  We study the bending rigidity of highly charged stiff polyelectrolytes, for
both a single chain and many chains forming a bundle. A theory is developed to
account for the interplay between competitive binding of counterions and charge
correlations in softening the polyelectrolyte (PE) chains. The presence of even
a small concentration of multivalent counterions leads to a dramatic reduction
in the bending rigidity of the chains that are nominally stiffened by the
repulsion between their backbone charges. The variation of the bending rigidity
as a function of $f_{0}$, the fraction of charged monomers on the chain, does
not exhibits simple scaling behavior; it grows with increasing $f_{0}$ below a
critical value of $f_{0}$. Beyond the critical value, however, the chain
becomes softer as $f_{0}$ increases. The bending rigidity also exhibits
intriguing dependence on the concentration of multivalent counterion $n_{2}$;
for highly charged PEs, the bending rigidity decreases as $n_2$ increases from
zero, while it increases with increasing $n_{2}$ beyond a certain value of
$n_{2}$. When polyelectrolyte chains form a $N$-loop condensate (e.g., a
toroidal bundle formed by $N$ turns (winds) of the chain), the inter-loop
coupling further softens the condensate, resulting in the bending free energy
of the condensate that scales as $N$ for large $N$.
