On the use of ³J-coupling NMR data to derive structural information on proteins

Abstract

Values of ³J-couplings as obtained from NMR experiments on proteins cannot easily be used to determine protein structure due to the difficulty of accounting for the high sensitivity of intermediate ³J-coupling values (4–8 Hz) to the averaging period that must cover the conformational variability of the torsional angle related to the ³J-coupling, and due to the difficulty of handling the multiple-valued character of the inverse Karplus relation between torsional angle and ³J-coupling. Both problems can be solved by using ³J-coupling time-averaging local-elevation restraining MD simulation. Application to the protein hen egg white lysozyme using 213 backbone and side-chain ³J-coupling restraints shows that a conformational ensemble compatible with the experimental data can be obtained using this technique, and that accounting for averaging and the ability of the algorithm to escape from local minima for the torsional angle induced by the Karplus relation, are essential for a comprehensive use of ³J-coupling data in protein structure determination.

Introduction

Structural information on proteins can be derived from a variety of observable quantities, such as X-ray diffraction intensities, NMR, CD, Raman or infrared spectra to mention a few (van Gunsteren et al. 2016). For proteins in aqueous solution, NMR measurements provide the highest information density, i.e. ratio of the number of independent, measured values of observable quantities for a molecule and the number of independent molecular degrees of freedom. Quantities observable using NMR techniques are ³J-couplings, chemical shifts, nuclear Overhauser enhancement intensities (NOEs), S² order parameters and residual dipolar couplings (RDCs). The measured value of such an observable quantity Q is an average 〈Q〉_space,time of Q over the molecules (space) in the test tube and over a time window determined by the particular measurement technique. This means that 〈Q〉  constitutes an average over a Boltzmann-weighted set, i.e. a statistical-mechanical ensemble, of configurations $\vec{r}$. The weights are proportional to $\exp(-V(\vec{r})/(k_{B}T)),$ where V($\vec{r}$) indicates the energy of a molecular configuration or structure $\vec{r},\;k_B$ is Boltzmann’s constant and T is the temperature.

If an observable quantity Q is dependent on the molecular configuration $\vec{r}$, one may try to derive an expression or function $Q(\vec{r})$ that approximates the relation between Q and $\vec{r},$ which expression may then be used to derive molecular structures that are compatible with measured values of Q, i.e.

$$\langle Q \rangle =\int Q(\vec{r})\mathrm{exp}(-V(\vec{r})/({k}_{B}T)){\rm d}\vec{r}/\int \mathrm{exp}(-V(\vec{r})/({k}_{B}T)){\rm d}\vec{r},$$

(1)

i.e. Boltzmann averages over the high-dimensional molecular configuration space.

When deriving molecular structures $\vec{r}$ from a set of measured values Q^exp of Q, the following problems may be met.

• 1. Q^exp values are subject to uncertainty or error.

• 2. It is not possible to fully account for averaging over space and time inherent in the experimental measurement: inversion of the averaging operation in Eq. (1) is impossible.

• 3. For most bio-molecular systems the number of independent Q^exp values available is much smaller than the number of degrees of freedom of the system. This means that the structure determination problem is underdetermined and can only be addressed by using an atomic model, i.e. a function V($\vec{r}$) specifying likely structural parameters (e.g. bond lengths and bond angles) of a system. The function V($\vec{r}$) may yield low-energy values for configurations that are physically most reasonable. The fewer Q^exp values are available, the larger the influence of the choice of molecular model and interaction function V($\vec{r}$) and its (in)accuracy on the generated structures will be.

• 4. The function Q($\vec{r}$) is not known or its accuracy is uncertain or low.

• 5. The inverse function $\vec{r}$(Q) of the function Q($\vec{r}$) may not exist, or if it does, it may be multiple-valued, as in the case of the Karplus relation or function (Karplus 1959, 1963) linking a ³J-coupling (Q) to a torsional angle θ($\vec{r}$) in a molecule,

  $${}^{3}J(\theta ) = a\cos^{2} (\theta ) + b\cos (\theta ) + c,$$

  (2)

• 6. The generation or sampling of molecular configurations $\vec{r}$ must be biased, i.e. guided towards those that are (on average) compatible with Q^exp. This is particularly challenging when the inverse function $\vec{r}$(Q) of the function Q($\vec{r}$) is multiple-valued. The inverse function θ(³J) of Eq. (2) is multiple-valued. For a range of ³J-values there are four different θ-values satisfying Eq. (2).

Although ³J-couplings are relatively easily measurable, their use to derive molecular structure is hampered by all six mentioned problematic aspects of procedures to derive molecular structure from measured Q^exp values: (i) Small values of ³J-couplings, such as occurring for ³J_HNHα-couplings in helical structures, are not easily precisely measured; (ii) The averaging of ³J-couplings may cover very long time periods and is not limited by the rotational tumbling time of the molecule, as for NOEs; (iii) ³J-couplings can only be measured for particular parts of proteins, backbone ³J_HNHα-couplings and side-chain ³J_HαHβ-couplings only address values for the φ and χ₁ torsional angles respectively; Although a range of other ³J-couplings can be measured, particularly with isotopically labelled samples, in general these show a smaller range of values and often have less well defined Karplus relation parameters. They are therefore not so useful for structure determination (Wang and Bax 1996; Perez et al. 2001); (iv) The parameters a, b and c of the Karplus relation Eq. (2) are of empirical nature and their values are commonly derived by matching X-ray diffraction derived crystal structures ($\vec{r}$) to solution NMR measured ³J-couplings (Q) for a set of (protein) molecules (Schmidt et al. 1999; Schmidt 2007). This leads to an uncertainty of 1–2 Hz in Eq. (2) (Dolenc et al. 2010; Steiner et al. 2012); (v) The inverse function θ(³J) of Eq. (2) is up to fourfold multiple-valued.

A few years ago, a procedure has been proposed to effectively handle the multiple-valuedness of θ(³J) by searching for all possible local minima of the biasing restraining function V_k^J,restr(J_k(θ_k($\vec{r}$(t))), 〈J_k〉_t; K^Jr, N_le, J_k⁰, ΔJ^fb) that keeps the value of 〈J_k〉 of the kth ³J-coupling close to J_k⁰ = J_k^exp (Smith et al. 2016). This restraining function has a flat-bottom on the interval [J_k⁰ − ΔJ^fb, J_k⁰ + ΔJ^fb] and is beyond this interval harmonic, see Fig. 1 of Smith et al. (2016) with ΔQ^h = ∞. Time is denoted by t and K^Jr is the overall weight or force constant of the penalty or restraining function. The local-elevation parameter N_le defines the number of intervals N_le of the torsional angle θ around the θ_i⁰-values

$$\theta_{i}^{0} \equiv { 2}\pi {i}/N_{le} \quad i = {1},{2}, \ldots ,N_{le}$$

(3)

and their widths

$$\Delta \theta^{0} \equiv { 2}\pi /N_{le} ,$$

(4)

used in the local-elevation search and sampling algorithm (Christen et al. 2007). The restraining function, biquadratic in J_k(θ_k($\vec{r}$(t))) and 〈J_k〉_t, only yields a non-zero energy and restraining force when both, J_k(θ_k($\vec{r}$(t))) and 〈J_k〉_t, adopt values on the same side outside the flat-bottom region. When either the instantaneous value J_k(θ_k($\vec{r}$(t))) or the average 〈J_k〉_t satisfies J⁰ = J^exp within a given uncertainty ΔJ^fb, there is no need for restraining.

The local-elevation searching and sampling technique (Huber et al. 1994) is used, in which the potential energy at already visited parts of configuration space is raised in order to avoid repetitive sampling of the same parts of configuration space in a simulation. The basic idea of local-elevation structure refinement based on adaptive restraints (Christen et al. 2007) is that whenever the simulated average  〈J〉 _t of the ³J-coupling and the current value J(t) of the ³J-coupling do not match the measured target value J⁰ = J^exp to within an uncertainty ΔJ^fb, the force constant or weight ${\omega }_{{\theta }_{ki}}(t)$ of the penalty or restraining function V_k^J,restr(J_k(θ_k($\vec{r}$(t)))), 〈J_k〉_t; K^Jr, N_le, J_k⁰, ΔJ^fb), acting on the current value θ_k(t) of the torsional angle θ_k corresponding to the ³J-coupling, is increased. The restraining potential-energy term of a given ³J-coupling J_k is a sum of N_le terms, local with respect to a particular value θ_i⁰ of θ_k,

$$V_{k}^{J,restr} \left( {\theta_{k} \left( {\vec{r}\left( t \right)} \right);K^{Jr} ,N_{le} ,J_{k}^{0} ,\Delta J^{fb} } \right) = \sum\limits_{i = 1}^{{N_{le} }} {K^{Jr} \omega_{{\theta_{ki} }} (t)\exp \left( { - (\theta_{k} (t) - \theta_{i}^{0} )^{2} /(2(\Delta \theta^{0} )^{2} )} \right)}$$

(5)

with the weight ${\omega }_{{\theta }_{ki}}\left(t\right)$ of the ith penalty function changing during the simulation according to

$$\omega_{{\theta_{ki} }} \left( t \right) = t^{ - 1} \mathop \smallint \limits_{0}^{t} \delta_{{\theta_{k} \left( {\vec{r}\left( {t^{\prime}} \right)} \right)\theta_{i}^{0} }} (J_{k} \left( {t^{\prime}} \right) - J_{k}^{0} - \Delta J^{fb} )^{2} H\left( {J_{k} \left( {t^{\prime}} \right);J_{k}^{0} + \Delta J^{fb} } \right)( \langle J_{k} \rangle_{t^{\prime}} - J_{k}^{0} - \Delta J^{fb} )^{2} H( \langle J_{k} \rangle_{t^{\prime}} ;J_{k}^{0} + \Delta J^{fb} ){\rm d}t^{\prime}$$

