Microcanonical versus Canonical Analysis of Protein Folding

Abstract

The microcanonical analysis is shown to be a powerful tool to characterize the protein folding transition and to neatly distinguish between good and bad folders. An off-lattice model with parameter chosen to represent polymers of these two types is used to illustrate this approach. Both canonical and microcanonical ensembles are employed. The required calculations were performed using parallel tempering Monte Carlo simulations. The most revealing features of the folding transition are related to its first-order-like character, namely, the S-bend pattern in the caloric curve, which gives rise to negative microcanonical specific heats, and the bimodality of the energy distribution function at the transition temperatures. Models for a good folder are shown to be quite robust against perturbations in the interaction potential parameters.

First-order coil-globule transition driven by vibrational entropy

“…Figure 1e-h reveal that the derivative of S(E) is no longer monotonic when vibrations are included in the picture. The region where β grows with E, which would correspond to a negative microcanonical specific heat, is canonically forbidden and is one of the hallmarks of a discontinuous transition with coexistence between two different phases [26][27][28] . The coexistence temperature can be extracted from the β(E) curve using the customary Maxwell construction; this value β coex 3.18 agrees well with the temperature of the (canonical) specific heat maximum β max = 3.20 (2).…”

Generalized simulated tempering for exploring strong phase transitions

“…In this paper, the applicability of ST is further extended to diverse first-order phase transition systems by analyzing how conventional ST struggles to sample metastable states associated with the backbending ͑or S-loop͒ in the statistical temperature T S ͑E͒ ͑or microcanonical caloric curve͒ 46 characteristic to first-order phase transitions in finite size systems. [47][48][49][50][51][52][53][54] Our analysis reveals that an intrinsic instability of the canonical ensemble to a negative slope region in T S ͑E͒ is a primary cause for poor acceptance for temperature transitions in ST.…”

“…[47][48][49][50][51][52][53][54] The backbending in T S ͑E͒ manifests an inherent instability of the canonical ensemble to phase-coexistent states, which invokes a bimodal structure in the canonical PDF P ␤ ͑E͒ = e −F ST ͑E;␤͒ / Z͑␤͒, F ST ͑E ; ␤͒ = E − TS͑E͒ being the Helmholtz free energy density. Noting that F ST ͑E ; ␤ c ͒ has three extrema at energies satisfying When the system size L is small, both free energy minima at E 1 ‫ء‬ and E 3 ‫ء‬ can be sampled across a free energy barrier at E 2 ‫ء‬ ͓see Fig.…”

Generalized Replica Exchange Method

“…1͑b͒, the so called backbending or S-loop. 32,33,35,36 The existence of the backbending has been verified in recent experiments on nuclear fragmentation 37 and cluster melting, 38 and its physical origin has been attributed to avoiding a "static" phase coexistence due to the free energy cost forming interfaces. The backbending in T S ͑E͒ manifests a bimodal structure in P T ͑E͒ ϰ e −␤F͑E,T͒ , F͑E , T͒ = E − TS͑E͒ being the Helmhotz free energy density and ␤ = ͓k B T͔ −1 ͑k B =1͒.…”

“…In many finite size systems, such as spins, 29 nuclei fragmentations, 30,31 model proteins, [32][33][34] and atomic clusters, 8,35,36 S͑E͒ shows a convex dip, i.e., ‫ץ‬ 2 S / ‫ץ‬E 2 Ͼ 0, 30 across the transition region, as sketched in Fig. 1͑a͒.…”