Hydration Effect on Amide I Infrared Bands in Water: An

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Hydration Effect on Amide I Infrared Bands in Water: An Interpretation Based on an Interaction Energy Decomposition Scheme Marwa H. Farag,† Manuel F. Ruiz-López,*,‡,§ Adolfo Bastida,† Gérald Monard,‡,§ and Francesca Ingrosso*,‡,§ †

Departamento de Química Física, Regional Campus of International Excellence “Campus Mare Nostrum”, Universidad de Murcia, 30100 Murcia, Spain ‡ Université de Lorraine, SRSMC UMR 7565, Vandœuvre-lès-Nancy, F-54506, France § CNRS, SRSMC UMR 7565, Vandœuvre-lès-Nancy, F-54506, France S Supporting Information *

ABSTRACT: The sensitivity of some infrared bands to the local environment can be exploited to shed light on the structure and the dynamics of biological systems. In particular, the amide I band, which is specifically related to vibrations within the peptide bonds, can give information on the ternary structure of proteins, and can be used as a probe of energy transfer. In this work, we propose a model to quantitatively interpret the frequency shift on the amide I band of a model peptide induced by the formation of hydrogen bonds in the first solvation shell. This method allows us to analyze to what extent the electrostatic interaction, electronic polarization and charge transfer affect the position of the amide I band. The impact of the anharmoniticy of the pontential energy surface on the hydration induced shift is elucidated as well.

1. INTRODUCTION Understanding the effect of the environment on the vibrational properties of peptides represents a necessary step in the analysis of the infrared (IR) absorption spectrum of proteins. This spectroscopic technique has been shown to be extremely helpful not only to investigate the vibrational properties of complex systems, but also to gain precious insights into the local structure and the dynamics of molecules ranging from polypeptides to large proteins. Some IR bands are in fact very sensitive to their surroundings. In particular, the amide I band, corresponding mainly to the stretch of the carbonyl bond coupled with the -NH bend in the peptide linkage, has been extensively used as a probe of protein conformation, of conformational dynamics and folding, and as a tool for time-resolved experiments to follow the path of vibrational energy relaxation.1−9 The most popular molecular model for the peptide link is Nmethylacetamide (NMA), whose IR spectrum has been extensively studied both by experiments and theory.10−42 As in larger systems, the amide I band of NMA is very sensitive to the environment. In particular, in aqueous solution the carbonyl oxygen atom behaves as an acceptor of hydrogen bond (Hbond), involving on average two H-bond donor water molecules. The -NH group is a potential hydrogen bond donor, interacting with one water molecule.22,26 In NMA, having to deal with just one peptide bond, strongly simplifies © XXXX American Chemical Society

the analysis of the solvent effect on the amide I band, which is complicated in di- and polypeptides by the presence of different local environments and of different conformations. To understand the solvatochromic effect of water on the amide I band, NMA−water complexes have been widely used as model systems in theoretical work. The effect of the first neighbors has often been decoupled with respect to that of farther solvent molecules, to be able to separately analyze cooperative effects. It is indeed reasonable to think that those water molecules that are H-bonded to the solute have the largest effect on the vibrational solvatochromism of the amide I band. On the other hand, water molecules in farther solvation shells and in the bulk also contribute to the total shift, as a result of collective interactions, where cooperative effects play a role. Mirkin and Krimm20 explored the effect of two water molecules (a H-bond donor and a H-bond acceptor) on the NMA harmonic frequencies, and they evaluated the solvent effect using a reaction field in a medium having a dielectric constant of 20. They calculated a splitting of the amide I band, which is enhanced by the presence of the dielectric medium. The dielectric solvent effect was also included in later work by Special Issue: Branka M. Ladanyi Festschrift Received: August 27, 2014 Revised: September 15, 2014

A

dx.doi.org/10.1021/jp508675a | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

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Torii and collaborators26 and by Mennucci and Martinez.21 In the former case, the experimental shift in water is well reproduced by considering NMA and three water molecules immersed in a continuum solvent, using the Hartree−Fock method and a 6-31G++(d,p)basis set. In the latter case, DFT calculations using a B3LYP hybrid functional43 and a 6-31G +(d,p) basis set were used to calculate the harmonic frequencies of NMA surrounded by one to three water molecules and immersed in a continuum described at the Polarizable Continuum Model level with the Integral Equation Formalism,44,45 finding a reasonably good agreement for the solvent-induced shift on the amide I band, but much worse results for the shift on the amide II and III bands. Guo and Karplus22 focused on the effect of H-bonding on the structure and on the rotations around the N−Cα bonds. By means of quantum chemistry calculations at the Hartree−Fock and Möller-Plesset second-order perturbation theory levels, they showed that there is a cooperative effect, given by a water molecule H-bonded on the N−H bond side, strengthening the H-bond on the carbonyl side. Hydrogen bonds with water also rise the barrier for the rotation of the methyl groups of NMA, hinting that the inclusion of polarization effects may be necessary in the development of force fields for proteins. The effect of hydration in clusters of NMA with D2O molecules was analyzed by Cho and collaborators. The amide I frequency in the complexes was shown to be proportional to the elongation of the C−O bond due to the presence of the D2O molecules: the frequency shift compared to the gas phase is therefore explained as the combination of the molecular distortion and of the anharmonicity of the gas phase potential energy surface46−48 (PES). Some work on the vibrational solvatochromism of molecules containing a carbonyl group in clusters with water molecules showed that the shift on the C− O stretch is linearly correlated to the projection of the electric field onto the stretching bond axis.49 This result was confirmed in a work by Reppert and Tokmakoff,40 in which an empirical parametrization is presented allowing to correlate the experimental frequency shifts with the frequency map evaluated from molecular dynamics simulations of peptides. Using a distributed multipole analysis of the gas phase electronic density of NMA, it was possible to predict solvent-induced shifts in aqueous solution with an accuracy that is comparable with DFT calculations.41 An extension of the distributed multipole approach provided an interesting analysis of the interaction energy between NMA and a cluster of surrounding water molecules.42 This method allows to take into account electrostatic, exchange-repulsion, polarization, and chargetransfer interactions, and the frequency shifts thus obtained are in agreement with those obtained when a Hartree−Fock level of theory is used. The anharmonicity of the amide I oscillator for NMA−water clusters was examined by Torii27 using a mixed (internalnormal) coordinate system to calculate the cubic force constants. The electric anharmonicity, related to the second derivative of the dipole moment, was also evaluated, taking into account cooperative interactions. Empirical expressions to correct force fields that are based on gas phase calculations were also derived by Bour and Keiderling33 based on DFT calculations at the BPW91/631G** level. These calculations were performed on NMA− water cluster geometries extracted from classical MD simulations. In later work, Bour and collaborators34 show the deficiency in DFT methods to reproduce correcly the out-of-

