Depolarizing Effects in Hydrogen Bond Energy in 310-Helices Revealed by Quantum Chemical Analysis

Hydrogen-bond (H-bond) energies in 310-helices of short alanine peptides were systematically examined by precise DFT calculations with the negative fragmentation approach (NFA), a modified method based on the molecular tailoring approach. The contribution of each H-bond was evaluated in detail from the 310-helical conformation of total energies (whole helical model, WH3-10 model), and the results were compared with the property of H-bond in α-helix from our previous study. The H-bond energies of the WH3-10 model exhibited tendencies different from those exhibited by the α-helix in that they depended on the helical position of the relevant H-bond pair. H-bond pairs adjacent to the terminal H-bond pairs were observed to be strongly destabilized. The analysis of electronic structures indicated that structural characteristics cause the destabilization of the H-bond in 310-helices. We also found that the longer the helix length, the more stable the H-bond in the terminal pairs of the WH3-10 model, suggesting the action of H-bond cooperativity.


Introduction
Proteins are macromolecules essential for sustaining life and are also known to perform diverse biochemical functions in nature, such as molecular recognition, chemical catalysis, molecular switching, and the structural maintenance of cells [1][2][3][4]. They typically comprise 20 different amino acids linked by peptide bonds. In aqueous solutions, the polypeptide chains in proteins fold according to their amino acid sequencing and form a three-dimensional structure. The remarkable functional versatility of proteins results from the chemical diversity of the side chains of the constituent amino acids, flexibility of the polypeptide chains, and excellent variety of structures rendered possible by the wide range of amino acid sequences.
Approximately 90% of the amino acid residues in protein structures are found in locally ordered secondary structures, such as an α-helix or a β-sheet [5]. These secondary structures assemble and fold to form three-dimensional structures, also known as tertiary structures. In other words, secondary structures are the building blocks of protein structures.
The helix is the most commonly observed secondary structure and can be classified into different helical conformations [3,4]. Of these, the α-helix is predominant and is found in 80% of the helical structures [6]. The α-helix comprises a remarkably rigid arrangement of polypeptide chains and is a common secondary structural element in fibrous and globular proteins. It is arranged such that the peptide C=O group of the i-th residue along the helix faces the peptide N-H group of the (i + 4)-th residue, which results in the formation of In the present study, we systematically investigated the H-bond energies and associated electron density changes in 3 10 -helices using QM calculations and compared them with those observed for α-helices in our previous study [35]. The contribution of each H-bond was evaluated from the total conformational energy (whole-helical and WH 3-10 models). To understand the characteristics of the H-bond energy in 3 10 -helices, we additionally evaluated the H-bond energies using the following simplified models: minimal hydrogen bond (MH 3-10 ) models, wherein only H-bond donors and acceptors were present with capping methyl groups, and the single tern (ST 3-10 ) model, which includes a single helical turn. The characteristic interactions essential for 3 10 -helices are quantitatively discussed.
The H-bond distances and torsion angles were analyzed for the optimized WH 3-10 models. The H-bond distance was defined as the distance between the oxygen atom in the backbone C=O group and the hydrogen atom in the backbone N-H group, which form an H-bond. A histogram of the H-bond distances and a plot of the H-bond distances for each pair in the WH 3-10 and the corresponding α-helix models (WH a ) are shown in Figure 1.
The mean values and standard deviations of the H-bond distances for the WH 3-10 and WH a models were found to be 2.05 ± 0.04 and 2.30 ± 0.07 Å, respectively. The WH 3-10 models generally exhibited shorter H-bonds than those exhibited by the WH a models. A characteristic feature of WH [3][4][5][6][7][8][9][10] is that the H-bond distances tend to be greater in the H-bond pairs adjacent to the terminal ones ( Figure 1b and Table 1). The 4-2 H-bond pair in WH3-10, sandwiched between the N-and C-terminal H-bond pairs as indicated in the figure of Methods and Materials, showed the longest H-bond distance.
bonyl oxygen and the participation of amide hydrogen in the H-bond. Such depolarizations redistribute the electron density and are caused by local short-ranged electrostatic interactions with neighboring species in the α-helical structure [35].
In the present study, we systematically investigated the H-bond energies and associated electron density changes in 310-helices using QM calculations and compared them with those observed for α-helices in our previous study [35]. The contribution of each Hbond was evaluated from the total conformational energy (whole-helical and WH3-10 models). To understand the characteristics of the H-bond energy in 310-helices, we additionally evaluated the H-bond energies using the following simplified models: minimal hydrogen bond (MH3-10) models, wherein only H-bond donors and acceptors were present with capping methyl groups, and the single tern (ST3-10) model, which includes a single helical turn. The characteristic interactions essential for 310-helices are quantitatively discussed.

