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, 8 (12), e83438
eCollection

Critical Mutation Rate Has an Exponential Dependence on Population Size in Haploid and Diploid Populations

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Critical Mutation Rate Has an Exponential Dependence on Population Size in Haploid and Diploid Populations

Elizabeth Aston et al. PLoS One.

Abstract

Understanding the effect of population size on the key parameters of evolution is particularly important for populations nearing extinction. There are evolutionary pressures to evolve sequences that are both fit and robust. At high mutation rates, individuals with greater mutational robustness can outcompete those with higher fitness. This is survival-of-the-flattest, and has been observed in digital organisms, theoretically, in simulated RNA evolution, and in RNA viruses. We introduce an algorithmic method capable of determining the relationship between population size, the critical mutation rate at which individuals with greater robustness to mutation are favoured over individuals with greater fitness, and the error threshold. Verification for this method is provided against analytical models for the error threshold. We show that the critical mutation rate for increasing haploid population sizes can be approximated by an exponential function, with much lower mutation rates tolerated by small populations. This is in contrast to previous studies which identified that critical mutation rate was independent of population size. The algorithm is extended to diploid populations in a system modelled on the biological process of meiosis. The results confirm that the relationship remains exponential, but show that both the critical mutation rate and error threshold are lower for diploids, rather than higher as might have been expected. Analyzing the transition from critical mutation rate to error threshold provides an improved definition of critical mutation rate. Natural populations with their numbers in decline can be expected to lose genetic material in line with the exponential model, accelerating and potentially irreversibly advancing their decline, and this could potentially affect extinction, recovery and population management strategy. The effect of population size is particularly strong in small populations with 100 individuals or less; the exponential model has significant potential in aiding population management to prevent local (and global) extinction events.

Conflict of interest statement

Competing Interests: The authors have declared that no competing interests exist.

Figures

Figure 1
Figure 1. Two-peak fitness landscape.
There is one narrow peak of high fitness (peak 0), and one broader, flatter peak of lower fitness (peak 1).
Figure 2
Figure 2. Verification of the method against analytical models for the error theshold.
Nowak and Schuster present an analytical expression for the population size dependence of the error threshold (Equation 3). Ochoa et al. , include a reformulation of the Nowak and Schuster analytical expression (Equation 4), in which they make explicit the reduction in the critical mutation rate when moving from infinite populations to those of size formula image (see section 3 for the detailed derivation). The observed consistency between our results and the analytical models provides verification for our results and the algorithmic method as a whole. It should be noted that the formula image axis represents the mutation rate by which 95% of runs have lost the lower, flatter peak (peak 1).
Figure 3
Figure 3. The results of the simulation can be approximated by an exponential function.
This applies to both peak 0 (high, narrow peak) and peak 1 (lower, flatter peak). formula image (with formula image being population size). The parameters (and their standard error in brackets) obtained by curve-fitting using a least squares method were, for the high, narrow peak (peak 0): formula image = 1.221% (0.0033%), formula image = 7.001% (1.4390%), formula image = 1.440 (0.1701), formula image = 0.3250 (0.02739), and for the lower, flatter peak (peak 1): formula image = 2.184% (0.0122%), formula image = 5.438% (1.0466%), formula image = 7.721 (0.2734), formula image = 0.3978 (0.0476).
Figure 4
Figure 4. Transition from survival-of-the-fittest to survival-of-the-flattest and subsequently to the error catastrophe.
Each point represents the number of generations it took to lose the high, narrow peak (peak 0) and the number to lose the lower, flatter peak (peak 1), in a single run of the GA for population size 100, sequence length 30. Where a peak was not lost within 10,000 generations, a value of −1 was assigned for that particular run of the genetic algorithm: all points on the negative side of either axis should be taken to have a value greater than 10,000.
Figure 5
Figure 5. The relationship between population size and critical mutation rate is consistent across haploids and diploids.
Here formula image is the dominance parameter, as described in the section entitled Fitness Calculation. The simulation was run using the formula image values listed. The points show the results obtained, which can be approximated by exponential functions as shown by the lines (obtained by curve-fitting using a least squares method). The left graph shows the curve obtained for the critical mutation rate and the right graph shows the error threshold, both for a diploid population. Refer to Figure 3 for the equivalent curves for a haploid population.
Figure 6
Figure 6. Percentage of runs losing the peaks at different mutation rates and population sizes.
The results shown are for the diploid method with formula image, for peak 0 (a, left) and peak 1 (b, right). In the two lower projections the axis coming out of the page is the percentage of runs. The lower dashed line across these projections indicates, for population sizes of several hundred individuals, approximately where the percentage loss of peak 0 begins to rise steeply and that of peak 1 begins to fall steeply as mutation rate is increased: the transition from survival-of-the-fittest to survival-of-the-flattest. Likewise, the upper dashed line indicates approximately where the percentage loss of peak 0 has reached 100% and that of peak 1 has reached its minimum before rising back upward as mutation rate is increased further: the transition from survival-of-the-flattest to the error catastrophe.

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