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QTAIM study of the closed-shell interactions in peptide secondary structures: A cluster treatment of oligo- and polyalanines
Chemical Physics Letters 440 (2007) 279–285 www.elsevier.com/locate/cplett
QTAIM study of the closed-shell interactions in peptide secondary structures: A cluster treatment of oligo- and polyalanines M.V. Vener *, A.N. Egorova, D.P. Fomin, V.G. Tsirelson Department of Quantum Chemistry, Mendeleev University of Chemical Technology, Miusskaya Square 9, 125047 Moscow, Russia Received 18 January 2007; in final form 4 April 2007 Available online 19 April 2007
Abstract The closed-shell interactions in oligo- and polyalanines are studied by the quantum theory of atoms in molecules (QTAIM) using electron densities derived from the B3LYP/6-31+G** ground-state electronic wave-functions. The QTAIM enabled us to identify a large number of the intraturn closed-shell stabilizing interactions in the b-turns, which were presented by several conformers of the tetrapeptide model compound. We found that only b-turn type IVa exhibits a 10-member pseudocycle. The intrachain H-bonds between the adjacent N–H and C@O groups in the antiparallel b-sheet conformation of polyalanine have not been found. At the same time, these interactions do exist in the parallel conformation and are even stronger than the interchain N–H. . .O bonds. A weak interaction between the C@O group at the position i and the side-chain C–H group at the position i + 3 was detected in the a-helical conformation of polyalanine. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction The local geometry of the secondary structures of polyaminoacids is influenced by several weak closed-shell interactions, as the intra- and intermolecular hydrogen bonding (H-bonding) [1–4] and the stacking interactions in base duplexes [5]. In this Letter, we aim to provide insight into the features of the closed-shell bonding interactions driving the intra- and intermolecular recognition in the secondarystructure elements of the alanine-based oligo- and polypeptides responsible for their mutual arrangement (the local geometry). While much literature exists about the geometry and energy of the closed-shell interactions in the cluster [6– 9] and periodical-crystal [10–12] models of oligo- and polypeptides, no previous reports discussing the electron density characteristics underlying the H-bonding in the secondary structures have been published yet, to our best knowledge. Therefore, we present the first systematic quantum theory of atoms-in-molecules (QTAIM) [13] study of
*
Corresponding author. Fax: +7 495 609 2964. E-mail address: [email protected] (M.V. Vener).
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.04.044
the closed-shell interactions in the alanine-based oligopeptides (c-turns and b-turns) as well as in the a-helix and bsheet conformations of polyalanines. 2. The secondary-structure models and computational methods Two sets of the secondary-structure models are used in this study. The first one includes the gas-phase models of the c- and b-turns presenting by the corresponding conformers of the tripeptide and tetrapeptide model compounds. The cluster models of the a-helix and b-sheets conformations of polyalanines display the second set. All the DFT calculations in this work were carried out using the PC version [14] of the GAMESS(US) program package [15]. Geometries of the gas-phase models of the c-turns and b-turns were optimized at the B3LYP/ 6-31+G** level of approximation. The minimum-energy states of the structures have been confirmed by calculating the harmonic frequencies. Single-point B3LYP/6-31+G** computations have been performed for each cluster model of the a-helix and b-sheets conformations of polyalanine frozen at the geometries taken from the DFT calculations,
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which employ the periodic boundary conditions [11,16]. The cluster models were derived from the corresponding infinite structures in a following way. (i) All the atoms entering the unit cell were included into consideration. (ii) Terminal N–H and C@O fragments were saturated with hydrogen atoms and hydroxyl groups, respectively. Electron-density properties were evaluated with the AIMPAC computer program suite [17]. The following QTAIM features at the H. . .A (A = O, N, C, H) bond critical point were considered: the values of electron density, qb, the Laplacian of electron density, $2qb, as well as the ratio |vb|/gb, where vb and gb are the potential and kinetic energy densities, respectively. According to [13,18,19], this set of parameters allows to quantify the atomic interactions. The closed-shell interactions show qb 6 0.05 a.u., $2qb > 0 and |vb|/gb < 1, and are presented graphically by atomic interaction lines [20] which may be, in principle, very curved and winding, e.g. see [5,21]. The number and type of the critical points revealed in our structures have fit to the Poincare–Hopf relation n b + rc = 1 [13], which links the critical points of different type with the molecular graph elements (n, b, r, c are the numbers of nuclear, bond, ring and cage critical points, respectively). 3. Results and discussion Two conformers of N-acetyl-L-alanine N 0 -methylamide (the tripeptide model compound), C7ax and C7eq [4,22], are shown in Fig. 1. They are stabilized by the intramolecular (intraturn) H-bond between the N–H group at the position i and the C@O group at the position i + 2 [23]. The number 7 stands for the number of atoms involved in the pseudocycle generated by the intraturn H-bond. An alternative nomenclature of common use for these structures is c-turn [24]. The energy difference between the C7eq and C7ax conformers is 2.51 kcal/mol (B3LYP/ 6-31+G**), which is close to 2.28 kcal/mol reported in [25] as the result of the MP2/aug-cc-pVDZ calculation. Geometrical parameters of the H-bonded fragments computed for the considered conformers at the B3LYP/6-
