Cells adapt their shapes with their experienced stiffness: A geometrical approach to differentiation

The elasticity-directed differentiation of mesenchymal stem cells has been widely studied since mid 2000s. Over nearly linear-elastic materials the differentiation of the stem cells are shown to be related to the Young's modulus of the substrate. While it is found that constraining the stem cells to some pre-patterned shape and size affects their differentiation, it has been also well recognized that the cell morphology intimately relates to its type and functionality such that the cell morphology has been accepted as an indicator for the differentiation path. This paper conjectures the importance of geometry in differentiation and claims that the elasticity indicator for differentiation is indeed "the stiffness" which contains not only the elasticity coefficient but also the geometrical information. An elasticity model is derived for a singular cell with different shapes over a thick substrate, which almost resembles the condition for most of the tests with sparse distribution of the cells. Analysis of principle shapes such as square, rectangle, and hexagon (with and without dendrites) suggest that the larger the aspect ratio i.e. the further the shape to the roundness, the larger the substrate stiffness experienced by the cells. By moving towards a more round shape such as a hexagon with or without dendrites the substrate stiffness falls off rapidly. Then by including the stiffness of the cell body itself we arrive at a more important finding; the cells at free culture condition prefer the shape which best equalizes their experienced stiffness of the substrate and that of their own body. The body-to-substrate stiffness ratio, hence, explains why a slender rectangle is the preferred shape at myogenic differentiation range and a hexagon with dendrites is the preferred one at neurogenic range.