Improved Whitten -Rabinovitch Approximation for the Rice-Ramsperger-Kassel-MarcusCalculation of Unimolecular Reaction Rate Constants for Proteins

Meiling Sun,† Jeong Hee Moon,‡ and Myung Soo Kim*,†

Department of Chemistry, Seoul National UniVersity, Seoul 151-742, Korea, and Korea Research Institute ofBioscience and Biotechnology, Daejon 305-806, Korea

ReceiVed: October 2, 2006; In Final Form: January 16, 2007

The Whitten-Rabinovitch (WR) approximation used in the semi-classical calculation of the Rice-Ramsperger-Kassel-Marcus (RRKM) unimolecular reaction rate constant was improved for reliableapplication to protein reactions. The state sum data for the 10-mer of each amino acid calculated by theaccurate Beyer-Swinehart (BS) algorithm were used to obtain the residue-specific correction functions (w).The correction functions were obtained down to a much lower internal energy range than reported in theoriginal work, and the cubic, rather than quadratic, polynomial was used for data fitting. For a specifiedsequence of amino acid residues in a protein, an average was made over these functions to obtain the sequence-specific correction function to be used in the rate constant calculation. Reliability of the improved methodwas tested for dissociation of various peptides and proteins. Even at low internal energies corresponding tothe RRKM rate constant as small as 0.1 s-1, the rate constant calculated by the present method differed fromthe accurate BS result by 60% only. In contrast, the result from the original WR calculation differed from theaccurate result by a factor of 3000. Compared to the BS method, which is difficult to use for proteins, themain advantage of the present method is that the RRKM rate constant can be calculated instantly regardlessof the protein mass.

I. Introduction

In the study of a unimolecular reaction, it is often useful tohave a rate constant evaluated theoretically. For a unimolecularreaction occurring under the microcanonical condition, theRice-Ramsperger-Kassel-Marcus (RRKM) theory1-6 is widelyused to estimate the statistically expected rate constant. Thetheoretical rate constant is useful not only for direct comparisonwith the experimental one but also for various other purposes.For example, RRKM calculation is done in the field of massspectrometry7-9 as an aid for understanding the structure anddissociation dynamics of molecular ions, estimating their internalenergy contents, etc.

When the molecular rotation is ignored, the expression for aunimolecular reaction rate constant derived by the RRKM theoryis as follows.

Here F(E) is the vibrational state density of the reactant atthe internal energyE, E0 is the critical energy of the reaction,Nq(E - E0) is the vibrational state sum from 0 toE - E0 at thetransition state (TS),h is the Planck constant, andσ is thereaction path degeneracy, which is usually 1 for reactionsinvolving large molecules.

Even though the expression for the RRKM rate constant looksdeceptively simple, various difficulties are encountered in actualcalculation. The parameters needed for a RRKM calculation

are the critical energy, the complete vibrational frequency setfor the reactant, and that at the TS geometry. The majordifficulty arises from the fact that the parameters related to theTS, namely E0 and the frequencies at the TS, cannot bemeasured experimentally. The data obtained via quantumchemical calculations can be used when the TS can be foundthrough computation. The less rigorous but more widely usedapproach is to treat these as adjustable parameters. It is well-known that the RRKM rate constant remains nearly the sameregardless of the changes in individual frequencies as long asthe entropy of activation,∆Sq, is kept the same.7,10-12 Hence,the RRKM calculation of the rate-energy relation is oftentreated as a two parameter (E0 and ∆Sq) problem. In thisapproach, the frequency set at the TS is obtained from thereactant set by adjusting some of the frequencies in the latterset such that the postulated value of∆Sq results. The fact thatreasonable estimates forE0 and ∆Sq are needed for reliableestimation of the rate constant is the main drawback of thisapproach.

When one attempts a RRKM calculation for large biologicalmolecules such as peptides and proteins, additional difficultiesarise due to the very large number of degrees of freedominvolved. One of these difficulties is that it is virtually impossibleat the moment to obtain the complete set of reactant frequencieseither via experiment or via computation. In our previouspaper,13 we reported a systematic and efficient method toestimate the frequency set for a peptide or a protein with anyamino acid sequence and presented its utility in a RRKMcalculation. The method started with the vibrational frequencysets for twenty amino acids calculated at the density functionaltheory (DFT) level. Then, the frequencies disappearing uponpeptide bond formation were deleted from each set to obtainthe fictitious sets for an amino acid residue at the N- or

* To whom correspondence should be addressed. Telephone:+82-2-880-6652. Fax:+82-2-889-1568. E-mail: myungsoo@snu.ac.kr.

