Strategy for Modeling the Electrostatic Responses of the

Article pubs.acs.org/JPCB

Cite This: J. Phys. Chem. B XXXX, XXX, XXX−XXX

Strategy for Modeling the Electrostatic Responses of the Spectroscopic Properties of Proteins Hajime Torii* Department of Chemistry, Faculty of Education and Department of Optoelectronics and Nanostructure Science, Graduate School of Science and Technology, Shizuoka University, 836 Ohya, Shizuoka 422-8529, Japan S Supporting Information *

ABSTRACT: For better understanding and more efficient use of the spectroscopic probes (vibrational and NMR) of the local electrostatic situations inside proteins, appropriate modeling of the properties of those probes is essential. The present study is devoted to examining the strategy for constructing such models. A more well-founded derivation than the ones in previous studies is given in constructing the models. Theoretical analyses are conducted on two representative example cases related to proteins, i.e., the peptide group of the main chains and the CO and NO ligands to the Fe2+ ion of heme, with careful treatment of the behavior of electrons in the electrostatic responses and with verification of consistency with observable quantities. It is shown that, for the stretching frequencies and NMR chemical shifts, it is possible to construct reasonable electrostatic interaction models that encompass the situations of hydration and uniform electric field environment and thus are applicable also to the cases of nonuniform electrostatic situations, which are highly expected for inside of proteins.

1. INTRODUCTION It is widely recognized that electrostatic interactions play an important role in the structural formation and functional dynamics of proteins.1−11 There are a few spectroscopic probes (vibrational and NMR) that are expected to be useful measures of the local electrostatic situations inside protein molecules.12−27 The responses of vibrational modes to electrostatic situations are also important for correct interpretation of the measured vibrational spectroscopic signals in both the frequency and time domains, including two-dimensional infrared spectra.28−36 However, the quantitative aspects of these responses and the relations among different spectroscopic measures are rather controversial. For example, for the relation between the CN stretching frequency and the 13C NMR chemical shift of the nitrile group, it was discussed that the high-frequency deviation observed for the hydrogen-bonding cases compared with the dipolar solvation cases arises from a nonelectrostatic factor.37 However, in a recent theoretical study,38 we have successfully constructed quantitative electrostatic interaction models that encompass both the hydrogenbonding cases and the dipolar solvation cases, showing that the nonelectrostatic factor for the high-frequency deviation as previously thought is actually arising from the spatially inhomogeneous (nonuniform) nature of the electrostatic situations in the hydrogen-bonding cases. In the present study, the extent of applicability of the strategy for constructing electrostatic interaction models developed in our recent studies38,39 is examined by conducting theoretical analyses (with further theoretical extension) on two repre© XXXX American Chemical Society

sentative example cases related to proteins, i.e., the peptide group of the main chains and the CO and NO ligands to the Fe2+ ion of heme. With regard to the former, there are a few characteristic vibrational modes known for the peptide group,40,41 and among them, the amide I mode is widely utilized as a measure of the secondary structures of the peptide chains and the solvation of the individual peptide groups.28−36,42−56 We have shown in a recent study39 that the apparently peculiar angular configuration dependence in the effect of the CO hydration on the amide I frequency is reasonably well reproduced by a carefully constructed electrostatic interaction model. In the present study, it is extended to include the effect of the N−H hydration and dipolar solvation as well as the 13C and 17O NMR chemical shifts. A more wellfounded derivation than the ones in previous studies28−36,39,57 is given in constructing the models. With regard to the CO and NO ligands to the Fe2+ ion of heme, it has been discussed that they are useful measures of the electrostatic situations around them.14−18,22,24,58−60 The present study is devoted to elucidating the extent that they are useful for this purpose and to constructing electrostatic interaction models that can quantitatively convert the measured spectroscopic signals to the quantities related to electrostatic situations. Received: November 1, 2017 Revised: December 1, 2017 Published: December 1, 2017 A

DOI: 10.1021/acs.jpcb.7b10791 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B

Figure 1. Structures of (a) N-methylacetamide-d1 (NMA-d1), (b) FeII···porphine···imidazole···carbon monoxide [FeII(P)(ImH)(CO)] complex, and (c) FeII···porphine···imidazole···nitric oxide [FeII(P)(ImH)(NO)] complex. The Cartesian coordinate system employed for these species is shown on the lower side.

