Chemical Reaction Network Theory

Chemical Reaction Network Theory ( CRNT ) examines the qualitative behavior of steady state concentrations of a chemical reaction network without using the kinetic parameters . It defines a general relationship between the network structure and the set of fixed points of the corresponding system of ordinary differential equations . For a subclass of chemical systems, this approach is able to algebraically predict or exclude the existence of multiple steady states , and thus does not require the use of numerical methods .

The system of ordinary differential equations, which corresponds to a chemical reaction network, consists of polynomials with in principle any degree. As a result, an analytical investigation of the fixed points of such a problem cannot generally be carried out with linear algebra . In addition, there is the problem that every reaction that follows the law of mass action is assigned a kinetic parameter that is often not known or not exactly known. Knowledge of the kinetic parameters is necessary for the numerical solution of the differential equation system. As a result, in the worst case, there are two sets of unknowns: (i) the kinetic parameters and (ii) the concentrations of the individual species at a fixed point. Even with knowledge of the kinetic parameters and the numerical determination of a fixed point, it is not clear whether multiple fixed points exist; d. H. whether another fixed point exists if the starting concentrations are selected differently (which are in the same linear subspace as the previous ones, see stoichiometric subspace and stoichiometric compatibility class ). CRNT can answer this question by calculating an index, the so-called deficiency (see below), for a subset of the chemical reaction networks without knowledge of the kinetic parameters or the concentrations. The deficiency is calculated in O notation when the chemical reaction network is created in . ${\displaystyle {\mathcal {O}}(1)}$ ${\displaystyle {\mathcal {O}}(n)}$

history

The first foundations of the CRNT were developed by Horn and Jackson and worked out and further developed by Martin Feinberg and colleagues.

Basics

The CRNT describes chemical reaction networks on which the law of mass action is based. In this paragraph, the term “reaction network” is used as a set of reactions that can be found in a textbook on biochemistry (e.g. all reactions in glycolysis ). In the Classic CRNT section , the term is precisely defined in the sense of the CRNT.

Reversible reactions , i.e., reactions that can run in forward and reverse directions must in this case two, irreversible be split reactions: an irreversible reaction for each direction. Accordingly, CRNT only describes reaction networks that consist of irreversible reactions. Such a network is divided into four sets :

• 1. The set of species : ${\displaystyle {\mathcal {S}}}$

The set of species consists of the individual substrates and products of the reactions of the reaction network.

• 2. The set of complexes : ${\displaystyle {\mathcal {C}}}$

This amount is made up of the totality of the species that are consumed or produced by a reaction. The elements from are multisets . That is, it consists of the total of the quantities of the species to the left and right of the reaction arrows. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}}$

• 3. The amount of reactions : ${\displaystyle {\mathcal {R}}}$

This set consists of all reactions of the chemical reaction network under consideration.

• 4. The set of kinetic rate constants : ${\displaystyle {\mathcal {K}}}$

This set consists of the rate constants of all reactions of the chemical reaction network under consideration.

Note on the notation: There are two equivalent representations of a complex: (i) as an element of the set as defined above; and (ii) as the vector from Let be the function that returns the stoichiometric coefficient of species in complex , i.e. H. if and otherwise. The index in is then to be understood as a function that supplies the entry of species in vector , i.e. ${\displaystyle {\mathcal {C}}}$ ${\displaystyle \mathbb {N} ^{\vert {\mathcal {S}}\vert }.}$ ${\displaystyle y(s)}$ ${\displaystyle s\in {\mathcal {S}}}$ ${\displaystyle y\in {\mathcal {C}}}$ ${\displaystyle y(s)\geq 1,}$ ${\displaystyle s\in y}$ ${\displaystyle y(s)=0}$ ${\displaystyle s}$ ${\displaystyle y_{s}}$ ${\displaystyle s\in {\mathcal {S}}}$ ${\displaystyle y\in \mathbb {N} ^{\vert {\mathcal {S}}\vert }}$

${\displaystyle y_{s}={\begin{cases}y(s),&{\text{wenn }}s\in y\\0,&{\text{andernfalls}}\end{cases}}.}$

