell proliferation and cell migration are two potential mechanisms that could conceivably drive melanoma nest formation. However, it is unclear which one of these two putative mechanisms plays a dominant role in driving nest formation.ResultsWe use a suite of three-dimensional (3D) experiments in human skin tissue and a parallel series of 3D individual-based simulations to explore whether cell migration or cell proliferation plays a dominant role in nest formation. In the experiments we measure nest formation in populations of irradiated (non-proliferative) and non-irradiated (proliferative) melanoma cells, cultured together with primary keratinocyte and fibroblast cells on a 3D experimental human skin model. Results show that nest size depends on initial cell number and is driven primarily by cell proliferation rather than cell migration.ConclusionsNest size depends on cell number, and is driven primarily by cell proliferation rather than cell migration. All experimental results are consistent with simulation data from a 3D individual based model (IBM) of cell migration and cell proliferation.

Standard melanoma-associated markers do not identify the MM127 metastatic melanoma cell line

Methods: We conducted a retrospective collaborative study involving centers from 11 countries and 10 institutions in the US. One hundred and four ASNs were analyzed by Mass Spectrometry Imaging. We divided the patients into 4 clinical groups with Groups 1 to 4 representing best to worst clinical behavior. We then correlated the clinical outcome with the Mass Spectrometry Imaging and histopathologic diagnosis.