† Seoul National University.‡ Korea Research Institute of Bioscience and Biotechnology.

k ) σNq(E - E0)

hF(E)(1)

2747J. Phys. Chem. B2007,111,2747-2751

10.1021/jp066453t CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 02/15/2007

C-terminus or inside the chain. For a specified sequence of aprotein, these residue frequencies were collected and thefrequencies appearing upon peptide bond formation were addedto obtain the complete set for the reactant. The frequenciesappearing upon protonation of a protein were added as neededto handle the reaction of protonated proteins that are of greatinterest in the field of mass spectrometry.14-16 The methodtreated all the vibrations as harmonic and ignored anharmonicity.Also ignored was the fact that some modes are better treated asinternal rotations rather than vibrations.1,5 Such simplificationsand the estimation of the frequencies at the TS using thepostulated value of∆Sq adopted in this method are causes foruncertainty in the RRKM calculations.

Once the frequency sets for the reactant and TS are provided,various methods can be used to calculate the vibrational statesum and density. Several approximate methods had beendeveloped in early days of RRKM calculations such as theWhitten-Rabinovitch (WR) semi-classical approximation17 andthe steepest descent method18 to name a few. Even though thesemethods are efficient, all of these provide erroneous results atlow internal energy range.1 For example, Derrick et al.19 showedthat the WR method overestimated the dissociation rate constantof a small protein by orders of magnitude in the low internalenergy. Virtually all of these approximate methods becameobsolete after the invention of an efficient direct countingalgorithm by Beyer and Swinehart (BS algorithm).20 When weattempted the RRKM calculation for proteins with relativemolecular masses (RMM) as large as 10 000 or larger usingthe BS algorithm, however, we found that several hours or evendays of computation were needed.

In this paper, we will present a modification of the Whitten-Rabinovitch method for proteins. Its reliability in RRKMcalculation for proteins will be demonstrated by comparing withthe results obtained by the BS algorithm.

II. Method

In the Whitten-Rabinovitch (WR) approximation,17 the fol-lowing expression is used for the vibrational state sum assuggested by Rabinovitch and Diesen.21

HereE is the vibrational internal energy as before,Ez is thezero-point energy,s is the number of the vibrational degrees offreedom, andνi is the frequency of theith vibrational mode.This expression differs from the original semi-classical expres-sion suggested by Marcus and Rice2 in that the parametera isregarded as a molecule-dependent function of the internal energyrather than a constant (a ) 1) in the latter expression. To reducethe molecule dependence in the calculation, Whitten andRabinovitch suggested to use another functionw defined asbelow.

with

By comparing with the state sums for various organic andinorganic molecules calculated by direct count, Whitten andRabinovitch obtained the following expression forw.

Here, ε is the internal energy scaled with the zero-pointenergy.

The expression for the state density is obtained from thederivative of eq 2 as follows.

Because both the sum and density are expressed in explicitanalytical forms, the rate constant is calculated instantly as thefrequencies andE0 are provided.

As has been mentioned already, the rate constant calculatedby the WR method deviates significantly from the direct countresult in the low internal energy range. It is well-known in thefield of mass spectrometry that the critical energies for manydissociation reactions of peptide and protein ions are ratherlow.22-24 Such an ion may dissociate on the mass spectrometrictime scale (1µs to 1 s) even when the internal energy is onlya few electronvolts above the critical energy. Because the zero-point energy increases in proportion to the reactant mass, a fewelectronvolts of internal energy correspond toε less than 0.1,which is below the limit optimized by Whitten and Rabinovitch.Also to be mentioned is that thew data used to derive thefunctions in eqs 5 and 6 in the original work showed asignificant scatter even atε larger than 0.1 because these werethe values obtained for compounds with widely differentstructures.17 The corresponding scatter is expected to be smallerwhen only the data for peptides and proteins are taken intoaccount. The method used to calculate the RRKM rate constantfor proteins in this work is essentially the same as in the originalWR method. The only difference is that the sequence-specificw function derived from peptide data is used in the present work.Details of the method are as follows.

