Blog - network theory (examples)

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

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 $L$ for “ligand”), the IL-1 receptor (denoted by $R$), the human IL-1 receptor antagonist (denoted by $A$), a decoy receptor or trap“ (denoted by $T$) which, by binding to the ligand, helps block IL-1 signaling, and the four possible dimers $R L, R A, A T,$ and $L T$. The model consists of four reversible reactions:

$R + L \leftrightarrow R L$

$R + A \leftrightarrow R A$

$A + T \leftrightarrow A T$

$L + T \leftrightarrow L T$

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 $R$, $L$, $A$, and $T$ are conserved, giving rise to a matrix $A$ with 4 rows.

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 $t$ is $P$, the model says

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

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

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

$\frac{d F}{d t} = \Lambda - \nu F - \frac{b}{\gamma} F P$

$\frac{d P}{d t} = b F M - \mu M$

This *is* the rate equation of a stochastic Petri net. If $\mu \ll \nu$ and $\Lambda b/ \nu \mu$ 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:

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

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

$\beta = \frac{\Lambda}{\nu} b, \alpha = \frac{\beta}{\Lambda \gamma}$

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