A computational model of cell migration coupling the growth of focal adhesions with oscillatory cell protrusions

Learning dynamical models of single and collective cell migration: a review

Single and collective cell migration are fundamental processes critical for physiological phenomena ranging from embryonic development and immune response to wound healing and cancer metastasis. To understand cell migration from a physical perspective, a broad variety of models for the underlying physical mechanisms that govern cell motility have been developed. A key challenge in the development of such models is how to connect them to experimental observations, which often exhibit complex stochastic behaviours. In this review, we discuss recent advances in data-driven theoretical approaches that directly connect with experimental data to infer dynamical models of stochastic cell migration. Leveraging advances in nanofabrication, image analysis, and tracking technology, experimental studies now provide unprecedented large datasets on cellular dynamics. In parallel, theoretical efforts have been directed towards integrating such datasets into physical models from the single cell to the tissue scale with the aim of conceptualising the emergent behaviour of cells. We first review how this inference problem has been addressed in both freely migrating and confined cells. Next, we discuss why these dynamics typically take the form of underdamped stochastic equations of motion, and how such equations can be inferred from data. We then review applications of data-driven inference and machine learning approaches to heterogeneity in cell behaviour, subcellular degrees of freedom, and to the collective dynamics of multicellular systems. Across these applications, we emphasise how data-driven methods can be integrated with physical active matter models of migrating cells, and help reveal how underlying molecular mechanisms control cell behaviour. Together, these data-driven approaches are a promising avenue for building physical models of cell migration directly from experimental data, and for providing conceptual links between different length-scales of description.

Modeling cell protrusion predicts how myosin II and actin turnover affect adhesion-based signaling

Orchestration of cell migration is essential for development, tissue regeneration, and the immune response. This dynamic process integrates adhesion, signaling, and cytoskeletal subprocesses across spatial and temporal scales. In mesenchymal cells, adhesion complexes bound to extracellular matrix mediate both biochemical signal transduction and physical interaction with the F-actin cytoskeleton. Here, we present a mathematical model that offers insight into both aspects, considering spatiotemporal dynamics of nascent adhesions, active signaling molecules, mechanical clutching, actin treadmilling, and nonmuscle myosin II contractility. At the core of the model is a positive feedback loop, whereby adhesion-based signaling promotes generation of barbed ends at, and protrusion of, the cell's leading edge, which in turn promotes formation and stabilization of nascent adhesions. The model predicts a switch-like transition and optimality of membrane protrusion, determined by the balance of actin polymerization and retrograde flow, with respect to extracellular matrix density. The model, together with new experimental measurements, explains how protrusion can be modulated by mechanical effects (nonmuscle myosin II contractility and adhesive bond stiffness) and F-actin turnover.

Enhanced Piezoelectric Fibered Extracellular Matrix to Promote Cardiomyocyte Maturation and Tissue Formation: A 3D Computational Model

Mechanical and electrical stimuli play a key role in tissue formation, guiding cell processes such as cell migration, differentiation, maturation, and apoptosis. Monitoring and controlling these stimuli on in vitro experiments is not straightforward due to the coupling of these different stimuli. In addition, active and reciprocal cell-cell and cell-extracellular matrix interactions are essential to be considered during formation of complex tissue such as myocardial tissue. In this sense, computational models can offer new perspectives and key information on the cell microenvironment. Thus, we present a new computational 3D model, based on the Finite Element Method, where a complex extracellular matrix with piezoelectric properties interacts with cardiac muscle cells during the first steps of tissue formation. This model includes collective behavior and cell processes such as cell migration, maturation, differentiation, proliferation, and apoptosis. The model has employed to study the initial stages of in vitro cardiac aggregate formation, considering cell-cell junctions, under different extracellular matrix configurations. Three different cases have been purposed to evaluate cell behavior in fibered, mechanically stimulated fibered, and mechanically stimulated piezoelectric fibered extra-cellular matrix. In this last case, the cells are guided by the coupling of mechanical and electrical stimuli. Accordingly, the obtained results show the formation of more elongated groups and enhancement in cell proliferation.