The state sum for a 10-mer, X10, of each amino acid wascalculated with the BS algorithm as a function ofε in the rangeε ) 0.0005-2. Then, thew data for each peptide were calculatedusing eqs 2-4. These were taken as the representative data forthis amino acid residue because thew data obtained from higherpolymers were essentially the same. These data were used toobtain thew function of the following form.

The first term in this function was added to achieve fits withthe correlation coefficient better than 0.99999. Two improvedWR methods, IWR′ and IWR, were developed and tested. Inthe IWR′ method, a universalw function for proteins wasestimated by averaging thew functions of the 10-mers of twentyamino acids using the natural abundances25 of the amino acidsas the weighting factors. In the IWR method, theci parametersfor each amino acid residue in a protein were collected andaveraged term by term, for example,C3 ) ⟨c3⟩, to estimate thew function for this protein. Namely, thew function used in thelatter method differed for proteins with different amino acidcomposition unlike in the former method. It will be shown inthe next section that IWR is our method of choice. The functionin eq 6 turned out to be an excellent fit to thew data atε > 1

w ) (5.00ε + 2.73ε0.5 + 3.51)-1 0.1< ε < 1.0 (5)

w ) exp(-2.4191ε0.25) 1.0e ε (6)

ε ) E/Ez (7)

F(E) )(E + aEz)

s-1

(s - 1)! ∏ hνi[1 - â(dw

dε )] (8)

w ) (c3ε1.5 + c2ε + c1ε

0.5 + c0)-1 (9)

N(E) )(E + aEz)

s

s!∏hνi

(2)

w ) (1 - a)/â (3)

â ) s - 1s

⟨ν2⟩⟨ν⟩2

(4)

2748 J. Phys. Chem. B, Vol. 111, No. 10, 2007 Sun et al.

for all the cases investigated even though calculations aboveε

) 1 were hardly needed in practical cases.Details of the method to calculate a rate-energy relation for

a protein reaction with the BS algorithm were explained in aprevious report13 and will not be repeated here. For the improvedWhitten-Rabinovitch (IWR) calculation, the array (c3, c2, c1,c0) for each amino acid residue has been stored in the softwaretogether with the residue frequencies. When the amino acidsequence of the reactant is specified, the residue frequenciesare collected and thew function is calculated from the arrays.The frequencies at the TS are estimated from the reactantfrequencies and∆Sq, which is designated by a user as one ofthe input parameters of the software. These and theE0 valueare used to calculateF(E) andNq(E - E0) and finally k(E).

III. Results and Discussion

The inverse cubic fittings of thew data were made for the10-mers of twenty amino acids as described in the previoussection. The 0.001e ε e 1.0 ranges of the twentyw functionsthus obtained are shown in Figure 1 together with the originalw function drawn with eq 5. Thew data atε > 1.0 are notshown because eq 6 is an excellent description in this rangeregardless of the peptides. Twow functions, namely those forthe 10-mers of methionine (M10) and threonine (T10), are markedin the figure. As the internal energy decreases, the peptidewdata begin to deviate more and more from eq 5. Deviation isthe largest for M10, and it is the smallest for T10. The (c3, c2, c1,c0) arrays determined for the twenty 10-mers through curvefitting are listed in Table 1.

Figure 2 compares the rate-energy relations for M10 calcu-lated by various methods using anE0 of 0.5 eV and∆Sq of 0eu (1 eu) 4.184 J K-1 mol-1). The zero-point energy for thispeptide calculated with the reactant frequencies used in the ratecalculation is 39.6 eV. Hence, the internal energy of 3.96 eV isequivalent to the scaled energy (ε) of 0.1. The rate constantcalculated with the BS algorithm at this energy is 1.48× 106

sec-1, corresponding to the half-life of 0.48µs. On the otherhand, the rate constant calculated by the original WR methodis 5.79× 106 s-1, which is larger than the correct (BS) valueby a factor of 3.9. Such a difference is understandable becausethe correctw value for M10 at this energy is 0.2121, which is alittle larger than the 0.2052 calculated by eq 5. It is to beemphasized thatε ) 0.1 was the lower limit for the internalenergy in the derivation of thew function in the original work,that the discrepancy between the WR and BS rate constants is

