Calculation of Proteins Total Side-Chain Torsional ...geisslergroup.org/wp-content/uploads/2019/06/Pub2009-3.pdf · organic crystals.1 This observation motivated in early studies

doi:10.1016/j.jmb.2009.05.068 J. Mol. Biol. (2009) 391, 484–497

Available online at www.sciencedirect.com

Calculation of Proteins’ Total Side-Chain TorsionalEntropy and Its Influence on Protein–Ligand Interactions

Kateri H. DuBay and Phillip L. Geissler⁎

Department of Chemistry,University of California atBerkeley, Berkeley, CA 94720,USA

Chemical Sciences Division,Lawrence Berkeley National Lab,Berkeley, CA 94720, USA

Physical Biosciences Division,Lawrence Berkeley National Lab,Berkeley, CA 94720, USA

Received 12 January 2009;received in revised form20 May 2009;accepted 22 May 2009Available online28 May 2009

*Corresponding author. E-mail [email protected] used: MC, Monte C

dynamics; LJ, Lennard–Jones; SB, sahydrogen bond; IS, implicit solvent;accessible surface area; CaM, calmophotoactive yellow protein; WL, Wa

0022-2836/$ - see front matter © 2009 E

Despite the high density within a typical protein fold, the ensemble ofsterically permissible side-chain repackings is vast. Here, we examine theextent of this variability that survives energetic biases due to van der Waalsinteractions, hydrogen bonding, salt bridges, and solvation. Monte Carlosimulations of an atomistic model exhibit thermal fluctuations among adiverse set of side-chain arrangements, even with the peptide backbonefixed in its crystallographic conformation. We have quantified the torsionalentropy of this native-state ensemble, relative to that of a noninteractingreference system, for 12 small proteins. The reduction in entropy perrotatable bond due to each kind of interaction is remarkably consistentacross this set of molecules. To assess the biophysical importance ofthese fluctuations, we have estimated side-chain entropy contributionsto the binding affinity of several peptide ligands with calmodulin.Calculations for our fixed-backbone model correlate very well withexperimentally determined binding entropies over a range spanning morethan 80 kJ/(mol·308 K).

© 2009 Elsevier Ltd. All rights reserved.

Keywords: side-chain entropy; configurational entropy; side-chain fluctuations;protein–ligand binding; protein thermodynamics
Edited by M. Levitt
Introduction

Native protein conformations are extremely dense,with packing fractions comparable to those oforganic crystals.1 This observation motivated inearly studies of protein structure and dynamics ajigsaw-puzzle notion, in which amino acid sidechains of a folded structure become fixed in a uniquespatial arrangement by steric interactions with theirneighbors. Still today, many computational proce-dures that explore side-chain packing strive toidentify a single native configuration.2

The three-dimensional structure of a protein, how-ever, can fluctuate considerably. Large-scale mo-tions involve partial or full unfolding and backbonehinge motions, but subtle structural variation onsmaller scales has been highlighted by severalexperimental measurements. NMR relaxation tech-

ress:

arlo; MD, molecularlt bridge; HB,SASA, solvent-dulin; PYP,ng–Landau.

lsevier Ltd. All rights reserve

niques in particular resolve fluctuations at the levelof single bond vectors in both the backbones andside-chains of folded proteins.3–5 The Lipari–Szaboorder parameters6 they determine, which increasefrom Saxis

2 = 0 to Saxis2 = 1 as rotational motion

becomes restricted, report on the range of picose-cond to nanosecond dynamics for backbone amideand side-chain methyl groups. Computational stu-dies suggest that order parameters lower than 0.8point to transitions between multiple rotamericstates in addition to the inevitable vibrations aboutoptimal torsional angles.7,8

Side-chain methyl group order parameters often liein the range 0.2bSaxis2 b0.8,3 indicating extensiveexploration of different rotameric states. These resultsare corroborated by dipolar coupling measurements,suggesting that side chains substantially populatedifferent rotameric states within the ensemble offolded configurations.9,10 Evidence for alternativeside-chain conformations has even been found inelectron density maps from crystallography expe-riments.11 The data accumulating from such studiespaint a consistent picture: Residual side-chain fluc-tuations in the native-state ensemble are distributedheterogeneously throughout the protein; side-chainbond vectors fluctuate more significantly than dothose along the backbone; and the entropy associated

d.

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http://dx.doi.org/10.1016/j.jmb.2009.05.068

485Calculation of Side-Chain Torsional Entropy

with such fluctuations is likely to be a significantplayer in protein thermodynamics.4,12

Computational studies focusing on geometricaspects of side-chain packing have reconciled theevidence for significant torsional fluctuations withconstraints due to steric interactions in a denseenvironment.13 Much as in a dense liquid, volumeexclusion reduces the diversity of accessible config-urations greatly, but by no means completely.Nearly 1020 distinct side-chain conformations weredetermined to satisfy hard-core constraints in a 125-residue protein with native backbone structure.13 Towhat degree non-steric interactions further reducethis variability is not at all clear a priori. Populatingeven a very small fraction of the geometricallyacceptable arrangements would be sufficient toallow for significant contributions to free energiesof folding and ligand binding.Mean field theories,14 various interpretations of

molecular dynamics (MD) simulations,15–18 andseveral Monte Carlo (MC) approaches13,19–21 haveall been used to estimate the residual entropy of side-chain rotations in folded proteins. Each ofthese approaches, however, is limited by underlyingapproximations or formidable practical challenges.Mean field approaches, by definition, do not accountfor a complete range of thermal fluctuations; straight-forward MD simulations can explore only rearrange-ments that occur on computationally accessible timescales. MC methods are similarly hindered bysampling difficulties intrinsic to such tightly packedsystems. As a compromise, entropies are sometimescalculated separately for single residues or smallgroups of neighboring residues while keeping otherresidues fixed.19 Studies that do confront the fullcombinatorial problem, allowing all side chains torotate simultaneously, have neglected potentiallyimportant contributions from intra-rotameric mo-tions20 or have considered geometric effects indepen-dent of non-steric interactions.13,21

In this article, we present a new approach forestimating side-chain torsional entropy. Building onalgorithms developed by Kussell et al.,13 our calcula-tions are enabled by enhanced MC methods and aschematic treatment of forces due to sterics, van derWaals interactions, hydrogen bonding, salt bridges(SBs), and solvation. Through this combination, weachieve thorough sampling of thermal fluctuations,incorporate fully coupled rotations of all residues,and address a comprehensive set of physical interac-tions. Model outlines our approach and the physicalperspectives underlying it. Results and Discussiondescribes applications to a series of small globularproteins, quantifying and comparing the ways inwhich various forces act to limit rotational freedom.Within the model we have developed, substantial

freedom remains in the packing of side-chains,even in the presence of strong, anisotropic attrac-tions such as hydrogen bonding. The correspondingentropy can, therefore, in principle, strongly in-fluence the thermodynamics of folding, protein–protein binding, and protein–ligand interactions.Indeed, it now appears from calorimetric data that,

in several systems, entropy changes figure promi-nently in tuning protein binding affinities.22,23 In thecase of stromelysin 1 binding to the N-terminal do-main of tissue inhibitor of metalloproteinases 1, theyeven overcome a substantially unfavorable enthalpyof binding.24 Implicating the involvement of sidechains in these phenomena, entropies inferred fromNMR order parameters correlate strongly with calo-rimetrically determined binding entropies for cal-modulin (CaM) and several peptide ligands.12 Wefind even better agreement between binding en-tropy measurements and calculated values based onthe methods we have developed. This comparison isdiscussed in detail in Results and Discussion.

