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Additive Model for the Second Harmonic Generation Hyperpolarizability Applied to a Collagen-Mimicking Peptide (Pro-Pro-Gly)10 C. Loison* and D. Simon Laboratoire de Spectrome´trie Ionique et Mole´culaire (UMR 5579), UniVersite´ de Lyon, UniVersite´ Lyon 1, CNRS, Domaine Scientifique de la Doua, F-69622 Villeurbanne, France ReceiVed: February 2, 2010; ReVised Manuscript ReceiVed: May 6, 2010

Second harmonic generation (SHG) spectrometry has been recently applied to investigate the structure of proteins and sugars (collagen, myosin, starch, etc.). The interpretation of experimental data at the molecular length-scale remains often qualitative because of the difficulty to model the SHG signal of such large molecules. Simpson and co-workers proposed to estimate the hyperpolarizability of the peptide backbone as the sum of the individual hyperpolarizabilities of the peptide bonds.49 This article discusses the hyperpolarizabilities obtained using such an additive model for a peptide (Pro-Pro-Gly)10 modeling collagen, for which experimental hyperpolarizabilities have been measured and modeled recently.46 To investigate possible parameters for the model, we performed time-dependent density functional theory (TDDFT) calculations of the hyperpolarizability of a few molecules containing one peptide bond. In a second step, the additive model is applied. The results produced using different input parameters are compared to each other and to experimental data. For the chosen peptide, the additive model using N-methylacetamide as a building block agrees qualitatively with hyper-Rayleigh scattering data. The results emphasize the need for more reference data to test the additivity hypothesis and the transferability of the parameters to other secondary structure of proteins. The second harmonic generation (SHG) is a nonlinear optical process during which two photons of energy pω are converted into a single photon of energy 2pω. SHG by bulk solids was first observed, but interfaces1,2 or isolated entities may also generate second harmonic light.3-7 Second harmonic generation has been exploited for various applications among which the investigation of interfaces and adsorption,8,9 the probe of small particles,10-13 and the imaging of biological materials.14-20 Various endogenous SHG signals have been observed in highly ordered biological samples, including connective tissues, muscles, and brain. The molecular origin of the signal among the tissue is difficult to elucidate, but type-I collagen, myosin bundles, and microtubules have clearly been identified as strong harmonophores.21-26 As SHG are highly sensitive to the molecular spatial organization of the harmonophores and to their environment, it is a powerful technique to investigate protein structures or activity, either at a micrometer scale27-33 or at the molecular scale.26,34-43 The quantitative interpretation of a protein hyperpolarizability tensor at the molecular scale remains a challenge. The SHG responses of endogenous biological harmonophores are not yet fully characterized,44 and the theoretical modeling of the optical response of a harmonophore assembly is involved.11,15,21,45,46 For large polymers such as proteins, precise quantum chemical calculations of the hyperpolarizability are beyond the reach of today’s computer power. Terenziani et al. proposed a bottomup strategy where a molecule or an aggregate is decomposed into individual chromophores interacting via electrostatic forces.47,48 For not too large aggregates, exact NLO responses and spectra can be calculated and account for interchromophore couplings. A few experimental and theoretical studies suggested that the SHG hyperpolarizability can be interpreted as the sum * To whom correspondence should be addressed: E-mail: cloison@ lasim.univ-lyon1.fr.

of hyperpolarizabilities of elementary building blocks.26,33,40,41,49-53 The first analysis of experimental SHG signal of peptides in terms of additive contribution of building blocks was proposed by Mitchell et al. for R-helical peptides adsorbed at air-water interfaces.40,41 Deniset-Besseau et al. concluded that the sizedependence of the molecular hyperpolarizability of collagenlike molecules is well described by an additive model but the nature of the building block remains unclear.46 Simpson and co-workers proposed an additive model of protein hyperpolarizability where the building block is a small molecule containing a single peptide bond: N-methylacetamide (NMA).49,51 Their approach is based on a perturbation theory in which the exciton coupling effects are neglected. The predictive power of such an additive model obviously depends on the validity of the additivity hypothesis and on the choice of the input parameters (an averaged hyperpolarizability tensor per peptide bond within the protein environment). To sustain the additivity hypothesis, they showed that the first hyperpolarizability of a molecule containing two peptide bonds is approximately reproduced by the sum of the hyperpolarizabilities of the two individual peptide bonds.49 This group implemented a data analysis and visualization program to calculate the nonlinear optical properties of proteins based on this additive model (NLOPredict), but to our knowledge, only a few comparisons with experiments are available yet.52,54,55 The purpose of the present work is to study Simpson’s model for a helical peptide mimicking collagen triple-helical coiledcoils: [(Pro-Pro-Gly)10]3.56 The choice of collagen as a case study of SHG of the protein backbone is relevant because of (1) its biological importance and its relevance for SHG biological imaging, (2) its rigid helical structure, which sustains the validity of the additivity hypothesis, and (3) the high probability that the peptide bond is the main chromophore of the nonlinear optical response. First, the choice of the input hyperpolarizability tensor of the peptide bond is discussed. Indeed, the input

10.1021/jp100997q  2010 American Chemical Society Published on Web 06/30/2010

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Loison and Simon

Figure 1. Structure of the studied amides (Cs symmetry) and axis orientation for the hyperpolarizability calculations. 1, formamide; 2, acetamide; 3, N-methylformamide; 4, N-methylacetamide; 5, N-azolidineformamide.

