An iterative method to calculate forces exerted by single cells and multicellular assemblies from the detection of deformations of flexible substrates
- 100 Downloads
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 σ 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 Δ=20 μm (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 Δ=4 μm. Our iterative method allows, without using explicit regularization, the detailed reconstruction of the two investigated patterns when noise is not too high (σ XY /u max≤6%, where u max is the maximal deformation), and when the plane of recorded deformations is close to the surface (Δ/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.
KeywordsDictyostelium slugs Elastic substrates Traction forces Regularization methods Iterative methods
J.P.R. acknowledges support from the Japan Society for the Promotion of Science (Invitation Fellowship for Research in Japan, Long Term, FY2002, FY2003, FY2004). The warm hospitality of Pr. S. Iwasaki at Tohoku Institute of Technology is acknowledged. Part of this work was done at Photodynamics Research Center (PRC). We acknowledge Prof. J. Nishizawa and Prof. S. Ushioda for permitting us to use the optical facilities at PRC. We would like also to thank S. Sawai, C. Cottin-Bizonne, L. Bocquet and C. Ybert for helpful comments on the manuscript.
- Landau LD, Lifshitz EM (1986) Theory of elasticity, 3rd edn (J. B. Sykes, W. H. Reid, translators). Pergamon Press, Oxford, pp 37–41Google Scholar
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in FORTRAN 77: the art of scientific computing (vol 1 of Fortran numerical recipies), 2nd edn. Cambridge University Press, Cambridge. http://www.lib-lanl.gov/numerical/index.html
- Reinhart-King CA, Dembo M, Hammer DA (2005) The Dynamics and mechanics of endothelial cell spreading. Biophys J. BioFAST, published on April 22, 2005 as DOI 10.1529/biophysj.104.054320Google Scholar