ARTICLE: Polyominoes: mapping genotypes to phenotypes

A tractable genotype–phenotype map modelling the self-assembly of protein quaternary structure

• Proteins in our cells have intricate structures built by genetic instructions, and these structures are vital for life: we produce a computational model to explore the relationship between genetic instructions and structure, and how evolution and mutation may change proteins.

Biological evolution sculpts the natural world and relies on the conversion of genetic information (stored as sequences, usually of DNA, called genotypes) into functional physical forms (called phenotypes). The complicated nature of this conversion, which is called a genotype-phenotype (or GP) map, makes the theoretical study of evolution very difficult. It is hard to say how a population of individuals may evolve without understanding the underlying GP map.

This is due to the two fundamental forces of evolution -- mutations and natural selection -- acting on different aspects of an organism. Mutations occur to genotypes (G), while natural selection, the ultimate adjudicator of the fate of mutations in the population, acts on the phenotype (P). Without understanding the link between these two -- the GP map -- we can't easily say, for example, how many mutations we expect important proteins within a virus strain to undergo with time, and thus how quickly the virus will evolve to be unrecognised by our immune systems.

Simple models for the mapping of genotype to phenotype have helped answer important questions for some model biological systems, such as RNA molecules and a coarse-grained model of protein folding. One important class of biological structure which has not yet been modelled in this way are protein complexes: structures formed through proteins binding together, fulfilling vital biological functions in living organisms. In this work, we introduce the "polyomino" model, based on the self-assembly of interacting square tiles to form polyomino structures. The square tiles that make up a polyomino are assigned different "sticky patches", modelling the interactions between different proteins that form a complex. A huge range of structures can be formed by varying the details of these patches, mimicking the range of protein complexes that exist in biology (though there are some obvious differences in the shapes of structures that can be formed).

Our simple model explores the interactions between protein subunits, and how these interactions shape a surface that evolution explores. (top) Sickle-cell anemia involves a mutation that changes the way proteins interact, making normally independent units form a dangerous extended structure. (bottom) Our polyomino model models this effect. The resultant dramatic effects on structure, fitness, and evolution can then be explored.

Despite its abstraction we show that the polyomino model displays several important features which make it a potentially useful model for the GP map underlying protein complex evolution. On top of this, we demonstrate that our model possesses similar properties to RNA and protein folding models, interestingly suggesting that universal features may be present in biological GP maps and that the "landscapes" upon which evolution searches may thus have general properties in common. You can find the paper free here and you can play with polyominoes here! Iain [blog article also here]