## Task 2: Exhaustive FFT Search With Symmetries

Introduction
Many of the protein complexes in the protein Data bank (PDB) are symmetric homo-oligomers. According to the 3D-Complex database [1], C2 homo-dimers comprise the majority of known homo-oligomers. However, many complexes have higher order rotational symmetry (i.e. Cn>2), and a good number have multiple rotational symmetry axes, namely those with dihedral (Dn), tetrahedral (T), octahedral (O), and icosahedral (I) point group symmetries. Although symmetrical complexes are often solved directly by X-ray crystallography, it would still be very useful to be able to predict whether or not a given monomer might self-assemble into a symmetrical structure.
We present a new point group symmetry docking algorithm. In the last few years, several protein-protein docking programs have been adapted to predict symmetrical pair-wise docking orientations for Cn and Dn symmetries [2,3,4]. However, to our knowledge, there does not yet exist an algorithm which can automatically generate perfectly symmetrical protein complexes for arbitrary point group symmetry types.

Methods
We introduce the notion of a "docking equation" in which the notation ￼represents an interaction between proteins A and B in 3D space. It is also useful introduce the operators ￼ and ￼ which represent the actions of translating an object by an amount ￼ and rotating it according to the three Euler rotation angles ￼. Then, guided by Figure 1, and assuming that we start with two identical monomers at the origin, we can write down a Cn docking equation for the two monomers as $$\hat T(0,y,0) \hat R(\alpha,\beta,\gamma) A(\underline{x}) \leftrightarrow \hat R(0,0,\omega) \hat T(0,y,0) \hat R(\alpha,\beta,\gamma) B(\underline{x})$$Then, we perform a series of fast Fourier transform (FFT) correlation searches using the Hex spherical polar Fourier docking algorithm [5] to determine the four parameters ￼. For higher symmetries, Dn, T, O, and I, we introduce two more parameters and perform a series of FFT in a similar way. The calculation for each structure takes less than a minute on a modern workstation.

Figure 1. Illustrations of the C3 and D3 point group symmetries.

Results & Conclusions
We validated our method on protein structures from the 3D-Complex database [1], which contains 17,183 protein complexes with assigned biological unit and symmetry type. It mostly contains cyclic and dihedral proteins, as well as 86 tetrahedral, 47 octahedral, and 6 icosahedral complexes (excluding all viral structures). Starting from the structures of monomers, we generated symmetric biological units based on the symmetry type for each complex provided by 3D-Complex. Figure 2 summarizes the performance of our method on these proteins, where we say that the model is correct if all pair-wise RMSDs are smaller than 10 Ångstroms. On average, we found about 55% of correct predictions ranked first.

Figure 2. Summary of the correctly predicted complexes found on the first place (blue) and in the top ten solutions (green).

Figure 3 shows correctly predicted examples from each of the symmetry types. Each complex is perfectly symmetrical, although due to the soft docking function in Hex it is possible that some interfaces might contain minor steric clashes.

Figure 3. Illustrations of the correctly predicted complexes. For each complex, the group symbol and the PDB code are shown.

References
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2. S. R. Comeau & C. J. Camacho. Journal of Structural Biology 150, 233–244 (2004).
3. B. Pierce, W. Tong, & Z. Weng. Bioinformatics 21, 1472–1478 (2005).
4. D. Schneidman-Duhovny, Y. Inbar, R. Nussinov, & H. J. Wolfson. Nucleaic Acids Research 33, W363–W367 (2005).
5. D. W. Ritchie & G. J. L. Kemp. Proteins: Structure, Function, Genetics 39, 178–194 (2000).
6. D. W. Ritchie and S. Grudinin. Some Recent Progress on Generating Symmetric Protein Complexes Using Spherical Polar Fourier Docking Correlations. In Proceedings of the 14th Open Days in Biology, Computer Science and Mathematics (JOBIM), July 1 - 4, 2013, Toulouse (France), 2013.