Probing the role of electrostatics of polypeptide main-chain in protein folding by perturbing N-terminal residue stereochemistry: DFT study with oligoalanine models

Abstract

Expression of the genome rests primarily on proteins adopting folds in the specificity of their sequence code over side-chains. The underlying basis, despite intense scrutiny of the apparent code, remains largely unclear as a protein-folding problem. The interactions internal to the polypeptide main-chain are ubiquitous to folding but their contribution to the energetics of folding remains uncertain. Given the uncertain roles of the solvent and sequence, and uncertain energetics of folded proteins, simple models are required to study the interactions between the backbone peptide units of polypeptide main-chain by exclusion of solvent and sequence effects. Thus, the oligoalanine peptides Ac–^LAla₄–NHMe (Ia), Ac–^DAla–^LAla₃–NHMe (Ib), Ac–^LPro–^LAla₃–NHMe (IIa), Ac–^DPro–^LAla₃–NHMe (IIb), Ac–^LPro₂–^LAla₂–NHMe (IIIa) and Ac–^DPro–^LPro–^LAla₂–NHMe (IIIb) were chosen as the N-terminal alanine or proline and L- or D-residue stereochemically perturbed models to scrutinize the role of electrostatics of backbone peptide units on the energetics of folding with density functional theory (DFT). DFT calculations revealed that the end-protected tetraalanine isomers, Ia and Ib, fold to identical specificity in hydrogen bonds, but with a strong contrast of energetics. Ib with a DLLL-stereochemical structure folds with apparent strength of hydrogen bonds twofold than Ia without, surprisingly, a notable change in geometry of the interactions involved. DFT calculations demonstrated that the energetics of folding from the extended structure to the folded structure in oligoalanine peptides critically depend on the geometrical relationship between backbone peptide units of the polypeptide structure. The results of the present study will provide key insights into the protein-folding problem and helix stability.

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Introduction

Proteins fold by maximizing main-chain in the mutual hydrogen bonds of backbone peptide units and unfold by maximizing the backbone peptide units in solvation. With solvation of backbone peptide units, polypeptide chain adopt a polyproline-II (PPII) conformation^1–6 whereas, with hydrogen bonds between backbone peptide units, the chain adopt a helical fold of an α conformation or extended fold of a β conformation. The conformational options in correspondence of folded and unfolded proteins and of α and β folds are specific for the poly-L stereochemistry of the structure, and are selectable naturally under sequence and solvent control. The basis for the stereochemical specificity of the conformational options of the natural structure are clear,^7–10 but the basis for their selection under solvent and sequence control define the protein-folding problem.^11–21 The protein-folding problem may lie in the interactions internal to the polypeptide main-chain. This possibility has been addressed by examining polypeptide main-chain for the effect of modifying the structure stereochemically.^22–27 Oligoalanine of a natural poly-L structure has been examined with computation and experiments for the effect of modifying the structure stereochemically.^22–27 In addition, stereochemistry has been explored as a sequence-level variable in approaching protein design.^28–31

In order to overcome the configurational entropy lost upon folding of a random coil structure into an α-helix structure, a number of electrostatic forces contribute to the favourable enthalpy of helix folding. The hydrogen bonds between backbone peptide units, hydrogen bonds and salt bridge interactions between side-chains, backbone–side-chain interactions at N- and C-termini of the helical structure, and through-space electrostatic interactions between charged side-chains and the helix macrodipole contribute towards electrostatic forces. However, energetic contributions and stability by the electrostatics of backbone peptide units to the polypeptide structure remain unclear. The role of electrostatic forces in the study of equilibria between unfolded states and α-helices in peptides is a topic of current interest and has been highlighted in recent studies.^32–38

