Abstract

The mapping between biological genotypes and phenotypes is central to the study of biological evolution. We introduce a rich, intuitive and biologically realistic genotype– phenotype (GP) map that serves as a model of self-assembling biological structures, such as protein complexes, and remains computationally and analytically tractable. Our GP map arises naturally from the self-assembly of polyomino structures on a two-dimensional lattice and exhibits a number of properties: redundancy (genotypes vastly outnumber phenotypes), phenotype bias (genotypic redundancy varies greatly between phenotypes), genotype component disconnectivity (phenotypes consist of disconnected mutational networks) and shape space covering (most phenotypes can be reached in a small number of mutations). We also show that the mutational robustness of phenotypes scales roughly logarithmically with phenotype redundancy and is positively correlated with phenotypic evolvability. Although our GP map describes the assembly of disconnected objects, it shares many properties with other popular GP maps for connected units, such as models for RNA secondary structure or the hydrophobic-polar (HP) lattice model for protein tertiary structure. The remarkable fact that these important properties similarly emerge from such different models suggests the possibility that universal features underlie a much wider class of biologically realistic GP maps. A further property of the polyomino GP map is that the modularity, symmetry and structural complexity of the phenotype can be quantified rigorously. We can therefore use our GP map to study the emergence of these phenotypic properties in the course of biological evolution, and present results on the distribution of complexity in phenotypes, which are intimately linked with the properties of genotype-phenotype maps.