What Factors Inﬂuence the Metal−Proton Spin−Spin Coupling Constants in Mercury- and Cadmium-Substutited Rubredoxin? Małgorzata Kauch and Magdalena Pecul* Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland S Supporting Information *

ABSTRACT: The indirect metal−proton spin−spin coupling constants between protons in cysteine groups and the mercury or cadmium nucleus have been calculated for a small model of Me−rubredoxin complex (Me = Cd, Hg) by means of density functional theory with zeroth-order regular approximation Hamiltonian (DFT-ZORA). The calculated spin−spin coupling constants, in spite of the moderate size of the model system, are in good agreement with the values measured in NMR experiment, which are in the 0.29− 0.56 Hz range for the Cd complex and in the 0.57−2.20 Hz range for the Hg complex. The robustness of the chosen method has been veriﬁed by calculations with a number of diﬀerent exchange-correlation functionals and basis sets. Additionally, it has been shown that the short- and long-distance metal−proton coupling constants are aﬀected mainly by the values of the metal−proton distance and the H−N−C−C dihedral angle.

1. INTRODUCTION Indirect nuclear spin−spin coupling constants derived from nuclear magnetic resonance spectra are valuable probes of molecular structure because of their extreme sensitivity to conformational changes and relative simplicity of their dependence on molecular geometry (in some cases, they can be expressed as a function of one structural variable, as exempliﬁed in the Karplus relation1,2). The scope of application of the spin−spin coupling constants for structural studies has been extended by measurements of so-called through-space indirect nuclear spin−spin coupling constants,3−9 which allow the probing of supramolecular structure and intermolecular interactions (primarily hydrogen bonds). Rapid progress in this area of NMR has been made possible by the fact that, parallel to improvements in the experimental techniques of NMR measurements of small spin−spin coupling constants, considerable advances have been made in ab initio calculations of spin−spin coupling constants.10−19 Thus, through-space indirect spin−spin coupling constants have become an area of fruitful cooperation between computational and experimental NMR spectroscopies.20−29 Many such joint investigations have been carried out in particular for proteins.20−25 The calculations of the spin−spin coupling constants in proteins reported in the literature are almost entirely limited to the spin−spin coupling constants of light nuclei (proton, carbon, or nitrogen). There are some examples of ab initio calculations of heavy metal coupling constants in biologically relevant systems, but they are focused on large, short-range coupling constants (such as those between metal nucleus intercalated in DNA and nucleobase nitrogen nucleus30,31). The relative scarcity of the calculations of the heavy metal spin−spin coupling constants in large molecular systems in general is a consequence of the fact that they require accounting for relativistic eﬀects by means of explicitly © XXXX American Chemical Society

relativistic Hamiltonians. (Because spin−spin coupling constants are connected with electron density in the core region, simulations using relativistic eﬀective core potentials on heavy atoms are useless29 for this purpose.) Such calculations are resource-consuming and still not standard, although the computational codes enabling them32−34 are getting more popular nowadays. Experimental measurements of small through-space spin− spin coupling constants started in the early 1990s. Among the ﬁrst works reporting on such measurements in proteins were reports of the coupling constants between protons of cysteine moieties surrounding the metal ion and 113Cd or 199Hg nuclei in rubredoxin substituted with cadmium or mercury cations, respectively.3,35 At the time the experiments were carried out, there were no tools available for calculations of the spin−spin coupling constants, especially in such a large system involving a heavy nucleus (and therefore requiring accounting for relativistic eﬀects). Nowadays, suitable methods have been developed36−38 and implemented as computational codes.32−34 The calculations of spin−spin coupling constants for a heavy metal ion interacting with a molecular system mimicking a protein environment, although challenging because of the size of the system, are feasible. We have therefore carried out such calculations for a model of rubredoxin−metal complexes to understand the origin of relatively large values of these couplings and the diﬀerences between them. We have also simulated the inﬂuence of conformational changes on them to identify the geometry parameters that determine the values of the couplings. Received: February 23, 2014 Revised: April 27, 2014

