  The main s-process taking place in low mass stars produces about half of the
elements heavier than iron. It is therefore very important to determine the
importance and impact of nuclear physics uncertainties on this process. We have
performed extensive nuclear reaction network calculations using individual and
temperature-dependent uncertainties for reactions involving elements heavier
than iron, within a Monte Carlo framework. Using this technique, we determined
the uncertainty in the main s-process abundance predictions due to nuclear
uncertainties link to weak interactions and neutron captures on elements
heavier than iron. We also identified the key nuclear reactions dominating
these uncertainties. We found that $\beta$-decay rate uncertainties affect only
a few nuclides near s-process branchings, whereas most of the uncertainty in
the final abundances is caused by uncertainties in neutron capture rates,
either directly producing or destroying the nuclide of interest. Combined total
nuclear uncertainties due to reactions on heavy elements are in general small
(less than 50%). Three key reactions, nevertheless, stand out because they
significantly affect the uncertainties of a large number of nuclides. These are
$^{56}$Fe(n,$\gamma$), $^{64}$Ni(n,$\gamma$), and $^{138}$Ba(n,$\gamma$). We
discuss the prospect of reducing uncertainties in the key reactions identified
in this study with future experiments.
