Biomedical EPR Part-B Methodology Instrumentation and Dynamics - Sandra R. Eaton
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0.005, 0.01, 0.02 and 0.04; 
in all cases and the integrated intensity is normalised to unity in the absence of rotational diffusion [ordinate in units of 
Inset gives the diagnostic ST-EPR lineheight ratio measured at point P', relative to the turning point P at the right-hand extremum of the spectrum (see Marsh and Horváth, 1992a).
(cf. Eq. 19):
is the value of 
at 
corresponding to the diagnostic spectral position 
and 
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6.APPLICATIONS: SLOW ROTATION
This section gives examples of the classical application of saturation transfer EPR to the study of slow rotational diffusion on the submillisecond timescale. Applications are chosen to illustrate specific aspects of rotational diffusion in membranes such as the effects of protein concentration, aqueous viscosity, hydrophobic matching and anisotropic rotation, rather than giving a comprehensive review.
6.1Dependence on Protein Density
A systematic study of the dependence of the ST-EPR rotational correlation times on lipid/protein ratio, LP, was undertaken by Fajer et al. (1989) with spin-labelled cytochrome c oxidase reconstituted in bilayer membranes of dimyristoyl phosphatidylcholine. Fig. 5 gives the effective rotational relaxation rate, 
deduced from the central ST-EPR lineheight ratio, C'/C, as a function of lipid/protein ratio in the reconstituted membranes. Protein rotational diffusion is drastically reduced in gel-phase membranes at 1°C where the lipid chains are largely frozen. Rotation is much faster in fluid-phase membranes, with effective correlation times in the tens of microsecond regime. The rate at which cytochrome oxidase rotates in the membrane decreases (i.e., the correlation time increases) progressively with increasing protein packing density.
The hindering of cytochrome oxidase rotation by protein crowding can be described with a simple collisional model based on random proteinprotein contacts. The observed diffusion coefficient, 
is a statistical average of that for freely rotating species that do not experience any influence from other proteins 
and that for highly hindered species that have other proteins immediately adjacent 
where 
denotes the probability for the freely rotating species. This model assumes that the lifetime of protein-protein contacts is shorter than the rotation period. For a translational diffusion coefficient of 
this is likely to be the case (Fajer et al., 1989). The probability 
is obtained from a lattice model designed to calculate the frequency of lipid-protein contacts in random dispersions (Hoffmann et al., 1981). Each lipid occupies one lattice site and each protein occupies R lattice sites, where 
for the cytochrome c oxidase monomer (Deatherage et al., 1982) 
is then the probability that all N lattice sites at the protein perimeter are occupied by lipid molecules:
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where LP is the lipid/protein mole ratio (see Fig. 5).
Figure 5. Dependence of the rotational relaxation rate, 
on lipid/protein molar ratio for cytochrome c oxidase in membranes of dimyristoyl phosphatidylcholine at 30°C (fluid phase, 
and 1°C (gel phase, 
Solid line is a non-linear least squares fit of the random collision model (Eqs. 25 and 26) to the data at 30°C with fixed R = 27 (see Fajer et al., 1989). The inset shows the lattice model used to calculate the probability, 
that a protein (given by the hexagons that occupy R lipid lattice sites) does not contact any other protein. The probability that a lipid (circle) does not occupy a site (e.g., the asterisked position) that is adjacent to a protein is given by R/(LP+R) where LP is the lipid/protein mole ratio.
A non-linear least squares fit with R fixed and N as the parameter to be optimised is given by the solid line in Fig. 5. This depicts the time-averaged protein-protein interactions taking place in the fluid phase. The fitted value of N = 18 corresponds to 36 first-shell lipid sites at the perimeter of the protein, in both bilayer halves of the membrane. This is somewhat smaller than the number of boundary lipids found by EPR measurements with spinlabelled lipids (Knowles et al., 1979). Presumably, the latter reflects the invaginated nature of the intramembranous surface of the protein, which is effectively smoothed when considering protein-protein contacts.
at a lipid/protein ratio of 60:1 (mol/mol). Effective correlation times 
are deduced from the low-field 
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Figure 6 gives the effective rotational correlation times of rhodopsin in the lipids of different chainlength that are deduced from the low-field and high-field regions of the ST-EPR spectra. These measurements are all made in the fluid membrane state at equivalent temperatures above the chainmelting transition for the different lipids. Mostly, the correlation times for L"/L and H"/H are comparable, which is expected because both are reflecting the same anisotropic rotation. The longest effective rotational correlation times are obtained from recombinants with dilauroyl phosphatidylcholine, 
with a steep decrease on increasing the lipid chainlength, through a minimum at chainlengths of C14 to C15, and a subsequent rise on increasing the lipid chainlength to dipalmitoyl and diheptadecanoyl phosphatidylcholines, 
and 
The pronounced increases in rotational correlation time for rhodopsin in the long and short chainlength lipids can be attributed to protein aggregation. This is driven by hydrophobic mismatch in both cases, as illustrated diagrammatically in the inset to Fig. 6. A lipid that is too short exposes part of the hydrophobic domain of the protein to a polar environment. A lipid that is too long forces contact of the hydrophobic lipid chains with polar groups on the protein. In each case, these energetically unfavourable interactions are alleviated by segregation of the protein from the lipids.
