Fig 1.
An overview of the genotype-phenotype map.
The gene regulatory module has as input a morphogen concentration [M](x) that varies approximately exponentially across a 1-dimensional embryo of length L, and outputs a transcription factor [T](x). Spatial gene regulation of T is achieved via a simple Hamming model of protein DNA binding as shown. Fitness is determined by weighting any gene expression in the anterior half of the embryo as positive, and any in the posterior half as negative (as shown bottom left). We show idealised expression profiles that give maximum fitness when gene expression is confined only to the anterior half (bottom middle), and minimum fitness when there is no discrimination between anterior and posterior (bottom right); these translate to log-fitnesses of F = 0 and F = −∞, respectively (see Methods).
Fig 2.
Histogram of the fitness of populations over single long runs as a function of the population scaled fitness contribution 2NκF.
Fig 3.
Plot of the times series of two hybrids Rmca(a & b) and Rmca(c & d) at population scaled fitness contributions of 2NκF = 1 (a & c) and 2NκF = 10 (b & d).
Times when the fitness of a hybrid drop below the critical value F* (dashed line) correspond to DMIs.
Fig 4.
Decomposition of hybrid DMIs on a Boolean hyper 4-cube.
Each point on the 4-cube represent each possible hybrid genotype across 4 loci, including the genotype of each parental lineage, where each red cross represents an incompatible hybrid genotype. As shown in a) the fundamental types of DMIs, where a blue square or face identify a subspace of genotypes that correspond to a 2-way DMI, a single green edge or line corresponds to a 3-way DMI and a red open circle corresponds to a single isolated 4-way DMI. b) A more complicated pattern of hybrid DMIs and their decomposition into fundamental types.
Fig 5.
Plot of the total number of DMIs vs divergence time, together with their decomposition into the total number of 2-way, 3-way, 4-way DMIs, for various scaled populations sizes.
For 2NκF ≤ 20 the solid lines correspond to fits of the simulation data to Eq 1, while for 2NκF ≥ 50 correspond to fits to Eq 2.
Table 1.
Table of values of the exponent γ characterising the power law of growth of DMIs at short times and small scaled population sizes.
Table 2.
Table of values of the parameters characterising the sub-diffusive growth of DMIs for large scaled population sizes; β = 1 corresponds to normal diffusive motion, β < 1 to sub-diffusion and β > 1 super-diffusion, while K* corresponds roughly to the number of substitutions required to reach the inviable region.
Fig 6.
Plot of the spectrum 2-way DMIs vs scaled divergence time for different scaled population sizes.
Here the divergence time for each pair-wise DMI is scaled by the number of base-pairs involved in each interaction in order to remove the effect that interactions with a larger number of base-pairs mutate at a larger rate in proportion to their length; for Irm, ℓ′ = 5, and for Irc and Imc, ℓ′ = 10.