A generalized transport model for biased cell migration in an anisotropic environment

Semi-autonomous wound invasion via matrix-deposited, haptotactic cues

Proper wound healing relies on invasion of fibroblasts via directed migration. While the related experimental and mathematical modeling literature has mainly focused on cell migration directed by soluble cues (chemotaxis), there is ample evidence that fibroblast migration is also directed by insoluble, matrix-bound cues (haptotaxis). Furthermore, numerous studies indicate that fibronectin (FN), a haptotactic ligand for fibroblasts, is present and dynamic in the provisional matrix throughout the proliferative phase of wound healing. In the present work, we show the plausibility of a hypothesis that fibroblasts themselves form and maintain haptotactic gradients in a semi-autonomous fashion. As a precursor to this, we examine the positive control scenario where FN is pre-deposited in the wound matrix, and fibroblasts maintain haptotaxis by removing FN at an appropriate rate. After developing conceptual and quantitative understanding of this scenario, we consider two cases in which fibroblasts activate the latent form of a matrix-loaded cytokine, TGFβ, which upregulates the fibroblasts' own secretion of FN. In the first of these, the latent cytokine is pre-patterned and released by the fibroblasts. In the second, fibroblasts in the wound produce the latent TGFβ, with the presence of the wound providing the only instruction. In all cases, wound invasion is more effective than a negative control model with haptotaxis disabled; however, there is a trade-off between the degree of fibroblast autonomy and the rate of invasion.

Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis

Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.

Phagocytic glioblastoma-associated microglia and macrophages populate invading pseudopalisades

Hypoxic pseudopalisades are a pathological hallmark of human glioblastoma, which is linked to tumour malignancy and aggressiveness. Yet, their function and role in the tumour development have scarcely been explored. It is thought that pseudopalisades are formed by malignant cells escaping from the hypoxic environment, although evidence of the immune component of pseudopalisades has been elusive. In the present work, we analyse the immunological constituent of hypoxic pseudopalisades using high-resolution three-dimensional confocal imaging in tissue blocks from excised tumours of glioblastoma patients and mimic the hypoxic gradient in microfluidic platforms in vitro to understand the cellular motility. We visualize that glioblastoma-associated microglia and macrophages abundantly populate pseudopalisades, displaying an elongated kinetic morphology across the pseudopalisades, and are oriented towards the necrotic focus. In vitro experiments demonstrate that under hypoxic gradient, microglia show a particular motile behaviour characterized by the increase of cellular persistence in contrast with glioma cells. Importantly, we show that glioblastoma-associated microglia and macrophages utilize fibres of glioma cells as a haptotactic cue to navigate along the anisotropic structure of the pseudopalisades and display a high phagocytic activity at the necrotic border of the pseudopalisades. In this study, we demonstrate that glioblastoma-associated microglia and macrophages are the main immune cells of pseudopalisades in glioblastoma, travelling to necrotic areas to clear the resulting components of the prothrombotic milieu, suggesting that the scavenging features of glioblastoma-associated microglia and macrophages at the pseudopalisades serve as an essential counterpart for glioma cell invasion.

Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.

Simple mechanical cues could explain adipose tissue morphology

The mechanisms by which organs acquire their functional structure and realize its maintenance (or homeostasis) over time are still largely unknown. In this paper, we investigate this question on adipose tissue. Adipose tissue can represent 20 to 50% of the body weight. Its investigation is key to overcome a large array of metabolic disorders that heavily strike populations worldwide. Adipose tissue consists of lobular clusters of adipocytes surrounded by an organized collagen fiber network. By supplying substrates needed for adipogenesis, vasculature was believed to induce the regroupment of adipocytes near capillary extremities. This paper shows that the emergence of these structures could be explained by simple mechanical interactions between the adipocytes and the collagen fibers. Our assumption is that the fiber network resists the pressure induced by the growing adipocytes and forces them to regroup into clusters. Reciprocally, cell clusters force the fibers to merge into a well-organized network. We validate this hypothesis by means of a two-dimensional Individual Based Model (IBM) of interacting adipocytes and extra-cellular-matrix fiber elements. The model produces structures that compare quantitatively well to the experimental observations. Our model seems to indicate that cell clusters could spontaneously emerge as a result of simple mechanical interactions between cells and fibers and surprisingly, vasculature is not directly needed for these structures to emerge.

Emergence of spatial patterns in a mathematical model for the co-culture dynamics of epithelial-like and mesenchymal-like cells

Accumulating evidence indicates that the interaction between epithelial and mesenchymal cells plays a pivotal role in cancer development and metastasis formation. Here we propose an integro-differential model for the co-culture dynamics of epithelial-like and mesenchymal-like cells. Our model takes into account the effects of chemotaxis, adhesive interactions between epithelial-like cells, proliferation and competition for nutrients. We present a sample of numerical results which display the emergence of spots, stripes and honeycomb patterns, depending on parameters and initial data. These simulations also suggest that epithelial-like and mesenchymal-like cells can segregate when there is little competition for nutrients. Furthermore, our computational results provide a possible explanation for how the concerted action between epithelial-cell adhesion and mesenchymal-cell spreading can precipitate the formation of ring-like structures, which resemble the fibrous capsules frequently observed in hepatic tumours.

