# Chemical Reaction Network Theory

## Martin Feinberg Group for Chemical Reaction Network Theory

*Mathematics of Complex Chemical Systems*

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*Mathematics of Complex Chemical Systems*

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## ABOUT

Reaction systems with complex chemistry give rise to complicated systems of nonlinear equations that don’t lend themselves to analytic solution. What’s more, increased complexity in the governing equations can give rise to complicated new phenomena that simple textbook reactors don’t admit. Even in the constant temperature case common in biology there can, for example, be unstable steady states, multiple steady states, sustained composition oscillations, and wild, chaotic dynamics.

Since each new network of chemical reactions gives rise to its own complicated system of differential equations, it becomes apparent that, in the absence of an overarching theory, we would be forced to study reaction systems on a case-by-case basis, and each new case would be fraught with terrible analytical difficulties. What’s needed is a way of looking at things from a broader and more general perspective.

That’s what *Chemical Reaction Network Theory* tries to do. The aim of the theory is to tie aspects of reaction network structure in a precise way to the kinds of dynamics the network might admit. A lot of progress has been made along these lines, but there is also much that remains unknown.

This website has resources both for those interested in the mathematics behind Chemical Reaction Network Theory and for those simply interested in seeing results that the theory delivers. For those that want to learn more about the underlying mathematics, *Lectures on Chemical Reaction Networks* provide a somewhat dated but nevertheless useful introduction. For those interested in seeing user-friendly computer implementation of the theory, *The Chemical Reaction Network Toolbox* provides means to make otherwise difficult deductions about reaction network behavior. More generally, there is an annotated bibliography of work with students and collaborators over the years. The bibliography is intended as a guide not only to work on reaction network theory but also to loosely related work on foundations of classical thermodynamics and the theory of reactor design.

The book "Foundations of Chemical Reaction Network Theory" was published by Springer in early 2019.

B.Ch.E., Cooper Union (1962)

M.S., Purdue University (1963)

Ph.D., Princeton University (1968)

Richard M. Morrow Professor Emeritus Feinberg has been a pioneer in the rapidly emerging field of chemical reaction network theory, which aims to tie aspects of reaction network structure in a precise way to the variety of qualitative behaviors that might be engendered.

Prized by his students for his clear elucidation of complex concepts, he won Ohio State's highest teaching award, the Alumni Award for Distinguished Teaching, in 2014.

He is the author of Foundations of Chemical Reaction Network Theory (Springer, 2019). The book will be of value to mathematicians and mathematically-inclined biologists, chemists, physicists, and engineers who want to contribute to chemical reaction network theory or make use of its powerful results. Peer reviewers have described the book as "awe-inspiring" and highly original: "There is nothing even remotely comparable in the literature." Since its publication, there have been over 21,000 chapter downloads.

He is no longer accepting students.

- Ohio State University Alumni Award for Distinguished Teaching (highest teaching honor at Ohio State; sole recipient), 2014
- Wilhelm Lectures, Princeton University, 2011
- Amundson Lectures, University of Houston, 2006
- John Von Neumann Lecture in Theoretical Biology, Institute for Advanced Study, 1997
- Wilhelm Award, American Institute of Chemical Engineers, 1996
- MacQuigg Award for Undergraduate Teaching, Ohio State University, 1999
- Edward Peck Curtis Award for Excellence in Undergraduate Teaching, University of Rochester, 1994
- Camille & Henry Dreyfus Teacher- Scholar, 1974

Visit the Book, Lectures, and Bibliography sections below or Google Scholar for information about publications.

## RESEARCH

* Chemical Reaction Network Theory.* My students and I are interested in complex chemical systems in which several reactions occur simultaneously. Real systems are almost always of this kind, so it becomes important to understand reactors with complicated chemistry in a systematic way.

