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.     

Mechanical model of cytoskeleton structuration during cell adhesion and spreading

The biomechanical behavior of an adherent cell is intimately dependent on its cytoskeleton structure. Several models have been proposed to study this structure taking into account its existing internal forces. However, the structural and geometrical complexities of the cytoskeleton's filamentous networks lead to difficulties for determining a biologically realistic architecture. The objective of this paper is to present a mechanical model, combined with a numerical method, devoted to the form-finding of the cytoskeleton structure (shape and internal forces) when a cell adheres on a substrate. The cell is modeled as a granular medium, using rigid spheres (grains) corresponding to intracellular cross-linking proteins and distant mechanical interactions to reproduce the cytoskeleton filament internal forces. At the initial state (i.e., before adhesion), these interactions are tacit. The adhesion phenomenon is then simulated by considering microtubules growing from the centrosome towards transmembrane integrin-like receptors. The simulated cell shape changes in this process and results in a mechanically equilibrated structure with traction and compression forces, in interaction with the substrate reactions. This leads to a compressive microtubule network and a corresponding tensile actin-filament network. The results provide coherent shape and forces information for developing a mechanical model of the cytoskeleton structure, which can be exploitable in future biomechanical studies of adherent cells.     

Computational Tension Mapping of Adherent Cells Based on Actin Imaging

Forces transiting through the cytoskeleton are known to play a role in adherent cell activity. Up to now few approaches haves been able to determine theses intracellular forces. We thus developed a computational mechanical model based on a reconstruction of the cytoskeleton of an adherent cell from fluorescence staining of the actin network and focal adhesions (FA). Our custom made algorithm converted the 2D image of an actin network into a map of contractile interactions inside a 2D node grid, each node representing a group of pixels. We assumed that actin filaments observed under fluorescence microscopy, appear brighter when thicker, we thus presumed that nodes corresponding to pixels with higher actin density were linked by stiffer interactions. This enabled us to create a system of heterogeneous interactions which represent the spatial organization of the contractile actin network. The contractility of this interaction system was then adapted to match the level of force the cell truly exerted on focal adhesions; forces on focal adhesions were estimated from their vinculin expressed size. This enabled the model to compute consistent mechanical forces transiting throughout the cell. After computation, we applied a graphical approach on the original actin image, which enabled us to calculate tension forces throughout the cell, or in a particular region or even in single stress fibers. It also enabled us to study different scenarios which may indicate the mechanical role of other cytoskeletal components such as microtubules. For instance, our results stated that the ratio between intra and extra cellular compression is inversely proportional to intracellular tension.     

Cooperativity between integrin activation and mechanical stress leads to integrin clustering

Integrins are transmembrane receptors involved in crucial cellular biological functions such as migration, adhesion, and spreading. Upon the modulation of integrin affinity toward their extracellular ligands by cytoplasmic proteins (inside-out signaling) these receptors bind to their ligands and cluster into nascent adhesions. This clustering results in the increase in the mechanical linkage among the cell and substratum, cytoskeleton rearrangements, and further outside-in signaling. Based on experimental observations of the distribution of focal adhesions in cells attached to micropatterned surfaces, we introduce a physical model relying on experimental numerical constants determined in the literature. In this model, allosteric integrin activation works in synergy with the stress build by adhesion and the membrane rigidity to allow the clustering to nascent adhesions independently of actin but dependent on the integrin diffusion onto adhesive surfaces. The initial clustering could provide a template to the mature adhesive structures. Predictions of our model for the organization of focal adhesions are discussed in comparison with experiments using adhesive protein microarrays.     