2. COMPUTATIONAL PROCEDURE 2.1. General Aspects. The calculations were carried out for (1) N-methylacetamide-d1 (NMA-d1, Figure 1a, see also Note 1 in the Supporting Information) in various hydrogen-bonding and other solvation situations and (2) the FeII···porphine··· imidazole···carbon monoxide and FeII···porphine···imidazole··· nitric oxide complexes [abbreviated as FeII(P)(ImH)(CO) and FeII(P)(ImH)(NO), shown in Figure 1b,c, respectively] in various situations of hydrogen-bonding solvation and uniform electric field environment. With regard to NMA-d1 and its complexes, theoretical calculations and analyses were carried out for the following species: (i) an isolated NMA-d1 molecule, (ii) NMA-d1 in uniform electric field in the −z direction (with the z axis being taken along the CO bond with +z in the C → O direction, as shown in Figure 1a), (iii) the NMA-d1···D2O 1:1 complexes with various angular configurations of the hydrogen bond on the CO or N−D group of NMA-d1,39,61 (iv) the NMA-d1··· nD2O (n = 1 and 2) optimized complexes with all combinations of hydrogen bonding out of the three hydrogen-bonding sites on NMA-d1 (two on CO and one on N−D),62 (v) the NMA-d1···3D2O and NMA-d1···3TFE-d1 optimized complexes (TFE = 2,2,2-trifluoroethanol) occupying all of the three hydrogen-bonding sites on NMA-d1, (vi) the NMA-d1···acetone and NMA-d1···dimethyl sulfoxide (DMSO) optimized complexes with a hydrogen bond on N−D, (vii) the above two types of complexes (species v and vi) in dielectric medium modeling the respective solvent surrounding the complex. The density functional theory (DFT) calculations of these species were done at the B3LYP/6-31+G(2df,p) level using the Gaussian 03 and 09 programs.63,64 To support the discussion, the calculations were also carried out for the (viii) NMA-d1··· nBrF (n = 1 and 2) optimized halogen-bonding complexes65 and (ix) NMA-d1···CS2 and NMA-d1···CCl4 1:1 complexes with the solvent molecule interacting with CO of NMA-d1 (similarly to the halogen-bonding complexes). The DFT calculations for species viii and ix (and for an isolated NMAd1 molecule as a reference for them) were done with the same 6-31+G(2df,p) basis set but with the ωB97XD functional, which is less popular in analyzing vibrational properties but is a long-range and dispersion-corrected functional and is demonstrated in a previous study66 to show good performance for halogen-bonded complexes. The analyses after all of these calculations were carried out with our original programs. To focus the analysis on the responses of spectroscopic properties

to (local) electrostatic situations, the complexes with significant solute−solvent vibrational mixing (such as the complex with urea and the NMA-d1 oligomers) were not included, although the NMA-d1 oligomers are also interesting target systems for studying the vibrational couplings between peptide groups through hydrogen bonds.28,67−75 For all of the species except for iii, the structures were first fully optimized in a usual sense and then the amide I mode frequencies and the 13C and 17O NMR chemical shifts of CO of the peptide group were calculated. To obtain NMR chemical shifts, the NMR isotropic shielding tensors were calculated with the gauge-independent atomic orbital method,76 and the tetramethylsilane molecule and liquid water were adopted as the references for the 13C and 17O NMR chemical shifts, respectively. For the latter, the NMR isotropic shielding tensor calculated for an isolated water molecule and the observed gas− liquid difference (−36.1 ppm) of the 17O NMR chemical shift77 were used to estimate the reference value of liquid water. For species ii, six values of uniform electric field from 0.002 to 0.012 Eh a0−1 e−1 and four values from 0.016 to 0.028 Eh a0−1 e−1 were considered. For species vii, the integral equation formalism polarizable continuum model78 was used to represent the dielectric medium. For species iii, various angular configurations were derived by optimizing the structures with θ and φ (of the spherical polar coordinate system defined around CO or N−D of NMA-d1, with the O or D atom being taken as the origin) of the hydrogen bond being fixed at specified values. For the CO··· D hydrogen-bonded species,39 these values were (a) θ (180° minus the CO···D angle) ≅ 0°, standing for a linear hydrogen bond, (b) φ (N−CO···D dihedral angle) taken with the interval of 30° in the range of 0° ≤ φ ≤ 180° for θ = 15, 30, and 45°, and (c) φ taken with the interval of 15° in the range of 0° ≤ φ ≤ 180° for θ = 60 and 75° and 45° ≤ φ ≤ 180° for θ = 90°. For the N−D···O hydrogen-bonded species,61 they were (a) θ (180° minus the N−D···O angle) = 0°, (b) φ (C− N−D···O dihedral angle) taken with the interval of 30° in the range of 0° ≤ φ ≤ 180° for θ = 15° and 30° and 30° ≤ φ ≤ 180° for θ = 45°, supplemented by (θ, φ) = (45°, 165°), and (c) φ taken with the interval of 15° in the range of 45° ≤ φ ≤ 180° for θ = 60°, 60° ≤ φ ≤ 150° for θ = 75°, and 90° ≤ φ ≤ 135° for θ = 90°, respectively. [For the angular configurations out of the ranges of the above sets (for example, 0° ≤ φ ≤ 30° for θ = 90° of the CO···D species), optimized hydrogenbonded structures could not be obtained even though we tried B


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Figure 2. Plots of the (a) 13C NMR chemical shift (of the CO group), (b, d) 17O NMR chemical shift, and (c, e, f) amide I frequency against the (a−c) CO bond length, (d, e) 13C NMR chemical shift, and (f) 17O NMR chemical shift calculated for an isolated NMA-d1 molecule (black open circle), NMA-d1 in uniform electric fields in the −z direction (with the z axis being taken as shown in Figure 1, dark red squares), the NMA-d1···D2O 1:1 complexes with various angular configurations of the hydrogen bond on the CO (small black filled circles) or N−D (green +’s) group of NMA-d1, the NMA-d1···nD2O (n = 1 and 2) optimized complexes with all combinations of hydrogen bonding out of the three hydrogen-bonding sites on NMA-d1 (blue ×’s), the NMA-d1···3D2O and NMA-d1···3TFE-d1 optimized complexes (blue and red open circles, respectively) occupying all of the three hydrogen-bonding sites on NMA-d1, and the NMA-d1···acetone and NMA-d1···DMSO optimized complexes with a hydrogen bond on N−D (dark yellow and green open triangles, respectively). For the NMA-d1···3D2O, NMA-d1···3TFE-d1, NMA-d1···acetone, and NMA-d1··· DMSO optimized complexes, the values calculated for the complexes in dielectric media modeling respective solvent are also shown with filled symbols.