Complex chemistry gives rise to intricate systems of nonlinear equations that don’t lend themselves to analytic solution. What’s more, increased complexity in the governing equations can give rise to complicated new phenomena that simple systems don’t admit. Even in the isothermal setting normally studied in biology there can, for example, be unstable steady states, multiple steady states, sustained composition oscillations, and wild, chaotic dynamics—possibilities we need to take into account.

Since each new network of chemical reactions gives rise to its own complicated system of differential equations, it becomes apparent that, in the absence of an overarching theory, we would be forced to study complex chemical systems on a case-by-case basis, and each new case would be fraught with terrible analytical difficulties. What’s needed is a way of looking at things from a broader and more general perspective.

That’s what Chemical Reaction Network Theory tries to do. The aim of the theory is to tie aspects of reaction network structure in a precise way to the variety of qualitative behaviors that might be engendered. A lot of progress has been made along these lines, but there is also much that remains unknown. For more on chemical reaction network theory, see the annotated bibliography.

* Other Areas of Study.* Although my attention now is largely focused on chemical reaction network theory, with particular reference to biology, I maintain an interest in two other areas with which I was intensely occupied in the past. One of these is mathematical foundations of classical thermodynamics. Another is a general theory of reactor-separator design, which has some ties, at least in spirit, with classical thermodynamics and with reaction network theory. In particular, I am interested in understanding theoretical limits to what can be achieved, consistent with certain design constraints, over all possible steady-state designs (even unimagined ones) that are consistent with those constraints. Articles about both topics can be found in the annotated bibliography.

## BOOK

*Foundations of Chemical Reaction Network Theory* was published by Springer in 2019.

**Availability**

At Springer: If your institution subscribes to SpringerLink you can download the book here. Otherwise it can be obtained from Springer here.

At Amazon the book can be obtained here.

* Corrections: *A list of corrections can be found here

## TOOLBOX

Welcome to the Chemical Reaction Network Toolbox download page. The Toolbox is available as its latest release, Version 2.35 (which was written for Windows), or as its DOS version. The Windows version can run on Mac or Linux operating systems with the aid of the freeware Winebottler or a similar program. The DOS version, which has a powerful simulation module absent in the Windows version, can also run on any system under the auspices of the freely available DOSBox program. For a more detailed explanation of the two Toolbox versions' capabilities and differences, please see the text below, Toolbox History and Explanation.

You can download the current version along with the Guide here: The Chemical Reaction Network Toolbox Ver. 2.35. Programming for Versions 2.X was the result of dedicated work by Phillipp Ellison, Haixia Ji, and Daniel Knight.

Here is a link to the older DOS version: The Chemical Reaction Network Toolbox, Version 1.1.

**New in Version 2.35:** Version 2.35 is the same as Version 2.3 with the addition of one module, accessible from the Reports menu (after running the Basic Report). When I was writing *Foundations of Chemical Reaction Network Theory* (Springer), it occurred to me that it would be great if I could enter a network in the Toolbox and then get LaTeX code for a network display along with code for the corresponding system of differential equations. These could then be copied and pasted into LaTeX manuscripts for articles and books. After I finished *Foundations*, I decided to write such a module and add it to the Toolbox for the benefit of others. This should save you lots of work!

Best wishes,

Marty

**Toolbox History and Explanation**

The Toolbox is a work in progress. We are making available a copy of The Chemical Reaction Network Toolbox, Version 2.35. Versions 2.X were intended to implement in Windows various parts of Chemical Reaction Network Theory that appeared over time in the literature. Naturally, literature articles are the primary source for detailed information about what the Toolbox is intended to do, but the Guide that comes packaged with the program does provide some information for newcomers.

