Matrix stoichiometric

For a systematic treatment of mechanisms, we need a suitable mathematical device. On possibility is to use a stoichiometric matrix to represent the mechanisms in symbolic form. 

We write the reaction mechanism using a the stoichiometric matrix, a. For a mechanism consisting of G gases, S adsorbates including free sites, and R reactions, a is a i by G+S matrix. 

We will frequently need products or sums running over subsets of the molecules. We will use the convention that the molecules are enumerated with gases number 1.G, free sites is number G+1 and adsorbates are number G+2.G+S. 

As an example we will consider the mechanism for ammonia synthesis. We include Ar in the gas phase to illustrate the effect of inerts. This mechanism is rich enough to illustrate most of the features discussed below. 

The stoichiometry for a chemical system, in which only one chemical reaction takes place, is described by 

The above reaction equations can also be written in the form 

It is easy to understand that the stoichiometric Equation 2.1 can equally well be written using the vector notation 

For a system in which several (S) chemical reactions take place simultaneously, for every reaction, we can write an equation analogous to Equation 2.1 here the stoichiometric coefficient for component i in reaction j is denoted by Vij, 

Instead of a vector for the stoichiometric coefficients, we now have a stoichiometric matrix 

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Here 8kj is the Kronecker delta-symbol. Apart from components fka of the matrix of functionalities f and conversion p of the groups A, (37) to the right-hand part of Eqs. (40), there also enter the elements and Kja of the matrix of bonds p and the stoichiometric matrix k, respectively... 

To establish the dependence of quantities py on time it is necessary to determine the dimensionless concentrations Q,-- bonds per monomeric unit. It is sufficient for this purpose to solve Eqs. (33) and to calculate the integral (36). The element Kja of the stoichiometric matrix, according to (41), represents the fraction of all groups Aj in the initial mixture which belong to monomer Ma. 

A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59 62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63 69]. 

Figure 5. A minimal model of glycolysis One unit of glucose (G) is converted into two units of pyruvate (P), generating a net yield of 2 units of ATP for each unit of glucose. Gx, Px, and Glx are considered external and are not included into the stoichiometric matrix. A A graphical depiction of the network. B The stoichiometric matrix. Rows correspond to metabolites, columns correspond to reactions. C A list of individual reactions. D The corresponding system of differential equations. Abbreviations G, glucose (Glc) TP, triosephosphate, P, pyruvate.

Note that Eq. (6) includes thermodynamic equilibrium (v° = 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions. 

Aiming to construct explicit dynamic models, Eqs. (5) and (6) provide the basic relationships of all metabolic modeling. All current efforts to construct large-scale kinetic models are based on an specification of the elements of Eq (5), usually involving several rounds of iterative refinement For a schematic workflow, see again Fig. 4. In the following sections, we provide a brief summary of the properties of the stoichiometric matrix (Section III.B) and discuss the most common functional form of enzyme-kinetic rate equations (Section III.C). A selection of explicit kinetic models is provided in Table I. TABLE I Selected Examples of Explicit Kinetic Models of Metabolisin 1 ... 

The stoichiometric matrix N is one of the most important predictors of network function [50,61,63,64,68] and encodes the connectivity and interactions between the metabolites. The stoichiometric matrix plays a fundamental role in the genome-scale analysis of metabolic networks, briefly described in Section V. Here we summarize some formal properties of N only. 

The stoichiometric matrix N consists of m rows, corresponding to m metabolic reactants, and r columns, corresponding to r biochemical reactions or transport processes (see Fig. 5 for an example). Within a metabolic network, the number of reactions (columns) is usually of the same order of magnitude as the number of metabolites (rows), typically with slightly more reactions than metabolites [138]. Due to conservation relationships, giving rise to linearly dependent rows in N, the stoichiometric matrix is usually not of full rank, but... 

The stoichiometric matrix is characterized by its four fundamental subspaces, two of which are described in more detail below. An examples of each subspace is given in Section III.B.3. 

The left nullspace E of the stoichiometric matrix N is defined by a set of linearly independent vectors ej that are arranged into a matrix E that fulfills [50, 96]... 

As a simple example, consider the minimal glycolytic pathway shown in Fig. 5. The stoichiometric matrix N has m = 5 rows (metabolites) and m = 6 columns (reactions and transport processes). The rank of the matrix is rank(A) = 4, corresponding to m — ran l< (TVj = 1 linearly dependent row in N. The left nullspace E can be written as... 

Starting with an evaluation of the stoichiometric matrix, we obtain the null space matrix K and the link matrix L,... 

Following the workflow depicted in Fig. 26, we start with an analysis of the stoichiometric matrix. The model consists of m = 9 (internal) metabolites and r = 9 reactions, interconnected according to the stoichiometry specified below ... 

The rank of the stoichiometric matrix is rank (A) = 7, corresponding to two mass conservation relationships, namely,... 

The construction of the structural kinetic model proceeds as described in Section VIII.E. Note that in contrast to previous work [84], no simplifying assumptions were used the model is a full implementation of the model described in Refs. [113, 331]. The model consists of m = 18 metabolites and r = 20 reactions. The rank of the stoichiometric matrix is rank (N) = 16, owing to the conservation of ATP and total inorganic phosphate. The steady-state flux distribution is fully characterized by four parameters, chosen to be triosephosphate export reactions and starch synthesis. Following the models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331], 11 of the 20 reactions were modeled as rapid equilibrium reactions assuming bilinear mass-action kinetics (see Table VIII) and saturation parameters O1 1. 

Non-negative integers a,y, are stoichiometric coefficients of component / in reaction i. Matrix F = (y,y) is a stoichiometric matrix of some rank (rk), with elements jij — fiij—ocij. 

Up to the scaling, they are co-factors Ag of elements of any column of stoichiometric matrix F (see, for instance, Bykov et ah, 1998 Lazman and Yablonskii, 1991). We can always assign the directions of elementary reactions so that all stoichiometric coefficients are non-negative and this will be assumed later. 

In this case we don t make any approximations in the reaction scheme. We consider a Langmuir Hinshelwood mechanism with stoichiometric matrix ttrc. 

A stoichiometric matrix is one whose elements are the stoichiometric coefficients of the reacting substances. Its rows correspond to the reactions and its columns are the reacting substances. 

The size of a stoichiometric matrix is (s-by-N), where s is the number of reactions and N the number of reactants in the system. Stoichiometric equations for a complex reaction can be represented as... 

What requirements must the stoichiometric matrix T fit Chemists choose stoichiometric coefficients such that, in each reaction, the number of atoms on the left-hand and right-hand sides are the same for every element. Hence the law of constant mass for atoms of a given type must hold over the reaction steps. In matrix representation, this requirement is of the form... 

We obtain a matrix in which all the elements are zero (a zero matrix). Hence the stoichiometric matrix is written correctly. 

Equation (25) makes it possible to construct correctly the stoichiometric matrix r for a given number of substances and, hence, molecular matrix A. Among the rows of this matrix one can find those that are linearly dependent. Thus, in matrix (23), the third row will be obtained if one adds the two upper rows and multiplies the sum by 1/2. 

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