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

Understanding the regulation of chronic wounds by tissue inhibitors of matrix metalloproteinases through mathematical modelling

Understanding the biochemistry and pharmacodynamics of chronic wounds is of key importance, due to the millions of people in the UK affected and the significant cost to the NHS. Chronic wounds are characterised by elevated concentrations of matrix metalloproteinases (MMPs) that destroy the surrounding extracellular matrix (ECM). However, fibroblasts can produce tissue inhibitors of MMPs (TIMPs) in order to regulate wound healing. Therefore, the role of TIMPs in both acute and chronic wounds needs to be properly understood in order to develop therapeutic treatments. In this work, we propose a reaction-diffusion system of four partial differential equations that describe the interaction of the ECM, fibroblasts, MMPs, and TIMPs in a wound. We observe that, subject to parameter sets corresponding to both acute and chronic wound healing, this mathematical model gives rise to travelling wave solutions. Using bifurcation analysis, we demonstrate that excessive degradation of the ECM results in the emergence of chronic wounds, and the reversal of these chronic wounds is prohibited for lower TIMP production values. These results are replicated within a simplified model obtained via a parameter sensitivity analysis. This model is further extended to more realistic spatial domains where we demonstrate the effectiveness of a therapeutic hydrogel containing TIMPs as a treatment for chronic wounds.

Genesis of intimal thickening due to hemodynamical shear stresses

This paper investigates intimal growth in arteries, induced by hemodynamical shear stress, through finite element simulation using the FEniCS computational environment. In our model, the growth of the intima depends on cross-section geometry and shear stress. In this work, the arterial wall is modeled as three distinct layers: the intima, the media and the adventitia, each with different mechanical properties. We assume that the cross-section of the vessel does not change in the axial direction. We further assume that the blood flow is steady, non-turbulent and unidirectional. Blood flow induces shear stress on the endothelium and stimulates the release of platelet derived growth factor (PDGF) which drives the growth. We simulate intimal growth for three distinct arterial cross section geometries. We show that the qualitative nature of intimal thickening varies depending on arterial geometry. For cross section geometries that are annular, the growth of the intima is uniform in the angular direction, and the endothelium stays circular as the intima grows. For non-annular cross section geometries, the intima grows more quickly where it is thicker, and shear stress and intimal thickening are negatively correlated with the distance from the flow center, where the flow velocity is maximal. Over time, the maxima and minima of the curvature increase and decrease, respectively, the PDGF concentration increases and the lumen becomes more polygonal. The model provides a framework for coupling hemodynamics simulations to mathematical descriptions of atherosclerosis, both of which have been modeled separately in great detail.

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.

Stability of a two-dimensional biomorphoelastic model for post-burn contraction

We consider the stability analysis of a two-dimensional model for post-burn contraction. The model is based on morphoelasticity for permanent deformations and combined with a chemical-biological model that incorporates cellular densities, collagen density, and the concentration of chemoattractants. We formulate stability conditions depending on the decay rate of signaling molecules for both the continuous partial differential equations-based problem and the (semi-)discrete representation. We analyze the difference and convergence between the resulting spatial eigenvalues from the continuous and semi-discrete problems.

Sensitivity of a two-dimensional biomorphoelastic model for post-burn contraction

We consider a two-dimensional biomorphoelastic model describing post-burn scar contraction. This model describes skin displacement and the development of the effective Eulerian strain in the tissue. Besides these mechanical components, signaling molecules, fibroblasts, myofibroblasts, and collagen also play a significant role in the model. We perform a sensitivity analysis for the independent parameters of the model and focus on the effects on features of the relative surface area and the total strain energy density. We conclude that the most sensitive parameters are the Poisson's ratio, the equilibrium collagen concentration, the contraction inhibitor constant, and the myofibroblast apoptosis rate. Next to these insights, we perform a sensitivity analysis where the proliferation rates of fibroblasts and myofibroblasts are not the same. The impact of this model adaptation is significant.

A self-generated Toddler gradient guides mesodermal cell migration

The sculpting of germ layers during gastrulation relies on the coordinated migration of progenitor cells, yet the cues controlling these long-range directed movements remain largely unknown. While directional migration often relies on a chemokine gradient generated from a localized source, we find that zebrafish ventrolateral mesoderm is guided by a self-generated gradient of the initially uniformly expressed and secreted protein Toddler/ELABELA/Apela. We show that the Apelin receptor, which is specifically expressed in mesodermal cells, has a dual role during gastrulation, acting as a scavenger receptor to generate a Toddler gradient, and as a chemokine receptor to sense this guidance cue. Thus, we uncover a single receptor-based self-generated gradient as the enigmatic guidance cue that can robustly steer the directional migration of mesoderm through the complex and continuously changing environment of the gastrulating embryo.

Stability of a one-dimensional morphoelastic model for post-burn contraction

To deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {O}}}(h^2)$$\end{document}O(h2). Next we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results.

