Robustness of Boolean dynamics under knockouts

The response to a knockout of a node is a characteristic feature of a networked dynamical system. Knockout resilience in the dynamics of the remaining nodes is a sign of robustness. Here we study the effect of knockouts for binary state sequences and their implementations in terms of Boolean threshold networks. Besides random sequences with biologically plausible constraints, we analyze the cell cycle sequence of the species Saccharomyces cerevisiae and the Boolean networks implementing it. Comparing with an appropriate null model we do not find evidence that the yeast wildtype network is optimized for high knockout resilience. Our notion of knockout resilience weakly correlates with the size of the basin of attraction, which has also been considered a measure of robustness.

Figure 1

(Color online) (a,b) The distribution of the number of inputs which can be compensated by all other nodes are shown for the first null model (a) and the more realistic null model (b). The blue (light gray in print) bars represent the distribution of the random cell cycles and the single green (dark gray in print) bar represents the knockout resilience of the wildtype network. (c,d) The distribution of the number of nodes which are robust with respect to all single node knockouts for the first null model (c) and the more realistic null model (d). As above, the blue (light gray in print) bars represent the random cell cycle sequences while the single green (dark gray in print) bar shows the system robustness for the wildtype network. Note the logarithmic scale of the four diagrams.

Figure 2

(Color online) The wildtype network of the cell cycle of the yeast species Saccharomyces cerevisiae [5]. The edges of the network are directed and can be activating (dashed green arrow) or inhibiting (solid red arrow). All self-couplings (solid yellow) are inhibiting. The network comprises 34 different interactions, 15 of which are activating and 19 are inhibiting.

Figure 3

The distribution of the specific knockout schemes is shown. The filled bars represent the wildtype component. The open bars represent all components and the single dashed bar represents the resilience combination of the wildtype itself. Each column represents one of the 24 different resilience combinations which occur among the functional networks. The nodes of the networks are shown as squares (see legend for names). If the knockout of a particular node is functional the square is filled, otherwise it is open. The resilience combinations are ordered such that the average basin size increases.

Figure 4

(Color online) The distribution of positive (dashed lines), negative (dotted lines), and all edges (straight lines) for each of the corresponding resilience combinations are shown. The lighter lines (light gray) represent networks from all components, while the darker lines (dark gray) with squares represent networks only reachable by mutations from the wildtype. The resilience combinations are depicted as in Fig. 3.

Figure 5

(Color online) The averages of the basin sizes corresponding to the specific resilience combination are shown. The straight line (dark gray) represents networks from all components, while the line with squares (light gray) represents networks only from the wildtype component. Additionally, the area between the 10% and 90% quantiles is inked. The resilience combinations are depicted as in Fig. 3. For each resilience combination ${10}^5$ networks were sampled.