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An Assessment of the Accuracy ofthe RRIGS Hydration Potential:Comparison to Solutions of thePoisson]Boltzmann Equation

JOSEPH D. AUGSPURGER,U HAROLD A. SCHERAGACornell University, Baker Laboratory of Chemistry, Ithaca, New York 14853-1301

Received 11 June 1996; accepted 28 October 1996

ABSTRACT: A rapid, pairwise hydration potential, the reduced radiusŽ .independent Gaussian sphere RRIGS approximation, has been presented

recently. Because experimental values of the conformational dependence of thehydration free energy are unavailable, this hydration potential is testable by

Ž .comparison to a presumably more accurate yet more computationally intensivemodel. One such method is the electrostatic hydration approach, which modelsthe protein as a collection of point charges in a low-dielectric medium and thesolvent as a high-dielectric continuum. The electrostatic free energy can bedetermined by solving the Poisson]Boltzmann equation, which is carried outwith the program DelPhi. The total free energy of hydration is calculated byadding a free energy of cavity formation term to this electrostatic term.Comparison is made for many conformations of two proteins, bovine pancreatic

Ž .trypsin inhibitor BPTI and the carboxy-terminal fragment of the L7rL12Ž .ribosomal protein CTF . Thirty-nine near-native structures of BPTI, previously

generated by Ripoll and coworkers, and 150 conformations of CTF, generated bya threading algorithm to cover a wide range of conformational space, were usedin these comparisons. It is shown that, for the neutral forms of these proteins,the RRIGS hydration potential correlates very well with the electrostatic modelhydration free energy, although the correlation is better for the CTF

* Special Fellow of the Leukemia Society of America.Correspondence to: H. A. ScheragaContract grant sponsor: National Institutes of Health; con-

tract grant number A: GM-14312Contract grant sponsor: National Science Foundation; con-

tract grant number: MCB95-13167Contract grant sponsor: National Foundation for Cancer

ResearchContract grant sponsor: Association for International Cancer

ResearchContract grant sponsor: Leukemia Society of America

Q 1997 by John Wiley & Sons, Inc. CCC 0192-8651 / 97 / 081072-07

RRIGS HYDRATION POTENTIAL

conformations than for the near-native BPTI conformations. For charged forms,the correlation is much poorer. These results serve as evidence that solvent-exposure models of hydration, which leave out cooperative effects betweendifferent groups, may be appropriate for modeling neutral or slightly chargedspecies, because these cooperative effects are likely to be small. However, forhighly charged species where cooperative effects are surely large, such anapproach will be less accurate. Q 1997 by John Wiley & Sons, Inc. J ComputChem 18: 1072]1078, 1997

Introduction

ne approach to the prediction of proteinO structure is minimization of a conformation-ally dependent energy function. This approach isbased on the widely accepted notion that proteinstructures are thermodynamically stable; that is,that they adopt the structure which corresponds tothe global minimum of the conformational energyhypersurface. The successful application of thisapproach to protein structure prediction requiresthat the energy function be computationally effi-

Žcient so that the search of the energy hypersurface.can be accomplished in a reasonable time and that

it accurately reflect the energetics of the protein insolution as a function of its conformation.

We have recently introduced a method to modelthe conformationally dependent free energy of hy-dration that satisfies the first of these two criteria.The reduced radius, independent Gaussian sphereŽ . 1RRIGS hydration potential was developed foruse with the empirical conformational energy pro-

Ž . 2 ] 5gram for peptides ECEPP , which uses fixedbond lengths and bond angles. It was shown to bequite rapid, because it is based on a pairwisecalculation of the exposed volume of the hydration

Ž .shell VHS about each atom. RRIGS is based onthe approximation that the total hydration freeenergy is a sum of the hydration free energies ofthe individual atoms, and that the hydration freeenergy of each atom is proportional to its solvent

Ž .exposure as measured by the exposed VHS . TheVHS is related to the hydration free energy ofdifferent atom types by empirically determinedparameters. The present work assesses the accu-racy of the RRIGS potential, that is, the aforemen-tioned second criterion.

