Loss of conformational entropy in protein foldingcalculated using realistic ensembles and itsimplications for NMR-based calculationsMichael 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 Instituteand 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 thethermodynamics of protein folding. However, accurate determi-nation of the quantity has proven challenging. We calculate thisloss using molecular dynamic simulations of both the native proteinand a realistic denatured state ensemble. For ubiquitin, the totalchange in entropy is TΔSTotal = 1.4 kcal·mol−1 per residue at 300 Kwith only 20% from the loss of side-chain entropy. Our analysisexhibits mixed agreement with prior studies because of the use ofmore accurate ensembles and contributions from correlated motions.Buried side chains lose only a factor of 1.4 in the number of confor-mations available per rotamer upon folding (ΩU/ΩN). The entropy lossfor helical and sheet residues differs due to the smaller motions ofhelical residues (TΔShelix−sheet = 0.5 kcal·mol−1), a property not fullyreflected in the amide N-H and carbonyl C=O bond NMR order param-eters. The results have implications for the thermodynamics of foldingand binding, including estimates of solvent ordering and microscopicentropies obtained from NMR.

NMR order parameters | molecular dynamics | helix propensity |sheet propensity | denatured state

An accurate determination of the loss of conformational en-tropy is critical for dissecting the energetics of reactions

involving protein motions, including folding, conformationalchange, and binding (1–6). Given the difficulty of directly mea-suring the conformational entropy, most early estimates reliedon 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 ofeither the native state ensemble (NSE) or the denatured state en-semble (DSE) and invoked assumptions about the entropy ofthe other ensemble [e.g., assuming the NSE is a single state or thatthe DSE is a composite of all side-chain (SC) rotameric states in theProtein Data Bank (PDB)]. Most previous approaches focused onhelices and omitted contributions from vibrations and correlatedmotions (12, 13), thereby partly accounting for the spectrum ofcalculated values.We address these issues by calculating the chain’s conforma-

tional entropy from the distributions of the backbone (BB) (ϕ,ψ)and SC rotametric angles, [χn], obtained from all-atom simu-lations of the NSE and DSE for mammalian ubiquitin (Ub). Thisstudy extends our previous calculation of the loss of BB entropythat used an experimentally validated DSE (14). The calculatedangular distributions reflect both the Ramachandran (Rama)basin populations and the torsional vibrations. Correlatedmotions are accounted for through the use of joint probabilitydistributions [e.g., P(ϕ,ψ ,χ1,χ2)].The computed loss of BB entropy is 80% of the total entropy

loss at 300 K. The BB entropy is independent of burial andresidue type (excluding Pro, Gly, and pre-Pro residues) butdepends on the secondary structure. Helical residues lose moreBB entropy than sheet residues, TΔShelix−sheet = 0.5 kcal·mol−1 at300 K, a difference not fully reflected by either amide N-H orcarbonyl C=O bond NMR order parameters. The SC entropyloss, TΔSSC ∼ 0.2 kcal·mol−1·rotamer−1, is largely independent of

2° structure and weakly correlated with burial. Combining thiscorrelation with the average loss of BB entropy for each 2° structuretype provides a site-resolved estimate of the entropy loss for aninput structure (godzilla.uchicago.edu/cgi-bin/PLOPS/PLOPS.cgi).This estimate can assist in thermodynamic studies and coarse-grained modeling of protein dynamics and design.

ResultsThe NSE and DSE of Ub (15) are generated from moleculardynamics (MD) simulations using the CHARMM36 force field(14, 16) and the TIP3P water model (17). The DSE is producedfrom short trajectories initiated from 315 members of a largerensemble constructed using BB (ϕ,ψ) dihedral angles found ina PDB-based coil library (18). The DSE’s NMR chemical shifts(19, 20), N-H residual dipolar couplings (RDCs) (18) (Fig. S1),and radius of gyration, Rg (14, 18, 21), agree with data forchemically denatured Ub. The average helicity for residues lo-cated in the helix and the rest of the residues is 2% in water at293 K (22). Hydrogen exchange studies are also consistent withall of the H-bonds being broken upon global unfolding (23),further supporting the use of the statistical coil model to de-scribe the DSE.To maintain agreement with experiment, the (ϕ,ψ) angles are

constrained in the simulations to remain in their Rama basins viareflecting walls that separate the five major basins (Fig. S2).Without these constraints, the MD would undergo interbasintransitions, altering the angles to produce an ensemble that nolonger agrees with the experimental RDCs. Additionally, the