The statistical mechanical and experimental studies of equilibria in model peptides have highlighted that sequentially neighbouring backbone peptide units “unfold” the polypeptide chain if they have a poly-L structure and “fold” the chain if they have an alternating-L,D structure on the basis of the electrostatics of backbone peptide units.^3,24–27 Thus, sequence-non-local electrostatics—mainly hydrogen bonds—will fold polypeptide structures either synergistically or under conflict with sequence-local electrostatics on the basis of the stereochemistry of the structure, poly-L, viz., homochiral, or alternating-L,D, viz., heterochiral. Thus, folding of polypeptide structures involves either the frustrated or harmonious energetics of the structure. The stereochemistry, poly-L or alternating-L,D, provides the basis for the effect which is specific for homochiral and heterochiral polypeptide structures. The notion that protein “folding” and “unfolding” may manifest conflicted electrostatics in the poly-L chain of backbone peptide units is reinforced by the observation that shorter helices of poly-L structure, having fewer hydrogen bonds to counter balance unfavourable C_local (Coulomb energy over the peptide groups within a residue and its immediate sequence neighbours summed over the entire chain), are unstable folds in vacuum.³⁹

The frustration in the energetics that a poly-L structure manifests in the interaction of backbone peptide units may define the protein-folding problem. In 1965, Flory found a poly-L random coil to be three-fold higher in characteristic ratio, viz., in the spatial extension of the polymer chain, in a polypeptide structure than in a poly-L-lactic-acid structure, being the cognate polyester structure identical in the stereochemistry of the structure.^40,41 Flory concluded on the basis of calculations that backbone peptide units “unfold” a polypeptide structure, relative to a polyester structure, electrostatically, and implied that the effect was stronger than the role peptide hydrogen bonds have to “fold” a polypeptide structure because this could be expected to diminish the characteristic ratio in a polypeptide structure relative to that in a polyester structure. Next, Flory examined the effect of mutating a polypeptide structure stereochemically. He found that the proportion of D-residues increased progressively, the poly-L random coil diminished in characteristic ratio gradually and eventually ten-fold in alternating-L,D structure to a value that, surprisingly, was lower than the limit of steric possibility for the structure.^42–44

Following up on the conclusion of Flory that backbone peptide units “unfold” a poly-L structure to a β conformation electrostatically, Avbelj proposed in 1995 that C_local, implicitly due to the effect of a poly-L structure, provides for α versus β conformational selection to involve steric interferences of side-chains in the solvent-mediated screening of C_local.⁴⁵ Unifying Flory's puzzle and Avbelj's hypothesis,^45,46 we highlighted that ordering of a poly-L structure to an α conformation given the energetic frustration of the structure mandates critical involvement of the solvent as a screen of unfavourable C_local.^26,27 Affirming the notion that a poly-L structure may provide in C_local the basis for a sequence role has been the observations of Avbelj, Baldwin, and co-workers. According to their observations, solvation in the main-chain, calculated with empirical parameters using DelPhi,⁴⁷ correlates with the solvent-blocking effect in the side-chains manifest in the polypeptide main-chain^48,49 and, with propensities the side-chains manifest for α and β conformations in the protein main-chain.^50–52

The protein-folding problem has become amenable to scrutiny without requirement of assumptions about the geometry or interaction of the structure involved.^53–66 The oligoalanine structure of a chain-length of four residues, tetraalanine, is adopted as a promising model to address the role of the geometrical relationship between dipoles of backbone peptide units on the energetics of folding using density functional theory (DFT) studies. DFT calculations offer an advantage by taking into account polarization effects compared with a fixed point charge interaction lacking a polarization effect in classical force fields such as AMBER, CHARMM, GROMOS, etc. During recent years, significant advances have been made in the models that account for polarization effects^67–69 into force fields that include the fluctuating charge model,^70,71 Drude oscillator,^72,73 and induced multipole.^74–78 However, quantum chemical calculations can reliably model the structure, dynamics of peptides, and are free from empirical assumptions. Significant efforts in the first-principle approaches for sampling of the conformational space of peptides have been made.^54,79–81 Accordingly, we address the protein-folding problem with DFT and by mutating appropriate models of the natural structure stereochemically. Thus, oligoalanine models are chosen for DFT calculations to address the role of mutual interactions of backbone peptide units in the energetics of polypeptide main-chain folding. The oligoalanine models of LLLL- and DLLL-structures having alanine or proline as N-terminal residue are examined for the energetics of folding in vacuum and in the presence of a solvent which is modeled as a dielectric. The models are also examined for interactions of backbone peptide units that do and do not qualify according to well accepted geometric criteria as hydrogen bonds using empirical force fields. The mutual interactions of backbone peptide units are dissected for a specific effect in the energetics of folding of polypeptide main-chain.

