Validation tool for traction force microscopy

Traction force microscopy (TFM) is commonly used to estimate cells' traction forces from the deformation that they cause on their substrate. The accuracy of TFM highly depends on the computational methods used to measure the deformation of the substrate and estimate the forces, and also on the specifics of the experimental set-up. Computer simulations can be used to evaluate the effect of both the computational methods and the experimental set-up without the need to perform numerous experiments. Here, we present one such TFM simulator that addresses several limitations of the existing ones. As a proof of principle, we recreate a TFM experimental set-up, and apply a classic 2D TFM algorithm to recover the forces. In summary, our simulator provides a valuable tool to study the performance, refine experimentally, and guide the extraction of biological conclusions from TFM experiments.     

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.     

High Resolution,Large Deformation 3D Traction Force Microscopy

Traction Force Microscopy (TFM) is a powerful approach for quantifying cell-material interactions that over the last two decades has contributed significantly to our understanding of cellular mechanosensing and mechanotransduction. In addition, recent advances in three-dimensional (3D) imaging and traction force analysis (3D TFM) have highlighted the significance of the third dimension in influencing various cellular processes. Yet irrespective of dimensionality, almost all TFM approaches have relied on a linear elastic theory framework to calculate cell surface tractions. Here we present a new high resolution 3D TFM algorithm which utilizes a large deformation formulation to quantify cellular displacement fields with unprecedented resolution. The results feature some of the first experimental evidence that cells are indeed capable of exerting large material deformations, which require the formulation of a new theoretical TFM framework to accurately calculate the traction forces. Based on our previous 3D TFM technique, we reformulate our approach to accurately account for large material deformation and quantitatively contrast and compare both linear and large deformation frameworks as a function of the applied cell deformation. Particular attention is paid in estimating the accuracy penalty associated with utilizing a traditional linear elastic approach in the presence of large deformation gradients.     

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.     

Improved-Throughput Traction Microscopy Based on Fluorescence Micropattern for Manual Microscopy

Traction force microscopy (TFM) is a quantitative technique for measuring cellular traction force, which is important in understanding cellular mechanotransduction processes. Traditional TFM has a significant limitation in that it has a low measurement throughput, commonly one per TFM dish, due to a lack of cell position information. To obtain enough cellular traction force data, an onerous workload is required including numerous TFM dish preparations and heavy cell-seeding activities, creating further difficulty in achieving identical experimental conditions among batches. In this paper, we present an improved-throughput TFM method using the well-developed microcontact printing technique and chemical modifications of linking microbeads to the gel surface to address these limitations. Chemically linking the microbeads to the gel surface has no significant influence on cell proliferation, morphology, cytoskeleton, and adhesion. Multiple pairs of force loaded and null force fluorescence images can be easily acquired by means of manual microscope with the aid of a fluorescence micropattern made by microcontact printing. Furthermore, keeping the micropattern separate from cells by using gels effectively eliminates the potential negative effect of the micropattern on the cells. This novel design greatly improves the analysis throughput of traditional TFM from one to at least twenty cells per petri dish without losing unique advantages, including a high spatial resolution of traction measurements. This newly developed method will boost the investigation of cell-matrix mechanical interactions.     

Traction force microscopy by deep learning

Cells interact mechanically with their surroundings by exerting and sensing forces. Traction force microscopy (TFM), purported to map cell-generated forces or stresses, represents an important tool that has powered the rapid advances in mechanobiology. However, to solve the ill-posed mathematical problem, conventional TFM involved compromises in accuracy and/or resolution. Here, we applied neural network-based deep learning as an alternative approach for TFM. We modified a neural network designed for image processing to predict the vector field of stress from displacements. Furthermore, we adapted a mathematical model for cell migration to generate large sets of simulated stresses and displacements for training and testing the neural network. We found that deep learning-based TFM yielded results that resemble those using conventional TFM but at a higher accuracy than several conventional implementations tested. In addition, a trained neural network is appliable to a wide range of conditions, including cell size, shape, substrate stiffness, and traction output. The performance of deep learning-based TFM makes it an appealing alternative to conventional methods for characterizing mechanical interactions between adherent cells and the environment.     

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.     

