Lipid Bilayer-Membrane Protein Coupling - Membrane Biophysics, Biological and Medical Physics, Biomedical Engineering - page 89

channels. The binding energy between monomers in a channel mentioned above
(Eq. 5.16 ) is always a standard condition, no matter how we derive this energy, and
the monomer-monomer binding is kept constant throughout this study by not dis-
turbing the membrane's inner region where the binding occurs. In the presence of a
hydrophobic bilayer thickness gramicidin A channel length mismatch, τ observed
in other studies was mainly seen not to follow the modest change in U gA , gA due to
a slight change of the gramicidin A monomer's charge profile in the case of bind-
ing of amphiphiles, anti-fusion, or antimicrobial peptides with channels in a varied
membrane environment. All these observations, taken together, suggest that a change
of (the already formed) gramicidin A channel stability is mainly due to the change
of the gramicidin A channel bilayer coupling energy ( U gA , bilayer ), though the total
potential energy between two gramicidin A monomers in a membrane-associated
gramicidin A channel is given by

U ( r ) = U gA , gA ( r ) + 2 U gA , bilayer ( r ). (5.19)

Here, U gA , bilayer (

)
(Eqs. 5.14 and 5.15 ) for the hydrophobic mismatch to be filled by single, double etc.
lipids representing the first-, second-, etc. order screening in the screened Coulomb
interaction. The protein-protein interaction energy

)

is the first-, second-, etc. order term in the expansion of V sc ( r r

G prot , and the bilayer defor-

G def (mentioned in an earlier section) are proportional to U gA , gA
and U gA , bilayer (in Eq. 5.19 ), respectively. The zeroth-order term in the expansion of
V sc (

mation energy

(Eqs. 5.14 and 5.15 ) represents the direct Coulomb interaction when the gram-
icidin A channel length exactly matches the bilayer thickness. In practice, gramicidin
A channels appear with some level of hydrophobic mismatch between the bilayer
thickness and the gramicidin A channel length, so there is some amount of screened
Coulomb interaction to be expected. The coefficients in the interaction terms can
be calculated using an energy minimum criterion ∂ U

r )

( r )/∂ r = 0 resulting in the

condition

r ∗ 3 −

A = 2 r ∗ 2 C B = 4 r ∗ C ,

r 0 ,

(5.20)

r ∗ −

where r 0 ( ≈ ( represents one-half of the hydrophobic mismatch of the

bilayer thickness and the channel length.
According to Eqs. 5.14 - 5.19 and the description here, changes in F dis (recall
the definition from Eq. 5.8 ) could arise largely from changes in bilayer thickness
(determined mainly by lipid acyl chain lengths), from changes in lipid geometry
(mainly lipid curvature), changes in relative charges between lipids and gramicidin
A monomers, bilayer dielectric condition, and the bilayer elastic moduli. In an exper-
imental protocol we can vary d 0 −

d 0 − l )/ 2 )

l by choosing bilayers with different thickness
or gramicidin A monomers with different lengths, or both, which consequently
changes

G def and F dis and, as a result, the stability of the gramicidin A chan-
nels becomes regulated. We discuss the experimental techniques used and a few test
cases investigated in the next section. In the case when d 0 − 0, the channel expe-

riences negligible destabilization due to bilayer deformation at the channel bilayer

l ≈