plane bend potential of the N atom belonging to the peptide bond. Combination bands of the CO stretch with lower frequency motions in the condensed phase and the anharmonicity of the amide I potential result from nonspecific solute−solvent interactions as well as from H-bonding. IR solvent-induced shifts have also been modeled by la Cour Jansen and Knoester28 using a map of the electrostatic field generated on the solute by the presence of the solvent. DFTbased calculations are used to parametrize the frequency of the normal mode, the anharmonicity, and the transition dipole for the first three vibrational states. This method allows to reproduce both the solvent shift and the line width of IR experimental spectra obtained with the Fourier Transform technique. Molecular mechanics force-field based electrostatic maps have also been developed recently, showing that it is possible to achieve a good level of transferability when modeling the effect of highly polar solvents.50 On the other hand, the effects of solvents of lower polarity on the IR spectrum of NMA are poorly reproduced, since the effect of dispersion interactions is not well described by classical molecular mechanics force fields. In this work, we develop a method allowing a qualitative (or semiquantitative) interpretation of the effect of a H-bond on the amide I band, based on an the use of approximated Molecular Orbital Theory methods and on an interaction energy decomposition (IED) scheme that is well suited to treat systems of very large size, such as a protein. To achieve this goal, we developed an expression based on a perturbative approach, allowing to link the hydration induced shift of the vibrational band to each term of the IED in a straightforward way. To further analyze the factors playing a key role in the shift of the amide I vibrational frequency, we focus on the model system NMA−H2O. Particular attention is payed to inspecting in depth the potential energy surface of this mode, for the isolated NMA molecule and for three 1:1 complexes, in which one water molecule behaves whether as H-bond donor or H-bond acceptor. The interaction energy terms stemming from the IED scheme are evaluated along the potential energy surface of the complexes. A rich variety of methods has been developed in the last 40 years to decompose the interaction energy of a supermolecular complex, using different types of quantum chemistry calculations.51−59 However, in order to treat systems of large size, such as biomolecules (proteins, enzymes, etc.), as well as the solvent, linear scaling techniques, and approximated quantum methods need to be employed, like for instance the one developed by van der Vaart and Merz within the Divide and Conquer approach implemented for semiempirical Hamiltonians.56 This method has been successfully applied to investigate the nature of protein−protein interactions.60 Besides, the PM3 Hamiltonian61 has shown to be a reasonable choice to describe the vibrational properties of NMA in water,37,62 and in a preceding work63 we applied it to analyze the bulk solvent effect on the infrared bands of cis and trans NMA using Born−Oppenheimer molecular dynamics simulations (SEBOMD simulations64). The solvent-induced shift on these bands was shown to be in very good agreement with the one measured experimentally and with ab initio and DFT calculations carried out in this work, confirming that the PM3 method is a reasonable choice for this type of studies. Therefore, the decomposition scheme by Van der Vaart and Merz was chosen here, but it is worth noting that our theoretical development does not depend on the choice of the B



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Divide and Conquer approach,68−70 according to which the Fock matrix of the system is partitioned into submatrices during the process of diagonalization, and the overlap between the subsystems together with the Fermi energy are used to control the charge flow between subunits. According to the scheme proposed in ref 56, interaction energies can be decomposed into electrostatic, polarization, and charge transfer components. The decomposition is achieved by performing three distinct calculations after evaluating the total interaction energy. First, the system is brought to the equilibrium distance in the complex starting from the infinitely separated molecules, and no charge transfer is allowed. Second, intramolecular charge redistribution is activated. Finally, intermolecular charge transfer is allowed as well. For more details about the calculation we refer the reader to the original work. As in any other interaction decomposition scheme, one should pay some caution in the interpretation of each term. As it was noted by Kitaura and Morokuma,51 the electrostatic contribution represents the interaction between occupied molecular orbitals (MOs) without mixing, the polarization term is related to mixing between occupied and virtual MOs on the same molecule, and finally the charge transfer term contains also a contribution arising from the electron exchange and delocalization. When applying this method to the study of our system, we need to take into account an additional term, related with the changes in the geometrical structure arising from intermolecular interactions, compared to the gas phase equilibrium geometry of each molecule in the complex (deformation). To summarize, we assumed that the total interaction energy Eint can be partitioned into the following terms:

interaction energy decomposition approach and that it can be easily extended to other schemes. This paper is organized as follows. After presenting the theoretical background of our method, developed to decompose the solvent-induced shift, in section 2, we describe the technical details of our calculations in section 3. Results are presented and discussed in section 4, and concluding remarks are provided in section 5.