Model

In developing a theoretical approach, we areguided by the notion that side-chain rearrangementswithin a protein's native state are not strongly me-diated by motions of the peptide backbone. Physi-cally, we expect that once themolecule has folded, it issubject to global constraints of high packing fractionthat vary little with small-amplitude backbone fluc-tuations. Empirically, we note that correlationsobserved between backbone NMR order parameters,S2, and their associated side-chain parameters, Saxis

2 ,areweak.25 FollowingKussell et al.,13 we thus adopt amodel in which the peptide backbone is fixed in itscrystallographically determined conformation. As aresult, applications of our methods are limited toproteins whose native structures have been deter-mined with high resolution.The sole degrees of freedom in our calculations are

dihedral anglesχ for rotatable side-chain bonds withheavy-atom (i.e., non-hydrogen) substituents. Othervariables are known to influence side-chain entropy,15

but torsional entropy alone is thought to provide agood approximation.19 Natural amino acids possessno more than a handful of such dihedral degrees offreedom.Alanine, for example, has none, while lysineand argininepossess the largest number (four).As in asimple molecule such as propane, local bondingenergetics bias such angles to lie in one of typicallythree ranges. For classification purposes, we considerthese ranges as discrete rotameric states, each with anideal angular value θ. We do, however, permitdeviations from these ideal angles, ϕ=χ−θ. We andothers have found them to be essential for accom-modating tightly packed rearrangements.13,26 Theintrinsic energetic penalty Edihedrals limiting suchfluctuations in our model is quadratic in ϕ, exceptfor dihedrals between sp2 and sp3 hybridized carbons,where Edihedrals=0 and is therefore χ-independent.Correspondingly, these bonds possess a single dis-crete rotamer state.It is well known from studies of microscopic struc-

ture in liquids27 and polymeric materials28 that themost essential feature of non-covalent interactions indense environments is the harsh repulsion betweenoverlapping moieties. Energetic models that discardconstraints of volume exclusion in favor of slowlyvarying potentials for computational convenience29

486 Calculation of Side-Chain Torsional Entropy

are therefore not suitable for our purpose of quanti-fying side-chain entropy. Nonetheless, the precisedependence of steric interactions on inter-atomicdistances is likely unimportant,13,30 provided thatpenetration becomes prohibitively costly at theappropriate length scale. We employ a Lennard–Jones (LJ) potential between all pairs of heavy atomsseparated by at least three bonds,which describes vander Waals attractions in addition to imposing stericconstraints.31,32 This interaction is truncated at bothsmall and large distances: For separations larger thantwice the LJ diameter, we set the potential energy tozero (and shift the entire potential to maintaincontinuity at the cutoff); separations smaller than3/4 the van der Waals contact distance are assignedinfinite energy and thus disallowed entirely. Thislatter modification, introduced for practical reasons,has no physical consequences at reasonable tempera-tures and densities.The pairwise interactions we expect to exert the

largest influence on side-chain packing are electro-static in nature, namely, SBs and hydrogen bonds(HBs). We model these energetics based on previouscoarse-grained approaches.33,34 Although we makeno effort to represent electrostatic forces betweenresidues in great detail, their strength and aniso-tropy should be appropriate to the chemical varietyof natural amino acids.Finally, we treat hydrophobic effects in terms of the

relative amounts of polar and non-polar surface areaexposed to solvent. This simplistic implicit solvent(IS) description does not address the sensitivity ofaqueous solvation to the spatial distribution ofhydrophobic and hydrophilic moieties at the proteinsurface,35,36 but it does roughly account for themany-body nature of such effects. For this purpose, weutilized an inexpensive but faithful approximation tostandard procedures for determining solvent-acces-sible surface area (SASA).37,38 SeeMethods for details.The full potential energy function governing our

model sums these various interactions,

E Q;Að Þ = Edihedrals + Enon−bonded + Eimplicit solvent:ð1ÞIt depends on the set of N torsional angles for all

rotatable bonds described above, which we specifythrough the nearest ideal values, Θ={θ1, θ2,…, θN},and deviations about them, Φ={ϕ1, ϕ2,…, ϕN}. Notethat we have collected LJ, SB, and hydrogen-bondingcontributions into a total potential Enon-bonded forpairwise-additive, non-bonded interactions. Freeparameters in the energies of Eq. (1) were tunedexclusively for the purpose of ensuring that side-chain packing in crystallographic configurationsyields energies not much larger than those ofalternative arrangements generated in the course ofcomputer simulations. Their values liewellwithin therange of analogous parameters appearing in othermodels that attempt a similar level of resolution.Because it represents steric constraints realistically,

our model shares with many other approachessevere challenges to thorough sampling of thermalfluctuations. From typical configurations, it is diffi-

cult to rotate a side-chain bond through the ∼120°needed to transit from the neighborhood of one idealangle to another without introducing steric overlaps.In real systems, such an isolated rotation wouldincur great energetic cost; in our model, the price isoften not even finite. We circumvent this problemwith MC sampling procedures that preserve theBoltzmann distribution determined by Eq. (1).Specifically, we employ a modified energy functionin which the singular hard core of our van der Waalspotential is replaced by a finite constant energyɛtunnel. Correcting exactly for the resulting bias istrivial, since the relativeweights of sterically allowedconfigurations are unchanged. For many purposes,one need only discard sampled configurations thatviolate steric constraints (see Methods). The advan-tage of this artifice is an ability to “tunnel” throughdisallowed regions of configuration space. If ɛtunneldoes not greatly exceed the energy kBT of typicalthermal excitations, simulations can move muchmore readily through the free-energy barriers thatfrustrate MD. An optimal value of ɛtunnel must alsoensure that the proportion of sterically inadmissiblestates generated by MC simulations is not over-whelmingly large. This procedure can enhancesampling efficiency considerably. Several of thecalculations we present nonetheless additionallyrequired adaptive umbrella sampling39 and/orstaging through multiple ensembles in which side-chain interactions are gradually introduced (seeMethods) to obtain well-converged results withavailable computing resources.Themodel energetics andMetropolisMCmethods

we have described provide a straightforward andcomputationally manageable way to characterizeside-chain fluctuations quantitatively. By design, oursampling scheme is not dynamically realistic on thetime scale of torsional vibrations. Individual trialmoves that advance these simulations often switchdirectly between distinct rotameric states. In thecourse of natural dynamics, such transitions occuron time scales of picoseconds to milliseconds.40,41