Significance

Despite 40 years of study, no consensus has been achieved onthe magnitude of the loss of backbone (BB) and side-chain (SC)entropies upon folding, even though these quantities are es-sential for characterizing the energetics of folding and con-formational change. We calculate the loss using experimentallyvalidated denatured and native state ensembles, avoiding thedrastic assumptions used in many past analyses. By also ac-counting for correlated motions, we find that the loss of BBentropy is three- to fourfold larger than the SC contribution.Our values differ with some calculations by up to a factor of 3 anddepend strongly on 2° structure. These results have implicationsupon other thermodynamic properties, the estimation of entropyusing NMR methods, and coarse-grained simulations.

Author contributions: M.C.B., K.F.F., and T.R.S. designed research; M.C.B. and E.J.H. per-formed research; M.C.B., J.M.J., and T.R.S. analyzed data; and M.C.B., J.M.J., K.F.F., andT.R.S. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.1To whom correspondence may be addressed. Email: freed@uchicago.edu or trsosnic@uchicago.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1407768111/-/DCSupplemental.

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removal of constraints would produce chain collapse [e.g.,<Rg>MD

unconstrained < 15 Å (24, 25), whereas Rgexp ∼ 25 Å (21)].

Entropies are calculated as before using S = R ΣPilnPi, wherePi is the probability of being in microstate i (9, 12–14, 26–30).Although entropy is defined in terms of the probability dis-tributions over all atomic positions, it can be decomposed intothe protein entropy and the average solvent entropy conditionalon the protein’s microstates. Conversion to (ϕ,ψ ,[χn]) angularcoordinates from (x,y,z) coordinates normally requires a Jacobian.However, the Jacobian is constant across phase space (assumingother bond angles and lengths are constant); hence, it is notcalculated. Similarly, the size of a microstate in phase space isaffected by bin size, but this effect cancels for entropy differencesso that our values can be compared with other values of ΔS in theliterature. The restriction to BB and SC angles ignores otherdegrees of freedom, such as bond length vibrations, that are as-sumed to cancel in entropy differences.To account for correlated motions, entropies are calculated

using joint probability distributions for up to four angles at atime, where the effects of limited sampling statistics are, on av-erage, TΔSsampling ∼ 0.1 kcal·mol−1 (Fig. S3A). For the loss ofchain entropy upon folding per residue, TΔSTotal equals 1.380.46kcal·mol−1 (the subscript represents the SD across the sequencerather than the uncertainty). Rigorously, the total entropy can-not be separated into BB and SC values due to correlations.Nevertheless, such a separation is useful even if not fully valid.Accordingly, we compute the individual BB entropy and then thecorrected SC entropy from the difference between the total and BBentropies. The average reduction of 0.14 kcal·mol−1 due to corre-lated BB/SC motions is assigned to the SC entropy. The BB entropyloss, 1.10.3 kcal·mol−1, is fourfold larger than the SC loss, TΔSSC =0.240.30 kcal·mol−1 (0.150.16 kcal·mol−1·rotamer−1). If the mutualentropy between the BB and SC is split evenly between the two, theBB term still is threefold larger than the SC term.The errors due to finite sampling and bin size are estimated

from four tests (Fig. S3). First, the NSE and DSE ensembles aresplit into halves, quarters, eighths, and sixteenths. The de-pendence of the total entropy on ensemble size indicates that ourcalculation has converged to within ∼2%. In addition, the resi-due-level entropies are very similar whether calculated usingeither a 1/2:1/2 or a 3/4:1/4 decomposition of the total ensemble,or with 10° or 15° bins (using the entire ensemble). These fourcalculations indicate a sampling uncertainty of TΔS ∼ 0.1kcal·mol−1·residue−1 (SI Text).

