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Unformatted text preview: This is a scanned copy of written versions of 4.5 of 9 lectures delivered at the Mathematics Research Center,
University of WisconsinMadison in the autumn of 1 9 79. © 1980 Martin F einberg Lee/tuna on Chenicaz Reaction Netwaalvs Martin Feinberg Department of Chemical Engineering
University of Rochester
Rochester,“ 1462? Present address:
Departments of Chemical Engineering & Mathematics
The Ohio State Univerisity
140 W. 19th Avenue
Columbus, OH 43210 USA
email .‘ feinberg. [email protected] Content» Pheﬁace Lectuhe l} Inthoduetton 1.A. Motivatton
1.3. Notation Lectuhe 2; Reaetton Netwohhb, Ktnettcb and the Induced Dtﬁéehenttat Eguattonb 2.A. Reaction Netwohhé
2.8. Ktnettcb
2.C. The Dtﬁﬁehehttat Equattonb 60k a Reaetton SyAtem 2.0. An Etementahy Connectton betveen Reactton Netvohh
Sthuetuhe and the Natute 05 CompoAttion Thajeetohteb 2.E. Open Sybtemb: why Study "Funny" Reaction Netwohhb?
Exampﬂe 2.E.1. Cohttnuoub Ftow Stthhed Tank Reactohb Exampte 2.E.2. Homogeneoub Reactohb wtth Centatn
Specteb Coneenthattonb Regahded Conatant Exampte 2.5.3. Intehconnected CettA Lectuhe §5 Igg_Theohemé 3.A. Some Qoeéttoné.
Phobtem 3.A.l. The extétence 06 pobtttue equtttbhta
Phobtem 3.A.2. The untqueneéb 05 pOAtttve equtttbhta
Phobtem 3.A.3. The Atabtttty 06 poottxve equtttbhta Phobtem 3.A.4. The extAtenee 06 pehtodtc compobttton
eyeteb 3.8. A Ltttte Vocabutahg
3.C. The Deﬁtctency Zeho Theohem
3.0. The Deﬁtctency One Theohem Lectuhe if Some Deétntttonb Eﬂg'Paopobtiionb 4.A. Some Mottoatton
4.8. Some Ghaphtcat Aépectb 06 Reaetton Netwohhb 1—1
11 ‘110 21
2—1 28 212
220
222 224
227 31 32
34
35
310 312
313
319
325 41 42
45 4.C. Some In/te/Lplag 06 Stoichiommy and Gwyn/dad same/tune 420 4.17. A P/LopaAi/téon Concunéng the Na/tu/Le 06 Equ/Lubua 432
Lee/tune 3: Wood gﬁ ihe Deﬁioéencg Zeno Thea/Lem 51 ,
5.A. P1006 . 53
5.A.1. Pnooﬁ 06 pan/u (i) and (LU 53 5.A.2. Pnooﬁ 06 loam (iii), given the exA'A/tence 06 a
WAX/ave equiLéb/uium 55 5.A.3. Moots 06 the exutence 06 a poultéve equ/Lbéblu'um 516 LECTURE 7: INTRODUCTION This lecture is divided into two parts. In the first part I shall try
to provide some motivation for everything that follows. In particular,
I shall try to explain, at least in an informal way, how chemists and
chemical engineers arrive at the differential equations they work with and
how these differential equations are tied to reaction network structure.
Once this is done we can begin to understand why a reasonably general theory
of chemical reaction networks is necessary. Moreover, we can begin to
understand why, despite the great complexity of the differential equations
involved, such a theory should even be possible in principle. In the
second part of this lecture I shall discuss the important but more mundane subject of notation. 7 . A. Motivation Ihese lectures will be about a special but rather large class of
ordinary differential equations — those that derive from chemical reaction
networkS. In order that I might provide some sense of how these equations
come about it will be useful if I write down an example of a reaction net—
work and indicate informally how it induces a system of ordinary differential
equations. Then I can discuss the kinds of problems we will want to
consider. Suppose that A, B, C, D and E are chemical species, and suppose I
believe that the chemical reactions occurringanwng these species are reasonably well reflected in the following diagram: A 222:: 2B A+c :—__'> D (1.1) B+E What I have written down is a diagram of a chemical reaction network.
It indicates that a molecule of A can decompose into two molecules of B,
that two molecules of B can react to form one molecule of A, that a
molecule of A can react with a molecule of C to form a molecule of D,
and so on. Now suppose that I throw various amounts of my species into a pot.
