INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

In recent years, evidence has been growing for an important role of mechanical force in regulating the behavior of single cells and their communities (Chicurel et al., 1998; Galbraith and Sheetz, 1998; Geiger et al., 2001). Force on cells can be either external (e.g., resulting from blood flow or traction of other cells) or internal. For animal cells, internal force is mostly generated by the actin cytoskeleton and transmitted to the extracellular matrix (ECM) through cell-matrix adhesions. For stationary animal cells cultured on flat substrates, the most prominent type of cell-matrix adhesion are focal adhesions (FAs) (Burridge and Chrzanowska-Wodnicka, 1996; Geiger and Bershadsky, 2001). FAs are large supramolecular assemblies, consisting of a submembrane plaque with more than 50 different proteins (including vinculin and paxillin) and a transmembrane part provided by receptors of the integrin family. They can be detected as dark areas in interference reflection microscopy (Abercrombie and Dunn, 1975) and as regions of close approach in transmission electron microscopy (Chen and Singer, 1982). FAs are only one variant of the different types of cell-matrix adhesions, which develop in different situations; in particular, FAs have different morphologies and composition than cell-matrix adhesions in a physiological context (Cukierman et al., 2002). Nevertheless, they are excellent model systems for studying integrin-mediated crosstalk between extracellular matrix and cytoskeleton, which is not only ubiquitous under physiological conditions, but also relevant in the biotechnological context, e.g., when culturing cells on biochips. Forces exerted at FAs allow the cell to probe the mechanical properties of its environment (Pelham and Wang, 1997; Lo et al., 2000; Zamir et al., 2000). Successful adhesion of stationary cells implies forces being sustained at FAs, and several studies indicate that FAs function as mechanosensors, which feed directly into cellular regulation (Choquet et al., 1997; Riveline et al., 2001). In particularly, it has been shown that there is a close relationship between external forces applied to cell-matrix adhesions and their state of aggregation: applying force by optical tweezers (Choquet et al., 1997) or micropipette manipulation (Riveline et al., 2001) stimulates signaling from and growth of FAs. Recently, we have found that for stationary cells there is a linear relationship between the internal forces exerted at a FA and its lateral size (Balaban et al., 2001). Another recent study has shown that this relationship is inverse for focal complexes close to the advancing edge of locomoting fibroblasts (Beningo et al., 2001).

Until recently, quantitative measurements of force have been hardly possible on the level of FAs, in contrast to the level of single molecules, which have been investigated extensively with a variety of quantitative methods, including optical tweezers (Finer et al., 1994), atomic force microscopy (Rief et al., 1997), and the biomembrane force probe (Merkel et al., 1999). The main technique to measure cellular forces is the elastic substrate method (Beningo and Wang, 2002), which was introduced by Harris and coworkers in the early 1980s (Harris et al., 1980, 1981). Until today, there are few alternative methods to the elastic substrate method. One of them is the use of a micromachined device, that measures cellular forces acting on cantilevers etched into a solid substrate (Galbraith and Sheetz, 1997); another is the use of centrifugal forces to induce rupture of adhesion (Thoumine et al., 1996).

In the seminal work by Harris and coworkers (Harris et al., 1980, 1981), the highly viscous, polymeric fluid polydimethylsiloxane (PDMS) was crosslinked at the surface by exposing it to heat. A thin elastic film over a fluid is obtained that under cell traction yields a wrinkled pattern, which is characteristic of the pattern of forces exerted. Major improvements of the wrinkling substrates method include the tuning of the elastic compliance (Burton and Taylor, 1997; Burton et al., 1999). However, deformation data can be analyzed only semiquantitatively with this technique, because the buckling of thin polymer films is a nonlinear phenomenon that is very difficult to treat in elasticity theory. Wrinkling can be suppressed by prestressing the film, thus allowing only for tangential deformation, which can be tracked by fluorescent latex beads (Lee et al., 1994). Quantitative analysis of elastic substrate data was pioneered by Dembo and coworkers. Using linear elasticity theory for thin elastic films and numerical algorithms for solving inverse problems, the forces exerted by keratocytes on the substrate could be reconstructed (Dembo et al., 1996; Oliver et al., 1999). One key ingredient of this method is the use of a regularization scheme, because the inverse problem is ill-posed (that means highly sensitive to noise in the displacement data).