Cell traction force and measurement methods

Cell traction forces (CTFs) are crucial to many biological processes such as inflammation, wound healing, angiogenesis, and metastasis. CTFs are generated by actomyosin interactions and actin polymerization and regulated by intracellular proteins such as alpha-smooth muscle actin (α-SMA) and soluble factors such as transforming growth factor-β (TGF-β). Once transmitted to the extracellular matrix (ECM) through stress fibers via focal adhesions, which are assemblies of ECM proteins, transmembrane receptors, and cytoplasmic structural and signaling proteins (e.g., integrins), CTFs direct many cellular functions, including cell migration, ECM organization, and mechanical signal generation. Various methods have been developed over the years to measure CTFs of both populations of cells and of single cells. At present, cell traction force microscopy (CTFM) is among the most efficient and reliable method for determining CTF field of an entire cell spreading on a two-dimensional (2D) substrate surface. There are currently three CTFM methods, each of which is unique in both how displacement field is extracted from images and how CTFs are subsequently estimated. A detailed review and comparison of these methods are presented. Future research should improve CTFM methods such that they can automatically track dynamic CTFs, thereby providing new insights into cell motility in response to altered biological conditions. In addition, research effort should be devoted to developing novel experimental and theoretical methods for determining CTFs in three-dimensional (3D) matrix, which better reflects physiological conditions than 2D substrate used in current CTFM methods.     

Eigenstrain as a mechanical set-point of cells

Cell contraction regulates how cells sense their mechanical environment. We sought to identify the set-point of cell contraction, also referred to as tensional homeostasis. In this work, bovine aortic endothelial cells (BAECs), cultured on substrates with different stiffness, were characterized using traction force microscopy (TFM). Numerical models were developed to provide insights into the mechanics of cell–substrate interactions. Cell contraction was modeled as eigenstrain which could induce isometric cell contraction without external forces. The predicted traction stresses matched well with TFM measurements. Furthermore, our numerical model provided cell stress and displacement maps for inspecting the fundamental regulating mechanism of cell mechanosensing. We showed that cell spread area, traction force on a substrate, as well as the average stress of a cell were increased in response to a stiffer substrate. However, the cell average strain, which is cell type-specific, was kept at the same level regardless of the substrate stiffness. This indicated that the cell average strain is the tensional homeostasis that each type of cell tries to maintain. Furthermore, cell contraction in terms of eigenstrain was found to be the same for both BAECs and fibroblast cells in different mechanical environments. This implied a potential mechanical set-point across different cell types. Our results suggest that additional measurements of contractility might be useful for monitoring cell mechanosensing as well as dynamic remodeling of the extracellular matrix (ECM). This work could help to advance the understanding of the cell-ECM relationship, leading to better regenerative strategies.     

Regulation of SMC traction forces in human aortic thoracic aneurysms

Smooth muscle cells (SMCs) usually express a contractile phenotype in the healthy aorta. However, aortic SMCs have the ability to undergo profound changes in phenotype in response to changes in their extracellular environment, as occurs in ascending thoracic aortic aneurysms (ATAA). Accordingly, there is a pressing need to quantify the mechanobiological effects of these changes at single cell level. To address this need, we applied Traction Force Microscopy (TFM) on 759 cells coming from three primary healthy (AoPrim) human SMC lineages and three primary aneurysmal (AnevPrim) human SMC lineages, from age and gender matched donors. We measured the basal traction forces applied by each of these cells onto compliant hydrogels of different stiffness (4, 8, 12, 25 kPa). Although the range of force generation by SMCs suggested some heterogeneity, we observed that: 1. the traction forces were significantly larger on substrates of larger stiffness; 2. traction forces in AnevPrim were significantly higher than in AoPrim cells. We modelled computationally the dynamic force generation process in SMCs using the motor-clutch model and found that it accounts well for the stiffness-dependent traction forces. The existence of larger traction forces in the AnevPrim SMCs were related to the larger size of cells in these lineages. We conclude that phenotype changes occurring in ATAA, which were previously known to reduce the expression of elongated and contractile SMCs (rendering SMCs less responsive to vasoactive agents), tend also to induce stronger SMCs. Future work aims at understanding the causes of this alteration process in aortic aneurysms.

     

Validation of the Boussinesq equation for use in traction field determination

Cell traction force plays an important role in many biological processes. Several traction force microscopy methods have been developed to determine cell traction forces based on the Boussinesq solution. This approach, however, is rooted in a half-space assumption. The purpose of this study was to determine the error induced in the half-space assumption using a finite element method (FEM). It demonstrates that displacement error between the FEM and the Boussinesq equation can be used to measure the accuracy of the Boussinesq equation, although singularity exists in the loading point. For one concentrated force, significant difference between the FEM and the Boussinesq equation occurs in the whole field; this difference decreases with an increase in the plate thickness. However, in the case of the balanced forces, the offset of the balanced forces decreases the errors in the middle area. Overall, this study demonstrates that increasing the thickness of the polyacrylamide gel is important for reducing the error of the Boussinesq equation when determining the displacement field of the gel under loads.     

