Dynamics of Cell Shape and Forces on Micropatterned Substrates Predicted by a Cellular Potts Model

Micropatterned substrates are often used to standardize cell experiments and to quantitatively study the relation between cell shape and function. Moreover, they are increasingly used in combination with traction force microscopy on soft elastic substrates. To predict the dynamics and steady states of cell shape and forces without any a priori knowledge of how the cell will spread on a given micropattern, here we extend earlier formulations of the two-dimensional cellular Potts model. The third dimension is treated as an area reservoir for spreading. To account for local contour reinforcement by peripheral bundles, we augment the cellular Potts model by elements of the tension-elasticity model. We first parameterize our model and show that it accounts for momentum conservation. We then demonstrate that it is in good agreement with experimental data for shape, spreading dynamics, and traction force patterns of cells on micropatterned substrates. We finally predict shapes and forces for micropatterns that have not yet been experimentally studied.     

The dynamics and mechanics of endothelial cell spreading

Cell adhesion to extracellular matrix is mediated by receptor-ligand interactions. When a cell first contacts a surface, it spreads, exerting traction forces against the surface and forming new bonds as its contact area expands. Here, we examined the changes in shape, actin polymerization, focal adhesion formation, and traction stress generation that accompany spreading of endothelial cells over a period of several hours. Bovine aortic endothelial cells were plated on polyacrylamide gels derivatized with a peptide containing the integrin binding sequence RGD, and changes in shape and traction force generation were measured. Notably, both the rate and extent of spreading increase with the density of substrate ligand. There are two prominent modes of spreading: at higher surface ligand densities cells tend to spread isotropically, whereas at lower densities of ligand the cells tend to spread anisotropically, by extending pseudopodia randomly distributed along the cell membrane. The extension of pseudopodia is followed by periods of growth in the cell body to interconnect these extensions. These cycles occur at very regular intervals and, furthermore, the extent of pseudopodial extension can be diminished by increasing the ligand density. Measurement of the traction forces exerted by the cell reveals that a cell is capable of exerting significant forces before either notable focal adhesion or stress fiber formation. Moreover, the total magnitude of force exerted by the cell is linearly related to the area of the cell during spreading. This study is the first to monitor the dynamic changes in the cell shape, spreading rate, and forces exerted during the early stages (first several hours) of endothelial cell adhesion.     

Spatial and temporal coordination of traction forces in one-dimensional cell migration

ABSTRACT

Migration of a fibroblast along a collagen fiber can be regarded as cell locomotion in one-dimension (1D). In this process, a cell protrudes forward, forms a new adhesion, produces traction forces, and releases its rear adhesion in order to advance itself along a path. However, how a cell coordinates its adhesion formation, traction forces, and rear release in 1D migration is unclear. Here, we studied fibroblasts migrating along a line of microposts. We found that when the front of a cell protruded onto a new micropost, the traction force produced at its front increased steadily, but did so without a temporal correlation in the force at its rear. Instead, the force at the front coordinated with a decrease in force at the micropost behind the front. A similar correlation in traction forces also occurred at the rear of a cell, where a decrease in force due to adhesion detachment corresponded to an increase in force at the micropost ahead of the rear. Analysis with a bio-chemo-mechanical model for traction forces and adhesion dynamics indicated that the observed relationship between traction forces at the front and back of a cell is possible only when cellular elasticity is lower than the elasticity of the cellular environment.     

Characterization of the nuclear deformation caused by changes in endothelial cell shape

We investigated the mechanotransduction pathway in endothelial cells between their nucleus and adhesions to the extracellular matrix. First, we measured nuclear deformations in response to alterations of cell shape as cells detach from a flat surface. We found that the nuclear deformation appeared to be in direct and immediate response to alterations of the cell adhesion area. The nucleus was then treated as a neo-Hookean compressible material, and we estimated the stress associated with the cytoskeleton and acting on the nucleus during cell rounding. With the obtained stress field, we estimated the magnitude of the forces deforming the nucleus. Considering the initial and final components of this adhesion-cytoskeleton-nucleus force transmission pathway, we found our estimate for the internal forces acting on the nucleus to be on the same order of magnitude as previously measured traction forces, suggesting a direct mechanical link between adhesions and the nucleus.     

