Rigidity effects and mechanical unfolding of proteins

We describe a new method that shows promise for evaluating the partition function for a protein under an applied external force within a Distance Constraint Model (DCM). This approach is based on an approximate account for the rigidity effects due to hydrogen bond crosslinking using Maxwell constraint counting. Within a mean-field treatment, the free energy is estimated accurately over an ensemble of accessible conformations conditional upon the breaking of various weakest-link distance constraints, as they successively break due to a series of mini structural transitions. These calculations are performed using an exact transfer matrix approach combined with a combinatorial partitioning of the structure into different parts based on separating lines of unfolding pathways. The various shortest paths over an ensemble of structures that ``crack'' open in different ways are used to obtain the appropriate Boltzmann weight, related to the work done by the external pulling force. For structures with beta-hairpin geometry, all permutations of unfolding pathways are enumerated exactly. For a simple minimal DCM, results for extension-force curves agree markedly well with experiment. Using computational methods, this approach can be used to describe single-molecule experiments on mechanical protein unfolding under different settings, such as fixed extension, or constant force conditions. This work is supported by NIH R01 GM073082.