Fig. 1. Molecular graphs and the critical point pattern in the C7eq and C7ax conformers of the tripeptide model compound: (a) inverse c-turn, (b) classic c-turn. The small dots indicate the position of the critical points in the electron density: red stands for the bond critical points, while yellow stands for the ring critical points. The intraturn H-bonds (Table 1) are given by the broken lines. The color coding of this caption also applies to Figs. 2–4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
31+G** level of approximation are in reasonable agreement with those obtained using MP2/aug-cc-pVDZ (Table 1). The most stable conformer C7eq (the inverse
Table 1 Structural parameters for H-bonds in B3LYP/6-31+G** (in regular letters) and MP2/aug-cc-pVDZa (in italics) geometries of the C7eq and C7ax conformers of the tripeptide model compound Conformer
H-bonded fragment
Structural parameters ˚ ˚ R(X. . .O), A R(H. . .O), A
\(XHO), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
C7eqc
N–H. . .O
2.97 2.91
2.09 2.02
143 145
0.020 0.007–0.041
0.059 0.029–0.199
1.03
C7ax
N–H. . .O
2.86 2.82 3.14 3.08
1.93 1.88 2.55 2.48
150 152 113 114
0.028 0.007–0.041 0.010 0.004–0.009
0.081 0.029–0.199 0.037 0.029–0.062
1.03
Cb–H. . .O
0.85
Topological properties at the H. . .O bond critical point in these structures computed using B3LYP/6-31+G** (in regular Letters) and the corresponding experimental valuesb, obtained for the crystal structures of the six amino acids (in italics). The H-bonds are defined in Fig. 1. a Ref. [25]. b Ref. [26]. c Total energy is 495.898557 a.u.
M.V. Vener et al. / Chemical Physics Letters 440 (2007) 279–285
c-turn) has only one intraturn H-bond of the N–H. . .O type, while the local-minimum structure C7ax (the classic c-turn) exhibits an additional H-bond of the C(b)–H. . .O type (Fig. 1). The latter H-bond generates the 6-member pseudocycle with the ring point located quite close to the C(b)–H. . .O bond critical point, see Fig. 1b. It evidences that stability of the 6-member pseudocycle seems to be much less than that of the 7-member one. Computed qb and $2qb values at the H. . .O bond critical point are found to be in reasonable agreement with the available experimental data derived from the electron density of the amino-acid crystals [26] (Table 1) and the computed topological properties of the intramolecular H-bond between a methyl group and a negatively charged oxygen in the biomolecules creatine and carbamoyl sarcosine [27]. Calculated |vb|/gb ratio is around 0.9 for the considered Hbonds. It fits the 0.5 6 |vb|/gb 6 1.0 ratio reported earlier
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for the backbone–backbone interactions in different dinucleotides [5]. Summing up, the use of the B3LYP/ 6-31+G** approximation for the QTAIM study of the oligo- and polyalanines seems to be verified. The C10 conformers of N-acetyl-(L)-alanyl-(L)-alanine methylamide (the tetrapeptide model compound) [4] are shown in Fig. 2. They are stabilized by the intraturn Hbond between the N–H group at the position i and the C@O group at the position i + 3 [23]. The number of atoms involved in the pseudocycle generated by the intraturn Hbond is 10. An alternative nomenclature of common use for these structures is b-turn [24]. Local-minimum structures B (type II, b-turn) and D (type IVa, b-turn) are only 2.88 and 1.11 kcal/mol higher than the global-minimum structure A (type I, b-turn). We were unable to localize the C (type III, b-turn) conformer, see Fig. 2 in [4]; it seems to be unstable in the gas-phase model.
Fig. 2. Molecular graphs and the critical point pattern in the C10 conformers of the tetrapeptide model compound: (a) A (type I, b-turn), (b) B (type II, bturn), (c) D (type IVa, b-turn). See the captions of Fig. 1 for the color coding. The broken lines schematically present the atomic interaction lines. The latter describe the intraturn H-bonds given in Table 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Table 2 Geometrical parameters of the X–H. . .Y fragments, where X = C, N, Y = O, N, H, and topological properties at the H-bond critical point, computed for the C10 conformers of the tetrapeptide model compound Conformer a
H-bonded fragment or interaction groups
Structural parameters ˚ ˚ R(X. . .Y), A R(H. . .Y), A
\(XHY), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
A
Ni-Hb. . .Oi+3 Ni–Hb. . .Ni+1 Ni+1–H. . .Ni+2
3.143 2.768 2.793
2.172 2.353 2.332
159.7 103.4 106.5
0.015 0.016 0.016
0.047 0.077 0.063
0.98 0.77 0.86
B
Ni–Hb. . .Oi+3 Ni+1–Hb. . .Oi+3 Cbiþ1 –H. . .Oi+2 Ni–Hb. . .Hb–Ni+1
3.378 3.054 3.176 –
2.433 2.031 2.553 1.930
155.1 139.2 92.6 107.3b
0.009 0.024 0.010 0.013
0.032 0.070 0.035 0.064
0.89 1.03 0.84 0.76
D
Ni–Hb. . .Oi+3 Caiþ2 –Hb. . .Oi+3 Caiþ2 –Hb. . . H–Caiþ1
3.005 3.434 –
2.026 2.597 1.901
160.8 133.0 –
0.020 0.009 0.015 0.003–0.017c
0.063 0.029 0.056 0.009–0.054c
0.99 0.88 0.81 0.75d
The H-bonds are defined in Fig. 2. Hb indicates that H-atom participates in a bifurcated H-bond. Available theoretical data on quantum-topological properties of the C–H. . .H–C bonds [5,28] are given in italics. a Total energy is 743.243722 a.u. b The Hb. . .Hb–Ni+1 angle. c Ref. [28]. d Ref. [5].
