Electron density based analysis of N–H⋯O[double bond, length as m-dash]C hydrogen bonds and electrostatic interaction energies in high-resolution secondary protein structures: insights from quantum crystallographic approaches

Suman K. Mandal a, Benoît Guillot b and Parthapratim Munshi *a
aChemical and Biological Crystallography Laboratory, Department of Chemistry, School of Natural Sciences, Shiv Nadar University, Dadri, Uttar Pradesh – 201314, India. E-mail: parthapratim.munshi@snu.edu.in
bLaboratoire de Cristallographie, Institut Jean Barriol, Université de Lorraine, 34 Cours Léopold, Nancy 54000, France

Received 16th April 2020 , Accepted 1st June 2020

First published on 1st June 2020


In proteins, the main-chain N–H⋯O[double bond, length as m-dash]C hydrogen bonds (HBs) play a crucial role in the formation of α-helices and β-sheets. Accurate analysis of such hydrogen bonds and their electrostatic interaction energies is essential for studying binding interactions and for better understanding of the energetics involved in protein folding. Here, we studied 22 high-resolution (0.87 Å to 0.48 Å) secondary protein structures (4.7 kDa to 54.5 kDa) from the RCSB PDB and performed topological analyses of 1443 N–H⋯O[double bond, length as m-dash]C HBs (750 in α-helices and 693 in β-sheets) using the multipole analysis based experimental electron densities as transferred from the ELMAM2 database. This is the first study of its kind involving by far the largest number of high-resolution protein structures and HBs from both α-helices and β-sheets. Further, based on the accurate estimation of the electrostatic interaction energies, the excellent correlations with various topological parameters have been demonstrated. The excellent correlations have also been observed between the topological parameters. Thereby, we identified the limiting values of the topological parameters and the electrostatic interaction energies to establish the presence of the true N–H⋯O[double bond, length as m-dash]C HBs in protein main-chains via quantitative and qualitative analyses of electron densities using quantum crystallographic approaches – the quantum theory of atoms in molecules (QTAIM) and the noncovalent interaction (NCI) index.


Introduction

The folding of polypeptide chains to adopt a three-dimensional protein structure upon the formation of α-helices and β-sheets is essentially governed by the interatomic interactions among the amino acids. The two main stabilizing forces which contribute to the protein folding and its stability are the hydrophobic effects and the hydrogen bonds (HBs).1 During protein folding, the nonpolar side chains are buried due to hydrophobic effects and the main-chains form HBs via N–H donors and C[double bond, length as m-dash]O acceptors. However, the debate on which of these two energetic factors predominates is still active. While some earlier studies2,3 direct towards the hydrophobic effects, other recent experimental4–6 and theoretical7 studies point to HBs. The crucial role of HBs in the formation of α-helices and β-sheets has been elucidated long ago by Mirsky and Pauling.8 The key building blocks of both α-helix and β-sheet structures are N–H⋯O[double bond, length as m-dash]C HBs and they contribute about 5–10 kcal mol−1 to the stability of proteins, as estimated by Pauling's group.9,10 The electrostatic interaction energy, which dictates the structure–function correlation for proteins,11 is believed to be the major contributor to the HB energy.12 Accurate analysis of HBs and knowledge of the local electrostatics in proteins are essential for their proper thermodynamic modelling, estimation of energies for the folding and binding interactions.

However, an accurate study of HBs demands for precise atomic locations, especially for the H-atoms, which can be located13 using high-resolution (sub-atomic) single-crystal X-ray diffraction (SCXRD) and complementary neutron diffraction experiments.14 Although achieving high-resolution (better than ∼0.40 Å) SCXRD data on small molecule crystals has now become trivial,15–17 this remains an uphill task for protein crystals. This is because growing good quality protein crystals, poor stability, high atomic thermal motions, high solvent contents, large number of H-atoms etc. are some of the common factors, which limit the resolution of the diffraction data. However, the number of protein structures determined at high resolution is increasing steadily. Currently, only 82 unique structures (excluding peptides, hormones and oligonucleotides) with a resolution of 0.87 Å or better have been deposited in the Protein Data Bank (PDB: http://www.rcsb.org/pdb/home/home.do). A few of the HB studies using the protein structures from the PDB have provided a wealth of information. However, these studies were based on the lower number of data points, only the distance–angle criteria or low structural resolutions (1.8 Å to 1.4 Å and 0.95 Å to 0.87 Å).18,19 Moreover, electrostatic interaction energies (Eelec) derived purely from geometrical parameters provide information on dipole–dipole interactions only and ignore significant contributions from higher order multipoles. Further, molecular modeling studies based on point-charge potentials have some inherent drawbacks, such as modeling of lone pairs and aromatic rings.20 Alternatively, the multipole modeling based charge density analysis21 allows extracting precise structural information in terms of its detailed electron density distributions. The electron density, ρ(r), can be utilized to calculate electrostatic potential, ϕ(r). The derived ρ(r) and ϕ(r) can be treated together to estimate Eelec.22 Topological analysis of ρ(r) can be performed using Bader's quantum theory of atoms in molecules (QTAIM),23 based on which one can identify the bond critical point (BCP), where the gradient of ρ(r) vanishes, i.e.ρ(r) = 0. The bond path (BP), the length of which is represented as Rij, not necessarily the interatomic distance, can be traced along the electron density gradient. Essentially, the BP represents a path of maximum electron density between two bonded nuclei.24,25 The second derivative of ρ(r), i.e.2ρ(r), the Laplacian, which can be decomposed into the contributions from the three principle axes, represented by the three eigenvalues (λ1, λ2 and λ3) of the electron density Hessian matrix.26 The signs of the eigenvalues and their sum (λ1 + λ2 + λ3) can be used for the characterization of the chemical bonds.

