The simplest model of protein hydration is a two-state one, where water (i.e., D20) exchanges chemically between a bound and a free state. Using the general approach to equilibrium problems described in Chapter 6 and making the reasonable assumption that for bound water is much less than that of free water, the observed longitudinal relaxation time is 
where the symbols are defined as follows:
fi = fraction of bound water,
/o = fraction of free water, ru = relaxation time of 2D for bound water,
T\,o = relaxation time of 2D for free water, tm = lifetime of a water molecule in the bound state.
where the relaxation times have been replaced by the corresponding relaxation rates (Ri = l/7\). The frequency (oj)-dependent relaxation rates 0bs and /?i i are distinguished from the frequency independent Rifl ones for free water. In other words, the frequency dependent component of the measured deuterium relaxation rate is directly proportional to the longitudinal relaxation rate of the deuteriums in water bound to the protein.
Three independent dynamic processes with separate correlation times (r) need to be considered for the "bound" water molecules. These are radial diffusion perpendicular to the protein surface involving "bound"-"free" chemical exchange (rrad), lateral diffusion parallel to the protein surface (rlat), and rotation of the protein itself (rrot) . Then, the correlation time associated with the nanosecond time scale motion of the protein-bound water (rc) is given by
A consequence of eq. (8.8) is that the fastest of the dynamic processes dominates the observed field dispersion of the relaxation; that is, it governs 1/tc. For a moderately sized globular protein rml ~ 10 8 s < Tiat ~ 10~7 s < trad ~ 10-6 s. Thus, rc is approximately the same as rrot. As expected from this relation, an increase of tc is observed with increasing protein size ,
Models for the influence of frequency («) on the longitudinal relaxation rate were derived from the theory of quadrupolar relaxation . The model is outlined in Table 8.2 for a system with a single correlation time. This model holds for a solution of monomeric protein in D20, characterized by a single correlation time resolutions of protein in equilibrium with their aggregates are characterized by more than one correlation time. Each distinct y'th aggregate has its own correlation time tcj. The general model for the frequency dependence of Ri that accounts for such aggregates is given in Table 8.3.
Table 8.2 Model for Frequency Dependence of the NMR Longitudinal Relaxation Rate
Assumptions: Single correlation time, tc Regression equations:
Special instructions: Also test models with more than one tc (see Table 8.3)
Table 8.3 General Model for Frequency Dependence of NMR Longitudinal Relaxation for Aggregating Protein Solutions
Assumptions: Multiple correlation time, tcj Regression equations:
Regression parameters: Data:
Special instructions: Test for goodness of fit by comparing residual plots and using the extra sum of squares F test (see Section 3.C.1)
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