Rigidity Theory-Based Approximation of Vibrational EntropyChanges upon Binding to BiomoleculesHolger Gohlke,*,† Ido Y. Ben-Shalom,† Hannes Kopitz,† Stefania Pfeiffer-Marek,‡

and Karl-Heinz Baringhaus§

†Institute for Pharmaceutical and Medicinal Chemistry, Department of Mathematics and Natural Sciences, Heinrich Heine UniversityDusseldorf, 40225 Dusseldorf, Germany‡R&D/Pre-Development Sciences, Sanofi-Aventis Deutschland GmbH, Industriepark Hochst, 65926 Frankfurt am Main, Germany§R&D Resources/Site Direction, Sanofi-Aventis Deutschland GmbH, Industriepark Hochst, 65926 Frankfurt am Main, Germany

*S Supporting Information

ABSTRACT: We introduce a computationally efficientapproximation of vibrational entropy changes (ΔSvib) uponbinding to biomolecules based on rigidity theory. Fromconstraint network representations of the binding partners,ΔSvib is estimated from changes in the number of lowfrequency (“spongy”) modes with respect to changes in thenetworks’ coordination number. Compared to ΔSvib computedby normal-mode analysis (NMA), our approach yieldssignificant and good to fair correlations for data sets ofprotein−protein and protein−ligand complexes. Our approachcould be a valuable alternative to NMA-based ΔSvib computation in end-point (free) energy methods.

End-point (free) energy methods such as MM-PBSA1 orMM-GBSA2,3 are widely used in the early stages of drug

discovery to rank potential ligands4−6 and for identifyinginteraction hot spots in epitopes of protein−protein com-plexes.7,8 The methods’ computational efficiency results fromseveral approximations, including the estimate of changes in theconfigurational entropies (ΔSconfig) of the binding partners.9,10

However, likely the largest uncertainty in MM-PB(GB)SA-typecalculations originates from these estimates.1,11

In MM-PB(GB)SA-type calculations, ΔSconfig is usuallyestimated in the rigid rotor, harmonic oscillator (RRHO)approximation;12 there, translational, rotational, and vibrationalcontributions toΔSconfig are computed separately for each speciesin the binding equilibrium by gas-phase statistical mechanics.13

Estimating changes in the vibrational entropy (ΔSvib) uponcomplex formation is particularly challenging. As to fundamentalchallenges, within the RRHO approximation, changes in thevibrational entropy (ΔSvib) are estimated by normal-modeanalysis (NMA).14 For rigid small molecules, compared toreference values, absolute average percentage deviations up to2.5% were found in this context when using unscaled harmonicfrequencies.15 Applying NMA likely also leads to systematicerrors because anharmonic contributions are neglected.15,16 Thismay become prominent when neglecting the anharmonicity ofthe torsional potential surface because, due to the anharmonicity,energy levels are more congested than predicted by a harmonicpotential.16 In addition, for individually minimized structuresfrom the same trajectory, variations in ΔSvib upon complexationof ∼5 kcal mol−1 have been observed.17 Still, NMA-based ΔSvibestimates are widely used18−20 and can be considered the gold

standard. As to technical challenges, estimating ΔSvib by NMA iscomputationally very burdensome, which precludes applyingsuch calculations to more than a few protein−ligand complexesin general.21 Consequently, alternative methods for computingΔSvib have been applied including the use of quasi-harmonicanalysis (QHA),22 variations of the NMA approach,23,24 or asolvent-accessible surface area-based model.25 Issues of con-vergence have been found to pose a limitation to thestraightforward applicability of QHA to problems of binding tobiomolecules, however.26

Here, we introduce a computationally highly efficientapproximation of ΔSvib upon binding to biomolecules based onrigidity theory. We compare its results for data sets of protein−protein and protein−small molecule complexes to thoseobtained with NMA-based ΔSvib and find significant and goodto fair correlations. The principle idea underlying our approach isthat, rather than estimating ΔSvib from changes in the vibrationalfrequencies of normal modes and, hence, the width of energywells upon binding, we estimate ΔSvib from changes in thevariation of the number of low frequency modes. This is describedin detail in the following.As to the theoretical background, in normal-mode analysis, a

potential energy function V(x) is expanded in a Taylor seriesexpansion about some point x0.

