physiological ionic strength, which is reminiscent of DNA–histone complexation in
nucleosomes [218]. For salt concentrations above 1.5 M, when the electrostatic
interactions are well screened, no DNA–nanosphere electrostatically-driven compaction could be achieved, whereas for low salt concentrations, when the electrostatic
contribution to the DNA persistence length grows, compaction is suppressed by the
bending energy penalty.
Note that typically electro-neutral aggregates are also observed in other dense
self-assembled DNA nanostructures, e.g., DNA sandwich-like lamellar complexes
with cationic lipids [191, 219–221], where complexation is suggested to be driven
by the release of condensed counterions of DNA and of lipid head groups [222].
In general, electrostatic interactions in spherical complexes with arbitrary polyelectrolyte wrapping patterns and for arbitrary orientations of the complexes in
space, pose a complicated mathematical problem. One can expect that the
modulations of the electrostatic potential emerging in a solution near the complexes
due to nonhomogeneous charge distributions will modify a DLVO-like pure repulsion between effective like-charged spheres [242]. Correlation-induced electrostatic attraction between complexes can occur when these potential modulations
on the complexes are in phase: the potential patches of different sign face each other
near the contact of the two complexes, forming an electrostatic zipper. The effective
screening length of this electrostatic attraction ~ (k
2 + (2p/P)
2
)
À1/2 is a combination
of k and of a typical period P of alternating positive–negative patches on the
surfaces. It is shorter than the decay length 1/k of repulsive interactions between
uniformly charged spheres. Thus, in order to overcome the net charge repulsion, the
complexes should be neutralized to a large extent by the adsorbed polyelectrolytes
ð ^ y 1Þ and their separation should be smaller than P (short-range attraction and
long-range repulsion).
Such zipper-like electrostatic attraction could be one of the reasons for condensation of DNA molecules and of nucleosome core particles in solutions of some
multivalent counterions (for a discussion see [25, 223]). Another possibility is internucleosomal attraction mediated by bridging interactions of flexible polypeptide
histone tails [213, 224, 225]. The theoretical basis for such attraction driven by
sharing of polyelectrolyte chains adsorbed simultaneously on two nucleosomes/
macroions has been developed in [46, 175, 226, 227].
7 Limitations and Further Studies
The described theoretical concepts are certainly limited in their predictive power by
the various underlying approximations that are adopted to derive analytical solutions.
The following aspects apply (more or less) to weak and strong adsorption.
Uniformly Charged Surface and Polymer Chain—The implications of charge
nonuniformities on surfaces and charge discreteness of adsorbing polyelectrolytes
are not evident. Muthukumar studied the influence of surface-charge patchiness on
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
47
nucleosomes [218]. For salt concentrations above 1.5 M, when the electrostatic
interactions are well screened, no DNA–nanosphere electrostatically-driven compaction could be achieved, whereas for low salt concentrations, when the electrostatic
contribution to the DNA persistence length grows, compaction is suppressed by the
bending energy penalty.
Note that typically electro-neutral aggregates are also observed in other dense
self-assembled DNA nanostructures, e.g., DNA sandwich-like lamellar complexes
with cationic lipids [191, 219–221], where complexation is suggested to be driven
by the release of condensed counterions of DNA and of lipid head groups [222].
In general, electrostatic interactions in spherical complexes with arbitrary polyelectrolyte wrapping patterns and for arbitrary orientations of the complexes in
space, pose a complicated mathematical problem. One can expect that the
modulations of the electrostatic potential emerging in a solution near the complexes
due to nonhomogeneous charge distributions will modify a DLVO-like pure repulsion between effective like-charged spheres [242]. Correlation-induced electrostatic attraction between complexes can occur when these potential modulations
on the complexes are in phase: the potential patches of different sign face each other
near the contact of the two complexes, forming an electrostatic zipper. The effective
screening length of this electrostatic attraction ~ (k
2 + (2p/P)
2
)
À1/2 is a combination
of k and of a typical period P of alternating positive–negative patches on the
surfaces. It is shorter than the decay length 1/k of repulsive interactions between
uniformly charged spheres. Thus, in order to overcome the net charge repulsion, the
complexes should be neutralized to a large extent by the adsorbed polyelectrolytes
ð ^ y 1Þ and their separation should be smaller than P (short-range attraction and
long-range repulsion).
Such zipper-like electrostatic attraction could be one of the reasons for condensation of DNA molecules and of nucleosome core particles in solutions of some
multivalent counterions (for a discussion see [25, 223]). Another possibility is internucleosomal attraction mediated by bridging interactions of flexible polypeptide
histone tails [213, 224, 225]. The theoretical basis for such attraction driven by
sharing of polyelectrolyte chains adsorbed simultaneously on two nucleosomes/
macroions has been developed in [46, 175, 226, 227].
7 Limitations and Further Studies
The described theoretical concepts are certainly limited in their predictive power by
the various underlying approximations that are adopted to derive analytical solutions.
The following aspects apply (more or less) to weak and strong adsorption.
Uniformly Charged Surface and Polymer Chain—The implications of charge
nonuniformities on surfaces and charge discreteness of adsorbing polyelectrolytes
are not evident. Muthukumar studied the influence of surface-charge patchiness on
Strong and Weak Polyelectrolyte Adsorption onto Oppositely Charged Curved. . .
47