Cell-substrate mechanics guide collective cell migration through intercellular adhesion: a dynamic finite element cellular model

During the process of tissue formation and regeneration, cells migrate collectively while remaining connected through intercellular adhesions. However, the roles of cell-substrate and cell-cell mechanical interactions in regulating collective cell migration are still unclear. In this study, we employ a newly developed finite element cellular model to study collective cell migration by exploring the effects of mechanical feedback between cell and substrate and mechanical signal transmission between adjacent cells. Our viscoelastic model of cells consists many triangular elements and is of high resolution. Cadherin adhesion between cells is modeled explicitly as linear springs at subcellular level. In addition, we incorporate a mechano-chemical feedback loop between cell-substrate mechanics and Rac-mediated cell protrusion. Our model can reproduce a number of experimentally observed patterns of collective cell migration during wound healing, including cell migration persistence, separation distance between cell pairs and migration direction. Moreover, we demonstrate that cell protrusion determined by the cell-substrate mechanics plays an important role in guiding persistent and oriented collective cell migration. Furthermore, this guidance cue can be maintained and transmitted to submarginal cells of long distance through intercellular adhesions. Our study illustrates that our finite element cellular model can be employed to study broad problems of complex tissue in dynamic changes at subcellular level.

Cell motility dependence on adhesive wetting

Adhesive cell-substrate interactions are crucial for cell motility and are responsible for the necessary traction that propels cells. These interactions can also change the shape of the cell, analogous to liquid droplet wetting on adhesive substrates. To address how these shape changes affect cell migration and cell speed we model motility using deformable, 2D cross-sections of cells in which adhesion and frictional forces between cell and substrate can be varied separately. Our simulations show that increasing the adhesion results in increased spreading of cells and larger cell speeds. We propose an analytical model which shows that the cell speed is inversely proportional to an effective height of the cell and that increasing this height results in increased internal shear stress. The numerical and analytical results are confirmed in experiments on motile eukaryotic cells.

Center or periphery? Modeling the effects of focal adhesion placement during cell spreading

Focal adhesions are often observed at the cell's periphery. We provide an explanation for this observation using a system-level mathematical model of a cell interacting with a two-dimensional substrate. The model describes the biological cell as a hypoelastic continuum material whose behavior is coupled to a deformable, linear elastic substrate via focal adhesions that are represented by collections of linear elastic attachments between the cell and the substrate. The evolution of the focal adhesions is coupled to local intracellular stresses which arise from mechanical cell-substrate interactions. Using this model we show that the cell has at least three mechanisms through which it can control its intracellular stresses: focal adhesion position, size, and attachment strength. We also propose that one reason why focal adhesions are typically located on the cell periphery instead of its center is because peripheral focal adhesions allow the cell to be more sensitive to changes in the microenvironment. This increased sensitivity is caused by the fact that peripherally located focal adhesions allow the cells to modulate its intracellular properties over a much larger portion of the cell area.

Modeling, signaling and cytoskeleton dynamics: integrated modeling-experimental frameworks in cell migration

Cell migration is a complex and multistep process involved in homeostasis maintenance, morphogenesis, and disease development, such as cancer metastasis. Modeling cell migration and the relevant cytoskeleton dynamics have profound implications for studying fundamental development and disease diagnosis. This review focuses on some recent models of both cell migration and migration-related cytoskeleton dynamics, addressing issues such as the difference between amoeboid and mesenchymal migration modes, and between single-cell migration and collective cell migration. The review also highlights the computational integration among variable external cues, especially the biochemical and mechanical signaling that affects cell migration. Finally, we aim to identify the gaps in our current knowledge and potential strategies to develop integrated modeling-experimental frameworks for multiscale behavior integrating gene expression, cell signaling, mechanics, and multicellular dynamics. WIREs Syst Biol Med 2017, 9:e1365. doi: 10.1002/wsbm.1365 For further resources related to this article, please visit the WIREs website.

Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

Complex matrix remodeling and durotaxis can emerge from simple rules for cell-matrix interaction in agent-based models

Using a top-down approach, an agent-based model was developed within NetLogo where cells and extracellular matrix (ECM) fibers were composed of multiple agents to create deformable structures capable of exerting, reacting to, and transmitting mechanical force. At the beginning of the simulation, long fibers were randomly distributed and cross linked. Throughout the simulation, imposed rules allowed cells to exert traction forces by extending pseudopodia, binding to fibers and pulling them towards the cell. Simulated cells remodeled the fibrous matrix to change both the density and alignment of fibers and migrated within the matrix in ways that are consistent with previous experimental work. For example, cells compacted the matrix in their pericellular regions much more than the average compaction experienced for the entire matrix (696% versus 21%). Between pairs of cells, the matrix density increased (by 92%) and the fibers became more aligned (anisotropy index increased from 0.45 to 0.68) in the direction parallel to a line connecting the two cells, consistent with the "lines of tension" observed in experiments by others. Cells migrated towards one another at an average rate of ∼0.5 cell diameters per 10,000 arbitrary units (AU); faster migration occurred in simulations where the fiber density in the intercellular area was greater. To explore the potential contribution of matrix stiffness gradients in the observed migration (i.e., durotaxis), the model was altered to contain a regular lattice of fibers possessing a stiffness gradient and just a single cell. In these simulations cells migrated preferentially in the direction of increasing stiffness at a rate of ∼2 cell diameter per 10,000 AU. This work demonstrates that matrix remodeling and durotaxis, both complex phenomena, might be emergent behaviors based on just a few rules that control how a cell can interact with a fibrous ECM.

Motility efficiency and spatiotemporal synchronization in non-metastatic vs. metastatic breast cancer cells

Metastatic breast cancer cells move not only more rapidly and persistently than their non-metastatic variants but in doing so use the mechanical work of the cytoskeleton more efficiently. The efficiency of the cell motions is defined for entire cells (rather than parts of the cell membrane) and is related to the work expended in forming membrane protrusions and retractions. This work, in turn, is estimated by integrating the protruded and retracted areas along the entire cell perimeter and is standardized with respect to the net translocation of the cell. A combination of cross-correlation, Granger causality, and morphodynamic profiling analyses is then used to relate the efficiency to the cell membrane dynamics. In metastatic cells, the protrusions and retractions are highly "synchronized" both in space and in time and these cells move efficiently. In contrast, protrusions and retractions formed by non-metastatic cells are not "synchronized" corresponding to low motility efficiencies. Our work provides a link between the kinematics of cell motions and their energetics. It also suggests that spatiotemporal synchronization might be one of the hallmarks of invasiveness of cancerous cells.

Asymmetry between pushing and pulling for crawling cells

Eukaryotic cells possess motility mechanisms allowing them not only to self-propel but also to exert forces on obstacles (to push) and to carry cargoes (to pull). To study the inherent asymmetry between active pushing and pulling we model a crawling acto-myosin cell extract as a one-dimensional layer of active gel subjected to external forces. We show that pushing is controlled by protrusion and that the macroscopic signature of the protrusion dominated motility mechanism is concavity of the force-velocity relation. In contrast, pulling is driven by protrusion only at small values of the pulling force and it is replaced by contraction when the pulling force is sufficiently large. This leads to more complex convex-concave structure of the force-velocity relation; in particular, competition between protrusion and contraction can produce negative mobility in a biologically relevant range. The model illustrates active readjustment of the force generating machinery in response to changes in the dipole structure of external forces. The possibility of switching between complementary active mechanisms implies that if necessary "pushers" can replace "pullers" and vice versa.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

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.

Mathematical modeling of invadopodia formation

In invasive cancer cells, specialized sub-cellular membrane structures which carry out a pivotal process in cancer invasion, termed invadopodia, are observed. Invadopodia appear irregularly within the cytoplasm and their general shape is small punctuated finger-like protrusions with dimension up to several μm long. They may exist and persist on a timescale between several tens of minutes to one hour. The formation of invadopodia requires the integration of several processes that include actin reorganization, extracellular matrix (ECM) degradation, signaling processes through receptors such as the epidermal growth factor receptor (EGFR) and matrix metalloproteinase (MMP) synthesis and delivery to the location of the invading front. In this paper, we consider a mathematical model investigating the coupling of these fundamental processes, and we investigate how invadopodia appear in this model. We investigate the spatio-temporal dynamics of the model in two spatial dimensions by using numerical computational simulations. We show that in a special parameter region of the model, random fluctuations of ECM degradation and a positive feedback loop regarding the up-regulation of MMPs allow us to reproduce finger-like protrusions which have similar size and lifetime as invadopodia. This study provides a new insight into how invadopodia appear in cancer cells and why space and time scales exist for invadopodia.

