Computer Programs in Biomedicine 11 (1980) 3-8 © Elsevier/North-Holland Biomedical Press A PROGRAM FOR THE COMPUTATION OF HELICAL PARAMETERS FROM INTERNAL COORDINATES Silvano COLOMBANO and Robert REIN Roswell Park Memorial hzstitute, 666 Elm Street, Buffalo, NY 14263 and Robert D. MacELROY NASA Ames Research Center, Moffett Field, CA 94035, USA A method for the calculation of helical parameters from internal coordinates of an arbitrary repeat unit, has been imple- mented in a I:ORTRAN IV program. The coordinates of all atoms in a cylindrical coordinate system are also obtained. An appli- cation of the program for a systematic approach to the problem of protein-DNA recognition is introduced. Helical parameters Internal coordinates Protein-DNA recognition I. Introduction It is often convenient to describe a helical polymer chain in terms of the helical parameters d and 0, (respectively, the translation along the helical axis, and the angle of rotation around the same axis per repeat unit) and Pi (distance of the ith atom from the axis). Expressions for helical parameters as functions of internal coordinates, (bond lengths, bond angles, and dihedral angles) have been developed by several workers (e.g.. Shimanouchi and Mizushima [ 1], Miyazawa [2], Sugeta and Miyazawa, [31). The meth- od of Sugeta and Miyazawa seemed most readily adaptable to a computer algorithm. In this paper, we briefly describe a FORTRAN implementation of their method, and introduce an application to the problem of protein-DNA recognition for the case of helical repeat units comprised of dipeptides. ~.,. Principles Let rip ¢i, and 7i/represent bond lengths, bond angles and dihedral angles, respectively (fig. 1), and let the atoms of any backbone repeat unit be num- bered consecutively from 1, then the cartesian posi- tion vector of the ith atom with respect of atom no 1 is given by: Xi=BI2 +AI2B23 +AI2A23B34 +...+ +A12A23...Ai-2,i-lBi-l,i i>2 (1) Where --COS ~j Ai/ = ]sin ~/cos ri/ lsin ~/sin ri/ -sin ¢,/ -cos ~/cos Ti/ "--COS~i sin ~'# and [',,l Osin r i] 1 (2) cos T o _i (3) These expressions follow very closely the formal- ism of Sugeta and Miyazawa [3 ]. For more details, their wcrk may be consulted, or that of Go and Scheraga [4], who derive similar expressions, but do not apply them to the calculation of helical param- eters. If the ruth repeat unit is considered, from the