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correction proposed in 1998 by Zhang−Wang and the correlation term presented in 1996 by Perdew−Burke− Ernzerhof)60,61 for the generalized gradient approximation (GGA) part have been used. 2.2. Calculations of Spin−Spin Coupling Constants. The calculations of spin−spin coupling constants have been carried out using one-component and two-component (including the spin−orbit coupling term) ZORA Hamiltonians implemented in the ADF 2010.01 program package.32,56,57 Three diﬀerent all-electron basis sets (TZP, a core double-ζ, valence triple-ζ, polarized basis set; TZ2P, a core double-ζ, valence triple-ζ, doubly polarized basis; and QZ4P, quadruple-ζ basis set) have been tested. For comparison, also nonrelativistic calculations have been carried out using the same basis sets. Several exchange-correlation functionals have been chosen to check how their selection inﬂuences the value of the calculated coupling constants. Two popular hybrid funtionals have been tested: Becke, three-parameter, Lee−Yang−Parr (B3LYP)62 and hybrid form of Perdew−Burke−Ernzerhof exchangecorrelation functional (PBE0).63−65 The BVP86 (modiﬁed BP86 with VWN parametrization for LDA, the gradient correction proposed in 1988 by Becke and the correlation term presented in 1986 by Perdew);66,67 PW91 (corrections proposed in 1991 by Perdew−Wang);68 KT1 and KT2 (functionals of Keal and Tozer);69 revPBE;60,61 and BEE functionals (the Bayesian error estimation)70 have also been applied for the calculations. Because ADF does not allow for the LDA functional to be used in the perturbative part of the calculations, the VWN parametrization has been used instead in the case of the KT2 and BEE functionals. For the calculations of the eﬀects of geometry distortion, only revPBE with the TZ2P basis set has been used.

The calculations have been carried out by means of density functional theory (DFT) with zeroth-order regular approximation (ZORA) Hamiltonian.36,39−44 The usefulness of this approach has been proven many times for diﬀerent types of nuclear spin−spin coupling constants.45−54 In particular, it has been shown that the coupling constants obtained using the ZORA Hamiltonian with spin−orbit coupling term are very close to those calculated with four-component Dirac−Coulomb Hamiltonian,54 while the calculations are less time-consuming. Thus, ZORA-DFT seems to be the best tool for calculations of spin−spin coupling constants in relatively large systems, such as the models of the complexes of rubredoxin with cadmium and mercury under consideration.

2. COMPUTATIONAL DETAILS 2.1. Model Structure and Geometry Optimization. As a model system, the structure consisting of four cysteine molecules and a metal cation was chosen (Figure 1).

3. RESULTS 3.1. Geometry of the Systems. The geometry parameters of rubredoxin, and in particular of the metal binding site, optimized for the model complexes with mercury, cadmium, and zinc are in quite good agreement with the crystallographic data for the nonsubstituted rubredoxin structures71 and with the parameters obtained for the zinc−rubredoxin complex from the NMR data.55 (For more detailed information about selected geometry parameters, see Tables S1−S4 in Supporting Information, and for comparison between crystallographic and NMR data, see ref 72.) Apparently, intercalation with a metal cation does not cause drastic changes in the protein structure. The metal−ligand distances (deﬁned as the distance between the metal cation and the nearest ligand atom) for the Hg and Cd complexes are close to each other and on the average larger than that for the Zn complex by about 0.15 Å (max. 0.32 Å) with geometry optimized in the same fashion as the Cd and Hg complexes. It is worth nothing that the Zn− ligand distance in the optimized complex is on the average overestimated by about 0.15 Å in comparison with the NMR data, which may be an indication of a similar overestimation in the Cd and Hg complexes. In all metal-substituted rubredoxines there are two types of cysteine ligands, located opposite to each other in the rubredoxin molecule. For one type, the distance between H and Me is large (about 6.0 Å) and the dihedral angles H−N− C−C, H−C−C−S, H−C−S−Me, and H−S−Me−C are about 300°, 160°, 200°, and 180°, respectively. For the other type of the ligand, the H−Me distance is much smaller (about 3.0 Å) and the dihedral angles are about 330°, 60°, 300°, and 160°,

Figure 1. Model system of metal-substituted rubredoxin. The protons to which the couplings are calculated are printed in dark green.