Computational model for cell migration in three-dimensional matrices

Although computational models for cell migration on two-dimensional (2D) substrata have described how various molecular and cellular properties and physiochemical processes are integrated to accomplish cell locomotion, the same issues, along with certain new ones, might contribute differently to a model for migration within three-dimensional (3D) matrices. To address this more complicated situation, we have developed a computational model for cell migration in 3D matrices using a force-based dynamics approach. This model determines an overall locomotion velocity vector, comprising speed and direction, for individual cells based on internally generated forces transmitted into external traction forces and considering a timescale during which multiple attachment and detachment events are integrated. Key parameters characterize cell and matrix properties, including cell/matrix adhesion and mechanical and steric properties of the matrix; critical underlying molecular properties are incorporated explicitly or implicitly. Model predictions agree well with experimental results for the limiting case of migration on 2D substrata as well as with recent experiments in 3D natural tissues and synthetic gels. Certain predicted features such as biphasic behavior of speed with density of matrix ligands for 3D migration are qualitatively similar to their 2D counterparts, but new effects generally absent in 2D systems, such as effects due to matrix sterics and mechanics, are now predicted to arise in many 3D situations. As one particular sample manifestation of these effects, the optimal levels of cell receptor expression and matrix ligand density yielding maximal migration are dependent on matrix mechanical compliance.     

Simulations of the contractile cycle in cell migration using a bio-chemical-mechanical model

Cell migration relies on traction forces in order to propel a cell. Several computational models have been developed that help explain the trajectory that cells take during migration, but little attention has been placed on traction forces during this process. Here, we investigated the spatiotemporal dynamics of cell migration by using a bio-chemical-mechanical contractility model that incorporates the first steps of cell migration on an array of posts. In the model, formation of a new adhesion causes a reactivation of stress fibre assembly within a cell. The model was able to predict the spatial distribution of traction forces observed with previous experiments. Moreover, the model found that the strain energy exerted by the traction forces of a migrating cell underwent a cyclic relationship that rose with the formation of a new adhesion and fell with the release of an adhesion at its rear.     

A three-dimensional mathematical model of the human masticatory system predicting maximum possible bite forces

A three-dimensional mathematical model of the human masticatory system, containing 16 muscle forces and two joint reaction forces, is described. The model allows simulation of static bite forces and concomitant joint reaction forces for various bite point locations and mandibular positions. The system parameters for the model were obtained from a cadaver head. Maximum possible bite forces were computed using optimization techniques; the optimization criterion we used was the minimizing of the relative activity of the most active muscle. The model predicts that at each specific bite point, bite forces can be generated in a wide range of directions, and that the magnitude of the maximum bite force depends on its direction. The relationship between bite force direction and its maximum magnitude depends on bite point location and mandibular position. In general, the direction of the largest possible bite force does not coincide with the direction perpendicular to the occlusal plane.     

Continuum model of fibroblast-driven wound contraction: inflammation-mediation.

We propose a mathematical model to aid the understanding of how events in wound healing are orchestrated to result in wound contraction. Ultimately, a validated model could provide a predictive means for enhancing or mitigating contraction as is appropriate for managing a particular wound. The complex nature of wound healing and the lack of a modeling framework which can account for both the relevant cell biology and biomechanics are major reasons for the absence of models to date. Here we adapt a model originally proposed by Murray and co-workers to show how cell traction forces can result in spatial patterns of cell aggregates since it offers a framework for understanding how traction exerted by wound fibroblasts drives wound contraction. Since it is a continuum model based on conservation laws which reflect assumed cell and tissue properties, it is readily extended to account for emerging understanding of the cell biology of wound healing and its relationship to inflammation. We consider various sets of assumed properties, based on current knowledge, within a base model of dermal wound healing and compare predictions of the rate and extent of wound contraction to published experimental results.     

A novel cell force sensor for quantification of traction during cell spreading and contact guidance

In this work, we present a ridged, microfabricated, force sensor that can be used to investigate mechanical interactions between cells exhibiting contact guidance and the underlying cell culture substrate, and a proof-of-function evaluation of the force sensor performance. The substrates contain arrays of vertical pillars between solid ridges that were microfabricated in silicon wafers using photolithography and deep reactive ion etching. The spring constant of the pillars was measured by atomic force microscopy. For time-lapse experiments, cells were seeded on the pillared substrates and cultured in an on-stage incubator on a microscope equipped with reflected differential interference contrast optics. Endothelial cells (ECs) and fibroblasts were observed during attachment, spreading, and migration. Custom image analysis software was developed to resolve cell borders, cell alignment to the pillars and migration, displacements of individual pillars, and to quantify cell traction forces. Contact guidance classification was based on cell alignment and movement angles with respect to microfabricated ridges, as well as cell elongation. In initial investigations made with the ridged cell force sensor, we have observed contact guidance in ECs but not in fibroblast cells. A difference in maximal amplitude of mechanical forces was observed between a contact-guided and non-contact-guided, but mobile, EC. However, further experiments are required to determine the statistical significance of this observation. By chance, we observed another feature of cell behavior, namely a reversion of cell force direction. The direction of forces measured under rounded fibroblast cells changed from outwards during early cell attachment to inwards during further observation of the spreading phase. The range of forces measured under fibroblasts (up to 138 nN) was greater than that measured in EC (up to 57 nN), showing that the rigid silicon sensor is capable of resolving a large range of forces, and hence detection of differences in traction forces between cell types. These observations indicate proof-of-function of the ridged cell force sensor to induce contact guidance, and that the pillared cell force sensor constructed in rigid silicon has the necessary sensitivity to detect differences in traction force vectors between different cell phenotypes and morphologies.     

