Diffusion trajectory of an asymmetric object: Information overlooked by the mean square displacement

Diffusion of an asymmetric object is characterized by its translational and rotational diffusion coefficients. Until now, anisotropic diffusion studies have been based on ensemble averages. Here we present a theoretical basis for the analysis of the trajectories of a single particle with anisotropic diffusion coefficients. We discuss the relevance of this method for motion of biomolecules in the membrane of living cells.

Figure 1

Definition of principal axes of the translational diffusion coefficients, $D_{\Vert }$ and $D_{\perp }$, and the rotational diffusion coefficient, $D_r$, around the center of diffusion ($\mathsf{C}$; position ${\vec{X}}^{\left(0\right)}$). The marker (asterisk, position $\vec{X}$) is at a distance $d$ off the center along the longitudinal direction.

Figure 2

Examples of trajectories of symmetric (a) and asymmetric (b) particles generated numerically. The marker on the particle is placed at the center of diffusion $(d=0)$. The thick black symbols represent the position of the particle at $t=0$ [at the origin (0,0)], $t=\tau ∕2$, and $t=\tau$. The principal diffusion coefficients are chosen to be $(D_{\Vert },D_{\perp },D_{\mathrm{r}})=(1,1,1∕2)$ (a) or $(1,1∕10,1∕2)$ (b), in the length units of $\ell$ and the time unit of $\tau$. Spatial coordinates are also represented in units of $\ell$. The time step of calculation is chosen to be $\mathrm{\Delta }t=\tau ∕200$.

Figure 3

Comparison of the simulated mean square displacements (MSD), $⟨\mid {\vec{X}}_t-{\vec{X}}_0{\mid }^2⟩$ $(d\ne 0)$ using 600 trajectories (thick curve) with the analytical result (dashed curve). The dashed lines are the initial and final asymptotes. Except for the off-center distance of the marker chosen, $d=2\ell$, the result is completely scaled by the characteristic length $\ell$ and time $\tau$, irrespective to the anisotropy parameter, $\eta$. The time step of calculation is chosen to be $\mathrm{\Delta }t=\tau ∕20$.

Figure 4

Cross correlation between the displacement and particle orientation, $\widehat{C}\left(t\right)$ (a), and persistence of the diffusing direction $C(0,t,2t)$ (b). See (6, 7), respectively, for the definitions. The functions are presented vs $t∕\tau$ for various values of the anisotropy parameter, $\eta =0$ (bottom), $1∕3$ (middle), $1$ (top). The off-center distance of the marker was chosen to be $d=0$. Analytical results (dashed curves) are compared with the simulated data over 600 trajectories (thick curves). In (b) the calculation is truncated at $t=0.4\tau$ since, with the present time step of calculation $(\mathrm{\Delta }t=\tau ∕600)$, the accuracy becomes poor beyond this point [due mainly to the fact that the function $C(0,t,2t)$ is a subtle difference between two growing terms of $\sim t^2$].