Scaling of the Bicoid morphogen gradient: the effect of state dependent diffusion

The mechanisms underlying the scaling of the Bicoid morphogen gradient in Drosophila with the embryo size is not clearly understood. We propose a model with a spatially varying diffusion coefficient along the anterior-posterior axis, that is proportional to the nearly periodic spatial distribution of the nucleo-cytoplasmic domains and alternating between regions of fast and slow diffusion. We postulate that for specific interpretation of the heterogeneous environment, where the space available for free diffusion within an energid is assumed to be proportional to the embryo size, a change in the embryo size can lead to a size-dependent scaling of the gradient lengthscale. We further study a two component model with slow and fast diffusing Bicoid, and identify this model to be equivalent to the heterogeneous diffusion model further postulating that the fast-state occupancy which is a measure of the fraction of time spent in the fast diffusing state, should also scale with the embryo size via the energid size. Finally, we incorporate nuclear shuttling into our model to understand the effect of shuttling on the gradient lengthscale and scaling with embryo size. We argue that for the particular case where the degradation within the nucleus is low, nuclear shuttling does not perturb the Bicoid gradient. Our study suggests that scaling of the gradient length scale is possible due to spatial heterogeneity and does not depend on nuclear trapping as suggested previously.