Structural properties of genotype-phenotype maps

Abstract

The map between genotype and phenotype is fundamental to biology. Biological information is stored and passed on in the form of genotypes, and expressed in the form of phenotypes. A growing body of literature has examined a wide range of genotype-phenotype (GP) maps and has established a number of properties that appear to be shared by many GP maps. These properties are 'structural' in the sense that they are properties of the distribution of phenotypes across the point-mutation network of genotypes. They include: a redundancy of genotypes, meaning that many genotypes map to the same phenotypes, a highly non-uniform distribution of the number of genotypes per phenotype, a high robustness of phenotypes and the ability to reach a large number of new phenotypes within a small number of mutational steps. A further important property is that the robustness and evolvability of phenotypes are positively correlated. In this review, I give an overview of the study of GP maps with particular emphasis on these structural properties, and discuss a model that attempts to explain why these properties arise, as well as some of the fundamental ways in which the structure of GP maps can affect evolutionary outcomes.

Keywords: RNA secondary structure; evolvability; genotype; neutral evolution; phenotype; robustness.

Figure 1.

An illustration of three GP maps. (a) RNA secondary structure [18,19], which folds from a sequence of the four bases A, C, G and U. Base pairs are shown as double bonds. The folding process can be predicted computationally with a high level of accuracy. (b) The HP model [20] of protein folding. The example shown here is a sequence of hydrophobic (H) and polar (P) residues that folds onto a two-dimensional lattice. (c) The Polyomino model [21,22] is a self-assembly model in which sequences encode the interfaces of square two-dimensional tiles. These self-assemble on a lattice in a stochastic process. The interfaces (sometimes termed ‘colours’) are written in clockwise order and concatenated for the tiles. In this case, interface 1 binds to interface 2, and 0 denotes a neutral interface that does not interact. The two tiles self-assemble into the five-tile cross shape shown on the right.

Figure 2.

The three basic structural properties of GP maps, illustrated on a simple network of genotypes (nodes) that map to particular phenotypes (colours), and are connected by single-point mutations (edges). The redundancy of GP maps means that there is a much larger number of genotypes than phenotypes. The bias of a GP map is the extent to which the distribution of the number of genotypes per phenotype is non-uniform. In many real GP maps, this distribution is approximately Zipfian. The presence of robustness is one way of describing the fact that the different genotypes of a given phenotype are often close to each other in genotype space. As we discuss in the main text, the presence of large and robust neutral networks does not simply follow from bias.

Figure 3.

Schematic illustration of the relationship between phenotypic robustness and phenotypic frequency for the RNA, HP and Polyomino GP maps. In these maps, robustness scales logarithmically with frequency. If the phenotypes were randomly distributed (according to the biased distribution), we would expect to see the null model ρ_p = f_p (red line). The three shaded areas show the location of the vast majority of values of ρ_p versus f_p for the three maps. For the full data and background, see [23].

Figure 4.

Illustration of the relationship between robustness and evolvability, which can be defined for a genotype or a phenotype. At the genotypic level (a) a single genotype (highlighted by a black ring) can either be evolvable, meaning that it is a single mutation away from genotypes of many different phenotypes (a, left), or it can be robust, meaning that it is surrounded by genotypes that map to the same phenotype as itself (a, right). A genotype therefore faces a trade-off between these two quantities, and cannot be evolvable and robust at the same time. By contrast, at the phenotypic level (b) these quantities are positively correlated, which means that a phenotype (shown as the set of genotypes highlighted by black rings) can be both evolvable and robust (b, right). For phenotypes, the evolvability is defined as the total number of different phenotypes that lie within the point-mutation neighbourhood of a phenotype, and robustness is defined as the average fraction of genotypic neighbours that leave the phenotype unchanged, taken across all genotypes of that phenotype.