Versions 2.X are descendants of Versions 1.X, which were written many years ago for the Microsoft DOS operating system. (Yes, DOS!) In some limited respects, the DOS versions remain more powerful. In particular, the DOS versions contained a ChemLab component, which provided numerical solutions (and their graphical display) for the network differential equations that derive from mass action systems. People who want those capabilities can still download the final DOS version. It can be run very nicely in Windows, on a Mac, or on a Unix machine -- all under the freely available DOSBox emulator

One advantage of Versions 2.X is, of course, that they are written for Windows, not DOS. But they also extend considerably the power of the old Network Analyst component of Version 1.X, which was centered around deficiency-oriented parts of Chemical Reaction Network Theory. (The deficiency is a non-negative integer index with which reaction networks can be classified.)

The very first DOS version of the Toolbox -- Version 1.0 -- was aimed at implementing theory for networks of deficiencies zero and one. For those networks it could determine whether there could be *any* set of rate constants for which the corresponding differential equations admit two distinct positive steady states that are stoichiometrically compatible. Version 1.1 was intended to also implement work in Phillipp Ellison's Ph.D. thesis, which extended deficiency one theory to large classes of higher deficiency networks.

The Windows versions (Versions 2.X) contain successive improvements resulting from advances in Chemical Reaction Network Theory. The earliest versions implemented theory in the Ph.D. thesis of Haixia Ji, which extends Phillipp Ellison's work.

Somewhat later ones implemented a different strand of theory that had its origins in work of Ph.D. student Paul Schlosser and that took another direction in the Ph.D. work of Gheorghe Craciun. With Gheorghe's work a central question became whether a network has "the mass action injectivity property." If the network does have the mass action injectivity property, then multiple positive stoichiometrically compatible steady states are impossible, no matter what the rate constants are. In those later versions, the Toolbox could tell you whether a network does (or does not) have the mass action injectivity property.

Still later, another strand of reaction network theory was implemented in the Toolbox, this one the result of work with Guy Shinar. Unlike the others, this strand is *not* tied to the presumption of mass action kinetics. If a network is endowed with a subtle structural property called *concordance*, then -- so long as the kinetics falls within a large and natural class (which embraces the mass action class as a special case) -- the resulting dynamical equations *must* have certain attributes. In particular, there is no possibility of switch-like transitions between two distinct stoichiometrically compatible positive steady states. As of Version 2.1, there were modules that determine whether or not a network has the concordance property (or the more powerful strong concordance property). In Version 2.2 there were improvements to the user interface and also an additional module that implements still more general aspects of concordance theory (related to *concordance with respect a species influence specification* -- that is, a user-provided specification of which species are to be regarded as inducers or inhibitors of the various reactions). Version 2.35 has further enhancements and provides more incisive reports.

Network entry in Versions 2.X is fairly intuitive. Networks can be saved in *.net files. After entering the network, you should go to the Reports menu and run the Basic Report. That report will guide you to other reports that might be relevant for the network at hand. (The Mass Action Injectivity Report and the various Concordance Reports are always options after the Basic Report has been run.)

## LECTURES

You'll find below links to scanned copies of typewritten versions of 4½ out of 9 lectures that I gave at the Mathematics Research Center of the University of Wisconsin-Madison in the autumn of 1979. (Typing was by the wonderful Donna Porcelli.) The occasion was a semester-long in-gathering of people interested in the behavior of complex chemical systems. At the end of that period there was a large meeting, the proceedings of which were published by Academic Press in a book, "Dynamics and Modeling of Reactive Systems," edited by W. Stewart, W. H. Ray and C. Conley. My chapter amounted to a summary of some of the things I talk about during the course of the nine lectures.

It was an exciting time, which began when Charles Conley called me at the University of Rochester. He explained the MRC plans for 1979 and asked if I could spend a semester in Wisconsin. He said that they would pay my salary, provided that my salary wasn't too high. I told him my salary. Conley asked if I could come for a year.

The mix of long-term visitors and permanent Wisconsin people made for a very stimulating mini-sabbatical. Rutherford Aris, Harmon Ray, Warren Stewart, Manfred Morari, and Klavs Jensen were among the active chemical engineers. Among the mathematicians were Charles Conley, Chris Jones, Neil Fenichel, and Mike Crandall. At the time, John Nohel was the gracious director of the MRC. I couldn't have asked for a better group of people to whom I might expose recent thinking about reaction networks.