parameters may depend on the sequence of the protein, on its conformation, and on the frequency of the exciting light. To get insight in the choice of reasonable input parameters, we investigated several building blocks containing a single peptide bond using quantum chemistry calculations (formamide, ethanamide, N-methylformamide, N-methylethanamide, Section 2.1). The influences of methyl substitutions and laser wavelength on the hyperpolarizability have been studied. In a second step (Section 2.2), the hyperpolarizability of (Pro-Pro-Gly)10 modeling collagen has been calculated using Simpson’s model, and some ratios of hyperpolarizability components were compared to experimental data.46 The influences of the input building block and of the conformation of the peptide have been investigated. 1. Methods The following section gathers the computational details: quantum calculations are described in Section 1.1, followed by the hyperpolarizability calculation model (Section 1.2). For completeness, Section 1.3 is devoted to some molecular dynamics simulations done to describe the conformations of the denaturated peptide. 1.1. Quantum Calculations. The quantum calculations have been performed using the Amsterdam density functional (ADF2007)57 and the MOLCAS758 packages. Ground-state electronic structure calculations and geometry optimizations have been performed by density functional theory (DFT) for the molecules presented in Figure 1. Both Cs or C1 geometries are given in the Supporting Information, but only the Cs geometries are studied in the article. The geometries were optimized with the local density approximation treated in the Volsko-Wilk-Nusair parametrization with the nonlocal correction to exchange by Becke59 and to correlation by Perdew60 (BP86) as implemented in ADF. The basis set used was ZORA-QZ4P (see discussion below). The maximum differences for chemical bond lengths and angles between the theoretical and gas-phase electron diffraction61 structures were 3 pm and 2.3° (see the Supporting Information).

Time-dependent DFT (TDDFT) was used to calculate the excited-state energies, transition momenta, and hyperpolarizabilities with linear response theory as implemented in ADF,62,63 for which the exchange correlation kernel is the adiabatic linear density approximation (ALDA). We compare the formamide results calculated with the BP86 and the statistical average of orbital potential (SAOP) functionals in Section 2.1.1. The latter functional, SAOP64 has a correct asymptotic behavior in 1/r when r tends to infinity and has proven to be well adapted for the calculation of the excitation energies and hyperpolarizability of small organic molecules.65 Excitations energies were calculated for 30 singlets states, and the convergence relative to this number was achieved within 10-2 eV. The hyperpolarizabilities are given in the B convention, which is based on a perturbation series expansion of the dipole moment in term of an external field.66 Basis set influence was tested using three Slater-type basis sets of increasing precision including polarized and diffuse functions from the standard ADF libraries (ATZ2P, ET-pVQZ, ZORA-QZ4P), using the default correction for linear dependencies. The differences between basis sets for the excitation energies and transition strengths of the most intense πfπ* transition and for the hyperpolarizability components were respectively less than 0.03 eV, 0.02 and 10% (see the Supporting Information). The largest basis set (QZ4P) is a core triple ζ, valence quadruple ζ, with four polarization functions (2d and 2f for C, N, and O; 2p and 2d for H), which is a very safe option. For comparison, the (hyper)polarizabilities of formamide were also calculated using finite-field (FF) perturbation. Derivations were calculated using a five-point stencil, with a perturbative electric field of 0, ( e, ( 2e in x and y directions, and e varying from 0.001 to 0.01 au. The calculation precision was then increased relative to ADF default settings (keyword inputs: converge 10-8, integration 8, and linearscaling 12). The values given in Section 2.1.1 correspond to e ) 0.005 au, the variations calculated for different perturbing fields were less than 0.05 au

SHG Applied to Collagen-Mimicking Peptide for the polarizabilities and less than 10% au for hyperpolarizability components. Pictures were prepared using Gabedit,67 Gnuplot,68 and Inkscape69 softwares. 1.2. Weak Coupling Model for Hyperpolarizability of Peptides. Simpson and co-workers implemented a data analysis and visualization program to calculate the nonlinear optical properties of biopolymers (NLOPredict54,55). Their approach is based on a perturbation theory in which the exciton coupling effects are neglected. It has been applied to the calculation of SHG and two-photon absorption response calculations of R-helices.41,52 The total hyperpolarizability tensor is calculated as a sum of the hyperpolarizability tensors of each single peptide bond in its particular orientation. Such an approach obviously excludes peptides containing polarizable residues such as tryptophan and tyrosine. This supposes that the responses of the individual peptide bond all interfere constructively and limits the application of the model to proteins much smaller than the exciting wavelength, typically between 750 and 1000 nm. For larger proteins, retardation effects have to be included.46 The input of the program is the hyperpolarizability tensor of the single peptide bond (the building block) and the orientation of the peptide backbone of the protein. The input hyperpolarizability tensor provided with their package by a restricted ZINDO calculations for the NV1 excitation of NMA at its resonance and is therefore noted βNV1 (190 nm). They estimated that this approach describes about 95% of the nonlinear optical response resulting from the NV1 electronic transition. NLOPredict was used to calculate the hyperpolarizability of different conformations of the collagen-like peptide. We did not use βNV1 as input but other tensors: either simplistic phenomenological models or tensors calculated by linear response approach for an exciting wavelength of 800 nm, noted βTDDFT. The main component of βNV1 is along the transition moment of the NV1 electronic transition, that is, parallel to the CN bond, whereas βTDDFT has a larger component along the CO bond. This distinction comes from the different excitation energies (Section 2.1.3). 1.3. Molecular Mechanics and Decorrelation Time. Recent hyper-Rayleigh scattering (HRS) data on collagen of type I extracted from a rat-tail tendon permitted to compare the hyperpolarizabilities of native and denaturated collagen.46 HRS measurements were also performed on the collagen-like peptide [(Pro-Pro-Gly)10]3. To compare some of these experimental data with the results of the additive model, we chose to study the model peptide (Pro-Pro-Gly)10 in its native and denatured states. The X-ray scattering 1K6F structure from the Protein Data Bank was used to characterize the native state.70 The denaturated state was described by an ensemble of 20 conformations obtained from two molecular dynamics simulations of a single (Pro-ProGly)10 strand in water with two different ionization states. The experimental data to which the theoretical results on the peptide (Pro-Pro-Gly)10 would be compared were solutions of (Pro-ProGly)10 in aqueous acetic acid (pH ) 2.5) so that the peptide should be present in about the same amount of the cationic and zwitterionic forms.46 Therefore, the conformations of both the cationic and the zwitterionic peptides were studied. The cation was simulated with the charge on the N terminus with one Clcounterion and the zwitterion with one Na+ and one Clcounterions. Calculations were performed using the GROMOS G53a6 united atom force field71 and the SPC water model72 using GROMACS3.3.73 Berendsen weak coupling was applied with constants of 0.1 ps for the temperature of 300 K and 0.5 ps for the pressure of