This naturally requires a fixed order (e.g. lexicographical order ) of the species in the vector . ${\displaystyle y\in \mathbb {N} ^{\vert {\mathcal {S}}\vert }}$

example 1

The chemical reaction network consisting of the single reaction with rate constant has: ${\displaystyle {\text{A}}+{\text{A}}\rightarrow {\text{B}}}$ ${\displaystyle k_{{\text{A}}+{\text{A}}\rightarrow {\text{B}}},}$

• the species ${\displaystyle {\mathcal {S}}=\{{\text{A}},{\text{B}}\};}$
• the complexes ...
  • ... as a set: (here the complexes and are equivalent (different notation )); ${\displaystyle {\mathcal {C}}=\{\{{\text{A}},{\text{A}}\},\{{\text{B}}\}\}}$ ${\displaystyle \{{\text{A}},{\text{A}}\}}$ ${\displaystyle \{2{\text{A}}\}}$
  • ... as vectors: (with lexicographical order of the species); ${\displaystyle \{[2,0]^{T},[0,1]^{T}\}}$
• the reactions ${\displaystyle {\mathcal {R}}=\{{\text{A}}+{\text{A}}\rightarrow {\text{B}}\};}$
• the rate constants ${\displaystyle {\mathcal {K}}=\{k_{{\text{A}}+{\text{A}}\rightarrow {\text{B}}}\}.}$

The most important amount is here. Two complexes that belong to the same reaction can now e.g. B. can be described as follows: with ${\displaystyle {\mathcal {C}}.}$ ${\displaystyle y,y'\in {\mathcal {C}}}$ ${\displaystyle y\rightarrow y'\in {\mathcal {R}}.}$

Classic CRNT

See also. Let the set of all real numbers be greater than zero and the set of all real numbers greater than or equal to zero. ${\displaystyle \mathbb {P} :=\{x\in \mathbb {R} \ \vert \ x>0\}}$ ${\displaystyle {\overline {\mathbb {P} }}:=\mathbb {P} \cup \{0\}}$

Definitions

Positive steady state

Let be the vector of the concentrations of the chemical reaction network (law of mass action). The system is in a positive steady state if and . ${\displaystyle c}$ ${\displaystyle {\text{d}}c/{\text{d}}t=0}$ ${\displaystyle c\in \mathbb {P} }$

Response network

A reaction network is a triple with the set of the species; with as a set of complexes; with as a set of reactions. ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {R}}}$

For example 1 then applies ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})=(\{{\text{A}},{\text{B}}\},\{\{{\text{A}},{\text{A}}\},\{{\text{B}}\}\},\{{\text{A}}+{\text{A}}\rightarrow {\text{B}}\}).}$

Chemical reaction network

A chemical reaction network is a reaction network that is equipped with kinetics . In other words, a positive rate constant is associated with every reaction of the reaction network. ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}},{\mathcal {K}})}$

For example 1 then applies ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}},{\mathcal {K}})=(\{{\text{A}},{\text{B}}\},\{\{{\text{A}},{\text{A}}\},\{{\text{B}}\}\},\{{\text{A}}+{\text{A}}\rightarrow {\text{B}}\},\{k_{{\text{A}}+{\text{A}}\rightarrow {\text{B}}}\}).}$

A complex is directly linked to , where , also written if either or . That means two complexes are directly linked if a reaction exists in which connects them. ${\displaystyle y}$ ${\displaystyle y'}$ ${\displaystyle y,y'\in {\mathcal {C}}}$ ${\displaystyle y\leftrightarrow y'}$ ${\displaystyle y\rightarrow y'\in {\mathcal {R}}}$ ${\displaystyle y'\rightarrow y\in {\mathcal {R}}}$ ${\displaystyle {\mathcal {R}}}$

Linkage class

Be . Complex is linked to complex , indicated by if either or it exists such that . The equivalence relation induces a partition of into equivalence classes , which are referred to as linkage classes . ${\displaystyle y,y'\in {\mathcal {C}}}$ ${\displaystyle y}$ ${\displaystyle y'}$ ${\displaystyle y\sim y'}$ ${\displaystyle y=y'}$ ${\displaystyle y_{1},\ldots ,y_{m}\in {\mathcal {C}}}$ ${\displaystyle y=y_{1}\leftrightarrow \ldots \leftrightarrow y_{m}=y'}$ ${\displaystyle \sim }$ ${\displaystyle {\mathcal {C}}}$

