Resultants and loop closure

The problem of tripeptide loop closure is formulated in terms of the angles {τ_i}_i=1³ describing the orientation of each peptide unit about the virtual axis joining the C_α atoms. Imposing the constraint that at the junction of two such units the bond angle between the bonds C_α-N and C_α-C is fixed at some prescribed value θ results in a system of three bivariate polynomials in u_i := tan τ_i/2 of degree 2 in each variable. The system is analyzed for the existence of common solutions by making use of resultants, determinants of matrices composed of the coefficients of two (or more) polynomials, whose vanishing is a necessary and sufficient condition for the polynomials to have a common root. Two resultants are compared: The classical Sylvester resultant and the Dixon resultant. It is shown that when two of the variables are eliminated in favor of the third, a polynomial of degree 16 results. To each one of its real roots, there is a corresponding common zero of the system. To each such zero, there corresponds a consistent conformation of the chain. The Sylvester method can find these zeros among the eigenvalues of a 24 × 24 matrix. For the Dixon approach, after removing extraneous factors, an optimally sized eigenvalue problem of size 16 × 16 results. Finally, the easy extension to the more general problem of triaxial loop closure is presented and an algorithm for implementing the method on arbitrary chains is given.