Computational model for cell migration in three-dimensional matrices

Protrusion of a Virtual Model Lamellipodium by Actin Polymerization: A Coarse-grained Langevin Dynamics Model

We report the development of a coarse-grained Langevin dynamics model of a lamellipodium featuring growing F-actin filaments in order to study the effect of stiffness of the F-actin filament, the G-actin monomer concentration, and the number of polymerization sites on lamellipodium protrusion. The virtual lamellipodium is modeled as a low-aspect-ratio doubly capped cylinder formed by triangulated particles on its surface. It is assumed that F-actin filaments are firmly attached to a lamellipodium surface where polymerization sites are located, and actin polymerization takes place by connecting a G-actin particle to a polymerization site and to the first particle of a growing F-actin filament. It is found that there is an optimal number of polymerization sites for rapid lamellipodium protrusion. The maximum speed of lamellipodium protrusion is related to competition between the number of polymerization sites and the number of available G-actin particles, and the degree of pulling and holding of the lamellipodium surface by non-polymerizing actin filaments. The lamellipodium protrusion by actin polymerization displays saltatory motion exhibiting pseudo-thermal equilibrium: the lamellipodium speed distribution is Maxwellian in two dimensions but the lamellipodium motion is biased so that the lamellipodium speed in the direction of the lamellipodium motion is much larger than that normal to the lamellipodium motion.

Contact guidance mediated three-dimensional cell migration is regulated by Rho/ROCK-dependent matrix reorganization

Cells generate mechanical force to organize the extracellular matrix (ECM) and drive important developmental and reparative processes. Likewise, tumor cells invading into three-dimensional (3D) matrices remodel the ECM microenvironment. Importantly, we previously reported a distinct radial reorganization of the collagen matrix surrounding tumors that facilitates local invasion. Here we describe a mechanism by which cells utilize contractility events to reorganize the ECM to provide contact guidance that facilitates 3D migration. Using novel assays to differentially organize the collagen matrix we show that alignment of collagen perpendicular to the tumor-explant boundary promotes local invasion of both human and mouse mammary epithelial cells. In contrast, organizing the collagen matrix to mimic the ECM organization associated with noninvading regions of tumors suppresses 3D migration/invasion. Moreover, we demonstrate that matrix reorganization is contractility-dependent and that the Rho/Rho kinase pathway is necessary for collagen alignment to provide contact guidance. Yet, if matrices are prealigned, inhibiting neither Rho nor Rho kinase inhibits 3D migration, which supports our conclusion that Rho-mediated matrix alignment is an early step in the invasion process, preceding and subsequently facilitating 3D migration.

Microarchitecture of three-dimensional scaffolds influences cell migration behavior via junction interactions

Cell migration plays a critical role in a wide variety of physiological and pathological phenomena as well as in scaffold-based tissue engineering. Cell migration behavior is known to be governed by biochemical stimuli and cellular interactions. Biophysical processes associated with interactions between the cell and its surrounding extracellular matrix may also play a significant role in regulating migration. Although biophysical properties of two-dimensional substrates have been shown to significantly influence cell migration, elucidating factors governing migration in a three-dimensional environment is a relatively new avenue of research. Here, we investigate the effect of the three-dimensional microstructure, specifically the pore size and Young's modulus, of collagen-glycosaminoglycan scaffolds on the migratory behavior of individual mouse fibroblasts. We observe that the fibroblast migration, characterized by motile fraction as well as locomotion speed, decreases as scaffold pore size increases across a range from 90 to 150 mum. Directly testing the effects of varying strut Young's modulus on cell motility showed a biphasic relationship between cell speed and strut modulus and also indicated that mechanical factors were not responsible for the observed effect of scaffold pore size on cell motility. Instead, in-depth analysis of cell locomotion paths revealed that the distribution of junction points between scaffold struts strongly modulates motility. Strut junction interactions affect local directional persistence as well as cell speed at and away from the junctions, providing a new biophysical mechanism for the governance of cell motility by the extracellular microstructure.