For strong mammalian cells like fibroblasts, the nonwrinkling PDMS-films are too weak. By replacing PDMS with polyacrylamide (PAA) gel, a thick elastic substrate was achieved, which is soft enough to deform under cell traction (Pelham and Wang, 1997). Like any isotropic elastic medium, it is characterized by two elastic constants. In several recent studies, a thick PAA film with Young modulus E  6 to 24 kPa and Poisson ratio v  0.5 was used to quantitatively investigate traction of fibroblasts (Dembo and Wang, 1999; Lo et al., 2000; Beningo et al., 2001). Because the marker bead displacements near the substrate surface are much smaller than the film thickness, they can be evaluated under the assumption that the thick film behaves like an elastic halfspace, whose elastic Green function is well known (Landau and Lifshitz, 1970). This allowed to reconstruct a continuous force field emanating from underneath the cell by using standard techniques for the solution of ill-posed inverse problems. Very recently, is has been suggested by Butler and coworkers that the inverse problem becomes computationally more efficient when being solved in Fourier space and that regularization is not needed when reconstructing the force pattern (Butler et al., 2002).

Recently, we developed a novel elastic substrate technique to measure cellular forces at the level of single FAs (Balaban et al., 2001). A thick polymer film made from PDMS with a Young modulus E  10 to 20 kPa and Poisson ratio v  0.5 was micropatterned by standard lithographic techniques. Due to the regularity of the surface pattern, its deformation can be easily extracted from microscope pictures by an automatic procedure. Cell traction was generated by stationary, yet mechanically active cells (human foreskin fibroblasts, cardiac fibroblasts, or cardiac myocytes) expressing green fluorescent protein (GFP)-vinculin. Vinculin is one of the major proteins of the submembrane plaque of FAs and can be tagged with GFP at its amino terminal. GFP-vinculin localizes at FAs and has good overlap with the dark areas in interference reflection microscopy (Riveline et al., 2001). In our setup, GFP-vinculin marks FAs with very high optical quality. The cells studied in our experiments show mature adhesion with well-developed FAs and stress fibers and with little ruffling activity. We never observed traction near an area deprived of FAs, which allows us to assume that FAs are the main sites of application of force by the cells and to develop a numerical procedure, which reconstructs discrete forces at sites of FA. Correlation with the lateral size of the FAs showed that there exists the following linear relationship between force F and area A of a single FA: A  1 µm² + 0.2 µm²/nN F. In detail, we found a stress constant of 5.5 ± 2 nN/µm². For close packing of integrins, this finding translates into a force of few pN per receptor, which is consistent with recent experiments on strength of single molecular bonds at slow loading (Merkel et al., 1999).

The main difference between our new method and previous work on force reconstruction on thick elastic substrates (Dembo and Wang, 1999; Lo et al., 2000; Beningo et al., 2001) is the assumption of localized force, which necessitates several changes to the standard procedure. In this paper, we address the details of our new computational method and show how the elastic substrate method is affected by the assumption of localized force and the need for regularization. We use systematic simulation of data to confirm that the inverse problem of linear elasticity theory is ill-posed for reasonable levels of noise and to show that regularization in general cannot be neglected. Data simulation is also used to estimate both the spatial and force resolution of our method. The concept of a force multipolar expansion is used to show under which experimental conditions one can neglect the details of the force distribution close to FAs and the finite thickness of the elastic substrate.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Experimental method

The details of the experimental method have been described previously (Balaban et al., 2001). Briefly, the preparation of the micropatterned surfaces was carried out in two steps. First, the negative pattern (typically a grid of 0.5-µm diameter dots with pitch 2 µm) was prepared using standard optical lithography on solid substrates (Si or GaAs wafers). The solid substrates and their photoresist (Microposit S1805, Shipley) pattern were then used as a mold for patterning the surface of the PDMS elastomer. The PDMS elastomer (Sylgard 184, Dow Corning) was poured on glass coverslips, partially cured, put in contact with the photoresist molds, and cured again. After peeling off the mold, the photoresist pattern resulted in a topographic modulation of 0.3-µm depth on the surface of the 40-µm-thick PDMS layer. The topographic pattern was visualized with phase-contrast microscopy. Alternatively, pretreatment of the mold resulted in a fluorescent pattern, which was peeled off with and remained in the elastomer, which then has no topographic modulation.

