The near-symmetry of proteins

Imperfect symmetry facilitated the evolution of specificity and high-order stoichiometry in vertebrate hemoglobin

Many proteins form paralogous multimers - molecular complexes in which evolutionarily related proteins are arranged into specific quaternary structures. Little is known about the mechanisms by which they acquired their stoichiometry (the number of total subunits in the complex) and heterospecificity (the preference of subunits for their paralogs rather than other copies of the same protein). Here we use ancestral protein reconstruction and biochemical experiments to study historical increases in stoichiometry and specificity during the evolution of vertebrate hemoglobin (Hb), a α2β2 heterotetramer that evolved from a homodimeric ancestor after a gene duplication. We show that the mechanisms for this evolutionary transition was simple. One hydrophobic substitution in subunit β after the gene duplication was sufficient to cause the ancestral dimer to homotetramerize with high affinity across a new interface. During this same interval, a single-residue deletion in subunit α at the older interface conferred specificity for the heterotetrameric form and the trans-orientation of subunits within it. These sudden transitions in stoichiometry and specificity were possible because the interfaces in Hb are isologous - involving the same surface patch on interacting subunits, rotated 180° relative to each other - but the symmetry is slightly imperfect. This architecture amplifies the impacts of individual mutations on stoichiometry and specificity, especially in higher-order complexes, and allows single substitutions to differentially affect heteromeric vs homomeric interactions. Many multimers are isologous, and symmetry in proteins is always imperfect; our findings therefore suggest that elaborate and specific molecular complexes may often evolve via simple genetic and physical mechanisms.

CSM Software: Continuous Symmetry and Chirality Measures for Quantitative Structural Analysis

We present a comprehensive and updated Python-based open software to calculate continuous symmetry measures (CSMs) and their related continuous chirality measure (CCM) of molecules across chemistry. These descriptors are used to quantify distortion levels of molecular structures on a continuous scale and were proven insightful in numerous studies. The input information includes the coordinates of the molecular geometry and a desired cyclic symmetry point group (i.e., Cs, Ci, Cn, or Sn). The results include the coordinates of the nearest symmetric structure that belong to the desired symmetry point group, the permutation that defines the symmetry operation, the direction of the symmetry element in space, and a number, between zero and 100, representing the level of symmetry or chirality. Rather than treating symmetry as a binary property by which a structure is either symmetric or asymmetric, the CSM approach quantifies the level of gray between black and white and allows one to follow the course of change. The software can be downloaded from https://github.com/continuous-symmetry-measure/csm or used online at https://csm.ouproj.org.il.

Continuous symmetry and chirality measures: approximate algorithms for large molecular structures

Quantifying imperfect symmetry of molecules can help explore the sources, roles and extent of structural distortion. Based on the established methodology of continuous symmetry and chirality measures, we develop a set of three-dimensional molecular descriptors to estimate distortion of large structures. These three-dimensional geometrical descriptors quantify the gap between the desirable symmetry (or chirality) and the actual one. They are global parameters of the molecular geometry, intuitively defined, and have the ability to detect even minute structural changes of a given molecule across chemistry, including organic, inorganic, and biochemical systems. Application of these methods to large structures is challenging due to countless permutations that are involved in the symmetry operations and have to be accounted for. Our approach focuses on iteratively finding the approximate direction of the symmetry element in the three-dimensional space, and the relevant permutation. Major algorithmic improvements over previous versions are described, showing increased accuracy, reliability and structure preservation. The new algorithms are tested for three sets of molecular structures including pillar[5]arene complexes with Li+, C100 fullerenes, and large unit cells of metal organic frameworks. These developments complement our recent algorithms for calculating continuous symmetry and chirality measures for small molecules as well as protein homomers, and simplify the usage of the full set of measures for various research goals, in molecular modeling, QSAR and cheminformatics.

