Theoretical Insights into Sub-Terahertz Acoustic Vibrations of

Subscriber access provided by UB + Fachbibliothek Chemie | (FU-Bibliothekssystem)

Letter

Theoretical Insights into sub-Terahertz Acoustic Vibrations of Proteins Measured in Single Molecule Experiments Adrien Nicolaï, Patrice Delarue, and Patrick Senet J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b01812 • Publication Date (Web): 28 Nov 2016 Downloaded from http://pubs.acs.org on November 28, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry Letters

Theoretical Insights into sub-Terahertz Acoustic Vibrations of Proteins Measured in Single Molecule Experiments Adrien Nicola¨ı, Patrice Delarue, and Patrick Senet∗ Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Univ. Bourgogne Franche-Comté, 9 Av. A. Savary, BP 47 870, F-21078 Dijon Cedex, France. E-mail: [email protected]

1

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Proteins are an important class of nanobioparticles with acoustical modes in the sub-THz frequency range. There is a considerable interest to measure and establish the role of these acoustical vibrations for the biological function. So far, the technique providing the most detailed information about the acoustical modes of proteins is the very recent Extraordinary Acoustic Raman (EAR) spectroscopy. In this technique, proteins are trapped in nanoholes and excited by two optical lasers of slightly different wavelengths producing an electric field at low-frequency (< 100 GHz). We demonstrate that the acoustical modes of proteins studied by EAR spectroscopy are both infrared and Raman active modes and we provided the interpretation of the spectroscopic fingerprints measured at the single molecule level. Combination of the present calculations with techniques based on the excitation of a single nanobioparticle by an electric field, as the EAR spectroscopy, should provide a wealth of information on the role of molecular dynamics for the biological function.

2


Page 2 of 26

Page 3 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


TOC graphic

3



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Proteins are nanobioparticles for which all vibrational modes with a frequency smaller c than about 7 THz (˜ ν ≡ ≃ 230 cm−1 ) are collective modes, i.e. modes which cannot be atν tributed to the vibrations of specific bonds or chemical groups. 1 According to the Boltzmann law, these collective modes contribute significantly to the protein free-energy at the physiological temperature. 2 Since the seminal work of McCammon, Gelin, Karplus and Wolynes in 1976, 3 there has been a considerable theoretical interest to establish the possible role of the collective vibrational modes for the protein biological function. 3–22 Indeed, computational simulations demonstrated that low-frequency dynamics of proteins correlates with the structural changes induced by ligand or protein binding 17,20,21 and that correlated motions of distant residues might play a role in enzyme catalysis. 16,18,22 Experimentally, collective vibrational modes of proteins have been investigated by Neutron scattering, 1,23–32 Raman spectroscopy 33–38 and by Far-Infrared (FIR) spectroscopy 39–43 since about four decades. The lowest typical frequencies of the collective modes were predicted to be in the subTHz frequency range, 3 typically between 10 (˜ ν = 0.3 cm−1 ) and 300 GHz (˜ ν = 10 cm−1 ) and are named acoustical modes by analogy with elastic modes of solid nanoparticles. 44 Because the acoustical modes involve large scale displacements of protein segments, they are not depending on the atomistic details of the protein structure but are mainly dependent on the protein shape and on the connectivity of its main-chain. The acoustical modes of proteins are thus the modes which are the most sensitive to a change of the global shape/size of these nanobioparticles but are also the most difficult to measure. Only a few measurements provide an evidence of resonances at sub-THz frequencies in proteins. To the best of our knowledge, the lowest frequency of a normal mode measured in proteins is about 10 GHz (˜ ν = 0.3 cm−1 ) and corresponds to the frequency of a longitudinal acoustical phonon observed by Brillouin scattering in a fibrous protein of micrometric length (collagen). 24 Modes with frequency as low as 200 GHz (˜ ν = 7 cm−1 ) have been extracted also from the nonlinear Raman optical responses of globular proteins in solution using a Brownian oscillator model. 45,46 THz spectroscopy revealed also a spectroscopic feature at very low frequency

4


Page 4 of 26

Page 5 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(around 200 GHz) in bacteriorhodopsin. 42 FIR measurements using synchrotron radiation detected absorption at 590 GHz (˜ ν = 19.6 cm−1 ) in low hydrated lysozyme. 39 So far, the technique providing the most detailed information of protein excitations at frequencies below 100 GHz is the very recent single-molecule spectroscopy named Extraordinary Acoustic Raman (EAR) spectroscopy. 47 In this technique, a single protein molecule is trapped in a nanohole and then excited by two optical lasers of slightly different wavelengths which produce a beat signal at low-frequency (< 100 GHz). 47 The beat signal is an electromagnetic field which can interact with the protein acoustical modes. Vibrational resonances are detected by measuring the increase of the molecule fluctuations when the frequency of the beat field matches the frequency of an acoustical mode. The mechanism of excitation of the acoustical (Raman) active modes of proteins in EAR spectroscopy is not fully explained 47 but is believed to be due to the modulation of the electrostriction force at the trapping site of the molecule. Electrostriction is a nonlinear phenomena in which the strain induced by an electrostatic electric field applied to a dielectric body is proportional to the square of the applied electric field. 48 At the microscopic level, it is related to the anharmonicity of the interaction potential between the atoms of a molecule and to the nonlinearity of its electronic polarisability. 48,49 Electrostriction is a general nonlinear phenomena occurring for all dielectrics to which an electric field is applied. Motivated by this recent EAR technique, we explored theoretically the possibility to excite acoustical modes of large proteins using a low-frequency electric field. In the present work, we demonstrate that the nonlinearity of the protein response, i.e. electrostriction, is not required to excite their acoustical modes by an electric field because the protein acoustical modes induce a variation of the molecular dipole moment. In a linear response theory, the low-frequency dipole moment is proportional to the intensity of the field and the mechanism of excitation corresponds to an infrared absorption. The fact that the acoustical modes of (large) proteins are infrared active is due to the strong polar character of these nanoobjects. In addition, we demonstrate that the excitation of the acoustical modes of proteins induce