J. Phys. Chem. A, Vol. 114, No. 29, 2010 7771 1 bar, with a compressibility of 4.5 × 10-5 bar-1. van der Waals interactions were handled using a cutoff of 1.0 nm. Long-range interactions were updated every 10th time step during neighbor searching. Analytic corrections to the energy and pressure beyond van der Waals cutoff were used. Integration time step was set to 1 fs. The first conformation of the single strand was extracted from the 1K6F Protein Data Bank structure file and embedded in a water box of about 10 000 water molecules with periodic boundary conditions. Prior to the MD simulation, the energy of the system was minimized using the steepest descent algorithm until the maximum force was less than 200 au. After this procedure, constraints on all bonds were fixed using LINCS algorithm every iteration step.74 A first position-restraint molecular dynamics simulation was done to equilibrate the peptide-water interactions without large change of the peptide conformation. Then, the peptide position restraint was eliminated, and an equilibration run was done for 20 ns. During this equilibration run, the initial helical structure was destroyed, and the polymer adopted a coiled conformation within 5 ns. After this equilibration period, production runs of 200 ns with a storage every 10 ps were performed. Lyman et al. proposed a protocol to estimate the smallest time delay between two statistically independent conformations (the structural decorrelation time).75 Their method is based on the time-analysis of all the coordinates, not of a single arbitrary observable, the gyration radius for example. The conformations of the simulated trajectory are first distributed into S separate bins according to their relative distance in phase space. This procedure, called a structural decomposition, is arbitrary, but the following analysis is done for hundreds of different structural decompositions and averaged. For a given structural decomposition, subsamples of the trajectory containing n conformations separated by a given time τ are distributed in the bins. The distribution of the n conformations in the S bins is analyzed through the occupation of each bin i (ni): the variance σ of the fractional occupation (fi ) ni/n) is calculated. For very large time separation τ, the n conformations of a subsample should be decorrelated and their distribution among the bins should have the same variance σ as the variance σ0 of a random distribution of n states in S bins. The structural decorrelation time is defined as the smaller τ for which the subsample distribution equals the random distribution ((σ/σ0)2 ) 1). The rescaled variance (σ/σ0)2 approaches the unity with increasing time τ for both the zwitterionic and cationic peptides (see the Supporting Information). The solvent and counterion positions were not considered in the analysis. The variances were calculated with subsamples of length n ) 2, 3, 4, 5, and the trajectory was decomposed into 10 bins. The curves were averaged over 400 different structural decompositions. The analyses for different subsamples lengths n agree, and the decorrelation time is about 10 ns for the cation and 15 ns for the zwitterion. Finally, 10 conformations were extracted every 20 ns from the production runs of 200 ns. 2. Results and Discussion 2.1. Hyperpolarizabilities of Single Peptide Bonds. Various optical spectroscopies were developed to study the structure and reactivity of proteins: absorbance, fluorescence, circular dichro¨ısm, and so forth. In this context, the optical properties of the small molecules containing a single peptide bond have been a subject of interest for several decades.61,76-79 The excitation energies of peptide bonds have been particularly studied with various experimental and theoretical methods. Even beyond the properties of a single peptide bond, couplings between peptide bonds play an important role in the optical spectra.80-82

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Figure 2. Frontier orbitals of formamide (see Figure 1 for orientation). The lobes are isovalue surfaces of the electronic density density of the B3LYP/6-311+ G** orbitals (0.15 or 0.17 au), and the color indicates the sign of the real wave function. Some lobes are completely hidden in the chosen representation, but the orbitals of a′′ symmetry are indeed antisymmetric relative to the {x, y} plane.

TABLE 1: Comparison of Experimental and Calculated Vertical Excitation Energies (in eV) and Oscillator Strengths for S1 and S2 Excited States of Formamide S1 (W band) Method a

CASPT2/ANO-L (8,7) MRCIb EOM-CCSDc MR-CISDd TDDFT (SAOP/QZ4P)e TDDFT (BP86/QZ4P)e Experiment (VUV)f

S2 (NV1 band)

Eexc

f (×104)

Eexc

f

5.61 5.80 5.71 5.72 5.81 5.34 5.8

10

7.41 7.94 7.66 7.60 7.81 7.41 7.0-8.0

0.37

4 10 8 7 10

0.21 0.34 0.29 0.24 0.31

a From Serrano-Andre`s et al.83 b From Hisrt et al.76 c From Fogarasi et al.85 d From Antol et al.84 e This work. f From Gingell et al.86

For the hyperpolarizability calculations, the excitation energy of the most intense πfπ* transition is a key parameter that is relatively difficult to determine precisely. Therefore, we first discuss and compare different theoretical methods in the case of formamide in Section 2.1.1. The comparison validates our TDDFT (SAOP/QZ4P) calculations, which are used in the rest of the article. In Section 2.1.2, the hyperpolarizabilities obtained with TDDFT (SAOP/QZ4P) calculations are presented for acetamide, propanamide, N-methylformamide, N-methylacetamide, and N-azolidineformamide. The results permit to discuss the effects of methyl substitution on the molecular hyperpolarizability. The influence of the laser wavelength is studied on N-methylacetamide as a representative of the family in Section 2.1.3. 2.1.1. Formamide. Amides contain a carbonyl bond (CO) conjugated with the lone pair of a nitrogen atom linked to the carbon atom. Schematically, the amide group is planar with three molecular orbitals of π -type localized on the CNO group (Figure 2): a bonding orbital (1a′′), a nonbonding (2a′′), and an antibonding orbital (3a′′). Among the σ-type valence orbitals, the nonbonding doublet of oxygen (n) lies close in energy to the nonbonding 2a′′. Two amide electronic transitions appear in the far-UV optical response of peptides and proteins: at longer wavelengths is S1, a transition which corresponds in formamide to the nf3a′′ transition and at shorter wavelengths is S2, the first πfπ* transition which is mainly the HOMO-LUMO transition. The next πfπ* excitation noted S3 (1a′′ f 3a′′ transition of formamide) is much higher in energy, close to 10 eV. Historically, the three UV absorption bands associated with these transitions have been respectively noted W, NV1, and NV2. First, we compare the vertical excitation energies obtained for S1 and S2 to available published data (see Table 1). Theoretical investigations as involved as CASPT2, CCSD, and MRCI76,83-85 resulted in excitation energies from 7.41 to 7.94 eV (Table 1). The discrepancies between the results were attributed by Serrano-Andres and collaborators to different