significant at this energy, and that the rate constant is quite largeeven at such a low internal energy. As expected from the largerdeviation of the correctw data from eq 5 at lower energy asseen in Figure 1, the WR rate constant in Figure 2 differs morefrom the BS rate constant at lower internal energy. When anion cyclotron resonance mass spectrometer is used, the dis-sociation of a protein ion occurring on the time scale as longas 10 s may be observed, even though canonical rate constantsrather than microcanonical ones calculated in this work maybe more appropriate in some cases. Hence, let us compare therate constants at the internal energy corresponding to the BSrate constant of 0.1 s-1 from now on. In the present case, thisoccurs at 1.38 eV. At this energy, the WR rate constant is aslarge as 300 s-1, corresponding to a factor of 3000 differencefrom the correct result. It is evident that the original WR methodcannot provide a reasonable estimate of a rate constant atε <0.1, even though physically meaningful reactions may occur atsuch a low internal energy. To summarize, the WR method israpid but inaccurate in the low-energy range and the BS methodis accurate but slow in the high-energy range. A way to get outof this dilemma may be to calculate the rate-energy relationsusing both methods, the BS calculation in the low-energy rangeand the WR calculation over the entire energy range of interest,and use the two together. Certainly, a better way is to devise aunified method that is efficient and accurate at the same time,as attempted in the present work.

Figure 1. The w functions for the 10-mers of twenty amino acidsobtained by comparing the vibrational state sums calculated by the BSalgorithm with eq 2. Thew functions for M10 and T10 are marked, whichform the upper and lower boundaries of the data, respectively. Alsodrawn is thew function in the original Whitten-Rabinovitch ap-proximation. The abscissa is the internal energy scaled by the zero-point energy.

Figure 2. The rate-energy relations for M10 calculated by the BS(s), WR (- ‚ -), IWR′ (-O-), and IWR (----) methods using anE0

of 0.5 eV and a∆Sq of 0 eu.

TABLE 1: List of the c Parameters Derived from the10-mers of Twenty Amino Acids

c3 c2 c1 c0

Ala(A) 0.763603 3.422871 3.911022 3.166518Cys(C) 0.831468 3.203076 4.014916 3.201546Asp(D) 1.138528 2.643839 4.308187 3.206942Glu(E) 1.329782 2.226432 4.641321 3.068151Phe(F) 1.236776 2.461058 4.385026 3.152760Gly(G) 1.095241 2.802168 4.215922 3.150475His(H) 1.347598 2.266191 4.474229 3.176003Ile(I) 0.919614 3.068504 4.184229 3.044556Lys(K) 1.128924 2.624142 4.441859 3.012481Leu(L) 1.039792 2.839106 4.299746 3.024606Met(M) 1.215631 2.452458 4.548003 2.988617Asn(N) 1.262370 2.378021 4.448872 3.149506Pro(P) 1.349337 2.226658 4.544044 3.083142Gln(Q) 1.222632 2.460278 4.463422 3.080824Arg(R) 1.305170 2.346235 4.582904 3.021681Ser(S) 1.067582 2.816962 4.217357 3.166776Thr(T) 0.886543 3.266588 3.900925 3.215289Val(V) 0.881758 3.171401 4.115937 3.068043Trp(W) 1.285702 2.468111 4.353988 3.194517Tyr(Y) 1.222054 2.492761 4.348006 3.185401

Improved WR Approximation for Proteins J. Phys. Chem. B, Vol. 111, No. 10, 20072749

Initially, we hoped to achieve the above goal by using auniversalw function that could be used for proteins with anysequence. The IWR′ method described in the previous sectionwas devised for this purpose. The rate-energy relationscalculated by the IWR′ method were in excellent agreementwith the BS results for many peptides consisting of a variety ofamino acid residues. In the IWR′ calculations, the largest erroris expected for the peptides consisting of only one type of aminoacid residue, especially M10 and T10. The rate-energy relationfor M10 calculated by the IWR′ method with anE0 of 0.5 eVand a∆Sq of 0 eu is shown in Figure 2 also. The IWR′ rateconstant is in much better agreement with the BS result than isthe WR result. At the internal energy of 1.38 eV chosen in theprevious comparison, the IWR′ rate constant is 1.3 s-1

,

corresponding to a factor of 13 difference from the correct value.Namely, the difference factor of 3000 in the original WR methodhas been reduced to 13 in IWR′. We are not satisfied with theIWR′ method, however, because our aim is to develop a methodthat can reproduce a BS rate constant within a factor of 2.

