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

We address the problem of diffusion on a comb whose teeth display varying lengths. Specifically, the length $\ell$ of each tooth is drawn from a probability distribution displaying power law behavior at large $\ell ,P\left(\ell \right)\sim {\ell }^{-(1+\alpha )}$ ($\alpha >0$). To start with, we focus on the computation of the anomalous diffusion coefficient for the subdiffusive motion along the backbone. This quantity is subsequently used as an input to compute concentration recovery curves mimicking fluorescence recovery after photobleaching experiments in comblike geometries such as spiny dendrites. Our method is based on the mean-field description provided by the well-tested continuous time random-walk approach for the random-comb model, and the obtained analytical result for the diffusion coefficient is confirmed by numerical simulations of a random walk with finite steps in time and space along the backbone and the teeth. 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, we show that recovery curves obtained with the help of the analytical expression for the anomalous diffusion coefficient 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 comblike systems.

Figure 1

Scheme of a random-comb structure with equally spaced teeth of varying length.

Figure 2

Double logarithmic plots for the time evolution of $\langle x^2\rangle /t^{\gamma }$ as obtained from numerical simulations. The anomalous diffusion exponent is assumed to be given by the theoretical prediction, i.e., $\gamma =(1+\alpha )/2$ for $\alpha <1$ and $\gamma =1$ for $\alpha \ge 1$. All particles are initially located on the $x$ axis [$y\left(0\right)=0$]. The different curves correspond, from top to bottom, to $\alpha =0.2,0.5,0.6,0.8\text{and}1$ [Fig. 2] and to $\alpha =99.0,2.0,1.7,1.5,1.2,1.1,\text{and}1$ [Fig. 2]. The additional dashed curve corresponding to the behavior of $1/lnt$ is seen to match the asymptotic long-time behavior when $\alpha =1$. For $\alpha =0.5$, the data represented by the curve denoted by “r” correspond to the case where the particles are initially distributed at random along the teeth, and no particles are placed on backbone sites.

Figure 3

Anomalous diffusion coefficient $D_0\left(\alpha \right)$ as a function of $\alpha$. The dots correspond to simulation results. Those dots marked with vertical bars correspond to $\alpha$-values for which the simulation time is not sufficient for $\langle x^2\rangle /t^{\gamma }$ to reach a plateau. In such cases, the dots provide upper bounds for $D_0\left(\alpha \right)$. The solid curve corresponds to the theoretical expression given by Eq. (23) for $0<\alpha <1$, and by Eq. (28) for $\alpha >1$. The dashed horizontal line denotes the asymptotic value $D_0(\alpha \to \infty )=1/4$, corresponding to a uniform teeth with $\ell =1$ everywhere.

Figure 4

Graphical representation of the equivalence between ${\psi }_n\left(\ell \right)$ and $U_n(\ell +1)$. The probability $1/2$ of the walker jumping from $(x=0,y=0)$ to either $(x=+1,y=0)$ or $(x=-1,y=0)$ in the left figure is equal to the probability of the walker jumping from $y=0$ to $y=-1$ along the extended tooth shown on the right figure.

Figure 5

Log-log plot of simulation results for ${\psi }_n(\ell ,\theta )(1-\theta )/\theta$ vs $n$ for $\ell =50$ (squares), $\ell =100$ (diamonds), and $\ell =200$ (circles) for $\theta =1/2$ (open symbols), $\theta =1/3$ (solid symbols), and $\theta =2/3$ (crossed symbols) for ${10}^9$ realizations. Note the excellent collapse of the simulation results for the three different values of $\theta$ and sufficiently large $n$ ($n\ge 50$, say). The solid curves are the asymptotic values of ${\psi }_n\left(\ell \right)$ obtained by means of Eqs. (7), (9), and (15).

Figure 6

Average fraction of bound particles $E\equiv N_b/N$ as a function of $\kappa$.

Figure 7

Normalized diffusion coefficient $D_{\kappa }/D_0$ vs $\kappa$ [$D_0$ is the diffusion coefficient for $K_{\mathrm{on}}=0$]. The dashed curves represent the theoretical prediction given by Eq. (31), whereas the symbols represent simulation results.

Figure 8

Concentration $C_L\left(t\right)$ of particles in the bleached spot as a function of the rescaled time for $\kappa =0$ for different ratios $L/M$ obtained by fixing the value of $L$ and using three different values of $M$. The $L$ values were set to $20,40$, and 80. Curves for different values of $L$ but for the same value of $L/M$ are seen to collapse into a single curve. The $L/M$ values corresponding to collapsing data sets are, from top to bottom, $0.2,0.1$ and $0.05$. The dotted recovery curve corresponds to the case of an infinite system [Eq. (40)]. The inset shows a zoom-in of the data in the region where changes in the time derivative of the recovery curve are largest (different symbols are used to distinguish different data sets corresponding to the same value of $L$). The solid curves are plots obtained from the numerical inversion of Eq. (38).

Figure 9

Concentration $C_L\left(t\right)$ of particles in the bleached spot as a function of the rescaled time in the $\kappa >0$ case. The collapse of data sets corresponding to different values of $\kappa$ is due to the fact that the recovery curves are plotted in terms of the $\kappa$-dependent rescaled time. The solid curve again corresponds to the inversion of Eq. (38).

Figure 10

Semi-logarithmic plot representing $C_L\left(t\right)$ vs. ${log}_{10}{\left(K_{\gamma }{t}^{\gamma }\right)}^{1/2}/L$ for the two extreme cases of a comb model with an infinitely long backbone ($M\to \infty$), namely, (i) the case where all the teeth are infinite, i.e., the case with $\alpha =0$ (implying $\gamma =1/2$, see solid curve), and (ii) the case where all the teeth are finite, i.e., the case of normal diffusion with $\alpha >1$ (implying $\gamma =1$, see dashed curve) and $K_1\equiv D$.