Redundant mechanisms for stable cell locomotion revealed by minimal models

Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly constant shape and velocity. Here we use moving-boundary simulations to explore four different minimal mechanisms for cell locomotion: 1), a biophysical model for myosin contraction-driven motility; 2), a G-actin transport-limited motility model; 3), a simple model for Rac/Rho-regulated motility; and 4), a model that assumes that microtubule-based transport of vesicles to the leading edge limits the rate of protrusion. We show that all of these models, alone or in combination, are sufficient to produce half-moon steady shapes and movements that are characteristic of keratocytes, suggesting that these mechanisms may serve redundant and complementary roles in driving cell motility. Moving-boundary simulations demonstrate local and global stability of the motile cell shapes and make testable predictions regarding the dependence of shape and speed on mechanical and biochemical parameters. The models shed light on the roles of membrane-mediated area conservation and the coupling of mechanical and biochemical mechanisms in stabilizing motile cells.

Cell-biomaterial mechanical interaction in the framework of tissue engineering: insights, computational modeling and perspectives

Tissue engineering is an emerging field of research which combines the use of cell-seeded biomaterials both in vitro and/or in vivo with the aim of promoting new tissue formation or regeneration. In this context, how cells colonize and interact with the biomaterial is critical in order to get a functional tissue engineering product. Cell-biomaterial interaction is referred to here as the phenomenon involved in adherent cells attachment to the biomaterial surface, and their related cell functions such as growth, differentiation, migration or apoptosis. This process is inherently complex in nature involving many physico-chemical events which take place at different scales ranging from molecular to cell body (organelle) levels. Moreover, it has been demonstrated that the mechanical environment at the cell-biomaterial location may play an important role in the subsequent cell function, which remains to be elucidated. In this paper, the state-of-the-art research in the physics and mechanics of cell-biomaterial interaction is reviewed with an emphasis on focal adhesions. The paper is focused on the different models developed at different scales available to simulate certain features of cell-biomaterial interaction. A proper understanding of cell-biomaterial interaction, as well as the development of predictive models in this sense, may add some light in tissue engineering and regenerative medicine fields.

Spatial organization of mesenchymal stem cells in vitro--results from a new individual cell-based model with podia

Therapeutic application of mesenchymal stem cells (MSC) requires their extensive in vitro expansion. MSC in culture typically grow to confluence within a few weeks. They show spindle-shaped fibroblastoid morphology and align to each other in characteristic spatial patterns at high cell density. We present an individual cell-based model (IBM) that is able to quantitatively describe the spatio-temporal organization of MSC in culture. Our model substantially improves on previous models by explicitly representing cell podia and their dynamics. It employs podia-generated forces for cell movement and adjusts cell behavior in response to cell density. At the same time, it is simple enough to simulate thousands of cells with reasonable computational effort. Experimental sheep MSC cultures were monitored under standard conditions. Automated image analysis was used to determine the location and orientation of individual cells. Our simulations quantitatively reproduced the observed growth dynamics and cell-cell alignment assuming cell density-dependent proliferation, migration, and morphology. In addition to cell growth on plain substrates our model captured cell alignment on micro-structured surfaces. We propose a specific surface micro-structure that according to our simulations can substantially enlarge cell culture harvest. The 'tool box' of cell migratory behavior newly introduced in this study significantly enhances the bandwidth of IBM. Our approach is capable of accommodating individual cell behavior and collective cell dynamics of a variety of cell types and tissues in computational systems biology.

Mechanisms of Cell Propulsion by Active Stresses

The mechanisms by which cytoskeletal flows and cell-substrate interactions interact to generate cell motion are explored using a simplified model of the cytoskeleton as a viscous gel containing active stresses. This model yields explicit general results relating cell speed and traction forces to the distributions of active stress and cell-substrate friction. It is found that 1) the cell velocity is given by a function that quantifies the asymmetry of the active-stress distribution, 2) gradients in cell-substrate friction can induce motion even when the active stresses are symmetrically distributed, 3) the traction-force dipole is enhanced by protrusive stresses near the cell edges or contractile stresses near the center of the cell, and 4) the cell velocity depends biphasically on the cell-substrate adhesion strength if active stress is enhanced by adhesion. Specific experimental tests of the calculated dependences are proposed.

