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In this paper, we systematically presented the orientation determination of protein helical secondary structures using vibrational spectroscopic methods, particularly, nonlinear sum frequency generation (SFG) vibrational spectroscopy, along with linear vibrational spectroscopic techniques such as infrared spectroscopy and Raman scattering. SFG amide I signals can be collected using different polarization combinations of the input laser beams and output signal beam to measure the second-order nonlinear optical susceptibility components of the helical amide I modes, which are related to their molecular hyperpolarizability elements through the orientation distribution of these helices. The molecular hyperpolarizability elements of amide I modes of a helix can be calculated based on the infrared transition dipole moment and Raman polarizability tensor of the helix; these quantities are determined by using the bond additivity model to sum over the individual infrared transition dipole moments and Raman polarizability tensors, respectively, of the peptide units (or the amino acid residues). The computed overall infrared transition dipole moment and Raman polarizability tensor of a helix can be validated by experimental data using polarized infrared and polarized Raman spectroscopy on samples with well-aligned helical structures. From the deduced SFG hyperpolarizability elements and measured SFG second-order nonlinear susceptibility components, orientation information regarding helical structures can be determined. Even though such orientation information can also be measured using polarized infrared or polarized Raman amide I signals, SFG has a much lower detection limit, which can be used to study the orientation of a helix when its surface coverage is much lower than a monolayer. In addition, the combination of different vibrational spectroscopic techniques, for example, SFG and attenuated total reflectance Fourier transform infrared spectroscopy, provides more measured parameters for orientation determination, aiding in the deduction of more complicated orientation distributions. In this paper, we discussed two types of helices, the alpha-helix and 3-10 helix. However, the orientation determination method presented here is general and thus can be applied to study other helices as well. The calculations of SFG amide I hyperpolarizability components for alpha-helical and 3-10 helical structures with different chain lengths have also been performed. It was found that when the helices reached a certain length, the number of peptide units in the helix should not alter the data analysis substantially. It was shown in the calculation, however, that when the helix chain is short, the SFG hyperpolarizability component ratios can vary substantially when the chain length is changed. Because 3-10 helical structures can be quite short in proteins, the orientation determination for a short 3-10 helix needs to take into account the number of peptide units in the helix.

Orientation Determination of Protein Helical Secondary Structures Using Linear and Nonlinear Vibrational Spectroscopy