The bulk elastomer was characterized by suspending known masses to the end of stripes, as described by Pelham and Wang (1997). The elastomer stripes return to their original length even after applying a force that induces an elongation of 70% for 24 h. The Poisson ratio was found to be v  0.5, by following changes in volume upon stretching. The Young modulus varied between 12 to 1000 kPa, as the ratio of the silicone elastomer to curing agent varied from 50:1 to 10:1. Note that a Young modulus of 1 kPa is considerably less than the value for houseware rubber, which is 1 MPa. In a simple scaling picture the Young modulus is kTc = kT/a³, in which c is the effective crosslinker concentration and a is the effective mesh size. Thus, a difference of three orders of magnitude in Young modulus E corresponds only to a difference of one order of magnitude for mesh size a (from 1.6 nm to 16 nm for a change in E from MPa to kPa).

Additional calibration of the surface properties was performed in situ under the microscope: the patterned elastomer surfaces were immersed in culture medium for several days and a calibrated micropipette was used to deflect the surface. The deflection of the micropipette was measured and translated into force. Knowing the force applied by the calibrated micropipette, the Young modulus of the elastomer was calculated from the displacements as described below and found to be consistent with the value measured in the bulk. The relaxation time of the patterned elastomer after mechanical perturbation was measured by phase contrast microscopy, using a video system (25 frames/s). The typical time for rapid recoiling to 80% of the distance to the original position was 100 ms, whereas full relaxation (>95%) occurred within 400 ms.

Before plating, the substrates were coated with fibronectin. Cells (human foreskin fibroblasts, cardiac fibroblasts, or cardiac myocytes) from primary cultures were transfected with GFP-vinculin and plated on the substrates. Observation was done between 10 to 60 h after plating. Image acquisition was done as previously described using the DeltaVision acquisition system (Applied Precision, Issaquah, WA) (Zamir et al., 1999, 2000). Images were processed with Priism software.

Computational method

Linear elasticity theory

(1)

(2)

Inverse problem

(3)

(4)

Resolution and bootstrap method

Force distribution at focal adhesions

(5)

(6)

(7)

View larger version (26K): [in this window] [in a new window]   FIGURE 1   Theoretical model for force distribution over a finite-sized adhesion plaque like a focal adhesion: we assume that force points in x direction and that its magnitude is distributed in a Hertzian manner over an ellipse. (a, b, and c) Long axis of ellipse parallel to direction of force. (d, e, and f) Long axis of ellipse perpendicular to direction of force. (a and d). Displacement following from the distributed force. (b, c, e, and f) Dashed lines: magnitude of displacement following from the distributed force along the x and y directions (numerical result). Solid lines: magnitude of displacement following from equivalent point-like forces exerted at the origin (Boussinesq Green function). Vertical bars mark the edge of the focal adhesion. For Young modulus E = 6 kPa and overall force F = 10 nN, all distances are in micrometers. The crossover from the distributed description to the point-like description occurs outside the focal adhesion boundaries on the scale of the ellipse dimensions.

Finite size effects

11

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Simulated data

Data simulation allows an accurate check of our method and to estimate its resolution. In Fig. 2 a we show an artificial force pattern F₀ that mimics traction by a polarized fibroblast as monitored in our experiments. The cell is assumed to be elongated with FAs occuring close to the rim. Forces are assumed to be exerted only at the FAs at the lower and upper sides, which can be considered to be connected by stress fibers running parallel to the long axis of the cell. One test of our force reconstruction will be whether forces are generated at the focals at the sides, which in the original pattern do not exert force. Neighboring forces along the upper and lower sides are separated by a distance of 4 µm and are assumed to alternate in magnitude, because this allows to test the resolution of our force reconstruction. Typical force is assumed to be 20 nN per FA. Fig. 2 a also shows the displacement resulting from this force pattern. The relation between force and displacement is governed by the Young modulus E, which we assume to be 12 kPa (this is the smallest value obtained in our experiments). Displacements are calculated on a grid of dots with pitch 2 µm (like for the micropatterned substrates) and are assumed to be subject to Gaussian noise with standard deviation  = 1 pixel = 0.133 µm (this is the level of noise resulting from image processing with the water algorithm). Then the largest displacement picked up is 1.3 µm. Fig. 2, b and c, show two of the several changes to this reference case that we will discuss below: in Fig. 2 b, the distance between microfabricated dots has been increased from 2 to 4 µm, and in Fig. 2 c, the number of FAs has been increased from 9 to 13 (thus, the distance between FAs has been decreased from 4 to 2 µm).

