Replication distributed rates

Figure 6. Calculated percentages (based on distributed rate model optimal simulations) of TCE removed from the CFSTRs flushed with the non-surfactant solution, the 30 mg/L Triton X-100 solution, the 300 mg/L Triton X-100 solution, and the 3,000 mg/L Triton X-100 solution. The removal profiles shown are averages of the replicate experiments.

Due to autocatalytic reactions, there is a tendency for further increase of the molecules that are in the majority. This leads to competition for replication between molecular types of the same species. Since the total number of Y molecules is small, this competition leads to all-or-none behavior for the survival of molecules. As a result, only a single type of species Y remains, while for species X the numbers of molecules of different types are statistically distributed as guaranteed by the uniform replication error rate. 

Probabilistic methods can be applied in dose-response assessment when there is an understanding of the important parameters and their relationships, such as identification of the key determinants of human variation (e.g., metabolic polymorphisms, hormone levels, and cell replication rates), observation of the distributions of these variables, and valid models for combining these variables. With appropriate data and expert judgment, formal approaches to probabilistic risk assessment can be applied to provide insight into the overall extent and dominant sources of human variation and uncertainty. 

Figure 4. The technique of serial transfer. An RNA sample which is capable of replication in the assay is transferred into a test-tube containing stock solution. This medium contains the four nucleoside triphosphates (ATP, UTP, GTP and CTPJand a virus specific RNA polymerase, commonly QP-replicase because of the stability of this protein, in a suitable buffer solution. RNA replication starts instantaneously. After a given period of time a small sample is transferred to the next test-tube and this procedure is repeated about one hundred times. The transfer has two consequences (i) the material consumed in the replication is replaced, and (ii) the distribution of RNA variants is subjected to a constraint selecting for the fastest replicating species. Indeed, the rate of replication is increased by several orders of magnitude in serial transfer experiments starting out from natural QB RNA and leading to variants that are exclusively suited for fast replication and hence are unable to infect their natural hosts, Escherichia coli.

Figure 10. The molecular quasispecies and its support in sequence space. Due to unavoidable non-zero mutation rates, replicating populations form distributions of genotypes or polynucleotide sequences. As shown in the sketch these distributions are centered around a most frequent genotype called the master sequence. A population thus occupies a connected region in sequence space which, according to usual mathematical terminology, is called the support of the population.

Figure 11. The error threshold of replication and mutation in genotype space. Asexually reproducing populations with sufficiently accurate replication and mutation, approach stationary mutant distributions which cover some region in sequence space. The condition of stationarity leads to a (genotypic) error threshold. In order to sustain a stable population the error rate has to be below an upper limit above which the population starts to drift randomly through sequence space. In case of selective neutrality, i.e. the case of equal replication rate constants, the superiority becomes unity, Om = 1, and then stationarity is bound to zero error rate, pmax = 0. Polynucleotide replication in nature is confined also by a lower physical limit which is the maximum accuracy which can be achieved with the given molecular machinery. As shown in the illustration, the fraction of mutants increases with increasing error rate. More mutants and hence more diversity in the population imply more variability in optimization. The choice of an optimal mutation rate depends on the environment. In constant environments populations with lower mutation rates do better, and hence they will approach the lower limit. In highly variable environments those populations which approach the error threshold as closely as possible have an advantage. This is observed for example with viruses, which have to cope with an immune system or other defence mechanisms of the host.

Figure 12. The error threshold of replication and mutation in phenotype space. The genotypic error threshold approaches zero in the case of selective neutrality. Despite changing genotypes a phenotype may be conserved in evolution whenever it has higher fitness than the other phenotypes in the population. The concept of error threshold can easily be extended to competition between phenotypes. The distribution of phenotypes is stationary provided the error rate does not exceed the maximum value pmax which is a function of the mean fraction of nearest neighbors, X, and the superiority of the master phenotype, a. The illustration shows the position of the phenotypic error threshold in the X, p plane. Selective neutrality allows more errors to be tolerated and pmax increases accordingly with increasing X. If X approaches the inverse superiority, X — a-1, the tolerated error may grow to pmax = 1, and this means the phenotype will never be lost, no matter how many errors are made in replication.

Figure 9. Optimal distributions of mass-transfer rate coefficients for the labelled CFSTR experiments. The distributions shown are the average distributions of the replicate experiments.

