We present a reconstruction algorithm that resolves cellular tractions in diffraction-limited nascent adhesions (NAs). The enabling method is the introduction of sparsity regularization to the solution of the inverse problem, which suppresses noise without underestimating traction magnitude. We show that NAs transmit a distinguishable amount of traction and that NA maturation depends on traction growth rate. A software package implementing this numerical approach is provided.
At a glance
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- Supplementary Figure 1: Workflow of a synthetic experiment using simulated traction fields. (262 KB)
Designed traction field (a) is used to calculate a displacement field (d) using Boussinesq-based forward calculation. Exact displacement was applied to individual virtual beads whose images are generated by a 2D Gaussian matching the microscope’s point spread function. Using the pair of bead images from undeformed (b) and deformed (e) substrate, bead displacements were tracked (f) and used for traction reconstruction (c) with selected regularization scheme.
- Supplementary Figure 2: Gel swelling and its effect for traction measurement from a cell. (507 KB)
Fluorescent beads are imaged on the silicone gel substrate before and after trypsin application, from which displacement field (the first row) and traction fields (the second row) are obtained. Note that after t=10 min, the gel starts to show swelling: some parts of the gel bulge out generating an artifactual displacement field of max 0.7 pixel, or 50 nm at t = 30 min, which is the time we typically use for imaging beads in their relaxed state. In an 8 kPa-gel this displacement corresponds to ~100 Pa traction peaks. Note that these values are small compared to the traction generated by cells. Nevertheless, since the swelling of a gel can affect the displacement measurement, some of the traction peaks in the traction image (right in bottom row) can be attributed to the gel swelling. Thus, we regard traction maxima more than 100 Pa as significant traction from cell adhesions.
- Supplementary Figure 3: Choice of the regularization parameters λL-corner and λoptimal in the traction reconstruction shown in Figure 1 using L2 regularization. (138 KB)
(a-c) and L1 regularization (d-f). (a,d) L-curve with λL-corner (black circle) and λoptimal (red circle). (b,e) First derivative of L-curve (slope) with respect to residual norm. (c,f) Second derivative of L-curve (curvature) with respect to residual norm. λoptimal is determined heuristically by finding an inflection point before (c) or after (f) λL-corner.
- Supplementary Figure 4: L-curve analysis with λFGmin, and λBGmin in L1 regularization. (181 KB)
(a) Log-scale plot of traction RMS error on (blue) and outside (green) traction impact regions, and of residual norm (magenta) and L1 self-norm (brown) as a function of λ. Local minima in traction RMS error curves (filled circles) define λFGmin, and λBGmin. Black filled circles represent λL-corner on residual norm and self-norm. (b) L-curve of L1 regularization with λFGmin, λBGmin, λoptimal and λL-corner indicated. (c) Traction maps reconstructed under L1 regularization using λFGmin (left), λBGmin (middle) and λoptimal (right). Traction maps confirm λoptimal as the most accurate choice. Note that λoptimal is derived from the inflection point of the L-curve larger than the λL-corner (see Suppl. Fig. 3). Tractions reconstructed under λBGmin and λFGmin are underestimated, especially on small traction regions (inset). Scale bar: 10 μm. Scale bar in inset: 2 μm.
- Supplementary Figure 5: Analysis of sparsity limit in L1 regularization. (176 KB)
Original traction maps are created by randomly distributing random sized adhesions (top row), which are applied to synthetic bead images as discussed in online methods. Beads are tracked by subpixel correlation by image interpolation (SCII) and traction fields reconstructed using L1 regularization (shown in the second row). From sparse adhesions (the number of adhesions, n, is 5) to dense adhesions (n=22 in 160x160 pixel), tractions are recovered consistently. A minimum distance (1 μm) between neighboring adhesions was set according to average spacing in the experimental paxillin image of cell protrusion. L-curve analysis (third row) shows that all three conditions yield a distinct λoptimal from λL-corner for the regularization parameter selection. Root-mean-square error between the original and reconstructed traction field, averaged by the number of pixel in the entire field, shows that the deviation does not increase but rather decreases with adhesion density.
