Lecture 4, Solution Structures of Native and Denatured Proteins Using RDCs 1 Lecture 4 Solu.pdf

Lecture 4, Solution Structures of Native and Denatured Proteins Using RDCs 1 Lecture 4 Solution Structures of Native and Denatured Proteins Using RDCs In the previous lectures, we introduced the principles of the Residual Dipolar Coupling (RDC), which can provide global orientational restrains on internuclear vectors. In this lecture, we introduce some RDC applications, that is, using RDCs to solve the solution structures of native [1, 2] and denatured [3] proteins. 1 Determining Exact Protein Native Structure In the section, we first introduce a quartic equation and two simple trigonometric equa- tions that can compute, exactly and in constant time, the backbone dihedral angles for a residue from RDCs in two media on any single backbone vector type. Furthermore, based on these exact solutions we introduce a systematic algorithm for determining protein backbone substructure consisting of both α-helices and β-sheets [1]. 1.1 Theoretical Background Based on the equations of NH RDCs measured in two media, we derived a quartic equation which can compute the vector orientation, for instance, with respect to x, we have f4u 4 + f3u 3 + f2u 2 + f1u+ f0 = 0, (1) u = 1? 2 (x a )2 where the coefficients f0, f1, f2, f3, f4, a as well as y and z can be computed from the equations of NH RDCs (Appendix A in [1]). As shown in Figure 1, the number of real solutions of Equation 1 is at most 8, or in other words, given NH RDCs measured in two media, we can obtain up to 8 possible vector orientations, one of which is the real orientation. In order to compute the dihedral angles for a single residue, we showed that if the directions of any two vectors vi and vi+1 in consecutive peptide planes i and i+1 are known, the intervening backbone angles (φi, ψi) can be computed from the following two trigonometric equations: sin(φi + a1) = b1, sin(ψi + a2) = b2, (2) where a1, b1, a2 and b2 are computed from the six angles between two consecutive residues, vi and vi+1. The full expressions for al