I am going to presume that the pot is stirred constantly So that its
contents remain spatially homogeneous for all time, and I shall also
suppose that the contents of the pot arev forever maintained at fixed
temperature and total volume; This, of.course, is not to say that the
chemical composition of the mixture within the pot will remain constant
in time, for the occurrence of chemical reactions will serve to consume
certain species and generate others. In fact it is the temporal evolution
of the composition that we wish to investigate. With this in mind we
denote the (instantaneous) values of the molar concentrations of the
species by cA(t), CB(t)’.CC(t)’ cD(t) and cg(t), and we abbreviate this
list of numbers by the “composition vector" c(t).* Thus the picture we
are thinking about, at least for the moment, looks something like that shown in Figure 1.1.1‘ A molar concentration, say cA, specifies the number of A molecules per unit volume of mixture. More precisely, CA is the number of A molecules
per unit volume divided by Avogardro's number, 6.0231(1023. We shall be
somewhat more precise about what we mean by the "composition vector" in .Section 1.3. , +The reactor depicted in Figure 1.1 is closed with respect to the exchange
of matter with its environment. Our focus on such reactors is temporary and is merely intended to illustrate in a simple context how chemists and
engineers formulate differential equations based upon a set of reactions believed to approximate the true chemistry. We will begin to consider
"open" reactors in the next lecture (Section 2.E). There we shall indicate how open reactors can be modelled in terms of reaction networks and how the
appropriate differential equations, like those for closed reactors, bear a
definite relationship to reaction network structure. memosTAr , l  CJtLCeﬁLCJtLCJtLCe“) Figure 1.1 We would like to write down differential equations that describe the
evolution of the five molar concentrations. Since chemical reactions are
the source of composition changes, the key to understanding how to write
down differential eQuations lies in knowing how rapidly each of the several
reactions occurs._ What is generally assumed is that the instantaneous
occurrence rate of each reaction depends in its own way on the instantaneous
mixture composition vector, c. Thus; we presume, for example, the existence (e) of a nonnegative realvalued rate function ﬁﬁA*2B(.) such that 7% A*ZB is the instantaneous occurrence rate Of reaction A +23 (per unit volume of mixture) when the instantaneous mixture composition is given by the 1—4 ink .
vector c. Similarly, we preSume the existence of a rate function £28+A %A+C>D(.) for the reaction A+C>D, and so on. A kinetics for a reaction network is (') for the reaction 23+A, a rate function an assignment of a rate function to each reaction in the network. Once we presume that network (1.1) is endowed with a kinetics we are
in a position to write down the system of differential equations that govern
our reactor. Suppose that, at some instant, the reactor is in some composi
tion state c. Let us begin by thinking about the instantaneous rate of change of c Every time the reaction A>2B occurs we lose a molecule of A, and thgt reaction has an occurrence rate %A>2B(C)' 0n the other
hand, every time the reaction 2B+A occurs we _g_a_in a molecule of A,
and that reaction occurs at rate %2B+A(C)' Similarly, the reactions
B+E +A+C and D+A+C produce amolecule of A with each occurrence, while each occurrence of the reaction A+C>D results in the loss of a molecule of A. Thus we write (C) + K ' (C). (1.2) (C) + B+E+A+C E (c) .+ 1CZB+A(C)  1C A=T7( A+213 A+c'—>D 1CD+A+C If we turn our attention to species B we notice that whenever the
reaction A+2B occurs we gain two molecules of B, and whenever ZB+A occurs we lose two molecules of B. When D+B+E occurs we gain one B, and when B+E+A+C occurs we lose one B. Thus, we write E=27€ B A+2B (C)  it (e) (1.3) (c) +1€ (c) — 27C ZB+A D+B+E B+E+A+C Continuing in this way we can write down equations for ac, ED, era to generate the full system of differential equations that govern our reactor:
*9: More precisely, K (c) is the number of occurrences of A+2B A+2B
per unit time per unit volume divided by Avogadro's number 15 &A = “kmgm +16 (c)  1c (c) +f< (c) + 76. (c) ZB+A A+C+D D+A+C B+E+A+C a = 2KA+2B(C‘) ' 212nm“) + KD+B+E(C) ’ KB+E+A+C(C) (C) + £B+E+A+C(C) (1. 4) (e) +71! 0
II c ' k A+C+D D+A+C 5 =16 («9—76 (c) A+C+D(C) ' {Comm D+B+E c K: (c)  X: E ‘D+B+E B+E+A+c ( C.) . Thus far we haven't said anything about the nature of rate functions,
and that is what we shall do now. More often than not chemists and engineers presume the kinetics to be of mass action type. With mass action kinetics one can merely look at a reaction and write down its rate function up to a
