^{1}and Harden McConnell

^{2}

### Abstract

A thermodynamic model of cholesterol-phospholipid complexes is used as a starting point for calculating fluctuations in membranes containing cholesterol and phospholipids. The calculations describe fluctuations in the concentration of complexes formed between cholesterol and phospholipids with longer saturated fatty acid chains. The fluctuations in complex concentrations arise by two distinct mechanisms. In one, the chemical composition of the sample varies from point to point, and the concentration of the complexes varies according to local chemical equilibrium. In the second, the composition remains fixed, and the complexes form and dissociate according to chemical reaction kinetics. In both cases the nuclear resonance frequency of a deuterium labeled phospholipid undergoes fluctuations and line broadening as a consequence of the formation and dissociation of complexes. For a specific ternary lipid mixture at its critical composition, deuterium nuclear resonance line broadening of chain labeled phospholipids is calculated for temperatures up to 10°–20° above the miscibility critical temperature. This line broadening is associated with fluctuations in the degree of phospholipid chain ordering related to the formation and dissociation of complexes.

The authors are greatly indebted to Sarah Veatch, Klaus Gawrisch, Sarah Keller, and their collaborators for discussions of their ongoing NMR studies of bilayers composed of cholesterol and phospholipids. They are also indebted to Sunney Chan for helpful comments.

I. INTRODUCTION

II. BACKGROUND THEORY

III. NUCLEAR RELAXATION BY FLUCTUATIONS IN COMPLEX CONCENTRATIONS

IV. THERMODYNAMIC COMPOSITION FLUCTUATIONS

V. DIFFUSION AND COMPOSITION CORRELATION FUNCTIONS

VI. CHEMICAL EXCHANGE

VII. CHEMICAL EXCHANGE INCLUDING A GRADIENT ENERGY

VIII. DEUTERIUM NMRLINEWIDTHS

IX. PARAMETER VALUES

X. DISCUSSION

### Key Topics

- Nuclear magnetic resonance
- 26.0
- Diffusion
- 24.0
- Rotational correlation time
- 20.0
- Critical point phenomena
- 15.0
- Dissociation
- 11.0

## Figures

(Color) Theoretical phase diagram. The diagram simulates the results obtained experimentally by Veatch and Keller using fluorescence microscopy (Ref. 11) and NMR spectroscopy (Ref. 3) for the ternary lipid mixture, cholesterol, DPPC, and DOPC. The theoretical diagram uses a thermodynamic model (Refs. 8 and 10) involving the formation of a complex of one molecule of cholesterol and two molecules of DPPC, along with a mean-field repulsion between the complex and DOPC.

(Color) Theoretical phase diagram. The diagram simulates the results obtained experimentally by Veatch and Keller using fluorescence microscopy (Ref. 11) and NMR spectroscopy (Ref. 3) for the ternary lipid mixture, cholesterol, DPPC, and DOPC. The theoretical diagram uses a thermodynamic model (Refs. 8 and 10) involving the formation of a complex of one molecule of cholesterol and two molecules of DPPC, along with a mean-field repulsion between the complex and DOPC.

Molecular free energies as a function of composition. The free energy parameters employed to construct the phase diagram in Fig. 1 are used to calculate the molecular free energy changes for composition variations along three illustrative directions indicated by the dotted lines. Small composition fluctuations along a line parallel to the stoichiometric tie line (b) have the lowest free energy, particularly near the critical temperature, as can be seen in panel (b). Composition fluctuations calculated in this paper only include composition fluctuations along this direction (b) where the initial mole fractions of cholesterol and DPPC are in a 1:2 ratio. The intersection point of the three lines is the ternary critical composition.

Molecular free energies as a function of composition. The free energy parameters employed to construct the phase diagram in Fig. 1 are used to calculate the molecular free energy changes for composition variations along three illustrative directions indicated by the dotted lines. Small composition fluctuations along a line parallel to the stoichiometric tie line (b) have the lowest free energy, particularly near the critical temperature, as can be seen in panel (b). Composition fluctuations calculated in this paper only include composition fluctuations along this direction (b) where the initial mole fractions of cholesterol and DPPC are in a 1:2 ratio. The intersection point of the three lines is the ternary critical composition.

