Reaction graphs reactions

Kaufmann s group developed the program GRAMS, a project for machine learning of generic reactions, starting from examples described in the literature. For that they developed the concept of the maximal common substructures between reaction graphs. " Reactions are coded by an extended bond table. 45 000 structures were tested with this system. 

Figure 1. Five representations of the same chemical information. The canonical chemical reaction graph (a) can be represented in linear notation (b, see Appendix) or as a bond-centered labeled graph (c) by using time-variant bonds. The labeled graph affords an adjacency table (d) and a LISP list representation (e).

Reaction graphs are encoded in the same way as static structures. Bonds which change during the reaction are coded as x y where x is the bond type before the reaction and y is the bond type after the reaction. Thus c-c -c represents the reduction of propene to propane and (c-o -cl-c-c-), cl- i represents the formation of tetrahydrofuran and an iodine atom from 4-iodobutan-l-ol. 

A multiscale system where every two constants have very different orders of magnitude is, of course, an idealization. In parametric families of multiscale systems there could appear systems with several constants of the same order. Hence, it is necessary to study effects that appear due to a group of constants of the same order in a multiscale network. The system can have modular structure, with different time scales in different modules, but without separation of times inside modules. We discuss systems with modular structure in Section 7. The full theory of such systems is a challenge for future work, and here we study structure of one module. The elementary modules have to be solvable. That means that the kinetic equations could be solved in explicit analytical form. We give the necessary and sufficient conditions for solvability of reaction networks. These conditions are presented constructively, by algorithm of analysis of the reaction graph. 

A nonempty set V of graph vertexes forms a sink, if there are no oriented edges from At eV to any Ay V. Eor example, in the reaction graph A] <— A2 A3 the one-vertex sets A] and A3 are sinks. A sink is minimal if it does not contain a strictly smaller sink. In the previous example, A] and A3 are minimal sinks. Minimal sinks are also called ergodic components. 

For second order reactions, graphs showing the fractional conversion for various residence times and reactant feed ratios have been drawn up by Eldridge and PlRET(7). These graphs, which were prepared from numerical calculations based on equation 1.46, provide a convenient method for dealing with sets of equal sized tanks of up to five in number, all at the same temperature. 

Recently, structurized forms for steady-state kinetic equations have been obtained that can be written directly from the reaction graph [7-9], Equation (69) is a particular case for these structurized forms. 

Fig. 1. Simple examples for bipartite graphs of reaction mechanisms. , Reaction nodes O, substance nodes.

The presence (or absence) of autonomous groups of substances is easily checked. We assume they are absent. As usual, a more rigorous condition compared with the absence of two autonomous groups is fulfilled. It is the condition of an orientally connected reaction graph. (Here we speak about graphs of linear mechanisms when nodes are substances and edges are elementary reactions.)... 

Let us examine the properties of eqn. (152) under the assumption of oriented connectivity. Let us fix some co-invariant simplex D0 zt 0, , 2,- = C > 0. Da has a unique steady state z°. Vector z° is positive since, due to the connectivity of the reaction digraph, no steady-state points exist on the boundary Da. Indeed, if we assume the opposite (some components z° are zero), we obtain kJt for such i and j as 2° 0 and z° = 0. But from this it follows that, moving along the direction of arrows in the graph of the reaction mechanism, we cannot get from the substances for which 2° 0 to those for which z° = 0, and this is contrary to oriented connectivity (the arrows in the reaction graph correspond, naturally, to the elementary reactions with non-zero rate constants). 

If the reaction graph is orientally connected, the phase space of a linear system (a balance polyhedron) has a metric (154) in which all trajectories of the system monotonically converge and the distance between them tends to zero at t - oo. This holds true for both constant and variable coefficients (rate constants), if in the latter case it is demanded that all rate constants have upper and positive lower limits (0 < a < k(t) < / < oo, a, / = const). 

This equation is independent of the order in which the steps are numbered. Temkin suggested an algorithm on the basis of eqn. (30) to obtain an explicit form of the steady-state kinetic equations. For linear mechanisms in this algorithm it is essential to apply a complex reaction graph. In some cases the derivation of a steady-state equation for non-linear mechanisms on the basis of eqn. (30) is also less difficult. 

Here C(u) is the set of simple cycles in the reaction graph passing through... 

This algorithm permits us to determine the number of parameters "manually on the basis of the reaction graph without derivation of a steady-state kinetic equation. For large-sized and complex-structure graphs it is recommended that the corresponding sets of spanning trees are selected using computations [60]. 

Calculation of the coefficients dt for a given matrix is a very laborious process. We will give a method to calculate these coefficients proceeding directly from the complex reaction graph. Like a steady-state kinetic equation, a characteristic polynomial will be represented in the general (struc-turalized) form ... 

Proof It suffices to prove that the sum of all the feth order minors amounting to the coefficient of /. is at the same time equal to the sum of the weights for all the (n - )-spanning trees of the reaction graphs. At k = 0 the coefficient of 1° amounts to the B(c) matrix determinant. Since, according to the conservation law, any diagonal element of B(c) satisfies the equality... 

The concepts of "graph-molecule and "graph-reaction are natural for chemistry, which is a science which pays much attention to the order of arrangement, bonds, and sequences of transformations. 

By associating the reaction graph with a surface we obtain a minimal crosssection of the actual multidimensional potential energy hypersurface, which still has... 

In what follows, any graph used to study a reaction network will be termed a reaction graph. Although there have been many such uses over the years, there are three general categories which largely cover all uses of graphs ... 

Finally two previous reviews of reaction graphs should be cited in general Bonchev and Mekenyan (1994)1 and Temkin, Zeigamik and Bonchev (1996).2... 

The subscripts which distinguish the steps honor, respectively, Tafel, Volmer and Heyrovsky. Unlike the MCFC cathodic reaction mechanisms, however, these steps combine pairwise to yield the overall reaction. The reaction mechanism graphs for each of the three reaction mechanisms are shown in Figure 6. Notice that it is not possible to represent the entire mechanism by a single reaction mechanism graph. This is because, unlike in the MCFC case, there are now independent full reaction routes which yield the over all reaction. In both of the MCFC examples, there was only one. Still the three separate graphs do clearly convey the three HER reaction routes. 

Joseph D. Fehribach reviews and discusses in Chapter 3 the uses of graphs in the study of chemical reaction network, particularly electrochemical reaction networks for electrochemical systems. He defines any graph used to study a reaction network as a reaction graph. He mentions three categories that cover die uses... 

But the slope of the second graph is zero The rate-determining step does not involve NaOH so adding more of it docs not speed up the reaction. The reaction shows first-order kinetics (the rate is proportional to one concentration only) and the mechanism is called S l, that is, Substitution, Nucleophilic, 1st order. 

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