5



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

also a variation of their molecular electronic polarizability. This is the first theoretical evidence that the acoustical modes of proteins are both infrared and Raman active. The implication for the trapping mechanism is discussed below. Although that nonlinear and anharmonic effects are neglected in the present study, we show that the linear response calculations reveal well-defined fingerprints of the acoustical modes of proteins in excellent agreement with the ones observed in EAR spectroscopy. This does not exclude the contribution of nonlinear effects in large proteins but we expect the linear response to be the dominant mechanism. The structures of three proteins of different size and shape studied by EAR 47 spectroscopy were extracted from the Protein Data Bank and their structural characteristics are given in Table 1. For each protein, one all-atom MD simulations in explicit water (TIP3P model 50 ) was conducted using the GROMACS 4.6.3 software package 51,52 and the AMBER99sb-ILDN force-field 53 at T = 300 K and P = 1.0 bar for a duration of 1 ns (See Supporting Information for more details). Then, an arbitrary frame was selected at the end of each MD trajectory and the coordinates of the protein structure and of its first hydration shell (i.e. by selecting all water molecules within a radius of 0.3 nm from any protein atom) were recorded, as shown in Fig. 1(a). Indeed, because of the strong coupling between a protein and its solvent, the protein and its first solvation shell must be considered as an integrated nanobioparticle 54 for the study of its vibrational modes. It corresponds to a total number of atoms of 1516, 6060 and 15090 for aprotinin, carbonic anhydrase and conalbumin, respectively. Finally, the structure of each nanobioparticle shown in Fig. 1(a) was optimized using a BroydenFletcher-Goldfarb-Shanno (BFGS) algorithm. 55 The frequencies of the vibrational modes of each nanobioparticle as well as the directions of the atomic displacements within each mode were computed by diagonalizing the mass-weighted Hessian of the optimized structure, computed by finite-difference of forces, using the GROMACS software package (in

6


Page 6 of 26

Page 7 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Structural properties of the three nanobioparticles studied in the present work. M (in kDa) the molecular weight of the non hydrated protein, Natom represents the total number of atoms of the protein in the AMBER99sb-ILDN topology, N the number of water molecule in the first shell of hydration, Rg (in nm) the radius of gyration of the hydrated protein, L (in nm) the largest dimension of the hydrated protein, V (in nm3 ) the volume of the hydrated protein. Name

PDB ID

Sequence

M

Natom

Nw

Rg

L

V

aprotinin

4PTI 56

1-58

6.5

892

208

1.1

3.5

15.5

carbonic anhydrase

57

4-260

29.0

892

676

1.9

6.1

59.6

5-686

75.0

10254

1612

3.0

9.9

147.6

conalbumin

1G6V 2D3I

58

double precision). For each system, it was verified that the first 6 eigenfrequencies were equal to zero and there was no imaginary frequency. The classical (infrared) absorption spectra P (ν) of each nanobioparticle in an applied ~ electric field E(ν), oscillating at frequency ν, was calculated from its computed vibrational modes by using the following formula adapted from a previous work (see Supporting Information for more details): 59 −6 2 3N X ~ π |E(ν)| d(W/hν) γl ν ~ l |2 , = |∆ρ P (ν) ≡ 2 2 2 )2 + ν 2 γ 2 ] dhν 2 h [(ν − ν l l l=7

(1)

where h is the Planck constant, W is the energy absorbed by the molecule, νl and γl are respectively the vibrational frequency and damping of the lth vibrational mode, N is the total number of atoms and ∆~ ρl is the variation of the molecular dipole moment in the vibrational mode l, with N X qκ~eκl ∆~ ρl = √ mκ κ=1

(2)

In Eq. (2), qκ and mκ are the charge and the mass of the atom κ of the protein, respectively. The vector ~eκl is the eigenvector component of the atom κ of the lth mode. Eq. (1) was implemented in C language from the output files generated by GROMACS. 52 The damping factor γl was taken arbitrarly identical for all acoustical modes because their frequencies and

