Fig 1.
Distribution of modularity in random networks with different number of interactions.
Each panel shows the distribution of a modularity score for sparse (20 interactions) and dense (50 interactions) networks with ten nodes. Each sample contains 3,000 random networks. (A) Distribution of the raw modularity score Qopt. Mean ± SD Qopt equals 0.278 ± 0.057 for sparse and 0.135 ± 0.029 for dense networks. Kolmogorov-Smirnov test: D = 0.922; p < 2.2 × 10−16. (B) Distribution of the normalized modularity score QN. Mean ± SD QN equals 0.008 ± 0.981 for sparse and 0.052 ± 0.991 for dense networks. Kolmogorov-Smirnov test: D = 0.064; p = 0.257. (C) Distribution of the raw modularity score QP for a specific partition P. Mean ± SD QP equals −0.002 ± 0.101 for sparse and 3.7 × 10−4 ± 0.05 for dense networks. Kolmogorov-Smirnov test: D = 0.22; p = 6.18 × 10−11. (D) Distribution of the normalized modularity score for a specific partition P. Mean ± SD
equals −0.023 ± 1.02 for sparse and 0.007 ± 0.999 for dense networks. Kolmogorov-Smirnov test: D = 0.062; p = 0.29.
Fig 2.
Gene network dynamics, mutation and evaluation of fitness.
(A) A gene network can be described as a graph or as a matrix in which positive entries (green squares) represent activatory interactions (arrows) and negative entries (red squares) represent inhibitory interactions (blunt-end lines). (B) Each row with six squares represent a system state. White and black squares represent active and inactive genes, respectively. An initial pattern of gene activity (at t = 0) is updated according to the interactions described in panel (A) until dynamics lead to a GAP, a sequence of system states that is repeated indefinitely. The gene network in (A) is able to yield two different GAPs when it starts its dynamics from different initial system states. (C) Summary of the procedure to evaluate a network’s fitness in evolutionary simulations. Fitness is higher in networks that produce GAPs, from different initial conditions, similar to target GAPs that selection favours. (D) Mutations in the model change interactions between genes. They thus modify the matrix that specifies whether a gene i regulates a second gene j. The activatory interaction that mutation wrecks is dashed in the top network and its corresponding entry is marked with an X.
Table 1.
Modularity in developmental multistable regulatory networks.
Fig 3.
Modularity evolves after selection for an additional gene activity phenotype.
(A) Target GAPs I and II. White and black squares represent active and inactive genes, respectively. Genes 1-5 are grouped in set and genes 6-10 are grouped in set
. Note that genes in set
have the same activity state in both target GAPs. In contrast, genes in set
have a different activity state in both target GAPs. Networks evolve in a first stage under selection to produce target GAP I. In a second stage, selection favours networks that produce target GAPs I and II from distinct initial system states, that may occur, for example, in cells in different parts of an organism. (B)
increases after selection for an additional gene activity phenotype. Each dot represents an independently evolving population. The horizontal axis indicates the
score for a network with the highest fitness before starting selection for the two GAPs. The vertical axis denotes the same score but for a network after selection for both GAPs. The solid diagonal is the identity line. (C) After selection for GAP I alone, interactions between any pair of genes occur with similar frequency. Grayscale indicates the fraction of independently evolved networks that have a certain source-target regulatory interaction. (D) After selection for both target GAPs I and II interactions occur mainly either between genes in set
or between genes in set
.
Fig 4.
Selection for two GAPs produces a greater increase in modularity in sparser networks.
The selection regimes are the same as in Fig 3. In a first stage, selection favoured networks that produced target GAP I. In a second stage, networks with highest fitness were those that produced GAPs I and II in cells with different initial system states (Fig 3A). The score refers to a partition P that assigns the first five genes to one set and the rest of the genes to another. The number of expected interactions is modulated through γ, the relative propensity to acquire interactions.
Fig 5.
The rate κ of perturbation of initial system states contributes to the increase in modularity due to selection for two GAPs.
The increase in modularity after selection for two target GAPs is greater when evolution occurs under higher rates of non-genetic perturbation. The selection regimes are the same as in Fig 3. In a first stage, selection favoured networks that produced target GAP I. In a second stage, networks with highest fitness were those that produced GAPs I and II from different initial system states (Fig 3A). refers to a partition P that assigns the first five genes to one set and the rest of the genes to another.