The lectures have origins in work initiated by Fritz Horn and Roy Jackson. Much has happened since then, only a small part of it reflected in the lectures, but the early work of Fritz and Roy planted seeds without which little fruit would have come into being. For me, Fritz, Roy, and Rutherford Aris have been towering examples of chemical engineering scholars and, more importantly, friends. Fritz died before these lectures were given. He was brilliant. Even now, when I do something I think is nice, I wish Fritz were here to see it.

People who knew Charles Conley -- and those who know the work he left behind -- agree that he was a dazzling, remarkably innovative mathematician. His great interest in my lectures was an extremely important source of encouragement. (I didn't know at the time that Charlie was interested in just about everything!)

Here, then, are links to the lectures.

Lecture 1. Introduction (1500KB)

Lecture 2. Reaction Networks, Kinetics, and the Induced Differential Equations (1900KB)

Lecture 3. Two Theorems (2190KB)

Lecture 4. Some Definitions and Propositions (1963KB)

Lecture 5. Proof of the Deficiency Zero Theorem (1103KB)

References. Partial Bibliography for Lectures 1-5 (93KB)

Martin Feinberg

Chemical and Biomolecular Engineering & Mathematics

The Ohio State University

feinberg.14@osu.edu

## ANNOTATED BIBLIOGRAPHY

I thought it would be useful to divide this partial reaction network theory bibliography into three parts:

The first section, a) EARLY REVIEWS, below, contains some expository articles on early results in chemical reaction network theory, mostly written for a mixed audience of chemical engineers, chemists, and mathematicians.

The second section, b) LESS MATHEMATICAL ARTICLES, below, contains more recent papers which, although they have mathematical content, are are also aimed at a wide audience. Some of these are explicitly biological in flavor.

The third section, c) MORE MATHEMATICAL ARTICLES, below, contains articles that have proofs. However, in these articles it is often the case that results are stated in friendly terms at the beginning, and only later do the more mathematical parts begin. Papers [1.C6] and [1.C10] are examples.

**If you have trouble obtaining any of these, especially those without links, please contact me at feinberg.14@osu.edu.**

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.

[1.A5] M. Feinberg. Mathematical aspects of mass action kinetics. In N. Amundson and L. Lapidus, editors, *Chemical Reactor Theory: A Review*, pages 1–78. Prentice-Hall, Englewood Cliﬀs, NJ, 1977. *(Peason Publishing, unlike other publishers, refused permission for me to provide, for download, a scanned copy of this very old, out-of-print article. Pearson now owns Prentice-Hall.)*

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.

Networks.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*.

1.B. Chemical Reaction Network Theory: Less Mathematical Articles

Although the following articles have mathematical content, they were not written with an audience of mathematicians in mind. Some are biological in ﬂavor, while others (the ones with “CFSTR” in the title) were meant more for chemical engineers. In each case, though, the writing is directed only a little toward the nominal target audience.

It will help non-chemical engineers to know that “CFSTR” means *continuous ﬂow stirred tank reactor*. In effect, this is just a stirred cell with a feed stream of ﬁxed composition continuously entering it and a exit stream continuously leaving it. In the exit stream every species is withdrawn at a rate proportional to its concentration within the cell. The mathematics is roughly the same as that for a biochemical system in which every species is synthesized at a ﬁxed rate (perhaps zero) and every species degrades at a rate proportional to its current concentration.

I’ll begin with a “CFSTR” article, based on remarkable PhD work by Paul Schlosser, that moved chemical reaction network theory results in a direction substantially different from the deﬁciency-oriented ones. The newer results, organized around the *Species-Complex-Linkage Graph*, were a precursor of later results centered on the *Species-Reaction Graph*, which in turn came from Gheorghe Craciun’s (also remarkable) PhD thesis.

