Elasticity theory for self-assembled protein lattices with application to the martensitic phase transition in bacteriophage T4 tail sheath.

We propose an elasticity theory for one- and two-dimensional arrays of globular proteins for which the free energy is affected by relative position and relative rotation between neighboring molecules. The kinematics of such assemblies is described, the conditions of compatibility are found, a form of the free energy is given, and formulas for applied forces and moments are developed. It is shown that fully relaxed states of sheets consist of helically deformed sheets which themselves are composed of helical chains of molecules in rational directions. We apply the theory to the fascinating contractile deformation that occurs in the tail sheath of the virus bacteriophage T4, which aids its invasion of its bacterial host. Using electron density maps of extended and contracted sheaths, we approximate the domains of each molecule by ellipsoids and then evaluate our formulas for the position and orientation of each molecule. We show that, with the resulting kinematic description, the configurations of extended and contracted tail sheaths are generated by a simple formula. We proposed a constrained version of the theory based on measurements on extended and contracted sheath. Following a suggestion of Pauling [Discuss. Faraday Soc. 13, 170 (1953)], we develop a simple model of the molecular interaction. The resulting free energy is found to have a double-well structure. Certain simple deformations are studied (tension, torsion inflation); the theory predicts a first-order Poynting effect and some unexpected relations among moduli. Finally, the force of penetration is given, and a possibly interesting program of epitaxial growth and patterning of such sheets is suggested.