Revisiting Robustness and Evolvability: Evolution in Weighted Genotype Spaces

<div><p>Robustness and evolvability are highly intertwined properties of biological systems. The relationship between these properties determines how biological systems are able to withstand mutations and show variation in response to them. Computational studies have explored the relationship between these two properties using neutral networks of RNA sequences (genotype) and their secondary structures (phenotype) as a model system. However, these studies have assumed every mutation to a sequence to be equally likely; the differences in the likelihood of the occurrence of various mutations, and the consequence of probabilistic nature of the mutations in such a system have previously been ignored. Associating probabilities to mutations essentially results in the <i>weighting</i> of genotype space. We here perform a comparative analysis of weighted and unweighted neutral networks of RNA sequences, and subsequently explore the relationship between robustness and evolvability. We show that assuming an equal likelihood for all mutations (as in an unweighted network), underestimates robustness and overestimates evolvability of a system. In spite of discarding this assumption, we observe that a negative correlation between sequence (genotype) robustness and sequence evolvability persists, and also that structure (phenotype) robustness promotes structure evolvability, as observed in earlier studies using unweighted networks. We also study the effects of base composition bias on robustness and evolvability. Particularly, we explore the association between robustness and evolvability in a sequence space that is AU-rich – sequences with an AU content of 80% or higher, compared to a normal (unbiased) sequence space. We find that evolvability of both sequences and structures in an AU-rich space is lesser compared to the normal space, and robustness higher. We also observe that AU-rich populations evolving on neutral networks of phenotypes, can access less phenotypic variation compared to normal populations evolving on neutral networks.</p></div

Minimum Free Energies of structures formed by RNA sequences of varied GC content.

<p>Data shown are for one million sequences of varied GC-content in comparison to normal unbiased sequences. The MFEs were calculated using viennaRNAFold routine of Vienna RNA package <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0112792#pone.0112792-Gruber1" target="_blank">[10]</a>. We observe that the Minimum Free Energies of sequences become less negative as their GC content decreases, reflecting a decrease in thermal stability.</p

Mean genotype robustness and evolvability of 10⁶ AU-rich sequences and 10⁶ normal sequences.

<p>The genotype space was weighted using κ = 2.5. We observed that the mean genotype robustness is higher for AU-rich sequences, while mean genotype evolvability is lesser, in comparison to normal space. In a pair-wise Wilcoxon signed rank test between the two datasets, the <i>p</i>-value was less than 10⁻⁸. Spearman rank correlation values mentioned are between genotype robustness and genotype evolvability.</p><p>Mean genotype robustness and evolvability of 10⁶ AU-rich sequences and 10⁶ normal sequences.</p

Population evolution on a highly robust phenotype's neutral network with AU-rich and normal starting populations.

<p>We observe that AU-rich starting population can access lesser variation: lesser number of cumulative novel phenotypes and unique phenotypes in 1-neighbourhood during 10 generations of mutations. Data shown are for a population of 100 identical sequences for each of 20 inversely folded normal sequences and 20 inversely folded AU-rich sequences for the phenotype. Mutations occur at the rate of µ = 1 (one nucleotide per sequence per generation), thereby leading to a rate of evolution of Nµ = 100.</p

Comparison of genotype robustness and evolvability of normal and AU-rich sequences.

<p>(a) Histogram of genotype evolvability of one million normal sequences and one million AU-rich sequences. We observed that a majority of AU-rich sequences have lesser genotype evolvability in comparison to normal sequences. b) Histogram of genotype robustness of one million normal sequences and one million AU-rich sequences. We observed that a majority of AU-rich sequences have higher genotype robustness in comparison to normal sequences. The genotype space is weighted using κ = 2.5.</p

Comparison of 1-neighbourhoods of sequences of varied GC-content.

<p>Data shown are the histogram of fractions of folded sequences in 1-neighbourhood for each of 1000 randomly chosen normal, AU-rich (GC content ≤20%) and GC-rich (GC content ≥80%) sequences. We observe that a majority of AU-rich sequences have a lower fraction of folded neighbours, compared to normal sequences and GC-rich sequences, in their 1-neighbourhood. The sequences were folded using viennaRNAFold routine of Vienna RNA package <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0112792#pone.0112792-Gruber1" target="_blank">[10]</a>.</p

Populations evolving at the rate Nµ = 100.

<p>(a), (b) The more robust phenotype has higher structural diversity in its 1-neighbourhood. We observe that the number of unique phenotypes encountered in the 1-neighbourhood by both the phenotypes is higher for κ = 0.5 than κ = 2.5. (c), (d) More robust phenotypes evolving on larger neutral networks have greater access to variation. We observe that the cumulative novel phenotypes encountered by both phenotypes are higher for κ = 0.5 than κ = 2.5. Data shown are for a population of 100 identical sequences for each of 40 inversely folded sequences from the two phenotypes. Mutations occur at the rate of µ = 1 (one nucleotide per sequence per generation). The neutral networks of these phenotypes are weighted using κ = 0.5 and 2.5. Refer <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0112792#pone.0112792.s003" target="_blank">Figure S3</a> for data corresponding to κ = 10.</p

Comparison of phenotype robustness and evolvability of normal and AU-rich sequences.

<p>(a) Histogram of phenotype evolvability of AU-rich and normal neutral networks (NNs) of 2.5×10⁴ structures. We observe that AU-rich neutral networks, in general, have lesser phenotype evolvability in comparison to normal neutral networks. (b) Histogram of phenotype robustness of AU-rich and normal neutral networks (NNs) of 2.5×10⁴ structures. We observe that AU-rich neutral networks, in general, have higher phenotype robustness in comparison to normal neutral networks. These neutral networks are weighted using κ = 2.5.</p

Mean phenotype robustness and evolvability of AU-rich and normal neutral networks of 2.5×10⁴ structures.

<p>These networks were weighted using κ = 2.5. We observed that the mean phenotype robustness is higher for AU-rich neutral networks, while mean phenotype evolvability is lesser, in comparison to normal neutral networks. In a pair-wise Wilcoxon signed rank test between the two datasets, the <i>p</i>-value was less than 10⁻⁸. Spearman rank correlation values mentioned are between genotype robustness and genotype evolvability.</p><p>Mean phenotype robustness and evolvability of AU-rich and normal neutral networks of 2.5×10⁴ structures.</p

Mean phenotype robustness and evolvability of 2.5×10⁴ structures (phenotypes).

<p>The neutral networks of these phenotypes were weighted using three different values of κ. We observed that with increasing κ, the mean phenotype robustness increases while mean phenotype evolvability decreases (in a pair-wise Wilcoxon signed rank test for all pairs of data sets, the <i>p</i>-values were less than 10⁻¹⁷). Spearman rank correlation values mentioned are between phenotype robustness and phenotype evolvability.</p><p>Mean phenotype robustness and evolvability of 2.5×10⁴ structures (phenotypes).</p