Ideally, a direct comparison should be made toexperimental data to assess the accuracy of a theo-retical model. In this case, there are no direct,experimental data for conformationally dependent

hydration free energies of proteins. Therefore, wechose to evaluate RRIGS by comparison to a widely

Žused and presumably more accurate although.more computationally intensive method of calcu-

lating conformationally dependent hydration freeenergies. The electrostatic model of hydration of aprotein treats the protein as a collection of pointcharges in a low-dielectric medium, immersed in ahigh-dielectric continuum; the electrostatic free en-ergy is calculated by solving the Poisson]Boltz-mann equation. This is augmented by a term thatreflects the free energy of cavity formation in wa-ter. Whereas this method is widely used, it iscomputationally too intensive to be appropriatewith currently available computers to carry out asearch for the global minimum of the very com-plex energy hypersurface of a protein. Thus, wecompare the hydration free energy predicted bythe rapid RRIGS potential to those of the electro-static hydration model to assess its accuracy.

Theory and Methods

To make a significant test of the RRIGS hydra-tion potential requires that many conformationswhich span a large portion of conformational spacebe examined. To generate such an ensemble ofstructures, a threading algorithm was used. In thisthreading algorithm, the amino acid sequence of agiven protein is made to adopt the structures ofmany different proteins. Obviously, the sequenceto be threaded must possess fewer residues thanthe target structures into which it will be threaded;furthermore, the side-chain conformations cannot

Ž .be defined as they will be mostly different. Thepresence of disulfide bonds complicates threading,because the individual cysteine residues of thesequence to be threaded are not likely to be closeto each other in the target structures. Thus, wechoose to apply this threading approach to gener-ate an ensemble of structures of the carboxy-termi-

Ž .nal fragment CTF of the L7rL12 ribosomal pro-Ž .tein; this fragment contains 68 residues 53]120 ,

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possesses no disulfide bonds, and a high-resolu-tion crystal structure is known.6

A data base of 197 nonhomologous proteins wastaken from the protein data bank for which high-

Žresolution structures were available resolution˚.- 2.0 A . For each of 150 structures taken from

this data base, both the structure into which theamino acid sequence of CTF would be threaded as

Ž .well as the starting position residue number werechosen randomly. The following protocol was usedto locate a low-energy local minimum of theECEPP q RRIGS conformational energy functionfor CTF which was close to the threaded proteinstructure:

1. A terminally blocked, 68-residue poly-L-alanine backbone structure was generated bystarting from the dihedral angles, f, c, v, ofthe experimental target structure and mini-mizing the rms deviation between the back-bone atoms of poly-L-alanine and the back-bone atoms of the target structure. This typi-cally resulted in an rms deviation from thetarget structure for the backbone atoms of

˚0.3]0.6 A.2. Starting with these resulting backbone dihe-

dral angles, f, c, v, for CTF, each side-chaindihedral angle of CTF was minimized se-quentially while the backbone dihedral an-gles were held fixed. Three cycles of mini-mization of all side-chain dihedral angleswere carried out.

3. Finally, from the resulting minimized side-chain structure, a local minimization with allbackbone and side-chain dihedral angles al-lowed to vary was carried out by means ofthe SUMSL minimization routine7 to arrive atthe final, local minimum conformation.

When this protocol was applied to the experimen-tal CTF structure, the resulting structure possessed

˚a backbone rms deviation of 0.97 A from the exper-imental structure and an ECEPPr35 energy ofy413.3 kcalrmol. In a small number of cases,

Žunphysical structures e.g., with a side chain over-.lapping the backbone resulted when one of the

ŽCTF prolines which in ECEPP have fixed dihedral.angles, f was positioned just before a turn in the

target structure, and such high-energy conforma-tions were discarded. This threading algorithmwas parallelized to take advantage of the IBM SP2parallel architecture, using up to 32 processors inone run, and was used to generate 150 differentCTF conformations.

A second set of structures was also examined.This consisted of 39 structures of BPTI which arelocal minima of the ECEPP function. They weregenerated by application of the self-consistent elec-

Ž .trostatic field SCEF and electrostatically drivenŽ .Monte Carlo EDMC methods starting from a

near-native conformation; the largest rms devia-tion of the backbone atoms of these conformations

˚ 8from one of the crystal structures was about 2 A.These represent a different type of ensemble ofconformations, namely a cluster of closely relatedstructures. They were included in this assessmentto test whether the RRIGS potential will accuratelydiscriminate among native and near-native struc-tures in terms of the hydration free energy.