A topic of equal interest is the oligomer state of rhodopsin in the C15 chainlength lipids for which hydrophobic matching is best. The effective rotational correlation time, 
deduced from the ST-EPR spectra using calibrations from isotropic solutions is related to the diffusion coefficient,
for uniaxial rotation by (Robinson and Dalton, 1980; Marsh and Horváth, 1989):
where 
is the orientation of the spin-label z-axis relative to the membrane normal, and 
The rotational diffusion coefficient is related to the cross-sectional dimensions, 
and 
and the intramembrane height, 
of the rotating species by the Stokes-Einstein equation (see e.g., Marsh and Horváth, 1989):
where 
is the rotational frictional coefficient, 
is the effective intramembrane viscosity and 
is a shape factor that depends weakly
(see also following section 6.3). The corresponding true rotational correlation time is: 
As usual, 
is Boltzmann’s constant and 
is 
(Fig. 6); an interpolated value for the membrane thickness is 
(Tardieu, 1972); and the effective membrane viscosity is in the region of 
(Cherry and Godfrey, 1981). This yields a value of 
for the intramembranous diameter of the rotating species. This is an upper estimate because it is assumed that 
and 
Cross-sectional dimensions of the dimer of frog rhodopsin are 
(Corless et al., 1982). Thus rhodopsin is most likely a monomer in 
as it is in rod outer segment disc membranes (Downer, 1985).
equal to those of the different sections, 
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where 
are the elliptical semi-axes of the different protein cross-sections (see Fig. 7 inset). The rotational correlation time of the protein therefore is given from Eqs. 28-30 by:
Specifically, the dependence on external viscosity is given by:
where the summation (as indicated by the prime) now extends only over the extramembrane sections of the protein, and 
is the rotational correlation time given by Eq. 28 above, when external viscosity can be neglected.
Figure 7 gives the dependence of the effective rotational correlation time of membranous spin-labelled Na,K-ATPase on viscosity of the external glycerol-containing medium. From the slope and intercept of the linear viscosity dependence, together with Eqs. 28 and 33, it is concluded that 5070% of the Na,K-ATPase protein is external to the membrane (Esmann et al., 1994). This conclusion obtained from hydrodynamics is consistent with the results of low-resolution structural studies on this protein (Maunsbach et al., 1989). Fig. 7 also shows that, with polyethylene glycol solutions, a pronouncedly non-linear dependence on the viscosity, 
is found that is much larger than the viscosity dependence obtained with glycerol solutions. This greater effect of polyethylene glycol undoubtedly corresponds to a dehydration-induced aggregation of the membrane proteins that may be related to the ability of polyethylene glycol to induce membrane fusion. The rotational correlation time reached at 50% polyethylene glycol corresponds to a degree of aggregation of the membrane proteins between two and five, depending on whether the ethylene glycol polymer is excluded from the membrane surface region (Esmann et al., 1994). Clearing of proteins from areas of apposing membrane, by aggregation, is a prerequisite for the close approach of the lipid bilayers that is needed for effective membrane fusion.
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Figure 7. Dependence on extramembrane viscosity, 
of the effective rotational correlation time, 
deduced from the high-field diagnostic lineheight ratio in the ST-EPR spectra of Na,K-ATPase spin-labelled with a chloromercuri-reagent. The solid line is a linear regression for the glycerol data with intercept 
and gradient 
The dashed line for the PEG data is simply to guide the eye (see Esmann et al., 1994). The inset illustrates the different dimensions of the extramembrane and intramembrane sections of the protein that experience viscosities
and 
respectively.
6.4Anisotropic Rotational Diffusion
Rotational correlation times are routinely deduced from experimental STEPR spectra by comparing the diagnostic lineheight ratios in the low-field, central and high-field regions of the spectrum with those obtained from isotropically rotating spin-labelled haemoglobin in solutions of known viscosity (see Table 1). The outer lineheight ratios, L"/L and H"/H, are sensitive to rotation of the nitroxide z-axis, via modulation of the hyperfine interaction, and the central lineheight ratio, C'/C, is sensitive to rotation about all three nitroxide axes, via modulation of the g-value anisotropy (see Eq. 7 and Fajer and Marsh, 1983). For anisotropic rotation, the different lineheight ratios will therefore have differential sensitivities, as illustrated in Table 2 for the rotational diffusion of a spin-labelled phospholipid in gelphase lipid bilayer membranes (Marsh, 1980). The nitroxide z-axis is oriented along the lipid long molecular axis for this particular spin probe. At low temperatures the effective correlation times, 
deduced from
of the nitroxide 
NONLINEAR EPR DISPLAYS