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194 Cell-Penetrating Peptides: Processes and Applications
N
∑
i=
1
Eint S E C i tr i z
=− () () ( )
(9.4)
The general behavior of Equation 9.4 is that E int increases when solvent-accessible
hydrophilic atoms (i.e., E tr > 0) penetrate the membrane and decreases when
solvent-accessible hydrophobic atoms do. The more atoms are accessible, the larger
is the effect.
E int decreases when two hydrophobic or hydrophilic atoms come close together
in the hydrophobic and hydrophilic phases, respectively. IMPs and water-soluble
proteins tend to form compact structures. For water-soluble proteins this is explained
by the fact that the protein minimizes its hydrophobic surface in contact with water.
For IMPs, Rees and colleagues 48 suggested that interactions between adjacent lipids
are disrupted and replaced by weaker interactions between the protein and lipids.
E lip accounts for perturbation of the lipid bilayer due to peptide insertion. It is defined
as
N
∑
i=
1
Elip = αlip
SC i ( Z )
(9.5)
where α lip is an empirical factor fixed to + 0.018. The concept of this equation is
very simple: E lip increases with the surface of the protein in contact with lipids. The
assumption is made that lipids act as a pool of free-solvating CH 2 groups, although
these groups are covalently linked in acyl chains.
9.2.4 CHARGE SIMULATION (THIRD RESTRAINT)
More sophisticated methods exist to simulate electrostatic energy. Starting from an
atomistic model of phosphatidyl choline phospholipid (PC) bilayers, La Rocca et al.
calculated an average electrostatic potential along the bilayer. 32 This method exists
for PC, a globally neutral lipid, but does not fit for charged monolayers. We have
developed a method to mimic charge distribution of bilayers.
The electrostatic potential across the bilayer may be obtained by solving Poisson’s
equation:
∇ ()∇ ()=− () ρ z
ε z Φ z
(9.6)
where Φ(z), ε(z), and ρ(z) are the electrostatic potential, dielectric constant, and
charge density, respectively, at position z. Water is treated as a z-dependent dielectric
constant. Although this is a simplification, it captures the essentials of variation of
the electrostatic potential along the bilayer normal at a similar level of approximation
as the other elements of the empirical energy function used to represent the bilayer.
At any point, the electrostatic potential resulting from Equation 9.6 will depend on
ε
i
0
i