The RRIGS hydration potential is defined to be:

RIGGS Ž . Ž .DG s d VHS 1iÝhyd ii

Žwhere i represents all atoms in the protein exceptfor the nonpolar hydrogens because a united-atomapproach was used to treat the hydration parame-

1.ters and d is an empirically determined parame-iter. A set of d parameters, appropriate for model-iing all 20 naturally occurring amino acids in theirneutral states, using 24 different atom types, wasreported previously.1 These parameters were de-

Ž .termined by a least squares fit of eq. 1 to 140experimentally determined free energies of hydra-tion for small organic molecules. An enlarged dataset was generated by including nine additionalŽ .ionic species, for which experimentally derivedvalues of DG have been reported.9 Two newhydparameter sets were generated, in which four andsix, respectively, new atom types were added todescribe the charged species, by least squares fit-ting.

The free energy of hydration based on the elec-trostatic model can be described by:

ELEC Ž .DG s DG q DG 2hyd PB cav

where DG represents the electrostatic free en-PBergy found by solving the Poisson]Boltzmannequation and DG is the free energy of cavitycavformation. DG was calculated by means of thePBDelPhi program, version 5.0.10 The inner and outerdielectric constants were set to 2.0 and 80.0, re-spectively. The number of grid elements was de-termined by setting the box filled parameter to

˚80% with a grid size of 0.5 A. Convergence was˚Ž .tested by using a smaller grid size 0.33 A as well

as by adding a slight offset to the grid origin;neither of these led to changes of more than 0.5%

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in the values of DG . Two sets of atomic chargesPBwere examined. The PARSE set of charges11 hadbeen determined empirically to reproduce experi-mental free energies of hydration, and AMBER

Ž .charges from the AMBER force field had beengenerated by fitting point charge models to elec-trostatic potential maps calculated by ab initiomethods.12

The cavity formation term of Sitkoff et al.11 isgiven by

Ž .DG s gS q b 3cav acc

˚2where g s 0.005 kcalrmolrA , S is the totalaccsolvent accessible surface area, and b s 0.86kcalrmol; the g and b parameters had been deter-mined by fitting experimental alkane transfer freeenergies to the computed accessible surface area.11

The molecular surface program of Connolly13, 14

Ž .was used to calculate the surface area S of theaccdifferent protein conformations.

Results and Discussion

Figure 1 depicts a comparison between DGRRIGShyd

and DGELEC for the 39 BPTI structures and the 150hydCTF structures for both the PARSE and AMBERcharge sets, with both proteins in the neutral state.As can be seen from the figure, there is a strongcorrelation between the RRIGS hydration free en-ergies and the electrostatic hydration free energiesfor both charge sets, with correlation coefficients of0.904 and 0.908 with the PARSE and AMBERcharge sets, respectively. The DGRRIGS values arehyd

ELEC Ž .seen to match the DG PARSE results morehydclosely. This result is not unexpected because boththe RRIGS potential and the PARSE charge setwere developed empirically to reproduce experi-mental hydration free energies, whereas the AM-BER charge set was not.

Ž .These results for CTF depicted in Fig. 1 showthat, for a wide range of conformational spacewhich spans a range of hydration free energies ofseveral hundred kilocalories per mole, the RRIGShydration model correlates strongly with the elec-trostatic hydration model. This is encouraging, in-dicating that, for widely different conformations ofneutral species, the RRIGS hydration potential canprovide essentially the same information as theelectrostatic hydration model, but much faster interms of computational time. On a single thin node

Žof an IBM SP2 computer the equivalent of an.RS6000r390 , DelPhi calculations required about

FIGURE 1. The free energy of hydration of 150conformations of CTF and 39 conformations of BPTI as

( ELEC)predicted by the electrostatic model DG and byhyd( RRIGS)the RRIGS potential DG for both proteins in theirhyd

neutral states are compared for both the PARSE charge( )set and the AMBER charge set. Filled circles v

( )represent CTF with PARSE charges, open circles `( )BPTI with PARSE charges, filled triangles ' CTF with

( )AMBER charges, and open triangles ^ BPTI withAMBER charges. The solid line would represent exact

( ELEC RRIGS )agreement DG = DG .hyd hyd

80 seconds for CTF and about 40 seconds for BPTI,whereas a single RRIGS evaluation required only0.8 and 0.6 seconds, respectively. This represents aspeedup of almost two orders of magnitude by theRRIGS hydration potential compared to DelPhi.