Toward single cell traction microscopy within 3D collagen matrices

Mechanical interaction between the cell and its extracellular matrix (ECM) regulates cellular behaviors, including proliferation, differentiation, adhesion, and migration. Cells require the three-dimensional (3D) architectural support of the ECM to perform physiologically realistic functions. However, current understanding of cell–ECM and cell–cell mechanical interactions is largely derived from 2D cell traction force microscopy, in which cells are cultured on a flat substrate. 3D cell traction microscopy is emerging for mapping traction fields of single animal cells embedded in either synthetic or natively derived fibrous gels. We discuss here the development of 3D cell traction microscopy, its current limitations, and perspectives on the future of this technology. Emphasis is placed on strategies for applying 3D cell traction microscopy to individual tumor cell migration within collagen gels.     

3D Traction Stresses Activate Protease-Dependent Invasion of Cancer Cells

Cell invasion and migration that occurs, for example, in cancer metastasis is rooted in the ability of cells to navigate through varying levels of physical constraint exerted by the extracellular matrix. Cancer cells can invade matrices in either a protease-independent or a protease-dependent manner. An emerging critical component that influences the mode of cell invasion is the traction stresses generated by the cells in response to the physicostructural properties of the extracellular matrix. In this study, we have developed a reference-free quantitative assay for measuring three-dimensional (3D) traction stresses generated by cells during the initial stages of invasion into matrices exerting varying levels of mechanical resistance. Our results show that as cells encounter higher mechanical resistance, a larger fraction of them shift to protease-mediated invasion, and this process begins at lower values of cell invasion depth. On the other hand, the compressive stress generated by the cells at the onset of protease-mediated invasion is found to be independent of matrix stiffness, suggesting that 3D traction stress is a key factor in triggering protease-mediated cancer cell invasion. At low 3D compressive traction stresses, cells utilize bleb formation to indent the matrix in a protease independent manner. However, at higher stress values, cells utilize invadopodia-like structures to mediate protease-dependent invasion into the 3D matrix. The critical value of compressive traction stress at the transition from a protease-independent to a protease-dependent mode of invasion is found to be ∼165 Pa.     

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.     

Effect of segmentation errors on 3D-to-2D registration of implant models in X-ray images

In many biomedical applications, it is desirable to estimate the three-dimensional (3D) position and orientation (pose) of a metallic rigid object (such as a knee or hip implant) from its projection in a two-dimensional (2D) X-ray image. If the geometry of the object is known, as well as the details of the image formation process, then the pose of the object with respect to the sensor can be determined. A common method for 3D-to-2D registration is to first segment the silhouette contour from the X-ray image; that is, identify all points in the image that belong to the 2D silhouette and not to the background. This segmentation step is then followed by a search for the 3D pose that will best match the observed contour with a predicted contour. Although the silhouette of a metallic object is often clearly visible in an X-ray image, adjacent tissue and occlusions can make the exact location of the silhouette contour difficult to determine in places. Occlusion can occur when another object (such as another implant component) partially blocks the view of the object of interest. In this paper, we argue that common methods for segmentation can produce errors in the location of the 2D contour, and hence errors in the resulting 3D estimate of the pose. We show, on a typical fluoroscopy image of a knee implant component, that interactive and automatic methods for segmentation result in segmented contours that vary significantly. We show how the variability in the 2D contours (quantified by two different metrics) corresponds to variability in the 3D poses. Finally, we illustrate how traditional segmentation methods can fail completely in the (not uncommon) cases of images with occlusion.     

CompuCell3D Simulations Reproduce Mesenchymal Cell Migration on Flat Substrates

Mesenchymal cell crawling is a critical process in normal development, in tissue function, and in many diseases. Quantitatively predictive numerical simulations of cell crawling thus have multiple scientific, medical, and technological applications. However, we still lack a low-computational-cost approach to simulate mesenchymal three-dimensional (3D) cell crawling. Here, we develop a computationally tractable 3D model (implemented as a simulation in the CompuCell3D simulation environment) of mesenchymal cells crawling on a two-dimensional substrate. The Fürth equation, the usual characterization of mean-squared displacement (MSD) curves for migrating cells, describes a motion in which, for increasing time intervals, cell movement transitions from a ballistic to a diffusive regime. Recent experiments have shown that for very short time intervals, cells exhibit an additional fast diffusive regime. Our simulations’ MSD curves reproduce the three experimentally observed temporal regimes, with fast diffusion for short time intervals, slow diffusion for long time intervals, and intermediate time -interval-ballistic motion. The resulting parameterization of the trajectories for both experiments and simulations allows the definition of time- and length scales that translate between computational and laboratory units. Rescaling by these scales allows direct quantitative comparisons among MSD curves and between velocity autocorrelation functions from experiments and simulations. Although our simulations replicate experimentally observed spontaneous symmetry breaking, short-timescale diffusive motion, and spontaneous cell-motion reorientation, their computational cost is low, allowing their use in multiscale virtual-tissue simulations. Comparisons between experimental and simulated cell motion support the hypothesis that short-time actomyosin dynamics affects longer-time cell motility. The success of the base cell-migration simulation model suggests its future application in more complex situations, including chemotaxis, migration through complex 3D matrices, and collective cell motion.     