27 If x0 denotes the location of aminimum of V(x), the gradient of V(x) vanishes. If also third andhigher-order derivatives of V(x) are ignored, the dynamics of the

Received: January 7, 2017Published: March 29, 2017

Letter

pubs.acs.org/JCTC

© 2017 American Chemical Society 1495 DOI: 10.1021/acs.jctc.7b00014J. Chem. Theory Comput. 2017, 13, 1495−1502

system can be described in terms of linearly independent normalmodes obtained from diagonalizing the Hessian matrix, each oneassociated with a frequency νi. From νi, the vibrationalcontributions to thermodynamic properties can be deter-mined.28,29 For Svib, one obtains (eq 1)

∑ ν=

−− −ν

ν−

=

−−⎡

⎣⎢⎤⎦⎥S T

he

k T e1

ln(1 )i

Ni

h k Th k T

vib1

1

3 6

/ B/

i

i

B

B

(1)

Note that Svib is particularly sensitive to the frequencies of thelowest modes of vibration;12,28 in contrast, for νi →∞, both thefirst and second term in the square brackets approach zero. Thelow-frequency modes reflect the presence of weak forces in thebiomolecular system, encoded, e.g., as torsion angle and van derWaals potentials in current state-of-the-art biomolecular forcefields.30

To now make the connection to approximating Svib based onrigidity theory, we first neglect weak forces in V(x), resulting in aKirkwood31 or Keating32 potential VK, schematically written as(eq 2)33

α β θ= Δ + ΔV l2

( )2

( )K2 2

(2)

Here, VK describes small displacements from an equilibriumstructure in a bond-bending network in terms of changes in bondlength (Δl) and bond angle (Δθ), with α and β being the forceconstants for bond stretching and bending, respectively.Diagonalizing the Hessian from eq 2 ascertains a number F ofvibrational modes with zero frequency.33 These so-called floppymodes correspond to the ways in which the network can becontinuously deformed at no cost in energy by rotations aroundbonds; F decreases with an increasing mean coordination ⟨r⟩ inthe network (Figure 1A).33 For a comprehensive review onrigidity theory for biomolecules see ref 34.As to the methodological development, we now describe how Svib

can be estimated from the variation of the number of Fwith ⟨r⟩. Instudies on random central-force networks and general lattices,the negative of the number of floppy modes,−F, has been shownto act as a free energy:39,40 −F is a concave function of ⟨r⟩ (i.e.,−F(2) = −d2F/d⟨r⟩2 ≤ 0) (Figure 1A), as required of a freeenergy, such that if there is an ambiguity, the system will alwaysbe in the lowest free energy, i.e., maximum floppy modes,state.35,36 Note that the floppy modes were considered to havezero frequency in this context. However, in “real” molecularsystems, floppy modes will become “spongy”,33 i.e., will have asmall finite frequency, if weak forces are present in thenetwork.37,38 Assuming that only these “spongy” modescontribute to the estimate of Svib (see the comment below eq 1for a justification) and that the modes have the same small finitefrequency νF (see eqs S1−S8 in the Supporting Information, SI,as to the adequacy of this assumption) then yields for thecontribution to the free energy due to vibrations, Gvib (eq 3; seeeqs S9−S12 in the Supporting Information for details):

υ≈ − −⎜ ⎟⎛⎝

⎞⎠G F NkT Nh

32 Fvib (3)

Employ ing the defining equat ion for ent ropyS = −(∂G/∂T)N,p

29 and considering that F = f(⟨r⟩) (see above)and ⟨r⟩ = f(T) (see below) results in (eq 4)

υ

= −∂∂

≈ − +⎜ ⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝

⎞⎠

SG

T

Fr

rT

NkT Nh F Nkd

ddd

32

( )

N p

F

vibvib

,

(4)

Here, ⟨r⟩ has been considered a temperature-related quantityas in flexibility and rigidity analysis before;39 in thermal unfoldingsimulations of constraint network representations of biomole-cules, ⟨r⟩ was decreased with increasing temperature.40,41 Toproceed with eq 4, we thus make the assumption (eq 5)

= −rT

cdd (5)

Figure 1. (A) Number of floppy modes (F) as a function of the meancoordination ⟨r⟩ is shown exemplarily for one MD simulations-generated conformation of the trypsin-ligand complex (PDB ID1K1N, solid line) and the protein only (dashed line). Along the lines,rigid cluster decompositions of both structures computed at identicalEcut values (mentioned under the structures, in kcal mol