multiplicative positive constant. Here is the way things work: For the reaction A+ZB we presume that
the more A there is in the reactor the more occurrences of the reaction
there will be. In fact, we presume that the instantaneous occurrence rate of 'A*ZB is proportional to the instantaneous value of cA . Thus, we write where a is a positive constant. For the reaction A+C~+D the situation is a little different. An
occurrence requires that a molecule of A meet a molecule of C in the
reactor, and we take the probability of such an encounter to be propor tional to the product c c Although we do not presume that every such A C'
encounter yields a molecule of D, we nevertheless take the occurrence rate of A+C+D to be given by %AH+D(C) = YCAce ’
where Y is a positive constant; Similarly, an occurrence of the
reaction 2B+A requires that two molecules of B have an encounter, and we take 2 2B_>A(c)  8(CB) ,
where B is a positive constant. Thus, with mass action kinetics the rate functions for network (1.1) take the form KA+2B(C) = occA
2
123 +A(C) = BCCB)
KA+C+D(C) = Y CACC
' (1.5)
1CD+B+E(C) = 6°13
zD+A+C(C) = 6CD
KB+E+A+C(C) = “13“}: The positive numbers a, B, y, E, 6 and g, called the rate constants for the corresponding reactions, are sometimes estimated on the basis of
chemical principles or else one makes an attempt to deduce them from
experiments. When a reaction network is presumed to be endowed with mass
action kinetics it is the custom to indicate the rate constants (or
symbols for them) alongside the corresponding reaction arrows in the net—
work diagram. Thus for our example we might have a display like that
shown in (1.6). (1.6) 17 If we assume mass action kinetics for the network we have been
studying, the appropriate differential equations are obtained by inserting (1.5) into (1.4): . 2 cA—0LcA+ 8(cB) v‘YcACC+ 6CD+ EchE E= 20Lc 28(c)2+ 8c  EC B A B D BCE CC =  YcAcC + 6CD + chcE (1.7)
CD = 'YcAcC — (5+€)cD rd”
II EcDEch'E . We have arrived at a fairly concrete system of ordinary differential
equations, and we can begin to pose questions about them. Here are some of the questions we might like to ask: (a) Does the system (1.7) admit a positive equilibrium — that is, an equilibrium at which all species concentrations are positive? (b) Does the system (1.7) admit multiple positive equilibria (in a sense to be made precise in the next lecture*)?
(c) Does the system (1.7) admit an unstable positive equilibrium? (d) Does the system (1.7) admit a periodic (positive) composition. trajectory? These are not easy questions, and the answers to them might of course
depend on the particular (positive) values taken by the rate constants
a, 8, y, 5, E, and g . Even if we could answer these questions for all positive values of the rate constants what would we have accomplished? * y
We shall want to know whether there can exist multiple positive equilibria within a stoichiometric compatibility class. In rough terms a stoichio
metric compatibility class is a certain set of compositions which remains
invariant under the flow given by (1.7). 18 We would have understood one model chemical system fairly well, at least
with respect to certain qualitative issues. There are, however, thousands of distinct reaction networks that
might, on one occasion or another, command our attention. Each has its
own system of differential equations, perhaps more compliCated by far
than the system we have been considering. How, then, are we to proceed?
It is clear that we cannot rely forever on purely §d_hgg studies of what—
ever systems might present themselves for examination. Even if we cast
aside the long—term enormity of such an undertaking, there are still two
problems that must be faced in the short run. First, questions of the
kind we have posed will, for the most part, be confronted by engineers
and chemists, not mathematicians. Second, it is by no means clear that ‘mathematicians, even the most expert, are currently in a position to
provide much help. The fact is that even moderately large systems of
nonlinear differential equations —— in particular polynomial systems like
those displayed in (1.7) —— remain poorly understood* in general. It seems to me that what is required is a rather broadbased theory of those systems of differential equations that derive from reaction net works, a theory which would in some sense cuts across the fine details of individual problems to provide qualitative information about large classes of systems all at once. Moreover, we would like the results of such a theory to be of the kind that engineers and chemists can use easily in addressing questions like those we have posed.
This seems like a lot to ask, and I should try to explain why a theory of the kind I have in mind should even be possible in principle.
