Error threshold stochastic

The quasispecies model defines an optimal mutation rate for evolving populations (Eigen et al., 1988). At the critical mutation rate pmml (referred to as the error threshold), the distribution becomes too broad for selection to withstand the dispersion and it wanders stochastically on the fitness landscape. The optimal mutation rate for evolvability should be as close to pm Crit as possible without exceeding it. Indeed, it was found that viral mutation rates are very close to pm m,. By assuming that the mutation probability is the same at each residue, the error threshold in terms of mutation rate pm ai, was derived as... 

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. 

Figure 30. Error threshold as function of population size. Stochastic replication-mutation dynamics in ensemble of polynucleotide sequences with chain length v = 20 simulated by Gillespie s algorithm [95]. Critical single-digit accuracy of replication (q in) at which ordered quasi-species is converted into changing population of sequences with finite lifetimes is plotted as function of 1/N, reciprocal population size (lower curve). For further details see ref. 96. Upper curve is theoretical prediction of Eqn. (V.l) based on ref. 51.

In this section we analyze experimental data and make comparisons with theory. Data were obtained for 100 CdSe-ZnS nanocrystals at room temperature.1 We first performed data analysis (similar to standard approach) based on the distribution of on and off times and found that a+= 0.735 0.167 and v = 0.770 0.106,2 for the total duration time T = T = 3600 s (bin size 10 ms, threshold was taken as 0.16 max I(t) for each trajectory). Within error of measurement, a+ a k 0.75. The value of a 0.75 implies that the simple diffusion model with a = 0.5 is not valid in this case. An important issue is whether the exponents vary from one NC to another. In Fig. 13 (top) we show the distribution of a obtained from data analysis of power spectra. The power spectmm method [26] yields a single exponent apSd for each stochastic trajectory (which is in our case a+ a apSd). Figure 13 illustrates that the spread of a in the interval 0 < a < 1 is not large. Numerical simulation of 100 trajectories switching between 1 and 0, with /+ (x) = / (x) and a = 0.8, and with the same number of bins as the experimental trajectories, was performed and the... 

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