Computed values of the geometrical parameters of the H-bonded fragments and topological properties at the Hbond critical point are given in Table 2. According to it, all the H-bonds in the C10 conformers fall into the region of the closed-shell interactions [19] characterizing by relatively small values of qb (<0.03 a.u.), small positive values of $2qb (<0.08 a.u.) and |vb|/gb < 1.0. In contrast to cturns, the b-turns have bifurcated or ‘three-center’ Hbonds [29], cf. Figs. 1 and 2. The nature of these interactions is considered below in a separate paragraph. Bifurcated H-bond between the Ca–H groups in the conformer D involves the H. . .H interaction, see Fig. 2c. Its topological properties are in reasonable agreement with the available theoretical data (Table 2). The global-minimum conformer A exhibits two additional H-bonds of the N–H. . .N type between the adjacent N–H groups (Fig. 2a). The N–H. . .N fragments are strongly non-linear, however, their strength is comparable to that of the ‘standard’ intraturn N–H. . .O fragment, because the qb values are close to each other (Table 2). Due to the N–H. . .N interactions, the 10-member pseudocycle in the conformer A splits into 5- and 7-member pseudocycles and the additional 5-member pseudocycle is formed (Fig. 2a). The ring critical points of the 5-member pseudocycles locate very close to the bond critical points of the corresponding H. . .N bonds, see Fig. 2a. This is why their stability is expected to be less than the 7-member pseudocycle. Several additional H-bonds are detected for the conformers B and D, see Fig. 2b and c. The X– H. . .Y fragments, X = N, C and Y = N, O, formed by these bonds, are strongly non-linear, with the XHY angle varying from 93° to 133° (Table 2). The Ni+1–H. . .Oi+3 interaction is found to be the strongest closed-shell interaction in the conformer B, qb = 0.024 a.u. (Table 2). It generates two consecutive 7-member pseudocycles, one of which,
however, splits into 6- and 3-member pseudocycles by the weak interaction between the adjacent NH groups, see Table 2 and Fig. 2b. It should be noted that the conformer B is found to be energetically less favorable than conformer A for some model peptides in the low-polarity solvents [23]. However, the conformer B becomes preferred in the solid state because it allows the N–H groups to participate in the nearest-neighbor contacts yielding two consecutive cturns, i.e. the two 7-member pseudocycles [23], in accord with the results of the present study. Special attention should be paid to the nature of the closed-shell interactions between the Ni–H and Ni+1–H groups in the conformers A and B. The corresponding atomic interaction lines are very curved and winding. In the conformer A it starts on the Ni+1 atom and goes toward the adjacent Ni–H bond. It curves near the Ni–H bond critical point and ends on the H atom of the latter group (Fig. 2a). In the conformer B the atomic interaction line starts on the Ni atom and runs along the Ni–H bond. It changes the direction near the Ni–H bond critical point, goes toward the Ni+1–H bond critical point and ends on the H atom of the latter group (Fig. 2b). It means that these closed-shell interactions might be treated as the week interactions between adjacent N–H groups. It is known [2,9], that the antiparallel and parallel bsheets share one common feature: each carbonyl oxygen atom that juts into the region between the two strands has access to two sorts of proton donors (Figs. 3 and 4). This oxygen atom forms two types of the interchain H-bonds: N–H. . .O and Ca–H. . .O. According to the computed values of qb, $2qb and |vb|/gb (Table 3), the N–H. . .O bonds are stronger than Ca–H. . .O in the antiparallel conformation but only by a relatively small margin. In the parallel conformation, the Ca–H. . .O bonds are found to be even stronger than N–H. . .O, see Table 3. We note that
M.V. Vener et al. / Chemical Physics Letters 440 (2007) 279–285
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Fig. 3. Molecular graphs and the critical point pattern in the parallel b-sheet conformation of polyalanine. See the captions of Fig. 1 for the color coding. The intra- and interchain H-bonds (Table 3) are given by the broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Molecular graphs and the critical point pattern in the antiparallel b-sheet conformation of polyalanine. See the captions of Fig. 1 for the color coding. The interchain H-bonds (Table 3) are given by the broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 3 Geometrical parameters of the X–H. . .Y fragments, X = C, N, Y = O, N, H, and topological properties at critical point of the H-bonds, computed for the b-sheet conformers of polyalanine Conformer
H-bonded fragment
Structural parameters ˚ ˚ R(X . . . Y), A R(H . . . Y), A
\(XHY), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
Parallel
N–H. . .O N–H. . .O N–H. . .O (intra) N–H. . .O (intra) Ca–H. . .O Ca–H. . .O
3.490 3.486 2.642 2.666 3.180 3.186
2.615 2.609 2.132 2.209 2.244 2.251
143.1 143.4 108.4 105.1 141.6 141.9
0.006 0.006 0.022 0.019 0.014 0.014
0.024 0.024 0.084 0.084 0.048 0.048
0.81 0.80 0.91 0.87 0.91 0.92
Anti-parallel
N–H. . .O N–H. . .O Ca–H. . .O Ca–H. . .O Ca–H. . .H–aC (inter) Cb–H. . .H–bC (inter)
3.067 3.074 3.260 3.248 – –
2.094 2.101 2.354 2.348 2.289 2.787
157.3 156.9 138.7 138.0 – –
0.016 0.016 0.012 0.013 0.005 0.003
0.056 0.052 0.040 0.040 0.020 0.008
0.96 0.96 0.94 0.94 0.67 0.65
The H-bonds are defined in Figs. 3 and 4.
