Vibrational Spectroscopy of Biomolecules by mixed

Quantum / Classical Molecular Dynamics

Daniele Bovi,1 Riccardo Spezia,2 Rodolphe Vuilleumier,3 Alberto

Mezzetti,4 Marie-Pierre Gaigeot,2 and Leonardo Guidoni1, 5

1Universita di Roma “La Sapienza”,

Dipartimento di Fisica - P.le Aldo Moro 5, 00185 Roma, Italy

2Laboratoire Analyse et Modelisation pour la Biologie et l’Environnement,

CNRS UMR 8587, Universite d’Evry Val d’Essonne, 91025 Evry Cedex.

3Ecole Normale Superieure - Chemistry Department

- 24 rue Lhomond, F-75231 Paris Cedex 05, France

4Laboratoire de Spectrochimie Infrarouge et Raman - UMR CNRS 8516,

Universite de Sciences et Technologies de Lille, France.

5Universita degli studi dell’Aquila - Dipartimento

di Chimica, Ingegneria Chimica e Materiali

1

I. INTRODUCTION

Vibrational spectroscopy is a valuable experimental probe to investigate the static and

dynamical behaviour of biomolecules in their complex environments. In this respect, mixed

Quantum Mechanics/Molecular Mechanics (QM/MM), based on ab initio molecular dy-

namics methods1,2 offer a computational tool to help the interpretation of the experimental

data. Within the QM/MM scheme, the quantum mechanical calculations, which represent

the most computationally demanding, part can be restricted to the portion of the system

that is directly involved in the vibrations, whereas all the rest of the system is treated

at the classical force-field level. Temperature, solvent (or protein) environment effects are

therefore explicitly considered. Vibrational frequencies and effective normal modes were ob-

tained directly from QM/MM dynamics at finite temperature using the approach described

in reference3. It consists in spectrum decomposition of localized peaks on frequency. This

procedure is performed by the molsim code, which will be soon released as a free code to

the community under GPL licence. During the HPC-Europa2 project, we introduced in the

molsim code a module to support the user in the the definition of the Pulay coordinates4 of

complex molecules. This user-friendly wizard uses standard topology files of biomolecules

as input and automatically assigns symmetries and internal coordinates, a crucial procedure

to process large biomolecules.

We applied the above methodology to calculate Infrared and Raman scattering frequen-

cies of the Peridinin molecule by first principles QM/MM dynamics at room temperature.

Peridinin molecules are Light-harvesting (LH) complexes, involved in the light collection

process into the peridinin-chlorophyll-a-protein (PCP). Step-scan FTIR results in the PCP

protein5,6 have pinpointed the difficulties in the precise assignment of bands of peridinin in

the complex. One strategy is to rely on comparison with IR or Raman data for isolated

peridinin. If these data are available for several solutions of peridinin in solvents with differ-

ent physicochemical properties (notably concerning polarity and capability to give hydrogen

bonds) an additional (and crucial) piece of information is added, i.e. a measurement of the

effect of the surrounding environment on band positions. To this aim we studied three differ-

ent Peridinin solutions: Perdinin in cyclohexane (an apolar/aprotic solvent), in acetonitrile

(a polar/aprotic solvent), and in methanol (a polar/protic solvent). It is important to men-

tion that a full understanding of the influence of the surrounding environment on vibrational

2

bands may help to rationalise why Resonance Raman bands for peridinin in PCP complexes

in vivo recorded on different microorganisms are placed at different wavenumbers7,8 . We

have performed QM/MM simulations at 300 K temperature for each system using Density

Functional Theory to describe the Peridinin molecule and Amber force-field to describe the

different solvents. We used the QM/MM version of the program CPMD9 using parallel runs

up to 64 cores at the CINES (Jade) supercomputing centre in Montpellier (France). The

computational results were successfully compared with the experimental spectra obtained

at the University of Lille.

II. METHODS

A. QM/MM colculations

In order to describe the environment influence on electronic displacement regarding the

Peridinin molecule we used a multiscale model, in which the whole system is divided into a re-

strict atomic group (QM region), composed by only one solute molecule, and a solvent (MM

region). This method is able to concentrate the computational effort on the subsystem that

requires the ab-initio description, without dispersion of resources and lost of local interac-

tions induced by the surrounding environment. We performed QM/MM Born-Oppenheimer

dynamics, using the QM/MM Hamiltonian1,2 defined by

H = HQM + HMM + HQM/MM (1)

where the three terms are respectively the hamiltonian of the isolated quantum system, the

hamiltonian of the isolated classical system and the interface hamiltonian describing the

interaction between them. The latter gives an additional potential to both QM and MM

systems.

