Motivation: Despite the importance of beta-sheets as building blocks in proteins and also toxic elements in the pathological disorders, ranging from Alzheimer's disease to mad cow disease, the principles underlying their stability are not well understood. Non-random beta-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 beta-sheet conforms to a minimal surface with zero mean curvature, like the physical model of soap films.
Results: We here suggest that the stability of a residue in the all beta-sheet proteins can be measured with its mean curvature parameter, using discrete differential geometry. The sharply decreasing mean curvature with increasing number of beta-strands identifies a significant cooperative effect whereby the interstrand interaction increases in strength with the number of beta-strands. Furthermore, strong correlations of mean curvatures with previous beta-sheet propensities of amino acids show that their intrinsic differences in adopting the ideal beta-sheet structure are affected by the water-accessible area of side-chains, and result in the distinct statistical and thermodynamic beta-sheet propensities. Therefore, we conclude that mean curvature should be considered as the significant stability index of a beta-sheet structure.