Secondary Structure and Amino Acid Dependence. The loss of SCentropy is largely independent of 2° structure (Fig. 1B). In con-trast, the change in BB entropy varies for helices, sheets, and coilresidues, being 1.5, 1.0, and 1.2 kcal·mol−1, respectively. This

difference emerges because of a higher entropy reduction in theNSE resulting from a tighter (ϕ,ψ) distribution for residues inhelices and, to a lesser extent, in turns, which are considered tobe coil residues (Fig. 2A). Because helical residues, regardless ofamino acid type, sample a very similar (ϕ,ψ) distribution in theNSE, variations in the BB entropy loss upon helix formationarise from residue-dependent differences in the DSE (Fig. 1Dand Fig. S4). However, as discussed below, the net entropy lossdoes not correlate well with helical propensity.The DSE entropies of the β-branched residues Val and Ile are of

lower entropy than the average of other non-Gly, pre-Pro, and Proresidues by TΔS = 0.3 kcal·mol−1 (Fig. S4C). The spread for theother residues is due to sequence dependence and closely matchesthe spread calculated from Dunbrack’s coil library (31) (Fig. S4D).

NMR Order Parameters and Entropy. Lipari–Szabo NMR orderparameters (SLZ) have been used to estimate BB and SC en-tropies in folding and binding (3, 11–13, 26, 32–37). The orderparameter reflects the angular distribution of a bond vector [e.g.,the peptide N-H) relative to the molecular (reference) frame for

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 andcan be compared with values in the literature. Absolute entropy in D should only be compared with other calculations using 10° bins. Points are spread outhorizontally for ease of viewing.

Fig. 2. Rama distributions and order parameters in the NSE vary with 2°structure. (A) Native distribution for I30 (helix) is tighter than for I3 (sheet) andhas 0.44 kcal·mol−1 less entropy. (B) Probability distributions for the N-H, C=O,and peptide plane vectors for I3 and I30. The motions of their N-H vectors aremore similar to each other compared with the motions of the other two pairs.(C) Entropy calculated using S2LZ with the diffusion-in-a-cone model under-estimates BB entropy of sheet residues in the NSE. Points are colored accordingto 2° structure as in Fig. 1. (D) Entropy differences calculated from the probabilitydistributions of the vectors in B are less than those entropy differences calculatedfrom the BB, indicating that none of the three vectors fully reflects the BB en-tropy. ILE, isoleucine (Ile).

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subnanosecond motions (32, 33)]. Our NSE BB entropy is similarto the entropy calculated using the N-H order parameter witha diffusion-in-a-cone model (32), albeit with a notable discrepancy(Fig. 2 and Fig. S5). Generally, S2LZ is slightly higher for helical thanfor sheet residues (S2LZ,helix ∼ 0.90–0.92 and S2LZ,sheet ∼ 0.8–0.91),but their BB entropy often differs significantly (i.e., TΔShelix−sheetBB =0.52 kcal·mol−1, even for sheet residues having high order param-eters, S2LZ ≥ 0.89). As a result, no single relationship between theN-H order parameter and entropy can simultaneously describeboth helical and sheet residues (Fig. 2C).We also calculated TΔShelix−sheet directly from the motions of

the N-H, C=O, and peptide plane vectors for I3 (sheet) and I30(helix) (Fig. 2B). For these three vectors, TΔShelix−sheetvector = 0.18,0.29, and 0.35 kcal·mol−1, respectively, whereas our Rama-basedcalculation yields 0.44 kcal·mol−1. Because the calculations arebased on the same data, they should agree if they report on thesame quantity (38). Given that they produce disparate values, weconclude that the three vectors are reporting on differentmotions (38) and are not true proxies of the total BB entropy.Nevertheless, a single model relating each vector’s SLZ

2 value toits own entropy, rather than the BB entropy, successfullydescribes both helical and sheet residues (32) (Fig. S5).

Correlated Motions. The values listed above are adjusted for corre-lated motions, a moderate effect on average (−0.120.26 kcal·mol−1)(Fig. S6). The contributions from correlations between the BB andSC angles are deduced from STotal − (SBB + SSC), the differencebetween the total entropy and the sum of the BB and SC entropies,independently evaluated from the distributions Pi(ϕ,ψ) and Pi([χn]),respectively. These correlations differ in the NSE and DSE, −0.05and −0.19 kcal·mol−1·residue−1, respectively (Fig. S6A). Thedifference arises because the BB samples multiple Rama basins inthe DSE, and the BB angles are correlated with χ1. In contrast,the BB angles typically remain in a single basin in the NSE, so theeffect of these correlations is absent. Hence, this correlationproduces a measurable reduction for the difference between theNSE and DSE, TΔSSC correction = −0.14 kcal·mol−1.The contribution from correlated BB rotations of adjacent