The purely classical energy EMM of the corresponding term in hamiltonian (1) is described

by standard Force Fields. In our case we used Amber Force Field18,19 that can be divided

in bonded atoms and non-bonded ones.

EMM = Ebonded + Enon−bonded (2)

3

The first one assumes the general parametric form

Ebonded =∑

r∈bonds

Kb(r − r0)2 +

∑θ∈angles

Kθ(θ − θ0)2 +∑

φ∈dihedrals

Vn2

[1 + cosnφ− γ] (3)

These are effective harmonic potentials describing bond, angle and dihedral strengths.

Whereas the interaction between non-bonded atoms equals to

Enon−bonded =∑

i < j

∈ MM

[(AijR12ij

− Bij

R6ij

)+qiqjεRij

](4)

where the first term represents the Lennard-Jones potential, describing the van der Waals

interactions, and the latter the Coulombic electrostatic interactions.

The QM system, described by HQM in eq. (1), obeys to the quantum mechanics laws.

This has been solved by using DFT in the framework proposed by Kohn and Sham10. The

valence electrons equations are described by KS orbitals, expanded into the plane-waves

basis set, with a cut-off radius of 70 Ry, using the PBE functional. Trouiller-Martins

normconserving Pseudo-potentials11 were employed by Kleinman-Bylander scheme12 in

order to include the effects of the core electrons.

The additional term HQM/MM in eq. 1 defines the coupling interaction between the QM

and MM system, behaving as external potential in both systems. On the whole the QM/MM

method is able to perform both interface situations: between bonded atoms and between

non-bonded atoms1. In our special case we are only interested in the interface description

of non-bonded atoms. Therefore we have to take in account the electrostatic interaction

between the electron density of the quantum atoms and the effective charges of the classical

environment as well as the van der Waals interations as described by the classical force field.

These assume the form

EQM/MM =∑i∈MM

qi

∫d~r ρ (~r) νi (|~r − ~ri|) +

∑i ∈ MM

j ∈ QM

vvdw (rij) (5)

4

in which qi is the classical charge pointed on his atomic center ri and νi (|~r − ~ri|) the Coulom-

bic potential modified at short-range to avoid the spill-out of the electron density1 in the

presence of nearby positively charged classical atom. Nevertheless the direct evaluation of

the integral (5) is an heavy calculation, since it scales as the number of classical atoms.

Therefore we divide the MM system in two or more concentric shells around the QM atoms.

Only the inner area, the nearest one to the QM region, interacts by eq.(5). In contrast we

calculate the electrostatic interaction between the QM system and the outer area by fitting

the electron density on each quantum atom or even on multipole charge development of the

whole quantum system.

The van der Waals interaction is explicitly treated as Lennard-Jones potential as shown in

eq. (4).

B. Vibrational Analysis

We have obtained vibrational frequencies from both optimized geometries, in gas phase

and implicit solvents, as well as molecular dynamics. From optimized geometries vibrational

bands can be easily determined by the Hessian diagonalization. This method corresponds to

the standard normal mode analysis which is always considered a good description of isolated

systems at zero temperature. Thereby it contains the harmonic approximation, therefore it

can create problems at room temperature in condensed phase. In particular, anharmonicities

are not considered. Moreover the atomic details and dynamical features of the solvent are

not taken into account.

Molecular dynamics can circumvent these problems, as was shown in both liquid and

gas phases20 . Vibrational information can be extracted from Fourier transform of atomic

velocity auto-correlation function

VDOS(ω) =N∑i=1

∫ ∞−∞dt 〈vi(t)vi(0)〉 exp (iωt) (6)

where i runs over all atoms of the investigated system. Considering a partial VDOS, cal-

culated over a subsystem of one or more atoms, we can obtain specific information on

vibrational properties. This kind of analysis can highlight the contribution of each atom

to vibrational bands. In a similar way one is able to generalize this approach to internal

molecular variables (bonds, angles, dihedrals) as well as to linear combinations of these,

