Figure 1

Schematic illustration of a Boolean circuit. (A) Each circuit has N=3 genes, shown as labeled open circles (a,b,c). Genes can be in one of two states: expressed (1) or not expressed (0). Regulatory interactions are depicted as directed edges a→b, which indicate that the gene product of a regulates the expression of b. The signal-integration logic of each gene is captured in a lookup table, where each entry encodes the gene’s regulatory response to one of the 2^N possible combinations of the states of its regulating gene products. These lookup tables also specify the circuit’s wiring diagram. For example, the expression of gene a is independent of gene c because its lookup table specifies the Boolean logic function “a and b.” The regulatory interaction c→a is thus inactive, as indicated by the gray color of this edge. (B) By concatenating the rightmost columns of each lookup table, the signal-integration logic and wiring diagram of a circuit can be represented as a single vector G of length L = N × 2^N. We consider G to be the circuit’s genotype. (C) The circuit shown in panel (A) is a member of the genotype set of the bifunction F ⁽¹⁾:〈0,0,0〉↦〈0,0,0〉, F ⁽²⁾:〈0,0,1〉↦〈0,1,0〉. (D) Each circuit with a given multifunction may map initial states that are not part of the multifunction to new equilibrium states. We consider these to be latent phenotypes. For example, of the five states that are not part of the multifunction shown in (C), three map to the new equilibrium expression states shown in (D), while the other two map to equilibrium expression states that are already part of the multifunction. This circuit therefore has f = 3 latent phenotypes, and each is a fixed-point. Other circuits with this bifunction may have more or fewer latent phenotypes.