  We prove that operators of the form $A=-a(x)^2\Delta^{2}$, with suitable
growth conditions on the coefficient $a(x)$, generate analytic semigroups in
$L^1(\mathbb{R}^N)$. In particular, we deduce generation results for the
operator $A :=- (1+|x|^2)^{\alpha} \Delta^{2}$, $0\leq\alpha\leq2$. Moreover,
we characterise the maximal domain of $A$ in $L^1(\mathbb{R}^N)$.
