Biology Reference
In-Depth Information
determined by considering the solution to be divided into laminae
paralleltotheelectrode.Thelaminaeareinthermalequilibrium,but
at differing energies due to the potential
ϕ
, so the concentration
n
i
of species i with valence
z
i
is related to its bulk concentration
n
i
by
the Boltzmann factor
n
i
=
n
i
exp(
−
z
i
e
ϕ/
kT
)
(6.3)
The net charge density
ρ
(
x
) is related to the potential by the
Poissonequation
ρ
(
x
)
=
εε
0
d
2
ϕ
d
x
2
(6.4)
where
ε
istherelativedielectricpermittivity,
ε
0
isthepermittivityof
freespace,and
x
isthedistancefromtheelectrode.Useofboundary
conditions leads to the non-linear Poisson-Boltzmann equation.
For
ϕ
k
T
/
e
, the linearized Poisson-Boltzmann equation results.
Alternatively, the non-linear Poisson-Boltzmann equation may be
solved for a symmetrical electrolyte that contains only one cationic
and one anionic species, both with charge magnitude
z
,givingthe
Grahame equation for the charge per unit area on the electrode
σ
1
:
OHP
=
8k
T
εε
0
n
0
sinh
|
z
|
e
ϕ
OHP
2k
T
σ
1
=−
εε
0
d
d
x
(6.5)
6.3.1.2 DNA charge fraction
dsDNA is a semi-flexible chain with persistence length
∼
100 nm,
where the persistence length is the distance in which tangent
vectors decorrelate, a measure of the rigidity of a polymer. Short
duplexes can be considered as cylinders of 2
.
0nmdiameterand
axiallengthperbasepairof0.34nm.Thecorrespondingparameters
for ssDNA have not been established. Stacking interactions between
hydrophobic bases tend to produce a stiff single-stranded helix and
ssDNAhasbeenmodeledasacylinderofdiameter
∼
1.4nmandaxial
length per monomer of 0.34 nm [19]. However, if ssDNA is assumed
to be a freely jointed chain with a length per base of 0.43 nm [20],
itspersistencelengthvariesfrom5nmat1mMionicstrengthto0.8
nm at 100 mM ionic strength [21]. This is consistent with a much
stronger rigidity of dsDNA compared to ssDNA.