We have found that MC trajectories comprising50,000 sweeps are sufficient (but not excessive) forsampling a representative set of side-chain rearran-gements in small globular proteins (including on theorder of 250 rotatable bonds). Exploring the samerange of fluctuations using straightforward MDsimulations of detailed atomistic models such asCHARMM or AMBER, which proceed in roughlyfemtosecond steps, would be extremely taxing if notunfeasible. Indeed, previous MDwork suggests thatthe breadth of side-chain motions cannot be reliablygauged from nanosecond trajectories even for verysmall proteins.42

Results and Discussion

Entropy of side-chain configurations

Absolute entropies are not well defined for contin-uous classical variables. It is therefore necessary in


computing torsional entropy of a model such as oursto specify a standard state. For this purpose,we choosea noninteracting reference system where all dihedralangles are equally likely, E(ref)=0. All entropies wereport are given relative to this maximally flexiblesystem, Δ(ref)S=Sconfig−Sconfig(ref) , where Sconfig is theconfigurational entropy associated with fluctuationsbothwithin and between distinct rotamericwells. Thischoice of reference state has severalmerits. First, a stateinwhichmotions of one residue are independent fromall others serves as a crude proxy for side-chain fluc-tuations of an unfolded protein. In otherwords,Δ(ref)Scould be thought of as a rough estimate for the changein torsional entropy upon folding. Second, by settingE(ref) equal to a constant, we remove all chemicaldetails distinguishing between different rotatingmoieties. The reference state consequently has anentropy per rotatable bond, s(ref), that is consistentacross proteins of arbitrary composition. Thus, whilewe can determine side-chain entropies only up to anadditive constant, Ns(ref), where N is the total numberof rotatable bonds, we ensure that s(ref) has the samevalue for all proteins we consider. Finally, a noninte-racting standard state facilitates ligand affinity calcu-lations based on the thermodynamic cycle shown inFig. 1. Because non-translational free-energy contribu-tions are invariant when two molecules A and B bindin their noninteracting reference states, associationentropies can be computed via

D bindingð ÞS = SA: : :B � SA + SBð Þ=D refð ÞSA: : :B � D refð ÞSA + D refð ÞSB

� �:ð2Þ

We compute these entropy differences using thecorresponding changes in energy and partitionfunction Q,

D refð ÞS = kBlnQconfig

Q refð Þconfig

0@

1A +

1T

hEi � hE refð Þi refð Þ� �

ð3Þ

Angled brackets denote equilibrium averagesover canonical ensembles at temperature T. (Weperform most calculations at T=300 K.) Lackingsuperscripts, these brackets refer to the Boltzmanndistributions determined by the full energy functionof Eq. (1); the superscript “(ref)” refers to statistics ofthe noninteracting reference system. The ratio of

Fig. 1. Thermodynamic cycle relating the change inentropy upon protein–ligand binding, Δ(binding)S, to theentropic differences between the interacting and thenoninteracting reference cases for the bound and unboundspecies. Note thatΔS=0 for the binding of the ligand to theprotein within the reference system, allowingΔ(binding)S tobe calculated as shown in Eq. (2).

partition functions in Eq. (3) could be evaluatedusing Zwanzig's formula,

Qconfig

QðrefÞconfig

=P

Q

RdAexp �βE Q;Að Þð ÞP

Q

RdAexp �βE refð Þ� �

= hexp �β E Q;Að Þ � E refð Þ� �� i refð Þ;

ð4Þ

where β− 1 =kBT. It is therefore necessary in principleonly to sample configurations from the noninterac-ting system. This approach is not practical, however,since the ensembles defined by E and E(ref) overlapweakly. We overcome this problem with a stagingprotocol that introduces several intervening ensem-bles. In these intermediate states, IS and non-bondedinteractions are scaled by a parameter 0bλb1 (seeMethods).Side-chain configurational entropy is commonly

discussed in terms of separate contributions fromvibrations within a rotameric state (Svib) and fromconformational transitions between discrete rota-meric states (Sconf).

43 Many computational effortsfocus exclusively on Sconf, even though recent theore-tical studies highlight the importance of vibrationalentropy changes in ligand binding.18,44 That a largeset of rotameric states becomes accessible only whensuch vibrations are allowed indicates that thesemotions are in fact strongly interdependent.13,26 Ourcalculations of Δ(ref)Smake no attempt to treat vibra-tional and conformational contributions separately.We will, however, describe ways to quantify the va-riability of one motion, while fixing or integrating outthe other.

Entropic losses due to side-chain interactions

We have applied the techniques outlined in theprevious section to determine side-chain entropies of12 small proteins, ranging in size from 46 to 143residues and exhibiting a diverse set of secondarystructures. For each molecule, we have also per-formed calculations with model energetics thatinclude only a subset of the interaction types des-cribed by Enon-bonded and Eimplicit solvent. In this way,we quantify the extent to which different kinds offorces limit torsional freedom in the dense environ-ment of a folded protein. Results for the entropyreduction per rotatable bond,Δ(ref)S/N, are shown inFig. 2. For each variant of the model, the similarity ofΔ(ref)S/N values across the entire set of proteins isstriking. Local energetic biases due to covalentbonding, described by Edihedrals and included in allof the interacting systems, result in a significantreduction in entropy. Althoughweakly dependent onthe specific amino acid makeup of the protein, it isfound to be quite consistent across the 12 proteins. Ofthe various interactions considered in isolation,electrostatics yields the largest entropy reduction inmost cases. Sterics and van der Waals attractionseffect changes similar in magnitude but typicallysomewhat smaller. Solvation forces, in effect actingonly at the periphery of themolecule, contribute least,even though approximately 60% of the residues in

Fig. 2. Total side-chain dihedral entropy per rotatable bond of 12 small proteins, relative to that of a noninteractingreference system [see Eq. (3)]. Results are shown for various combinations of interaction types at 300 K. D refers to anoninteracting reference system that includes only the intrinsic dihedral energy Edihedrals. All other cases include thisdihedral potential together with subsets of side-chain interactions: S indicates steric energetics due to the repulsive part ofthe LJ potential; LJ indicates the full Lennard–Jones potential; IS indicates the implicit solvent; and HBSB indicates thehydrogen bonding and SB interactions. The proteins studied here are barstar (1a1945), calmodulin (3cln46), crambin(1cbn47), eglin c (1cse48), GB3 (1igd49), protein L (1hz650), PYP (1f9i51), PZD2 (1r6j52), SH2 (1d1z53), CspA (1mjc54),ubiquitin (1ubq55), and tenascin (1ten56). These results were calculated using Metropolis MC.57 Five trials starting fromdifferent randomly chosen side-chain configurations were run for 50,000MC sweeps each when calculating ⟨E(Θ,Φ)⟩ andfor 17 stages of three 20,000-sweep trials each when calculating Q/Qref. Error bars represent one standard deviation.


