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Biophysical Studies of Cell-Penetrating Peptides 229
number of H-bonding water molecules are removed, therefore promoting intramolecular
H-bonds. The most common alcohols used in such studies are trifluoroethanol
(TFE) and hexafluoroisopropanol (HFP), which are water mixible; however, it has
also been suggested that some aggregated solvent clusters could exist in aqueous
mixtures. 25 Whether this type of solvent mixture should be considered a good
membrane mimetic can be debated.
10.4 INTERFACIAL PHENOMENA — ELECTROSTATICS
Outside the charged membrane surface is the so-called interphase region, a region
of solvent more strongly associated with the membrane than the bulk region, which
is far away (∞) from the surface. In the theoretical treatment of the diffuse double
layer one assumes that the interphase is also composed of two regions. Close to the
surface is the Stern layer, where ions are attracted or repelled. Outside this region
is a diffuse layer of ions, within which all the ionic particles, including charged
peptides like CPPs, are assumed to obey a Boltzmann distribution. With an electrical
potential φ(r) at a position r, the mean-field electrical interaction energy becomes
z ieφ(r), where z i is valency of the ion (±) and e is electronic charge (F/N A). For each
ion, concentration (mol/dm 3 ) is assumed to become
c i(r) = c i(∞) exp [–z ieφ(r)/kT]
The net charge density (charges/m 3 ) in the aqueous phase at a point r is then
A
all
∑ i i A
all
∑ i i i
i
i
()= ( ) ()= ( ) ∞
3 3
ρ r 10 N e zc r 10 N e zc exp zeφ r kT
The Poisson equation is a fundamental result (from Gauss laws) in electrostatics
connecting the electrical potential φ(r) and charge density ρ(r):
∆φ(r) = – ρ(r)/ε οε r
[ ]
( ) − ()
In the one-dimensional case, the Laplace operator becomes ∆ = d 2 /dx 2 , where x is
the distance from the Stern layer. With x = ∞, the limit is out in the bulk phase.
When the medium is not a perfect dielectricum, variation in the permittivity (ε r) can
exist.
Even in the one-dimensional case, the combined Poisson–Boltzmann equation
has no simple analytical solution; we must assume certain boundary conditions. A
useful relation between the surface charge density, σ 0, and the surface potential φ(0),
at x = 0, was derived by Grahame: 26
2 3
σ = 210 ⋅ ε ε RT c ( ∞)
exp(
− z eφ( 0) kT)−1
0
all
∑
o r i
i
{ i
}