Stoichiometric Models
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1 1.1. MASS-BALANCE EQUATION Stoichiometric Models 1.1 Mass-Balance Equations According to the law of conservation of mass, any observed net change in the amount of a species must be due to the difference between the inward and outward flows from the species pool (Figure 1.1). The Inflows X Outflows dx= D P Inflow P Outflows Figure 1.1: Mass Balance: The rate of change in species X is equal to the difference between the sum of the inflows and the sum of the outflows equations which describe such flows are called the mass balance equations and are central to building mathematical models of cellular networks: ds i D X X Inflows Outflows (1.1) A reaction rate which contributes to a flow is given by the term, c ij v j, where c ij is the stoichiometry and v j the reaction rate. Therefore the balance equation can be written (under constant volume conditions) in the more formal way as: ds i D X j c ij v j (1.2) that is the sum over all flows into and out and of a particular species pool. In the equation, S i is the concentration of species i, c ij is the stoichiometric mole number for species i with respect to reaction j
2 2 and v j is the rate of reaction for reaction j. Since the stoichiometries have appropriate signs depending on whether they refer to products or reactants the summation correctly accounts for the difference between in and out flows. Consider a simple linear chain of reactants from to S 5 shown in Figure 1.2. The mass-balance equations for this simple system can be written as shown in equation 1.3. S 3 S 4 S 5 Figure 1.2: Simple Straight Chain Pathway. d ds 3 D d D ds 4 D D ds 5 D (1.3) Each species in the network is assigned a mass-balance equation which accounts for the flows into and out of the species pool. For a branched system such as the following: v 5 Figure 1.3: Multi-Branched Pathway.
3 1.1. MASS-BALANCE EQUATIONS 3 the mass-balance equations are given by: d D d D v 5 Finally consider a more complex pathway such as: A C X! 2X X C Y! Z Z! Y C B This example is more subtle because we must be careful to take into account the stoichiometry change between the reactant and product side in the first reaction ( ). In reaction, the overall stoichiometry for X is C1 because two X molecules are made for every one consumed. Taking this into account the rate of change of species X can be written as: dx D C 2 or more simply as. The full set of mass-balance equations can therefore be written as: da dy D dx D dz db D D D It is therefore fairly straight forward to derive the balance equations from a visual inspection of the network. Many software tools exist that will assist in this effort by converting network diagrams, either represented visually on a computer screen or provided as a text file listing the reactions in the network.
4 4 1.2 Stoichiometry Matrix Stoichiometry refers to the molar proportions of reactants and products in a chemical reaction. Given a hypothetical reaction such as: 3A C 4B! 2C C D with reactants A and B and products C and D, the stoichiometry is indicated by the number of participating reactant and product molecules. Thus the stoichiometry for A is three, for B, four, for C, two and for D, one. When describing multiple reactions in a network, it is convenient to represent the stoichiometries in a compact form called the stoichiometry matrix, N. This matrix is a m row by n column matrix where m is the number species and n the number reactions. The columns of the stoichiometry matrix correspond to the distinct chemical reactions in the network, the rows to the molecular species, one row per species. Thus the intersection of a row and column in the matrix indicates whether a certain species takes part in a particular reaction or not, and, according to the sign of the element, whether there is a net loss or gain of substance, and by the magnitude, the relative quantity of substance that takes part in that reaction. The elements of the stoichiometry matrix thus concern the relative mole amounts of chemical species that react in a particular reaction; it does not concern itself with the rate of reaction. For example, consider the simple chain of reactions which has five molecular species and four reactions. The four reactions are labeled, to. S 3 S 4 S 5 The stoichiometry matrix for this simple system is given by:
5 1.2. STOICHIOMETRY MATRIX 5 N D S 3 S 4 S 5 Entries in the stoichiometry matrix are computed as follows. Given a species S i and reaction v j, the corresponding entry in the stoichiometry matrix at ij (row i and column j ) is given by the total stoichiometry for S i on the product side minus the total stoichiometry of S i on the reactant side. Thus, considering species in reaction, we note that the total stoichiometry on the product size is zero (no molecules are formed on the product side) and the total stoichiometry of on the reactant side is C1. Subtracting one from the other (0 1) we obtain 1, which is entered into the stoichiometry matrix. This rather long winded approach to computing stoichiometries avoids errors that arise when a species occurs as both reactant and product. For example, for a more complex reaction such as: 3A! 2A C B the stoichiometry entry for A is.2 3/ that is 1, because the stoichiometry for A on the product side is 2 and on the reactant side is The System Equation Equation 1.2, which describes the mass balance equation, can be reexpressed in terms of the stoichiometry matrix to form the system equation. ds D Nv (1.4) where N is the m n stoichiometry matrix and v is the n dimensional rate vector, whose ith component gives the rate of reaction i as a function of the species concentrations.
6 6 Looking again at the model depicting the simple chain of reactions, the system equation can be written down as: ds D Nv D If stoichiometry matrix is multiplied into the rate vector, the massbalance equations show earlier (1.3) are recovered. As already stated, the stoichiometry matrix represents the connectivity of the network and contains information on the network s structural characteristics. These characteristics fall into two groups, relationships among the species and relationships among the reaction rates. Each will be considered in turn. Let us first consider the relationships among the species, that is relationships between the rows of the stoichiometry matrix Row and Column Properties The system equation 1.4 gives a consist representation of a stoichiometry model. More importantly it separates two distinct types of information, topology and kinetics. The stoichiometry matrix, N, describes the topological characteristics of the network while the v vector describes the kinetics of the individual steps. The remainder of this chapter will be concerned with the properties of the stoichiometry matrix and what useful information can be extracted from it. Information in the stoichiometry matrix is stored in the matrix rows and columns. In particular the rows include information on the constraints related to the species concentrations, while the columns contain information on constraints among the pathway fluxes.
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