geometry optimization.39,61] Then, the amide I frequency and the 13C and 17O NMR chemical shifts of the CO of the peptide group were calculated as described above. Here, it is supposed that constraints on softer modes [the bending modes of CO···D and N−D···O and the torsion modes of N−C O···D and C−N−D···O] do not essentially have any harmful effect on the properties of a harder mode (amide I).38,39,61,65 In addition to these, the dependence of the amide I frequency and the 13C and 17O NMR chemical shifts on the hydrogen-bond distance was also examined for some typical angular configurations, at θ ≅ 0° and at (θ, φ) = (45°, 0°), (60°, 90°), and (60°, 180°) for the CO···D hydrogen-bonded species and at (θ, φ) = (15°, 0°), (60°, 90°), and (45°, 180°) for the N−D···O hydrogen-bonded species, to support the discussion on the quality of the electrostatic interaction model. As described below in Section 3.1, the electrostatic situations of NMA-d1 in species ii and iii were employed to construct electrostatic interaction models. The electrostatic potentials and electric fields on the atomic sites of NMA-d1 were evaluated by treating the NMA-d1 molecule as a set of ghost atoms (i.e., the atoms in the NMA-d1 molecule were removed,

but the basis sets of all of those atoms were retained). In constructing electrostatic interaction models, the DFTcalculated vibrational frequencies were scaled by 0.9860.38,39,65 With regard to the FeII(P)(ImH)(CO) and FeII(P)(ImH)(NO) complexes (Figure 1b,c), calculations and analyses were carried out for (i) the isolated complexes, (ii) the complexes in uniform electric field, (iii) the complexes with the O atom of the ligand being hydrogen-bonded to a water molecule [FeII(P)(ImH)(XO)···H2O, X = C and N] in various angular configurations, and (iv) the FeII(P)(ImH)(XO)···H2O and FeII(P)(ImH)(XO)···nCH3OH (n = 1 and 2) optimized complexes. For species ii, six values of uniform electric field from 0.002 to 0.012 Eh a0−1 e−1 were considered. The direction of the field was set as −z (with the z axis being taken along the CO bond with +z in the C → O direction, as shown in Figure 1b) for FeII(P)(ImH)(CO), and along the NO bond or along the Fe···N bond for FeII(P)(ImH)(NO). For species iii, various angular configurations were derived by optimizing the structures with θ and φ (of the spherical polar coordinate system defined around CO or NO of the ligand, with the O atom being taken as the origin) of the hydrogen bond being C


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The Journal of Physical Chemistry B fixed at specified values. In the case of FeII(P)(ImH)(CO)··· H2O, because the structure is sufficiently symmetric, only one value of φ [N (porphine)···CO···H (water) dihedral angle] was considered for θ (180° minus the CO···H angle) = 0, 30, 45, 50, 55, 60, 70, and 80°. In the case of FeII(P)(ImH)(NO)··· H2O, because the Fe···NO angle is bent, a limited range of (θ, φ) was considered to avoid steric hindrance between water and porphine. The values were (a) θ (180° minus the NO··· H angle) = 0°, (b) φ (Fe···NO···H dihedral angle) taken with the interval of 30° in the range of 90° ≤ φ ≤ 180° for θ = 15°, and (c) φ taken with the interval of 15° in the range of 90° ≤ φ ≤ 180° for θ = 30°, 105° ≤ φ ≤ 180° for θ = 45°, 120° ≤ φ ≤ 180° for θ = 60°, 135° ≤ φ ≤ 180° for θ = 75°, and 150° ≤ φ ≤ 180° for θ = 90°, as well as a few other supplementary sets of (θ, φ). The DFT calculations were done at the B3LYP/631+G(2df,p) level. The initial structure in the geometry optimization of the isolated FeII(P)(ImH)(NO) complex was taken from a previous study.79 All of the other aspects of the calculations not explicitly stated here (such as the calculations of the NMR chemical shifts) are in parallel to those explained for NMA-d1 and its complexes. 2.2. Electron Density Analysis. To examine the mechanism giving rise to the enhancement of the dipole derivative of the CO stretch of FeII(P)(ImH)(CO) and of the NO stretch of FeII(P)(ImH)(NO), as well as the factor determining the direction of the dipole derivative in the latter case, the modulation in the electron density induced by the CO or NO stretching mode was examined by calculating the electron density derivative61,80,81 ∂ρ(el)(r)/∂Q, where Q is the vibrational coordinate of interest. This is related to the electronic contribution of the dipole derivative ∂μ(el)/∂Q as

∫ dr r(∂ρ(el)(r )/∂Q )

∂μ(el) /∂Q = −e

electrostatic responses of the vibrational frequencies. It is worth noting that this linear correlation includes not only the CO··· D hydrogen-bonded species but also the N−D···O hydrogenbonded species and the molecules in uniform electric field and in other solvation situations, indicating that the frequency changes are controlled through the changes in the CO bond length. (This is seen also in the plot that includes the halogenbonded and related species, focusing on the changes of the values from those of an isolated molecule to offset the necessity of using different functionals, shown in Figure S1 in the Supporting Information). In contrast to the case of the nitrile group,38 the NMR chemical shifts (13C and 17O in the present case) are also almost linearly correlated to the CO bond length, as shown in Figure 2a,b. However, some deviations from the linear correlations are also noticeable in the plots involving the 13C NMR chemical shift shown in Figure 2a,d,e, and this will be a possible reason for the independence among the electrostatic interaction models of the three spectroscopic signals derived below. Elucidation of the origin of those deviations will be deferred to later studies. The above results suggest that any models of vibrational frequency changes should be consistent with the mechanism that generates the force along the mode, which is the sign inverted derivative of the potential energy (−∂V/∂Q). Here, we confine ourselves to the atomic-site framework. Decomposing the atomic partial charges (qn, of atomic site n) into three parts, the intermolecular electrostatic interaction energy (V) is expressed as V=

