Figure 1. Motion of a single bond vector expressed as autocorrelation and spectral density functions. The time domain of the autocorrelation function, C(t), and the frequency domain of the spectral density function, J(), interconvert upon a real Fourier transformation. In both panels, the motion shown results from an overall rotational correlation time of 4 ns, an internal correlation time of 100 ps, and an order parameter (S²) of 0.5. (a) The blue line corresponds to the autocorrelation function for overall tumbling; the red line corresponds to internal reorientation in the molecular frame. The dashed line corresponds to the composite autocorrelation function resulting from both overall and internal motions. Functions are normalized to a value of 1 at time zero. (b) The spectral density function, J(), is decomposed into its overall and internal components, shown in blue and red, respectively. Vertical lines are drawn at frequencies sampled by ¹⁵N and/or ¹³C relaxation at a field strength corresponding to a ¹H frequency of 600 MHz. Short and long dashed lines correspond to frequencies probed by ¹⁵N and ¹³C relaxation, respectively. Green lines are drawn at the zero and ¹H frequencies, and are sampled by both nuclei.

Figure 2. Model-free order parameters for backbone and side-chain bond vectors in the human ubiquitin protein. Backbone amide S² values are derived from ¹⁵N relaxation and are shown as solid circles. Methyl side-chain order parameters are derived from ¹³C relaxation and are shown as open circles and triangles. Each methyl side-chain order parameter corresponds to the threefold symmetry axis, which extends from the C–CH₃ bond vector (see Lee and Wand, (1999); Lee et al., 1999).