photoactive yellow protein (PYP) are consideredsolvent-exposed.13 The CaM structure, consistingof two globular regions connected by an extendedα-helix,46 retains the most entropy. The steric con-tribution for CaM is among the smallest in this set ofmolecules, as might be expected from its relativelyopen structure, but the isolated effects of otherinteraction types are not at all atypically weak.Averaging the entropy reduction per rotatable

bond over this set of proteins yields Δ(ref)S/N=−5.2 kJ/(mol·300 K). As we have noted, one mightregard the noninteracting reference system as aschematic representation of the unfolded state,whose side-chain rotations should be considerablyless restricted than in a native fold. A more faithfuldescription of the unfolded state would include thelocal biases of Edihedrals, which operate regardless ofnon-covalent structure. Accounting for the corre-sponding entropic reduction, averaged over the setof proteins we consider, of −1.8 kJ/(mol·300 K), weobtain a typical difference of Δ(ref,dihedrals)S/N=−3.6 kJ/(mol·300 K) between the noninteractingstate with restrained dihedrals, denoted by a super-script “(ref,dihedrals),” and the fully interactingstate. The change in entropy upon folding due toside-chain conformational fluctuations has beenestimated from several different approaches, lea-ding to a consensus figure of ≈−2.1 kJ/(mol·300 K)per rotatable bond.43 That Δ(ref,dihedrals)S/N exceedsthis value in magnitude is not surprising. Viewed asan approximation of an unfolded protein, our refe-rence state, even with dihedral restraints, certainly

overestimates torsional freedom. Further, rigidity ofthe peptide backbone would likely cause our modelto underestimate torsional freedom of the foldedstate. Despite these limitations, the two values arenonetheless well within kB of one another. We consi-der this correspondence an assuring sign that ourmodel captures the basic physical determinants ofside-chain entropy correctly.Over the set of fully globular proteins (which

excludes CaM) we have studied, results forΔ(ref)S/Nrange from −5.02 kJ/(mol·300 K) for protein L to−5.66 kJ/(mol·300 K) for eglin C. Since the standardstate is equivalent in all cases, the range of absolutetorsional entropies per rotatable bond, Sconfig/N, isidentical in breadth to the range inΔ(ref)S/N. Judgingfrom these 12 proteins, natural variations in nativeside-chain environments can easily shift torsional en-tropies by an amount δSconfig/N≈0.6 kJ/(mol·300 K).[Note that when the values of Δ(ref,dihedrals)S/N forprotein L and eglin C are compared, the difference is≈0.7 kJ/(mol·300 K), indicating that this differenceis not simply due to differing numbers of rotatablesp3–sp3 hybridized bonds.] In the context of proteinbinding thermodynamics, this result provides arough gauge for the potential strength of entropicdriving forces. If, for example, two globular proteinsform a complex whose interface is comparable tothe internal structure of typical native folds, overallside-chain entropy may nonetheless change by asmuch as N (0.6 kJ/(mol·300 K)). For a complex with100 residues, this maximum change in total entropywould amount to a substantial 102 kJ/(mol·300 K).

Fig. 3. Entropic contributions to the binding freeenergies −TΔ(binding)S for four CaM–peptide complexes.Results of MC simulations are plotted against correspon-ding calorimetric measurements from Ref. 12. UnboundCaM is shown on the plot as the reference point at (0,0). Forthe CaM–peptide complexes, ⟨E(Θ, Φ)⟩ was calculatedusing a WL bias in six sets of 10 trials, each with 90,000–100,000 sweeps. Average values were calculated withineach set and errors were calculated across the six sets. ForunboundCaMand peptides,MetropolisMCwas sufficientto calculate ⟨E(Θ,Φ)⟩, and 5 trials of at least 50,000 sweepseachwere performed. AWL bias was also used to calculate⟨Δ(C)⟩(tunnel) and the first ratio of partition functions on theright-hand side of Eq. (20) in three sets of 10 trials for eachof the CaM–peptide complexes. Metropolis MC was usedto calculate the remaining 22 stages in the Q/Qrefcalculation of the CaM–peptide complexes, as well as thefullQ/Qref calculation for unbound CaM (in 26 stages) andthe unbound peptides (in 17 stages). Each stage included 3trials of 20,000 sweeps. Averages and errors werecalculated between the three independent calculations ofQ/Qref. Error bars represent one standard deviation.Errors in the calorimetric measurements of TΔ(binding)Sare ≤ 1.0 kJ/mol.12


Side-chain entropic contributions to CaM–ligandbinding

Calorimetry provides unambiguous evidence forstrong entropic contributions to protein binding equi-libria.12,24 CaM, for example, binds a series of pep-tides with similar affinities, but with widely varyingentropies of association.12 For the specific ligandCaMKKα(p), the contribution to the free energy ofbinding due to entropy alone is nearly 100 kJ/mol,but it is not clear how such entropic changes aredistributed among the degrees of freedom associatedwith solvent, peptide backbone, and amino acid side-chains. The role of side-chain rotations in CaM bin-ding thermodynamics has recently been explored byestimating torsional entropy from NMR order para-meters.12 Although the connection between Saxis

2 andside-chain entropy is not precise, and although thisestimate, of necessity, neglects correlated fluctuationsof different residues and fluctuations that take placeon time scales longer than those detected in therelaxation experiment, thermodynamic trends weresuccessfully predicted.12 Specifically, Frederick et al.found a linear correlation between calorimetric resultsfor Δ(binding)S and those computed from NMR data,with a slope of 0.51 and correlation coefficient r=0.88.Side-chain contributions to CaM affinity thus appearconsiderable.Our approach provides a way to estimate side-

chain contributions to Δ(binding)S without the as-sumptions inherent when inferring thermodynamicbehavior from NMR order parameters. This CaM–peptide system thus serves as a test both of themethods we have developed and of the notion thattorsional fluctuations can play an essential rolein peptide binding. We focus on the four peptidesconsidered in Ref. 12 for which calorimetricdata12,58,59 and high-resolution structures are avail-able: CaMKKα (1ckk60), smMLCK (1cdl61), CaMKI(1mxe62), and eNOS (1niw63). The thermodynamiccycle in Fig. 1 was used to calculate Δ(binding)S fromour Δ(ref)S calculations of the bound and unboundCaM and ligand species. The entropy of unboundCaM was computed using the globular structure ofRef. 64 (1prw). The backbone conformation of eachligand when co-crystallized with CaM was used aswell for the unbound peptide.Binding entropies determined by our model

match the trend of experimental data as well astheir overall scale, as shown in Fig. 3. In particular,the experimental orderΔ(binding)SCaMKKα(p)bΔ

(binding)

SsmMLCK(p)bΔ(binding)SCaMKI(p)bΔ(binding)SeNOS(p)is correctly reproduced, although the differencebetween smMLCK(p) and CaMKKα(p) cannot beresolvedwithin statistical errors. Correlation betweencomputed values and experimental measurementsexceeds reasonable expectations, given our exclusivefocus on side-chain contributions and neglect ofbackbone fluctuations. We emphasize that modelparameters were not adjusted to obtain this agree-ment. Neither is the correspondence a trivial conse-quence of peptide size and composition; thecomplexes we have studied possess similar numbers

of rotatable bonds (betweenN=281 and N=286) andrank differently byN and byΔ(binding)S. Furthermore,calculations employing reduced sets of interactiontypes inmany cases compare poorlywith experiment.Our results thus bolster the conclusion of Ref. 12 thatside-chain torsional rearrangements constitute amajor,if not dominant, source of CaM binding entropy.