2. ANALYSIS OF THE SOLVENT-INDUCED SHIFT Our goal is to relate the solvent-induced shift on the amide I vibration to the different contributions arising from the interaction energy decomposition. As a first step toward this goal, we study a very simple system, comprising one NMA molecule and one water molecule. In the harmonic approximation, the relative solvent-induced shift can be written in terms of the corresponding normal mode force constant k, obtained by diagonalizing the mass-weighted Hessian, in the gas phase isolated molecule (g) and in the complex (c) modeling the condensed phase. Making use of a first order expansion, we can express the relative solvent induced shift, Δνrel as Δν rel =

νc − νg νg

=

kc − kg kg

+1 −1≃

1 kc − kg 2 kg

(1)

where the quantity (kc − kg)/kg is considered to be small compared to one. For instance, in the case of the amide I band of NMA in water solution, the (absolute) solvent induced shift is just about 6% of the gas phase frequency.63 In the following, we define ENMA as the energy of the isolated NMA molecule, Ew as the energy of the isolated water molecule, Ec as the energy of the 1:1 NMA−water complex, and Eint as the interaction energy, given by the expression: 0 Ec = E NMA + Ew0 + E int

E int = Eele + Epol + ECT + Edef

being Eele, Epol, ECT, and Edef the electrostatic, polarization, charge transfer, and deformation contributions, respectively. The latter is evaluated as the difference between the energy of each molecule calculated using the geometry that they have in the complex and the energy of the corresponding isolated molecule in the equilibrium configuration:

(2)

where the superscript 0 refers to the equilibrium geometry of the isolated molecules. Besides, the normal coordinate amide I in the isolated NMA molecule and in the NMA−water complex will be named Qg and Qc, respectively. In the expressions below, all the derivatives are calculated at the equilibrium configuration of the complex. However, to simplify the notation we shall omit to explicitly specify this in the development of the mathematical expressions. The force constant kc is the second derivative of Ec with respect to Qc, and therefore, using eq 2, we can write

0 Edef = E NMA − E NMA + Ew − Ew0

kc =

∂ E int ∂Q c2

(5)

We note that, since the second and the fourth term do not depend on Qc, they disappear when calculating the derivatives in eq 3. Substituting eq 4 into eq 3, we obtain

2

kc =

(4)

(3)

∂ 2Eele ∂Q c2

+

∂ 2Epol ∂Q c2

+

∂ 2ECT ∂Q c2

+

∂ 2Edef ∂Q c2

= kele + k pol + k CT + kdef

since the first two terms on the right-hand side of eq 2 are constant values. Following an IED scheme approach, we can decompose Eint into the sum of different terms, each of which gives information on a particular kind of intermolecular interaction. As we already mentioned, we employ the IED scheme by van der Vaart and Merz.56 This approach was inspired by those methods that achieve the interaction energy decomposition through selectively deleting some elements of the Fock matrix connecting occupied and unoccupied orbitals. In particular, a similar scheme is used by the Natural Bond Order Analysis65−67 and by the Kitaura−Morokuma method.51 In this particular formulation, the authors take advantage of some features of the

(6)

where the four quantities on the right-hand side have been introduced for simplicity, but they do not correspond to actual force constants. The second derivatives in eq 6 are evaluated by exploring the corresponding PESs (see section 3 for more details). By substitution of eq 6 into eq 1, we finally have 1 rel rel rel rel [Δνele + Δνpol + ΔνCT + Δνdef ] 2 kdef − kg ⎤ k pol k 1⎡k ⎥ = ⎢ ele + + CT + 2 ⎢⎣ kg kg kg kg ⎥⎦

Δν rel ≃

C

(7)



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The numerical results of this analysis will be presented in section 4.

3. COMPUTATIONAL DETAILS The Divide and Conquer approach implemented in the Amber code, version 9,71 was used to find the minima for isolated NMA, both in the cis and the trans conformation, and for NMA−water complexes with one, two, and three water molecules. Calculations were run at the PM361 level of quantum chemistry. We chose complexes between one water molecule H-bonded on the H-bond donor and on the H-bond acceptor groups of NMA. An analysis of partial charges was also carried out using Mulliken charges and the CM1 and CM2 models,72,73 both for the gas phase molecules and for the complexes. Calculations on the isolated molecules and on the three complexes were also carried out at different ab initio and DFT levels using an aug-cc-PVTZ basis set.74 In particular, Hartree−Fock (HF) and 75 Møller−Plesset second order perturbation theory (MP2),76 as well as DFT calculations using the B3LYP,43 the M062X,77 the ωB97XD,78 and the PBE079 functionals were performed. A normal-mode analysis was done after geometry optimization. The geometries of the complexes presented in the following are characterized by real frequencies. The equilibrium normal modes (ENMs) for the isolated NMA conformers and for the three complexes were calculated by diagonalizing the corresponding mass weighted Hessian of the system. The amide I mode was identified and the corresponding potential energy surface was explored by evaluating the PE on a total of 13 points along the normal coordinate. For each structure corresponding to one of these points in the case of the complexes, an energy decomposition was performed for the NMA−water interaction energy, by using the method by van der Vaart and Merz.56 A numerical fit of the corresponding energy surfaces was then performed.

