Evolved Genotype-Phenotype Maps

Figure 3 shows typical genotype-phenotype maps produced during runs with and without constructional selection. The run without constructional selection reflects the underlying distribution of pleiotropy vectors sampled for each new gene. In the run with constructional selection, during the evolution of the first few genes, the discovery of new fitness components selects for high pleiotropy, but as these fitness components evolve toward their optima, selection becomes strong against new genes affecting them.

  

Figure: Two genotype-phenotype maps evolved through genome growth, with (left) and without (right) constructional selection. Dark squares indicate that fitness component j depends on gene i. The columns in the right map reflect the sampling distribution of the pleiotropy vectors, in which the number of fitness components affected is uniform on [1,f]. The left map shows how under constructional selection, later genes have lower pleiotropy as the genome grows and becomes more adapted.

This increasing selection for low pleiotropy can be seen in Figure 4, which shows the distribution of pleiotropies k_n as the genome grows, over repeated runs of genome growth. The mode for k_n is always 1 after the first few genes, but as shown in Figure 5, the mean k_n tends toward 1 from initial values of around 16, or half of the maximum possible, f=31.

  

Figure: The distribution, from repeated runs of the genome growth algorithm, of pleiotropy values k_n, from each gene's pleiotropy vector $\protect\mbox{\boldmath\(p\)\unboldmath}_n$, as the genome grows; with (left) and without (right) selection.

  

Figure: The average pleiotropy values k_n for each gene as the genome grows, from the runs in Figure 4, with selection.

The progress in adaptation can be compared between runs with and without constructional selection. Figure 6 shows plots for a number of runs. Without constructional selection, disruptive new genes are not filtered out, and adaptation shows little progress once the fitness components are saturated with genes that affect them. With constructional selection, however, fitness continues to increase with each new gene throughout the genome growth.

  

Figure: Fitness as a function of genome size for several runs of the genome growth algorithm. Dark lines are with, and light lines without constructional selection.

As the genome grows, the trajectories of individual fitness components can be seen in Figure 7. With constructional selection, once a fitness component has reached a high value (low points in graph), only new genes that leave it alone are likely to be incorporated in the genome. Occasionally, however, one component is sacrificed for the improvement in another, which show up as spikes in the graph. By the time the genome has reached a size of 31 genes, most of the components have reached values well above their expected value of 1/2. Without constructional selection, the jumble of spikes represents the continuing randomization of the fitness components as genes with random pleiotropy are incorporated into the genome.

  

Figure: Fitness components during genome growth, for one genome evolved with (left) and one without (right) constructional selection. Fitness components are sorted according to their value at the end of the run.

Here, most of the adaptation under constructional selection occurs during the incorporation of new genes, rather than during the adaptive walks (through allelic substitution) between gene additions. This is because there is a much larger pool of new pleiotropy vectors to sample from than the pool of genotypes in the 1-mutant neighborhood of an existing genotype (2^f vs. n). The evolutionary process under constructional selection is figuratively the ``building'' of a fitness peak, gene by gene, rather than the climbing of a fitness peak.