Sensitivity and feasibility of a one-dimensional morphoelastic model for post-burn contraction

We consider a one-dimensional morphoelastic model describing post-burn scar contraction. Contraction can lead to a limited range of motion (contracture). Reported prevalence of burn scar contractures are 58.6% at 3-6 weeks and 20.9% at 12 months post-reconstructive surgery after burns. This model describes the displacement of the dermal layer of the skin and the development of the effective Eulerian strain in the tissue. Besides these components, the model also contains components that play a major role in the skin repair after trauma. These components are signaling molecules, fibroblasts, myofibroblasts, and collagen. We perform a sensitivity analysis for many parameters of the model and use the results for a feasibility study. In this study, we test whether the model is suitable for predicting the extent of contraction in different age groups. To this end, we conduct an extensive literature review to find parameter values. From the sensitivity analysis, we conclude that the most sensitive parameters are the equilibrium collagen concentration in the dermal layer, the apoptosis rate of fibroblasts and myofibroblasts, and the secretion rate of signaling molecules. Further, although we can use the model to simulate significant distinct contraction densities in different age groups, our results differ from what is seen in the clinic. This particularly concerns children and elderly patients. In children we see more intense contractures if the burn injury occurs near a joint, because the growth induces extra forces on the tissue. Elderly patients seem to suffer less from contractures, possibly because of excess skin.

Mathematical Modeling Can Advance Wound Healing Research

Significance: For over 30 years, there has been sustained interest in the development of mathematical models for investigating the complex mechanisms underlying each stage of the wound healing process. Despite the immense associated challenges, such models have helped usher in a paradigm shift in wound healing research. Recent Advances: In this article, we review contributions in the field that span epidermal, dermal, and corneal wound healing, and treatments of nonhealing wounds. The recent influence of mathematical models on biological experiments is detailed, with a focus on wound healing assays and fibroblast-populated collagen lattices. Critical Issues: We provide an overview of the field of mathematical modeling of wound healing, highlighting key advances made in recent decades, and discuss how such models have contributed to the development of improved treatment strategies and/or an enhanced understanding of the tightly regulated steps that comprise the healing process. Future Directions: We detail some of the open problems in the field that could be addressed through a combination of theoretical and/or experimental approaches. To move the field forward, we need to have a common language between scientists to facilitate cross-collaboration, which we hope this review can support by highlighting progress to date.

A multiphase model of growth factor-regulated atherosclerotic cap formation

Atherosclerosis is characterised by the growth of fatty plaques in the inner artery wall. In mature plaques, vascular smooth muscle cells (SMCs) are recruited from adjacent tissue to deposit a collagenous cap over the fatty plaque core. This cap isolates the thrombogenic plaque content from the bloodstream and prevents the clotting cascade that leads to myocardial infarction or stroke. Despite the protective role of the cap, the mechanisms that regulate cap formation and maintenance are not well understood. It remains unclear why some caps become stable, while others become vulnerable to rupture. We develop a multiphase PDE model with non-standard boundary conditions to investigate collagen cap formation by SMCs in response to diffusible growth factor signals from the endothelium. Platelet-derived growth factor stimulates SMC migration, proliferation and collagen degradation, while transforming growth factor (TGF)-[Formula: see text] stimulates SMC collagen synthesis and inhibits collagen degradation. The model SMCs respond haptotactically to gradients in the collagen phase and have reduced rates of migration and proliferation in dense collagenous tissue. The model, which is parameterised using in vivo and in vitro experimental data, reproduces several observations from plaque growth in mice. Numerical and analytical results demonstrate that a stable cap can be formed by a relatively small SMC population and emphasise the critical role of TGF-[Formula: see text] in effective cap formation. These findings provide unique insight into the mechanisms that may lead to plaque destabilisation and rupture. This work represents an important step towards the development of a comprehensive in silico plaque model.

Mechanistic models of PLC/PKC signaling implicate phosphatidic acid as a key amplifier of chemotactic gradient sensing

Chemotaxis of fibroblasts and other mesenchymal cells is critical for embryonic development and wound healing. Fibroblast chemotaxis directed by a gradient of platelet-derived growth factor (PDGF) requires signaling through the phospholipase C (PLC)/protein kinase C (PKC) pathway. Diacylglycerol (DAG), the lipid product of PLC that activates conventional PKCs, is focally enriched at the up-gradient leading edge of fibroblasts responding to a shallow gradient of PDGF, signifying polarization. To explain the underlying mechanisms, we formulated reaction-diffusion models including as many as three putative feedback loops based on known biochemistry. These include the previously analyzed mechanism of substrate-buffering by myristoylated alanine-rich C kinase substrate (MARCKS) and two newly considered feedback loops involving the lipid, phosphatidic acid (PA). DAG kinases and phospholipase D, the enzymes that produce PA, are identified as key regulators in the models. Paradoxically, increasing DAG kinase activity can enhance the robustness of DAG/active PKC polarization with respect to chemoattractant concentration while decreasing their whole-cell levels. Finally, in simulations of wound invasion, efficient collective migration is achieved with thresholds for chemotaxis matching those of polarization in the reaction-diffusion models. This multi-scale modeling framework offers testable predictions to guide further study of signal transduction and cell behavior that affect mesenchymal chemotaxis.

Cell movement directed by external gradients of chemical composition is critical for immune responses, wound healing, and development. Although theoretical concepts explaining how shallow external gradients might definitively polarize a cell’s motility have been offered over the past two decades, mathematical models cast in terms of defined molecules and mechanisms are uncommon in this context. Based on both recent and older insights from the literature, we offer mechanistic models that are able to explain experimentally observed polarization of signal transduction elicited by shallow attractant gradients. A novel insight of our models is the implicated role of phosphatidic acid, a membrane lipid produced by at least two enzymatic pathways, in two positive feedback loops that amplify signal transduction locally. In separate simulations, we explored the implications of polarization for efficient cell invasion during wound healing. We expected that the ability to polarize in response to shallow gradients would enhance the speed of wound invasion, but an unexpected finding is that this property can promote intermittent polarization throughout the wound.