Figure 1. Atom−atom distances in the peptide bond region and atomic partial charges (according to three different charge schemes) for isolated trans NMA, cis NMA, and water computed at the PM3 level.

V (Q ) = a + bQ +

4. RESULTS 4.1. Gas Phase Calculations. We start our analysis by presenting in Figure 1 some geometrical parameters and partial charges for the isolated molecules in their minimum energy geometry: cis NMA, trans NMA, and water. These data will be used later for a comparison with the results obtained in the complexes. The potential energy surface obtained for cis and trans NMA along the amide I normal mode in the gas phase Qg is reported on the top side of Figure 2, computed with respect to the equilibrium energy of cis NMA. Starting from the equilibrium structure, we moved backward and forward along Qg within the interval bounded by the classical turning points of the amide I mode, considered as a classical harmonic oscillator, defined by the condition that the total energy at these points must be equal to the thermal energy at room temperature (kBT = 208.5 cm−1). Thirteen points were thus generated, for which we calculated the potential energy. We note that the PM3 method predicts the cis isomer to be slightly more stable than the trans isomer, which is in disagreement with higher level calculations.63 This is probably due to the poor description of the repulsion interactions between methyl groups by semiempirical Hamiltonians but it should not influence the properties of the amide I vibrational mode significantly. The two data sets have been fitted to a third order polynomial of the form:

1 2 1 cQ + dQ 3 2! 3!

(8)

where a, b, c, and d are the parameters obtained through the fit. This analytical form was chosen over a second order and a fourth order polynomial based on least-squares regression analysis. In the case of the gas phase PES, the c parameter represents the second derivative of the curve evaluated in the equilibrium position Qg = 0, thus, c = kg. The parameters corresponding to the PES of the amide I normal mode for cis and trans NMA are reported in Table 1, and the corresponding curves are displayed as dashed lines in Figure 2. As a comparison, we present in the last two columns of Table 1 the results obtained for trans NMA by using the HF and the MP2 method. We note that the third order term, d, is close to the one obtained using the PM3 method in the case of HF, whereas is slightly smaller in the case of MP2. The linear term b is larger for the HF and MP2 PESs than in the case of PM3. The discussion that we shall present in the following will be devoted to the results obtained for the water−NMA complexes. The structural and electronic changes induced by the interactions between the NMA molecule and H2O will be discussed and compared. 4.2. NMA−Water Complexes: Structure, Energy and PESs. The minimum energy structures obtained for one water molecule hydrogen-bonded to the carbonyl group of cis and trans NMA (Complexes 1 and 2) and for one water molecule D



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Figure 3. Structure of the three NMA−water complexes on which we performed an interaction energy decomposition.

isolated molecules and in the complexes are reported as Supporting Information. As expected, the charge transfer between NMA and water changes its direction when the H2O molecule switches its behavior from H-bond donor (in Complexes 1 and 2) to acceptor (in Complex 3). As for the NMA molecule, what can be observed in all cases is that the charge transfer is not exclusively localized on the atoms that are closer to the water molecule. Indeed, complex formation favors the zwitterionic form of the peptide bond, so that the atoms change their charges in a way that increases the value of the local dipoles in the -NH and -CO groups. We turn now to the analysis of the amide I vibration. In Table 2 we report a comparison of the normal-mode frequencies obtained using the PM3 method with those obtained from ab initio and DFT calculations using an augcc-PVTZ basis set. The absolute frequencies are reported as well as the shift compared with the gas phase data. Concerning the gas phase results, we note that the frequencies obtained using the MP2 and the DFT-B3LYP levels are the closest to experiments (the amide I band is situated between 1714 and 1731 cm−1 for both trans and cis NMA13,14). However, apart from the PM3 case, the theoretical results give a different amide I frequency for cis and trans NMA (i.e., higher for the cis than for the trans form). We note that the absolute frequencies from the HF calculations are very close to those from PM3. When we analyze the frequency shifts induced in the presence of one water molecule, in the case of the DFT calculations we observe a strong dependence of the results on the chosen functional. The smallest shift is predicted for Complex 3 in all cases, while the ordering between the shifts of Complexes 1 and 2 varies with the method. The results obtained using PM3 are in reasonable agreement with the MP2 results, the quality of the agreement being comparable to that obtained with HF and DFT. This comparison allows us to

