Elasticity and Strength of Beta-Sheet Protein Materials: Geometric Confinement and Size Effects

Elasticity and strength of proteins influence their biological functions. Under external forces, many proteins exhibit entropic elasticity with a characteristic stiffening elastic behavior and unravel due to the rupture of interstrand H-bonds. We develop a fracture mechanics based theoretical framework that considers the free energy competition between entropic elasticity of polypeptide chains and rupture of peptide hydrogen bonds, which we use here to provide an explanation for the intrinsic strength limit of protein domains at vanishing rates [1, 2]. Our analysis predicts that individual protein domains stabilized only by hydrogen bonds cannot exhibit rupture forces larger than 100–300 pN in the asymptotic quasi-static limit. This result explains earlier experimental and computational observations that suggest such a universal, asymptotic strength limit at vanishing pulling rates. We show that the rupture strength of H-bond assemblies in beta-sheets is governed by geometric confinement effects, suggesting that clusters of at most 3–4 H-bonds break concurrently, even under uniform shear loading of a much larger number of H-bonds. These strength, elasticity and size effect predictions all agree well with recent experimental findings and proteomics data. Our model confirms that fracture mechanics concepts, previously primarily applied to macroscale fracture phenomena, can also be directly applied at nanoscale, to be used for describing failure mechanisms in protein materials. Our strength and optimal size predictions lead to a key hypothesis: confined H-bond clusters are prevalent in alpha helices, beta-sheets and beta-solenoids, perhaps as an evolutionary design principle that derives from generic mechanical properties of the fundamental building blocks of life.