Classical Physics
Circular Dichroism Calculations
General molecular and ensemble-level spectroscopic techniques, accessible in the solvated environment, provide insight into structural properties such as secondary structure content for these systems.14,15 One such technique, widely used for studying proteins, is circular dichroism (CD) spectroscopy. CD measures the difference in absorption of left- and right- circularly polarized light in the absence of a magnetic field. As electrons in the molecule absorb light, the transition of electrons into local excited states induce dipole moments within the molecule. These induced dipole moments interact with one another electronically and magnetically. The amide groups in proteins possess characteristic pi-pi* transitions (180-210 nm, ~140 nm) n-pi* transitions (220 nm) whose locations and intensities indicate secondary structural features of the peptide sequence.16 The molar elipticity, epsilon, and the CD spectrum, De, of a molecule arise from the dipole strength, D (Equation 3) and rotational strength, R (Equation 4) which arise from interaction of the electronic (e) and magnetic dipole moments (m) of each specific transition from ground state, 0, to excited state a, in some wavelength of applied light, l.[3][4]
NA is Avogadro's number, h is Planck's constant, and c is the speed of light. Three parameters describe each CD spectrum: the position of maximum absorption (nmax), intensity of the absorption (Demax), and the shape of the band.
There are several methods to predict CD spectra for a molecule with knowledge of the structure. Quantum mechanics allows for direct solution of the dipole and rotational strength, although this is computationally infeasible for very large systems. One approach to handle larger molecules requires division of the molecule into a number of separate model chromophores and treating those chromophores quantum mechanically. Coupled with solution of the Schrödinger equation for isolated model chromophores over ground and excited states, this splitting yields the method of Tinoco17and the matrix method[18-20].
Another approach is to calculate CD using classical physics. The dipole interaction model[21]-[26]is one such classical physics-based method for predicting the CD of peptides and molecules built from peptides. This model, which is capable of treating the amide chromophore well, includes all atoms except the amide group as points having nondispersive polarizability, and the amide group as a single point possessing dispersive polarizability. The relationship between the rotational strength, R, and the measured difference in absorption of left- and right- circularly polarized light is given by Equation 5 for the classical dipole interaction model, assuming a Lorentzian band shape.[22,24,27-29]
In the dipole interaction model, the sum over all dispersive oscillators (light-absorbing units, where there are q dispersive oscillators) of the interaction of the rotational strength (Rk) at each wavelength describes the CD (De) spectrum. NA is Avogadro's number, lambda is the half peak bandwidth, q is the number of dispersive oscillators, n is the number of peptide residues, and nðk is the normal mode wavenumber. The rotational strength of each segment of the molecule is obtained by dividing the molecule into atoms with isotropic and anisotropic polarizability. The nondispersive oscillators (those with isotropic polarizability) have constant polarizability factors. These atomic polarizabilities were obtained experimentally from fits to molecular polarizabilities of simple organics beginning with experimental atomic polarizability data determined at the NaD line (589 nm)[26,30]. Dispersive oscillators (those whose strength is wavelength-dependant), such as the amide chromophore, have been optimized to reproduce mean polarizabilities and Kerr constants at 589.3 nm for the pi-pi* transition in a variety of simple amides[31]. These original parameters were the first to reproduce CD for alpha-helical structures[22]. More recently, the parameters have been reoptimized to include molecular anisotropies to create three new sets of dipole interaction parameters: (1) general peptide systems, (2) alpha-helical systems ,and (3) poly-L-Pro-II. [23,32,33]
The dipole interaction model has proven successful for a variety of applications, including the prediction of CD spectra for beta-sheets,34 beta-turns[35,36], alpha-helices[22,37] beta-peptides[38-40], and is the only method published to obtain the correct pi-pi* spectrum for poly-L-proline II[41,42] and collagen[43]. The model has also proven successful on whole proteins including alpha-spectrin, tropomyosin[37] and lactate dehydrogenase[44].
"Theoretical UV Circular Dichroism of Cyclo(L-Proline-L-Proline)". Carlson, K. L.; Lowe, S. L.; Hoffmann, M. R.; Thomasson, K. A. Journal of Physical Chemistry A (2006) 110, 1925-1933.
"Theoretical UV Circular Dichroism of Aliphatic Cyclic Dipeptides". Carlson, K.L.; Lowe, S.L.; Hoffmann, M.R.; and Thomasson, K.A. . Journal of Physical Chemistry A (2005) 109, 5463-5470.
"Two Helical Conformations from a Single Foldamer Backbone: "Split Personality" in Short Alpha/Beta-Peptides." Hayden, A.; Schmitt, M.A.; Ngassa, F.N.; Thomasson, K.A.; and Gellman, S.H. Angewandte Chemie (Intl. Ed. in English) (2004) 43, 505-510.
"Dipole Interaction Model Predicted pi-pi* Circular Dichroism of Cyclo(L-Pro)3 Using Structures Created by Semi-empirical, Ab Initio, and Molecular Mechanics Methods.". S. L. Lowe, K. S. Pierce, J. Czlapinski, G. Kie-Adams, Rajeev Pandey, M. R. Hoffmann, K. A. Thomasson.. Journal of Peptide Research (2003) 61, 189-201.
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