Chemical Reaction Network Theory

Chemical Reaction Network Theory ( CRNT ) studied the qualitative behavior of the steady state concentrations of a chemical reaction network without the use of 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 capable of the existence of multiple steady states algebraically predict or exclude, and is therefore without the use of numerical methods.

  • 4.1 Definitions 4.1.1 Positive steady-state
  • 4.1.2 reaction network
  • 4.1.3 Chemical reaction network
  • 4.1.4 Linkageklasse 4.1.4.1 Example 2
  • 4.2.1 Deficiency Zero Theorem
  • 4.2.2 Deficiency -One Theorem
  • 5.1 Definitions 5.1.1 basis vectors of the complex space
  • 5.1.2 The non-linear mapping ψ
  • 5.1.3 matrix Ik
  • 5.1.4 Matrix Ia
  • 5.1.5 matrix Y

Introduction

The system of ordinary differential equations, which corresponds to a chemical reaction network is composed of, in principle, with any degree polynomials. As a result, can not be done with linear algebra, an analytical study of the fixed points of such a problem in general. In addition, the difficulty arises that any reaction that follows the law of mass action, a kinetic parameter is assigned, which is not, or not exactly known often. For the numerical solution of the differential equation system but the knowledge of the kinetic parameters is necessary. As a result, exist in the worst case, two levels of unknowns: ( i) the kinetic parameters and (ii) the concentration of the individual species at a fixed point. Even with knowledge of the kinetic parameters and the numerical calculation of a fixed point, it is not clear whether multiple fixed points exist; i.e., at a different selection of the starting concentrations ( which are in the same linear subspace as the previous ones, see stoichiometric and stoichiometric subspace compatibility class ), there is another fixed point. CRNT, this question through the calculation of an index, called the deficiency ( see below), answer for a subset of chemical reaction networks without the knowledge of the kinetic parameters or concentrations. In O notation, the calculation of the deficiency in is when the creation of the chemical reaction network is done in.

History

Getting basics of CRNT developed by Horn and Jackson and developed by Martin Feinberg and employees and further developed.

Basics

The CRNT describes chemical reaction networks that are based on the law of mass action. In this paragraph, the term " reaction network " is understood as a set of reactions as it can be found in a textbook of biochemistry (eg all reactions of glycolysis ). In paragraph Classic CRNT the term is defined exactly in the sense of CRNT.

Reversible reactions, that is, reactions which can proceed in the forward and reverse direction, in this case have to be split into two irreversible reactions: an irreversible reaction for each direction. Accordingly CRNT describes only reaction networks, which consist of irreversible reactions. Such a network is divided into four levels:

  • 2 The amount of the complexes:
  • 3 The set of reactions:
  • 4 The amount of kinetic rate constants:

Note on Notation: There are two equivalent representations of a complex: (i ) is defined as element of the set above; and (ii) a vector from Let the function that returns the stoichiometric coefficients of the species in the complex, i.e., if and otherwise. The index is then to be understood as a function that returns the entry of species in vector, so

For this purpose, a fixed natural order (eg lexicographic ordering ) of the species is provided in the vector.

Example 1

The chemical reaction network, consisting of the single reaction rate constant with having:

  • The species
  • The complexes ... As ... Quantity: ( here are the complexes and equivalent ( different notation ) );
  • As vectors ...: ( with lexicographical ordering of the species );

Here, the main quantity of two complexes, which are in the same reaction, for example, can now be described as follows: with

Classic CRNT

See also. Be the set of all real numbers greater than zero and the set of all real numbers greater than or equal to zero.

Definitions

Positive steady-state

Be the vector of concentrations of the chemical reaction network ( law of mass action ). The system is in a positive steady state if and.

Reaction network

A reaction network is a triple with a lot of the species; with the amount of complexes; with a set of reactions.

Then for Example 1

A chemical reaction network

A chemical reaction network is a response network which is equipped with a kinetics. That with each reaction of the reaction network is associated a positive rate constant.

Then for Example 1

A complex is directly linked with, where, also written when either or. That two complexes are directly linked if there is a reaction in which it connects.