Heterogeneous distribution of side-chainentropy

Though ordered, a folded protein is structurallyheterogeneous on all scales from atomic to macro-molecular. One might expect that the rotationalfreedom of side chains is similarly nonuniform.Indeed, the fluctuation spectrumof a protein's interiorhas been likened to that of a solid, while the exposedsurface is often considered fluid.65 Our calculationsreveal spatial patterns of torsional variability that arenot nearly as simple as this conjecture would suggest.


We do find, on average, that side chains of surfaceresidues are less tightly constrained by native inter-actions than are those of the interior. However, thereare many exceptions, and simple features of crystal-lographic structures such as secondary structure andpacking density do not reliably foreshadow the extentof local side-chain fluctuations.25

As a measure of local torsional variability, weconsider the Gibbs entropy S

ðresÞassociated with a

single residue's notionally discrete rotameric states,

Sresð Þi = � kB

XQi

p Qið Þlnp Qið Þ; ð5Þ

whereΘi={θ1(i),…, θ ið Þ

Ni} denotes the set of ideal torsion

angles for each of the Ni rotatable sp3−sp3 hybri-dized side-chain bonds belonging to residue i. Thepopulations p(Θi) of these 3Ni states are determinedin simulations by constructing a histogram oversterically allowed configurations. Effectively inte-grating out torsional fluctuations Φi={ϕ1

(i),…, f ið ÞNi}

within each ideal rotameric state, we focus ondiscrete degrees of freedom with a manageable setof possible realizations. As a result, we can calculateconverged, absolute values of S

ðresÞi . This analysis

focuses explicitly on “conformational” contributionsto entropy. Others have calculated analogous quan-tities for different models,43 some lacking vibra-tional fluctuations altogether.21 In our calculations,coupling between conformational and vibrationalmotions, and between rearrangements of differentresidues, is implicit in the weights p(Θi).We have computed S

ðresÞi for all residues in each of

the small proteins listed in Fig. 2, and for severalsubsets of the interaction types in Eq. (1). Here, we

Fig. 4. Side-chain conformational entropy, SðresÞi [see Eq. (

interactions. The side-chains are color coded according to eachmaximum entropy and blue indicating its minimum. S

ðresÞi

hybridized rotatable bonds. (a) Noninteracting reference systbonds), (b) LJ interactions, (c) IS interactions, (d) both LJ aninteractions. All interacting runs include the effects of Edihedrindependent trials, each run for 50,000 MC sweeps. Images w

present and discuss in detail results only for PYP,whose behavior is typical of the entire set ofmolecules. Figure 4 illustrates the complex spatialdistributions of rotational freedom generated by ourmodel. It also demonstrates that different interactiontypes limit side-chain rearrangements in differentways.Values of S

ðresÞi are indicated in Fig. 4 by the

coloring of residues within PYP's three-dimensionalstructure. Although side chains are shown in theircrystallographic configurations, it is fluctuationsaway from this ideal packing that determine thelocal entropies depicted. The color scale varies byresidue according to its maximum possible value ofSðresÞi . Bright red corresponds to this maximum valueSðresÞi = kBNiln3

� �, while dark blue signifies an

absence of rotamer variability SðresÞi = 0

� �. Residues

that possess no rotatable bonds are colored blue,though Eq. (5) is not well defined in this case.Results for our noninteracting reference system

are shown in Fig. 4a. Lacking any bias on side-chainconfiguration, all residues with rotatable bondsexhibit their maximum local entropy and are thuscolored red. Figure 4b–e correspond to differentsubsets of interaction types, each including thebasic local energetics Edihedrals of torsional rotations.Figure 4f shows results for the full model potentialof Eq. (1).Of the interaction types we consider, the combina-

tion of steric constraints and van der Waals attrac-tions effects local entropy in ways most similar to thesolid/liquid caricatures of a protein's interior/ex-terior (see Fig. 4b). However, even in this case, theentropic distinction between exposed and buried

5)], for all residues i in PYP (1f9i51) for various kinds ofresidue's value of S

ðresÞi , with red indicating the residue's

values have been normalized by the number of sp3–sp3

em (blue residues indicate amino acids without rotatabled IS interactions, (e) HB and SB interactions, and (f) allals. Results were calculated using Metropolis MC for fiveere made using MacPyMOL.66

Fig. 5. Side-chain NMR order parameters, Saxis2 , and χ1

rotameric populations for eglin c (1cse48). Results of MCsampling plotted against NMR-derived measurements. (a)Comparison of the MC and NMR10 methyl group orderparameters. (b) Comparison of χ1 rotamer state popula-tions determined from MC sampling and experimentalthree-bond J-coupling constants.10 Five independent trialswere run for 50,000 sweeps each using Metropolis MC.Error bars represent one standard deviation.


residues is not clear-cut. By itself, the IS energy hasan opposing effect, significantly limiting the motionof only those residues that can be readily accessed bysolvating water molecules (see Fig. 4c). Electrostaticinteractions exert a rather different influence onpatterns of torsional freedom (see Fig. 4e), inisolation affecting only those residues that donate/accept HBs or participate in SBs. The directionality ofthese forces, as well as the fact that HB partners mayreside on the peptide backbone, begets restrictionson side-chain motion that are, in general, muchmorelocalized and anisotropic than those due to otherinteraction types. The net effect of all these interac-tions, when operating simultaneously, is a localentropy much reduced from that of the referencestate and distributed throughout the structure muchless smoothly than would be expected from thenotion that buried residues adopt unique rotamerstates (see Fig. 4f).These same interactions restrict torsional vibra-

tions as well, whose variety is essentially overlookedby the local entropy of Eq. (5). This neglect isreasonable for assessing rotamer flexibility inqualitative terms but does not suffice for quantifyingthe magnitude of entropic driving forces. As anexample, the total side-chain rotational entropy of aprotein can be estimated by summing local entropiesover all residues, S =

Pi S

ðresÞi . Discarding contribu-

tions from vibrational fluctuations in this mannerdiminishes computed CaM-peptide binding entro-pies by nearly 60%. Nevertheless, the reduction ofTD bindingð ÞS estimates is consistent in magnitudeacross the peptide ligands we have studied, so thatcorrelation with experimental data remains strong,with a correlation coefficient of r=0.96.In a separate approach to quantifying the impor-

tance of torsional vibrations, we consider a newreference state, denoted by “(ref′,dihedrals),” inwhich rotamers do not interact but are nonethelessconstrained to a single set of ideal dihedral angles Θand are governed by the dihedral potential. We canthus estimate the loss of entropy Δ(ref′,dihedrals)Sconfigsolely due to restrictions on vibrational motionresulting from interactions. These values closelymirror the results shown in Fig. 2, but on a scalesmaller by roughly a quarter.Entropies of the reference systems we have

considered are simply related, S refð Þconfig � S ref0ð Þ

config =kBN sp3ð Þln3, where N sp3ð Þ is the total number ofrotatable sp3–sp3 hybridized bonds. A thermody-namic cycle can thus be used to connect interactingsystems differing by constraints on ideal dihedralangles. In this way, we find that fixing Θ in the fullyinteracting model effects an entropy loss of ≈0.8 kJ/(mol·300 K) per rotatable sp3–sp3 hybridized bond.

