Evolution of Developmental Canalization in Networks of Competing Boolean Nodes

Developmental canalization, which leads to a reduction in the variation of phenotype expression relative to the complexity of the genome, has long been thought to be an important property of evolving biological systems. We demonstrate that a highly canalized state develops in the process of self-organization recently discovered in $N-K$ Boolean networks that evolve based on a competition between the nodes. The model provides a simplified description of the evolution of genetic regulatory networks in developmental systems. The mechanism responsible for the evolution is shown to be a balance of two dynamical effects which compete to bring the network to a nonrandom critical steady state. Unlike other proposed evolutionary mechanisms that select for canalization, this mechanism does so while maintaining the system's capacity for further evolution in the steady state.

Figure 1

The average internal homogeneity $\langle P\rangle$ as a function of $t$, the number of evolutionary changes of the network (epochs), illustrating the process of self-organization. Data shown are the average of 10 000 realizations of networks with in-degree $K=3$ and $N=999$ nodes. The inset shows that the initial self-organization is toward a smaller $\langle P\rangle$, and therefore toward a more chaotic network.

Figure 2

The distribution of the average fraction of updates that “1” is the majority output state of long attractors shown at different stages in the evolutionary process ($t$ is the number of evolutionary changes that have occurred). Results are calculated from 5000 different realizations. The initial broad distribution that occurs in RBNs ($t=0$) evolves to a sharply peaked distribution.

Figure 3

Selection for canalization. The average relative probability of occurrence of the 256 different $K=3$ Boolean strategies shown at different stages in the evolution­ary process. The strategies are grouped according to the classification in Table , and the groups are ordered with decreasing canalization. Results are the average of 10 000 different realizations.