∑ [qn(nuc+ faith)Φ(rn) + qn(CF)Φ(rn) + qn(stbn)Φ(rn)] n

(2)

where Φ(rn) is the electrostatic potential at location rn of atom n. In the first term on the right-hand side, qn(nuc+faith) stands for the electric charge of the atomic nucleus (of atom n) and the electrons faithfully following it [denoted as an effective charge Zn(eff) in previous studies39,65] and is invariant upon the atomic displacement [∂qn(nuc+faith)/∂Q = 0]. This term contributes to −∂V/∂Q through the displacement of rn. In the second term, qn(CF) is the electric charge of the electrons contributing to the charge flux, which is the interatomic transfer of electron density upon displacement along Q. In the third term, qn(stbn) stands for the electric charge of the electrons that are invariant in location and magnitude (being stubborn in this sense) and, hence, do not contribute to −∂V/∂Q. Then, we obtain

(1)

For this purpose, ρ(el)(r) was calculated for r in a rectangular box of 19.6 × 19.6 × 18.1 Å3 (which was determined so that each boundary of the box is at least 5 Å from any atom in the system) with the interval of 0.02 Å, for the equilibrium (Q = 0) and displaced (Q = 0.03 Å amu1/2) structures, and a numerical differentiation was carried out. To remove the contribution of the electrons simply following the vibrating atomic nuclei and to clearly see the effects of the coordination of the ligand to the Fe2+ ion, the difference in ∂ρ(el)(r)/∂Q between the modes of the complex and the isolated ligand, defined as δ(∂ρ(el)(r)/∂Q)  (∂ρ(el)(r)/∂Q)complex − (∂ρ(el)(r)/∂Q)isolated, was calculated. Here, in calculating (∂ρ(el)(r)/∂Q)isolated, the atomic positions of the ligand were fixed to those in the complex, and the basis sets of all of the other atoms in the complex were also retained (i.e., those atoms were treated as ghost atoms),61,80,81 similarly to the calculations of intermolecular interaction energies based on the Counterpoise method.

−

∂V = ∂Q

⎡

∑ ⎢⎢−qn(nuc + faith) n

⎣

⎤ ∂q (CF) ∂Φ(rn) ∂rn · − n Φ(rn)⎥ ⎥⎦ ∂Q ∂rn ∂Q

⎡ ⎤ ∂q (CF) ∂r =∑ ⎢qn(nuc + faith) n ·E(rn) − n Φ(rn)⎥ ⎢ ⎥⎦ ∂Q ∂Q n ⎣

3. RESULTS AND DISCUSSION 3.1. Peptide Group. The correlations among the CO bond length, the frequency of the amide I mode (which may be more exactly denoted as amide I′ because the peptide group is deuterated), and the 13C and 17O NMR chemical shifts calculated for NMA-d1 in various hydrogen-bonding and other solvation situations are shown in Figure 2. It is recognized from Figure 2c that, similarly to many other hydrogen-bondaccepting stretching modes,65 the amide I mode frequency is linearly correlated to the CO bond length.28,62,74,82,83 As described below, this is a key property for modeling the

(3)

where E(rn) is the electric field at location rn of atom n. The first term is called the vector component, whereas the second term is called the scalar component, controlled by E(rn) and Φ(rn) from the surroundings of the mode, respectively. Note that Φ(rn) (electrostatic potential from the environment) does not explicitly depend on Q (intramolecular mode of the solute). In the first term, dealing with the atomic nuclei and the electrons faithfully following them, Φ(rn) varies upon displacement along Q through its dependence on rn and the atomic D


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Table 1. Parameters of the 4P4F Maps for the Solvation-Induced Changes in the Amide I Frequency and the NMR Chemical Shifts 13

amide I frequency atom

(dn)z/cm

−1

(Eh a0

−1

−1 −1

e )

C chemical shift

(cn)z/ppm

(Eh a0−1

−1 −1

e )

17

O chemical shift

(cn)z/ppm (Eh a0−1 e−1)−1

2941.3 −2663.4 572.1 548.0 amide I frequency

−264.9 470.3 1106.8 670.4 13 C chemical shift

−3101.8 −1428.4 5610.3 1230.9 17 O chemical shift

atom

ln/cm−1 (Eh e−1)−1

kn/ppm (Eh e−1)−1


C O N D

1414.9 −2021.6 1722.1 −1115.4

−401.8 413.4 818.5 −830.1

2604.5 −2709.5 2861.6 −2756.6

C O N D

Figure 3. Solvation-induced changes in the (a) amide I frequency, (b) 13C NMR chemical shift, and (c) 17O NMR chemical shift predicted by the 4P4F electrostatic interaction model (4P4F map) proposed in this study (eqs 4 and 5, the parameter values shown in Table 1) plotted against the values directly calculated at the B3LYP/6-31+G(2df,p) level (the frequencies being scaled by 0.9860) for the NMA-d1···D2O 1:1 complexes with various angular configurations of the hydrogen bond on the CO (blue ×’s) or N−D (green +’s) group of NMA-d1 and for NMA-d1 in uniform electric fields in the −z direction (multiplied by a factor of 1.28, dark red squares). The black dotted line in each panel is the line of gradient unity passing through the origin.

