Hydration time-average properties

Eq. (14), which was originally postulated by Zimmerman and Brittin (1957), assumes fast exchange between all hydration states (i) and neglects the complexities of cross-relaxation and proton exchange. Equation (15) is consistent with the Ergodic theorem of statistical thermodynamics, which states that at equilibrium, a time-averaged property of an individual water molecule, as it diffuses between different states in a system, is equal to a... 

The process of protein hydration is the stepwise addition of water to dry protein, until the hydration end point is reached. Heat capacity measurements (Yang and Rupley, 1979) serve as a framework on which to develop a picture. Figure 38 gives the dependence on hydration level of various time-average properties of lysozyme, over the hydration range 0-0.4 h, from the dry protein to slightly beyond the end point of the process. Curve d shows the dependence of the apparent specific heat on hydration level. It is directly related to the extent to which the thermal response of the lysozyme-water system deviates from ideal behavior. The nonideality of the system shows three discontinuities at 0.07, 0.25, and 0.38 h. 

Fig. 38. Comparison of heat capacity and spectroscopic properties. Effect of hydration on lysozyme time-average properties. (Curve a) Carboxylate absorbance (1580 cm" ) (curve b) amide I shift ( 1660 cm" ) (curve c) OD stretching frequency (—2570 cm ) (curve d) apparent specific heat capacity (curve e) diamagnetic susceptibility. From Rupley elal. (1983).

There is an impression, which is likely correct, that the hydration dependence is more complex for dynamic properties than for time-average properties. Several examples follow. 

Careri et al. (1980) and Rupley et al. (1983) compared the above data with other time-average and dynamic properties, to develop a picture of the hydration process. 

The dependence of protein and solvent dynamics on hydration fits well into the above three-stage picture for some, but not all, properties. For dynamic properties that do not fit well, analysis on a case-by-case basis within the framework of the time-average picture can be informative. For example, consider protonic conduction, measured by the megahertz frequency dielectric response for partially hydrated powders of lysozyme. The capacitance grows explosively above a hydration level of 0.15 A, in a way characteristic of a phase transition (Section HI, A). The hydration dependence of thermodynamic properties shows, however. 

Some apparent conflicts between the hydration dependence of dynamic properties and the time-average picture may be treated relatively simply, as noted above. There is additional discussion in Section III. 

Now, the question is how to get information on the more subtle quantity, the hydration numbers. Some confusion arises here, for in some research papers the coordination number (the average number of ions in the first layer around the ion) is also called the hydration number However, in the physicochemical literature, this latter term is restricted to those water molecules that spend at least one jump time with the ion, so that when its dynamic properties are treated, the effective ionic radius scans to be that of the ion plus one or more waters. A startling difference between co-ordination number and solvation number occurs when the ionic radius exceeds about 0.2 nm (Fig. 2.23a). 

Usha and Wittebort (1989) studied the NMR of crystalline cram-bin. At 140 K the protein hydrate is stationary, with t = 1 msec. Above 200 K changes in the signal with temperature are consistent with a glass transition or melting of the hydration water. This broad transition parallels closely the changes with temperature found for the heat capacity, Mossbauer spectroscopic, and other properties of hydrated protein crystals. At room temperature no more than 12 water molecules are orien-tationally ordered. The average rotational correlation time of the hydration water is about 40 times longer than that for bulk water. 

Figure 1 shows the dielectric relaxation properties of the Tween microemulsions plotted on the complex permittivity plane (from Foster et al ( 1). The mean relaxation frequency (corresponding to the peak of each semicircle) decreases gradually from 20 GHz for pure water at 25°C to ca. 2 GHz for a concentrated microemulsion containing 20% water. Since the permittivity of the suspended oil/ emulsifier is 6 or less at frequencies above 1 GHz, this relaxation principally arises from the dipolar relaxation of the water in the system. Therefore, the data shown in Figure 1 clearly show that the dielectric relaxation times of the water in the microemulsions are slower on the average than those of the pure liquid. The depressed semicircles indicate a distribution of relaxation times (9), and were analyzed assuming the presence of two water components (free and hydration) in our previous studies. 

FIGU RE 16.6 Temperature dependence of protein and water dynamical properties from MD simulations of a hydrated powder of MBP [8]. (a) MSFs of protein nonexchangeable H atoms averaged over 1 ns blocks of the trajectories, (b) Temperature dependence of the inverse of the correlation times, of the protein-water hydrogen bond correlation functions [73]. (c) Value of... 

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