Although we shall also be interested in the more general situation, let me temporarily restrict my attention to reaction networks endowed
with mass action kinetics. Thinking back to the source of the system
(1.7), we recall that it derived in a rather orderly way from the
network (1.6). In fact, we knew how to write down the appropriate *
Consider, for example, the remarkable complexity of the seemingly innocuOus Lorenz system [L1], which is composed of three polynomial
ordinary differential equations in three variables. differential equations (up to values of the rate constants) merely from
inspection of the reaction diagram. Had we begun with a different net—
work we would have arrived at a different system of differential equations,
but again the essential shape of those equations (up to values of the rate constants) would have derived from the reaction network in a precise
way. Indeed, it is the close connection between reaction network
structure and the shape of the induced differential equations that lends
thesubject of chemical reactor theory its coherency. This is our source of hope. If reactor behavior is determined by‘a
system of differential equations which, in turn, is determined by the
underlying reaction network in a precise way, then perhaps one can prove
theorems which tie qualitative aspects of reactor behavior directly to
reaction network structure. Can this in fact be done? I hope that these lectures will help to
demonstrate that one can proceed surprisingly far in this direction. In
Lecture 3 I will state two theorems, one of which will immediately answer
all four questions that we posed about the system (1.7). The answers
are ygp, pp, pp, and pp; these answers hold for all positive values of
the rate constants a, B, y, 6, 6, Hand 5. Moreover, these answers can
be obtained merely from inspection of the reaction diagram (1.6); one need
not even write out the differential equations. The fact is that one can
delineate a large class of networks —— some extremely complicated — for
which the corresponding differential equations admit solutions of a very
limited variety, regardless of the (positive) values the rate constants take. Our objectives will be rather broad, and I should try to make clear
what these are. Results of the kind I have just described are typical of those we are after. We seek to classify reaction networks according to
their capacity to induce differential equations which admit behavior of a
specified type. .When we restrict our attention to networks endowed with
mass action kinetics we will ppp_ask, for example, whether the differential
equations for a particular network taken with specified rate constants
admit periodic orbits. Rather, we will ask if the network is such that the induced differential equations admit periodic orbits for §£_least one set of rate constants — that is, if the network has the capacity to 1—10 admit periodic orbits. The network itself will be our object of study,
*
not the network endowed with a particular set of rate constants. 1 . 8. Notation It is not difficult to see that the differential equations induced
by a reaction network can be rather cumberSome. With this in mind I want
to spend a little time talking about notation. Although the notation
I shall use is quite natural to the problems we shall address, it is not
entirely traditional. With each reaction network we can associate three sets. The first is
the set A; of chemical species — {A, B, C, D, E} in network (1.1). The second is the set of objects that appear before and after the reaction
arrows —— {A, 2B, A+C, D, B+E}' in (1.1). These objects are called the complexes of the network, and the set of complexes will be denoted by
the symbol g'. The third is the set 52 of reactions — {A*ZB, ZB+A,
A+C+D, D+A+c, D+B+E, B+E+A+c} in network (1.1). With each of these sets we shall want to associate a (finitedimensional)
vector space so that we can, for example, speak of a "vector of species
concentrations" or a "vector of reaction rate constants." If m is the
number of species, if n is the number of complexes and if r is the
number of reactions in a network we can, of course, work in the vector in n Spaces 1R , P. , and 2er . In this way we can speak of the "composition vector c a B9“, c being the molar concentration of the 1th species, 1
i = 1, 2,...,m. And we can speak of the "rate constant vector k 6 Eg,'
kj being the rate constant of the "jth reaction," j = 1,2,...,r. This is what tradition would seem to require. * It is perhaps worth mentioning here that, in practice, complete sets of
rate constants for intricate networks are hardly ever known with great
precision. It is often the case that chemists have a very good sense of
what reactions are occurring but can estimate or measure rate constants
only to within a considerable margin of uncertainty. For a discussion
of the relationship between reaction network structure and the extent to
which rate constants can be determined uniquely from certain classes of
experiments see [F3] and, for more detail, [F1] and [K]. In [Fl] there
is also a discussion of how information about the reaction network itself
can, in principle, be inferred from nearequilibrium experiments. 1—11 It turns out, hOwever, that Hf“, IR“, and IRr are somewhat awkward
media in which to work; At the very least these spaces require that we
number everything in sight so that we can speak of the "ith species,"
"kth ( I the "jth reaction,‘ or the complex.’ Thus, we must impose an
artificial ordering on each of the three sets of objects even before we
begin to work, and we must carry that order around thereafter, suppressing
or rearranging it whenever it becomes intrusive. There is a much better
and far more natural way to do things, and that is what we shall discuss ROW. We denote the real numbers by Hi, the positive real numbers by E
and the non—negative real numbers by i; . If I is a set we denote by IRI the vector space of real~va1ued
functions with domain I . (Addition of functions and multiplication of
a function by a real number are defined in the usual way.) By H’I
[resp., lﬁz ] we mean the subset of IRI consisting of those functions
which take exclusively positive [resp., non~negative] values. Henceforth in this.section we shall suppose that I is a finite
set. In this case if x is a vector i...
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