title: Optimal simultaneous superpositioning of multiple structures with missing data.
abstract: BACKGROUND
  Superpositioning is an essential technique in structural biology that facilitates the comparison and analysis of conformational differences among topologically similar structures. Performing a superposition requires a one-to-one correspondence, or alignment, of the point sets in the different structures. However, in practice, some points are usually 'missing' from several structures, for example, when the alignment contains gaps. Current superposition methods deal with missing data simply by superpositioning a subset of points that are shared among all the structures. This practice is inefficient, as it ignores important data, and it fails to satisfy the common least-squares criterion. In the extreme, disregarding missing positions prohibits the calculation of a superposition altogether.
  
  
  RESULTS
  Here, we present a general solution for determining an optimal superposition when some of the data are missing. We use the expectation-maximization algorithm, a classic statistical technique for dealing with incomplete data, to find both maximum-likelihood solutions and the optimal least-squares solution as a special case.
  
  
  METHODS
  The methods presented here are implemented in THESEUS 2.0, a program for superpositioning macromolecular structures. ANSI C source code and selected compiled binaries for various computing platforms are freely available under the GNU open source license from http://www.theseus3d.org.
  
  
  BACKGROUND
  dtheobald@brandeis.edu
  
  
  BACKGROUND
  Supplementary data are available at Bioinformatics online.
authors.affiliation: Department of Biochemistry, Brandeis University, MS009, Waltham, MA 02454, USA. dtheobald@brandeis.edu
authors.forename: Douglas L
authors.initials: DL
authors.lastname: Theobald
authors.forename: Phillip A
authors.initials: PA
authors.lastname: Steindel
mesh_terms: D000465:Algorithms; D000595:Amino Acid Sequence; D016018:Least-Squares Analysis; D016013:Likelihood Functions; D008958:Models, Molecular; D017434:Protein Structure, Tertiary; D016415:Sequence Alignment / Q000379:methods*; D012984:Software