ria

gassz

sszoDenTR-

Received 30 January 2014In nal form 12 March 2014Available online 20 March 2014

ated

tions have been analyzed by Fourier expansions. The ndings show that rotation around bonds connect-

studies.

1.2. Historic background

For several decades, during the second half of the 20th century,it was an unexplained phenomenon that the barrier to internalrotation (torsion) about a CAC single bond in ethane (CH3ACH3)

n potential usingst of the iuse it re

extent of intramolecular interactions [6]. The potentialcurves and surfaces can be represented by suitably chosenical functions. Such functions can describe the atomic motioin a molecule, or within complexes formed by the interaction of themolecules [7]. If this function is in a suitable form then it can beused as an approximate analytic solution of the Schrdinger equa-tion. Such equations can also be used for estimating rates of inter-nal rotation [8,9].

Several functional groups have a unique internal rotational de-gree of freedom around a symmetric axis [10]. In such cases xed

Corresponding author. Fax: +45 8619 6199.E-mail address: kemskj@chem.au.dk (S.J. Knak Jensen).

Chemical Physics Letters 599 (2014) 169174

Contents lists availab

y

.e lbe accurate but impractical to use and truncated expansion may bepractical but not sufciently accurate. Thus some compromise isneeded. The present Letter aims to analyse what functions arepractical to use yet may give accurate enough results for such

electron pair.The mathematical representation of the rotatio

analytic functions has its own story. The forecarotation and its energy changes is essential becahttp://dx.doi.org/10.1016/j.cplett.2014.03.0290009-2614/ 2014 Elsevier B.V. All rights reserved.nternalect theenergyanalyt-n with-1.1. Perspective

Protein folding is a century old problem. With the aid of confor-mational analysis the biological problem became a chemical prob-lem. However, due to the complexity of the molecular system forceeld methods have been developed. Initially such molecularmechanics methods were based on spectroscopic data and laterfurther development utilized optimized potential energy functions.Yet the reliability of the analytic potentials used today may still bean open question. On the one hand very extensive expansion may

hydrogen peroxide (HOAOH) was very difcult to compute. Theapparent discrepancy was claried in 1978 when Cremer [1]studied hydrogen peroxide with a large polarized basis set whichproduced remarkably accurate results. Moreover, Peterson andCsizmadia [2] demonstrated that all critical points of thePotential Energy Hypersurface (PEHS) of three independencevariables of n-butane (H3CACH2ACH2ACH3) could be reproducedat a modest level of theory. Further studies at that time [35]indicated that the presence of lone pairs made the descriptionmore difcult than the modest distortion of CAH bonding1. Introductioning atoms without lone pairs can be described with a one term Fourier-series. In contrast, two or threeterms are needed if the connected atoms have lone pairs. The analysis inspires adoption of a simpliedFourier expansion that reproduces the data well, suggesting that Fourier-type-series with few termsare useful in describing any internal rotation analytically.

2014 Elsevier B.V. All rights reserved.

could be successfully calculated by just about any method,whereas the lower anti barrier along the OAO single bond intopology of the chemical environment. Energies obtained from electron structure and force-eld calcula-Fourier type potential energy function foof selected organic functional groups

Anita Rgyanszki a, Attila Surnyi a, Imre G. CsizmadSelma Yarligan Uysal e, Bla Viskolcz a

aDepartment of Chemical Informatics, Faculty of Education, University of Szeged, BoldobDepartment of Chemistry, University of Toronto, M5S 3H6 Toronto, Ontario, CanadacDepartment of Applied Informatics, Faculty of Education, University of Szeged, BoldogadDepartment of Chemistry, Langelandsgade 140, Aarhus University, DK-8000 Aarhus C,eDepartment of Chemistry, Faculty of Arts and Science, Eskisehir Osmangazi University,

a r t i c l e i n f o

Article history:

a b s t r a c t

The energy changes associ

Chemical Ph

journal homepage: wwwconformational change

a,b, Andrs Kelemen c, Svend J. Knak Jensen d,,

ony sgt. 6, H-6725 Szeged, Hungary

ny sgt. 6, H-6725 Szeged, Hungarymark26480 Eskisehir, Turkey

with internal rotation of a functional group in a molecule depend on the

le at ScienceDirect

sics Letters

sevier .com/ locate /cplet t

bond angles may be assumed for the internal torsional movements,thus only a single bond is engaged in the internal rotation [11] andthe potential energy depends on the dihedral angle only [12,13].