The characterization of HBs via topological analysis of ρ(r) using the QTAIM has been commonly employed in small molecule crystals17,27–29 or in model polypeptides30 and is becoming popular in the cases of protein systems.31–38 Although the experimental charge density analysis (ECDA) using high-resolution SCXRD data is a well-established methodology in the case of small molecules but it has remained challenging for biological macromolecules. Consequently, so far, only six of the high-resolution X-ray protein structures were subjected to ECDA (Table 1).31–36,39 However, for all of these, the initial charge density models were constructed based on the experimental ρ(r) transferred from ELMAM/ELMAM2 databases40,41 using MoPro.42,43 For modeling the deformation of the atomic densities, due to the interatomic interactions in a crystal, the aspherical description of ρ(r) is essential and for macromolecules, this can be conveniently achieved using an electron density database transfer approach. Liebschner et al. have utilized the ELMAM database for the topological analysis of HBs and weak interactions in six selected proteins, out of which five structures with resolutions ranging from 0.99 Å to 0.89 Å were retrieved from the PDB.37 Their study also included protein human aldose reductase (hAR) at a resolution of 0.66 Å and, however, focused only on the helices. Recently, some of us performed topological analysis of main-chain N–H⋯O[double bond, length as m-dash]C HBs in both α-helices and β-sheets in hAR based on the electron densities transferred from the ELMAM2 database for the estimation of the Eelec.39 The energies were found to be comparable to those measured experimentally.9 That insightful study motivated us to perform quantitative and qualitative analysis of the main-chain N–H⋯O[double bond, length as m-dash]C HBs in several high-resolution secondary protein structures for accurate characterization and estimation of their Eelec.

Table 1 List of selected high-resolution secondary protein structures from the RCSB PDB
No. Protein systems PDB ID Resolution (Å) Molecular weight (kDa) Main-chain average B-factor (Å2) Year of publication
a ECDA has been performed.
1 High-potential iron–sulfur protein (HIPIP) 5D8V 0.48 8.8 3.43 2016
2 Crambin 1EJG 0.54 4.7 2.65 2000
3 Hen egg white lysozyme (HEWL) 2VB1 0.65 14.3 4.77 2007
4 Human aldose reductase (hAR) 1US0 0.66 35.9 5.58 2004
5 Pyrococcus abyssi rubredoxin 1YK4 0.69 5.8 4.95 2005
6 Cholesterol oxidase 4REK 0.74 54.5 8.72 2015
7 Serine protease 1GCI 0.78 26.7 7.33 1998
8 Cytochrome b5 reductase 5GV8 0.78 30.8 8.68 2017
9 Trypsin 1PQ7 0.80 22.2 5.81 2003
10 Human parvulin 14 3UI4 0.80 11.2 4.19 2011
11 α-Lytic protease 2H5C 0.82 19.9 4.78 2006
12 Triose-phosphate isomerase 2VXN 0.82 27.2 6.27 2010
13 Proteinase K 2PWA 0.83 29.0 7.39 2007
14 Ponsin 2O9S 0.83 7.7 5.12 2007
15 Cytochrome c6 4EIC 0.84 9.4 7.58 2013
16 Acutohaemonlysin 1MC2 0.85 14.1 7.93 2003
17 Diisopropyl fluorophosphatase (DFP) 1PJX 0.85 35.1 8.05 2003
18 1,2-Alpha-mannosidase 4AYO 0.85 50.5 5.15 2012
19 Cyclophilin D 4O8H 0.85 17.7 5.11 2014
20 Glutaredoxin NrdH 4HS1 0.87 9.3 7.86 2013
21 Cu-Containing nitrite reductase 5AKR 0.87 40.8 8.01 2015
22 Fatty acid-binding protein 4TJZ 0.87 14.9 8.8 2015


The noncovalent interaction (NCI) descriptor, which can be derived from ρ(r) and its derivative and it is based on the maps of the reduced density gradient (RDG), has been proved to be an extremely useful approach for the qualitative analysis of HBs.44,45 The NCI approach complements the QTAIM approach. Further, the combination of these two approaches for the reliable analysis of weaker interactions appeared to be an efficient strategy in quantum crystallography.46–48 The program NCIPLOT can be utilized for plotting reduced electron density gradient isosurfaces in the interaction regions – the BCP and its surroundings.49 The concept of a sudden change in the RDG, as implemented in the NCI, helps in visualizing intermolecular interactions.

In this study, from the RCSB PDB, we retrieved 22 protein structures ranging from 4.7 kDa to 54.5 kDa and with resolutions of 0.87 Å to 0.48 Å for studying main-chain N–H⋯O[double bond, length as m-dash]C HBs in both α-helices and β-sheets (Table 1). Accordingly, we performed topological analyses of 1443 N–H⋯O[double bond, length as m-dash]C HBs (750 in α-helices and 693 in β-sheets) using multipole-based electron densities as transferred from the ELMAM2 database. Subsequently, we estimated Eelec and examined its relationships with various topological parameters. The relationships between the topological parameters have also been discussed. Thereby, we identified the limiting values of the various topological parameters and Eelec of the protein main-chain N–H⋯O[double bond, length as m-dash]C HBs via quantitative and qualitative analyses of electron densities using quantum crystallographic approaches – the QTAIM and the NCI index.

Materials and methods

Structure selection and preparation

Only monomeric (not multimeric) structures, which were deposited in the RCSB PDB as a complete functional protein and not a subdomain of a larger protein, published in scientific journals with a resolution of 0.87 Å or better and with a main-chain average B-factor of lower than 9 Å2 were selected for this study. In the case of the structures with multiple entries, the one with the highest resolution satisfying the above criteria was considered in this analysis. Thus, 22 protein structures were identified as listed in Table 1. As mentioned earlier, the previous report on the topological properties of HBs using electron densities from the ELMAM database included five out of six structures with resolutions ranging from 0.99 Å to 0.89 Å.37 Further, our PDB search with the aforementioned selection criteria did not result in any structures between resolutions of 0.89 Å and 0.87 Å. Therefore, in this study, we have chosen 0.87 Å as the lower limit.

