Simulation of the chiral SFG response of supramolecular structures requires vibrational couplings

Abstract

Chiral vibrational sum frequency generation (SFG) spectroscopy probes the structure of the solvation shell around chiral macromolecules. The dominant theoretical framework for understanding the origin of chiral SFG signals is based on analysis of molecular symmetry, which assumes no interaction between molecules. However, water contains strong intermolecular interactions that significantly affect its properties. Here, the role of intermolecular vibrational coupling in the chiral SFG response of the O-H stretch of water surrounding an antiparallel β-sheet at the vacuum-water interface is investigated. Both intramolecular and intermolecular couplings between O-H groups are required to simulate the full lineshape of the chiral SFG signal. This dependence is also observed for a chiral water dimer, illustrating that this phenomenon is not specific to larger systems. We also find that a dimer of C_3v molecules predicted to be chiral SFG inactive by the symmetry-based theory can generate a chiral SFG signal when intermolecular couplings are considered, suggesting that even highly symmetric solvent molecules may produce chiral SFG signals when interacting with a chiral solute. The consideration of intermolecular couplings extends the prevailing theory of the chiral SFG response to structures larger than individual molecules and provides guidelines for future modeling.

Graphical Abstract

Introduction

Biological processes occur in aqueous solution or at an interface involving water, and this water does not always behave as a passive medium. In particular, water molecules can bind to biomacromolecules and modulate their structure and function. These water molecules can interact with a solute tightly enough to adopt large-scale structure, but loosely enough to form a structural ensemble and exchange rapidly with bulk solvent. A complete understanding of the behavior of biomacromolecules requires a thorough description of these water molecules. Chiral vibrational sum frequency generation (SFG) spectroscopy is an emergent technique to probe the solvation shell of biomacromolecules.^1–3 As a second-order nonlinear chirality-sensitive technique, chiral SFG can suppress background signal from water at non-centrosymmetric interfacial environments and isotropic bulk water. Thus, it can selectively detect solvent molecules forming chiral supramolecular structures around chiral macromolecules, such as DNA and protein.^{1, 2} Vibrational spectroscopy can resolve subpopulations exchanging on the picosecond timescale and does not require molecular labels that perturb chemical structures. Importantly, chiral SFG can detect chiral solvent structure induced by interactions with biomacromolecules even if the individual solvent molecules in that structure exchange rapidly.^{1, 2, 4, 5}

The last two decades have seen extensive applications of chiral SFG to biological systems.^6–8 In 2003, a chiral SFG spectrum of the amide I response of a protein at the air-water interface was published.⁷ Our group extended the application to probe various protein secondary structures using the amide I, C-H, and N-H stretch modes.^{6, 9, 10} Achiral spectra of DNA in the O-D and C-H stretch regions showed that SFG can detect the effects of solvation in D₂O.¹¹ Chiral SFG spectra of the C-H stretch in DNA were reported.^{12, 13} Others have applied femtosecond time-resolved SFG to monitor the relaxation of the amide I vibrational mode in the presence of water solvent.¹⁴ Reports of the chiral SFG response of water surrounding DNA and antiparallel β-sheet protein demonstrated that even highly dynamic solvation structures of a biological system can be probed with SFG.^{1, 2, 5} We recently used the internal heterodyne phase-sensitive chiral SFG method¹⁵ and observed that the phase of the water O-H stretching bands is correlated with the absolute handedness of (L-) and (D-) proteins, further establishing the method for probing hydration structures of biomacromolecules.⁵

Simpson’s chiral SFG theory¹⁶ has served as a foundation for predicting and interpreting chiral SFG studies of interfaces. This theory takes intramolecular couplings into account by considering molecular units larger than single bonds when assigning the symmetry of vibrational modes.¹⁶ Because this symmetry-based theory has been derived and applied mainly for chiral macromolecules (e.g., various protein secondary structures), intermolecular coupling is not relevant and thus not considered. However, the hydrogen-bonding interactions between water molecules are paramount. Thus, the recent reports of chiral SFG signals from water solvent surrounding biomacromolecules^{1, 2, 5} call for examination of intermolecular couplings.

In this paper, we examine the impact of intermolecular coupling as well as intramolecular coupling on chiral SFG signals¹⁶ using an electric field mapping method.^17–19 We show that intramolecular or intermolecular vibrational coupling is necessary to produce nonzero chiral SFG signals in the simulation and that the signal intensity contributed by these two couplings are comparable. In contrast, neither of these couplings is needed to generate an achiral SFG response, and the achiral SFG response is only mildly affected when such couplings are included. We demonstrate these effects using the antiparallel β-sheet system formed by the LK₇β (LKLKLKL) peptide at the vacuum-water interface (Figure 1).²⁰ To examine the fundamental origin of the coupling effect, we also study a small model system composed of a dimer of water frozen in a chiral configuration. We find that couplings are necessary to observe a chiral SFG signal in this minimal system as well. Similarly, we examine a dimer of ammonia with C_3v symmetry in contrast to the C_2v symmetry of water to further illustrate the interplay of molecular symmetry and vibrational couplings for generating chiral SFG responses. Finally, we examine a dimer of methane to show that the trends seen with the ammonia system translate to other highly symmetric point groups. We discuss our results in the context of Simpson’s symmetry-based theory.¹⁶ This work will be helpful to experimentalists considering SFG spectra of isotopically labeled water molecules in biological systems (e.g., HOD largely lacks intramolecular coupling but still has intermolecular coupling) and solvent systems other than water for chiral polymers and macromolecules. The results will also guide theorists in proposing approximations to the dipole-polarizability time correlation function formalism²¹ for calculating SFG responses. In particular, although couplings can be omitted to save computational time for calculating achiral SFG spectra, they must be included to model chiral SFG responses.²² We recognize that couplings have not always been reported to be essential for a chiral SFG signal.²³ We discuss the intermolecular and intramolecular couplings in these previous studies in the context of our findings in the Discussion.

