Gramicidin Lateral Distribution in Phospholipid Membranes: Fluorescence Phasor Plots and Statistical Mechanical Model

When using small mole fraction increments to study gramicidins in phospholipid membranes, we found that the phasor dots of intrinsic fluorescence of gramicidin D and gramicidin A in dimyristoyl-sn-glycero-3-phosphocholine (DMPC) unilamellar and multilamellar vesicles exhibit a biphasic change with peptide content at 0.143 gramicidin mole fraction. To understand this phenomenon, we developed a statistical mechanical model of gramicidin/DMPC mixtures. Our model assumes a sludge-like mixture of fluid phase and aggregates of rigid clusters. In the fluid phase, gramicidin monomers are randomly distributed. A rigid cluster is formed by a gramicidin dimer and DMPC molecules that are condensed to the dimer, following particular stoichiometries (critical gramicidin mole fractions, Xcr including 0.143). Rigid clusters form aggregates in which gramicidin dimers are regularly distributed, in some cases, even to superlattices. At Xcr, the size of cluster aggregates and regular distributions reach a local maximum. Before a similar model was developed for cholesterol/DMPC mixtures (Sugar and Chong (2012) J. Am. Chem. Soc. 134, 1164–1171) and here the similarities and differences are discussed between these two models.

where s g  is the lateral interaction energy of a gramicidin (located in one of the layer of a rigid cluster) with nearest neighbor condensed lipids plus half of the gramicidin-gramicidin interaction energy between the elements of the gramicidin dimer, u g  is the lateral interaction energy of a gramicidin (located in one of the layer of a fluid unit) with nearest neighbor fluid lipids plus the interaction energy between the gramicidin and fluid lipids (located in the other layer of the fluid unit).
is the energy of a phospholipid molecule in the fluid phase. is the average energy of a phospholipid molecule in a rigid cluster.
Half of the interaction energy between the nearest neighbor lattice units, is calculated by assuming periodic boundary conditions [1].  [1]]. Here we note that very few models of statistical mechanics have been solved exactly; in most of the cases one has to rely on approximative methods. Among them, the mean-field approximation (such as the Bragg-Williams approximation) is one of the most widely used. The advantage of the mean-field theory is its simplicity and it correctly predicts the qualitative features of a system in most cases [2].
The half of the internal entropies of the s and u state lattice units s S and u S are: where g  and are the intramolecular entropy of a gramicidin and the intramolecular entropy of a phospholipid molecule in the fluid phase, respectively. Since the gramicidin conformation in the fluid and rigid units are similar [3], the same value of g  is taken in the fluid and rigid unit.
is the average intramolecular entropy of a phospholipid molecule in a rigid cluster.
The mixing entropy of the molecules of one layer of the bilayer within the fluid phase, i.e. the mixing entropy of the gramicidin molecules and the phospholipid molecules in the fluid phase, is: where is the Boltzmann constant, and in the second equality Stirling's approximation [1] was utilized.
is the sum of the mixing entropies of the u state lattice units located in one layer of the bilayer. On the other hand, the mixing entropy of the molecules belonging to an s state lattice unit is zero because the gramicidin dimer is assumed to be located at the same place within each rigid cluster and thus the number of microstates within a rigid cluster is 1.
The s and u state lattice units can be arranged along the lattice on different ways and thus half of the mixing entropy of the s and u state lattice units is: where in the second equality Stirling's approximation [1] was again utilized.

On the energy and entropy parameters of the model.
Our model will be applied to gramicidin/DMPC mixtures. Since a lattice unit represents several molecules, the parameters associated with it depend on the actual number of the molecules and their physical and geometrical properties.
Let us consider first the intramolecular energy difference: . In an s state lattice unit if the condensing effect of the gramicidin dimer brings all the nearby phospholipid molecules close to the gel state then where is the gel-to-fluid transition enthalpy of DMPC bilayers (see Table S1). In reality the average state of the phospholipid molecules in the rigid cluster is between the fluid and gel state.
and g p A is the cross section of a DMPC in gel state.