For the convenience of transferring of the multipole modelling based electron densities from the ELMAM2 database, for the structures with multiple conformers, only the major conformers were selected using the module ‘pdbset’ as implemented in the CCP4 suite50 (Scheme S1). The module pdbset was also used to remove the solvents including the water molecules. Moreover, most of these high-resolution structures were lacking the information on H-atom positions. This could be because of the use of inconsistent methods of refinement using diffraction data from various sources. Therefore, in our analysis, for consistency, all the H-atoms were removed with the tool pdbset. Subsequently, in the cleaned structures (Scheme S1), the H-atoms were added to the standard neutron distances (e.g. N–H = 1.02 Å)51 after maintaining the appropriate geometries using MoPro. In the absence of neutron diffraction data, this approach is generally adopted while dealing with the multipole modelling based electron densities of the H-atoms as the experimentally observed electron density peak does not truly correspond to the position of the H-atom nucleus. Since the non H-atom coordinates were determined using very high resolution (0.87 Å to 0.48 Å) X-ray diffraction data, further structural refinement using a common refinement scheme was felt not necessary.52 The modified structures with the new H-atom positions and with the non H-atom coordinates as deposited in the PDB were then considered for transferring the multipole modelling based electron densities using MoPro. Some of these structures contained metals in ionic or complexed forms in addition to some heavy elements as a part of the ligands/substrates (Table S1). However, due to the unavailability of the multipole information of such ions and atoms in the ELMAM2 database, the electron densities were transferred only on the protein atoms (e.g. chain A, i.e. monomer) and not on the metal ions/atoms and on the ligands. The levels of multipoles up to which the electron densities transferred were quadrupole for H-atoms, octupole for C-, N- and O-atoms and hexadecapole for S-atoms. In this context, it is noteworthy that our present study focuses on the main-chain N–H⋯O[double bond, length as m-dash]C HBs only.

In the case of α-helices, we consider the standard ii + 4 type of N–H⋯O[double bond, length as m-dash]C HBs only. However, no such N–H⋯O[double bond, length as m-dash]C HBs are identified in the α-helices of proteins ponsin and DFP. Similarly, no N–H⋯O[double bond, length as m-dash]C HBs are identified in the β-sheet of protein cytochrome C6 (Table 2). The main-chain N–H⋯O[double bond, length as m-dash]C HBs in this study were identified based on the following criteria: H⋯O distances of ≤2.65 Å and the N–H⋯O[double bond, length as m-dash]C and C[double bond, length as m-dash]O⋯H–N angles of 120–180°, similar criteria to those considered in some earlier studies.19,37,53 Subsequently, based on the QTAIM approach, the BCPs were located for all such N–H⋯O[double bond, length as m-dash]C HBs present in these 22 protein structures.

Table 2 Average electrostatic energies (Eelec) of N–H⋯O[double bond, length as m-dash]C HBs in α-helices and β-sheets
Sr. No. Proteins α-Helix HBs β-Sheet HBs
No. Average Eelec (kcal mol−1) No. Average Eelec (kcal mol−1)
1 High-potential iron–sulfur protein (HIPIP) 5 −8.60 6 −10.05
2 Crambin 10 −8.42 2 −10.75
3 Hen egg white lysozyme (HEWL) 31 −8.90 4 −9.88
4 Human aldose reductase (hAR) 77 −9.06 20 −9.37
5 Pyrococcus abyssi rubredoxin 1 −3.03 5 −10.04
6 Cholesterol oxidase 91 −8.95 57 −10.23
7 Serine protease 45 −8.91 31 −9.76
8 Cytochrome b5 reductase 36 −8.44 60 −10.56
9 Trypsin 12 −8.99 50 −9.65
10 Human parvulin 14 22 −8.10 15 −10.80
11 α-Lytic protease 4 −7.85 61 −10.04
12 Triose-phosphate isomerase 73 −9.07 22 −10.21
13 Proteinase K 50 −9.42 40 −9.41
14 Ponsin 14 −9.83
15 Cytochrome c6 35 −9.37
16 Acutohaemonlysin 38 −9.03 3 −9.95
17 Diisopropyl fluorophosphatase (DFP) 100 −10.00
18 1,2-Alpha-mannosidase 159 −9.27 23 −9.20
19 Cyclophilin D 16 −8.89 33 −9.31
20 Glutaredoxin NrdH 22 −8.98 12 −11.60
21 Cu-Containing nitrite reductase 10 −10.10 61 −10.38
22 Fatty acid-binding protein 13 −9.27 74 −10.82
Total number of HBs and average energies 750 −9.04 693 −10.08


Electrostatic interaction energy

For the N–H⋯O[double bond, length as m-dash]C HBs, the Eelec values based on the ρ(r) at the BCPs and the corresponding ϕ(r) were calculated following the same procedure as discussed earlier and using the formula as given below.37,39 The C[double bond, length as m-dash]O group and the N–H group of a HB were treated as two different entities (A and B) for the derivations of the corresponding ρ(r) and ϕ(r).
 
Eelec = ∫ ϕB(r)ρA(r)dr = ∫ ϕA(r)ρB(r)dr(1)

The integrations of the product of ρ(r) and ϕ(r) over the whole space with ρ as non-zero were performed using program VMoPro.

Noncovalent interactions analysis

The methods utilized and the protocol followed for the generation of the NCI isosurfaces are given in the form of a flowchart in Scheme S1. For NCI analysis, the 3D electron density grids generated around the main-chain N–H⋯O[double bond, length as m-dash]C HBs using ELMAM2 and VMoPro were fed into the program NCImilano54 to calculate the RDG of the electron densities and Abramov's energy densities. Subsequently, the module MolIso as implemented in MolecoolQT55 was used for the visualization of the NCI isosurfaces. Additionally, the electron densities obtained upon quantum-mechanical calculations on the N–H⋯O[double bond, length as m-dash]C HBs using Gaussian0956 at the MP2/6-311G(d,p) level of theory and experimental geometry were used for the generation of NCI isosurfaces using NCIPLOT.49 The NCI isosurfaces were also generated using the program NCIPLOT based on the promolecular densities – the sum of the spherically averaged atomic charge densities. The electron density isosurfaces thus obtained from the latter approaches were visualized using VMD.57

Results and discussion

Based on our HB criteria, from the 22 high-resolution secondary protein structures, we identify 1443 N–H⋯O[double bond, length as m-dash]C HBs, out of which 750 are from the α-helices and 693 are from the β-sheets (166 parallel and 527 antiparallel). We list the total number of N–H⋯O[double bond, length as m-dash]C HBs in the α-helices and the β-sheets for each protein system in Table 2.