S1 (W) S2 (NV1) S3 (NV2)

a

Functional

Label

Transition

Weight (%)

SAOP BP86 SAOP BP86 SAOP BP86

1A′′ 1A′′ 3A′ 4A′ 10A′ 13A′

nf3a′′ nf3a′′ 2a′′f3a′′ 2a′′f 3a′′ 1a′′f3a′′ 1a′′f3a′′ 10a′f18a′ 10a′f19a′

99.8 99.7 83.5 83.7 77.1 37.3 34.7 11.3

Only transitions with weights higher than 10% are detailed.

TABLE 3: Influence of the Calculation Method and Functional on Static (Hyper)polarizabilities for Formamide with the Geometry and Orientation Defined in Figure 1a Method b

LR-BP 86 FF-BP 86b LR-SAOPb FF-SAOPb FF-CCSD(T)c

|µ|

〈R〉

βyyy

βxxx

βyyx

βzzx

βxxy

βzzy

1.55 1.52 1.68 1.46 1.51

29.0 26.2 27.6 25.1 28.4

76 80 52 41

48 50 25 18

28 34 17 16

15 16 10 7

13 12 9 7

8 9 6 8

a Dipole moment and (hyper)polarizabilities are given in atomic units. The acronyms LR refer to the linear expansion of the density, and FF to explicit finite field perturbation. b This work, QZ4P basis set. c From Alparone et al.50

treatments of the Rydberg/valence excited-state interactions, and they judged that none of the approaches was clearly more precise than the other.83 Unfortunately, the πfπ* vertical absorption energy of the peptide bond is also relatively difficult to determine precisely experimentally. The experimental absorption spectrum of gaseous formamide in the V-UV region shows five overlapping bands emerging from many transitions toward excited valence and Rydberg states.86,87 The πfπ* transition is associated to the large NV1 absorption band corresponding to energies from 7.0 to 8.4 eV with a maximum at 7.36 eV. It would be a very hard task to resolve theoretically the vibrational structure in the band: at least two conical intersections were reported along the dominant vibration of the Franck-Condon region (the C-N bond stretching).84,88,89 The present TDDFT πfπ* NV1 excitation energies are in the same range as the one obtained with wave function-based calculations, with an important effect of the functional: (7.4 and 7.8 eV for BP86 and SAOP, respectively). On the contrary, the weight on the HOMOfLUMO transition (about 84%) is relatively insensitive to the functional. For the NV2 excitation, the two functionals also depict a different decomposition on the orbital excitations (Table 2). The SAOP functional yields a larger weight on the transition 1a′′f3a′′ (77%) whereas the BP86 functional also involves 10a′f18a′′ and 10a′f19a′ transitions. The static (hyper)polarizability results are summarized in Table 3. On the basis of the discrepancies between different calculation methods, we estimate the uncertainties on the hyperpolarizability components at about 20%. The averaged polarizability values (〈R〉 ) 1/3ΣiRii) obtained by the TDDFT methods are 〈R〉 ) 27.6 au and 29.0 au with SAOP and BP86 functional, respectively, in good agreement with the theoretical value of 〈R〉 ) 28.4 au obtained from CCSD(T)/HyPol calculation50 and close to the experimental data of 〈R〉 ) 28.6 au obtained from molar Kerr constants of formamide in dioxane.90

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TABLE 4: Excited-States Energies (∆E, in eV), Oscillator f, in au) for NV1 Strength (f), and Transition Moment (µ Transition of the Studied Amides (Figure 1)a ∆E Mol.

TDDFTb

CASPT2c

expc

fb

µx

µy

1 2 3 4 5

7.81 7.65 7.24 7.01 6.01

7.41 7.21 6.71 6.76

7.4 7.4 7.0 6.8

0.29 0.18 0.20 0.16 0.11

1.0 1.2 1.0 1.0 0.9

0.1 0.4 0.3 -0.1 0.2

a The excitations were selected according to their main orbital transition (Table 2). b This work TDDFT (SAOP/QZ4P). c Serrano-Andre`s and references therein.83

Because of the planar symmetry of the molecule, all except the static hyperpolarizability components βxxx, βyyy, βxyy, βxxy, βxzz, and βyzz vanish. The value of βvec obtained by Alparone et al. using CCSD(T)/HyPol finite-field perturbation of -49 au (eq 1) is in very good agreement with the values obtained using LR-SAOP calculations (βvec(0) ) -49.6 au).

βvec

f f β · µ0 ) |µ0 |

(1)

f where f µ0 and β ) {βx, βy, βz} are the permanent dipole and hyperpolarizability vectors, respectively, with

βi )

∑

1 (β + βjij + βjji) 3 j)x,y,z ijj

(2)

In more details, the largest static value is βyyy. A static twolevel model considering exclusively the NV1 excited state (SOS2) would lead to hyperpolarizability components proportional the transition moment f µ01 squared and to the dipole f ) (µ f -µ f ) (eq 3). moment difference ∆µ 01 1 0

βijk(0) ∝ [µi01µj01∆µk01 + µj01µk01∆µi01 + µk01µi01∆µj01]

(3) For the NV1 excitation, the transition moment is almost f ) {-1.1, -0.3, 0} au with SAOP/QZ4P), aligned along x (µ f ) and the dipole moment difference is estimated at ∆µ 01 {-0.6, -1.0, 0} Debye (with SAOP/QZ4P). The static twolevel model predicts that the largest component is βxxx, with βxxy ) 38%, βyyx ) -33% and βyyy ) 12% of its value; the comparison with the hyperpolarizability components obtained with preciser methods shows that the two-level model is insufficient for the formamide (Table 3). So the TDDFT excitation energies and hyperpolarizabilities of formamide are in agreement with available experimental and theoretical data, especially using the SAOP functional. 2.1.2. Effect of Methyl Substitution. In the following section, the effect of methyl substitution on the πfπ* excitation energies and the hyperpolarizability of small amides is discussed. First, Table 4 lists the excited-state energies (∆E, in eV), oscillator f, in au) of the amides strengths (f), and transition moments (µ defined in Figure 1. As already discussed by Serrano-Andres, the substitution on the nitrogen atom leads to a considerable stabilization of the

TABLE 5: Same as Table 4 for NV2 Transition ∆E Mol.