Structures of H-Bonds of the Optimized Whole-Helical Models of 310-Helices
We constructed three models of the 310-helices: WH3-10, which is oligoalanine peptides capped with the acetyl (Ace) and N-methyl amide groups (Nme), referred to as Ace-(Ala)n-Nme (n = 2 to 7) and the simplified models: single turn model (ST3-10) and minimal H-bond model (MH3-10). We denote WH3-10 model of Ace-(Ala)n-Nme by WH3-10-n and represent the s-th H-bond in WH3-10-n counting from the N-terminus by n-s. For comparison, we used the previously reported whole-helical structure models (WHa) and simplified models (STa and MHa) of the α-helices. The details of these models are described in Methods and Materials.
The H-bond distances and torsion angles were analyzed for the optimized WH3-10 models. The H-bond distance was defined as the distance between the oxygen atom in the backbone C=O group and the hydrogen atom in the backbone N-H group, which form an H-bond. A histogram of the H-bond distances and a plot of the H-bond distances for each pair in the WH3-10 and the corresponding α-helix models (WHa) are shown in Figure 1. The mean values and standard deviations of the H-bond distances for the WH3-10 and WHa models were found to be 2.05 ± 0.04 and 2.30 ± 0.07 Å , respectively. The WH3-10 models generally exhibited shorter H-bonds than those exhibited by the WHa models. A characteristic feature of WH3-10 is that the H-bond distances tend to be greater in the H-bond pairs adjacent to the terminal ones ( Figure 1b and Table 1). The 4-2 H-bond pair in WH3-10, sandwiched between the N-and C-terminal H-bond pairs as indicated in the figure of Methods and Materials, showed the longest H-bond distance.  (a) Histograms of the H-bond distances in the WH 3-10 and WH a models. Plots of the H-bond distances in the (b) WH 3-10 and (c) WH a models of each length. The H-bond distances of Ace-(Ala) n -Nme, which forms one-to-six H-bonds, are shown in maroon, red, black, blue, green, and violet-red, respectively. The stars and circles indicate the H-bond pairs adjacent to the terminal pair and the other pairs, respectively. The H-bond torsion angle was evaluated as the angle between the vectors of the C=O and N-H atoms on the backbone. This corresponds to the dihedral angle of the C, O, N, and H atoms comprising the H-bond. As shown in Figure 2b, the N-terminal H-bonding pairs in the WH a model exhibited significantly different torsion angles than those exhibited by the other H-bonding pairs. In the WH 3-10 models, the H-bond torsion angles of individual H-bond pairs varied with the length of the helix and their position in it ( Figure 2a). In addition, the H-bond torsion angles of the WH 3-10 models are narrowly distributed, indicating that the molecular backbone of the WH 3-10 models imposes stronger structural constraints than those of the WH a models. These constraints can possibly lead to considerably shorter H-bond distances in the WH 3-10 models than in the WH a models, as shown in Figure 1a.  The H-bond torsion angle was evaluated as the angle between the vectors of the C and N-H atoms on the backbone. This corresponds to the dihedral angle of the C, O, and H atoms comprising the H-bond. As shown in Figure 2b, the N-terminal H-bondi pairs in the WHa model exhibited significantly different torsion angles than those exh ited by the other H-bonding pairs. In the WH3-10 models, the H-bond torsion angles individual H-bond pairs varied with the length of the helix and their position in it (Figu 2a). In addition, the H-bond torsion angles of the WH3-10 models are narrowly distribute indicating that the molecular backbone of the WH3-10 models imposes stronger structu constraints than those of the WHa models. These constraints can possibly lead to cons erably shorter H-bond distances in the WH3-10 models than in the WHa models, as show in Figure 1a.