The static deformation of electron density and the Laplacian maps as plotted for the representative strong and weak N–H⋯O[double bond, length as m-dash]C HBs in α-helices and β-sheets demonstrate the accuracy of electron density features around the N–H and C[double bond, length as m-dash]O groups of the protein main-chain (Fig. S1 and S2). The directionality of the lone pair of electrons of the O-atoms and for the H-atoms, the different degrees of polarization of the electron densities due to the formation of HBs are clearly noticeable in these maps.

The bond path along the H⋯O contacts and the corresponding ρBCP are found to exist for all the 1443 N–H⋯O[double bond, length as m-dash]C HBs. The succession of N–H⋯O[double bond, length as m-dash]C HBs in terms of the bond path and the corresponding BCPs for an α-helix and a β-sheet of protein hAR is shown in Fig. S3. The details of the topological properties and the ∠C[double bond, length as m-dash]O⋯H and ∠N–H⋯O of the ii + 4 α-helix and β-sheet hydrogen bonds are listed in Tables S2 and S3, respectively. Further, the distributions of the ∠N–H⋯O and the ∠C[double bond, length as m-dash]O⋯H across the ranges of Rij and Eelec for the α-helix and the β-sheet hydrogen bonds are shown in Fig. S4–S7. The ranges of these ∠N–H⋯O and ∠C[double bond, length as m-dash]O⋯H are listed in Tables S4 and S5, respectively. The overall ranges (considering the extremities of the Rij and Eelec based ranges) of ∠N–H⋯O and ∠C[double bond, length as m-dash]O⋯H for the α-helix, β-sheet and both (α-helices and β-sheets together) HBs are also listed in Table S6.

The population of N–H⋯O[double bond, length as m-dash]C HBs and their distributions in α-helices and β-sheets in the different shells of Rij and ρBCP with shell widths of 0.05 Å and 0.01 e Å−3, respectively, are shown in Fig. 1a and b. There are 50 short HBs with 1.72 Å < Rij ≤ 1.80 Å and with ∼0.30 e Å−3 > ρBCP ≥ 0.25 e Å−3 and the majority (89.3%) of the HBs are found to populate in 1.8 Å < Rij ≤ 2.2 Å which corresponds to the ρBCP in the range of 0.25 e Å−3 > ρBCP ≥ ∼0.1 e Å−3. However, the shell of 1.9 Å < Rij ≤ 1.95 Å exhibited the maximum number of HBs (17.7%). Meanwhile, the peak of the number of HBs vs. the ρBCP distribution is noticed between ∼0.20 e Å−3 to 0.17 e Å−3. There are 117 HBs populated in 2.2 Å < Rij ≤ 2.45 Å and the corresponding ρBCP values are in the range of ∼0.1 e Å−3 to 0.06 e Å−3. Thereafter and up to an Rij value of 2.7 Å, the population diminished to ∼10 HBs or less per shell (<1%), which have ρBCP < 0.06 e Å−3. The aforementioned population analysis based on the large number of data points (total 1443) clearly shows the trend of the N–H⋯O[double bond, length as m-dash]C HBs in protein systems.


image file: d0ce00577k-f1.tif
Fig. 1 Distributions of (a) Rij (Å) and (b) ρBCP (e Å−3) of N–H⋯O[double bond, length as m-dash]C HBs in α-helices and β-sheets.

The Rij and the ρBCP values in both α-helices and β-sheets are found to follow an exponential relationship with a regression coefficient (R2) better than 99% (Fig. 2). A similar exponential relationship was also observed in the case of small molecules17,28,58 and NADH-cytochrome b5 reductase.36 While the Rij and ρBCP values for the 750 N–H⋯O[double bond, length as m-dash]C HBs, in the α-helices, are in the range of 1.720 Å to 2.670 Å and 0.297 e Å−3 to 0.038 e Å−3, respectively, those of the 693 N–H⋯O[double bond, length as m-dash]C HBs, in the β-sheets, are varying from 1.715 Å to 2.650 Å and 0.297 e Å−3 to 0.039 e Å−3, respectively. For the α-helices, the ranges of Rij and ρBCP are found to be in good agreement with those reported earlier.37 In our study, covering both α-helices and β-sheets, the Rijvs. ρBCP plot (Fig. 2) not only highlights the variation of the HB population across the Rij values but also demonstrates the demarcation of N–H⋯O[double bond, length as m-dash]C HBs. However, the ρBCP for the α-helices and the β-sheets appeared to follow the same trend as can be noticed from their indistinguishable exponential fitting curves. While the shorter N–H⋯O[double bond, length as m-dash]C HBs exist with high ρBCP values (∼0.30 e Å−3 to 0.25 e Å−3), the majority of the N–H⋯O[double bond, length as m-dash]C HBs have ρBCP values between 0.25 e Å−3 and ∼0.10 e Å−3. Meanwhile, the longer N–H⋯O[double bond, length as m-dash]C HBs have ρBCP values between ∼0.10 e Å−3 and ∼0.06 e Å−3 and the longest ones have ρBCP values < ∼0.06 e Å−3. These demarcations are in accordance with the observations made from the distribution plots as shown in Fig. 1a and b. Further, as expected, all of the 1443 N–H⋯O[double bond, length as m-dash]C HBs have positive Laplacian values (Tables S2 and S3), negative λ1 and λ2 values and positive λ3 values (Table 3).


image file: d0ce00577k-f2.tif
Fig. 2 Exponential relationship between the Rij (Å) and ρBCP (e Å−3) values of the 1443 N–H⋯O[double bond, length as m-dash]C HBs in the α-helices (750) and β-sheets (693). The dashed lines denote the demarcations of the shortest, most populated, longer and longest N–H⋯O[double bond, length as m-dash]C HBs in the 22 high-resolution proteins.
Table 3 Topological parameters of N–H⋯O[double bond, length as m-dash]C HBs across the strongest to the weakest regions as shown in the Laplacian maps (Fig. S10†) and NCI isosurfaces (Fig. 6a–i)
Fig. 6 PDB ID C[double bond, length as m-dash]O H–N ρ BCP R ij E elec 2ρBCP λ 1 λ 2 λ 3
Of residue e Å−3 Å kcal mol−1 e Å−5
(a) 2pwa phe_113 asp_117 0.297 1.719 −16.75 2.188 −1.92 −1.88 5.99
(b) 1us0 val_258 ala_208 0.282 1.739 −15.66 2.190 −1.76 −1.74 5.69
(c) 1us0 val_27 val_31 0.179 1.949 −9.89 1.450 −0.98 −0.95 3.38
(d) 1pjx tyr_203 leu_190 0.179 1.949 −10.00 1.520 −0.98 −0.94 3.44
(e) 1us0 ile_74 leu_106 0.060 2.441 −4.38 0.670 −0.23 −0.23 1.13
(f) 1us0 arg_232 ile_236 0.061 2.453 −4.28 0.620 −0.25 −0.24 1.11
(g) 1us0 glu_53 ala_57 0.061 2.457 −4.41 0.610 −0.25 −0.24 1.10
(h) 2pwa gly_92 lys_57 0.048 2.545 −3.85 0.579 −0.16 −0.13 0.87
(i) 1us0 ala_143 leu_147 0.044 2.585 −3.50 0.520 −0.14 −0.14 0.81