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.

A model of fibroblast motility on substrates with different rigidities

To function efficiently in the body, the biological cells must have the ability to sense the external environment. Mechanosensitivity toward the extracellular matrix was identified as one of the sensing mechanisms affecting cell behavior. It was shown experimentally that a fibroblast cell prefers locomoting over the stiffer substrate when given a choice between a softer and a stiffer substrate. In this article, we develop a discrete model of fibroblast motility with substrate-rigidity sensing. Our model allows us to understand the interplay between the cell-substrate sensing and the cell biomechanics. The model cell exhibits experimentally observed substrate rigidity sensing, which allows us to gain additional insights into the cell mechanosensitivity.

Stochastic model of integrin-mediated signaling and adhesion dynamics at the leading edges of migrating cells

Productive cell migration requires the spatiotemporal coordination of cell adhesion, membrane protrusion, and actomyosin-mediated contraction. Integrins, engaged by the extracellular matrix (ECM), nucleate the formation of adhesive contacts at the cell's leading edge(s), and maturation of nascent adhesions to form stable focal adhesions constitutes a functional switch between protrusive and contractile activities. To shed additional light on the coupling between integrin-mediated adhesion and membrane protrusion, we have formulated a quantitative model of leading edge dynamics combining mechanistic and phenomenological elements and studied its features through classical bifurcation analysis and stochastic simulation. The model describes in mathematical terms the feedback loops driving, on the one hand, Rac-mediated membrane protrusion and rapid turnover of nascent adhesions, and on the other, myosin-dependent maturation of adhesions that inhibit protrusion at high ECM density. Our results show that the qualitative behavior of the model is most sensitive to parameters characterizing the influence of stable adhesions and myosin. The major predictions of the model, which we subsequently confirmed, are that persistent leading edge protrusion is optimal at an intermediate ECM density, whereas depletion of myosin IIA relieves the repression of protrusion at higher ECM density.

Stochastic Dynamics of Membrane Protrusion Mediated by the DOCK180/Rac Pathway in Migrating Cells

Cell migration is regulated by processes that control adhesion to extracellular matrix (ECM) and force generation. While our fundamental understanding of how these control mechanisms are actuated at the molecular level (signal transduction) has been refined over many years, appreciation of their dynamics has grown more recently. Here, we formulate and analyze by stochastic simulation a quantitative model of signaling mediated by the integrin family of adhesion receptors. Nascent adhesions foster the activation of the small GTPase Rac by at least two distinct signaling pathways, one of which involves tyrosine phosphorylation of the scaffold protein paxillin and formation of multiprotein complexes containing the guanine nucleotide exchange factor DOCK180. Active Rac promotes protrusion of the cell's leading edge, which in turn enhances the rate of nascent adhesion nucleation; we call this feedback mechanism the core protrusion cycle. Protrusion is antagonized by stable adhesions, which form by myosin-dependent maturation of nascent adhesions, and we propose here a feedforward mechanism mediated by the tyrosine kinase c-Src by which this antagonism is regulated so as to allow transient protrusion at higher densities of ECM. We show that this "buffering of inhibition" mechanism, coupled with the core protrusion cycle, is capable of tuning the frequencies of protrusion and adhesion maturation events.