(6)

for an attractive (both, J_k(t) and  〈J_k〉 _t larger than J_k⁰) ³J-coupling restraint, and according to

$$\omega _{{\theta _{{ki}} }} \left( t \right) = t^{{ - 1}} \mathop \smallint \limits_{0}^{t} \delta _{{\theta _{k} \left( {\vec{r}\left( {t^{\prime}} \right)} \right)\theta _{i}^{0} }} (J_{k} \left( {t^{\prime}} \right) - J_{k}^{0} + \Delta J^{{fb}} )^{2} (1 - H\left( {J_{k} \left( {t^{\prime}} \right);J_{k}^{0} - \Delta J^{{fb}} } \right))(\langle J_{k} \rangle _{{t^{\prime}}} - J_{k}^{0} + \Delta J^{{fb}} )^{2} (1 - H(\langle J_{k} \rangle _{{t^{\prime}}} ;J_{k}^{0} - \Delta J^{{fb}} )){\rm d}t^{\prime}$$

(7)

for a repulsive (both, J_k(t) and  〈J_k〉 _t smaller than J_k⁰) ³J-coupling restraint. The Kronecker delta δ is defined using finite differences,

$$\begin{aligned} \delta_{{\theta_{k} \left( {\vec{r}\left( {t^{\prime}} \right)} \right)\theta_{i}^{0} }} &= 1\quad {\text{ if }}\left( {\theta_{i}^{0} - \Delta \, \theta_{i}^{0} /{2}} \right) \, \le \theta_{i} (t) < \left( {\theta_{i}^{0} + \Delta \, \theta_{i}^{0} /{2}} \right) \hfill \\ \delta_{{\theta_{k} \left( {\vec{r}\left( {t^{\prime}} \right)} \right)\theta_{i}^{0} }} &= 0\quad {\text{ otherwise}} \hfill \\ \end{aligned}$$

(8)

and the Heaviside step function H(x;x₀) is defined by

$$\begin{aligned} H\left( {x;x_{0} } \right) & = 0{\text{ for }}x < x_{0} \\ & {\text{ = 1 for }}x \ge x_{0} . \\ \end{aligned}$$

(9)

This means that the torsional angle θ is pushed away from the value θ(t) when both the current J(t) and the averaged  〈J〉 _t values are deviating more than ± ΔJ^fb from the target value J⁰. In this way the whole range of θ values is searched for those values that yield ³J-couplings close to J^exp (Allison and van Gunsteren 2009). The time-averaged value  〈J〉 _t of J(t) is commonly (Torda et al. 1989) exponentially damped

$$\langle {J}_{k}({\theta }_{k}(\vec{r}(t))){\rangle}_{t}={\left[{\tau }_{J}(1-\text{exp}(-t/{\tau }_{J}))\right]}^{-1}{\int }_{0}^{t}\text{exp}(-(t-t{^{\prime}})/{\tau }_{J}){J}_{k}({\theta }_{k}(\vec{r}(t{^{\prime}}))){\rm d}t{^{\prime}}$$

(10)

in the restraining function V_k^J,restr(J_k(θ_k($\vec{r}$(t))), 〈J_k〉_t; K^Jr, N_le, J_k⁰, ΔJ^fb), Eqs. (6) and (7), in order to avoid that the restraining force progressively approaches zero with time.

This procedure to refine protein structure using measured ³J-couplings was applied (Smith et al. 2016) to sets of 95 backbone ³J_HNHα-couplings and 62 side-chain ³J_HαHβ-couplings (58 stereo-specifically assigned ³J_HαHβ-couplings plus those of Glu 7 and Arg 45 were included in the set of restraints) measured for hen egg white lysozyme (HEWL) (Smith et al. 1991). It was concluded that (i) the weight or force constant K^J,restr of the restraining function should be chosen as small as possible while large enough to bring the average ³J-couplings close to the target values in order to approximate as good as possible the properly (using V($\vec{r}$)) Boltzmann-weighted conformational probability distribution, (ii) the averaging time τ_J should match the experimental one but should not be larger than 1/10 of the total simulation time in order to secure sufficient statistics when averaging, and (iii) the flat-bottom Δ³J^fb should represent the uncertainty or inaccuracy of the Karplus relation ³J(θ).

In the present article the biquadratic time-averaging local-elevation restraining (BQ-TA-LER) method (Smith et al. 2016) is further investigated using NMR and X-ray data on HEWL, particularly concentrating on side chains whose ³J_HαHβ-couplings are significantly averaged.

Fig. 1

Ribbon pictures of the structure of HEWL with residues for which experimentally derived backbone ³J_HNHα-coupling values (left panel, experimentally stereo-specifically assigned (set bb1) in blue, computationally stereo-specifically assigned (set bb2) in magenta) and side-chain ³J_Hα-Hβ-coupling values (right panel, experimentally stereo-specifically assigned (set sc1) in green, computationally stereo-specifically assigned (set sc2) in red) are available

Materials and methods

Energy minimisations and molecular dynamics simulations were performed using the GROMOS bio-molecular simulation software (Schmid et al. 2011a, 2012; van Gunsteren et al. 2019).

Molecular model

The protein was modelled using the GROMOS bio-molecular force field 54A7 (Poger et al. 2010; Schmid et al. 2011b). In view of the pH used in the experimental NMR measurements, pH = 3.5, only Glu 35 was protonated and His was doubly protonated (Bartik et al. 1994). The simple point charge (SPC) model (Berendsen et al. 1981) was used to describe the solvent molecules in the rectangular periodic box. To compensate for the overall positive charge of the protein, 10 Cl⁻ ions were included in the solution. All bond lengths and the bond angle of the water molecules were kept rigid with a relative geometric precision of 10^–4 using the SHAKE algorithm (Ryckaert et al. 1977), allowing for a 2 fs MD time step in the leap-frog algorithm (Hockney and Eastwood 1981) used to integrate the equations of motion. For the non-bonded interactions a triple-range method (van Gunsteren et al. 1986) with cut-off radii of 0.8/1.4 nm was used. Short-range (within 0.8 nm) van der Waals and electrostatic interactions were evaluated every time step based on a charge-group pair list (van Gunsteren et al. 2019). Medium-range van der Waals and electrostatic interactions, between pairs at a distance larger than 0.8 nm and shorter than 1.4 nm, were evaluated every fifth time step (10 fs), at which time point the pair list was updated, and kept constant between updates. Outside the larger cut-off radius (1.4 nm) a reaction-field approximation (Barker and Watts 1973; Tironi et al. 1995) with a relative dielectric permittivity of 61 (Heinz et al. 2001) was used. Minimum-image periodic boundary conditions were applied.

Simulation set-up

Four X-ray crystal structures were used as initial structures for the energy minimisations followed by MD simulations.

• 1. Structure “2VB1” of the Protein Data Bank (PDB) (Berman et al. 2000), derived from a triclinic unit cell at 0.065 nm resolution at T = 100 K. It contains multiple side-chain conformations for 46 residues.

• 2. Structure “4LZT” of the PDB, derived from a triclinic unit cell at 0.095 nm resolution at T = 295 K. It contains multiple side-chain conformations for 8 residues.

• 3. Structure “1IEE” of the PDB, derived from a tetragonal unit cell at 0.094 nm resolution at T = 110 K. It contains multiple side-chain conformations for 33 residues.

• 4. Structure “1AKI” of the PDB, derived from an orthorhombic unit cell at 0.15 nm resolution at T = 298 K. It contains no multiple side-chain conformations.

For the initial structures the side-chain conformation with the highest occupancy was chosen. The atom-positional root-mean-square differences (RMSD) between these four initial structures are for 2VB1/4LZT 0.086 nm for all atoms and 0.027 nm for the backbone atoms, for 2VB1/1IEE 0.191 nm (all) and 0.075 nm (bb), for 2VB1/1AKI 0.165 nm (all) and 0.056 nm (bb), for 4LZT/1IEE 0.183 nm (all) and 0.073 nm (bb), for 4LZT/1AKI 0.158 nm (all) and 0.049 nm (bb), and for 1IEE/1AKI 0.151 nm (all) and 0.047 nm (bb).

An initial structure was first energy minimised in vacuo to release possible strain induced by small differences in bond lengths, bond angles, improper dihedral angles, and short distance non-bonded contacts between the force-field parameters and the X-ray structure. Subsequently, the protein was put into a rectangular box filled with water molecules, such that the minimum solute-wall distance was 1.0 nm, and water molecules closer than 0.23 nm from the solute were removed. This resulted in boxes with 12,157 water molecules for the initial protein structures. In order to relax unfavourable contacts between atoms of the solute and the solvent, a second energy minimisation was performed for the protein in the periodic box with water while keeping the atoms of the solute harmonically position-restrained (van Gunsteren et al. 2019) with a force constant of 25,000 kJ mol⁻¹ nm⁻².

The resulting protein-water configuration was used as initial configuration for the MD simulation. In order to avoid artificial deformations in the protein structure due to relatively high-energy atomic interactions still present in the system, the MD simulation was started at T = 60 K and then the temperature was slowly raised to T = 308 K. Initial atomic velocities were sampled from a Maxwell distribution at T = 60 K. The equilibration scheme consisted of five short 20 ps simulations at temperatures 60, 120, 180, 240 and 308 K at constant volume. During the first four of the equilibration periods, the solute atoms were harmonically restrained to their positions in the initial structures with force constants of 25,000, 2500, 250, and 25 kJ mol⁻¹ nm⁻². The temperature was kept constant using the weak-coupling algorithm (Berendsen et al. 1984) with a relaxation or coupling time τ_Τ = 0.1 ps. Solute and solvent were separately coupled to the heat bath. Following this equilibration procedure, the simulations were performed at a reference temperature of 308 K and a reference pressure of 1 atm. The pressure was kept constant using the weak-coupling algorithm (Berendsen et al. 1984) with a coupling time τ_p = 0.5 ps and an isothermal compressibility κ_T = 4.575 × 10^–4 (kJ mol⁻¹ nm⁻³)⁻¹. The centre of mass motion of the system was removed every 1000 time steps (2 ps).

