Random Time-Scale Invariant Diffusion and Transport Coefficients

Single particle tracking of mRNA molecules and lipid granules in living cells shows that the time averaged mean squared displacement $\overline{{\delta }^2}$ of individual particles remains a random variable while indicating that the particle motion is subdiffusive. We investigate this type of ergodicity breaking within the continuous time random walk model and show that $\overline{{\delta }^2}$ differs from the corresponding ensemble average. In particular we derive the distribution for the fluctuations of the random variable $\overline{{\delta }^2}$. Similarly we quantify the response to a constant external field, revealing a generalization of the Einstein relation. Consequences for the interpretation of single molecule tracking data are discussed.

Figure 1

Simulations of the subdiffusive CTRW process with $\alpha =3/4$ and free boundary conditions show that the TA MSD is a random variable depending on individual trajectories. The solid curve is the averaged behavior Eq. (3). The measurement time is $t={10}^8$ and $\psi (\tau )=\alpha {\tau }^{-(1+\alpha )}$ for $\tau >1$.

Figure 2

(a) $⟨\overline{{\delta }^2}⟩$ versus $\Delta$ for $\alpha =1/2$ and $t={10}^8$. Stars are simulations and the solid curve is theory Eq. (3) without fitting. (b) The EB parameter converges slowly to the asymptotic value $\mathrm{EB}=0.5708$ given by Eq. (8). Here $\Delta =10$ (dots), $\Delta =2500$ (stars), and $\alpha =1/2$.

Figure 3

PDF of the scaled random variable $\xi =\overline{{\delta }^2}/⟨\overline{{\delta }^2}⟩$ for $\alpha =1/2$ and $\alpha =3/4$ with $t={10}^8$ and $t={10}^7$, respectively. The full line is Eq. (7). Stars $(\Delta =2500)$ and circles $(\Delta =4\times {10}^4)$ are simulations.

Figure 4

Time average $\overline{{\delta }^2}$ vs $\Delta$ for unbiased CTRW on a lattice of size $L=62$, mimicking a particle bounded in a finite domain, as found in the cell. Compared with the unbounded case in Fig. 1 the diffusion is slower. The trajectory averaged MSD follows $\overline{{\delta }^2}\sim {\Delta }^{\beta }$ with $\beta =3/4$ similar to what is found in 1, 2. The measurement time was $t={10}^8$, $\alpha =3/4$.