7



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 26

the scale of their motions are similar. We choose a value of γ = 3 GHz (corresponding to a wavenumber of 0.1 cm−1 ) which is comparable to the spectral resolution of EAR. 47 Because the all-atom computed acoustical frequencies are much larger than γ/2 (undamped regime), the effect of the damping is only to broaden the peaks at the resonance frequencies. It is worth noting that the damping of the acoustical modes in EAR is much smaller than the ones deduced from bulk previous measurements 45,54 and calculations, 60 for which γ ∼ = 150 GHz (critically damped regime) 45 to 600 GHz (overdamped regime), 54,60 but comparable to the resolution of the four-photon laser spectroscopy, for which γ = 6 GHz. 61 Due to the fact that the studied nanobioparticles do not have a particular symmetry [Fig. 1(a)], their vibrational modes may be both infrared and Raman active. Therefore, the Raman activity of the vibrational modes of the three proteins was also computed, as follows. In a Raman active mode, the elastic deformation of the molecule induces a variation of the molecular electronic polarisability α 62 and the Raman intensity is proportional to the square of the derivative of the molecular polarisability relative to the collective normal coordinate Q 62 [see Eq. (S7) in SI]. In a previous work, we demonstrated that the electronic polarisability of an amino acid, computed ab initio (at the MP2/6-311G(d,p) level), is simply proportional to its number of electrons. 63 Assuming an average electronic density for all amino acids, this implies that the polarisability of an amino acid is simply proportional to its steric volume. 63 This property is demonstrated numerically here (Fig. S1) where the volume of the amino acid was computed using the algorithm 64 implemented in the GROMACS software package 52 and the polarisabilities were taken from Ref. 63 Using this property, the Raman activity A of each mode l of frequency νl can be estimated by computing the following quantity: 2 ∂α 2 ∂α ∂V 2 2 ∂V ∼ A(νl ) ≡ C = = ∂V ∂Ql ∂Ql , ∂Ql

(3)

where V is the steric volume of the nanobioparticle and the constant C is 353.34 a.u./nm3 (1 a.u = 1.649 10−41 C2 m2 J−1 , see Fig. S1). The derivative in Eq. (3) was computed by finite difference using Ql = ± 0.1 and the steric volume V computed using GROMACS. Finally, to 8


Page 9 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


compare the Raman activities [Eq. (3)] with the absorption spectrum [Eq. (2)], we defined a continuous spectrum P ′ (ν) using a Lorentzian broadening:

P ′ (ν) =

A(νl ) , (ν − νl )2 + (γ/2)2

(4)

where γ = 3 GHz. The computed spectra of infrared active modes [Eq. (1)] and Raman active modes [Eq. (4)] of the nanobioparticles are presented and compared below 300 GHz in Fig. 1(b) and (c), respectively. Extended spectra up to 2 THz are shown in Supporting Information (Fig. S2). As the size increases from aprotinin to conalbumin (Table 1), the spectra P (ν) and P ′ (ν) are shifted to lower frequencies [Fig. 1(b)]. Another interesting property is that the lowest frequency acoustical modes of the nanobioparticles are separated by a gap from the rest of the normal modes as it is more clearly observed for conalbumin. As shown in Fig. 1(b) and (c), signatures of the low-frequency modes occurred both in the infrared and Raman spectra of active modes. In Fig. 2, we compare the spectra of the lowest frequency-modes of the three nanobioparticles. In the absorption spectra P (ν) [Fig. 2(a)], we clearly distinguish two low-frequency acoustical modes for aprotinin and a shoulder, one and a very weak shoulder for carbonic anhydrase and three for conalbumin. Similar signatures are found in the spectra of the Raman active modes, P ′ (ν), except for aprotinin for which one mode (label 2) and a shoulder were observed. Because of its small size, the vibrational spectra of aprotinin should be the most sensitive to the particular conformational state selected to compute its vibrational modes. We repeated the calculation of the infrared and Raman spectra P (ν) and P ′ (ν) of each nanobioparticle for another hydrated protein structure selected randomly from a second MD trajectory at T = 300 K and P = 1.0 bar performed with different initial conditions. The average of the infrared and Raman spectra of the two hydrated structures of each nanobioparticle are very similar to the infrared and Raman spectra shown in Fig. 2, except for the average Raman spectrum P ′ (ν) of aprotinin, which shows two peaks as in the

9



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

infrared spectrum P (ν) (by comparison between Fig. 2 and Fig. S3 in SI). There is a striking similarity between the computed spectra of the acoustical modes of the nanobioparticles and those measured by EAR except for a frequency shift of the whole

Figure 1: (a) All-atom structures of the three different nanobioparticles studied in the present work. Structures are shown using a cartoon representation and the first layer of water surrounding the proteins is shown using a surface representation. The figure was prepared with PyMOL (https://www.pymol.org/). (b) Classical absorption spectra P (ν) and (c) Raman activity spectra P ′ (ν) of aprotinin (red), carbonic anhydrase (green) and conalbumin (blue) computed up to 300 GHz from Eq. (2) and Eq. (4), respectively. 2-columns figure 10