³ J-coupling restraining

Two sets of backbone ³J_HN-Hα couplings and two sets of side-chain ³J_Hα-Hβ couplings for restraining (Smith et al. 2016) were used, see Tables 1, 2, 3 and 4.

• 1. A set (bb1) of 95 backbone ³J_HN-Hα-coupling values, see Table II of Smith et al. (1991) from which the values for 11 glycine residues were omitted, because these had not been stereo-specifically assigned.

• 2. A set (bb2) of 22 experimentally stereo-specifically unassigned backbone ³J_HN-Hα-coupling values for the 11 glycine residues, see Table II of Smith et al. (1991). 10 of these were stereo-specifically assigned based on the ³J_HN-Hα-coupling values calculated from the four unrestrained MD simulations starting from the four mentioned X-ray structures (see below) as either 4 or 3 of the unrestrained MD simulations suggested the same stereo-specific assignment. Gly 104 could not be stereo-specifically assigned using this criterion. Instead it was stereo-specifically assigned based on the ³J_HN-Hα-coupling values calculated for the four X-ray structures.

• 3. A set (sc1) of 58 ³J_Hα-Hβ-coupling values, see Tables III and IV of Smith et al. (1991), which were stereo-specifically assigned using experimental data.

• 4. A set (sc2) of 38 out of 40 experimentally stereo-specifically unassigned ³J_Hα-Hβ-coupling values, see Table III of Smith et al. (1991), which were stereo-specifically assigned based on the ³J_Hα-Hβ-coupling values calculated from the four unrestrained MD simulations starting from the four mentioned X-ray structures (see below) as either 4 or 3 of the unrestrained MD simulations suggested the same stereo-specific assignment. Glu 7 could not be stereo-specifically assigned using this criterion. It could also not be stereo-specifically assigned using the ³J_Hα-Hβ-coupling values calculated for the four X-ray structures.

Table 1 Backbone ³J_HNHα-coupling values (95) in Hz derived and assigned based on NMR measurements (set bb1) and from four unrestrained MD simulations starting from four X-ray crystal structures, and the mean of the latter four values and the root-mean-square deviation (RMSD) from it

The distribution of these ³J_HN-Hα-coupling and ³J_Hα-Hβ-coupling values over the protein is shown in Fig. 1 (backbone experimentally assigned (bb1): blue; backbone computationally stereo-specifically assigned (bb2): magenta; side chain experimentally stereo-specifically assigned (sc1): green; side chain computationally stereo-specifically assigned (sc2): red).

For the calculation of the backbone ³J_HN-Hα-couplings, the Karplus relation Eq. (2) was used with the parameter values a = 6.4 Hz, b = − 1.4 Hz and c = 1.9 Hz (Pardi et al. 1984), see Fig. 2, left panel (black lines). The side-chain ³J_Hα-Hβ-couplings were calculated using the parameter values a = 9.5 Hz, b = − 1.6 Hz and c = 1.8 Hz (deMarco et al. 1978), see Fig. 2, right panel (black lines).

Fig. 2

Left panel: ³J_HNHα-coupling Karplus curves as function of the φ-angle (black lines: from Pardi et al. 1984; blue lines: from Brüschweiler and Case 1994; solid lines: α hydrogen and α2 Re hydrogen for Gly; dashed lines: α3 Si hydrogen for Gly). Right panel: ³J_Hα-Hβ-coupling Karplus curves as function of the χ₁-angle from deMarco et al. 1978 (solid line: β hydrogen for Ile and Thr and β₂ hydrogen, dashed line: β hydrogen for Val and β₃ hydrogen)

The experimentally derived ³J_HN-Hα-coupling values for Val 2, Thr 51, Asp 66, Cys 115, Thr 118 and Ile 124 lie outside the Karplus curve, so were set to 9.7 Hz, which is the maximum of the Karplus curve used (Pardi et al. 1984). None of the experimentally derived ³J_Hα-Hβ-coupling values lie outside the Karplus curve used (deMarco et al. 1978). The nomenclature for the H_α2 and H_α3 atoms in Gly residues and the H_β, H_β2 and H_β3 atoms in the side chains was defined as in Markley et al. (1998), see Fig. 3.

Fig. 3

Figure taken from Markley et al. (1998, Fig. 1)

Recommended atom identifiers for the 20 common amino acids follow the 1969 IUPAC-IUB guidelines (IUPAC-IUB Commission on Biochemical Nomenclature 1970). Backbone atoms are shown for Pro, Gly, and Ala but not for the other L-amino acids (where they correspond to those bounded by the dashed line in the Ala structure). Greek letters are used as atom identifiers.

The initial implementation of the algorithm proposed in Smith et al. (2016) in the GROMOS MD program was incorrect. The factors t⁻¹ and dt in the expressions of Eqs. (6) and (7) (Eqs. (21) and (22) in Smith et al. 2016) were not implemented, which meant that the local-elevation weight factors ω_φ or ω_θ were not time-averaged and of different magnitude than intended. They would thus only stay constant instead of decrease when the restraints were satisfied. This implementation error has been corrected. As a consequence, the values of the restraining force constants K^Jr used in the previous work (Smith et al. 2016) are much smaller than the value of the force constant K^Jr used in the present work. In Smith et al. (2016), K^Jr was varied between 5 and 50⋅10^–4 kJ mol⁻¹ Hz⁻⁴. Here, K^Jr is set to 50 kJ mol⁻¹ Hz⁻⁴, because the weights ω_θ are much smaller. In Smith et al. (2016), the memory relaxation time was varied between 5 and 50 ps, because of the length of 2 ns of the many test simulations. Here it could be extended to τ_Q ≡ τ_J = 500 ps, in view of the 20 ns length of the MD simulations. In Smith et al. (2016), the flat-bottom parameter of the restraining potential energy term was varied between 0.5 and 1 Hz. Here the value Δ³J^fb = 1.0 Hz is used, which means a flat bottom of 2 Hz width (Smith et al. 2016), representing the uncertainty of the Karplus relation used.

MD simulations performed

Four unrestrained MD simulations, starting from the four mentioned X-ray crystal structures, were performed:

• 1. MD_2VB1,

• 2. MD_4LZT,

• 3. MD_1IEE,

• 4. MD_1AKI,

each 20 ns long. The average solute temperatures were 311 K and the solvent temperatures 312 K.

Starting from the 2VB1 X-ray crystal structure, four ³J-restraining MD simulations were performed:

• 5. MD_2VB1_bb1 + bb2, applying ³J_HN-Hα-coupling restraining to sets bb1 and bb2, 117 restraints,

• 6. MD_2VB1_sc1, applying ³J_Hα-Hβ-coupling restraining to set sc1, 58 restraints,

• 7. MD_2VB1_sc1 + sc2, applying ³J_Hα-Hβ-coupling restraining to sets sc1 and sc2 (without Glu 7), 96 restraints,

• 8. MD_2VB1_bb1 + bb2 + sc1 + sc2, applying ³J_HN-Hα-coupling and ³J_Hα-Hβ-coupling restraining to sets bb1, bb2, sc1 and sc2 (without Glu 7), 213 restraints, again each 20 ns long. The average solute and solvent temperatures were as mentioned above.

Analysis of atomic trajectories and X-ray structures

Trajectory energies and atomic coordinates were stored at 5 ps intervals and used for analysis (Eichenberger et al. 2011).

In view of the various factors contributing to an uncertainty of about 2 Hz inherent to the Karplus relation linking structure and ³J-couplings, as discussed in the Introduction, a deviation of less than 2 Hz between ³J-coupling values calculated from X-ray or MD trajectory structures and ³J-coupling values derived from experiment is considered insignificant.

The GROMOS force fields treat aliphatic carbons as united CH, CH₂ and CH₃ atoms. So inter-hydrogen distances involving the aliphatic hydrogen atoms were calculated using virtual atomic positions for CH and pro-chiral CH₂ (van Gunsteren et al. 1985) and pseudo-atomic positions for CH₃ (Wüthrich et al. 1983) for those hydrogen atoms (van Gunsteren et al. 2019). The pseudo-atom NOE distance bound corrections of Wüthrich et al. (1983) were used (van Gunsteren et al. 2016). The set of NOE distance bounds (Smith et al. 1993; Schwalbe et al. 2001) can be found in Table S1 of Supporting Information, together with values of the five simulations starting from the 2VB1 X-ray crystal structure. The NOE between Trp 28 HZ3 and Leu 56 HG was reassigned as between Trp 28 HZ3 and Leu 56 HD* following reassessment of the experimental spectra. Inter-hydrogen distances were calculated as  〈r⁻³〉 ^−1/3, i.e. using r⁻³ averaging over the trajectory structures, where r indicates the actual hydrogen–hydrogen distance. In view of the uncertainty inherent to the calculation of NOE bounds and r⁻³ averaged distances, deviations from experiment of less than 0.1 nm are considered insignificant.

