Changes in Bond-Orientational Order of Residues are Associated with Shifts in Energy Landscapes

We show that local order, measured not only by the number density distribution around a given node, but also as the geometric preferences of neighbors quantified by bond orientational order (BOO)[1] identifies subtle local structural changes in proteins.We first establish a correspondence between topological and geometrical quantities utilized in protein physics[2]. In the topological case, local parameters are represented as moments of the adjacency matrix. For the geometrical counterpart, local information is encoded in BOO parameters, showing up in series expansion of bond density on a unit sphere. Of special importance, the respective third moments are measures of local compactness; both clustering coefficient C and third-order rotational invariant W use the coupling of three vectors/edges.We then generalize a topological index[3] that measures the propensity of residues to find alternative routes to communicate with function-related destinations. The average number of alternative n-step paths a given residue generates to its neighbors (equal to 2C for two-step paths), normalized by the reachability of that residue by all others in the structure successfully measures the degree of collectivity of motions in a protein.We finally demonstrate the utility of these concepts by showing that W is a good descriptor for identifying local structural changes between the apo and holo forms of ferric-binding and maltose-binding proteins. In both cases, though the holo form resides close to the apo form on the free energy surface as a weakly populated conformation, BOO changes between the states are clearly detectable. Expanding in terms of BOOs, we offer an alternative method for calculating the free-energy change.[1]Steinhardt et al. Physical ReviewB, 28, 784(1983).[2]Atilgan et al. Annual Review of Biophysics, 2012, to appear.[3]Atilgan et al. Biophysical Journal, 99, 933(2010).