https://hal.inria.fr/hal-01779893/document

Analytical symmetry detection in protein assemblies. I. Cyclic symmetries.

The majority of protein oligomers form clusters which are nearly symmetric. Understanding of that imperfection, its origins, and perhaps also its advantages requires the conversion of the currently used vague qualitative descriptive language of the near-symmetry into an accurate quantitative measure that will allow to answer questions such as: “What is the degree of symmetry deviation of the protein?,” “how do these deviations compare within a family of proteins?,” and so on. We developed quantitative methods to answer this type of questions, which are capable of analyzing the whole protein, its backbone or selected portions of it, down to comparison of symmetry-related specific amino-acids, and which are capable of visualizing the various levels of symmetry deviations in the form of symmetry maps. We have applied these methods on an extensive list of homomers and heteromers and found that apparently all proteins never reach perfect symmetry. Strikingly, even homomeric protein clusters are never ideally symmetric. We also found that the main burden of symmetry distortion is on the amino-acids near the symmetry axis; that it is mainly the more hydrophilic amino-acids that take place in symmetry-distortive interactions; and more. The remarkable ability of heteromers to preserve near-symmetry, despite the different sequences, was also shown and analyzed. The comprehensive literature on the suggested advantages symmetric oligomerizations raises a yet-unsolved key question: If symmetry is so advantageous, why do proteins stop shy of perfect symmetry? Some tentative answers to be tested in further studies are suggested in a concluding outlook. Proteins 2014; 83:722–734. © 2014 Wiley Periodicals, Inc.

The near-symmetry of proteins.

https://hal.inria.fr/hal-01816449/document

Analytical symmetry detection in protein assemblies. II. Dihedral and cubic symmetries

Symmetry is a fundamental property of nature, used extensively in physics, chemistry, and biology. The Continuous symmetry measures (CSM) is a method for estimating the deviation of a given system from having a certain perfect symmetry, which enables us to formulate quantitative relation between symmetry and other physical properties. Analytical procedures for calculating the CSM of all simple cyclic point groups are available for several years. Here, we present a methodology for calculating the CSM of any complex point group, including the dihedral, tetrahedral, octahedral, and icosahedral symmetry groups. We present the method and analyze its performances and errors. We also introduce an analytical method for calculating the CSM of the linear symmetry groups. As an example, we apply these methods for examining the symmetry of water, the symmetry maps of AB4 complexes, and the symmetry of several Lennard-Jones clusters. © 2014 Wiley Periodicals, Inc.

Continuous symmetry measures for complex symmetry group.

The majority of oligomeric proteins form clusters which have rotational or dihedral symmetry. Despite the many advantages of symmetric packing, protein oligomers are only nearly symmetric, and the origin of this phenomenon is still in need to be fully explored. Here we apply near-symmetry analyses by the Continuous Symmetry Measures methodology of protein homomers to their natural state, namely their structures in solution. NMR-derived structural data serves us for that purpose. We find that symmetry deviations of proteins are by far higher in solution, compared to the crystalline state; that much of the symmetry distortion is due to amino acids along the interface between the subunits; that the distortions are mainly due to hydrophilic amino acids; and that distortive oligomerization processes such as the swap-domain mechanism can be identified by the symmetry analysis. Most of the analyses were carried out on distorted C2-symmetry dimers, but C3 and D2 cases were analyzed as well. Our NMR analysis supports the idea that the crystallographic B-factor represents non-classical crystals, in which different conformers pack in the crystal, perhaps from the conformers which the NMR analysis provides.

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The near-symmetry of protein oligomers: NMR-derived structures

The continuous symmetry methodology has been developed to provide a quantitative estimation for the degree of a selected symmetry point-group in a molecular structure. Previous developed measures included the C(S), C(n), S(n) (including C(i)) symmetries, several polyhedral symmetries, and the related chirality measure. Motivated by the abundance and importance of C(nv) symmetry in many key molecules and by the abundance of D(n) symmetry in many organometallic complexes and in many protein clusters, measures for these symmetries, based on the general definition of the continuous symmetry measure, were developed. Detailed description of the derivation of the measures, along with plenty of examples for their application, is provided.

Maayan Bonjack

Papers

The near-symmetry of protein oligomers: NMR-derived structures

Continuous symmetry analyses: C(nv) and D(n) measures of molecules, complexes, and proteins.