Anomalous diffusion and dynamics of fluorescence recovery after photobleaching in the random-comb model

Abstract

We address the problem of diffusion on a comb whose teeth display a varying length. Specifically, the length ℓ of each tooth is drawn from a probability distribution displaying the large-ℓ behaviorOur 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. PACS numbers: 05.40.Fb,

Diffusion–Advection Equations on a Comb: Resetting and Random Search

“…Here, h(u, t) is the so-called subordination function. The latter is the PDF, which subordinates the process governed by Equation (21) to the process governed by Equation (13). By the Laplace transform of Equation ( 23), and by using the subordination function h(u, s) = 1 sη(s) e −u/η(s) , (…”

“…Different generalizations of the comb geometry have been considered. For example, various diffusion processes have been considered in a comb with a finite finger length [4][5][6], diffusion on cylindrical [7,8] and circular combs [9][10][11], more complex branched structures [12], random comb models [13], and a comb with ramified teeth [14], as well as the problem of first encounters for two workers [15]. Diffusion processes in fractal mesh and grid structures have been considered as well: In this case, anomalous diffusion of a particle is affected by the fractal structure of the infinite numbers of backbones and fingers [16].…”

“…It has been shown that these models are useful for description of anomalous transport through porous solid pellets with various porous geometries [17]. Comb models are also applicable for describing diffusion in percolation clusters [2,18,19], anomalous transport of inert compounds in spiny dendrites [20][21][22], modeling electron transport in disordered nanostructured semiconductors [23,24], dispersive transport of charge carriers in two-layer polymers [25], percolative phonon-assisted hopping in twodimensional disordered systems [26,27], and anomalous diffusion of fluorescence recovery after photobleaching in a random-comb model [13]. Another interesting realization is that turbulent diffusion in a comb appears to be due to multiplicative noise [28,29].…”

Richardson diffusion in neurons

“…Here we consider a diffusion process on the times t > h. In this case, there is a simple relation between these PDFs P (x, y = 0, t) ∼ hρ(x, t), which reduces to the equality at the asymptotically large time scale, t ≫ h. As it follows from Ref. [15], this relation should be also true for random finger's length with the finite mean length of the order of h.…”

“…It has been shown in experimental and numerical studies that the transport of inert particles along dendrite structure of neurons corresponds to anomalous diffusion (namely subdiffusion), when the temporal behavior of the MSD is of the power law t γ , where the transport exponent 0 < γ < 1 depends on the geometry and density of the dendritic spines [9][10][11]. Dendritic spines are the basic functional units in pre-and post synaptic activity of neurons [12], and further studies have shown that the comb model can be used to describe the movement and binding dynamics of particles, including reaction transport of Ca 2+ ions inside the spines [13][14][15]. A comb model has been suggested as a simplified toy model, which reflects this property of anomalous diffusion, resulted from the geometry, which mimics geometry of spiny dendrites, such that the backbone is the dendrite and the fingers are the spines, see Fig.…”