Ranking migration cue contributions to guiding individual fibroblasts faced with a directional decision in simple microfluidic bifurcations

Directed cell migration in complex micro-environments, such as in vivo pores, is important for predicting locations of artificial tissue growth and optimizing scaffold architectures. Yet, the directional decisions of cells facing multiple physiochemical cues have not been characterized. Hence, we aim to provide a ranking of the relative importance of the following cues to the decision-making of individual fibroblast cells: chemoattractant concentration gradient, channel width, mitosis, and contact-guidance. In this study, bifurcated micro-channels with branches of different widths were created. Fibroblasts were then allowed to travel across these geometries by following a gradient of platelet-derived growth factor-BB (PDGF-BB) established inside the channels. Subsequently, a combination of statistical analysis and image-based diffusion modeling was used to report how the presence of multiple complex migration cues, including cell-cell influences, affect the fibroblast decision-making. It was found that the cells prefer wider channels over a higher chemoattractant gradient when choosing between asymmetric bifurcated branches. Only when the branches were symmetric in width did the gradient become predominant in directing which path the cell will take. Furthermore, when both the gradient and the channels were symmetric, contact guidance became important for guiding the cells in making directional choices. Based on these results we were able to rank these directional cues from most influential to the least as follows: mitosis > channel width asymmetry > chemoattractant gradient difference > and contact-guidance. It is expected that these results will benefit the fields of regenerative medicine, wound healing and developmental biology.

Advanced 2D/3D cell migration assay for faster evaluation of chemotaxis of slow-moving cells

Considering the essential role of chemotaxis of adherent, slow-moving cells in processes such as tumor metastasis or wound healing, a detailed understanding of the mechanisms and cues that direct migration of cells through tissues is highly desirable. The state-of-the-art chemotaxis instruments (e.g. microfluidic-based devices, bridge assays) can generate well-defined, long-term stable chemical gradients, crucial for quantitative investigation of chemotaxis in slow-moving cells. However, the majority of chemotaxis tools are designed for the purpose of an in-depth, but labor-intensive analysis of migratory behavior of single cells. This is rather inefficient for applications requiring higher experimental throughput, as it is the case of e.g. clinical examinations, chemoattractant screening or studies of the chemotaxis-related signaling pathways based on subcellular perturbations. Here, we present an advanced migration assay for accelerated and facilitated evaluation of the chemotactic response of slow-moving cells. The revised chemotaxis chamber contains a hydrogel microstructure–the migration arena, designed to enable identification of chemotactic behavior of a cell population in respect to the end-point of the experiment. At the same time, the assay in form of a microscopy slide enables direct visualization of the cells in either 2D or 3D environment, and provides a stable and linear gradient of chemoattractant. We demonstrate the correctness of the assay on the model study of HT-1080 chemotaxis in 3D and on 2D surface. Finally, we apply the migration arena chemotaxis assay to screen for a chemoattractant of primary keratinocytes, cells that play a major role in wound healing, being responsible for skin re-epithelialization and a successful wound closure. In direction of new therapeutic strategies to promote wound repair, we identified the chemotactic activity of the epithelial growth factor receptor (EGFR) ligands EGF and TGFα (transforming growth factor α).

Cell Sequence and Mitosis Affect Fibroblast Directional Decision-Making During Chemotaxis in Microfluidic Mazes

Introduction

Directed fibroblast migration is central to highly proliferative processes in regenerative medicine and developmental biology. However, the mechanisms by which single fibroblasts affect each other's directional decisions, while chemotaxing in microscopic pores, are not well understood.

Methods

We explored effects of cell sequence and mitosis on fibroblast platelet-derived growth factor-BB (PDGF-BB)-induced migration in microfluidic mazes with two possible through paths: short and long. Additionally, image-based modeling of the chemoattractant's diffusion, consumption and decay, was used to explain the experimental observations.

Results

It both cases, the cells displayed behavior that is contradictory to expectation based on the global chemoattractant gradient pre-established in the maze. In case of the sequence, the cells tend to alternate when faced with a bifurcation: if a leading cell takes the shorter (steeper gradient) path, the cell following it chooses the longer (weaker gradient) path, and vice versa. Image-based modeling of the process showed that the local PDGF-BB consumption by the individual fibroblasts may be responsible for this phenomenon. Additionally, it was found that when a mother cell divides, its two daughters go in opposite directions (even if it means migrating against the chemoattractant gradient and overcoming on-going cell traffic).

Conclusions

It is apparent that micro-confined fibroblasts modify each other's directional decisions in a manner that is counter-intuitive to what is expected from classical chemotaxis theory. Consequently, accounting for these effects could lead to a better understanding of tissue generation in vivo, and result in more advanced engineered tissue products in vitro.

A two-phase model of early fibrous cap formation in atherosclerosis

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) are recruited from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and dangerously vulnerable to rupture. We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using data from a range of in vitro and in vivo studies, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development.

Collective gradient sensing and chemotaxis: modeling and recent developments

Cells measure a vast variety of signals, from their environment's stiffness to chemical concentrations and gradients; physical principles strongly limit how accurately they can do this. However, when many cells work together, they can cooperate to exceed the accuracy of any single cell. In this topical review, I will discuss the experimental evidence showing that cells collectively sense gradients of many signal types, and the models and physical principles involved. I also propose new routes by which experiments and theory can expand our understanding of these problems.