- Supplementary Figure 6: Analysis for sparsity limit in L2-regularized force solution, which is performed with the same protocol as in Supplementary Figure 5. (169 KB)
The reconstructed traction maps vary a lot depending on traction distribution, which is mainly due to different shapes in L-curve and thereby inconsistent λoptimal determination. RMS error per pixel also varies depending on the adhesion density.
- Supplementary Figure 7: Resolution analysis in traction reconstruction using L2 and L1 regularization. (380 KB)
(a) Original traction field is designed with two circular regions with 2 pixels radius and a peak magnitude of 400 Pa. The edge-to-edge distance between the regions is varied from 0 to 26 pixels. The displacement field is calculated analytically and applied to synthetic bead images with a bead density of 0.03 beads per pixel2, similar to experimental conditions. Beads are detected and tracked using subpixel correlation tracking and traction fields reconstructed using L2- and L1- regularization. The reconstructed traction at the left adhesion (fw/) was measured within the vicinity of one mesh length (5 pixel) and compared to the measurement (fw/o) obtained in absence of a force impact in the right adhesion (b) Measurements of fw/ and fw/o reconstructed under L2 regularization as a function of distance. Thick lines depict mean of all measurements (N = 30) (b) Measurements of fw/ and fw/o reconstructed under L1 regularization. Note that L2 regularization significantly underestimates the force impact. (d) Ratio of fw/ over the fw/o for tractions reconstructed under L2 and L1 regularization. The critical distances dc50L1 and dc50L2 define the distances at which the influence from the neighboring adhesion is less than half of the single-adhesion force, reconstructed by either L1-regularized or L2-regularized reconstruction schemes. The critical distances dc,L1 and dc,L2 define the distances at which the influence from the neighboring adhesion becomes negligible. Examples of the synthetic experiments with and without a neighboring adhesion are shown in Supplementary Figs. 8 and 9.
- Supplementary Figure 8: Examples of traction reconstruction with two adhesions at varying edge-to-edge distances between adhesions. (381 KB)
Five out of 30 examples are shown.
- Supplementary Figure 9: Examples of traction reconstruction with one adhesion. (347 KB)
The same bead distribution is used for the synthetic experiment with two adhesions illustrated in Supplementary Fig. 8. Traction at the left adhesion, red-circled, was measured and compared to the traction measured at the location of the second adhesion.
- Supplementary Figure 10: Proximity of beads to small force-impact region as a function of bead density. (94 KB)
The distance threshold of 8 pixels is derived from the criterion that a small force (200 Pa) on an impact region of 8 pixels in diameter yields bead displacement of ≥ 0.1 pixels. That is, if there is no bead within 8 pixel radius, the force from this region may not be reconstructed. Pixel resolution is assumed to be 72 nm/px. Generation of random bead placement and random impact region placement was repeated 1000 times, after which the number of regions with at least one bead closer than 8 px were quantified using k-d tree-based nearest neighbor search. Red dotted line indicates our experimental bead density, which produces ~13 percent of adhesions with no force.
- Supplementary Figure 11: Quantification of traction in a live PtK1 cell using the L2-λoptimal method. (75 KB)
(a) Comparison of tractions in emerging NAs (n = 158) to the local traction maxima measured in a 1 μm-wide band outside the cell edge (n = 332), and in the entire area outside the cell edge (n = 735). (b) Time courses of traction in maturing NA tracks (green lines, n = 40) and failing NA tracks (orange lines, n = 242); thick lines represent average time course. (c,d) Comparison of rate of traction increase (c) and traction magnitude in the time point of initial appearance (d) between failing NA tracks and maturing NA tracks. Sample numbers are the same as in b. ***: P < 0.001.