displacement ∂rn/∂Q. In the second term, dealing with the electrons contributing to the charge flux, it is supposed that those electrons are transferred over atomic sites, so that the changes in the interaction sites for the corresponding portions of electron density are included through the changes in the electric charges, ∂qn(CF)/∂Q. As a result, the first term is vectorial in nature at the atomic level and cannot be substituted (even effectively) by the scalar component. The displacement, ΔQ, along the mode is obtained as ΔQ = (1/k)(−∂V/∂Q), where k is the quadratic force constant of the mode. Note that because ΔQ is the displacement from the situation without the electrostatic interaction from the environment, −∂V/∂Q appearing here includes only the force from the environment, as defined in eq 2, and not the intramolecular force that already exists even without the electrostatic perturbation from the environment. Then, the electrostatic interaction model for the frequency change is expressed as Δν ̃ =

∑ [dn·E(rn) + lnΦ(rn)] n

(CF) b ∂qn ln = − k ∂Q

and b (typically negative if Q is taken along bond stretching) is the coefficient of the correlation between Δν̃ and ΔQ. In the same way, it would be most reasonable to suppose that the changes in the NMR chemical shifts are modeled (but rather phenomenologically in this case) as Δδ =

b (nuc + faith) ∂rn q k n ∂Q

∑ [cn·E(rn) + knΦ(rn)] n

(7)

The parameters of the electrostatic interaction models (usually called “maps” in this context) for the changes in the amide I frequency and the 13C and 17O NMR chemical shifts of the peptide group obtained in this study are shown in Table 1. They may be called 4P4F maps to represent that the electrostatic potentials and the electric fields on four atomic sites (C, O, N, D) are employed. Referring to the previous results that the frequency changes occurring upon hydration on CO and other hydrogen-bond-accepting groups are almost axially symmetric,39,65 only the z component taken along the CO bond (Figure 1a) is considered for dn and cn. The parameters (eight for each spectroscopic signal) are obtained by the fitting to the DFT-calculated values for the NMA-d1··· D2O 1:1 complexes with various angular configurations of the

(4)

where

dn =

(6)

(5) E


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in Figure 2c indicates that the frequency change per 0.001 Eh a0−1 e−1 of uniform electric field is −3.817 cm−1 [which is (if multiplied by 1.28) correctly represented by the present electrostatic interaction model as shown in Figure 3a]. This corresponds to the Stark tuning rate of 0.7423 cm−1 (MV cm−1)−1, in agreement with the values [0.65−0.78 cm−1 (MV cm−1)−1] observed (assuming a local field factor12,85 of ∼2) or calculated in previous studies.57,85,86 3.2. CO and NO Ligands to the Fe2+ Ion of Heme. The correlations among the CO or NO bond length, the frequency of the CO or NO stretching mode, and the magnitude of the applied uniform electric field calculated for FeII(P)(ImH)(CO), FeII(P)(ImH)(NO), and their complexes are shown in Figure 4. There are three things worth noting: (1) The correlations are nearly linear for FeII(P)(ImH)(CO) but are significantly nonlinear for FeII(P)(ImH)(NO), suggesting that carbon monoxide is a better probe of the electrostatic situations around it. (2) In the correlations between the stretching frequency and the bond length, the points of the hydrogen-bonding complexes are on the same correlation line (curve) as those in uniform electric field environment, indicating that the frequency changes are controlled through the changes in the bond length. (3) For FeII(P)(ImH)(NO), the fields along Fe···N are more effective than those along N O in the changes of bond length and stretching frequency. The third point, as well as the enhanced sensitivity to the electric field of the CO and NO molecules upon ligation,14,17,18 is related to the enhancement and its mechanism of the dipole derivative ∂μ/∂Q appearing in eq 8. Magnitudes |∂μ/∂Q| of the CO and NO stretching modes in the complexes are calculated as 3.821 and 3.649 D Å−1 amu−1/2, respectively, which are significantly enhanced from those of isolated molecules (1.375 and 1.004 D Å−1 amu−1/2). (See also Note 2 in the Supporting Information.) To see the origin of this enhancement, an electron density analysis has been carried out. The results are shown in Figure 5. It is seen that, although only the CO or NO bond is vibrating, the electron densities are modulated also in the spatial region around the Fe2+ ion. Furthermore, in this spatial region, the modulation of the electron densities is negative in total, as clearly recognized from the large negative value of the running integral (shown with a light blue line), meaning that as the CO or NO bond stretches, part of the electron density is transferred from the spatial region around the Fe2+ ion to the CO or NO bond, i.e., a charge flux is induced between the Fe2+ ion and the ligand. The value of this running integral at the center of the Fe···C or Fe···N bond (shown with a pink dotted vertical line), which is −2.15 × 10−3 a0−1me−1/2 for FeII(P)(ImH)(CO) and −2.55 × 10−3 a0−1me−1/2 for FeII(P)(ImH)(NO), is an estimate of the magnitude of this charge flux and is responsible for the enhancement of the dipole derivative of 1.97 and 2.20 D Å−1 amu−1/2 (taking the distance between Fe2+ and the bond center of the ligand as the length of the charge flux) out of the total enhancement of 2.446 and 2.840 D Å−1 amu−1/2 of the −z component (taken along CO and Fe···N bonds) of the dipole derivative. As a result of this charge flux, ∂μ/∂Q makes an angle of only 7.3° from the Fe···N bond, although the Fe··· NO angle in the optimized structure of FeII(P)(ImH)(NO) is 140.6°. This result resolves the contradiction between X-ray and vibrational Stark effect spectroscopy in the estimated configuration of the NO ligand that was pointed out in a previous study.17