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Scheiner [9] reached the same conclusion in his recent theoretical study of the energy contributions of the N–H. . .O and Ca–H. . .O H-bonds to stability of the b-sheet conformations of polyglycine. The values of electron density at the O. . .H bond critical point evaluated in the present study for the b-sheets agree with the corresponding qb values calculated for the alanine octapeptide [30]. According to Table 4 of Ref. [30], qb varies from 0.003 to 0.019 a.u. The QTAIM enabled us to detect the nearest-neighbor intrachain interactions that stabilize the parallel b-sheet conformation. Indeed, the intrachain H-bonds exist between the adjacent N–H and O@C groups (Fig. 3) yielding formation of the 5-member pseudocycles. As it follows from the QTAIM parameters (Table 3), they are even stronger the interchain H-bonds of the N–H. . .O in the parallel b-sheet conformation. The point is that the 5-member pseudocycles were not detected for the antiparallel conformation. Possible explanation of this finding is the different structural parameters of the intrachain N–H O fragments in the parallel and antiparallel conformations. In the parallel b-sheet conformation the considered fragments are strongly nonlinear, the XHO angle equals to 105.1° for the fragments located in the region between the two strands and to 108.4° for the fragments located outside the region (Table 3). In the case of the antiparallel b-sheet conformation, these fragments are even more non-linear, than those in the parallel conformation. The O. . .H distances are also larger than those in the parallel conformation. On the other hand, the interchain H-bonds of the Cb–H. . .H–bC and Ca–H. . .H–aC type are observed in the antiparallel b-sheet conformation (Fig. 4), they are very weak as it is seen from Table 3. We like to note that the interchain H-bonds of the Ca– H. . .H–aC type were also recently reported for the antiparallel b-sheet conformation of polyglycine, see Fig. 11 in [31]. Each carbonyl in the a-helix conformation of polyalanine forms the two types of H-bonds: Ci = O. . .H–Ni+4 and Ci = O. . .H–Cbiþ3 (Fig. 5). According to our calculations, the C@O. . .H–Cb interactions (qb = 0.003 a.u.) are
much weaker than N–H. . .O (qb = 0.025 a.u.). We note that Morozov and Lin [32] reached the similar conclusion in their recent all-atom force-field molecular dynamics simulations of folding and unfolding mechanisms in the alanine-based a-helical polypeptide. It means that omitting of the nearest-neighbor interactions in classical LifsonRoig thermodynamic helix-coil transition theory [33] has no solid physical foundation. 4. Conclusions The QTAIM approach enabled us to detect a large number of the intraturn stabilizing interactions in b-turn structures of the tetrapeptide model compound. We found that only the conformer D (type IVa, b-turn) exhibits a 10member pseudocycle. Intrachain H-bonds between adjacent N–H and O@C groups exist in the parallel b-sheet conformation of polyalanine. They are even stronger than the interchain N– H. . .O bonds. Intrachain H-bonds, yielding formation of the 5-member pseudocycles, are not detected in the antiparallel b-sheet conformation. It looks as the geometry prevents the intrachain H-bond formation of the antiparallel b-sheet conformation of the polyalanine. The weak interactions of the Ci = O. . .H–Cbiþ3 type exist in the alanine-based a-helical polypeptide. These nearestneighbor interactions play a significant structure-forming role; therefore, they should be included into the model potentials explored in the all-atom force-field molecular dynamics simulations of folding and unfolding mechanisms in the a-helical polypeptides. Acknowledgements This study was supported by the Russian Federal Agency for Education (Program ‘Development of the Highest-School Scientific Potential: 2006–2008, Project 2.1.1.5051) and Russian Foundation for Basic Research, Grant 07-03-00702. M.V.V. thanks Dr. J. Rossmeisl for providing the coordinates of the b-sheet conformations of polyalanine and A.A. Rykounov for useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Fig. 5. Schematic 2D representation of the a-helix conformation of polyalanine. The intrachain H-bonds corresponding to this scheme are listed in the text.
