Distances as degrees of freedom

The flexible alignment of molecular structures using simulated annealing with weighted Lagrangian multipliers

A framework for superimposing small molecules is presented. The proposed method consists of a simple atom-based, flexible alignment. The optimization procedure used in the alignment is based on a recently published variant of the simulated annealing whereby nonlinear constraints are accommodated using Lagrangian multipliers. It differs from other published superposition algorithms in that any number of nonlinear constraints can be readily imposed on the structural alignment directly through the objective function without assuming an a priori trade-off between competing conditions. These can include equality and equality constraints on distances, angles, and energy states. Examples illustrating the use of the proposed approach are also provided.

Explicit distance geometry: identification of all the degrees of freedom in a large RNA molecule

An alternative approach to distance geometry ("explicit" distance geometry) is being developed for problems, such as the modeling of RNA folding in the ribosome, where relatively few distances are known. The approach explicitly identifies minimal sets of additional distances that can be added to a distance matrix in order to calculate structures that are consistent with all the known information without distorting the original input data. These additional distances are bounded to the extent possible by the known distances. These explicitly added distances can be treated as degrees of freedom and used to explore the full range of alternative foldings consistent with the original input in an organized way. The present paper establishes that it is practical to explicitly determine such degrees of freedom for even very large RNAs. To demonstrate the feasibility of the approach tRNA was represented as a simple undirected graph containing all relevant information represented in the usual cloverleaf secondary structure and nine base-base tertiary interactions. Using a three atom representation for each residue a total of 206 degrees of freedom are explicitly identified. To accomplish this a graph theoretic approach was used in which a minimal covering cycle basis was determined.