Nonlinear continuum description of the E. coli cell wall under high turgor pressure

The principal stress-bearing component of bacterial cell walls is the tough, covalently linked peptidoglycan network. It is an active material that maintains shape and integrity of bacterial cells in the presence of high turgor pressures while constantly growing. It is not well understood how its extremal material properties derive from its molecular structure. The large stress in the cell wall demands a description going beyond small-strain approximations, an approach that has not been taken in the biomaterials community. We here derive a continuum elastic model of E. coli peptidoglycan (PG) in two steps: We first reduce the complex microscopic structure of the PG layer to the spring elements of a 2D hyperelastic triangular network, and then interpret those results in the context of finite-strain continuum theory. We present a self-contained formalism to study stress-strain relationships for large deformations and arbitrary anisotropies. We obtain the minimal model compatible with the available data on E. coli morphology and mechanics. We show that the strain hardening of PG leads to a stiffness matrix that is linearly proportional to the turgor pressure of the cell.