The model replacing peptide groups by free amino acids may seem rather primitive, but as it turned out, it is suﬃcient for the purpose. We have also performed trial calculations with larger systems (metal cation and two sequences: Cys-Ile-Pro-Cys and Cys-Lys-Ile-Cys), but they have been found unsuitable for DFT calculations because of a very small highest occupied molecular orbital−lowest unoccupied molecular orbital (HOMO− LUMO) gap (about 0.1 eV). As far as we know, there are no crystallographic structures of rubredoxin intercalated with Cd or Hg available; thus, the model systems had to be constructed by means of computational methods. Four cysteine molecules and a metal atom were cut from the structure of rubredoxin−zinc complex (determined by Blake and co-workers on the basis of solution NMR experiment55), and the metal ion (zinc cation) was substituted by mercury or cadmium cation; the “free ends” of cysteine molecules have been replaced by hydrogen atoms. For those two structures, the geometry optimization has been carried out by means of the Amsterdam Density Functional (ADF) 2010.01 program package32,56,57 using DFT method with the scalar ZORA Hamiltonian39,40,42 and the TZP basis set (Slatertype orbitals)58 with eﬀective core potentials (Hg 4f, Cd 4p, S 2p, O 1s, N 1s, C 1s). The Vosko−Wilk−Nusair (VWN) parametrization59 of local-density approximation (LDA) and the revPBE functional (with the revised PBE exchange B

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Table 1. Comparison of the Indirect Spin−Spin Coupling Constants 1K(Hg,Hn) (×1019 kg m−2 s−2 A−2) and 1J(Hg,Hn) (Hz) Calculated Using the TZ2P Basis Set and the revPBE Functionala n

1

K(Hg,H )

1

J(Hg,Hn)

experimental Hn−Hg distance (Å) a

nonrel sc-ZORA so-ZORA nonrel sc-ZORA so-ZORA 1 J(Hg,Hn)

H2

H7

H12

H17

−0.165 −0.257 −0.260 −0.358 −0.558 −0.565 − 6.00

−0.099 −0.932 −1.096 −0.216 −2.027 −2.385 (−)2.20 ± 0.1 3.30

−0.231 −0.514 −0.534 −0.503 −1.119 −1.163 (−)0.57 ± 0.1 6.22

0.161 −1.240 −1.432 0.350 −2.697 −3.116 (−)1.90 ± 0.2 3.22

The signs of the experimental values are assigned on the basis of the calculations.

Table 2. Comparison of the Indirect Spin−Spin Coupling Constants 1K(Cd,Hn) (×1019 kg m−2 s−2 A−2) and 1J(Cd,Hn) (Hz) Calculated Using the TZ2P Basis Set and the revPBE Functionala n

1

K(Cd,H )

1

J(Cd,Hn)

experimental Hn−Cd distance (Å) a

nonrel sc-ZORA so-ZORA nonrel sc-ZORA so-ZORA 1 J(Cd,Hn)

H2

H7

H12

H17

−0.093 −0.114 −0.113 0.248 0.304 0.302 − 5.98

−0.105 −0.228 −0.245 0.281 0.609 0.655 (+)0.56 ± 0.03 3.29

−0.111 −0.148 −0.150 0.296 0.396 0.402 (+)0.30 ± 0.03 6.20

−0.070 −0.233 −0.252 0.186 0.622 0.674 (+)0.56 ± 0.03 3.24

The signs of the experimental values are assigned on the basis of the calculations.

