Nonhelical intermediate structures from pPA 12-mer simulations: (a) knotted conformation, (b) -like strand structure. These structures do not contribute to trapped states in 12-mers, but topologically similar structures in pPA 20-mer chains do lead to trapped kinetic macrostates.
Chemical representation of the backbone of poly-phenylacetylene. We show the atoms involved (1-2-5-6) in the proper dihedral potential added across the acetylene bond in order to bias the chain toward coplanar configurations.
(a) The cumulative folding time distribution illustrates the long time scale trapped kinetic phase. The data are fit to a four-state model (smooth curves) which includes a trapped state in equilibrium with the unfolded state. The inset shows the folding time distribution at shorter times to help visualize the short time scale kinetic phases. Also shown is the four-state kinetic model with the values for the fundamental rate constants extracted from the nonlinear least-squares fit of the model to the concentration curves. (b) An equally valid kinetic model contains five states with an additional trapped state in equilibrium with the intermediate state. We chose the four-state model for simplicity, although our kinetic analysis suggests that a fifth unique, trapped state does indeed exist. This does not preclude the possibility of even more states. However, the four-state model contains the essential features of the kinetic behavior of the 20-mer chains and is sufficient for demonstrating the obvious length dependent kinetic features of the phenylene ethynylene class of Foldamers. (c) Three-state kinetic model representing the folding dynamics of the pPA 12-mer chains. This model has previously been reported and is included strictly for comparison purposes to highlight the differences in folding mechanism of phenylene ethynylene oligomers as the chain length increases. The noticeable difference between the 20-mer and the 12-mer chains is the lack of a trapped kinetic state in the 12-mer chains, despite similar topologically diverse intermediate conformational states to those observed in the 20-mer chains.
The “gap statistic,” calculated from the -means clustering solutions, assists in uncovering the optimal number of clusters in the data set. The optimal number of clusters should be the value of for which the gap is maximized. Unfortunately, the gap does not reach a maximum, so we take the most prominent local maximum at as the optimal number of clusters in the data set.
Classification of clusters into five kinetic classes, two of which are kinetically trapped classes of conformations. These two classes of trapped conformations are combined together into one trap macrostate for the kinetic model shown in Fig. 3(a). (a) Structural characterization of the clusters obtained from the -means clustering method. Those clusters with low helical content, as measured by , are typically in trapped conformational states. (b) Kinetic map, in vs MFPT space, of the clusters obtained from -means clustering. These figures showing the structural characterization and kinetic map of the pPA 20-mers demonstrate a very high correlation between conformations with low helical content also being in trapped kinetic states. (c) Stereo view of the cluster centers of the clusters obtained from -means clustering. There are 16 images shown, although, the gap statistic produced 15 clusters. -means could not distinguish the native cluster from a helical intermediate cluster. We know the folding times for each trajectory; therefore, we split the intermediate cluster into two clusters, one with helical intermediates and the other with the native conformations, by reassigning any conformation in the intermediate cluster to the folded cluster, if the given conformation appeared after the chain folded.
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