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 setups. Lattice-free simulation results are compared to these mean-field descriptions and to a corresponding lattice-based model. Data from a proliferation experiment is 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.

Mean-field descriptions of collective migration with strong adhesion

Random walk models based on an exclusion process with contact effects are often used to represent collective migration where individual agents are affected by agent-to-agent adhesion. Traditional mean field representations of these processes take the form of a nonlinear diffusion equation which, for strong adhesion, does not predict the averaged discrete behavior. We propose an alternative suite of mean-field representations, showing that collective migration with strong adhesion can be accurately represented using a moment closure approach.

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.

Fractional models for the migration of biological cells in complex spatial domains

Travelling wave phenomena are observed in many biological applications. Mathematical theory of standard reaction-diffusion problems shows that simple partial differential equations exhibit travelling wave solutions with constant wavespeed and such models are used to describe, for example, waves of chemical concentrations, electrical signals, cell migration, waves of epidemics and population dynamics. However, as in the study of cell motion in complex spatial geometries, experimental data are often not consistent with constant wavespeed. Non-local spatial models have successfully been used to model anomalous diffusion and spatial heterogeneity in different physical contexts. In this paper, we develop a fractional model based on the Fisher-Kolmogoroff equation and analyse it for its wavespeed properties, attempting to relate the numerical results obtained from our simulations to experimental data describing enteric neural crest-derived cells migrating along the intact gut of mouse embryos. The model proposed essentially combines fractional and standard diffusion in different regions of the spatial domain and qualitatively reproduces the behaviour of neural crest-derived cells observed in the caecum and the hindgut of mouse embryos during in vivo experiments.

Lattice-free models of cell invasion : discrete simulations and travelling waves

Invasion waves of cells play an important role in development, disease and repair. Standard discrete models of such processes typically involve simulating cell motility, cell proliferation and cell-to-cell crowding effects in a lattice-based framework. The continuum-limit description is often given by a reaction–diffusion equation that is related to the Fisher–Kolmogorov equation. One of the limitations of a standard lattice-based approach is that real cells move and proliferate in continuous space and are not restricted to a predefined lattice structure. We present a lattice-free model of cell motility and proliferation, with cell-to-cell crowding effects, and we use the model to replicate invasion wave-type behaviour. The continuum-limit description of the discrete model is a reaction–diffusion equation with a proliferation term that is different from lattice-based models. Comparing lattice based and lattice-free simulations indicates that both models lead to invasion fronts that are similar at the leading edge, where the cell density is low. Conversely, the two models make different predictions in the high density region of the domain, well behind the leading edge. We analyse the continuum-limit description of the lattice based and lattice-free models to show that both give rise to invasion wave type solutions that move with the same speed but have very different shapes. We explore the significance of these differences by calibrating the parameters in the standard Fisher–Kolmogorov equation using data from the lattice-free model. We conclude that estimating parameters using this kind of standard procedure can produce misleading results.

Design and interpretation of cell trajectory assays

Cell trajectory data is often reported in the experimental cell biology literature to distinguish between different types of cell migration. Unfortunately, there is no accepted protocol for designing or interpreting such experiments and this makes it difficult to quantitatively compare different published data sets and to understand how changes in experimental design influence our ability to interpret different experiments. Here, we use an individual based mathematical model to simulate the key features of a cell trajectory experiment. This shows that our ability to correctly interpret trajectory data is extremely sensitive to the geometry and timing of the experiment, the degree of motility bias and the number of experimental replicates. We show that cell trajectory experiments produce data that is most reliable when the experiment is performed in a quasi 1D geometry with a large number of identically{prepared experiments conducted over a relatively short time interval rather than few trajectories recorded over particularly long time intervals.