Comparison of H-Bond Energy in Helical Model Systems
To investigate the effects of the helical backbone atoms linking the H-bond accep and donor on the H-bond energies, we compared the H-bond energies for the WH3-10, S 10, and MH3-10 models of the 310-helices as well as those for the α-helices from our previo study, namely WHa, STa, and MHa [35]. In the MH3-10 and MHa models, the two pepti groups of hydrogen donors and acceptors were separated without linking the heli backbone atoms. The H-bond energies were calculated with the NFA (Details are show in Section 4.2). In Figure 3a, the H-bond energies of the WH3-10 (black) and ST3-10 (re models are plotted versus those of the MH3-10 models for the 310-helices. Those of the WH STa, and MHa models for α-helices [35] are also shown in Figure 3b. In the α-helices, the STa model reproduced the H-bond energies of the WHa mod (Figure 3b), indicating that the adjacent residue destabilized the H-bond with respect the MHa model [35]. In the 310-helices, the ST3-10 models also destabilized the H-bond w

Comparison of H-Bond Energy in Helical Model Systems
To investigate the effects of the helical backbone atoms linking the H-bond acceptor and donor on the H-bond energies, we compared the H-bond energies for the WH 3-10 , ST 3-10 , and MH 3-10 models of the 3 10 -helices as well as those for the α-helices from our previous study, namely WH a , ST a , and MH a [35]. In the MH 3-10 and MH a models, the two peptide groups of hydrogen donors and acceptors were separated without linking the helical backbone atoms. The H-bond energies were calculated with the NFA (Details are shown in Section 4.2). In Figure 3a, the H-bond energies of the WH 3-10 (black) and ST 3-10 (red) models are plotted versus those of the MH 3-10 models for the 3 10 -helices. Those of the WH a , ST a , and MH a models for α-helices [35] are also shown in Figure 3b. 10-3, 3-1, is adjacent to the C-terminal H-bond pair 3-2, and vice versa.
Considering the H-bond energies of the ST3-10 model, only the H-bond pairs next to the N-or C-terminal were destabilized by the adjacent residue, as shown in Figure 3a. The pair sandwiched between the N-and C-termini were also the most affected (4-2 of WH3- , which causes a difference in the linear regression coefficient between the WH3-10 and ST3-10 models, as shown in Figure 3a. To further examine the effect of neighboring residues on the H-bond energy of the 310-helices, we constructed additional models as follows: STEC, in which the ST3-10 model, Ace-(Ala)2-Nme, was extended to the C-terminal (Ace-(Ala)2-Ala-Nme); STEN, in which the ST3-10 model was extended to the N-terminal (Ace-Ala-(Ala)2-Nme); STECN, in which the ST3-10 model was extended to both N-and C-termini (Ace-Ala-(Ala)2-Ala-Nme); and STE2N, in which the ST3-10 model was extended by two residues to the N-terminal (Ace-Ala-Ala-(Ala)2-Nme). All peptide structures were generated based on the WH3-10 model, and their H-bond energies were computed by using the NFA.
As expected, in the STECN model, both the N-and C-terminal H-bond pairs were adjacent to the target H-bond, thereby making the H-bond energy unstable ( Figure 4). The STEC and STEN models were more stable than the STECN models. The STE2N model showed nearly the same results as the ST3-10 model and higher energy values than those of the terminal pairs in the WH3-10 model, as shown in the crosses in Figure 4. In the α-helices, the ST a model reproduced the H-bond energies of the WH a model (Figure 3b), indicating that the adjacent residue destabilized the H-bond with respect to the MH a model [35]. In the 3 10 -helices, the ST 3-10 models also destabilized the H-bond with respect to the MH 3-10 models, similar to the α-helices, but failed to provide the equivalent H-bond energies. In particular, the H-bond pairs adjacent to the N-or C-terminus were strongly destabilized in energy (underlined in Table S1). This indicates that the helical backbone atoms participating in the H-bond are partly involved in the destabilization of the H-bond but that other factors also lead to unstable H-bond energies. We also found a tendency that the longer the helix length, the more stable the H-bond energy of the N-and C-terminal pairs in the WH 3-10 model (shown in Table S1). However, there is an exception: The H-bond energies of the N-and C-terminal pairs in WH 3-10 -3 were observed to be higher than those of WH 3-10 -2. They are close to those of the H-bond pairs adjacent to the terminal H-bond pair in the other systems. This would be because the N-and C-termini are adjacent to each other in WH 3-10 -3. For example, the N-terminal H-bond pair of WH 3-10 -3, 3-1, is adjacent to the C-terminal H-bond pair 3-2, and vice versa.
Considering the H-bond energies of the ST 3-10 model, only the H-bond pairs next to the N-or C-terminal were destabilized by the adjacent residue, as shown in Figure 3a. The pair sandwiched between the N-and C-termini were also the most affected (4-2 of WH 3-10 -4), which causes a difference in the linear regression coefficient between the WH 3-10 and ST 3-10 models, as shown in Figure 3a.
To further examine the effect of neighboring residues on the H-bond energy of the 3 10 -helices, we constructed additional models as follows: ST EC , in which the ST 3-10 model, Ace-(Ala) 2 -Nme, was extended to the C-terminal (Ace-(Ala) 2 -Ala-Nme); ST EN , in which the ST 3-10 model was extended to the N-terminal (Ace-Ala-(Ala) 2 -Nme); ST ECN , in which the ST 3-10 model was extended to both N-and C-termini (Ace-Ala-(Ala) 2 -Ala-Nme); and ST E2N , in which the ST 3-10 model was extended by two residues to the N-terminal (Ace-Ala-Ala-(Ala) 2 -Nme). All peptide structures were generated based on the WH 3-10 model, and their H-bond energies were computed by using the NFA.
As expected, in the ST ECN

