  We address the problem of diffusion on a comb whose teeth display a varying
length. Specifically, the length $\ell$ of each tooth is drawn from a
probability distribution displaying the large-$\ell$ behavior $P(\ell) \sim
\ell^{-(1+\alpha)}$ ($\alpha>0$). Our method is based on the mean-field
description provided by the well-tested CTRW approach for the random comb
model, and the obtained analytical result for the diffusion coefficient is
confirmed by numerical simulations. We subsequently incorporate retardation
effects arising from binding/unbinding kinetics into our model and obtain a
scaling law characterizing the corresponding change in the diffusion
coefficient. Finally, our results for the diffusion coefficient are used as an
input to compute concentration recovery curves mimicking FRAP experiments in
comb-like geometries such as spiny dendrites. We show that such curves cannot
be fitted perfectly by a model based on scaled Brownian motion, i.e., a
standard diffusion equation with a time-dependent diffusion coefficient.
However, differences between the exact curves and such fits are small, thereby
providing justification for the practical use of models relying on scaled
Brownian motion as a fitting procedure for recovery curves arising from
particle diffusion in comb-like systems.
