Calculation of the force field required for nucleus deformation during cell migration through constrictions
Authors: Estabrook Ian D. , Thiam Hawa Racine , Piel Matthieu , Hawkins Rhoda J. 
Published: December 2020

During cell migration in confinement, the nucleus has to deform for a cell to pass through small constrictions. Such nuclear deformations require significant forces. A direct experimental measure of the deformation force field is extremely challenging. However, experimental images of nuclear shape are relatively easy to obtain. Therefore, here we present a method to calculate predictions of the deformation force field based purely on analysis of experimental images of nuclei before and after deformation. Such an inverse calculation is technically non-trivial and relies on a mechanical model for the nucleus. Here we compare two simple continuum elastic models of a cell nucleus undergoing deformation. In the first, we treat the nucleus as a homogeneous elastic solid and, in the second, as an elastic shell. For each of these models we calculate the force field required to produce the deformation given by experimental images of nuclei in dendritic cells migrating in microchannels with constrictions of controlled dimensions [1]. These microfabricated channels provide a simplified confined environment mimicking that experienced by cells in tissues. We extract the nuclear shape from the boundary of the fluorescently stained region in each consecutive image over time. From this we calculate the deformation field between images and use our elastic models to calculate the traction force field. Our calculations therefore predict the forces felt by a deforming nucleus as a migrating cell encounters a constriction. Since a direct experimental measure of the deformation force field is very challenging and has not yet been achieved, our numerical approaches can make important predictions motivating further experiments, even though all the parameters are not yet available. In addition, the algorithm we have developed could be adapted to analyse experimental images of deformation in other situations.

Author summary

Many cell types are able to migrate and squeeze through constrictions that are narrower than the cell’s resting radius. For example, both immune cells and metastatic cancer cells change their shape to migrate through small holes in the complex tissue media they move in. During migration the cell nucleus is more difficult to deform than the cell cytoplasm and therefore significant forces are required for a cell to pass through spaces that are smaller than the resting size of the nucleus. Experimental measurements of these forces are extremely challenging but experimental images of nuclear deformation are regularly obtained in many labs. Therefore we present a computational method to analyse experimental images of nuclear deformation to deduce the forces required to produce such deformations. A mechanical model of the nucleus is necessary for this analysis and here we present two different models. The first treats the nucleus as a homogeneous elastic solid and the second treats the nucleus as an elastic shell. Our computational tool enables us to obtain detailed information about forces causing deformation from microscopy images.