# 1.A. Chemical Reaction Network Theory: Early Review Articles

Although they are now a little outdated, the following reviews for a general audience give a fairly good summary of some early results in chemical reaction network theory. These early results tie the qualitative behavior of mass action systems to a property of the underlying network called the network’s *deﬁciency*. (The deﬁciency is a non-negative integer index with which reaction networks can be classiﬁed.) A very early deﬁciency-oriented article can be found here:

The following review is, I think, a good place to begin. It was written in connection with a symposium attended by a nice mix of chemists, engineers, and mathematicians.

The following articles constitute a two-part review, written a little later than the 1980 review. They take things somewhat further and provide much more detail.

The next review was written even earlier than any of the previous ones. Among other things, it attempts to explain for a chemical engineering audience a little bit about the proof of the Deﬁciency Zero Theorem. This review highlights the crucial work of Horn and Jackson, but it also uses a line of argument that’s a little diﬀerent from theirs.

If, however, you are a mathematician seeking proofs, then I’d strongly recommend *Lectures on Chemical Reaction Networks*, which are a written version of about half the lectures I gave in 1979 at the Mathematics Research Center of the University of Wisconsin. (I got distracted by foundations of classical thermodynamics before I wrote the remaining half. You’ll see some of that work, appearing at about the same time, elsewhere in this bibliography.) *Lectures* will take you up through a proof of the Deﬁciency Zero Theorem.

Here is a link to the written lectures:

Although it was written substantially later, I am including with this set of articles the following one because it travels further down the same road. The article describes progress, based on the PhD work of Phillipp Ellison, toward extending previous results about deﬁciency one networks to networks of higher deﬁciency. In particular, Phillipp’s work, like deﬁciency one theory, aims to translate questions about a network's capacity for multiple steady states (and therefore questions about *systems of nonlinear equations*) into questions about *systems of linear inequalities*, which are far more tractable. Phillipp’s work in both theory and software development provided the basis for a major improvement in *The Chemical Reaction Network Toolbox*.