TDDFT

1 2 3 4 5

10.80 9.10 8.39 8.79 7.71

a

CASPT2b

fa

µx

µy

10.50 10.08 9.70 9.60

0.10 0.11 0.06 0.16 0.06

-0.1 -0.6 0.5 -0.6 0.2

0.6 0.4 -0.3 0.7 0.5

a

This work TDDFT (SAOP/QZ4P). erences therein.83

b

Serrano-Andre`s and ref-

TABLE 6: Non-Zero Components and Absolute Value of the First Static Hyperpolarizability Tensor (SAOP/QZ4P) of the Amides Defined in Figure 1, in Atomic Units Mol.

βyyy

βxxx

βxyy

βxzz

βyxx

βyzz

|β|

1 2 3 4 5

50 39 55 44 50

27 8 8 -13 1

18 7 11 15 20

6 2 15 7 20

8 7 9 12 16

6 9 -1 2 4

50 32 32 23 43

NV1 state. The TDDFT transition energies decrease as 7.81, 7.24, and 6.01 eV for formamide (1), N-methylformamide (3), and N-azolidineformamide (5), and 7.65 and 7.01 eV for acetamide (2) and N-methylacetamide (4), respectively. The NV1 transition is mainly a nitrogen-to-carbon electron transfer. The excited state can therefore be stabilized by methyl substituents providing electrons to the nitrogen. Within a simplified SOS2 model, this diminution of the excitation energies would increase the hyperpolarizability. A reduction of the oscillator strength in the alkylated amides as compared to formamide is also observed. The NV1 transition is mainly a HOMO to LUMO transition, and the decrease of the oscillator strength can be attributed to the diminution of the HOMO|LUMO overlap accompanying the electron delocalization on the substituents. Within a simplified SOS2 model, this diminution of the transition moment would lower the hyperpolarizability. Similar substitution effects are observed on the NV2 transition (Table 5), but no simple electrostatic interpretation appears in this case since the transition is an intracarbonyl excitation. Table 6 lists the selected components of the static first hyperpolarizability tensor of the amides defined in Figure 1. The value |β|, defined as (β2x + β2y + βz2)1/2, diminishes when the number of substituents increases on the carbon, but no simple evolution is observed for the substitution on the nitrogen. Regarding the different components in detail, the substitution strongly affects all components, but βyyy always remains the most important. 2.1.3. Influence of Laser WaWelength. The frequency dependence of the hyperpolarizability tensor is large close to the resonance conditions. For NMA for instance, the resonance with the S0 f S1 transition is expected around 360 and 180 nm, and the resonance with the S0 f S2 transition is expected at 140 and 280 nm. Figure 3 illustrates the variations of the two largest components of the SHG tensor of NMA with varying the exciting laser wavelength. For larger wavelengths corresponding to the usual laser used in SHG microscopy or spectroscopy (λ g 500 nm), the photon frequency is too low to enter in resonance with any electronic excitation, so that the hyperpolarizability components vary only slightly in a large spectral domain. In this domain, depicted by a gray rectangle in Figure 3, the larger component of the SHG hyperpolarizability tensor is βyyy, followed by βxxx ∼ -βyyy/3. The main component of the

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Figure 4. (A) Ribbon representation of folded conformation of the [(Pro-Pro-Gly)10]3 in the crystallographic structure 1K6F.70 (B1-B3) Three conformations of the denaturated peptide (Pro-Pro-Gly)10 arbitrarily chosen among the ensemble generated by molecular dynamics (Section 1.3). Figure 3. Absolute values of some components of the hyperpolarizability β(-2ω, ω, ω) of NMA as a function of laser wavelength (λ ) 2πc/ω), calculated using LR-DFT (QZ4P/SAOP) in T convention. The arrows emphasize the increases of βxxx near by the NV1 resonances (see text for details). The gray inset specifies the spectral domain used for usual SHG spectroscopy or microscopy.

hyperpolarizability tensor at 800 nm is not aligned with the CN bond (the x-direction). This is no longer valid for the lower wavelength in the UV domain (λ e 400 nm), where resonances play an important role. Close to resonances, the calculations correctly predict large variations of the hyperpolarizability, even if the linear response approximation fails. As the dipole moment of the NV1 transition is mainly along the x-direction, the resonances with NV1 excitation can be detected by a divergence of βxxx (see arrows in Figure 3). At these particular wavelengths, the main component of the β tensor is βxxx, and the two-level model becomes suitable. The large difference between Simpson’s input tensor (at λ ∼ 180 nm) and the present one (βTDDFT at 800 nm) originates in the different excitation energies. The main component of their hyperpolarizability tensor is parallel to the transition moment of the NV1 electronic transition of the peptide bond (i.e., parallel to the CN bond, along the x-direction), while the tensors obtained with TDDFT at 800 nm have a larger component along the y-direction. Finally, the TDDFT calculations permit to discuss the validity of the Kleinman symmetries,91 which are often taken as granted to simplify analytical calculations of SHG signals. The Kleinman rules, stipulating that the indices of the susceptibility may be freely permuted (βijk ) βikj ) βjki ) βjik ) βkij ) βkji) are strictly valid only for the static limit (ω ) 0). For the reference molecule NMA, the Kleinman symmetries are verified within 5% at 800 nm. To conclude, the dispersion effects are large in the UV domain, but in the visible and near-IR domain usually available for the lasers used in SHG applications, the dispersion effects are relatively mild. In the following, only the results obtained for an exciting wavelength of 800 nm are presented. 2.2. Application of the Additive Model to (Pro-Pro-Gly)10. Collagens are the major proteins of the extracellular matrix of mammals (about 25% of all proteins) and play a central role in the formation of networks and fibrils involved in the architecture of organs. Its molecular and supramolecular structures have been studied for more than 50 years.92 The molecule contains many helical domains, composed of three chains wrapped around each other in a right-handed helix. Each of these chains is itself a left-handed helix similar to poly-proline-II helices. Close packing of the chains in the ropelike superhelix is ensured by the presence of glycine in the sequence, which follows the pattern (Gly-Pro-Y)n or (Gly-X-Hyp)n, where X and Y may be any of various other amino acid residues. The conformation is known with high precision from X-ray scattering patterns of

