  We study the electronic transport through short DNA chains with various
sequences of base pairs between voltage-biased leads. The strong coupling of
the charge carriers to local vibrations of the base pairs leads to the
formation of polarons, and in the relevant temperature range the transport is
accomplished by sequential polaron hopping. We calculate the rates for these
processes, extending what is known as the $P(E)$-theory of single-electron
tunneling to the situation with site-specific local oscillators. The
non-equilibrium charge rearrangement along the DNA leads to sequence-dependent
current thresholds of the `semi-conducting' current-voltage characteristics
and, except for symmetric sequences, to rectifying behavior. The current is
thermally activated with activation energy approaching for voltages above the
threshold the bulk value (polaron shift or reorganization energy). Our results
are consistent with some recent experiments.
