Second, through the use of SOPs, the struc ture of a Boolean network is usually represented and depicted intuitively being a hypergraph. every hyperarc level ing into a node i is surely an AND clause of other nodes and rep resents one method of activating i. hence, all hyperarcs ending in i are ORed together. A hyperarc carries a signal flow to its end node as well as the binary worth from the movement relies on the state of all its start out nodes. In the following, such a hypergraph induced by a minimal SOP representation of the Boolean network are going to be termed a logical interaction hyper graph. In Figure 8 a possible instance of the LIH compatible together with the interaction graph of TOYNET in Figure 3 is depicted. In every single of the 4 nodes with extra than 1 incoming arc, the logical concatenation has now been specified. By way of example, B is now activated if A AND I1 are energetic simul taneously. In contrast, C is activated if B OR E is current.
and F is energetic if E OR G are in an active state. Hence, C and F retain their graph like struc ture. Wnt-C59 dissolve solubility Inhibiting arcs within the interaction graph are interpreted during the corresponding LIH as NOT operations. Thus, arc 7 is now interpreted as being a is energetic if D is just not current. Seeing that arc 2 and 3 in Figure three have already been combined with an AND in Figure eight, we interpret this new hyperarc as E gets to be activated if I2 is current AND I1 NOT. Consequently, in contrast to inhibiting arcs in interaction graphs, usually we usually do not assign a minus signal towards the complete hyperarc, but to its adverse branches. whereas all other branches get positive signs. Because of the assignment of signs LIHs can formally be observed as signed directed hypergraphs. The pure logical description of a signaling or regulatory network functions very well when the activation of the species by other individuals follows a sigmoid curve.
Complications that may arise when describing a real network inside of the logical framework and probable solutions are mentioned within a later part. LIHs can be formally represented and stored inside a comparable way as interaction graphs. The underlying hypergraph is stored by an m ? n incidence matrix B through which the rows correspond to your species along with the columns to the n hyper arcs. If species i is contained in the set of begin nodes of a hyperarc Doripenem k then Bik one, if i is definitely the endpoint of hyperarc k then Bik one, and if i is not really concerned in k we’ve Bik 0. For storing the NOTs operating on specified species in a hyperarc we may possibly use one more m ? n matrix U that merchants in Uik a one if species i enters the hyperarc k with its negated worth and 0 else. Accordingly, the incidence matrix B to the LIH of TOYNET reads To be concise, the two non zeros entries of U are indicated by an asterisk inside the incidence matrix. Representing a Boolean network as a LIH we will conveniently reconstruct the underlying interaction graph from the matrices B and U.