Finite-element analysis of the adhesion-cytoskeleton-nucleus mechanotransduction pathway during endothelial cell rounding: axisymmetric model

Endothelial cells possess a mechanical network connecting adhesions on the basal surface, the cytoskeleton, and the nucleus. Transmission of force at adhesions via this pathway can deform the nucleus, ultimately resulting in an alteration of gene expression and other cellular changes (mechanotransduction). Previously, we measured cell adhesion area and apparent nuclear stretch during endothelial cell rounding. Here, we reconstruct the stress map of the nucleus from the observed strains using finite-element modeling. To simulate the disruption of adhesions, we prescribe displacement boundary conditions at the basal surface of the axisymmetric model cell. We consider different scenarios of the cytoskeletal arrangement, and represent the cytoskeleton as either discrete fibers or as an effective homogeneous layer When the nucleus is in the initial (spread) state, cytoskeletal tension holds the nucleus in an elongated, ellipsoidal configuration. Loss of cytoskeletal tension during cell rounding is represented by reactive forces acting on the nucleus in the model. In our simulations of cell rounding, we found that, for both representations of the cytoskeleton, the loss of cytoskeletal tension contributed more to the observed nuclear deformation than passive properties. Since the simulations make no assumption about the heterogeneity of the nucleus, the stress components both within and on the surface of the nucleus were calculated. The nuclear stress map showed that the nucleus experiences stress on the order of magnitude that can be significant for the function of DNA molecules and chromatin fibers. This study of endothelial cell mechanobiology suggests the possibility that mechanotransduction could result, in part, from nuclear deformation, and may be relevant to angiogenesis, wound healing, and endothelial barrier dysfunction.     

Physical model for self-organization of actin cytoskeleton and adhesion complexes at the cell front

Cell motion is driven by interplay between the actin cytoskeleton and the cell adhesions in the front part of the cell. The actin network segregates into lamellipodium and lamellum, whereas the adhesion complexes are characteristically distributed underneath the actin system. Here, we suggest a computational model for this characteristic organization of the actin-adhesion system. The model is based on the ability of the adhesion complexes to sense mechanical forces, the stick-slip character of the interaction between the adhesions and the moving actin network, and a hypothetical propensity of the actin network to disintegrate upon sufficiently strong stretching stresses. We identify numerically three possible types of system organization, all observed in living cells: two states in which the actin network exhibits segregation into lamellipodium and lamellum, whereas the cell edge either remains stationary or moves, and a state where the actin network does not undergo segregation. The model recovers the asynchronous fluctuations and outward bulging of the cell edge, and the dependence of the edge protrusion velocity on the rate of the nascent adhesion generation, the membrane tension, and the substrate rigidity.     

Cytoskeletal regulation of pulmonary vascular permeability.

The endothelial cell (EC) lining of the pulmonary vasculature forms a semipermeable barrier between the blood and the interstitium of the lung. Disruption of this barrier occurs during inflammatory disease states such as acute lung injury and acute respiratory distress syndrome and results in the movement of fluid and macromolecules into the interstitium and pulmonary air spaces. These processes significantly contribute to the high morbidity and mortality of patients afflicted with acute lung injury. The critical importance of pulmonary vascular barrier function is shown by the balance between competing EC contractile forces, which generate centripetal tension, and adhesive cell-cell and cell-matrix tethering forces, which regulate cell shape. Both competing forces in this model are intimately linked through the endothelial cytoskeleton, a complex network of actin microfilaments, microtubules, and intermediate filaments, which combine to regulate shape change and transduce signals within and between EC. A key EC contractile event in several models of agonist-induced barrier dysfunction is the phosphorylation of regulatory myosin light chains catalyzed by Ca(2+)/calmodulin-dependent myosin light chain kinase and/or through the activity of the Rho/Rho kinase pathway. Intercellular contacts along the endothelial monolayer consist primarily of two types of complexes (adherens junctions and tight junctions), which link to the actin cytoskeleton to provide both mechanical stability and transduction of extracellular signals into the cell. Focal adhesions provide additional adhesive forces in barrier regulation by forming a critical bridge for bidirectional signal transduction between the actin cytoskeleton and the cell-matrix interface. Increasingly, the effects of mechanical forces such as shear stress and ventilator-induced stretch on EC barrier function are being recognized. The critical role of the endothelial cytoskeleton in integrating these multiple aspects of pulmonary vascular permeability provides a fertile area for the development of clinically important barrier-modulating therapies.     

