The diphenylalanine peptide self-assembles to form nanotubular structures of remarkable mechanical, piezolelectrical, electrical, and optical properties. The tubes are unexpectedly stiff, with reported Young's moduli of 19-27 GPa that were extracted using two independent techniques. Yet the physical basis for the remarkable rigidity is not fully understood. Here, we calculate the Young's modulus for bulk diphenylalanine peptide from first principles, using density functional theory with dispersive corrections. The calculation demonstrates that at least half of the stiffness of the material is the result of dispersive interactions. We further quantify the nature of various inter- and intramolecular interactions. We reveal that despite the porous nature of the lattice, there is an array of rigid nanotube backbones with interpenetrating "zipper-like" aromatic interlocks that result in stiffness and robustness. This presents a general strategy for the analysis of bioinspired functional materials and may pave the way for rational design of bionanomaterials.