Mathematical model for the effects of adhesion and mechanics on cell migration speed

Matrix strains induced by cells: Computing how far cells can feel

Many tissue cells exert contractile forces that mechanically couples them to elastic matrices and that influence cell adhesion, cytoskeletal organization, and even cell differentiation. However, strains within the depths of matrices are often unclear and are likely relevant not only to the fact that some matrices such as so-called basement membranes are thin relative to cell dimensions but also to defining how far cells can 'feel'. Here we briefly present experimental results for cell spreading on thin, ligand-coated gels and for prestress in stem cells in relation to gel stiffness. We then introduce a finite element computation in which a cell is placed on an elastic matrix, while matrix elasticity and thickness are varied in order to compute and compare elastostatic deformations within the matrix. Average interfacial strains between cell and matrix show large deviations only when soft matrices are a fraction of the height and width of a cell, proving consistent with experiments. Three-dimensional (3D) cell morphologies that model stem cell-derived neurons, myoblasts, and osteoblasts show that a cylinder-shaped myoblast induces the highest strains, consistent with the prominent contractility of muscle. Groups of such cells show a weak crosstalk in matrix strains, but the cells must be much closer than a cell-width. Cells thus feel on length scales closer to that of adhesions than on cellular scales or higher.

Positively and negatively modulating cell adhesion to type I collagen via peptide grafting

The biophysical interactions between cells and type I collagen are controlled by the level of cell adhesion, which is dictated primarily by the density of ligands on collagen and the density of integrin receptors on cells. The native adhesivity of collagen was modulated by covalently grafting glycine-arginine-glycine-aspartic acid-serine (GRGDS), which includes the bioactive RGD sequence, or glycine-arginine-aspartic acid-glycine-serine (GRDGS), which includes the scrambled RDG sequence, to collagen with the hetero-bifunctional coupling agent 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide. The peptide-grafted collagen self-assembled into a fibrillar gel with negligible changes in gel structure and rheology. Rat dermal fibroblasts (RDFs) and human smooth muscle cells demonstrated increased levels of adhesion on gels prepared from RGD-grafted collagen, and decreased levels of adhesion on RDG-grafted collagen. Both cell types demonstrated an increased ability to compact free-floating RGD-grafted collagen gels, and an impaired ability to compact RDG-grafted gels. RDF migration on and within collagen was increased with RDG-grafted collagen and decreased with RGD-grafted collagen, and dose-response experiments indicated a biphasic response of RDF migration to adhesion. Smooth muscle cells demonstrated similar, though not statistically significant, trends. The ability to both positively and negatively modulate cell adhesion to collagen increases the versatility of this natural biomaterial for regenerative therapies.

Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications

Resorting to a multiphase modelling framework, tumours are described here as a mixture of tumour and host cells within a porous structure constituted by a remodelling extracellular matrix (ECM), which is wet by a physiological extracellular fluid. The model presented in this article focuses mainly on the description of mechanical interactions of the growing tumour with the host tissue, their influence on tumour growth, and the attachment/detachment mechanisms between cells and ECM. Starting from some recent experimental evidences, we propose to describe the interaction forces involving the extracellular matrix via some concepts coming from viscoplasticity. We then apply the model to the description of the growth of tumour cords and the formation of fibrosis.

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.

Optimum size of a molecular bond cluster in adhesion

The strength of a bonded interface is considered for the case in which bonding is the result of clusters of discrete bonds distributed along the interface. Assumptions appropriate for the case of adhesion of biological cells to an extracellular matrix are introduced as a basis for the discussion. It is observed that those individual bonds nearest to the edges of a cluster are necessarily subjected to disproportionately large forces in transmitting loads across the interface, in analogy with well-known behavior in elastic crack mechanics. Adopting Bell's model for the kinetics of bond response under force, a stochastic model leading to a dependence of interface strength on cluster size is developed and analyzed. On the basis of this model, it is demonstrated that there is an optimum cluster size for maximum strength. This size arises from the competition between the nonuniform force distribution among bonds, which tends to promote smaller clusters, and stochastic response allowing bond reformation, which tends to promote larger clusters. The model results have been confirmed by means of direct Monte Carlo simulations. This analysis may be relevant to the observation that mature focal adhesion zones in cell bonding are found to have a relatively uniform size.

Changes in the magnitude and distribution of forces at different Dictyostelium developmental stages

The distribution of forces exerted by migrating Dictyostelium amebae at different developmental stages was measured using traction force microscopy. By using very soft polyacrylamide substrates with a high fluorescent bead density, we could measure stresses as small as 30 Pa. Remarkable differences exist both in term of the magnitude and distribution of forces in the course of development. In the vegetative state, cells present cyclic changes in term of speed and shape between an elongated form and a more rounded one. The forces are larger in this first state, especially when they are symmetrically distributed at the front and rear edge of the cell. Elongated vegetative cells can also present a front-rear asymmetric force distribution with the largest forces in the crescent-shaped rear of the cell (uropod). Pre-aggregating cells, once polarized, only present this last kind of asymmetric distribution with the largest forces in the uropod. Except for speed, no cycle is observed. Neither the force distribution of pre-aggregating cells nor their overall magnitude are modified during chemotaxis, the later being similar to the one of vegetative cells (F(0) approximately 6 nN). On the contrary, both the force distribution and overall magnitude is modified for the fast moving aggregating cells. In particular, these highly elongated cells exert lower forces (F(0) approximately 3 nN). The location of the largest forces in the various stages of the development is consistent with the myosin II localization described in the literature for Dictyostelium (Yumura et al.,1984. J Cell Biol 99:894-899) and is confirmed by preliminary experiments using a GFP-myosin Dictyostelium strain.