S² order parameters were calculated using the ensemble averaging expression (Henry and Szabo 1985)

$$S_{XY}^{2} = \frac{1}{2}\left\{ {3\sum\limits_{\alpha = 1}^{3} {\sum\limits_{\beta = 1}^{3} {\left\langle {\frac{{\mu_{XY\alpha } (t)\mu_{XY\beta } (t)}}{{r_{XY}^{3} (t)}}} \right\rangle_{\tau}^{2} } } - \left\langle {\frac{1}{{r_{XY}^{3} (t)}}} \right\rangle_{\tau}^{2} } \right\}(r_{XY}^{eff} )^{6} ,$$

(11)

where τ indicates the time-averaging window, here 1 ns, shorter than the rotational correlation time of 5.7 ns of HEWL in solution (Smith et al. 1993),

$$\mu_{XY1} \equiv \, \left( {x_{X} {-}x_{Y} } \right)/r_{XY} ,\mu_{XY2} \equiv \, \left( {y_{X} {-}y_{Y} } \right)/r_{XY} ,\mu_{XY3} \equiv \, \left( {z_{X} {-}z_{Y} } \right)/r_{XY} ,$$

(12)

are the normalised x-, y-, and z-components of the vector r_XY ≡ r_X − r_Y connecting atoms X and Y, and r_XY ≡ |r_XY| its length (Hansen et al. 2014). To obtain a dimensionless quantity the term in curly brackets is multiplied with the 6th power of the effective length (r^eff_XY) of the vector r_XY. Because in the present work bond length constraints are used, the length of r_XY is essentially constant over time and thus equal to its effective value r^eff_XY. The set of 79 experimentally derived S² values (Buck et al. 1995; Moorman et al. 2012) together with the S² values calculated from the five MD simulations started from the 2VB1 X-ray crystal structure (Smith et al. 2020) can be found in Supporting Information, Table S2.

Before calculating S²_XY, the protein trajectory structures are superimposed using the backbone atoms (N, C_α, C) of residues 3–126 in the fit in order to eliminate the effect of overall rotation of the protein upon the S²_XY-values. Use of only the backbone atoms of four of the five α-helices and two β-strands in HEWL (residues 4–15, 24–36, 41–45, 50–53, 89–99, and 108–115) did not lead to significantly different S²_XY-values.

For the Asn and Gln residues, one S²_NH(exp) value per NH₂ group was available. This required the assignment to one of the two NH1 and NH2 bond vectors. This was done by calculating S²_NH1(MD)- and S²_NH2(MD)-values from the unrestrained simulation MD_2VB1 starting from the 2VB1 X-ray structure and then selecting the N–H vector with its S²_NH(MD) value closest to S²_NH(exp) for restraining. A corresponding procedure was used to assign experimentally unassigned S²_CG1- and S²_CG2-values for Val and S²_CD1- and S²_CD2-values for Leu residues (Smith et al. 2020).

In view of the uncertainty inherent to the derivation of S²_XY(exp)-values from relaxation experiments and inherent to the calculation of S²_XY(MD)-values from MD simulation, a deviation of less than 0.2 between simulation and experiment is considered insignificant.

Atom-positional root-mean-square fluctuations, i.e. around their average positions, in the MD trajectories were calculated using the above-mentioned superposition too.

The secondary structure assignment was done with the program DSSP, based on the Kabsch–Sander rules (Kabsch and Sander 1983).

Hydrogen bonds were identified according to a geometric criterion: a hydrogen bond was assumed to exist if the hydrogen-acceptor distance was smaller than 0.25 nm and the donor-hydrogen-acceptor angle was larger than 135°.

Results and discussion

Comparison of ³ J-coupling values calculated from X-ray structures or MD trajectories with NMR measured values

Table 1 lists 95 backbone ³J_HNHα-coupling values derived and stereo-specifically assigned based on NMR measurements (set bb1), as calculated from the four unrestrained MD simulations starting from four X-ray crystal structures, and as calculated from the four X-ray structures. The mean of the four MD or X-ray values and the root-mean-square deviation (RMSD) from it are also presented. Deviations from the experimental values of more than 2 Hz are denoted in italics. The mean values of the MD simulations show 14 deviations larger than 2 Hz, while for the X-ray structures there is only one such case (Trp 111). The great majority of the large differences between MD simulation and experiment are found for residues (11), for which large ³J_HNHα-coupling values (> 9 Hz) have been observed (Ser 36, Gln 41, Asn 65, Arg 68, Thr 69, Leu 84, Trp 108, Arg 114, Cys 115, Thr 118 and Ile 124). This can be explained by considering the Karplus curve in Fig. 2 (left panel). Its maximum of 9.7 Hz lies at φ ≈ 240° or − 120° and ³J_HNHα-coupling values larger than 8 Hz are only found for φ-angle values between 210° and 270°. Any motion around φ ≈ 240° will lower the calculated ³J_HNHα-coupling value. We note that other parametrisations of the Karplus curve show maxima of 10 Hz (Wang and Bax 1996) or even 11 Hz (Brüschweiler and Case 1994), see Fig. 1 in Dolenc et al. (2010). The remaining three residues with ³J_HNHα-coupling value differences with experiment larger than 2 Hz are found for Ser 50, Ile 78 and Asn 103, whose experimental values are 7.8, 8.0 and 8.2 Hz, respectively. For the X-ray structures, the only ³J_HNHα-coupling value differing more than 2 Hz from experiment (7.1 Hz) is for Trp 111, where all four values are smaller than 4.9 Hz. Use of the Karplus relation of Brüschweiler and Case (1994) (Fig. 2, left panel, blue lines) slightly lowers the ³J_HNHα-coupling values smaller than 6.3 Hz and increases values larger than 6.3 Hz, the increase being about 1 Hz for ³J_HNHα-coupling values larger than 8.5 Hz. It does not improve significantly the agreement of the ³J_HNHα-coupling values obtained from the MD simulations and the X-ray structures with the experimental ones.

Table 2 lists 22 experimentally stereo-specifically unassigned backbone ³J_HNHα-coupling values derived from NMR measurements (set bb2), as calculated from the four unrestrained MD simulations starting from four X-ray crystal structures, and as calculated from the four X-ray structures. The mean of the four MD or X-ray values and the root-mean-square deviation (RMSD) from it are also presented. The stereo-specific assignment of the experimental values for the α2 Re and the α3 Si hydrogens in glycine residues is based on the criterion that the ³J_HN-Hα-coupling values calculated from the four unrestrained MD simulations starting from the four X-ray structures do suggest in 4 or 3 of the unrestrained MD simulations the same assignment. Only Gly 104 could not be stereo-specifically assigned using this criterion. For this residue the stereo-specific assignment was based on the ³J_HN-Hα-coupling values calculated from the four X-ray structures using a corresponding criterion. For only two residues the X-ray structures would suggest stereo-specific assignments different from the ones based on the MD trajectories, for Gly 67, which shows a rather small difference of 0.7 Hz between the two experimental ³J_HN-Hα-coupling values, and for Gly 126, with a somewhat larger difference of 1.2 Hz between the two experimental ³J_HN-Hα-coupling values. Deviations of the MD or X-ray calculated ³J_HN-Hα-coupling values from the experimental ones of more than 2 Hz are denoted in italics. The MD trajectories show four of such deviations, for Gly 49 α2 Re in the MD_2VB1 simulation, for Gly 104 α2 Re and α3 Si in the MD_4LZT simulation and α3 Si in the MD_1AKI simulation. The X-ray structures show seven of such deviations, for Gly 102 α2 Re and α3 Si in the 1IEE and 1AKI structures, for Gly 104 α3 Si in the 2VB1 structure and for Gly 126 α3 Si in the 4LZT and 1AKI structures.

Table 2 Experimentally stereo-specifically unassigned backbone ³J_HNHα-coupling values (22) in Hz derived from NMR measurements (set bb2) and values from four unrestrained MD simulations starting from four X-ray crystal structures, and the mean of the latter four values and the root-mean-square deviation (RMSD) from it

Table 3 lists 58 side-chain ³J_HαHβ-coupling values derived and stereo-specifically assigned based on NMR measurements (set sc1), as calculated from the four unrestrained MD simulations starting from four X-ray crystal structures, and as calculated from the four X-ray structures. The mean of the four MD or X-ray values and the root-mean-square deviation (RMSD) from it are also presented. Deviations from the experimental values larger than 2 Hz are denoted in italics. Out of 58*4 = 232 ³J_HαHβ-coupling values, the MD trajectories yield 58 values 2 Hz larger than experiment, and the X-ray structures 55 values. The MD trajectories show a variation in ³J_HαHβ-coupling values larger than 2 Hz for seven hydrogens in five residues, Tyr 23 β₂, Asn 27 β₂, Phe 34 β₂ and β₃, Asn 46 β₂ and β₃, and Cys 127 β₂. The X-ray structures show two such cases, for Val 99 and Val 109. In all these cases the mean of the four MD or X-ray values lies within 2 Hz from experiment, but the four values contributing to the mean show a large variation of 2.2–3.7 Hz for the MD values and 3.4–4.6 for the X-ray values. Apparently, MD trajectories and different X-ray structures contain different side-chain χ₁-angle conformers for these residues.

