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crc press - E-Lib FK UWKS

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194 Cell-Penetrating Peptides: Processes and Applications<br />

N<br />

∑<br />

i=<br />

1<br />

Eint S E C i tr i z<br />

=− () () ( )<br />

(9.4)<br />

The general behavior of Equation 9.4 is that E int increases when solvent-accessible<br />

hydrophilic atoms (i.e., E tr > 0) penetrate the membrane and decreases when<br />

solvent-accessible hydrophobic atoms do. The more atoms are accessible, the larger<br />

is the effect.<br />

E int decreases when two hydrophobic or hydrophilic atoms come close together<br />

in the hydrophobic and hydrophilic phases, respectively. IMPs and water-soluble<br />

proteins tend to form compact structures. For water-soluble proteins this is explained<br />

by the fact that the protein minimizes its hydrophobic surface in contact with water.<br />

For IMPs, Rees and colleagues 48 suggested that interactions between adjacent lipids<br />

are disrupted and replaced by weaker interactions between the protein and lipids.<br />

E lip accounts for perturbation of the lipid bilayer due to peptide insertion. It is defined<br />

as<br />

N<br />

∑<br />

i=<br />

1<br />

Elip = αlip<br />

SC i ( Z )<br />

(9.5)<br />

where α lip is an empirical factor fixed to + 0.018. The concept of this equation is<br />

very simple: E lip increases with the surface of the protein in contact with lipids. The<br />

assumption is made that lipids act as a pool of free-solvating CH 2 groups, although<br />

these groups are covalently linked in acyl chains.<br />

9.2.4 CHARGE SIMULATION (THIRD RESTRAINT)<br />

More sophisticated methods exist to simulate electrostatic energy. Starting from an<br />

atomistic model of phosphatidyl choline phospholipid (PC) bilayers, La Rocca et al.<br />

calculated an average electrostatic potential along the bilayer. 32 This method exists<br />

for PC, a globally neutral lipid, but does not fit for charged monolayers. We have<br />

developed a method to mimic charge distribution of bilayers.<br />

The electrostatic potential across the bilayer may be obtained by solving Poisson’s<br />

equation:<br />

∇ ()∇ ()=− () ρ z<br />

ε z Φ z<br />

(9.6)<br />

where Φ(z), ε(z), and ρ(z) are the electrostatic potential, dielectric constant, and<br />

charge density, respectively, at position z. Water is treated as a z-dependent dielectric<br />

constant. Although this is a simplification, it captures the essentials of variation of<br />

the electrostatic potential along the bilayer normal at a similar level of approximation<br />

as the other elements of the empirical energy function used to represent the bilayer.<br />

At any point, the electrostatic potential resulting from Equation 9.6 will depend on<br />

ε<br />

i<br />

0<br />

i

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