Quasi-species models

The quasi-species model rests on essentially three prerequisites ... 

Stochastic theory must supplement deterministic theory as in the classical wild-type model. This is a formidable task for the complete quasi-species model due to the nonlinear type of relations involved (cf. section 5.). 

The results of classical neutral theory are valid only for systems of relatively low population numbers and large genomes. If the genome is large enough that even the 3v one-error mutants cannot be populated because the population number n is smaller than 3v, one may expect the results of so-called neutral theory to be representative. Otherwise, modifications due to the reproducible population of (nearly) neutral mutants, as indicated by the deterministic quasi-species model, pertain and finally destroy the basic assumption of the blind production of mutants at the periphery of the mutant spectrum. 

There are, however, more principal limitations, and it is important to stress that the quasi-species model is a particular model holding only where its prerequisites are fulfilled. The linear autocatalytic rate law is one of the prerequisities typical for the quasi-species nature. If rates become independent of the concentrations of growing substraces, coexistence may replace competition, or if, on the other hand, the autocatalytic rate law depends more... 

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. 

The quasi-species model describes processes related to the Darwinian evolution of certain self-replicating entities within the framework of physical chemistry, the origin of life, and viral evolutionary dynamics. Catalytic... 

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. 

It was shown by 1. Leuthausser that the quasi-species model corresponds to a problem in equilibrium statistical mechanics. Indeed, the matrix... 

This analogy was recently exploited by P. Tarazona" to solve the quasi-species model for several choices of the fitness landscape A(S). Since the properties which affect the reproducing efficiency of RNA molecules in experiments are rather complex, it is costumary to represent A(S) by a random function, such as a spin-glass Hamiltonian. The simplest choice is to assume that A(S) takes on independent, identically distributed, random values for each different genotype S. This corresponds to identify the fitness landscape with the Random Energy Model of spin-glasses introduced by B. Derrida. ... 

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