Table 3 Side-chain ³J_HαHβ-coupling values (58) in Hz derived and stereo-specifically assigned based on NMR measurements (set sc1) and from four unrestrained MD simulations starting from four X-ray crystal structures, and the mean of the latter four values and the root-mean-square deviation (RMSD) from it

Table 4 lists 40 experimentally stereo-specifically unassigned side-chain ³J_HαHβ-coupling values derived from NMR measurements (set sc2), as calculated from the four unrestrained MD simulations starting from four X-ray crystal structures, and as calculated from the four X-ray structures. The mean of the four MD or X-ray values and the root-mean-square deviation (RMSD) from it are also presented. The stereo-specific assignment of the experimental values for the β₂ and β₃ hydrogens is based on the criterion that the ³J_HαHβ-coupling values calculated from the four unrestrained MD simulations starting from the four X-ray structures do suggest in 4 or 3 of the unrestrained MD simulations the same stereo-specific assignment. Only Glu 7 could not be stereo-specifically assigned using this criterion. For four residues the X-ray structures would suggest stereo-specific assignments different from the ones based on the MD trajectories, for Asn 19, which shows a rather small difference of 0.9 Hz between the two experimental ³J_HαHβ-coupling values, for Asn 37, with a larger difference of 3.9 Hz between the two experimental ³J_HαHβ-coupling values, for Asn 74, with a difference of 6.6 Hz between the two ³J_HαHβ-coupling values, and for Asn 77, with a difference of 3.4. Hz between the two ³J_HαHβ-coupling values. Deviations of the MD or X-ray calculated ³J_HαHβ-coupling values from the experimental ones of more than 2 Hz are denoted in italics. The MD trajectories show 45 of such deviations, for 13 residues, Glu 7, Lys 13, Arg 68, Ser 72, Asn 74, Asn 77, Ser 85, Ser 86, Ser 100, Asp 101, Asn 106, Arg 125 and Arg 128. The X-ray structures show 104 of such deviations, for 17 residues, Glu 7, Lys 13, Asn 19, Trp 28, Asn 37, Arg 45, Arg 68, Ser 72, Asn 74, Asn 77, Ser 85, Ser 86, Ser 100, Asp 101, Asn 106, Arg 125 and Arg 128. The MD trajectories show a variation in ³J_HαHβ-coupling values larger than 2 Hz for four hydrogens in three residues, Ser 72 β₃ (2.6 Hz), Asn 74 β₂ (2.7 Hz), and Asp 101 β₂ and β₃ (both 3.2 Hz). The X-ray structures show 15 of such cases in nine residues, Glu 7 β₂ (4.8 Hz) and β₃ (5.2 Hz), Arg 45 β₂ (4.1 Hz) and β₃ (4.7 Hz), Arg 68 β₂ (4.4 Hz), Ser 85 β₂ (4.7 Hz) and β₃ (3.9 Hz), Ser 86 β₂ (3.5 Hz), Ser 100 β₂ (2.9 Hz), Asp 101 β₂ (4.1 Hz) and β₃ (4.5 Hz), Arg 125 β₂ (3.7 Hz) and β₃ (5.4 Hz), and Arg 128 β₂ (3.9 Hz) and β₃ (4.3 Hz). In 6 of these 19 cases the mean of the four MD or X-ray values lies within 2 Hz from experiment, while the four values contributing to the mean show a large variation of 2.6–3.2 Hz for the MD values and 2.9–5.4 for the X-ray values. Apparently, MD trajectories and different X-ray structures contain different side-chain χ₁-angle conformers for these residues.

Table 4 Experimentally stereo-specifically unassigned side-chain ³J_HαHβ-coupling values (40) in Hz derived from NMR measurements (set sc2 plus Glu 7) and values from four unrestrained MD simulations starting from four X-ray crystal structures, and the mean of the latter four values and the root-mean-square deviation (RMSD) from it

This analysis shows that neither the X-ray crystal structures nor the MD trajectories of HEWL in aqueous solution are wholly compatible with the 117 backbone ³J_HNHα-coupling values and the 96 side-chain ³J_HαHβ-coupling values as obtained from NMR experiments of this protein in aqueous solution. Application of biquadratic time-averaging local-elevation (BQ-TA-LER) ³J-coupling restraining in MD simulation should be able to produce atomic trajectories compatible with the experimental ³J-coupling values.

Comparison of ³ J-coupling values calculated from ³ J-coupling time-averaging local-elevation restraining MD trajectories with NMR measured values

Table 5 lists 95 backbone ³J_HNHα-coupling values derived and stereo-specifically assigned based on NMR measurements (set bb1) and calculated from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure using different sets of backbone and side-chain restraints. The unrestrained MD simulation shows 17 ³J_HNHα-coupling values (in italics) that deviate more than 2 Hz from the experimental values (residues 41, 46, 50, 56, 65, 68, 69, 78, 84, 87, 103, 108, 114, 115, 118, 124, 127). ³J_HNHα-coupling time-averaging local-elevation restraining towards the sets bb1 and bb2 of 95 and 22 target backbone ³J_HNHα-coupling values leads, as expected, to good agreement between simulation and experiment for the 95 backbone ³J_HNHα-couplings. No deviations larger than 2 Hz are observed. Restraining towards the sets sc1 or sc1 and sc2 of 58 and 38 side-chain ³J_HαHβ-coupling values yields 18 or 20 deviations larger than 2 Hz, respectively. Side-chain ³J_HαHβ-coupling restraining does not improve the agreement between simulation and experiment for the backbone ³J_HNHα-couplings.

Table 5 Backbone ³J_HNHα-coupling values (95) in Hz derived and assigned based on NMR measurements (set bb1) and from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 6 lists 22 experimentally stereo-specifically unassigned backbone ³J_HNHα-coupling values derived from NMR measurements (set bb2) and values calculated from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure using different sets of backbone and side-chain restraints. The unrestrained MD simulation shows one ³J_HNHα-coupling value (in italics), for the α2 Re hydrogen in residue Gly 49, that deviates more than 2 Hz from the experimental value. ³J_HNHα-coupling time-averaging local-elevation restraining towards the sets bb1 and bb2 of 95 and 22 target backbone ³J_HNHα-coupling values leads, as expected, to good agreement between simulation and experiment for the 22 backbone ³J_HNHα-couplings. No deviations larger than 2 Hz are observed.

Table 6 Experimentally stereo-specifically unassigned backbone ³J_HNHα-coupling values (22) in Hz derived from NMR measurements (set bb2) and values from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 7 lists 58 side-chain ³J_HαHβ-coupling values in Hz derived and stereo-specifically assigned based on NMR measurements (set sc1) and from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure using different sets of backbone and side-chain restraints. The unrestrained MD simulation shows 14 ³J_HαHβ-coupling values (in italics) that deviate more than 2 Hz from the experimental values (residues 18, 20(2), 27, 30, 46(2), 51, 59, 61, 66, 69, 89, 99). Restraining towards the sets bb1 and bb2 of 95 and 22 backbone ³J_HNHα-coupling values yields 18 deviations larger than 2 Hz, and so no improvement of the agreement between simulated and experimental ³J_HαHβ-coupling values, as one would expect. Backbone ³J_HNHα-coupling restraining does not improve the agreement between simulation and experiment for the side-chain ³J_HαHβ-couplings. ³J_HαHβ-coupling time-averaging local-elevation restraining towards the sets sc1 or sc1 and sc2 of 58 and 38 target side-chain ³J_HαHβ-coupling values leads, as expected, to good agreement between simulation and experiment for the 58 side-chain ³J_HαHβ-couplings. No deviations larger than 2 Hz are observed. This is also the case when restraining towards all 213 experimental backbone and side-chain ³J-coupling values (sets bb1, bb2, sc1 and sc2).

Table 7 Side-chain ³J_HαHβ-coupling values (58) in Hz derived and stereo-specifically assigned based on NMR measurements (set sc1) and from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 8 lists 40 experimentally stereo-specifically unassigned side-chain ³J_HαHβ-coupling values derived from NMR measurements (set sc2 plus Glu 7) and values from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure using different sets of backbone and side-chain restraints. The unrestrained MD simulation shows 13 ³J_HαHβ-coupling values (in italics) that deviate more than 2 Hz from the experimental values (residues 68, 72(2), 77(2), 85, 86, 101(2), 106, 125(2), 128). Restraining towards the sets bb1 and bb2 of 95 and 22 backbone ³J_HNHα-coupling values yields 13 deviations larger than 2 Hz, and so no improvement of the agreement between simulated and experimental ³J_HαHβ-coupling values, as one would expect. Backbone ³J_HNHα-coupling restraining does not improve the agreement between simulation and experiment for the side-chain ³J_HαHβ-couplings, apart from a few cases. This is also observed using the sc1 set of 58 restraints (14 deviations larger than 2 Hz). ³J_HαHβ-coupling time-averaging local-elevation restraining towards the sets sc1 and sc2 of 58 and 38 target side-chain ³J_HαHβ-coupling values leads, as expected, to good agreement between simulation and experiment for the 40 side-chain ³J_HαHβ-couplings. Only two deviations (Glu 7 β₂ and β₃) larger than 2 Hz are observed. This is also the case when restraining towards all 213 experimental backbone and side-chain ³J-coupling values (sets bb1, bb2, sc1 and sc2), only for Glu 7 β₂ the deviation is larger than 2 Hz. Note that Glu 7 is not part of the set sc2 of restraints.

Table 8 Experimentally stereo-specifically unassigned side-chain ³J_HαHβ-coupling values (40) in Hz derived from NMR measurements (set sc2 plus Glu 7) and values from the unrestrained and ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Tables 9, 10, 11 and 12 summarise the agreement between experimental ³J-coupling values and those calculated from the four X-ray structures, from the four unrestrained MD simulation trajectories and from the ³J-coupling time-averaging local-elevation restrained MD simulation trajectories. Using backbone ³J_HNHα-coupling restraints (sets bb1 and bb2) only, the experimental ³J_HNHα-coupling values are reproduced and using ³J_HαHβ-coupling restraints (sc1 or sc1 and sc2) only, the experimental ³J_HαHβ-coupling values are reproduced. Not surprisingly, there appears to be insignificant mutual influence between the different sets of restraints. Using all 213 experimental backbone and side-chain ³J-coupling restraints (sets bb1, bb2, sc1 and sc2) no significant deviations from experimental ³J-coupling values are observed.

