Revisiting Robustness and Evolvability: Evolution in Weighted Genotype Spaces

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 weighting 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.

SUPPLEMENTARY TEXT S1

Definitions
In this section, we rationalize the need for new definitions of robustness and evolvability at the level of both genotype and phenotype, so as to incorporate the weighting of the network. We show that the new definitions essentially concur with the earlier definitions put forth by Wagner [1], for an unweighted network.
Let us consider an unweighted network with all edges having the same probability (p=0.33). The genotype space consists of 6 nodes (G1, G2 ... G6) and the phenotype space is given by P1, P2, P3 and P4.

Genotype Robustness
In order to calculate genotype robustness of G1 (RG1), we first identify the neutral neighbour(s) of G1: {G4}.
We see that the values as given by the two definitions are identical.

Genotype Evolvability
We calculate the genotype evolvability of G1 (EG1) as (new definition) the mean probability of evolving from G1 (with phenotype P1) to a different phenotype, summed over all the different phenotypes found in the 1-neighbourhood of the genotype.
In order to evaluate this, we first identify phenotypes different from G1 (forming phenotype P1) in

Phenotype
Shape P1 P2 P3 P4 Thus, number of unique structures in the 1-neighbourhood U = 2. Table S1.1 discusses the probabilities and occurrences of these unique structures in the 1-neighbourhood. Finally, we define genotype evolvability as, E G1 = ∑ Mean probability of evolving to a different structure = 2×0.33 = 0.66

Old definition:
Genotype evolvability = Fraction of unique structures in 1-neighbourhood of G1 Here also we observe that the two definitions yield identical values.

Phenotype Robustness
Phenotype robustness of P1 (R P1 ): We define phenotype evolvability as (new definition) probability of evolving to a neutral neighbour from a genotype averaged over all the genotypes with phenotype P1 In order to calculate this value, we first identify all genotypes forming phenotype P1: G1 and G4 Phenotype robustness is therefore given by average genotype robustness of G1 and G4 RP1 = (0.33+0.33)/2 = 0.33.
According to the old definition, phenotype robustness is the fraction of neutral neighbours of a genotype, averaged over all genotypes that form a given phenotype.
We observe that the values resulting from both definitions are identical.
Thus, the number of unique structures in the 1-neighbourhood of P1 is U = 3. Table S1.2 discusses the probabilities and occurrences of these unique structures in the 1-neighbourhood of P1. Finally, phenotype evolvability is defined as, EP1= ∑ Mean probability of evolving to a different structure = 3×0.33 = 1

Phenotype in 1neighbourhood
Old definition: Number of unique phenotypes found in the 1-neighbourhood of the phenotype P1: To summarise, the values of the robustness and evolvability as given by both the new and old definitions are:

. Values of the robustness and evolvability (both new and old definitions).
We see that if all edges have the same probability (p), as in an unweighted network, then ( ) ( ) = p, a constant.
In the above example network, p =0.33.
We observe that both the new and old definitions of genotype robustness and evolvability and phenotype robustness yield identical values for an unweighted network. The new definition of phenotype evolvability essentially reduces to the earlier definition, only scaled by a constant value, in the case of an unweighted network.

Weighted Neutral Networks
Let us now consider a weighted network with the same genotypes and phenotypes. However, the probabilities corresponding to different mutations are no longer assumed to be the same. We see that weighting the network essentially does not change the values of robustness and evolvability (both genotype and phenotype), according to the earlier (old) definitions. Thus, we modified these definitions, and came up with a new set of definitions for the system properties that tries to incorporate edge (mutation) probability. Also, for an unweighted network, these modified definitions of robustness and evolvability yield identical values to those given by the earlier definitions.