As has been mentioned already in the previous section, ourmethod of choice (IWR) is to calculate a sequence-specificwfunction by averaging thew functions (w-1 actually) of theresidues contained in a protein. The rate-energy relation forM10 calculated by IWR using the sameE0 and∆Sq values asbefore is also shown in Figure 2. Because all the residues arethe same in M10, the sequence-specificw function evaluatedfrom the residue functions is exactly the same as the residuefunction that was derived using the state sum calculated by theBS algorithm. Hence, one may expect to obtain an IWR rateconstant that is essentially identical to the BS result. Even thoughthe IWR and BS rate-energy relations in Figure 2 are nearlythe same, a tiny difference can be observed. For example, theIWR rate constant at 1.38 eV is 0.160, namely 60% larger thanthe BS result. To see if the functional fitting of thew data isthe source of such an error, we calculated the state density at1.38 eV using thew function and compared it with the BS statedensity. The BS density turned out to be larger than the IWRdensity only by 0.2%, which is the fitting error for the statedensity. We also calculated the state sum at 0.88 eV, which isthe internal energy at the TS. Here, the difference between theIWR and BS results was only 7% when the reactant frequencieswere used. Hence, it is unlikely that the fitting error is the mainsource for the 60% error in the rate constant. The main error inthe present IWR method may arise from the fact that thevibrational frequencies at the TS are different from the reactantfrequencies that are used to obtain thew function. To checksuch a possibility, we carried out RRKM calculations for M10

with the sameE0 but increasing∆Sq to 10 eu, which is a typicalvalue for a loose transition state5 reaction. Here again, the rate-energy data calculated by IWR were much better than those byWR, even though the deviation from the BS results was largerthan in theE0 ) 0.5 eV and∆Sq ) 0 eu case, as expected. Therate constant in the 0.1 s-1 range differed by a factor of 6. TheRRKM calculations were done with anE0 of 1.0 eV and a∆Sq

of 10 eu also. Here, the IWR and BS results were in excellentagreement as in theE0 ) 0.5 eV and∆Sq ) 0 eu case.

The RRKM calculations were done for peptides consistingof various amino acids also. These included well-known peptidessuch as angiotensin I (DRVYIHPFHL, monoisotopic relativemolecular mass (RMM) of 1295.7) and 100 randomly generatedpeptides consisting of ten amino acid residues. We also carriedout calculations for various fictitious proteins such as thefictitious 8-mer of angiotensin I, (DRVYIHPFHL)8, with amonoisotopic RMM of 10 239.4 to check the reliability of the

present IWR method for higher mass proteins. The general trendin all these cases was the same as in the M10 case: overestima-tion of k near 0.1 s-1 by 60-80% in the calculations withE0 )0.5 eV and∆Sq ) 0 eu and overestimation by a factor of 6with E0 ) 0.5 eV and∆Sq ) 10 eu.

From the results presented so far, it is obvious that a rate-energy relation calculated by the IWR method reproduces thecorresponding BS result satisfactorily except whenE0 is smalland ∆Sq is large at the same time. In the present work, thevibrational frequency set at the TS is estimated by adjustingsome of the frequencies of the reactant such that the postulated∆Sq results. As∆Sq gets larger, these adjusted frequencies getsmaller. Then, the state sum evaluated using thew functionderived with the reactant frequency data deviates more fromthe correct value, resulting in larger deviation in the rateconstant. WhenE0 gets larger also, the agreement between theIWR and BS results gets better even when∆Sq is large. Theexplanation for the better agreement at largerE0 is as follows.As E0 increases, the rate constant at a given internal energydecreases rapidly. In fact, the internal energy needed to maintainthe rate constant at the same value, and hence the internal energyat the TS also, increases almost in proportion toE0. At highinternal energy, however, the influence of the above low-frequency vibrations at the TS becomes less important, resultingin better agreement between the IWR and BS rate constants.

