  The effect of sequence heterogeneity on polynucleotide translocation across a
pore and on simple models of molecular motors such as helicases, DNA
polymerase/exonuclease and RNA polymerase is studied in detail. Pore
translocation of RNA or DNA is biased due to the different chemical
environments on the two sides of the membrane, while the molecular motor motion
is biased through a coupling to chemical energy. An externally applied force
can oppose these biases. For both systems we solve lattice models exactly both
with and without disorder. The models incorporate explicitly the coupling to
the different chemical environments for polymer translocation and the coupling
to the chemical energy (as well as nucleotide pairing energies) for molecular
motors. Using the exact solutions and general arguments we show that the
heterogeneity leads to anomalous dynamics. Most notably, over a range of forces
around the stall force (or stall tension for DNA polymerase/exonuclease
systems) the displacement grows sublinearly as t^\mu with \mu<1. The range over
which this behavior can be observed experimentally is estimated for several
systems and argued to be detectable for appropriate forces and buffers. Similar
sequence heterogeneity effects may arise in the packing of viral DNA.
