Fast determination of the optimal rotational matrix for macromolecular superpositions

Finding the rotational matrix that minimizes the sum of squared deviations between two vectors is an important problem in bioinformatics and crystallography. Traditional algorithms involve the inversion or decomposition of a 3 x 3 or 4 x 4 matrix, which can be computationally expensive and numerically unstable in certain cases. Here, we present a simple and robust algorithm to rapidly determine the optimal rotation using a Newton-Raphson quaternion-based method and an adjoint matrix. Our method is at least an order of magnitude more efficient than conventional inversion/decomposition methods, and it should be particularly useful for high-throughput analyses of molecular conformations.