Determining the Magnitude of the Fully Asymmetric Diffusion Tensor from Heteronuclear Relaxation Data in the Absence of Structural Information

Heteronuclear relaxation data have been used to study internal dynamics and characterize the rotational anisotropy of macromolecules.1 Recently, it has been shown that the dependence of heteronuclear relaxation times on rotational diffusion anisotropy can provide structural restraints for simulated annealing structure refinement that characterize long-range order a priori.2 While a detailed analysis of internal dynamics requires that both the magnitude and orientation of the diffusion tensor be known,3,4 only the magnitude of the tensor is required to derive information on the shape of the molecule and for structure refinement.2 (In the latter case knowledge of the orientation of the diffusion tensor is not required since it is allowed to float during the calculations). If the structure is already known to high accuracy (e1.8 A resolution crystal structure), the six parameters that describe the magnitude and orientation of the diffusion tensor can be obtained by least squares optimization on the basis of the measured relaxation times and the orientations of the internuclear vectors.3,4 In this paper, we describe a simple and robust procedure, based on an examination of histograms of heteronuclear T1/T2 ratios, to obtain the magnitude of the fully asymmetric diffusion tensor. For the general case of rigid body anisotropic reorientation, the spectral density function, J(ω), in the limit of very fast internal motions, is given to a good approximation5 by J(ω) ) SCk[τk/(1 + ωτk)] where ω is the angular frequency; S is the generalized order parameter; τk are time constants that are functions of the elements Dzz, Dyy, and Dxx of the second rank diagonal diffusion tensor D, with Dzz defined as the unique component of D; and the coefficients Ck ) fk(Dxx, Dyy,Dzz, cos2θ,sin2φ), where θ is the angle between the interactomic vector and the z axis of the diffusion tensor, and φ the angle between the projection of the interatomic vector on the x-y plane and the x axis.6 The effective correlation time τc,eff is given by (2Dzz + 2Dyy + 2Dxx), the diffusion anisotropy A by 2Dzz/(Dxx + Dyy), and the rhombicity η by 3/2(Dyy Dxx)/[Dzz 0.5(Dyy + Dxx)]. The heteronuclear T2 is a function of frequency-dependent and -independent terms, while the heteronuclear T1 is only a function of the frequency dependent terms.7 Thus the heteronuclear T1/T2 ratio is given by f(ω, τc,eff, A, η, cos2 θ, sin2 φ). For an oblate ellipsoid, (i.e., A Dzz), the minimum and maximum heteronuclear T1/T2 ratios always correspond to vectors lying along the z (θ ) 0°) and x (θ ) 90°, φ ) 0°) axes of the diffusion tensor, respectively. For a prolate ellipsoid (i.e., A > 1 and Dzz > Dyy g Dxx), the maximum heteronuclear T1/T2 ratio always corresponds to vectors lying along the z (θ ) 0°) axis of the diffusion tensor, but the value of θ corresponding to the minimum heteronuclear T1/T2 ratio depends on ω, τc,eff, and A and very weakly on η. The ω and τc,eff dependence of the cutoff value of A below which the minimum 15N T1/T2 ratio corresponds to vectors lying along the x axis is depicted in Figure 1. Thus, for τc,eff ) 4 ns, the cutoff value of A is 3.6, 4.1, and 4.7 at 1H frequencies of 500, 600, and 750 MHz, respectively; for τc,eff ) 10 ns, the cutoff value of A is 5.9, 6.2, and 6.5 at 1H frequencies of 500, 600, and 750 MHz. As A increases above these values, the minimum 15N T1/T2 value shifts from vectors lying along θ ) 90°, φ ) 0° to vectors lying at θ ) 54.7°, φ ) 0° (i.e., θ is at the magic angle). Thus, for globular proteins (which invariably have anisotropies below these cutoff values) the minimum 15N T1/T2 ratio will always correspond to vectors lying along the x axis. For this class of proteins, how can one find the heteronuclear T1/T2 ratio that corresponds to vectors lying along the third (i.e., y) principal axis of the diffusion tensor? By analogy with solidstate NMR line shapes, if the distribution of vectors is isotropic, one would expect this T1/T2 ratio to correspond to the maximum of the T1/T2 distribution function. This has been confirmed by numerical simulations. The distribution of heteronuclear T1/T2 values obtained by generating a million random, isotropically distributed vectors, does have a powder pattern-like appearance, and the most probable T1/T2 value does indeed coincide with the T1/T2 value at θ ) 90°, φ ) 90°. This is illustrated in Figure 2. The values of (T1/T2)zz and (T1/T2)xx are obtained from the extreme values of the experimentally observed heteronuclear T1/ T2 ratios. Thus, for a prolate ellipsoid (A > 1) the value of (T1/ T2)zz is determined by taking the average of the cluster of high T1/T2 values within one experimental standard deviation of the maximum observed T1/T2 value. The same procedure is used to estimate (T1/T2)xx from the cluster of observed low T1/T2 values. The value of (T1/T2)yy is obtained from the maximum of the † Laboratory of Chemical Physics. ‡ Laboratory of Biophysical Chemistry. (1) (a) Torchia, D. A.; Nicholson, L. K., Cole, H. B. R.; Kay, L. E. In NMR of Proteins; Clore, G. M., Gronenborn, A. M., Eds.; MacMillan Press: London, 1993; pp 190-219. (b) Wagner, G.; Hyberts, S.; Peng, J. W. In NMR of Proteins; Clore, G. M., Gronenborn, A. M., Eds.; MacMillan Press: London, 1993; pp 220-257. (c) Palmer, A. G. III Curr. Opin. Biotechnol. 1993, 4, 385-391. (2) Tjandra, N.; Garrett, D. S.; Gronenborn, A. M.; Bax, A.; Clore, G. M. Nature Struct. Biol. 1997, 4, 443-449. (3) (a) Tjandra, N.; Feller, S. E.; Pastor, R. W.; Bax, A. J. Am. Chem. Soc. 1995, 117, 12562-12566. (b) Bruschweiler, R.; Liao, X.; Wright, P. E. Science 1995, 268, 886-889. (c) Lee, L. K.; Rance, M., Chazin, W. J.; Palmer, A. G. III. J. Biomol. NMR 1997, 9 287-298. (d) Phan, I. Q. H.; Boyd, J.; Campbell, I. D. J. Biomol. NMR 1996, 8, 369-378. (4) Tjandra, N.; Wingfield, P. T., Stahl, S. J.; Bax, A. J. Biomol. NMR 1996, 8, 273-284. (5) Lippari, G.; Szabo, A. J. Am. Chem. Soc. 1982, 104, 4546-4559. (6) Woessner, D. E. J. Chem. Phys. 1962, 36, 647-654. (7) Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, 1961. Figure 1. τc,eff and ω dependence of the cutoff value of the diffusion anisotropy A for a prolate ellipsoid (A > 1) below which the minimum and most probable value of 15N T1/T2 coincide with the x and y axes of the diffusion tensor, respectively. Curves are shown for 1H frequencies of 750 (____), 600 (_ _ _), and 500 (-) MHz and are calculated for the axially symmetric case (η ) 0). The introduction of rhombicity results in a very small increase in the cutoff values of A. Thus, as τc,eff f ∞, the cutoff value for A is 6.99 for η ) 0 and 7.12 for η ) 1. 4889 J. Am. Chem. Soc. 1998, 120, 4889-4890