Example 2

The reaction network is given ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})=(\{{\text{A}},{\text{B}},{\text{C}}\},\{\{2{\text{A}}\},\{{\text{B}}\},\{2{\text{B}}\},\{{\text{C}}\},\{2{\text{C}}\}\},\{R_{1},R_{2},R_{3},R_{4}\})}$

${\displaystyle {\begin{array}{crcl}R_{1}:=&2{\text{A}}&\rightarrow &{\text{B}},\\R_{2}:=&{\text{B}}&\rightarrow &2{\text{A}},\\R_{3}:=&{\text{B}}&\rightarrow &{\text{C}},\\R_{4}:=&2{\text{C}}&\rightarrow &2{\text{B}}.\end{array}}}$

The graph which represents the reaction network from Example 2 . The linkage classes consist of the connected components of the graph (l = 2). The nodes of the graph are equivalent to the complexes (n = 5). The strong linkage classes are marked by the nodes in the framed subgraphs. The terminal strong linkage classes are marked by double-framed nodes.

The linkage classes then consist of and . An intuitive way to determine the linkage classes is to graph the set of reactions , with reactions "assembling" at the ends where they have equal complexes (see figure ). ${\displaystyle {\mathcal {L}}_{1}=\{\{2{\text{A}}\},\{{\text{B}}\},\{{\text{C}}\}\}}$ ${\displaystyle {\mathcal {L}}_{2}=\{\{2{\text{B}}\},\{2{\text{C}}\}\}}$

Strong linkage class

Be . Complex reacts ultimately to complex written when either or there are so . Complex is strongly linked to , written , if and . The equivalence relation induces a partition of into equivalence classes, which are referred to as strong linkage classes . ${\displaystyle y,y'\in {\mathcal {C}}}$ ${\displaystyle y}$ ${\displaystyle y'}$ ${\displaystyle y\Rightarrow y'}$ ${\displaystyle y=y'}$ ${\displaystyle y_{1},\ldots ,y_{m}\in {\mathcal {C}}}$ ${\displaystyle y=y_{1}\rightarrow \ldots \rightarrow y_{m}=y'}$ ${\displaystyle y}$ ${\displaystyle y'}$ ${\displaystyle y\approx y'}$ ${\displaystyle y\Rightarrow y'}$ ${\displaystyle y'\Rightarrow y}$ ${\displaystyle \approx }$ ${\displaystyle {\mathcal {C}}}$

The strong Linkageklassen of Example 2 are given by , , and . The strong linkage classes can again be easily determined if the reaction network is represented as a graph. It then holds for every pair of nodes of a strong linkage class that a directed path from to and back exists. In the lower graph, the strong linkage classes are marked by frames (see figure ). ${\displaystyle {\overline {\mathcal {L}}}_{1}=\{\{2{\text{A}}\},\{{\text{B}}\}\}}$ ${\displaystyle {\overline {\mathcal {L}}}_{2}=\{\{{\text{C}}\}\}}$ ${\displaystyle {\overline {\mathcal {L}}}_{3}=\{\{2{\text{B}}\}\}}$ ${\displaystyle {\overline {\mathcal {L}}}_{4}=\{\{2{\text{C}}\}\}}$ ${\displaystyle y,y'\in {\mathcal {C}}}$ ${\displaystyle y}$ ${\displaystyle y'}$

Terminal strong linkage class

A terminal strong linkage class is a strong linkage class in which no complex reacts to a complex of another strong linkage class.

The terminal strong linkage classes of Example 2 are given by and . If the reaction network is again understood as a graph, then a terminal strong linkage class is a linkage class from which no reaction arrow points to another strong linkage class. In the lower graph, the strong linkage classes are marked by double frames (see figure ). ${\displaystyle {\overline {\mathcal {L}}}_{2}=\{\{{\text{C}}\}\}}$ ${\displaystyle {\overline {\mathcal {L}}}_{3}=\{\{2{\text{B}}\}\}}$

The following definition and the statements derived from it only apply to reaction networks which contain exactly one terminal strong linkage class per linkage class.

