SMALL-AMPLITUDE PROTEIN CONFORMATIONAL DYNAMICS - 2ND-ORDER ANALYTIC RELATION BETWEEN CARTESIAN COORDINATES AND DIHEDRAL ANGLES

An analytic expression for protein atomic displacements in Cartesian c oordinate space (CCS) against small changes in dihedral angles is deri ved. To study time-dependent dynamics of a native protein molecule in CCS from dynamics in the internal coordinate space (ICS), it is necess ary to convert small changes of internal coordinate variables to Carte sian coordinate variables. When we are interested in molecular motion, six degrees of freedom for translational and rotational motion of the molecule must be eliminated in this conversion, and this conversion i s achieved by requiring the Eckart condition to hold. In this article, only dihedral angles are treated as independent internal variables (i .e., bond angles and bond lengths are fixed), and Cartesian coordinate s of atoms are given analytically by a second-order Taylor expansion i n terms of small deviations of variable dihedral angles. Coefficients of the first-order terms are collected in the K matrix obtained previo usly by Noguti and Go (1983) (see ref. 2). Coefficients of the second- order terms, which are for the first time derived here, are associated with the (newly termed) L matrix. The effect of including the resulti ng quadratic terms is compared against the precise numerical treatment using the Eckart condition. A normal mode analysis (NMA) in the dihed ral angle space (DAS) of the protein bovine pancreatic trypsin inhibit or (BPTI) has been performed to calculate shift of mean atomic positio ns and mean around the mean positions. The analysis shows that the inv olving the L matrix have significant contributions to at room temperat ure. This indicates that NMA in CCS errors when applied for such large molecules as proteins. avoided by carrying out NMA in DAS and by cons idering terms up to second order in the conversion of atomic motion fr om DAS to CCS. (C) 1995 by John Wiley and Sons, Inc.