Figure 2. Potential energy surfaces for isolated trans and cis NMA computed with respect to the equilibrium value for trans NMA (top panel). In the bottom panel, we display the PESs for the three complexes with water computed with respect to the equilibrium value for Complex 2. Qg and Qc represent the amide I normal mode in the gas phase and in the complexes, respectively. In dashed lines, we report the fits obtained according to the expression in eq 8

hydrogen bonded to the -NH group of trans NMA (Complex 3) are reported in Figure 3. For Complexes 1 and 3, we found another minimum lying at slightly higher energies (their structure is provided as Supporting Information, Figure 1). In the former case, the water molecule is still hydrogen bonded to the carbonyl group of NMA, but it is located on the side N-methyl group. In the latter case, the H atoms of the water molecule bonded to the -NH group lie out-of-plane. In Figure 4 we show the effect of the mutual water−NMA interaction on the geometry of the peptide bond and of water, and on their atomic charges (we report the charge shift with respect to the isolated molecules, according to the three different schemes used in this work). The absolute values of the atomic charges on the most representative atoms for the

Table 1. Fit Parameters for the Potential Energy Surface of the Amide I Mode in Isolated Cis and Trans NMA and PM3 Calculationsa b

a bc cd de

trans

cis

trans (HF)

trans (MP2)

−21213.6 −6.8022 × 10−5 315.549 −599.507

−21213.8 −6.8022 × 10−5 315.405 −414.569

−155059.4 −0.008006291 307.747 −609.964

−155665.2 +0.00205508 256.937 −481.791

In the case of trans NMA, the results obtained using the HF and the MP2 method are shown as well. bkcal/mol. ckcal Å−1 g−1/2 mol−1/2. dkcal Å−2 g−1. ekcal mol1/2 Å−3 g−3/2. a

E



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Figure 4. NMA−water complexes: distances affected by complex formation and shift in partial charges (on some representative atoms) according to three different charge schemes computed at the PM3 level.

Table 2. Computed Amide I Frequencies for Isolated NMA and Absolute Frequencies (Frequency Shifts in Parentheses) for the 1:1 Complexes with Watera

a

method

trans NMA

cis NMA

complex 1

complex 2

complex 3

PM3 HF MP2 B3LYP M062X ωB97XD PBE0

1928.7 1904.9 1740.6 1734.6 1788.6 1781.8 1777.3

1928.3 1915.1 1750.1 1742.3 1803.9 1789.2 1782.2

1902.7(−26.0) 1879.1(−25.8) 1720.4(−20.2) 1709.0(−25.6) 1749.2(−39.4) 1752.8(−29.0) 1748.2(−29.1)

1900.5(−27.8) 1885.5(−29.6) 1734.2(−16.8) 1717.8(−24.5) 1763.2(−40.7) 1759.4(−29.8) 1751.0(−31.2)

1920.4(−8.3) 1894.8(−10.1) 1735.0(−5.6) 1727.4(−7.2) 1777.7(−10.9) 1772.7(−9.1) 1765.9(−11.4)

The displayed values are expressed in cm−1.

conclude that the semiempirical PM3 level is able to capture the correct behavior for a description of the water induced shift on the amide I band of NMA in small clusters. The PM3 potential energy surface for the amide I mode for each of the three complexes is shown on the bottom of Figure 2. Each curve is reported with respect to the equilibrium value for Complex 2. The parameters resulting from a fit of each of the three curves to the expression in eq 8 are reported as Supporting Information. As in the case of the gas phase curves, these fits gave the best results based on regression analysis. To further investigate the PESs obtained by the fitting procedure, we can wonder whether or not the parameters from the fits have the correct physical meaning, in particular, regarding the c parameter. We thus compared the amide I frequencies as calculated from the diagonalization of the Hamiltonian and the corresponding values resulting from the fits. In Table 3 we report the frequencies obtained from the fits, which can be compared with those in Table 2, resulting from the normal-mode analysis.

Table 3. Amide I Frequency for NMA in the Gas Phase and in the Complexes as Evaluated from the Fit in Eq 8 (see Figure 2) νc (Δν; cm−1)

trans NMA cis NMA

νg (cm−1)

complex 1

1929.0

1903.0 (−26.0)

1928.5

complex 2

complex 3 1920.6 (−8.4)

1901.5 (−28.0)

We can conclude that the two sets of results are in remarkable agreement. We have now all the basic ingredients that are necessary to introduce the results obtained with our IED approach. 4.3. Interaction Energy Decomposition. We start our discussion by analyzing the behavior of each term resulting from the interaction energy decomposition provided in eq 4, that is, how the electrostatic, the polarization, the charge transfer, and the deformation terms vary along the normal F



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Figure 5. Variation of the four different terms in eq 4 (electrostatic, polarization, charge transfer, and deformation terms) along the amide I normal coordinate for the three complexes considered in this work (black circles). The computed values are computed with respect to the corresponding equilibrium value. The fits to the expression in eq 8 are shown as red dashed lines.

mode amide I in the water−NMA complexes. In Figure 5 we show the computed values of each energy contribution and the corresponding fit curves to eq 8, the parameters of which are provided as Supporting Information. The interaction energy Eint (eq 4), calculated as the sum of different terms as in eq 4, was compared with the interaction energy computed through eq 2, which we named Eint (eq 2). The two interaction energy curves and the corresponding polynomial fits (eq 8) are displayed in Figure 6 (computed with respect to the equilibrium value for Complex 1), whereas the parameters from the fits are reported as Supporting Information. The electrostatic, polarization, and charge transfer terms are quite similar in shape (being very close to linear), the only remarkable change being the different sign of the slope for the curve related to charge transfer in Complex 3. As we already mentioned, the sign of the charge flux in this case is opposite to the one obtained for Complexes 1 and 2. The deformation energy, in contrast, roughly exhibits a quadratic form, expected from its definition in eq 2. Indeed, in eq 5 the changes of this term along the normal coordinate are related with the changes in the total potential energy of the complex. Based on their fitted analytical expressions, we can obtain the second derivative of each interaction term in Qc = 0 (the c parameter of the fit), which allows us to evaluate the respective contribution to the hydration-induced shifts in H2O−NMA complexes. In Table 4 we report the results for the corresponding relative frequency shifts obtained by using eq 7. The total shift obtained as the sum of the different contributions and the one evaluated based on the frequency values from normal-mode analysis (Table 2) are in very good agreement (the absolute relative error is less than 2%). First of all, as a general remark, we observe that the contribution due to polarization is almost negligible in all cases. On the other hand, the largest contribution to the relative shift