Linkageklasse

Be. Complex is linked to the complex, characterized in that when either or there are so. The equivalence relation induces a partition into equivalence classes, which are called Linkageklassen.

Example 2

Consider the reaction network with

The Linkageklassen then consist of and. An intuitive method to determine the Linkageklassen is to record the amount of responses as a graph, with reactions at the ends are " assembled ", where they have the same complexes ( see figure).

Strong Linkageklasse

Be. Complex reacts ultimately to complex, written when either or there are making. Complex is strongly linked with, written when and. The equivalence relation induces a partition into equivalence classes, which are referred to as strong Linkageklassen.

Terminals strong Linkageklasse

One terminal is a strong strong Linkageklasse Linkageklasse in which no complex react to form a complex of other strong Linkageklasse.

The terminal strong Linkageklassen of Example 2 are given by and. When one conceives the reaction network again as a graph, then a terminal strong Linkageklasse is a Linkageklasse from which shows no reaction arrow on another strong Linkageklasse. In the lower graph the strong Linkageklassen are marked by a double frame ( see illustration).

The following definition and the derived statements speak only as reaction networks containing exactly one terminal strong Linkageklasse per Linkageklasse.

Deficiency

The deficiency of a response network ( abbreviated ) is defined by

Wherein is the number of the complexes, the number of Linkageklassen and the rank of the stoichiometric matrix of the given reaction network.

The stoichiometric matrix of Example 2 is given by

Consequently, results. The deficiency of Example 2 is then.

Weak reversibility

A reaction network is called weakly reversible if each Linkageklasse consists of a terminal strong Linkageklasse.

In Example 2 is not weakly reversible reaction network.

Stoichiometric subspace

The stoichiometric subspace ( abbreviated ) of a reaction network is the linear hull of its reaction vectors. That is,

Since the amount of the reaction vectors are identical to the columns of the stoichiometric matrix, the sub-space is equivalent to the stoichiometric of the column space.

Stoichiometric compatibility class

Two vectors are stoichiometrically compatible if. Stoichiometric compatibility is an equivalence relation, which in equivalence classes, the stoichiometric compatibility classes divides.

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

Theorems

Deficiency Zero Theorem

Be a reaction network with deficiency zero.

  • ( i) If the network is not weakly reversible, then takes the corresponding system of ordinary differential equations neither a positive nor a steady-state periodic orbit at ( independent of the choice of the kinetic rate constants ).
  • ( ii ) If the network is weakly reversible, then the corresponding system of ordinary differential equations for an arbitrary choice of the kinetic rate constants following properties: each positive stoichiometric compatibility class contains exactly one positive steady state; This positive steady state is asymptotically stable; and there are no non-trivial periodic orbits in.

See or for a proof.

Deficiency -One Theorem

Be a reaction network with deficiency. And be with the deficiencies of the Linkageklassen. Furthermore, it is assumed:

  • ( i);
  • ( iii ) any Linkageklasse contains only one terminal strong Linkageklasse.

If the corresponding ordinary differential equations for a choice of the kinetic rate constants a positive steady state assume then there exists a unique positive steady state in each stoichiometric compatibility class. If the network is weakly reversible, then take the corresponding ordinary differential equations a positive steady state for any choice of the kinetic rate constants.

See or for a proof.

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 decomposed into four independent illustrations, a non-linear and three linear maps

Which are defined below.

Definitions

Basis vectors of the complex space

If, then be

The basis vectors of the complex space are then given by the set.

The amount of the basis vectors represents a matrix and a corresponding sorting the vectors provided is the identity matrix.

The non-linear mapping ψ

Be a chemical reaction network. The nonlinear mapping is given by

With

Matrix Ik

Is supplemented

Matrix Ia

Be a chemical reaction network. The linear map is given by

With.

Matrix Y

Be a chemical reaction network. The linear map is defined by with.

It is in this case so that it can be written in simplified form as (see also stoichiometric matrix).

Example

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

Chemical kinetics Product (chemistry) Function (mathematics)
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