Figure 1.

(a) The typical folding temperature T_f for different polymer lengths with a two-state-like transition. The solid curve indicates T_f for peptides without heterogeneities. Above n_max (indicated by ○), the curve is replaced by a thin one where the transition is between a partially folded ensemble and the coil state. Here c represents the “coil” state, and h^(c), h^(p) stand for the “complete,” “partial” hairpin structure, respectively. In the presence of a hydrophobic cluster (with locations indicated in the box), T_f is increased (the dashed curve) with modified n_max (◊, central; ▹, distal-end; ◃, β-turn); the transition curves above n_max for these cases are not shown. Here σ = 1, V₂ = 0.15 V₁, δV₁ = 0.05V₁, γ = 3, and ν₀ = 10⁻⁴. (b) The free energy − ln [Z_n,Q] of different ensembles at T_f for n = 14 polymers with a distal-end hydrophobic cluster. Here the parameters aside from V₂ are the same as in a. The symbols “○,” “∗,” and “+” stand for the coil, completely folded, and partially folded ensembles. Note that in the strong heterogeneity case, there is a highly folded ensemble (Q = 13) as an intermediate. Curves of this kind allow for the determination of n_max. (c) The dependence of n_max on the variation of V₂, in the case of having a distal-end hydrophobic cluster.