Figure 6. Potential energy surface for the interaction energy along the normal coordinate amide I in the three complexes. Displayed are the results obtained for Eint (eq 2; black dots) and for Eint (eq 4; light blue crosses). The corresponding fits to the expression in eq 8 are shown as red lines and as blue dashed lines, respectively. The PESs are reported as computed with respect to the equilibrium value for Complex 1.

is related to the deformation term. Further analysis on this term will be provided in the next section. The sign of Δνrel CT changes when going from the complexes where the water molecule is on the carbonyl side (Complex 1) to the complex where it is on the -NH side (Complex 3) of trans NMA. This is most likely related to the charge flux from NMA to water, which changes its sign as well, as shown above. For Complexes 1 and 2, the electrostatic and charge transfer terms have opposite sign. When interpreting this result, we need to take into account that, strictly speaking, the CT term does not arise solely from charge transfer, since it is not easy to separate this contribution from the total interaction energy. G



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Table 4. Amide I Relative Solvent-Induced Shift: Relative Contribution for Each Term in the Interaction Energy Decomposition (Eq 7), Resulting Total Relative Shift, and Comparison with the Values Obtained from Table 2 complex 1 complex 2 complex 3

Δνrel ele

Δνrel pol

Δνrel CT

Δνrel def

Δνrel

Δνrel (Table 2)

0.00777 0.00870 0.00106

−0.00083 −0.00070 0.00009

−0.00312 −0.00425 0.00005

−0.03043 −0.03254 −0.00967

−0.01333 −0.01439 −0.00423

−0.01348 −0.01442 −0.00430

modes in the gas phase and in the complexes are approximately equivalent and differ only by a constant value that is associated with the different geometry of NMA at the minima of the corresponding PESs. Results for the term (∂Qg/∂Qc)2 for the three complexes are reported in Table 6, which shows that this

Delocalization effects are included in this term as well. Our results can be qualitatively compared to those obtained in ref 42, where the authors perform an interaction energy decomposition analysis of the hydration induced shift. These calculations are performed on constrained trans NMA−water clusters and the deformation term is not included, but an approximate expression is used that accounts for mechanical and electric anharmonicities. In agreement with our results, the polarization term is smaller compared to the electrostatic term, and the induced shift changes its sign in going from Complex 1 to Complex 3. However, in our case, the electrostatic term is related to a blue shift, where a red shift is observed in ref 42. In addition, in the latter case the charge transfer term is found to be negligible. These differences are probably related to the scheme used to perform the partition of the interaction energy. For instance, in ref 42 an exchange-repulsion term is included, which gives an important contribution to the total shift and roughly cancels out the polarization term. 4.4. Anharmonicity and the Deformation Term. In our definition of the deformation term, provided in eq 5, we note two terms that do not vary along the normal coordinate (E0NMA and E0w) and a term representing the energy of water in the complex (Ew). We have assumed that the variation of the latter along the amide I normal coordinate is negligible. We showed that this is a reasonable assumption by calculating the contribution of the water atoms displacements to Qc. The results are reported in Table 5, and they confirm that neglecting water displacements is indeed a good approximation.

Table 6. Derivatives of the Amide I Normal Coordinate in the Gas Phase with Respect to the Corresponding Mode in the Complex (∂Qg/∂Qc)2 complex 1 complex 2 complex 3

term is very close to one (with a slightly larger difference in the case of Complex 2). This is not surprising since, in all cases, the normal modes correspond mostly to the CO bond stretch. Under the preceding assumptions, we can then rewrite eq 9b in the following way: kdef ≃

0.0223 0.0254 0.0002

Hence, Ew can be considered to be constant along the amide I normal mode in these complexes, and therefore, the deformation energy can be simply related to the potential energy surface of NMA (ENMA), giving

kdef =

∂ 2Edef ∂Q c2

≃

(9a)

∂ 2E NMA ∂Q c2

(9b)

For the sake of convenience, through very simple algebra we can recast the definition in eq 9b in the following way: kdef ≃

∂ 2E NMA ∂Q c2

⎛ ∂Q g ⎞2 ∂ 2E NMA ⎟⎟ ≃ ⎜⎜ 2 ⎝ ∂Q c ⎠ ∂Q g

∂Q g2

(11)

ENMA)/(∂Q2g))