model triple-helical peptides such as [(Pro-Pro-Gly)10]393,94 (Figure 4 A). Various changes of collagen structures come along during organ aging or pathologies. To propose and evaluate relevant therapies, it is crucial to characterize the collagen 3D distribution and structure. Nonlinear optical microscopy has been successfully developed for this purpose. As no staining and no conjugated residues are needed to observe the second harmonic signal, the backbone is most probably the harmonophore. The following section reports theoretical calculations of the SHG signal of the peptide (Pro-Pro-Gly)10. As detailed in Section 1.2, the theoretical model is based on the assumption of the additivity of the hyperpolarizabilities of the peptide bonds of the protein backbone. Such an investigation may be seen as a first step toward more accurate approaches that account for the possible contributions of excitonic or polarization effects. The discussion does not focus on the validity of the additivity hypothesis but on the influence of the input parameters and of the backbone conformation on several hyperpolarizability ratios. We first compare the conformations of a collagen model peptide in its native state (X-ray crystallography data93,94) to a denaturated state (molecular dynamics simulations). Then, the theoretical nonlinear optical properties of the different conformers are described and confronted with experimental data by Deniset-Besseau et al.46 2.2.1. Link between Experimental and Theoretical Data. To permit a more quantitative interpretation of the SHG microscopy images, Deniset-Besseau et al. have measured the molecular hyperpolarizability of type I collagen in its native state and after thermal denaturation and of the model peptide (Pro-Pro-Gly)10 in its triple-helical conformation.46 They performed hyperRayleigh scattering experiments, which have become a standard to measure second order hyperpolarizabilities of solubilized molecules.95 When the solvent is used as an internal reference, the hyperpolarizability of the protein βp is deduced from the hyperpolarizability of the solvent βs and the SHG intensity I(2ω) at different solute concentrations using the relationship

I(2ω) ) G(Nsβs2 + Npβ2p)I(ω)2 where G is a geometrical factor, I(ω) the exciting light intensity, and Ns and Np the number of solvent and solute molecules in the excited volume. The collected intensity depends on the polarization orientation of exciting and detected lights. In the work by Deniset-Besseau et al.,46 the exciting beam is polarized along Z and propagates along Y, and the second harmonic signal is collected at an incidence of 90 degrees (X direction). Then βp2 can be rewritten as 〈|βYZZ|2 + |βZZZ|2〉, where βYZZ and βZZZ are the tensor components of the molecular hyperpolarizability



in the laboratory frame, and the notation 〈 · 〉 stands for the averaging over molecular orientations in the excited volume. The dependence of the signal intensity on the exciting light polarization also permitted to determine the ratio between tensor components in the laboratory frame, that is, the macroscopic depolarization ratio D ) 〈|βYZZ|2〉/|〈βZZZ|2〉. In general, no complete characterization of the hyperpolarizability tensor is expected. But for symmetric molecules, the number of independent tensor components is reduced, and all the relevant ratios may be experimentally determined. Moreover, for symmetrical molecules, the relationship between the tensor in the laboratory frame {X, Y, Z} and in the molecular frame {X, Y, Z}96-98 is simplified. For example, if the molecule belongs to the C2V symmetry group, its hyperpolarizability tensor has only three independent tensor elements βZZZ, βZXX, and βZYY, where Z is the C2 axis. The depolarization ratio D can then be written as a function of uX ) βZXX/βZZZ and uY ) βZYY/βZZZ:

D)

3 - 2(uX + uY) + 11(uX2 + uY2 ) - 2uXuY 15 + 18(uX + uY) + 27(uX2 + uY2 ) + 18uXuY

(4) If X and Y directions are equivalent as in the C∞V point group (uX ) uY ) u), eq 4 simplifies to

D(u, u) )

3 - 4u + 20u2 15 + 36u + 72u2

(5)

This equation has been applied to deduce u from the experimentally measured D in case of the collagen model peptide [(Pro-Pro-Gly)10]3.46 Absolute values of hyperpolarizability obtained experimentally and theoretically are difficult to compare because of different reasons already discussed in the literature.66,99 Nevertheless, ratios of hyperpolarizability can be compared more easily, for example, the ratio of hyperpolarizability |β| between the native and denaturated states, the macroscopic depolarization ratio D,96,98 and the in-plane anisotropy u ) βZXX/βZZZ of the folded triple-helix.97 2.2.2. NatiWe and Denaturated Conformations. The collagen-model peptide chosen is (Pro-Pro-Gly)10, which crystalline structure is referenced in the Protein Data Bank (PDB) as 1K6F.70 Its triple-helix conformation is similar to the collagen one, with Ramachandran angles averaging to ΦR ) -75° and ΨR ) 170°. The values are slightly different for the prolines and for glycines, but they are well conserved along the chain and among the other collagen-like peptides with varying sequences available in the PDB (1CDG, 2CUO, 1WZB, 2D3F, 2D3H). For a single helix in the native state, the values of the asphericity (δ) and acylindricity (S) as proposed by Aronovitz100 are δ ) 0.9 and S ) 1.8, very close to the ideal values of a rod (δ ) 1, S ) 2). For the denaturated conformations produced by molecular dynamics simulation, the mean values of the Ramachandran angles are slightly different and characteristics of an all-trans polyproline-II conformations (PPII: ΦR ) -75° and ΨR ) 155°). The denaturation of the triple helix also comes with a broadened distribution of the Ramachandran angles (not shown). Three representative conformations of the denaturated peptide depicted in Figure 4 B illustrate that the rodlike helical structure is completely lost. The asphericity (δ) and acylindricity (S) of the zwitterionic and cationic polymers are similar: δ ) 0.36

