Implicit representation of extended Petri nets by controled reactions. A controled reaction Rc = (r,fr) is composed of the reaction vector r and the associated control function fr. The arcs of an extended Petri net can be thought of to consist of two sets. (1) The standard arcs and (2) the control arcs (bidirected arcs and inhibitory arcs). A reaction vector describes how the marking of the connected places changes upon firing of a transition. The control function defines the conditions under which firing of at least one of all transitions with the same reaction vector may occur. The marking of the places of all four Petri nets shown in panels a) to d) has been choosen such that all transitions can fire. If there is a token in place A, the transition in a) can always fire, the transition in b) can only fire if there is at least one token in place C, while the transition in c) can only fire if there is at least one token in place C and if place D is empty. In the Petri net of panel d), the token has two options to move from place A to place B. It can move through firing of T1 if there is at least one token in place C and at least one token in place D. The alternative path, firing of T2, requires that there is at least one token in place D while place E must be empty. In panel d), the reaction vector of the controled reaction Rc = (r,fr) represents the set of two transitions T1 and T2 each of which connects the places A and B in the same direction through standard arcs. Note that if places A in panels a) to d) do not contain any token, none of the transitions could fire. Symbols: ∧, logic AND; ∨, logic OR; -, logic NOT.
Durzinsky et al. BMC Systems Biology 2011 5:113 doi:10.1186/1752-0509-5-113