In the 1970s the Fourier-series were investigated by Radomand Pople to determine the internal rotation around a single bond[9,14]. In most cases Fourier-series, with relative few terms wereused for the tting of the curves.

In the late 1980s Chung developed a method for calculatingtorsional energies with n-term Fourier-series. The full term expan-sion Fourier-series is recommended for higher accuracy in contrastto minimum number of terms [13]. Nevertheless the practicalquestion still remains; how many terms are needed to obtain reli-able results for r-bonds.

1.3. Scope

We are considering the dihedral torsion or internal rotation oftypical organic functional groups, like a methyl group (H3C).The simplistic view-point is that the rotation is always the sameno matter wherever it is located in any molecule. A more sophisti-cated point of view is that the characteristic of the methyl rotationis inuenced somewhat by its environment and most certainly bythe group it is attached to. The rst question we wish to investigate

As a third question we wish to examine the rotation around asingle bond (CAO or OAO) in compounds containing more thanone r-bonds resulting in variable chemical environments. In thesecase studies we would examine the effect

of the presence of a methyl group in an OAO torsion(III) and in aH3CAO torsion(I) and the introduction of an OH group in aH3CAO torsion(II).

In this Letter we explore rotations around bonds in small mol-ecules. However, the perspective is to extend the study to side-chain rotations in amino acid residues. As a modest step in that

where DE0 is a constant, n is the number of terms in the expansion

EC)term

a2

Relax 4.97 4.83 6.42 101

138.111141712

118.817.0 8.16 101 1.97 3.47 101 0.9994

170 A. Rgyanszki et al. / Chemical Physics Letters 599 (2014) 169174CAO Rigid 1.96 1.97 1.04 104Relax 2.97 2.99 2.50 101

CAS Rigid 1.34 1.35 3.79 103Relax 2.72 2.75 4.30 105

X-Y family (n = 2)OAO Rigid 14.0 10.1 7.34 101

Relax 12.3 17.1 1.62SAS Rigid 11.5 3.16 2.28 101

Relax 15.2 5.65 5.80 101SAO Rigid 17.0 1.82 3.92 102

Relax 13.5 7.02 6.789 101

X-Y family (n = 3)NAO Rigid 27.0 2.07 1.03

Relax 28.7 3.33 1.37NAS Rigid 24.8 2.17 101 3.46 101

Relax 19.6 4.31 5.44 101is how large such a nearest neighbour interaction is in compoundsH3C-X, where X may be chosen horizontally or vertically along theperiodic table:

The second questionwe will explore is if it is possible to quantifythe changes in energy for the rotation about homonuclear diatomicunits such as (XX) as the number of lone pairs of X increases, as inthe series:

Table 1Fits of the Fourier series (1) to the rigid- and relaxed torsional potential energy curves (PX and X-Y systems, (X,Y = O, N, S). The data are arranged according to the number of

DE0 a1 b1

C-X family (n = 1)CAC Rigid 6.68 6.63 1.12 102

Relax 5.86 5.99 1.07 101CAN Rigid 4.05 4.04 6.31 103NAN Rigid 13.9 18.2 1.34 7.3Relax 12.1 12.2 2.52 101 11and k is a constant indicating the periodicity of the rotating group(i.e., k = 3 for a CH3 group).

, DE (kJ/mol), obtained by DFT [B3LYP/6-31G(d)] calculations for fully hydrogenated C-s, n, in the Fourier series.

b2 a3 b3 R2

1 0.9936

1 0.9963 1 0.9996 0.9999 0.9968

.4 1.95 0.99972 1.55 0.9994.3 1.64 0.9997.2 2.95 0.9972.1 7.42 101 0.9997.7 2.48 0.9987

4.9 4.24 0.99921.5 2.06 0.99941 101 1.82 22.6 2.54 0.9995direction we have included the simplest chiral amino acid residue(alanine) among the considered species.

