Various time-scales

Fig. 5. Theory vs. experiment rupture forces computed from rupture simulations at various time scales (various pulling velocities Vcant) ranging from one nanosecond (vcant = 0.015 A/ps) to 40 picoscconds (vcant = 0.375 A/ps) (black circles) compare well with the experimental value (open diamond) when extrapolated linearly (dashed line) to the experimental time scale of milliseconds.

Abstract This chapter reviews the theoretical background for continuum models of solvation, recent advances in their implementation, and illustrative examples of their use. Continuum models are the most efficient way to include condensed-phase effects into quantum mechanical calculations, and this is typically accomplished by the using self-consistent reaction field (SCRF) approach for the electrostatic component. This approach does not automatically include the non-electrostatic component of solvation, and we review various approaches for including that aspect. The performance of various models is compared for a number of applications, with emphasis on heterocyclic tautomeric equilibria because they have been the subject of the widest variety of studies. For nonequilibrium applications, e.g., dynamics and spectroscopy, one must consider the various time scales of the solvation process and the dynamical process under consideration, and the final section of the review discusses these issues. 

To summarize, in this article we have discussed some aspects of a semiclassical electron-transfer model (13) in which quantum-mechanical effects associated with the inner-sphere are allowed for through a nuclear tunneling factor, and electronic factors are incorporated through an electronic transmission coefficient or adiabaticity factor. We focussed on the various time scales that characterize the electron transfer process and we presented one example to indicate how considerations of the time scales can be used in understanding nonequilibrium phenomena. 

Driscoll et al. (155) summarized the relationship between N export and N deposition that is indicated by input-output budget data from a large number of watersheds in the United States and Canada these data are augmented in Figure 17b with results from recently published reports. Driscoll et al. (155) stressed that the data illustrated in Figure 17b were collected by using widely differing methods and various time scales (from 1 year to... 

Figure 2. Illustration of the various time scales in cluster dynamics. Times are drawn versus temperature T to accomodate the two processes which strongly depend on temperature. From [6],...

Polymer Backbone Motion. Alternate descriptions of molecular motion utilize an effectively non-exponential autocorrelation function to describe polymer dynamics. One formalism is the use of a log-/2 distribution of correlation times in place of a single correlation time(14). Such a description may simulate the various time scales for overall and internal motions in polymers. 

These basic properties have been exploited in the experiments described here. Various time scales have been investigated. The static or dynamic nature of the observed local anisotropy has been specified. It has been demonstrated that 2H NMR is sensitive to very low degrees or very small variations in the magnitude of the elastic constraints stored in elastic chains. 

The description of the averaging in Equation (1) is by no means trivial, and the development of corresponding models for an effective inclusion of flexibility of molecules on various time scales is still an ongoing field of research. We therefore will start out with the most simple case, the averaging of a completely rigid molecule, before going into effects of conformational averaging. 

Eli Ruckenstein Various time scales have to be compared to obtain an answer to this question. If the relaxation time to attain equilibrium is short compared to the time scale determined by the fluid mechanics, local thermodynamic equilibrium can be assumed. Such an assumption can be made for micellar solutions because the surfactant aggregation and the dissociation of aggregates are relatively rapid processes. It is, however, expected that the size of the aggregate will depend on the stress. For concentrated dispersions, the problem may be more complicated because the processes of aggregation and dissociation of aggregates probably are much slower. 

In this review, we present NMR spectroscopic techniques currently used to study protein dynamics at various time scales. Instead scrutinizing each technique, we put emphasis on their fundamentals. On the other hand, we enumerate a number of NMR-derived parameters and discuss their relation and relevance to macromolecular motions. As a complement, we briefly describe several other techniques capable of capturing protein dynamics, as synthesis of different methods is the most fruitful way to understand biomolecular processes. 

From the simpler resonance line-shape and H/D-exchange analysis to the more complex studies of inherent dynamics, occurring on various time scale of motion, NMR remains a good choice to investigate protein flexibility and plasticity. If linebroadening due to exchange and inhomogeneity is minimized (or completely eliminated), then half-width, Aom, of a line becomes proportional to R, the transverse relaxation rate constant. 

In order to elucidate the structural and, possibly, the dynamic features that make these artificial monomeric species less active than the native enzyme, selected solution (and crystal) structures have been determined and their mobility has been characterized on various time scales (Band et al., 1998, 1999a, 2000 Ferraroni ct a/., 1999). 

We saw in the last section that disparate length scales and time scales exist for turbulent flows. Various time scales also are associated with the... 

Distinction of the various time scales at which subprocesses proceed is useful to determine the necessary complexity or possible simplification of the reactor model. A reactor study under dynamic conditions asks for a detailed kinetic model on the level of... 

Analysis of the decay associated spectra (DAS) with 10-nm resolution confirms the physical picture of the various time scales. The shortest time scales correspond to energy flow out of highest-energy Chls. The 0.3-ps component appears as a decay in the 650- to 670-nm windows and as a major rise at 680-700 nm (680 nm is the maximum of the absorption spectrum). The 2- to 3-ps... 

The lifetime is 4H0 ns at pH 4.0, yielding a ka value of 1.2 0.2 x 10i M"i s" in approximate aRreement with the radiolysis value quoted above (161. The value of kq for H quenchinR was estimated from the Stern-Volmer law and lifetimes of 480 ns (at pH 4.0) and 1130 ns (at pH 7.0). A comparison of values ofAAo from measurements made indicates static quenchinR on various time scales by a factor of 1.5 - 2 on RoinR from pH 7.0 to pH 4.0. Inspection of the spectra at 20 ps directly show the static quenchinR effect (See FIGURE 3). 

There is naturally a close connection between the different pictures provided by the various time scales. The vibration-averaged structure differs at every position at a given instant, and at every instant at a given position. In both cases, however, its average value gives the diffusion-averaged structure. 

Since various motions mentioned occur on various time scales, it is worthwhile to cover a wide range of frequencies. An important advantage of DRS over other spectroscopic techniques is the capability to explore a range of timescales from as slow as 10" s to as fast as 10 " s. Therefore, this single technique allows to study a broad spectrum of motions as a function of temperature, pressure, or composition. Decomposition of DRS spectra into constituents corresponding to various processes is needed. A procedure for doing this which handles also the temperature dependencies of relaxational processes has been devised by Schlosser and his colleagues [91]. 

Dynamic pictures of globular and membrane proteins undergoing motions with various time-scales, as determined by solution and sohd-state NMR measurements, prove to be an indispensable means for the interpretation of their biological functions as well as particular physical properties, besides providing knowledge about their static 3D structures. 

Until rather recently, these reduced schemes were mainly developed by hand using QSSA and an a priori definition of which species were to be considered in steady state, a particularly time-consuming procedure. Before presenting the various time scale separation methods, the basic step of the QSSA reduction procedure will be presented in the subsection below followed by the theory rate-controlled constraints equilibria (RCCE). Thereafter are the time scale separation methods outlined. 

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