Nanotopography enhanced mobility determines mesenchymal stem cell distribution on micropatterned semiconductors bearing nanorough areas

Surface micropatterns are relevant instruments for the in vitro analysis of cell cultures in non-conventional planar conditions. In this work, two semiconductors (Si and TiO2) have been micropatterned by combined ion-beam/chemical-etching processes leading to selective areas bearing nanorough features. A preferential affinity of human mesenchymal stem cells (hMSCs) for planar areas versus nanotopographic ones is observed. Fluorescence microscopy after β-catenin staining suggests that hMSCs adhesion is inhibited on nanostructured porous silicon areas. This has a direct impact in the development of actin fibers and suggests different cell migration mechanisms on the materials of a micropattern. hMSCs organization on nanotopographic micropatterns has been modeled by using a simplified random walk approach. The model attributes preferential cell mobilities on the nanotopographic areas with respect to the planar and considers purely stochastic movement with no inertial term. Simulations of the cell distribution have been run on 1D and 2D micropatterns and compared with the real hMSC cultures. The simulations allow defining two regimes for cell organization as a function of cell density. hMSCs ordering on planar areas is diffusion-induced in most micropatterns but constriction forced disorder appears for high cell densities. The relative mobility on the planar versus nanotopographic areas can be used as a quality indicator of the nanotopography contrasts in the diffusion induced ordering regime. It is shown that the relative mobility is favorable for the TiO2 versus the Si based system, and allows envisaging its use for the calibrated design of nanotopography based micropatterned materials.

A force based model of individual cell migration with discrete attachment sites and random switching terms

A force based model of cell migration is presented which gives new insight into the importance of the dynamics of cell binding to the substrate. The main features of the model are the focus on discrete attachment dynamics, the treatment of the cellular forces as springs, and an incorporation of the stochastic nature of the attachment sites. One goal of the model is to capture the effect of the random binding and unbinding of cell attachments on global cell motion. Simulations reveal one of the most important factor influencing cell speed is the duration of the attachment to the substrate. The model captures the correct velocity and force relationships for several cell types.

The Effect of Exogenous Zinc Concentration on the Responsiveness of MC3T3-E1 Pre-Osteoblasts to Surface Microtopography: Part I (Migration)

Initial cell-surface interactions are guided by the material properties of substrate topography. To examine if these interactions are also modulated by the presence of zinc, we seeded murine pre-osteoblasts (MC3T3-E1, subclone 4) on micropatterned polydimethylsiloxane (PDMS) containing wide (20 µm width, 30 µm pitch, 2 µm height) or narrow (2 µm width, 10 µm pitch, 2 µm height) ridges, with flat PDMS and tissue culture polystyrene (TC) as controls. Zinc concentration was adjusted to mimic deficient (0.23 µM), serum-level (3.6 µM), and zinc-rich (50 µM) conditions. Significant differences were observed in regard to cell morphology, motility, and contact guidance. We found that cells exhibited distinct anisotropic migration on the wide PDMS patterns under either zinc-deprived (0.23 µM) or serum-level zinc conditions (3.6 µM). However, this effect was absent in a zinc-rich environment (50 µM). These results suggest that the contact guidance of pre-osteoblasts may be partly influenced by trace metals in the microenvironment of the extracellular matrix.

On some models for cancer cell migration through tissue networks

We propose some models allowing to account for relevant processes at the various scales of cancer cell migration through tissue, ranging from the receptor dynamics on the cell surface over degradation of tissue fibers by protease and soluble ligand production towards the behavior of the entire cell population. For a genuinely mesoscopic version of these models we also provide a result on the local existence and uniqueness of a solution for all biologically relevant space dimensions.

Modelling cell migration strategies in the extracellular matrix

The extracellular matrix (ECM) is a highly organised structure with the capacity to direct cell migration through their tendency to follow matrix fibres, a process known as contact guidance. Amoeboid cell populations migrate in the ECM by making frequent shape changes and have minimal impact on its structure. Mesenchymal cells actively remodel the matrix to generate the space in which they can move. In this paper, these different types of movement are studied through simulation of a continuous transport model. It is shown that the process of contact guidance in a structured ECM can spatially organise cell populations. Furthermore, when combined with ECM remodelling, it can give rise to cellular pattern formation in the form of "cell-chains" or networks without additional environmental cues such as chemoattractants. These results are applied to a simple model for tumour invasion where it is shown that the interactions between invading cells and the ECM structure surrounding the tumour can have a profound impact on the pattern and rate of cell infiltration, including the formation of characteristic "fingering" patterns. The results are further discussed in the context of a variety of relevant processes during embryonic and adult stages.