Page 10 of 26

Page 11 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


computed vibrational spectrum to higher frequencies compared to the experimental ones. At the time of writing, Fig. 2 was the first comparison of computed lowest frequency modes of different large proteins to experimental data. As nanobioparticles do not have any welldefined symmetry, there is no means to predict even the number of acoustical modes in the frequency ranges represented in panel (a) without an all-atom model. Owing to the approximations of the present calculations, namely the harmonic approximation, a classical force-field with rigid atomic charges, and the model of the nanobioparticle which neglects the effect of bulk solvent, the agreement between the theory and the experiment is remarkable. During the process of publication of the present work, we became aware of another attempt to model the acoustical modes of the proteins studied by EAR. 65 In this recent work, the authors use a simplified atomistic protein model in which all the atoms (except hydrogens) were connected by the same scalar force constant within a sphere of 0.79 nm. The force constant for each protein was fitted onto the B-factors extracted from X-ray crystallographic data. 65 This model, named Anisotropic Network Model (ANM) 66 is wellknown to reproduce the large scale motions of proteins. 44 We showed that ANM reproduces the density of vibrational states of a hydrated large protein computed from an all-atom forcefield up to about 10 cm−1 (300 GHz). 44 The authors computed the Raman activity of each ANM vibrational mode by calculating the change of the electronic polarisability due to the deformation of a dielectric ellipsoid fitted on the protein atomic structure. 65 We implemented the ANM model of the authors and reproduced the published values of the normalized Raman intensities (nRI) 65 of the lowest frequency modes in Fig. 2 (b). As it can be seen, the ANM model predicts two modes for aprotinin, one mode for carbonic anhydrase and three modes for conalbumin in a frequency range similar to the experimental one. However the frequency shift between the ANM model and the experiment is larger than the present force-field calculations except for conalbumin for which it is comparable. However, Fig. 2 (b) does not correspond to the results discussed in Ref. 65 Indeed, to reduce the frequency disagreement between the theoretical and the experimental frequencies, the authors of Ref. 65 apply a

11



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2: (a) Calculated spectra P (ν) and P ′ (ν) from all-atom force-field calculations. (b) Calculated Raman spectra (nRI: normalized Raman Intensity) extracted from ANM allatom model of Ref. 65 without applying the scaling factor ξ of the ANM vibrational frequencies (see main text). (c) EAR experimental data extracted from Ref. 47 All data are shown in the same frequency range for aprotinin (red), carbonic anhydrase (green) and conalbumin (blue). The inset in panel (a) represents the frequency of the lowest frequency mode computed with the all-atom force-field as a function of the radius of gyration of the nanobioparticles (filled circles). Frequencies of acoustical modes measured for lysozyme 45 (Rg = 1.7 nm, PDB ID: 1DPX) and hydrated α-chymotrypsin 61 (Rg =2.5 nm, PDB ID: 1YPH) are represented by empty circles. Lowest frequencies observed in EAR data are shown as filled circles. Full and dotted lines in the inset represent a 1/Rg law. 2-columns figure scaling factor ξ to the ANM vibrational frequencies. They divide each ANM vibrational frequency by a constant factor different for each protein, and varying between 10 and 20. The scaling factor ξ is therefore chosen in order to fit the frequency of the ANM mode

12


Page 12 of 26

Page 13 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


having the highest Raman intensity with the frequency of the first experimental peak. We disagree with this approach which drastically modify the initial ANM density of states. The maximum of the ANM density of states shifts from about 2 THz (as in the all-atom classical force-field model) to an irrealistic value of 200 GHz (see Fig. S4 in SI). It is well-known from Neutron scattering experiments that a maximum in the vibrational density of states of all proteins studied occurs around 1.5-2.0 THz, in agreement with the present calculations. 44 Our interpretation is that the lowest frequency modes computed approximately with the elegant ANM model, without the ξ scaling factor applied, as shown here in Fig. 2 (b), must be compared to the EAR signatures. Moreover, from our all-atom calculations, we observe that the lowest frequency ν1 of the absorption spectrum P (ν) varies as the inverse of the gyration radius Rg of the protein, as it also does in EAR experiments (see the inset in panel a of Fig. 2). This behavior would be expected for spherical particles of increasing size as the acoustical modes of an elastic sphere of a material varies as the inverse of the sphere radius. 44 In fact, this 1/R law was observed in EAR experiments for polystyrene nanospheres with increasing size. 47 In addition, the typical frequencies computed using the all-atom force-field are comparable to recent measurements of acoustical modes of hydrated lysozyme 45 (Rg = 1.7 nm, PDB ID: 1DPX) and hydrated α-chymotrypsin 61 (Rg = 2.5 nm, PDB ID: 1YPH). As shown in the inset of Fig. 2(a), these experimental frequencies nearly follow the 1/Rg curve calculated with the present all-atom classical force-field. As mentioned above, we observe a frequency shift ∆ν between the theory and the experiment, which decreases as the size of the nanobioparticle increases (Fig. 2). More precisely, the frequency shift between the computed spectrum and the experimental one is 150 GHz (˜ ν = 5.0 cm−1 ), 120 GHz (˜ ν = 4.0 cm−1 ) and 30 GHz (˜ ν = 1.0 cm−1 ) for the aprotinin, carbonic anhydrase and conalbumin proteins, respectively. The anharmonicity, which is not included in the present calculations, might be the key to understand the frequency shift between theory and experiment. Another hypothesis is the possible softening of the acoustical

