Trajectories diffusive modes

The cumulative functions of the diffusive modes can be constructed by using Eq. (60). The trajectories start from the border of a disk with an initial position... 

Illustration of the diffusive mode of transport near the substrate resulting in randomized molecular trajectories in the vicinity of the mask, ultimately leading to more diffuse patterns than in the case of VTE. (From Shtein, M. et at, /. Appl. Phys., 93,4005,2003. With permission.)... 

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)...

In addition, to detect the various types of motion displayed by a moving particle within a trajectory, the MSD must be taken over subregions of the trajectory. Otherwise, the MSD over the full trajectory would result in an averaging effect over all modes of motion. The careful description of the various modes of motion within one trajectory requires the separation of the trajectory in several parts, e.g., manually according to morphological differences or by velocity thresholds [37,41]. A careful trajectory analysis also includes a morphological analysis of the trajectory pattern and should include more information than the shape of the MSD or effective diffusion coefficient curves. Particles showing hop diffusion may fulfil all analysis criteria for diffusive motion whilst the hop diffusion pattern is only visible in the trajectory [41]. 

A more sophisticated method for automated trajectory analysis and mode of motion detection provides the use of a rolling-window algorithm. The algorithm described by Arcizet et al. [45,46] reliably separates the active and passive states of particles and extracts the velocity during active states as well as the diffusion coefficients during passive states (Fig. 5). It takes into account that active transport by microtubules is characteristically directional over a certain time and measures... 

Shalashilin and Thompson [46-48] developed a method based on classical diffusion theory for calculating unimolecular reaction rates in the IVR-limited regime. This method, which they referred to as intramolecular dynamics diffusion theory (IDDT) requires the calculation of short-time ( fs) classical trajectories to determine the rate of energy transfer from the bath modes of the molecule to the reaction coordinate modes. This method, in conjunction with MCVTST, spans the full energy range from the statistical to the dynamical limits. It in essence provides a means of accurately... 

Figure 1 Modes of diffusion of individual membrane proteins as revealed by single-molecule tracking techniques. The hypothetical trajectory of an individual plasma membrane protein as traced by single-particle tracking techniques is shown. An individual protein can switch between several different modes of over time, which include confined diffusion (region 1), free diffusion (region 2), and immobilization (region 3).

For specific values of K, one has the coexistence of many accelerator modes (i.e., ballistic trajectories), and one observes anomalous diffusion in the... 

In this section experiments on the diffusion of d-PS and d-PMMA chains in PS and PMMA hosts, respectively, are discussed in light of the Doi-Edwards model. In this model the motion of a labelled chain in an entangled melt is imagined to occur along an average trajectory called the primitive path which is defined as the center line of the "tube". The dynamical modes of this chain in its "tube" are assumed to be described by the Rouse model therefore the diffusion coefficient of the chain along the primitive path is denoted by ... 

If one tries to develop a perturbation theory proceeding directly from the reaction-diffusion equations this meets with serious difficulties. They arise because translation and rotation perturbation modes for the spiral wave are not spatially localized. We bypass such difficulties by using the quasi-steady approximation formulated in the previous section. In this approximation the trajectory of the tip motion can be calculated by solving a system of ordinary differential equations which depend only on the local properties of the medium in the vicinity of the tip. The perturbations which originate outside a small neighbourhood of the end point propagate quickly to the periphery and do not influence the motion of the tip. The evolution of the entire curve can then be calculated in the WR approximation using the known trajectory of the tip motion as a dynamic boundary condition. 

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