Journal Club Theme of March 1: Measuring Cellular Tractions
Cell tractions are the outcome of the complex process of cytoskeletal force generation that cell uses to maintain structural stability, to sense the physical environment and to propel itself. We are only now beginning to understand the process of cytoskeletal force generation, and we cannot yet say much about the losses in transmission through focal adhesion/integrin complexes (attachment ‘islands’ at the cell-substrate interface), but we can definitely measure the tractions that result from cytoskeletal force generation. The mechanics behind the measurement method might be of interest to the wider audience of iMechanica, as it involves an interesting inverse problem and different solution methods that have incited lively discussions in past years.
The main idea is to calculate tractions from the measurement of the deformation they caused in a substrate to which the cell is attached. It is assumed that the in-plane forces and out-of-plane displacements are not coupled, and that, effectively, the out-of-plane forces are negligible.
Among the discrete methods, the simplest one uses arrays of microfabricated elastomeric posts whose protein-coated tips act as attachment points for the cells:
Tan JL, Tien J, Pirone DM, Gray DS, Bhadriraju K and Chen CS. Cells lying on a bed of micro-needles: An approach to isolate mechanical force. PNAS 100 (4): 1484-1489 (2003).
Force at the post tip is calculated using Euler-Bernoulli beam theory, but the post's short length (compared to diameter) and considerable deflection do not warrant EB theory’s neglect of transverse shear strain, so using EB overestimates the force. Also, the inward forces bring post tips in contact with each other and EB theory breaks down—that this happens frequently can be seen in the recent movie . Optimizing the post stiffness to prevent contact while providing measurable deflection, involves repeated expensive microfabrication procedure, which is the main drawback for this technique.
In contrast, the substrate for all the continuum methods is an isotropic linearly elastic solid, usually a hydrogel or a silicon rubber film. Fluorescent markers (beads) are embedded in the substrate surface and their motions are recorded using fluorescence microscopy. Substrate thickness (usually ~100 microns) is at least order of magnitude greater than the maximum surface displacement in order to satisfy the key assumption: that the substrate is a semi-infinite body. This assumption was needed in order to use the Boussinesq Equations, which simplify the solution of this inverse problem. Displacements are given as the convolution of Green’s tensor and traction and the challenge is to solve the inverse problem given the limitation of the noisy displacement field obtained from the measurements.
In the first method that launched the cell traction assays, tractions are explicitly set to 0 outside the cell boundary while the noisy displacement field is specified everywhere, effectively providing for an ill-posed inverse problem that requires somewhat arbitrary smoothing in order to obtain stable solution:
Dembo M and Wang Y-L. Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys J 76: 2307-2316 (1999).
Furthermore, the problem kernel is not diagonal in real space, which leads to bulky matrices and computationally intensive inversion.
In the competing traction calculation method, inversion is performed in Fourier space because the kernel is diagonal in Fourier space and inversion is exact and considerably less computationally intensive:
Butler JP, Tolić-Nørrelykke IM, Fabry B and Fredberg JJ. Traction fields, moments, and strain energy that cells exert on their surroundings. Am J Physiol Cell Physiol 282: C595-C605 (2002).
Lack of periodicity in measured data results in artifact tractions at the boundary of the computational region, but these artifacts are contained only at the boundary and do not affect tractions in the region of interest (cell-covered area) provided that the distance to the boundary is equal to several cell lengths.
When tested on identical data set, the real-space and the Fourier-space method provided reasonably similar results .
An interesting detail remains unnoticed by cell biologists: the small-strain assumption is implied in the Boussinesq Equations but apparently unsupported by the experimental data. Substrate stiffness is fine-tuned to allow measurable displacements at given imaging conditions but that results with large strains which can be easily inferred from the time-lapse movies of substrate deformations (strain data is not reported in the references). It seems that the error of small-strain assumption should be considerable but it is not clear if it results in overestimates of tractions, as my intuition suggests, or it is of a more complex nature. iMechanica is perhaps the best forum for discussions of this specific issue and this class of inverse problems in elasticity.
 Supporting Movie 1 for: du Roure O, Saez A, Buguin A, Austin RH, Chavrier P, Silberzan P and Ladux B. Force mapping in epithelial cell migration. PNAS 102 (7): 2390-2395 (2005).
 Ning Wang, personal communication