Electronic Structures around the H-Bond Donors and Acceptors
In addition to the H-bond energies, NFA can approximately represent the cha electronic structures upon H-bond formation by Equation (6) in Section 4.2. To ex the difference in the H-bond energy between the ST3-10 and MH3-10 models and be the WH3-10 and ST3-10 models in the context of their electronic structures, the differen the change in electron density were computed using Equations (1) and (2)

Electronic Structures around the H-Bond Donors and Acceptors
In addition to the H-bond energies, NFA can approximately represent the change of electronic structures upon H-bond formation by Equation (6) in Section 4.2. To examine the difference in the H-bond energy between the ST 3-10 and MH 3-10 models and between the WH 3-10 and ST 3-10 models in the context of their electronic structures, the differences in the change in electron density were computed using Equations (1)  As illustrated in Figure 5a-c, we found that the electron density increases in the vicinity of the oxygen atom of the C=O group and that it decreases in the vicinity of the hydrogen atom of the N-H group, thus demonstrating the formation of the H-bond. The ∆ρ WH 3−10 HB of 4-2 appears to be slightly smaller than that of 4-1, indicating a larger depolarization of 4-2 than that of 4-1. This larger depolarization results in a weaker H-bond at 4-2, as illustrated in Figure 5a-c. A remarkable difference was observed between 4-2 and 4-1 and between 4-2 and 4-3: the ∆∆ρ WH 3−10 −ST 3−10 HB of 4-1 and that of 4-3 was considerably smaller than that of 4-2, implying that an effect other than the helical backbone atoms between the acceptor and donor of H-bond pair was at play in the 4-2 pair (Figure 5g,i). Namely, the backbone atoms locating at the N-and C-terminal sides could provide the depolarization effect. This could be the reason behind the difference in the H-bond energy between the WH 3-10 and ST [3][4][5][6][7][8][9][10] in the H-bond pairs adjacent to the terminal pairs.

Dependence of Helix Length on H-Bond Energies
We investigated the dependence of helix length on the mean value of the H-bond energies in the WH3-10 and WHa models. The mean H-bond energies of these models were plotted as functions of the minimum length of the corresponding helices, as shown in Figure 6. In WHa models, the energy of the H-bond gradually stabilized with an increase in the length of the helix, demonstrating the well-known "H-bond cooperativity" phenomenon [27][28][29][30], where long-range interaction could make more stable helices. The mechanism behind the cooperativity in helix formation can be deconstructed into two parts, namely, electrostatic interactions between residues and nonadditive many-body effects caused by the redistribution of electron density with increasing helix length [29]. In the WH3-10 model series, the H-bond of ST3-10 was destabilized in the first increment of the minimum length of the helix, while subsequent increments gradually stabilized it. This is because the first increase in helix makes the H-bond adjacent to the terminal H-bond pair, leading to considerable destabilization of the H-bond, as discussed above. Subsequent elongation of the helix results in the stabilization of the terminal H-bond through H-bond cooperativity [28,30].

Dependence of Helix Length on H-Bond Energies
We investigated the dependence of helix length on the mean value of the H-bond energies in the WH 3-10 and WH a models. The mean H-bond energies of these models were plotted as functions of the minimum length of the corresponding helices, as shown in Figure 6. In WH a models, the energy of the H-bond gradually stabilized with an increase in the length of the helix, demonstrating the well-known "H-bond cooperativity" phenomenon [27][28][29][30], where long-range interaction could make more stable helices. The mechanism behind the cooperativity in helix formation can be deconstructed into two parts, namely, electrostatic interactions between residues and nonadditive many-body effects caused by the redistribution of electron density with increasing helix length [29]. In the WH 3-10 model series, the H-bond of ST 3-10 was destabilized in the first increment of the minimum length of the helix, while subsequent increments gradually stabilized it. This is because the first increase in helix makes the H-bond adjacent to the terminal H-bond pair, leading to considerable destabilization of the H-bond, as discussed above. Subsequent elongation of the helix results in the stabilization of the terminal H-bond through H-bond cooperativity [28,30].