We remind that the term ((∂ is still evaluated at the equilibrium configuration in the complex. The validity of this equation is indeed confirmed by the comparison of the deformation curves in Figure 5 with those for the isolated NMA energy in Figure 2. These curves are very similar, the only main difference being the position of the coordinate origin. For instance, if we calculate the value of the frequency corresponding to the force constant at the minimum for the three deformation curves in Figure 5, we find 1926.5, 1922.7, and 1923.4 cm−1, which are quite close (within 0.3%) to the frequency of amide I for isolated NMA (Table 2). Based on the approximated expression in eq 11 we note that, if the energy of NMA were harmonic (or approximately harmonic), kdef ≃ kg, since the force constant would not depend on the point at which the second derivative is evaluated. Therefore, using eq 7, Δνrel def ≃ 0. However, we have shown that Δνrel def provides the largest contribution to the total frequency shift (see Table 4). In other words, we can conclude that the largest part of the amide I frequency shift under formation of a hydrogen bonded complex is due to the modification of the equilibrium geometry (CO elongation) along an anharmonic PES. As a corollary, one can expect a correlation between the frequency shift observed for the amide in a specific molecular environment and the modification of the amide I normal mode, nearly connected with the CO stretch. We can explore this issue by comparing the frequency shifts obtained through normal-mode analysis on NMA−water complexes with the ones calculated by using the carbonyl bond elongation. In fact, from the CO bond length in the complex we can deduce the value of the normal coordinate,

H2O contribution (%)

Edef = E NMA + const

∂ 2E NMA 2

Table 5. Total Atomic Displacement Contribution of the Water Molecule in the Amide I Normal Mode of the Complexes complex 1 complex 2 complex 3

0.999 0.989 0.996

(10)

The full development leading to eq 10 is reported as Supporting Information. We found that the amide I normal H



Article

Figure 7. Structure of the complexes of NMA with two and three water molecules.

corresponding to which we can calculate the second derivative of the gas phase PES. When we apply this hypothesis to the HF and MP2 calculations on Complex 1, we find a frequency shift on the amide I mode of −12.6 and −33.5 cm−1, respectively, to be compared with the values in Table 2. The relative error on the estimated frequencies are computed to be 0.7% in the case of the HF calculation and 0.8% in the case of MP2, corresponding to about 10 cm−1, which can considered to be a reasonable error on frequencies.42 Our next step is to compare the results obtained using the PM3 method on the three 1:1 complexes with those stemming from larger systems. First, we performed normal-mode analysis on three more complexes of trans NMA with two and three water molecules (the optimized structures are shown in Figure 7). In addition, we extended this comparison by using the condensed phase data from our SEBOMD simulations of NMA in water.63 In the latter case, we evaluated the CO distance in the condensed phase and in the isolated molecule as an ensemble average along the molecular dynamics runs. The frequencies are obtained from the IR absorption spectra calculated as the Fourier transforms of the molecular dipole self-correlation function. Results are reported in Figure 8, from which we can conclude that our hypothesis is reasonably well confirmed in the case of the system under study.

Figure 8. Correlation between the frequency shifts calculated for the amide I mode in the NMA molecule and those evaluated based on the CO bond elongation (the gas phase PM3 values for cis and trans NMA are 1928.3 and 1928.7 cm−1, respectively). Frequency shifts for the mono- to trihydrates were obtained from normal-mode analysis, whereas in the condensed phase they were calculated from the IR absorption spectra of NMA from semiempirical Born−Oppenheimer MD (ref 63).

We adopted an interaction energy decomposition scheme56 based on quantum mechanical calculations and linear scaling techniques that allowed us to take into account electrostatics, polarization, and charge transfer terms. In addition, we explicitly consider the energy contribution related to the deformation of the molecular geometry following the formation of the complex between water and NMA. The study was carried out by using a semiempirical PM3 Hamiltonian since the longterm goal is, in fact, to generalize the application of our method to biological system of large size in the presence of explicit

5. CONCLUDING REMARKS In this work we developed a new methodology to quantitatively determine how the hydration induced shift on the amide I vibrational band is related to the intermolecular interactions between solute and solvent. As a first step and in order to focus on the formation of one intermolecular hydrogen bond, we applied our method to 1:1 complexes between a model peptide (NMA) and a water molecule. I