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.

The role of integrin alpha D beta2 (CD11d/CD18) in monocyte/macrophage migration

Integrin alpha(D)beta(2) (CD11d/CD18) is a multiligand macrophage receptor with recognition specificity identical to that of the major myeloid cell-specific integrin alpha(M)beta(2) (CD11b/CD18, Mac-1). Despite its prominent upregulation on inflammatory macrophages, the role of alpha(D)beta(2) in monocyte and macrophage migration is unknown. In this study, we have generated model and natural cell lines expressing different densities of alpha(D)beta(2) and examined their migration to various extracellular matrix proteins. When expressed at a low density, alpha(D)beta(2) on the surface of recombinant HEK293 cells and murine IC-21 macrophages cooperates with beta(1)/beta(3) integrins to support cell migration. However, its increased expression on the alpha(D)beta(2)-expressing HEK293 cells and its upregulation by PMA on the IC-21 macrophages result in increased cell adhesiveness and inhibition of cell migration. Furthermore, ligation of alpha(D)beta(2) with anti-alpha(D) blocking antibodies restores beta(1)/beta(3)-driven cell migration by removing the excess alpha(D)beta(2)-mediated adhesive bonds. Consistent with in vitro data, increased numbers of inflammatory macrophages were recovered from the inflamed peritoneum of mice after the administration of anti-alpha(D) antibody. These results demonstrate that the density of alpha(D)beta(2) is critically involved in modulating macrophage adhesiveness and their migration, and suggest that low levels of alpha(D)beta(2) contribute to monocyte migration while alpha(D)beta(2) upregulation on differentiated macrophages may facilitate their retention at sites of inflammation.

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.

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.

Kinetic model for lamellipodal actin-integrin 'clutch' dynamics

In migrating cells, with especial prominence in lamellipodial protrusions at the cell front, highly dynamic connections are formed between the actin cytoskeleton and the extracellular matrix through linkages of integrin adhesion receptors to actin filaments via complexes of cytosolic "connector" proteins. Myosin-mediated contractile forces strongly influence the dynamic behavior of these adhesion complexes, apparently in two counter-acting ways: negatively as the cell-generated forces enhance complex dissociation, and at the same time positively as force-induced signaling can lead to strengthening of the linkage complexes. The net balance arising from this dynamic interplay is challenging to ascertain a priori, rendering experimental studies difficult to interpret and molecular manipulations of cell and/or environment difficult to predict. We have constructed a kinetics-based model governing the dynamic behavior of this system. We obtained ranges of parameter value sets yielding behavior consistent with that observed experimentally for 3T3 cells and for CHO cells, respectively. Model simulations are able to produce results for the effects of paxillin mutations on the turnover rate of actin/integrin linkages in CHO cells, which are consistent with recent literature reports. Overall, although this current model is quite simple it provides a useful foundation for more detailed models extending upon it.

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.

Surface rheology of hydrophobically modified PEG polymers associating with a phospholipid monolayer at the air-water interface

Surface rheology of irreversibly bound hydrophobically modified poly(ethylene glycol) (PEG) polymers (HMPEG) on a dipalmitoylphosphatidylcholine (DPPC) monolayer is investigated to determine attributes that may contribute to immune recognition. Previously, three comb-graft polymers (HMPEG136-DP3, HMPEG273-DP2.5, and HMPEG273-DP5) adsorbed on liposomes were examined for their strength of adsorption and protection from complement binding. The data supported an optimal ratio between the hydrophilicity of the PEG polymer and the number of hydrophobic anchors. The HMPEG polymers have different polymer brush thicknesses (4.2-5.9 nm) and levels of cooperativity (2.5-5 hydrophobes). The results indicate that an increased viscous force (above 0.25 mN s/m) at the surface may enable the polymers to shield liposomes from protein interactions. Similar rheological behavior is shown for all polymer architectures at low polymer surface coverage (0.5 mg/m2, in the mushroom regime), whereas at high surface coverage (>0.5 mg/m2, in the brush regime), we observe a structural dependence of the surface viscous forces at 40 mN/m. This threshold correlates with a 92% decrease in complement protein binding for liposomes coated with 1 mg/m2 HMPEG273-DP5. This may suggest that surface viscous forces play a role in reducing complement protein binding.

