Determining substrate displacement and cell traction fields--a new approach

This paper presents a new approach for the traction force microscopy (TFM) method which determines traction forces exerted by adherent cells on a thin, elastic polyacrylamide gel embedded with fluorescent microbeads. In this enhanced TFM method, a pattern recognition technique is first applied to match the pair of microbead embedded images before and after deformation, which subsequently provides the displacement field of the elastic substrate. Once the displacement field is obtained, the 3-D finite element method (FEM) is used to compute cell traction forces. The new TFM has been applied to determine traction forces of human tendon fibroblasts. Compared to existing TFM methods, the present method has the following advantages: (1) its displacement field obtained is associated with microbead movements; (2) it considers the finite thickness of the thin polyacrylamide gel and is therefore free from the infinite half-space approximation adopted by existing TFM methods; and (3) its computation procedure for determining cell traction forces is fast.     

Traction patterns of tumor cells

The traction exerted by a cell on a planar deformable substrate can be indirectly obtained on the basis of the displacement field of the underlying layer. The usual methodology used to address this inverse problem is based on the exploitation of the Green tensor of the linear elasticity problem in a half space (Boussinesq problem), coupled with a minimization algorithm under force penalization. A possible alternative strategy is to exploit an adjoint equation, obtained on the basis of a suitable minimization requirement. The resulting system of coupled elliptic partial differential equations is applied here to determine the force field per unit surface generated by T24 tumor cells on a polyacrylamide substrate. The shear stress obtained by numerical integration provides quantitative insight of the traction field and is a promising tool to investigate the spatial pattern of force per unit surface generated in cell motion, particularly in the case of such cancer cells.     

A Novel Method for Localizing Reporter Fluorescent Beads Near the Cell Culture Surface for Traction Force Microscopy

PA gels have long been used as a platform to study cell traction forces due to ease of fabrication and the ability to tune their elastic properties. When the substrate is coated with an extracellular matrix protein, cells adhere to the gel and apply forces, causing the gel to deform. The deformation depends on the cell traction and the elastic properties of the gel. If the deformation field of the surface is known, surface traction can be calculated using elasticity theory. Gel deformation is commonly measured by embedding fluorescent marker beads uniformly into the gel. The probes displace as the gel deforms. The probes near the surface of the gel are tracked. The displacements reported by these probes are considered as surface displacements. Their depths from the surface are ignored. This assumption introduces error in traction force evaluations. For precise measurement of cell forces, it is critical for the location of the beads to be known. We have developed a technique that utilizes simple chemistry to confine fluorescent marker beads, 0.1 and 1 µm in diameter, in PA gels, within 1.6 μm of the surface. We coat a coverslip with poly-D-lysine (PDL) and fluorescent beads. PA gel solution is then sandwiched between the coverslip and an adherent surface. The fluorescent beads transfer to the gel solution during curing. After polymerization, the PA gel contains fluorescent beads on a plane close to the gel surface.     

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.     

A Novel Cell Traction Force Microscopy to Study Multi-Cellular System

Traction forces exerted by adherent cells on their microenvironment can mediate many critical cellular functions. Accurate quantification of these forces is essential for mechanistic understanding of mechanotransduction. However, most existing methods of quantifying cellular forces are limited to single cells in isolation, whereas most physiological processes are inherently multi-cellular in nature where cell-cell and cell-microenvironment interactions determine the emergent properties of cell clusters. In the present study, a robust finite-element-method-based cell traction force microscopy technique is developed to estimate the traction forces produced by multiple isolated cells as well as cell clusters on soft substrates. The method accounts for the finite thickness of the substrate. Hence, cell cluster size can be larger than substrate thickness. The method allows computing the traction field from the substrate displacements within the cells'' and clusters'' boundaries. The displacement data outside these boundaries are not necessary. The utility of the method is demonstrated by computing the traction generated by multiple monkey kidney fibroblasts (MKF) and human colon cancerous (HCT-8) cells in close proximity, as well as by large clusters. It is found that cells act as individual contractile groups within clusters for generating traction. There may be multiple of such groups in the cluster, or the entire cluster may behave a single group. Individual cells do not form dipoles, but serve as a conduit of force (transmission lines) over long distances in the cluster. The cell-cell force can be either tensile or compressive depending on the cell-microenvironment interactions.     

