Conformation of proline ring from C nuclear magnetic spin lattice relaxation times

Abstract

Apart from the natural fascination that the dynamics of molecules generate, the important role that molecular dynamics play in the functioning of biological molecules is well recognized. On the other hand, information about solution conformations of molecules is sought with great interest to establish structure–activity relationships. Naturally, considerable importance is attached to the conformations of side chains of biological molecules such as peptides and proteins, since side chain conformations influence the overall molecular structure and constitute potential sites of interaction with other molecules. The proline ring of the proline residue occurring in peptides and proteins is one such side chain whose conformations are of great interest. Because of its unique ring structure, the proline residue can impose conformational restrictions and decisively affect the overall molecular conformation. While nuclear magnetic relaxation has been a time tested tool for probing molecular dynamics, the analysis of NMR vicinal coupling constants using Karplus type equations has traditionally been the main method for determining solution conformations. This thesis presents an unconventional approach, using nuclear magnetic relaxation data to develop a method for determining solution conformations of the proline ring from ¹³C spin–lattice relaxation times (T ). Normally, relaxation data are used only to derive amplitudes and rates of motion. These ideas are elaborated in the Introduction (Chapter 1), emphasizing the usefulness of relaxation data for studying molecular dynamics in solution. The key point may be summarized as follows: the generally observed differences in T values of proline ring carbons strongly suggest differences in internal motions of the dipole–dipole (d–d) vectors of the corresponding ¹³C nucleus and its bonded proton. This internal motion occurs with respect to a reference frame fixed in the molecule, which itself undergoes isotropic rotational Brownian motion. What is the nature of the internal motion of the proline d–d vectors Energy calculations and crystallographic studies show that the proline ring is essentially bistable, i.e., it has two stable conformational states. Vicinal coupling studies also show that the proline ring undergoes rapid interconversion between these two states in solution. Therefore, the natural model for the internal motion of d–d vectors is their reorientation caused by transitions between the two conformational states-the so called bistable jump model. ________________________________________ Chapter 2 - T Expression for Bistable Jump Dynamics Chapter 2 derives the T relaxation equation for a system undergoing overall isotropic rotational motion along with independent internal motion. The internal motion is then modelled as a bistable jump, yielding an expression involving: • the jump angle (difference between orientations in the two states), and • the lifetimes of the two states. ________________________________________ Chapter 3 - Relating T Values to Proline Ring Conformations Chapter 3 describes how the T equation for the bistable model connects experimental T values with proline ring torsion angles. Key steps: 1. For any d–d vector in the ring, a local coordinate system involving only proline ring atoms can be chosen such that its components depend on one torsion angle. 2. Since the two conformational states have different orientations, two coordinate systems are defined. 3. Vector components are transformed to a common coordinate system (transformations derived in Appendix 2). 4. Thus, the scalar product of d–d vectors from the two states becomes a function of ring torsions, enabling direct connection between torsion angles and the T equation. ________________________________________ Chapter 4 - Application to Experimental Data Chapter 4 applies the theory to 45 sets of experimental T data from proline containing peptides, using three approximation methods: Method I Assumes: • torsions in one state are equal in magnitude but opposite in sign to those of the other state, • state lifetimes are equal. The T equation simplifies and can be solved analytically, using two experimental T values to determine the ring torsions. A search algorithm (Method I) finds torsions that minimize RMS deviation. Method II Similarity to Method I, but does not assume equal lifetimes. Lifetimes are allowed to vary. Method III The assumption of torsion sign reversal is removed; torsions in the two states are independent. A full conformational space search is required, iterated across a range of lifetimes (10 –10 s). The T relationship is used to identify best fit torsions. Results of all three methods are tabulated. ________________________________________ Chapter 5 - Discussion and Comparison with Other Studies Chapter 5 compares the derived torsions with those from crystal structures, energy calculations, and vicinal coupling analyses. Important results: • The torsion angle ranges obtained from all three methods are consistent with known crystal structure and theoretical data. • Histograms of torsions from Methods I–III and from crystal structures show similar symmetric distributions, centred around 0°, reflecting symmetry in the T equation with respect to the two states. • This symmetry also explains why state lifetimes emerge as nearly equal, even when not assumed. The consistency across Methods I, II, and III validates the approach. Finally, it is noted that for larger peptides, vicinal coupling data often cannot be obtained due to spectral complexity, whereas ¹³C spectra remain simple, allowing T measurement. Hence, the method developed here provides a practical alternative for deriving solution conformations of proline rings.