Table 9 Number of deviations, |³J_HNHα (exp) − ³J_HNHα (MD or X-ray)|, for the 95 backbone ³J_HNHα-coupling values derived and assigned based on NMR measurements (set bb1), in four X-ray crystal structures, in the four unrestrained MD simulations starting from these, and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 10 Number of deviations, |³J_HNHα (exp) − ³J_HNHα (MD or X-ray)|, for the 22 backbone ³J_HNHα-coupling values derived but stereo-specifically unassigned from NMR measurements (set bb2), in four X-ray crystal structures, in the four unrestrained MD simulations starting from these, and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 11 Number of deviations, |³J_HαHβ (exp) − ³J_HαHβ (MD or X-ray)|, for the 58 side-chain ³J_HαHβ-coupling values derived and stereo-specifically assigned based on NMR measurements (set sc1), in four X-ray crystal structures, in the four unrestrained MD simulations starting from these, and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Table 12 Number of deviations, |³J_HαHβ (exp) − ³J_HαHβ (MD or X-ray)|, for the 38 side-chain ³J_HαHβ-coupling values derived but stereo-specifically unassigned from NMR measurements (set sc2), in four X-ray crystal structures, in the four unrestrained MD simulations starting from these, and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints

Other quantities and agreement with NOE atom–atom distance bounds and S ² order-parameter values

The application of backbone ³J_HNHα-coupling time-averaging local-elevation restraining in MD simulation does not significantly influence the secondary structure of the protein, with the four main α-helices, the two 3₁₀-helices and the triple-stranded anti-parallel β-sheet in the protein all being maintained (see Fig. S1 and S2 of Supporting Information). However, there are some subtle differences in loop regions, particularly around residues Gly 102–Gly 104 and around Gly 117–Thr 118. For example, there are increases in the populations of the hydrogen bonds 107 NH–104 O and 118 NH–115 O and a decrease in the population of the hydrogen bond 104 NH–101 O compared to the unrestrained simulations. These regions are known to be mobile in solution, Gly 102, Asn 103 and Thr 118 all having lower backbone NH order parameters (0.72, 0.52 and 0.72, respectively; Buck et al. 1995) and these regions were less well defined in the NMR structure of HEWL (Schwalbe et al. 2001) having few longer range NOE identified for them. They provide an example of the extra conformational insights that could be obtained with time-averaging local-elevation restraining to glycine ³J_HNHα-couplings although further experimental data would be needed to confirm the details of the hydrogen bond population changes observed here. As expected, the time-averaging local-elevation restraining does slightly enhance the atomic mobility, as observed from the backbone atom-positional fluctuations, see Figure S3 in Supporting Information.

Table 13 summarises for 1630 NOE atom–atom distance bounds of HEWL derived from NMR experiments (Smith et al. 1993; Schwalbe et al. 2001) the agreement between experimental NOE atom–atom distance bounds and the corresponding distances calculated from the four X-ray structures, from the four unrestrained MD simulation trajectories and from the ³J-coupling time-averaging local-elevation restrained MD simulation trajectories using different sets of backbone and side-chain restraints. Only ³J-coupling restraining to all 213 backbone and side-chain restraints does improve the agreement with experiment for the 1630 NOE atom–atom distance bounds.

Table 13 Number of NOE distance bound violations in four X-ray crystal structures, in the four unrestrained MD simulations starting from these, and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints. Number of NOE distance bounds: 1630, see Table S1 in Supporting Information

Table 14 summarises for 51 S²_CH and 28 S²_NH side-chain order parameters of HEWL derived from NMR experiments (Buck et al. 1995; Moorman et al. 2012) the agreement between experimental S² order parameter values and those calculated from the four unrestrained MD simulation trajectories and from the ³J-coupling time-averaging local-elevation restrained MD simulation trajectories using different sets of backbone and side-chain restraints. ³J-coupling restraining does not significantly change the agreement with experiment for the 79 S² order parameters. This is not surprising, because the 79 S² order parameters reflect motions along degrees of freedom that are different from the ones for which ³J-couplings are available.

Table 14 Number of deviations, |S²(exp) − S²(MD)|, for the 51 S²_CH-values, the 11 S²_NH-values of Trp and Arg residues and the 17 S²_NH2-values of Asn and Gln residues, respectively (Smith et al. 2020), in the four unrestrained MD simulations starting from four different X-ray crystal structures and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints. Order parameter values are reported in Table S2 in Supporting Information

Table 15 summarises for 121 backbone S²_NH order parameters of HEWL derived from NMR experiments (Buck et al. 1995) the agreement between experimental S² order parameter values and those calculated from the four unrestrained MD simulation trajectories and from the ³J-coupling time-averaging local-elevation restrained MD simulation trajectories using different sets of backbone and side-chain restraints. ³J-coupling restraining does reduce the number of deviations larger than 0.4, but the number of deviations larger than 0.2 is not reduced.

Table 15 Number of deviations, |S²(exp) − S²(MD)|, for the 121 backbone S²_NH-values, (Buck et al. 1995), in the four unrestrained MD simulations starting from four different X-ray crystal structures and in the ³J-coupling time-averaging local-elevation restrained MD simulations starting from the 2VB1 X-ray crystal structure and using different sets of backbone and side-chain restraints. Order parameter values are reported in Table S3 in Supporting Information

Importance of time-averaging

Values of measured ³J-couplings may the result from considerable conformational averaging. This is illustrated in Figs. 4, 5, 6, 7 and 8 for one backbone φ-angle and four side-chain χ₁-angles.

Fig. 4

Variation of the backbone φ-angle (degree) determining the ³J_HNHα-coupling, of this ³J_HNHα-coupling (Hz) with an experimental value of 8.2 Hz (green line), and the presence of the Asn 103 NH–Asp 101 OD1 hydrogen bond for residue Asn 103 as function of time from the unrestrained MD simulation MD_2VB1 (left panels) and from the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 (right panels), each starting from the 2VB1 X-ray structure. Red lines: average ³J-coupling value in the MD simulations

Fig. 5

Variation of the side-chain χ₁-angle (degree) determining the ³J_Hα-Hβ-coupling, of this ³J_Hα-Hβ-coupling (Hz) with an experimental value of 9.5 Hz (green line), and the presence of the Thr 89 OG1-HG1–Asp 87 OD1 hydrogen bond for residue Thr 89 as function of time from the unrestrained MD simulation MD_2VB1 (left panels) and from the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_VB1_bb1 + bb2 + sc1 + sc2 (right panels), each starting from the 2VB1 X-ray structure. Red lines: average ³J-coupling value in the MD simulations

Fig. 6

Variation of the side-chain χ₁-angle (degree) determining the ³J_Hα-Hβ-coupling, and of this ³J_Hα-Hβ-coupling (Hz) with an experimental value of 6.3 Hz (green line), for residue Val 99 as function of time from the unrestrained MD simulation MD_VB1 (left panels) and from the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_VB1_bb1 + bb2 + sc1 + sc2 (right panels), each starting from the 2VB1 X-ray structure. Red lines: average ³J-coupling value in the MD simulations

Fig. 7

Variation of the side-chain χ₁-angle (degree) determining the two, β₂ and β₃, ³J_Hα-Hβ-couplings, of these two ³J_Hα-Hβ-couplings (Hz) with experimental values of 5.6 Hz and 6.6 Hz respectively (green lines), and the presence of the Asn 103 NH–Asp 101 OD1 hydrogen bond for residue Asp 101 as function of time from the unrestrained MD simulation MD_2VB1 (left panels) and from the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 (right panels), each starting from the 2VB1 X-ray structure. Red lines: average ³J-coupling value in the MD simulations

Fig. 8

Variation of the side-chain χ₁-angle (degree) determining the two, β₂ and β₃, ³J_Hα-Hβ-couplings, of these two ³J_Hα-Hβ-couplings (Hz) with experimental values of 10.5 Hz and 3.6 Hz respectively (green lines), and the presence of the Ala 107 NH–Asn 106 OD1 hydrogen bond for residue Asn 106 as function of time from the unrestrained MD simulation MD_2VB1 (left panels) and from the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 (right panels), each starting from the 2VB1 X-ray structure. Red lines: average ³J-coupling value in the MD simulations

Figure 4 shows for Asn 103 the variation of the backbone φ-angle determining the ³J_HNHα-coupling, of this ³J_HNHα-coupling with an experimental value of 8.2 Hz (green line), and the presence of the Asn 103 NH–Asp 101 OD1 hydrogen bond as function of time for the unrestrained (left panels) and the ³J-coupling time-averaging local-elevation restraining to all 213 experimental ³J-coupling values (right panels) MD simulations starting from the 2VB1 X-ray structure. A variation of the φ-angle of about 30° around its average value of − 69° leads to the ³J_HNHα-coupling covering the range 2–10 Hz (RMSF 1.4 Hz) and an average value of 5.4 Hz (red line) in the unrestrained MD simulation, and in the ³J-coupling time-averaging local-elevation restraining simulation to a distribution of ³J_HNHα-couplings shifted to larger values (RMSF 1.5 Hz) with an average of 8.1 Hz. The restraining shifts the average ³J_HNHα-coupling more than 2 Hz towards the experimental value and slightly increases its fluctuation. The Asn 103 NH–Asp 101 OD1 hydrogen bond regularly populated in the unrestrained simulation disappears upon ³J_HNHα-coupling restraining. The 2VB1 X-ray structure contains two φ-angle conformations, − 79° (main conformation) and − 93° (alternative conformation), with ³J_HNHα-couplings of 6.6 Hz and 8.2 Hz respectively, of which only one matches the experimental ³J_HNHα-coupling value. Both φ-angle values are covered in the MD simulations.

Figure 5 shows for Thr 89 the variation of the side-chain χ₁-angle determining the ³J_Hα-Hβ-coupling, of this ³J_Hα-Hβ-coupling with an experimental value of 9.5 Hz (green line), and the presence of the Thr 89 OG1-HG1–Asp 87 OD1 hydrogen bond as function of time from the unrestrained (left panels) and ³J-coupling time-averaging local-elevation restraining to all 213 experimental ³J-coupling values (right panels) MD simulations starting from the 2VB1 X-ray structure. In the unrestrained simulation, the χ₁-angle stays around + 60° with an occasional excursion to − 50°, leading to ³J_Hα-Hβ-coupling values between 2 and 6 Hz, with an occasional excursion to 10–12 Hz. The average ³J_Hα-Hβ-coupling of 4.8 Hz (RMSF 3.4 Hz) (red line) deviates substantially from the experimental value of 9.5 Hz. Restraining completely changes the χ₁-angle and ³J_Hα-Hβ-coupling distributions. The average χ₁-angle becomes − 40° with large fluctuations (RMSF 31°) and the average ³J_Hα-Hβ-coupling becomes 10.4 Hz (red line) with a much-reduced variation (RMSF 1.7 Hz). The presence of the Thr 89 OG1-HG1–Asp 87 OD1 hydrogen bond is somewhat reduced by the restraining. The 2VB1 X-ray χ₁-angle values of − 67° or − 69° result in ³J_Hα-Hβ-couplings of 12.8 Hz and 12.7 Hz respectively, deviations of more than 3 Hz from the experimental value of 9.5 Hz.

