Fig. 3
FN and guarded FN examples. a FN modeling an exponential decay with uncertain rate. Let v:a = r and s:0.9a ≤ r ≤ 1.1a, then the FN models the exponential decay of a molecule of type A whose exact rate is uncertain but known to be in the interval [0.9[A],1.1[A]], where [A] is the concentration of A. b Evolution of two objective functions with m0[A] = 5 and λ0[R] = 0. The dashed and the solid lines correspond to the maximization and the minimization of m[A], respectively. The trajectories provide an upper and a lower bound for the potential evolutions of m[A]. The plot has been obtained by model predictive control with sample time of 0.1 units and a prediction horizon of one step. c Guarded FN modeling an activation process. Let v1:a = r, v2:b = r, s1:a = r and s2:r = 0 if m[A]>1, s2:r = 1 otherwise. Notice that the equations associated with s2 depend on the state, more precisely on m[A]. Moreover, note that the tokens in A are not used to produce intensity in R2, and hence A is not connected to s2. The net models an exponential decay of molecule A, and a constant production rate of molecule B of 0.1 (this is modeled by λ0[R2] = 0.1) when m[A]>1 and of 1.1 when m[A] ≤ 1. Thus, A can be seen as a repressor that only allows a residual production of B when m[A] is higher than 1. There are only two regions in this net, one is defined as m[A]>1, the other as m[A] ≤ 1. d Marking evolution with m0[A] = 5, m0[B] = 0, λ0[R1] = 0, λ0[R2] = 0.1, and the objective function is to minimize m[B]. It can be seen that the linear growth of m[B] changes from rate 0.1 to 1.1 when m[A] falls below 1. The plot has been obtained through model predictive control with a sample time of 0.1 and a prediction horizon of one step
