Network growth models and genetic regulatory networks

We study a class of growth algorithms for directed graphs that are candidate models for the evolution of genetic regulatory networks. The algorithms involve partial duplication of nodes and their links, together with the innovation of new links, allowing for the possibility that input and output links from a newly created node may have different probabilities of survival. We find some counterintuitive trends as the parameters are varied, including the broadening of the in-degree distribution when the probability for retaining input links is decreased. We also find that both the scaling of transcription factors with genome size and the measured degree distributions for genes in yeast can be reproduced by the growth algorithm if and only if a special seed is used to initiate the process.

Figure 1

Simulated (circles) and predicted (squares) degree distribution functions for different values of $c$. (a) $c=0.1$, (b) $c=0.3$, (c) $c=0.5$, and (d) $c=0.7$. For models with no innovation and $c_i=c_o$ the in-degree and out-degree distributions are identical.

Figure 2

In-degree (circles) and out-degree (squares) distribution functions for $c_o=0.5$ and different values of $c_i$: (a) $c_i=0.1$, (b) $c_i=0.3$, (c) $c_i=0.5$, and (d) $c_i=0.7$. Each data point represents the average number of nodes with a given number of inputs in an ensemble of 100 networks.

Figure 3

In-degree (circles) and out-degree (squares) distribution functions for $c_o=0.5$ and $c_i=0.1$, with various values of $p$: (a) $p=0.001$, (b) $p=0.005$, and (c) $p=0.01$. Each different value of $p$ results in a different percentage of innovated links: (a) $39.1\%$, (b) $59.8\%$, and (c) $64.6\%$.

Figure 4

In-degree (circles) and out-degree (squares) distribution functions for $c_o=0.5$ and $c_i=0.5$, with various values of $p$: (a) $p=0.001$, (b) $p=0.005$, and (c) $p=0.01$. Each different value of $p$ results in a different percentage of innovated links: (a) $16.2\%$, (b) $36.5\%$, and (c) $43.2\%$.

Figure 5

In-degree (circles) and out-degree (squares) distribution functions for $c_o=0.5$ and different values of $c_i$: (a) $c_i=0.1$, (b) $c_i=0.3$, (c) $c_i=0.5$, and (d) $c_i=0.7$, with $n_0=3000$ in the rich-gets-richer innovation model. The percentage of innovated links is (a) $48.3\%$, (b) $41.4\%$, (c) $37.5\%$, and (d) $35.3$%.

Figure 6

Growth rate of the number of transcription factors for the parameters $c_i=0.2$ and $c_o=0.5$, with (a) $p=0$ and (b) $p=0.005$. The networks are grown from a seed of one regulator linked to 99 other nodes. Solid lines show the scaling laws consistent with van Nimwegen's analysis of (a) bacteria and (b) eukaryotes.

Figure 7

(a) In-degree (circles) and out-degree (squares) distribution functions for the model parameters equal to those of Fig. 6b: $c_o=0.5$, $c_i=0.2$, and $p=0.005$. (b) Same as (a) except $c_i=0.3$. The networks are grown from a seed of one regulator linked to itself and 99 other nodes.