Free Form Deformation–Based Image Registration Improves Accuracy of Traction Force Microscopy

Traction Force Microscopy (TFM) is a widespread method used to recover cellular tractions from the deformation that they cause in their surrounding substrate. Particle Image Velocimetry (PIV) is commonly used to quantify the substrate’s deformations, due to its simplicity and efficiency. However, PIV relies on a block-matching scheme that easily underestimates the deformations. This is especially relevant in the case of large, locally non-uniform deformations as those usually found in the vicinity of a cell’s adhesions to the substrate. To overcome these limitations, we formulate the calculation of the deformation of the substrate in TFM as a non-rigid image registration process that warps the image of the unstressed material to match the image of the stressed one. In particular, we propose to use a B-spline -based Free Form Deformation (FFD) algorithm that uses a connected deformable mesh to model a wide range of flexible deformations caused by cellular tractions. Our FFD approach is validated in 3D fields using synthetic (simulated) data as well as with experimental data obtained using isolated endothelial cells lying on a deformable, polyacrylamide substrate. Our results show that FFD outperforms PIV providing a deformation field that allows a better recovery of the magnitude and orientation of tractions. Together, these results demonstrate the added value of the FFD algorithm for improving the accuracy of traction recovery.     

Traction Force Measurements of Human Aortic Smooth Muscle Cells Reveal a Motor-Clutch Behavior

The contractile behavior of smooth muscle cells (SMCs) in the aorta is an important determinant of growth, remodeling, and homeostasis. However, quantitative values of SMC basal tone have never been characterized precisely on individual SMCs. Therefore, to address this lack, we developed an in vitro technique based on Traction Force Microscopy (TFM). Aortic SMCs from a human lineage at low passages (4-7) were cultured 2 days in conditions promoting the development of their contractile apparatus and seeded on hydrogels of varying elastic modulus (1, 4, 12 and 25 kPa) with embedded fluorescent microspheres. After complete adhesion, SMCs were artificially detached from the gel by trypsin treatment. The microbeads movement was tracked and the deformation fields were processed with a mechanical model, assuming linear elasticity, isotropic material, plane strain, to extract the traction forces formerly applied by individual SMCs on the gel. Two major interesting and original observations about SMC traction forces were deduced from the obtained results: 1. they are variable but driven by cell dynamics and show an exponential distribution, with 40% to 80% of traction forces in the range 0-10 μN. 2. They depend on the substrate stiffness: the fraction of adhesion forces below 10 μN tend to decrease when the substrate stiffness increases, whereas the fraction of higher adhesion forces increases. As these two aspects of cell adhesion (variability and stiffness dependence) and the distribution of their traction forces can be predicted by the probabilistic motor-clutch model, we conclude that this model could be applied to SMCs. Further studies will consider stimulated contractility and primary culture of cells extracted from aneurysmal human aortic tissue.     

Measurements of elastic moduli of silicone gel substrates with a microfluidic device

Thin layers of gels with mechanical properties mimicking animal tissues are widely used to study the rigidity sensing of adherent animal cells and to measure forces applied by cells to their substrate with traction force microscopy. The gels are usually based on polyacrylamide and their elastic modulus is measured with an atomic force microscope (AFM). Here we present a simple microfluidic device that generates high shear stresses in a laminar flow above a gel-coated substrate and apply the device to gels with elastic moduli in a range from 0.4 to 300 kPa that are all prepared by mixing two components of a transparent commercial silicone Sylgard 184. The elastic modulus is measured by tracking beads on the gel surface under a wide-field fluorescence microscope without any other specialized equipment. The measurements have small and simple to estimate errors and their results are confirmed by conventional tensile tests. A master curve is obtained relating the mixing ratios of the two components of Sylgard 184 with the resulting elastic moduli of the gels. The rigidity of the silicone gels is less susceptible to effects from drying, swelling, and aging than polyacrylamide gels and can be easily coated with fluorescent tracer particles and with molecules promoting cellular adhesion. This work can lead to broader use of silicone gels in the cell biology laboratory and to improved repeatability and accuracy of cell traction force microscopy and rigidity sensing experiments.