Comparisons to experimentally determinedside-chain fluctuations

Comparisons between these detailed local entro-pies and experimental data are ambiguous in severalrespects. As we have noted, the quantity S

ðresÞi

discards contributions of torsional vibrations wehave found to be numerically significant. On theexperimental side, currently feasiblemeasurements atthis level of resolution can only be related to thermo-dynamics in approximate ways. NMR order para-meters, for example, are sensitive only to the range ofrotational fluctuations that function on picosecond tonanosecond time scales.3 Dipolar coupling and J-coupling experiments report on longer time scalemotions, but for side-chains, they are generally onlyapplied to the rotatable bond closest to the peptidebackbone (whose dihedral angle is denoted χ1).

67

Despite these limitations, we employ methyl orderparameters and χ1 rotamer populations as roughpoints of comparison for our computer simulations.


Figure 5 presents results for the specific protein eglinc, for which bothmethyl group order parameters andχ1 rotamer populations have been experimentallydetermined.10Values of Saxis

2 derived from NMR data10 and thosecalculated from MC simulations (using the approachdescribed in Ref. 68), both shown in Fig. 5a, aremodestly correlated (r=0.66). Previous simulationresults obtained from 50-ns MD trajectories for adetailed model of calbindin have matched experi-mental measurements more closely, but not drama-tically so (r=0.8).42 Numerical calculations for eglin cthat utilize a sampling procedure inconsistent withBoltzmann statistics generate still stronger corre-lation,69 perhaps highlighting the sensitivity ofSaxis2 to very sluggish rearrangements. The result of

such comparisons indicates that side-chain fluctua-tions are overly restricted in our model, as might beexpected from the neglect of backbone flexibility.Alanine orientation, for example, is completely fixedin our model, yielding Saxis

2 =1 identically. The NMRresult for alanine in eglin c, Saxis

2 =0.8,10 points to non-negligible effects of backbone motion, although sucheffects appear to correlate weakly with measuredside-chain fluctuations.25

Populations of distinct χ1 rotamer states inferredfrom experiment10 also agree reasonably (but notstrikingly) well with results from our simulations(see Fig. 5b). The dearth of probabilities between0.1 and 0.9 indicate that these bonds are stronglybiased toward one rotameric state. This factshould not, however, be taken as a sign of overalltorsional rigidity. Bonds that are not proximal tothe backbone show greater variability. Indeed, in atypical configuration of our model, roughly one-sixth of the rotatable sp3–sp3 hybridized side-chainbonds in eglin c adopt an ideal dihedral angle θidifferent from the most probable.

Importance of model interactions and thoroughsampling

The high correlation between experimental andcalculated TΔ(binding)S values in the CaM–ligandsystem suggests that our model includes the inter-actions most essential for describing side-chainfluctuations within the folded protein. We empha-size the importance of considering energetics beyondthose imposing steric constraints, despite the denseenvironment; when non-steric interactions areomitted, calculated entropies correlate only moder-ately with calorimetric measurements. Similarly,neglecting inter-residue correlations and intra-rota-meric fluctuations substantially reduces the quanti-tative correspondencewith experimental data. Theseresults strongly recommend models of side-chainthermodynamics that include intra-rotameric fluc-tuations18,44 and respect not only constraints ofpacking but also the diversity and broad energyspectrum of sterically allowed configurations.Our MC sampling methods probe diverse side-

chain configurations that may be difficult to accessusing more straightforward sampling methods.

Notably, NMR order parameters, Saxis2 , estimated

from a 5-ns MD trajectory of barstar resemble ourMC results more closely than do those determinedfrom only 250 ps of time evolution.4 The diversity ofside-chain packings we have identified suggests animportant role for still slower fluctuations. Even withour MC sampling procedure, obtaining convergedresults for CaM–peptide TΔ(binding)S values requiresthe implementation of advanced techniques such asstaging and the use of Wang–Landau (WL) proce-dures. This necessity highlights the limitations asso-ciatedwith calculating entropies fromMDsimulationsalone, as has been attempted previously.15–18 In onestudy on protein–protein binding, several shorterMDtrajectories were run in order to improve the con-vergence of calculated binding entropies, but theerrors were still quite large.70Combining MC and MD techniques might pro-

vide an optimal approach for exploring structuralexcursions broadly while preserving the dynamicalcharacter of short-time relaxation.71 Capturing thetime dependence of the slowest side-chain rearran-gements, which in our model must navigate severedynamical bottlenecks, will likely require impor-tance sampling in trajectory space.72

Conclusions

We have examined spontaneous side-chain fluctua-tions in several folded proteins using computersimulations that sample all side-chain torsionaldegrees of freedom simultaneously. Overall, our MCmethod facilitates exploration of rearrangements thatproceed sluggishly in the course of natural dynamics,and our model appears to successfully capture thephysical character of these variations. Their extent islikely underestimated due to backbone constraints,rendering conclusions about their thermodynamicsignificance conservative.We have assessed the impact of various interaction

types in restricting the range of side-chainmotions, byquantifying entropy reductions relative to a noninter-acting reference system. The ability to probe theseinteractions separately is a strength of our computa-tional approach that would be difficult to mimicexperimentally. These reductions, normalized by thenumber of rotatable bonds, are remarkably consistentamong the 12 proteins we have considered, despitesignificantly heterogeneous distributions of rotationalfreedom. Under the collected influence of steric,dispersive, and electrostatic forces, globular proteinsin our model possess, on average, an entropy perrotatable bond of 5.2±0.2 kJ/(mol·300 K) less thantheir noninteracting counterparts.Our binding entropy calculations for CaM–pep-

tide complexes, which correlate strongly withcalorimetric measurements, underscore the thermo-dynamic importance of side-chain torsional free-dom. They also hint at the possibility that correlatedside-chain fluctuations could communicate struc-tural change over significant distances. Indeed,NMRstudies show that effects of side-chain mutation or


ligand binding on side-chain methyl dynamics canextend far from the site of perturbation.10,73 Thecomputational tools we have presented are wellsuited to explore this unconventional mechanism forprotein allostery.