TABLE 7: Hyperpolarizability (SHG β in 10-30 esu, in B Convention for a Laser Wavelength of 800 nm), Depolarization Ratio D and In-Plane Anisotropy u of a Single Peptide Chain (Pro-Pro-Gly)10 for Input β Tensors of Molecules 1, 4, and 5a Denaturated Mol. 1 4 5

|β| D u |β| D u |β| D u

Helical

Zwitterion

Cation

2.7 0.51 -0.3 3.5 0.15 0.9 1.85 0.46 -1.3

0.9 ( 0.2 0.25 ( 0.1

0.82 ( 0.2 0.25 ( 0.05

1.3 ( 0.2 0.4 ( 0.1

0.9 ( 0.2 0.5 ( 0.1

1.3 ( 0.15 0.45 ( 0.2

1.10 ( 0.20 0.45 ( 0.2

a The helical conformation was extracted from the PDB crystalline structure 1K6F. For the denatured conformations, average and variance were calculated on 10 independent conformations obtained by molecular dynamics simulations (Section 1.3).

and 0.29, respectively, and S ∼ 0. These values are neither those of a spherical object (δ ) 0, S ) 0) nor those of a self-avoiding random coil (δ ) 0.43, S ) 0.55101). 2.2.3. Hyperpolarizabilities Using TDDFT Input Parameters. The additive method proposed by Simpson and co-workers to calculate the hyperpolarizability tensors of protein backbones was applied to the peptide (Pro-Pro-Gly)10. To get insight on the influence of the parameters, the β values at 800 nm of molecules 1, 4, and 5 of Figure 1 have been used as input. The values of the hyperpolarizability |β| and the hyperpolarizability ratios D and u are summarized in Table 7. A tendency is observed in the theoretical prediction of the additive model, independently of the input β of the peptide bond: the |β| of the peptide in the helical conformation is two to three times the mean values obtained with the ensemble of denaturated conformations. This tendency is also observed experimentally for the rat-tail collagen (triple helices of (Pro-Pro-Gly)337): the experimental value of |β| of a single chain is divided by about 2.5 upon denaturation. Nevertheless, a quantitative comparison between data concerning peptides of different lengths should be avoided because of intermolecular destructive interference effects present in the (Pro-Pro-Gly)337, which are negligible in the (Pro-Pro-Gly)10 and neglected in the present additive model. Given the length differences of the molecule, no quantitative agreement is expected. Nevertheless, the qualitative tendency is clear: denaturation lowers hyperpolarizability, and the SHG intensity, which is proportional to the hyperpolarizability squared, almost disappears. This signal decrease was interpreted by Deniset-Besseau et al. as a loss of orientation correlation between the harmonophores during denaturation.46 The present simulations sustain this interpretation, as the helical structure is completely lost after 5 ns of simulation (Figure 4 B). The results obtained for the macroscopic depolarization ratio D and in-plane anisotropy u are very sensitive to the input β tensor: the depolarization ratio of the helical peptide varies from 0.13 to 0.38 depending on the model, and the in-plane anisotropy varies from -1.3 to 1.0. The experimental depolarization ratio obtained for native [(Pro-Pro-Gly)10]3 is 0.11 ( 0.02. The theoretical results obtained with the input tensors of 1 and 5 are not compatible with the experimental data. On the contrary, the results obtained using 4 are in reasonable agreement with experimental data: The depolarization ratio of the triple-helical conformation is calculated as low as 0.15, with a u value of 0.9. The increase of depolarization ratio upon denaturation is

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TABLE 8: Components of the First Hyperpolarizability Tensor of the Triple Helical Collagen Model [(Pro-Pro-Gly)10]3 (in Atomic Units, at 800 nm, with NMA as the Building Block) βZZZ

βXXZ

βYYZ

βXZX

βYZY

βZXX

βZYY

-893

-970

-967

-970

-967

-985

-982

βYYY

βYXZ

βXXX

βZYX

βYZZ

βZZX

βXXY

-170

92

59

33

-13

-14

2

also well reproduced using NMA as a building block. Using the hypothesis of a C∞V object for the folded triple helix [(ProPro-Gly)10]3, Schanne-Klein and co-workers deduced from the depolarization ratio D a value of u in the range of [0.2:0.7].46 The agreement between the experimental data and the theoretical result is not quantitative, but the order of magnitude and the sign are well reproduced. Thus, in the following, we focus on the results obtained with the β tensor of NMA as an input. The different hyperpolarizability components of the triple helix from 1K6F are detailed in Table 8, with Z as axis of the triple helix. As expected, the calculated response of the triple-helix is close to the one of an infinitely long rod. Such a C∞V object has only two independent hyperpolarizability components βZZZ and βXXZ, with

βXXZ ) βYYZ ) βXZX ) βYZY ) βZXX ) βZYY

Figure 5. Molecular hyperpolarizability |β|, macroscopic depolarization ratio D, and in-plane anisotropy u obtained by the additive model for the triple helical structure [(Pro-Pro-Gly)10]3 (1K6F) using a simplified β input tensor with only two nonzero components βyyy and βxxx with βyyy/βxxx ) tan(θ) as a function of θ.