−1), respectively,are depicted, with each colored body indicating one rigid cluster; thelargest rigid cluster is colored in blue. (B)−F(1) =−dF/d⟨r⟩, introducedhere as an entropy-like quantity, is shown as a function of the meancoordination ⟨r⟩, corresponding to the curves in panel A. Here, Ecutvalues corresponding to ⟨r⟩ are depicted along the abscissa for thecomplex (Ecut

com) and the protein (Ecutrec). To compare −F(1) for

different biomolecular systems, the respective ⟨r⟩ at a fixed Ecut value wasdetermined. The vertical dashed and dotted lines depict −F(1) forcomplex and protein at Ecut = −1.0 kcal mol−1. Note that in this Ecutrange, the rigid cluster decompositions (panel A) differ the most.

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that is, ⟨r⟩ decreases linearly with increasing T. This assumptionwas inspired by a consensus view on protein dynamics, whichrevealed that proteins as a whole behave like dense liquids42 andthat the mean coordination in the liquid water has been proposedto decrease linearly with increasing temperature.43 Furthermore,Figure 1 exemplarily depicts that −dF/d⟨r⟩ is larger by a factorO(101) than F. We feel it is safe to assume that this example canbe generalized for the type of constraint networks considered inthis work, proteins and protein−ligand complexes, given that for26 proteins with diverse architectures, including monomers andoligomers, very similar F(⟨r⟩) and −dF/d⟨r⟩(⟨r⟩) curves werefound, leading to the notion of a universal behavior in terms ofrigidity loss upon unfolding.44 Finally, for the “spongy” modesconsidered here, NkT ≫ 3/2NhνF at T ≈ 300 K and vF ≲ 150cm−1;45 accordingly, (NkT − 3/2NhνF) is larger by a factorO(102 to 103) thanNk. If |c| (eq 5) > 1/(101 × (102 to 103)) K−1,the second term on the right side of eq 4 can then be neglectedcompared to the first one. For the liquid water, |c| = 0.004 K−1 hasbeen reported,43 fulfilling this condition. Taken together, weintroduce here that the negative of the change in the number offloppy modes with respect to a change in the mean coordinationis related to the vibrational entropy

≈ − ′ = − ′SFr

c F cd

dvib(1)

(6)

with c′ = c × (NkT − 3/2NhνF).Considering that the idea behind vibrational entropy is that a

phase space is explored by atoms as they vibrate and that thevibrational entropy is the larger the larger the phase space is, eq 6yields that −F(1) is a measure for the size of the phase space.Notably, it thereby matters how the network reacts to changes(F(1)), rather than how the network’s floppyness is in absoluteterms (F) (see also next paragraph). We note that, in addition to−F, a thermodynamic interpretation has been provided for F(2) inthat F(2) has been regarded as a specific heat and used tocharacterize the order of transitions of constraint networksswitching between rigid and flexible states.35 However, to thebest of our knowledge, no thermodynamic interpretation of F(1)

has yet been presented.For F(1), a relation with respect to F and the total number of

bonds NB in constraint networks with a f ixed number of nearestneighbors has been derived (eq 7)38,39

− ∼ − ≥F N F N(3 )/ 0(1)B (7)

Equation 7 yields that −F(1) ≥ 0, as required of an entropy.Furthermore, −F(1) depends on the actual network state (Figure1B): IfNB is low, related to a very flexible network (Figure 1A),−F(1) approaches a positive limit, indicating the maximum entropyof the system; ifNB is high, related to a very rigid network (Figure1A), −F(1) approaches zero, as expected for a system for whichonly one state of realization exists. Note that adding constraints,e.g., due to binding of a ligand, to a constraint networkrepresentation of a biomolecule with either low or high NB willnot lead to marked changes in −F(1) (Figure 1B). In contrast,marked changes are to be expected when constraints are added toa network with intermediate NB (see vertical lines in Figure 1Band the large change in the related rigid cluster decompositionsin Figure 1A), reminiscent of a biomolecule with marginalstability.46 The change in vibrational entropy upon binding to abiomolecule is then approximated as (eq 8)