Methods

Model

The potential energy function governing side-chainfluctuations in our model is a sum of three physicallydistinct contributions: from the local torsional bias ofcovalent bonding (Edihedrals), from direct interactionsbetween non-bonded moieties (Enon-bonded), and from thefree energy of aqueous solvation (Eimplicit solvent).The local dihedral energy Edihedral,i of a rotatable bond i

depends on its hybridized geometry. Since ideal angles aredifficult to identify for sp3−sp2 hybridized rotatablebonds,74 we impose no intrinsic bias on the correspondingrotations, that is, Edihedral,i=0 independent of χi for thesebonds. For the more prevalent sp3–sp3 rotatable bonds, thedihedral energy function is constructed so that theBoltzmann weight exp(−βEdihedral,i) is a sum of (un-normalized) Gaussian distributions centered at idealrotamer angles θi,

Edihedral;i = � kBTlnXθi

exp � χi � θið Þ22σ2

" # !ð6Þ

We parameterize this function through the approximatewidth of empirical distributions of side-chain torsionalrotations, σ=12.7°, as found in the rotamer library.74 Foreach sp3–sp3 bond, three ideal values of θi are assignedusing data from Ref. 74 (see Supplemental Material). Sincethe range of χi is unbounded in our simulations, each ofthese three ideal values is in fact repeated with a period of2π; that is, Gaussian distributions in Eq. (6) are centered atθi, θi±2π, θi±4π,…. The strongest overlap among thesedistributions is between neighboring ideal angles; how-ever, in practice, σ is sufficiently small compared to thespacing between ideal values that the overlap is extremelyweak. Neglecting this overlap entirely, we could considerEdihedral,i as a piecewise continuous superposition ofquadratic functions centered at each ideal rotamer angle.With this approximation, a protein's total intrinsicdihedral energy can be written

Edihedrals Að ÞckBTXi

f2i

2j2 hi; ð7Þ

where fi =minui mi � uið Þ is the deviation of dihedral angleχi from its nearest ideal rotamer angle, θi. The indicatorfunction hi takes values of hi=1 if bond i is sp3–sp3

hybridized and hi=0 if bond i is sp3−sp2 hybridized. Theexact function Edihedrals=∑iEdihedral,i is a slightly smoothedversion of Eq. (7), more closely resembling the detaileddihedral potentials used in CHARMM and AMBER.Our model includes non-bonded interactions due to

sterics and van der Waals attractions, due to SBs, and dueto hydrogen bonding:

Enon−bonded Q;Að Þ =Xipj

Lij rij� �

+qiqj

K rij� �

rij+Hij rij;C

� �" #:

ð8Þ

We denote the distance between heavy atoms i and j asrij, their charges as qi and qj, and the set of anglesdescribing their HB geometry asΨ. The factor K(r)=0.124rÅmol/kJ accounts empirically for the screening of ionicinteractions in the heterogeneous environment of a pro-tein's interior.31,34

We represent steric as well as dispersion interactionsbetween heavy atoms using a modified LJ potential

Lij rij� �

=

l; rijbr4ij

eijrminij

rij

!12

�2rminij

rij

!6

+ a

24

35; r4ijVrijb2jij

0; rijz2jij:

8>>>><>>>>:

ð9Þ

We set the distance rijmin of minimum energy to be the

sum of van der Waals radii for atoms i and j (taken fromRef. 32). Attraction strengths ɛij are taken from Ref. 31. Thesmooth decay at long distances is truncated at rij=2σij,where σij=(1/2)

1/6rijmin, and the entire potential is shifted

by the constant α=0.0615 so that the potential iscontinuous; that is, limrijY2j�

ijLij = 0. We describe the

harsh repulsion at short distances (rij⁎=0.75rij

min) with ahard sphere potential (rather than the sharp but smoothr−12 of LJ) for sampling purposes as described below. Also,toward that end, we define a non-singular version of thesteric interaction

L tunnelð Þij rij

� �=

etunnel; rijbr4ijLij rij� �

; rijzr4ij:

(ð10Þ

The superscript “(tunnel)” accompanying other quan-tities indicates usage of L(tunnel) in place of L. Stericrepulsions and dispersion attractions are only consideredfor atoms separated by at least three bonds.Hydrogen bonding between the donors and acceptors

specified in Table 1 of Ref. 33 is described by a potentialadapted from Ref. 31,

Hij rij;C� �

= D0 5rHBminij

rij

!12

�6rHBminij

rij

!10

+ D

24

35F Cð Þ; rHB4

ij Vrijb4:0)

0; otherwise:

8>><>>:

ð11ÞThe strength D0=18 kJ/mol of a perfectly aligned HB

was chosen such that the total energies of crystallographicstructures lie within the energy range of typical repackedstructures. Averaged over fluctuations in donor–acceptorgeometry, the resulting dissociation energy amounts toroughly 11 kJ/mol when the full potential is consideredfor PYP. The donor–acceptor distance rij

HBmin=2.75 Å ofminimum hydrogen-bonding energy was taken from Ref.31. rij

HB⁎ is set to 2.52 Å, and the entire potential is againshifted by a constant η=0.0858 to preserve continuity.Orientation dependence of this model potential is deter-mined by a set Ψ of three angles. In terms of the unitvectors uDA pointing from the donor D to the acceptor A,uDD′ pointing from the donor to its nearest bonded heavyatom D′, and u AA′ pointing from the acceptor to its nearestbonded heavy atom A′, these angles are defined asψD=cos

−1(uDA·uDD′) and ψA=cos−1(uAD·uAA′). ψn is the

angle between the normals of the planes defined by (D,D′,D″) and (A,A′,A″), where A″ and D″ are the nextantecedent heavy atoms, bound either to the acceptor'sor the donor's nearest bound neighbor or to the acceptor


or donor itself. See Ref. 33 and Fig. 1 for details.31,33,34 Wetake their influence to be multiplicatively separable,

F cD;cA;cnð Þ = fD cDð ÞfA cAð Þgn cnð Þ; ð12Þ

where

fD cDð Þ = cos2 cD � c4D

� �90BVcDV180

B

0; otherwise:

�ð13Þ

For donors that are sp2 hybridized, ψD⁎ =120°, while for

sp3 donors, ψD⁎ =109.5°. The function fA(ψA) differs from fD

(ψD) only in the range 60°≤ψA≤180° over which it isnonzero, and only for sp3 hybridized acceptors. Finally,

gn cnð Þ = 0; cnN60Band the donor is sp2 hybridized

1; otherwise:

�ð14Þ

Hydrogen bonding between protein and solvent isallowed when a side-chain donor or acceptor has notformed its maximum number of HBs33 with other proteindonors or acceptors. Contributions of these bonds to thepairwise interaction energy Enon-bonded are small, favor-able by exactly 2 kJ/mol in all cases, with no distance orangular component. More substantial effects of thesebonds are subsumed in Eimplicit solvent, whose strength isdetermined by the energy of sequestering non-polar atomsfrom solvent by exposing polar moieties instead.We represent solvent–protein interactions primarily

according to the composition of SASA

Eimplicit solvent Q;Að Þ = γAnon−polar Q;Að Þ; ð15Þwhere γ=0.3 kJ/molÅ2 is the surface tension of ahydrocarbon–water interface.37 The exposed non-polararea Anon-polar(Θ,Φ) is calculated using a computationallyinexpensive implementation of the Shrake–Rupleyalgorithm.38 Fifty points are placed at uniform densityon a sphere centered at each heavy atom, with a radius Requal to the sum of its van der Waals radius and that of awater molecule. We then determine the fraction x of suchpoints that lie outside all spheres centered on neighboringatoms. A non-polar atom's contribution to SASA iscomputed as 4πR2x. For the crystal structure of PYP, thisestimate differs from values obtained with the more taxingbut exact method GETAREA75 by only 1.5% of the totalsurface area for typical heavy atoms and by 1.0% of thetotal surface area for the final value of Anon-polar.Within the framework of this model, glycines, alanines,

and prolines possess no degrees of freedom and thereforecannot contribute to the overall entropy. In addition,residues that participate in disulfide bridges, thoseresidues binding to Ca2+ in CaM, and the residue attachedto the chromophore in PYP are considered to have norotatable bonds. All bond lengths and angles are takendirectly from the Protein Data Bank structures for eachprotein, and in all cases when more than one structure isresolved, the first structure is always used. Within thecrystalline unit, themost complete structures were used. Incrystal structures where non-standard amino acids areused to assist in crystallization or phasing,wemutate thoseresidues back to standard amino acids before sampling.Unresolved residues at the N- or C-termini were notincluded in the modeling, while unresolved side chains oratoms in among the resolved portion of each protein werearbitrarily assigned appropriate initial positions.We developed this model using only PYP and protein L

for testing and refining. No potential refinement wasdone to optimize results for eglin c, CaM, or CaM–ligandcomplexes.