13



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

modes of the nanobioparticles by the bulk solvent, which is ignored in the present simulations. For a nanobioparticle, there is a strong coupling between the biopolymer and water, which decreases with the particle size. Indeed, we observe that the hydration water contributes to 65, 50 and 40% to the atomic displacements in the computed acoustical vibrational modes [Fig. 2(a)] of aprotinin, carbonic anhydrase and conalbumin, respectively. What insights from the theoretical spectra shown in Fig. 2 do we gain about the excitation mechanisms of a single protein in EAR experiments ? The trapping force for particles of size much smaller than the electromagnetic wavelength of the laser is dominated by a gradient ~ 67 mainly due to the interaction of the particle-induced dipole moment p~ with force (~p.∇E), ~ In EAR experiments, due to the interference of the two laser beams, the electric field E. the time-dependent electric field has a component at an acoustical frequency [ν < 200 GHz in Fig. 2(c)]. This beat signal induces two things: first, a variation of the molecular dipole moment due to the excitation of the infrared acoustical modes which is an absorption, and second, a variation of the real part of the molecular electronic polarisability because the active infrared modes are also Raman active, as shown in Fig. 2(a). This phenomenon modulates the trapping gradient force and the root-mean-square fluctuations of the nanobioparticle measured in EAR. According to this interpretation, the product P (v)P ′ (v) might be the best descriptor of the signatures of the acoustical modes of the absorbing nanobioparticles observed in EAR (Fig. 2). For non absorbing particles, as the polystyrene nanoparticles studied in Ref., 47 the EAR mechanism might be different and due only to electrostriction. Two questions remain from a theoretical point of view: to what motions correspond the acoustical modes shown in Fig. 2(a) and what is the origin of their large dipole moment variations? As shown in Fig. 3 for the most intense modes (and in Fig. S5 for the others), the global motions described by the lowest-frequency acoustical modes of each nanobioparticle correspond to torsional motions of secondary structures (aprotinin), of subdomains (carbonic anhydrase) or of entire domains (conalbumin), depending on the size and on the shape of the nanobioparticle. To decipher the origin of the dipole moments of the acoustical modes

14


Page 14 of 26

Page 15 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3: Cartoon representation of the acoustic collective modes extracted from classical spectra shown in Fig. 2 for aprotinin (modes 1’ and 2), carbonic anhydrase (mode 1) and conalbumin (mode 1). Black arrows represent the direction and the strength of the atomic displacement vector in the corresponding mode for the Cα atoms. Colored arrows represent the global motion of each protein in the corresponding mode. Spheres represent the position of the Cα and the color code corresponds to the strength of the displacement per residue. 2-columns figure

15



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

in P (ν), we computed the distribution of the variation of the molecular dipole moment, ∆~ ρl , along the amino-acid sequence of the three proteins [Eq. (2)]. As shown in Fig. 4 (Fig. S6) for the modes described in Fig. 3 (Fig. S5), the largest contributions to ∆~ ρl of each acoustical mode are due almost exclusively to the positively charged residues (Arginine and Lysine). The percentage of positively charged residues for each protein is similar: 17, 12 and 12% for aprotinin, carbonic anhydrase and conalbumin, respectively. The total charge of each protein is +6, -2 and -4 for aprotinin, carbonic anhydrase and conalbumin, respectively. Finally, compared to negatively charged residues, Arginine and Lysine amino acids are characterized by longer side-chains, i.e. L ∼ 6.4 ˚ A (for comparison L ∼ 2.6 ˚ A for aspartic acid and L ∼ 4.0 ˚ A for glutamic acid residues). This new result explains why the acoustical modes of nanobioparticles are infrared active. In conclusion, we provide the first interpretation of excitations of acoustical modes of nanobioparticles of different sizes and shapes by a low-frequency electric field, as measured for single biomolecules in EAR experiments. The type of motions and the origin of the dipolar character of the modes are identified: they are torsional large-scale vibrational modes producing significant local variation of the molecular dipole moment due to the motions of the charged residues, i.e. Arginine and Lysine residues, with the longest side chains. The combination of simulations presented here with measurements of acoustical modes of proteins of therapeutic interest, for example Hsp70 chaperone proteins, 44 should provide new informations on the dynamics of conformational changes. We hope that the present results will stimulate new experimental investigations of acoustical modes of nanobioparticles as proteins, protein nanomachines and viruses.

Acknowledgement The authors acknowledge Dr. Fabrice Neiers (INRA, Dijon) for helpful discussions on the proteins studied in the present work. Dr. Aymeric Leray (ICB, Dijon), and Dr. Lucien Saviot

16


Page 16 of 26

Page 17 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(ICB, Dijon) are gratefully acknowledged for useful discussions on single molecule Raman spectroscopy. The calculations were performed using HPC resources from DSI-CCuB (Université de Bourgogne). This work was partially supported by a grant from the Air Force Office of Scientific Research (AFOSR), as part of a joint program with the Directorate for Engineering of the National Science Foundation (NSF), Emerging Frontiers and Multidisciplinary Office. The authors also thank the Conseil Régional de Bourgogne for the funding (PARI Nano2bio).

Figure 4: Dipole moment variation k∆~ ρl k along the amino-acid sequence of (a) aprotinin, (b) carbonic anhydrase and (c) conalbumin for each normal mode shown in Fig. 3. Positively and negatively charged residues are shown with blue and red dots along the sequence, respectively. Other residues are shown by black dots.1-column figure

17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Supporting Information Supplementary materials including the derivation of the classical spectrum formula presented in the main text as well as details about computational methods and supplemental figures are included.