View larger version (51K): [in this window] [in a new window]   FIGURE 2   Artificial force patterns F0 mimicing traction by polarized fibroblasts and the resulting displacement fields u. We assume that focal adhesions form a rim, but forces are exerted only at the upper and lower sides. Force magnitude is alternating to test the resolution of the force reconstruction. The assumed Young modulus is E = 12 kPa. (a) Nine focals adhesions on each side with average distance 4 µm. The distance between the microfabricated dots is 2 µm. Typical magnitude of force is 20 nN. Displacement is subject to Gaussian noise with standard deviation  = 1 pixel = 0.133 µm, largest displacement is 1.3 µm. (b) Same parameter values as in a, but doubled distance between the micro-fabricated dots. (c) Same parameter values as in a, but now there are 13 focal adhesions with distance 2 µm on each side. Typical force has been slightly decreased as to achieve same maximal displacement.

In Fig. 3, we reconstruct the force pattern from the displacement data shown in Fig. 2 a. In Fig. 3 a, we plot residual norm R = |GF()  u|² (in absolute units) and deviation from original force F = |F()  F₀| (normalized to 100) as a function of regularization parameter . For small regularization (small ), maximal agreement with the data is achieved; the residual norm R nevertheless attains a finite value, because there is no force field that can exactly reproduce the displacements due to Gaussian noise. For large regularization (large ), the force field vanishes and the residual norm levels off at the value |u|². The solid and dotted straight lines indicate expectation value and confidence interval, respectively, for a ² estimate. Its intersection with the R curve suggests  = 0.04 for the regularization. In fact this is also the value of  for which R starts to rise as a function of , so this result agrees nicely with the L-curve criterion. More important, it also agrees with the minimum in F, the deviation from the original force pattern. It is important to note that even the optimal choice of  cannot reproduce the original force pattern. Fig. 3 a shows that F has its minimum at 24%, that means a considerable part of the original information has been lost by the smoothing operation of the elastic kernel and cannot be retrieved by the inversion. This corresponds to a error of 4 nN for the reconstruction of the 20 nN original single force. The fact that F rises again for smaller values of  indicates the need for regularization: without regularization ( = 0, F = 30%), the agreement between reconstructed and original force is worse than for the proper value of regularization ( = 0.04, F = 24%). In Fig. 3 b we plot the reconstructed (solid) and original (dashed) force pattern. Note that our method nicely reproduced the overall characteristics of the pattern: only small forces are generated at the sides, and for the forces at the upper and lower sides, both the directions and the alternating magnitudes are reproduced. In Fig. 3 c, we show an example of larger regularization ( = 0.1), which is still within the ²-interval and consistent with a noise level of  = 1 pixel = 0.133 µm. Yet the resolution in the force magnitude is lost, and their values are estimated as being too low.

View larger version (21K): [in this window] [in a new window]   FIGURE 3   (a) Residual norm R = |GF()  u|2 (in absolute units) and deviation from original force F = |F()  F0| (normalized to 100) as a function of regularization parameter  for the force and displacement data from Fig. 2 a. The solid and dotted straight lines indicate expectation value and confidence interval for the corresponding 2-estimate, respectively. For the choice of the regularization parameter , 2-criterion, L-curve criterion and the minimum in F all suggest  = 0.04. (b and c) Dashed and solid arrows are original and reconstructed forces, respectively. (b) Force reconstruction with  = 0.04. Even for optimal reconstruction, some information is inevitably lost. (c) Force reconstruction with  = 0.1. Regularization is too strong and spatial resolution is lost.

Until now we showed that for parameter values corresponding to our experiments, the spatial resolution can be considered to be better than 4 µm and the force resolution will be around 4 nN. We now demonstrate that for data containing less information than assumed here, force reconstruction will worsen considerably. In Fig. 4, we show the effect of a noise level increased to  = 2 pixel = 0.266 µm. Monitoring R and F as a function of  (Fig. 4 a) determines  = 0.09 for optimal regularization, but this time F is considerably higher (37% compared with 24%), and goes up to 60% for the case without regularization (compared with 30% for the reference case). As was to be expected, with increased noise, regularization becomes more relevant. Fig. 4, b and c, compare reconstructed and original force patterns for  = 0.09 and vanishing , respectively. In the first case of optimal regularization, force reconstruction is worse than in Fig. 3 b for less noise, and in the second case without regularization, the force pattern becomes rather erratic. In particular, now larger forces are generated at the sides.