The following article gives a description of results in Gheorghe’s thesis, centered around the Species-Reaction Graph, this time with an explicitly biological ﬂavor:

The next review article is much more recent than the preceding ones and chronologically out of sequence. It is in the same vein as [1.B1] and [1.B2], but there is no reliance on the presumption of mass action kinetics. Although the underlying mathematical machinery is not made explicit, the results derive from the theory of *concordan**t chemical reaction networks* described elsewhere in this bibliography and developed in collaboration with Guy Shinar. Much like prior work by Banaji and Craciun, the theory of concordant reaction networks extends earlier Species-Reaction Graph results to broad categories of kinetics.

The next articles are different from the preceding ones because they deal with a very different issue: *concentration robustness*. It seems that certain reaction networks have the capacity to maintain the steady-state concentration of a certain critical species within very narrow bounds even against very large changes in the total supplies of the various network constituents. The ﬁrst article [1.B4] connects that capacity to reaction network structure (and draws on deﬁciency one theory). The second article [1.B5] sets the ﬁrst article within a broader context.

**Martin Feinberg Group for****Chemical Reaction Network Theory**

1.C. Chemical Reaction Network Theory: More Mathematical Articles

It’s worth repeating here that, for mathematicians who want to go deeply into chemical reaction network theory, I think the best entry point is

The following article was my ﬁrst venture into the mathematics of complex chemical systems. Its aim was to determined what can be said about rate constants and the reaction network itself when one has information about composition dynamics near an equilibrium point. (It’s assumed that the kinetics is mass action and exhibits detailed balance.) Item [1.C2] is a “pure math” article that was a consequence of things in [1.C1].

The papers [1.C3]–[1.C5] below provide the basis for what is now known as the Deﬁciency Zero Theorem. (Although they do not derive from work by me or my students, the articles by Fritz Horn and Roy Jackson are listed here because they provide the foundation for the Deﬁciency Zero Theorem story.)

In a seminal and beautiful paper [1.C3], Horn and Jackson invented the idea of complex balancing at an equilibrium as a generalization of what chemists call detailed balancing. They showed in [1.C3] that if a mass action system admits a complex balanced equilibrium at which all species concentrations are positive, then the corresponding differential equations admit only behavior of a special dull kind. In [1.C4] Horn went further; in particular, he examined conditions on rate constants in a mass action system (analogous to the Wegscheider conditions for detailed balancing) that ensure the existence of a positive complex balanced equilibrium. In [1.C5] I showed that, when what has come to be called the deﬁciency of the underlying reaction network is zero, then complex balancing must obtain at every equilibrium, independent of what the kinetics might be.

The following paper is out of chronological sequence, but it is close in spirit to the immediately preceding ones. The paper was in response to a question put to me by Roy Jackson. He wanted to know of a systematic way to state conditions on mass action rate constants that are necessary and sufficient for the existence of a positive equilibrium at which detailed balance obtains. It turns out that there is a nice connection of this with the underlying network’s deﬁciency and its cycle structure.

The next paper is largely a technical one, but it is a workhorse in many of the reaction network theory papers that followed it. Especially important is the Appendix. Readers familiar with the mathematics of reaction network theory will perhaps know that, when the kinetics is mass action, the species formation rate function is often written in the decomposed form YA_{k}Ψ(c) — see for example §4.A in *Lectures on Chemical Reaction Network*s. The Appendix of [1.C7] deals with the connection between reaction network structure and qualitative properties of the kernel (nullspace) of A_{k}.