Figure 1.

The LK₇β (LKLKLKL) peptide model system. (A) Each LK₇β peptide consists of alternating leucine (black) and lysine (blue) residues and is capped with an acetyl group (magenta) on the N-terminus and an NH₂ group at the C-terminus. (B) An antiparallel β-sheet is constructed at the vacuum-water interface from five LK₇β peptides arranged such that leucine sidechains point toward the vacuum and lysine sidechains point into the water. The system is neutralized with Cl⁻ ions (green). Approximately 1350 water molecules surround the peptide. The simulation box is roughly 45 × 45 × 45 Å. Half of the box is composed of the system and the other half is vacuum.

Symmetry-based chiral SFG theory

We interpret the simulated SFG spectra of water in this study based on Simpson’s symmetry-based chiral SFG theory.¹⁶ Herein, (x, y, z) will represent the lab frame coordinates, and (a, b, c) will represent the molecular frame coordinates. The indices I, J, and K will be used to indicate x, y, or z, whereas the indices i, j, and k will be used to indicate a, b, or c.

Chiral SFG signals can be generated from a chiral interface, which is defined as a molecular system with C_∞ symmetry with the rotational axis aligned with the surface normal. Within the electric dipole approximation, SFG signals arise from the second-order nonlinear response of materials

$$\mathbf{P}={\mathbf{P}}_0+{\chi }^{(1)}{\mathbf{E}}_1+{\chi }^{(2)}{\mathbf{E}}_1{\mathbf{E}}_2+\dots$$ (1)

where P is the material’s polarization, E₁ and E₂ are incident electric fields, and χ⁽ⁿ⁾ is the n^th-order susceptibility tensor.²⁴ In vibrational SFG, one beam is in the mid-infrared (IR) range and the other is in the visible range but typically far off electronic resonance.^25–27 The χ⁽²⁾ tensor contains 27 elements ${\chi }_{IJK}^{(2)}$ as in

$$E_{SFG}^I\propto \sum _{JK}{\chi }_{IJK}^{(2)}{E}_{vis}^J{E}_{IR}^K$$ (2)

Where $E_{SFG}^I$, $E_{vis}^J$, and $E_{IR}^K$ are the optical fields of the SFG, visible, and IR beams, respectively, with the indices I, J, and K specifying the components of these optical fields along a Cartesian direction (x, y, or z) in the lab reference frame (Figure 2). Manipulating the polarization of the IR and visible beams and the detector of the SFG signal isolates different elements of the χ⁽²⁾ tensor. Among the 27 elements, those with all three indices that are different (I≠J≠K) are called orthogonal. Orthogonal χ⁽²⁾elements for a uniaxial assembly can be nonzero only for a chiral interface.

Figure 2.

Schematic depiction of a vibrational SFG experiment with the psp polarization setting, where p indicates polarization on the plane of incidence (the xz plane) and s indicates polarization perpendicular to that plane (i.e., along y). The polarization is specified in the order of sum frequency (detector), visible, and infrared. The psp polarization setting isolates four elements of the second order susceptibility tensor (χ⁽²⁾). Among these four elements, ${\chi }_{xyx}^{(2)},{\chi }_{zyz}^{(2)}=0$ due to the C_∞ symmetry of a non-centrosymmetric interface (xy-plane) about the z-axis and ${\chi }_{xyz}^{(2)}=0$ given the symmetry of the Raman polarizability tensor in the absence of electronic resonance. Ignoring Fresnel factors, the chiral response is given by ${\chi }_{psp}^{(2)}\propto {\chi }_{zyx}^{(2)}$.

To perform a chiral SFG experiment, the psp polarization setup can be used (Figure 2), where s polarization is taken to lie along the y-axis, and p polarization lies along x and z on the plane of incidence. Figure 2 shows that the effective second-order susceptibility, ${\chi }_{psp}^{(2)}$, depends only on ${\chi }_{xyx}^{(2)}$, ${\chi }_{zyz}^{(2)}$, ${\chi }_{xyz}^{(2)}$, and ${\chi }_{zyx}^{(2)}$. The first two elements vanish due to isotropy in the xy plane, as is found at a non-centrosymmetric interface.¹⁶ The last two elements are orthogonal and ${\chi }_{xyz}^{(2)}=0$ under the condition of electronic nonresonance due to the symmetric polarizability tensor.^6,16 Hence, the experimental measurable of the chiral SFG response, ${\chi }_{psp}^{(2)}$, is only related to ${\chi }_{zyx}^{(2)}$:

$${\chi }_{psp}^{(2)}\propto {\chi }_{zyx}^{(2)}-{\chi }_{xyz}^{(2)}={\chi }_{zyx}^{(2)}$$ (3)

In the symmetry-based theory, the system of interest has C_∞ symmetry. Although this is true for many experimental systems, it is not true for molecular dynamics trajectories with limited sampling. To avoid this issue, the ${\chi }_{xyz}^{(2)}$ term in equation 3 is kept in our calculation for extracting chiral SFG signal from MD trajectories (see equations 9–10 and S4).