Prognostic value of angiogenesis in operable non-small cell lung cancer

Tumour angiogenesis is an important factor for tumour growth and metastasis. Although some recent reports suggest that microvessel counts in non-small cell lung cancer are related to a poor disease outcome, the results were not conclusive and were not compared with other molecular prognostic markers. In the present study, the vascular grade was assessed in 107 (T1,2-N0,1) operable non-small cell lung carcinomas, using the JC70 monoclonal antibody to CD31. Three vascular grades were defined with appraisal by eye and by Chalkley counting: high (Chalkley score 7-12), medium (5-6), and low (2-4). There was a significant correlation between eye appraisal and Chalkley counting (P < 0.0001). Vascular grade was not related to histology, grade, proliferation index (Ki67), or EGFR or p53 expression. Tumours from younger patients had a higher grade of angiogenesis (P = 0.05). Apart from the vascular grade, none of the other factors examined was statistically related to lymph node metastasis (P < 0.0001). A univariate analysis of survival showed that vascular grade was the most significant prognostic factor (P = 0.0004), followed by N-stage (P = 0.001). In a multivariate analysis, N-stage and vascular grade were not found to be independent prognostic factors, since they were strongly related to each other. Excluding N-stage, vascular grade was the only independent prognostic factor (P = 0.007). Kaplan-Meier survival curves showed a statistically significant worse prognosis for patients with high vascular grade, but no difference was observed between low and medium vascular grade. These data suggest that angiogenesis in operable non-small cell lung cancer is a major prognostic factor for survival and, among the parameters tested, is the only factor related to cancer cell migration to lymph nodes. The integration of vascular grading in clinical trials on adjuvant chemotherapy and/or radiotherapy could substantially contribute in defining groups of operable patients who might benefit from cytotoxic treatment.

Invading edge vs. inner (edvin) patterns of vascularization : an interplay between angiogenic and vascular survival factors defines the clinical behaviour of non-small cell lung cancer

Neo-angiogenesis during neoplastic growth involves endothelial mitogenic and migration stimuli produced by cancer or tumour stromal cells. Although this active angiogenesis takes place in the tumour periphery, the process of vessel growth and survival in inner areas and its clinical role remain largely unexplored. The present study compared the microvessel score (MS) as well as the single endothelial cell score (ECS) in the invading edge and in inner areas of non-small cell lung carcinomas (NSCLCs). Three different patterns of vascular growth were distinguished: the edvin (edge vs. inner) type 1, where a low MS was observed in both peripheral and inner tumour areas; the edvin type 2, where a high MS was noted in the invading front but a low MS in inner areas; and the edvin type 3, where both peripheral and inner tumour areas had a high MS. The ECS was high in the invading edge in edvin type 2 and 3 cases and was sharply decreased in both types in inner areas, suggesting that endothelial cell migration is unlikely to contribute to the angiogenic process in areas away from the tumour front. Expression of the vascular endothelial growth factor (VEGF) and of thymidine phosphorylase (TP) was associated with a high MS in the invading edge. VEGF was associated with a high MS in inner areas (edvin 3), while TP expression was associated with edvin type 2, showing that VEGF (and not TP) contributes to the preservation of the inner vasculature. Both edvin type 2 and 3 cases showed an increased incidence of node metastasis, but edvin type 3 cases had a poorer prognosis, even in the N1-stage group. The present study suggests that tumour factors regulating angiogenesis and vascular survival are not identical. A possible method is reported to quantify these two parameters by comparing the MS in the invading edge and inner areas (edvin types). This observation may contribute to the evaluation of the effectiveness of different therapeutic approaches, namely vascular targeting vs. anti-angiogenesis. Copyright (C) 2000 John Wiley and Sons, Ltd.

bcl-2 and c-erbB-2 proteins are involved in the regulation of VEGF and of thymidine phosphorylase angiogenic activity in non-small-cell lung cancer

