Traction force microscopy with optimized regularization and automated Bayesian parameter selection for comparing cells

Abstract

Adherent cells exert traction forces on to their environment which allows them to migrate, to maintain tissue integrity, and to form complex multicellular structures during developmental morphogenesis. Traction force microscopy (TFM) enables the measurement of traction forces on an elastic substrate and thereby provides quantitative information on cellular mechanics in a perturbation-free fashion. In TFM, traction is usually calculated via the solution of a linear system, which is complicated by undersampled input data, acquisition noise, and large condition numbers for some methods. Therefore, standard TFM algorithms either employ data filtering or regularization. However, these approaches require a manual selection of filter- or regularization parameters and consequently exhibit a substantial degree of subjectiveness. This shortcoming is particularly serious when cells in different conditions are to be compared because optimal noise suppression needs to be adapted for every situation, which invariably results in systematic errors. Here, we systematically test the performance of new methods from computer vision and Bayesian inference for solving the inverse problem in TFM. We compare two classical schemes, L1- and L2-regularization, with three previously untested schemes, namely Elastic Net regularization, Proximal Gradient Lasso, and Proximal Gradient Elastic Net. Overall, we find that Elastic Net regularization, which combines L1 and L2 regularization, outperforms all other methods with regard to accuracy of traction reconstruction. Next, we develop two methods, Bayesian L2 regularization and Advanced Bayesian L2 regularization, for automatic, optimal L2 regularization. Using artificial data and experimental data, we show that these methods enable robust reconstruction of traction without requiring a difficult selection of regularization parameters specifically for each data set. Thus, Bayesian methods can mitigate the considerable uncertainty inherent in comparing cellular tractions in different conditions.

Figure 1

Schematic representation of a typical traction force microscopy (TFM) setup and different reconstruction methods for TFM. (a) Cells are plated on a planar gel substrate containing fiducial markers. Tracking the markers allows to infer the deformations u in the surrounding of the cell. These deformations are linearly related to the cellular traction forces f. The problem of calculating traction f from displacement u is associated with inverting an ill-conditioned matrix M. This problem can be solved with different reconstruction methods. (b) In this work, we test five regularization methods for traction reconstruction: L2 regularization (L2), L1 regularization (L1), EN regularization (EN), Proximal Gradient Lasso (PGL) and Proximal Gradient Elastic Net (PGEN). Furthermore, we develop two Bayesian approaches that do not have any free parameters, namely Bayesian L2 regularization (BL2) and Advanced Bayesian L2 regularization (ABL2).

Figure 2

Systematic tests illustrate substantial ambiguity in the choice of regularization parameters. (a) Schematic of the employed procedure to test the reconstruction methods. (a,i) Artificial traction pattern consisting of circular spots that uniformly exert a traction of 100 Pa. (a,ii) Analytical calculation of the gel displacements. (a,iii) The displacement field is sampled at random positions representing measurements of motion of fiducial markers. (a,iv) Reconstruction of the traction. (b) Central formula summarizing different regularization approaches. (c) Dependence of various error measures on the regularization parameters. (b,i)–(b,v) Error measures defined in Eqns. (9–12) exhibit various extrema and turning points, making the definition of an optimal parameter challenging. Note that the minima of the errors do not correspond to values of regularization parameters suggested by the L-curve criterion (Green dotted lines vs. black dotted lines). DTMA: Deviation of traction magnitude at adhesion, DTMB: deviation of traction magnitude in background. (b,I)–(b,V) Traction fields calculated with regularization parameters that correspond to the error minima at the black dotted lines. Space bar: 5 μm.

Figure 3

The elastic net (EN) outperforms other reconstruction methods in the presence of noise and when applied to undersampled data. (a) Artificial test data with uniform traction spots and 4% noise in the displacements. Traction maps in (ii–vi) result from usage of different regularization methods. Space bar: 5 μm; displacements are sampled on average every 0.5 μm (b) Comparison of errors resulting from undersampled data. Undersampling is realized by reducing the number of displacement vectors m. (c) L-curves with chosen regularization parameters (gray boxes) for a data set containing 2% noise and m/n = 0.4. (d) Comparison of errors for the regularization parameters shown in (c). EN regularization shows a favorable tradeoff between error and background signal.

Figure 4

Bayesian L2 regularization (BL2) and Advanced Bayesian L2 regularization (ABL2) are robust methods for automatic, optimal regularization. (a) Schematic diagram of the procedure employed to infer ${\hat{\lambda }}_2$ in BL2 and ABL2. BL2 requires the variance of the displacement measurements 1/β that can be obtained by analyzing displacement noise far away from any cell. ABL2 estimates this noise strength directly from the data. (b) Artificial test data. For the shown results, 5% Gaussian noise is added to the displacements. Space bars: 5 μm. (c) For BL2, the optimal regularization parameter is located at the maximum of a one-dimensional plot of the evidence Eq. (8). (d) Reconstruction of traction force using BL2. (e) For ABL2, the optimal regularization parameter is located at the maximum of a two-dimensional plot of the data evidence vs. α and β. (f) Reconstruction of traction force using ABL2. (g,i)–(g,iv) Comparison of reconstruction accuracy for L2, BL2 and ABL2. Different levels of traction forces were applied to change the signal-to noise ratio. Here, σ_n is the standard deviation of the noise and ${\sigma }_{\overline{\mathbf{u}}}$ is the standard deviation of the noise-free traction field. Note that BL2 outperforms the other methods for high noise levels.

Figure 5

Test of all reconstruction methods using experimental data. (a) Image of an adherent podocyte with substrate displacements shown as green vectors. (b–h) Reconstructed traction forces using L2, L1, EN, BL2, PGL, PGEN and ABL2, respectively. Reconstruction with L2-type regularization exhibits a comparatively high background noise. L1-regulation shows very high, localized traction. Based on tests with artificial data, we expect that these peaks overestimate the traction. The EN method combines the advantages of L1 and L2 regularization, namely a clean background and localized traction of reasonable magnitude. PGL and PGEN have smooth traction forces at adhesion and background. (g,h) The Bayesian methods BL2 and ABL2 yield very similar results as the standard L2 regularization without requiring a search for the optimal regularization parameters. For better visibility, only every fourth traction vector is shown. Space bar: 25 μm.

Figure 6

Bayesian L2 regularization robustly adapts to different traction levels allowing quantitative analysis of time series. (a) Image of a spontaneously beating heart muscle cell on an elastic, micropatterned substrate. (b) Overall norm of traction magitude in successive image frames. The maximum corresponds to one contraction of the heart muscle cell. Traction is calculated with BL2 or, for comparison, via L2 regularization where λ₂ is either selected manually for every frame using the L-curve criterion or held constant throughout the image sequence. (c) Optimal regularization parameter suggested by BL2 and the norm displacement field correlate. (d,i)–(d,iii) Cell images with displacement field at frames 1, 4, and 6. (e,i)–(e,iii) Snapshots of the traction fields resulting from L2 regularization with a manually chosen parameter λ_L-curve and a constant parameter λ_L-const. in an intermediate range. (f,i)–(f,iii) Snapshots of the traction fields resulting from BL2. Note the different scaling of displacement and tractions for the different frames. Frame 1 (I) illustrates that BL2 yields a smaller traction magnitude than L2 in the presence of large noise, where the L-curve criterion is hard to employ. As a result, BL2 allows to differentiate real traction from noise outside of the cell. For better visibility, only every fourth traction vector is shown.