Biology Reference
In-Depth Information
Manning's counterion condensation (CC) theory [22] is the
asymptotic Poisson-Boltzmann solution for straight polyelec-
trolytes of infinite length at infinite separation and zero salt
concentration. If the axial charge spacing
b
is less than the Bjerrum
length
l
B
(the distance at which two unit charges have a Coulomb
interaction energy equal to the thermal energy k
T
), a fraction
of
the polyelectrolyte charge is compensated by counterions localized
to the polyelectrolyte, reducing its netcharge:
θ
=
1
−
ξ
−
1
θ
(6.6)
where
is the Manning parameter, i.e., the number of unit charges
per Bjerrum length, given by
ξ
=
l
B
/
b
=
e
2
ξ
/εε
0
k
Tb
(6.7)
If
b
<
l
B
, counterion condensation occurs and the net axial
charge density of the polyelectrolyte is reduced to one charge per
Bjerrum length (equal to 0.714 nm for water at 25
◦
C). CC remains
valid as long as the polyelectrolyte length is greater than the Debye
screening length
λ
D
. At greater salt concentrations,
excessive counterion condensation is expected. CC holds for helical
chargelattices,withthecounterionfractionstilldependentuponthe
axial charge spacing [23]. For dsDNA
b
=
0.17 nm, giving a charge
fraction of 24%. For ssDNA
b
≈
0.43 nm, giving a charge fraction
of 60%, and the same effective charge per unit length. Due to the
reduction in length upon duplex formation, dsDNA is expected to
havealowernetchargethanssDNA.Thisisvalidaslongas
b
λ
D
;
so counterion condensation is expected to remain valid at ionic
strengths much less than 500 mM, corresponding to a Debye length
of 0.43 nm. This charge fraction value of
∼
25% for dsDNA has been
confirmed experimentally [24].
Moleculardynamicssolutionshaveshownthatastheseparation
ofpolyelectrolytesisdecreasedfrominfinite,thecounterionfraction
increasesslightlyfromtheManninglimit.Atlowsaltconcentrations,
CC is qualitatively unchanged. The layer of condensed counterions
contracts, but the amount of condensation is only marginally
increased. Increasing salt concentration leads to a crossover
between Manning condensation and charge screening when the
Debye length becomes smaller than the radius of the condensed
λ
D
and
b