Close inspection of Figure 1 reveals that, for theBPTI conformations which are a set of near-nativeconformations, there are more discrepancies in therelative ordering of the hydration free energy ascalculated by the RRIGS hydration potential andby DelPhi. The conformation that possesses thelowest electrostatic hydration free energy has aRRIGS hydration energy about 50 kcalrmol higherthan the lowest RRIGS hydration energy. This in-dicates that the RRIGS hydration potential may bemost suited for rapid screening of large numbersof widely varying conformations, as may be en-countered and examined with a low-resolutionconformational energy function.

However, the neutral charge state for these twoproteins is not a physically realistic one; at pH 7,the GLU, ASP, LYS, and ARG side chains wouldall be expected to be ionized. A more realistic testwould involve the examination of the two proteins

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AUGSPURGER AND SCHERAGA

in their charged states. Thus, we now compare theRRIGS potential to the electrostatic hydration

Ž .model for the fully charged states of BPTI q5Ž .and CTF q2 . We first examined the charged

states using parameter sets generated by fittingexperimental free energies of hydration of smallmolecules, including the few known ionic species.9

Ž .The results not shown exhibit essentially no cor-relation between DGRRIGS and DGELEC.hyd hyd

A difficulty in this approach is the dearth ofexperimental data for charged species, leading to agreat imbalance between the number of neutralŽ . Ž .140 and charged 9 species in the data set usedto parameterize the hydration free energies. Be-cause of this lack of experimental data, we choseto use the data from the electrostatic hydrationmodel in place of experimental hydration free en-ergies to make a different kind of test expressed interms of the following question: Can the DGELEC

hyd

values for a training subset from the 150 CTF and39 BPTI test conformations be used to generate aset of d values which would allow DGRRIGS toi hydreproduce DELEC for the remaining conformations?hyd

First, 50 CTF conformations and all 39 BPTIconformations were used as a training set to gener-ate a set of d values, which were then used toi

calculate DGRRIGS for the remaining 100 CTF con-hyd

formations. This result is shown in Figure 2. Thecorrelation between DGRRIGS and DGELEC is muchhyd hyd

less than the results for neutral proteins. The re-sulting correlation coefficients were 0.375, y0.040,and 0.118 for the BPTI conformations, the 50 CTFconformations included in the training set, and the100 test CTF conformations, respectively.

A second fitting was carried out using as thetraining set the 100 CTF conformations and 20BPTI conformations. The results for the fitting of50 conformations of CTF and 19 conformations ofBPTI with this training set are given in Figure 3.The CTF results are inaccurate, as in the firstfitting shown in Figure 2; moreover, the BPTIconformations now exhibit a negative correlation.The resulting correlation coefficients were 0.455,0.322, y0.731, and 0.173, for the BPTI and CTFconformations in the training set and the BPTI andCTF test conformations, respectively. These resultsindicate that the RRIGS hydration method cannotreproduce the hydration energies of the electro-static hydration model for the proteins in thesehighly charged states.

To try to identify the inadequacy of the RRIGShydration potential in charged proteins, two more

FIGURE 2. Comparison of the free energy of hydrationfor 100 charged CTF conformations, indicated by open

( )squares I , as predicted by the electrostatic model( ELEC)DG using PARSE charges, and by the RRIGShyd

( RRIGS)potential DG , where the d parameters werehyd idetermined by fitting the DGELEC values from a traininghydset of 50 CTF conformations and 39 BPTI conformations.The fitted values for the training set are also depicted,

( )with filled circles v indicating the CTF and open circles( )` the BPTI conformations.

fitting procedures were carried out, using all of theconformations but treating BPTI and CTF sepa-rately. These results are shown in Figures 4 and 5,respectively, and the correlation coefficients are0.947 for BPTI and 0.455 for CTF. There is a signifi-cant difference between the capability of the RRIGSmethod to reproduce the electrostatic hydrationenergies for these two different data sets; the fit ismuch better for the BPTI than for the CTF struc-tures.