Tumour angiogenesis has been recently recognised as one of the most important prognostic factors in lung cancer. Although a variety of angiogenic factors have been identified, the angiogenesis process remains poorly understood. Bcl-2, c-erbB-2 and p53 are well-known oncogenes involved in non- small-cell lung cancer pathogenesis. A direct correlation of thymidine phosphorylase (TP) and of vascular endothelial growth factor (VEGF) with intratumoural angiogenesis has been reported. In the present study we investigated the possible regulatory role if bcl-2, c-erB-2 proteins in angiogenesis and in VEGF and TP expression in non-small-cell lung cancer. Two hundred sixteen specimens from T1,2-NO, 1 staged patients treated with surgery alone were immunohistochemically examined. Bcl-2 and c-erbB-2 were significantly inversely related to each other (P = 0.04) and both were inversely associated with microvessel density (P < 0.02). High TP and VEGF reactivity was statistically related to loss of bcl-2 expression (P < 0.01). A significant co-expression of c-erbB-2 with TP was noted (P = 0.01). However, TP expression was related to high angiogenesis only in cases with absence of c-erB-2 expression (P < 0.0001). c-erbB-2 expression in poorly vascularised tumours was linked with poor outcome (P = 0.03). The present study provides strong evidence that the bcl-2 gene has a suppressive function over genes involved in both angiogenesis (VEGF and TP) and cell migration (c- erbB-2) in NSCLC. TP and c-erbB-2 proteins are significantly, and often simultaneously, expressed in bcl-2 negative cases. However, expression of the c-erbB-2 abolishes the TP-related angiogenic activity. Whether this is a result of a direct activity of the c-erbB-2 protein or a consequence of a c- erbB-2-related immune response remains to be further investigated.

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, ?, 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 ?. 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 - 243 ?m2 hour-1, q = 0.3 - 0.5 (low to moderate strength) and ? = 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.

Comparing methods for modelling spreading cell fronts

Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation; however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of individual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performace of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.

In vitro assessment of migratory behavior of two cell populations in a simple multichannel microdevice

Recent literature suggests that mesenchymal stem/stromal cells (MSC) could be used as Trojan Horses to deliver “death-signals” to cancer cells. Herein, we describe the development of a novel multichannel cell migration device, and use it to investigate the relative migration rates of bone marrow-derived MSC and breast cancer cells (MCF-7) towards each other. Confluent monolayers of MSC and MCF-7 were established in adjacent chambers separated by an array of 14 microchannels. Initially, culture chambers were isolated by air bubbles (air-valves) contained within each microchannel, and then bubbles were displaced to initiate the assay. The MCF-7 cells migrated preferentially towards MSC, whilst the MSC did not migrate preferentially towards the MCF-7 cells. Our results corroborate previous literature that suggests MSC migration towards cancer cells in vivo is in response to the associated inflammation rather than directly to signals secreted by the cancer cells themselves.

Chemotaxis and chemokinesis of human prostate tumor cell lines in response to human prostate stromal cell secretory proteins containing a nerve growth factor-like protein

The migration of three human prostate tumor epithelial cell lines (TSU-pr1, PC-3, DU-145) in response to secreted protein from a human prostate stromal cell line was investigated by using the modified blind-well Boyden chamber assay. Migrated cells were quantified by spectrophotometrically measuring the concentration of crystal violet stain extracted from their nuclei. Cell number was correlated linearly with the concentration of extracted crystal violet stain. All three tumor cell lines showed intrinsic migratory ability in the absence of chemoattractants, such that approximately 1-7% of plated cells migrated across the filter of the Boyden chambers during a 5-h incubation period. Prostate tumor cell migration was significantly enhanced (3-13-fold) in response to stromal cell secretory protein in a dose-dependent manner, whereas bovine serum albumin had no effect on stimulating tumor cell migration. Immunoprecipitation of the stromal cell secreted protein with a nerve growth factor antibody partially and significantly reduced its stimulatory activity for tumor cell migration. A Zigmond-Hirsch matrix assay of tumor cell migration in response to various concentration gradients of stromal cell secreted protein demonstrated both chemotaxis and chemokinesis by all three cell lines. These results are consistent with the stromal cell secretory protein stimulation of chemokinetic tumor cell migration through the capsule of the prostate. Outside of the prostate gland metastasis of tumor cells may occur by chemotaxis to preferential sites containing chemoattractants similar to or related to maintenance factors that can substitute for components of stromal cell secretory protein.

