Many of the most important molecules of life are polymers. In animals, the most abundant of the proteinaceous polymers are the collagens, which constitute the fibrous matrix outside cells and which can also self-assemble into gels. The physically measurable stiffness of gels, as well as tissues, increases with the amount of collagen, and cells seem to sense this stiffness. An understanding of this mechanosensing process in complex tissues, including fibrotic disease states with high collagen, is now utilizing 'omics data sets and is revealing polymer physics-type, nonlinear scaling relationships between concentrations of seemingly unrelated biopolymers. The nuclear structure protein lamin A provides one example, with protein and transcript levels increasing with collagen 1 and tissue stiffness, and with mechanisms rooted in protein stabilization induced by cytoskeletal stress. Physics-based models of fibrous matrix, cytoskeletal force dipoles, and the lamin A gene circuit illustrate the wide range of testable predictions emerging for tissues, cell cultures, and even stem cell-based tissue regeneration. Beyond the epigenetics of mechanosensing, the scaling in cancer of chromosome copy number variations and other mutations with tissue stiffness suggests that genomic changes are occurring by mechanogenomic processes that now require elucidation.