Papers [1.C8] and [1.C9] provide the foundations for deﬁciency one theory. In particular, [1.C8] has a proof of the *Deﬁciency One Theorem*, and [1.C9] gives the mathematics behind the *Deﬁciency One Algorithm*, which was implemented in even the earliest versions of *The Chemical Reaction Network Toolbox*. The algorithm converts questions about the capacity of deﬁciency one mass action systems to admit multiple steady states (essentially questions about systems of nonlinear polynomial equations) into questions about systems of linear inequalities, for which there is a well-developed theory.

The next article [1.C10] is out of chronological order, but it has resonances with papers dealing with detailed balancing, complex balancing, the Deﬁciency Zero Theorem, and the Deﬁciency One Theorem. The central question in [1.C10] is about the robustness of steady states: How does reaction network structure alone inﬂuence the sensitivity of the steady state concentration of a particular species to a perturbation in the supply of another species?

It turns out that such sensitivities often have bounds that depend in a striking way on the extent to which the various molecular species are constructed from a large number of distinct building blocks (as are proteins) and, also, the extent to which those building blocks combine gregariously. I have great affection for this paper because it was the beginning of my collaboration with Guy Shinar. The paper is an elaboration on work done by Guy when he was a student in Uri Alon’s lab at Weizmann Institute. (The relationship between [1.B4] and [1.C10] is discussed in [1.B5].)

Another robustness/sensitivity article can be found here:

In one way or another the articles [1.C12]-[1.C17] provide mathematical foundations for [1.B.2], the more friendly expository *PNAS* paper that draws connections between the capacity of a (mass action) network to give multiple steady states and the nature of the network’s Species-Reaction Graph. I now think that the self-contained articles [1.C16] and [1.C17] substantially transcend [1.C12]-[1.C15] in terms of results, while at the same time proceeding in a different, more economical way. Some readers might want to start there. However, [1.C12]-[1.C15] give an alternative route, based largely on Gheorghe Craciun’s remarkable PhD work — a route that remains highly compelling.

Both [1.C12] and [1.C13] presume mass action kinetics. In this context, [1.C12] provides a determinant condition which, for a given reaction network, serves to ensure *injectivity*. Injectivity is a network property that, among other things, precludes the possibility of multiple positive steady states, regardless of rate constant values. In turn, [1.C13] provides conditions on a network’s Species-Reaction Graph that suﬃce to ensure that the determinant condition of [1.C12] is satisﬁed. From this, injectivity and multiple-steady-state-preclusion follow.

In very surprising subsequent work, Gheorghe Craciun and Murad Banaji subsequently proceeded in a different way to show that Species-Reaction Graph conditions that suffice for injectivity (and, therefore, the preclusion of multiple steady states) in the context of mass action kinetics also suffice for injectivity for the far wider class they called *non-autocatalytic (NAC) kinetics*.

However, [1.C12], [1.C13] — and the Craciun-Banaji work — have limitations: Crucial to the mathematics is the presumption that the reaction network under consideration is *fully open.* This amounts to saying that every species is “removed” (perhaps through an unspeciﬁed degradation reaction) at a rate proportional to its current concentration. Although this presumption is natural for continuous ﬂow stirred tank reactors (CFSTRs) in a chemical engineering setting, it is less compelling for models of cellular reaction systems, in which only certain species (or none at all) might suffer degradation. Removal of the fully open presumption makes the mathematics substantially more delicate.

The papers [1.C14] and [1.C15] are aimed at establishing conditions under which results in [1.C12] and [1.C13] can, in fact, be invoked in the absence of the “fully open” presumption. The analysis in [1.C.14] shows that results of the kind given in [1.C12] and [1.C13] can be invoked in quite general settings, provided that one restricts attention to steady states that are, in a certain sense, *non-degenerate*. The argument in [1.C14] for extending “fully open” results to more general settings does not rely on mass action kinetics. The paper [1.C15] does invoke mass action kinetics, but it gives far sharper results, without limiting considerations to non-degenerate steady states. Among other things, it shows that the results in [1.C12] and [1.C13] do apply even in consideration of systems that are not fully open, provided that the reaction network in question falls within the large class of *normal* networks — in particular, if it is *reversible* or even *weakly reversible*. (A weakly reversible network is one in which every reaction arrow is in at least one directed arrow cycle.)