What makes cells move: requirements and obstacles for spontaneous cell motility

Movement of individual cells and of cellular cohorts, chains or sheets requires physical forces that are established through interactions of cells with their environment. In vivo, migration occurs extensively during embryonic development and in adults during wound healing and tumorigenesis. In order to identify the molecular events involved in cell movement, in vitro systems have been developed. These have contributed to the definition of a number of molecular pathways put into play in the course of migratory behaviours, such as mesenchymal and amoeboid movement. More recently, our knowledge of migratory modes has been enriched by analyses of cells exploring and moving through three-dimensional (3D) matrices. While the cells' morphologies differ in 2D and 3D environments, the basic mechanisms that put a cellular body into motion are remarkably similar. Thus, in both 2D and 3D, the polarity of the migrating cell is initially defined by a specific subcellular localization of signalling molecules and components of molecular machines required for motion. While the polarization can be initiated either in response to extracellular signalling or be a chance occurrence, it is reinforced and sustained by positive feedback loops of signalling molecules. Second, adhesion to a substratum is necessary to generate forces that will propel the cell engaged in either mesenchymal or ameboid migration. For collective cell movement, intercellular coordination constitutes an additional requirement: a cell cohort remains stationary if individual cells pull in opposite directions. Finally, the availability of space to move into is a general requirement to set cells into motion. Lack of free space is probably the main obstacle for migration of most healthy cells in an adult multicellular organism. Thus, the requirements for cell movement are both intrinsic to the cell, involving coordinated signalling and interactions with molecular machines, and extrinsic, imposed by the physicochemical nature of the environment. In particular, the geometry and stiffness of the support act on a range of signalling pathways that induce specific cell migratory responses. These issues are discussed in the present review in the context of published work and our own data on collective migration of hepatocyte cohorts.

On the effect of substrate curvature on cell mechanics

Cell movement on a substrate or within the extracellular matrix is the phenomenological response to a biochemical signals' cascade transcripted into biophysical processes and viceversa. The process is complex in nature, including different length scales from the whole cell to organelle and protein levels, where substrate/ECM curvature has been shown to play an important role on cell's behavior. From a macroscopic perspective, the cytoskeleton may be modeled as a continuum body unbalanced by internal protein motors. In this work, we propose a cell constitutive model to simulate cell attachment on curved substrates, activated by contractile forces. We first analyze a single fiber bundle composed by microtubules, actin filaments and myosin machinery. Then, the model is macroscopically extended to the cytoskeletal level using homogenization. Substrate curvature has two implications in our model: (i) it forces fibers to work in a curved (bent) position and (ii) it eventually creates a pre-deformation state in the cytoskeleton. Interestingly, the model shows higher contractile force inhibition as curvature increases when implemented over different substrate morphologies, being this consistent with experimental results. The presented model may result useful in many new regenerative medicine techniques, miniaturized experimental tests, or to analyze cell behavior on manufactured nanoscaffolds for tissue engineering.

Multi-scale modelling of cancer cell intravasation: the role of cadherins in metastasis

Transendothelial migration is a crucial process of the metastatic cascade in which a malignant cell attaches itself to the endothelial layer forming the inner wall of a blood or lymph vessel and creates a gap through which it enters into the bloodstream (or lymphatic system) and then is transported to distant parts of the body. In this process both biological pathways involving cell adhesion molecules such as VE-cadherin and N-cadherin, and the biophysical properties of the cells play an important role. In this paper, we present one of the first mathematical models considering the problem of cancer cell intravasation. We use an individual force-based multi-scale approach which accounts for intra- and inter-cellular protein pathways and for the physical properties of the cells, and a modelling framework which accounts for the biological shape of the vessel. Using our model, we study the influence of different protein pathways in the achievement of transendothelial migration and give quantitative simulation results comparable with real experiments.

Cortical factor feedback model for cellular locomotion and cytofission

Eukaryotic cells can move spontaneously without being guided by external cues. For such spontaneous movements, a variety of different modes have been observed, including the amoeboid-like locomotion with protrusion of multiple pseudopods, the keratocyte-like locomotion with a widely spread lamellipodium, cell division with two daughter cells crawling in opposite directions, and fragmentations of a cell to multiple pieces. Mutagenesis studies have revealed that cells exhibit these modes depending on which genes are deficient, suggesting that seemingly different modes are the manifestation of a common mechanism to regulate cell motion. In this paper, we propose a hypothesis that the positive feedback mechanism working through the inhomogeneous distribution of regulatory proteins underlies this variety of cell locomotion and cytofission. In this hypothesis, a set of regulatory proteins, which we call cortical factors, suppress actin polymerization. These suppressing factors are diluted at the extending front and accumulated at the retracting rear of cell, which establishes a cellular polarity and enhances the cell motility, leading to the further accumulation of cortical factors at the rear. Stochastic simulation of cell movement shows that the positive feedback mechanism of cortical factors stabilizes or destabilizes modes of movement and determines the cell migration pattern. The model predicts that the pattern is selected by changing the rate of formation of the actin-filament network or the threshold to initiate the network formation.