An iterative method to calculate forces exerted by single cells and multicellular assemblies from the detection of deformations of flexible substrates

We present a new method for quantification of traction forces exerted by migrating single cells and multicellular assemblies from deformations of flexible substrate. It is based on an iterative biconjugate gradient inversion method. We show how the iteration and the solution are influenced by experimental parameters such as the noise on deformations sigma (XY) , and the mean depth of recorded deformations Z(M). In order to find the validity range of our computational method, we simulated two different patterns of force. The first artificial force pattern mimics the forces exerted by a migrating Dictyostelium slug at a spatial resolution of Delta = 20 mu m (Rieu et al. in Biophys J 89:3563-3576, 2005) and corresponds to a large and spread force field. The second simulated force pattern mimics forces exerted by a polarized fibroblast at discrete focal adhesion sites separated by Delta = 4 mu m. Our iterative method allows, without using explicit regularization, the detailed reconstruction of the two investigated patterns when noise is not too high (sigma(XY) /u(max) <= 6%, where u(max) is the maximal deformation), and when the plane of recorded deformations is close to the surface (Delta/Z(M)>= 4). The method and the required range of parameters are particularly suitable to study forces over large fields such as those observed in multicellular assemblies.