Raman intensity and spectra predictions for cylindrical viruses

A theoretical framework for predicting low frequency Raman vibrational spectra of viral capsids is presented and applied to the M13 bacteriophage. The method uses a continuum elastic theory for the vibrational modes and a bond-charge polarizability model of an amorphous material to roughly predict the Raman intensities. Comparison is made to experimental results for the M13 bacteriophage virus.

Figure 1

(Color online) General experimental setup for Raman scattering. Incident light propagates along the $z$ axis and is then scattered toward the detector located at polar angles $\Theta$ and $\Phi$. $\Phi$ (not shown) is the standard polar angle in the $x$-$y$ plane.

Figure 2

(Color online) The 50 amino acid peptide building block for the M13 viral capsid.

Figure 3

(Color online) Views of the M13 bacteriophage showing the assembly of the protein building blocks into the completed capsid. Electrostatically negative (red) and positive (blue) regions are shown. (a) Schematic diagram of a cross section of the M13 capsid. (b) Axial view of M13. (c) View of the inside surface of the M13 capsid. One periodic unit of the capsid is shown which is approximately $16\mathrm{nm}$ in length. (d) View of the outside surface of the M13 capsid.

Figure 4

Frequency dispersion relation $\omega \left(k\right)$ for wave vectors in the EWT model of a cylindrical shell appropriate to M13. The shell has inner and outer radii of 1.9 and $3.4\mathrm{nm}$, respectively. (a) $n=0$, (b) $n=1$, (c) $n=2$.

Figure 5

Log-log plots of the $10.4\text{−}{\mathrm{cm}}^{-1}$, $n=1$ mode (at $R_{in}=1.9\mathrm{nm}$ and $R_{out}=3.4\mathrm{nm}$) of the cylindrical shell in the EWT model vs thickness $T$ $(T=R_{out}-R_{in})$. (a) Varying the thickness $T$ by changing the inner radius $R_{in}$ only. (b) Varying the thickness $T$ by changing the outer radius $R_{out}$ only. (c) Varying the thickness $T$ by scaling both the inner and outer radii by $X$ where $T=X(R_{out}-R_{in})$.

Figure 6

Theoretical relative Raman intensities of M13 for modes with (a) $n=0$, (b) $n=1$, (c) $n=2$. Dashed lines correspond to radial-torsional modes while solid lines correspond to axial modes. An arbitrary full width at half maximum of $1{\mathrm{cm}}^{-1}$ is used. We hypothesize that the radial-torsional modes (dashed) are highly damped.

Figure 7

(Color online) Theoretical displacement patterns for various modes. Axial modes are shown as cross sections through the $x$-$z$ plane of the virus while radial modes are shown as cross sections through the $x$-$y$ plane of the virus: (a) axial mode at $10.4{\mathrm{cm}}^{-1}$, (b) radial-torsional mode at $20.4{\mathrm{cm}}^{-1}$, (c) high frequency axial mode at $30.6{\mathrm{cm}}^{-1}$.

Figure 8

Theoretical Raman scattering profile of M13 bacteriophage after applying a broadening of $5{\mathrm{cm}}^{-1}$ to account for the inhomogeneous broadening seen in experiment. Two theoretical profiles are given in order to analyze the possible effects of damping due to the surrounding water. (a) Theoretical Raman profile if no water damping is accounted for, and all modes are included. (b) Theoretical Raman profile if radial-torsional modes are severely damped when compared to the damping of axial modes.

Figure 9

Experimental Raman intensity of M13 virus obtained from [23]. Three different concentrations of viral particles are shown. The intensity of the peak at $8.5{\mathrm{cm}}^{-1}$ grows in proportion to viral concentration.