Characterization and novel activation of 72-kDa metalloproteinase in retinal interphotoreceptor matrix and Y-79 cell culture medium

Analysis of bovine interphotoreceptor matrix and conditioned medium from human Y-79 retinoblastoma cells by gelatin SDS-PAGE zymography reveals abundant activity of a 72-kDa M(r) gelatinase. The 72-kDa gelatinase from either source is inhibited by EDTA but not aprotinin or NEM, indicating that it is a metalloproteinase (MMP). The 72-kDa MMP is converted to a 62-kDa species with APMA treatment after gelatin sepharose affinity purification typical of previously described gelatinase MMP-2. The latent 72-kDa gelatinase from either bovine IPM or Y-79 media autoactivates without APMA in the presence of calcium and zinc after 72 hr at 37°C, producing a fully active mixture of proteinase species, 50 (48 in Y-79 medium), 38 and 35 kDa in size. The presence of inhibitory activity was examined in both whole bovine IPM and IPM fractions separated by SDS-PAGE. Whole IPM inhibited gelatinolytic activity of autoactivated Y-79-derived MMP in a dose-dependent manner. Inhibitory activities are observed in two protein fractions of 27-42 and 20-25 kDa. Western blots using antibodies to human tissue inhibitor of metalloproteinase 1 and 2 (TIMP-1 and -2) reveal the presence of two TIMP-1-like proteins at 21 and 29 kDa in inhibitory fractions of the bovine IPM. TIMP-2 was not detected in the inhibitory IPM fractions, consistent with the observed autoactivation of bovine IPM 72-kDa gelatinase. Potential roles for this IPM MMP-TIMP system include physiologic remodelling of the neural retina-RPE cell interface and digestion of shed rod outer segment as well as pathological processes such as retinal detachment, PE cell migration, neovascularization and tumor progression. Cultured Y-79 cells appear to be a good model for studying the production and regulation of this proteinase system.

Elevated YKL40 is associated with advanced prostate cancer (PCa) and positively regulates invasion and migration of PCa cells

Chitinase 3-like 1 (CHI3L1 or YKL40) is a secreted glycoprotein highly expressed in tumours from patients with advanced stage cancers, including prostate cancer (PCa). The exact function of YKL40 is poorly understood, but it has been shown to play an important role in promoting tumour angiogenesis and metastasis. The therapeutic value and biological function of YKL40 are unknown in PCa. The objective of this study was to examine the expression and function of YKL40 in PCa. Gene expression analysis demonstrated that YKL40 was highly expressed in metastatic PCa cells when compared with less invasive and normal prostate epithelial cell lines. In addition, the expression was primarily limited to androgen receptor-positive cell lines. Evaluation of YKL40 tissue expression in PCa patients showed a progressive increase in patients with aggressive disease when compared with those with less aggressive cancers and normal controls. Treatment of LNCaP and C4-2B cells with androgens increased YKL40 expression, whereas treatment with an anti-androgen agent decreased the gene expression of YKL40 in androgen-sensitive LNCaP cells. Furthermore, knockdown of YKL40 significantly decreased invasion and migration of PCa cells, whereas overexpression rendered them more invasive and migratory, which was commensurate with an enhancement in the anchorage-independent growth of cells. To our knowledge, this study characterises the role of YKL40 for the first time in PCa. Together, these results suggest that YKL40 plays an important role in PCa progression and thus inhibition of YKL40 may be a potential therapeutic strategy for the treatment of PCa.