Mean curvature as a major determinant of β-sheet propensity

Motivation: Despite the importance of β-sheets as building blocks in proteins and also toxic elements in the pathological disorders, ranging from Alzheimer's disease to mad cow diease, the principles underlying their stability are not well understood. Non-random β-sheet propensities of amino acids have been revealed both by their distinct statistical preferences within known protein structures and by the relative thermodynamic scales through the experimental host-guest systems. However, recent fitting analysis has proved that a native β-sheet conforms to a minimal surface with zero mean curvature, like the physical model of soap films. Results: We here sugg est that the stability of a residue in the all β-sheet proteins can be measured with its mean curvature parameter, using discrete differential geometry. The sharply decreasing mean curvature with increasing number of β-strands identifies a significant cooperative effect whereby the interstrand interaction increases in strength with the number of β-strands. Furthermore, strong correlations of mean curvatures with previous β-sheet propensities of amino acids show that their intrinsic differences in adopting the ideal β-sheet structure are affected by the water-accessible area of side-chains, and result in the distinct statistical and thermodynamic β-sheet propensities. Therefore, we conclude that mean curvature should be considered as the significant stability index of a β-sheet structure.