Conformational Entropy from Slowly Relaxing Local Structure Analysis of ¹⁵N–H Relaxation in Proteins: Application to Pheromone Binding to MUP‑I in the 283–308 K Temperature Range

The slowly relaxing local structure (SRLS) approach is applied to ¹⁵N–H relaxation from the major urinary protein I (MUP-I), and its complex with pheromone 2-<i>sec</i>-butyl-4,5-dihydrothiazol. The objective is to elucidate dynamics, and binding-induced changes in conformational entropy. Experimental data acquired previously in the 283–308 K temperature range are used. The N–H bond is found to reorient globally with correlation time, τ_1,0, and locally with correlation time, τ_2,0, where τ_1,0 ≫ τ_2,0. The local motion is restricted by the potential <i>u</i> = −<i>c</i>₀²<i>D</i>₀₀², where <i>D</i>₀₀² is the Wigner rotation matrix element for <i>L</i> = 2, <i>K</i> = 0, and <i>c</i>₀² evaluates the strength of the potential. <i>u</i> yields straightforwardly the order parameter, ⟨<i>D</i>₀₀²⟩, and the conformational entropy, <i>S</i>_k, both given by <i>P</i>_eq = exp­(−<i>u</i>). The deviation of the local ordering/local diffusion axis from the N–H bond, given by the angle β, is also determined. We find that <i>c</i>₀² ≅ 18 ± 4 and τ_2,0 = 0<i>–</i>170 ps for ligand-free MUP-I, whereas <i>c</i>₀² ≅ 15 ± 4 and τ_2,0 = 20<i>–</i>270 ps for ligand-bound MUP-I. β is in the 0<i>–</i>10° range. <i>c</i>₀² and τ_2,0 decrease, whereas β increases, when the temperature is increased from 283 to 308 K. Thus, SRLS provides physically well-defined structure-related (<i>c</i>₀² and ⟨<i>D</i>₀₀²⟩), motion-related (τ_2,0), geometry-related (β), and binding-related (<i>S</i>_k) local parameters, and their temperature-dependences. Intriguingly, upon pheromone binding the conformational entropy of MUP-I decreases at high temperature and increases at low temperature. The very same experimental data were analyzed previously with the model-free (MF) method which yielded “global” (in this context, “relating to the entire 283–308 K range”) amplitude (<i>S</i>²) and rate (τ_e) of the local motion, and a phenomenological exchange term (<i>R</i>_ex). <i>S</i>² is found to decrease (implying implicitly “global” increase in <i>S</i>_k) upon pheromone binding