BioJava can be used to detect, analyze, and visualize symmetry and pseudo-symmetry in the quaternary (biological assembly) and tertiary (internal) structural levels.
The quaternary symmetry of a structure defines the relations between its individual chains or groups of chains. For a more extensive explanation about symmetery visit the [PDB help page] (http://www.rcsb.org/pdb/staticHelp.do?p=help/viewers/jmol_symmetry_view.html).
In the quaternary symmetry detection problem, we are given a set of chains
with its Atom coordinates and we are asked to find the higest overall symmetry that
relates them. The solution is divided into the following steps:
- First, we need to identify the chains that are identical (or similar in the pseudo-symmetry case). For that, we perform a pairwise alignment of all chains and determine clusters of identical chains.
- Next, we reduce the each chains to a single point, its centroid (center of mass).
- After that, we try different symmetry relations to superimpose the chain centroids and obtain their RMSD.
- At last, based on the parameters (cutoffs), we determine the overall symmetry of the structure, with the symmetry relations obtained in the previous step.
- In case of asymmetric structure, we discard combinatorially a number of chains and try to detect any local symmetries present.
The quaternary symmetry detection algorithm is implemented in the biojava class QuatSymmetryDetector. An example of how to use it programatically is shown below:
//First download the structure in the biological assembly form
Structure s;
//Set some parameters if needed different than DEFAULT - see descriptions
QuatSymmetryParameters parameters = new QuatSymmetryParameters();
parameters.setVerbose(true); //print information
//Instantiate the detector
QuatSymmetryDetector detector = QuatSymmetryDetector(structure, parameters);
//The getters calculate the quaternary symmetry automatically
List<QuatSymmetryResults> globalResults = detector.getGlobalSymmetry();
List<List<QuatSymmetryResults>> localResults = detector.getLocalSymmetries();The return type are List because there can be multiple valid options for the
quaternary symmetry. The local results List is empty if there exist no local
symmetry in the structure, and the global results List has always size bigger
than 1, returning a C1 point group in the case of asymmetric structure.
The QuatSymmetryResults object contains all the information of the symmetry.
This object will be used later to obtain axes of symmetry, point group name,
stoichiometry or even display the results in Jmol.
In global symmetry all chains have to be part of the symmetry description.
In a point group a single or multiple rotation axes define the overall symmetry operations, with the property that all the axes coincide in the same point.
In helical symmetry there is a single axis with rotation and translation components.
In local symmetry a number of chains is left out, so that the symmetry only applies to a subset of chains.
In pseudo-symmetry the chains related by the symmetry are not completely identical, but they share a sequence similarity above the pseudo-symmetry similarity threshold.
If we consider hemoglobin, at a 95% sequence identity threshold the alpha and beta subunits are considered different, which correspond to an A2B2 stoichiometry and a C2 point group. At the structural similarity level, all four chains are considered homologous (~45% sequence identity) with an A4 pseudostoichiometry and D2 pseudosymmetry.
Internal symmetry refers to the symmetry present in a single chain, that is, the tertiary structure. The algorithm implemented in biojava to detect internal symmetry is called CE-Symm.
The CE-Symm algorithm was originally developed by [Myers-Turnbull D., Bliven SE.,
Rose PW., Aziz ZK., Youkharibache P., Bourne PE. & Prlić A. in 2014]
(http://www.sciencedirect.com/science/article/pii/S0022283614001557) 
By a procedure called refinement, the subunits of the chain that are part of the symmetry are defined and a multiple alignment is created. This process can be thought as to divide the chain into other subchains, and then superimposing each subchain to each other to create a multiple alignment of the subunits, respecting the symmetry axes.
The internal symmetry detection algorithm is implemented in the biojava class CeSymm. It returns a MultipleAlignment, see the explanation of the model in Data Models, that describes the internal subunits multiple alignment. In case of no symmetry detected, the returned alignment represents the optimal self-alignment produced by the first step of the CE-Symm algorithm.
//Input the atoms in a chain as an array
Atom[] atoms = StructureTools.getRepresentativeAtomArray(chain);
//Initialize the algorithm
CeSymm ceSymm = new CeSymm();
//Choose some parameters
CESymmParameters params = ceSymm.getParameters();
params.setRefineMethod(RefineMethod.SINGLE);
params.setOptimization(true);
params.setMultipleAxes(true);
//Run the symmetry analysis - alignment as an output
MultipleAlignment symmetry = ceSymm.analyze(atoms, params);
//Test if the alignment returned was refined with
boolean refined = SymmetryTools.isRefined(symmetry);
//Get the axes of symmetry from the aligner
SymmetryAxes axes = ceSymm.getSymmetryAxes();
//Display the results in jmol with the SymmetryDisplay
SymmetryDisplay.display(symmetry, axes);
//Show the point group, if any of the internal symmetry
QuatSymmetryResults pg = SymmetryTools.getQuaternarySymmetry(symmetry);
System.out.println(pg.getSymmetry());To enable some extra features in the display, a SymmetryDisplay
class has been created, although the StrucutreAlignmentDisplay
and MultipleAlignmentDisplay methods can also be used for that
purpose (they will not show symmetry axes or symmetry menus).
Lastly, the SymmetryGUI class in the structure-gui package
provides a GUI to trigger internal symmetry analysis, equivalent
to the GUI to trigger structure alignments.
The symmetry display is similar to the quaternary symmetry, because part of the code is shared. See for example this beta-propeller (1U6D), where the repeated beta-sheets are connected by a linker forming a C6 point group internal symmetry:
One additional feature of the internal symmetry display is the representation of hierarchical symmetries and repeats. Contrary to point groups, some structures have different levels of symmetry. That is, the whole strucutre has, e.g. C2 symmetry and, at the same time, each of the two parts has C2 symmetry, but the axes of both levels are not related by a point group (i.e. they do not cross to a single point).
A very clear example are the beta-gamma-crystallins, like 4GCR:
Another feature of the display is the option to show the multiple alignment of the symmetry related subunits created during the refinement process. Search for the option Subunit Superposition in the symmetry menu of the Jmol window. For the previous example the display looks like that:
The subunit display highlights the differences and similarities between the symmetry related subunits of the chain, and helps the user to identify conseved and divergent regions, with the help of the Sequence Alignment Panel.
Finally, the internal and quaternary symmetries can be combined to obtain the global overall combined symmetry. As we have seen before, the protein 1VYM is a DNA-clamp that has three chains relates by C3 symmetry. Each chain is internally C2 symmetric, and each part of the C2 internal symmetry is C2 symmetric, so a case of hierarchical symmetry (C2 + C2). Once we have divided the whole structure into its asymmetric parts, we can analyze the global symmetry that related each one of them. The interesting result is that in some cases, the internal symmetry multiplies the point group of the quaternary symmetry. What seemed a C3 + C2 + C2 is combined into a D6 overall symmetry, as we can see in the figure below:
This results can give hints about the function and evolution of proteins and biological structures.
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