Selection quasi-species model

Several attempts to describe replication-mutation networks by stochastic techniques were made in the past. We cannot discuss them in detail here, but we shall brieffy review some general ideas that are relevant for the quasispecies model. The approach that is related closest to our model has been mentioned already [51] the evolutionary process is viewed as a sequence of stepwise increases in the populations mean fitness. Fairly long, quasi-stationary phases are interrupted by short periods of active selection during which the mean fitness increases. The approach towards optimal adaptation to the environment is resolved in a manner that is hierarchical in time. Evolution taking place on the slow time scale represents optimization in the whole of the sequence space. It is broken up into short periods of time within which the quasi-species model applies only locally. During a single evolutionary step only a small part of sequence space is explored by the population. There, the actual distributions of sequences resemble local quasispecies confined to well-defined regions. Error thresholds can be defined locally as well. 

In the case of selective neutrality—this means that all variants have the same selective values—evolution can be modeled successfully by diffusion models. This approach is based on the analysis of partial differential equations that describe free diffusion in a continuous model of the sequence space. The results obtained thereby and their consequences for molecular evolution were recently reviewed by Kimura [2]. Differences in selective values were found to be prohibitive, at least until now, for an exact solution of the diffusion approach. Needless to say, no exact results are available for value landscapes as complicated as those discussed in Section IV.3. Approximations are available for special cases only. In particular, the assumption of rare mutations has to be made almost in every case, and this contradicts the strategy basic to the quasi-species model. 

We have recently developed a quasi-species approach for analyzing mutation and selection in catalytic reactions of varying order. We discussed how the error catastrophe that reflects the transition from localized to delocalized quasi-species population is affected by catalytic replication of different reaction orders. Specifically, the second-order mechanisms lead to a discontinuity in the mean fimess of the popnlation at the error threshold. This is in contrast to the behavior of the first-order, antocatalytic replication mechanism, considered in the standard quasi-species model. This suggested that quasi-species models with higher order of replication mechanisms produce discontinuities in the mean fitness and, hence, in the viable population fraction, at the error threshold, while lower-order replication mechanisms yield a continuous mean fitness fnnction. 

The typical outcome of quasi-species behavior is natural selection in the Darwinian sense. We believe that it should be possible to adapt the quasispecies model to any situation where natural selection in the Darwinian sense plays a major role. Hence the model may be generalized so as to include all kinds of horizontal gene transfer typical for recombination. The model as presented is essentially a deterministic model that holds only for a sufficiently large population size. This limitation and a possible way to overcome it will be the subject of the final section. 

It is, indeed, just this native complexity which for our today s chemistry provides provocation and stimulation, intimidation and temptation, love and hate and fate together. The present artificial systems still remain utterly outclassed by even the most primitive life forms such as RNA-viruses. The possibilities of describing natural selection behavior according to quasi-species distributions in the extreme multidimensionalities of sequence spaces [3d, e] are, for artificial systems, at best a very distant utopia. With all its early primi-tivity, but also with its promising inherent potential of minimal models of self-replication [6,7,9] and - just to follow - minima vita models [8], chemistry, nevertheless, is gaining new qualities by retracing transitions to life. 

In order to compare the finite element model with the one-dimensional Chiao model, an extremely simple mesh of only five elements extending in a column from the laminate centerline to the outer surface was used to model the gradients in the laminate through-thickness direction. Figure 7 shows the reaction history (fractional concentration of reactive species, C, versus time) obtained from this run, selected at the location nearest the heated surface. This figure also shows the comparison with the quasi-isothermal and Chiao models. 

Thus, the mechanism of catalytic processes near and far from the equilibrium of the reaction can differ. In general, linear models are valid only within a narrow range of (boundary) conditions near equilibrium. The rate constants, as functions of the concentration of the reactants and temperature, found near the equilibrium may be unsuitable for the description of the reaction far from equilibrium. The coverage of adsorbed species substantially affects the properties of a catalytic surface. The multiplicity of steady states, their stability, the ordering of adsorbed species, and catalyst surface reconstruction under the influence of adsorbed species also depend on the surface coverage. Non-linear phenomena at the atomic-molecular level strongly affect the rate and selectivity of a heterogeneous catalytic reaction. For the two-step sequence (eq.7.87) when step 1 is considered to be reversible and step 2 is in quasi-equilibria, it can be demonstrated for ideal surfaces that... 

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