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represented by any coordinate linear transform

qk =N∑i=1

Zkixi (7)

where the coefficients Zki are the elements of the transformation matrix. In order to find

that set of variables qk describing the effective normal modes, we begin by writing the VDOS

in terms of the transformed coordinates

VDOS(ω) =N∑i=1

∑k, l

ZikZil

∫ ∞−∞dt 〈qk(t)ql(0)〉 exp (iωt) (8)

and modifying the matrix elements Z at the aim to localize each partial VDOS on only one

resonating frequency. For this purpose it is useful to introduce the functional

Ω(n) =∑k

[β

2π

∫ ∞−∞

dω|ω2n|P qk (ω) −

(β

2π

∫ ∞−∞

dω|ωn|P qk (ω)

)2]

(9)

with P qk (ω) being the partial VDOS of the transformed variables qk. From eq. 9, it

becomes clear that the minimum value of Ω(n) corresponds with a set of localized spectra.

In this case the variables qk are the normal coordinates. It is possible to replace the

minimum research by a generalized eigenvalue problem3,20 . This calculation has been

made by the computational code molsim. Nevertheless we need to underline that the

definition of the variable basis set is a tricky choice, since a wrong set could lead to

a fictitious convergence not in the absolute minimum. For large systems (a few tens

of atoms) it is not evident how to choose a complete basis set. Therefore we insert in

the code a helpful wizard which uses some rules4 to reach a reasonable definition of variables.

From molecular dynamics it is furthermore possible to obtain the Infra-red spectrum

through the Fourier transform of the dipole-dipole correlation function

IIR(ω) =2πβω2

3cV

∫ ∞−∞

dt〈D(t) ·D(0)〉 exp[iωt] (10)

We should note that vibrational spectra, obtained from DFT-based molecular dynamics,

do not need scaling factors to recover experimental data, but just a constant shift over the

whole spectrum. At least this is true for the correctly sampled frequency region. In fact when

using scaling factors this is because of errors regarding the potential energy surface deriving

from the chosen functional and the anharmonicity. In molecular dynamics anharmonicity

6

is considered, thus the gap with respect to experiments derives from those potential energy

surface errors which are not so largely frequency dependent as those ones deriving from

anharmonicity.

III. RESULTS AND DISCUSSION

In order to study the Peridinin vibrational properties by QM/MM molecular dynamics in

different solvents, we have introduced a Peridinin Model System (PMS) (fig. 2(b)), allowing

us to reduce the QM system size. Referring to figure 2(b) we point out the bond groups

composing the vibrational modes on which we will refer later on.

Figure 1

(a) Peridinin Molecule structure

(b) Peridinin Model System structure

- The Carbonyl, which is the most environmental interacting bond as an effect of its

high polarity, corresponds to the stretching between C5 and O20;

- The Allene is the group of three Carbon atoms (C15, C16, C17), interesting for the

asymmetric stretching between the bonds C15-C16 and C16-C17;

- The Polyene chain, composed by the sequence of 14 alternating single and double bonds

(from C15 to C6 and C2-3-4-18-19), establishes a conjugated system corresponding to

7

several collective modes. In particular we are interested in 7 modes of them, indexed

as Chain(n) later on.

A. Gas phase and PCM calculations

The PMS (fig. 2(b)), used in the simulations, is able to catch key vibrational properties5 of

the Peridinin (fig. 2(a)), although the frequencies of this model system may have some shift.

In order to have control over our model results, we have analyzed the vibrational properties of

the Peridinin and PMS in gas phase. For thus reason, Peridinin and PMS were optimized in

gas phase using DFT and 6-31g basis set. We employed two functionals: the popular hybrid-

GGA B3LYP13,14 and PBE15 . The latter was employed also in the QM/MM molecular

dynamics calculations. Vibrational frequencies were obtained by standard diagonalization

of Hessian matrix. PBE data results are displayed in the table I.

Table I. Gaussian frequencies (cm−1) of the Peridinin and the PMS in gas phase.

System Chain(1) Chain(2) Chain(3) Chain(4) Chain(5) Chain(6) Chain(7) Carbonyl Allene

Peridinin 1525.4 1542.2 1560.4 1606.1 1612.8 1633.9 1656.9 1711.5 1963.8

PMS 1528.3 1548.1 1561.6 1612.2 1617.3 1635.2 1655.7 1716.7 1994.2

As expected, the larger effect is observed in the Allene frequency shift. This is reasonable

because the position of this group is close to the missing part of the Peridinin.