High resolution traction force microscopy based on experimental and computational advances

Cell adhesion and migration crucially depend on the transmission of actomyosin-generated forces through sites of focal adhesion to the extracellular matrix. Here we report experimental and computational advances in improving the resolution and reliability of traction force microscopy. First, we introduce the use of two differently colored nanobeads as fiducial markers in polyacrylamide gels and explain how the displacement field can be computationally extracted from the fluorescence data. Second, we present different improvements regarding standard methods for force reconstruction from the displacement field, which are the boundary element method, Fourier-transform traction cytometry, and traction reconstruction with point forces. Using extensive data simulation, we show that the spatial resolution of the boundary element method can be improved considerably by splitting the elastic field into near, intermediate, and far field. Fourier-transform traction cytometry requires considerably less computer time, but can achieve a comparable resolution only when combined with Wiener filtering or appropriate regularization schemes. Both methods tend to underestimate forces, especially at small adhesion sites. Traction reconstruction with point forces does not suffer from this limitation, but is only applicable with stationary and well-developed adhesion sites. Third, we combine these advances and for the first time reconstruct fibroblast traction with a spatial resolution of ∼1 μm.     

Detection of the third and fourth heart sounds using Hilbert-Huang transform

Background

Magnetic nanoparticles are gaining great roles in biomedical applications as targeted drug delivery agents or targeted imaging contrast agents. In the magnetic nanoparticle applications, quantification of the nanoparticle density deposited in a specified region is of great importance for evaluating the delivery of the drugs or the contrast agents to the targeted tissues. We introduce a method for estimating the nanoparticle density from the displacement of tissues caused by the external magnetic field.

Methods

We can exert magnetic force to the magnetic nanoparticles residing in a living subject by applying magnetic gradient field to them. The nanoparticles under the external magnetic field then exert force to the nearby tissues causing displacement of the tissues. The displacement field induced by the nanoparticles under the external magnetic field is governed by the Navier's equation. We use an approximation method to get the inverse solution of the Navier's equation which represents the magnetic nanoparticle density map when the magnetic nanoparticles are mechanically coupled with the surrounding tissues. To produce the external magnetic field inside a living subject, we propose a coil configuration, the Helmholtz and Maxwell coil pair, that is capable of generating uniform magnetic gradient field. We have estimated the coil currents that can induce measurable displacement in soft tissues through finite element method (FEM) analysis.

Results

From the displacement data obtained from FEM analysis of a soft-tissue-mimicking phantom, we have calculated nanoparticle density maps. We obtained the magnetic nanoparticle density maps by approximating the Navier's equation to the Laplacian of the displacement field. The calculated density maps match well to the original density maps, but with some halo artifacts around the high density area. To induce measurable displacement in the living tissues with the proposed coil configuration, we need to apply the coil currents as big as 10⁴A.

Conclusions

We can obtain magnetic nanoparticle maps from the magnetically induced displacement data by approximating the Navier's equation under the assumption of uniform-gradient of the external magnetic field. However, developing a coil driving system with the capacity of up to 10⁴A should be a great technical challenge.     

A novel patterned magnetic micropillar array substrate for analysis of cellular mechanical responses

Traction forces generated at cellular focal adhesions (FAs) play an essential role in regulating various cellular functions. These forces (1–100 nN) can be measured by observing the local displacement of a flexible substrate upon which cells have been plated. Approaches employing this method include using microfabricated arrays of poly(dimethylsiloxane) (PDMS) micropillars that bend by cellular traction forces. A tool capable of applying a force to FAs independently, by actively moving the micropillars, should become a powerful tool to delineate the cellular mechanotransduction mechanisms. Here, we developed a patterned magnetic micropillar array PDMS substrate that can be used for the mechanical stimulation of cellular FAs and the measurement of associated traction forces. The diameter, length, and center-to-center spacing of the micropillars were 3, 9, and 9 µm, respectively. Iron particles were embedded into the micropillars, enabling the pillars to bend in response to an external magnetic field, which also controlled their location on the substrate. Applying a magnetic field of 0.3 T bent the pillars by ∼4 µm and allowed transfer of external forces to the actin cytoskeleton through FAs formed on the pillar top. Using this approach, we investigated the traction force changes in cultured aortic smooth muscle cells (SMCs) after local compressive stimuli to release cell pretension. The mechanical responses of SMCs were roughly classified into two types: almost a half of the cells showed a little decrease of traction force at each pillar following compressive stimulation, although cell area increased significantly; and the rest showed the opposite, with increased forces and a simultaneous decrease in area. The traction forces of SMCs fluctuated markedly during the local compression. The root mean square of traction forces significantly increased during the compression, and returned to the baseline level after its release. These results suggest that the fluctuation of forces may be caused by active reorganization of the actin cytoskeleton and/or its dynamic interaction with myosin molecules. Thus, our magnetic micropillar substrate would be useful in investigating the mechanotransduction mechanisms of cells.     

