Interconnections of neutral spaces in RNA influence evolutionary trajectories.

<p><i>A</i>) RNA neutral component for phenotype with genotypes (drawn in blue). Lines depict single mutations to itself, or to two alternative phenotypes (grey) and (red). The genotypes were ordered using the Fruchterman-Reingold algorithm <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635-Fruchterman1" target="_blank">[30]</a>. <i>B</i>) Illustration of the fitness landscape.</p

Illustration of the mean field approximation.

<p><i>A</i>) An example genotype space: Each point corresponds to a unique genotype; shape and color of the marker indicate the phenotype. Genotypes joined by edges can be interconverted by single mutations. Edges for neutral mutations share the color of the (conserved) phenotype, non-neutral mutations are shown as black dashed lines. The shading of the genotypes illustrates the number of individuals carrying the respective genotype in a hypothetical population. The mutations away from the genotypes occupied by the population determine the accessible phenotypes. <i>B</i>) Our meanfield approximation averages over the internal structure of neutral spaces. So neutral spaces are represented by the markers of their phenotypes only, with the size representing the neutral space size (ie. number of genotypes in the space). The uniform shading of the blue neutral space implies that in the meanfield approximation, the population is assumed to continually explore the neighbourhood of its entire neutral space. Mutational outcomes are thus determined from the local frequencies of phenotypes around the neutral space, as measured by the coefficients. This mean field approximation allows us to derive analytic forms that can be compared to simulations of the full GP map.</p

The arrival of the frequent.

<p>Probability that phenotype is discovered (dotted lines) or is fixed (dashed lines) as a function of mutation rate for different relative selection coefficients for . The probability that is discovered is independent of relative fitness (within statistical simulation errors). Phenotype is much more likely to fix than phenotype , even when the latter is much more fit, due to an “arrival of the frequent” phenomenon.</p

Test of the meanfield model.

<p><i>A</i>) Median discovery times for the random GP map averaged over 100 simulations with and varying mutation rates. Note that the y-axis is scaled with . In the the polymorphic limit (), <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635.e113" target="_blank">Eq. (4)</a> (dashed line) describes discovery times well for . Phenotypes with larger are part of the standing variation typically found in the first generation (yellow dash-dotted line). In the monomorphic limit (), <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635.e168" target="_blank">Eq. (7)</a> (dotted line) quantitatively describes for , whereas <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635.e113" target="_blank">Eq. (4)</a> tracks the simulation data with just one fit parameter multiplying for the intermediate regime with (solid line). For the curves follow <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635.e113" target="_blank">Eq. (4)</a>, for reasons described in the text. <i>Inset</i>: For the random GP map the local phenotype frequency correlates very well with the global frequency . <i>B</i>) Local frequency ranked for the phenotypes that link with single point mutations from the genotypes that map to this RNA structure; an example sequence from is shown in the figure. <i>Inset</i>: The local connections are roughly proportional to the global frequency , but there is significant scatter due to the internal correlations of the RNA neutral spaces. Organge points depict the phenotypes for which . Light blue points depict the phenotypes that are discovered in our simulations, and the dark blue points depict the accessible phenotypes that are not found ( itself is shown in green). <i>C</i>) Simulations of (blue dots) versus for the RNA phenotype shown in B), compared to <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0086635#pone.0086635.e113" target="_blank">Eq. (4)</a> (solid line) with a factor multiplying . Here , and the simulations were run for generations. Also shown are the purely polymorphic (dashed) and monomorphic (dotted) predictions. Dark blue dots above (dot-dashed line) depict some of the accessible phenotypes that are not found (as can be seen in see the inset of B). We estimate that about generations would be needed to find the phenotypes with the smallest .</p

Biological GP maps have much larger and fewer neutral components than their random counterparts due to neutral correlations.