18


Page 18 of 26

Page 19 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


References (1) Middendorf, H. D. Biophysical Applications of Quasi-Elastic and Inelastic Neutron Scattering. Annu. Rev. Biophys. Bio. 1984, 13, 425–451. (2) G¯o, N.; Scheraga, H. A. Analysis of the Contribution of Internal Vibrations to the Statistical Weights of Equilibrium Conformations of Macromolecules. J. Chem. Phys. 1969, 51, 4751–4767. (3) McCammon, J. A.; Gelin, B. R.; Karplus, M.; Wolynes, P. G. The Hinge-Bending Mode in Lysozyme. Nature 1976, 262, 325–326. (4) Colonna-Cesari, F.; Perahia, D.; Karplus, M.; Eklund, H.; Brdn, C. I.; Tapia, O. Interdomain Motion in Liver Alcohol Dehydrogenase. Structural and Energetic Analysis of the Hinge Bending Mode. J. Biol. Chem. 1986, 261, 15273–15280. (5) G¯o, N.; Noguti, T.; Nishikawa, T. Dynamics of a Small Globular Protein in Terms of Low-Frequency Vibrational Modes. P. Natl. Acad. Sci. USA 1983, 80, 3696–3700. (6) Brooks, B.; Karplus, M. Harmonic Dynamics of Proteins: Normal Modes and Fluctuations in Bovine Pancreatic Trypsin Inhibitor. P. Natl. Acad. Sci. USA 1983, 80, 6571–6575. (7) Levitt, M.; Sander, C.; Stern, P. S. Protein Normal-Mode Dynamics: Trypsin Inhibitor, Crambin, Ribonuclease and Lysozyme. J. Mol. Biol. 1985, 181, 423–447. (8) Brooks, B.; Karplus, M. Normal Modes for Specific Motions of Macromolecules: Application to the Hinge-Bending Mode of Lysozyme. P. Natl. Acad. Sci. USA 1985, 82, 4995–4999. (9) Hayward, S.; Berendsen, H. J. Systematic Analysis of Domain Motions in Proteins from Conformational Change: New Results on Citrate Synthase and T4 Lysozyme. Proteins 1998, 30, 144–154. 19



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(10) Gerstein, M.; Krebs, W. A Database of Macromolecular Motions. Nucleic Acids Res. 1998, 26, 4280–4290. (11) Gerstein, M.; Lesk, A. M.; Chothia, C. Structural Mechanisms for Domain Movements in Proteins. Biochemistry 1994, 33, 6739–6749. (12) Bennett, W. S.; Huber, R. Structural and Functional Aspects of Domain Motions in Proteins. CRC Cr. Rev. Bioch. Mol. 1984, 15, 291–384. (13) Karplus, M.; Petsko, G. A. Molecular Dynamics Simulations in Biology. Nature 1990, 347, 631–639. (14) Berendsen, H. J.; Hayward, S. Collective Protein Dynamics in Relation to Function. Curr. Opin. Struc. Biol. 2000, 10, 165–169. (15) Tama, F.; Sanejouand, Y.-H. Conformational Change of Proteins Arising from Normal Mode Calculations. Protein Eng. 2001, 14, 1–6. (16) Rod, T. H.; Radkiewicz, J. L.; Brooks, C. L. Correlated Motion and the Effect of Distal Mutations in Dihydrofolate Reductase. P. Natl. Acad. Sci. USA 2003, 100, 6980–6985. (17) Tobi, D.; Bahar, I. Structural Changes Involved in Protein Binding Correlate with Intrinsic Motions of Proteins in the Unbound State. P. Natl. Acad. Sci. USA 2005, 102, 18908–18913. (18) Benkovic, S. J.; Hammes-Schiffer, S. Enzyme Motions Inside and Out. Science 2006, 312, 208–209. (19) Henzler-Wildman, K.; Kern, D. Dynamic Personalities of Proteins. Nature 2007, 450, 964–972. (20) Dobbins, S. E.; Lesk, V. I.; Sternberg, M. J. E. Insights into Protein Flexibility: The Relationship Between Normal Modes and Conformational Change upon Proteinprotein Docking. P. Natl. Acad. Sci. USA 2008, 105, 10390–10395. 20


Page 20 of 26

Page 21 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(21) Bakan, A.; Bahar, I. The Intrinsic Dynamics of Enzymes Plays a Dominant Role in Determining the Structural Changes Induced upon Inhibitor Binding. P. Natl. Acad. Sci. USA 2009, 106, 14349–14354. (22) Nashine, V. C.; Hammes-Schiffer, S.; Benkovic, S. J. Coupled Motions in Enzyme Catalysis. Curr. Opin. Chem. Biol. 2010, 14, 644–651. (23) Middendorf, H. D.; Hayward, R. L.; Parker, S. F.; Bradshaw, J.; Miller, A. Vibrational Neutron Spectroscopy of Collagen and Model Polypeptides. Biophys. J. 1995, 69, 660– 673. (24) Harley, R.; James, D.; Miller, A.; White, J. W. Phonons and the Elastic Moduli of Collagen and Muscle. Nature 1977, 267, 285–287. (25) Cusack, S.; Doster, W. Temperature Dependence of the Low Frequency Dynamics of Myoglobin. Measurement of the Vibrational Frequency Distribution by Inelastic Neutron Scattering. Biophys. J. 1990, 58, 243–251. (26) Berney, C. V.; Renugopalakrishnan, V.; Bhatnagar, R. S. Collagen. An Inelastic Neutron-Scattering Study of Low-Frequency Vibrational Modes. Biophys. J. 1987, 52, 343–345. (27) Bartunik, H. D.; Jolls, P.; Berthou, J.; Dianoux, A. J. Intramolecular Low-Frequency Vibrations in Lysozyme by Neutron Time-Of-Flight Spectroscopy. Biopolymers 1982, 21, 43–50. (28) Middendorf, H. D. Neutron Studies of the Dynamics of Globular Proteins. Physica B 1992, 182, 415–420. (29) Diehl, M.; Doster, W.; Petry, W.; Schober, H. Water-Coupled Low-Frequency Modes of Myoglobin and Lysozyme Observed by Inelastic Neutron Scattering. Biophys. J. 1997, 73, 2726–2732. 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 26

