Minimum density of functional proteins to make a system evolvable

To make an evolutionary system highly evolvable, the genotype space of a protein must be occupied by a number of functional proteins. In addition, these proteins have to be interconnected, so that a mutation may be able to create another functional protein and explore the genotype space. The genotype space of a fixed-length protein is mathematically analyzed to formulate this condition quantitatively. A graph whose node represents a set of adjacent genotypes mapped to the same phenotype, and whose edge represents a mutational transition between a pair of nodes, is introduced. We then apply the random graph theory to this graph, and formulate the minimum density of functional proteins for high connectivity of the graph. The minimum density is approximately the reciprocal of the product of the average number of adjacent genotypes mapped to the same phenotype and the number of different genotypes created from one genotype through a unit mutational step. The formula derived is tested using data for a fictional two-dimensional protein model. This work was presented in part at the Fifth International Symposium on Artificial Life and Robotics, Oita, Japan, January 26–28, 2000