Discussion
We systematically investigated the H-bond energies of various 310-helices and found them to exhibit tendencies different from those exhibited by α-helices. Here, we discuss the following three issues: (i) why the H-bond energies in the ST3-10 model are destabilized compared with those in the MH3-10 model, (ii) why the H-bond pairs adjacent to the terminal pair are largely destabilized compared with other H-bond pairs, and (iii) why the terminal H-bond pairs are stabilized, particularly for long 310-helices.
For the issue (i), as mentioned in Section 2.3, the C=O and N-H groups participating in the H-bond were depolarized in the ST3-10 model in comparison to the MH3-10 model. This depolarization could be caused by the helical backbone atoms linking the H-bond pair (residue 2 of Figure 7). In α-helices, the adjacent C=O group is involved in depolarization [35]. However, in the 310-helices, the C=O group of the H-bond pair is closer to the adjacent N-H group (~2.8 Å ) than to another adjacent C=O group (~3.4 Å ). Therefore, the destabilization of the H-bond is attributed to the depolarization caused by the N-H group.
For the issue (ii), when ST3-10 was extended to the N-terminus by a single residue (STEN), an additional H-bond was formed between the C=O group of residue −1 and the N-H group of residue 2. This H-bond formation causes polarization of the N-H group of residue 2, leading to further depolarization of the C=O group of residue −1. Thus, the Hbond could be destabilized. When STEN is further extended to the N-terminal (STE2N), an additional H-bond is formed between the N=H group of residue 1 and C=O group of residue -2 (back side of the helix in the right panel of Figure 7). This H-bond formation induces polarization of the N-H group of residue 1, causing polarization of the adjacent C=O group of residue 0. This effect could cancel the depolarization effect by the N-H group of residue 2 on the C=O group of residue 0, thereby strengthening the H-bond. At the C-terminal, the similar depolarization effect could destabilize the H-bond.