(30) Lushnikov, S. G.; Svanidze, A. V.; Sashin, I. L. Vibrational Density of States of Hen Egg White Lysozyme. J. Exp. Theor. Phys. 2005, 82, 30–33. (31) Bellissent-Funel, M. C.; Teixeira, J.; Chen, S. H.; Dorner, B.; Middendorf, H. D.; Crespi, H. L. Low-Frequency Collective Modes in Dry and Hydrated Proteins. Biophys. J. 1989, 56, 713–716. (32) Paciaroni, A.;

Orecchini, A.;

Haertlein, M.;

Moulin, M.;

Conti Nibali, V.;

De Francesco, A.; Petrillo, C.; Sacchetti, F. Vibrational Collective Dynamics of Dry Proteins in the Terahertz Region. J. Phys. Chem. B 2012, 116, 3861–3865. (33) Brown, K. G.; Erfurth, S. C.; Small, E. W.; Peticolas, W. L. Conformationally Dependent Low-Frequency Motions of Proteins by Laser Raman Spectroscopy. P. Natl. Acad. Sci. USA 1972, 69, 1467–1469. (34) Genzel, L.; Keilmann, F.; Martin, T. P.; Wintreling, G.; Yacoby, Y.; Frhlich, H.; Makinen, M. W. Low-Frequency Raman Spectra of Lysozyme. Biopolymers 1976, 15, 219– 225. (35) Painter, P. C.; Mosher, L. E.; Rhoads, C. Low-Frequency Modes in the Raman Spectra of Proteins. Biopolymers 1982, 21, 1469–1472. (36) Urabe, H.; Sugawara, Y.; Ataka, M.; Rupprecht, A. Low-Frequency Raman Spectra of Lysozyme Crystals and Oriented DNA Films: Dynamics of Crystal Water. Biophys. J. 1998, 74, 1533–1540. (37) Hédoux, A.; Ionov, R.; Willart, J.-F.; Lerbret, A.; Affouard, F.; Guinet, Y.; Descamps, M.; Prvost, D.; Paccou, L.; Dande, F. Evidence of a Two-Stage Thermal Denaturation Process in Lysozyme: A Raman Scattering and Differential Scanning Calorimetry Investigation. J. Chem. Phys. 2006, 124, 014703.

22


Page 23 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(38) Crupi, C.; D’Angelo, G.; Wanderlingh, U.; Vasi, C. Raman Spectroscopic and LowTemperature Calorimetric Investigation of the Low-Energy Vibrational Dynamics of Hen Egg-White Lysozyme. Philos. Mag. 2011, 91, 1956–1965. (39) Moeller, K. D.; Williams, G. P.; Steinhauser, S.; Hirschmugl, C.; Smith, J. C. Hydration-Dependent Far-Infrared Absorption in Lysozyme Detected Using Synchrotron Radiation. Biophys. J. 1992, 61, 276–280. (40) Markelz, A. G.; Roitberg, A.; Heilweil, E. J. Pulsed Terahertz Spectroscopy of DNA, Bovine Serum Albumin and Collagen Between 0.1 and 2.0 THz. Chem. Phys. Lett. 2000, 320, 42–48. (41) Markelz, A.; Whitmire, S.; Hillebrecht, J.; Birge, R. THz Time Domain Spectroscopy of Biomolecular Conformational Modes. Phys. Med. Biol. 2002, 47, 3797–3805. (42) Whitmire, S. E.; Wolpert, D.; Markelz, A. G.; Hillebrecht, J. R.; Galan, J.; Birge, R. R. Protein Flexibility and Conformational State: A Comparison of Collective Vibrational Modes of Wild-Type and D96N Bacteriorhodopsin. Biophys. J. 2003, 85, 1269–1277. (43) Falconer, R. J.; Markelz, A. G. Terahertz Spectroscopic Analysis of Peptides and Proteins. J. Infrared Millim. Te. 2012, 33, 973–988. (44) Nicola¨ı, A.; Delarue, P.; Senet, P. In Computational Methods to Study the Structure and Dynamics of Biomolecules and Biomolecular Processes; Liwo, A., Ed.; Springer Series in Bio-/Neuroinformatics 1; Springer Berlin Heidelberg, 2014; pp 483–524. (45) Giraud, G.; Karolin, J.; Wynne, K. Low-Frequency Modes of Peptides and Globular Proteins in Solution Observed by Ultrafast OHD-RIKES Spectroscopy. Biophys. J. 2003, 85, 1903–1913. (46) Giraud, G.; Wynne, K. Time-Resolved Optical Kerr-Effect Spectroscopy of Low-