Programmed biomolecule delivery to enable and direct cell migration for connective tissue repair

Dense connective tissue injuries have limited repair, due to the paucity of cells at the wound site. We hypothesize that decreasing the density of the local extracellular matrix (ECM) in conjunction with releasing chemoattractive signals increases cellularity and tissue formation after injury. Using the knee meniscus as a model system, we query interstitial cell migration in the context of migratory barriers using a novel tissue Boyden chamber and show that a gradient of platelet-derived growth factor-AB (PDGF-AB) expedites migration through native tissue. To implement these signals in situ, we develop nanofibrous scaffolds with distinct fiber fractions that sequentially release active collagenase (to increase ECM porosity) and PDGF-AB (to attract endogenous cells) in a localized and coordinated manner. We show that, when placed into a meniscal defect, the controlled release of collagenase and PDGF-AB increases cellularity at the interface and within the scaffold, as well as integration with the surrounding tissue.

A Reaction-Diffusion Model Explains Amplification of the PLC/PKC Pathway in Fibroblast Chemotaxis

During the proliferative phase of cutaneous wound healing, dermal fibroblasts are recruited into the clotted wound by a concentration gradient of platelet-derived growth factor (PDGF), together with other spatial cues. Despite the importance of this chemotactic process, the mechanisms controlling the directed migration of slow-moving mesenchymal cells such as fibroblasts are not well understood. Here, we develop and analyze a reaction-diffusion model of phospholipase C/protein kinase C (PKC) signaling, which was recently identified as a requisite PDGF-gradient-sensing pathway, with the goal of identifying mechanisms that can amplify its sensitivity in the shallow external gradients typical of chemotaxis experiments. We show that phosphorylation of myristoylated alanine-rich C kinase substrate by membrane-localized PKC constitutes a positive feedback that is sufficient for local pathway amplification. The release of phosphorylated myristoylated alanine-rich C kinase substrate and its subsequent diffusion and dephosphorylation in the cytosol also serves to suppress the pathway in down-gradient regions of the cell. By itself, this mechanism only weakly amplifies signaling in a shallow PDGF gradient, but it synergizes with other feedback mechanisms to enhance amplification. This model offers a framework for a mechanistic understanding of phospholipase C/PKC signaling in chemotactic gradient sensing and can guide the design of experiments to assess the roles of putative feedback loops.

Physical models of collective cell motility: from cell to tissue

In this article, we review physics-based models of collective cell motility. We discuss a range of techniques at different scales, ranging from models that represent cells as simple self-propelled particles to phase field models that can represent a cell's shape and dynamics in great detail. We also extensively review the ways in which cells within a tissue choose their direction, the statistics of cell motion, and some simple examples of how cell-cell signaling can interact with collective cell motility. This review also covers in more detail selected recent works on collective cell motion of small numbers of cells on micropatterns, in wound healing, and the chemotaxis of clusters of cells.

Local modulation of chemoattractant concentrations by single cells: dissection using a bulk-surface computational model

Chemoattractant gradients are usually considered in terms of sources and sinks that are independent of the chemotactic cell. However, recent interest has focused on 'self-generated' gradients, in which cell populations create their own local gradients as they move. Here, we consider the interplay between chemoattractants and single cells. To achieve this, we extend a recently developed computational model to incorporate breakdown of extracellular attractants by membrane-bound enzymes. Model equations are parametrized, using the published estimates from Dictyostelium cells chemotaxing towards cyclic AMP. We find that individual cells can substantially modulate their local attractant field under physiologically appropriate conditions of attractant and enzymes. This means the attractant concentration perceived by receptors can be a small fraction of the ambient concentration. This allows efficient chemotaxis in chemoattractant concentrations that would be saturating without local breakdown. Similar interactions in which cells locally mould a stimulus could function in many types of directed cell motility, including haptotaxis, durotaxis and even electrotaxis.

Self-generated chemotactic gradients-cells steering themselves

Chemotaxis is a fundamentally important part of biology, but we know very little about how gradients of chemoattractant are formed. One answer is self-generated gradients, in which the moving cells break down the attractant to provide their own gradient as they migrate. Here we discuss where self-generated gradients are known, how they can be recognized, and where they are likely to be found in the future.

Nox4 and Duox1/2 Mediate Redox Activation of Mesenchymal Cell Migration by PDGF

Platelet derived growth factor (PDGF) orchestrates wound healing and tissue regeneration by regulating recruitment of the precursor mesenchymal stromal cells (MSC) and fibroblasts. PDGF stimulates generation of hydrogen peroxide that is required for cell migration, but the sources and intracellular targets of H2O2 remain obscure. Here we demonstrate sustained live responses of H2O2 to PDGF and identify PKB/Akt, but not Erk1/2, as the target for redox regulation in cultured 3T3 fibroblasts and MSC. Apocynin, cell-permeable catalase and LY294002 inhibited PDGF-induced migration and mitotic activity of these cells indicating involvement of PI3-kinase pathway and H2O2. Real-time PCR revealed Nox4 and Duox1/2 as the potential sources of H2O2. Silencing of Duox1/2 in fibroblasts or Nox4 in MSC reduced PDGF-stimulated intracellular H2O2, PKB/Akt phosphorylation and migration, but had no such effect on Erk1/2. In contrast to PDGF, EGF failed to increase cytoplasmic H2O2, phosphorylation of PKB/Akt and migration of fibroblasts and MSC, confirming the critical impact of redox signaling. We conclude that PDGF-induced migration of mesenchymal cells requires Nox4 and Duox1/2 enzymes, which mediate redox-sensitive activation of PI3-kinase pathway and PKB/Akt.