Role of myosin II activity and the regulation of myosin light chain phosphorylation in astrocytomas

The generation of contractile force mediated by actin-myosin interactions is essential for cell motility. Myosin activity is promoted by phosphorylation of myosin light chain (MLC). MLC phosphorylation in large part is controlled by kinases that are effectors of Rho family GTPases. Accordingly, in this study we examined the effects of ROCK and Rac1 inhibition on MLC phosphorylation in astrocytoma cells. We found that low concentrations of the ROCK inhibitor Y27632 increased the phosphorylation state of the Triton X-100 soluble fraction of MLC, whereas higher concentrations of Y27632 decreased soluble phospho-MLC. These effects of Y27632 were dependent on Rac1. The soluble form of phospho-MLC comprises about 10% of total phospho-MLC in control cells. Interestingly, ROCK inhibition led to a decrease in the phosphorylation state of total MLC, whereas Rac1 inhibition had little effect. Thus, the soluble form of MLC is differentially regulated by ROCK and Rac1 compared with MLC examined in a total cell extract. We also observed that astrocytoma migration is stimulated by low concentrations of the myosin II inhibitor blebbistatin. However, higher concentrations of blebbistatin inhibit migration leading us to believe that migration has a biphasic dependence on myosin II activity. Taken together, our data show that modulation of myosin II activity is important in determining optimal astrocytoma migration. In addition, these findings suggest that there are at least two populations of MLC that are differentially regulated.

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.

Experimental characterization and computational modelling of two-dimensional cell spreading for skeletal regeneration

Limited cell ingrowth is a major problem for tissue engineering and the clinical application of porous biomaterials as bone substitutes. As a first step, migration and proliferation of an interacting cell population can be studied in two-dimensional culture. Mathematical modelling is essential to generalize the results of these experiments and to derive the intrinsic parameters that can be used for predictions. However, a more thorough evaluation of theoretical models is hampered by limited experimental observations. In this study, experiments and image analysis methods were developed to provide a detailed spatial and temporal picture of how cell distributions evolve. These methods were used to quantify the migration and proliferation of skeletal cell types including MG63 and human bone marrow stromal cells (HBMSCs). The high level of detail with which the cell distributions were mapped enabled a precise assessment of the correspondence between experimental results and theoretical model predictions. This analysis revealed that the standard Fisher equation is appropriate for describing the migration behaviour of the HBMSC population, while for the MG63 cells a sharp front model is more appropriate. In combination with experiments, this type of mathematical model will prove useful in predicting cell ingrowth and improving strategies and control of skeletal tissue regeneration.

Oligodendrocyte lineage transcription factor 2 inhibits the motility of a human glial tumor cell line by activating RhoA

The basic helix-loop-helix transcription factor, oligodendrocyte lineage transcription factor 2 (OLIG2), is specifically expressed in the developing and mature central nervous system and plays an important role in oligodendrogenesis from neural progenitors. It is also expressed in various types of glial tumors, but rarely in glioblastoma. Although we previously showed that OLIG2 expression inhibits glioma cell growth, its role in tumorigenesis remains incompletely understood. Here, we investigated the effect of OLIG2 expression on the migration of the human glioblastoma cell line U12-1. In these cells, OLIG2 expression is controlled by the Tet-off system. Induction of OLIG2 expression inhibited both the migration and invasiveness of U12-1 cells. OLIG2 expression also increased the activity of the GTPase RhoA as well as inducing the cells to form stress fibers and focal adhesions. Experiments using short interfering RNA against p27(Kip1) revealed that up-regulation of the p27(Kip1) protein was not essential for RhoA activation, rather it contributed independently to the decreased motility of OLIG2-expressing U12-1 cells. Alternatively, semiquantitative reverse transcription-PCR analysis revealed that mRNA expression of RhoGAP8, which regulates cell migration, was decreased by OLIG2 expression. Furthermore, expression of C3 transferase, which inhibits Rho via ADP ribosylation, attenuated the OLIG2-induced inhibition of cell motility. Imaging by fluorescence resonance energy transfer revealed that in U12-1 cells lacking OLIG2, the active form of RhoA was localized to protrusions of the cell membrane. In contrast, in OLIG2-expressing cells, it lined almost the entire plasma membrane. Thus, OLIG2 suppresses the motile phenotype of glioblastoma cells by activating RhoA.

Intelligent behaviors of amoeboid movement based on complex dynamics of soft matter

We review how soft matter is self-organized to perform information processing at the cell level by examining the model organism Physarum plasmodium. The amoeboid organism, Physarum polycephalum, in the class of true slime molds, exhibits the intelligent behavior of foraging in complex situations. When placed in a maze with food sources at two exits, the organism develops tubular structures with its body which connect the food sources along the shortest path so that the rates of nutrient absorption and intracellular communication are maximized. This intelligent behavior results from the organism's control of a dynamic network through which mechanical and chemical information is transmitted. We review experimental studies that explore the development and adaptation of structures that make up the network. Recently a model of the dynamic network has been developed, and we review the formulation of this model and present some key results. The model captures the dynamics of existing networks, but it does not answer the question of how such networks form initially. To address the development of cell shape, we review existing mechanochemical models of the protoplasm of Physarum, present more general models of motile cells, and discuss how to adapt existing models to explore the development of intelligent networks in Physarum.