Cellular traction force recovery: An optimal filtering approach in two-dimensional Fourier space

Quantitative estimation of cellular traction has significant physiological and clinical implications. As an inverse problem, traction force recovery is essentially susceptible to noise in the measured displacement data. For traditional procedure of Fourier transform traction cytometry (FTTC), noise amplification is accompanied in the force reconstruction and small tractions cannot be recovered from the displacement field with low signal-noise ratio (SNR). To improve the FTTC process, we develop an optimal filtering scheme to suppress the noise in the force reconstruction procedure. In the framework of the Wiener filtering theory, four filtering parameters are introduced in two-dimensional Fourier space and their analytical expressions are derived in terms of the minimum-mean-squared-error (MMSE) optimization criterion. The optimal filtering approach is validated with simulations and experimental data associated with the adhesion of single cardiac myocyte to elastic substrate. The results indicate that the proposed method can highly enhance SNR of the recovered forces to reveal tiny tractions in cell-substrate interaction.     

Three-dimensional traction force microscopy: a new tool for quantifying cell-matrix interactions

The interactions between biochemical processes and mechanical signaling play important roles during various cellular processes such as wound healing, embryogenesis, metastasis, and cell migration. While traditional traction force measurements have provided quantitative information about cell matrix interactions in two dimensions, recent studies have shown significant differences in the behavior and morphology of cells when placed in three-dimensional environments. Hence new quantitative experimental techniques are needed to accurately determine cell traction forces in three dimensions. Recently, two approaches both based on laser scanning confocal microscopy have emerged to address this need. This study highlights the details, implementation and advantages of such a three-dimensional imaging methodology with the capability to compute cellular traction forces dynamically during cell migration and locomotion. An application of this newly developed three-dimensional traction force microscopy (3D TFM) technique to single cell migration studies of 3T3 fibroblasts is presented to show that this methodology offers a new quantitative vantage point to investigate the three-dimensional nature of cell-ECM interactions.     

Mouvements des diatomées VI. Les efforts pendant le déplacement apical Mesures,analyses, relations: longueur,vitesse, force

Experiments involving diatom movements were conducted on 27 species to determine the maximum force developed during apical displacement of each species. The results are compared with previously published data, and the force developed by each of the 27 species also has been analyzed in relation to the speed and length of these diatoms. Speed and length are shown to be totally independent variables, and no consistent correlation between speed and force developed was found. By constrast, a direct correlation was established between diatom length and force developed. Results for various species, however, are quite divergent. For some species a strong correlation occurred between force developed and speed. Calculation of the force developed in absolute values and comparison with real weight in water of the same diatom species show that maximum forces developed are 109 to 5000 times greater than the weight capacity of the species in water. This clearly demonstrates that diatoms are capable of extremely quick movements. Comparisons between maximum forces developed during a polar pivoting and the force per unit length of the raphe shows that the latter one is three times greater. Owing to measurements of the resistance to bending of organic matter, effects of weight traction force and of diatom thrust on organic matter can be calculated, and it can be demonstrated that diatoms can undertake complex movements without requiring exterior forces. Moreover, general mechanics calculations can be applied to the study of forces developed by diatoms without major problems.     

Cell Traction Forces Direct Fibronectin Matrix Assembly

Interactions between cells and the surrounding matrix are critical to the development and engineering of tissues. We have investigated the role of cell-derived traction forces in the assembly of extracellular matrix using what we believe is a novel assay that allows for simultaneous measurement of traction forces and fibronectin fibril growth at discrete cell-matrix attachment sites. NIH3T3 cells were plated onto arrays of deformable cantilever posts for 2-24 h. Data indicate that developing fibril orientation is guided by the direction of the traction force applied to that fibril. In addition, cells initially establish a spatial distribution of traction forces that is largest at the cell edge and decreases toward the cell center. This distribution progressively shifts from a predominantly peripheral pattern to a more uniform pattern as compressive strain at the cell perimeter decreases with time. The impact of these changes on fibrillogenesis was tested by treating cells with blebbistatin or calyculin A to tonically block or augment, respectively, myosin II activity. Both treatments blocked the inward translation of traction forces, the dissipation of compressive strain, and fibronectin fibrillogenesis over time. These data indicate that dynamic spatial and temporal changes in traction force and local strain may contribute to successful matrix assembly.     

