  The production of prompt $D^0$ mesons in proton-lead collisions in the
forward and backward configurations at a center-of-mass energy per nucleon pair
of $\sqrt{s_\mathrm{NN}} = 8.16~\mathrm{TeV}$ is measured by the LHCb
experiment. The nuclear modification factor of prompt $D^0$ mesons is
determined as a function of the transverse momentum $p_\mathrm{T}$, and
rapidity in the nucleon-nucleon center-of-mass frame $y^*$. In the forward
rapidity region, significantly suppressed production with respect to $pp$
collisions is measured, which provides significant constraints of nuclear
parton distributions and hadron production down to the very low Bjorken-$x$
region of $\sim 10^{-5}$. In the backward rapidity region, a suppression with a
significance of 2.0 - 3.8 standard deviations compared to nPDF expectations is
found in the kinematic region of $p_\mathrm{T}>6~\mathrm{GeV}/c$ and
$-3.25<y^*<-2.5$, corresponding to $x\sim 0.01$.