Computational methods

Modeling of peptides

The peptides were modeled using an in-house software package: computer aided peptide modeling (CAPM).³¹ CAPM can handle D-amino acids effectively. An in-house program, PDBmake, was used to generate coordinates of a CAPM modeled structure.

DFT calculations

Molecular orbital calculations were performed with the DFT method at the B3LYP/6-31+G(d) level^82,83 using Gaussian 09.⁸⁴ This method combines Becke's three parameter functional with nonlocal correlation provided by the correlation functional of Lee, Yang, and Parr.^82,83 The B3LYP functional has been employed for the conformational analysis of polypeptides in a number of studies.^81,85 All optimizations were implemented in vacuum, whereas those involving Ia and Ib were also implemented in the presence of water as a solvent, which was modeled implicitly as a dielectric. The geometry optimization in the presence of the solvent water was performed using the SMD model by Truhlar and co-workers⁸⁶ as implemented in Gaussian 09.⁸⁴

Electrostatic calculations

Coulomb calculations were performed with atomic partial charge assignments drawn from GROMOS96 (ref. 87) and with dielectric constant (ε) one. Coulomb energy over all peptide groups in the polypeptide main-chain gave C_total. Coulomb energy over the peptide groups within a residue and its immediate sequence neighbours, as defined in Fig. S1 (ESI†), summed over the entire chain, gave Coulomb local energy (C_local). Coulomb energy over the peptide groups satisfying cutoff criteria of distance 0.35 nm and angle of ≤30° for hydrogen bonds, summed over the entire chain, gave the Coulomb hydrogen bond energy (C_Hb). Coulomb non-local energy (C_non-local) was computed from the relationship C_non-local = C_total − (C_local + C_Hb).

Results and discussion

The isomeric oligoalanine models, Ac–^L/DXxx₁–^LXxx₂–^LAla₃–^LAla₄–NHMe, are assessed in ^L/DAla₁–^LAla₂ variants Ia and Ib, ^L/DPro₁–^LAla₂ variants IIa and IIb, and ^L/DPro₁–^LPro₂ variants IIIa and IIIb for the energetics of folding as a function of having the N-terminal residue Ala or Pro and being L or D in structure (Table 1).

Table 1 The end-protected tetraalanine, proline, and diproline peptides varied in N-terminal residue stereochemistry chosen for DFT calculations

Model	Oligopeptides
Ia	Ac–^LAla–^LAla–^LAla–^LAla–NHMe
Ib	Ac–^DAla–^LAla–^LAla–^LAla–NHMe
IIa	Ac–^LPro–^LAla–^LAla–^LAla–NHMe
IIb	Ac–^DPro–^LAla–^LAla–^LAla–NHMe
IIIa	Ac–^LPro–^LPro–^LAla–^LAla–NHMe
IIIb	Ac–^DPro–^LPro–^LAla–^LAla–NHMe

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The LLLL and DLLL-diastereomer, a and b, respectively, of tetraalanine (I), proline (II), and diproline (III) peptides are modeled in extended (E), semi-extended (S), and folded (F) options specifically as ^Lβ₄, ^LP₄, ^Lα₄, ^Dβ^Lβ₃, ^DP^LP₃, and ^DP^Lα₃ folds (^L/DP stands for the polyproline-II conformation of a specific stereochemical structure) as shown in the stick models in Fig. 1 and S2–S6 (ESI†) and illustrated as φ, ψ plots (Fig. S2–S6, ESI†).

Fig. 1 Stick model for the LLLL- and DLLL-isomer of end-protected tetraalanine (Ia, Ib) displaying fully-extended (^LE and ^DE), semi-extended (^LS and ^DS), and fully-folded (^LF and ^DF) structures as β, polyproline-II (PPII; P), 3₁₀-helix, and ^DPPII-capped-3₁₀-helix folds. ^L/Dβ, ^L/DP, and ^L/Dα notation in parenthesis refer to a L- or D-residue of β, P, or α conformation (φ, ψ's given in Fig. S2–S6, ESI†), with the subscript displaying the number of residues in each conformation. The hydrogen bonds between backbone CO and NH groups in the fully folded structures (^LF and ^DF) are shown as a dashed line in brown.