The current work assumes a psp setup, but the methods and results are directly applicable to the other two polarization setups (spp and pps) for chiral SFG experiments.

The macroscopic chiral SFG response, ${\chi }_{psp}^{(2)}$, is the ensemble average of microscopic responses given by molecular hyperpolarizabilities (β).²⁶ The hyperpolarizability arises from the transition dipole and polarizability derivatives as in

$${\beta }_{ijk}\propto \left(\frac{\partial {\alpha }_{ij}}{\partial Q}\right)\left(\frac{\partial {\mu }_k}{\partial Q}\right)$$ (4)

where α is the transition polarizability matrix, μ is the transition dipole vector, and Q is a normal mode coordinate. The ensemble average can be expressed as

$${\chi }_{IJK}^{(2)}\propto \sum _{i,j,k}\langle R_{Ii}(\psi )R_{Jj}(\theta )R_{Kk}(\varphi )\rangle {\beta }_{ijk}$$ (5)

where R is an Euler rotation matrix, ϕ, θ, and ψ are Euler rotation angles, (i, j, k) represents the molecular frame, (I, J, K) represents the laboratory frame, and the brackets indicate averaging over molecular orientations. Applying equation 5 and integrating over all rotations in the xy-plane (ϕ) leads to

$$\begin{gathered} {\chi }_{zyx}^{(2)}\propto \frac{1}{2}\{\left(-{\beta }_{cab}+{\beta }_{cba}\right){\cos }^2(\theta ) \\ +\cos (\theta )\sin (\theta )[\left({\beta }_{aab}-{\beta }_{aba}+{\beta }_{cbc}-{\beta }_{ccb}\right)\cos (\psi ) \\ +\left(-{\beta }_{bab}+{\beta }_{bba}+{\beta }_{cac}-{\beta }_{cca}\right)\sin (\psi )] \\ +{\sin }^2(\theta )(\left(-{\beta }_{abc}+{\beta }_{acb}\right){\cos }^2(\psi ) \\ -\left({\beta }_{aac}-{\beta }_{aca}-{\beta }_{bbc}+{\beta }_{bcb}\right)\cos (\psi )\sin (\psi )+\left({\beta }_{bac}-{\beta }_{bca}\right){\sin }^2(\psi )\} \end{gathered}$$ (6)

The c-axis points along the primary rotation axis of the molecule. Although all chiral molecular β_ijk (i≠j≠k) elements are zero for water molecules with C_2v symmetry, the achiral β elements can combine to give rise to chiral χ⁽²⁾ elements if the water molecules are arranged in a chiral configuration at an interface. Eliminating all zero-valued β elements in equation 6^{16, 28} based on symmetry arguments results in the expression of the chiral ${\chi }_{zyx}^{(2)}$ element as

$${\chi }_{zyx}^{(2)}\propto \frac{1}{4}{\sin }^2(\theta )\sin (2\psi )\left({\beta }_{aac}-{\beta }_{bbc}+{\beta }_{bcb}-{\beta }_{aca}\right)$$ (7)

Therefore, chiral SFG can be sensitive to water when the water molecules adopt a chiral supramolecular structure in the solvation shells of biomacromolecules (e.g., protein and DNA) at interfaces. We are not the first to notice this possibility. Simpson theorized the appearance of chiral spectra from achiral chromophores with certain symmetries as early as 2004,¹⁶ and in 2012, Laaser and Zanni published a theory of chiral 2D-SFG signals arising from pairs of achiral but coupled chromophores.²⁹

Derivation of Equation 7 from Equation 6 suggests that water is quite a unique solvent with respect to generating the chiral SFG response. Its C_2v symmetry is high enough (with two reflection planes) to eliminate all microscopic chiral (orthogonal) β elements, but low enough to preserve some achiral β elements that can still contribute to the macroscopic chiral ${\chi }_{zyx}^{(2)}$, as shown in equation 7. Hence, the symmetry-based theory predicts that chiral supramolecular assemblies of water situated in an interfacial environment with C_∞ symmetry are chiral SFG active. However, when the molecule becomes just slightly more symmetric (such as C_3v), the prediction can be drastically different because the higher the symmetry, the more β elements become zero. In particular, all β elements in equation 6 are zero or cancel out for a C_3v molecular system. Thus, the symmetry-based theory predicts that a C_3v molecule (e.g., ammonia, NH₃) cannot generate chiral signals (e.g., N-H stretching in NH₃) even when the molecules are assembled into chiral supramolecular structures at an interface.^{16, 28} Below, we will discuss how introducing intermolecular vibrational couplings can alter this prediction, leading to a chiral SFG response in simulated spectra.

Computational Methods

MD simulations.

A starting structure of five antiparallel LK₇β (LKLKLKL) strands (Figure 1A) was placed into a box with at least 10 Å of TIP4P-Ew water (a modification of the original TIP4P model that is optimized for Ewald summation treatment of long-range interactions)²³ next to each of the four edges of the protein and above and below it.⁵ Half of the water was removed and Cl⁻ ions were added to the aqueous half to neutralize the system, creating a vacuum above the neutral solvated protein system.^{2, 5, 30} The protein was placed at the interface with the lysine sidechains pointing into the water and the leucine side chains facing the vacuum (Figure 1B) and was modeled with the AMBER ff14SB force field.³¹ The N-termini were acetylated, and the C-termini were amidated (−NH₂). The energy of the system was minimized, followed by equilibration in an NVT (constant number of particles, volume, and temperature) ensemble for 6 ns at 298 K. Langevin dynamics was propagated with a friction coefficient of 1 ps⁻¹ and a timestep of 1 fs. Long-range electrostatic interactions were treated with the particle-mesh Ewald method.³² Water molecules were kept rigid with the SETTLE algorithm, and bonds involving protein hydrogens were constrained with the SHAKE method.^{33, 34} Configurations were saved every 10 fs. For the calculation of achiral SFG spectra, trajectories were wrapped in the z-direction to consolidate the slab into a single unit, and spectral calculations took periodic boundary conditions in the x- and y- directions into account. Wrapping was not necessary for chiral SFG calculations (see SI for explanation). The simulations were performed with the OpenMM 7.4 library on a Tesla V100 GPU using CUDA 9.2.³⁵ See the SI for full equilibration details.