hydrogen bond on CO and N−D as well as NMA-d1 in uniform electric field (mimicking dipolar solvation) of 0.002− 0.012 Eh a0−1 e−1 in the −z direction (107 points in total). As shown in Figure 3, the maps can reasonably well reproduce the DFT-calculated values. (Dependence on the hydrogen-bond distance at some typical angular configurations is also reasonably well reproduced, as shown in Figures S2−S4 in the Supporting Information.) In the course of the fitting, it has been found that the values of the changes in the frequencies and the NMR chemical shifts for NMA-d1 in uniform electric fields should be multiplied by 1.28 to construct the models that encompass the situations of both hydration and uniform electric field environment. This is probably because the parameters implicitly include the effect of polarization56 of water by the peptide group in the hydration cases, and this effect should be offset for the values in uniform electric field environment. It is also worth noting that to reproduce the angular configuration dependence of the amide I frequency the vector component (dn in eq 4) is essential.65 (See also Figures S5 and S6 in the Supporting Information.) As a result, the present 4P4F map is better in reproducing the DFT-calculated frequency changes than many of the maps proposed previously28−34,36,75 (Figure S7). Because of the negative nature of (dn)z for the O atom, which is robust in the fitting procedure, the frequency changes (negative) induced by the scalar component are significantly canceled by the vector component when the electric field in the −z direction is strong (which is typical for rather linear CO···H configurations), clarifying the relation of the electrostatic situation of the hydrogen-bond-accepting and vibrating O atom to the frequency change of the amide I mode. This negative nature of (dn)z also means that qn(nuc+faith) is positive (estimated as ∼2.8 e) for this atom because (∂rn/∂Q)z is positive (and b is negative), although the total atomic partial charge is supposed to be negative. In other words, significant amount of electron density (amounting to about −3 e) around the O atom behaves more or less separately from the atomic nucleus in the amide I mode. The overall good performance of the present 4P4F map implies that the hydration-induced changes in the amide I frequency and the 13C and 17O NMR chemical shifts of the peptide group mainly arise from the electrostatic origin. However, it is also possible that the effect of the van der Waals interaction55 is recognizable in some cases, especially for the peptide groups in hydrophobic environment. In uniform electric field environment (of field E), the frequency change is also expressed as84 Δν ̃ =

b ∂μ ·E k ∂Q

(8)

where ∂μ/∂Q is the dipole derivative of the mode, so that the parameters appearing in eq 4 are related to ∂μ/∂Q as

∑ (dn − lnrn) = n

b ∂μ k ∂Q

(9)

The parameters shown in Table 1 correspond to the value of ∂μz/∂Q = −3.3625 D Å−1 amu−1/2 (the sum of the scalar and vector components calculated in Table S1 in the Supporting Information), which is about 1.28 times the DFT-calculated dipole derivative (−2.6139 D Å−1 amu−1/2), as naturally expected from the fitting procedure explained above, where the frequency changes induced by uniform electric fields are multiplied by 1.28. Discussing in another way, the result shown F


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Figure 5. (a) Two-dimensional (xz) contour plots of ∫ dy δ(∂ρ(el)(r)/ ∂QCOstr), one-dimensional plot of ∫ ∫ dx dy δ(∂ρ(el)(r)/∂QCOstr) (black curve), and its running integral along the z axis (light blue curve) calculated for the change in the electron density derivative δ(∂ρ(el)(r)/ ∂Q COstr ) occurring upon complex formation [≡ (∂ρ (el) (r)/ ∂QCOstr)complex − (∂ρ(el)(r)/∂QCOstr)isolated] of the CO stretching mode (QCOstr) of the FeII(P)(ImH)(CO) complex. The z axis is taken along the CO bond, and the imidazole molecule is placed approximately on the xz plane, as shown in Figure 1b. The contours in the two-dimensional plot are drawn with the interval of 0.8 × 10−4 a0−3me−1/2 in the range from −20 × 10−4 a0−3me−1/2 to 20 × 10−4 a0−3me−1/2, with the color code shown on the left-hand side. The positions of the C and O atoms of the ligand and the Fe atom are indicated with black filled circles in the two-dimensional plot and with purple dotted vertical lines in the one-dimensional plot. The black open circles in the two-dimensional plot stand for the positions of all atoms of imidazole and the N atoms of porphine. The pink dotted vertical line in the one-dimensional plot stands for the location of the center of the Fe···C bond. (b) Same type of plots for δ(∂ρ(el)(r)/ ∂QNOstr) of the NO stretching mode (QNOstr) of the FeII(P)(ImH)(NO) complex. The z axis is taken along the Fe···N bond, and the x axis is taken so that the NO ligand and the imidazole molecule are approximately on the xz plane, as shown in Figure 1c. The contours in the two-dimensional plot are drawn in the same way as in part (a).

Figure 4. Plots of the (a) CO bond length and (b, c) CO stretching frequency against the (a, b) magnitude of the electric field and (c) CO bond length calculated for the FeII(P)(ImH)(CO) complexes in uniform electric fields in the −z direction (with the z axis being taken as shown in Figure 1, dark red squares), the FeII(P)(ImH)(CO)···H2O complexes in various angular configurations (small black filled circles), the FeII(P)(ImH)(CO)···H2O (blue ×’s) and FeII(P)(ImH)(CO)···nCH3OH (n = 1 and 2, green open circles) optimized complexes, and an isolated FeII(P)(ImH)(CO) complex (black open circle). The dark red dotted line in each panel connects the points of isolated FeII(P)(ImH)(CO) and in the uniform electric field of 0.002 E h a 0 −1 e−1 . (d−f) Same type of plots for FeII(P)(ImH)(NO). The dark red squares and dotted lines are for the field along the NO bond, and the orange squares and dotted lines are for the field along the Fe···N bond.