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QTAIM study of the closed-shell interactions in peptide secondary structures: A cluster treatment of oligo- and polyalanines M.V. Vener *, A.N. Egorova, D.P. Fomin, V.G. Tsirelson Department of Quantum Chemistry, Mendeleev University of Chemical Technology, Miusskaya Square 9, 125047 Moscow, Russia Received 18 January 2007; in final form 4 April 2007 Available online 19 April 2007
Abstract The closed-shell interactions in oligo- and polyalanines are studied by the quantum theory of atoms in molecules (QTAIM) using electron densities derived from the B3LYP/6-31+G** ground-state electronic wave-functions. The QTAIM enabled us to identify a large number of the intraturn closed-shell stabilizing interactions in the b-turns, which were presented by several conformers of the tetrapeptide model compound. We found that only b-turn type IVa exhibits a 10-member pseudocycle. The intrachain H-bonds between the adjacent N–H and C@O groups in the antiparallel b-sheet conformation of polyalanine have not been found. At the same time, these interactions do exist in the parallel conformation and are even stronger than the interchain N–H. . .O bonds. A weak interaction between the C@O group at the position i and the side-chain C–H group at the position i + 3 was detected in the a-helical conformation of polyalanine. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction The local geometry of the secondary structures of polyaminoacids is influenced by several weak closed-shell interactions, as the intra- and intermolecular hydrogen bonding (H-bonding) [1–4] and the stacking interactions in base duplexes [5]. In this Letter, we aim to provide insight into the features of the closed-shell bonding interactions driving the intra- and intermolecular recognition in the secondarystructure elements of the alanine-based oligo- and polypeptides responsible for their mutual arrangement (the local geometry). While much literature exists about the geometry and energy of the closed-shell interactions in the cluster [6– 9] and periodical-crystal [10–12] models of oligo- and polypeptides, no previous reports discussing the electron density characteristics underlying the H-bonding in the secondary structures have been published yet, to our best knowledge. Therefore, we present the first systematic quantum theory of atoms-in-molecules (QTAIM) [13] study of
*
Corresponding author. Fax: +7 495 609 2964. E-mail address: [email protected] (M.V. Vener).
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.04.044
the closed-shell interactions in the alanine-based oligopeptides (c-turns and b-turns) as well as in the a-helix and bsheet conformations of polyalanines. 2. The secondary-structure models and computational methods Two sets of the secondary-structure models are used in this study. The first one includes the gas-phase models of the c- and b-turns presenting by the corresponding conformers of the tripeptide and tetrapeptide model compounds. The cluster models of the a-helix and b-sheets conformations of polyalanines display the second set. All the DFT calculations in this work were carried out using the PC version [14] of the GAMESS(US) program package [15]. Geometries of the gas-phase models of the c-turns and b-turns were optimized at the B3LYP/ 6-31+G** level of approximation. The minimum-energy states of the structures have been confirmed by calculating the harmonic frequencies. Single-point B3LYP/6-31+G** computations have been performed for each cluster model of the a-helix and b-sheets conformations of polyalanine frozen at the geometries taken from the DFT calculations,
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which employ the periodic boundary conditions [11,16]. The cluster models were derived from the corresponding infinite structures in a following way. (i) All the atoms entering the unit cell were included into consideration. (ii) Terminal N–H and C@O fragments were saturated with hydrogen atoms and hydroxyl groups, respectively. Electron-density properties were evaluated with the AIMPAC computer program suite [17]. The following QTAIM features at the H. . .A (A = O, N, C, H) bond critical point were considered: the values of electron density, qb, the Laplacian of electron density, $2qb, as well as the ratio |vb|/gb, where vb and gb are the potential and kinetic energy densities, respectively. According to [13,18,19], this set of parameters allows to quantify the atomic interactions. The closed-shell interactions show qb 6 0.05 a.u., $2qb > 0 and |vb|/gb < 1, and are presented graphically by atomic interaction lines [20] which may be, in principle, very curved and winding, e.g. see [5,21]. The number and type of the critical points revealed in our structures have fit to the Poincare–Hopf relation n b + rc = 1 [13], which links the critical points of different type with the molecular graph elements (n, b, r, c are the numbers of nuclear, bond, ring and cage critical points, respectively). 3. Results and discussion Two conformers of N-acetyl-L-alanine N 0 -methylamide (the tripeptide model compound), C7ax and C7eq [4,22], are shown in Fig. 1. They are stabilized by the intramolecular (intraturn) H-bond between the N–H group at the position i and the C@O group at the position i + 2 [23]. The number 7 stands for the number of atoms involved in the pseudocycle generated by the intraturn H-bond. An alternative nomenclature of common use for these structures is c-turn [24]. The energy difference between the C7eq and C7ax conformers is 2.51 kcal/mol (B3LYP/ 6-31+G**), which is close to 2.28 kcal/mol reported in [25] as the result of the MP2/aug-cc-pVDZ calculation. Geometrical parameters of the H-bonded fragments computed for the considered conformers at the B3LYP/6-