Δ− = − − − − −F F F F( ) ( ) ( )(1) (1)com

(1)rec

(1)lig (8)

where com, rec, and lig refer to the complex, receptor, and ligand,respectively.As to the computational ef f iciency of approximating the change

in vibrational entropy upon binding by eq 8, three comments arein order: (I) If α and β become (infinitely) large in eq 2, the bondstretching and bending forces become bond and angleconstraints, leading to a constraint network. A representationof biomolecules in terms of constraint networks has beensuccessfully used in the analysis of bimolecular rigidity andflexibility previously; there, in addition to covalent interactions,noncovalent interactions (hydrogen bonds, salt bridges, andhydrophobic tethers) are modeled via bond and angleconstraints.47−49

(II) Rather than by diagonalizing the Hessian from eq 2, F canalso be determined by an advanced constraint (Maxwell)counting33,50 on the constraint network as implemented in thecombinatorial “pebble game” algorithm.51,52 This algorithmperforms with a time complexity of, on average, O(N), providingfor a dramatic speed up for large (biomolecular) systemscompared to the time complexity of O(N3) for matrixdiagonalization. In fact, for a normal-sized protein (∼250residues), the computing time for determining F is ∼8 s on asingle CPU core with 2.5 GHz.(III) For the description of a system’s dynamics by NMA, the

system must reside at a local minimum on the potential energyhypersurface. In MM/PB(GB)SA-type applications, typically,structures have been minimized to a root mean-square gradient(RMSG) of the potent i a l energy f rom 10− 5 to10−3 kcal mol−1 Å−1 before applying NMA.26 Performingminimizations of biomolecules to such low RMSG is computa-tionally demanding. For a protein of ∼250 residues, energyminimization and diagonalization of the Hessian require ∼2.5 hon a single CPU core with 2.5 GHz. In contrast, no minimizationis required prior to applying the “pebble game” algorithm toconstraint networks to compute F.As to computing −F(1) for eq 8, three steps are followed (see SI

for details). (I) A structural ensemble of the complex is generatedby all-atom molecular dynamics (MD) simulations. Performingthe subsequent analyses on an ensemble rather than a singlestructure overcomes the problem that results from constraintcounting are sensitive to the input structural information.53,54

(II) A constraint network is generated for each complexconformation, as done in previous studies of biomolecularrigidity and flexibility.47−49 In addition, a constraint network isgenerated for the receptor conformation extracted from therespective complex, as is for the extracted “ligand” in the case ofprotein−protein complexes. In contrast, small molecule ligandslack the typical network character and, thus, are not suitable forevaluation by constraint counting. For such ligands,−F(1)lig in eq8 is replaced by a scaled s× Svib,lig value, as detailed in the SI. In all,a so-called single-trajectory approach is pursued, as often appliedin end-point (free) energy methods, which neglects possibleconformational changes of the unbound structures but usuallygives less noisy results than the three-trajectory alternative.1

(III) A “constraint dilution trajectory” of network states {σ} isgenerated from each initial constraint network by successivelyremoving noncovalent constraints.40,41,44,55,56 Here, hydrogenbond constraints (including salt bridges) are removed in theorder of increasing strength40,44,57 such that for network state σonly those hydrogen bonds are retained that have an energy EHB≤ Ecut. The hydrogen bond energy EHB is determined from anempirical energy function58 successfully used by us59−62 andothers41,44,55,63,64 in this context. For each σ, F(⟨r⟩) is computed

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by constraint counting with the program f irst,47,51 and from that−F(1) at a given ⟨r⟩ by numerical differentiation. Using eq 7instead is not possible because atoms in constraint networksgenerated from biomolecules have a variable number of nearestneighbors. To compare −F(1) for different biomolecular systems,the respective ⟨r⟩ at a fixed Ecut value was determined. Here, Ecut= −1.0 kcal mol−1 was used unless otherwise noted, motivatedfrom previous studies.53