View larger version (23K): [in this window] [in a new window]   FIGURE 4   (a) Residual norm R and deviation from original force F as a function of regularization parameter  for the force pattern from Fig. 2 a with the noise level increased to  = 2 pixel = 0.266 µm. (b) Force reconstruction with  = 0.09. Due to larger noise, more information is lost even for optimal regularization. (c) Force reconstruction without regularization ( = 0) yields a rather erratic force pattern.

We now return to a noise level of  = 1 pixel = 0.133 µm, but decrease the density of micropatterned dots, that is, we pick up less displacements. The corresponding displacement data is shown in Fig. 2 b: the distance between dots has been doubled from 2 to 4 µm. Fig. 5 a shows that now the force reconstruction is even worse than in the case of increased noise: optimal regularization now corresponds to a 45% deviation in reconstructed from original force and goes up to over 70% for the case without regularization. This drastic effect had to be expected, because the relevant information is stored in the displacements that are above the noise level, that is in the displacements close to FAs, of which now many are lost. We also confirmed that the reconstruction is not considerably improved when adding displacements farer away from the cell (data not shown). Note, however, that the procedure of choosing  is not affected by adding data with little additional information.

View larger version (13K): [in this window] [in a new window]   FIGURE 5   (a) Residual norm R and deviation from original force F as a function of regularization parameter  for the data from Fig. 2 b, that is distance between micropatterned dots is increased to 4 µm. (b) Same for data from Fig. 2 c, that is there are 13 focal adhesions with distance 2 µm on each side. (c) Same for the case of five focal adhesions with distance 7 µm on each side. The more displacement is picked up and the larger the distance between focal adhesions, the better is the force reconstruction.

In Fig. 5, b and c, we show the effect of changing the distance between FAs from 4 to 2 µm and 7 µm, respectively (the displacement data for the first case is shown in Fig. 2 c). To be able to compare the different case for the same level of noise,  = 1 pixel = 0.133 µm, we adjusted the typical force in such a way that the largest displacement picked up remains close to 1.3 µm. This amounts to decreasing and increasing the typical force of 20 nN by ~5 nN, respectively. Then the deviation from original force at optimal regularization, which was 24% in the reference case, goes up to 31% and down to 15% for the two other cases, respectively. Although the corresponding standard deviations for single forces remain in the range of 4 nN, in the first case the spatial resolution is worsened, whereas in the second case it is improved. Moreover, in the case of well-separated focal adhesions, regularization becomes less relevant: in Fig. 5 c, there is only little difference in F for all values of  up to  = 0.04, which is the level of optimal regularization.

In Fig. 6, we show the results of a simple bootstrap analysis for the force reconstruction presented in Fig. 4, that is the same patterns of forces and dots like in Fig. 2 a, but a noise level in the displacement data that is increased to  = 2 pixel = 0.266 µm. The average force pattern resulting from this bootstrap analysis is stronger regulated than the initial force estimate, because we did not adjust the regularization parameter  when doing the bootstrap simulations. However, the bootstrap analysis now allows us to obtain error intervals for the components of the single forces. In the case of optimal regularization with  = 0.09, Fig. 6 a, the error intervals are of more or less constant size around 2 nN. In the case without regularization, Fig. 6 b, the error bars are increased to an average size of 6 nN, and are larger if FAs with significant forces are nearby. Note that the error intervals resulting from this kind of bootstrap analysis do not include the original force pattern shown in Fig. 2 a, because they reflect only the effect of noise in the displacement data on the force resolution and do not deal with the spatial resolution for which the above analysis showed that it has been lost in this particular case due to the smoothing action of the elastic kernel (compare Fig. 4). In general, more complicated bootstrap procedures could be developed to get more precise estimates of the errors involved.

View larger version (34K): [in this window] [in a new window]   FIGURE 6   Simple bootstrap analysis for the case of Fig. 4, that is for data from Fig. 2 a, but with noise level increased to  = 2 pixel = 0.266 µm. (a) Case of optimal regularization, force resolution around 2 nN. (b) Case without regularization, force resolution around 6 nN. This kind of bootstrap analysis indicates the effect of noise in the displacement data on the force resolution but does not reflect the spatial resolution.

