Eigen model

Similarly, the proton transfer on the hydroxo oxo complex is illustrated by Eqs. (13)-(15), based on the Eigen model (Scheme 4), and can again be due to protolysis, hydrolysis, or direct proton exchange... 

The exchange mechanism of proton transfer in these systems is explained in terms of the Eigen model (Scheme 4) as discussed above by either hydrolysis (Mo(IV) and W(IV)) or protolysis (Tc(V) and Re(V)) pathways, coupled with direct proton transfer in the intermediate pH... 

Wojcik M, Tachiya M (2009) Accuracies of the empirical theories of the escape probability based on Eigen model and Braun model compared with the exact extension of Onsager theory. J Chem Phys 130 104107... 

The first step is diffusion controlled while the second represents the fast formation of the outer sphere complex. The final step involves the conversion of the outer to the inner sphere complex. This is the rate determining step and is dependent on the equilibrium concentration of the outer sphere complex. Consequently, calculations of rate constants by the Eigen model involves estimation of the formation constant of the outer sphere species. 

The Eigen model requires two different hydrogen-bonded intermediates eorresponding to the two alkene produets that are formed. Eaeh intermediate leads to a different transition state, structures 2 and 3 drawn in Scheme 2. The former, 2, leads ultimately to 1-methylcyclopentene as the final product. This transition state has no overall molecular symmetry, but retains the methyl rotor as well as free rotation about the carbon-nitrogen single bond for a net symmetry number a = 9. Because transition state 2 is chiral, the reaction path degeneracy is equal to (18/ ) = 4. 

It should be noted that this relation is formally identical to the spectral resolution of the 2 - RDM. That is, in this model, all happens as if the eigen-vectors of the 2-SRH were natural geminals. 

This dilemma could be overcome by the hypercycle model hypercycles are in fact not theoretical concepts, but can be observed (in a simple form) in today s organisms, where an RNA virus transfers the information for an enzyme in the host cell, which is able to carry out the preferred synthesis of new virus RNA. This RNA synthesis is supported by host factors, and an RNA minus-strand is formed. The following RNA replication affords a plus-strand. The process corresponds to a double feedback loop and involves the enzyme coded by the RNA matrix and the information present in the matrix in the form of a nucleotide sequence. Both factors contribute to the replication of the matrix, so that there is second-order autocatalysis (Eigen et al., 1982). 

The hypercycle models developed later by Eigen were much more complex. Since both protein enzymes and nucleic acids contribute to hypercycles, the latter could only have come into operation at a later stage of the (hypothetical) RNA world. It seems possible that the protein enzymes on the primeval Earth could have been replaced by ribozymes. 

As expected, a response to the hypercycle criticisms appeared, in fact in the same issue of the Journal of Theoretical Biology (Eigen et al., 1980). According to this, the Freiburg investigations refer to one particular evolution model, in which the occurrence of mutants with different, selective values is ignored. In such realistic models, the error threshold loses its importance for the stability of the wild type. If the latter reaches a finite fitness value, it can always be the subject of selection, as no rivals are present. 

Hans Kuhn, who described his own models in an article on the Self-organisation of Molecular Systems and the Evolution of the Genetic Apparatus (Kuhn, 1972), also worked in the Max Planck Institute for Biophysical Chemistry in Gottingen. Eigen... 

Genuine self-organisation, i.e., self-organisation as a property of the system. Here, a system with a high degree of complexity organises itself under certain conditions. A typical example is Eigen s hypercycle model (see Sect. 8.3). 

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.

Fig. 3. Classification of human prion diseases. Sporadic the transformation from PrPc (circle) to PrPSc (square) occurs without apparent cause. Familial a point mutation ( ) is thought to facilitate the transformation. Infectious the transformation arises via PrPSc which acts as a template. The kinetic equations are defined by Eigen (1996). The infectious form includes kuru, iatrogenic CJD (iCJD), variant CJD (vCJD first reported in 1996), bovine spongiform encephalopathy (BSE first reported in 1985), and scrapie. In the nucleation-dependent model, monomeric PrPc and PrPSc are in chemical equilibrium.

Figure 8 also shows values of f that have been calculated by two other methods. In the first, Jaswal (19) has used lattice-vibration eigen-frequencies and eigenvectors which have been calculated in the first Brillouin zone using the deformation-dipole model for the lattice. This... 

A second important application of CMD has been to study the dynamics of the hydrated proton. This study involved extensive CMD simulations to determine the proton transport rate in on our Multi-State Empirical Valence Bond (MS-EVB) model for the hydrated proton. = Shown in Fig. 4 are results for the population correlation function, (n(t)n(O)), for the Eigen cation, HsO, in liquid water. Also shown is the correlation function for D3O+ in heavy water. It should be noted that the population correlation function is expected to decay exponentially at long times, the rate of which reflects the excess proton transport rate. The straight line fits (dotted lines) to the semi-log plots of the correlation functions give this rate. For the normal water case, the CMD simulation using the MS-EVB model yields excellent agreement with the experimental proton hopping... 

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