SFG calculations of the aqueous protein system.

The chiral SFG response was calculated using the dipole-polarizability time correlation function approach with electric field mappings.^{17–19, 21, 36} This approach uses empirical mappings between the electric field measured at the water hydrogens in the direction along the O-H bond and quantities such as the dipole derivative, the vibrational frequency, and the vibrational couplings. The inhomogeneous limit approximation was used as in the expression

$${\chi }_{IJK}^{(2)}(\omega )\approx \langle \sum _{a=1}^{N_{\text{OH}}}\frac{\sum _{b=1}^{N_{\text{OH}}}{\alpha }_{IJ}^b{U}_{ba}\sum _{b=1}^{N_{\text{OH}}}{\mu }_K^b{U}_{ba}}{{\lambda }_a-\omega -\frac{i}{2\tau }}\rangle$$ (8)

where α is the transition polarizability tensor of each O-H bond computed with a bond-polarizability approach, μ is the O-H stretch transition dipole moment, U is the eigenvector matrix of the array of vibrational frequencies and couplings, λ is the corresponding set of eigenvalues of the Hamiltonian matrix, and τ is the vibrational lifetime, which is assumed to be 1.3 ps.^{17, 18, 37} The indices a and b refer to particular O-H bonds, while the indices I, J, and K each refer to the dimensions x, y, z making up a component of χ⁽²⁾.

In the inhomogeneous limit, the response is a simple average over conformations with no explicit time dependence.¹⁷ This approach was used to maximize the number of frames averaged during the procedure for an optimal signal-to-noise ratio. The computationally less expensive alternative, the time-averaging approximation, correlates the dipole and polarizability across a short time interval, with the frequencies and couplings averaged over that interval. As shown in Figure S1, this approach did not significantly change the spectrum other than increasing the noise.¹⁷ A fully time-dependent correlation-function approach such as that introduced previously¹⁹ is not feasible for a system of this size and a signal that requires long trajectories to converge.

The calculation of intermolecular couplings using this approach requires a dipole-dipole distance. To find this distance, each O-H bond was assigned an effective dipole location 0.67 Å from the oxygen along the bond.¹⁹ For achiral SFG, the contribution of the bottom half of the system was suppressed with a sigmoidal damping function to avoid contributions from a second surface opposite the surface containing the protein.¹⁹ The z-component of the effective dipole location of each O-H bond was used to assign the bond’s position along the z-axis for the damping function. No damping was necessary for chiral SFG calculations because the protein stayed on the top surface.

The polarizability is assumed to be localized at each O-H bond, and the ratio of the component of the polarizability derivative along the bond to the components perpendicular to the bond is assumed to be constant.^{36, 38} We use a ratio of 5.6, as the Skinner group has in the past.³⁶ The value of this constant affects the magnitude and sign of the chiral SFG response but does not affect the position of the peaks (see SI, Figure S2).

For each SFG calculation, 100 ns of molecular dynamics data was analyzed, corresponding to 10,000,000 frames. The electric fields needed for the frequency mapping were calculated using the previously optimized charges for the atoms of TIP4P water,¹⁹ while AMBER ff14sb charges were used for the protein and ions.^{19, 31} The protein was assumed not to have any vibrational coupling with water molecules and therefore influenced the system only geometrically and as a source of electric field. The frequency resolution was 1 cm⁻¹. The calculations were implemented in Python and Cython, using the MDAnalysis and NumPy libraries.^{39, 40}

Extracting chiral SFG signal from MD trajectories.

The chiral SFG theory is derived based on the assumption that the system of interest has C_∞ symmetry. This is not true for molecular dynamics trajectories with limited sampling. However, it is not necessary to sample every orientation equally.²³ Instead, one can simply assume that each frame of the trajectory is rotated by a random angle about the z-axis. One can then integrate over all values of the angle. In the rotated frame of reference denoted by the prime symbol, each component of χ⁽²⁾ is transformed according to

$$\begin{gathered} {\chi }_{z^{\prime }{y}^{\prime }{x}^{\prime }}^{(2)}\to {\chi }_{zyx}^{(2)}-{\chi }_{zxy}^{(2)} \\ {\chi }_{x^{\prime }{y}^{\prime }{z}^{\prime }}^{(2)}\to {\chi }_{xyz}^{(2)}-{\chi }_{yxz}^{(2)} \end{gathered}$$ (9)

and therefore, according to equation 3, the psp response is given by

$${\chi }_{psp}^{(2)}\propto {\chi }_{zyx}^{(2)}-{\chi }_{zxy}^{(2)}+{\chi }_{yxz}^{(2)}-{\chi }_{xyz}^{(2)}$$ (10)

Where ${\chi }_{yxz}^{(2)}-{\chi }_{xyz}^{(2)}$ in the original frame, which comes from ${\chi }_{x^{\prime }{y}^{\prime }{z}^{\prime }}^{(2)}$, is approximately equal to zero because the Raman polarizability tensor, which corresponds to the first two indices in χ⁽²⁾, is approximately symmetric when the incident beam frequencies are far from an electronic resonance.²³ This leaves

$${\chi }_{psp}^{(2)}\propto {\chi }_{zyx}^{(2)}-{\chi }_{zxy}^{(2)}$$ (11)

SFG calculation of model systems.