The structures model the canonical folds of a protein structure in extended and helical conformations, and an unfolded structure in PPII conformation and the diastereomeric equivalent having a D-residue in the conformation specific for the stereochemistry of the structure. Of particular note are ^LF (^Lα₄) and ^DF (^DP^Lα₃) folds as 3₁₀-helix and D-PPII-capped-3₁₀-helix structures being canonical LLLL and isomeric DLLL-structures. The folds are identical in the donor–acceptor specificity of ^3→NC₁₀, ^4→1C₁₀, and ^C→2C₁₀ hydrogen bonds as defined in Fig. 2, but vary in the orientation of the backbone peptide units on the basis of N-terminal residue stereochemistry. Thus, ^LF (^Lα₄) and ^DF (^DP^Lα₃) as 3₁₀-helix and D-PPII-capped-3₁₀-helix folds, respectively, are ideal structures for assessing the likely role of the electrostatics of backbone peptide units in polypeptide main-chain folding. The 3₁₀-helix is a less common helix in protein structures than the α-helix,⁸⁸ but is an ideal model for the objectives of the present study given the congruity of hydrogen bonds in the F folds and given the ideality of φ, ψ's of isomeric structures. Moreover, the α-helix at this chain-length is known to be an unstable fold in vacuum according to reported calculations.^39,65

Fig. 2 The hydrogen bonds in the model peptides are defined on the basis of the sequential position of the donor followed by the acceptor residue (N and C represent the terminal blocking groups) and the number of atoms in the hydrogen bonded cycle.

The ^L/DF folds model protein helices in parallel alignment of their individual dipoles of backbone peptide units with the exception of the N-terminal dipole in the ^DF fold that is tilted ∼90° relative to a dipole in the ^LF fold, as illustrated in Fig. 3 and S7 (ESI†). In the poly-L structure, the individual dipoles of backbone peptide units are in mutually parallel alignment in helices due to natural homochiral stereochemistry whereas, in the alternating-L,D structure, the dipoles are antiparallel in helices, as observed in the gramicidin-A structure,^89–91 due to the heterochiral structure. With the β conformation in all participating residues, gramicidin helices are the conformational equivalent of β-sheet elements of a protein structure, which are also antiparallel in the alignment of peptide dipoles due to the poly-L stereochemistry of the structure. Thus, mutual alignment of dipoles of backbone peptide units characterizing the folds of protein structure, parallel in helical structures and antiparallel in extended β-sheet structures, are specific for the poly-L stereochemistry of the natural structure; the structure merits scrutiny for a possible role in protein folding. Varying in the N-terminal residue stereochemical structure, the F folds are congruous in hydrogen bonds with respect to their specificity in donor and acceptor backbone peptide units, but are distinct in dipolar alignment due to the specific stereochemistry in the structure. Thus, the structures are ideal models to test dipoles of backbone peptide units for a likely role in the energetics of folding on the basis of their alignment and their role due to participation in hydrogen bonds.

Fig. 3 3₁₀-Helix (^LF) and ^DPPII-capped-3₁₀-helix (^DF) folds for end-protected tetraalanine (Ia, Ib) are identical in donor–acceptor specificity of backbone hydrogen bonds, but distinct in the geometry of dipoles of backbone peptide units due to a ∼90° tilt in the dipole of a backbone peptide unit caused by the L- to D-stereochemical mutation of the N-terminal residue. The model peptides are shown in stick representation with hydrogen bonds between backbone CO and NH groups shown as a dashed line in brown.

