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These are some examples of stochastic Petri nets appearing in applications. See also Experiments in chemical reaction networks.

A receptor antagonist model

This is an example from biology discussed by Sontag and Zeilberg, who cite:

  • G. Gnacadja, A. Shoshitaishvili, M.J. Gresser, B. Varnum, D. Balaban, M. Durst, C. Vezina and Y. Li, Monotonicity of interleukin-1 receptor-ligand binding with respect to antagonist in the presence of decoy receptor, J. Theoret. Biol. 244 (2007), 478-488.

This paper analyzes a model involving the chemical interleukin-1 (IL-1). This chemical is a cytokine: a small protein molecule that is used for communication between cells. IL-1 is involved with immune responses, inflammatory reactions, and hematopoiesis. The species in the model are IL-1 (denoted as LL for “ligand”), the IL-1 receptor (denoted by RR), the human IL-1 receptor antagonist (denoted by AA), a decoy receptor or trap“ (denoted by TT) which, by binding to the ligand, helps block IL-1 signaling, and the four possible dimers RL,RA,AT,R L, R A, A T, and LTL T. The model consists of four reversible reactions:

R+LRL R + L \leftrightarrow R L
R+ARA R + A \leftrightarrow R A
A+TAT A + T \leftrightarrow A T
L+TLT L + T \leftrightarrow L T

This reaction system is complex-balanced because it is weakly reversible. It has 8 complexes, 4 strongly connected components, and rank 4, and hence deficiency zero. The total amounts of RR, LL, AA, and TT are conserved, giving rise to a matrix AA with 4 rows.

The Beverton-Holt model

This is a population model which illustrates how stochastic Petri nets can, in certain limits, give rise to differential equations that do not look like rate equations of the usual form:

  • Horst R. Thieme, Mathematics in Population Biology, Section 5.2: The Beverton-Holt and Smith Differential Equations, Princeton Series in Theoretical and Computational Biology, Princeton U. Press, Princeton, 2003.

If the population at time tt is PP, the model says

dPdt=(β1+αPμ)P \frac{d P}{d t} = \left(\frac{\beta}{1 + \alpha P} - \mu\right) P

The fact that we are dividing by 1+αP1 + \alpha P makes this equation not of the usual form.

However, we can start with this pair of equations, which involve not only the population PP but also the amount FF of some food resource:

dFdt=ΛνFbγFP \frac{d F}{d t} = \Lambda - \nu F - \frac{b}{\gamma} F P
dPdt=bFMμM \frac{d P}{d t} = b F M - \mu M

This is the rate equation of a stochastic Petri net. If μν\mu \ll \nu and Λb/νμ\Lambda b/ \nu \mu is not much smaller than 1, the dynamics of the food is much faster than that of the population, so we can assume the food is approximately in equilibrium at any time:

ΛνFbγFP=0 \Lambda - \nu F - \frac{b}{\gamma} F P = 0

If we use this to solve for FF in terms of PP, and plug this into the equation for dPdt\frac{d P}{d t}, we get the Beverton-Holton equations with

β=Λνb,α=βΛγ \beta = \frac{\Lambda}{\nu} b, \alpha = \frac{\beta}{\Lambda \gamma}

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