Table 3. Comparison of the Indirect Spin−Spin Coupling Constants 1J(Hg,Hn) (Hz) Calculated Using the revPBE Functional and the Scalar ZORA Approach with Diﬀerent All-Electron Basis Sets J(Hg,Hn)

H2

H7

H12

H17

TZP TZ2P QZ4P experimental

−0.593 −0.558 −0.563 −

−1.864 −2.027 −3.386 (−)2.20 ± 0.1

−1.169 −1.119 −1.100 (−)0.57 ± 0.1

−2.439 −2.697 −3.973 (−)1.90 ± 0.2

1

which the diﬀerences are larger). The sign of the calculated coupling is independent of the level of theory, except for 1 J(Hg,H17), where the nonrelativistic value for 1J(Hg,H17) has a sign opposite that of the relativistic values. The reduced coupling constants (K) for mercury are as a rule signiﬁcantly larger than those for cadmium. This is entirely a relativistic eﬀect, because for the nonrelativistic values the diﬀerence is much smaller (and in the case of H7, the cadmium coupling is actually larger). The relativistic eﬀects are dominated by the scalar (spin-free) terms. The inclusion of scalar eﬀects inﬂuences mainly the Fermi contact term (the paramagnetic term changes very slightly). The importance of the spin−orbit coupling eﬀects depends on the proton−metal distance. The eﬀects are negligible for long-distance couplings, but they are substantial (especially for mercury couplings) when the proton−metal distance is below 4 Å. Taking into account the spin−orbit eﬀects increases mainly the pure Fermi contact contribution, unlike for example the case of halogen couplings, where the spin−orbit eﬀect is dominated by the cross-term between Fermi contact−spin−dipole and paramagnetic spin−orbit term.36,73 The spin−orbit contribution for cadmium is about 10 times smaller than that for mercury. The change in the Fermi contact contribution (by inclusion of spin−orbit eﬀects) is about 0.145 × 1019 kg m−2 s−2 A−2 for the mercurysubstituted system and about 0.014 × 1019 kg m−2 s−2 A−2 for

respectively. The coupling constants of interest are to protons H2 and H12 (belonging to one type of cysteine) and to protons H7 and H17 (belonging to the other type of cysteine). Considering their structural similarity, we expect the coupling constants of the H2 and H12 protons and of the H7 and H17 protons to exhibit similar behavior. 3.2. Spin−Spin Coupling Constants. 3.2.1. Relativistic and Nonrelativistic Results. Table 1 contains the comparison of the mercury coupling constants 1J(Hg,H) as computed using the scalar ZORA (denoted sc-ZORA), spin−orbit ZORA (denoted so-ZORA), and Schrödinger Hamiltonian (denoted nonrel). It also lists the experimental values (with the sign assigned on the basis of the calculations because it was not determined in the experiment). The relevant data for the cadmium−rubredoxin complex are shown in Table 2. To facilitate the comparison between cadmium and mercury couplings, we list also their reduced values. The calculated coupling constants match well the experimental data if the relativistic eﬀects are accounted for. The relativistic eﬀects dominate the total values of the mercury coupling constants: the nonrelativistic absolute values are usually about 2−10 times smaller than the values obtained using the relativistic methods and are further apart from experiment. In the case of cadmium couplings, the absolute relativistic values are as a rule about 1.5 times greater than the nonrelativistic values (except for 1J(Cd,H7) and 1J(Cd,H17), for C

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Table 4. Comparison of the Indirect Spin−Spin Coupling Constants 1J(Cd,Hn) (Hz) Calculated Using the revPBE Functional and the Scalar ZORA Approach with Diﬀerent All-Electron Basis Sets J(Cd,Hn)

H2

H7

H12

H17

TZP TZ2P QZ4P experimental

0.322 0.304 0.278 −

0.540 0.609 0.973 (+)0.56 ± 0.03

0.419 0.396 0.346 (+)0.30 ± 0.03

0.534 0.622 0.940 (+)0.56 ± 0.03

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Table 5. Comparison of the Indirect Spin−Spin Coupling Constants 1J(Hg,Hn) (Hz) Calculated Using the All-Electron TZ2P Basis Set, the Scalar ZORA Hamiltonian, and Diﬀerent Exchange-Correlation Functionals J(Hg,Hn)