<p>A) The logarithm of the largest neutral component for a given phenotype is plotted as a function of frequency for random null models (with <i>K</i> = 4, <i>L</i> = 12) and three biological GP maps, RNA12, <i>S</i>_2,8 and HP24. The vertical dotted line denotes the giant component threshold <i>δ</i> ≈ 1/36, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e020" target="_blank">Eq (5)</a>, for the schematic random model with <i>K</i> = 4, <i>L</i> = 12. The vertical dashed line denotes the single component threshold <i>λ</i> ≈ 0.37, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e022" target="_blank">Eq (6)</a>, for the schematic random model. The biological GP maps show much larger connected components below these thresholds, due to the presence of positive neutral correlations. B) The logarithm of the total number of neutral components against frequency is plotted for the same models. The theoretical thresholds <i>δ</i> and <i>λ</i> work well for random model but again the number of components in the biophysical models differ greatly from the random model expectation due to the presence of correlations. In both plots, error bars represent a single standard deviation from the 100 independent realisations of the random null model used to derive the neutral component statistics.</p

Schematic depiction of the GP map properties of redundancy, phenotype bias and neutral correlations.

<p>Phenotypes are represented by colours, genotypes as nodes and mutations as edges. A) Each colour appears multiple times with uniform redundancy. B) Some colours appear more often than others, demonstrating a phenotype bias. C) A rearrangement of the colours from the middle plot illustrates positive neutral correlations where the same colours are more likely to appear near each other than would be expected by random chance arrangement. The black box surrounding the six orange genotypes depicts a single component (a set of genotypes connected by neutral point mutations, also called a neutral network) of the orange phenotype. Such positive neutral correlations enhance the probability that such neutral networks occur.</p

Phenotype mutation probabilities scale with global frequency.

<p>We present results for the three GP maps: A) RNA20, B) <i>S</i>_3,8 and C) HP5x5. We plot the relationship between <i>ϕ</i>_<i>qp</i> (circles) and <i>f</i>_<i>q</i> for the largest non-deleterious phenotype <i>p</i> in <i>S</i>_3,8 and HP5x5, and for the second largest in RNA20 (not the largest due to computational expense). We see in each case a strong positive correlation (<i>p</i>-value ≪ 0.05 in all cases), very similar to the expectation for the null model (not shown here, but for which the correlation is exact to within statistical fluctuations, see ref. [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.ref014" target="_blank">14</a>] and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.s001" target="_blank">S1 Text</a>). Spearman rank correlation coefficients are shown in the top-left of each plot. Differences from <i>ϕ</i>_<i>qp</i> = <i>f</i>_<i>q</i> are relatively small compared to the overall range of variation, except for sets of phenotypes that are not connected at all, which typically arise due to biophysical constraints. These are shown as downward triangles along the lower horizontal dotted line which represents <i>ϕ</i>_<i>qp</i> = 0. For each plot, the upward triangle indicates <i>ϕ</i>_<i>pp</i> = <i>ρ</i>_<i>p</i>, the phenotype robustness, which is always over-represented (<i>ρ</i>_<i>p</i> ≫ <i>f</i>_<i>p</i>) due to neutral correlations.</p

Non-neutral local mutational neighbourhood correlations result in mutational neighbourhoods of neutral neighbours being more similar than randomly selected neutral pairs.

<p>We present results for the three GP maps: A) RNA20, B) <i>S</i>_3,8 and C) HP5x5. Using the ratio of Bhattacharyya coefficients defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e041" target="_blank">Eq (10)</a>, we show that neutral neighbours (<i>g</i> and <i>h</i>) have a closer phenotype probability distribution than a randomly chosen neutral pair (<i>g</i> and <i>g</i>₂). This is seen through the ratio being skewed with a mean (coloured vertical dashed lines) larger than unity (black vertical dashed lines). The standard error on this mean is negligible compared to the distance of the mean from one.</p

Illustration of further non-neutral correlations.

<p>A) On the right, the orange phenotype is over-represented relative to the null model: The red genotype in the centre has more orange neighbours than would be expected by the global frequency of orange. B) The phenotypes that appear in the mutational neighbourhood of two neutral neighbours are expected to be more similar (right) than two non-neighbouring genotypes of the same phenotype (left).</p

GP map nomenclature.

<p>GP map nomenclature.</p