Fig. 1. Molecular graphs and the critical point pattern in the C7eq and C7ax conformers of the tripeptide model compound: (a) inverse c-turn, (b) classic c-turn. The small dots indicate the position of the critical points in the electron density: red stands for the bond critical points, while yellow stands for the ring critical points. The intraturn H-bonds (Table 1) are given by the broken lines. The color coding of this caption also applies to Figs. 2–4. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
31+G** level of approximation are in reasonable agreement with those obtained using MP2/aug-cc-pVDZ (Table 1). The most stable conformer C7eq (the inverse
Table 1 Structural parameters for H-bonds in B3LYP/6-31+G** (in regular letters) and MP2/aug-cc-pVDZa (in italics) geometries of the C7eq and C7ax conformers of the tripeptide model compound Conformer
H-bonded fragment
Structural parameters ˚ ˚ R(X. . .O), A R(H. . .O), A
\(XHO), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
C7eqc
N–H. . .O
2.97 2.91
2.09 2.02
143 145
0.020 0.007–0.041
0.059 0.029–0.199
1.03
C7ax
N–H. . .O
2.86 2.82 3.14 3.08
1.93 1.88 2.55 2.48
150 152 113 114
0.028 0.007–0.041 0.010 0.004–0.009
0.081 0.029–0.199 0.037 0.029–0.062
1.03
Cb–H. . .O
0.85
Topological properties at the H. . .O bond critical point in these structures computed using B3LYP/6-31+G** (in regular Letters) and the corresponding experimental valuesb, obtained for the crystal structures of the six amino acids (in italics). The H-bonds are defined in Fig. 1. a Ref. [25]. b Ref. [26]. c Total energy is 495.898557 a.u.
M.V. Vener et al. / Chemical Physics Letters 440 (2007) 279–285
c-turn) has only one intraturn H-bond of the N–H. . .O type, while the local-minimum structure C7ax (the classic c-turn) exhibits an additional H-bond of the C(b)–H. . .O type (Fig. 1). The latter H-bond generates the 6-member pseudocycle with the ring point located quite close to the C(b)–H. . .O bond critical point, see Fig. 1b. It evidences that stability of the 6-member pseudocycle seems to be much less than that of the 7-member one. Computed qb and $2qb values at the H. . .O bond critical point are found to be in reasonable agreement with the available experimental data derived from the electron density of the amino-acid crystals [26] (Table 1) and the computed topological properties of the intramolecular H-bond between a methyl group and a negatively charged oxygen in the biomolecules creatine and carbamoyl sarcosine [27]. Calculated |vb|/gb ratio is around 0.9 for the considered Hbonds. It fits the 0.5 6 |vb|/gb 6 1.0 ratio reported earlier
281
for the backbone–backbone interactions in different dinucleotides [5]. Summing up, the use of the B3LYP/ 6-31+G** approximation for the QTAIM study of the oligo- and polyalanines seems to be verified. The C10 conformers of N-acetyl-(L)-alanyl-(L)-alanine methylamide (the tetrapeptide model compound) [4] are shown in Fig. 2. They are stabilized by the intraturn Hbond between the N–H group at the position i and the C@O group at the position i + 3 [23]. The number of atoms involved in the pseudocycle generated by the intraturn Hbond is 10. An alternative nomenclature of common use for these structures is b-turn [24]. Local-minimum structures B (type II, b-turn) and D (type IVa, b-turn) are only 2.88 and 1.11 kcal/mol higher than the global-minimum structure A (type I, b-turn). We were unable to localize the C (type III, b-turn) conformer, see Fig. 2 in [4]; it seems to be unstable in the gas-phase model.
Fig. 2. Molecular graphs and the critical point pattern in the C10 conformers of the tetrapeptide model compound: (a) A (type I, b-turn), (b) B (type II, bturn), (c) D (type IVa, b-turn). See the captions of Fig. 1 for the color coding. The broken lines schematically present the atomic interaction lines. The latter describe the intraturn H-bonds given in Table 2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Table 2 Geometrical parameters of the X–H. . .Y fragments, where X = C, N, Y = O, N, H, and topological properties at the H-bond critical point, computed for the C10 conformers of the tetrapeptide model compound Conformer a
H-bonded fragment or interaction groups
Structural parameters ˚ ˚ R(X. . .Y), A R(H. . .Y), A
\(XHY), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
A
Ni-Hb. . .Oi+3 Ni–Hb. . .Ni+1 Ni+1–H. . .Ni+2
3.143 2.768 2.793
2.172 2.353 2.332
159.7 103.4 106.5
0.015 0.016 0.016
0.047 0.077 0.063
0.98 0.77 0.86
B
Ni–Hb. . .Oi+3 Ni+1–Hb. . .Oi+3 Cbiþ1 –H. . .Oi+2 Ni–Hb. . .Hb–Ni+1
3.378 3.054 3.176 –
2.433 2.031 2.553 1.930
155.1 139.2 92.6 107.3b
0.009 0.024 0.010 0.013
0.032 0.070 0.035 0.064
0.89 1.03 0.84 0.76
D
Ni–Hb. . .Oi+3 Caiþ2 –Hb. . .Oi+3 Caiþ2 –Hb. . . H–Caiþ1
3.005 3.434 –
2.026 2.597 1.901
160.8 133.0 –
0.020 0.009 0.015 0.003–0.017c
0.063 0.029 0.056 0.009–0.054c
0.99 0.88 0.81 0.75d
The H-bonds are defined in Fig. 2. Hb indicates that H-atom participates in a bifurcated H-bond. Available theoretical data on quantum-topological properties of the C–H. . .H–C bonds [5,28] are given in italics. a Total energy is 743.243722 a.u. b The Hb. . .Hb–Ni+1 angle. c Ref. [28]. d Ref. [5].