Spatial and temporal traction response in human airway smooth muscle cells

Tractions that cells exert on theirsubstrates are essential in cell spreading, migration, and contraction.These tractions can be determined by plating the cells on a flexiblegel and measuring the deformation of the gel by using fluorescent beadsembedded just below the surface of the gel. In this article we describe the image correlation method (ICM) optimized for determining the displacement field of the gel under a contracting cell. For the calculation of the traction field from the displacement field we usethe recently developed method of Fourier transform traction cytometry(FTTC). The ICM and FTTC methods are applied to human airway smoothmuscle cells during stimulation with the contractile agonist histamineor the relaxing agonist isoproterenol. The overall intensity of thecell contraction (the median traction magnitude, the energy transferredfrom the cell to the gel, and the net contractile moment) increasedafter activation with histamine, and decreased after treatment withisoproterenol. Cells exhibited regional differences in the time courseof traction during the treatment. Both temporal evolution and magnitudeof traction increase induced by histamine varied markedly amongdifferent cell protrusions, whereas the nuclear region showed thesmallest response. These results suggest that intracellular mediatorsof cell adhesion and contraction respond to contractile stimuli withdifferent rates and intensities in different regions of the cell.

     

The Poisson Ratio of the Cellular Actin Cortex Is Frequency Dependent

Cell shape changes are vital for many physiological processes such as cell proliferation, cell migration, and morphogenesis. They emerge from an orchestrated interplay of active cellular force generation and passive cellular force response, both crucially influenced by the actin cytoskeleton. To model cellular force response and deformation, cell mechanical models commonly describe the actin cytoskeleton as a contractile isotropic incompressible material. However, in particular at slow frequencies, there is no compelling reason to assume incompressibility because the water content of the cytoskeleton may change. Here, we challenge the assumption of incompressibility by comparing computer simulations of an isotropic actin cortex with tunable Poisson ratio to measured cellular force response. Comparing simulation results and experimental data, we determine the Poisson ratio of the cortex in a frequency-dependent manner. We find that the Poisson ratio of the cortex decreases in the measured frequency regime analogous to trends reported for the Poisson ratio of glassy materials. Our results therefore indicate that actin cortex compression or dilation is possible in response to acting forces at sufficiently fast timescales. This finding has important implications for the parameterization in active gel theories that describe actin cytoskeletal dynamics.     

Galectin-3 secretion and tyrosine phosphorylation is dependent on the calpain small subunit, Calpain 4

Cell adhesion and migration are important events that occur during embryonic development, immune surveillance, wound healing and in tumor metastasis. It is a multi-step process that involves both mechanical and biochemical signaling that results in cell protrusion, adhesion, contraction and retraction. Each of these events generates mechanical forces into the environment measured as traction forces. We have previously found that the calpain small subunit, Calpain 4, is required for normal traction forces, and that this mechanism is independent of the catalytic activities of the holoenzymes that are formed between Calpain 4 and each of the proteolytic heavy chains of Calpain 1 and 2. To define a potential mechanism for the Calpain 4 regulation of traction force, we have evaluated the levels of tyrosine phosphorylation, a hallmark of force dependent signaling within focal adhesions. Using 2D gel electrophoresis we compared tyrosine phosphorylation profiles of Calpain 4 deficient mouse embryonic fibroblasts (MEFs) to the levels in wildtype MEFs and MEF’s deficient in the large catalytic subunits, Capn1 and Capn2. Of particular interest, was the identification of Galectin-3, a galactose binding protein known to interact with integrins. Galectin-3 has previously been shown to regulate cell adhesion and migration in both normal and tumor cells; however its full mechanism remains elusive. We have found that Calpain 4 is essential for the tyrosine phosphorylation of galectin-3, and its ultimate secretion from the cell, and speculate that its secretion interferes with the production of traction forces.     

