Traction microscopy to identify force modulation in subresolution adhesions

Journal name:
Nature Methods
Volume:
12,
Pages:
653–656
Year published:
DOI:
doi:10.1038/nmeth.3430
Received
Accepted
Published online

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

Figures

  1. L-curve analysis for L2 and L1 regularization.
    Figure 1: L-curve analysis for L2 and L1 regularization.

    (a) Simulated input traction field. Inset, rescaled traction map displaying small traction-impact regions in the dashed window. The orientation of the traction field is downward. (b) Top, minimization problem in inverse problem; the regularization parameter λ determines the weight of the regularization semi-norm against the residual norm; p denotes the type of norm (for example, L2 norm if p = 2, or L1 norm if p = 1). Bottom, traction maps resulting from qualitative choices for the indicated parameters applied to L2 regularization. FG, foreground (on adhesions); BG, background (between adhesions). Insets, rescaled traction maps of small traction-impact regions in the dashed windows; red arrowheads indicate noise spikes, yellow arrowheads show traction signals. (c) L-curve for L2 regularization indicating how the λL-corner and λoptimal parameters are chosen (Supplementary Fig. 3). (d,e) Traction maps reconstructed under L2-λL-corner (d) and L2-λoptimal (e) regularization. Insets as in b. (f) L-curve for L1 regularization indicating how the λoptimal parameter is chosen. (g,h) Traction maps reconstructed under L1-λL-corner (g) and L1-λoptimal (h) regularization. Insets as in b. (i) Magnitude ratio of reconstructed to original tractions in the indicated regions (n = 9 NAs, n = 26 FAs); error bars, s.d. (j) Average traction stress background outside traction impact regions. n = 229,255 pixels in nonadhesion area; error bars, s.e.m. Dotted line indicates the noise level in original traction field. ***P < 0.001, Student's t-test. n.s., not significant. (k) Detectability of traction at NAs with a diameter of 6 pixels and load of transmit equal to 600 Pa. Five replicate simulations; error bars, s.e.m. (l) Regimes of traction detectability as a function of adhesion diameter (72 nm/pixel) and traction magnitude for reconstructions under the indicated schemes. Five replicate simulations for each pair of traction magnitude and adhesion diameter. Scale bars (a,b,d,e,g,h), 10 μm (main images) and 2 μm (insets).

  2. Colocalization of tractions with adhesions.
    Figure 2: Colocalization of tractions with adhesions.

    (a) Snapshot of HaloTag-TMR-paxillin at the leading edge of a protruding PtK1 cell. (bd) Traction maps reconstructed under L2-λL-corner (b), L2-λoptimal (c) and L1-λoptimal (d) regularization. (ei) Colocalization analysis in an FA-rich region (box i in ad). (jn) Colocalization analysis in an NA-rich region (box ii in ad). (e,f,j,k) Inverted HaloTag-TMR-paxillin image without (e,j) and with (f,k) tracked NAs (red circles), FCs (orange outlines) and FAs (black outlines). See also Supplementary Video 1. (gi,ln) Traction maps (overlaid by tracked adhesion regions) reconstructed under L2-λL-corner (g,l), L2-λoptimal (h,m), and L1-λoptimal (i,n) regularization. (o) Traction magnitude in NAs compared to the magnitude of local stress maxima outside the cell. n = 10, 27, 14 background peaks in b, c and d, respectively, n = 67 each for NAs in bd; error bars, s.e.m. (p) Fraction of NAs colocalized with local traction stress maxima in L2-λL-corner, L2-λoptimal and L1-λoptimal reconstruction. n = 7,653 NAs from 100 frames; error bars, s.e.m. **P < 0.01, ***P < 0.001, Student's t-test. n.s., not significant. Scale bars, 2 μm.

  3. Traction analysis in NAs of a migrating cell.
    Figure 3: Traction analysis in NAs of a migrating cell.

    (a,b) Time-lapse images of paxillin (Pax) and traction stress map reconstructed under L1 (Trac) for representative examples of a maturing NA (a) and a failing NA (b). Black solid lines in a show segmentation of FCs and FAs associated with tracked adhesion. Dotted red circles in the traction maps indicate the positions of detected adhesions. Scale bars, 1 μm. Time codes show minutes:seconds. (c,d) Fluorescence intensity (black) and traction stress (orange) as a function of time in a and b, respectively. Continuous lines in c and d represent data during nascent adhesion state; dotted lines in c represent data during focal complex and focal adhesion state. a.u., arbitrary units. (e) Comparison of tractions in emerging NAs (n = 158), in a 1-μm-wide band outside the cell edge (n = 332), and in the entire area outside the cell edge (n = 735). Boxes extend from the 25th to 75th percentiles, with a line at the median; whiskers extend to 1.5× the interquartile range; outliers are plotted as asterisks. (f) Time courses of traction in maturing NA tracks (n = 40) and failing NA tracks (n = 242); thick lines represent average time course. (g,h) Comparisons of rate of traction increase (g) and traction magnitude at initial appearance (h) between failing and maturing NA tracks. Sample numbers as in f. Error bars, s.e.m. (ik) Spatial clustering analysis of failing and maturing NAs. (i) Failing and maturing adhesion tracks at a particular time point overlaid on a paxillin image. Scale bar, 2 μm. (j) The ratio of maturing adhesions to all NA tracks present at each frame was averaged over all frames and compared to the ratio of either maturing or failing adhesions present in the circular vicinity of each adhesion at the time point of appearance. (k) Ratios of maturing adhesions for the total population (n = 787) and for neighbors of failing NAs (n = 787) and of maturing NAs (n = 779) for vicinities of varying radii. ***P < 0.001, **P < 0.01, ANOVA. The ratio for neighbors of failing adhesions is similar for all distances, as determined by Tukey's honest significant difference test. Error bars, s.e.m.