2. Methods

Let DE be the energy change associated with rotation of a func-tional group relative to the lowest energy. DE can be expanded in aFourier-series with the general form:

DE DE0 Xnj1

aj cosj2pk/360

bj sin j2pk/360

19 1.09 3.41 7.66 101 0.9977.7 4.85 101 4.42 2.74 101 0.9904

and a1 for n = 1 inspired us to adopt a simple even tting functionfor DE as

DE DEmaxn 1

Xnj1

DEmax2j

cosj2pku360

2

where DEmax is determined from the t. For example, the H3C-Xfamily can be represented by an expansion with one term as

DE DEmax2

DEmax2

cos0:052u 3

For the X-Y family -excluding the NAN and NAS members- thefunction is:

DE DEmax3

DEmax2

cos0:0175u DEmax22

cos0:0175 2u 4

In the case of NAN and NAS we obtain the function:

hysics Letters 599 (2014) 169174 171DE was calculated in two ways, (i): by quantum mechanicsusing the B3LYP/6-31G(d) implementation of the density func-tional theory [15] in the GAUSSIAN09 software package [16] amethod extensively used for small and medium sized moleculesand (ii) by force eld methods, Amber/GAFF and Amber99SB[1719] tools routinely used for big molecules like proteins.The side chain methyl rotation in N-acetyl Alanine-methylamidewas also studied at the B3LYP/6-31G(d) level of theory and bythe force eld method, Amber99SB. Using force-eld methods forthe molecules of the sizes studied here may seem like overkillbut is it useful for estimation of the associated energy changeswhen proteins are studied. DE was calculated for a range of dihe-dral angles, /, in interval [0,2p].

The calculation of DE was done as a Potential Energy Curve (PEC) ofrigid rotation, DErigid, as well as a PEC of relaxed rotation, DErelax.The former is considered the less useful because in rigid rotationthe functional groups are rotated like mechanically rigid wheelswithout geometrical adjustment. In contrast, in relaxed rotationthe groups involved in the internal rotation would have the oppor-tunity to adjust their geometries during rotation. The PEC curve islower for the relaxed rotation than for the rigid rotation. However,the energy lowering needs not be the same for the transition statesas for the potential minima. DErigid and DErelax were both calculatedat the B3LYP/6-31G(d) level. DErelax was also calculated using forceeld methods to assess the quality of the analysis of data obtainedfrom the two levels of description.

3. Results

A Fourier series (1) was tted to the DErigid and DErelax data forthe two families of compounds (H3C-X and X-Y where X, Y = O, N,S). The results are shown in Table 1. It appears that the t is verysimple in the case of the H3C-X family as a single term (n = 1) willpresent the data very well. We note that the b1 coefcient is muchsmaller than the a1 coefcient, suggesting the PEC is close to beingan even function. It is also worth noting that a1 is very close to DE0.The variation of DE with / is shown in Figure 1. The quality of thet is estimated by calculating the deviation between the ttedfunction and the raw data as a function of the dihedral angle. Wend the maximum deviation to be smaller than 0.1 kJ/mol.

For the second family (X-Y) two or three terms are required.Again we note from Table 1 that the b-coefcients are much smal-ler than the a-coefcients. The DE plots for the X-Y family areshown in Figure 2.

Inuences of substituents on DE were studied using the com-pounds I, II and III. The variations of DE in these cases are shownin Figure 3. This gure shows, that H3CAO rotations (left hand sideof Figure 3) can be represented fairly accurately by single term(n = 1) while the rotation about a OAO bond requires a two term(n = 2) expansion in the Fourier-series.

It appears from Figure 2 that hydrazine has a very shallow min-imum at the anti conformation [20], which increases the demandfor more terms in the Fourier expansion. Figure 4 illustrates howthe quality of the t increases for hydrazine as the number of terms

A. Rgyanszki et al. / Chemical Pincreases. The case of n = 3 does a perfect job indeed.The observation that the b-coefcients in (1) are much smaller

than the a-coefcients in most cases along with the closeness of E0Figure 1. Fits of (1) with n = 1 to the torsional potential energy curves (PEC), DE (kJ/mol), for fully hydrogenated C-X families, (X = C, N, O, S). Open (blue) and lled(red) symbols indicate data for rigid- and relaxed rotation, respectively. Data

obtained at the B3LYP/6-31G(d) level of theory. (For interpretation of the referencesto colour in this gure legend, the reader is referred to the web version of thisarticle.)

hysi172 A. Rgyanszki et al. / Chemical PDE DEmax4

DEmax2

cos0:0175u DEmax22

cos0:0175 2u

DEmax23

cos0:0175 3u 5

In Table 2 we compare the results obtained from tting (2) to theDE data derived from the relaxed dft calculations to the force eldresults.