We submitted E, S, and F folds to full optimization with DFT using the Gaussian 09 suite of the program⁷¹ at the B3LYP/6-31+G(d) level of theory.^82,83 All optimizations were implemented in vacuum whereas those involving Ia and Ib were also implemented in the presence of water as solvent that was modeled implicitly as a dielectric.⁸⁶ In vacuum, ^L/DE (^Lβ₄ and ^Dβ^Lβ₃) and ^L/DF (^Lα₄ and ^DP^Lα₃) folds are recovered as stable minima of energy whereas, with participation of solvent, the ^L/DS (^LP₄ and ^DP^LP₃) folds are also recovered as stable minima of energy. Specific for the folding condition, in vacuum or in solvent, minima were unique in φ, ψ's (Tables S1 and S2, ESI† and energetics of the structures in Table S3, ESI†), which implied the robustness of convergence because initial structures were deliberately varied in φ, ψ's (Fig. S8 and S9, ESI†). Some optimizations converged in the E structure having a terminal residue trapped in a hydrogen bonded γ-turn structure, as shown in Fig. S8 (ESI†). These structures were rejected in favour of the E structures that did not feature a hydrogen bonded γ-turn in the calculation of folding energy. The E structures in proline and diproline peptides, II and III, respectively, were always recovered with a proline locked in a γ-turn structure (Fig. S3 and S4, ESI†). The folding energy for these structures was calculated with a hydrogen bonded E fold as the reference structure.

The results of optimization show that the ^DF structures are more stable folds than the ^LF structures (Fig. 4 and Table 2). Thus, mutated from a LLLL- to a DLLL-structure, the F fold in tetraalanine, proline, and diproline peptides is stabilized by 0.82, 1.30, and 8.55 kcal mol⁻¹, respectively, in vacuum.

Fig. 4 Enthalpy change (kcal mol⁻¹) in folding of end-protected tetralanine (Ia, Ib) from ^L/DE to ^L/DF structures in vacuum and from ^L/DE to ^L/DS to ^L/DF structures in the presence of solvent, in transfer of ^L/DE and ^L/DF folds from vacuum to solvent, and in mutation of LLLL- to DLLL-isomer structures is shown in panel (a). Enthalpy change (kcal mol⁻¹) in folding of end-protected proline (IIa, IIb) and diproline (IIIa, IIIb) peptides from ^L/DE to ^L/DF structures, and in the mutation of LLLL- to DLLL-isomer structures in vacuum is shown in panel (b). The fully-extended (^LE and ^DE), semi-extended (^LS and ^DS), and fully-folded (^LF and ^DF) structures for the LLLL- and DLLL-stereochemical structures are in blue and red, respectively.

Table 2 Enthalpy change (ΔH_E→F) and free energy change (ΔG_E→F) (at 298 K in units of kcal mol⁻¹) for end-protected tetraalanine (Ia, Ib), proline (IIa, IIb) and diproline (IIIa, IIIb) peptides optimized in vacuum, and for end-protected tetraalanine (Ia, Ib) optimized in the presence of solvent

Model peptides	Optimization in vacuum		Optimization in presence of solvent
	ΔHE→F	ΔGE→F	ΔHE→S	ΔGE→S	ΔHS→F	ΔGS→F
Ia	−0.83	3.55	−2.77	−1.93	−0.84	1.3
Ib	−1.65	2.72	−2.94	−2.25	−1.46	0.7
IIa	−1.62	−1.51
IIb	−2.92	−1.17
IIIa	1.19	3.08
IIIb	−6.36	−2.77

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As noted from the geometric parameters of hydrogen bonds listed in Table 3, the basis does not involve hydrogen bonds because the hydrogen bonds appear to be comparable in average strength in ^LF and ^DF folds. In tetraalanine peptides (Ia, Ib), the ^DF fold stabilizes relative to the ^LF fold with an effect that is internal to the folds because the ^DE and ^LE folds of the structures are noted to be practically identical in stability (because enthalpy change is only −0.05 kcal mol⁻¹), as shown in Fig. 4 (panel (a)). Thus, in Ia and Ib, either the ^DF fold stabilizes or the ^LF fold destabilizes due to a ∼90° tilt in the dipole of a backbone peptide unit caused by the L- to D-stereochemical mutation of the N-terminal residue. As noted in Table 3, the tilt has practically no effect on the average length of hydrogen bonds between backbone peptide units (^3→NC₁₀ hydrogen bond shortens, but ^4→1C₁₀ and ^C→2C₁₀ hydrogen bonds elongate). Thus, the effect stabilizing the ^DF structure relative to the ^LF structure is independent of the backbone peptide units that participate in hydrogen bonds. The observed effect that is not definable because hydrogen bonds also stabilize the DLLL-diastereomer in proline and diproline peptides, II and III, respectively.

