High resolution traction force microscopy based on experimental and computational advances - PubMed (original) (raw)

High resolution traction force microscopy based on experimental and computational advances

Benedikt Sabass et al. Biophys J. 2008.

Abstract

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 approximately 1 microm.

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Figures

FIGURE 1

FIGURE 1

(A) Schematic representation of traction force microscopy on flat elastic substrates. Marker beads in the substrate and the corresponding displacement vector field are shown in blue. Sites of adhesion and the corresponding force vector field are shown in red. (B) If force is assumed to be strongly localized, one can use the concept of point forces, which leads to a divergent displacement field at the site of force application. Here we plot the magnitude of the displacement in two perpendicular directions. When relating force to displacement, the mathematical divergence can be avoided by using a simple cutoff rule. (C) If force is assumed to be spatially extended (here we show constant traction over a circular site of adhesion), then displacement is finite inside the adhesion area. Again we plot the magnitude of the displacement in two perpendicular directions. Traction of 2 kPa (2 nN/_μ_m2) at an adhesion of 2 _μ_m in diameter corresponds to a maximum displacement of 0.3 _μ_m on a 10 kPa substrate. At a distance larger than roughly twice the adhesion size, the displacements resulting from the two assumptions are identical.

FIGURE 2

FIGURE 2

Extracting the displacement field from movement of two types of nanobeads with correlation-based particle tracking velocimetry. (A) The two different nanobeads (each with diameter 40 nm) are shown in red and blue. A fibroblast fluorescently marked with GFP-paxillin is shown in green. (B) Combination of both channels (bottom) shows more, but less distinct features than one channel alone (top). (C) Displacement field extracted from one channel only. Open circles at the base of an arrow indicate that this bead could only be tracked by enlarging the correlation window, effectively lowering the spatial resolution. Solid circles indicate beads which could not be tracked. The resulting mesh size is 700 nm. (D) Displacement field extracted from both channels simultaneously. Despite the displacement density being twice as high as in panel C, the number of enlarged windows and discarded beads is significantly lower. The resulting mesh size is 500 nm. Space bar 5 _μ_m.

FIGURE 3

FIGURE 3

Reconstruction of traction at pointlike adhesions. (A) Random configuration of point forces (red) and resulting displacement field (blue). Space bar 10 mesh sizes (≈5 _μ_m). (B and C) Reconstructed traction magnitude using the boundary element method (BEM). (B) BEM, 0th order regularization. (C) BEM, first-order regularization. (D_–_G) Reconstructed traction magnitude using Fourier-transform traction cytometry (FTTC). (D) FTTC, Gaussian filter. (E) FTTC, Wiener filter. (F) FTTC, 0th order regularization. (G) FTTC, second-order regularization. Solid arrows in panel A mark weak forces which cannot be detected at all. Open arrows mark positions where the reconstruction cannot distinguish between close-by forces in the original pattern. The performance is best for panels B, E, and F, with a spatial resolution of 1 _μ_m. (H) Traction background is a statistical measure (n = 10) for the detection limit and is plotted here as a function of absolute noise (in pixel) for different absolute magnitudes of displacement (median and maximum value given in pixel). Traction background increases stronger with displacement magnitude than with noise.

FIGURE 4

FIGURE 4

Reconstruction of traction at finite-sized adhesion sites. (A) Circular adhesion sites with constant traction (red) and resulting displacement field (blue). Space bar 10 mesh sizes (≈5 _μ_m). (B) Traction reconstruction with BEM and 0th order regularization for 5% noise. (C) Traction reconstruction with FTTC and Wiener filtering for 5% noise. (D) Deviation of traction magnitude (DTM) as a function of adhesion size (measured in units of mesh size) for 0% noise. DTM is optimal at zero and worst for −1 (complete underestimation) or +1 (complete overestimation). Here DTM is negative, i.e., the traction is systematically underestimated. The inverse scaling (dotted line) of magnitude deviation as a function of adhesion size indicates sampling problems for adhesion sites which are smaller than two mesh-sizes (∼1 _μ_m). (E) Same plot but with 10% noise. Necessary filtering shifts the effective sampling frequency and only adhesion sites larger than four mesh sizes (∼2 _μ_m) can be properly examined (n = 10).

FIGURE 5

FIGURE 5

Traction reconstruction with point forces (TRPF). (A) Displacement field includes 10% noise. Point forces as calculated with TRPF (green) are localized in the center of adhesion sites (red). Space bar 10 mesh sizes (≈5 _μ_m). (B) Deviation of traction magnitude (DTM) as a function of adhesion size as in Fig. 4, D and E. The underestimation of traction magnitude is small and does not depend strongly on adhesion size with TRPF. (C) Reconstruction of traction when point forces (orange) are not localized exactly in the center of adhesions, as it may occur with small and ill-defined adhesion sites. (D) Standard deviation of traction magnitude and directional error are much higher if point forces are not localized properly (orange) compared to correct localization (green) (n = 10).

FIGURE 6

FIGURE 6

Traction forces at adhesions of a stationary fibroblast. (A) Image section of an extension of a fibroblast marked by GFP-paxillin (green). (B) Overlay with displacement field extracted with two differently colored nanobeads. (C) Traction vector reconstruction using BEM and 0th order regularization. The computational mesh inside the wedge-shaped region is chosen such that the cell contour is well included. (D) Traction vector reconstruction using FTTC with 0th order regularization. The computational mesh is a simple square lattice required for the FFTs. (E) Traction magnitude for the BEM-result. (F) Traction magnitude for the FTTC-result. Units of color bar given in Pascal. Both methods give similar results with a spatial resolution at ∼1 _μ_m and a lower bound for traction detection of ∼500 Pa. (G) Traction vector reconstruction with TRPF. For each adhesion, one point has been selected (for very large adhesions, two points have been selected). (H) Comparison of TRPF, BEM, and FTTC for the region of interest marked in panel A around one large adhesion.

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