The excellent correlation between the Rij and the ρBCP and the estimated error of 0.05 e Å−3 (see the ESI) for the ρBCP clearly suggest the accuracy of electron density modelling in protein systems via an experimental database transfer approach, at least for these 22 high-resolution proteins. Given these superior results, we have estimated the electrostatic interaction energy, Eelec, for all of the 1443 HBs using eqn (1). The average Eelec values for the N–H⋯O[double bond, length as m-dash]C HBs in the α-helices and β-sheets for each protein system are listed in Table 2. The estimated Eelec values vary from −16.75 kcal mol−1 to −2.85 kcal mol−1 for the α-helices and −16.99 kcal mol−1 to −3.26 kcal mol−1 for the β-sheets (Tables S2 and S3). The average Eelec values of −9.04 kcal mol−1 for the 750 HBs in the α-helices and −10.08 kcal mol−1 for the 693 HBs in the β-sheets (Table 2) are slightly overestimated but lie almost within the range of the experimental values (−5 kcal mol−1 to −10 kcal mol−1) for the overall energy of the N–H⋯O[double bond, length as m-dash]C HBs.9 For the 750 HBs in the α-helices and the 693 HBs in the β-sheets, the Eelec (Fig. 3) values show an exponential relationship with their corresponding Rij values and once again, find exceptional correlations. From the Eelecvs. Rij plot, it can be inferred that the majority of the N–H⋯O[double bond, length as m-dash]C HBs with Rij values between 1.8 Å and 2.2 Å have Eelec values in the range of about −6 kcal mol−1 to −14 kcal mol−1 with an estimated error of 1.9 kcal mol−1, which are almost within the range of the experimental values.9 Moreover, in this range, the Eelec values for the two secondary structures are indistinguishable. The stronger HBs have Eelec values approximately between −14 kcal mol−1 and −17 kcal mol−1 and those of the weaker and weakest HBs vary approximately between −6 kcal mol−1 and −4.5 kcal mol−1 and up to approximately −3 kcal mol−1, respectively. In these three regions, the Eelec values for the α-helices are found to be systematically slightly lower than those of the β-sheets. The error on Eelec (over the entire range) is estimated to be of 2.52 kcal mol−1.


image file: d0ce00577k-f3.tif
Fig. 3 Exponential relationship between the Rij (Å) and Eelec (kcal mol−1). The dashed lines denote the demarcations of the strongest, most populated, weaker and weakest N–H⋯O[double bond, length as m-dash]C HBs.

Although both the Eelec and the ρBCP of the HBs in the α-helices and the β-sheets vary exponentially with Rij but they themselves found to follow a quadratic relationship (Fig. 4) with a slightly better R2 than a linear relationship (Fig. S8a). However, the Eelec and the ρBCP follow a linear relationship for the shorter HBs with ρBCP > ∼0.06 e Å−3 (Fig. S8b and S8c). The relationships between the various topological parameters and their values are found to agree well with those reported earlier for small molecules17,28,58 and protein systems.36,37 No significant differences in the topological parameters of the N–H⋯O[double bond, length as m-dash]C HBs were noticed between the parallel and anti-parallel β-sheets (Table S3).


image file: d0ce00577k-f4.tif
Fig. 4 Quadratic relationship between the ρBCP (e Å−3) and Eelec (kcal mol−1).

Further, to investigate if the N–H⋯O[double bond, length as m-dash]C HBs follow a different trend along the α-helices, we studied the terminal HBs and focused on the population of the weaker and the weakest HBs. Upon inspection of the results, as summarized in Table S7, we notice that out of the total of 20 weakest HBs in the α-helices (Fig. 1a), 12 (60%) HBs belong to the terminal residues. Meanwhile, in the weaker HB region, only ∼32% HBs are found to be originated from the terminal residues. There are almost equal numbers of HBs identified from the N-terminal (96) and C-terminal (95) residues. However, the Eelecvs. Rij plot (Fig. S9) for these entire terminal HBs (total 191) shows a similar exponential relationship (R2 = 98.4%) to that noticed for the entire HBs in the α-helices (Fig. 2, R2 = 99.9%), which also includes the HBs from the central part of the α-helices. In this context, it is noteworthy that in this study the HBs have been identified purely based on the QTAIM approach and not based on the distance–angle criteria, which often employed for studying the HBs in proteins.

Moreover, to verify if the populations of the HBs are affected by the resolutions of the structures, we analyzed the percentage distribution of the population of HBs in the different regions (strongest, most populated, weaker and weakest) of the HBs, in terms of both Rij and Eelec for two systems with a resolution difference of 0.22 Å and having almost the same number (∼35) of N–H⋯O[double bond, length as m-dash]C HBs: HEWL (PDB ID: 2VB1) with a resolution of 0.65 Å and glutaredoxin NrdH (PDB ID: 4HS1) with a resolution of 0.87 Å. Interestingly, the distributions display a similar trend in HB populations for both structures (Table S8).

