Reaction networks linear

Now we have the complete theory and the exhaustive construction of algorithms for linear reaction networks with well-separated rate constants. There are several ways of using the developed theory and algorithms ... 

Fig. 7.10 Linear reaction network. All reactions are first-order mass action kinetics. The back reactions (A i to A g) are always 0.1 s fci is set to 1.0 s . The remainder of the forward rate constants are broken into three groups, as shown. Within a group, all the forward coefficients are assumed to be identical. These rate constants are chosen from three possible rates 0.7, 70.0, and 7,000.0 s (slow (S), medium (M), and fast (F). (From [1].)...

The task now is to select the linear combinations that will most probably correspond to independent parts of the reaction network with easily interpretable stoichiometry. A simplification of the data in the matrix can be achieved by such a rotation that the axes go through the points in Fig. A-2 (this is equivalent to some zero-stoichiometric coefficients) and that the points of Fig. A-3 are in the first quadrant (this corresponds to positive reaction extents) if possible. Rotations of the abscissa through 220° and the ordinate through 240° lead to attaining both objectives. The associated rotation matrix is ... 

For a complex system, determination of the stoichiometry of a reacting system in the form of the maximum number (R) of linearly independent chemical equations is described in Examples 1-3 and 14. This can be a useful preliminary step in a kinetics study once all the reactants and products are known. It tells us the minimum number (usually) of species to be analyzed for, and enables us to obtain corresponding information about the remaining species. We can thus use it to construct a stoichiometric table corresponding to that for a simple system in Example 2-4. Since the set of equations is not unique, the individual chemical equations do not necessarily represent reactions, and the stoichiometric model does not provide a reaction network without further information obtained from kinetics. 

It can be straightforwardly verified that indeed NK = 0. Each feasible steady-state flux v° can thus be decomposed into the contributions of two linearly independent column vectors, corresponding to either net ATP production (k ) or a branching flux at the level of triosephosphates (k2). See Fig. 5 for a comparison. An additional analysis of the nullspace in the context of large-scale reaction networks is given in Section V. 

Even though the governing phenomena of coupled reaction and mass transfer in porous media are principally known since the days of Thiele (1) and Frank-Kamenetskii (2), they are still not frequently used in the modeling of complex organic systems, involving sequences of parallel and consecutive reactions. Simple ad hoc methods, such as evaluation of Thiele modulus and Biot number for first-order reactions are not sufficient for such a network comprising slow and rapid steps with non-linear reaction kinetics. 

If the reader can use these properties (when it is necessary) without additional clarification, it is possible to skip reading Section 3 and go directly to more applied sections. In Section 4 we study static and dynamic properties of linear multiscale reaction networks. An important instrument for that study is a hierarchy of auxiliary discrete dynamical system. Let A, be nodes of the network ("components"), Ai Aj be edges (reactions), and fcy,- be the constants of these reactions (please pay attention to the inverse order of subscripts). A discrete dynamical system

dynamical system for a given network we find for each A,- the maximal constant of reactions Ai Af k ( i)i>kji for all j, and — i if there are no reactions Ai Aj. Attractors in this discrete dynamical system are cycles and fixed points. 

The catalytic cycle is one of the most important substructures that we study in reaction networks. In the reduced form the catalytic cycle is a set of linear... 

For kinetic systems with well-separated constants the left and right eigenvectors can be explicitly estimated. Their coordinates are close to +1 or 0. We analyzed these estimates first for linear chains and cycles (5) and then for general acyclic auxiliary dynamical systems (34), (36) (35), (37). The distribution of zeros and +1 in the eigenvectors components depends on the rate constant ordering and may be rather surprising. Perhaps, the simplest example gives the asymptotic equivalence (for of the reaction network A,+2 with... 

In this chapter, we study networks of linear reactions. For any ordering of reaction rate constants we look for the dominant kinetic system. The dominant system is, by definition, the system that gives us the main asymptotic terms of the stationary state and relaxation in the limit for well-separated rate constants. In this limit any two constants are connected by the relation or... 

The extreme pathways are a basis (in the sense of vector spaces) for all routes through the reaction network, i.e., ary route can be written as a linear combination of the extreme pathways. When all the reaction steps are irreversible, these two sets are equivalent, i.e., they have the same elements. When some of the steps are reversible, there are more (perhaps far more) elementary modes, though this set is still much smaller than the countably infinite set of all routes. In any event, all routes can be obtained as combinations of the elements of either set, and thus if one studies these relatively small sets of routes, one can hope to obtain results for all routes. 

At this point, we should mention the difference between independent chemical equations and independent chemical reactions. The former are of mathematical significance, being helpful to carry out consistent material balance. The latter are useful for describing the chemical steps implied in a chemical-reaction network. They may be identical with the independent stoichiometric equations, or derived by linear combination. This approach is useful in formulating consistent kinetic models. 

Analogous to the question of maximal ATP production, we can use linear programming to compute the maximum possible production of other cofactors. For example, this reaction network may be used in the cell to generate NADPH to be used in other metabolic pathways. To compute the maximal NADPH yield, we have to add a transport flux for NADPH to by adding a new column with non-zero entry 59,43 = — 1. 

The four materials are (1) polymer 1 linear and polymer 2 crosslinked (2) polymer 1 crosslinked and polymer 2 linear (3) and (4) are obtained by interchanging polymers 1 and 2. In each case the second polymerized material is grafted to the first polymerized material. Equations 9-14 and the related discussion result in 72 possibilities. The difference lies in the counting or omission of the time sequence of events. The reaction network scheme does not consider the importance of the time-order of events. Thus Equation 31 yields the minimum number of distinguishable materials. 

Thus four protons are produced when going through the cycle once. In chanical oscillators usually the proton is the autocatalyst species, too (that is why oscillators as a rule do operate in protic solvents (water, methanol, formic acid) only), with > 5 in several cases (Pota and Stedman 1994). From this value, stoichiometries of such processes must be substracted which wiU decompose, bind or remove (from the reaction network) the autocatalyst species. Non-linear modes of chemical... 

To construct the pertraction network, the particular reaction networks should be added via 0 junctions to three linear networks corresponding with the processes of diffusion of C2A, C2B, and CH carrier species. The resulting reaction-diffusion network, as presented in Figure 13.6, consists of four coupled loops representing the pertraction of A " ", and H" " cations. The loops are coupled by common capacitances Aj, Bj, and Hj (i = f, s) and by the capacitances CHj and CH for a reacting or diffusing acidic form of the carrier. From the network in Figure 13.6, all the model equations used further in numerical calculations can be deduced with the help of Kirchhoff s law for a 0 junction (KCL) ... 

The structured character of biochemical reaction networks is exploited by the synthesis algorithm in early pruning and abstraction, with significant gains in computational efficiency. This happens because the algorithm processes first those compounds and reactions that lead to few or no new combinations. This is shown schematically in Fig. 19, where some irrelevant portions of the network are pruned and some linear chains of reactions are compacted. We should note that this kind of reaction sequences are very common in biochemical reaction systems. 

Complex Rate Equations The examples above are for special cases amenable to simple treatment. Complex rate equations and reaction networks with complex kinetics require individual treatment, which often includes both numerical solvers for the differential and algebraic equations describing the laboratory reactor used to obtain the data and linear or nonlinear parameter estimation. 

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