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
 
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