Locomotion of fish epidermal keratocytes on spatially selective adhesion patterns

Cell migration results from forces generated by assembly, contraction, and adhesion of the cytoskeleton. To address how these forces integrate in space and time, novel assays are required that allow spatial separation of the different force categories. We used micro-contact printing of fibronectin on glass substrates to study the effect of adhesion patterns on fish epidermal keratocytes locomotion. Cells migrated at similar speeds on homogeneously adhesive substrates and on patterns with 5 microm-wide adhesive stripes interleaved by non-adhesive stripes with a width varied between 5 and 13 microm. The leading edge protruded on adhesive stripes and lagged behind on non-adhesive stripes. On patterns with non-adhesive stripes wider than 13 microm cells halted, although the lamellipodium did not collapse. High correlation was found between the widths of protruding and lagging edge segments and the widths of the underlying stripes. We explain our data by the force balances between actin polymerization, contraction and adhesion on fibronectin stripes; and between actin polymerization, contraction and lamellipodium-internal elastic tension on non-adhesive stripes. We tested our model further by blocking lamellipodium actin network contraction and polymerization. In both experiments we observed that cells eventually lost their ability to move. However, the two perturbations induced distinct morphological responses. The data suggested that forces powering forward motion of keratocytes are largely associated with network assembly whereas contraction maintains cell polarity. This study establishes spatially selective adhesion substrates and cell morphological readouts as a means to elucidate the mechanical balance between substrate adhesion and cytoskeleton-internal tension in cell migration.

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.

Nonlinear mechanical modeling of cell adhesion

Cell adhesion, which is mediated by the receptor-ligand bonds, plays an essential role in various biological processes. Previous studies often described the force-extension relationship of receptor-ligand bond with linear assumption. However, the force-extension relationship of the bond is intrinsically nonlinear, which should have significant influence on the mechanical behavior of cell adhesion. In this work, a nonlinear mechanical model for cell adhesion is developed, and the adhesive strength was studied at various bond distributions. We find that the nonlinear mechanical behavior of the receptor-ligand bonds is crucial to the adhesive strength and stability. This nonlinear behavior allows more bonds to achieve large bond force simultaneously, and therefore the adhesive strength becomes less sensitive to the change of bond density at the outmost periphery of the adhesive area. In this way, the strength and stability of cell adhesion are soundly enhanced. The nonlinear model describes the cell detachment behavior better than the linear model.

Endothelial cell migration on RGD-peptide-containing PEG hydrogels in the presence of sphingosine 1-phosphate

Sphingosine 1-phosphate (S1P) is a potent chemokinetic agent for endothelial cells that is released by activated platelets. We previously developed Arg-Gly-Asp (RGD)-containing polyethylene glycol biomaterials for the controlled delivery of S1P to promote endothelialization. Here, we studied the effects of cell adhesion strength on S1P-stimulated endothelial cell migration in the presence of arterial levels of fluid shear stress, since an upward shift in optimal cell adhesion strengths may be beneficial for promoting long-term cell adhesion to materials. Two RGD peptides with different integrin-binding specificities were added to the polyethylene glycol hydrogels. A linear RGD bound primarily to beta(3) integrins, whereas a cyclic RGD bound through both beta(1) and beta(3) integrins. We observed increased focal adhesion formation and better long-term adhesion in flow with endothelial cells on linear RGD peptide, versus cyclic RGD, even though initial adhesion strengths were higher for cells on cyclic RGD. Addition of 100 nM S1P increased cell speed and random motility coefficients on both RGD peptides, with the largest increases found on cyclic RGD. For both peptides, much of the increase in cell migration speed was found for smaller cells (<1522 microm(2) projected area), although the large increases on cyclic RGD were also due to medium-sized cells (2288-3519 microm(2)). Overall, a compromise between high cell migration rates and long-term adhesion will be important in the design of materials that endothelialize after implantation.

One-dimensional elastic continuum model of enterocyte layer migration

Necrotizing enterocolitis is the leading cause of death from gastrointestinal disease in preterm infants. It results from an injury to the mucosal lining of the intestine, leading to translocation of bacteria and endotoxin into the circulation. Intestinal mucosal defects are repaired by the process of intestinal restitution, during which enterocytes migrate from healthy areas to sites of injury. In this article, we develop a mathematical model of migration of enterocytes during experimental necrotizing enterocolitis. The model is based on a novel assumption of elastic deformation of the cell layer and incorporates the following effects: i), mobility promoting force due to lamellipod formation, ii), mobility impeding adhesion to the cell matrix, and iii), enterocyte proliferation. Our model successfully reproduces the behavior observed for enterocyte migration on glass coverslips, namely the dependence of migration speed on the distance from the wound edge, and the finite propagation distance in the absence of proliferation that results in an occasional failure to close the wound. It also qualitatively reproduces the dependence of migration speed on integrin concentration. The model is applicable to the closure of a wound with a linear edge and, after calibration with experimental data, could be used to predict the effect of chemical agents on mobility, adhesion, and proliferation of enterocytes.

