FIG. 3. Comparing the optimized energies from multiple simulated annealing runs for coupled and uncoupled simulations. Dark bars, coupled runs; light bars, uncoupled runs.

Another important feature of the present protocol is of averaging in sequence space. We return to the N different sequences. Each of the proteins has a unique energy surface. By virtue of experimental observations, we know that all the homologous protiens share similar structures at their native fold. At approximately the same coordinate Xnative, we expect

FIG. 4. A schematic drawing of potential smoothing using discrete, local averaging: ⟨V(X)⟩=1/N V(XiδX,Xi·

all the proteins to be in an energy minimum. Therefore, all of the energies are correlated and the result of the sum is a quantity, which increases linearly with N.

On the other hand, for unfolded states the energies of the different homologous structures are not necessarily correlated. Consider for example a correlated mutation at the hydrophobic core of the protein. To maintain the compactness of the hydrophobic core of the native state, valine and tryptophan may replace a pair of phenylalanines. At unfolded conformations, it is not necessary to assume that the contacts and the energies of the above residues are still correlated. It is more likely that the energies are not correlated. We therefore estimate This estimate is in the spirit of the Random Energy Model as applied to proteins (14).

The new energy surface Etotal is therefore distorted in a favorable way when comparing it to the original εi. The shared minimum (which databases of protein structures support its existence) is deeper compared with other portions of the energy surface. The enhancement of the well depth of the shared minimum may make it the global energy minimum of the new average energy even if originally it was not. This enhancement suggest the new protocol as possibly effective for kinetically stable proteins.

It is the combination of the spatial and sequence averaging, that provides significant improvement in structure prediction of ab initio algorithms as discussed above.

This research was supported by a Binational Science Foundation grant (to R.E. and J.S.) and by National Institutes of Health Grant GM37408 (to J.S.).

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