4. Discussion

It appears from Table 1 that not all r-bonded functional groupsbehave in an identical way during internal rotation (torsion). Someof them like H3C-X exhibit rather high symmetry with three iden-tical transition states [13]. These can be represented with a single(n = 1) cosine function. Rotation about other r-bonds, where thethree TS are not identical or where only a pair of non-identicalTS is in the potential, require a longer Fourier expansion (n = 2 or3). All of the potentials however were tted very accurately withR2 values ranging from 0.9904 to 1 (Table 1).

The TS obtained by molecular mechanics (MM) simulationsagreed reasonably well (Table 2) for the C-X family. In the caseof the X-Y family noticeable deviations were observed in the casesof the SAO, SAS, NAO and NAN linkages, 33.9 kJ/mol, -18.6 kJ/mol,13.9 kJ/mol and 26.3 kJ/mol, respectively. This is understandablebecause traditional force eld software were written to simulateprotein structure and protein folding. Thus discrepancies may be

Figure 2. Fits of (1) with n = 2, 3 to the torsional potential energy curves (PEC), DE (kJ/msymbols indicate data for rigid- and relaxed rotation, respectively. Data obtained at the Bgure legend, the reader is referred to the web version of this article.)cs Letters 599 (2014) 169174expected in cases of bonds that are not common in proteins. SinceMM simulations, in general, aim to reproduce geometries and rel-ative stabilities of minima, the achieved TS energy approximationare really very good.

5. Conclusion

Fourier series, comprised by sin and cos terms are capable to tany function. However, the potential energy functions consideredhere are close to being even functions which makes a pure cosexpansion satisfactory. Depending on the structural complexityof a given molecule, a 1-term or 2-term or 3-term expansionturned out to be adequate, using only cos expansion. For CAC,CAN, CAO and CAS bonds the 1-term expansion was accurate en-ough. For OAO, SAS and SAO bonds the 2-term expansion was sat-isfactory. In contrast, 3-term expansion was necessary to t thepotential energy function associated with the internal rotationabout the NAO, NAS and NAN bonds. The CAC rotation potential,tted by Molecular Mechanics (MM), agreed very well with thepresent Fourier series t for the methyl rotation in ethane as wellas in alanine-diamide.

The ultimate purpose is to build a very accurate multi variablemathematical function for the Potential Energy Hypersurface ofexible molecules, such as peptides, of several internal bond-rota-tions. The present study made for a single dihedral rotation sug-gests that such an ultimate goal is achievable.

ol), for fully hydrogenated X-Y families, (X, Y = N, O, S). Open (blue) and lled (red)3LYP/6-31G(d) level of theory. (For interpretation of the references to colour in this

Figure 3. Fits of (1) to the potential energy curves for case studies of the environmental effects. Open (blue) and lled (red) symbols indicate data for rigid- and relaxedrotation, respectively. Data obtained at the B3LYP/6-31G(d) level of theory. (For interpretation of the references to colour in this gure legend, the reader is referred to theweb version of this article.)

Figure 4. Fits of (1) to torsional potential energy curves (PEC), DE (kJ/mol), for hydrazine with n = 1, 2, and 3 terms. Open and lled symbols indicate data for rigid- andrelaxed rotation, respectively. Data obtained at the B3LYP/6-31G(d) level of theory.

A. Rgyanszki et al. / Chemical Physics Letters 599 (2014) 169174 173

Acknowledgement

The authors would like to thank Balzs Jjrt for helpful contri-butions and Mikls Krsz for helpful discussions.