A model for the interplay of receptor recycling and receptor-mediated contact in T cells

Orientation of organelles inside T cells (TC) toward antigen-presenting cells (APC) ensures that the immune response is properly directed, but the orientation mechanisms remain largely unknown. Structural dynamics of TC are coupled to dynamics of T-cell receptor (TCR), which recognizes antigen on the APC surface. Engagement of the TCR triggers its internalization followed by delayed polarized recycling to the plasma membrane through the submembrane recycling compartment (RC), which organelle shares intracellular location with the TC effector apparatus. TCR engagement also triggers TC-APC interface expansion enabling further receptor engagement. To analyze the interplay of the cell-cell contact and receptor dynamics, we constructed a new numerical model. The new model displays the experimentally observed selective stabilization of the contact initiated next to the RC, and only transient formation of contact diametrically opposed to the RC. In the general case wherein the TC-APC contact is initiated in an arbitrary orientation to the RC, the modeling predicts that the contact dynamics and receptor recycling can interact, resulting effectively in migration of the contact to the TC surface domain adjacent to the submembrane RC. Using three-dimensional live-cell confocal microscopy, we obtain data consistent with this unexpected behavior. We conclude that a TC can stabilize its contact with an APC by aligning it with the polarized intracellular traffic of TCR. The results also suggest that the orientation of TC organelles, such as the RC and the effector apparatus, toward the APC can be achieved without any intracellular translocation of the organelles.

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.

Comparative maps of motion and assembly of filamentous actin and myosin II in migrating cells

To understand the mechanism of cell migration, one needs to know how the parts of the motile machinery of the cell are assembled and how they move with respect to each other. Actin and myosin II are thought to be the major structural and force-generating components of this machinery (Mitchison and Cramer, 1996; Parent, 2004). The movement of myosin II along actin filaments is thought to generate contractile force contributing to cell translocation, but the relative motion of the two proteins has not been investigated. We use fluorescence speckle and conventional fluorescence microscopy, image analysis, and computer tracking techniques to generate comparative velocity and assembly maps of actin and myosin II over the entire cell in a simple model system of persistently migrating fish epidermal keratocytes. The results demonstrate contrasting polarized assembly patterns of the two components, indicate force generation at the lamellipodium-cell body transition zone, and suggest a mechanism of anisotropic network contraction via sliding of myosin II assemblies along divergent actin filaments.

Surface modification and property analysis of biomedical polymers used for tissue engineering

The response of host organism in macroscopic, cellular and protein levels to biomaterials is, in most cases, closely associated with the materials' surface properties. In tissue engineering, regenerative medicine and many other biomedical fields, surface engineering of the bio-inert synthetic polymers is often required to introduce bioactive species that can promote cell adhesion, proliferation, viability and enhanced ECM-secretion functions. Up to present, a large number of surface engineering techniques for improving biocompatibility have been well established, the work of which generally contains three main steps: (1) surface modification of the polymeric materials; (2) chemical and physical characterizations; and (3) biocompatibility assessment through cell culture. This review focuses on the principles and practices of surface engineering of biomedical polymers with regards to particular aspects depending on the authors' research background and opinions. The review starts with an introduction of principles in designing polymeric biomaterial surfaces, followed by introduction of surface modification techniques to improve hydrophilicity, to introduce reactive functional groups and to immobilize functional protein molecules. The chemical and physical characterizations of the modified biomaterials are then discussed with emphasis on several important issues such as surface functional group density, functional layer thickness, protein surface density and bioactivity. Three most commonly used surface composition characterization techniques, i.e. ATR-FTIR, XPS, SIMS, are compared in terms of their penetration depth. Ellipsometry, CD, EPR, SPR and QCM's principles and applications in analyzing surface proteins are introduced. Finally discussed are frequently applied methods and their principles to evaluate biocompatibility of biomaterials via cell culture. In this section, current techniques and their developments to measure cell adhesion, proliferation, morphology, viability, migration and gene expression are reviewed.

Analysis of directional cell migration on defined FN gradients: role of intracellular signaling molecules

Directional cell migration on extracellular matrix (ECM) plays important roles in embryonic development and adult organisms. To study the mechanisms and signaling pathways involved in the regulation of directional cell migration, we created defined fibronectin (FN) gradients by using microfluidic systems. We found that fibroblasts exhibited haptotaxis towards higher FN concentration on the gradient. Furthermore, the net movements in the direction of FN gradients correlated with the increase in the slope of the gradient although the overall rate of migration was not correlated. Consistent with previous observations on the uniformly coated surface, local higher FN concentration led to reduced migration rate due to increased spreading. Upon transfection of N-WASP or activated Cdc42, but not FAK or Grb7, the cells showed increased directional migration. However, transfection of FAK, but not the other signaling molecules, led to an increase in the persistence of directional cell migration, which is dependent on the slope of the gradient as well as FAK interaction with PI3K. Together, these studies reveal some novel properties of directional cell migration on defined FN gradient and suggested a role for FAK signaling and N-WASP and Cdc42 in the differential regulation of the persistence and rate of directional cell migration.