The Laplacian of electron density, ∇2ρBCP, provides a measure of the local charge depletion (positive value) or concentration (negative value) in the interatomic region, i.e. for the noncovalent interactions (Fig. S10) or covalent bonds, respectively. However, the demarcation (strong, weak, etc.) of HBs from the ∇2ρBCPvs. Rij plot (Fig. 5), which follows an exponential relationship, appeared to be not as obvious as that noticed from the ρBCPvs. Rij plot. Therefore, we have employed the complementary NCI approach for the qualitative analysis of the N–H⋯O[double bond, length as m-dash]C HBs, especially focusing on the weaker HBs. To monitor the variation in NCI isosurfaces across the regions of the strongest to the weakest N–H⋯O[double bond, length as m-dash]C HBs, we have chosen some representative N–H⋯O[double bond, length as m-dash]C HBs from both α-helices and β-sheets (Table 3). NCI isosurfaces are usually plotted using NCIPLOT based on either the promolecular densities49 or the densities based on fully quantum-mechanical calculations.56 However, recently, extremely localized molecular orbital (ELMO) based NCI isosurfaces are used to detect the NCI in biosystems.59 Here, to the best of our knowledge, for the first time, we report the NCI isosurfaces in protein systems using ELMAM2 based experimental electron densities. The comparison of NCI isosurfaces plotted using the aforementioned methodologies, except for the ELMO based electron densities, is shown in Fig. S11. The NCI-ELMAM2 isosurfaces are found to compare well with those based on the molecular densities. The QTAIM based topological parameters along with the Eelec values of these representative HBs are listed in Table 3. The NCI isosurfaces for the N–H⋯O[double bond, length as m-dash]C HBs in the shortest or strongest (Rij ≤ 1.8 Å), most populated (1.8 Å < Rij ≤ 2.2 Å), longer or weaker (2.2 Å < Rij ≤ 2.45 Å) and longest or weakest (Rij ≥ 2.45 Å) regions are shown in Fig. 6a–i. The NCI isosurfaces with well-defined shapes are clearly visible for the strongest and up to the weaker N–H⋯O[double bond, length as m-dash]C HBs (Fig. 6a–f). Subsequently, at ∼2.46 Å, the NCI isosurface starts diffusing and merging with the other isosurfaces in the vicinity (Fig. 6g). Afterwards, the isosurfaces disappear for the weakest N–H⋯O[double bond, length as m-dash]C HBs (Fig. 6h and i). These observations also suggest that the limit of Rij for the protein main-chain N–H⋯O[double bond, length as m-dash]C HB is ∼2.45 Å. Further, these NCI isosurface analyses suggest that the limits of ∇2ρBCP and λ3 (curvature along the bond) for the N–H⋯O[double bond, length as m-dash]C HB are ∼0.62 e Å−5 and ∼1.1 e Å−5, respectively (Table 3). The linear dependence of ∇2ρBCP (e Å−5) with λ3 (e Å−5) is shown in Fig. S12.


image file: d0ce00577k-f5.tif
Fig. 5 Exponential relationship between the Rij (Å) and ∇2ρBCP (e Å−5).

image file: d0ce00577k-f6.tif
Fig. 6 NCI isosurfaces of N–H⋯O[double bond, length as m-dash]C HBs (a) in the strongest region (∼1.72 Å) in one of the α-helices of proteinase K (2pwa), (b) in the strongest region (∼1.74 Å) in one of the β-sheets of human aldose reductase (1us0), (c) in the most populated region (∼1.95 Å) in one of the α-helices of 1us0, (d) in the most populated region (∼1.95 Å) in one of the β-sheets of DFP-ase (1pjx), (e) in the weaker region (∼2.44 Å) in one of the β-sheets of 1us0, (f) at the interface of the weaker and weakest regions (∼2.45 Å) in one of the α-helices of 1us0, (g) at the interface of the weaker and weakest regions (∼2.46 Å) in one of the α-helices of 1us0, (h) in the weakest region (∼2.55 Å) in one of the β-sheets of 2pwa and (i) in the weakest (∼2.6 Å) regions in one of the α-helices of 1us0. The color scale for sign(λ2)ρ au is given on the left.

Conclusion

The quantitative and qualitative analyses of the topological parameters and the electrostatic interaction energies have been performed on 1443 main-chain N–H⋯O[double bond, length as m-dash]C hydrogen bonds (750 in α-helices and 693 in β-sheets) in 22 high-resolution (0.87 Å to 0.48 Å) secondary protein structures ranging from 4.7 kDa to 54.5 kDa, as retrieved from the RCSB PDB. This study stands out to be the first of its kind involving by far the largest number of high-resolution protein structures and HBs both from the α-helices and β-sheets. The topological analyses have been performed based on the experimental electron densities as transferred from the ELMAM2 database. Further, the estimated values of the Eelec compare well with the experimental values as reported in the literature and demonstrate excellent correlations with the topological parameters. Moreover, the relationships and the excellent correlations between the topological parameters are found to be in good agreement with those observed in the cases of small molecules and some protein systems. The limiting values of the topological parameters (Rij ≤ 2.45 Å; ρ(r) = ∼0.06 e Å−3; ∇2ρ(r) = ∼0.62 e Å−5; λ3 = ∼1.1 e Å−5) and the corresponding Eelec (≃4.3 kcal mol−1) to establish the presence of true N–H⋯O[double bond, length as m-dash]C HBs in protein main-chains are thus identified upon employing the QTAIM in conjunction with the NCI index – the quantum crystallographic approaches. To the best of our knowledge, the NCI-ELMAM2 approach using NCImilano has been employed for the first time for the qualitative analysis of HBs. Further, similar studies for the quantitative and qualitative analysis of other types of HBs (N–H⋯N, C–H⋯O etc.) present in these (and possibly some more) protein systems are planned. Lastly, this study based on a large set of high-resolution protein structures demonstrates that the QTAIM approach can be employed not only for the accurate estimation of the electrostatic interaction energies but also to establish the limiting values of the topological parameters of the true HBs in proteins, when studied together with the NCI approach.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

SKM thanks UGC and SNU for research fellowships. We are grateful to SNU for the infrastructure and high-performance computer cluster facility. PM thanks SERB, India Govt. for the research grant (EMR/2014/000491). We thank Dr. Gabriel Saleh and Dr. Christian Jelsch for their help on the use of NCIMilano and the MoPro utility programs. PM dedicates this paper to the memory of his beloved father, Pijush Kanti Munshi, who passed away recently.

