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How do cells tune emergent properties at the scale of tissues? One class ofsuch emergent behaviors are rigidity transitions, in which a tissue changesfrom a solid-like to a fluid-like state or vice versa. Here, we introduce a newway that cells can tune rigidity using "adaptive degrees of freedom." We studyrigidity in vertex models, using the elastic energy as a cost function and thecell stiffnesses, target shapes, and target areas as sets of adaptive degreesof freedom that can be introduced to minimize the cost function. We show thatthe rigidity transition is unaffected when cell stiffnesses are treated asadaptive degrees of freedom. When preferred shapes or areas are treated asadaptive degrees of freedom, however, induced spatial correlations in targetcell shapes or areas shift the rigidity transition. These observations suggestthat tissues can tune properties at the cell-scale to obtain desiredtissue-scale behaviors.

Rigidity of Epithelial Tissues as a Double Optimization Problem