Deficiency

The deficiency of a reaction network (abbreviated to ) is defined by ${\displaystyle \delta }$

${\displaystyle \delta :=n-l-q,}$

where stands for the number of complexes, for the number of linkage classes and for the rank of the stoichiometric matrix of the given reaction network. ${\displaystyle n}$ ${\displaystyle l}$ ${\displaystyle q}$ ${\displaystyle N}$

The stoichiometric matrix of Example 2 is given by ${\displaystyle N}$

${\displaystyle N=\left[{\begin{array}{rrrr}-2&2&0&0\\1&-1&-1&2\\0&0&1&-2\end{array}}\right].}$

Hence it follows . The deficiency of Example 2 is then . ${\displaystyle q={\text{rang}}(N)=2}$ ${\displaystyle \delta =5-2-2=1}$

Weak reversibility

A reaction network is called weakly reversible if every linkage class consists of a terminal strong linkage class.

In Example 2 is not a weak reversible reaction network.

Stoichiometric subspace

The stoichiometric subspace (abbreviated to ) of a reaction network is the linear envelope of its reaction vectors. Ie ${\displaystyle S}$ ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})}$

${\displaystyle S:={\text{span}}(\{y-y'\in \mathbb {R} ^{\vert {\mathcal {S}}\vert }{\text{ }}\vert {\text{ }}y\rightarrow y'\in {\mathcal {R}}\}).}$

Since the set of reaction vectors are identical to the columns of the stoichiometric matrix , the stoichiometric subspace is equivalent to the column space of . ${\displaystyle N}$ ${\displaystyle N}$

Stoichiometric compatibility class

Two vectors are stoichiometrically compatible if . Stoichiometric compatibility is an equivalence relation which is divided into equivalence classes, the stoichiometric compatibility classes . ${\displaystyle c,c'\in {\overline {\mathbb {P} }}^{\vert {\mathcal {S}}\vert }}$ ${\displaystyle c-c'\in S}$ ${\displaystyle {\overline {\mathbb {P} }}^{\vert {\mathcal {S}}\vert }}$

Accordingly, the trajectory of the development of the concentrations over time must always be in the same stoichiometric compatibility class as the concentrations at time t = 0.

Theorems

Deficiency-Zero Theorem

Let a reaction network with deficiency zero. ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})}$

• (i) If the network is not weakly reversible, then the corresponding system of ordinary differential equations assumes neither a positive steady state nor a periodic orbit in (regardless of the choice of the kinetic rate constants). ${\displaystyle \mathbb {P} ^{\vert {\mathcal {S}}\vert }}$
• (ii) If the network is weakly reversible, then the corresponding system of ordinary differential equations has the following properties for any choice of the kinetic rate constants: Every positive stoichiometric compatibility class contains exactly one positive steady state; this positive steady state is asymptotically stable; and there are no nontrivial periodic orbits in . ${\displaystyle \mathbb {P} ^{\vert {\mathcal {S}}\vert }}$

See or for some evidence.

Deficiency-One Theorem

Be a deficient response network . And be with the deficiencies of the linkage classes. Furthermore it is assumed: ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}})}$ ${\displaystyle \delta }$ ${\displaystyle \delta _{{\mathcal {L}}_{i}}}$ ${\displaystyle 1\leq i\leq l}$

• (i) with ; ${\displaystyle \delta _{{\mathcal {L}}_{i}}\leq 1}$ ${\displaystyle 1\leq i\leq l}$
• (ii) ; ${\displaystyle \sum _{i=1}^{l}\delta _{{\mathcal {L}}_{i}}=\delta }$
• (iii) each linkage class contains only one terminal strong linkage class.

If the corresponding ordinary differential equations assume a positive steady state for a choice of the kinetic rate constant, then there is exactly one positive steady state in each stoichiometric compatibility class. If the network is weakly reversible, then the corresponding ordinary differential equations assume a positive steady state for each choice of kinetic rate constant.

See or for some evidence.

Relationship to the differential equations

The system of ordinary differential equations of a chemical reaction network is given by the function . The function of each chemical reaction network can now be broken down into four independent maps , one non-linear and three linear maps ${\displaystyle f(c):={\text{d}}c/{\text{d}}t}$ ${\displaystyle f(c)}$

${\displaystyle f(c)=YI_{a}I_{k}\psi (c),}$

which are defined below.

