Eigen curves

Figure 5. Rate constants (logJfcq) for quenching of dimethoxycarbene (197) with ROH (Table 3)" as a function of pAfa (ROH).156 A theoretical Eigen curve (solid line) was fitted to the data points (circles). The dotted lines are extrapolated from the portions of the Eigen curve where a = 0 and a = —1. These dotted lines cross at pKa (ROH) = pKa [(MeO C ].155...

Most enzyme kinetic studies assume that proton transfer is fast compared with catalysis, but this is also not necessarily so. It has long been known that bimolecular rate constants for proton transfer between electronegative atoms follow an Eigen curve , with the rate in the thermodynamically favourable direction being dilfusion controlled (10 ° M s at ambient temperature) and in the thermodynamically unfavourable direction being s ... 

In such a mechanism, what is the effect of increasing the nucleophilicity of the nucleophile and the basicity of the catalyst For a constant nucleophile changing the basicity of the catalyst would affect the rate as shown in the Eigen curve of Fig. 2a. When proton transfer from to B is thermodynamically favourable the rate of proton transfer is diffusion controlled and hence independent of the basicity of the catalyst B. When it is thermodynamically unfavourable the rate decreases proportionally to the decreased basicity of the catalyst, and is given by Kk K] /, where ATj and AT are the acid dissociation constants of T- and BH", respectively. 

Fig. 8.3 The proliferation curves of RNA strands (the Q beta system) for decreasing concentrations of added matrix molecules. If the number of matrix molecules is larger than that of the enzymes, a linear proliferation is observed (first curve). This slows down at high concentrations, due to product inhibition. RNA proliferation is exponential if the amount of enzyme is larger than that of the matrix. If no matrix is added, the system goes through an incubation phase and then forms an RNA sequence which is related to certain Q beta fragments (Eigen et al., 1982)...

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Separation of the subspace. Now let A be an isolated eigenvalue of H0 with finite multiplicity m, and let En be the projection on the associated eigen-space. Then we can draw a curve F in the complex plane enclosing A0 in its interior but containing inside or on it no other point of the spectrum of... 

Complex eigen-frequencies of overstable convective modes coupled with envelope g modes as a function of the ratio q (thick curves). heal frequencies of high order envelope g modes are also given (thin curves). Resonance couplings find themselves in the wavy features in the imaginary part of the frequency. 

Eigen (1964) found that a plot of ApR against the rate constant for proton transfer between acetylacetone and a series of bases gave a curved plot. It should be noted, however, that Eigen s explanation for curvature is quite different from the one based on Marcus theory and the reactivity-selectivity principle. The curvature discussed by Eigen is attributed to a change from a rate-determining proton transfer to a diffusion controlled reaction which is independent of the catalyst p. 

However, the experimental studies relate to spatial growth of disturbances as the flow system is always excited by fixed frequency sources. Hence a spatial theory is preferred to study the stability of non-isothermal flows. Despite the distinction between temporal and spatial methods, the neutral curve, however, is identical. Iyer Kelly (1974) reported results using linear spatial theory under parallel flow approximation for free-convection flow past heated, inclined plates. Tumin (2003) also reports the spatial stability of natural convection flow on inclined plates providing the eigen spectrum. 

Fig. 4. Vibrational frequencies of a diatomic lattice. CuBr is taken as an example. Shown are experimental results (circles) due to neutron diffraction by Prevot et al.12s) and theoretical curves from calculations of Vardeny et al. 1S7). Eigen-frequencies along three principal directions in reciprocal-space k are shown. Points of high symmetry appear above the figure...

Provided also that the concentration changes are small compared with the equilibrium concentrations the differential equations governing the approach to the new equilibrium can always be linearised such that the kinetic progress curves are exponential or the sums of exponentials. In principle, the number of relaxations which may be observed, equals the number of independent equilibria involved in the mechanism of the reaction. Each relaxation is characterised by a relaxation time (t) with (1/t) being equivalent to /c bs (see above). The theoretical basis of chemical relaxation has been extensively discussed by Eigen and de Maeyer [18] and the temperature jump method by Brunori [19]. In this section we will briefly illustrate some applications of temperature-jump methods to proteins and enzymes. 

FIGURE 3.8 The quantum harmonic oscillator eigen-function probabilities (density) representation (thick continuous curves) for ground state ( = 0), and few excited vibronic states ( = 2, 5, and 10) for the working case of HI molecule (respecting the coordinated centered on its mass center) the classical potential is as well illustrated (by the dashed curve in each instant) for facihtating the correspondence principle discussion. 

Figure 8. Two solutions for the ordinary kinetic equations of the Eigen mechanism [3], eq 1 (dashed curves), as compared with the exact numerical solution for the Smoluchowski equation (full curves). Both models have the same /Cj and /Cr, but different values for the complex separation rate constant in the kinetic scheme were employed (a) giving the same area or (b) the same initial transient behavior as compared with the exact solution for the R OH decay [10b].

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