Article

(5) Peterson, K. A.; Rella, C. W.; Engholm, J. R.; Schwettman, H. A. Ultrafast Vibrational Dynamics of the Myoglobin Amide I Band. J. Phys. Chem. B 1999, 103, 557−561. (6) Ganim, Z.; Chung, H. S.; Smith, A. W.; DeFlores, L. P.; Jones, K. C.; Tokmakoff, A.; Amide, I. Two-Dimensional Infrared Spectroscopy of Proteins. Acc. Chem. Res. 2008, 41, 432−441. (7) Woutersen, S.; Mu, Y.; Stock, G.; Hamm, P. Subpicosecond Conformational Dynamics of Small Peptides Probed by TwoDimensional Vibrational Spectroscopy. Proc. Natl. Acad. Sci. U.S.A. 2001, 98, 11254−11258. (8) Kim, Y. S.; Hochstrasser, R. M. Applications of 2D IR Spectroscopy to Peptides, Proteins, and Hydrogen-Bond Dynamics. J. Phys. Chem. B 2009, 113, 8231−8251. (9) Paulson, L. O.; Anderson, D. T. Infrared Spectroscopy of the Amide I Mode of N-Methylacetamide in Solid Hydrogen at 2−4 K. J. Phys. Chem. B 2011, 115, 13659−13667. (10) Hayashi, T.; Zhuang, W.; Mukamel, S. Electrostatic DFT Map for the Complete Vibrational Amide Band of NMA. J. Phys. Chem. A 2005, 109, 9747−9759. (11) Rodrigo, M. M.; Tarazona, M. P.; Saiz, E. Experimental Determination and Analysis of the Direction of the Dipole Moments of Some Substituted Amides. J. Phys. Chem. 1986, 90, 2236−2243. (12) Meighan, R. M.; Cole, R. H. Dielectric Properties of Alkyl Amides. I. Vapor Phase Dipole Moments and Polarization in Benzene Solution. J. Phys. Chem. 1964, 68, 503−508. (13) Ataka, S.; Takeuchi, H.; Tasumi, M. Infrared Studies of the Less Stable Cis Form of N-Methylformmaide and N-Methylacetamide in Low-Temperature Nitrogen Matrices and Vibrational Analyses of the Trans and Cis Forms of These Molecules. J. Mol. Struct. 1984, 113, 147−160. (14) Kubelka, J.; Keiderling, T. A. Ab Initio Calculation of Amide Carbonyl Stretch Vibrational Frequencies in Solution with Modified Basis Sets. 1. N-Methylacetamide. J. Phys. Chem. A 2001, 105, 10922− 10928. (15) Herrebout, W. A.; Clou, K.; Desseyn, H. O. Vibrational Spectroscopy of N-Methylacetamide Revisited. J. Phys. Chem. A 2001, 105, 4865−4881. (16) Chen, X. G.; Schweitzer-Stenner, R.; Asher, S. A.; Mirkin, N. G.; Krimm, S. Vibrational Assignments of Trans-N-Methylacetamide and Some of its Deuterated Isotopomers from Band Decomposition of IR, Visible, and Resonance Raman Spectra. J. Phys. Chem. 1995, 99, 3074− 3083. (17) Yang, S.; Cho, M. IR Spectra of N-Methylacetamide in Water Predicted by Combined Quantum Mechanical/Molecular Mechanical Molecular Dynamics Simulations. J. Chem. Phys. 2005, 123, 134503− 134503. (18) Schultheis, V.; Reichold, R.; Schropp, B.; Tavan, P. A Polarizable Force Field for Computing the Infrared Spectra of the Polypeptide Backbone. J. Phys. Chem. B 2008, 112, 12217−12230. (19) Chen, X. C.; Schweitzer-Stenner, R.; Krimm, S.; Mirkin, N. G.; Asher, S. A. N-Methylacetamide and its Hydrogen-Bonded Water Molecules are Vibrationally Coupled. J. Am. Chem. Soc. 1994, 116, 11141−11142. (20) Mirkin, N. G.; Krimm, S. Ab Initio Vibrational Analysis of Isotopic Derivatives of Aqueous Hydrogen Bonded Trans-NMethylacetamide. J. Mol. Struct. 1996, 337, 219−234. (21) Mennucci, B.; Martinez, J. M. How to Model Solvation of Peptides? Insights from a Quantum-Mechanical and Molecular Dynamics Study of N-Methylacetamide. 1. Geometries, Infrared, and Ultraviolet Spectra in Water. J. Phys. Chem. B 2005, 109, 9818−9829. (22) Guo, H.; Karplus, M. Ab Initio Studies of Hydrogen Bonding of N-Methylacetamide: Structure, Cooperativity, and Internal Rotational Barriers. J. Phys. Chem. 1992, 96, 7273−7286. (23) Jorgensen, W. L.; Gao, J. Cis-Trans Energy Difference for the Peptide Bond in the Gas Phase and in Aqueous Solution. J. Am. Chem. Soc. 1988, 110, 4212−4216. (24) Du, Q.; Wei, D. Solvation and Polarization of the N-Methyl Amine Molecule in Aqueous Solution: A Combined Study of

solvent molecules. Our calculations gave a satisfying agreement with ab initio and DFT calculations regarding the prediction of the hydration-induced shift. We showed that the deformation contribution to the hydration induced shift of the amide I band is the dominant one for the 1:1 complexes considered. Through a simple mathematical development, we analytically describe this contribution in terms of a few critical quantities related to the curvature of the potential energy surface of NMA. This is possible since, in the system under study and in a first order approximation, the target vibrational mode in the hydrated solute is approximately the same as the corresponding mode in the isolated solute (the amide I motion is mainly characterized by the carbonyl bond stretch). The interesting conclusion of this analysis is that, with a reasonable level of accuracy, one could simply infer the hydration induced shift on amide I based on the PES of the isolated NMA molecule and on the elongation of the CO bond when the molecule is solvated, in agreement with previous studies derived in the context of continuum models of solvation.80−83 This finding was further confirmed by calculations for NMA−water complexes with 2−3 water molecules and for NMA in water solution.

■

ASSOCIATED CONTENT

* Supporting Information S

Details about atomic partial charges, the fits of the complexes potential energy surfaces, the structure of some additional complexes considered in this work and the derivation of eq 10 are provided as Supporting Information.This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was partially supported by the Ministerio de Educación y Ciencia of Spain under Projects CTQ2011-25872 and CONSOLIDER CSD2009-00038, by the Fundatción Séneca del Centro de Coordinación de la Investigación de la Región de Murcia under Project 08735/PI/08. M.H.F. gratefully acknowledges a fellowship from the Ministerio de Educación y Ciencia of Spain.

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