Actin cytoskeleton dynamics and the cell division cycle

The network of actin filaments is one of the crucial cytoskeletal structures contributing to the morphological framework of a cell and which participates in the dynamic regulation of cellular functions. In adherent cell types, cells adhere to the substratum during interphase and spread to assume their characteristic shape supported by the actin cytoskeleton. This actin cytoskeleton is reorganized during mitosis to form rounded cells with increased cortical rigidity. The actin cytoskeleton is re-established after mitosis, allowing cells to regain their extended shape and attachment to the substratum. The modulation of such drastic changes in cell shape in coordination with cell cycle progression suggests a tight regulatory interaction between cytoskeleton signalling, cell–cell/cell–matrix adhesions and mitotic events. Here, we review the contribution of the actin cytoskeleton to cell cycle progression with an emphasis on the effectors responsible for the regulation of the actin cytoskeleton and integration of their activities with the cell cycle machinery.     

Interplay between adhesion turnover and cytoskeleton dynamics in the control of growth cone migration

The migration of neuronal growth cones, driving axon extension, is a fascinating process which has been subject of intense investigation over several decades. Many of the key underlying molecules, in particular adhesion proteins at the cell membrane which allow for target recognition and binding, and cytoskeleton filaments and motors which power locomotion, have been identified. However, the precise mechanisms by which growth cones coordinate, in time and space, the transmission of forces generated by the cytoskeleton to the turnover of adhesion proteins, are still partly unresolved. To get a better grasp at these processes, we put here in relation the turnover rate of ligand/receptor adhesions and the degree of mechanical coupling between cell adhesion receptors and the actin rearward flow. These parameters were obtained recently for N-cadherin and IgCAM based adhesions using ligand-coated microspheres in combination with optical tweezers and photo-bleaching experiments. We show that the speed of growth cone migration requires both a fairly rapid adhesion dynamics and a strong physical connection between adhesive sites and the cytoskeleton.     

A 3D Motile Rod-Shaped Monotrichous Bacterial Model

We introduce a 3D model for a motile rod-shaped bacterial cell with a single polar flagellum which is based on the configuration of a monotrichous type of bacteria such as Pseudomonas aeruginosa. The structure of the model bacterial cell consists of a cylindrical body together with the flagellar forces produced by the rotation of a helical flagellum. The rod-shaped cell body is composed of a set of immersed boundary points and elastic links. The helical flagellum is assumed to be rigid and modeled as a set of discrete points along the helical flagellum and flagellar hook. A set of flagellar forces are applied along this helical curve as the flagellum rotates. An additional set of torque balance forces are applied on the cell body to induce counter-rotation of the body and provide torque balance. The three-dimensional Navier–Stokes equations for incompressible fluid are used to describe the fluid dynamics of the coupled fluid–microorganism system using Peskin’s immersed boundary method. A study of numerical convergence is presented along with simulations of a single swimming cell, the hydrodynamic interaction of two cells, and the interaction of a small cluster of cells.     

A mechanoresponsive cadherin-keratin complex directs polarized protrusive behavior and collective cell migration

Collective cell migration requires maintenance of adhesive contacts between adjacent cells, coordination of polarized cell protrusions, and generation of propulsive traction forces. We demonstrate that mechanical force applied locally to C-cadherins on single Xenopus mesendoderm cells is sufficient to induce polarized cell protrusion and persistent migration typical of individual cells within a collectively migrating tissue. Local tension on cadherin adhesions induces reorganization of the keratin intermediate filament network toward these stressed sites. Plakoglobin, a member of the catenin family, is localized to cadherin adhesions under tension and is required for both mechanoresponsive cell behavior and assembly of the keratin cytoskeleton at the rear of these cells. Local tugging forces on cadherins occur in vivo through interactions with neighboring cells, and these forces result in coordinate changes in cell protrusive behavior. Thus, cadherin-dependent force-inducible regulation of cell polarity in single mesendoderm cells represents an emergent property of the intact tissue.     

Continuum model of cell adhesion and migration

The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.     

