## The Realistic Error Threshold of the Neurospora VS Ribozyme

As previously pointed out, the formal description by Eigen (1971) of the mutation-selection dynamics of a population of biological sequences led to the realization of one of the greatest paradox of prebiotic evolution: the error rate poses a limit to the length of the information that can be selectively maintained within the system. A stable cloud of mutants (that is, a quasi-species) can form around a master sequence as long as the maximum chain length (N) is below the critical error rate per site per replication («*) as determined by the following expression:

ln s

where s is the selective superiority of the master. Very roughly then, assuming as customary that lns z 1, Eq. (2) states that the maximum selectively maintainable amount of information (N) is about the inverse of the mutation rate per base per replication. Experimental evidence suggests that without the aid of peptide enzymes the upper bound of copying fidelity per nucleotide per replication could not be higher than 0.99 (Johnston et al., 2001), and is quite likely significantly lower than this figure. Accordingly, the maximum N would be lower than 100 nu-cleotides.

However, Eigen's model is based on the assumption that the whole genotype -the master sequence - has to be maintained for functionality. This assumption might be justifiable in a DNA-protein world, but in the RNA world the enzymatic activity of a molecule was mainly based on its three-dimensional structure rather than on the exact order of its building blocks (except for a few critical sites). It is a well-established fact that the number of possible RNA secondary structures is considerably less than the number of possible sequences (Schuster et al., 1994; Stadler and Haslinger, 1999), and it is also possible that two molecules with completely different sequences share the same secondary structure (Huynen, 1996; Huynen et al., 1996; van Nimwegen et al., 1999). In other words, when RNA structure is considered it might be feasible to maintain ribozyme functionality (phenotype) at mutational rates that would not allow the preservation of the master sequence.

In order to determine the "realistic" error threshold for the VS ribozyme we have explored the dynamics of a population of RNA molecules with N = 144 at various mutation rates per nucleotide per replication (m). At the replication step a sequence is picked at random according to its fitness. Thus, the probability pj of choosing sequence i with enzymatic activity A; is:

where SAj is the sum of activities for all sequences in the population. The next step is to copy the chosen sequence with error rate m (only point mutations were considered). Because a quarter of the times (assuming equal probability for each nucleotide) no effective change will occur in the position even though there is a mutational event, the effective mutation rate is m* = 0.75 m.

The new sequence then replaces a randomly chosen sequence, which allows keeping a constant population of molecules and is also equivalent to the assumption that the rate of degradation is the same for all molecules and independent of enzymatic activity (Bonhoeffer and Stadler, 1993). As a final point we should emphasize here that the occurrence of thresholds for error propagation was originally derived as a deterministic kinetic theory that is only valid in the limited case of an infinite number of molecules. Alves and Fontanari (1998) have extended it to finite populations and found that the critical error rate per site per

Effective Error Rate per Digit (n ) Fig. 3.5 Error threshold for the VS ribozyme. was 5000 molecules. A generation is defined as Maximum (downward triangles), mean a number of replication steps equal to the

(square) and minimal (upward triangles) ob- population size. For each error rate 10 repli-served times to extinction are plotted as a cates were obtained. A line is fitted to the last function of the error rate. The population size four data points (solid line; R = -0.904).

replication decreases linearly with 1/N. In our present case we extrapolated to an infinite population size by recording the time to extinction (that is, the number of generations when no functional ribozymes remained in the population) at various error rates and fitting a straight line to those last few points which still showed a downward trend. The error threshold is then the intersection of the line with the error rate axis.

As shown in Fig. 3.5 the "realistic" error threshold for the VS ribozyme was estimated to be m* = 0.052 (this figure refers to the "effective" mutation rate; see above). To compare this figure with the error threshold that would be obtained without considering secondary structure we have considered two different fitness landscapes: Mount Fuji and single-peaked landscapes.

In Mount Fuji landscape we have assigned an activity value to all possible nucleotides at a given position, with the wild-type nucleotides at each position having a value A; =1 for the enzymatic activity. For those positions where experimental data are available we considered that value, otherwise we either used the value for the same position derived for another nucleotide (if more than one such value were available, we used the lowest of the two) or used a predefined value. In the last case we considered two scenarios: those mutants have uniformly either A; = 0.8 or A = 0.2. Therefore, in the first case we assumed that those positions are not functionally important, whereas in the second case they are. In no case can the enzymatic activity of the molecule be higher than 1. The fitness value of a sequence was then calculated as the product of the individual activities, and the resulting "Mount Fuji error thresholds" were m* = 0.032 (A; = 0.2) and m* = 0.025 (A; = 0.8), which are substantially lower than the "realistic" error threshold from the functional landscape. Mount Fuji landscape retains some characteristics of the functional landscape - the fitness effect of the single-stranded regions - but it is no longer possible to have compensatory mutations by changing one base pair into another. In fact, every mutation affecting a helical region counts as a mispair, which causes the error catastrophe to occur at lower value.

For the single-peaked landscape we used the same assumption as in Eigen's model. Thus, the wild-type sequence has A; = 1 at each position and all other sequences have A = 0.217, which is the average activity of all experimentally tested one-point mutants. The "single-peaked error threshold" was found to be m* = 0.023, lower than in either ofthe previous cases. The reason is that this landscape retains no information about the structure, and no neutral mutations are possible.

By using Eq. (2) above and assuming lns = 1, the maximal error rate for the VS ribozyme would be m* z 0.007. This figure is nearly an order of magnitude lower than the one we got by using a realistic fitness landscape. Furthermore, according to the Eigen's model the error rate of 0.052 would permit a ribozyme of maximum length 20 to be maintained. In summary, it is quite obvious that the inclusion of structural information, as well as information derived from experimental data, crucially alleviates the burden imposed by Eigen's (1971). This is the first report ofa realistic error threshold calculated for an existing ribozyme. To the best ofour knowledge there is only one other indication of an error threshold calculated using a structural landscape. Huynen and co-workers (1996) used the secondary structure of the phenylalanine tRNA (N = 76) as their object of investigation, and assumed that the fitness decrease of a mutant is proportional to the difference between the target structure and the structure of the mutant sequence. They reported that the error threshold for the tRNAPhe is 0.0031. This seems to be too low, as even the Eigen's model predicts a higher error threshold (m* z 0.01). This low value might be an artefact of the folding algorithm, which for tRNAs often predicts a minimum free energy structure unlike the known cloverleaf structures.

There is as yet no reported replicase ribozyme. The most promising result thus far is a ribozyme that can extend a sequence by 14 nucleotides according to a template (Johnston et al., 2001). This ribozyme works with a 0.967 copying fidelity. If a functional replicase ribozyme had the same fidelity, then it could replicate the VS ribozyme without the threat of the error catastrophe.

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