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

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Abstract

Adherent biological cells generate traction forces on a substrate that play a central role for migration, mechanosensing, differentiation, and collective behavior. The established method for quantifying this cell–substrate interaction is traction force microscopy (TFM). In spite of recent advancements, inference of the traction forces from measurements remains very sensitive to noise. However, suppression of the noise reduces the measurement accuracy and the spatial resolution, which makes it crucial to select an optimal level of noise reduction. Here, we present a fully automated method for noise reduction and robust, standardized traction-force reconstruction. The method, termed Bayesian Fourier transform traction cytometry, combines the robustness of Bayesian L2 regularization with the computation speed of Fourier transform traction cytometry. We validate the performance of the method with synthetic and real data. The method is made freely available as a software package with a graphical user-interface for intuitive usage.

Program summary

Program Title: Easy-to-use TFM software

Program Files doi: http://dx.doi.org/10.17632/229bnpp8rb.1

Licensing provisions: GNU General Public License v3.0

Programming language: Matlab version R2010b or higher

Supplementary material: A user manual for the software and a test data set.

Nature of problem: Calculation of the traction forces on the surface of an elastic material from observed displacements.

Solution method: Traction forces are efficiently calculated by combining L2 regularization in Fourier space with Bayesian inference of the regularization parameter.

Introduction

Traction force microscopy (TFM) is a technique for measuring surface traction forces on an elastic substrate. Due to the unique possibilities offered by the technique, TFM enjoys wide popularity among biologists, materials scientists, and experimental physicists, see Refs. [1], [2], [3], [4], [5], [6], [7] for a non-comprehensive list of recent reviews. With TFM, one can record “images” and “movies” of the spatial distribution of traction forces on a surface. Moreover, TFM is essentially an imaging technique and does not require a perturbation of the sample. Therefore, the technique complements other techniques such as atomic force microscopy, optical tweezers, or the surface force apparatus. In materials science and physics, TFM has been used to measure interfacial forces during wetting, adhesion, rupture, and friction processes, see, e.g., Refs. [8], [9], [10]. However, the most important application of TFM is in biology, where it is being employed extensively for studying the mechanobiology of adherent cells. The traction patterns generated by adherent cells vary typically on a length scale of about one micrometer. The studied phenomena include cell migration and adhesion regulation [11], [12], [13], [14], [15], [16], three-dimensional collective cell organization [17], [18], [19], [20], and cell migration in wound-healing assays [21], [22], to name just a few. Overall, the length scales of traction patterns that have been studied with TFM range from below one micrometer in bacterial adhesion [23], [24] to centimeters in propulsion waves of slugs and snails [25].

A typical TFM setup for studying cell adhesion is sketched in Fig. 1. Cells are placed on a flat elastic substrate which is usually synthetic, for example a soft polyacrylamide (PAA) gel, and contains fluorescent beads as fiducial markers. [26] The live cells are imaged together with the fluorescent markers on a microscope. Surface traction results in local substrate deformations that can be monitored by tracking the motion of the beads in the substrate. The extraction of a discrete displacement field U describing the deformation of the substrates is usually done by comparing images of the fluorescent markers before and after removal of the adherent cells. Established methods for calculating the displacement fields include particle image velocimetry, single particle tracking, or optical flow tracking [2], [3], [27]. Here and in the following, positions on the two-dimensional surface are described in a Cartesian system with coordinates x=(x1,x2). Using the planar deformations (U1,U2) as input, a spatial map of traction (F1,F2) on the gel surface can be mathematically reconstructed if out-of-plane forces are assumed to be negligible. For the reconstruction, the substrate is typically assumed to be a homogeneous, isotropic and linearly elastic half-space. Thus, the relation between the continuous displacement field Ui(x) and the traction force field Fj(x) on the surface of substrates can be expressed as [28] Ui(x)=Ωj=12Gij(xx)Fj(x)d2x,where Gij(x) is the Green’s function and Ω covers the whole substrate surface. The traction forces can be calculated in real space with finite element methods [29], [30] or boundary element methods [26], [31]. To calculate the tractions based on Eq. (1) numerically, the integral equation needs to be discretized. In real space, one can employ linear shape functions [26], [31], [32] to write Eq. (1) as a linear matrix equation u=Mf, where the lower case letters u and f denote the discretized displacements and tractions. The coefficient matrix M results from integration of the shape functions. Such real-space methods are very flexible since they permit the study of various linear material responses encoded in the Green’s function and spatial constraints are easily incorporated. However, accurate construction of the matrix M requires significant computation time on desktop machines. Alternatively, Eq. (1) can be solved in Fourier space by making use of the convolution theorem. This approach is called Fourier transform traction cytometry (FTTC) [12], [26]. We employ a spatial wave vector k=(k1,k2) with absolute value k=|k|. In standard FTTC, the integral Eq. (1) is written as ũik={jG̃ijf̃j}k, where the tilde denotes the Fourier-transformed quantity. Using a matrix formulation analogous to the real-space expression, we have ũ=M̃f̃ with M̃ having a tri-diagonal structure. For conceptual clarity, in the following we will write the measurement noise in the recorded displacement explicitly as s in the real-space domain and as s̃ in Fourier space. This noise can be estimated in the experiment by quantifying the variance of the measured displacements in absence of traction. The discretized equations then read u=Mf+sin real space,ũ=M̃f̃+s̃in Fourier space.For traction force microscopy, either of these equations is employed to determine the tractions f. The removal of noise is critical in most TFM methods. In real-space TFM calculations, the condition number of M, defined as the ratio of the largest singular value to the smallest, is almost always much larger than unity, typically above 105. M is therefore ill-conditioned which implies that small noise produces drastic changes in the calculated traction forces. For FTTC, spatially varying random noise occurs mainly at high spatial wave numbers. Hence, noise suppression can be achieved by suppressing high frequency data. In Ref. [32], we systematically tested a variety of traction reconstruction approaches based on Eq. (2). The standard approach for solving the equation in real space is L2 regularization [2], [26], [33], [34], which invokes a penalty on the traction magnitude to robustly suppress the effects of noise. With Fourier space methods, a low-pass filter is frequently employed to suppress noise in the displacement field before direct inversion of Eq. (2) [12]. Alternatively, Fourier-space traction reconstruction can also be combined with L2 regularization, which conveys additional robustness [3], [26], [27], [35].

