Dynamical regimes in non-ergodic random Boolean networks

Abstract

Random boolean networks are a model of genetic regulatory networks that has proven able to describe experimental data in biology. Random boolean networks not only reproduce important phenomena in cell dynamics, but they are also extremely interesting from a theoretical viewpoint, since it is possible to tune their asymptotic behaviour from order to disorder. The usual approach characterizes network families as a whole, either by means of static or dynamic measures. We show here that a more detailed study, based on the properties of system’s attractors, can provide information that makes it possible to predict with higher precision important properties, such as system’s response to gene knock-out. A new set of principled measures is introduced, that explains some puzzling behaviours of these networks. These results are not limited to random Boolean network models, but they are general and hold for any discrete model exhibiting similar dynamical characteristics

Identifying Critical States through the Relevance Index

“…It has also been observed elsewhere that the Derrida plots computed by perturbing a subset of all of the possible initial states (for example, those belonging to some attractor) can be very different from the theoretical ones [23]. The phenomena related to the peculiarities of restricting the set of states of an RBN require further studies, as the differences shown here confirm.…”

“…While this might seem surprising, it should be recalled that critical states share some properties, but there may be differences. Indeed, it has already been shown elsewhere [23] that RBNs can show heterogeneous behaviors in different positions along the critical line (even maintaining critical dynamics). Figure 7b (the interpolating curve has been obtained by fitting a quadratic function to the measured points, and it is only a visual aid).…”

“…Moreover, our analysis also supports the statement that the information provided by the dynamics of an RBN along its transients might considerably differ from that of its attractors. Preliminary results in this direction were obtained in [23,25].…”

Evolving Always-Critical Networks

“…we should mention that Boolean networks are nonergodic systems, and different notions of criticality can be proposed, as discussed in [29]. The classical Derrida parameter is measured by computing the distance between initially close states, which are chosen at random, without any constraints.…”

“…This can be achieved by considering the downstream effect of a single flip on the input of the various Boolean functions. The choice of the most representative value of the sensitivity may be complicated [29] but, in the case of ergodic systems, the dynamic and static methods asymptotically provide the same estimate (while the effective values computed on finite numbers of cases may of course differ).…”