Dynamical regimes

We have previously calculated conformational free energy differences for a well-suited model system, the catalytic subunit of cAMP-dependent protein kinase (cAPK), which is the best characterized member of the protein kinase family. It has been crystallized in three different conformations and our main focus was on how ligand binding shifts the equilibrium among these ([Helms and McCammon 1997]). As an example using state-of-the-art computational techniques, we summarize the main conclusions of this study and discuss a variety of methods that may be used to extend this study into the dynamic regime of protein domain motion. 

The following two techniques were developed to expand such static calculations into a pseudo-dynamic regime by calculating higher derivatives of the potential energy and by introducing an additional degree of freedom. 

The discretized symbol-sequence defined in equation 8.39 suggests that we might use two other familiar measures of complexity to characterize the various dynamical regimes of behavior namely, the pattern entropy and dynamical entropy. 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

The standard effective spectroscopic Fermi resonant Hamiltonian allows more complicated types of behavior. The full three-dimensional aspects of the monodromy remain to be worked out, but it was shown, with the help of the Xiao—KeUman [28, 29] catastrophe map, that four main dynamical regimes apply, and that successive polyads of a given molecule may pass from one regime to another. 

The dynamical regimes that may be explored using this method have been described by considering the range of dimensionless numbers, such as the Reynolds number, Schmidt number, Peclet number, and the dimensionless mean free path, which are accessible in simulations. With such knowledge one may map MPC dynamics onto the dynamics of real systems or explore systems with similar characteristics. The applications of MPC dynamics to studies of fluid flow and polymeric, colloidal, and reacting systems have confirmed its utility. 

M. Ripoll, K. Mussawisade, R. G. Winkler, and G. Gompper, Dynamic regimes of fluids simulated by multiparticle-collision dynamics, Phys. Rev. E 72, 016701 (2005). 

The reptation model thus predicts four dynamic regimes for segment diffusion. They are summarized in Fig. 18. 

The 15N spectral peaks of fully hydrated [15N]Gly-bR, obtained via cross-polarization, are suppressed at 293 K due to interference with the proton decoupling frequency, and also because of short values of T2 in the loops.208 The motion of the TM a-helices in bR is strongly affected by the freezing of excess water at low temperatures. It is shown that motions in the 10-j-is correlation regime may be functionally important for the photocycle of bR, and protein-lipid interactions are motionally coupled in this dynamic regime. 

To validate the model developed in the present study, the simulations are first conducted and compared with the experimental results of Wachters and Westerling (1966). In their experiments, water droplets impact in the normal direction onto a hot polished gold surface with an initial temperature of 400 °C. Different impact velocities were applied in the experiment to test the effect of the We number on the hydrodynamics of the impact. The simulation of this study is conducted for cases with different Weber numbers, which represent distinct dynamic regimes. 

Abstract. The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples. 

Fig. 7 presents partial results of dynamic regime experiments for chromate adsorption and desorption by ODA-clinoptilolite. As shown by breakthrough curves, ODA-clinoptilolite column quantitatively removes chromate species from simulated waste water , apparently more efficiently by lower flow rate. Consequently to similar configuration of chromate and sulfate molecules, such loaded column was more efficient to regenerate with Na2S04 than NaCl solution, as elution curves at the Fig. 7 illustrate. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

Figure 20. The stability diagram. Depending on the trace Tr and determinant A ad be of the Jacobian matrix M, the steady state can be classified intodistinct dynamic regimes. The parabola indicates the line Tr2 4A, corresponding to the occurrence of imaginary eigenvalues. For an interpretation of the different dynamic regimes, see text.

Interestingly, the simple model is already sufficient to exhibit a variety of dynamic regimes, including bistability and oscillations. 

Figure 28. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence of ATP on the first reaction V (ATP) (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for > 0 the state is unstable. Transitions occur via a saddle node (SN) and a Hopf (HO) bifurcation. Parameters are v° 1, TP° 1, ATP0 0.5, At 1, and 6 0.8. See color insert.

Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter 0 TP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. 

To demonstrate the applicability of the described approach to a system of a reasonable complexity, we briefly consider a (parametric) model of the CO2-assimilating Calvin cycle. In particular, we seek to detect and quantify the possible dynamic regimes of the model without specifying a set of explicit differential equations. 

Fig. 3.57. Chromatograms of CBWD (Reactive black 5) dye solution after 15 min electrolysis (b) under dynamic regime with flow at 3 ml/min in comparison with the solution before electrolysis (a). Working electrode Fe52 column OCTYL flow rate 0.8 ml/min mobile phase (20 80) aqueous phosphate buffer, pH 5-methanol (2.5 min) and linear gradient buffer-methanol (20 80) to (50 50) temperature 25°C detection wavelength 220 nm. Reprinted with permission from M. Ceron-Rivera et al. [127].

Figure 1. The different dynamic regimes for a polmer system. The notation for the regimes is given in the text. (Reproduced from Ref. 20. Copyright 1983 American Chemical Society.)...

The dialysis membrane employed is usually hydrophilic and isolates two aqueous solutions in a static or dynamic regime depending on the particular purpose. While these sensors are formally similar to those discussed in the previous section, it is molecules or ions that are separated (by virtue of a concentration gradient), the process being aided both by the dynamic character of the acceptor solution and the reaction involved, which removes the species transferred across the membrane. 

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