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Structure Prediction of CPPs and Iterative Development of Novel CPPs 195
x, y, and z (where the xy plane is that of the membrane). However, on the time scale
of peptide–bilayer interactions the peptide is assumed to be influenced by a potential
averaged in the membrane plane:
(9.7)
where the integral in Equation 9.7 is extended on the surface of the xy plane over
an area S. When the dielectric function is dependent upon z, the average potential
Φ(z) is the solution of Equation 9.6 with an average charge distribution:
where
1
Φ()= z Φ()
r dxdy
S
2
d Φ() z dε
Φ z ρ z
ε()
z + 2
dz dz dz ε
(9.8)
(9.9)
and where the definition of the integral and of S are the same as in Equation 9.7.
To make an average charge distribution, the mean and standard deviations of the z
coordinate of each phospholipid atom type were calculated. The contribution from
each atom type to the overall charge density of the bilayer was approximated as a
Gaussian function with mean and standard deviations as calculated, and normalized
to the partial charge per unit area of that atom type:
(9.10)
where A is the area per head group, 49 q i is the partial charge of atom type i, G is a
Gaussian function of the z coordinate for atom type i, and the sum is extended over
all atom types of the phospholipid molecule. The resultant charge density of the
bilayer is localized in a plane at –15.75 and +15.75 Å.
Imagine that a charge is distributed uniformly over the entire xy-plane
(Figure 9.3), with a charge per unit area or surface density of charge, δ. We wish to
calculate the electric intensity at the point P. Let the charge be subdivided into narrow
strips parallel to the y-axis and of dx width. Each strip can be considered as a charged
line. The area of a portion of a strip of length L is L dx, and the charge dq on the
strip is
dq = δLdx
The charge per length unit, dλ, is therefore
∫
() = ()
1
ρ()= z ρ()
z dxdy
S
∑
1
ρ()= z qG i i()
z
A
i
0