Trajectory analysis mean-square displacement

Trajectory Analysis and Mean-Square Displacement Plots. 288... 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

The resulting trajectories are usually analyzed for the mode of motion executed by the particle as the motion provides information on the location and status of the particle as described in this review. The most common analysis starts with the calculation of the so-called mean-square displacement (MSD). Then, the time dependence of the MSD is plotted. This plot allows a mode of motion analysis [35]. A simplified way to calculate the MSD is depicted in Fig. 2b. The mean square displacement describes the average of the squared distances between a particle s start and end positions for all time-lags of certain length At within one trajectory. 

Sampling and Analysis We continued the canonical ensemble simulation for another 10 to 20 ns, while the trajectories of the sorbate molecules were written out every 100 fs. The analysis of these trajectories enabled ns to compnte the mean square displacements and therefore the diffusivity. 

---