Table 3 Hydrogen bond parameters in ^L/DF folds of end-protected tetraalanine (Ia, Ib), proline (IIa, IIb) and diproline (IIIa, IIIb) peptides optimized in vacuum and in the presence of solvent

H-Bond	Distance (Å)/angle (in °)	Optimization in vacuum						Optimization in presence of solvent
		Ia	Ib	IIa	IIb	IIIa	IIIb	Ia	Ib
^3→NC10	O⋯H	2.167	2.146	2.171	2.139	2.150	2.158	2.380	2.873
	O⋯N	3.175	3.119	3.181	3.115	3.145	3.145	3.375	3.753
	O⋯H–N	170.5	159.3	171.3	159.8	165.2	162.8	165.2	145.3
^4→1C10	O⋯H	2.255	2.264	2.301	2.242	2.206	2.218	2.381	2.438
	O⋯N	3.256	3.258	3.300	3.237	3.209	3.218	3.379	3.427
	O⋯H–N	167.9	165.7	167.7	165.7	168.8	167.4	166.3	163.7
^C→2C10	O⋯H	2.225	2.240	2.203	2.240	2.262	2.220	2.230	2.366
	O⋯N	3.199	3.211	3.182	3.210	3.230	3.191	3.214	3.344
	O⋯H–N	160.7	160.1	161.6	159.7	159.2	159.7	162.6	161.3
Average	O⋯H	2.216	2.217	2.225	2.207	2.206	2.199	2.330	2.559
	O⋯N	3.210	3.196	3.221	3.187	3.195	3.185	3.323	3.508
	O⋯H–N	166.4	161.7	166.9	161.7	164.4	163.3	164.7	156.8

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As shown in Fig. 4 (panel (b)) and reported in Table 2, these folds are stabilized with proline(s) on the basis of stereochemical-, positional-, and conformational-specific effects manifesting at the level of the ^DE or ^DF folds, or both. The effects are reminiscent of the observed proline effects in protein helices and in their synthetic models.^92–96 The likely basis for the proline effects considering the present results may include the: (1) noted entrapment of the E structure in proline peptides in a hydrogen bonded γ-turn structure; (2) enforcement of consecutive L-proline residues to a sterically crowded helical conformation in the ^LF fold of IIIa; (3) possible interaction of the electron cloud characterizing the proline side-chain in the ^DF folds of IIb and IIIb with the electrostatics of backbone peptide units in a polarizing interaction.

As shown in Fig. 4 (panel (a)) and listed in Table 2, the enthalpy change (ΔH_E→F) in folding from the extended structure to the folded structure is −0.83 and −1.65 kcal mol⁻¹ for Ia and Ib, respectively. For Ib, the extra gain in stability by 0.82 kcal mol⁻¹ in folding from the ^DE structure to the ^DF structure is practically identical to the folding of Ia from the ^LE structure to the ^LF structure over its three hydrogen bonds, i.e. 0.83 kcal mol⁻¹ (Fig. 4). It is remarkable that a ∼90° tilt in the dipole of a backbone peptide unit in the ^DF fold affects the fold nearly as strongly as the combined effect of three hydrogen bonds in the ^LF fold. The energetics of folding of a polypeptide main-chain structure is commonly interpreted on the basis of the strength of hydrogen bonds that are internal to the polypeptide main-chain. This interpretation needs critical investigation considering that the F folds manifest an effect—comparable to the cumulative strength in their hydrogen bonds—that does not qualify as hydrogen bonds. The specific effect is inaccessible with DFT, and we address it empirically at the level of the electrostatics of the structures. Thus, in Ia and Ib, we calculated Coulomb energy over all the NH and C=O dipoles of the polypeptide chain in the E and F structures using an assignment of charges from GROMOS96.⁸⁷ On this basis, C_total in the structures is determined and partitioned into specific effects on the basis of specific cutoffs based on geometry and sequence. One effect that was considered was C_Hb determined with a well-accepted geometrical cutoff to identify donor and acceptor backbone peptide units.^97,98 Another effect that was considered was C_local, which was determined with a sequence-based cutoff over four peptides contiguous in sequence (Fig. S1, ESI†). It is important to note that the considered peptides span a α-helical turn and enclose a 3₁₀-helical turn and the generic β-turn of a poly-L structure (Fig. S1, ESI†). Thus, C_local merits consideration for a possible role in the observed strength and specificity of the important folds in protein structure. Electrostatics that were non-local in sequence, defined as sequence-non-local electrostatics (C_non-local), were computed from the relationship C_non-local = C_total − (C_local + C_Hb).^26,27