Discussion
We systematically investigated the H-bond energies of various 3 10 -helices and found them to exhibit tendencies different from those exhibited by α-helices. Here, we discuss the following three issues: (i) why the H-bond energies in the ST 3-10 model are destabilized compared with those in the MH 3-10 model, (ii) why the H-bond pairs adjacent to the terminal pair are largely destabilized compared with other H-bond pairs, and (iii) why the terminal H-bond pairs are stabilized, particularly for long 3 10 -helices.
For the issue (i), as mentioned in Section 2.3, the C=O and N-H groups participating in the H-bond were depolarized in the ST 3-10 model in comparison to the MH 3-10 model. This depolarization could be caused by the helical backbone atoms linking the H-bond pair (residue 2 of Figure 7). In α-helices, the adjacent C=O group is involved in depolarization [35]. However, in the 3 10 -helices, the C=O group of the H-bond pair is closer to the adjacent N-H group (~2.8 Å) than to another adjacent C=O group (~3.4 Å). Therefore, the destabilization of the H-bond is attributed to the depolarization caused by the N-H group. For understanding the electronic structures in the more quantitative manner, Hirshfeld population analysis [36][37][38] was performed for WH3-10, ST3-10, and MH3-10 models of 4-1, 4-2, and 4-3, respectively. We calculated the local dipole moments of the C=O and For the issue (ii), when ST 3-10 was extended to the N-terminus by a single residue (ST EN ), an additional H-bond was formed between the C=O group of residue −1 and the N-H group of residue 2. This H-bond formation causes polarization of the N-H group of residue 2, leading to further depolarization of the C=O group of residue −1. Thus, the H-bond could be destabilized. When ST EN is further extended to the N-terminal (ST E2N ), an additional H-bond is formed between the N=H group of residue 1 and C=O group of residue -2 (back side of the helix in the right panel of Figure 7). This H-bond formation induces polarization of the N-H group of residue 1, causing polarization of the adjacent C=O group of residue 0. This effect could cancel the depolarization effect by the N-H group of residue 2 on the C=O group of residue 0, thereby strengthening the H-bond. At the C-terminal, the similar depolarization effect could destabilize the H-bond.
For understanding the electronic structures in the more quantitative manner, Hirshfeld population analysis [36][37][38] was performed for WH 3-10 , ST 3-10 , and MH 3-10 models of 4-1, 4-2, and 4-3, respectively. We calculated the local dipole moments of the C=O and N-H groups of the backbone H-bond acceptor and donor, respectively, as follows [35]: Here, q i C and q i O are the Hirshfeld atomic charges of the C and O atoms in the C=O group of the i-th residue, and q i H and q i N are those of the N-H group of the i-th residue, respectively.  Table 2A,B. In the former, the H-bond pairs were formed, and in the latter, the H-bonds were not formed between the C=O group of the i-th residue and the N-H group of the (i + 3)-th residue.
As clearly shown in Table 2A,B, all Ratio (ST 3-10 /MH 3-10 ) and Ratio (WH 3-10 /MH 3-10 ) were less than 1. Namely, µ i CO and µ i+3 HN of ST 3-10 models and those of WH 3-10 models were always smaller than the corresponding dipole moments of MH 3-10 models, in which no neighboring carbonyl (C=O) or amide (N-H) groups exist. The amplitude of the depolarization effects relating to the above issue (i) was about 3% to 5% on average from Table 2A,B, independently of whether the H-bonds are formed or not.
Relating to the issue (ii), we found µ i CO of WH 3-10 model of 4-2 was 2% to 3% smaller than those of 4-1 and 4-3, as indicated in Table 2A,B. In contrast, µ i+3 HN of the WH 3-10 model of 4-2 had similar values to those of 4-1 and 4-3. These phenomena correlate well with the differences of electron densities shown in Figure 5.
The H-bond energies for the WH 3-10 and WH a models were compared using QM (E HB in Equation (5) in Section 4.2) and MM (E HB_MM in Equation (7) in Section 4.2) calculations. Figure 8a exhibits the H-bond energies calculated with QM plotted versus those with MM for individual pairs of the WH 3-10 and WH a models. In the WH a models, the H-bond energies obtained via QM calculations were shown to be strongly correlated to those obtained via MM, with a correlation coefficient of 0.89. However, the MM calculations overestimated the magnitude of the H-bond energies by~1 kcal/mol (the mean energy values obtained via QM and MM were −3.21 ± 0.39 and −4.24 ± 0.45 kcal/mol, respectively). Our previous study, wherein the energies were obtained via QM calculation, showed that the destabilization of the H-bond energies in the WH a model was attributed to the depolarization of the H-bond donors and acceptors caused by adjacent residues [35].   (7) in Section 4.2. Here, we used the AMBER ff99SB force-field rameters [16] for the atomic partial charges, which were originally determined by mu ple-conformation models fitting to the local conformations of a single amino acid for b the α-helical and extended structures [39]. Thus, the parameters may well reproduce bond energies for the α-helical conformations MHa but not for the 310-helices MH3-10. In the current study, it is revealed that the H-bond energy of 310-helix largely depe on its local conformation yielding the depolarization and on the long-ranged coopera ity effect. Those QM effects have not been included in the MM computations or for the bond energy of the α-helix [35]. In order to improve the MM force fields at least by incl ing the short-ranged interactions, there could be two approaches: (i) by modifying atomic partial charges, which are not constant values but depend on the local atomic c formation, and (ii) by creating new backbone dihedral parameters, which depends on In contrast, the H-bond energies for the WH 3-10 models obtained via QM calculation seem to be closer to those obtained with MM and were more stable than those of the WH a models (the mean energy values obtained via QM and MM for the WH 3-10 model were −4.96 ± 0.39 and −4.95 ± 0.24 kcal/mol, respectively). Unlike the WH a models, the correlation between the QM and MM calculations was weak: the correlation coefficient was evaluated to be 0.54, as shown in Figure 8a.
In Figure 8b, the H-bond energies calculated with QM are plotted versus those with MM for individual pairs of the MH 3-10 models and MH a models. Although the H-bond energies obtained via QM and MM for the MH a models almost coincided [35], the H-bond energies obtained by QM were significantly more stable than those obtained via MM for the MH 3-10 models. However, unlike the WH 3-10 models, the correlation between the H-bond energies obtained via QM and MM for the MH 3-10 models was acceptable, and the correlation coefficient was 0.88. The reason why the H-bond energies obtained using MM largely deviate from those obtained via QM may be the poor quality of the atomic partial charges in Equation (7) in Section 4.2. Here, we used the AMBER ff99SB forcefield parameters [16] for the atomic partial charges, which were originally determined by multiple-conformation models fitting to the local conformations of a single amino acid for both the α-helical and extended structures [39]. Thus, the parameters may well reproduce H-bond energies for the α-helical conformations MH a but not for the 3 10 -helices MH [3][4][5][6][7][8][9][10] .
In the current study, it is revealed that the H-bond energy of 3 10 -helix largely depends on its local conformation yielding the depolarization and on the long-ranged cooperativity effect. Those QM effects have not been included in the MM computations or for the H-bond energy of the α-helix [35]. In order to improve the MM force fields at least by including the short-ranged interactions, there could be two approaches: (i) by modifying the atomic partial charges, which are not constant values but depend on the local atomic conformation, and (ii) by creating new backbone dihedral parameters, which depends on not only a single amino acid residue but also on the parameter set including the neighboring residues, as suggested by our previous paper [35]. Those approaches may provide us more reliable MM parameters, although they would be difficult to attain. Whole-helical structure models of the 3 10 -helices (WH 3-10 ) were constructed using oligoalanine peptides capped with the acetyl (Ace) and N-methyl amide groups (Nme), referred to as Ace-(Ala) n -Nme. We used dipeptide-to-heptapeptide alanines (n = 2 to 7) for the WH 3-10 model, which consists of Ace-(Ala) n -Nme, is denoted by WH 3-10 -n. The backbone dihedral angles (ϕ, ψ) for each residue were set to ϕ = −49 • and ψ = −26 • . For comparison, we used the previously reported whole-helical structure models of the α-helices (WH a -n), composed of the alanine oligopeptide Ace-(Ala) n -Nme (n = 3 to 8), with ϕ = −57 • and ψ = −47 • [3]. These structures were optimized in the gas phase by the energy minimization of the electronic state while keeping the backbone dihedral angles fixed at the aforementioned values.