residues is evaluated from the difference of the joint and in-dividual entropies, SBB(ϕi,ψ i,ϕi+1,ψ i+1) − [(SBB(ϕi,ψ i) + SBB(ϕi+1,ψ i+1)] (Fig. S6A). This effect varies along the sequence (e.g.,TΔSNSEBB corr =−0:06 and −0.18 kcal·mol−1 for helices and sheets,respectively). However, the cumulative corrections are similar inthe NSE and the DSE, −0.14 and −0.13 kcal·mol−1, respectively.Consequently, their difference is small, hTΔSDSE−NSE

BB corr i= 0:02kcal ·mol�1. This near-cancellation primarily is due to the dif-ferent signs of the corrections for helical and sheet residues;hTΔSDSE−NSE

BB corr i=−0:08 and 0.06 kcal·mol−1, respectively. Helicalresidues have smaller corrections in the NSE compared with theDSE, whereas the opposite is true for sheet residues. The averagecontribution from correlated BB rotations for the 442 contactingpairs of nonsequence adjacent residues, defined by those residuesmaking at least one heavy atom contact during the entire NSEsimulations, is only −0.06 kcal·mol−1·residue−1 (Fig. S6C). The threelargest corrections, 0.30–0.34 kcal−1·mol−1, are in the N-terminal βturn (i,i + 2,3 contacts).The correlations between contacting SCs are determined using

the joint χ-angle distributions involving six or fewer angles ormore than six angles with bin sizes of 20° or 30°, respectively (Fig.S6B). We use an average of at least 3, 5, or 10 heavy-atom SCcontacts to define a pairwise interaction in the NSE trajectories(152, 113, and 41 pairs, respectively). After subtracting the corre-sponding correction for each pair in the DSE simulations, the re-duction in TΔS is −0.02 to −0.06 kcal·mol−1. This analysis indicatesthat motions of contacting SCs are nearly independent of eachother, in agreement with Hu and Kuhlman (6) and Li et al. (13).Our procedure of using joint probability distributions for up to

eight angles largely eliminates the need to calculate the higherorder corrections using a mutual information expansion or

maximum information spanning tree (39). Our calculation omitscorrelations across more than three pairs of BB dihedral angles,across more than three sets of SCs, or between BB and SC angleson different residues. The neglect of these terms should not af-fect our final values beyond the stated statistical uncertainties,especially considering that the correlations examined aboveyielded even smaller contributions (13).

Burial and Entropy. The loss of BB entropy is independent ofburial (r = −0.10), whereas the change in SC entropy is weaklycorrelated with the solvent accessible surface area (SASA) thatbecomes buried upon folding, <ΔSASADSE−NSE>/<SASADSE>(r = 0.47; Fig. S7). Non-Gly, Ala, and Pro residues that buryeither less or more than 50% of their SASA upon folding have anaverage TΔS per rotamer of 0.080.13 or 0.200.16 kcal·mol−1, re-spectively. The correlation slightly improves when the loss iscompared with the burial relative to the average for each residuetype (r = 0.52; Fig. 3A).Considerable site-to-site variation in the loss of SC entropy

exists for exposed and buried residues (Fig. 4). The surfaceresidues K33 and K48 (∼20% burial) have significantly differentSC entropy losses, 0.5 and −0.1 kcal·mol−1, respectively (Fig.4A). K33’s χ1 and χ2 angles are more restricted because of thethreefold greater number of contacts formed by SCs in the NSE,even though both residues have comparable exposure (135 vs.144 Å2) and similar DSE distributions (Fig. S8). Furthermore,some exposed Thr and Ser are restricted in the NSE (TΔSSC =0.30.2 kcal·mol−1). The S20 SC is only 10% buried in the NSE, yetits single rotamer loses 0.2 kcal·mol−1 of entropy on folding.Conversely, burial does not always imply the same reduction in

SC entropy even for the same residue type. Four core Iles with over90% burial have TΔSDSE−NSE

SC = 0.4–0.7 kcal·mol−1 due to varia-tions in the native state (χ1,χ2) distributions (Fig. 4B and Fig. S8),consistent with other studies (40). Conformational heterogeneityappears in the SC of a highly buried Arg while still maintainingordered contacts (41). These and other examples illustrate thatupon folding, surface and core residues lose an amount of entropythat depends on context. Hence, the common assumption that SCsadopt a single state when buried and an unfolded-like number ofstates when exposed to solvent is inaccurate in the aggregate.

