Coexistence of Flexibility and Stability of Proteins: An Equation of State

We consider a recently suggested "equation of state" for natively folded proteins, and verify its validity for a set of about 5800 proteins. The equation is based on a fractal viewpoint of proteins, on a generalization of the Landau-Peierls instability, and on a marginal stability criterion. The latter allows for coexistence of stability and flexibility of proteins, which is required for their proper function. The equation of state relates the protein fractal dimension d(f), its spectral dimension d(s), and the number of amino acids N. Using structural data from the protein data bank (PDB) and the Gaussian network model (GNM), we compute d(f) and d(s) for the entire set and demonstrate that the equation of state is well obeyed. Addressing the fractal properties and making use of the equation of state may help to engineer biologically inspired catalysts.