One-error mutants

One approach to calculating the stationary mutant distributions for longer sequences is to form classes of sequences within the quasi-species. These classes are defined by means of the Hamming distance between the master sequence and the sequence under consideration. Class 0 contains the master sequence exclusively, class 1 the v different one-error mutants, class 2 all v(v —1)/2 two-error mutants, and so on. In general we have all (JJ) fe-error mutants in class k. In order to be able to reduce the 2 -dimensional eigenvalue problem to dimension v 1, we make the assumption that all formation rate constants are equal within a given class. We write Aq for the master sequence in class 0, Ai for all one-error mutants in class 1, 4 2 for all two-error mutants in class 2, and in general A for all k error mutants in class k. 

Figure 10. Quasi-species as function of single-digit accuracy of replication (q) for chain v = 5. We plot relative stationary concentration of master sequence ( (,),fum of relative stationary concentrations of alt one-error mutants ((i), of all two-error mutants ( j), etc. Note that we have only one five-error mutant 7,5, = /s, in this particular example. We observe selection of master sequence at g = 1. Then relative concentration of master sequence decreases with decreasing q. At value q = 0.5 all sequences are present in equal concentrations. Hence, sums of concentrations of two- and three-error mutants are largest—they have statistical weight of 10—those of the one-and four-error mutants are half as large—they have statistical weight of 5—and finally master sequence 7q and its complementary sequence, the five-error mutant /ji, are present in relative concentration ofonly. At q = 0 we have selection o( master pair", which consists of/o and /31 in our example. Thus we have direct replication with errors in range 1 > g > 0.5 and complementary replication with errors in range 0 < q < 0.5. Rate constants chosen as Aq = 10[U ] and A = 1 [t ] for all mutants Ic 0. Here we denote arbitrary reciprocal time unit by [t" ]. All degradation rate constants were put equal 7>o = D, = Dj = = D31 = 0.

At large values of the accuracy of replication (g 1) we observe a quasispecies characteristic for direct replication, /n 2/ predominantly. The master sequence Iq is most frequent, followed by some one-error mutants, two-error mutants, and so on. 

The second example (Figure 15) considers two distant degenerate sequences Iq and /30, with d(0, 30) = 4. Accordingly, we observe selection in the limit q->l. The sequence with more efficient one-error mutants (/30) is selected. In the domain of complementary replication we are dealing with two... 

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. 

In order to characterize the distributions of selective values in the second and the third model, we explored the value landscape by a Monte Carlo search. We created three random samples of 38,000 different sequences each (one repeat with 76,000 sequences gave essentially the same results) with predetermined ratios of probabilities for (0/1) digits, Pi = 0.2857, p2 = 0.5, and p3 = 0.7143, which led to mutant distributions centered at the 20-, 35-, and 50-error mutants of the all-zero sequence Iq. Three different parts of the value landscapes determined by Eqs. (IV.9)-(IV.ll) were explored in that way. The results are shown in Figure 22. 

As expected, a response to the hypercycle criticisms appeared, in fact in the same issue of the Journal of Theoretical Biology (Eigen et al., 1980). According to this, the Freiburg investigations refer to one particular evolution model, in which the occurrence of mutants with different, selective values is ignored. In such realistic models, the error threshold loses its importance for the stability of the wild type. If the latter reaches a finite fitness value, it can always be the subject of selection, as no rivals are present. 

The WT lipase leads to an ee value of only 38% in favor of the (If ,45) enantiomer. The application of low-error epPCR increased the enantioselectivity slightly, but high-error rate epPCR turned out to be more successful, with several mutants showing ee values of 54-58% (45,137). The results are in line with the experience gained in the Pseudomonas aeruginosa lipase project (Section IV.A. 1). Of course, a library produced by high mutation rate can also contain hits that have only one amino acid exchange, and this was indeed observed in several cases. 

The activity of a very weak active mutant measured by steady state kinetics could result from traces of a wild-type or more active mutant in the preparation either as a contaminant or because of natural errors of misincorporation. The error rate in protein biosynthesis can be as high as one part in 100 or one part in 1000.10 The presence of a small amount of wild-type enzyme in an inactive mutant would give a low value of kcat (which is directly proportional to the concentration of wild type) but the KM value for the wild-type enzyme. Thus, the finding of a low value of kcat and the wild-type KM for a mutant is very suspicious. 

In Figure 5 two landscapes are shown that start at a binary master sequence (v = 50) and are followed up to the 12-error copies. One of the landscapes (left side) is the low-value plateau that has been considered already in previous examples, while the other resembles a mountain saddle as typical for any fractal type of hill country. The population numbers of mutants, relative to the population number of the wild type, were calculated by means of second-order perturbation theory [Eqs. (11.17) and (11.18)] for... 

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