Estimating Entropy from Structure. Our results suggest thatknowledge of the structure is sufficient to estimate the site-resolved BB and SC entropies. The BB entropy is estimatedusing the average values for α, β, and coil residues, corrected forthe dependence of the DSE entropy on residue type. The SCentropy’s sensitivity to burial is estimated using a linear corre-lation with the relative burial (ΔS·rot−1 = 0.4·ΔBurialrel; Fig. 3A).The sum provides a reasonable estimate of the total entropy for

Fig. 3. Estimating the entropy lost using burial levels and 2° structure. (A)Loss of SC entropy is mildly correlated with the relative burial level for eachamino acid (AA) type. (B) Total entropy is estimated from this correlation plusthe 2° structure dependence of the BB entropy, corrected for differences inthe DSE, and is compared with the values calculated from the simulations. Theunity line is provided as a visual guide.

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each Ub residue (Fig. 3B). We expect the values found for Ub,a typical globular protein, apply to other proteins when their DSEsare similarly unstructured.

DiscussionA method for determining the loss of entropy is essential fordissecting the energetics of folding, binding, and conformationalchange (1–3). Our method resembles a variety of previousstudies that use angular distributions to calculate entropies (9,12–14, 26–30), although we include features that have not beenpreviously considered or used in combination. MD simulationsare used to create realistic ensembles for both the NSE and DSE.We account for intrawell and interwell motions, obviating the needto invoke assumptions about the entropy in either state or the shapeof the wells. Our DSE includes Rama basin restraints so that theensemble has the proper Rg, chemical shifts, and distributions ofdihedral angles that are necessary to reproduce experimental RDCs(14, 18, 21). Without these constraints, the chain would unphysicallycollapse. We also evaluated contributions arising from correlatedmotions, TΔScorr ∼ 0.3 kcal·mol−1. An investigation of the depen-dence on burial and 2° structure finds that only the latter is signif-icant, TΔShelix−sheetBB = 0.5 kcal·mol−1.

Comparison with Other Studies. Our values for TΔSTotal, TΔSBB,TΔSSC = 1.4, 1.0–1.1, and 0.2–0.3 kcal·mol−1, respectively,overlap with some previous estimates but are smaller than others,particularly for the SC entropy (30). Our small value for TΔSSC isconsistent with the finding of Hu and Kuhlman (6) that includingSC entropy has minimal effect in amino acid preference forprotein design. Many previous methods similarly calculate SCentropy losses upon binding and folding by counting rotamericstates, sometimes using energy-weighted (9, 42) or Monte Carloapproaches (43–45), which are likely to have very good sampling.Doig and Sternberg (7) survey several works and report an av-erage loss of 0.5 kcal·mol−1·rotamer−1. Many approaches modelthe DSE’s rotamer distribution (e.g., using the diversity observedin the PDB), while assuming the NSE has one conformation (9).Using Monte Carlo sampling, DuBay and Geissler (8) find

TΔS = 0.9 kcal·mol−1 for the loss of SC entropy, although theynote that their reference state overestimates the freedom in theDSE. Their computed entropy loss of 0.14 kcal·mol−1·rot−1 forbinding is very similar to our SC value for folding. Other treat-ments assume single, sometimes quasiharmonic, energy wells (2,46), unlike our DSE’s multiwell energy surface.A variety of methods, typically applied to helices, produce

