Diffusion hopping

FIGURE 2-10 Tracking a gold particle attached to a single molecule of phosphatidyl ethanolamine. What appears to be simple Brownian diffusion at a time resolution of 33 ms per video frame (A) is revealed to actually consist of fast hop diffusion by recording 300 times faster (B) at 110 ps per video frame. In (A) each color represents 60 frames = 2 seconds. In (B) each color indicates an apparent period of confinement within a compartment and black indicates intercompartmental hops. The residency time for each compartment is indicated. The hypothetical explanations are illustrated in part (C) and discussed in the text. Adapted from [29]. 

The major part of the reports discussed above provides only a qualitative description of the catalytic response, but the LbL method provides a unique opportunity to quantify this response in terms of enzyme kinetics and electron-hopping diffusion models. For example, Hodak et al. [77[ demonstrated that only a fraction of the enzymes are wired by the polymer. A study comprising films with only one GOx and one PAH-Os layer assembled in different order on cysteamine, MPS and MPS/PAH substrates [184[ has shown a maximum fraction of wired enzymes of 30% for the maximum ratio of mediator-to-enzyme, [Os[/[GOx[ fs 100, while the bimolecular FADH2 oxidation rate constant remained almost the same, about 5-8 x 10 s ... 

Looks at the role of caveolins in the formation of membrane caveolae Covers the investigation of hop diffusion of membrane lipids using FRAP (fluorescence recovery after photobleaching)... 

After internalization in phase I, most polyplexes show anomalous or confined diffusion (phase II) followed by active transport (phase III) [37]. The anomalous diffusion and confinement displayed by the MSD analysis represent the local microenvironment of the particles in the cytoplasm where the cytoskeleton, organelles and large macromolecules are local obstacles for free diffusion. Suh et al. [41] tracked internalized polyplexes with a temporal resolution of 33 ms and found diffusive motion of polyplexes where the corresponding trajectories showed hop diffusion (Fig. 4). These hop diffusion patterns can be interpreted as the cages of the... 

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]...

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]. 

Now consider the case when a HOP diffuses into a sphere of radius a, initially at concentration C0, and the concentration outside the sphere is constant and equal to Q. Formally we can represent these boundary and initial conditions as follows ... 

The hop-diffusion pattern caimot be found in liposomes or membrane blebs. In these membranes, the membrane molecules show simple Brownian diffusion with a single diffusion coefficient (55). 

Accumulating evidence clearly points at involvement of the cell cytoskeleton in the compartmentalization of the membrane, in particular, the fine cytoskeleton filaments formed by actin in most eukaryotic cells or spectrin in mammalian red blood cells. However, single-particle tracking experiments show the same patterns of hop-diffusion for lipid molecules located in the extracellular leaflet of the plasma membrane. How can the membrane skeleton, which is located only on the cytoplasmic surface of the membrane, suppress the motion of lipids on the extracellular side ... 

Fujiwara T, Ritchie K, Murakoshi H, Jacobson K, Kusumi A. Phospholipids undergo hop diffusion in compartmentalized cell membrane. J. Cell Biol. 2002 157 1071-1081. 

Qdiffior double hop diffusion is 8 kJ/mol higher than for single hop. 

With these data in hand, the maximum value of the electron hopping diffusion constant, D = 1.8 x 10 cm s , can be interpreted to yield the unimolecular rate constant (Ai) of electron transfer between adjacent Os(DPP)3 sites according to the following equation ... 

A more accxuate measurement of the electron hopping diffusion constant, and of the related kgx, that did not require the assumption of 100% electroactivity of the Os(DPP)3 film, was obtained with a novel 2D electrochemical technique relying on a tandem of two collinear 40 gm long line microelectrodes spaced by 10 gm and acting as a generator probe and a collector probe. In this method, referred... 

We can now define the electron-hopping diffusion coefficient (units cm s ) as follows... 

Since it is well-known that in a microscopic sense, diffusion can be modeled in terms of a random walk, then in three dimensions, the electron-hopping diffusion coefficient can be expressed as follows... 

FIGURE 1.9. Variation of the hard-sphere electron-hopping diffusion coefficient with redox site concentration. Curves were calculated from Eqn. 28 using the A and k values obtained from Fig. 1.6. For the upper curve and inset, Tq = 1.3 nm, whereas for the lower curve, To = 0.6 nm. 

FIGURE 1.10. (a) Variation of with redox site concentration according to the He-Chen model [see Eqn. 38(a)]. (b) Comparison between the He-Chen prediction and experimental data for electron-hopping diffusion coefficients for Os(bpy)f loaded in Nafion films obtained via complex impedance spectroscopy by Sharp and coworkers (Ref. 40). 

FIGURE 1.94. Schematic representation of the twin-microelectrode polymer sandwich arrangement for determining electron-hopping diffusion coefficients. 

FIGURE 1.95. Schematic representation of the interdigitated array (IDA) arrangement for determining electron-hopping diffusion coefficients in the steady state, (a) the IDA configuration, (b) Polymer-overcoated IDA. (c) Finger dimensions, (d) A different perspective of the polymer-overcoated IDA device. 

Note that the lateral electron-hopping diffusion coeflScient can be related to the bimolecular electron self-exchange rate constant via the expression... 

The surface coverage F can be estimated by integrating the cyclic voltammogram at a slow sweep rate. The electron-hopping diffusion... 

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