Dependence of invadopodia function on collagen fiber spacing and cross-linking: computational modeling and experimental evidence

Invadopodia are subcellular organelles thought to be critical for extracellular matrix (ECM) degradation and the movement of cells through tissues. Here we examine invadopodia generation, turnover, and function in relation to two structural aspects of the ECM substrates they degrade: cross-linking and fiber density. We set up a cellular automaton computational model that simulates ECM penetration and degradation by invadopodia. Experiments with denatured collagen (gelatin) were used to calibrate the model and demonstrate the inhibitory effect of ECM cross-linking on invadopodia degradation and penetration. Incorporation of dynamic invadopodia behavior into the model amplified the effect of cross-linking on ECM degradation, and was used to model feedback from the ECM. When the model was parameterized with spatial fibrillar dimensions that closely matched the organization, in real life, of native ECM collagen into triple-helical monomers, microfibrils, and macrofibrils, little or no inhibition of invadopodia penetration was observed in simulations of sparse collagen gels, no matter how high the degree of cross-linking. Experimental validation, using live-cell imaging of invadopodia in cells plated on cross-linked gelatin, was consistent with simulations in which ECM cross-linking led to higher rates of both invadopodia retraction and formation. Analyses of invadopodia function from cells plated on cross-linked gelatin and collagen gels under standard concentrations were consistent with simulation results in which sparse collagen gels provided a weak barrier to invadopodia. These results suggest that the organization of collagen, as it may occur in stroma or in vitro collagen gels, forms gaps large enough so as to have little impact on invadopodia penetration/degradation. By contrast, dense ECM, such as gelatin or possibly basement membranes, is an effective obstacle to invadopodia penetration and degradation, particularly when cross-linked. These results provide a novel framework for further studies on ECM structure and modifications that affect invadopodia and tissue invasion by cells.

Cell mechanics of alveolar epithelial cells (AECs) and macrophages (AMs)

Cell mechanics provides an integrated view of many biological phenomena which are intimately related to cell structure and function. Because breathing constitutes a sustained motion synonymous with life, pulmonary cells are normally designed to support permanent cyclic stretch without breaking, while receiving mechanical cues from their environment. The authors study the mechanical responses of alveolar cells, namely epithelial cells and macrophages, exposed to well-controlled mechanical stress in order to understand pulmonary cell response and function. They discuss the principle, advantages and limits of a cytoskeleton-specific micromanipulation technique, magnetic bead twisting cytometry, potentially applicable in vivo. They also compare the pertinence of various models (e.g., rheological; power law) used to extract cell mechanical properties and discuss cell stress/strain hardening properties and cell dynamic response in relation to the structural tensegrity model. Overall, alveolar cells provide a pertinent model to study the biological processes governing cellular response to controlled stress or strain.

Mathematics of cell motility: have we got its number?

Mathematical and computational modeling is rapidly becoming an essential research technique complementing traditional experimental biological methods. However, lack of standard modeling methods, difficulties of translating biological phenomena into mathematical language, and differences in biological and mathematical mentalities continue to hinder the scientific progress. Here we focus on one area-cell motility-characterized by an unusually high modeling activity, largely due to a vast amount of quantitative, biophysical data, 'modular' character of motility, and pioneering vision of the area's experimental leaders. In this review, after brief introduction to biology of cell movements, we discuss quantitative models of actin dynamics, protrusion, adhesion, contraction, and cell shape and movement that made an impact on the process of biological discovery. We also comment on modeling approaches and open questions.

Modeling mechanosensing and its effect on the migration and proliferation of adherent cells

The behavior of normal adherent cells is influenced by the stiffness of the substrate they are anchored to. Cells are able to detect substrate mechanical properties by actively generating contractile forces and use this information to migrate and proliferate. In particular, the speed and direction of cell crawling, as well as the rate of cell proliferation, vary with the substrate compliance and prestrain. In this work, we present an active mechanosensing model based on an extension of the classical Hill's model for skeletal muscle behavior. We also propose a thermodynamical approach to model cell migration regulated by mechanical stimuli and a proliferation theory also depending on the mechanical environment. These contributions give rise to a conceptually simple mathematical formulation with a straightforward and inexpensive computational implementation, yielding results consistent with numerous experiments. The model can be a useful tool for practical applications in biology and medicine in situations where cell-substrate interaction as well as substrate mechanical behavior play an important role, such as the design of tissue engineering applications.

Modeling cell migration in 3D: Status and challenges

Cell migration is a multi-scale process that integrates signaling, mechanics and biochemical reaction kinetics. Various mathematical models accurately predict cell migration on 2D surfaces, but are unable to capture the complexities of 3D migration. Additionally, quantitative 3D cell migration models have been few and far between. In this review we look and characterize various mathematical models available in literature to predict cell migration in 3D matrices and analyze their strengths and possible changes to these models that could improve their predictive capabilities.