Micropipette manipulation

As a control experiment, we applied known forces to elastic substrates by lowering a micropipette onto the substrate and then shifting it tangentially. In Fig. 7, we show the numerical analysis of such an experiment in terms of a point-like force applied at the midpoint of the micropipette contact region. The -criterion suggests  = 0.05 (the L-curve criterion seems to suggest a somehow smaller value), which leads to a force estimate of F = 660 nN. A bootstrap analysis gives an error estimate of 13 nN. From the observed deflection of the micropipette, the applied force can be estimated to be 600 ± 90 nN, so the agreement is good. Note that the force applied by the micropipette is distributed, but because displacements are picked up only in the regions in which the field of view is not obscured by the micropipette, the force monopole approximation is appropriate. We also analyzed the same displacement data with increasing numbers of point forces distributed over the contact region and confirmed that this increases the estimate for the overall force only slightly. Moreover, the different forces turn out to be more or less parallel (no twist) and decay if one moves away from the midpoint of the contact region.

View larger version (13K): [in this window] [in a new window]   FIGURE 7   Analysis of a micropipette control experiment with one point-like force (Young modulus E = 12 kPa). (a) Residual norm R as function of regularization parameter . (b) Force reconstruction according to the -criterion ( = 0.05). The result F = 660 ± 13 nN for the overall force agrees with the experimental value F = 600 ± 90 nN inferred from the micropipette deflection.

Cell traction

The rigidity of our substrates has been optimized for studying traction from strong animal cells like fibroblasts and cardiac myocytes. In Fig. 8, we present the analysis for a whole human foreskin fibroblast. To resolve the displacement, such a high microscope resolution was needed that the cell did not fit into one single picture; the data presented here were assembled from two different pictures taken one after the other. Fig. 8 a shows the resulting fluorescence picture for the whole cell, which is strongly polarized. Most FAs are located along the rim of the cell, and more or less elongated along the long axis of the cell itself. The residual norm R as a function of regularization parameter  is shown in Fig. 8 b. We see that the -criterion suggests  = 0.01. However, the resulting regularization is too weak, as can be seen from the resulting force pattern shown in Fig. 8 c, which looks rather erratic. Therefore, we use the upper boundary of the confidence interval, that is  = 0.1, which is still consistent with the noise level (this choice also seems to be consistent with the L-curve criterion). The resulting force pattern is shown in Fig. 8 d. Because we have fitted ellipses to the FAs in Fig. 8, c and d, one sees clearly that for the stronger level of regularization, the forces in the upper part of the cell are more or less parallel to the elongation of the single FAs. This seems reasonable because one expects stress fibers to run in the same direction. In the lower part of the cell, the forces seem to be somehow rotated to the right. One reason for this might be that displacement data are rather scarce in this region, so too much information has been lost as to achieve a reliable force reconstruction (the drastic effect of too little displacement information has been shown in Fig. 5 a). We find that the force in the upper part can be as strong as 30 nN. In the lower part, most forces are in the order of 10 nN. Note that there are several small FAs at the sides that seem to carry only little force. In general, we find that the cell is highly polarized also in regard to the force pattern and that the two force bearing regions at the upper and lower sides more or less balance each other. Due to Newton's third law, the overall vector force is expected to vanish for a stationary cell, but in this analysis it amounts to 10% of the overall force magnitude, which is probably due to the unreliable force reconstruction in the lower part of the cell. From the viewpoint of a force multipolar expansion, one might say that the cell forms a force contraction dipole of strength P = 10¹¹ J; this corresponds to a pair of forces, separated by a distance of 60 µm and each 200 nN strong. A similar result, P = 3 10¹² J, was obtained by Butler and coworkers for a human airway smooth muscle cell (Butler et al., 2002) (there the force dipole tensor is called the moment matrix).

View larger version (42K): [in this window] [in a new window]   FIGURE 8   Cell traction from a stationary fibroblast (Young modulus E = 18 kPa). (a) Fluorescence image of the cell, which is transfected with GFP-vinculin. Vinculin is a major component of focal adhesions and its localization is used to identify the regions, which correspond to large forces. (b) Residual norm R as a function of regularization parameter  for a whole human foreskin fibroblast. (c) Force reconstruction for  = 0.01 (middle of -interval) and (d)  = 0.1 (upper boundary of -interval). In c, regularization is too weak, and the force pattern is erratic. In d, there is an unexplained drift in the lower part of the force pattern, but the overall force pattern is reasonable. Ellipses are fits to the focal adhesions as marked by GFP vinculin.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

