high salt concentrations. Our model also shows that, for a fixed number of strings
N s , the optimal H increases when k increases (see Fig. 16) because E el decreases
and the energy minimum shifts to larger H. In experiments, the pitch of DNA
wrapped around dendrimers indeed increases with k for spherical [186] dendrimers,
but it decreases for cylindrical dendrimers [187]. The helical pitch in salt-free
solution of DNA wrapped around a cylindrical dendrimer was reported to be
about 23 Æ 3 A ˚ [185]. This is smaller than the pitch of DNA in a nucleosome,
H % 24–27 A ˚ [13, 182], which is close to the equilibrium DNA–DNA separation in
hexagonal aggregates [188, 189] and in two-dimensional DNA lattices [190, 191].
Note that in the region of interaxial separations (approximately 20–25 A ˚ ) a very
strong DNA–DNA repulsion is detected [188]. The pitch of DNA–dendrimer
complexes was reported to further decrease at higher salt concentrations [187].
An accompanying swelling of the dendrimers at high salt, however, complicates
calculation of the real fraction of the neutralized dendrimer charge by wrapped
DNA by our theory.
One might argue that the DNA persistence length decreases at high salt
concentrations because the electrostatic persistence length l
el
p $ k
À2 decreases
[164, 165]. The DNA could then wrap more densely around the cylinder. However,
such effects should occur for spherical complexes also, but have not been observed
in experiments. For colloids much larger than the DNA persistence length, nearly
neutral complexes are often detected as a result of wrapping. For smaller colloids
with dimensions a % l p , when polymer bending severely impairs wrapping,
strongly undercharged complexes have been detected, consistent with our
predictions. This is illustrated in the article by Zinchenko et al. (see Fig. 3 in [192]).
Another unresolved issue is the charge of DNA–dendrimer complexes. Preferentially neutral complexes of DNA with spherical dendrimers were observed
[186, 193], in contrast to a pronounced (up to a factor of 2) overcharging of
cylindrical dendrimers by wrapped DNA [185]. The adsorption/attraction of a
cylindrical complex onto a positively charged silica surface was used as evidence
for overcharging. To have this attraction, however, the overcharging might not
be required because the negative DNA charges are closer to the silica surface than
the positive charges of the dendrimer and the net charge of the complex is not so
relevant. Hence, advanced experiments are required to understand the nature of a
possible overcharging of cylindrical DNA–dendrimer complexes.
6.2 Adsorption at a Spherical Surface
As mentioned in the Introduction, strongly adsorbed polyelectrolytes are able to
form various patterns on spherical surfaces. We will limit our presentation to
meridian-type polymer strings as shown in Fig. 12 (middle diagram), which cross
at two points – the north and south poles [59]. Other charge patterns can also be
40
R.G. Winkler and A.G. Cherstvy
N s , the optimal H increases when k increases (see Fig. 16) because E el decreases
and the energy minimum shifts to larger H. In experiments, the pitch of DNA
wrapped around dendrimers indeed increases with k for spherical [186] dendrimers,
but it decreases for cylindrical dendrimers [187]. The helical pitch in salt-free
solution of DNA wrapped around a cylindrical dendrimer was reported to be
about 23 Æ 3 A ˚ [185]. This is smaller than the pitch of DNA in a nucleosome,
H % 24–27 A ˚ [13, 182], which is close to the equilibrium DNA–DNA separation in
hexagonal aggregates [188, 189] and in two-dimensional DNA lattices [190, 191].
Note that in the region of interaxial separations (approximately 20–25 A ˚ ) a very
strong DNA–DNA repulsion is detected [188]. The pitch of DNA–dendrimer
complexes was reported to further decrease at higher salt concentrations [187].
An accompanying swelling of the dendrimers at high salt, however, complicates
calculation of the real fraction of the neutralized dendrimer charge by wrapped
DNA by our theory.
One might argue that the DNA persistence length decreases at high salt
concentrations because the electrostatic persistence length l
el
p $ k
À2 decreases
[164, 165]. The DNA could then wrap more densely around the cylinder. However,
such effects should occur for spherical complexes also, but have not been observed
in experiments. For colloids much larger than the DNA persistence length, nearly
neutral complexes are often detected as a result of wrapping. For smaller colloids
with dimensions a % l p , when polymer bending severely impairs wrapping,
strongly undercharged complexes have been detected, consistent with our
predictions. This is illustrated in the article by Zinchenko et al. (see Fig. 3 in [192]).
Another unresolved issue is the charge of DNA–dendrimer complexes. Preferentially neutral complexes of DNA with spherical dendrimers were observed
[186, 193], in contrast to a pronounced (up to a factor of 2) overcharging of
cylindrical dendrimers by wrapped DNA [185]. The adsorption/attraction of a
cylindrical complex onto a positively charged silica surface was used as evidence
for overcharging. To have this attraction, however, the overcharging might not
be required because the negative DNA charges are closer to the silica surface than
the positive charges of the dendrimer and the net charge of the complex is not so
relevant. Hence, advanced experiments are required to understand the nature of a
possible overcharging of cylindrical DNA–dendrimer complexes.
6.2 Adsorption at a Spherical Surface
As mentioned in the Introduction, strongly adsorbed polyelectrolytes are able to
form various patterns on spherical surfaces. We will limit our presentation to
meridian-type polymer strings as shown in Fig. 12 (middle diagram), which cross
at two points – the north and south poles [59]. Other charge patterns can also be
40
R.G. Winkler and A.G. Cherstvy