Self-Generated Chemoattractant Gradients: Attractant Depletion Extends the Range and Robustness of Chemotaxis

Chemotaxis is fundamentally important, but the sources of gradients in vivo are rarely well understood. Here, we analyse self-generated chemotaxis, in which cells respond to gradients they have made themselves by breaking down globally available attractants, using both computational simulations and experiments. We show that chemoattractant degradation creates steep local gradients. This leads to surprising results, in particular the existence of a leading population of cells that moves highly directionally, while cells behind this group are undirected. This leading cell population is denser than those following, especially at high attractant concentrations. The local gradient moves with the leading cells as they interact with their surroundings, giving directed movement that is unusually robust and can operate over long distances. Even when gradients are applied from external sources, attractant breakdown greatly changes cells' responses and increases robustness. We also consider alternative mechanisms for directional decision-making and show that they do not predict the features of population migration we observe experimentally. Our findings provide useful diagnostics to allow identification of self-generated gradients and suggest that self-generated chemotaxis is unexpectedly universal in biology and medicine.

Mechanical boundary conditions bias fibroblast invasion in a collagen-fibrin wound model

Because fibroblasts deposit the collagen matrix that determines the mechanical integrity of scar tissue, altering fibroblast invasion could alter wound healing outcomes. Anisotropic mechanical boundary conditions (restraint, stretch, or tension) could affect the rate of fibroblast invasion, but their importance relative to the prototypical drivers of fibroblast infiltration during wound healing--cell and chemokine concentration gradients--is unknown. We tested whether anisotropic mechanical boundary conditions affected the directionality and speed of fibroblasts migrating into a three-dimensional model wound, which could simultaneously expose fibroblasts to mechanical, structural, steric, and chemical guidance cues. We created fibrin-filled slits in fibroblast-populated collagen gels and applied uniaxial mechanical restraint along the short or long axis of the fibrin wounds. Anisotropic mechanical conditions increased the efficiency of fibroblast invasion by guiding fibroblasts without increasing their migration speed. The migration behavior could be modeled as a biased random walk, where the bias due to multiple guidance cues was accounted for in the shape of a displacement orientation probability distribution. Taken together, modeling and experiments suggested an effect of strain anisotropy, rather than strain-induced fiber alignment, on fibroblast invasion.

Melanoma cells break down LPA to establish local gradients that drive chemotactic dispersal

The high mortality of melanoma is caused by rapid spread of cancer cells, which occurs unusually early in tumour evolution. Unlike most solid tumours, thickness rather than cytological markers or differentiation is the best guide to metastatic potential. Multiple stimuli that drive melanoma cell migration have been described, but it is not clear which are responsible for invasion, nor if chemotactic gradients exist in real tumours. In a chamber-based assay for melanoma dispersal, we find that cells migrate efficiently away from one another, even in initially homogeneous medium. This dispersal is driven by positive chemotaxis rather than chemorepulsion or contact inhibition. The principal chemoattractant, unexpectedly active across all tumour stages, is the lipid agonist lysophosphatidic acid (LPA) acting through the LPA receptor LPAR1. LPA induces chemotaxis of remarkable accuracy, and is both necessary and sufficient for chemotaxis and invasion in 2-D and 3-D assays. Growth factors, often described as tumour attractants, cause negligible chemotaxis themselves, but potentiate chemotaxis to LPA. Cells rapidly break down LPA present at substantial levels in culture medium and normal skin to generate outward-facing gradients. We measure LPA gradients across the margins of melanomas in vivo, confirming the physiological importance of our results. We conclude that LPA chemotaxis provides a strong drive for melanoma cells to invade outwards. Cells create their own gradients by acting as a sink, breaking down locally present LPA, and thus forming a gradient that is low in the tumour and high in the surrounding areas. The key step is not acquisition of sensitivity to the chemoattractant, but rather the tumour growing to break down enough LPA to form a gradient. Thus the stimulus that drives cell dispersal is not the presence of LPA itself, but the self-generated, outward-directed gradient.

Chemotactic signaling in mesenchymal cells compared to amoeboid cells

Cell chemotaxis plays a pivotal role in normal development, inflammatory response, injury repair and tissue regeneration in all organisms. It is also a critical contributor to cancer metastasis, altered angiogenesis and neurite growth in disease. The molecular mechanisms regulating chemotaxis are currently being identified and key components may be pertinent therapeutic targets. Although these components appear to be mostly common in various cells, there are important differences in chemotactic signaling networks and signal processing that result in the distinct chemotactic behavior of mesenchymal cells compared to much better studied amoeboid blood cells. These differences are not necessarily predetermined based on cell type, but are rather chosen and exploited by cells to modify their chemotactic behavior based on physical constraints and/or environmental conditions. This results in a specific type of chemotactic migration in mesenchymal cells that can be selectively targeted in disease. Here, we compare the chemotactic behavior, signaling and motility of mesenchymal and amoeboid cells. We suggest that the current model of chemotaxis is applicable for small amoeboid cells but needs to be reconsidered for large mesenchymal cells. We focus on new candidate regulatory molecules and feedback mechanisms that may account for mesenchymal cell type-specific chemotaxis.

Directed migration of mesenchymal cells: where signaling and the cytoskeleton meet

Cell migration directed by spatial cues, or taxis, is a primary mechanism for orchestrating concerted and collective cell movements during development, wound repair, and immune responses. Compared with the classic example of amoeboid chemotaxis, in which fast-moving cells such as neutrophils are directed by gradients of soluble factors, directed migration of slow-moving mesenchymal cells such as fibroblasts is poorly understood. Mesenchymal cells possess a distinctive organization of the actin cytoskeleton and associated adhesion complexes as its primary mechanical system, generating the asymmetric forces required for locomotion without strong polarization. The emerging hypothesis is that the molecular underpinnings of mesenchymal taxis involve distinct signaling pathways and diverse requirements for regulation.

