Subdiffusive motion

Consistent with the fact that the longest relaxation time of the Zimm model is shorter than the Rouse model, the subdiffusive monomer motion of the Zimm model [(Eq. (8.70)] is always faster than in the Rouse model [Eq. (8.58)] with the same monomer relaxation time tq. This is demonstrated in Fig. 8.8, where the mean-square monomer displacements predicted by the Rouse and Zimm models are compared. Each model exhibits subdiffusive motion on length scales smaller than the size of the chain, but motion becomes diffusive on larger scales, corresponding to times longer than the longest relaxation time. ... 

We are interested in the effect of subdiffusive motion of the activator on Turing instabilities. We focus on this mechanism exclusively and consider the LE model in its original form without a substrate, i.e., a = 1 in (1.152) or (1.160). Equation (10.180) results in the following expression ... 

The death rate of the inhibitor in the Oregonator is linear. Subdiffusive motion of this species leaves the critical ratio 6 unchanged as y increases, 9 =... 

Busselez et al. [56] used neutron scattering with H/D substitution labeling to examine the structure and dynamics of glassy PVP and water in hydrated PVP systems, from which they identified two types of water motion. Consistent with the earlier simulations, they obtained structural evidence for the existence of water clusters, nanosegregation of PVP side-groups, and swelling and disorder within ring nanodomains in the presence of water. The Q-dependence observed for the relaxation time of water molecules indicated that water relaxation was a diffusive-like process in water-rich domains while between water clusters subdiffusive motions prevailed. 

The capability of combined nanoscale spatial and millisecond time resolution provided by SAXS is clearly revealed by a study involving the absorption of bovine serum albumin (BSA) onto spherical polyelectrolyte brushes (SPB). The experiment also highlighted the requirement of an advanced modeling capability for the complete exploration of the time-resolved SAXS data. The quantity of absorbed protein per brush as a function of time was provided from the radial electron density profile of SPB, which has been previously derived from the time-resolved SAXS intensities. Furthermore, an unexpected subdiffusive motion of proteins in the tethered polyelectrolyte brushes has been revealed. A quantitative explanation of this sub-diffusive mode can be approached in terms of a simple model involving direct motions of proteins enclosed in the effective interaction potential of the polyelectrolyte chains. 

In order to map the multichain model onto a simpler single-chain model, it is important to compare as many observables as possible. Indeed, there are many models that can reproduce a slowing down of G(t) with the plateau appearing in some window. There are less models that at the same time can also predict subdiffusive motion of the middle monomer as Si.mid However, these two fimaions do not directly... 

For highly concentrated polymer solutions, FCS measurements revealed subdiffusive motion as an additional mode on an intermediate timescale between the fast collective diffusion and the slow self-diffusion [24]. In such slow systems, however, FCS reaches its limits when probe motion becomes so slow that the number of molecules moving into or out of the confocal volume within the measurement time is too small to allow for reliable statistics. Increasing the measurement time is often not straightforward since all fluorescence dyes have only a limited photostability. If a dye bleaches within the confocal volume, it will fake a faster diffusional motion than its real value. Therefore, for the study of such concentrated systems, wide-field fluorescence microscopy and subsequent single molecule tracking is a much better method [120] and has been utilized to study the glass transition [87, 121]. 

The connection between anomalous conductivity and anomalous diffusion has been also established(Li and Wang, 2003 Li et al, 2005), which implies in particular that a subdiffusive system is an insulator in the thermodynamic limit and a ballistic system is a perfect thermal conductor, the Fourier law being therefore valid only when phonons undergo a normal diffusive motion. More profoundly, it has been clarified that exponential dynamical instability is a sufRcient(Casati et al, 2005 Alonso et al, 2005) but not a necessary condition for the validity of Fourier law (Li et al, 2005 Alonso et al, 2002 Li et al, 2003 Li et al, 2004). These basic studies not only enrich our knowledge of the fundamental transport laws in statistical mechanics, but also open the way for applications such as designing novel thermal materials and/or... 