H2

H7

H12

H17

KT1 KT2(+VWN) revPBE BEE(+VWN) BVP86 PW91 B3LYP PBE0 experimental

−0.719 −0.713 −0.558 −0.667 −0.653 −0.654 −0.863 −0.717 −

−1.359 −1.431 −2.027 −1.456 −1.547 −1.546 −1.839 −2.003 (−)2.20 ± 0.1

−1.353 −1.350 −1.119 −1.213 −1.208 −1.199 −1.268 −1.023 (−)0.57 ± 0.1

−2.240 −2.275 −2.697 −2.239 −2.377 −2.361 −2.528 −2.513 (−)1.90 ± 0.2

1

Table 6. Comparison of the Indirect Spin−Spin Coupling Constants 1J(Cd,Hn) (Hz) Calculated Using the All-Electron TZ2P Basis Set, the Scalar ZORA Hamiltonian, and Diﬀerent Exchange-Correlation Functionals J(Cd,Hn)

H2

H7

H12

H17

KT1 KT2(+VWN) revPBE BEE(+VWN) BVP86 PW91 B3LYP PBE0 experimental

0.320 0.319 0.304 0.305 0.306 0.306 0.403 0.315 −

0.584 0.569 0.609 0.550 0.596 0.589 0.359 0.397 (+)0.56 ± 0.03

0.452 0.453 0.396 0.410 0.418 0.413 0.456 0.359 (+)0.30 ± 0.03

0.696 0.672 0.622 0.635 0.696 0.683 0.424 0.466 (+)0.56 ± 0.03

1

for the complex with the mercury cation and in Table 6 for the complex with the cadmium cation. The results indicate that the choice of the functional does not aﬀect much the calculated couplings, so it seems that DFT is a robust method for such calculations. The sign of the calculated coupling is never aﬀected by the choice of the functional, and the magnitudes vary in a relatively small range. In particular, the results obtained using hybrid functionals (B3LYP and PBE0) do not diﬀer from each other much and are similar to those obtained by means of nonhybrid functionals; the diﬀerences are about 0.2 Hz for mercury and 0.1 Hz for cadmium. Even less variation is observed when the coupling constants calculated using diﬀerent nonhybrid functionals (KT1, KT2, revPBE, BEE, BVP86, and PW91) are compared. 3.3. Inﬂuence of the Geometry Changes. The metal− proton spin−spin coupling constants assume diﬀerent values for diﬀerent cysteine ligands. To elucidate which structural parameters are the main cause of such variation, we have carried out calculations with the system geometry distorted. The geometry deformations under study include the changes of metal−proton distance d(Me,H) and three diﬀerent types of internal rotation because two types of cysteine ligands diﬀer mostly in the d(Me,H) distance and the dihedral angles. The change of the metal−proton distance has been carried out by stretching of the S−Me bond between one of the cysteine molecules (one set of calculations for cysteine containing proton H12, and one for cysteine containing proton

the cadmium-substituted system; the changes in paramagnetic contribution are 5 times smaller for mercury and 3 times smaller for cadmium. The calculated couplings are small (especially the cadmium couplings), so a diﬀerent choice of a computational method may cause large relative changes in their values. Thus, we decided to investigate how the choice of a basis set and exchange-correlation functional inﬂuences the obtained results. 3.2.2. Choice of the Basis Set. The inﬂuence of the choice of the basis set on the calculated spin−spin coupling constants is presented in Table 3 for the system with the mercury cation and in Table 4 for the system with the cadmium cation. All employed basis sets are all-electron Slater-type basis sets. The change of the basis set does not aﬀect the signs of the calculated couplings. The extension of the basis set from TZ2P to QZ4P increases signiﬁcantly the absolute couplings values, moving them further apart from the experiment. Thus, good agreement of the calculated and experimental values for the TZ2P basis set results from error cancellation, a situation which is not infrequent in DFT calculations. Addition of polarization functions (extension from TZP to TZ2P) causes only a small modiﬁcation of the calculated couplings (less than 5.0%). The trends in the basis set eﬀects are the same for cadmium−proton and mercury−proton coupling constants. 3.2.3. Choice of the Exchange-Correlation Functional. The values of 1J(Me,Hn) coupling constants, as calculated using diﬀerent exchange-correlation functionals, are shown in Table 5 D

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Figure 2. Rotation of the −NH2 group in one of the four cysteine molecules (a), the rotation in cysteine around the C−C bond (b), and the cysteine around the S−Me bond (c).

