High resolution, large deformation 3D traction force microscopy

Abstract

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

Figure 1. Top-view displacement contours of a migrating Schwann cell measured by DVC and FIDVC.

Side by side comparison of the 3D cell displacements measured with (A) our previous DVC and (B) our new FIDVC algorithm . Cell outlines are shown in white. Scale bars = 40 m.

Figure 2. Displacement gradient comparison for large deformation.

and cross-sections of the calculated 3D displacement gradient for (A) a Schwann cell (scale bar = 40 m), (B) a polymorphonuclear leukocyte (scale bar = 20 m) and (C) a NIH 3T3 fibroblast (scale bar = 20 m). (D) Total force (), root mean squared tractions () and maximum traction () ratios plotted against the displacement gradient, under the application of a 3D Gaussian-shaped displacement field (inset). The numerator in the ratios is calculated using the new large deformation approach, whereas the denominator features the results from the traditional linear elastic, small deformation material approximation.

Figure 3. Undeformed and deformed surfaces due to a large deformation.

(A) Schematic of how a material deforms from a reference configuration, , at time , into a deformed configuration, , at time . (B) Angle change between the undeformed and deformed surface normals in (A) and , under the application of a cell-simulated Gaussian displacement field profile. The x-axis denotes the maximum value of the full-field 3D displacement gradient magnitude. The dot product represents the cosine of the angle between the two surface vectors. LSCM cross-sectional images (C) in the absence of a cell, and (D) directly underneath a locomoting Schwann cell. Scale bars = 5 m.

Figure 4. Analytical example of prescribed Gaussian displacement dipoles on the surface of a 3D LSCM imaging volume.

The (A) 3D surface displacement magnitude, , and (B) displacement gradient magnitude, . Profiles of calculated maximum 3D principal strains calculated from the (C) infinitesimal () and (D) Lagrangian () strains. The corresponding traction magnitudes calculated on the (E) undeformed surface, , using a linear elastic constitutive model, , and on (F) the deformed surface, using a large deformation (LD) constitutive model . Scale bars = 40 m.

Figure 5. Comparison of commonly reported metrics in TFM for the analytical example.

Side by side comparison of the (A) total force, (B) root mean squared (RMS) tractions and maximum tractions, and (C) strain energy for both the linear elastic, small deformation (SD) and non-linear, large deformation (LD) models. All of the values are normalized by the exact analytical solution.

Figure 6. Experimental example of a migrating Schwann cell on the surface of a 3D LSCM imaging volume.

(A) Magnitude of the 3D Schwann cell surface displacement field, , and its (B) resulting displacement gradient magnitude, (). Calculated maximum principal strains from the (C) infinitesimal (), and (D) Lagrangian () strains. The corresponding traction magnitudes calculated on the (E) undeformed surface, , using a linear elastic constitutive model, , and on the (F) deformed surface, using a large deformation (LD) constitutive model . Cell outlines are shown in white. Scale bars = 40 m.

Figure 7. Comparison of commonly reported metrics in TFM for the experimental example.

Side by side comparison of the (A) total force, (B) root mean squared (RMS) and maximum tractions, and (C) strain energy for both the linear elastic, small deformation (SD) and non-linear, large deformation (LD) models.

Figure 8. Flowchart of the large deformation high-resolution 3D TFM technique illustrating how cell surface tractions are being calculated.

Figure 9. Analytical benchmark validation examples of the free surface finder algorithm.

(A) shows the results of the surface finder given a perfectly flat surface, whereas in (B) the surface has regular imposed sinusoidal surface undulations. Scale bars = 20 m.