Spectral density factors and correlation times for no sample spinning. Panels (a) and (b) consider fluctuations in the total cholesterol concentration. The dependence of on temperature and on the cut-off wavelength is shown in (a). A plot of the average correlation time for is shown in (b) as a function of temperature. The zero frequency spectral density factors and the average correlation time are related to one another by the fluctuation in the concentration of complexes: . This correlation time diverges at the critical temperature (asymptote shown by dotted vertical line). Panels (c) and (d) consider complex formation (chemical exchange) kinetics, i.e., fluctuations in complex concentration at fixed total cholesterol concentration. The dependence of on temperature and on the cut-off wavelength is shown in (c). In this case the fluctuations arise from the kinetics of complex formation and dissociation, and are characterized by a single kinetic correlation time. A plot of this kinetic correlation time [Eq. (44)] is shown in (d) as a function of temperature.

Spectral density factors and correlation times for no sample spinning. Panels (a) and (b) consider fluctuations in the total cholesterol concentration. The dependence of on temperature and on the cut-off wavelength is shown in (a). A plot of the average correlation time for is shown in (b) as a function of temperature. The zero frequency spectral density factors and the average correlation time are related to one another by the fluctuation in the concentration of complexes: . This correlation time diverges at the critical temperature (asymptote shown by dotted vertical line). Panels (c) and (d) consider complex formation (chemical exchange) kinetics, i.e., fluctuations in complex concentration at fixed total cholesterol concentration. The dependence of on temperature and on the cut-off wavelength is shown in (c). In this case the fluctuations arise from the kinetics of complex formation and dissociation, and are characterized by a single kinetic correlation time. A plot of this kinetic correlation time [Eq. (44)] is shown in (d) as a function of temperature.

Magic angle spinning spectral density factors related to composition fluctuations and chemical exchange. Panels (a) and (b) give the spectral density factors due to composition fluctuations and complex formation (chemical exchange) kinetics, respectively. Both calculations include a gradient term in the free energy of the fluctuations.

Magic angle spinning spectral density factors related to composition fluctuations and chemical exchange. Panels (a) and (b) give the spectral density factors due to composition fluctuations and complex formation (chemical exchange) kinetics, respectively. Both calculations include a gradient term in the free energy of the fluctuations.

Calculated nuclear resonance linewidths. The linewidths refer to a half height linewidth (HHLW). This is for Lorentzian signals. Panel (a), solid line, gives the calculated Lorentzian HHLW due to fluctuations in complex concentration that arise from fluctuations in total cholesterol concentration. The linewidth refers to the quadrupole axis orientation , corresponding to the outer wings of the deuterium NMR spectra with no sample spinning. The Lorentzian (which diverges at the critical temperature) assumes rapid gradient diffusion so that Eq. (14) can be used. The dotted line gives the Gaussian HHLW corresponding to a zero gradient diffusion coefficient. The dashed curve gives the contribution to arising from chemical exchange kinetics. Panel (b) gives the sum of the two Lorentzian contributions in panel (a). Panels (c) and (d) give the results for the case of magic angle spinning at . In panel (c), the solid curve gives the Lorentzian HHLW due to fluctuations in total cholesterol concentration and the dashed curve gives the Lorentzian HHLW due to fluctuations in complex concentration with fixed total cholesterol concentration, arising from the kinetics of complex formation and dissociation. Panel (d) gives the sum of the two contributions in (c). All calculations include the gradient energy term [Eqs. (21) and (46)].

Calculated nuclear resonance linewidths. The linewidths refer to a half height linewidth (HHLW). This is for Lorentzian signals. Panel (a), solid line, gives the calculated Lorentzian HHLW due to fluctuations in complex concentration that arise from fluctuations in total cholesterol concentration. The linewidth refers to the quadrupole axis orientation , corresponding to the outer wings of the deuterium NMR spectra with no sample spinning. The Lorentzian (which diverges at the critical temperature) assumes rapid gradient diffusion so that Eq. (14) can be used. The dotted line gives the Gaussian HHLW corresponding to a zero gradient diffusion coefficient. The dashed curve gives the contribution to arising from chemical exchange kinetics. Panel (b) gives the sum of the two Lorentzian contributions in panel (a). Panels (c) and (d) give the results for the case of magic angle spinning at . In panel (c), the solid curve gives the Lorentzian HHLW due to fluctuations in total cholesterol concentration and the dashed curve gives the Lorentzian HHLW due to fluctuations in complex concentration with fixed total cholesterol concentration, arising from the kinetics of complex formation and dissociation. Panel (d) gives the sum of the two contributions in (c). All calculations include the gradient energy term [Eqs. (21) and (46)].

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