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Loss of Conformational Entropy in Protein Folding Calculated Using Realistic Ensembles and Its Implications for NMR-Based Calculations

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Loss of Conformational Entropy in Protein Folding Calculated Using Realistic Ensembles and Its Implications for NMR-Based Calculations Loss of conformational entropy in protein folding calculated using realistic ensembles and its implications for NMR-based calculations Michael C. Baxaa,b, Esmael J. Haddadianc, John M. Jumperd, Karl F. Freedd,e,1, and Tobin R. Sosnicka,b,e,1 aDepartment of Biochemistry and Molecular Biology, bInstitute for Biophysical Dynamics, cBiological Sciences Collegiate Division, dJames Franck Institute and Department of Chemistry, and eComputation Institute, The University of Chicago, Chicago, IL 60637 Edited* by Robert L. Baldwin, Stanford University, Stanford, CA, and approved September 12, 2014 (received for review April 28, 2014) The loss of conformational entropy is a major contribution in the 2° structure and weakly correlated with burial. Combining this thermodynamics of protein folding. However, accurate determi- correlation with the average loss of BB entropy for each 2° structure nation of the quantity has proven challenging. We calculate this type provides a site-resolved estimate of the entropy loss for an loss using molecular dynamic simulations of both the native protein input structure (godzilla.uchicago.edu/cgi-bin/PLOPS/PLOPS.cgi). and a realistic denatured state ensemble. For ubiquitin, the total This estimate can assist in thermodynamic studies and coarse- Δ = · −1 change in entropy is T STotal 1.4 kcal mol per residue at 300 K grained modeling of protein dynamics and design. with only 20% from the loss of side-chain entropy. Our analysis exhibits mixed agreement with prior studies because of the use of Results more accurate ensembles and contributions from correlated motions. The NSE and DSE of Ub (15) are generated from molecular Buried side chains lose only a factor of 1.4 in the number of confor- dynamics (MD) simulations using the CHARMM36 force field Ω Ω mations available per rotamer upon folding ( U/ N). The entropy loss (14, 16) and the TIP3P water model (17). The DSE is produced for helical and sheet residues differs due to the smaller motions of from short trajectories initiated from 315 members of a larger Δ = · −1 helical residues (T Shelix−sheet 0.5 kcal mol ), a property not fully ensemble constructed using BB (ϕ,ψ) dihedral angles found in reflected in the amide N-H and carbonyl C=O bond NMR order param- a PDB-based coil library (18). The DSE’s NMR chemical shifts eters. The results have implications for the thermodynamics of folding (19, 20), N-H residual dipolar couplings (RDCs) (18) (Fig. S1), and binding, including estimates of solvent ordering and microscopic and radius of gyration, Rg (14, 18, 21), agree with data for entropies obtained from NMR. chemically denatured Ub. The average helicity for residues lo- cated in the helix and the rest of the residues is 2% in water at NMR order parameters | molecular dynamics | helix propensity | 293 K (22). Hydrogen exchange studies are also consistent with sheet propensity | denatured state all of the H-bonds being broken upon global unfolding (23), further supporting the use of the statistical coil model to de- n accurate determination of the loss of conformational en- scribe the DSE. Atropy is critical for dissecting the energetics of reactions To maintain agreement with experiment, the (ϕ,ψ) angles are involving protein motions, including folding, conformational constrained in the simulations to remain in their Rama basins via change, and binding (1–6). Given the difficulty of directly mea- reflecting walls that separate the five major basins (Fig. S2). suring the conformational entropy, most early estimates relied Without these constraints, the MD would undergo interbasin on computational approaches (2, 7–10), although, more recently, transitions, altering the angles to produce an ensemble that no NMR methods have been used to measure site-resolved entropies longer agrees with the experimental RDCs. Additionally, the (11). The computational methods often calculated the entropy of either the native state ensemble (NSE) or the denatured state en- Significance semble (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 Despite 40 years of study, no consensus has been achieved on Protein Data Bank (PDB)]. Most previous approaches focused on the magnitude of the loss of backbone (BB) and side-chain (SC) helices and omitted contributions from vibrations and correlated entropies upon folding, even though these quantities are es- motions (12, 13), thereby partly accounting for the spectrum of sential for characterizing the energetics of folding and con- calculated values. formational change. We calculate the loss using experimentally We address these issues by