## Decomposition of Virus Normal Modes into Spherical Harmonics An Exploration of Symmetry Adapted Functions (SAFs)

##### Abstract

The normal modes for viruses, in which all of its atoms are oscillating at the same frequency, can be thought of as a very complicated system of masses and springs. Each atom has its own mass, and the bonds between atoms can be modeled as springs. The solution for normal modes of spherical viruses incorporates a sum of Laplacian spherical harmonics, based on the assumption that a viral capsid can be approximated by a thin-shelled homogenous mass distribution. However, the regular spherical harmonics by themselves do not exhibit the same type of symmetry as spherical viruses. In order to achieve icosahedral symmetry, we need to linearly combine specific spherical harmonics of the same order l, which generates the Symmetry Adapted Functions (SAFs). The SAFs are orthogonal functions that can be combined to build any function that is icosahedrally defined. These SAFs are not identical, however, and they predict different patterns of motion. Some SAFs predict max radial displacement about the symmetry axes, while others describe perfectly uniform contraction and expansion. As the order of the SAF is increased, the gradient between close regions increases. In other words, the SAF appears much spikier. The calculated normal modes for viruses are organized as a series of displacement instructions (in the form of a 3 dimensional vector) for every single atom. For a typical virus of 200,000 atoms, this means that the normal mode data is stored in the form of a 200,000 x 3 matrix. If we want to compare different calculated normal modes, we can build VMD animations based on the displacement instructions and compare them qualitatively. However, until now there has been no robust way to decompose and describe the normal modes with a set of parameters, which facilitates comparison and analysis. Instead of comparing two 200,000 x 3 matrices, the decomposition outlined in this paper allows us to compare the overlap weights of each mode with 13 SAFs. Because the SAFs are orthogonal, icosahedral functions, they can be used as a basis for icosahedral normal modes. Simply taking the calculated mode matrix and dotting it with an `artificial' mode predicted by a given SAF yields the overlap associated with that SAF. These overlap weights provide crucial details about the composition of a normal mode. If two different modes have the same exact decomposition but different mode number, we know that the two motions are exactly alike. One crucial piece of information regarding the SAFs is that they can only account for radial displacements. That is to say, if a mode is predominantly composed of twisting and stretching, the overlap values will be alarmingly low. There were two normal mode sets that were decomposed and compared in this paper. The first set, denoted the Hammond modes, was calculated by Rob Hammond for his SIP. The second set, denoted the Rizzolo modes, was calculated by Skylar Rizzolo for his SIP. Despite the fact that some of these calculated modes used the same viruses as input, they were outputting different motions. In order to see which modes describe similar motion, despite having a different mode number, both sets of modes were decomposed into the 13 SAFs and then compared. Modes that were visually similar when undergoing qualitative analysis showed similar decomposition patterns. Modes that were composed only of contraction and expansion showed strong overlap with SAF0, the breathing mode. We believe that an accurate set of calculated modes contains modes that show broad representation of SAFs, has strong overlap values with the SAFs, and does not contain multiple degenerate motions (e.g. having 4/10 modes being the breathing mode). Viruses that have convincing mode sets prove that they can be accurately approximated as a thin-shelled homogenous mass distribution. Viruses that showed poor overlap and tight distributions suggest that they cannot be described well by a thin-shelled model.