- Supplementary Figure 12: Bead tracking by subpixel correlation image interpolation (SCII). (337 KB)
(a-e) Pixel correlation with subpixel fitting (PCSF) tracking process. (a) 15 × 15 pixel template window displaying beads in an undeformed substrate. (b) Beads in a deformed substrate; dashed box indicates a template window at (ux, uy) = (0,0). For cross-correlation score calculation the template window is shifted by 1 pixel over interrogation range (e.g. -20 ≤ ux, uy ≤ 20). (c) Cross-correlation score. (d) Scores in 3 × 3 neighborhood around the peak found in c (orange dotted box). (e) Interpolation of maximum score position using 3 × 3 neighborhood around the peak. Due to interpolation error, the peak of the parabola (magenta dot) is 0.2 pixel away from ground truth. (f-j) SCII tracking process. (f) 150 × 150 pixel template window resampled from a using linear interpolation. (g) 10-fold resampled images of beads in deformed substrate. Thus, 1 pixel shift of the template corresponds to 0.1 pixel shift in the original image. (h) Cross-correlation scores on 400 × 400 grid for the same range as in c. (i) Scores in 21 × 21 neighborhood around the peak, corresponding to the 3 × 3 neighborhood shown in d. (j) Interpolation of maximum score position using 3 × 3 neighborhood around the peak. The remaining interpolation error is 0.002 pixel. (k-t) Experiment with a large (20 pixel in diameter, k-o) and a small (4 pixel in diameter, p-t) traction impact region. (k,p) Simulated traction fields. (l,m,q,r) Displacement field measured by PCSF (l,q) or SCII (m,r) tracking. (n,o,s,t) Traction fields reconstructed from displacement fields in l,m,q,r, respectively. (u,v) Traction RMS error (u) and detectability (ratio between peak traction and maximum traction in background, v) as a function of traction impact region diameter for PCSF (blue) and SCII (red) tracking. Data from five different simulations; thick lines represent average for each condition. (w,x) Displacement RMS error (w) and force detectability (x) as a function of template window side length for PCSF tracking (blue) and SCII (red), and bead images with 5 % (solid line) and 10 % (dotted line) white noise. Arrowheads highlight exceptional performance of SCII in 5 % noise regime. See Supplementary Note 1 for details.
- Supplementary Figure 13: RMS errors of tracking algorithms, pixel correlation with subpixel fitting (PCSF), correlation-based continuous window shift (CCWS), and subpixel correlation with image interpolation (SCII) tracking as a function of displacement. (195 KB)
Tracking was performed for a pair of synthetic bead images of undeformed and deformed virtual substrate with 5 % random noise, bead density of 3.5 beads per 100 pixel2, side length of interrogation window of 17 pixel.
- Supplementary Figure 14: Comparison between Boussinesq solution and Green’s function assuming a finite thickness of an elastic gel. (159 KB)
(a) G11 component, which determines the displacement in the direction of traction application (equation 5 in online methods), in both solutions as a function of a distance. Finite thickness solutions are adopted from Merkel et. al.. (b) Simulation of displacement field out of a single force distribution (top right) assuming an infinite (Boussinesq), 34 μm, 10 μm, 1 μm gel thickness. (c) Profile of uy on the middle section of the displacement fields in b.
- Video 1: Time-lapse images of paxillin (left) and traction (right) during Ptk1 cell protrusion (1.61 MB, Download)
- Time-lapse images of paxillin (left) and traction (right) during Ptk1 cell protrusion. Overlaid are computationally tracked NAs (red circles), segmented FCs (blue outlines), and segmented FAs (black outlines). For each NA, the track since initial detection is shown in a red line, ending in the center of the current NA. Once a NA track overlaps with a segmented FCs or FAs, the track color changes to blue (FC) or black (FA). Scale bar: 5 μm. Unit of traction stress: Pa. Time stamp: mm:ss
- Supplementary Text and Figures (2,548 KB)
Supplementary Figures 1–14 and Supplementary Notes 1 and 2
- Supplementary Software (23,782 KB)
Traction microscopy software consisting of stage drift correction, calculation of beads displacement, Outlier filtering of displacement field, and traction calculation under L1 or L2 regularization with L-corner or optimal regularization parameter. User instruction is embedded as PDF in the software GUI.