In many ways, the following papers with Guy Shinar and Daniel Knight subsume results of [1.C12]-[1.C15] and go further. As with the work of Craciun and Banaji, these papers embrace very wide and natural classes of kinetics. *But here, from the very beginning, systems that are not “fully open” are admitted for study alongside those that are.* In [1.C16] we describe and examine properties of a large class of reaction networks that possess a structural property called *concordance*. (Whether or not a particular network of interest is concordant can be determined by *The Chemical Reaction Network Toolbo*x.)

Among other things, we show in [1.C16] that, in at least some respects, concordance is *the* structural property that confers upon a network the assurance of certain dull behaviors against *every* assignment of kinetics within a very large class called *weakly monotonic*. In particular, a reaction network exhibits injectivity when taken with every weakly monotonic kinetics *if and only if* it is concordant. On the other hand, reaction network “discordance” provides a source for unstable behavior in the following sense: For any weakly reversible network that is *not* concordant, there exists a weakly monotonic kinetics such that the resulting kinetics system exhibits a positive *unstable* equilibrium. (In [1.C16] we also consider variants of the concordance property in connection with still wider classes of kinetics, in which certain species might inhibit the occurrence of a particular reaction while others might promote it.)

In [1.C17] and [1.C18] we connect the concordance of a reaction network to properties of its Species-Reaction Graph. In particular, we show that Species-Reaction Graph properties which, in [1.C13], sufficed for injectivity in the fully-open mass action context also suffice for network concordance — *and therefore for injectivity against all weakly monotonic (and still broader) kinetics*. This is true not only in the fully open context but also whenever the network under study is “nondegenerate,” in particular whenever it is weakly reversible.

[1.C17] Guy Shinar and Martin Feinberg. Concordant chemical reaction networks and the species-reaction graph. *Mathematical Biosciences*, 241:1–23, 2013. Corrigendum.

**Martin Feinberg Group for****Chemical Reaction Network Theory**

© 2021 The Ohio State University

The following little article shows how classical equations describing ideal gas mixtures and ideal solutions emerge in incredible detail from simple verbal postulates that seems to say very little at all. Unlike the other thermodynamics articles listed below, this one requires very little in the way of mathematics preparation. This article is recommended for people who like magic.

When I taught thermodynamics in the past, I was dissatisﬁed with arguments in support of Gibbs’s phase rule. (Gibbs seemed to have some reservations about his own arguments.) Then I came across an explicitly mathematical paper by Walter Noll that, for me, was beautiful and satisfying. The paper below is a variant of Noll’s, using weaker assumptions and, I think, a simpler argument.

One of the central problems in the foundations of classical thermodynamics is, loosely speaking, proof of the existence of entropy and thermodynamic temperature scales from more fundamental assertions about, for example, heat. And then one wants to examine properties of the derived notions of entropy and thermodynamic temperature. The classical 19th century arguments are brilliant, but they are also somewhat mysterious, typically relying on putative properties of engines, and they also raise questions about the full range of their applicability.

I am very proud of work with my friend Rick Lavine, in which we addressed these issues, using mathematical tools (e.g. the Hahn-Banach Theorem) that were not available to the pioneers. Unfortunately, the intersection of the class of people who are are familiar with these tools and the class of people who are interested in the thermodynamic questions is practically empty. At least in part, I think it is for this reason that the work didn’t receive much attention. It was gratifying to notice, however, that L. Craig Evans at the University of California, Berkeley, recently resurrected ideas from the papers below in his online notes, *Entropy and Partial Differential Equations* .

[2.3] Martin Feinberg and Richard B. Lavine. Foundations of the Clausius-Duhem Inequality. In James B. Serrin, editor, *New Perspectives in Thermodynamics*. Springer-Verlag, Berlin-Heidelberg-New York, 1986.