Even though the present IWR method is not adequate toreproduce the BS rate-energy relation for a simultaneouslysmallE0 and large∆Sq case such asE0 ) 0.5 eV and∆Sq ) 10eu, it is to be mentioned that such a case is unlikely to beencountered in protein dissociations. In dissociation reactions,a large value of∆Sq, or the occurrence of a loose transitionstate,5 is usually associated with simple bond cleavage reactions,which tend to have a largeE0 value. It is well-known in thefield of mass spectrometry that the b and y type fragment ionsare generated preferentially when the internal energy of aprotonated peptide is not high,26 which indicates that they aregenerated via reaction paths with smallE0 values. Accordingto the DFT calculation for the dissociation of [G3 + H]+ carriedout by Paizs and Suhai,22 generation of the b2 ion occurred withthe smallest value ofE0 (0.42 eV) and dominated in the lowinternal energy range. The value of∆Sq evaluated using thefrequencies at the reactant and TS geometries supplied by Paizsand Suhai was-2.65 eu. This is in agreement with ourspeculation that there is probably no dissociation channel withsimultaneously smallE0 and large∆Sq values for protein ions.If such a channel existed, it would have dominated the fragmention spectra over the entire internal energy range. However, it isknown26 that the major fragment ion types in the dissociationof peptide and small protein ions change from b and y to a, d,v, w, and x as the internal energy increases. Absence of reactionchannels with simultaneously smallE0 and large∆Sq valuesmeans that the IWR method developed in this work is probablyreliable for dissociation of protein ions. Without attachedprotons, the critical energy for the dissociation of a neutralprotein will be probably larger than the corresponding valuefor the protonated form.

The forgoing discussion on the influence of entropy on theaccuracy of the IWR method has arisen because thew functionderived with the reactant frequencies in this work is not quiteadequate at the TS. A related problem may arise from the factthat the reactant frequencies used in this work are thoseestimated for linear peptides and proteins. The frequencies of aprotein may be affected by hydrogen bonding even in the gasphase. Then, thew function derived in this work may be invalid

2750 J. Phys. Chem. B, Vol. 111, No. 10, 2007 Sun et al.

to deal with the dissociation of gas-phase proteins, renderingthe present method useless. Out of this concern, we attemptedsome primitive calculations for peptides with intramolecularhydrogen bonding. For this purpose, we assumed that thefrequencies affected by hydrogen bonding are those of the N-Hand CdO stretching modes. Then, the frequencies of thesemodes in the set stored for each amino acid residue was reducedby 5% based on the spectral correlation appearing in theliterature.27 The rate-energy relations thus obtained were hardlydistinguishable from the original data, suggesting that thewfunction derived by the present method can be used even whenintramolecular hydrogen bonding is present in a protein.

Finally, we compared the rate-energy relations calculatedwith E0 ) 0.5 eV and∆Sq ) 0 eu for the monomer, 8-mer, and80-mer of angiotensin I. The internal energies correspondingto, for example, 103 s-1 in these molecules were 1.97, 14.21,and 140.01 eV, respectively, increasing nearly in proportion tothe molecular mass. This is in agreement with our previoussuggestion28 that the internal energy needed to observe aparticular type of protein reaction with the same rate constantincreases almost in proportion to the number of degrees offreedom, which, in turn, is almost proportional to the molecularmass of proteins. Because the zero-point energies of proteinsincrease almost in proportion to the molecular mass also, 42.18,332.77, and 3321.73 eV for the monomer, 8-mer, and 80-mer,respectively, the three rate-energy curves can be brought intonear coincidence by drawing them with the scaled energy inthe abscissa.

IV. Conclusion

Even though the BS algorithm is efficient and accurate inevaluating the vibrational state sum and density needed in theRRKM calculation of a unimolecular reaction, its applicationto biopolymers is limited because the computational timeincreases rapidly with the molecular mass. In this work, theWR approximation used in the semi-classical calculation of arate constant has been improved such that the rate constant fora unimolecular reaction of a protein evaluated by this methodclosely reproduces the BS result. Its main advantage lies in thefact that a rate constant can be calculated instantly regardlessof the protein mass. It has been found that the rate constantcalculated by this method differs from the BS result significantlyfor reactions occurring with simultaneously smallE0 and large∆Sq values. Certainly, it is a weakness of the present methodeven though such a situation is unlikely to be encountered inactual protein dissociations. Another weakness of the presentapproach is that the method is valid only for peptides andproteins. Completely new functions must be derived for other

biopolymers such as nucleic acids and carbohydrates, eventhough the method to derive such functions would be the sameas in this work. In this regard, it may be worthwhile toinvestigate other approximate methods developed previouslysuch that an efficient and accurate method with generalapplicability to any biopolymer can be established.

Acknowledgment. Meiling Sun thanks the Ministry ofEducation for the Brain Korea 21 fellowship. The softwaredeveloped in this work will be made freely available.

References and Notes

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Improved WR Approximation for Proteins J. Phys. Chem. B, Vol. 111, No. 10, 20072751