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Experiments in proof of Feinberg's theorm in first order networks (Rev #2, changes)

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The papers

  • D. F. Anderson, G. Craciun and T.G. Kurtz,

Product-form stationary distributions for deficiency zero chemical reaction networks, arXiv:0803.3042.

  • 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}S = \{S_1, ..., S_m\}

  • (kth Reaction). v k,v kZ 0 mv_k, v_k'\in Z_{\geq 0}^m. v kv_k, consumed. v kv_k', created.

  • (Complex). Each v kv_k, v kv_k' is termed a complex.

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

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

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

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

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

  • (Reversible). v kv kRv_k\rightarrow v_k'\in R, \Rightarrow \exists v kv kRv_k'\rightarrow v_k\in R.

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

  • (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=span{v kv k|v kv kR}S= \text{span}\{v_k'-v_k|v_k\rightarrow v_k'\in R\}. Dim(SS):=s:=s.

  • (Compatibility Classes). Pick cR mc\in R^m. c+Sc+S is the stoichiometric compatibility class. (c+S)R 0 m(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). δ=|C|ls\delta = |C|-l-s. ll is the number of linkage classes of the graph. ss is Dim(SS):=s:=s. Note that we have thus far in our examples to date, consider the case with l=1l=1.

Definitions from Petri net field theory

Stochastic Petri Net

  • (States). S={1,...,k}S=\{1,...,k\}

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

τ\tau has inputs m(τ)=(m 1(τ),...,m k(τ))N km(\tau) = (m_1(\tau), ..., m_k(\tau))\in N^k

τ\tau has outputs n(τ)=(n 1(τ),...,n k(τ))N kn(\tau) = (n_1(\tau), ..., n_k(\tau))\in N^k

  • (Rate constant). r(τ)Rr(\tau)\in R.

Rate equation

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

dx i(t)dt= τTr(τ)x 1 m 1(τ)(t)x k m k(τ)(t)(n i(τ)m i(τ)) \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,

ψ i 1...i k\psi_{i_1...i_k} of having i 1i_1 things in state 1, i ki_k things in state kk etc. We write

ψ=ψ i 1...i kz 1 i 1z k i k\psi = \sum \psi_{i_1...i_k}z_1^{i_1}\cdots z_k^{i_k}

and a i ψ=z iψa_i^\dagger \psi = z_i \psi, a iψ=z iψa_i\psi = \frac{\partial}{\partial z_i} \psi then the master equation says

ddtψ=Hψ\frac{d}{dt}\psi = H\psi

Each transition corresponds to an operator. There is a term a 1 n 1a k n 1a 1 m 1a k m ka_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= τTr(τ)(a 1 n 1(τ)a k n k(τ)a 1 m 1(τ)a k m k(τ)N 1 m̲ 1N k m̲ k) 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 iR kX_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|{0,1}|v_k|\in \{0,1\} for each complex v kCv_k\in C. Hence, a network is first order if each entry of the v kv_k are zeros or ones. At most one entry can be a one.