Modeling the Strength of $\beta$-sheet Structures in Silk Crystals and Protein Molecules

The mechanical response of $\beta$-sheet structures to a tensile force directed along the axis of one chain can be modeled as an array of elastic springs. The \mbox{3-D} potential of H-bonds in $\beta$-sheets gives a shear stiffness of 4.5Nm$^{-1}$ and the chain repeat stiffness is 60Nm$^{-1}$. Nanocrystals $>$3.5nm long with $\geq$20 \mbox{H-bonds/chain} are the strong component of spider silk. They behave much like macro-scale objects, and two conditions must be met for pull-out failure: (1) the load on the most stressed H-bond exceeds the bond strength. (2) the energy of the system is lower after failure. (1) is the critical condition, and the predicted pull-out load is 3-4 times the H-bond strength. An energetically favorable `stick-slip' process is kinetically forbidden. Arrays within a single molecule such as titin have fewer bonds and can fail at low loads by the `stick-slip' process. The logarithmic rate dependence of failure load observed in AFM is \mbox{50pN/decade} and the stick-slip prediction is \mbox{30pN/decade}. Simulations at short times and high loads give slopes $>$10$\times$ higher, matching the prediction for failure at a single bond.