TΔSBB = 0.9–2.2 kcal·mol−1, with a median of 1.4 kcal·mol−1

(30), which is close to our helical value but greater than our sheetvalues, 1.5 and 1.0 kcal·mol−1, respectively. Calculations by Lee

et al. (9) and D’Aquino et al. (29) are in mixed agreement withour data. They use calorimetric data to estimate ΔSAla−Gly, whichis combined with a computation approach similar to ours to es-timate the DSE entropy. Their entropy loss is also amino acid-and burial-dependent. However, their Rama distributions in theDSE are broader than ours, and they assume one conformationfor buried SCs. Applying their analysis to Ub’s α-helix producesan average TΔSBB = 0.9 kcal·mol−1, which is lower than ourvalue of 1.5 kcal·mol−1. The corresponding SC calculationproduces TΔSSC = 0.7–1.0 kcal·mol−1 (using a 50–80% thresh-old for classifying a residue as buried), which is much higher thanour value of 0.3 kcal·mol−1. For the entire protein, TΔSSC = 0.5–0.7kcal·mol−1, which is again higher than our average value of 0.2kcal·mol−1. Nevertheless, the two methods produce comparabletotal losses of entropy for Ub’s helix (∼1.8 kcal·mol−1), but therelative contributions from the BB and SC are very different.Our BB entropy loss is fivefold larger than the SC entropy(1.5 vs. 0.3 kcal·mol−1), whereas theirs is nearly equal (0.9 vs.0.7–1.0 kcal·mol−1).Daggett and coworkers (28) also use MD to produce NSE and

DSE Rama maps from which they calculate the loss of BB en-tropy. Their values for the two states are smaller than ours byTΔS ∼ 0.6 kcal·mol−1. Their DSE maps (from 498 K simulations)are heavily dominated by helical conformations for Ala and, toa lesser extent, for Gly, whereas our maps are dominated byextended conformers, a necessary feature for matching experi-mental RDCs (18). Nevertheless, their calculations and ourcalculations produce similar values for the difference in the lossof BB entropy upon helix formation between an Ala and a Gly,ΔðTΔSDSE−NSE

BB ÞAla→Glyjhelix, 0.1 and 0.3 kcal·mol−1, respectively.Plaxco and coworkers (30) estimate the loss of BB entropy

from the work required to stretch an unfolded polypeptide to theforce at which the next attached protein unfolds (200–250 pN).This free energy, 1.4 kcal·mol−1, is equated to the loss of BBentropy under the implicit assumption that the native state hasone conformation or, more precisely, that the residual entropy ofthe extended chain is the same as the native state entropy. Al-though there is no a priori reason for this equivalence, a recentcomputational study provides support because the area sampledin the Rama map of an unfolded chain under 250 pN of force(47) is similar to the area sampled by helical residues in our NSEsimulations. The similarity explains why their approach producesa similar value for helical residues (but not for sheets).Coarse-grained simulations have been used to predict folding

pathways and conformational changes. These models generallyfall into two classes. In the first class, real-space dynamics areconducted and the conformational entropy is determined byΔS = Energy/Tmidpoint. For one Cα bead model, the entropy is1.6 kcal·mol−1, as calibrated using a variety of proteins (48). Thesecond class involves G�o-style Ising models that lack any explicitBB or SC degrees of freedom. As a result, an entropy term isadded for each unfolded residue and should correspond to ourtotal loss of entropy. Alm and Baker (49) and Muñoz and Eaton(50) use values of 1.8 and 1.0 kcal·mol−1, respectively. Althoughthese values are similar to our average, the models omit all de-pendence on 2° structure, amino acid type, or SC burial. This lackof dependence on 2° structure favors helix formation because it failsto account for the additional 0.5 kcal·mol−1 entropy loss for helixformation. This bias can influence the prediction of folding path-ways (e.g., it may explain the frequent misidentification of helix inthe transition state of protein L by coarse-grained models) (51). Ourentropies and their dependence on structure, residue type, andburial may be used to improve both classes of coarse-grainedmodels, as well as to assist in protein design.

NMR Order Parameters. Order parameters have been used to es-timate microscopic BB and SC entropies (3, 11–13, 26, 32–37).Yang and Kay (32) derived a popular equation relating S2LZ toentropy from a diffusion-in-a-cone model, whereas Palmer andcoworkers (52) showed that estimates of changes in stability are

Fig. 4. Entropy loss can vary even at the same burial level in the NSE. (A)Although Lys K33 and K48, are equally buried (21–22%), they have differentNSE (χ1,χ2) distributions and entropies, TΔSSC = 0.5 and −0.1 kcal·mol−1, re-spectively. (B) Four Iles are highly buried in the NSE (>90%), but I23 loses >0.3kcal·mol−1 more TΔSSC than the others due to its tighter (χ1,χ2) distribution.