The authors acknowledge the nancial support within TMOP-4.2.2.A-11/1/KONV-2012-0047 New functional material and theirbiological and environmental answers, TMOP-4.2.2.C-11/1/

KONV-2012-0010 Supercomputer the national virtual labora-tory, HUSRB/1002/214/193 Bile Acid Nanosystems as MoleculeCarriers in Pharmaceutical Applications and TMOP 4.2.4. A/2-11-1-2012-0001, National Excellence Program Elaborating andoperating an inland student and researcher personal support sys-tem, subsidized by the European Union and co-nanced by theEuropean Social Fund.

References

[1] D. Cremer, J. Chem. Phys. 69 (1978) 4440.[2] M.R. Peterson, I.G. Csizmadia, J. Am. Chem. Soc. 100 (1978) 100.[3] B. Viskolcz, R. Izsak, S.N. Fejer, I.G. Csizmadia, M. Szori, Int. J. Quantum Chem.

107 (2007) 1826.[4] B. Viskolcz, M. Szori, I.G. Csizmadia, S.N. Fejer, Mol. Phys. 104 (2006) 795.[5] B. Viskolcz, S.N. Fejer, I.G. Csizmadia, J. Phys. Chem. A 110 (2006) 3808.[6] M. Head-Gordon, J.A. Pople, J. Phys. Chem. 97 (1993) 1147.[7] G.H. Penner, J. Mol. Stuct. 137 (1986) 121.[8] L. Radom, W.J. Hehre, J.A. Pople, W.A. Lathan, J. Am. Chem. Soc. 95 (1973)

693.[9] L. Radom, J.A. Pople, W.J. Hehre, J. Am. Chem. Soc. 94 (1972) 2371.[10] J. Schroderus, I. Ozier, N. Moazzen-Ahmadi, J. Chem. Phys. 115 (2001) 1392.[11] R.H. Hunt, C.W. Peters, K.T. Hecht, R.A. Leacock, J. Chem. Phys. 42 (1965)

1931.[12] A. Chung-Phillips, J.W.H. Kao, J. Am. Chem. Soc. 101 (1979) 1087.[13] A. Chung-Philips, J. Chem. Phys. 88 (1988) 1764.[14] L. Radom, J.A. Pople, J. Am. Chem. Soc. 94 (1970) 4786.[15] A.D. Becke, J. Chem. Phys. 98 (1993) 5648.[16] M.J. Frisch et al. GAUSSIAN 09, Revision A.02. Gaussian Inc, Wallingford, CT, USA,

(2009).[17] J.M. Wang, R.M. Wolf, J.W. Caldwell, J. Comput. Chem. 25 (2004) 1157.[18] D.A. Case et al., AMBER 12, University of California, San Francisco, 2012.[19] A. Jakalian, D.B. Jack, C.I. Bayly, J. Comput. Chem. 23 (2002) 1623.[20] L. Song, M. Liu, W. Wu, Q. Zhang, Y. Mo, J. Chem. Theory Comput. 1 (2005) 394.

Table 2Comparison of the maxima (or transition states) obtained by tting (2) to the relaxedDE data (kJ/mol) from DFT [B3LYP/6-31G(d)] calculations to the force eld data forfully hydrogenated C-X and X-Y families, (X,Y = O, N, S). Data for N-acetyl Alanine-methylamide are also presented.

A: Higher TS B: Lower TS

Method DFT Force eld DFT Force eld

C-X family (n = 1)CAC 11.78 12.05 CAN 10.11 9.35 CAO 6.01 4.82 CAS 5.51 5.17 H3C-Ala 15.62 14.91

X-Y family (n = 2)SAO 47.86 13.96 18.77 11.42NAO 43.29 41.86 42.93 37.67SAS 34.42 53.02 23.08 47.86OAO 37.12 35.7 2.92 4.41

X-Y family (n = 3)NAS 36.66 22.74 36.66 21.81NAN 40.56 14.28 7.64 5.30

174 A. Rgyanszki et al. / Chemical Physics Letters 599 (2014) 169174

Fourier type potential energy function for conformational change of selected organic functional groups1 Introduction1.1 Perspective1.2 Historic background1.3 Scope

2 Methods3 Results4 Discussion5 ConclusionAcknowledgementReferences