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Frequency Dynamics in Di-L-Alanine, Poly-L-Alanine, and Lysozyme in Solution. J. Am. Chem. Soc. 2002, 124, 12110–12111. (47) Wheaton, S.; Gelfand, R. M.; Gordon, R. Probing the Raman-Active Acoustic Vibrations of Nanoparticles with Extraordinary Spectral Resolution. Nat. Photon. 2015, 9, 68–72. (48) Li, F.; Jin, L.; Xu, Z.; Zhang, S. Electrostrictive Effect in Ferroelectrics: An Alternative Approach to Improve Piezoelectricity. Appl. Phys. Rev. 2014, 1, 011103. (49) Achar, B. N. N.; Barsch, G. R.; Cross, L. E. Static Shell Model Calculation of Electrostriction and Third Order Elastic Coefficients of Perovskite Oxides. Ferroelectrics 1981, 37, 495–498. (50) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. Comparison of Simple Potential Functions for Simulating Liquid Water. J. Chem. Phys. 1983, 79, 926–935. (51) Hess, B.; Kutzner, C.; van der Spoel, D.; Lindahl, E. GROMACS 4: Algorithms for Highly Efficient, Load-Balanced, and Scalable Molecular Simulation. J. Chem. Theory Comput. 2008, 4, 435–447. (52) Pronk, S.; Páll, S.; Schulz, R.; Larsson, P.; Bjelkmar, P.; Apostolov, R.; Shirts, M. R.; Smith, J. C.; Kasson, P. M.; van der Spoel, D. et al. GROMACS 4.5: a HighThroughput and Highly Parallel Open Source Molecular Simulation Toolkit. Bioinformatics 2013, 29, 845–854. (53) Lindorff-Larsen, K.; Piana, S.; Palmo, K.; Maragakis, P.; Klepeis, J. L.; Dror, R. O.; Shaw, D. E. Improved Side-Chain Torsion Potentials for the Amber Ff99SB Protein Force Field. Proteins 2010, 78, 1950–1958.

24


Page 24 of 26

Page 25 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(54) Vinh, N. Q.; Allen, S. J.; Plaxco, K. W. Dielectric Spectroscopy of Proteins As a Quantitative Experimental Test of Computational Models of Their Low-Frequency Harmonic Motions. J. Am. Chem. Soc. 2011, 133, 8942–8947. (55) Zhu, C.; Byrd, R. H.; Lu, P.; Nocedal, J. Algorithm 778: L-BFGS-B: Fortran Subroutines for Large-Scale Bound-Constrained Optimization. ACM Trans. Math. Softw. 1997, 23, 550–560. (56) Marquart, M.; Walter, J.; Deisenhofer, J.; Bode, W.; Huber, R. The Geometry of the Reactive Site and of the Peptide Groups in Trypsin, Trypsinogen and Its Complexes with Inhibitors. Acta Crystallogr. B 1983, 39, 480–490. (57) Desmyter, A.; Decanniere, K.; Muyldermans, S.; Wyns, L. Antigen Specificity and High Affinity Binding Provided by One Single Loop of a Camel Single-Domain Antibody. J. Biol. Chem. 2001, 276, 26285–26290. (58) Mizutani, K.; Mikami, B.; Aibara, S.; Hirose, M. Structure of Aluminium-Bound Ovotransferrin at 2.15 Angstroms Resolution. Acta Crystallogr. D 2005, 61, 1636–1642. (59) Senet, P.; Lambin, P.; Vigneron, J. P.; Derycke, I.; Lucas, A. A. Phonon Surface Loss Function of Ionic-Crystal Films: A Comparison Between Microscopic and Macroscopic Approaches. Surf. Sci. 1990, 226, 307–321. (60) Meinhold, L.; Smith, J. C. Protein Dynamics from X-Ray Crystallography: Anisotropic, Global Motion in Diffuse Scattering Patterns. Proteins 2007, 66, 941–953. (61) Bunkin, A. F.; Gorchakov, A. P.; Lebedenko, S. I.; Nurmatov, A. A.; Pershin, S. M.; Chikishev, A. Y. Four-Photon Laser Spectroscopy of Liquid Water and Aqueous Solutions of α-Chymotrypsin in the Range ± 5 Cm−1 . Opt. Spectrosc. 99, 579–583. (62) Ferraro, J. R. Introductory Raman Spectroscopy, Second Edition, 2nd ed.; Academic Press: Amsterdam ; Boston, 2002. 25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(63) Krishtal, A.; Senet, P.; Van Alsenoy, C. Local Softness, Softness Dipole, and Polarizabilities of Functional Groups: Application to the Side Chains of the 20 Amino Acids. J. Chem. Phys. 2009, 131, 044312. (64) Eisenhaber, F.; Lijnzaad, P.; Argos, P.; Sander, C.; Scharf, M. The Double Cubic Lattice Method: Efficient Approaches to Numerical Integration of Surface Area and Volume and to Dot Surface Contouring of Molecular Assemblies. J. Comput. Chem. 1995, 16, 273–284. (65) DeWolf, T.; Gordon, R. Theory of Acoustic Raman Modes in Proteins. Phys. Rev. Lett. 2016, 117, 138101. (66) Atilgan, A. R.; Durell, S. R.; Jernigan, R. L.; Demirel, M. C.; Keskin, O.; Bahar, I. Anisotropy of Fluctuation Dynamics of Proteins with an Elastic Network Model. Biophys. J. 2001, 80, 505–515. (67) Chaumet, P. C.; Nieto-Vesperinas, M. Time-Averaged Total Force on a Dipolar Sphere in an Electromagnetic Field. Opt. Lett. 2000, 25, 1065–1067.

26


Page 26 of 26

Theoretical Insights into Sub-Terahertz Acoustic Vibrations of

Recommend Documents