Fig. 1

Graphical illustration of the exact calculation method for a 3-node Boolean network (BN). a Influence graph, list of n=3 nodes (V), Boolean functions (F) and state space (Σ) of the BN. States (S) within Σ have a decimal index, S₁...S₈b Generating the elements of the kinetic matrix by asynchronous updating. In asynchronous updating only one node of the BN changes its value at a given timepoint. This means from any state S there are n possible transitions, therefore each state S_i is repeated n=3 times. The table on the left shows the source states (S_i) of all possible n 2ⁿ=24 transitions. For each state (decimal indices on the left) the Boolean functions F are applied individually to the three nodes, updated node highlighted by the dashed black line. If there is a change in the node’s value, the dashed square is highlighted by the color of the transition’s rate, shown next to the arrow to the right representing the BN’s transitions. If there is no change in the node’s value, there is no transition (no arrow). The table on the right shows the target states (S^′) of the transitions with their decimal indices. In the case of no transition the target state is the same as the source and the decimal index not shown. The updated node of the target state is again highlighted in color. To the right of the table there is the corresponding element of the kinetic matrix K. For the transition S_i→S_j the corresponding element is K_j,i. c State transition graph (STG) and kinetic matrix K of the BN’s master equation. STG: numbers in the vertices refer to the decimal indices of the binary states of the BN’s state space Σ, as shown by the red-white table in panel A. Transition rates on arcs (arrows) are explained in Eq. 2. K is inserted into the master equation of the dynamics of probabilities, x(t), of the BN’s states as described in Eq. 3: dx(t)/dt=Kx(t). The colors of nonzero entries of K correspond to the transition rates on the STG’s arcs. Each transition rate has a separate color used both for the corresponding arc(s) of the STG and entries of the kinetic matrix K. As an example, the transitions from state 1 to 5 and 2 to 6 both have the transition rate u_A (as it is node A updated from 0 to 1) and these two arcs have a dark green color, same as the corresponding entries K_1,5 and K_2,6. The diagonal elements of K are equal to the sum of the off-diagonals in the given column, taken with negative sign (see Eq. 5), e.g. the entry K_2,2 contains −(u_A+u_B). Terminal vertices that are attractor states and their corresponding columns of K are in gray. d Topological sorting of the STG in c. The vertices of the STG are re-indexed by topological sorting with indices ascending from upstream to downstream vertices (color coding of arcs by the transition rates does not change from panel C). This entails reordering of the kinetic matrix K→K^′ and the columns of terminal vertices (attractor states of the BN, gray columns) being moved to the right of K^′, creating a block structure (Eq. 15) used for obtaining the stationary solution (Eq. 7). e Construction of the right (R) and left (L) kernels from K^′. R is a basis for the column null space of K^′ and has 3 columns as there are 3 terminal vertices (6,7,8 of the topologically sorted STG). Block Y of R corresponds to non-terminal vertices, therefore it has 5 rows and contains only 0s. Block V corresponds to the terminal vertices, therefore it has 3 rows with 1 in the rows of terminal vertices 6,7,8. The block U of L is constructed by transposing V of R and replacing nonzero elements by 1, so that U·V=I. Block X of L is calculated from the blocks B and N of K^′ as X=−U·B·N⁻¹. The nonzero terms of X of L, L_ij, are rational functions in the transition rates, encoding the conservations between non-terminal and terminal (attractor) states. The terms κ₁,κ₂ are κ₁=d_A+d_B+u_C, κ₂=(d_A+2d_B+u_C)