To calculate the SFG spectra of the model systems, the approach described above (equation 8) was adapted and implemented in a Mathematica notebook. Unlike our SFG calculations of the protein system, electric fields were ignored in the model system calculations, and the gas phase values of mapped quantities for the OH bond were used for all systems.³⁶ The water dimer structure coordinates were obtained from a pair of water molecules taken from near LK₇β in a frame of an MD trajectory. The C_3v (i.e., ammonia) molecule was built as a perfectly trigonal pyramidal structure (109.5° between adjacent bonds), and the dimer was built using simple geometric transformations that created a segment of a chiral helix. The same procedure was used to create a dimer of methane molecules. The chiral SFG spectrum of each model structure was calculated by averaging 200 conformations rotating around the z-axis. Here ${\chi }_{psp}^{(2)}={\chi }_{zyx}^{(2)}-{\chi }_{xyz}^{(2)}+{\chi }_{zyz}^{(2)}-{\chi }_{xyx}^{(2)}$ (see Figure 2), and the disappearance of the ${\chi }_{zyz}^{(2)}$ and ${\chi }_{xyx}^{(2)}$ components were monitored to ensure adequate rotational averaging (full integration over all rotations was impractical). The negative signs arise from the vectors specifying the orientations of the incoming and outgoing beams.²³ Unless otherwise stated, the O-H/N-H/C-H groups were treated as equivalent in terms of dipole and polarizability magnitudes, intramolecular and intermolecular couplings, and vibrational resonance frequency. Natural fluctuations in these quantities were not sufficient to create a chiral SFG signal in the absence of couplings in the protein-containing system (data not shown), so variation in these quantities was not considered in the model systems. The O-H/N-H/C-H stretch frequencies were set to 3400 cm⁻¹, roughly reflecting IR spectra of the O-H and N-H stretches, while the C-H frequency of methane was set to 2900, roughly reflecting IR spectra of liquid alkanes.^{36, 41} Note that the precise value of the resonance frequency does not affect the predicted spectra except by shifting them (see Figure S3). The vibrational lifetime (i.e., Lorentzian damping coefficient) was set to 0.01 seconds for the simple model dimer systems to achieve a Lorentzian peak broadness similar to that observed in the real system. If the lifetime is assumed to be in a more realistic range (~1 picosecond), the peaks become extremely sharp and less convenient to plot. The conclusions presented here are not affected by the choice of damping coefficient. The frequency resolution was 10 cm⁻¹. The polarizability was modeled as above, with a parallel-to-perpendicular polarizability derivative ratio of 5.6 assumed for both water and ammonia.^{36, 42}

Results

Biomacromolecules have been observed to organize surrounding water into dynamic chiral superstructures.^{1, 2, 5, 43} Here, we use a model system consisting of the antiparallel β-sheet formed by the short LK₇β peptide that is stabilized by the vacuum-water interface (Figure 1). Switching the chirality of the protein amino acids from (L-) to (D-) causes the experimental chiral SFG response to flip.⁵ This effect is clearly reproduced in our calculations (Figure 3). The chiral (psp) signal is around two orders of magnitude weaker than the achiral (ssp) signal and converges about ten times slower (Figure S4). These observations are likely the result of the complexity and variety of the chiral water supramolecular structures, the relatively slow dynamics of the protein scaffold compared to water, and the relatively small number of water molecules actively participating in the solvation shell. An interface containing pure water produces no significant chiral SFG signal (Figure 3). The small residual signal for pure water may be due to imperfect molecular dynamics sampling, which is limited by the high computational cost of the SFG spectrum calculations for systems of this size (~1350 water molecules). Figure S5 shows that two independent simulations of a pure water interface system produce extremely noisy and completely different signals, confirming that the chiral SFG response of a pure water surface is effectively zero. The small difference in intensity between (L-) and (D-) LK₇β signals is most likely also a result of limited sampling due to the high computational cost.

Figure 3.

(A) Imaginary and (B) real components of the psp SFG response of the O-H stretch of water around a pentamer of (L-) LK₇β (red) and its (D-) enantiomer (blue) at the vacuum-water interface (see Figure 1 for structure) as well as a pure-water interface with the same simulation box size (black). Spectra are averaged over 10,000,000 frames from 100 ns of molecular dynamics simulation. Spectra are unnormalized.

Understanding the role of vibrational couplings in chiral SFG signals is important for the interpretation of experimental data. We find that neglecting both intermolecular and intramolecular vibrational couplings completely removes the chiral SFG signal from water around LK₇β (Figure 4A), whereas the achiral SFG signal only blue-shifts slightly when both couplings are absent (Figure 4B). Note that we neglect couplings only in the SFG calculation and not in the MD simulation, where all interactions specified by the force field are preserved. Although both intermolecular and intramolecular couplings contribute to the chiral signal, either one is enough to make it appear. Intermolecular couplings cause a small red-shift in both the chiral and achiral SFG responses.⁴⁴ In real systems, this shift is likely caused by the vibrational delocalization of the O-H stretch over many molecules. Water molecules that interact with their neighbors have less electron density to devote to covalent bonds, which weakens the bonds and thus lowers the vibrational frequencies. In our model, the red shift is caused by the shift in the eigenspectrum of the coupling matrix due to significant off-diagonal components, thereby approximating the aforementioned physical effect. Because of the method by which these couplings were calculated, couplings between the O-H stretch and protein modes, for example the N-H stretch in the protein backbone, were not included.³⁶ However, the protein still contributed to the spectrum by modulating the local electric field experienced by the water molecules.³¹

Figure 4.