The spatially- and sequentially-resolved electrostatic effects of the backbone peptide units in folding from the extended (^L/DE) structure to the folded (^L/DF) structure for end-protected tetraalanine, Ia, and Ib, are reported in Table 4. According to DFT calculations in vacuum, the folding from the extended structure to the folded structure is favourable by −0.83 and −1.65 kcal mol⁻¹ for Ia and Ib, respectively (Table 2). However, according to electrostatic calculations, the folding from the extended structure to the folded structure is not favourable because the change in Coulomb total energy is 4.7 and 1.6 kcal mol⁻¹ for Ia and Ib, respectively (Table 4). This implies that hydrogen bond cooperativity effects⁶⁴ in the DFT calculations are important. As noted from Tables 2 and 4, the folding from the extended structure to the folded structure is more favourable for Ib with the DLLL-stereochemical structure as compared with Ia with a LLLL-stereochemical structure by 0.82 and 3.1 kcal mol⁻¹, according to DFT and electrostatic calculations, respectively. Thus, the stabilizing effect of the stereochemical mutation is independent of the method of calculation applied. As noted from Table 4, the end-protected tetralanine stereochemical isomers, LLLL and DLLL, in folding from the extended to the folded structure, are comparable in terms of ΔC_Hb and ΔC_non-local, but distinct in ΔC_local. The change in C_local in folding from the extended structure to the folded structure is 18.7 and 15.4 kcal mol⁻¹ for Ia and Ib, respectively (Table 4). This implies that sequence-local electrostatics, C_local, is destabilizing in effect and that the destabilization is more in Ia than Ib. Thus, sequence-local electrostatics is an effect clearly stereochemical in its basis because the destabilization due to sequence-local electrostatics is more in the LLLL-stereochemical structure than in the DLLL-stereochemical structure of the model end-protected tetraalanine. Thus, for the LLLL-stereochemical structure, the folds are “unfolded” with the sequence-local electrostatics of their backbone peptide units, and under conflict with their “folding” with sequence-non-local electrostatics—mainly hydrogen bonds—of their backbone peptide units.

Table 4 Change in Coulomb total energy, ΔC_total, (kcal mol⁻¹) in folding from the extended (E) structure to the folded (F) structure of end-protected tetraalanine (Ia, Ib) resolved into specific terms on the basis of geometrical and sequence-based cutoffs

Model peptides	^L/DE → ^L/DF (in vacuum)
	ΔCHb	ΔCnon-local	ΔClocal	ΔCtotal
Ia	−10.2	−3.8	18.7	4.7
Ib	−10.3	−3.5	15.4	1.6

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The solvent role is integral to the protein-folding problem. A basis for the ambiguity of the solvent role may reside in the ambiguity of the main-chain role. An investigation by perturbing models stereochemically has benefited studies of the role of the main-chain^26,27 as well as the role of the solvent in the conformational specificity of folding in the protein structure.^24,25 Varying in the balance of critical interactions, Ia and Ib offered an opportunity to address the solvent role with DFT. Given the polarity of backbone peptide units, the solvent invites attention to the possible role of molecular polarity and dielectric effect as a bulk medium. Based on the strength of dipolar solvation, the solvent may affect the mutual hydrogen bonds of backbone peptide units directly at the level of C_Hb. Based on its strength as a dielectric, the solvent may determine the apparent strength of hydrogen bonds in C_total and indirectly at the level of C_local. Thus, stereochemistry by determination of C_local in both magnitude and effect is suggested to be critical for the role of the main-chain as well as the solvent in protein folding.^24,25