New methods for automated phenotyping of complex cellular behaviors

Cellular shape change and movement are central to biologic processes that range from normal embryonic development to inflammatory diseases and cancer. Quantitative visual phenotyping of dynamic cellular behaviors creates unique challenges for image capture, analysis and storage. Despite substantial technological advances in molecular biology, biochemistry, genomics and proteomics, investigating cellular processes remains tremendously challenging and labor-intensive. We have developed algorithms and software implementations that allow for fully-automated analysis of experiments designed to investigate a range of cellular and organismal behaviors. By enabling cellular phenotyping, this automated approach creates a unique opportunity for investigators to perform large-scale experiments designed to determine gene function or to screen for small molecule modulators of important cellular behaviors.

Predictive oncology: a review of multidisciplinary, multiscale in silico modeling linking phenotype, morphology and growth

Empirical evidence and theoretical studies suggest that the phenotype, i.e., cellular- and molecular-scale dynamics, including proliferation rate and adhesiveness due to microenvironmental factors and gene expression that govern tumor growth and invasiveness, also determine gross tumor-scale morphology. It has been difficult to quantify the relative effect of these links on disease progression and prognosis using conventional clinical and experimental methods and observables. As a result, successful individualized treatment of highly malignant and invasive cancers, such as glioblastoma, via surgical resection and chemotherapy cannot be offered and outcomes are generally poor. What is needed is a deterministic, quantifiable method to enable understanding of the connections between phenotype and tumor morphology. Here, we critically assess advantages and disadvantages of recent computational modeling efforts (e.g., continuum, discrete, and cellular automata models) that have pursued this understanding. Based on this assessment, we review a multiscale, i.e., from the molecular to the gross tumor scale, mathematical and computational "first-principle" approach based on mass conservation and other physical laws, such as employed in reaction-diffusion systems. Model variables describe known characteristics of tumor behavior, and parameters and functional relationships across scales are informed from in vitro, in vivo and ex vivo biology. We review the feasibility of this methodology that, once coupled to tumor imaging and tumor biopsy or cell culture data, should enable prediction of tumor growth and therapy outcome through quantification of the relation between the underlying dynamics and morphological characteristics. In particular, morphologic stability analysis of this mathematical model reveals that tumor cell patterning at the tumor-host interface is regulated by cell proliferation, adhesion and other phenotypic characteristics: histopathology information of tumor boundary can be inputted to the mathematical model and used as a phenotype-diagnostic tool to predict collective and individual tumor cell invasion of surrounding tissue. This approach further provides a means to deterministically test effects of novel and hypothetical therapy strategies on tumor behavior.

Ameboid cell motility: A model and inverse problem, with an application to live cell imaging data

In this article a mathematical model for ameboid cell movement is developed using a spring-dashpot system with Newtonian dynamics. The model is based on the facts that the cytoskeleton plays a primary role for cell motility and that the cytoplasm is viscoelastic. Based on the model, the inverse problem can be posed: if a structure like a spring-dashpot system is embedded into the living cell, what kind of characteristic properties must the structure have in order to reproduce a given movement of the cell? This inverse problem is the primary topic of this paper. On one side the model mimics some features of the movement, and on the other side, the solution to the inverse problem provides model parameters that give some insight, principally into the mechanical aspect, but also, through qualitative reasoning, into chemical and biophysical aspects of the cell. Moreover, this analysis can be done locally or globally and in different media by using the simplest possible information: positions of the cell and nuclear membranes. It is shown that the model and solution to the inverse problem for simulated data sets are highly accurate. An application to a set of live cell imaging data obtained from random movements of a human brain tumor cell (U87-MG human glioblastoma cell line) then provides an example of the efficiency of the model, through the solution of its inverse problem, as a way of understanding experimental data.

Effects of applied DC electric field on ligament fibroblast migration and wound healing

Applied electric fields (static and pulsing) are widely used in orthopedic practices to treat nonunions and spine fusions and have been shown to improve ligament healing in vivo. Few studies, however, have addressed the effect of electric fields (EFs) on ligament fibroblast migration and biosynthesis. In the current study, we applied static and pulsing direct current (DC) EFs to calf anterior cruciate ligament (ACL) fibroblasts. ACL fibroblasts demonstrated enhanced migration speed and perpendicular alignment to the applied EFs. The motility of ligament fibroblasts was further modulated on type I collagen. In addition, type I collagen expression increased in ACL fibroblasts after exposure to pulsing EFs. In vitro wound-healing studies showed inhibitory effects of static EFs, which were alleviated with a pulsing EF. Our results demonstrate that applied EFs augment ACL fibroblast migration and biosynthesis and provide potential mechanisms by which EFs may be used for enhancing ligament healing and repair.

Cell Adhesion Strengthening: Measurement and Analysis

Cell adhesion to the extracellular matrix is a dynamic process involving numerous focal adhesion components, which act in coordination to strengthen and optimize the mechanical anchorage of cells over time. A method for systematically analyzing the cell adhesion strengthening process and the components involved in this process is described here. The method combines an adhesion strength assay based on applying fluid shearing to a population of cells and quantitative biochemical analyses.