  4. Workflow of a synthetic experiment using simulated traction fields.
    Supplementary Fig. 1: Workflow of a synthetic experiment using simulated traction fields.

    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.

  5. Gel swelling and its effect for traction measurement from a cell.
    Supplementary Fig. 2: Gel swelling and its effect for traction measurement from a cell.

    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.

  6. Choice of the regularization parameters [lambda]L-corner and [lambda]optimal in the traction reconstruction shown in Figure 1 using L2 regularization.
    Supplementary Fig. 3: Choice of the regularization parameters λL-corner and λoptimal in the traction reconstruction shown in Figure 1 using L2 regularization.

    (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.

  7. L-curve analysis with [lambda]FGmin, and [lambda]BGmin in L1 regularization.
    Supplementary Fig. 4: L-curve analysis with λFGmin, and λBGmin in L1 regularization.

    (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.

  8. Analysis of sparsity limit in L1 regularization.
    Supplementary Fig. 5: Analysis of sparsity limit in L1 regularization.

    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.

  9. Analysis for sparsity limit in L2-regularized force solution, which is performed with the same protocol as in Supplementary Figure 5.
    Supplementary Fig. 6: Analysis for sparsity limit in L2-regularized force solution, which is performed with the same protocol as in Supplementary Figure 5.

    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.

  10. Resolution analysis in traction reconstruction using L2 and L1 regularization.
    Supplementary Fig. 7: Resolution analysis in traction reconstruction using L2 and L1 regularization.

    (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.

  11. Examples of traction reconstruction with two adhesions at varying edge-to-edge distances between adhesions.
    Supplementary Fig. 8: Examples of traction reconstruction with two adhesions at varying edge-to-edge distances between adhesions.

    Five out of 30 examples are shown.

  12. Examples of traction reconstruction with one adhesion.
    Supplementary Fig. 9: Examples of traction reconstruction with one adhesion.

    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.

  13. Proximity of beads to small force-impact region as a function of bead density.
    Supplementary Fig. 10: Proximity of beads to small force-impact region as a function of bead density.

    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.

  14. Quantification of traction in a live PtK1 cell using the L2-[lambda]optimal method.
    Supplementary Fig. 11: Quantification of traction in a live PtK1 cell using the L2-λoptimal method.

    (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.

  15. Bead tracking by subpixel correlation image interpolation (SCII).
    Supplementary Fig. 12: Bead tracking by subpixel correlation image interpolation (SCII).

    (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.

  16. 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.
    Supplementary Fig. 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.

    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.

  17. Comparison between Boussinesq solution and Green/'s function assuming a finite thickness of an elastic gel.
    Supplementary Fig. 14: Comparison between Boussinesq solution and Green’s function assuming a finite thickness of an elastic gel.

    (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.

Videos

  1. Time-lapse images of paxillin (left) and traction (right) during Ptk1 cell protrusion
    Video 1: Time-lapse images of paxillin (left) and traction (right) during Ptk1 cell protrusion
    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

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Author information

Affiliations

  1. Department of Cell Biology, Harvard Medical School, Boston, Massachusetts, USA.

    • Sangyoon J Han,
    • Youbean Oak &
    • Gaudenz Danuser
  2. Department of Cell Biology, University of Texas Southwestern Medical Center, Dallas, Texas, USA.

    • Sangyoon J Han &
    • Gaudenz Danuser
  3. Department of Physics, University of California, San Diego, La Jolla, California, USA.

    • Alex Groisman

Contributions

S.J.H. designed and implemented the algorithms for TM and adhesion tracking, analyzed live-cell data and wrote the majority of the manuscript. Y.O. acquired live-cell images of PtK1 cells and Alexa Fluor 647–conjugated beads on gel surfaces. A.G. provided TM gel substrates. G.D. initiated the study and helped edit the manuscript.

Competing financial interests

The authors declare no competing financial interests.

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Supplementary information

Supplementary Figures

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. 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.

  10. 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.

  11. 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.

  12. 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.

  13. 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.

  14. 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. 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

PDF files

  1. Supplementary Text and Figures (2,548 KB)

    Supplementary Figures 1–14 and Supplementary Notes 1 and 2

Zip files

  1. 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.

Additional data