Boolean networks with veto functions

Boolean networks are discrete dynamical systems for modeling regulation and signaling in living cells. We investigate a particular class of Boolean functions with inhibiting inputs exerting a veto (forced zero) on the output. We give analytical expressions for the sensitivity of these functions and provide evidence for their role in natural systems. In an intracellular signal transduction network [Helikar et al., Proc. Natl. Acad. Sci. USA 105, 1913 (2008)], the functions with veto are over-represented by a factor exceeding the over-representation of threshold functions and canalyzing functions in the same system. In Boolean networks for control of the yeast cell cycle [Li et al., Proc. Natl. Acad. Sci. USA 101, 4781 (2004); Davidich et al., PLoS ONE 3, e1672 (2008)], no or minimal changes to the wiring diagrams are necessary to formulate their dynamics in terms of the veto functions introduced here.

Figure 1

Functional network using veto functions for the budding yeast cell cycle. Solid arrows represent activators; dashed arrows represent inhibitors. Departing from the wild type network based on threshold functions [25], the depicted network is obtained by deleting six interactions: the two activations from Mcm1/SFF to Swi5 and to Cdc20&Cdc14; and the four inhibitions from Clb5,6 to Cdh1, from Cdh1 to Clb1,2, from Cdc20&Cdc14 to Clb1,2, and from Clb1,2 to Swi5.

Figure 2

Functional network using veto functions for the fission yeast cell cycle. Solid arrows represent activators; dashed arrows represent inhibitors. The wiring is identical to the one given by Rybarsch and Bornholdt [25], bottom of Fig. 5 there.

Figure 3

Average sensitivity according to Eq. (26) for an ensemble of veto functions with inhibitor probability $\gamma$. The number of inputs $k$ is a fixed integer parameter of the ensemble (no averaging over $k$).