Contractile forces in tumor cell migration

Cancer is a deadly disease primarily because of the ability of tumor cells to spread from the primary tumor, to invade into the connective tissue, and to form metastases at distant sites. In contrast to cell migration on a planar surface where large cell tractions and contractile forces are not essential, tractions and forces are thought to be crucial for overcoming the resistance and steric hindrance of a dense three-dimensional connective tissue matrix. In this review, we describe recently developed biophysical tools, including 2-D and 3-D traction microscopy to measure contractile forces of cells. We discuss evidence indicating that tumor cell invasiveness is associated with increased contractile force generation.

Integrative mathematical oncology

Cancer research attracts broad resources and scientists from many disciplines, and has produced some impressive advances in the treatment and understanding of this disease. However, a comprehensive mechanistic view of the cancer process remains elusive. To achieve this it seems clear that one must assemble a physically integrated team of interdisciplinary scientists that includes mathematicians, to develop mathematical models of tumorigenesis as a complex process. Examining these models and validating their findings by experimental and clinical observations seems to be one way to reconcile molecular reductionist with quantitative holistic approaches and produce an integrative mathematical oncology view of cancer progression.

An integrated systems biology approach to understanding the rules of keratinocyte colony formation

Closely coupled in vitro and in virtuo models have been used to explore the self-organization of normal human keratinocytes (NHK). Although it can be observed experimentally, we lack the tools to explore many biological rules that govern NHK self-organization. An agent-based computational model was developed, based on rules derived from literature, which predicts the dynamic multicellular morphogenesis of NHK and of a keratinocyte cell line (HaCat cells) under varying extracellular Ca++ concentrations. The model enables in virtuo exploration of the relative importance of biological rules and was used to test hypotheses in virtuo which were subsequently examined in vitro. Results indicated that cell-cell and cell-substrate adhesions were critically important to NHK self-organization. In contrast, cell cycle length and the number of divisions that transit-amplifying cells could undergo proved non-critical to the final organization. Two further hypotheses, to explain the growth behaviour of HaCat cells, were explored in virtuo-an inability to differentiate and a differing sensitivity to extracellular calcium. In vitro experimentation provided some support for both hypotheses. For NHKs, the prediction was made that the position of stem cells would influence the pattern of cell migration post-wounding. This was then confirmed experimentally using a scratch wound model.

Computer simulations in connective tissue research: successes and challenges

Connective tissue research traditionally has been experimental in nature, with computational tools used primarily for data analysis and statistical treatment. However, recent developments in mechanics, chemistry, and physics have provided researchers with new and rich computational tools to study numerous high value problems in the field. Among the problems being tackled, complex cell-matrix interactions have attracted significant interest. Researchers have started to address issues at the molecular, macromolecular, cellular, and tissue level through a combination of theory, simulation, and analysis. While there have been numerous successes in this field, new research has highlighted newer and bigger challenges and a close connection between experiment and simulation is needed to tackle high value problems in connective tissue biology.

Cell responses to the mechanochemical microenvironment--implications for regenerative medicine and drug delivery

Soft-tissue cells are surprisingly sensitive to the elasticity of their microenvironment, suggesting that traditional culture plastic and glass are less relevant to tissue regeneration and chemotherapeutics than might be achieved. Cells grown on gels that mimic the elasticity of tissue reveal a significant influence of matrix elasticity on adhesion, cytoskeletal organization, and even the differentiation of human adult derived stem cells. Cellular forces and feedback are keys to how cells feel their mechanical microenvironment, but detailed molecular mechanisms are still being elucidated. This review summarizes our initial findings for multipotent stem cells and also the elasticity-coupled effects of drugs on cancer cells and smooth muscle cells. The drugs include the contractility inhibitor blebbistatin, the proliferation inhibitor mitomycin C, an apoptotis-inducing antibody against CD47, and the translation inhibitor cycloheximide. The differential effects not only lend insight into mechano-sensing of the substrate by cells, but also have important implications for regeneration and molecular therapies.

A simple immune system simulation reveals optimal movement and cell density parameters for successful target clearance

We report here a simple simulation of the immune system in which we analysed the behaviour of responder cells in the presence of target cells. Variable parameters determined the behaviour of the cells within the simulation, and many simulations using the same parameters ensured that statistical variability was achieved. The model demonstrated that high mobility of the target or responder cells produced a more robust response, and that clearance by the immune system was favoured when effector cells moved rapidly compared with the target cells. Therefore, the high motility coefficients exhibited by T cells studied in vivo may play a role in optimizing the effector response to pathogens. Surprisingly, when the number density of responding cells was increased, target cell numbers were limited more effectively, but there was an increased likelihood of a prolonged response.