Transport of a 1D viscoelastic actin-myosin strip of gel as a model of a crawling cell

Cell crawling is an important biological phenomenon because it underlies coordinated cell movement in morphogenesis, cancer and wound healing. This phenomenon is based on protrusion at the cell's leading edge, retraction at the rear, contraction and graded adhesion powered by the dynamics of actin and myosin protein networks. A few one-dimensional models successfully explain an anteroposterior organization of the motile cell, but don't sufficiently explore the viscoelastic nature of the actin-myosin gel. We develop and numerically solve a model of a treadmilling strip of viscoelastic actin-myosin gel. The results show that the strip translocates steadily as a traveling pulse, without changing length, and that protein densities, velocities and stresses become stationary. The simulations closely match the observed forces, movements and protein distributions in the living cell.

Cell adhesion molecule L1 modulates nerve-growth-factor-induced CGRP-IR fiber sprouting

Overexpression of nerve growth factor (NGF) using adenoviruses (Adts) after spinal cord injury induces extensive regeneration and sprouting of calcitonin-gene-related peptide immunoreactive (CGRP-IR) fibers, whereas overexpression of cell adhesion molecules (CAMs) has no effect on the normal distribution of these fibers. Interestingly, co-expression of cell adhesion molecule L1 and NGF significantly decreases (p<0.0001) CGRP-IR fiber sprouting within the spinal cord, when compared to NGF alone. Co-expression of cell adhesion molecules NCAM or N-cadherin had no effect on NGF-induced CGRP-IR fiber sprouting. These data demonstrate that reduced sprouting is specific to L1 co-expression and not other cell adhesion molecules. In vitro studies carried out to address potential mechanisms show that neurite outgrowth over astrocytes overexpressing L1 in the presence of NGF is comparable to controls, indicating that other factors present in vivo might be involved in the L1-mediated reduction in sprouting. One potential factor is semaphorin 3A (sema3A), which mediates growth cone collapse of CGRP-positive axons. Recent studies have shown that L1 is important in sema3A receptor signaling for cortical neurons. In our study, co-expression of sema3A indeed reduces neurite outgrowth from DRG neurons by about 40% on L1-expressing astrocytes. Based on these results, we hypothesize that overexpression of L1 potentiates sema3A signaling resulting in reduced sprouting.

Minute changes in composition of polymer substrates produce amplified differences in cell adhesion and motility via optimal ligand conditioning

We explored the interplay between substratum chemistry of polymeric materials and surface-adsorbed ligand concentration (human plasma fibronectin) in the control of cell adhesion and cell motility. We found that small changes in the chemical composition of a polymeric substratum had different effects on cellular motility--depending on the concentration of preadsorbed fibronectin. We used two tyrosine-derived polyarylates, poly(DTD diglycolate) and poly(DTD glutarate), as substrata for the seeding of NIH-3T3 fibroblasts. The only compositional difference between the two test polymers was that one single oxygen atom in the polymer backbone of poly(DTD diglycolate) had been substituted by a methylene group in the backbone of poly(DTD glutarate), The two polymers had closely matched hydrophobicity and physical properties. Flat, spin-coated surfaces of these polymers were pretreated with different concentrations of human plasma fibronectin (0-20 microg/ml). After seeding with NIH-3T3 fibroblasts, we examined the adhesion and motility behavior of these cells. We found that NIH-3T3 fibroblasts migrated significantly faster on poly(DTD diglycolate), but only when the polymer surfaces were pretreated with intermediate concentrations of fibronectin. Only at these intermediate levels of ligand conditioning, did the presence of an extra oxygen atom in the backbone of poly(DTD diglycolate) relative to poly(DTD glutarate) (i) alter the overall organization/concentration of the fibronectin; (ii) weaken cell attachment strength and inhibited excessive cell spreading; and (iii) promote cell motility kinetics. These findings indicate that the biological effect of minute changes in substratum chemistry is critically dependent on the level of surface-adsorbed cell-binding ligands.

Fibronectin organization under and near cells

Polymerization of soluble fibronectin molecules results in fibres that are visible as networks using fluorescently labelled fibronectin protomers or by antibody labelling. Displacement of fibres composed of modified protomers in living cells provides information regarding matrix structure, organization, and movement. A static analysis of fibronectin structures and patterns of organization provide insight into their reorganization during adhesion and motility. Confocal microscopy and atomic force microscopy (AFM) reveal fibronectin-containing networks aligned in arrays perpendicular to the retracting cell edge and in apparently disordered networks of fibres under the cell. The change in patterns suggests a reorganization of fibronectin from disordered arrays used for adhesion into ordered arrays during movement of the cell. Comparison of confocal images with corresponding AFM images confirms that the fibres left on the surface as the cell moves away do contain fibronectin. The orientation of these fibres relative to the tail (uropod) and the receding edges of the cell leads us to propose that cells generate a force on the fibres that exceeds the adhesion force of the fibres to the surface causing them to pull fibronectin fibres into straight arrays. However, when the fibres are parallel to the direction of pull, the fibres remain attached to the surface. The data supports the hypothesis that disorganized, linear fibres are the product of Fn polymerization induced by the cell beneath it and serve to adhere the cell to the substrate as the cell spreads, whereas arrays of fibres found outside the cell are formed as existing fibrils and reorganize during cell motility.

