  Derived from a biophysical model for the motion of a crawling cell, the
system \[(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta
v-kv+u\end{cases}\] is investigated in a finite domain
$\Omega\subset\mathbb{R}^n$, $n\geq2$, with $k\geq0$. While a comprehensive
literature is available for cases with $(*)$ describing chemotaxis systems and
hence being accompanied by homogeneous Neumann-type boundary conditions, the
presently considered modeling context, besides yet requiring the flux
$\partial_\nu u-u\partial_\nu v$ to vanish on $\partial\Omega$, inherently
involves homogeneous Dirichlet conditions for the attractant $v$, which in the
current setting corresponds to the cell's cytoskeleton being free of pressure
at the boundary.
  This modification in the boundary setting is shown to go along with a
substantial change with respect to the potential to support the emergence of
singular structures: It is, inter alia, revealed that in contexts of radial
solutions in balls there exist two critical mass levels, distinct from each
other whenever $k>0$ or $n\geq3$, that separate ranges within which (i) all
solutions are global in time and remain bounded, (ii) both global bounded and
exploding solutions exist, or (iii) all nontrivial solutions blow up in finite
time. While critical mass phenomena distinguishing between regimes of type (i)
and (ii) belong to the well-understood characteristics of $(*)$ when posed
under classical no-flux boundary conditions in planar domains, the discovery of
a distinct secondary critical mass level related to the occurrence of (iii)
seems to have no nearby precedent.
  In the planar case with the domain being a disk, the analytical results are
supplemented with some numerical illustrations, and it is discussed how the
findings can be interpreted biophysically for the situation of a cell on a flat
substrate.