Systems-based approaches toward wound healing

Wound healing in the pediatric patient is of utmost clinical and social importance because hypertrophic scarring can have aesthetic and psychological sequelae, from early childhood to late adolescence. Wound healing is a well-orchestrated reparative response affecting the damaged tissue at the cellular, tissue, organ, and system scales. Although tremendous progress has been made toward understanding wound healing at the individual temporal and spatial scales, its effects across the scales remain severely understudied and poorly understood. Here, we discuss the critical need for systems-based computational modeling of wound healing across the scales, from short-term to long-term and from small to large. We illustrate the state of the art in systems modeling by means of three key signaling mechanisms: oxygen tension-regulating angiogenesis and revascularization; transforming growth factor-β (TGF-β) kinetics controlling collagen deposition; and mechanical stretch stimulating cellular mitosis and extracellular matrix (ECM) remodeling. The complex network of biochemical and biomechanical signaling mechanisms and the multiscale character of the healing process make systems modeling an integral tool in exploring personalized strategies for wound repair. A better mechanistic understanding of wound healing in the pediatric patient could open new avenues in treating children with skin disorders such as birth defects, skin cancer, wounds, and burn injuries.

Modelling the interaction of keratinocytes and fibroblasts during normal and abnormal wound healing processes

The crosstalk between fibroblasts and keratinocytes is a vital component of the wound healing process, and involves the activity of a number of growth factors and cytokines. In this work, we develop a mathematical model of this crosstalk in order to elucidate the effects of these interactions on the regeneration of collagen in a wound that heals by second intention. We consider the role of four components that strongly affect this process: transforming growth factor-β, platelet-derived growth factor, interleukin-1 and keratinocyte growth factor. The impact of this network of interactions on the degradation of an initial fibrin clot, as well as its subsequent replacement by a matrix that is mainly composed of collagen, is described through an eight-component system of nonlinear partial differential equations. Numerical results, obtained in a two-dimensional domain, highlight key aspects of this multifarious process, such as re-epithelialization. The model is shown to reproduce many of the important features of normal wound healing. In addition, we use the model to simulate the treatment of two pathological cases: chronic hypoxia, which can lead to chronic wounds; and prolonged inflammation, which has been shown to lead to hypertrophic scarring. We find that our model predictions are qualitatively in agreement with previously reported observations and provide an alternative pathway for gaining insight into this complex biological process.

Mathematical model of intimal thickening in atherosclerosis: vessel stenosis as a free boundary problem

Atherosclerosis is an inflammatory disease of the artery characterized by an expansion of the intimal region. Intimal thickening is usually attributed to the migration of smooth muscle cells (SMCs) from the surrounding media and proliferation of SMCs already present in the intima. Intimal expansion can give rise to dangerous events such as stenosis (leading to stroke) or plaque rupture (leading to myocardial infarction). In this paper we propose and study a mathematical model of intimal thickening, posed as a free boundary problem. Intimal thickening is driven by damage to the endothelium, resulting in the release of cytokines and migration of SMCs. By coupling a boundary value problem for cytokine concentration to an evolution law for the intimal area, we reduce the problem to a single nonlinear differential equation for the luminal radius. We analyze the steady states, perform a bifurcation analysis and compare model solutions to data from rabbits whose iliac arteries are subject to a balloon pullback injury. In order to obtain a favorable fit, we find that migrating SMCs must enter the intima very slowly compared to cells in dermal wounds. This cell behavior is indicative of a weak inflammatory response which is consistent with atherosclerosis being a chronic inflammatory disease.

Chemotaxis: movement, direction, control

This review focuses on basic principles of motility in different cell types, formation of the specific cell structures that enable directed migration, and how external signals are transduced into cells and coupled to the motile machinery. Feedback mechanisms and their potential role in maintenance of internal chemotactic gradients and persistence of directed migration are highlighted.

Epithelial cell guidance by self-generated EGF gradients

Cancer epithelial cells often migrate away from the primary tumor to invade into the surrounding tissues. Their migration is commonly assumed to be directed by pre-existent spatial gradients of chemokines and growth factors in the target tissues. Unexpectedly however, we found that the guided migration of epithelial cells is possible in vitro in the absence of pre-existent chemical gradients. We observed that both normal and cancer epithelial cells can migrate persistently and reach the exit along the shortest path from microscopic mazes filled with uniform concentrations of media. Using microscale engineering techniques and biophysical models, we uncovered a self-guidance strategy during which epithelial cells generate their own guiding cues under conditions of biochemical confinement. The self-guidance strategy depends on the balance between three interdependent processes: epidermal growth factor (EGF) uptake by the cells (U), the restricted transport of EGF through the structured microenvironment (T), and cell chemotaxis toward the resultant EGF gradients (C). The UTC self-guidance strategy can be perturbed by inhibition of signalling through EGF-receptors and appears to be independent from chemokine signalling. Better understanding of the UTC self-guidance strategy could eventually help devise new ways for modulating epithelial cell migration and delaying cancer cell invasion or accelerating wound healing.