Methods and Materials
One-to-six H-bonds were present between the C=O and N-H groups in the backbone of the optimized WH 3-10 models, as H-bonds were formed between an i-th and (i + 3)-th peptide pair for WH [3][4][5][6][7][8][9][10] and between an i-th and (i + 4)-th peptide pair for WH a . The s-th H-bond in Ace-(Ala) n -Nme, counting from the N-terminus, is represented by n-s (Figure 9a shows WH 3-10 -4 as an example). The H-bond energies were individually calculated using DFT, as described below.
To analyze the origin of the H-bond interaction energy in 3 10 -helices, we designed two simplified models. One is an ST 3-10 model, composed of two successive alanine residues capped by Ace and Nme groups at the N-and C-termini, respectively (second column in Figure 9b). The other is an MH 3-10 model, which comprises two separated N-methyl acetamide molecules and mimics a single H-bond between the C=O and N-H groups in the backbone (third column in Figure 9b). The atomic positions of these two models were the same as those of the corresponding WH 3-10 models, except for the N-and C-terminal capping groups. The computation of the individual H-bond energies for these models was conducted in the same manner as that for each backbone H-bond in the WH 3-10 models, as described below. The H-bond energies of the three models were then compared. dues capped by Ace and Nme groups at the N-and C-termini, respectively (second column in Figure 9b). The other is an MH3-10 model, which comprises two separated N-methyl acetamide molecules and mimics a single H-bond between the C=O and N-H groups in the backbone (third column in Figure 9b). The atomic positions of these two models were the same as those of the corresponding WH3-10 models, except for the N-and C-terminal capping groups. The computation of the individual H-bond energies for these models was conducted in the same manner as that for each backbone H-bond in the WH3-10 models, as described below. The H-bond energies of the three models were then compared.

Calculation of H-Bond Energies Using the Negative Fragmentation Approach
The extraction of the H-bond energy from the total energy of a large molecule wherein the donor and acceptor atoms are linked through several covalent bonds, such as in α-and 310-helices, is not straightforward. In this study, we systematically computed the backbone H-bond energies in the WH3-10 models in the same manner as that computing the H-bond energies in the WHa models as reported previously [35], where we modified the MTA developed by Deshmukh et al. [34]. In the NFA, the H-bond energy, HB , in Ace-(Ala)n-Nme can be calculated using the following equation: where sys , A ̅ , D ̅ , and A∪D ̅̅̅̅̅̅̅ are the energies of the entire system, the system lacking the acceptor group, the system without the donor group, and the system lacking both acceptor and donor groups, respectively; detailed descriptions are available in our previous study [35]. In the original MTA, the energy of the entire system was estimated using the energies of all fragments [34]. In the NFA, we used the total energy of the entire system and showed that the difference between the results was negligible [35]. The change in the electron density upon H-bond formation, Δ HB , was evaluated as follows:

Calculation of H-Bond Energies Using the Negative Fragmentation Approach
The extraction of the H-bond energy from the total energy of a large molecule wherein the donor and acceptor atoms are linked through several covalent bonds, such as in αand 3 10 -helices, is not straightforward. In this study, we systematically computed the backbone H-bond energies in the WH 3-10 models in the same manner as that computing the H-bond energies in the WH a models as reported previously [35], where we modified the MTA developed by Deshmukh et al. [34]. In the NFA, the H-bond energy, E HB , in Ace-(Ala) n -Nme can be calculated using the following equation: where E sys , E A , E D , and E A∪D are the energies of the entire system, the system lacking the acceptor group, the system without the donor group, and the system lacking both acceptor and donor groups, respectively; detailed descriptions are available in our previous study [35]. In the original MTA, the energy of the entire system was estimated using the energies of all fragments [34]. In the NFA, we used the total energy of the entire system and showed that the difference between the results was negligible [35]. The change in the electron density upon H-bond formation, ∆ρ HB , was evaluated as follows: For comparison, we also computed the H-bond interaction energies via MM using the AMBER ff99SB force-field parameters [16], E HB_MM , for the corresponding H-bonds, as follows: where I and J are the four atoms constituting the peptide group, namely, C, O, N, and H, of an acceptor and a donor involved in the H-bond, respectively. B ij and C ij are the Lennard-Jones coefficients, r ij is the distance between the i-th and j-th atoms, and q i is the partial atomic charge of the i-th atom. The MM energy was calculated for the WH 3-10 and MH 3-10 models. Calculations for all models were performed using the Gaussian09 program package [40]. The B97D exchange-correlation functional was used with 6-31 + G(d) basis sets. This method is capable of correctly describing van der Waals interactions and is comparable with the MP2 method in the calculation of the H-bond interaction energies of the Ace-(Ala) n -Nme system in the gas phase [33]. Changes in electron density were computed using cube files provided in the Gaussian09 program packages [40], and molecules that exhibited such changes were depicted by using UCSF Chimera [41]. The other molecular structures were drawn by using the VMD software [42].

Conclusions
In this study, the H-bond energies and associated changes in the electron density of the atoms forming H-bond of the 3 10 -helices were systematically analyzed using the NFA method with high-quality DFT and MM computations and were compared with those of the α-helices. We prepared optimized structures of Ace-(Ala) n -Nme, where n ranged from 2 to 7 for the whole-helical structure models (WH 3-10 ). To quantitatively investigate the origin of the H-bond energy in each helical model, we also constructed single-turn models (ST 3-10 ), which comprised two successive alanine residues capped by Ace and Nme groups at the N-and C-termini, respectively, and minimum H-bond models (MH 3-10 ), which comprised only pairs of Ace-Nme forming a single H-bond. The structures of the ST 3-10 and MH 3-10 models were based on the WH 3-10 model. The individual H-bond energies were then computed using the NFA.
The distribution of the H-bond distance of the WH 3-10 models was narrow, and these models exhibited lower values than those exhibited by the α-helical models (WH a ). The shorter H-bond distance observed in the WH 3-10 model was due to restrictions imposed by the tight helical structure. The H-bond energy of the WH 3-10 model exhibited a tendency different from those exhibited by the ST 3-10 and MH 3-10 models; it depended on the location of the H-bond pair in the 3 10 -helices. Furthermore, the H-bonds in this model tended to be destabilized in the H-bond pairs adjacent to the terminal pairs and were stabilized at the terminal H-bond pairs. An analysis of changes in the electron density between the WH 3-10 and ST 3-10 models and between the ST 3-10 and MH 3-10 models suggested that the destabilization of the H-bond in the ST 3-10 model was attributed to the depolarization caused by the adjacent N-H group. It also suggested that the H-bond formation at this group causes polarization, leading to further depolarization of the C=O group participating in the H-bond pair and larger destabilization of the H-bond adjacent to the terminal H-bond pair. Except for the first increment, the elongation of the helix of the WH 3-10 model resulted in the stabilization of the terminal H-bond through H-bond cooperativity.