Sampling

All computer simulations were performed in canonicalensembles permissive of steric overlaps, that is, according tothe regularized potential E(tunnel). Physical quantities ofinterest must be calculated for the full potential E, which, ofcourse, precludes steric clashes. We have constructed thesetwo potentials such that converting computed averages⟨·⟩(tunnel) of an arbitrary observable · into physically realisticaverages ⟨·⟩ is a straightforward task. Let C be the number ofhard steric overlaps (instances of rijb rij⁎) in a given config-uration. It is simple to show that

hd i = hd D Cð Þi tunnelð Þ

hD Cð Þi tunnelð Þ ; ð16Þ

where the indicator function, Δ(x)=1 for x=0 and Δ(x)=0otherwise, effectively imposes steric constraints. Similarly,partition functions for the two ensembles are related byQ=Q(tunnel)⟨Δ(C)⟩(tunnel).Poor overlap between the canonical ensemble of interest

and that of the noninteracting reference state requires thatthe ratio of partition functions Q/Q(ref) in Eq. (4) becomputed in stages. To avoid performing simulations withhard steric constraints, we first make use of the aboveresult for the regularized partition function,

QQðrefÞ =

QQðtunnelÞ

QðtunnelÞ

QðrefÞ = hDðCÞiðtunnelÞ QðtunnelÞ

QðrefÞ ð17Þ

The statistical consequences of adding the dihedralpotential Edihedrals to our reference system can be evaluatedwith little computational effort, since no coupling amongdifferent rotatable bonds is involved. We can even cal-culate the corresponding ratio of partition functionsanalytically, Q(ref,dihedrals)/Q(ref) = (2πσ2)N

(sp3)/2. We intro-duce additional factors to exploit this simplicity,

QQ refð Þ = hD Cð Þi tunnelð Þ Q tunnelð Þ

Q ref; dihedralsð ÞQ ref; dihedralsð Þ

Q refð Þ : ð18Þ

Finally, we introduce non-bonded and IS interactions ina gradual way through the potential

E switchð Þ λð Þ = Edihedrals + λ EðtunnelÞnon−bonded + Eimplicit solvent

h i:ð19Þ

By varying the switching parameter λ between 0and 1, we interpolate between ensembles; in particular,E(switch)(0)=E(ref,dihedrals) and E(switch)(1)=E(tunnel). The non-interacting reference ensemble with dihedral bias can thenbe transformed into the fully interacting ensemble in aseries of M steps,

Q tunnelð Þ

Q ref;dihedralsð Þ =Q switchð Þ λ0ð ÞQ switchð Þ λ1ð Þ

Q switchð Þ λ1ð ÞQ switchð Þ λ2ð Þ N

Q switchð Þ λM�1ð ÞQ switchð Þ λMð Þ ;

ð20Þwhere λi=1− i/M. Partition function ratios are evaluatedaccording to

Q switchð Þ λi�1ð ÞQ switchð Þ λið Þ =hexp� h

ME tunnelð Þnon−bondedmEimplicit solvent

� � i switchð Þλi

;

ð21Þwhere hd i switchð Þ

λidenotes an average in the ensemble cor-

responding to energy function E(switch)(λi). By making Mlarge, the difference between consecutive ensembles can be


made arbitrarily small, ensuring convergence of numericalaverages in reasonable time.Our MC simulations proceed by steps that attempt to

reassign the value of a randomly selected side-chaindihedral angle χi. Trial values χi

(trial) are generated froma distribution p(gen) proportional to exp(−βEdihedrals),accounting for local dihedral biases. Specifically, for sp3–sp3 hybridized bonds,

p genð Þ χ trialð Þi

� �=13

2kj2� ��12exp �f

trialð Þ2i =2j2

h i: ð22Þ

In this case, two-thirds of the attempted MC movesinclude hopping to a different rotameric state. For sp3−sp2hybridized bonds, which lack intrinsic torsional bias inour model, trial values are selected from a distributionuniform in χi

(trial). These trial moves are accepted with aMetropolis probability57 p(acc) based on the Boltzmanndistribution determined by E(switch)(λ):

p accð Þ =min 1; exp �hλ DE tunnelð Þnon−bondedþDEimplicit solvent

h i� �h ið23Þ

Here, ΔEnon-bonded(tunnel) and ΔEimplicit solvent are changes in

interaction energies resulting from the trial move. Thisacceptance probability does not involve changes inEdihedrals, whose statistics are fully addressed by thegeneration probability of Eq. (22).Calculationswere performed formany different values of

λ (including λ=0 and λ=1, corresponding to the noninter-acting reference system with dihedral bias and theregularized full potential, respectively). All simulationswere repeated multiple times starting from randomlychosen initial side-chain configurations. Errors were esti-mated from variances among these trials or sets of trials.Straightforward Metropolis MC sampling was suffi-

cient to generate much of the data presented here. In thecase of the CaM–ligand complexes, however, preciseestimates could only be obtained with umbrella samplingtechniques. For this purpose, we employed the adaptivemethod of Wang and Landau (WL).39 Their originalprocedure was used to first construct a rough biasfunction, which was subsequently refined in severaladditional steps. During each refinement step, multipleindependent simulations were performed using the samebias, and their resulting energy distributions were pooledto obtain a new estimate for the density of states.76 Thisprocedure was repeated until the density of states couldbe confidently constructed over a range of energiesspanning those characteristic of physiological tempera-tures. Physical averages were finally computed in anonadaptive run according to

hd i =hd exp �hE Q;Að Þ½ �exp W Eð Þ½ �iW Eð Þhexp �hE Q;Að Þ½ �exp W Eð Þ½ �iW Eð Þ

; ð24Þ

where W(E) denotes the WL bias potential in units of kBTand ⟨·⟩W(E) indicates an average over the WL-biasedensemble.

Acknowledgements

This workwas supported by the Director, Office ofScience, Office of Basic Energy Sciences, MaterialsSciences and Engineering Division, of the U.S.

Department of Energy under Contract No. DE-AC02-05CH11231. K.H.D. was supported by aNational Science Foundation Graduate ResearchFellowship and the Berkeley Fellowship.

Supplementary Data

Supplementary data associated with this articlecan be found, in the online version, at doi:10.1016/j.jmb.2009.05.068

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Calculation of Proteins Total Side-Chain Torsional ...geisslergroup.org/wp-content/uploads/2019/06/Pub2009-3.pdf · organic crystals.1 This observation motivated in early studies