As shown in table II, PMS was further considered with three implict solvents by standard

PCM16: cyclohexane, acetonitrile and methanol.

Table II. Gaussian frequencies (cm−1) of the PMS in different PCM solvents.

System Chain(1) Chain(2) Chain(3) Chain(4) Chain(5) Chain(6) Chain(7) Carbonyl Allene

Cyclohexane 1525.2 1545.4 1558.3 1610.6 1614.8 1633.1 1654.7 1701.8 1990.6

Acetonitrile 1515.7 1538.4 1551.2 1606.3 1608.9 1628.6 1651.3 1672.3 1983.9

Methanol 1515.6 1538.4 1551.1 1606.2 1608.8 1628.6 1651.3 1671.5 1983.9

All calculations were performed with Gaussian09 package17.

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B. QM/MM Dynamics

QM/MM molecular dynamics were used to obtain microscopical and dynamical envi-

ronmental effects on Peridinin vibrational properties. We used the reduced model of Peri-

dinin shown in figure 2(b). The relationship between Peridinin and the model, employed in

QM/MM molecular dynamics, was investigated by comparing gas phase vibrational proper-

ties on the optimized geometries of the two systems (tab. I).

Our QM/MM partition was done by using a QM description, at DFT level, of the model

peridinin and a MM description of the different solvents.

The systems were build by using for the solute the optimized structure in gas phase

and then solvated with ∼ 103 solvent molecules. The systems were thermalized by classical

NVT simulations of about 0.5 ns where only the solvent molecules were allowed to move.

After they were further equilibrated via classical NpT dynamics with position restraints on

solute coordinates for about 2 ns, thus obtaining a box size compatible with standard pres-

sure. Periodic boundary conditions were employed and the box dimensions for the classical

part were in nm 4.1 x 4.2 x 4.7 (Cyclohexane), 5.9 x 5.7 x 7.1 (Acetonitrile) and 6.1 x 5.8 x 7.3

(Methanol).

Then QM/MM molecular dynamics was produced, via Born-Oppenheimer method for

QM region dynamics. The QM part is described via plane-waves, thus we employed the box

(17.0 x 13.4 x 28.4 A) for plane-waves representation. A time step of 0.36 fs was used and,

after a QM/MM NVT equilibration of 1.4 ps, the NVE production run was propagated for

about 12 ps. In figure 2 we show one snapshot of a produced dynamics.

C. Peridinin Vibrational Property

Referring to the method proposed by R.Vuilleumier and M.-P.Gaigeot20 and described in

sec. II B, VDOS decompositions were made to assign the vibrational bands to the effective

normal modes. The VDOS were obtained from molecular dynamics trajectories of all three

systems, PMS in cyclohexane, Acetonitrile and Methanol. Both the definition of internal

variables and the localization procedure of the spectra were carried out by the molsim code,

using the Ω functional parameter n = 2 (eq. 9). We obtained well localized peaks, as shown

in figure 3.

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Figure 2. Dynamics snapshot of the PMS in Methanol solution. The formation of an H-bond of the

carbonyl (mode of the blue arrows) with the Methanol (highlighted by the yellow) is shown. The electron

deformation, induced by the external field, is further shown in the difference density map between the

solution system and the isolated Peridinin (positive difference in green and negative one in tan).

Figure 3. Example of molsim result for the VDOS decomposition of the PMS in Acetonitrile solution

QM/MM molecular dynamics.

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While the assignment of the carbonyl and allene bands results clear enough, for the

polyene chain it is trickier because of its conjugated state that gives rise to several

interconnected modes delocalized over the carbon chain.

Our results point out the influence of the external electrostatic field, produced by the

solvent, on the vibrational spectrum of the Peridinin. The carbonyl band is shifted towards

lower frequencies as shown in figure 4.

Figure 4. Carbonyl peak shift, taken from the total VDOS of the three systems: PMS in cyclohexane,

deuterated acetonitrile and methanol.

The carbonyl frequency shifts also between the acetonitrile and the methanol solution, in

contrast with Gaussian results of tab. II. This is caused by the formation of a stable H-bond

between the methanol OH and the carbonyl CO, that cannot be taken into account by the

Gaussian PCM calculations. An explicit treatment of the dynamical effects of the solvent

is therefore important to properly describe the Peridinin vibrations in different solvents.

11

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