Measuring traction forces of motile dendritic cells on micropost arrays

Dendritic cells (DCs) migrate from sites of inflammation to secondary lymphoid organs where they initiate the adaptive immune response. Although motility is essential to DC function, the mechanisms by which they migrate are not fully understood. We incorporated micropost array detectors into a microfluidic gradient generator to develop what we consider to be a novel method for probing low magnitude traction forces during directional migration. We found migration of primary murine DCs is driven by short-lived traction stresses at the leading edge or filopodia. The traction forces generated by DCs are smaller in magnitude than found in neutrophils, and of similar magnitude during chemotaxis and chemokinesis, at 18 ± 1.4 and 16 ± 1.3 nN/cell, respectively. The characteristic duration of local DC traction forces was 3 min. The maximum principal stress in the cell occurred in the plane perpendicular to the axis of motion, forward of the centroid. We illustrate that the spatiotemporal pattern of traction stresses can be used to predict the direction of future DC motion. Overall, DCs show a mode of migration distinct from both mesenchymal cells and neutrophils, characterized by rapid turnover of traction forces in leading filopodia.     

Modelling Biological Gel Contraction by Cells: Mechanocellular Formulation and Cell Traction Force Quantification

Traction forces developed by most cell types play a significant role in the spatial organisation of biological tissues. However, due to the complexity of cell-extracellular matrix interactions, these forces are quantitatively difficult to estimate without explicitly considering cell properties and extracellular mechanical matrix responses. Recent experimental devices elaborated for measuring cell traction on extracellular matrix use cell deposits on a piece of gel placed between one fixed and one moving holder. We formulate here a mathematical model describing the dynamic behaviour of the cell-gel medium in such devices. This model is based on a mechanical force balance quantification of the gel visco-elastic response to the traction forces exerted by the diffusing cells. Thus, we theoretically analyzed and simulated the displacement of the free moving boundary of the system under various conditions for cells and gel concentrations. This modelis then used as the theoretical basis of an experimental device where endothelial cells are seeded on a rectangular biogel of fibrin cast between two floating holders, one fixed and the other linked to a force sensor. From a comparison of displacement of the gel moving boundary simulated by the model and the experimental data recorded from the moving holder displacement, the magnitude of the traction forces exerted by the endothelial cell on the fibrin gel was estimated for different experimental situations. Different analytical expressions for the cell traction term are proposed and the corresponding force quantifications are compared to the traction force measurements reported for various kind of cells with the use of similar or different experimental devices. This revised version was published online in June 2006 with corrections to the Cover Date.     

The mechanical basis of cell rearrangement. I. Epithelial morphogenesis during Fundulus epiboly

Many morphogenetic processes are accomplished by coordinated cell rearrangements. These rearrangements are accompanied by substantial shifts in the neighbor relationships between cells. Here we propose a model for studying morphogenesis in epithelial sheets by directed cell neighbor change. Our model describes cell rearrangements by accounting for the balance of forces between neighboring cells within an epithelium. Cell rearrangement and cell shape changes occur when these forces are not in mechanical equilibrium. We will show that cell rearrangement within the epidermal enveloping layer (EVL) of the teleost fish Fundulus during epiboly can be explained solely in terms of the balance of forces generated among constituent epithelial cells. Within a cell, we account for circumferential elastic forces and the force generated by hydrostatic and osmotic pressure. The model treats epithelial cells as two-dimensional polygons where the mechanical forces are applied to the polygonal nodes. A cell node protrudes or contracts when the nodal forces are not in mechanical equilibrium. In an epithelial sheet, adjacent cells share common boundary nodes; in this way, mechanical force is transmitted from cell to cell, mimicking junctional coupling. These junctional nodes can slide, and nodes may appear or disappear, so that the number of polygonal sides is variable. Computer graphics allows us to compare numerical simulations of the model with time-lapse cinemicroscopy of cell rearrangements in the living embryo, and data obtained from fixed and silver stained embryos. By manipulating the mechanical properties of the model cells we can study the conditions necessary to reproduce normal cell behavior during Fundulus epiboly. We find that simple stress relaxation is sufficient to account for cell rearrangements among interior cells of the EVL when they are isotropically contractile. Experimental observations show that the number of EVL marginal cells continuously decreases throughout epiboly. In order for the simulation to reproduce this behavior, cells at the EVL boundary must generate protrusive forces rather than contractile tension forces. Therefore, the simulation results suggest that the mechanical properties of EVL marginal cells at their leading edge must be quite different from EVL interior cells.     

