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Beta Sheet Formation

The cooperative folding of polypeptide strands into beta-sheets, stabilized by hydrogen bonds between adjacent strands.
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The statement of the theorem

Define two adjacent polypeptide segments, S1S_1 and S2S_2, with backbone atoms ri(1)\mathbf{r}_{i}^{(1)} and rj(2)\mathbf{r}_{j}^{(2)}, respectively. The formation of a β\beta-sheet is stabilized by inter-strand hydrogen bonds between the backbone carbonyl oxygen O(i)\text{O}(i) and the amide proton N(j)\text{N}(j) of the adjacent strand. The stability is maximized when the potential energy EH-bondE_{\text{H-bond}} is minimized, subject to the geometric constraints: \n\nEH-bond=∑i,j[Ad(O(i),N(j))2−Bd(O(i),N(j))]+Angle PenaltyE_{\text{H-bond}} = \sum_{i, j} \left[ \frac{A}{d(\text{O}(i), \text{N}(j))^2} - \frac{B}{d(\text{O}(i), \text{N}(j))} \right] + \text{Angle Penalty} \n\nwhere d(⋅,⋅)d(\cdot, \cdot) is the distance between atoms, and the angle penalty enforces the near-planarity and optimal dihedral angles characteristic of the extended β\beta-strand conformation.
Source: Wikipedia