Lipid Bilayer-Membrane Protein Coupling - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 86

in order to extend the applicability of the theory to a non-linear regime, we propose
to use the screened Coulomb interaction approximation instead of using the form
of energy (Eq. 5.11 ) found from elastic considerations of the bilayer model. This
screened Coulomb interaction approach has often been used in condensed matter
physics as the so-called Thomas-Fermi approximation [ 9 ] as well as in biophysics
for interacting systems of charged biomolecules in solution employing traditional
Debye screening to account for the presence of water and ions, first introduced in
the Debye-Hückel model [ 22 ]. Also, zwitterionic lipids having a dipole moment pro-
vide support for calculating localized (in the intermediate range) interaction energies
between channel-forming peptides and nearby lipid head group regions in a manner
equivalent to a free energy profile of interacting charged-zwitterionic lipid layers
using the Debye-Hückel theory [ 60 ]. Here, we wish to mention that the presence of
aqueous ions in the outer leaflet of the bilayer still leaves room for head group dipoles
to show considerable localized charge effects in the inner region where channel lipid
interactions take place. Moreover, the bilayer's spontaneous bending near channels
(see Fig. 5.2 ) is obtained by finding the energy required to bend a straight charged
chain where the screened Coulomb interactions lead to high values of induced stiff-
ness [ 69 ], which is an example where the elastic model [ 38 , 40 ] requires an extension
to a nonlinear regime. The interaction energy between a gramicidin A channel and
a host bilayer has been calculated based on experimentally observable parameters,
such as bilayer thickness d 0 [ 16 ], lipid head group cross-sectional area [ 35 ], chan-
nel length l [ 41 ], lipid charge q L [ 1 , 75 ], and dielectric parameters of the lipid
bilayer core [ 71 ], etc. Bilayer elastic parameters appear in the screened Coulomb
interaction as secondary ingredients. In this screened interaction we assume that the
gramicidin A channel couples with the lipid bilayer through a deformation of the
bilayer at the channel bilayer interaction interface (see Fig. 5.2 ). Considering that
the gramicidin A channel length is smaller than the thickness of the bilayer, the
channel extends its Coulomb interaction toward lipids sitting on the bilayer's rest-
ing thickness. The gramicidin A channel directly interacts with the nearest-neighbor
lipid (lp1) by the Coulomb interaction and this lipid interacts directly with its next-
nearest-neighbor lipid (lp2) but this second lipid experiences an interaction with the
channel which is screened due to the presence of the channel's nearest-neighbor lipid;
a first-order term in the extension of V sc (see Eq. 5.14 ). The interaction between the
third-nearest-neighbor lipid (lp3) and the channel is screened by both the nearest-
and next-nearest-neighbor lipids (the second-order term in the extension of V sc ).
Figure 5.6 illustrates this in a diagrammatic view. The chain peptide-lipid interaction
can be better explained by the curvilinear model diagram in Fig. 5.7 , but the real
condition is that the peptides interact with lipids in all directions on each monolayer
leaflet. The general form of the screened Coulomb interaction is as follows:

d 3 ke ( i k · r ) V sc ( V sc ( r ) = k ), (5.14)

where the screened Coulomb interaction in Fourier space is given by [ 9 ].