(A) Imaginary chiral (psp) SFG response of an (L-) LK₇β pentamer at the vacuum-water interface with different treatments of O-H/O-H vibrational coupling. (B) Imaginary achiral (ssp) SFG response of the same (L-) LK₇β system. (C) Intramolecular and intermolecular coupling magnitudes for the chiral SFG response between O-H groups among waters in the LK₇β system as a function of dipole-dipole distance, where each dot represents coupling between two OH groups for one configuration obtained from an MD trajectory. The mean and standard deviation are given for each type of coupling. The intramolecular couplings (red) are given for a single distance (1.14 Å) and thus correspond to a vertical line. (D) Fraction of total chiral SFG response (the sum of the absolute values of the responses for all frequencies) versus intermolecular coupling distance cutoff, where the distances were computed using effective dipole locations for each O-H bond.

Figure 4C shows the distribution of vibrational couplings calculated from a representative frame of an MD simulation plotted against the dipole-dipole distance. The average value of the intramolecular coupling (−17 cm⁻¹) is much larger in magnitude than the average value of the intermolecular coupling (−0.025 cm⁻¹), but a few very short-distance (< 3 Å) intermolecular couplings are larger in magnitude (> 100 cm⁻¹) than any intramolecular coupling. The standard deviation of the intermolecular coupling magnitudes (1.1 cm⁻¹) indicates that these larger couplings are extremely rare.³⁶ In Figure 4D, intramolecular couplings are included fully, but intermolecular couplings are only included up to a distance cutoff. This plot reveals that most of the contribution to the spectral intensity from the intermolecular couplings arises from interaction distances within 3 Å, that is, from the immediate neighbors of each molecule.

Although we have shown that short-range interactions produce most of the chiral SFG signal, we have not addressed the length scale at which these interactions originate. It may be that the observed effect of couplings is additive and only valid for large macromolecular systems and not for smaller chiral assemblies. To test this possibility, we calculated the chiral SFG response of a minimal water dimer system in a chiral configuration at an interface with the macroscopic symmetry of C_∞ (Figure 5A). Figure 5A (top) shows a pair of the enantiomeric water dimers. For simplicity we assumed that each O-H group has the same dipole moment and polarizability magnitude. Intramolecular couplings were calculated using gas-phase values³⁶ for mapped quantities such as the dipole derivative. Intermolecular couplings were calculated based on distance by using gas-phase values for the transition dipoles, as described in the Methods. We assumed a complex Lorentzian response with a damping factor chosen to produce peaks with a similar width as in the protein-water spectra, although the peak width does not affect our conclusions. As shown in the simulated spectra (Figure 5A) from top to bottom, when all couplings are included, the mirror-image water dimers produce signals with opposite phase. When only intramolecular couplings are considered, the intensity of the signals does not change significantly. Including just intermolecular couplings (as would be the case for a system such as HOD) produces a small but reliable signal as well. When there is no vibrational coupling, the signals disappear completely.

Figure 5.

Effect of coupling on model dimer systems: (A) water dimer, (B) ammonia dimer, (C) methane dimer. Top: Enantiomers of the dimers. Bottom: simulated chiral SFG spectra of the dimers in mirror images (black and red) for various combinations of vibrational couplings. The peak was set at 3400 cm⁻¹ for water and ammonia and 2900 cm⁻¹ for methane and the couplings were calculated using Skinner’s approach assuming no electric field, where the couplings still depend on dipole alignment and distance between dipoles (see Methods).

To further illustrate the interplay of vibrational couplings and molecular symmetry in determining the chiral SFG response, we performed the same calculation for two model systems with a high degree of symmetry: a dimer of ammonia molecules and a dimer of methane molecules, both arranged in a chiral configuration, shown as a pair of enantiomers in Figure 5B-C (top). We observed coupling effects similar to those of the water dimers but with a major difference – intramolecular couplings on their own produced no signal. This result aligns with Simpson’s prediction of no chiral signal from an ideal C_3v or T_d (tetrahedral) molecule.^{16, 28} In the next section, we will discuss how intermolecular couplings can in principle break symmetry and thereby produce chiral SFG signals in any molecule.

Discussion

Water has a uniquely strong and extensive network of hydrogen bonds. These bonds significantly delocalize the O-H stretch mode and therefore change the symmetry of vibrational chromophores. To illustrate this concept, Figure 6 schematically shows how couplings change the fundamental geometry of a water dimer that is frozen in an asymmetric configuration. When O-H stretches are uncoupled, the system reduces to a set of four O-H bonds. When intramolecular couplings are added, each pair of bonds is unified, and the smallest indivisible unit of the system becomes a whole water molecule with C_2v symmetry. When intermolecular couplings are considered, the entire system is interconnected, and the fundamental geometry behind the spectral response becomes the (chiral) geometry of the entire water superstructure interacting with the chiral protein or other macromolecules.

Figure 6.

Schematic showing how including vibrational couplings effectively changes the geometry (and symmetry) of a molecular system.