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not strongly model-dependent for values of S2LZ observed inproteins. Although order parameters are ill-defined in the DSEdue to the lack of a global reference frame, some estimates of thechange in BB entropy using N-H order parameters with a diffu-sion-in-a-cone model are close to our values, 0.8–1.6 kcal·mol−1

(32–34, 53).However, our calculation exhibits a notable discrepancy, with

S2LZ-derived values for the difference in entropy loss for helicaland sheet residues even for those positions having similarly highN-H order parameters. For example, the difference in the BBentropy between an Ile in a helix and in a sheet is TΔShelix−sheet =0.44 kcal·mol−1, compared with 0.18, 0.29, and 0.35 kcal·mol−1

for calculations based on the N-H, C=O, and peptide planevector motions, respectively (Fig. 2 B and D). The three vectorsare reporting on different motions, consistent with a study byWang et al. (38), and none fully reflects the BB entropy (Fig. 2Cand Fig. S5). This underreporting arises due to their being in-complete proxies of all (ϕ,ψ) combinations. The dominant BBmotion on the subnanosecond time scale is the rocking of thepeptide plane. This motion involves a counterrotation of the twoflanking dihedral angles, Δψ i−1 ∼ −Δϕi (13, 54), with the cor-relation being stronger for sheet residues because they undergolarger angular changes. Consequently, the three vectors do notsample the entire 2 degrees of (ϕ,ψ) angular freedom that definethe BB entropy (37), particularly for sheet residues.Consequently, no single relationship between S2LZ and the BB

entropy simultaneously describes both helices and sheet residueswith an accuracy better than 0.2–0.5 kcal·mol−1. This deficiencyis particularly true at S2LZ ≥ 0.9, where the entropy-order pa-rameter relationship is most sensitive to slight changes in S2LZ.Nevertheless, the S2LZ-entropy relationship is quite accurate forany one of the three vectors’ order parameter and its own en-tropy (32) (Fig. S5). This difference supports the application ofthe diffusion-in-a-cone model for a given vector motion and itsentropy but emphasizes that none of the three bond vectors’motions fully reflects the BB’s ϕ- and ψ-motions.SC order parameters have been used to estimate changes in

entropy upon binding (11). Wand and coworkers (55) finda strong correlation between the change in the SC methyl sym-metry axis order parameter and the loss in calmodulin’s con-formational entropy upon binding a series of target peptides.This relationship is used to calibrate an “entropy meter.” Later,Kasinath et al. (3) analyzed MD simulations and found thatmotions of methyl-bearing SCs are excellent proxies of the entireSC entropy once the individual values are weighted according tothe number of rotameric states. These and other calculationsstress the importance of making an in-depth comparison be-tween NMR order parameters and microscopic entropies.

Implications for Hydration Entropies. Estimates for the change inconformational entropy are used to estimate other thermody-namic properties for folding and binding reactions (1–3). Typi-cally, ΔS nearly vanishes for the entire reaction (e.g., near thetemperature of maximum protein stability), where the loss ofconformational entropy is balanced by the gain in solvent entropytypically associated with the burial of apolar and polar groups (1).Our revised value for the loss of conformational entropy canprovide an estimate of the entropy of hydration, ΔShydr = ΔSexp −ΔSconf. As a result, our study may help resolve inconsistencies inprevious estimates of hydration entropies using transfer studies ofmodel compounds that may be attributable to the use of alter-native reference states [e.g,. assumptions that the gas phase isa more appropriate reference state than the solvent phase (56)].

Secondary Structure Propensities. Residues in Ub’s helices, sheets,and coil regions lose, on average, 1.5, 1.0, and 1.2 kcal·mol−1 inBB entropy, but the average SC entropy loss is largely in-dependent of the 2° structure. Earlier studies find a strong cor-relation between helical propensity and the loss of SC entropy onfolding for apolar residues (10, 44). All 14 residues in Ub’s he-lices, or just the four apolar residues, display, at most, a mild

correlation between the Chou–Fasman helical propensities andeither the total, BB, or SC entropy, with the exception of the BBentropy for the four apolar residues (r = −0.91). Context could beresponsible for the difference between the present and prior stud-ies. The earlier results examine monomeric helices, whereas our Ubvalues are computed for a partially buried amphipathic helix.Our previous study examining (ϕ,ψ) preferences in a coil li-

brary found that both helical and sheet propensities correlatewell with a single factor, the probability that a residue in a PDB-based coil library adopts a helical angle and sheet angle, re-spectively, as given by ΔΔGhelix

