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.

A Bayesian traction force microscopy method with automated denoising in a user-friendly software package

“…(2) is L2 regularization, which is also called Tikhonov regularization or ridge regression. L2 regularization is a robust procedure that suppresses noise and produces a smoothed traction field [31]. Here, the residual u − Mf…”

“…Moreover, it has also been shown that the L-curve criterion can fail systematically [43,42]. Therefore, the L-curve criterion is often complemented with other methods for finding the regularization parameter, such as cross-validation [31]. In any case, a manual parameter variation is mandatory to check the validity of the solution.…”

“…Traction computations with Bayesian L2 regularization (BL2) were first introduced as a real-space approach in Ref. [31]. Here we describe the adaptation of this method to Fourier-space traction calculation.…”

“…In the second test series, we construct synthetic data to study the reconstruction quality for localized traction patterns. As in previous work [26,31], we assume that the traction is localized in circular spots, each having a constant traction magnitude. For every individual spot, the step-like traction profile can be integrated analytically to produce a displacement field.…”

Traction Force Microscopy by Deep Learning

“…To mitigate these problems, most conventional approaches include a regularization term in the fitting function, which decreases the noise and complexity of the solution while also limiting the 4 accuracy and resolution (2,17). An additional challenge is the difficulty to define the relative weight, represented by a parameter λ, between regularization and accuracy (18).…”