Cell shape characterization, alignment and comparison using FlowShape

Motivation The shape of a cell reflects, among other things, actomyosin activity and adhesion properties. Cell shape is further tightly linked to cell differentiation and can reveal important cellular behaviors such as polarization. Hence, it is useful and informative to link cell shape to genetic and other perturbations. However, most currently used cell shape descriptors capture only simple geometric features such as volume and sphericity. We propose FlowShape, a new framework to study cell shapes in a complete and generic way. Results In our framework a cell shape is first represented as a single function on a sphere. The curvature of the shape is measured and next mapped onto a sphere in a conformal manner. This special curvature map is then approximated by a series expansion: the spherical harmonics decomposition. This decomposition facilitates a wide range of shape analyses, including shape alignment, statistical cell shape comparison and inference of cell shape deformations over time. From this representation, we can reconstruct the cell shape using the Dirac equation. The new tool is applied to perform a complete, generic analysis of cell shapes, using the early Caenorhabditis elegans embryo as a model case. We distinguish and characterize the cells at the seven-cell stage. Next, a filter is designed to identify protrusions on the cell shape to highlight lamellipodia in cells. Furthermore, we use our framework to identify any shape changes following a gene knockdown of the Wnt pathway. Cells are first optimally aligned using the fast Fourier transform, followed by calculating an average shape. Shape differences between conditions are next quantified and compared to an empirical distribution. Finally, we put forward a highly performant implementation of the core algorithm, as well as routines to characterize, align and compare cell shapes, through the open-source software package FlowShape. Availability The data and code needed to recreate the results are freely available at https://doi.org/10.5281/zenodo.7391185. The most recent version of the software is maintained at https://bitbucket.org/pgmsembryogenesis/flowshape/. Author summary We present FlowShape, a framework for cell shape analysis, based on the concept of spherical harmonics decomposition. This decomposition allows for any function defined on a sphere to be rewritten as a weighted sum of basis functions. Contrary to previous work, we use a single function to describe a shape, the mean curvature, which implies that the decomposition weights can be used as a complete shape description. The expression of a shape in this manner allows for very efficient calculations, as we illustrate with the C. elegans embryo as a model. The decomposition permits efficient comparison and alignment of shapes. We demonstrate this by clustering the cells in the early embryo and illustrating the different shapes by cluster. The decomposition further facilitates averaging of shapes and searching for particular features on the shape by defining filters that can then be efficiently applied. Finally, we illustrate how the framework can facilitate statistical comparisons between shapes.