Based on the concept that vibrational couplings change the symmetry of vibrational chromophores (Figure 6), we can discuss the observations in the simulated chiral SFG spectra (Figures 4–5) in the context of Simpson’s symmetry-based theory. First, for Figure 4A, in the case of no vibrational coupling, each O-H bond in the protein-water system is an isolated linear oscillator, and the O-H vibrational mode belongs to the C_∞v symmetry group. For such a system, all tensor elements of the molecular hyperpolarizability are zero except β_ccc, β_aac, and β_bbc, where the c-axis points along the O-H bond, and all three elements are either proportional or equal to each other.²⁸ These three elements do not contribute to the chiral χ⁽²⁾ response according to equation 6 because β_aac = β_bbc. Thus, chiral SFG is forbidden¹⁶ (purple, Figure 4A). When only intramolecular couplings are considered (red, Figure 4A), the symmetry of the vibrational chromophores becomes C_2v. This lower symmetry leads to additional nonzero β elements (β_aca, β_bcb, β_caa, and β_cbb), where the c-axis is defined as the C₂ rotation axis of the molecule) which can contribute to chiral SFG as described by equation 7. Thus, the chiral SFG signal can be nonzero (red, Figure 4A). In the case of intermolecular couplings alone, each O-H group of individual water molecules communicates with O-H groups of other water molecules. This cross-talk further reduces the symmetry and results in additional nonzero β elements if the solvent surrounds a chiral template. In the extreme case, the symmetry is reduced to C₁ (with no symmetry elements), and thereby all β hyperpolarizability tensor elements are nonzero. In this case, β is associated with the delocalized vibrational chromophore due to intermolecular coupling, not individual molecules, such that β effectively becomes χ⁽²⁾ if the molecular frame is chosen to align with the laboratory frame. We note that β refers to the hyperpolarizability tensor of the smallest independent molecular unit of the system, which is larger than a single molecule if all couplings are considered. As long as not all β elements in equation 6 are zero or cancel out, ${\chi }_{zyx}^{(2)}$ does not vanish. Hence, chiral SFG is allowed (blue, Figure 4A). Finally, when both intramolecular and intermolecular couplings contribute, the net effect is a combination of the second and third cases, yielding larger chiral SFG signals (black, Figure 4A). We emphasize that both types of coupling are necessary to obtain the full lineshape, although we cannot generally predict the ratio of intramolecular versus intermolecular contributions in all systems.

The above arguments can also be applied to elucidate the simulated achiral spectra shown in Figure 4B, where the signals do not significantly depend on vibrational coupling. For achiral SFG experiments performed using the ssp polarization response, the effective ${\chi }_{ssp}^{(2)}$ arises from ${\chi }_{xxz}^{(2)}$ and ${\chi }_{yyz}^{(2)}$ according to the definition of coordinates presented in Figure 2, where ${\chi }_{xxz}^{(2)}={\chi }_{yyz}^{(2)}$ due to isotropy in the xy plane. Thus, ${\chi }_{ssp}^{(2)}$ can be expressed in terms of β elements as in equation S11 that contains β_ccc. This element does not vanish even for linear structures when β is averaged over all rotations about the z-axis.²⁸ Hence, in the case of no couplings, where the vibrational chromophore has C_∞v symmetry, the nonzero element β_ccc can contribute to the achiral signal (purple, Figure 4B). When intramolecular and/or intermolecular couplings are considered, the vibrational chromophores have less symmetry, which leads to more nonzero β elements that can contribute to ${\chi }_{ssp}^{(2)}$ as described in equation S11.

The differing sensitivity to couplings between chiral responses (Figure 4A) and achiral responses (Figure 4B) may also be due to the fundamentally different origins of the signals. Achiral signals are related to the magnitude of the z-component of the total system dipole, resolved by vibrational frequency, while chiral responses result from imperfect cancellation of hyperpolarizability terms and have no direct dependence on the overall dipole. Hence, achiral SFG signal intensity does not depend strongly on couplings or any other symmetry-breaking features unless those features increase the orientational anisotropy of the system as a whole along the z-axis.

Although the coupling effects observed for the protein-water system (Figure 4A) generally agree with those observed for the water dimer system (Figure 5A), intermolecular couplings alone produce a larger effect in the protein-water system than in the model water dimer. It seems that the large effect of intermolecular couplings shown in Figure 4A may require water assemblies larger than a dimer or a shorter inter-water distance than assumed in the model system. The intramolecular couplings may also be somewhat inflated relative to the intermolecular couplings in this simple model because gas-phase values of mapped quantities are used. The intramolecular coupling in the gas phase is around −55 cm⁻¹, whereas the average value in bulk water is approximately −30 cm⁻¹.³⁶ We decided to use gas-phase values for the model system (i.e., ignore the electric field) to demonstrate that the effect of couplings is mainly a geometric or topological effect, rather than the result of emergent chirality in local fields. Local fields may still play a role in the real system, but they are not necessary for the coupling effect.

Intramolecular and intermolecular couplings can have synergistic effects. For the C_3v dimer system, intermolecular coupling alone produces a weaker signal than when all couplings are included, even though intramolecular coupling alone produces no signal (Figure 5B). This observation suggests that intramolecular couplings contribute to the signal, but only if allowed to do so by a symmetry-breaking feature such as intermolecular couplings. This effect is not seen in the methane (T_d) dimer (Figure 5C), suggesting that the synergistic effect of intramolecular and intermolecular couplings is system dependent. Other symmetry-breaking features such as varying local electric fields may also be able to unlock chiral signals. However, all of our results suggest that intermolecular vibrational couplings alone are sufficient to unlock chiral signals in the simulated spectra.