As to results, the validity of eqs 6 and 8 was assessed on one dataset of protein−protein complexes and four data sets of protein−small molecule complexes by comparison to vibrationalentropies computed by NMA (see SI for details). The data setof protein−protein complexes comprises four antibody−antigen,four protease−protease inhibitor, and two signaling complexes,with diverse folds, protein sizes between 775 and 8398 atoms,and binding affinities from the μM to pM range. Initially, weprobed if −F(1) behaves as an extensive property, as required ofan entropy-like quantity, and confirmed in the case of insulindimerization for Svib computed by NMA.12 We computed −F(1)for each complex, receptor, and ligand of the data set members,resulting in 3 × 10 values. When plotted against the mass of theproteins, a significant and very good correlation results (R2 =0.92; bootstrapped 95% confidence interval (CI): 0.84 < R2 <0.96; p = 2.2× 10−15; Figure S5 in the SI), demonstrating a strongdependence of−F(1) on the system size, indicative of an extensiveproperty. Not surprisingly, −F(1) and Svib of the 3 × 10complexes, receptors, and ligands, respectively, also yield a verygood correlation (R2 = 0.95; data not shown). More importantly,we next correlated Δ−F(1) and ΔSvib (eq S13 in the SI), i.e.,estimates of changes in the vibrational entropy upon binding, forthe 10 protein−protein complexes, yielding a significant andgood correlation (R2 = 0.80; 95% CI: 0.19 < R2 < 0.96; p =0.0005; c′ (eq 6) = d(ΔSvib)/d(Δ−F(1)) = 0.045 cal mol−1 K−1;Figure 2A). In contrast, ΔSvib correlated against the area of theprotein−protein epitope buried upon complex formation yields aweak correlation (R2 = 0.36; data not shown). This demonstratesthat the good correlation of Δ−F(1) versus ΔSvib does not have atrivial, i.e., size-dependent, origin; rather, Δ−F(1) describesalterations in the density of vibrational states of the complexesrelative to the binding partners apparently with good accuracy.Note that significant and good correlations were also obtained if

Ecut was set to−0.6 or−1.4 kcal mol−1 (R2 = 0.63, 0.83; Figure S6in the SI), demonstrating that our approach is robust with respectto the choice of Ecut. Finally, we used structures directly extractedfrom the MD trajectories for computing Δ−F(1), rather than theminimized ones used as input for NMA (see SI for details). Notconsidering PDB ID 2JEL, Δ−F(1) of which deviates mostbetween nonminimized and minimized structures, resulted in acorrelation with R2 = 0.54 (95% CI: 0.01 < R2 < 0.96; p = 0.02;Figure S7 in the SI). Thus, despite structural deviations betweenrespective conformations used for constraint counting andNMA, still a good correlation is obtained.Computational alanine scanning allows from a single MD

simulation an estimate of the individual contribution of eachresidue of a protein−protein complex to the binding and hasproven valuable for identifying “hot spot” residues in protein−protein epitopes.2,7,65−67 For the vibrational entropy contribu-tion, the difference in the change in Svib upon binding (ΔΔSvib)was computed by NMA from the wild type and an alaninemutant; the mutant is generated from the wild typeconformations by removing respective atoms. Here, ΔΔ−F(1)was computed analogously. The correlation of ΔΔ−F(1) versusΔΔSvib for 30 alanine mutations in the interface of Ras-Raf (PDBID 1GUA)2 is significant and weak (R2 = 0.24; 95% CI: 0.01 < R2

< 0.58; p = 0.01; Figure S8 in the SI) if Ecut = −1.0 kcal mol−1 isused and three outliers are disregarded. The correlation can bemarkedly improved (R2 = 0.51; 95% CI: 0.22 < R2 < 0.72; p = 2.7× 10−5; Figure 2B) if Ecut = −0.2 kcal mol−1 is used, againdisregarding three outliers (residues E31Ras, E37Ras, and T68Raf).Apparently, analyzing stiffer constraint networks is favorablehere, likely because the ΔΔ−F(1) values for side chains on theprotein surface become less noisy when contributions from theprotein core become less pronounced. Side chains of residuesE37Ras and T68Raf are involved in salt bridges or a polar hydrogenbond across the center of the epitope, and their influence on thevibrational entropy change is underestimated by constraintcounting (Figure S9 in the SI). Residue E31Ras engages in a saltbridge interaction at the edge of the epitope, and its influence onthe vibrational entropy change is overestimated by constraintcounting (Figure S9 in the SI). Neglecting solvent influences orcooperative effects on the strength of the polar interactions in theenergy function EHB might cause these deviations. Note that the