calculating the chain’s conforma- validated denatured and native state ensembles, avoiding the tional entropy from the distributions of the backbone (BB) (ϕ,ψ) drastic assumptions used in many past analyses. By also ac- counting for correlated motions, we find that the loss of BB and SC rotametric angles, [χn], obtained from all-atom simu- lations of the NSE and DSE for mammalian ubiquitin (Ub). This entropy is three- to fourfold larger than the SC contribution. study extends our previous calculation of the loss of BB entropy Our values differ with some calculations by up to a factor of 3 and that used an experimentally validated DSE (14). The calculated depend strongly on 2° structure. These results have implications angular distributions reflect both the Ramachandran (Rama) upon other thermodynamic properties, the estimation of entropy basin populations and the torsional vibrations. Correlated using NMR methods, and coarse-grained simulations. motions are accounted for through the use of joint probability distributions [e.g., P(ϕ,ψ,χ1,χ2)]. Author contributions: M.C.B., K.F.F., and T.R.S. designed research; M.C.B. and E.J.H. per- The computed loss of BB entropy is 80% of the total entropy formed research; M.C.B., J.M.J., and T.R.S. analyzed data; and M.C.B., J.M.J., K.F.F., and loss at 300 K. The BB entropy is independent of burial and T.R.S. wrote the paper. residue type (excluding Pro, Gly, and pre-Pro residues) but The authors declare no conflict of interest. depends on the secondary structure. Helical residues lose more *This Direct Submission article had a prearranged editor. −1 BB entropy than sheet residues, TΔShelix−sheet = 0.5 kcal·mol at 1To whom correspondence may be addressed. Email: [email protected] or trsosnic@ 300 K, a difference not fully reflected by either amide N-H or uchicago.edu. carbonyl C=O bond NMR order parameters. The SC entropy This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. −1 −1 loss, TΔSSC ∼ 0.2 kcal·mol ·rotamer , is largely independent of 1073/pnas.1407768111/-/DCSupplemental. 15396–15401 | PNAS | October 28, 2014 | vol. 111 | no. 43 www.pnas.org/cgi/doi/10.1073/pnas.1407768111 Downloaded by guest on September 28, 2021 Fig. 1. Loss of conformational entropy upon folding for Ub. (A and B) Average value of TΔS is shown (line) [native 2° structure at top: strand (blue), helix (red), and coil (green)]. (C and D) Entropy and 2° structure. Data are calculated using 10° bins. Entropy differences (ΔS) in A–C are independent of bin size and can be compared with values in the literature. Absolute entropy in D should only be compared with other calculations using 10° bins. Points are spread out horizontally for ease of viewing. removal of constraints would produce chain collapse [e.g., difference emerges because of a higher entropy reduction in the unconstrained exp ϕ ψ <Rg>MD < 15 Å (24, 25), whereas Rg ∼ 25 Å (21)]. NSE resulting from a tighter ( , ) distribution for residues in Entropies are calculated as before using S = R ΣPilnPi, where helices and, to a lesser extent, in turns, which are considered to Pi is the probability of being in microstate i (9, 12–14, 26–30). be coil residues (Fig. 2A). Because helical residues, regardless of Although entropy is defined in terms of the probability dis- amino acid type, sample a very similar (ϕ,ψ) distribution in the tributions over all atomic positions, it can be decomposed into NSE, variations in the BB entropy loss upon helix formation the protein entropy and the average solvent entropy conditional arise from residue-dependent differences in the DSE (Fig. 1D on the protein’s microstates. Conversion to (ϕ,ψ,[χn]) angular and Fig. S4). However, as discussed below, the net entropy loss coordinates from (x,y,z) coordinates normally requires a Jacobian. does not correlate well with helical propensity. However, the Jacobian is constant across phase space (assuming The DSE entropies of the β-branched residues Val and Ile are of other bond angles and lengths are constant); hence, it is not lower entropy than the average of other non-Gly, pre-Pro, and Pro −1 calculated. Similarly, the size of a microstate in phase space is residues by TΔS = 0.3 kcal·mol (Fig. S4C). The spread for the affected by bin size, but this effect cancels for entropy differences other residues is due to sequence dependence and closely matches so that our values can be compared with other values of ΔS in the the spread calculated from Dunbrack’s coil library (31) (Fig. S4D). literature. The restriction to BB and SC angles ignores other degrees of freedom, such as bond length vibrations, that are as- NMR Order Parameters and Entropy. Lipari–Szabo NMR order BIOPHYSICS AND sumed to cancel in entropy differences. parameters (SLZ) have been used to estimate BB and SC en- COMPUTATIONAL BIOLOGY To account for correlated motions, entropies are calculated tropies in folding and binding (3, 11–13, 26, 32–37).
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