Figure 5. Comparison of prediction (4) with numerical data. Normal diffusion ( ). The ballistic motion ( ). Superdiffusion ID Ehrenfest gas channel (Li et al, 2005)(v) the rational triangle channel (Li et al, 2003) (empty box) the polygonal billiard channel with (i = (V > — 1)7t/4), and 2 = 7r/3 (Alonso et al, 2002)(A) the triangle-square channel gas(Li et al, 2005) (<>) / values are obtained from system size L e [192, 384] for all channels except Ehrenfest channel (Li et al, 2005). The FPU lattice model at high temperature regime (Li et al, 2005) ( ), and the single walled nanotubes at room temperature ( ). Subdiffusion model from Ref. (Alonso et al, 2002) (solid left triangle). The solid curve is f3 = 2 — 2/a.

Turning to the self-motion of PVE protons, Fig. 5.18 shows the NSE data obtained for Sseif(Q,t). If indeed there existed a second Gaussian subdiffusive regime at length scales shorter than that of the Rouse dynamics, then following Eq. 3.26 also for Q>Qr it should be possible to construct a Q-independent (r t)). This is shown in Fig. 5.19. For Q>0.20 A" a nearly perfect collapse of the experimental data into a single curve is obtained. The time dependence of the determined can be well described by a power law with an exponent of... 

Fig. 4 Example trajectories of diffusive (left) and subdiffusive (right) gene carriers recorded with a temporal resolution of 33 ms. The subdiffusive trajectory is characterized by confined motion in the MSD plot. However, the hop diffusion pattern of the trajectory can only be detected by its morphological pattern and not by the shape of MSD plot. Adapted with permission from the American Chemical Society and American Institute of Chemical Engineers [41]...

As gene carriers are internalized by endocytosis, the motion characteristics inside the cell resembles the movement of the endosomal compartments within the cell and the formed vesicles are transported along the microtubule network [38]. Suh et al. [41] quantified the transport of individual internalized polyplexes by multiple-particle tracking and showed that the intracellular transport characteristics of polyplexes depend on spatial location and time posttransfection. Within 30 min, polyplexes accumulated around the nucleus. An average of the transport modes over a 22.5 h period after transfection showed that the largest fraction of polyplexes with active transport was found in the peripheral region of the cells whereas polyplexes close to the nucleus were largely diffusive and subdiffusive. Disruption of the microtubule network by nocodazole completely inhibits active transport and also the perinuclear accumulation of polyplexes [37, 40, 47]. 

At longer times, monomers participate in collective motion of larger sections with smaller effective diffusion coefficient D(t). Therefore the mean-square displacement of monomers is not a linear function of time, but Instead subdiffusive ---------------------------------------------... 

There are four different regimes of monomer displacement in entangled linear polymer melts, shown in Fig. 9.20. The subdiffusive regime for the mean-square monomer displacement is a unique characteristic of Rouse motion of a chain confined to a tube, which has been found in both NMR experiments and computer simulations. 

A plot of this curve, 6 vs y, is shown in Fig. 10.1. The ratio of the diffusion constant of the inhibitor and the effective diffusion constant of the activator at the Turing threshold increases as the motion becomes more subdiffusive, y ... 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

Figure 15(c) shows the center-of-mass mean-square displacement gs t) normalized by 6 . The soft system shows slight deviations from the horizontal line for cross-correlation function that describes the anisotropy of the center-of-mass motion Rem with respect to the end-to-end vector Re. In general, we define the coefficient of anisotropy of motion of a point a with respect to a vector b as... 

Equation 49 is subdiffusive with y = 1/2 in Eq. 1. This is due to the fact that the trajectory of a bead in a chain is spatially confined by its neighbours. In the limit t tr the chain diffuses as one entity and the bead s trajectory follows mainly the motion of the chain s center of mass. The chain s center of mass obeys... 

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