In the present study, we modeled the solvent implicitly as a dielectric. Although the physical details of the solvent are more accurately captured by explicit solvent models, an enormous increase in computational cost accompanied with thermal sampling the configurations of the explicit solvent are significant challenges.^99,100 The results in Table 2 and Fig. 4 highlight that stereochemistry affects the solvent role and imply that the electrostatics of the oligoalanine model structure could be important. Ia and Ib are varied in the strength of solvation, which is practically identical in E folds. The strength of solvation is higher by 0.66 kcal mol⁻¹ in the ^LF fold than in the ^DF fold that can be correlated with C_local. Ia and Ib vary in the energy of relaxation from the E fold to the S fold, which is 0.44 kcal mol⁻¹ greater in Ia, and from the S fold to the F fold, which is 0.62 kcal mol⁻¹ greater in Ib. The stability of solvated structures is practically unaffected on mutating the ^LE structure to the ^DE structure, is diminished by 0.40 kcal mol⁻¹ on mutating the ^LS structure to the ^DS structure, and is increased by 0.22 kcal mol⁻¹ on mutating the ^LF structure to the ^DF structure. The specificity of energetics for the stereochemistry of the structures imply that the electrostatics of the backbone peptide units have a critical role. Considering the magnitude of C_local and its effect, we expected the solvent to strengthen both ^LF and ^DF folds by screening C_local. Counter to this expectation, we note in Table 3 that the solvated fold in the ^DF structure is elongated, and thus apparently weakened, in hydrogen bonds relative to the ^LF structure. Thus, an effect other than electrostatics and its screening may have a role in the strength of the specific folds in the model peptides. Overall, the role of the geometrical relationship between dipoles of backbone peptide units, with and without the involvement of solvent, manifested in the model peptides imply that folding of the polypeptide main-chain to specific hydrogen bonds involves the electrostatics of backbone peptide units in a critical way.

Conclusions

The effect of the orientation of backbone peptide units of the polypeptide main-chain on the energetics of folding has been examined with DFT by utilizing end-protected oligoalanine peptides of LLLL- and DLLL-stereochemical structures. The peptides fold to an identical number of hydrogen bonds between linked backbone peptide units, but with a distinct geometrical relationship among backbone peptide units. DFT calculations highlight that the energetics of folding critically depend on the geometrical relationship between the backbone peptide units of the polypeptide structure. The present study demonstrates that poly-L stereochemistry has a critical role by having mutual interactions of backbone peptide units (that qualify as hydrogen bonds) as opposed with the electrostatics of backbone peptide units (that do not qualify as hydrogen bonds). Thus, the specific folds of a protein structure are defined with the electrostatics of backbone peptide units that do not qualify as hydrogen bonds rather than hydrogen bonds between backbone peptide units, and are dependent on the solvent role as a dielectric. The solvent may be critical to protein conformation in its role not as a direct competitor of hydrogen bonds between backbone peptide units, but as a screen of the electrostatics of poly-L linked backbone peptide units that do not qualify as hydrogen bonds. The results of the present study will provide key insights into the helix stability and understanding of protein folding.

Acknowledgements

The authors acknowledge the Department of Science & Technology (09DST028), Government of India, for financial support and IIT Bombay, Mumbai, for the computing facility “Corona”. Bhupesh Goyal gratefully acknowledges the Science and Engineering Research Board (SERB), Department of Science & Technology, Government of India, for the award of a SERB Start-Up Research Grant (Young Scientists) (Sanction Number: SB/FT/CS-013/2014). The authors thank Dr Akhilesh Sharma for insightful discussions on the results of the present study.

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Footnotes

† Electronic supplementary information (ESI) available: Fig. S1–S9, Tables S1–S3 and Cartesian coordinates of the optimized geometries in vacuum and in presence of solvent. See DOI: 10.1039/c6ra22870d

‡ Present Address: Department of Chemistry, School of Basic and Applied Sciences, Sri Guru Granth Sahib World University, Fatehgarh Sahib–140406, Punjab, India.

§ Present Address: Life Sciences Institute, University of Michigan, Ann Arbor, MI, USA, 48105.

¶ Present Address: Department of Chemistry, University of Toronto, Toronto, ON, M5S 3H6.

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