Competitive coexistence in stoichiometric chaos

Classical predator-prey models, such as Lotka-Volterra, track the abundance of prey, but ignore its quality. Yet, in the past decade, some new and occasionally counterintuitive effects of prey quality on food web dynamics emerged from both experiments and mathematical modeling. The underpinning of this work is the theory of ecological stoichiometry that is centered on the fact that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P). The ratios of these elements can vary within and among species, providing simple ways to represent prey quality as its C:N or C:P ratios. When these ratios modeled to vary, as they frequently do in nature, seemingly paradoxical results can arise such as the extinction of a predator that has an abundant and accessible prey. Here, for the first time, we show analytically that the reduction in prey quality can give rise to chaotic oscillations. In particular, when competing predators differ in their sensitivity to prey quality then all species can coexist via chaotic fluctuations. The chaos generating mechanism is based on the existence of a junction-fold point on the nullcline surfaces of the species. Conditions on parameters are found for such a point, and the singular perturbation method and the kneading sequence analysis are used to demonstrate the existence of a period-doubling cascade to chaos as a result of the point.

New method for modeling connective-tissue cell migration: improved accuracy on motility parameters

Mathematical models of cell migration based on persistent random walks have been successfully applied to describe the motility of several cell types. However, the migration of slowly moving connective-tissue cells, such as fibroblasts, is difficult to observe experimentally and difficult to describe theoretically. We identify two primary sources of this difficulty. First, cells such as fibroblasts tend to migrate slowly and change shape during migration. This makes accurate determination of cell position difficult. Second, the cell population is considerably heterogeneous with respect to cell speed. Here we develop a method for fitting connective-tissue cell migration data to persistent random walk models, which accounts for these two significant sources of error and enables accurate determination of the cell motility parameters. We demonstrate the usefulness of this method for modeling both isotropic cell motility and biased cell motility, where the migration of a population of cells is influenced by a gradient in a surface-bound adhesive peptide. This method can discern differences in the motility of populations of cells at different points along the peptide gradient and can therefore be used as a tool to quantify the effects of peptide concentration and gradient magnitude on cell migration.

M5 mesoscopic and macroscopic models for mesenchymal motion

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.

Deterministic model of dermal wound invasion incorporating receptor-mediated signal transduction and spatial gradient sensing

During dermal wound healing, platelet-derived growth factor (PDGF) serves as both a chemoattractant and mitogen for fibroblasts, potently stimulating their invasion of the fibrin clot over a period of several days. A mathematical model of this process is presented, which accurately accounts for the sensitivity of PDGF gradient sensing through PDGF receptor/phosphoinositide 3-kinase-mediated signal transduction. Analysis of the model suggests that PDGF receptor-mediated endocytosis and degradation of PDGF allows a constant PDGF concentration profile to be maintained at the leading front of the fibroblast density profile as it propagates, at a constant rate, into the clot. Thus, the constant PDGF gradient can span the optimal concentration range for asymmetric phosphoinositide 3-kinase signaling and fibroblast chemotaxis, with near-maximal invasion rates elicited over a relatively broad range of PDGF secretion rates. A somewhat surprising finding was that extremely sharp PDGF gradients do not necessarily stimulate faster progression through the clot, because maintaining such a gradient through PDGF consumption is a potentially rate-limiting process.

Composition options for tissue-engineered bone

The logical assembly of tissue-engineered bone is ultimately directed by the clinical status of the patient. The basic elements for tissue-engineered bone should include signaling molecules, cells, and extracellular matrix. The assembly of these basic elements may need to be modified by tissue engineers to account for patient variables of age, gender, health, systemic conditions, habits, and anatomical implant. Moreover, different regions of the body will have different functional loads and vascularity. This review discusses several basic options that may be necessary to engineer bone, including spatial and temporal assembly of signaling factors, cells, and biomimetic extracellular matrices. Moreover, the importance of the health care status of the patient who may be receiving the tissue-engineered composition is emphasized.

'Reverse gear' cellular movement mediated by chemokines

We sought to model the mechanism by which leucocytes may be actively repulsed by a beta-chemokine signal. This model is used to interpret an apparent paradox in chemokine biology, whereby high levels of a T-cell chemoattractant, stromal cell derived factor-1 (SDF-1), are present in bone marrow and thymic tissues despite a paucity of mature T lymphocytes in these areas. We postulate the differential involvement in cell migration of the two binding sites on SDF-1 for its sole receptor, CXCR4, depending on whether high or low concentrations of SDF-1 are encountered by the cell. Site choice would be mediated by divergent affinities of the two binding interactions. We also propose differential signalling following SDF-1/CXCR4 interactions on the plasma membrane versus ligand/receptor complexes in endocytic vesicles. Preliminary data showing divergent susceptibility to kinase inhibitors depending on whether a cell is attracted to or repulsed by SDF-1, are consistent with this model. In terms of physical movement toward or away from a chemokine gradient, we compare the cycling of surface receptors during migration to the caterpillar drive of a tractor, which can change direction simply by altering the direction of rotation of its threads. Finally, the potential clinical implications of concentration-dependent, chemokine-based cell attraction and repulsion are discussed.