Traction Stresses and Translational Distortion of the Nucleus During Fibroblast Migration on a Physiologically Relevant ECM Mimic

Cellular traction forces, resulting in cell-substrate physical interactions, are generated by actin-myosin complexes and transmitted to the extracellular matrix through focal adhesions. These processes are highly dynamic under physiological conditions and modulate cell migration. To better understand the precise dynamics of cell migration, we measured the spatiotemporal redistribution of cellular traction stresses (force per area) during fibroblast migration at a submicron level and correlated it with nuclear translocation, an indicator of cell migration, on a physiologically relevant extracellular matrix mimic. We found that nuclear translocation occurred in pulses whose magnitude was larger on the low ligand density surfaces than on the high ligand density surfaces. Large nuclear translocations only occurred on low ligand density surfaces when the rear traction stresses completely relocated to a posterior nuclear location, whereas such relocation took much longer time on high ligand density surfaces, probably due to the greater magnitude of traction stresses. Nuclear distortion was also observed as the traction stresses redistributed. Our results suggest that the reinforcement of the traction stresses around the nucleus as well as the relaxation of nuclear deformation are critical steps during fibroblast migration, serving as a speed regulator, which must be considered in any dynamic molecular reconstruction model of tissue cell migration. A traction gradient foreshortening model was proposed to explain how the relocation of rear traction stresses leads to pulsed fibroblast migration.     

Hypergravity affects cell traction forces of fibroblasts

Cells sense and react on changes of the mechanical properties of their environment and, likewise, respond to external mechanical stress applied to them. However, whether the gravitational field as overall body force modulates cellular behavior is unclear. Different studies demonstrated that micro- and hypergravity influences the shape and elasticity of cells, initiate cytoskeleton reorganization, and influence cell motility. All these cellular properties are interconnected and contribute to forces that cells apply on their surrounding microenvironment. Yet, studies that investigated changes of cell traction forces under hypergravity conditions are scarce. Here, we performed hypergravity experiments on 3T3 fibroblast cells using the large-diameter centrifuge at the European Space Agency - European Space Research and Technology Centre. Cells were exposed to hypergravity of up to 19.5 g for 16 h in both the upright and the inverted orientation with respect to the g-force vector. We observed a decrease in cellular traction forces when the gravitational field was increased up to 5.4 g, followed by an increase of traction forces for higher gravity fields up to 19.5 g independent of the orientation of the gravity vector. We attribute the switch in cellular response to shear thinning at low g-forces, followed by significant rearrangement and enforcement of the cytoskeleton at high g-forces.     

A dynamical model for receptor-mediated cell adhesion to surfaces.

We present a dynamical model for receptor-mediated adhesion of cells in a shear field of viscous fluid to surfaces coated with ligand molecules complementary to receptors in the cell membrane. We refer to this model as the "point attachment model" because it considers the contact area between the cell and the surface to be a small, homogeneous region that mediates the initial attachment of the cell to the surface. Using a phase plane analysis of a system of nonlinear ordinary differential equations which govern the changes in free receptor density and bond density within the contact area with time, we can predict the conditions for which adhesion between the cell and the surface will take place. Whether adhesion occurs depends on values of dimensionless quantities that characterize the interaction of the cell and its receptors with the surface and its ligand, such as the bond formation rate, the receptor-ligand affinity, the fluid mechanical force, the receptor mobility, and the contact area. A key result is that there are two regimes in which different chemical and physical forces dominate: a rate-controlled high affinity regime and an affinity-controlled low-affinity regime. Many experimental observations can be explained by understanding which of these regimes is appropriate. We also provide simple approximate analytical solutions, relating adhesiveness to cell and surface properties as well as fluid forces, which allow convenient testing of model predictions by experiment.     