The SARS-CoV-2 spike protein structure: a symmetry tale on distortion trail

A preliminary step in the SARS-CoV-2 human infection process is a conformational change of the receptor binding domain (RBD) of its spike protein, characterized by a significant loss of symmetry. During this process, the residues which later on bind to the human angiotensin converting enzyme 2 (ACE2) receptor, become exposed at the surface of the protein. Symmetry analysis of a data set of 33 protein structures from experimental measurements and 32 structures from molecular dynamics simulation, show that the initial state carries clear indications on the structure of the final state, with respect to the local distortion along the sequence. This surprising finding implies that this type of analysis predicts the mechanism of change. We further show that the level of local distortion at the initial state increases with variant's transmissibility, for the wild type (WT) along with past and present variants of concern (WT ∼ alpha < beta < delta < Omicron BA.1), in accordance with the trend of their evolutionary path. In other words, the initial structure of the variant which is most infectious is also the most distorted, making its path to the final state shorter. It has been claimed that the RBD migration of the spike protein is allosterically controlled. Our analysis provides a quantitative support to a major theorem in this respect - that information about an allosteric process is encoded in the structure itself, suggesting that the path of local distortion is related to an allosteric information network.

Side chain flexibility and the symmetry of protein homodimers

A comprehensive analysis of crystallographic data of 565 high-resolution protein homodimers comprised of over 250,000 residues suggests that amino acids form two groups that differ in their tendency to distort or symmetrize the structure of protein homodimers. Residues of the first group tend to distort the protein homodimer and generally have long or polar side chains. These include: Lys, Gln, Glu, Arg, Asn, Met, Ser, Thr and Asp. Residues of the second group contribute to protein symmetry and are generally characterized by short or aromatic side chains. These include: Ile, Pro, His, Val, Cys, Leu, Trp, Tyr, Phe, Ala and Gly. The distributions of the continuous symmetry measures of the proteins and the continuous chirality measures of their building blocks highlight the role of side chain geometry and the interplay between entropy and symmetry in dictating the conformational flexibility of proteins.

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

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.

Improved algorithms for quantifying the near symmetry of proteins: complete side chains analysis

Symmetry of proteins, an important source of their elegant structure and unique functions, is not as perfect as it may seem. In the framework of continuous symmetry, in which symmetry is no longer a binary yes/no property, such imperfections can be quantified and used as a global descriptor of the three-dimensional structure. We present an improved algorithm for calculating the continuous symmetry measure for proteins that takes into account their complete set of atoms including all side chains. Our method takes advantage of the protein sequence and the division into peptides in order to improve the accuracy and efficiency of the calculation over previous methods. The Hungarian algorithm is applied to solve the assignment problem and find the permutation that defines the symmetry operation. Analysis of the symmetry of several sets of protein homomers, with various degrees of rotational symmetry is presented. The new methodology lays the foundations for accurate, efficient and reliable large scale symmetry analysis of protein structure and can be used as a collective variable that describes changes of the protein geometry along various processes, both at the backbone level and for the complete protein structure.

Analyzing the symmetrical arrangement of structural repeats in proteins with CE-Symm

Many proteins fold into highly regular and repetitive three dimensional structures. The analysis of structural patterns and repeated elements is fundamental to understand protein function and evolution. We present recent improvements to the CE-Symm tool for systematically detecting and analyzing the internal symmetry and structural repeats in proteins. In addition to the accurate detection of internal symmetry, the tool is now capable of i) reporting the type of symmetry, ii) identifying the smallest repeating unit, iii) describing the arrangement of repeats with transformation operations and symmetry axes, and iv) comparing the similarity of all the internal repeats at the residue level. CE-Symm 2.0 helps the user investigate proteins with a robust and intuitive sequence-to-structure analysis, with many applications in protein classification, functional annotation and evolutionary studies. We describe the algorithmic extensions of the method and demonstrate its applications to the study of interesting cases of protein evolution.