Computed values of the geometrical parameters of the H-bonded fragments and topological properties at the Hbond critical point are given in Table 2. According to it, all the H-bonds in the C10 conformers fall into the region of the closed-shell interactions [19] characterizing by relatively small values of qb (<0.03 a.u.), small positive values of $2qb (<0.08 a.u.) and |vb|/gb < 1.0. In contrast to cturns, the b-turns have bifurcated or ‘three-center’ Hbonds [29], cf. Figs. 1 and 2. The nature of these interactions is considered below in a separate paragraph. Bifurcated H-bond between the Ca–H groups in the conformer D involves the H. . .H interaction, see Fig. 2c. Its topological properties are in reasonable agreement with the available theoretical data (Table 2). The global-minimum conformer A exhibits two additional H-bonds of the N–H. . .N type between the adjacent N–H groups (Fig. 2a). The N–H. . .N fragments are strongly non-linear, however, their strength is comparable to that of the ‘standard’ intraturn N–H. . .O fragment, because the qb values are close to each other (Table 2). Due to the N–H. . .N interactions, the 10-member pseudocycle in the conformer A splits into 5- and 7-member pseudocycles and the additional 5-member pseudocycle is formed (Fig. 2a). The ring critical points of the 5-member pseudocycles locate very close to the bond critical points of the corresponding H. . .N bonds, see Fig. 2a. This is why their stability is expected to be less than the 7-member pseudocycle. Several additional H-bonds are detected for the conformers B and D, see Fig. 2b and c. The X– H. . .Y fragments, X = N, C and Y = N, O, formed by these bonds, are strongly non-linear, with the XHY angle varying from 93° to 133° (Table 2). The Ni+1–H. . .Oi+3 interaction is found to be the strongest closed-shell interaction in the conformer B, qb = 0.024 a.u. (Table 2). It generates two consecutive 7-member pseudocycles, one of which,
however, splits into 6- and 3-member pseudocycles by the weak interaction between the adjacent NH groups, see Table 2 and Fig. 2b. It should be noted that the conformer B is found to be energetically less favorable than conformer A for some model peptides in the low-polarity solvents [23]. However, the conformer B becomes preferred in the solid state because it allows the N–H groups to participate in the nearest-neighbor contacts yielding two consecutive cturns, i.e. the two 7-member pseudocycles [23], in accord with the results of the present study. Special attention should be paid to the nature of the closed-shell interactions between the Ni–H and Ni+1–H groups in the conformers A and B. The corresponding atomic interaction lines are very curved and winding. In the conformer A it starts on the Ni+1 atom and goes toward the adjacent Ni–H bond. It curves near the Ni–H bond critical point and ends on the H atom of the latter group (Fig. 2a). In the conformer B the atomic interaction line starts on the Ni atom and runs along the Ni–H bond. It changes the direction near the Ni–H bond critical point, goes toward the Ni+1–H bond critical point and ends on the H atom of the latter group (Fig. 2b). It means that these closed-shell interactions might be treated as the week interactions between adjacent N–H groups. It is known [2,9], that the antiparallel and parallel bsheets share one common feature: each carbonyl oxygen atom that juts into the region between the two strands has access to two sorts of proton donors (Figs. 3 and 4). This oxygen atom forms two types of the interchain H-bonds: N–H. . .O and Ca–H. . .O. According to the computed values of qb, $2qb and |vb|/gb (Table 3), the N–H. . .O bonds are stronger than Ca–H. . .O in the antiparallel conformation but only by a relatively small margin. In the parallel conformation, the Ca–H. . .O bonds are found to be even stronger than N–H. . .O, see Table 3. We note that
M.V. Vener et al. / Chemical Physics Letters 440 (2007) 279–285
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Fig. 3. Molecular graphs and the critical point pattern in the parallel b-sheet conformation of polyalanine. See the captions of Fig. 1 for the color coding. The intra- and interchain H-bonds (Table 3) are given by the broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Molecular graphs and the critical point pattern in the antiparallel b-sheet conformation of polyalanine. See the captions of Fig. 1 for the color coding. The interchain H-bonds (Table 3) are given by the broken lines. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Table 3 Geometrical parameters of the X–H. . .Y fragments, X = C, N, Y = O, N, H, and topological properties at critical point of the H-bonds, computed for the b-sheet conformers of polyalanine Conformer
H-bonded fragment
Structural parameters ˚ ˚ R(X . . . Y), A R(H . . . Y), A
\(XHY), °
Topological properties qb, a.u.
