Loss of conformational entropy in protein folding calculated using realistic ensembles and its implications for NMR-based calculations

Abstract

The loss of conformational entropy is a major contribution in the thermodynamics of protein folding. However, accurate determination of the quantity has proven challenging. We calculate this loss using molecular dynamic simulations of both the native protein and a realistic denatured state ensemble. For ubiquitin, the total change in entropy is TΔS Total = 1.4 kcal·mol −1 per residue at 300 K with only 20% from the loss of side-chain entropy. Our analysis exhibits mixed agreement with prior studies because of the use of more accurate ensembles and contributions from correlated motions. Buried side chains lose only a factor of 1.4 in the number of conformations available per rotamer upon folding (Ω U /Ω N ). The entropy loss for helical and sheet residues differs due to the smaller motions of helical residues (TΔS helix−sheet = 0.5 kcal·mol NMR order parameters | molecular dynamics | helix propensity | sheet propensity | denatured state A n accurate determination of the loss of conformational entropy is critical for dissecting the energetics of reactions involving protein motions, including folding, conformational change, and binding (1-6). Given the difficulty of directly measuring the conformational entropy, most early estimates relied on computational approaches (2, 7-10), although, more recently, NMR methods have been used to measure site-resolved entropies (11). The computational methods often calculated the entropy of either the native state ensemble (NSE) or the denatured state ensemble (DSE) and invoked assumptions about the entropy of the other ensemble [e.g., assuming the NSE is a single state or that the DSE is a composite of all side-chain (SC) rotameric states in the Protein Data Bank (PDB)]. Most previous approaches focused on helices and omitted contributions from vibrations and correlated motions (12, 13), thereby partly accounting for the spectrum of calculated values.We address these issues by calculating the chain's conformational entropy from the distributions of the backbone (BB) (ϕ,ψ) and SC rotametric angles, [χ n ], obtained from all-atom simulations of the NSE and DSE for mammalian ubiquitin (Ub). This study extends our previous calculation of the loss of BB entropy that used an experimentally validated DSE (14). The calculated angular distributions reflect both the Ramachandran (Rama) basin populations and the torsional vibrations. Correlated motions are accounted for through the use of joint probability distributions [e.g., P(ϕ,ψ,χ 1 ,χ 2 )].The computed loss of BB entropy is 80% of the total entropy loss at 300 K. The BB entropy is independent of burial and residue type (excluding Pro, Gly, and pre-Pro residues) but depends on the secondary structure. Helical residues lose more BB entropy than sheet residues, TΔS helix−sheet = 0.5 kcal·mol −1 at 300 K, a difference not fully reflected by either amide N-H or carbonyl C=O bond NMR order parameters. The SC entropy loss, TΔS SC ∼ 0.2 kcal·mol −1 ·rotamer −1 , is largely independent of 2°structure and weakly correlated with burial. Combinin...

Perplexing cooperative folding and stability of a low-sequence complexity, polyproline 2 protein lacking a hydrophobic core

“…The experimentally observed entropy loss upon folding often is near zero: ΔS exp ∼ 0 ∼ ΔS solvent + ΔS conf , implying that the gain in solvent entropy gain is largely counterbalanced by the loss of conformational entropy, ΔS solvent ∼ −ΔS conf (50,52). Indeed, our estimation of ΔS conf ∼ 1.4 kcal·mol −1 ·res −1 / (300 K) (36,37), is similar to the above estimate of hydrophobic stabilization. However, sfAFP buries only 40% of the hydrophobic surface area of a typical protein.…”

“…Explicit solvent molecular dynamics (MD) simulations were analyzed to calculate the conformational entropy of sfAFP in its native state for comparison with the model protein ubiquitin (Ub). Entropy is calculated according to S = -R Σ i P i lnP i , where P i is the population in 10°× 10°pixels in the Ramachandran map (36,37). We find that residues in sfAFP's PP2 helices and in Ub's β-sheets exhibit comparable backbone entropy (Fig.…”

“…Our prior studies (36,37) found that the backbone entropy of glycine in the unfolded state exceeds other residues by 0.3 kcal·mol −1 ·res −1 /(300 K). However, other residue types additionally lose side-chain entropy upon folding.…”

Free energies of solvation in the context of protein folding: Implications for implicit and explicit solvent models

“…As mentioned already, entropy has been one of the most difficult quantities to calculate directly from trajectories, and a consensus has not been reached on the best approach. To test whether the difference between and can be reduced by calculating with a different approach, we repeated our analysis using a relatively simple method described in a recently published paper, which we refer to as the additive joint entropy method. The entropies calculated with this method often differ by up to a factor of two from the ones calculated with the CC‐MLA method (Supporting Information Table S3), and generally led to better agreement between and (Supporting Information Fig.…”

“…CENCALC was used for the entropies calculated with the CC‐MLA method, while an in‐house implementation was used for the approach by Baxa et al, referred to here as the additive joint entropy method, with a bin size of 10°. These entropy calculations used the same number of frames as the energy calculations.…”