The third dimension bridges the gap between cell culture and live tissue

Moving from cell monolayers to three-dimensional (3D) cultures is motivated by the need to work with cellular models that mimic the functions of living tissues. Essential cellular functions that are present in tissues are missed by 'petri dish'-based cell cultures. This limits their potential to predict the cellular responses of real organisms. However, establishing 3D cultures as a mainstream approach requires the development of standard protocols, new cell lines and quantitative analysis methods, which include well-suited three-dimensional imaging techniques. We believe that 3D cultures will have a strong impact on drug screening and will also decrease the use of laboratory animals, for example, in the context of toxicity assays.

The emergence of ECM mechanics and cytoskeletal tension as important regulators of cell function

The ability to harvest and maintain viable cells from mammalian tissues represented a critical advance in biomedical research, enabling individual cells to be cultured and studied in molecular detail. However, in these traditional cultures, cells are grown on rigid glass or polystyrene substrates, the mechanical properties of which often do not match those of the in vivo tissue from which the cells were originally derived. This mechanical mismatch likely contributes to abrupt changes in cellular phenotype. In fact, it has been proposed that mechanical changes in the cellular microenvironment may alone be responsible for driving specific cellular behaviors. Recent multidisciplinary efforts from basic scientists and engineers have begun to address this hypothesis more explicitly by probing the effects of ECM mechanics on cell and tissue function. Understanding the consequences of such mechanical changes is physiologically relevant in the context of a number of tissues in which altered mechanics may either correlate with or play an important role in the onset of pathology. Examples include changes in the compliance of blood vessels associated with atherosclerosis and intimal hyperplasia, as well as changes in the mechanical properties of developing tumors. Compelling evidence from 2-D in vitro model systems has shown that substrate mechanical properties induce changes in cell shape, migration, proliferation, and differentiation, but it remains to be seen whether or not these same effects translate to 3-D systems or in vivo. Furthermore, the molecular "mechanotransduction" mechanisms by which cells respond to changes in ECM mechanics remain unclear. Here, we provide some historical context for this emerging area of research, and discuss recent evidence that regulation of cytoskeletal tension by changes in ECM mechanics (either directly or indirectly) may provide a critical switch that controls cell function.

Cell morphology and migration linked to substrate rigidity

A mathematical model, based on thermodynamics, was developed to demonstrate how substrate rigidity influences cell morphology and migration. The mechanisms by which substrate rigidity are translated into cell-morphological changes and cell movement are described. The model takes into account the competition between the elastic energies in the cell-substrate system and work of adhesion at the cell periphery. The cell morphology and migration are dictated by the minimum of the total free energy of the cell-substrate system. By using this model, reported experimental observations on cell morphological changes and migration can be better understood with a theoretical basis. In addition, these observations can be more accurately correlated with the variation of substrate rigidity. This study indicates that the activity of the adherent cell is dependent not only on the substrate rigidity but also is correlated with the relative rigidity between the cell and substrate. Moreover, the study suggests that the cell stiffness can be estimated based on the substrate stiffness corresponding to the change of trend in morphological stability.

Mathematical Models of Cell Motility

Cell motility is an essential biological action in the creation, operation and maintenance of our bodies. Developing mathematical models elucidating cell motility will greatly advance our understanding of this fundamental biological process. With accurate models it is possible to explore many permutations of the same event and concisely investigate their outcome. While great advancements have been made in experimental studies of cell motility, it now has somewhat fallen on mathematical models to taking a leading role in future developments. The obvious reason for this is the complexity of cell motility. Employing the processing power of today's computers will give researches the ability to run complex biophysical and biochemical scenarios, without the inherent difficulty and time associated with in vitro investigations. Before any great advancement can be made, the basics of cell motility will have to be well-defined. Without this, complicated mathematical models will be hindered by their inherent conjecture. This review will look at current mathematical investigations of cell motility, explore the reasoning behind such work and conclude with how best to advance this interesting and challenging research area.

The forces behind cell movement

Cell movement is a complex phenomenon primarily driven by the actin network beneath the cell membrane, and can be divided into three general components: protrusion of the leading edge of the cell, adhesion of the leading edge and deadhesion at the cell body and rear, and cytoskeletal contraction to pull the cell forward. Each of these steps is driven by physical forces generated by unique segments of the cytoskeleton. This review examines the specific physics underlying these phases of cell movement and the origins of the forces that drive locomotion.

Rational design of hydrogels for tissue engineering: Impact of physical factors on cell behavior

When designing suitable biomaterials for tissue-engineering applications, biological and chemical parameters are frequently taken into account, while the equally important physical design variables have often been neglected. For a rational design of biomaterials, however, all variables influencing cell function and tissue morphogenesis have to be considered. This review will stress the development of cross-linked hydrogels and outline the impact of their physical properties on cell function and tissue morphogenesis. In the first part, the principles of cellular mechanosensitivity, as well as the influence of substrate mechanics on cell behavior, will be discussed. Afterwards, methods to characterize the mechanical properties of biomaterials will be presented. The subsequent chapters will address hydrogels that allow for the control of their physical qualities followed by a discussion of their use in tissue-engineering applications.