It is important to reiterate the difference be-tween these two sets of conformations. The CTFset contains widely different conformations, withthe electrostatic hydration varying over a range ofnearly 500 kcalrmol. However, the BPTI set con-tains near-native conformations, with the largestbackbone rms deviation from the native structure

˚being 2 A or less. A significant difference betweenthe RRIGS method and the electrostatic method isthat the RRIGS approach neglects cooperative ef-fects; it is designed to be pairwise. These coopera-tive effects, however, are included in the electro-static method. Hence, it appears that the RRIGSmethod can reproduce DGELEC for neutral proteinshyd

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RRIGS HYDRATION POTENTIAL

FIGURE 3. Comparison of the free energy of hydrationfor 50 charged CTF conformations, indicated by filled

( )circles v , and 19 charged BPTI conformations,( )indicated by open circles ` , as predicted by the

( ELEC)electrostatic model DG using PARSE charges, andhyd( RRIGS)by the RRIGS potential DG , where the dhyd i

parameters were determined by fitting the DGELEC valueshydfrom a training set of 100 CTF conformations and 20BPTI conformations. The fitted values for the training set

( )are also depicted, with open triangles ^ indicating the( )CTF and filled triangles ' the BPTI conformations.

FIGURE 4. Comparison of DGRRIGS to DGELEC for allhyd hyd39 BPTI conformations when the d parameters wereigenerated by fitting the 39 DGELEC values of BPTI, usinghydthe RRIGS algorithm.

FIGURE 5. Comparison of DGRRIGS to DGELEC for allhyd hyd150 CTF conformations when the d parameters wereigenerated by fitting the 150 DGELEC values of CTF,hydusing the RRIGS algorithm.

Ž .Fig. 1 in which these cooperative effects wouldbe expected to be smaller than for charged pro-teins.

As seen in Figure 4, the RRIGS method alsoappears to reproduce DGELEC for charged BPTI. Ahydpossible reasons for this is that, because the con-formations are roughly similar but with an energyrange of about 100 kcalrmol, the cooperative ef-fects are likely to be nearly constant. On the otherhand, for the charged CTF conformations, wherethese cooperative effects are large and highly vari-able over the wide range of structures examined, a

Ž .pairwise approach is simply inadequate Fig. 5 .This conjecture regarding the importance of co-

operative effects on the ionizable side-chain groupsreceives support from the recent study of a 17-re-sidue peptide by Ripoll et al.15 They used a newmethod to compute the helix content of this pep-tide as a function of pH, whereby the degree ofionization of each ionizable group was calculatedfor different conformations. These calculationsshowed that the degree of ionization of the differ-ent ionizable side chains is strongly influenced bythe conformation of the peptide, which in turnstrongly influences the hydration energy. Theseresults,15 together with those of the present study,illustrate the necessity of including cooperativeeffects to model the interaction of highly chargedproteins with water accurately.

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Conclusion

The conformational hydration free energy of alarge number of conformations of two proteins, aspredicted by the RRIGS pairwise potential, hasbeen compared to the more computationally inten-sive electrostatic hydration model. It has beenshown to correlate quite strongly to the hydrationfree energies of the electrostatic hydration modelfor a neutral protein in a large sampling of confor-mational space, somewhat less so for a sampling ofnear-native conformations, and for roughly similar

Žconformations of a charged protein even as the.hydration energy varies by nearly 100 kcalrmol .

However, for very different conformations of acharged protein, it seems inadequate to model thehydration free energy accurately. We attribute thisto the importance of cooperative electrostatic ef-fects in this case. These results indicate that therapid RRIGS potential appears to be well suitedfor studying neutral or slightly charged peptidesin solution.

Acknowledgments

All computations reported in this study werecarried out on the IBM SP2 computer at the Cor-nell National Supercomputing Facility, a resourceof the Cornell Center for Theory and Simulation inScience and Engineering which receives majorfunding from the National Science Foundation, theState of New York, the IBM Corporation, andmembers of its Corporate Research Institute, with

additional funds from the National Institutes ofHealth. We thank Professor Barry Honig for pro-viding the most recent version of DelPhi, andDaniel R. Ripoll for the ECEPPAK version of theECEPPr3 program, which was modified to carryout the threading algorithm.

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