Because the NO stretching frequency is nonlinearly dependent on the applied electric field, hereafter, we focus only on the case of FeII(P)(ImH)(CO). The good linear correlations among the CO stretching frequency, CO bond length, and applied electric field (as well as the approximate linear correlations for the 13C and 17O NMR chemical shifts shown in Figure S8 in the Supporting Information) suggest that the same strategy for constructing

electrostatic interaction models explained above is applicable also to this case. In the course of the fitting procedure, it has been found sufficient to include only two atomic sites (C and O of the ligand) in the models so that the models may be called 2P2F maps. The parameters thus obtained are shown in Table G


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Table 2. Parameters of the 2P2F Maps for the Solvation-Induced Changes in the CO Stretching Frequency and the NMR Chemical Shifts 13

CO str. frequency atom

(dn)z/cm

−1

(Eh a0

−1

−1 −1

e )

C chemical shift

(cn)z/ppm

(Eh a0−1

−1 −1

e )

17

O chemical shift

(cn)z/ppm (Eh a0−1 e−1)−1

−133.8 −4524.9 CO str. frequency

1481.8 1214.4 13 C chemical shift

2736.1 411.7 17 O chemical shift

atom

ln/cm−1 (Eh e−1)−1



C O

5274.9 −5274.9

−1579.2 1579.2

−197.0 197.0

C O

Figure 6. Solvation-induced changes in the (a) CO stretching frequency, (b) 13C NMR chemical shift, and (c) 17O NMR chemical shift predicted by the 2P2F electrostatic interaction model (2P2F map) proposed in this study (eqs 4 and 5, the parameter values shown in Table 2) plotted against the values directly calculated at the B3LYP/6-31+G(2df,p) level (the frequencies being scaled by 0.9860) for the FeII(P)(ImH)(CO)···H2O complexes with various angular configurations of the hydrogen bond on the O atom of CO (blue ×’s) and for the FeII(P)(ImH)(CO) complexes in uniform electric fields in the −z direction (dark red squares). The black dotted line in each panel is the line of gradient unity passing through the origin.

4. CONCLUDING REMARKS In the present study, the strategy for appropriate modeling of the properties of the spectroscopic probes of the local electrostatic situations inside proteins is examined. The derivation starting from eq 2 clearly indicates that in the cases where the properties of those probes are strongly correlated to the force along the bond (inducing a bond length change) we need to keep both the scalar and vector components in the models. In this derivation, as well as in the discussion on the direction of the dipole derivative (in relation to Figure 5), careful treatment of the behavior of electrons is essential. The results shown in Figures 3 and 6 suggest that this way of modeling works well. For uniform electric field environment cases, the parameters of the models are consistent with the Stark tuning rates observed or calculated in previous studies. The merit of the present electrostatic interaction models is that they are applicable also to the cases of nonuniform electrostatic situations,38 which are highly expected for inside of proteins. Of course, with only a single observable quantity (such as the frequency change of a single vibrational mode), it is not possible to uniquely determine all of the parameter values of local electrostatic situations, for example, two values of electric field and one potential difference for the 2P2F map of FeII(P)(ImH)(CO). By combining a few nonredundant spectroscopic probes (vibrational and NMR), one would be able to narrow down the possible range of local electrostatic situations. By using a program of molecular dynamics (MD) simulations, one would also be able to derive some possible local electrostatic situations, among which one can select the

2. As shown in Figure 6, the maps can reasonably well reproduce the DFT-calculated values. (Dependence on the hydrogen-bond distance at some typical angular configurations is also reasonably well reproduced, as shown in Figures S9.) Note that two vectors (dn for the C and O atoms) are introduced in the present model (in addition to two scalar coefficients ln) to obtain the good agreement shown in Figure 6, instead of the distributed off-axis interaction points (explained as representing the electrostatic responses of the π electrons of the triple bond) in the previous studies.87,88 Indeed, those distributed off-axis interaction points work well to describe the axially symmetric angular configuration dependence of the frequency change of the triple-bond stretches. However, (almost) axially symmetric angular configuration dependence of the frequency change is also obtained for the double-bond stretches38,39 and cannot be attributed to the axially symmetric distribution of the π electrons. In this sense, the strategy explained in the present study enables us to model the electrostatic responses of the stretching modes of double and triple bonds in a unified way. On the basis of eq 9, which is applicable to uniform electric field environment, the parameters shown in Table 2 are consistent with the enhanced dipole derivative ∂μ/∂Q discussed above, as calculated in Table S2. The frequency change of −6.552 cm−1 per 0.001 Eh a0−1 e−1 of the uniform electric field shown in Figure 4b, corresponding to the Stark tuning rate of 1.274 cm−1 (MV cm−1)−1, is in good agreement with the value of ∼1.2 cm−1 (MV cm−1)−1 observed (assuming a local field factor12 of ∼2) or calculated in previous studies.14,16 H