Figure 6 shows for Val 99 the variation of the side-chain χ₁-angle determining the ³J_Hα-Hβ-coupling and of this ³J_Hα-Hβ-coupling with an experimental value of 6.3 Hz (green line), as function of time from the unrestrained (left panels) and ³J-coupling time-averaging local-elevation restraining to all 213 experimental ³J-coupling values (right panels) MD simulations starting from the 2VB1 X-ray structure. The unrestrained simulation shows a stable χ₁-angle value of − 62° with little variation (RMSF 10°) resulting in an average ³J_Hα-Hβ-coupling of 3.0 Hz (red line) (RMSF 1.6 Hz). It does not reproduce the experimental value of 6.3 Hz. The four X-ray structures deviate with ³J_Hα-Hβ-coupling values of 12.8 Hz (2VB1 and 4LZT), 2.2 Hz (1IEE) and 2.4 Hz (1AKI) even more from the experimental value. Using ³J-coupling local-elevation restraining other conformations of the χ₁-angle are accessed, resulting in a larger variation (RMSF 2.1 Hz) and raising the average to 5.3 Hz (red line). The 2VB1 X-ray structure contains two conformations, with χ₁-angles of 176° (main conformation) and − 53° (alternative conformation) and ³J_Hα-Hβ-couplings of 12.8 Hz and 4.2 Hz respectively. Averaging over different conformations seems to occur in aqueous solution.

Figure 7 shows for Asp 101 the variation of the side-chain χ₁-angle determining the two, β₂ and β₃, ³J_Hα-Hβ-couplings, of these two ³J_Hα-Hβ-couplings (Hz) with experimental values of 5.6 Hz and 6.6 Hz respectively (green lines), and the presence of the Asn 103 NH–Asp 101 OD1 hydrogen bond as function of time from the unrestrained (left panels) and ³J-coupling local-elevation restraining to all 213 experimental ³J-coupling values (right panels) MD simulations starting from the 2VB1 X-ray structure. The unrestrained simulation shows a stable χ₁-angle value of − 169° with little variation (RMSF 11°) resulting in average ³J_Hα-Hβ-couplings of 2.5 Hz (RMSF 0.8 Hz) and 12.2 Hz (RMSF 0.9 Hz) (red lines). These values deviate significantly from the experimental values of 5.6 Hz and 6.6 Hz. The four X-ray structures deviate with χ₁-angles of − 89° and ³J_Hα-Hβ-coupling values of 10.5 Hz (β₂) and 1.8 Hz (β₃) (2VB1 and 4LZT), a χ₁-angle of − 167° and ³J_Hα-Hβ-couplings of 2.2 Hz (β₂) and 12.4 Hz (β₃) (1IEE) and a χ₁-angle of − 139° and ³J_Hα-Hβ-couplings of 2.4 Hz (β₂) and 8.4 Hz (β₃) (1AKI) even more from the experimental values. Using ³J-coupling time-averaging local-elevation restraining other conformations of the χ₁-angle are accessed, resulting in a larger variations (RMSF 2.8 Hz (β₂) and 2.2 Hz (β₃)) of the ³J_Hα-Hβ-couplings and raising the β₂ average to 5.6 Hz and lowering the β₃ average to 5.9 Hz (red line).

Figure 8 shows for Asn 106 the variation of the side-chain χ₁-angle determining the two, β₂ and β₃, ³J_Hα-Hβ-couplings, of these two ³J_Hα-Hβ-couplings with experimental values of 10.5 Hz and 3.6 Hz respectively (green lines), and the presence of the Ala 107 NH–Asn 106 OD1 hydrogen bond as function of time from the unrestrained (left panels) and ³J-coupling local-elevation restraining to all 213 experimental ³J-coupling values (right panels) MD simulations starting from the 2VB1 X-ray structure. The unrestrained simulation shows after 4 ns a stable χ₁-angle value of about 60° with little variation (RMSF 9°) resulting in average ³J_Hα-Hβ-couplings of 4.7 Hz (RMSF 2.8 Hz) and 3.9 Hz (RMSF 2.7 Hz) (red lines). The β₂ value deviates significantly from the experimental value of 10.5 Hz, while the β₃ value is close to the experimental value of 3.6 Hz. The 2VB1 X-ray structure contains two χ₁-angle conformations, − 96° (main conformation) and − 169° (alternative conformation), with ³J_Hα-Hβ-coupling values of 9.3 Hz (β₂) and 2.1 Hz (β₃). The other three X-ray structures show only one conformation, with a χ₁-angle of − 70° and ³J_Hα-Hβ-couplings of 12.6 Hz (β₂) and 2.3 Hz (β₃) (4LZT), a χ₁-angle of − 64° and ³J_Hα-Hβ-couplings of 12.9 Hz (β₂) and 3.0 Hz (β₃) (1IEE) and a χ₁-angle of − 70° and ³J_Hα-Hβ-couplings of 12.6 Hz (β₂) and 2.4 Hz (β₃) (1AKI). Using ³J-coupling time-averaging local-elevation restraining other conformations of the χ₁-angle are accessed, resulting in smaller variations (RMSF 2.3 Hz (β₂) and 1.9 Hz (β₃)) of the ³J_Hα-Hβ-couplings, raising the β₂ average to 11.1 Hz and lowering the β₃ average to 3.1 Hz (red lines). The Ala 107 NH–Asn 106 OD1 hydrogen bond disappears when applying ³J-coupling restraining.

Importance of escaping from torsional-angle energy minima

In Figs. 9, 10, 11, 12 and 13 the local-elevation potential energies after the time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 for the φ-angle of Asn 103 and the χ₁-angles of Thr 89, Val 99, Asp 101 and Asn 106, are shown. The time series of these angles, the corresponding ³J-couplings and some hydrogen bonds in this simulation were shown in Figs. 4, 5, 6, 7 and 8 (right panels).

Fig. 9

Local-elevation ³J_HNHα-coupling restraining potential energy as function of the backbone φ-angle for residue Asn 103, built-up during the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 starting from the 2VB1 X-ray structure

Fig. 10

Local-elevation ³J_HαHβ-coupling restraining potential energy as function of the side-chain χ₁-angle for residue Thr 89, built-up during the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 starting from the 2VB1 X-ray structure

Fig. 11

Local-elevation ³J_HαHβ-coupling restraining potential energy as function of the side-chain χ₁-angle for residue Val 99, built-up during the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 starting from the 2VB1 X-ray structure

Fig. 12

Local-elevation ³J_HαHβ2-coupling (dashed line) and ³J_HαHβ3-coupling (dotted line) restraining potential energies and their sum (solid line) as function of the side-chain χ₁-angle for residue Asp 101, built-up during the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 starting from the 2VB1 X-ray structure

Fig. 13

Local-elevation ³J_HαHβ2-coupling (dashed line) and ³J_HαHβ3-coupling (dotted line) restraining potential energies and their sum (solid line) as function of the side-chain χ₁-angle for residue Asn 106, built-up during the ³J-coupling time-averaging local-elevation restraining (to all 213 experimental ³J-coupling values) MD simulation MD_2VB1_bb1 + bb2 + sc1 + sc2 starting from the 2VB1 X-ray structure

The backbone φ-angle of Asn 103 in the unrestrained simulation covers values in the range [− 90°, − 45°] (Fig. 3, upper left panel) yielding a ³J_HNHα-coupling of 5.4 Hz, much lower than the experimental value of 8.2 Hz. In the Karplus curve for this ³J-coupling (black solid line in the left panel of Fig. 2) this value corresponds to φ-angle values of about − 93° and − 147°. Time-averaging local-elevation ³J-coupling restraining shifts the φ-angle values towards the range [− 180°, − 90°] (Fig. 4, upper right panel), shifting the ³J_HNHα-coupling from 5.4 Hz in the unrestrained simulation to 8.1 Hz, close to the experimental value of 8.2 Hz. Figure 9 shows the local-elevation potential-energy term built-up in the ³J-coupling restraining simulation for φ-angle values in the range [− 90°, − 10°] that causes this shift. There is no build-up of local-elevation potential energy in the range [0°, + 160°], because these φ-angle values are not occurring in the ³J-coupling restraining simulation (Fig. 4, upper right panel).

The side-chain χ₁-angle of Thr 89 in the unrestrained simulation covers values in two ranges, [+ 30°, + 70°] and, less frequently, [− 70°, − 30°] (Fig. 5, upper left panel) yielding a ³J_Hα-Hβ-coupling of 4.8 Hz, much lower than the experimental value of 9.5 Hz. In the Karplus curve for this ³J-coupling (solid line in the right panel of Fig. 2) this value corresponds to χ₁-angle values of about − 25° and − 95° or of about + 112° and + 128°, respectively. Time-averaging local-elevation ³J-coupling restraining shifts the χ₁-angle values towards the range [− 10°, − 100°] (Fig. 5, upper right panel), shifting the ³J_Hα-Hβ-coupling from 4.8 Hz in the unrestrained simulation to 10.4 Hz, near the experimental value of 9.5 Hz. Figure 10 shows the local-elevation potential-energy term built-up in the ³J-coupling restraining simulation for χ₁-angle values in the range [− 10°, + 80°] and around − 55°, that causes this shift. The build-up of local-elevation potential energy around − 170° is caused by the χ₁-angle occasionally visiting this region in the ³J-coupling restraining simulation (Fig. 5, upper right panel). We note that the solvent accessibility of the side chain of Thr 89 in the 2VB1 X-ray structure is 76%.