References

  1. C. N. Pace, Nat. Struct. Mol. Biol., 2009, 16, 681–682 CrossRef CAS PubMed .
  2. J. D. Bernal, Nature, 1939, 143, 663–667 CrossRef .
  3. W. Kauzmann, Adv. Protein Chem., 1959, 14, 1–63 CrossRef CAS PubMed .
  4. C. N. Pace, S. Trevino, E. Prabhakaran and J. M. Scholtz, Philos. Trans. R. Soc., B, 2004, 359, 1225–1235 CrossRef CAS PubMed .
  5. P. J. Fleming and G. D. Rose, Protein Sci., 2005, 14, 1911–1917 CrossRef CAS PubMed .
  6. G. D. Rose, P. J. Fleming, J. R. Banavar and A. Maritan, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 16623–16633 CrossRef CAS PubMed .
  7. J. Kyte, Structure in protein chemistry, Garland Science, 2006 Search PubMed .
  8. A. E. Mirsky and L. Pauling, Proc. Natl. Acad. Sci. U. S. A., 1936, 22, 439–447 CrossRef CAS PubMed .
  9. L. Pauling, R. B. Corey and H. R. Branson, Proc. Natl. Acad. Sci. U. S. A., 1951, 37, 205–211 CrossRef CAS PubMed .
  10. L. Pauling and R. B. Corey, Proc. Natl. Acad. Sci. U. S. A., 1951, 37, 729–740 CrossRef CAS PubMed .
  11. A. Warshel, P. K. Sharma, M. Kato and W. W. Parson, Biochim. Biophys. Acta, Proteins Proteomics, 2006, 1764, 1647–1676 CrossRef CAS PubMed .
  12. S. Scheiner, in Reviews in Computational Chemistry, ed. D. Boyd and K. Lipkowitz, VCH Publishers, New York, 1991, pp. 165–218 Search PubMed .
  13. E. I. Howard, R. Sanishvili, R. E. Cachau, A. Mitschler, B. Chevrier, P. Barth, V. Lamour, M. Van Zandt, E. Sibley, C. Bon, D. Moras, T. R. Schneider, A. Joachimiak and A. Podjarny, Proteins: Struct., Funct., Bioinf., 2004, 55, 792–804 CrossRef CAS PubMed .
  14. P. Munshi, S.-L. Chung, M. P. Blakeley, K. L. Weiss, D. A. A. Myles and F. Meilleur, Acta Crystallogr., Sect. D: Biol. Crystallogr., 2012, 68, 35–41 CrossRef CAS PubMed .
  15. P. Coppens, X-ray charge densities and chemical bonding, International Union of Crystallography, 1997, vol. 4 Search PubMed .
  16. R. Herbst-Irmer and D. Stalke, Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater., 2017, 73, 531–543 CrossRef CAS PubMed .
  17. P. Munshi and T. N. Guru Row, CrystEngComm, 2005, 7, 608 RSC .
  18. E. N. Baker and R. E. Hubbard, Prog. Biophys. Mol. Biol., 1984, 44, 97–179 CrossRef CAS PubMed .
  19. R. S. Lipsitz, Y. Sharma, B. R. Brooks and N. Tjandra, J. Am. Chem. Soc., 2002, 124, 10621–10626 CrossRef CAS PubMed .
  20. D. J. Willock, S. L. Price, M. Leslie and C. R. A. Catlow, J. Comput. Chem., 1995, 16, 628–647 CrossRef CAS .
  21. N. K. Hansen and P. Coppens, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr., 1978, 34, 909–921 CrossRef .
  22. B. Fournier, E.-E. Bendeif, B. Guillot, A. Podjarny, C. Lecomte and C. Jelsch, J. Am. Chem. Soc., 2009, 131, 10929–10941 CrossRef CAS PubMed .
  23. R. F. W. Bader, Atoms in molecules, Wiley Online Library, 1990 Search PubMed .
  24. R. F. W. Bader, J. Phys. Chem. A, 1998, 102, 7314–7323 CrossRef CAS .
  25. G. R. Runtz, R. F. W. Bader and R. R. Messer, Can. J. Chem., 1977, 55, 3040–3045 CrossRef CAS .
  26. R. F. W. Bader and H. Essén, J. Chem. Phys., 1984, 80, 1943–1960 CrossRef CAS .
  27. P. Munshi and T. N. Guru Row, J. Phys. Chem. A, 2005, 109, 659–672 CrossRef CAS PubMed .
  28. P. R. Mallinson, G. T. Smith, C. C. Wilson, E. Grech and K. Wozniak, J. Am. Chem. Soc., 2003, 125, 4259–4270 CrossRef CAS PubMed .
  29. C. Gatti, E. May, R. Destro and F. Cargnoni, J. Phys. Chem. A, 2002, 106, 2707–2720 CrossRef CAS .
  30. R. Parthasarathi, S. S. Raman, V. Subramanian and T. Ramasami, J. Phys. Chem. A, 2007, 111, 7141–7148 CrossRef CAS PubMed .
  31. C. Jelsch, M. M. Teeter, V. Lamzin, V. Pichon-Pesme, R. H. Blessing and C. Lecomte, Proc. Natl. Acad. Sci. U. S. A., 2000, 97, 3171–3176 CrossRef CAS PubMed .
  32. Y. Hirano, K. Takeda and K. Miki, Nature, 2016, 534, 281–284 CrossRef CAS PubMed .
  33. J. Held and S. van Smaalen, Acta Crystallogr., Sect. D: Biol. Crystallogr., 2014, 70, 1136–1146 CrossRef CAS PubMed .
  34. B. Guillot, C. Jelsch, A. Podjarny and C. Lecomte, Acta Crystallogr., Sect. D: Biol. Crystallogr., 2008, 64, 567–588 CrossRef CAS PubMed .
  35. B. Zarychta, A. Lyubimov, M. Ahmed, P. Munshi, B. Guillot, A. Vrielink and C. Jelsch, Acta Crystallogr., Sect. D: Biol. Crystallogr., 2015, 71, 954–968 CrossRef CAS PubMed .
  36. K. Takaba, K. Takeda, M. Kosugi, T. Tamada and K. Miki, Sci. Rep., 2017, 7, 43162 CrossRef CAS PubMed .
  37. D. Liebschner, C. Jelsch, E. Espinosa, C. Lecomte, E. Chabrière and B. Guillot, J. Phys. Chem. A, 2011, 115, 12895–12904 CrossRef CAS PubMed .
