A general solution of a regular 2D spring network to model the E. coli peptidoglycan shell.

The cell wall of gram-negative E. coli bacteria derives its mechanical resilience from a 2D network of glycan strands covalently cross-linked by short peptides. The network is constantly growing by the insertion of new material while holding up to extreme stresses. It is only incompletely understood how the astonishing capabilities of this network derive from the molecular properties of its constituents and network geometry. We use a 2D triangular spring network to provide a bridge between molecular and continuum-elastic modeling. We derive general constitutive laws, valid for large deformations, non-linear responses, and arbitrary anisotropies. We then find the minimal model compatible with the available data on morphology and mechanics of E. coli cell walls. We show that, in the large-deformation regime, the stiffness of the network is always proportional to the magnitude of the stress, and therefore, to the turgor pressure of the cell.