The side-chain χ₁-angle of Val 99 in the unrestrained simulation initially covers values around 70° and 170° (Fig. 6, upper left panel) yielding ³J_Hα-Hβ-couplings between 2 and 13 Hz. After about 1 ns, the χ₁-angle stabilizes around − 62° with values in the range [− 50°, − 85°] yielding a ³J_Hα-Hβ-coupling of 3.0 Hz, much lower than the experimental value of 6.3 Hz. In the Karplus curve for this ³J-coupling (dashed line in the right panel of Fig. 2) this value corresponds to four χ₁-angle values of about − 129°, − 37°, + 37° and + 129°. The Karplus curve indicates that a slight shift of the χ₁-angle distribution towards less negative values, for example − 40°, would yield a ³J_Hα-Hβ-coupling of about 6 Hz. Time-averaging local-elevation ³J-coupling restraining indeed induces this slight shift towards χ₁-angle values in the range [− 40°, − 65°] (Fig. 6, upper right panel), but also makes the χ₁-angle repeatedly move over all angle values but those between − 40° and + 40°, thereby reaching very large ³J_Hα-Hβ-coupling values, in order to raise the average ³J_Hα-Hβ-coupling value from 3.0 Hz in the unrestrained simulation towards the experimental value of 6.3 Hz. Figure 11 shows the local-elevation potential-energy term built-up in the ³J-coupling restraining simulation for χ₁-angle values around − 80°, + 80° and 180°. It shows minima for − 126°, + 130° and for [− 35°, + 35°]. The χ₁-angle range [− 35°, + 35°] is disfavoured by the χ₁ dihedral-angle potential-energy term and the non-bonded van der Waals interaction of the force field. We note that the solvent accessibility of the side chain of Val 99 in the 2VB1 X-ray structure is only 7%. This side chain is surrounded by the side chains of Tyr 20, Trp 28, Ile 98 and Tyr 108.

The last two examples involve longer side chains for which the C_β-atom is connected to two hydrogens, H_β2 and H_β3. If their ³J_Hα-Hβ-couplings have been stereo-specifically assigned, the corresponding χ₁-angle can be restrained using two local-elevation ³J_Hα-Hβ-coupling restraining potential-energy terms. This imposes more restriction on the χ₁-angle motion than in the previously discussed cases.

The side-chain χ₁-angle of Asp 101 covers in the unrestrained simulation values around − 170° (Fig. 7, upper left panel) yielding average ³J_Hα-Hβ-couplings of 2.5 Hz for β₂ and 12.2 Hz for β₃, to be compared to experimental values of 5.6 Hz and 6.6 Hz respectively. In the Karplus curve for the ³J_Hα-Hβ2-coupling (solid line in the right panel of Fig. 2) the experimental value of 5.6 Hz corresponds to four χ₁-angle values of about − 116°, − 4°, + 76° and + 164°. In the Karplus curve for the ³J_Hα-Hβ3-coupling (dashed line in the right panel of Fig. 2) the experimental value of 6.6 Hz corresponds to four χ₁-angle values of about − 129°, − 37°, + 37° and + 129°. The Karplus curves suggest a χ₁-angle distribution dominantly around − 125°. χ₁-angle values around + 55° would bring both ³J_Hα-Hβ-couplings near their experimental values, but would require averaging of ³J_Hα-Hβ-couplings. Time-averaging local-elevation ³J-coupling restraining indeed shifts the χ₁-angle distribution towards values around − 125° (Fig. 7, upper right panel), but also makes the χ₁-angle repeatedly move over all angle values. Figure 12 shows the β₂ and β₃ local-elevation potential-energy terms (dashed and dotted lines, respectively) and their sum (solid line) built-up in the ³J-coupling restraining simulation. Major build-up is observed for χ₁-angle values around − 165°, − 70° and + 65°. The first two keep the χ₁-angle around − 125°. Yet, excursions to other χ₁-angle values seem required to push the average ³J_Hα-Hβ2- and ³J_Hα-Hβ3-couplings towards the corresponding experimental values of 5.6 Hz and 6.6 Hz, respectively. We note that the solvent accessibility of the side chain of Asp 101 in the 2VB1 X-ray structure is 42%.

The side-chain χ₁-angle of Asn 106 in the unrestrained simulation initially covers values around − 70° and − 170° (Fig. 8, upper left panel) yielding ³J_Hα-Hβ2- and ³J_Hα-Hβ3-couplings both fluctuating between 2 and 13 Hz. After about 4 ns, the χ₁-angle stabilizes around + 60° yielding an average ³J_Hα-Hβ2-coupling of 4.7 Hz and an average ³J_Hα-Hβ3-coupling of 3.9 Hz, the former much lower than the experimental value of 10.5 Hz, the latter close to the experimental value of 3.6 Hz. In the Karplus curve for the ³J_Hα-Hβ2-coupling (solid line in the right panel of Fig. 2) the experimental value of 10.5 Hz corresponds to two χ₁-angle values of about − 89° and − 31°. In the Karplus curve for the ³J_Hα-Hβ3-coupling (dashed line in the right panel of Fig. 2) the experimental value of 3.6 Hz corresponds to four χ₁-angle values of about − 111°, − 58°, + 58° and + 111°. The Karplus curves suggest a χ₁-angle distribution dominantly around − 75° with much averaging in the range [− 100°, − 45°]. Time-averaging local-elevation ³J-coupling restraining indeed shifts the χ₁-angle distribution towards values around between − 100° and − 45° (Fig. 8, upper right panel). Figure 13 shows the β₂ and β₃ local-elevation potential-energy terms (dashed and dotted lines, respectively) and their sum (solid line) built-up in the ³J-coupling restraining simulation. Major build-up is observed for χ₁-angle values between 0° and 90° and around − 170°. The small build-up at − 65° is due to the β₂ restraint, keeping the ³J_Hα-Hβ2-coupling below 12 Hz. The averaging between the two local-elevation energy minima in the range [− 100°, − 25°] results in average ³J_Hα-Hβ2- and ³J_Hα-Hβ3-couplings of 11.1 Hz and 3.1 Hz, close to the experimental values of 10.5 Hz and 3.6 Hz, respectively. We note that the solvent accessibility of the side chain of Asn 106 in the 2VB1 X-ray structure is 76%.

Use of measured ³ J-coupling values in force-field validation

Figures 4, 5, 6, 7 and 8 show examples of dihedral angles for which the ³J-couplings in the unrestrained simulations do not agree with the experimental values. The local-elevation potential-energy term used in the restraining simulations changes the dihedral-angle distribution such that the average ³J-coupling matches experiment. Thus such a potential-energy term contains information on a possible modification of the dihedral-angle term of the force field used. If a consistent picture of possible modifications would emerge from simulations of a collection of proteins, such a modification could be incorporated into the force field.

Conclusions

Although ³J-coupling constants are relatively easy to obtain from NMR experiments, their use in structure determination of proteins has been rather limited due to different aspects of the measurement and the relation between a ³J-coupling and molecular structure (van Gunsteren et al. 2016). The Karplus relation between structure and ³J-coupling value is multi-valued, with up to four torsional angle values mapping to a single ³J-coupling value. In addition, intermediate ³J-coupling values (4–8 Hz) are sensitive to the experimental averaging period, which is rather long. This is in contrast to some other NMR measurable quantities such as NOE intensities. The difficulty of accounting for conformational averaging and for the multi-valued function of a torsional angle in terms of a ³J-coupling has severely hampered the use of ³J-couplings in protein structure determination or refinement. However, with the advent of time-averaging local-elevation restraining MD simulation (Christen et al. 2007; Smith et al. 2016) both problems could be solved. Time-averaging can be taken into account in a simulation (Torda et al. 1989) and a molecular conformation can be induced to escape from an (incorrect) local minima due to restraining based on the Karplus relation by use of the local-elevation algorithm (Huber et al. 1994). This makes a comprehensive use of ³J-coupling data in structure determination possible (Smith et al. 2016).

The application of ³J-coupling time-averaging local-elevation restraining to the protein HEWL shows that this technique is able to produce a conformational ensemble compatible with the experimental ³J-coupling data. Analysis of the conformations underlying the ³J-couplings shows that conformational averaging plays an essential role in a number of cases and that finding an alternative minimum energy conformation for backbone φ or side-chain χ₁ angles is also of importance.

The ³J-coupling time-averaging local-elevation restraining does improve the agreement with 1630 NOE atom–atom distance bounds for HEWL, but only if all 213 backbone and side-chain ³J-coupling restraints are applied. It has no significant effect upon the agreement of the conformational ensemble with values of 121 backbone S²_NH and 79 side-chain S²_CH and S²_NH order parameters for HEWL. This one would more or less expect, considering the different degrees of freedom involved.

The results for the backbone ³J_HNHα-couplings based on the parametrisation of the Karplus relation due to Pardi et al. (1984) show that this parametrisation is not capable of reproducing large ³J_HNHα-coupling values, due to a maximum of 9.7 Hz of this Karplus relation. One could use alternative parametrisations, such as the one of Wang and Bax (1996) with a maximum of 10 Hz or the one of Brüschweiler and Case (1994) with a maximum of 11 Hz. However, in the present case of HEWL, no significant change of the agreement with experiment is observed by using the latter Karplus relation in the analysis.

Of particular interest from the work reported here, is the description of the behaviour of the side-chains with a high level of conformational mobility. Examples such as the data for the side-chains of Asp 101 and Asn 106 shown in Figs. 7 and 8 indicate that any hydrogen bonds involving flexible side-chains, which are present in crystal structures of the protein, may be very fluctuating or completely absent in solution. Indeed, the simulation results show just how conformationally disordered the surface of the protein really is. This needs to be kept in mind when X-ray structures are being used in areas such as drug design or to help with the interpretation of data from receptor binding or mutational studies.