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(7) Gunner, M. R.; Honig, B. Electrostatic Control of Midpoint Potentials in the Cytochrome Subunit of the Rhodopseudomonas viridis Reaction Center. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 9151−9155. (8) Müh, F.; Madjet, M. E.; Adolphs, J.; Abdurahman, A.; Rabenstein, B.; Ishikita, H.; Knapp, E.-W.; Renger, T. α-Helices Direct Excitation Energy Flow in the Fenna-Matthews-Olson Protein. Proc. Natl. Acad. Sci. U.S.A. 2007, 104, 16862−16867. (9) Polyansky, A. A.; Zagrovic, B. Protein Electrostatic Properties Predefining the Level of Surface Hydrophobicity Change upon Phosphorylation. J. Phys. Chem. Lett. 2012, 3, 973−976. (10) Melaccio, F.; Calimet, N.; Schapiro, I.; Valentini, A.; Cecchini, M.; Olivucci, M. Space and Time Evolution of the Electrostatic Potential During the Activation of a Visual Pigment. J. Phys. Chem. Lett. 2016, 7, 2563−2567. (11) Müh, F.; Plöckinger, M.; Renger, T. Electrostatic Asymmetry in the Reaction Center of Photosystem II. J. Phys. Chem. Lett. 2017, 8, 850−858. (12) Fried, S. D.; Boxer, S. G. Measuring Electric Fields and Noncovalent Interactions Using the Vibrational Stark Effect. Acc. Chem. Res. 2015, 48, 998−1006. (13) Pazos, I. M.; Ghosh, A.; Tucker, M. J.; Gai, F. Ester Carbonyl Vibration as a Sensitive Probe of Protein Local Electric Field. Angew. Chem., Int. Ed. 2014, 53, 6080−6084. (14) Dalosto, S. D.; Vanderkooi, J. M.; Sharp, K. A. Vibrational Stark Effects on Carbonyl, Nitrile, and Nitrosyl Compounds Including Heme Ligands, CO, CN, and NO, Studied with Density Functional Theory. J. Phys. Chem. B 2004, 108, 6450−6457. (15) Phillips, G. N., Jr.; Teodoro, M. L.; Li, T.; Smith, B.; Olson, J. S. Bound CO Is a Molecular Probe of Electrostatic Potential in the Distal Pocket of Myoglobin. J. Phys. Chem. B 1999, 103, 8817−8829. (16) Park, E. S.; Andrews, S. S.; Hu, R. B.; Boxer, S. G. Vibrational Stark Spectroscopy in Proteins: A Probe and Calibration for Electrostatic Fields. J. Phys. Chem. B 1999, 103, 9813−9817. (17) Park, E. S.; Thomas, M. R.; Boxer, S. G. Vibrational Stark Spectroscopy of NO Bound to Heme: Effects of Protein Electrostatic Fields on the NO Stretch Frequency. J. Am. Chem. Soc. 2000, 122, 12297−12303. (18) Franzen, S. An Electrostatic Model for the Frequency Shifts in the Carbonmonoxy Stretching Band of Myoglobin: Correlation of Hydrogen Bonding and the Stark Tuning Rate. J. Am. Chem. Soc. 2002, 124, 13271−13281. (19) Mankoo, P. K.; Keyes, T. Classical Molecular Electrostatics: Recognition of Ligands in Proteins and the Vibrational Stark Effect. J. Phys. Chem. B 2006, 110, 25074−25079. (20) Kaposi, A. D.; Wright, W. W.; Fidy, J.; Stavrov, S. S.; Vanderkooi, J. M.; Rasnik, I. Carbonmonoxy Horseradish Peroxidase as a Function of pH and Substrate: Influence of Local Electric Fields on the Optical and Infrared Spectra. Biochemistry 2001, 40, 3483− 3491. (21) Lehle, H.; Kriegl, J. M.; Nienhaus, K.; Deng, P.; Fengler, S.; Nienhaus, G. U. Probing Electric Fields in Protein Cavities by Using the Vibrational Stark Effect of Carbon Monoxide. Biophys. J. 2005, 88, 1978−1990. (22) Augspurger, J. D.; Dykstra, C. E.; Oldfield, E. Correlation of Carbon-13 and Oxygen-17 Chemical Shifts and the Vibrational Frequency of Electrically Perturbed Carbon Monoxide: A Possible Model for Distal Ligand Effects in Carbonmonoxyheme Proteins. J. Am. Chem. Soc. 1991, 113, 2447−2451. (23) de Dios, A. C.; Pearson, J. G.; Oldfield, E. Secondary and Tertiary Structural Effects on Protein NMR Chemical Shifts: An ab Initio Approach. Science 1993, 260, 1491−1496. (24) de Dios, A. C.; Earle, E. M. Electric Field Effects on 13C and 17O Chemical Shifts and CO Stretching Frequency of Carbon Monoxide Bound to Fe2+. J. Phys. Chem. A 1997, 101, 8132−8134. (25) Liu, C. T.; Layfield, J. P.; Stewart, R. J., III; French, J. B.; Hanoian, P.; Asbury, J. B.; Hammes-Schiffer, S.; Benkovic, S. J. Probing the Electrostatics of Active Site Microenvironments along the Catalytic Cycle for Escherichia coli Dihydrofolate Reductase. J. Am. Chem. Soc. 2014, 136, 10349−10360.

most probable one with the help of the electrostatic interaction models (maps) of spectroscopic probes by referring to the observed quantities. Combination of MD simulations with the electrostatic interaction models of spectroscopic probes is essential for calculating the whole band profiles of one- and two-dimensional vibrational spectra.28−36 In these ways, the present electrostatic interaction models will be helpful for interpreting solvation-induced (or, more generally, interactioninduced) changes in spectroscopic probes that work as measures of the local electrostatic situations inside the protein molecules.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.7b10791. Supplementary notes on studying NMA-d1 and its complexes with D2O and on the enhancement of dipole derivatives upon complex formation, supplementary figures on the correlations among the structural and spectroscopic properties, the dependence of the spectroscopic properties on the hydrogen-bond distances, the comparison of the performances of some electrostatic interaction models, and supplementary tables on dipole derivatives corresponding to the parameters of the electrostatic interaction models (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel:/Fax: +81-54-2384624. ORCID

Hajime Torii: 0000-0002-6061-9599 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS This study was supported in part by JSPS KAKENHI Grant Numbers JP16H00832 and JP16K05652.

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REFERENCES

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