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Symmetric Protein Structure Determination Using Arrangements of Circular Arcs

Abstract

Nuclear magnetic resonance (NMR) spectroscopy is a primary tool to perform structural studies of proteins in physiologically-relevant solution conditions. Restraints on distances between pairs of nuclei in the protein, derived from the nuclear Overhauser effect (NOE), provide information about the structure of the protein in its folded state. NMR studies of symmetric protein homo-oligomers present a unique challenge. Using X-filtered NOESY experiments, it is possible to determine whether an NOE restrains a pair of protons across different subunits or within a single subunit, but current experimental techniques are unable to determine in which subunits the restrained protons lie. Consequently, it is difficult to assign NOEs to particular pairs of subunits with certainty, thus hindering the structural analysis of the oligomeric state. Computational approaches are needed to address this subunit ambiguity, but traditional solutions often rely on stochastic search coupled with simulated annealing and simulations of simplified molecular dynamics, which have many tunable parameters that must be chosen carefully and can also fail to report structures consistent with the experimental restraints. In addition, these traditional approaches rarely provide guarantees on running time or solution quality. We reduce the structure determination of homo-oligomers with cyclic symmetry to computing geometric arrangements of unions of annuli in a plane. Our algorithm, DISCO, runs in expected O(n^2) time, where n is the number of distance restraints, potentially assigned ambiguously. DISCO is guaranteed to report the exact set of oligomer structures consistent with the distance restraints and also with orientational restraints from residual dipolar couplings (RDCs). We demonstrate our method using two symmetric protein complexes: the trimeric E. coli Diacylglycerol Kinase (DAGK), and a dimeric mutant of the immunoglobulin-binding domain B1 of streptococcal protein G (GB1). In both cases, DISCO computes oligomer structures with high precision and also finds distance restraints that are either mutually inconsistent or inconsistent with the RDCs.
restraintToAnnulus arrangementToGraph
 
A distance restraint (red dashed line) between two atoms p and q that lie in different subunits (colored ribbons) is satisfied when q lies between two spheres (blue) centered at p. Due to the molecular symmetry in the relative placement of the subunits, this distance restraint restricts the possible positions of the symmetry axis (black arrow) to the green annulus.
Analysis of all the possible assignments for a distance restraint yields a union of annuli in the plane. We compute the arrangement of the cirlcular curves bounding the unions of annuli for all the distance restraints. The dual graph of the arrangement reveals the Maximally Satisfying Regions (MSRs) which represent the set of symmetry axes that satisfy the greatest number of distance restraints.
crossing dagkMsrs
The MSRs are selected from the arrangement by computing the depth of each face using breadth-firstsearch on the dual graph. The search starts at the unbounded face (which has zero depth) and increments the depth each time the search enters a union of annuli. The faces of greatest depth are returned as the MSRs. The MSR and the arrangement of circular arcs using distance restraints from 12 disulfide bonds measured experimentally for Diacylglycerol Kinase, a trimeric protein complex. (click on figure to enlarge)>

References

  • Martin J, Yan T, Zhou P, Bailey-Kellogg C, Donald BR.
    A graphical method for analyzing distance restraints using residual dipolar couplings for structure determination of symmetric protein homo-oligomers.
    Protein Science, 20(6):970–985, 2011. [link]
  • Martin J, Yan T, Zhou P, Bailey-Kellogg C, Donald BR.
    A geometric arrangement algorithm for structure determination of symmetric protein homo-oligomers from NOEs and RDCs.
    In: Bafna V, Sahinalp SC, Eds. (2011) Proceedings of the Fifteenth Annual International Conference on Research in Computational Molecular Biology (RECOMB).  Springer, Berlin, pp 222–237.
  • Martin J, Yan T, Zhou P, Bailey-Kellogg C, Donald BR.
    A geometric arrangement algorithm for structure determination of symmetric protein homo-oligomers from NOEs and RDCs. 
    Journal of Computational Biology. (in press)
  • Bruce R. Donald.
    Algorithms in Structural Molecular Biology.
    MIT Press (2011). [link]

Contact

Jeffrey W. Martin
Bruce R. Donald
http://www.cs.duke.edu/~brd/

 

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