Embedding with a Rigid Substructure

Abstract: This paper presents a new distance geometry algorithm for calculating atomic coordinates from estimates of the interatomic distances, which maintains the positions of the atoms in a known rigid substructure. Given an $M \times 3$ matrix of coordinates for the rigid substructure $\mathbf X$, this problem consists of finding the $N \times 3$ matrix $\mathbf Y$ that yields of global minimum of the so-called STRAIN, i.e. [ \min_{\mathbf Y} \left| \begin{bmatrix} \mathbf{XX}^\top & \mathbf{XY}^\top \ \mathbf{YX}^\top & \mathbf{YY}^\top \end{bmatrix} \,-\, \begin{bmatrix} \mathbf A & \mathbf B \ \mathbf B^\top & \mathbf C \end{bmatrix} \right|_{\mathsf F}^2 ~, ] where $\mathbf A = \mathbf{XX}^\top$ , and $\mathbf B, \mathbf C$ are matrices of inner products calculated from the estimated distances. The vanishing of the gradient of the STRAIN is shown to be equivalent to a system of only six nonlinear equations in six unknowns for the inertial tensor associated with the solution Y . The entire solution space is characterized in terms of the geometry of the intersection curves between the unit sphere and certain variable ellipsoids. Upon deriving tight bilateral bounds on the moments of inertia of any possible solution, we construct a search procedure that reliably locates the global minimum. The effectiveness of this method is demonstrated on realistic simulated and chemical test problems.