The deterministic model with random fractional flow rates may be conceived on the basis of a deterministic transfer mechanism. In this formulation, a given replicate of the experiment is based on a particular realization of the random fractional flow rates and/or initial amounts 0. Once the realization is determined, the behavior of the system is deterministic. In principle, to obtain from the assumed distribution of 0 the distribution of common approach is to use the classical procedures for transformation of variables. When the model is expressed by a system of differential equations, the solution can be obtained through the theory of random differential equations [312-314]. 

Fig. 2.5. A quasi-species-type mutant distribution around a master sequence. The quasi-species is an ordered distribution of polynucleotide sequences (RNA or DNA) in sequence space. A fittest genotype or master sequence /m, which is commonly present at highest frequency, is surrounded in sequence space by a cloud of closely related sequences. Relatedness of sequences is expressed (in terms of error classes) by the number of mutations which are required to produce them as mutants of the master sequence. In case of point mutations the distance between sequences is the Hamming distance. In precise terms, the quasi-species is defined as the stable stationary solution of Eq. (2) [16,19, 20], In reality, such a stationary solution exists only if the error rate of replication lies below a maximal value called the error threshold. In this region, i.e. below...

Localization Threshold for Statistically Distributed Replication Rates... 

One possible option is to adopt a statistical description of the kinetic parameters and to ask how likely it is for the quasi-species to be localized about the wild type. This undertaking requires an analysis beyond the second order in perturbation theory since a distant mutant with a selective value very close to that of the wild type may jeopardize the stability of the latter in the population. We were however encouraged by the progress that had been made with a problem of similar difficulty in the very different area of electron or spin localization in disordered solids. Indeed, it turns out that an expression of the form of Eqn. (III.5) may be obtained, with an explicit expression for the superiority parameter Oq, dependent on the distribution of replication rates but not on any average involving population variables. 

If we ignore the destruction terms D, the different mutants are fully characterized through their rates for exact replication IV,. This assumption simplifies the following discussion by allowing a single distribution of value parameters f(W) to completely characterize the mutant spectrum, but this is not essential to the argument. Accordingly, we write... 

Effective Superiority of Wild Type for Various Replication Rate Distributions ... 

Here /(H represents an underlying probability distribution from which (in a possibly correlated manner) replication rates of different mutants are sampled. IV is average of this distribution and tVg the additionally specified maximum value (which truncates distribution in latter cases). This maximum value is assigned to master sequence m, so we may write iV = Wg. 

In comparison with Eqn. (III.4) the result of the statistical analysis is to provide an expression for the effective superiority parameter er ff of the wild type in terms of the distribution of replication rates of its mutants. 

Some values of the logarithmic average appearing in Eqn. (III.9) and of ln[Pg.182]

The remaining three distributions in Table I are appropriate when the wild-type replication rate is regarded as the largest value of many sequences sampled. We shall focus on this view in the following section, which allows rather general conclusions to be made with the aid of extreme-value theory. 

The evaluations of the error threshold and effective superiority discussed so far [Eqn. (III.IO)] assume specific knowledge of the wild-type net rate of exact replication It is clear, however, that the wild type is distinguished from the other mutants only insofar as it has the maximum selective value, and so it is sensible to regard this value as the random extreme of n trials from the probability distribution f W) [29]. We may then speak of the general... 

Here n is number of trials from probability distribution f W) of rates for exact replication having mean IP. is value of IV for which there is probability l/n of sampling higher replication rate. Numerical values of 1, and In are for n = 10 (see text). 

For the calculation of stationary mutant distributions we restrict attention to a uniform error rate per digit (1 — ) and assume equal degradation rate coefficients Dy = D2= =D = D. Since the addition of a constant to all diagonal elements of a matrix just shifts the spectrum of eigenvalues and has no influence on the eigenvectors, we need only consider the case D = 0 without loss of generality. Then the elements of the matrix W are determined by the replication rate coefficients (as in Section III.2) and are of the form... 

As outlined in previous sections, much depends on the properties of physically realistic sets of replication rate constants, despite the comparative insensitivity of error thresholds to details in their distributions. At present it is not feasible to measure or estimate real-valued landscapes empirically, not even in the most simple experimental systems like RNA replication in the Qj ... 

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