Molecular dynamics and forces of a motile cell simultaneously visualized by TIRF and force microscopies

Cells must exert traction forces onto the substratum for continuous migration. Molecular dynamics such as actin polymerization at the front of the cell and myosin II accumulation at the rear should play important roles in the exertion of forces required for migration. Therefore, it is important to reveal the relationships between the traction forces and molecular dynamics. Traction forces can be calculated from the deformation of the elastic substratum under a migrating cell. A transparent and colorless elastic substratum with a high refractive index (1.40) and a low Young's modulus (1.0 kPa) were made from a pair of platinum-catalyzed silicones. We used this substratum to develop a new method for simultaneous recording of molecular dynamics and traction forces under a migrating cell in which total internal refractive fluorescence (TIRF) and force microscopies were combined. This new method allows the detection of the spatiotemporal distribution of traction forces produced by individual filopodia in migrating Dictyostelium cells, as well as simultaneous visualization of these traction forces and the dynamics of filamentous myosin II.     

Propagation of force and the induced bending displacement along eukaryotic flagellar axoneme.

The mechanical forces responsible for inducing the bending movement of eukaryotic flagellar axonemes have components that propagate at velocities different from those for the displacement of the medium. These forces are subject to the Third Law of Mechanics which states the null-conservation of acting and reacting forces. Experimental demonstration of the propagating internal tensile force along the axoneme, when not accompanied by simultaneous bending displacement, demonstrates that the active force for the bending of the flagellar axoneme is part of the process of counterbalancing the corresponding reactive force to the ATP hydrolysis underlying the force generation.     

Imaging the traction stresses exerted by locomoting cells with the elastic substratum method.

The cytoskeletal activity of motile or adherent cells is frequently seen to induce detectable displacements of sufficiently compliant substrata. The physics of this phenomenon is discussed in terms of the classical theory of small-strain, plane-stress elasticity. The main results of such analysis is a transform expressing the displacement field of the elastic substrate as an integral over the traction field. The existence of this transform is used to derive a Bayesian method for converting noisy measurements of substratum displacement into "images" of the actual traction forces exerted by adherent or locomoting cells. Finally, the Monte Carlo validation of the statistical method is discussed, some new rheological studies of films are presented, and a practical application is given.     

Traction Force Screening Enabled by Compliant PDMS Elastomers

Actomyosin contractility is an essential element of many aspects of cellular biology and manifests as traction forces that cells exert on their surroundings. The central role of these forces makes them a novel principal therapeutic target in diverse diseases. This requires accurate and higher-capacity measurements of traction forces; however, existing methods are largely low throughput, limiting their utility in broader applications. To address this need, we employ Fourier-transform traction force microscopy in a parallelized 96-well format, which we refer to as contractile force screening. Critically, rather than the frequently employed hydrogel polyacrylamide, we fabricate these plates using polydimethylsiloxane rubber. Key to this approach is that the polydimethylsiloxane used is very compliant, with a lower-bound Young’s modulus of ～0.4 kPa. We subdivide these monolithic substrates spatially into biochemically independent wells, creating a uniform multiwell platform for traction force screening. We demonstrate the utility and versatility of this platform by quantifying the compound and dose-dependent contractility responses of human airway smooth muscle cells and retinal pigment epithelial cells. By directly quantifying the endpoint of therapeutic intent, airway-smooth-muscle contractile force, this approach fills an important methodological void in current screening approaches for bronchodilator drug discovery, and, more generally, in measuring contractile response for a broad range of cell types and pathologies.     

Coupling traction force patterns and actomyosin wave dynamics reveals mechanics of cell motion

Motile cells can use and switch between different modes of migration. Here, we use traction force microscopy and fluorescent labeling of actin and myosin to quantify and correlate traction force patterns and cytoskeletal distributions in Dictyostelium discoideum cells that move and switch between keratocyte‐like fan‐shaped, oscillatory, and amoeboid modes. We find that the wave dynamics of the cytoskeletal components critically determine the traction force pattern, cell morphology, and migration mode. Furthermore, we find that fan‐shaped cells can exhibit two different propulsion mechanisms, each with a distinct traction force pattern. Finally, the traction force patterns can be recapitulated using a computational model, which uses the experimentally determined spatiotemporal distributions of actin and myosin forces and a viscous cytoskeletal network. Our results suggest that cell motion can be generated by friction between the flow of this network and the substrate.