Experiments have shown that a semiflexible polyelectrolyte, such as a DNA,
can be condensed by multivalent counterions and the preferred form is toroid. By
using molecular dynamics simulations, a single DNA molecule is condensed into
a compact toroid-like structure. DNA is treated as a bead-spring chain, using
parameters of dsDNA. The influence of the counterion size and DNA monomer
size on the DNA structure is studied. We found that for a DNA monomer size
of ¾ and counterion size of 0.5¾, the complex (DNA plus condensed counterions)
forms a well-defined toroidal structure. The dependence of the structure of the
condensed DNA on the initial configuration is investigated. We observed that the
final conformation does not depend on the initial state. The condensed DNA toroid
is then stretched by pulling one end of the chain at various constant velocities to
investigate the effects of the pulling velocity on the force-extension curve (FEC).
We found that the pulling velocity influences the force profile and the internal
structure of the condensed DNA molecule. Moreover, the responses at both DNA
ends are different if the pulling velocity is larger than the reference Rouse velocity,
Vo. For velocities larger than Vo, the FEC’s dependence over the pulling velocity
is linear at the DNA end which is moving at constant velocity; nevertheless, these
FECs oscillate around a constant force (¼ 2.5KBT/¾) at the other end. We
found that a pulling velocity equals to 5×10−4¾/¿ does not perturb the complex.
Moreover, the influence of the pulling velocity on the bond length is linear. We
observed that the entropic behavior of the DNA molecule is strongly affected by
the condensed counterions. The FEC shows a series of “stick-release patterns”.
It gradually increases with increasing extension and then abruptly decreases; this
behavior appears repeatedly and becomes stronger and stronger as the condensed
DNA molecule is losing its turns. We showed that these ”stick-release patterns”
are a consequence of turn-by-turn unfolding of the condensed DNA toroid. The extensible worm-like chain (EWLC) model is found able to describe qualitatively
the behavior of the DNA molecule when its extention is close to the overall contour
length. We presented a clear evidence and described the mechanism of why the
condensed DNA molecule forms a “stick-release patterns”. Our results provide
new microscopic information about the internal structure of a single condensed
DNA toroid being stretched and are in qualitative agreement with experiments.

Experiments have shown that a semiflexible polyelectrolyte, such as a DNA,
can be condensed by multivalent counterions and the preferred form is toroid. By
using molecular dynamics simulations, a single DNA molecule is condensed into
a compact toroid-like structure. DNA is treated as a bead-spring chain, using
parameters of dsDNA. The influence of the counterion size and DNA monomer
size on the DNA structure is studied. We found that for a DNA monomer size
of ¾ and counterion size of 0.5¾, the complex (DNA plus condensed counterions)
forms a well-defined toroidal structure. The dependence of the structure of the
condensed DNA on the initial configuration is investigated. We observed that the
final conformation does not depend on the initial state. The condensed DNA toroid
is then stretched by pulling one end of the chain at various constant velocities to
investigate the effects of the pulling velocity on the force-extension curve (FEC).
We found that the pulling velocity influences the force profile and the internal
structure of the condensed DNA molecule. Moreover, the responses at both DNA
ends are different if the pulling velocity is larger than the reference Rouse velocity,
Vo. For velocities larger than Vo, the FEC’s dependence over the pulling velocity
is linear at the DNA end which is moving at constant velocity; nevertheless, these
FECs oscillate around a constant force (¼ 2.5KBT/¾) at the other end. We
found that a pulling velocity equals to 5×10−4¾/¿ does not perturb the complex.
Moreover, the influence of the pulling velocity on the bond length is linear. We
observed that the entropic behavior of the DNA molecule is strongly affected by
the condensed counterions. The FEC shows a series of “stick-release patterns”.
It gradually increases with increasing extension and then abruptly decreases; this
behavior appears repeatedly and becomes stronger and stronger as the condensed
DNA molecule is losing its turns. We showed that these ”stick-release patterns”
are a consequence of turn-by-turn unfolding of the condensed DNA toroid. The extensible worm-like chain (EWLC) model is found able to describe qualitatively
the behavior of the DNA molecule when its extention is close to the overall contour
length. We presented a clear evidence and described the mechanism of why the
condensed DNA molecule forms a “stick-release patterns”. Our results provide
new microscopic information about the internal structure of a single condensed
DNA toroid being stretched and are in qualitative agreement with experiments.

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