(6)

The other components are null. The large components in Table 8 almost fulfill eq 6 ((10%), and most other components had absolute values of less than 3% of |βZZZ|. A few components are not negligible, especially |βYYY| and |βYXZ|, but the SHG response of the helical peptide is indeed close to the one of a C∞V object. 2.2.4. Hyperpolarizabilities Using Simplified Input Tensors. The previous results show that the additivity model is highly dependent on the input parameters, at least in the case of collagen triple-helical coiled-coil. The origin of this dependency was investigated through the calculation of D and u obtained for the triple helical structure [(Pro-Pro-Gly)10]3 (1K6F, pdb structure) using simplified β input tensors. First, we choose a tensor with only two nonzero components βyyy and βxxx related through the equation βyyy/βxxx ) tan(θ) (see Figure 5). The three observables |β|, D, and u show a singular behavior near θ ) 58°: |β| approaches zero, D reaches a peak, and u diverges. Near the singularity, the contributions of the two peptide-bond components βyyy and βxxx to the triple-helix tensor component βZZZ cancel, so that the main component of the helix tensor βZZZ shrinks, the effective |β| collapses, and the in-plane anisotropy u ) βZXX/βZZZ diverges. The depolarization ratio also loses then its peculiar value of about 0.12. Far from this special point, the depolarization ratio D is relatively independent of the value of βyyy/βxxx. The input β tensor of 1 does not fit exactly in the model depicted by Figure 5 because some components other than βyyy and βxxx are not strictly zero. But if these are neglected, the θ value is 61°, which is very close to the singular point. Within this simplified model of input parameters, the angle domain which matches best the experimental data is θ = 40 ( 10°. The same type of analysis was done with another simplified input: βyyy ) cos(θ), βxyy ) βxzz ) βyzz ) sin (θ) (all the other components are zero except the ones imposed by Kleinman symmetries91). In this case, the variations of D, u, and |β| show a singularity point near θ ) 18° (Figure 6). The hyperpolariz-

Figure 6. Molecular hyperpolarizability |β|, macroscopic depolarization ratio D, and in-plane anisotropy u obtained by the additive model for the triple helical structure [(Pro-Pro-Gly)10]3 (1K6F) using a simplified β input tensor with βyyy ) cos(θ) and βxyy ) βxzz ) βyzz ) sin(θ), as a function of θ.

ability tensor of 5 happens to be close to this singularity (θ ) 20°). Noticeably, this second simplified model of input parameters with θ = 130 ( 20° also gives results in reasonable agreement with experimental data. Finally, the results permit to test the validity of eq 5 using various input parameters and various peptide structure. The relationship between D and u obtained for the collagen triple helix (1K6F) obtained within the additive model is compared to eq 5 in Figure 7. Additionally, the results obtained for a sequence extracted from the myosin tail forming R-helical coiled-coil (2FXM)102 have also been shown. For both types of helical structures, the additive model and eq 5 agree, even for cases near the singularity points where the in-plane anisotropy diverges. 3. Conclusion and Perspectives First, we have investigated, by using a TDDFT method, the linear and nonlinear optical response of small molecules



Figure 7. Macroscopic depolarization ratio D as a function of the in-plane anisotropy u obtained by the additive model for the helical structures of [(Pro-Pro-Gly)10]3 (1K6F) and SII-∆ myosin (2FXM) using a simplified β input tensor with only two nonzero components βyyy and βxxx. Comparison is made with the analytical formula obtained for a C∞V (eq 5). The lower panel is extracted from the upper panel and enlarged. Filled symbols emphasized with arrows are the results obtained with NMA as the building block.

containing one peptide bond. The evaluation of the effect of various chains on the optical response has been done by substituting methyl groups on the peptide bond. The various chains have a strong and complex effect on hyperpolariz-

abilities (|β| varies by more than a factor two among the five amides studied), but in the static limit, some general trends could be recognized: (1) the main component remains the βyyy, where y is directed in the peptide bond plane, perpen-

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dicular to the C-N bond; (2) substitution on the carbonyl side lowers the hyperpolarizability. For the N-methylacetamide, these conclusions remain valid for the wavelength domain usually used in SHG imaging (λ g 500 nm). These results on small amides were used as an input to calculate the hyperpolarizability of a triple-helical peptide modeling collagen using a simplistic additive model: it is assumed that the backbone makes the dominant contribution to the hyperpolarizability of the peptides and that its contribution can be decomposed into the sum of the individual contributions of the peptide bonds. This model had been proposed by Simpson and co-workers and applied to the study of the hyperpolarizabilities of R helices and β sheets near the resonance of the peptide bond (λ = 180 nm).49 The present study is complementary to their work because it treats a different secondary structure (the collagen triple-helix), for a different exciting wavelength (λ ) 800 nm). We have shown that the results at these two wavelengths are expected to differ substantially because of the different hyperpolarizability tensors of the peptide bond at NV1 resonance and far from any resonance. For the triple-helical peptide modeling collagen, the experimentally observed diminution of the signal upon denaturation is well reproduced by our calculations. Interestingly, the building block with the smallest |β| as an isolated molecule (NMA) leads to the largest one for the collagen additive model. Indeed, not only |β|, but also the repartition of the hyperpolarizability among the components influences the response: the depolarization ratio and anisotropy strongly depend on the input hyperpolarizability tensor used for the peptide bond. The investigations of these dependencies show that, within the additive model, the hyperpolarizability of the helices may show singularities (the in-plane anisotropy diverges and the depolarization ratio is near a maximum). The comparison of the theoretical results with experimental data published recently on the peptide [(Pro-ProGly)10]3 modeling collagen permitted to select N-methylacetamide as a reasonable first guess for the building block of Simpson’s model. The two other molecules used as building blocks (formamide and N-azolidine formamide) approach singularities of the additive model in the case of the collagen structure that are not physically relevant. Our results on the simple amides suggest that the hyperpolarizability of the peptide bond is sensitive to its environment. Therefore, further theoretical and experimental data would be needed to test the validity of the additivity hypothesis and the transferability of the model parameters for secondary structures different from the collagen triple-helices. Future studies are directed toward models including excitonic and polarization effects. Acknowledgment. The authors thank L. Haupert, G. Simpson, and R. Heiland for their help in using NLOPredict and J. Duboisset, P.-F. Brevet, A. Deniset-Besseau, and M.-C. SchanneKlein for stimulating discussions. Calculations were performed in PSMN (Poˆle Scientifique de Mode´lisation Nume´rique), CINES (Centre Informatique National de l’Enseignement Supe´rieur), and IDRIS (Institut du De´veloppement et des Ressources en Informatique Scientifique), which are acknowledged for generous computation time allocation (project x2009085142) and technical help. Supporting Information Available: Optimized geometries for the amides, hyperpolarizability tensor of NMA (SAOP/ QZ4P), and influence of the basis set and functional on several observables for formamide. This material is available free of charge via the Internet at http://pubs.acs.org.

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