A→X =−RT lnðPhelixX =Phelix

A Þ (57). Theindividual propensities can be viewed as the work or free energyrequired to restrict a residue’s BB to a specific subset ofϕ,ψ-angles in the NSE (e.g., helical angles) from the multitude ofangles adopted by the residue when the protein is unfolded(using a coil library as a proxy for the DSE). The total workcontains both entropic and enthalpic components, which mayexplain the absence of a correlation when considering only theconformational entropy.

ConclusionsWe determine the loss of protein conformation entropy uponfolding using realistic NSEs and DSEs and include contributionsfrom correlated motions. This protocol provides a nearly com-plete treatment for a thermodynamic parameter whose magni-tude has been debated for decades. The total loss of confor-mational entropy, 1.4 kcal·mol−1·residue−1, is largely due to theloss of BB entropy. The total conformational entropy lossequates to a factor of ∼5–13 reduction in the number of con-formations available per residue, with buried SCs losing onlya factor of 1.4 per rotamer. Helical residues lose a factor of 2.3more conformations than sheet residues due to more restrictedmotions in the native state. This difference should be accountedfor in coarse-grained simulations and NMR analyses of micro-scopic entropy changes upon folding and binding. These data areincorporated into a server that provides estimates of the entropyloss for an input native structure (godzilla.uchicago.edu/cgi-bin/PLOPS/PLOPS.cgi).

MethodsEntropy Calculations. The entropy is calculated from Pi(ϕ,ψ ,χ1,χ2) for residueswith up to two SC rotamers. The BB entropy is evaluated from Pi(ϕ,ψ ). The SCentropy is the difference between the total entropy and the BB entropy. Thetotal distributions for residues with more than three rotamers (K, R, E, Q) aredecomposed into two parts, Pi(ϕ,ψ ,χ1) and Pi([χn]). The total entropy is thesum of the entropies calculated from these two distributions minusthe entropy associated with Pi(χ1) to avoid double-counting. A furtherdescription of our methods is detailed in SI Text.

Creation of the Ensembles. The DSE is generated starting from 315 repre-sentative conformations of aDSE constructed using a PDB-based coil library ofnon–H-bonded residues in the PDB (18). Angle selection from the libraryaccounts for each residue’s chemical identity, as well as for those chemicalidentities of the neighboring residues and their conformation. Each residueis constrained to remain within its original Rama basin using a harmonicreflecting “wall” at the edge of the basin (14). This calculation decomposesthe total probability distribution into two components: the interbasin dis-tribution (established by the relative basin propensities in the coil library)and the distribution for intrabasin motions obtained with all-atom MDsimulations using NAMD (58). After reaching the target temperature, the BBrestraints are gradually released. Following an additional 1 ns of equilibra-tion with no BB restraints, a 2-ns trajectory is carried out while maintainingthe Rama basin restraints, resulting in ∼6 × 105 structures in the DSE.

The corresponding angular distributions in the NSE are generated from 10independent 100-ns trajectories at 300 K, starting from the energy-minimizedcrystal structure (59) and using the same equilibration protocol with BBrestraints described above. Structures after the first 10 ns of simulation aresaved every 1 ps (providing a total of 9 × 105 structures). MD simulations inboth states are run with the CHARMM36 (16) force field using the TIP3Pwater model (17).

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ACKNOWLEDGMENTS. We thank B. Roux, J. C. Gumbart, A. Palmer III,J. Wand, K. Sharp, R. L. Baldwin, and members of our group for commentsand helpful discussions. NAMD was developed by the Theoretical and Compu-tational Biophysics Group at the University of Illinois at Urbana–Champaign. Thiswork was completed using resources provided by The University of Chicago

Research Computing Center and National Institutes of Health (NIH) throughresources provided by the Computation Institute, the Bioscience Division ofThe University of Chicago, and the Argonne National Laboratory under GrantS10 RR029030 and by grants from the NIH (Grant GM055694) and NationalScience Foundation (Grants CHE-1111918 and CHE-1363012).

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