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I am currently reading up on the formation of $\beta$-sheets. The following text describes its formation from a free energy point of view:

The edge of a β-sheet consists of (a) edge β-strands, and (b) bends or loops connecting the β-strands [...]. Let the coil free energy be zero (i.e., the reference point); $f_{\beta}$ , the free energy of a residue in the center of the β-sheet; $f_{\beta}$ + $\Delta f_{\beta}$ , the free energy of an edge β-strand residue (i.e., $\Delta f_{\beta}$ is the edge effect); and U, the free energy of a bend. Since the β-sheet forms, it is stable (i.e., $f_{\beta}$ < 0), and the edge effects prevent it from falling into pieces (i.e., $\Delta f_{\beta}$ > 0 and U > 0) (Finkelstein, Protein Physics)

Why do $\Delta f_{\beta}$ and U need to be positive? Wouldn't this mean that the bends or folds of the sheet are unstable? Why would negative $\Delta f_{\beta}$ and U make the sheet fall to pieces?

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If you could decrease the free energy of the sheet by adding an edge (such that $\Delta f_\beta$ < 0) then edges would grow and form in the sheet to decrease its free energy. The free energy in this context is $not$ the free energy of an edge, since such it doesn't make sense to define the free energy of an edge in isolation. The free energy of an edge can only be defined with respect to the bulk (or sheet, in this case).

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