Figure 3. Dependence of the 1K(Me,H) reduced coupling constant on the internuclear distance for two types of cysteine molecules: (a) 1K(Hg,Hn) and (b) 1K(Cd,Hn). The values calculated for the optimized structures are marked as squares.

Figure 4. Dependence of the reduced 1K(Me,H12) and 1K(Me,H17) coupling constants (in 1019 kg m−2 s−2 A−2) on the relevant H−N−C−C dihedral angle during rotation of the −NH2 group in the Hg complex (a) and the Cd complex (b).

H17). The rotations under study are depicted in Figure 2: the rotation of the −NH2 group (rotation around the N−C bond) in one of the four cysteine molecules (Figure 2a), the rotation in cysteine around the C−C bond (Figure 2b), and the rotation of the cysteine around the S−Me bond (Figure 2c). For completeness of this study, the rotation of −NH2 group in one of the other cysteines molecule was also investigated (analogous to the rotation from Figure 2a, but for the other type of the cysteine ligand). 3.3.1. Distance Dependence. The diagrams in Figure 3 presents the inﬂuence of the change of the distance between Hn and the metal nucleus on the metal−proton coupling constant for two types of cysteine molecules: mercury−rubredoxine (Figure 3a) and cadmium−rubredoxine (Figure 3b). For better comparison of the trends for cadmium and mercury complexes, the couplings are given as reduced values.

The dependence of the metal−proton coupling on the internuclear distance has the same characteristic exponentiallike shape for the short-range H17 coupling constants in both systems. This resembles the diagrams obtained for the light nuclei coupling constants transmitted through intermolecular interactions26,74,75 for short internuclear distances. The plot for the H12 coupling exhibits a shallow minimum, resembling the trends observed for example for proton−ﬂuorine coupling for large internuclear distances.26 Similar to the through-space couplings for the light nuclei, the dependence of the metal coupling constants under study on the internuclear distance is determined mostly by the Fermi contact term. 3.3.2. Rotation of the −NH2 Group. The diagrams of the dependence of metal−proton coupling constants on the H− N−C−C dihedral angle (i.e., the inﬂuence of the rotation of the −NH2 group) in two types of cysteine molecules are shown in Figure 4 for mercury−rubredoxin (Figure 4a) and cadmium− E

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Figure 5. Dependence of the reduced 1K(Me,H17) coupling constant (in 1019 kg m−2 s−2 A−2) on the relevant dihedral angle N−C−C−S during the rotation of cysteine around the C−C bond in the Hg complex (a) and Cd complex (b).

Figure 6. Dependence of the reduced 1K(Me,H17) coupling constant (in 1019 kg m−2 s−2 A−2) on the relevant dihedral angle C−S−Me−S during the rotation of the cysteine around the S−Me bond in the Hg complex (a) and Cd complex (b).

4. SUMMARY AND CONCLUSIONS The indirect metal−proton nuclear spin−spin coupling constants have been calculated for complexes of mercury and cadmium with four cysteine molecules, mimicking the metal binding site in rubredoxin. The calculations have been carried out using density functional theory with two-component zeroth-order regular approximation Hamiltonian. The main conclusions can be summarized as follows. In spite of the small size of the model systems, the experimental values of 1J(113Cd,1H) and 1J(199Hg,1H) coupling constants have been well-reproduced by the calculations. For two short-distance mercury couplings, the experimental values are (−)2.20 ± 0.1 Hz and (−)1.90 ± 0.2 Hz (the signs are assigned on the basis of the calculations), whereas the calculated values are −2.07 and −2.70 Hz. For one of the long-distance coupling constants, the experimental value was measured as (−)0.57 ± 0.1 Hz and is about two times smaller than the calculated value of −1.12 Hz. For the fourth coupling, the experimental value is not available, but the calculated value is −0.56 Hz. Thus, assuming a similar behavior as for the other long-distance coupling constant, its actual value can be tentatively assigned as being between −0.2 and −0.3 Hz. The 1 113 J( Cd,1H) coupling constants are smaller than their mercury counterparts and better reproduced by the applied computational method. The diﬀerence in the corresponding reduced mercury and cadmium coupling constants can be attributed mainly to purely relativistic eﬀects because the nonrelativistic values are much closer to each other.