The C_2v point group is effectively just asymmetric enough that the surviving hyperpolarizability components can give rise to a chiral SFG signal (equation 7). This is under the condition that the C_2v molecules are assembled in chiral supramolecular assemblies at an interface with C_∞ symmetry. Molecules of the C_3v point group, for example, have relatively higher symmetry, so all components of the hyperpolarizability that can give rise to chiral signals disappear.¹⁶ The same is true for other highly symmetric solvents. Although chiral SFG studies have focused on water interacting with biomolecules, our work suggests that any solvent surrounding a chiral solute at an interface with the C_∞ symmetry may be able to generate a chiral SFG signal, provided that some symmetry-breaking features are present. Our results show that intermolecular vibrational couplings can be one of these features and generate chiral signals in simulated SFG spectra. This finding is consistent with the previously developed theory that predicts nonzero chiral 2D-SFG signals in pairs of achiral but coupled chromophores.²⁹

The results in Figures 5B and 5C have significant implications for the study of the chiral SFG responses of non-aqueous solvents, some of which are highly symmetric molecules or contain highly symmetric functional groups, such as the -CCl₃ group in chloroform and methyl group in methanol. One of the most common chemical groups in organic solvents is the methyl group, which has C_3v symmetry. Figure 5B shows that a C_3v molecule (i.e., ammonia) can produce chiral SFG signals if heterogeneous intermolecular couplings are present. As the calculations in Figure 5 assume little about each chemical system except geometry and the presence of couplings, the results for one C_3v molecule translate qualitatively to all C_3v molecules and functional groups. Similarly, the results for methane translate qualitatively to tetrahedral solvents such as carbon tetrachloride. Accordingly, the symmetry-breaking feature of intermolecular (or more precisely, inter-functional group) couplings may explain the prominent chiral peak assigned to the methyl asymmetric stretch observed by Wang et al. in the antiparallel β-sheet LK₇β peptides at the air-water interface.⁴⁵ Moreover, Geiger and coworkers observed a chiral SFG signal for methyl groups in DNA.^{12, 16} This observation has been a puzzle in the field because C_3v entities are forbidden to produce chiral SFG signals based on symmetry arguments. However, our results show that vibrational couplings or other environmental factors (e.g., local molecular interactions or electric fields) can potentially break the symmetry and induce chiral SFG signals.

In this paper, we have discussed coupling between individual O-H vibrational modes. We found that coupling of some kind is essential to break the symmetry and allow for a chiral SFG signal. There are many vibrational modes, however, that involve symmetry-breaking coupling within the mode itself. This includes the amide I vibrational mode, which consists of both a carbonyl stretch and an N-H bend. In this case, intramolecular couplings are taken into account. Hence, it is not surprising that Skinner reports only a modest effect of intermolecular couplings of amide groups on the chiral SFG spectrum in the amide I region.²³

Both the symmetric and asymmetric vibrational modes of water involve both O-H groups, but in isotopically labeled variants such as HOD, intramolecular coupling is minimal due to the difference in vibrational frequency of O-H (~3750 cm⁻¹) and O-D (~2750 cm⁻¹).^{19, 46} In this case, intramolecular coupling is ignored, and the symmetry-based theory predicts no chiral SFG signal. However, our results indicate that intermolecular couplings contribute substantially and alone can produce a basic lineshape that is similar to that for the combination of intramolecular and intermolecular couplings (Figure 4A). Hence, our simulations imply that the O-H stretch in HOD systems surrounding chiral solutes will produce a chiral response. This is in agreement with isotope dilution experiments by Perets et al.²

Conclusion

This work examines the origins of the chiral SFG signal of the O-H stretch from aqueous interfaces containing chiral biomacromolecular solutes. Although vibrational couplings change the achiral SFG spectrum only slightly,⁴⁴ they are absolutely essential to the simulation of a chiral response. Our analysis indicates that vibrational couplings change the effective geometry of vibrational chromophores and introduce crucial asymmetry to induce chiral SFG responses. Our calculations show that the dependence on coupling arises even for a simple model system composed of two water molecules in a fixed geometry. Therefore, the effect of couplings on chiral SFG is not a phenomenon arising only in complex macromolecular systems.

Our findings have implications for theorists modeling SFG responses. Importantly, researchers using any method that omits vibrational couplings between O-H groups (or other bonds) must exercise caution when modeling chiral SFG signals. Intramolecular and intermolecular couplings are found to have comparable contributions to the chiral SFG response of water. Including intermolecular couplings as well as intramolecular couplings is not critical to qualitatively capturing a chiral response, although it is necessary to obtain the full signal. In contrast, for molecules with higher symmetry than water, the symmetry-breaking effect of intermolecular couplings is absolutely required for the appearance of chiral SFG signals in simulated spectra. Nonetheless, experimental detection of such signals depends on many factors (e.g., signal-to-noise level) and thus remains to be examined. Other symmetry-breaking effects, such as local molecular interactions and applications of external electric fields, may also be effective in inducing chiral SFG signals, and these remain to be explored with experiments and simulations. This work provides the foundation for further investigations that may examine the chiral templating of non-aqueous solvents surrounding non-biological macromolecules as well as the interplay of asymmetric polarizability tensors (i.e., electronic resonance) and vibrational coupling in a variety of biological and non-biological systems.