  Morphogen gradients play an essential role in the spatial regulation of cell
patterning during early development. The classical mechanism of morphogen
gradient formation involves the diffusion of morphogens away from a localized
source combined with some form of bulk absorption. Morphogen gradient formation
plays a crucial role during early development, whereby a spatially varying
concentration of morphogen protein drives a corresponding spatial variation in
gene expression during embryogenesis. In most models, the absorption rate is
taken to be a constant multiple of the local concentration. In this paper, we
explore a more general class of diffusion-based model in which absorption is
formulated probabilistically in terms of a stopping time condition. Absorption
of each particle occurs when its time spent within the bulk domain (occupation
time) exceeds a randomly distributed threshold $a$; the classical model with a
constant rate of absorption is recovered by taking the threshold distribution
$\Psi(a)=\e^{-\kappa_0 a}$. We explore how the choice of $\Psi(a)$ affects the
steady-state concentration gradient, and the relaxation to steady-state as
determined by the accumulation time. In particular, we show that the more
general model can generate similar concentration profiles to the classical
case, while significantly reducing the accumulation time.