Integrins and extracellular matrix in mechanotransduction

Integrins bind extracellular matrix fibrils and associate with intracellular actin filaments through a variety of cytoskeletal linker proteins to mechanically connect intracellular and extracellular structures. Each component of the linkage from the cytoskeleton through the integrin-mediated adhesions to the extracellular matrix therefore transmits forces that may derive from both intracellular, myosin-generated contractile forces and forces from outside the cell. These forces activate a wide range of signaling pathways and genetic programs to control cell survival, fate, and behavior. Additionally, cells sense the physical properties of their surrounding environment through forces exerted on integrin-mediated adhesions. This article first summarizes current knowledge about regulation of cell function by mechanical forces acting through integrin-mediated adhesions and then discusses models for mechanotransduction and sensing of environmental forces.     

Keratocytes generate traction forces in two phases

Forces generated by goldfish keratocytes and Swiss 3T3 fibroblasts have been measured with nanonewton precision and submicrometer spatial resolution. Differential interference contrast microscopy was used to visualize deformations produced by traction forces in elastic substrata, and interference reflection microscopy revealed sites of cell-substratum adhesions. Force ranged from a few nanonewtons at submicrometer spots under the lamellipodium to several hundred nanonewtons under the cell body. As cells moved forward, centripetal forces were applied by lamellipodia at sites that remained stationary on the substratum. Force increased and abruptly became lateral at the boundary of the lamellipodium and the cell body. When the cell retracted at its posterior margin, cell-substratum contact area decreased more rapidly than force, so that stress (force divided by area) increased as the cell pulled away. An increase in lateral force was associated with widening of the cell body. These mechanical data suggest an integrated, two-phase mechanism of cell motility: (1) low forces in the lamellipodium are applied in the direction of cortical flow and cause the cell body to be pulled forward; and (2) a component of force at the flanks pulls the rear margins forward toward the advancing cell body, whereas a large lateral component contributes to detachment of adhesions without greatly perturbing forward movement.     

Boyden chamber-based method for characterizing the distribution of adhesions and cytoskeletal structure in HT1080 fibrosarcoma cells

A 2D model was previously presented that describes the gliding motility of human fibrosarcoma cells. The model was based on the observation that adhesions are present only on the outer rim of the leading lamella of the semicircular cell. The present model describes the organization of adhesions and the cytoskeleton of migrating HT1080 fibrosarcoma and LX2 hepatic stellate cells in three dimensions. The migration assays were performed in a modified Boyden chamber using fibronectin, Matrigel, or collagen I as chemoattractants. The distribution of the adhesions was analyzed by confocal laser scanning microscope, and following decoration with heavy meromyosin, the organization of actin filaments was analyzed by electron microscopy. Double labeling was performed to study the relationship of the actin and vimentin filament network in the moving cells. Vinculin containing adhesions were observed only at the front of the cell in the form of a ring while passing through a filter pore of the Boyden chamber. Actin filaments were present only below the plasma membrane, except the very tip of the leading lamella. Vimentin intermediate filaments were localized around the cell nucleus behind the actin filament-rich lamella.

This paper describes a model of the organization of adhesions and the cytoskeleton of migrating cells in the Boyden chamber. The model is based on the observation that adhesions are present only at the leading edge of the cell. The results extend the earlier 2D model of cell locomotion into 3D.     

Crosstalk between Cell Adhesion Complexes in Regulation of Mechanotransduction

Physical forces regulate numerous biological processes during development, physiology, and pathology. Forces between the external environment and intracellular actin cytoskeleton are primarily transmitted through integrin-containing focal adhesions and cadherin-containing adherens junctions. Crosstalk between these complexes is well established and modulates the mechanical landscape of the cell. However, integrins and cadherins constitute large families of adhesion receptors and form multiple complexes by interacting with different ligands, adaptor proteins, and cytoskeletal filaments. Recent findings indicate that integrin-containing hemidesmosomes oppose force transduction and traction force generation by focal adhesions. The cytolinker plectin mediates this crosstalk by coupling intermediate filaments to the actin cytoskeleton. Similarly, cadherins in desmosomes might modulate force generation by adherens junctions. Moreover, mechanotransduction can be influenced by podosomes, clathrin lattices, and tetraspanin-enriched microdomains. This review discusses mechanotransduction by multiple integrin- and cadherin-based cell adhesion complexes, which together with the associated cytoskeleton form an integrated network that allows cells to sense, process, and respond to their physical environment.     