Virtually all standard methods for traction calculation require the implicit or explicit choice of a parameter that suppresses noise and leaves as much of the true signal conserved as possible. Within a Bayesian framework, this parameter choice can be rationalized by relating filter- or regularization parameters to prior distributions that represent prior knowledge about the data. Maximizing the likelihood of the prior distributions yields the corresponding optimal trade-off between noise suppression and faithful data reconstruction. Bayesian regularization has been used for example in astrophysics [36], [37] and mechanical structure monitoring [38]. For inference of internal stress in a cell monolayer, an iterative maximum a posteriori estimation has been employed [39]. TFM with Bayesian L2 regularization (BL2) was introduced in Ref. [32] and is based on an established framework of Bayesian fitting [40]. Bayesian L2 regularization was first employed for real-space TFM methods since this variant allows comparison of a broad variety of approaches. For practical applications, however, calculations in Fourier domain have significant advantages in terms of robustness and speed. In this work, we present the corresponding method that we term Bayesian Fourier transform traction cytometry (BFTTC). We compare BFTTC with other methods such as classical L2 regularization, Bayesian L2 regularization in real-space (BL2), and regularized Fourier transform traction cytometry (FTTC). We find that BTTC is a computationally fast method that provides robust traction calculations without requiring manual adjustment of the noise-suppression level. We also provide a Matlab software package for BFTTC that is freely available for download. This software package is intended to provide a simple and robust solution for data analysis in the hands of experimentalists. A graphical user interface allows intuitive use of the program and little theoretical background knowledge is required.

Section snippets

Traction–displacement model

Assuming that the substrate is a semi-infinite half-space, the Green’s function in Eq. (1) is given by the standard expression Gij(x)=(1+ν)πE(1ν)δijr+νxixjr3,where E and ν represent the Young modulus and Poisson ratio, respectively. Here, r=|x| and δij is the Kronecker delta function. Denoting the wave vector by k=(k1,k2) with absolute value k=|k|, the Fourier-transformed Green’s function is given by G̃ijk=2(1+ν)E[δijkνkikjk3]. The continuous traction and displacement fields are discretized

Validation of the method with synthetic data

To check whether the proposed method actually finds the correct regularization parameter, synthetic data sets with exactly known underlying distributions are required. Therefore, we create random traction patterns with traction vectors at each grid point drawn from a Gaussian distribution. Exemplary data is shown in Fig. 2(a-i). The calculated displacement field is then corrupted with a controlled level of noise, see Fig. 2(a-ii). For the reconstruction, we search for the hyperparameter α that

Summary

Traction force microscopy is a popular technique for studying minute forces generated by biological cells, as well as wetting or frictional forces, on soft substrates. The technique is based on the measurement of substrate displacements below the specimen, which allows calculation of the traction forces. Usually, this calculation is done by solving an inverse linear problem involving elastic Green’s functions. The procedure requires methods for noise suppression. Dealing with noise

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

We thank S.V. Plotnikov (University of Toronto) and C. Schell (Albert-Ludwigs-University Freiburg) for providing the experimental test data.

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    The review of this paper was arranged by Prof. Stephan Fritzsche.

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