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Table 2 Computation of the transition probabilities associated to our simple regulatory network.

From: Dynamical modeling of the cholesterol regulatory pathway with Boolean networks

Attractors Steady State (SS) State Cycle (SC)
States 1010 0010 1100 1011
Perturbed states and their limit cycles 1010 → SS 1010 → SS 1100 → SC 0011 → SC
   0110 → SS 1000 → SC 1011 → SC
   0000 → SC 1110 → SS 1011 → SC
   0010 → SC 1101 → SC 1010 → SS
Resulting probability transition P(SS → SS) = 1 P(SC → SC) = 8/12 0.67
   P(SC → SS) = 4/12 0.33
  1. This table shows both the principle of our newly introduced stability analysis and its application on the simple regulatory network shown in figure 1. It has 2 main columns: the steady state column and the state cycle column. As the state cycle found during the dynamical synchronous analysis contains 3 states (see figure 2), the last column is divided into 3 sub-columns. In the line "Perturbed states and their limit cycles" we show the perturbation results of each state of each attractor by re-evaluating each species by its own Boolean function triggered asynchronously. Perturbations are done from species A to species D. There are 4 species in our simple regulatory model, therefore 4 new states are generated from 1 perturbed state. We then synchronously simulate each new state and note if their simulation leads to the attractor they are derived from or to the other attractor of the system. In other words, we watch in which basin of attraction are those new states (see figure 2). The arrows following by "SC" (state cycle) or "SS" (steady state) give those responses. For the steady state no perturbation has an effect because a steady state in synchronous analysis remains a steady state in asynchronous analysis. Thus we simplify the presentation, showing that all the perturbations applied to state [1010] leave this state unchanged.