Protein rigidity and flexibility: Applications to folding and thermostability

The mechanism of protein folding is an unsolved, difficult problem. Performing the inverse problem of unfolding a known protein structure has the advantage of known initial conditions. This study relates protein unfolding to a loss of structural stability and rigidity. Drawing on the wealth of knowledge about structural rigidity and flexibility from physics and mathematics, connections are made with proteins. Proteins are identified as a special case of amorphous (glassy) materials and are analyzed as such. The development of the FIRST software as a method to identify flexible and rigid regions in proteins along with a justification for its use to enumerate and partition the number of degrees of freedom (floppy modes) by constraint counting in networks (proteins) is presented. By removing hydrogen bonds in order from the weakest to strongest, protein unfolding by thermal dilution is simulated. This process also describes protein folding under the reasonable assumption (for two-state folders) that the problem is reversible. Along the simulated unfolding pathway two unique points are identified: the transition state and the folding core. The transition state occurs at the inflection point in the change in the fraction of floppy modes with respect to decreasing mean atomic coordination. The fraction of floppy modes as a function of mean coordination is similar to the fraction-folded curve for a protein as a function of denaturant concentration or temperature. Its second derivative, a specific heat-like quantity, shows a peak around a mean coordination of ⟨ r⟩ = 2.41 for the 26 diverse proteins we have studied. As the protein denatures, it loses rigidity at the transition state, proceeds to a state where only the initial folding core remains stable, then becomes entirely denatured or flexible. This universal behavior is found for proteins of diverse architecture, including monomers and oligomers, and is analogous to the rigid to floppy phase transition in network glasses. This approach provides a unifying principle for proteins and glasses, and identifies the mean coordination as the relevant structural variable, or reaction coordinate, for the unfolding pathway. The identification of the folding core is compared to a set of 10 structures that have hydrogen-deuterium exchange data. This computational procedure is shown to identify and predict biologically significant flexibility by comparison with experimental measures of flexibility for several proteins. In general, flexibility is observed to decrease upon ligand binding. Completing the study on structural flexibility and stability in proteins is an investigation into the role rigidity plays in thermostability. An increase in rigidity is shown to correlate with increased thermostability for eight families of homologous proteins. Comparisons are made between rigidity analysis from FIRST and experimental measures of thermostability, supporting rigidity as a general thermostabilizing mechanism.