Biomedical Engineering Reference
In-Depth Information
alignment along the surrounding electric fields [58]. Unfortunately, it is difficult to
use Equation 1.5 together with the experimental value of ε
r
to estimate the energy of
electrostatic interactions in water when charges are at short distance since (1) water
can no longer be approximated as a continuous medium, (2) water molecules may
be expelled by steric effects, (3) water structure is expected to be altered near sur-
faces, and (4) the relative dielectric constant may decrease in presence of a very high
electric field, a phenomenon known as dielectric saturation. The effective dielectric
constant may thus be much lower than 78.
In addition to the charge screening by water molecules, the interaction between
two fixed charges is decreased by surrounding ions. Indeed, according to Boltz-
mann's law, it is expected that free ions will get concentrated around ions of opposite
charge. This phenomenon is well accounted for by Poisson-Boltzmann's equation:
∑
i
Δ
V
+
(
ρ
+
c
i
q
i
exp
(
−
q
i
V
/
k
B
T
)
/
ε
(1.6)
where ρis the volume density of charges other than soluble ions,
c
i
and
q
i
are the
concentration and charge of ionic species
i
. A notable simplification is achieved if
the terms
q
i
V
k
B
T
are low enough to use a linearized form of this equation. The
interaction energy between two charges
q
and
q
at distance
r
may then be written as
/
qq
exp
(
−
κ
r
)
F
=
(1.7)
4πε
r
In a 1:1 electrolyte solution such as physiological saline, parameter κis equal to
2
cq
2
k
B
T
ε
1
/
2
κ
=
(1.8)
κis about 8
A
at room
temperature. Unfortunately, the linear approximation is not always fully valid in the
biological milieu. The increase of computer power led to a revival of interest in the
classical equations of electrostatic and numerical solution of Poisson-Boltzmann's
equations allowed investigators to build maps of the electrostatic potential of protein
surfaces, based on the surface distributions of charged amino acids such as glutamic
acid, aspartic acid, lysine, or arginine. This allowed clear visualization of active sites,
thus demonstrating the importance of electrostatic interactions in biomolecule recog-
nition [88]. The effective interaction energy between two opposite charges such as a
COO
−
and a
NH
3
group in a protein-protein interface was estimated at about 1.6
k
B
T
in an experimental study made on the high affinity interaction between thrombin
and hirudin, which involves four electrostatic bonds [181].
In a 0.15 M NaCl solution, the
Debye
−
H uckel
length 1
/
1.5.1.2 The Hydrogen Bond
In addition to charged ionic groups born by acidic amino acids, such as aspartic or
glutamic acid, or basic amino acids such as lysine or arginine the surface of any pro-
tein bears local charges resulting from the differential distribution of atomic nuclei
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