Normal morphogenesis of epithelial tissues and progression of epithelial tumors

Epithelial cells organize into various tissue architectures that largely maintain their structure throughout the life of an organism. For decades, the morphogenesis of epithelial tissues has fascinated scientists at the interface of cell, developmental, and molecular biology. Systems biology offers ways to combine knowledge from these disciplines by building integrative models that are quantitative and predictive. Can such models be useful for gaining a deeper understanding of epithelial morphogenesis? Here, we take inventory of some recurring themes in epithelial morphogenesis that systems approaches could strive to capture. Predictive understanding of morphogenesis at the systems level would prove especially valuable for diseases such as cancer, where epithelial tissue architecture is profoundly disrupted.

Phosphoinositides in chemotaxis

Phosphatidylinositol lipids generated through the action of phosphinositide 3-kinase (PI3K) are key mediators of a wide array of biological responses. In particular, their role in the regulation of cell migration has been extensively studied and extends to amoeboid as well as mesenchymal migration. Through the emergence of fluorescent probes that target PI3K products as well as the use of specific inhibitors and knockout technologies, the spatio-temporal distribution of PI3K products in chemotaxing cells has been shown to represent a key anterior polarity signal that targets downstream effectors to actin polymerization. In addition, through intricate cross-talk networks PI3K products have been shown to regulate signals that control posterior effectors. Yet, in more complex environments or in conditions where chemoattractant gradients are steep, a variety of cell types can still chemotax in the absence of PI3K signals. Indeed, parallel signal transduction pathways have been shown to coordinately regulate cell polarity and directed movement. In this chapter, we will review the current role PI3K products play in the regulation of directed cell migration in various cell types, highlight the importance of mathematical modeling in the study of chemotaxis, and end with a brief overview of other signaling cascades known to also regulate chemotaxis.

In chemotaxing fibroblasts, both high-fidelity and weakly biased cell movements track the localization of PI3K signaling

Cell movement biased by a chemical gradient, or chemotaxis, coordinates the recruitment of cells and collective migration of cell populations. During wound healing, chemotaxis of fibroblasts is stimulated by platelet-derived growth factor (PDGF) and certain other chemoattractants. Whereas the immediate PDGF gradient sensing response has been characterized previously at the level of phosphoinositide 3-kinase (PI3K) signaling, the sensitivity of the response at the level of cell migration bias has not yet been studied quantitatively. In this work, we used live-cell total internal reflection fluorescence microscopy to monitor PI3K signaling dynamics and cell movements for extended periods. We show that persistent and properly aligned (i.e., high-fidelity) fibroblast migration does indeed correlate with polarized PI3K signaling; accordingly, this behavior is seen only under conditions of high gradient steepness (>10% across a typical cell length of 50 μm) and a certain range of PDGF concentrations. Under suboptimal conditions, cells execute a random or biased random walk, but nonetheless move in a predictable fashion according to the changing pattern of PI3K signaling. Inhibition of PI3K during chemotaxis is accompanied by loss of both cell-substratum contact and morphological polarity, but after a recovery period, PI3K-inhibited fibroblasts often regain the ability to orient toward the PDGF gradient.

The effect of anisotropic collagen-GAG scaffolds and growth factor supplementation on tendon cell recruitment, alignment, and metabolic activity

Current surgical and tissue engineering approaches for treating tendon injuries have shown limited success, suggesting the need for new biomaterial strategies. Here we describe the development of an anisotropic collagen-glycosaminoglycan (CG) scaffold and use of growth factor supplementation strategies to create a 3D platform for tendon tissue engineering. We fabricated cylindrical CG scaffolds with aligned tracks of ellipsoidal pores that mimic the native physiology of tendon by incorporating a directional solidification step into a conventional lyophilization strategy. By modifying the freezing temperature, we created a homologous series of aligned CG scaffolds with constant relative density and degree of anisotropy but a range of pore sizes (55-243 μm). Equine tendon cells showed greater levels of attachment, metabolic activity, and alignment as well as less cell-mediated scaffold contraction, when cultured in anisotropic scaffolds compared to an isotropic CG scaffold control. The anisotropic CG scaffolds also provided critical contact guidance cues for cell alignment. While tendon cells were randomly oriented in the isotropic control scaffold and the transverse (unaligned) plane of the anisotropic scaffolds, significant cell alignment was observed in the direction of the contact guidance cues in the longitudinal plane of the anisotropic scaffolds. Scaffold pore size was found to significantly influence tendon cell viability, proliferation, penetration into the scaffold, and metabolic activity in a manner predicted by cellular solids arguments. Finally, the addition of the growth factors PDGF-BB and IGF-1 to aligned CG scaffolds was found to enhance tendon cell motility, viability, and metabolic activity in dose-dependent manners. This work suggests a composite strategy for developing bioactive, 3D material systems for tendon tissue engineering.

Signaling pathways that control cell migration: models and analysis

Dissecting the intracellular signaling mechanisms that govern the movement of eukaryotic cells presents a major challenge, not only because of the large number of molecular players involved, but even more so because of the dynamic nature of their regulation by both biochemical and mechanical interactions. Computational modeling and analysis have emerged as useful tools for understanding how the physical properties of cells and their microenvironment are coupled with certain biochemical pathways to actuate and control cell motility. In this focused review, we highlight some of the more recent applications of quantitative modeling and analysis in the field of cell migration. Both in modeling and experiment, it has been prudent to follow a reductionist approach in order to characterize what are arguably the principal modules: spatial polarization of signaling pathways, regulation of the actin cytoskeleton, and dynamics of focal adhesions. While it is important that we 'cut our teeth' on these subsystems, focusing on the details of certain aspects while ignoring or coarse-graining others, it is clear that the challenge ahead will be to characterize the couplings between them in an integrated framework.