Fluidization, resolidification, and reorientation of the endothelial cell in response to slow tidal stretches

Mechanical stretch plays an important role in regulating shape and orientation of the vascular endothelial cell. This morphological response to stretch is basic to angiogenesis, neovascularization, and vascular homeostasis, but mechanism remains unclear. To elucidate mechanisms, we used cell mapping rheometry to measure traction forces in primary human umbilical vein endothelial cells subjected to periodic uniaxial stretches. Onset of periodic stretch of 10% strain amplitude caused a fluidization response typified by attenuation of traction forces almost to zero. As periodic stretch continued, the prompt fluidization response was followed by a slow resolidification response typified by recovery of the traction forces, but now aligned along the axis perpendicular to the imposed stretch. Reorientation of the cell body lagged reorientation of the traction forces, however. Together, these observations demonstrate that cellular reorientation in response to periodic stretch is preceded by traction attenuation by means of cytoskeletal fluidization and subsequent traction recovery transverse to the stretch direction by means of cytoskeletal resolidification.     

Calculation of the force field required for nucleus deformation during cell migration through constrictions

During cell migration in confinement, the nucleus has to deform for a cell to pass through small constrictions. Such nuclear deformations require significant forces. A direct experimental measure of the deformation force field is extremely challenging. However, experimental images of nuclear shape are relatively easy to obtain. Therefore, here we present a method to calculate predictions of the deformation force field based purely on analysis of experimental images of nuclei before and after deformation. Such an inverse calculation is technically non-trivial and relies on a mechanical model for the nucleus. Here we compare two simple continuum elastic models of a cell nucleus undergoing deformation. In the first, we treat the nucleus as a homogeneous elastic solid and, in the second, as an elastic shell. For each of these models we calculate the force field required to produce the deformation given by experimental images of nuclei in dendritic cells migrating in microchannels with constrictions of controlled dimensions. These microfabricated channels provide a simplified confined environment mimicking that experienced by cells in tissues. Our calculations predict the forces felt by a deforming nucleus as a migrating cell encounters a constriction. Since a direct experimental measure of the deformation force field is very challenging and has not yet been achieved, our numerical approaches can make important predictions motivating further experiments, even though all the parameters are not yet available. We demonstrate the power of our method by showing how it predicts lateral forces corresponding to actin polymerisation around the nucleus, providing evidence for actin generated forces squeezing the sides of the nucleus as it enters a constriction. In addition, the algorithm we have developed could be adapted to analyse experimental images of deformation in other situations.     

A Simple Force-Motion Relation for Migrating Cells Revealed by Multipole Analysis of Traction Stress

For biophysical understanding of cell motility, the relationship between mechanical force and cell migration must be uncovered, but it remains elusive. Since cells migrate at small scale in dissipative circumstances, the inertia force is negligible and all forces should cancel out. This implies that one must quantify the spatial pattern of the force instead of just the summation to elucidate the force-motion relation. Here, we introduced multipole analysis to quantify the traction stress dynamics of migrating cells. We measured the traction stress of Dictyostelium discoideum cells and investigated the lowest two moments, the force dipole and quadrupole moments, which reflect rotational and front-rear asymmetries of the stress field. We derived a simple force-motion relation in which cells migrate along the force dipole axis with a direction determined by the force quadrupole. Furthermore, as a complementary approach, we also investigated fine structures in the stress field that show front-rear asymmetric kinetics consistent with the multipole analysis. The tight force-motion relation enables us to predict cell migration only from the traction stress patterns.     

Traction in smooth muscle cells varies with cell spreading

Changes in cell shape regulate cell growth, differentiation, and apoptosis. It has been suggested that the regulation of cell function by the cell shape is a result of the tension in the cytoskeleton and the distortion of the cell. Here we explore the association between cell-generated mechanical forces and the cell morphology. We hypothesized that the cell contractile force is associated with the degree of cell spreading, in particular with the cell length. We measured traction fields of single human airway smooth muscle cells plated on a polyacrylamide gel, in which fluorescent microbeads were embedded to serve as markers of gel deformation. The traction exerted by the cells at the cell-substrate interface was determined from the measured deformation of the gel. The traction was measured before and after treatment with the contractile agonist histamine, or the relaxing agonist isoproterenol. The relative increase in traction induced by histamine was negatively correlated with the baseline traction. On the contrary, the relative decrease in traction due to isoproterenol was independent of the baseline traction, but it was associated with cell shape: traction decreased more in elongated than in round cells. Maximum cell width, mean cell width, and projected area of the cell were the parameters most tightly coupled to both baseline and histamine-induced traction in this study. Wide and well-spread cells exerted larger traction than slim cells. These results suggest that cell contractility is controlled by cell spreading.