$2qb, a.u.
|vb|/gb
Parallel
N–H. . .O N–H. . .O N–H. . .O (intra) N–H. . .O (intra) Ca–H. . .O Ca–H. . .O
3.490 3.486 2.642 2.666 3.180 3.186
2.615 2.609 2.132 2.209 2.244 2.251
143.1 143.4 108.4 105.1 141.6 141.9
0.006 0.006 0.022 0.019 0.014 0.014
0.024 0.024 0.084 0.084 0.048 0.048
0.81 0.80 0.91 0.87 0.91 0.92
Anti-parallel
N–H. . .O N–H. . .O Ca–H. . .O Ca–H. . .O Ca–H. . .H–aC (inter) Cb–H. . .H–bC (inter)
3.067 3.074 3.260 3.248 – –
2.094 2.101 2.354 2.348 2.289 2.787
157.3 156.9 138.7 138.0 – –
0.016 0.016 0.012 0.013 0.005 0.003
0.056 0.052 0.040 0.040 0.020 0.008
0.96 0.96 0.94 0.94 0.67 0.65
The H-bonds are defined in Figs. 3 and 4.
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Scheiner [9] reached the same conclusion in his recent theoretical study of the energy contributions of the N–H. . .O and Ca–H. . .O H-bonds to stability of the b-sheet conformations of polyglycine. The values of electron density at the O. . .H bond critical point evaluated in the present study for the b-sheets agree with the corresponding qb values calculated for the alanine octapeptide [30]. According to Table 4 of Ref. [30], qb varies from 0.003 to 0.019 a.u. The QTAIM enabled us to detect the nearest-neighbor intrachain interactions that stabilize the parallel b-sheet conformation. Indeed, the intrachain H-bonds exist between the adjacent N–H and O@C groups (Fig. 3) yielding formation of the 5-member pseudocycles. As it follows from the QTAIM parameters (Table 3), they are even stronger the interchain H-bonds of the N–H. . .O in the parallel b-sheet conformation. The point is that the 5-member pseudocycles were not detected for the antiparallel conformation. Possible explanation of this finding is the different structural parameters of the intrachain N–H O fragments in the parallel and antiparallel conformations. In the parallel b-sheet conformation the considered fragments are strongly nonlinear, the XHO angle equals to 105.1° for the fragments located in the region between the two strands and to 108.4° for the fragments located outside the region (Table 3). In the case of the antiparallel b-sheet conformation, these fragments are even more non-linear, than those in the parallel conformation. The O. . .H distances are also larger than those in the parallel conformation. On the other hand, the interchain H-bonds of the Cb–H. . .H–bC and Ca–H. . .H–aC type are observed in the antiparallel b-sheet conformation (Fig. 4), they are very weak as it is seen from Table 3. We like to note that the interchain H-bonds of the Ca– H. . .H–aC type were also recently reported for the antiparallel b-sheet conformation of polyglycine, see Fig. 11 in [31]. Each carbonyl in the a-helix conformation of polyalanine forms the two types of H-bonds: Ci = O. . .H–Ni+4 and Ci = O. . .H–Cbiþ3 (Fig. 5). According to our calculations, the C@O. . .H–Cb interactions (qb = 0.003 a.u.) are
much weaker than N–H. . .O (qb = 0.025 a.u.). We note that Morozov and Lin [32] reached the similar conclusion in their recent all-atom force-field molecular dynamics simulations of folding and unfolding mechanisms in the alanine-based a-helical polypeptide. It means that omitting of the nearest-neighbor interactions in classical LifsonRoig thermodynamic helix-coil transition theory [33] has no solid physical foundation. 4. Conclusions The QTAIM approach enabled us to detect a large number of the intraturn stabilizing interactions in b-turn structures of the tetrapeptide model compound. We found that only the conformer D (type IVa, b-turn) exhibits a 10member pseudocycle. Intrachain H-bonds between adjacent N–H and O@C groups exist in the parallel b-sheet conformation of polyalanine. They are even stronger than the interchain N– H. . .O bonds. Intrachain H-bonds, yielding formation of the 5-member pseudocycles, are not detected in the antiparallel b-sheet conformation. It looks as the geometry prevents the intrachain H-bond formation of the antiparallel b-sheet conformation of the polyalanine. The weak interactions of the Ci = O. . .H–Cbiþ3 type exist in the alanine-based a-helical polypeptide. These nearestneighbor interactions play a significant structure-forming role; therefore, they should be included into the model potentials explored in the all-atom force-field molecular dynamics simulations of folding and unfolding mechanisms in the a-helical polypeptides. Acknowledgements This study was supported by the Russian Federal Agency for Education (Program ‘Development of the Highest-School Scientific Potential: 2006–2008, Project 2.1.1.5051) and Russian Foundation for Basic Research, Grant 07-03-00702. M.V.V. thanks Dr. J. Rossmeisl for providing the coordinates of the b-sheet conformations of polyalanine and A.A. Rykounov for useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Fig. 5. Schematic 2D representation of the a-helix conformation of polyalanine. The intrachain H-bonds corresponding to this scheme are listed in the text.
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