The paper just above had its genesis in some handwritten notes prepared in 1978 for James Serrin. Those notes were less ambitious in their goals and are far more succinct. (They also relied on the Hahn-Banach Theorem). Some readers might find them helpful. A scanned copy is available here:

In the 1990s I got interested in optimal reactor design, mainly because there was a resurgence of interest in old *attainable region* ideas of Fritz Horn, my late collaborator in chemical reaction network theory. I am very proud of the attainable region work I did — more on that later. However, I now think that the most important work to emerge from my temporary foray into design came from a different line of thought entirely, a line that (strangely) traces back to my seemingly unrelated paper [2.2] on Gibbs’s phase rule!

This work in design theory is summarized in [3.1] below, and its mathematical foundations are described in [3.2]. In rough terms, the central idea is this: Given a network of chemical reactions (with kinetics) and given a speciﬁed commitment of resources, there is an *absolute* and *computable* limit to what can be achieved in ANY steady-state reactor-separator design — a limit that is different from that afforded by stoichiometry or thermodynamics. Knowledge of such a limit is of practical importance, for it gives a benchmark against which actual designs can be judged. (Connections to the mathematics underlying the Gibbs phase rule are discussed in Remark A.1 of paper [3.2].)

As I said earlier, I am very proud of my attainable region work. There are some surprising and beautiful conclusions, and very different parts of mathematics come together in interesting ways. (For me it remains stunning that, associated with a given reaction network with kinetics and a given feed composition, there are certain exceptional numbers — something like eigenvalues — having special significance for reactor synthesis: A classical steady-state CFSTR design can have an optimal conversion relative to all other steady-state designs only if the CFSTR residence time assumes one of those exceptional values.) The review [3.3] was written after I had moved away from attainable region work, and it gives an indication of why I came to prefer ideas in [3.1] and [3.2].

The conference paper [3.4] is also friendly review, this time written for control theorists — I’m not one of those — to indicate why ideas in modern geometric control theory have somewhat surprising connections with Horn’s attainable region approach to optimal reactor design. Despite its intended control-theorist target, I think this article is suited to a much wider chemical engineering audience.

[3.4] Martin Feinberg. Geometric control theory and classical problems in chemical reactor design. In *Proceedings of 15th IFAC World Congress*, Barcelona, 2002.

Mathematical underpinnings of [3.3] and [3.4] are given in the three-article series [3.5]-[3.7]. The paper [3.8], based on Thomas Abraham’s PhD thesis, is an extension of these, viewed from a different perspective.

## PEOPLE

### Thomas Abraham

Graduated with a PhD at The Ohio State University in Chemical Engineering, 2005

Senior Research Engineer and Industry Technical Consultant, Nalco, 2005-present

### Gheorghe Craciun

Graduated with a PhD at The Ohio State University in Mathematics, 2002

Post-doctoral Research Fellow, Mathematical Biosciences Institute at The Ohio State University, 2002-2005

Professor, Departments of Mathematics and Biomolecular Chemistry, University of Wisconsin-Madison, 2005-present

### Yongjia Fan

Graduated with a master's degree from The Ohio State University in Chemical Engineering, 2010

PhD Tufts University, Chemical and Biological Engineering, 2015.

### Haixia Ji

Graduated with a PhD at The Ohio State University in Mathematics, 2011

Hedging Specialist, Nationwide Financial, 2011-present

### Daniel Knight

Graduated with a PhD at The Ohio State University in Chemical and Biomolecular Engineering, 2015.

Assistant Professor of Teaching, Chemical and Biomolecular Engineering, University of California at Irvine.

### Yangzhong Tang

Graduated at The Ohio State University with a PhD in Chemical Engineering, 2005

Research Fellow at Harvard Medical School, 2007-2011

Research Investigator, Sanofi, 2011-present