rubredoxin (Figure 4b) complex. As before, the couplings are depicted as reduced values to facilitate the comparison. The plots for both systems for corresponding protons have almost the same shape, resembling a typical Karplus curve.1,2 The amplitudes of the changes of the coupling constants with the dihedral angle are large (4.0 × 1019 kg m−2 s−2 A−2 for Hg complex and 0.9 × 1019 kg m−2 s−2 A−2 for Cd complex), and the couplings change their signs several times during the rotation. As expected, the coupling calculated for the proton H7 ﬁts the trend determined for H17 and that for the proton H2 matches the trends determined for H12 (see Figure 4a,b). It seems that the H−N−C−C dihedral angle is, apart from the internuclear distance, the main factor determining the values of the metal−proton coupling constants. We have also carried out the calculations of the coupling constants with cysteine ligands rotated around the C−C bonds (Figure 5a for mercury and 5b for cadmium) and S−Me bonds (Figure 6a for mercury and Figure 6b for cadmium). The changes span a smaller range (in the case of rotation around the S−Me bonds, the couplings do not change signs) and are much less regular. The dependence on the dihedral angle is similar for Cd and Hg coupling in the case of the rotation around the C− C bond, but it is diﬀerent for the rotation around the S−Me bond. All changes of the coupling constants with respect to geometry parameters are dominated by the Fermi contact term. F

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The relativistic eﬀects are dominated by the scalar (spin-free) eﬀects. The spin−orbit terms are negligible for long-distance metal−proton coupling constants, but they increase with the decreasing internuclear distance. The choice of the exchangecorrelation functional does not aﬀect much the calculated couplings. The basis set eﬀects are larger, and the good agreement with experiment obtained when the TZ2P basis set is used seems to be to some extent a result of error cancellation. The inﬂuence of diﬀerent geometry modiﬁcations (changes of the metal−proton distance and several dihedral angles) on the metal−proton coupling constants have been investigated to elucidate the origin of diﬀerent values of the couplings between metal nucleus and protons from diﬀerent cysteine types. The main factors determining the values of the coupling constants under study seem to be the internuclear distance (exponential dependence for short distances and minimum at about 6 Å for long distances) and the H−N−C−C dihedral angle. Because the spin−spin coupling constants calculated for the optimized model structure are overestimated in comparison with the experimental values, it seems likely that the metal−proton distance is too short. This overestimation of the calculated coupling will be at least partially negated if anharmonic vibrational averaging is carried out. Because at present it is outside of our computational abilities, we have carried a crude estimation of this eﬀect, assuming the elongation of the metal− proton distance by 0.1 Å. This would cause a decrease of the absolute values of the calculated short-distance couplings by 0.631 Hz in the case of 1J(Hg,H) and 0.146 Hz for 1J(Cd,H), bringing them somewhat closer to the experimental values. The conclusion from the changes of sign of the coupling observed during the internal rotation is that the sign of each metal− proton coupling must be assigned individually and not extrapolated from other measurements because it is very sensitive to structural changes.

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REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

Selected internal coordinates for cysteine molecules in the zinc−rubredoxin based on structures from solution NMR and for cysteine molecules in the zinc-, mercury-, and cadmiumsubstituted model systems (Tables S1−S4); xyz coordinates of the optimized complexes (Tables S5 and S6). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Phone: +48 22 822-02-11, ext. 501. Fax: +48 22 822-59-96. Notes

The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS This work has received support from the Polish Ministry of Science and Higher Education via the National Science Centre 2011/01/B/ST4/02984 (N N204 148565) grant, and from the Wrocław Centre for Networking and Supercomputing through a grant of computer time. The project has been carried out with the use of CePT infrastructure ﬁnanced by the European Union−The European Regional Development Fund within the Operational Programme “Innovative Economy” for 2007-2013. G

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