How large should proteins be? The minimal size of a good structure seeker

This work deals with the determination of the minimum length of a protein chain required for expeditious and robust folding into a specific target structural motif. Our goal requires that we explore the relationship between the size of a protein and its ability to fold from the statistical mechanics perspective. A coarse partition of the soft-mode conformation space in basins of attraction of the potential energy surface (PES) associated to different contact patterns (CPs) is assumed. This coarse description is consistent with low-temperature experimental probes, revealing the fine resolution of each PES basin, a feature known to be washed out at typical renaturation temperatures. This experimental observation provides the signature of an entrainment of fast-relaxing variables by a coarse CP-resolution of the dynamics. Our statistical analysis is thus focused on the size-dependent energy dispersion within basins and the observation that large energy-level dispersions prevent the entrainment of long-time torsional dynamics by CP transitions due to the breakdown of relaxation timescale separation. The entrainment by coarse dynamics is required by a good structure seeker, since it renders adiabatically the search in conformation space, a regime warranted by a relatively fast equilibration within each CP-basin. This drastic reduction in the number of dominant variables, probed for real proteins at low unphysiological temperatures, is reasonably assumed to be responsible for the robustness and expediency of the folding process, which is thus no longer dependent on the fine details of the torsional dynamics. Finding the minimal size of a protein chain fulfilling such requirements is precisely the goal of this work. If the chain is not long enough, the large relative dispersion of energy levels within a basin introduces correlations between substates of different basins and leads to a breakdown of the dynamic entrainment, rendering the folding process ineffective, and considerably decreasing its expediency. This is so since energy fluctuations within a CP-basin relax on timescales which are no longer incommensurably shorter than those of CP-transitions and intra-CP-basin relaxation interferes with the folding process as viewed from the coarser perspective.