Eigen and Schuster

To resolve this problem of inevitable loss of catalytic activities through replication errors, Eigen and Schuster proposed the hypercycle [7], where replicating chemicals catalyze each other and form a cycle A catalyzes the synthesis of B, B catalyzes the synthesis of C, C catalyzes the synthesis of A. In this case, each chemical mutually amplifies the synthesis of the corresponding chemical species in this cycle. There occur a variety of mutations to each species, but this mutant is not generally catalyzed in some other species in the cycle. Then, such a mutant is not catalyzed by C. This is also... 

This section proposes first steps towards the evaluation and description of these multiphase systems and should help to stimulate a better understanding of them, which does not seem possible without the use of nonequilibrium thermodynamics. Further, it would be interesting, and I think it should be possible, to use the concepts that Eigen and Schuster [87] worked out for the complex problem of evolution —in combination with the ideas proposed by Ebeling [77c]—to come to a more basic description and a deeper understanding of multiphase systems. 

Selection can be studied appropriately and in great detail under conditions which allow a straightforward mathematical analysis (Eigen and Schuster, 1979 Schuster et al., 1980). A convenient system, more elaborate than serial transfer experiments or stirred flow reactors, consists of an evolution reactor (Fig. 12) which allows to control the concentrations of all important reactants or products. In the simplest case we keep these concentrations constant as well as the sum of all concentrations of polynucleotides. Then, the relative concentrations of autocatalysts are the only remaining variables. 

For the sake of simplicity we omitted terms which result from mutations. The differential equation obtained by insertion of (16) into (9) has been studied in great detail (Hofbauer et al., 1980 Schuster et al., 1980). Co-operation between selfreplicating elements is observed in systems in which these elements are connected by a positive feedback loop of catalytic actions. In other words we require a closed loop of catalytic enhancement in order to stabilize the system against competition. Such a closed loop has been called an elementary hypercycle (Eigen and Schuster, 1979). In the context with the rate constants in equation (16) we find that some off-diagonal elements of the matrix of catalytic coefficients K = k, have to dominate all other elements including the di-... 

The detailed mathematical analysis of equation (18) has been given previously (Schuster et al., 1978 Eigen and Schuster, 1979 Schuster et al., 1979). A general proof was presented that the elements of equation (18) co-operate no selection occurs. The dynamics of higher dimensional elementary hypercycles (n 5) is of a certain interest. The individual concentrations oscillate in regular manner, contro-led by a stable limit cycle. 

We have modified this model slightly (Eigen and Schuster, 1979). Since an extensive discussion can be found there, we dispense here with all details. The essential features of our model are several predictions which are meaningful in the context of prebiotic chemistry and thus increase the plausibility of this approach ... 

Computer modeling of hypercycles (Kiippers, 1975 Eigen and Schuster, 1977, 1978) demonstrates the development of oscillatory behavior around a steady state which, through feedback and coupling of the members of the hypercycle, minimizes the effect of mistakes in reproduction and thus allows for selection and optimization of the dynamic structures. So viewed, hypercycles represent the minimum structural organization for a system to accumulate, maintain, and process the information in the genome. 

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