Spatiotemporal feedback between actomyosin and focal-adhesion systems optimizes rapid cell migration

Cells exhibit a biphasic migration-velocity response to increasing adhesion strength, with fast migration occurring at intermediate extracellular matrix (ECM) concentration and slow migration occurring at low and high ECM concentration. A simple mechanical model has been proposed to explain this observation, in which too little adhesion does not provide sufficient traction whereas too much adhesion renders cells immobile. Here we characterize a phenotype for rapid cell migration, which in contrast to the previous model reveals a complex interdependence of subcellular systems that mediates optimal cell migration in response to increasing adhesion strength. The organization and activity of actin, myosin II, and focal adhesions (FAs) are spatially and temporally highly variable and do not exhibit a simple correlation with optimal motility rates. Furthermore, we can recapitulate rapid migration at a nonoptimal ECM concentration by manipulating myosin II activity. Thus, the interplay between actomyosin and FA dynamics results in a specific balance between adhesion and contraction, which induces maximal migration velocity.

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.

Polarization and Movement of Keratocytes: A Multiscale Modelling Approach

Eukariotic cell motility is a complex phenomenon, in which the cytoskeleton and its major constituent, actin, play an essential role. Actin forms polymers of long, stiff filaments that are cross-linked into an anisotropic network inside a thin sheet-like cellular protrusion, the lamellipod. At the leading edge of this structure, polymerization of actin filaments creates the force that pushes out the membrane and leads to translocation of a motile cell. Dynamics of the actin network account for changes in cell shape, crawling motion and turning of the cell in response to external cues. Regulating the dynamics of the cytoskeleton, and playing a central role in signal transduction in the cell, are Cdc42, Rac and Rho (GTPases of the rho family, collectively known as the small G-proteins) and the actin nucleating complex, Arp2/3. In this paper, we use a multiscale modelling approach in a 2D model of a motile cell. We describe the mutual interactions of the small G-proteins, and their effects on capping and side-branching of actin filaments. We incorporate the pushing exerted by oriented actin filament ends on the cell edge, and a Rho-dependent contraction force. Combining these biochemical and mechanical aspects, we investigate the dynamics of a model epidermal fish keratocyte through in silico experiments. Our model gives insight into how, in response to some cue, a cell can polarize, form a leading edge, and move; concomitantly it explains how a keratocyte cell can maintain its shape and polarity, even after removal of the initial stimulus, and how it can change direction quickly in response to changes in its environment. We show that establishment of polarity stems from interactions of Cdc42, Rac and Rho, while maintenance and robustness of polarity is due to the rapid cytosolic diffusion of the inactive (GDI-bound) forms of the small G-proteins. Our model produces a cell shape that closely resembles the keratocytes and correct speeds for biologically reasonable parameter values. Movies of the simulations can be obtained from http://theory.bio.uu.nl/stan/keratocyte.

Fibronectin functional domains coupled to hyaluronan stimulate adult human dermal fibroblast responses critical for wound healing

Fibronectin (FN) facilitates dermal fibroblast migration during normal wound healing. Proteolytic degradation of FN in chronic wounds hampers healing. Previously, three FN functional domains (FNfd) have been shown to be sufficient for optimal adult human dermal fibroblast migration. Here we report the development of an acellular hydrogel matrix comprised of the FNfds coupled to a hyaluronan (HA) backbone to stimulate wound repair. Employing Michael-type addition, the cysteine- tagged FNfds were first coupled to a homobifunctional PEG derivative. Thereafter, these PEG derivative FNfd solutions, containing bifunctional PEG-derivative crosslinker were coupled to thiol-modified HA (HA-DTPH) to obtain a crosslinked hydrogel matrix. When evaluated in vitro, these acellular hydrogels were completely cytocompatible. While spreading and proliferation of adult human dermal fibroblasts plateaued at higher FNfd bulk densities, their rapid and robust migration followed a typical bell-shaped response. When implanted in porcine cutaneous wounds, these acellular matrices, besides being completely biocompatible, induced rapid and en masse recruitment of stromal fibroblasts that was not observed with RGD-tethered or unmodified hydrogels. Such constructs might be of great benefit in clinical settings where rapid formation of new tissue is needed.

Cell migration and polarity on microfabricated gradients of extracellular matrix proteins

This paper explores the effects of the surface density and concentration profiles of extra cellular matrix proteins on the migration of rat intestinal IEC-6 cells. Microfluidic devices were used to create linear, immobilized gradients of laminin. This study investigated both the impact of the steepness and local concentrations on the directedness of cell migration. The bulk concentrations of proteins in the feed streams in the mixing device determined the gradient profile and the local concentration of laminin in the device. Two sets of gradients were used to explore cell migration directedness: (i) gradients with similar change in local concentration, i.e., the same gradient steepness, and (ii) different gradients with similar local concentrations. Cells migrated up the gradients, independent of the steepness of the gradients used in this study. At the same local laminin concentration, the migration rate was independent of the gradient steepness. However, cell directedness decreased significantly at high laminin densities.