  38. M. Holcomb, R. Adhikary, J. Zimmermann and F. E. Romesberg, J. Phys. Chem. A, 2018, 122, 446–450 CrossRef CAS PubMed .
  39. S. K. Mandal and P. Munshi, in Understanding Intermolecular Interactions in the Solid State: Approaches and Techniques, ed. D. Chopra, The Royal Society of Chemistry, 2018, pp. 189–210 Search PubMed .
  40. B. Zarychta, V. Pichon-Pesme, B. Guillot, C. Lecomte and C. Jelsch, Acta Crystallogr., Sect. A: Found. Crystallogr., 2007, 63, 108–125 CrossRef CAS PubMed .
  41. S. Domagała, B. Fournier, D. Liebschner, B. Guillot and C. Jelsch, Acta Crystallogr., Sect. A: Found. Crystallogr., 2012, 68, 337–351 CrossRef PubMed .
  42. B. Guillot, L. Viry, R. Guillot, C. Lecomte and C. Jelsch, J. Appl. Crystallogr., 2001, 34, 214–223 CrossRef CAS .
  43. C. Jelsch, B. Guillot, A. Lagoutte and C. Lecomte, J. Appl. Crystallogr., 2005, 38, 38–54 CrossRef .
  44. E. R. Johnson, S. Keinan, P. Mori-Sánchez, J. Contreras-García, A. J. Cohen and W. Yang, J. Am. Chem. Soc., 2010, 132, 6498–6506 CrossRef CAS PubMed .
  45. G. Saleh, C. Gatti, L. Lo Presti and J. Contreras-García, Chem. – Eur. J., 2012, 18, 15523–15536 CrossRef CAS PubMed .
  46. L. Massa and C. F. Matta, J. Comput. Chem., 2018, 39, 1021–1028 CrossRef CAS PubMed .
  47. A. Genoni, L. Bučinský, N. Claiser, J. Contreras-García, B. Dittrich, P. M. Dominiak, E. Espinosa, C. Gatti, P. Giannozzi, J. Gillet, D. Jayatilaka, P. Macchi, A. Ø. Madsen, L. Massa, C. F. Matta, K. M. Merz, P. N. H. Nakashima, H. Ott, U. Ryde, K. Schwarz, M. Sierka and S. Grabowsky, Chem. – Eur. J., 2018, 24, 10881–10905 CrossRef CAS PubMed .
  48. S. Grabowsky, A. Genoni and H.-B. Bürgi, Chem. Sci., 2017, 8, 4159–4176 RSC .
  49. J. Contreras-García, E. R. Johnson, S. Keinan, R. Chaudret, J.-P. Piquemal, D. N. Beratan and W. Yang, J. Chem. Theory Comput., 2011, 7, 625–632 CrossRef PubMed .
  50. M. D. Winn, C. C. Ballard, K. D. Cowtan, E. J. Dodson, P. Emsley, P. R. Evans, R. M. Keegan, E. B. Krissinel, A. G. W. Leslie, A. McCoy, S. J. McNicholas, G. N. Murshudov, N. S. Pannu, E. A. Potterton, H. R. Powell, R. J. Read, A. Vagin and K. S. Wilson, Acta Crystallogr., Sect. D: Biol. Crystallogr., 2011, 67, 235–242 CrossRef CAS PubMed .
  51. F. H. Allen, Acta Crystallogr., Sect. B: Struct. Sci., 1986, 42, 515–522 CrossRef .
  52. K. A. Majorek, M. D. Zimmerman, M. Grabowski, I. G. Shabalin, H. Zheng and W. Minor, in Structural Biology in Drug Discovery, Wiley, 2020, pp. 253–275 Search PubMed .
  53. U. Koch and P. L. A. Popelier, J. Phys. Chem., 1995, 99, 9747–9754 CrossRef CAS .
  54. G. Saleh, L. Lo Presti, C. Gatti and D. Ceresoli, J. Appl. Crystallogr., 2013, 46, 1513–1517 CrossRef CAS .
  55. C. B. Hübschle and B. Dittrich, J. Appl. Crystallogr., 2011, 44, 238–240 CrossRef PubMed .
  56. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, Ö. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian09 Revision D.01 Search PubMed .
  57. W. Humphrey, A. Dalke and S. Klaus, J. Mol. Graphics, 1996, 14, 33–38 CrossRef CAS PubMed .
  58. E. Espinosa, M. Souhassou, H. Lachekar and C. Lecomte, Acta Crystallogr., Sect. B: Struct. Sci., 1999, 55, 563–572 CrossRef PubMed .
  59. D. Arias-Olivares, E. K. Wieduwilt, J. Contreras-García and A. Genoni, J. Chem. Theory Comput., 2019, 15, 6456–6470 CrossRef CAS PubMed .

Footnote

Electronic supplementary information (ESI) available: Details of NCI isosurface generation methods. List of metal ions/atoms and other heavy atoms present in the protein structures. 3D deformation density maps and 2D Laplacian maps of strong and weak HBs. The bond path along with the bond critical points of the HBs in α-helices and β-sheets for a specific protein. Topological parameters, Eelec and ∠C[double bond, length as m-dash]O⋯H and ∠N–H⋯O of i → i + 4 α-helix HBs and parallel and antiparallel β-sheets HBs. Details of error estimation. The quadratic and linear relationships of ρBCP and Eelec. Details of the terminal N–H⋯O[double bond, length as m-dash]C HBs in the α-helices. Rijvs. Eelec plot for the terminal HBs. Percentage distributions of the HB populations in HEWL and glutaredoxin NrdH. Laplacian maps of the representative N–H⋯O[double bond, length as m-dash]C HBs and the NCI isosurfaces of some selected α-helices and β-sheets from electron densities as derived using the various approaches. Plot of λ3 (e Å−5) and ∇2ρBCP (e Å−5). The file name is “CEC-ESI-20_05_20-revised”. See DOI: 10.1039/d0ce00577k

This journal is © The Royal Society of Chemistry 2020