Many protein structures show a great deal of regularity. Even within single polypeptide chains, about 25% of proteins contain self-similar repeating structures, which can be organized in ring-like symmetric arrangements or linear open repeats. The repeats are often related, and thus comparing the sequence and structure of repeats can give an idea as to the early evolutionary history of a protein family. Additionally, the conservation and divergence of repeats can lead to insights about the function of the proteins. This work describes CE-Symm 2.0, a tool for the analysis of protein symmetry. The method automatically detects internal symmetry in protein structures and produces a multiple alignment of structural repeats. The algorithm is able to detect the geometric relationships between the repeats, including cyclic, dihedral, and polyhedral symmetries, translational repeats, and cases where multiple symmetry operators are applicable in a hierarchical manner. These complex relationships can then be visualized in a graphical interface as a complete structure, as a superposition of repeats, or as a multiple alignment of the protein sequence. CE-Symm 2.0 can be systematically used for the automatic detection of internal symmetry in protein structures, or as an interactive tool for the analysis of structural repeats.

Chiral Ramachandran Plots I: Glycine

Ramachandran plots (RPs) map the wealth of conformations of the polypeptide backbone and are widely used to characterize protein structures. A limitation of the RPs is that they are based solely on two dihedral angles for each amino acid residue and provide therefore only a partial picture of the conformational richness of the protein. Here we extend the structural RP analysis of proteins from a two-dimensional (2D) map to a three-dimensional map by adding the quantitative degree of chirality-the continuous chirality measure (CCM)-of the amino acid residue at each point in the RP. This measure encompasses all bond angles and bond lengths of an amino acid residue. We focus in this report on glycine (Gly) because, due to its flexibility, it occupies a large portion of the 2D map, thus allowing a detailed study of the chirality measure, and in order to evaluate the justification of classically labeling Gly as the only achiral amino acid. We have analyzed in detail 4366 Gly residues extracted from high resolution crystallographic data of 160 proteins. This analysis reveals not only that Gly is practically always conformationally chiral, but that upon comparing with the backbone of all amino acids, the quantitative chirality values of Gly are of similar magnitudes to those of the (chiral) amino acids. Structural trends and energetic considerations are discussed in detail. Generally we show that adding chirality to Ramachandran plots creates far more informative plots that highlight the sensitivity of the protein structure to minor conformational changes.

The enigma of the near-symmetry of proteins: Domain swapping

The majority of proteins form oligomers which have rotational symmetry. Literature has suggested many functional advantages that the symmetric packing offers. Yet, despite these advantages, the vast majority of protein oligomers are only nearly symmetric. A key question in the field of proteins structure is therefore, if symmetry is so advantageous, why do oligomers settle for aggregates that do not maximize that structural property? The answer to that question is apparently multi-parametric, and involves distortions at the interaction zones of the monomer units of the oligomer in order to minimize the free energy, the dynamics of the protein, the effects of surroundings parameters, and the mechanism of oligomerization. The study of this problem is in its infancy: Only the first parameter has been explored so far. Here we focus on the last parameter-the mechanism of formation. To test this effect we have selected to focus on the domain swapping mechanism of oligomerization, by which oligomers form in a mechanism that swaps identical portions of monomeric units, resulting in an interwoven oligomer. We are using continuous symmetry measures to analyze in detail the oligomer formed by this mechanism, and found, that without exception, in all analyzed cases, perfect symmetry is given away, and we are able to identify that the main burden of distortion lies in the hinge regions that connect the swapped portions. We show that the continuous symmetry analysis method clearly identifies the hinge region of swapped domain proteins-considered to be a non-trivial task. We corroborate our conclusion about the central role of the hinge region in affecting the symmetry of the oligomers, by a special probability analysis developed particularly for that purpose.

Structural and evolutionary versatility in protein complexes with uneven stoichiometry

Proteins assemble into complexes with diverse quaternary structures. Although most heteromeric complexes of known structure have even stoichiometry, a significant minority have uneven stoichiometry--that is, differing numbers of each subunit type. To adopt this uneven stoichiometry, sequence-identical subunits must be asymmetric with respect to each other, forming different interactions within the complex. Here we first investigate the occurrence of uneven stoichiometry, demonstrating that it is common in vitro and is likely to be common in vivo. Next, we elucidate the structural determinants of uneven stoichiometry, identifying six different mechanisms by which it can be achieved. Finally, we study the frequency of uneven stoichiometry across evolution, observing a significant enrichment in bacteria compared with eukaryotes. We show that this arises due to a general increased tendency for bacterial proteins to self-assemble and form homomeric interactions, even within the context of a heteromeric complex.