Determination of Mean and Standard Deviation of Dihedral Angles Rold Do ¨ker,* Till Maurer,* Werner Kremer,* K.-P. Neidig,† and Hans Robert Kalbitzer* ,1 *Institut fu ¨ r Biophysik und Physikalische Biochemie, Universita ¨ t Regensburg, D-93040 Regensburg; and †Software Department, Bruker Analytik GmbH, D-76275 Ettlingen, Germany Received February 23, 1999 Backbone torsional angles are a characteristic and useful parameter for the description and characteri- sation of protein structures determined by x-ray crys- tallography or NMR spectroscopy. For the comparison of an ensemble of three-dimensional structures the calculation of the statistical parameters mean and standard deviation would be very useful. However, they are not defined unambiguously for periodic quan- tities such as the dihedral angles. In this paper a plau- sible and unique definition of these parameters is in- troduced and a straightforward method for their calculation is given. © 1999 Academic Press Key Words: backbone; protein structure; statistical treatment of angles/cyclic quantities; structure com- parison; torsion angles. Backbone torsion-angles are very useful for charac- terising protein structures and for assessing their qualities (1). If the bond lengths and bond angles are known, the secondary structure is determined only by the dihedral angles. If structures determined by NMR spectroscopy or x-ray crystallography are to be com- pared, these angles are of particular interest. On the one hand they allow the definition of one single struc- ture independent of its cartesian coordinates. On the other hand, an ensemble of structures calculated from NMR distance data can be further judged with the help of mean values and standard deviations of the torsion angles. Finally, homologous proteins can be compared in terms of their structural similarity and where along their sequence of amino acids they differ. The standard deviation of the torsion angles is a measure for these properties. However, because of the cyclic nature of the angles, the definitions normally used for non-cyclic quantities do not give unique values. These depend on the choice of the origin of the reference system and the range of values allowed (e. g. -180°/180° or 0°/360°). To overcome the problem Hyberts suggested the use of an order parameter S which can be taken as a measure for the variance of a set of N angles (2). In order to get S, the angles are regarded as vectors on the unit circle. They are added, and the result is divided by N. The length of the resulting vector is the parameter S with 0  S  1. However, S is only related in a complicated way to the variance and does not allow the calculation of a mean value. Since in our knowledge, a unique way to calculate mean and variance of a set of angles has not been published yet (although ‘trivial’ calculation routines exist in some software packages), we present here a method for their calculation with properties as close as possible to these quantities when calculated for non-cyclic quantities. METHODS The program CYCLIST for calculating the circular mean and circular standard deviation s c of angles is written in C and is available from the authors. It will also be implemented in the program package AURELIA (3) which can be obtained from Bruker, Karlsruhe. As input file the program CYCLIST requires a list of dihedral angles which can be calculated by AURELIA or entered manually. The three-dimensional NMR structures of HPr (Histidine-Containing Phosphocarrier Protein) from S. aureus (4), E. coli (5), and B. subtilis (6) were taken from the Brookhaven protein data base. RESULTS AND DISCUSSION In principle the mean x  and standard deviation s of a sample of N angles x i can be calculated by using the general definitions x  = 1 N   i=1 N x i [1] and s =  ¥ i=1 N  x i - x   2 N . [2] 1 To whom correspondence should be addressed. Fax: ++49 941/ 943-2479. E-mail: hans-robert.kalbitzer@biologie.uni-regensburg.de. Biochemical and Biophysical Research Communications 257, 348 –350 (1999) Article ID bbrc.1999.0462, available online at http://www.idealibrary.com on 348 0006-291X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.