Figure 2. (A) Correlation of Δ−F(1) versus ΔSvib computed for 10 protein−protein complexes. The average standard errors of the mean (SEM) ofΔ−F(1) andΔSvib are∼210 and∼10.0 cal mol−1 K−1, respectively. (B) Correlation ofΔΔ−F(1) versusΔΔSvib computed for 30 alanine mutations in theinterface of Ras-Raf (PDB ID 1GUA) 2 using Ecut = −0.2 kcal mol−1. The red symbols denote mutations E31Ras, E37Ras, and T68Raf considered outliers.The average SEM values ofΔΔ−F(1) andΔΔSvib are ∼70 and ∼1.3 cal mol−1 K−1, respectively. Dashed lines indicateΔ−F(1),ΔSvib = 0; the correlationline is represented by a straight line.

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vibrational entropy contributions of most of the side chains in theRas-Raf epitope disfavor binding, as indicated byΔΔSvib > 0. TheΔΔ−F(1) values mirror this finding for all but seven of the sidechains (disregarding the three outliers).Regarding the protein−small molecule complexes, the trypsin

data set encompasses 23 complexes with ligands ranging in sizefrom filling only the S1 pocket to those capturing the entire activesite and binding affinities covering a range of ∼3.4 log units. Thecorrelation of Δ−F(1) versus ΔSvib is significant and fair (R2 =0.40; 95% CI: 0.09 < R2 < 0.66; p = 0.001; c′ (eq 6) = d(ΔSvib)/d(Δ−F(1)) = 0.182 cal mol−1 K−1; Figure 3A and Table S1 in theSI). Some ligands lead to ΔSvib > 0, whereas others show ΔSvib <0. Ligands of the former group are usually small and make fewinteractions with the protein (Figure 3D), allowing for librationalmotions of the ligand;68 in contrast, those of the latter groupusually make many interactions with different parts of the protein(Figure 3C), stiffening the protein.69 Notably, this distinctionbetween ligands is almost perfectly reflected in theΔ−F(1) values(Figure 3A), revealing that constraint counting can distinguishbetween ligand binding that leads to favorable versus disfavorablevibrational entropy contributions to the binding affinity. Thisproperty is of high importance when ranking potential ligands.12

The factor Xa data set contains 20 complex structures with smallmolecule ligands that are more similar in size (400−600 Da) andshow a narrower distribution of binding affinities (range: ∼2.7log units). As an additional challenge, the data set contains bothligands that form the well-known salt bridge with Asp189 in theS1 pocket and those that place nonpolar moieties there. Thecorrelation of Δ−F(1) versus ΔSvib is significant and fair (R2 =

0.46; 95% CI: 0.06 < R2 < 0.74; p = 0.001; c′ (eq 6) = d(ΔSvib)/d(Δ−F(1)) = 0.243 cal mol−1 K−1; Figure 3B and Table S2 in theSI). Again, both Δ−F(1) and ΔSvib distinguish between ligandbinding that leads to favorable versus disfavorable vibrationalentropy contributions to the binding affinity (Figure 3B). Aligand of the former group is IIA (PDB ID 2BOH), which placesa chlorothiophen moiety into the S1 pocket (Figure 3F); one ofthe latter group is IMA (PDB ID 1LPG), which places abenzamidine moiety there (Figure 3E). As the ligands areotherwise similar in size and interaction pattern with the protein,one can speculate that it is the locking-in of protein and ligand bya salt bridge that leads to disfavorable vibrational entropycontributions in contrast to the less restrictive interactions of thechlorothiophen moiety. Finally, we investigated two additionaldata sets of Hsp90 and HIV-1 protease−small moleculecomplexes. These data sets were rather similar to the trypsinand factor Xa data sets with respect to the number of data pointsand the range of ligand sizes and binding affinities (Hsp90 dataset: 16 complex structures with small molecule ligands rangingfrom 150−500 Da and binding affinities spanning 4.7 log units;HIV-1 protease data set: 20/500−750 Da/4.2 log units). As amajor difference, however, the width of the distribution of ΔSvibcomputed by NMA across each data set is only ∼1/3 of that ofthe trypsin and factor Xa data sets (∼22−25 cal mol−1 K−1),being very similar in magnitude to the average standard deviationof the computed ΔSvib (∼23−29 cal mol−1 K−1). Therefore,according to Kramer et al.,70 the maximum possible squaredPearson correlation coefficient (R2