Definitions

Basis vectors of the complex space

If , then be ${\displaystyle {\mathcal {U}}\subseteq {\mathcal {C}}}$

${\displaystyle \omega _{\mathcal {U}}={\begin{cases}1,&{\text{wenn }}y\in {\mathcal {U}}\\0,&{\text{andernfalls.}}\end{cases}}}$

The basis vectors of the complex space are then given by the set . ${\displaystyle \mathbb {R} ^{\vert {\mathcal {C}}\vert }}$ ${\displaystyle \{\omega _{\{y\}}{\text{ }}\vert {\text{ }}y\in {\mathcal {C}}\}}$

The set of base vectors, represented as a matrix and provided that the vectors are sorted accordingly, is the identity matrix . ${\displaystyle E_{\vert {\mathcal {C}}\vert }}$

The nonlinear mapping ψ

Be a chemical reaction network. The non-linear mapping is given by ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}},{\mathcal {K}})}$ ${\displaystyle \psi :\mathbb {R} ^{\vert {\mathcal {S}}\vert }\rightarrow \mathbb {R} ^{\vert {\mathcal {C}}\vert }}$

${\displaystyle \psi (c):=\sum _{y\in {\mathcal {C}}}\omega _{\{y\}}c^{y},}$

With

${\displaystyle c^{y}:=\prod _{s\in {\mathcal {S}}}c_{s}^{y_{s}}.}$

Matrix I _k

will be added

Matrix I _a

Be a chemical reaction network. The linear mapping is given by ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}},{\mathcal {K}})}$ ${\displaystyle I_{a}:\mathbb {R} ^{\vert {\mathcal {R}}\vert }\rightarrow \mathbb {R} ^{\vert {\mathcal {C}}\vert }}$

${\displaystyle I_{a}(v):=\sum _{y\rightarrow y'\in {\mathcal {R}}}(\omega _{\{y'\}}-\omega _{\{y\}})v_{y\rightarrow y'},}$

with . ${\displaystyle v\in \mathbb {R} ^{\vert {\mathcal {R}}\vert }}$

Matrix Y

Be a chemical reaction network. The linear mapping is defined by with . ${\displaystyle ({\mathcal {S}},{\mathcal {C}},{\mathcal {R}},{\mathcal {K}})}$ ${\displaystyle Y:\mathbb {R} ^{\vert {\mathcal {C}}\vert }\rightarrow \mathbb {R} ^{\vert {\mathcal {S}}\vert }}$ ${\displaystyle Y(\omega _{\{y\}}):=y}$ ${\displaystyle y\in \mathbb {N} ^{\vert {\mathcal {S}}\vert }}$

It applies here and so that it can also be written in simplified form as (see also stoichiometric matrix ). ${\displaystyle N=YI_{a}}$ ${\displaystyle v(c)=I_{k}\psi (c)}$ ${\displaystyle f(c)}$ ${\displaystyle f(c)=Nv(c)}$

example

The system of ordinary differential equations of Example 2 is given by

${\displaystyle {\begin{array}{rcrrcrcrcr}{\frac {{\text{d}}c_{\text{A}}}{{\text{d}}t}}&=&-&2k_{R_{1}}c_{\text{A}}^{2}&+&2k_{R_{2}}c_{\text{B}}&&&&\\{\frac {{\text{d}}c_{\text{B}}}{{\text{d}}t}}&=&&k_{R_{1}}c_{\text{A}}^{2}&-&k_{R_{2}}c_{\text{B}}&-&k_{R_{3}}c_{\text{B}}&+&2k_{R_{4}}c_{\text{C}}^{2}\\{\frac {{\text{d}}c_{\text{C}}}{{\text{d}}t}}&=&&k_{R_{3}}c_{\text{B}}&-&2k_{R_{4}}c_{\text{C}}^{2}&&&&\end{array}}}$

Individual evidence

1. ^ ^{A b} F. Horn and R. Jackson: General mass action kinetics. Arch Rational Mech Anal 1972.
2. ↑ ^{a b c d} M. Feinberg: Lectures on chemical reaction networks. 1979.
3. ↑ ^{a b c d} M. Feinberg: The existence and uniqueness of steady states for a class of chemical reaction networks. Arch Rational Mech Anal 1995.
4. ↑ ^{a b c d e} J. Gunawardena: Chemical reaction network theory for in-silico biologists. 2003.
5. ↑ K. Gatermann, M. Eiswirth and A. Sensse: Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. Journal of Symbolic Computation 2005.
6. ↑ C. Conradi, D. Flockerzi, J. Raisch and J. Stelling: Subnetwork analysis reveals dynamic features of complex (bio) chemical networks. Proc Natl Acad Sci USA 2007.