Understanding effects of matrix protease and matrix organization on directional persistence and translational speed in three-dimensional cell migration

Recent studies have shown significant differences in migration mechanisms between two- and three-dimensional environments. While experiments have suggested a strong dependence of in vivo migration on both structure and proteolytic activity, the underlying biophysics of such dependence has not been studied adequately. In addition, the existing models of persistent random walk migration are primarily based on two-dimensional movement and do not account for the effect of proteolysis or matrix inhomogeneity. Using lattice Monte Carlo methods, we present a model to study the role of matrix metallo-proteases (MMPs) on directional persistence and speed. The simulations account for a given cell's ability to deform as well as to digest the matrix as the cell moves in three dimensions. Our results show a bimodal dependence of speed and persistence on matrix pore size and suggest high sensitivity on MMP activity, which is in very good agreement with experimental studies carried out in 3D matrices.

Migration of tumor cells in 3D matrices is governed by matrix stiffness along with cell-matrix adhesion and proteolysis

Cell migration on 2D surfaces is governed by a balance between counteracting tractile and adhesion forces. Although biochemical factors such as adhesion receptor and ligand concentration and binding, signaling through cell adhesion complexes, and cytoskeletal structure assembly/disassembly have been studied in detail in a 2D context, the critical biochemical and biophysical parameters that affect cell migration in 3D matrices have not been quantitatively investigated. We demonstrate that, in addition to adhesion and tractile forces, matrix stiffness is a key factor that influences cell movement in 3D. Cell migration assays in which Matrigel density, fibronectin concentration, and beta1 integrin binding are systematically varied show that at a specific Matrigel density the migration speed of DU-145 human prostate carcinoma cells is a balance between tractile and adhesion forces. However, when biochemical parameters such as matrix ligand and cell integrin receptor levels are held constant, maximal cell movement shifts to matrices exhibiting lesser stiffness. This behavior contradicts current 2D models but is predicted by a recent force-based computational model of cell movement in a 3D matrix. As expected, this 3D motility through an extracellular environment of pore size much smaller than cellular dimensions does depend on proteolytic activity as broad-spectrum matrix metalloproteinase (MMP) inhibitors limit the migration of DU-145 cells and also HT-1080 fibrosarcoma cells. Our experimental findings here represent, to our knowledge, discovery of a previously undescribed set of balances of cell and matrix properties that govern the ability of tumor cells to migration in 3D environments.

Capturing complex 3D tissue physiology in vitro

The emergence of tissue engineering raises new possibilities for the study of complex physiological and pathophysiological processes in vitro. Many tools are now available to create 3D tissue models in vitro, but the blueprints for what to make have been slower to arrive. We discuss here some of the 'design principles' for recreating the interwoven set of biochemical and mechanical cues in the cellular microenvironment, and the methods for implementing them. We emphasize applications that involve epithelial tissues for which 3D models could explain mechanisms of disease or aid in drug development.

Structure and dynamics of macrophage podosomes

Podosomes are short-lived, dynamic actin-rich adhesions classically formed in macrophages, osteoclasts and Src-transformed fibroblasts. Though related to the more commonly studied focal adhesions, sharing several structural and regulatory proteins, podosomes have distinct functional, structural, and dynamic characteristics. Here, we discuss current understanding of the function of podosomes in disparate cell types and how this relates to their structure and dynamics.

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.

Tissue cells feel and respond to the stiffness of their substrate

Normal tissue cells are generally not viable when suspended in a fluid and are therefore said to be anchorage dependent. Such cells must adhere to a solid, but a solid can be as rigid as glass or softer than a baby's skin. The behavior of some cells on soft materials is characteristic of important phenotypes; for example, cell growth on soft agar gels is used to identify cancer cells. However, an understanding of how tissue cells-including fibroblasts, myocytes, neurons, and other cell types-sense matrix stiffness is just emerging with quantitative studies of cells adhering to gels (or to other cells) with which elasticity can be tuned to approximate that of tissues. Key roles in molecular pathways are played by adhesion complexes and the actinmyosin cytoskeleton, whose contractile forces are transmitted through transcellular structures. The feedback of local matrix stiffness on cell state likely has important implications for development, differentiation, disease, and regeneration.