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

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

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

Greater mutational robustness indicates the presence of neutral correlations.

<p>A) The phenotype robustness <i>ρ</i>_<i>p</i> is plotted as a function of frequency <i>f</i>_<i>p</i> for all phenotypes in the RNA secondary structure models: RNA12, RNA 15, RNA20, the Polyomino models for protein quaternary structure: <i>S</i>_2,8 <i>S</i>_3,8 and the HP protein folding model HP24. Each model has an associated random model with the same frequencies, but we only show one example, with <i>K</i> = 4 and <i>L</i> = 12 and a set of phenotypes chosen with a broad range of frequencies to best illustrate the relationship (red points). All random models closely follow the expected theoretical curve <i>ρ</i>_<i>p</i> = <i>f</i>_<i>p</i> (grey line). The biophysical models exhibit a much larger robustness than the random models, which indicates the presence of positive neutral correlations. The red dotted line is <i>δ</i> (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e020" target="_blank">Eq (5)</a>) for <i>K</i> = 4, <i>L</i> = 12. If (<i>ρ</i> > <i>δ</i>) then large neutral networks are expected, which is much more likely for the biophysical models than for the random model. B) The average <i>n</i>-robustness 〈<i>ρ</i>^(<i>n</i>)〉, defined in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.e012" target="_blank">Eq 3</a>, for each of the three biological GP maps, along with the expected values 〈<i>ρ</i>^(<i>n</i>)〉 = 1/<i>N</i>_<i>P</i> for the associated random null models (flat coloured horizontal lines) is plotted against <i>n</i>. Across all three GP maps, we see a typical decay in robustness towards the random null model expectation with increasing mutational distance. From this decay a neutral correlation length can be defined which is shorter for the HP model than for the other two models. Error bars for HP24 are the standard error on the mean of the average <i>n</i>-robustness.</p

Enhanced robustness in Maynard Smith’s 4-letter word game.

<p>The single mutation path WORD → WORE → GORE → GONE → GENE is shown in red. All valid words within a one letter mutation of “WORD” and “GENE” are also depicted. According to the Merriam-Webster Official Scrabble Players Dictionary 2014, only 4,175 of the 456,976 possible 4-letter words are valid English words (at least for Scrabble). Since each word has 100 neighbours, for a random model, the expected number of valid words within a one letter mutation is < 1. Nevertheless, due to positive neutral correlations, the probability that a <i>valid</i> word has another valid word as a 1-mutant neighbour is more than ten times greater, as illustrated above for “WORD” and “GENE”. As pointed out by Maynard Smith, in a biological system, such correlations (in his case between “meaningful sequences”) can facilitate evolutionary dynamics [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.ref001" target="_blank">1</a>].</p

Non-neutral local over-representation correlations result in phenotypes being more likely to be found multiple times around genotypes.

<p>We present results for the three GP maps: A) RNA20, B) <i>S</i>_3,8 and C) HP5x5. We pick the same frequent phenotypes <i>q</i> in each of our biological GP maps as used in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004773#pcbi.1004773.g004" target="_blank">Fig 4</a>, and consider the prevalence of <i>q</i> around genotype <i>g</i> with phenotype <i>p</i>, <i><b>given that <i>q</i> occurs at least once</b></i> in the 1-mutation neighbourhood of <i>g</i>. The average of across the <i>n</i> = 10 most frequent phenotypes <i>p</i> in the neighbourhood of <i>q</i> (with <i>p</i> ≠ <i>q</i> and <i>p</i> ≠ del), is compared to the respective averages for random null expectations and defined in the text. The mean of each distribution is plotted as a dotted line in each case. Contiguous sections with a probability greater than 10⁻⁵ are joined with lines in order to guide the eye. The mean value of <i>m</i> for each of the biological GP maps and the two random controls are shown as respective dotted lines with the same colours. Compared to the two null expectations of occurrence, <i>q</i> is over-represented locally as demonstrated by the shift of the means to the right.</p

Non-neutral deleterious phenotype correlations: The deleterious phenotype is under-represented in the neighbourhood of folding or self-assembling phenotypes.

<p>We present results for the three GP maps: A) RNA20, B) <i>S</i>_3,8 and C) HP5x5. Histograms of the ratio of the phenotype mutation probability (<i>ϕ</i>_del,<i>p</i>) divided by the null model expectation of the global frequency (<i>f</i>_del) for the deleterious phenotype (non-folding for RNA/HP, non-assembling for Polyominoes). The distribution is clearly skewed to values < 1, as highlighted by the dashed vertical coloured lines representing the mean in each case.</p