# The Azimuth Project Experiments in proof of Feinberg&#x27;s theorm in first order networks (Rev #2, changes)

Showing changes from revision #1 to #2: Added | Removed | Changed

## The papers

• D. F. Anderson, G. Craciun and T.G. Kurtz,
• Feinberg, M. (1987). “Chemical reaction network structure and the stability of complex isothermal reactors: I. The deficiency zero and deficiency one theorems”. Chemical Engineering Science 42 (10): 2229–2268. doi:10.1016/0009-2509(87)80099-4.

• Feinberg, M. (1989). “Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity”. Chemical Engineering Science 44 (9): 1819–1827. doi:10.1016/0009-2509(89)85124-3.

• Schlosser, P. M. and M. Feinberg (June 1994). “A theory of multiple steady states in isothermal homogeneous CFSTRs with many reactions”. Chemical Engineering Science 49 (11): 1749–1767. doi:10.1016/0009-2509(94)80061-8.

• Feinberg, M. (December 1995). “The existence and uniqueness of steady states for a class of chemical reaction networks”. Archive for Rational Mechanics and Analysis 132 (4): 311–370. doi:10.1007/BF00375614.

## Definitions from Anderson’s paper

• (Chemical Species). $S = \{S_1, ..., S_m\}$

• (kth Reaction). $v_k, v_k'\in Z_{\geq 0}^m$. $v_k$, consumed. $v_k'$, created.

• (Complex). Each $v_k$, $v_k'$ is termed a complex.

• Notation for reactions. $v_k\rightarrow v_k'$. Source or reactant complex to the product complex.

• The set of all complexes. $\{v_k\}:= \bigcup_k(\{v_k\}\cup\{v_k'\})$.

• Notation for chemical reaction networks. Species: $S=\{S_i\}$. Complexes: $C=\{v_k\}$. Reactions: $R=\{v_k\rightarrow v_k'\}$.

• (Chemical Reaction Network). $\{S, C, R\}$.

• (Weakly reversible). $v_k\rightarrow v_k'\in R$, $\Rightarrow$ $\exists$ a sequence of reactions starting at $v_k'$ and leading to $v_k$.

• (Reversible). $v_k\rightarrow v_k'\in R$, $\Rightarrow$ $\exists$ $v_k'\rightarrow v_k\in R$.

• (Induced Directed Graph). Each $\{S, C, R\}$ gives rise uniquely to a directed graph, with nodes $C$ and edges in $R$.

• (Linkage Class). Each connected component of the resulting graph is termed a linkage class of the graph.

• (Connected Component). A subgraph in which any two components are connected.

• (Connected Graph). A graph that is itself connected has exactly one connected component, consisting of the whole graph.

• (Stoichiometric Subspace). $S= \text{span}\{v_k'-v_k|v_k\rightarrow v_k'\in R\}$. Dim($S$)$:=s$.

• (Compatibility Classes). Pick $c\in R^m$. $c+S$ is the stoichiometric compatibility class. $(c+S)\cap R^m_{\geq 0}$ is the positive stoichiometric compatibility class.

• (Related Fact). For stochastic and deterministic models, the state of the system remains within a single stoichiometric compatibility class for all time, assuming one starts in that class.

• (Deficiency). $\delta = |C|-l-s$. $l$ is the number of linkage classes of the graph. $s$ is Dim($S$)$:=s$. Note that we have thus far in our examples to date, consider the case with $l=1$.

## Definitions from Petri net field theory

### Stochastic Petri Net

• (States). $S=\{1,...,k\}$

• (Transitions). A set of transitions $T$. We consider $\tau \in T$ such that.

$\tau$ has inputs $m(\tau) = (m_1(\tau), ..., m_k(\tau))\in N^k$

$\tau$ has outputs $n(\tau) = (n_1(\tau), ..., n_k(\tau))\in N^k$

• (Rate constant). $r(\tau)\in R$.

## Rate equation

Let $x_i(t)$ be the amount of concentration of the ith state $(i=1,...,k)$

$\frac{d x_i(t)}{dt} = \sum_{\tau\in T} r(\tau) x_1^{m_1(\tau)}(t) \cdots x_k^{m_k(\tau)}(t)(n_i(\tau)-m_i(\tau))$

## Master equation

We can also arrive at a master equation. Now say we have a probability,

$\psi_{i_1...i_k}$ of having $i_1$ things in state 1, $i_k$ things in state $k$ etc. We write

$\psi = \sum \psi_{i_1...i_k}z_1^{i_1}\cdots z_k^{i_k}$

and $a_i^\dagger \psi = z_i \psi$, $a_i\psi = \frac{\partial}{\partial z_i} \psi$ then the master equation says

$\frac{d}{dt}\psi = H\psi$

Each transition corresponds to an operator. There is a term $a_1^{\dagger n_1}\cdots a_k^{\dagger n_1} a_1^{m_1}\cdots a_k^{m_k}$

In general, the master equation is written as

$H = \sum_{\tau \in T} r(\tau) (a_1^{\dagger n_1(\tau)}\cdots a_k^{\dagger n_k(\tau)}a_1^{m_1(\tau)}\cdots a_k^{m_k(\tau)}- N_1^{\underline m_1}\cdots N_k^{\underline m_k})$

## The Deficiency Zero Theorem

• (Feinberg). If the Petri net is weakly reversible and has deficiency 0, then $\exists$ ($X_i\in R^k$) such that the RHS of the rate equation vanishes identically.

## First Order Networks

• (First order network). We say a reaction network is a first order reaction network if $|v_k|\in \{0,1\}$ for each complex $v_k\in C$. Hence, a network is first order if each entry of the $v_k$ are zeros or ones. At most one entry can be a one.