Characterization of the Unfolded State and the Folding Process
The theoretically accessible conformational space of backbone conformations of polypeptide analogs depends on the number of easily rotatable torsional angles along the backbone. Assuming three (trans, gauche+, gauche-) conformations per torsional angle, the theoretical number of conformations of the 7-b-peptide discussed before would be 321 or about 109 conformers. Whether all these con-formers are accessible under physiological conditions can be investigated by clustering all peptide structures from a MD trajectory of a folding equilibrium into
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conformations which are characterized by a maximum atom-positional RMSD between their backbone atoms. This can be done as follows [17, 44]: the number of neighbors (that is the number of structures satisfying a given similarity criterion) is determined for each trajectory structure, with the criteria of similarity between two structures being the positional RMSD value of their main chain atoms. The structure with the highest number of neighbors is then taken as representating the first, most populated, conformation or cluster of structures. After removing the structures belonging to the first cluster from the trajectory, the procedure is repeated to find the second cluster or conformation, and so on. This clustering algorithm can also be applied to two trajectories representing different peptides of the same chain lengths or generated at different thermodynamic conditions for a single peptide. If the two trajectories sample the same part of configuration space and have similar conformational distributions, the resulting clusters will each have a comparable amount of structures from each of the two trajectories. If the two trajectories sample disjunct parts of configurational space, each cluster will only contain members of only one of the two trajectories. The former situation is illustrated in Fig. 6.13, which shows the result of a combined trajectory cluster analysis of a simulation of the 7-b-peptide at 360 K and 1 atm with one at 340 K and 1000 atm . So, the conformations characterizing the unfolded state at higher pressure are the same as those characterizing the unfolded state at higher temperature. This combined trajectory cluster analysis has also been
applied to two 6-b-peptides carrying different side chains, which made them adopt different stable folds: a 314-helix for one peptide (black bars in Fig. 14) and a hairpin for the other peptide (grey bars in Fig. 6.14) . In Fig. 6.14 the result of the combined cluster analysis of the two trajectories is shown. The conforma-tional distributions are rather different. The most populated cluster is a hairpin and the second most populated cluster a helix, as expected. Figure 6.15 shows an example of two completely disjunct folding equilibrium ensembles of two 6-b-peptides carrying identical side chains, but differing by the presence of two meth-ylgroups at all six a-carbon positions along the main chain. These methyl groups prevent helix formation leading to a completely different conformational ensemble from that of the unmethylated peptide in methanol .
Folding pathways can be determined by counting the number of transitions from and to each conformational cluster [41, 44]. Such an analysis has been applied to the dynamic folding equilibrium of the 7-b-peptide  which has a 314-helix as most populated cluster. At 340 K, more than one pathway leads to the helical fold. Figure 6.16 illustrates that these pathways are not necessarily downhill in free energy. We note, however, that in order to obtain statistically
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