A phenomenological cohesive model for the macroscopic simulation of cell–matrix adhesions

Cell adhesion is crucial for cells to not only physically interact with each other but also sense their microenvironment and respond accordingly. In fact, adherent cells can generate physical forces that are transmitted to the surrounding matrix, regulating the formation of cell–matrix adhesions. The main purpose of this work is to develop a computational model to simulate the dynamics of cell–matrix adhesions through a cohesive formulation within the framework of the finite element method and based on the principles of continuum damage mechanics. This model enables the simulation of the mechanical adhesion between cell and extracellular matrix (ECM) as regulated by local multidirectional forces and thus predicts the onset and growth of the adhesion. In addition, this numerical approach allows the simulation of the cell as a whole, as it models the complete mechanical interaction between cell and ECM. As a result, we can investigate and quantify how different mechanical conditions in the cell (e.g., contractile forces, actin cytoskeletal properties) or in the ECM (e.g., stiffness, external forces) can regulate the dynamics of cell–matrix adhesions.     

Locomotion and adhesion of polymorphonuclear leukocytes. Effects of the supporting substratum

Contact angle measurements have been used to correlate surface hydrophobicity of a supporting substratum with adhesion and locomotion of polymorphonuclear leukocytes. The binding of human serum albumin, a well-known chemokinetic substance, to hydrophilic glass slides gave rise to hydrophobic surfaces with adhesive properties conductive to cell polarization, thus allowing cell locomotion. Parallel contact angle and cell adhesion measurements suggested that albumin modified the cell-substratum interaction by increasing the van der Waals forces of attraction and reducing the electrostatic forces. By allowing cells to adhere to a hydrophobic surface (siliconized glass), it was found that protein could be omitted from in vitro test systems for leukocyte locomotion. It is suggested that quantitatively equal cell adhesion values may, depending on the type of attraction forces working in adhesion to the substratum, result in different locomotion patterns.     

Locomotion and adhesion of polymorphonuclear leukocytes effects of the supporting substratum

Contact angle measurements have been used to correlate surface hydrophobicity of a supporting substratum with adhesion and locomotion of polymorphonuclear leukocytes. The binding of human serum albumin, a well-known chemokinetic substance, to hydrophilic glass slides gave rise to hydrophobic surfaces with adhesive properties conducive, to cell polarization thus allowing cell locomotion. Parallel contact angle and cell adhesion measurements suggested that albumin modified the cellsubstratum interaction by increasing the van der Waals forces of attraction and reducing the electrostatic forces. By allowing cells to adhere to a hydrophobic surface (siliconized glass), it was found that protein could be omitted from in vitro test systems for leukocyte locomotion. It is suggested that quantitatively equal cell adhesion values may, depending on the type of attraction forces working in adhesion to the substratum, result in different locomotion patterns.     

The consequence of substrates of large-scale rigidity on actin network tension in adherent cells

Abstract

There is compelling evidence that substrate stiffness affects cell adhesion as well as cytoskeleton organization and contractile activity. This work was designed to study the cytoskeletal contractile activity of single cells plated on micropost substrates of different stiffness using a numerical model simulating the intracellular tension of individual cells. We allowed cells to adhere onto micropost substrates of various rigidities and used experimental traction force data to infer cell contractility using a numerical model. The model shows that higher substrate stiffness leads to an increase in intracellular tension. The strength of this model is its ability to calculate the mechanical state of each cell in accordance to its individual cytoskeletal structure. This is achieved by regenerating a numerical cytoskeleton based on microscope images of the actin network of each cell. The resulting numerical structure consequently represents pulling characteristics on its environment similar to those generated by the cell in-vivo. From actin imaging we can calculate and better understand how forces are transmitted throughout the cell.