In this paper, we presented a novel computational technique that allows to calculate forces at the level of single FAs from displacement data of elastic (micropatterned) substrates and fluorescence data of GFP-vinculin-labeled FAs. Our main assumption is that forces exerted at FAs marked by fluorescent GFP vinculin are appreciably higher than those developed in neighboring regions along the ventral cell membrane. This assumption is based on the fact that we never observed traction near an area deprived of FAs. Our finding that large force corresponds to large FAs seems to justify our assumption a posteriori. Because displacements can be measured only at discrete points, the Fredholm integral equation relating forces to displacements is converted into a system of linear equations. The Boussinesq solution for the Green function of an elastic isotropic halfspace is used as kernel for the Fredholm equation. We showed in the framework of a force multipole expansion that the assumption of point-like forces is reasonable as long as displacements are picked up at a distance to the FAs, which is similar to their lateral dimensions. The force multipolar expansion was also used to argue in detail why effects from the clamped boundary conditions at the bottom and at the sides of the polymer film can be neglected in our treatment.

It is well known that Fredholm integral equations of the first kind like the one of linear elasticity theory are ill-posed, irrespective of using the assumptions of localized or distributed force. By extensively simulating artificial data that mimic experimental conditions, we confirmed that in general the inverse elastic problem needs regularization to arrive at a reliable force estimate. In particular we showed that in most realistic cases, the deviation of reconstructed from original force F shows a clear minimum at finite regularization parameter . In the absence of this information, that is in real experiments, one has to estimate the optimal value for the regularization parameter . We used two different criteria, the - (or discrepancy) criterion and the L-curve criterion, which lead to identical results for simulated data. For real data, the agreement between the two criteria is less good (possibly due to the presence of non-Gaussian noise or imperfections of the elastic substrate) but still sufficient. In the rare cases that these criteria lead to erratic force patterns (compare Fig. 8 c), we used the upper limit of the -interval, because it is still consistent with the independently determined noise level.

It is important to note that spatial resolution for the force field is inherently restricted by the smoothing action of the Fredholm integral equation on the length scale µm, in which F = 10 nN is the typical force at FAs and E = 10 kPa a typical value for the Young modulus. Our simulations demonstrated that both spatial and force resolutions depend on the details of the displacement and force patterns. Although no generally valid values can be given, simulations for realistic situations showed that our spatial and force resolutions are better than 4 µm and 4 nN, respectively. This values have been derived above for a simulated reference pattern, which is somehow more difficult to reconstruct than experimental force patterns in which the high density of FAs of the reference pattern is realized only at certain regions of the cell. We conclude that calculated forces can be reliably attributed to single FAs if no other FAs are closer than a few micrometers. Simple bootstrap analysis like the one presented in Fig. 6 leads to an estimate for force resolution of 2 nN, but at the same time to a decrease in spatial resolution (which however is not quantified in this scheme). Although the smoothing action of the elastic kernel indicates a basic limitation of elastic substrate experiments, it is worth noting that it also benefits our quantitative analysis, because it allows to neglect corrections arising from the modulation of the micropattern.

The method presented here can now be used to analyse mechanically active cells in quantitative detail. Experimentally, it requires the use of (microstructured) elastic substrates and labeling of the force-transmitting system. We used GFP-vinculin to label FAs, but other possibilities include use of GFP-cDNA-constructs encoding other adhesion-associated proteins (like paxillin, zyxin, alpha-actinin, or actin) or specific antibodies. Numerically, it requires image analysis of the phase contrast and fluorescence pictures and use of the force reconstruction program. As the procedure described here is rather simple and robust, we expect that our protocol might become a standard tool for such a purpose. In contrast to the reconstruction of a continuous stress field (Dembo and Wang, 1999; Lo et al., 2000; Beningo et al., 2001), the reconstruction of a discrete force pattern is computationally rather cheap and needs only minutes on a standard PC. Therefore, it could be used to study mechanical activity of cells in real time. Note that if the force-transmitting system cannot be marked, the standard assumption of distributed force has to be used.

One main result of our analysis of simulated data is the confirmation that in general regularization cannot be neglected when reconstructing force patterns from elastic substrate data. The need for regularization was also demonstrated by real traction data presented in Fig. 8, where insufficient regularization leads to an erratic force pattern. This finding stands in marked contrast to recent work by Butler and coworkers, who suggested a new method that does not involve regularization (Butler et al., 2002). The starting point for this method is the observation that the Fredholm int