How cells use pseudopods for persistent movement and navigation

Pseudopods are cell extensions used for movement. The direction and trajectory of cell movement depend on how cells extend pseudopods. Experimental data show that new pseudopods are frequently formed by the splitting of an existing pseudopod, often as a series of left-right extensions. Here, these data on pseudopod extensions are discussed in the context of a theoretical model for pseudopod splitting.

Multimodal release of transforming growth factor-β1 and the BB isoform of platelet derived growth factor from PEGylated fibrin gels

We designed a growth factor release system to potentially stabilize neovascularization in the treatment of ischemic tissue. In this study, the release of PDGF-BB and TGF-β1 was controlled with distinct kinetics from injectable PEGylated fibrin gels. Growth factors can be loaded into PEGylated fibrin gels via 3 mechanisms: entrapment, conjugation through a homobifunctional amine reactive PEG linker, and physical affinity with the fibrin matrix. PDGF-BB was entrapped during thrombin-mediated crosslinking leading to a diffusion-controlled release over 2days. TGF-β1 was both conjugated through the PEG linker and bound to the matrix via physical affinity, delaying the release rate of TGF-β1 up to 10days. Further, the release rate was highly correlated to gel degradation rate indicating that TGF-β1 release is degradation-controlled. Therefore, by modulating the molar ratio of PEG to fibrinogen, we were able to control the release rate of TGF-β1 without altering the release kinetics of PDGF-BB. The bioactivity of loaded TGF-β1 was maintained upon release as evidenced by the inhibition of cell proliferation. This system could be expanded to incorporate growth factors loaded via 3 schemes with differing release rates from an injectable system allowing for a high degree of flexibility in other combinational drug delivery and tissue engineering systems.

Simulation of lung alveolar epithelial wound healing in vitro

The mechanisms that enable and regulate alveolar type II (AT II) epithelial cell wound healing in vitro and in vivo remain largely unknown and need further elucidation. We used an in silico AT II cell-mimetic analogue to explore and better understand plausible wound healing mechanisms for two conditions: cyst repair in three-dimensional cultures and monolayer wound healing. Starting with the analogue that validated for key features of AT II cystogenesis in vitro, we devised an additional cell rearrangement action enabling cyst repair. Monolayer repair was enabled by providing 'cells' a control mechanism to switch automatically to a repair mode in the presence of a distress signal. In cyst wound simulations, the revised analogue closed wounds by adhering to essentially the same axioms available for alveolar-like cystogenesis. In silico cell proliferation was not needed. The analogue recovered within a few simulation cycles but required a longer recovery time for larger or multiple wounds. In simulated monolayer wound repair, diffusive factor-mediated 'cell' migration led to repair patterns comparable to those of in vitro cultures exposed to different growth factors. Simulations predicted directional cell locomotion to be critical for successful in vitro wound repair. We anticipate that with further use and refinement, the methods used will develop as a rigorous, extensible means of unravelling mechanisms of lung alveolar repair and regeneration.

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.

Cell population-based model of dermal wound invasion with heterogeneous intracellular signaling properties

A deterministic model of dermal wound invasion, which accounts for the platelet-derived growth factor (PDGF) gradient sensing mechanism in fibroblasts mediated by cell surface receptors and the phosphoinositide 3-kinase (PI3K) signal transduction pathway, was previously described (Biophys J 2006; 90:2297-308). Here, we extend that work and implement a hybrid modeling strategy that treats fibroblasts as discrete entities endowed with heterogeneous properties, namely receptor, PI3K and 3' phosphoinositide phosphatase expression levels. Analysis of the model suggests that the wound environment fosters the advancement of cells within the population that are better fit to migrate and/or proliferate in response to PDGF stimulation. Thus, cell-to-cell variability results in a significantly higher rate of wound invasion as compared with the deterministic model, in a manner that depends on the way in which individual cell properties are sampled or inherited upon cell division.

Mechanisms of Gradient Detection: A Comparison of Axon Pathfinding with Eukaryotic Cell Migration

The detection of gradients of chemotactic cues is a common task for migrating cells and outgrowing axons. Eukaryotic gradient detection employs a spatial mechanism, meaning that the external gradient has to be translated into an intracellular signaling gradient, which affects cell polarization and directional movement. The sensitivity of gradient detection is governed by signal amplification and adaptation mechanisms. Comparison of the major signal transduction pathways underlying gradient detection in three exemplary chemotaxing cell types, Dictyostelium, neutrophils, and fibroblasts and in neuronal growth cones, reveals conserved mechanisms such as localized PI3 kinase/PIP3 signaling and a common output, the regulation of the cytoskeleton by Rho GTPases. Local protein translation plays a role in directional movement of both fibroblasts and neuronal growth cones. Ca(2+) signaling is prominently involved in growth cone gradient detection. The diversity of signaling between different cell types and its functional implications make sense in the biological context.

Multi-scale modeling of a wound-healing cell migration assay

A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhibit contact inhibition of migration and contact-limited proliferation. The continuum model includes the two dominant mechanisms and characteristics of cell migration and proliferation, using a cell diffusivity function that decreases with cell density and a logistic proliferative growth term. The discrete model arises naturally from the continuum model. Individual cells are simulated as continuous-time random walkers with nearest-neighbor transitions, together with a birth/death process. The migration and proliferation parameters are determined by analysing individual mice 3T3 fibroblast cell trajectories obtained during the development of a confluent cell monolayer and in a wound healing assay. The population-scale model successfully predicts the shape and speed of the traveling wave, while the discrete model is also successful in capturing the contact inhibition of migration effects.