max) on these data setsvanishes; in agreement, Δ−F(1) versus ΔSvib did not yield

Figure 3.Correlation ofΔ−F(1) versusΔSvib computed for protein−small molecule complexes of (A) the trypsin data set and (B) the factor Xa data set.Dashed lines indicateΔ−F(1),ΔSvib = 0; the correlation line is represented by a straight line. Data points of complexes that result in maximal favorable ordisfavorable vibrational entropy changes are circled, and the respective crystal structures of the complex are depicted: (C) ligands CRC200 (taken fromPDB ID 1K1N) and (D) nicotinamide (taken from PDB ID 2OTV); (E) ligands IMA (taken from PDB ID 1LPG) and (F) IIA (taken from PDB ID2BOH). The location of the S1 pocket in trypsin and factor Xa is indicated by a black arc; polar interactions between protein and ligand are depicted byred dashed lines. The average SEM of Δ−F(1) and ΔSvib are 0.1 and 1.0 cal mol−1 K−1, respectively.

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significant correlations (Tables S3 and S4 in the SI). Note thatthese last results restate the fundamental challenge of computingprecise changes in vibrational entropy in general,17 rather thanshowing a limitation of approximating them by Δ−F(1).In summary, we presented a computationally highly efficient

approximation of changes in the vibrational entropy (ΔSvib)upon binding to biomolecules based on rigidity theory.Compared to ΔSvib computed by NMA as a gold standard, ourapproach yields significant and good to fair correlations for datasets of protein−protein and protein−small molecule complexesas well as in alanine scanning. The slopes c′ = dΔSvib/dΔ−F(1)(eq 6) for the data sets of protein−protein and protein−ligandcomplexes differ by a factor of ∼5, and the ones for the twoprotein−ligand data sets are almost equal. While clearly furtherdata sets need to be investigated before a general conclusion canbe drawn, these results suggest that for the type of biomolecularcomplexes investigated here, a uniform c′ might exist. Withfurther data sets, it may also be worth investigating if includingthe term on the right side of the sum in eq 4 will lead to bettercorrelations withΔSvib. However, then linear regression needs tobe applied in order to determine the magnitude of c′. Overall, ourresults suggest that our approach is a valuable, computationallyefficient alternative to NMA-based ΔSvib computation, with apotential scope of application in end-point (free) energymethods.

■ COMPUTATIONAL METHODSDetails on the adequacy of the assumption that modescontributing to the estimate of Svib have the same frequency,the contribution to the free energy due to “spongy” modes, thedata set of protein−protein complexes, the data sets of protein−ligand complexes, the calculation of Svib by normal-modeanalysis, the constraint network generation and constraintcounting, and statistical evaluation are provided in theSupporting Information.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jctc.7b00014.

Supplemental tables:ΔSvib computed by NMA andΔ-F(1)computed by constraint counting for the protein−liganddata sets. Supplemental figures: Structures of the smallmolecules of the protein−ligand complexes and additionalcorrelations. (PDF)

■ AUTHOR INFORMATIONCorresponding Author*Phone: +49(0)211-81-13662. E-mail: gohlke@hhu.de.ORCIDHolger Gohlke: 0000-0001-8613-1447NotesThe authors declare the following competing financialinterest(s): S.P.-M. and K.-H.B. are employees of Sanofi-AventisDeutschland GmbH.

■ ACKNOWLEDGMENTSWe are grateful to Dr. Natasja Brooijmans for providingconformational ensembles of the four antibody−antigen andfour protease-protease inhibitor complexes, and to Dr.Christopher Pfleger, Dr. Denis Schmidt, and Christina Nutschel

for critically reading the manuscript. This work was supported byfunds from Sanofi-Aventis Deutschland GmbH. We are gratefulto the “Zentrum fur Informations- und Medientechnologie”(ZIM) at the Heinrich Heine University Dusseldorf forproviding computational support.

■ ABBREVIATIONS

CI: confidence intervalMD: molecular dynamicsMM-GBSA: molecular mechanics generalized Born surfaceareaMM-PBSA: molecular mechanics Poisson−Boltzmann sur-face areaNMA: normal-mode analysisRMSG: root mean-square gradientRRHO: rigid rotor harmonic oscillatorQHA: quasi-harmonic analysis

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