Self-sustaining autocatalytic networks within open-ended reaction systems

0.42 MB·English

by  Mike Steel

manuscript No.
(will be inserted by the editor)
Self-sustaining autocatalytic networks within open-ended
reaction systems
Mike Steel
Received:date/Accepted:date
5
1
0
2 Abstract Given any finite and closed chemical reaction system, it is possi-
ble to efficiently determine whether or not it contains a ‘self-sustaining and
n
collectively autocatalytic’ subset of reactions, and to find such subsets when
a
J theyexist.However,forsystemsthatarepotentiallyopen-ended(forexample,

3 when no prescribed upper bound is placed on the complexity or size/length
2 of molecules types), the theory developed for the finite case breaks down. We
investigate a number of subtleties that arise in such systems that are absent
]
N in the finite setting, and present several new results.
M Keywords chemical reaction system autocatalytic network ω-consistency
· ·
.
o
i
b 1 Introduction
-
q
Consider any system of chemical reactions, in which certain molecule types
[
catalysereactionsandwherethereisapoolofsimplemoleculetypesavailable
1
from the environment (a ‘food source’). One can then ask whether, within
v
this system, there is a subset of reactions that is both self-sustaining (each
1
3 moleculecanbeconstructedstartingjustfromthefoodsource)andcollectively
7 autocatalytic (every reaction is catalysed by some molecule produced by the
5 system or present in the food set) [8],[9]. This notion of ‘self-sustaining and
0
collectively autocatalytic’ needs to be carefully formalised (we do so below),
.
1 and is relevant to some basic questions such as how biochemical metabolism
0 began at the origin of life [2], [11], [14]. A simple mathematical framework
5
for formalising and studying such self-sustaining autocatalytic networks has
1
been developed – so-called ‘RAF (Reflexively-autocatalytic and F-generated)
:
v
Xi M.Steel
BiomathematicsResearchCentre,UniversityofCanterbury,Christchurch,NewZealand
r Tel.:[phone]ext7688
a
Fax:[phone]
E-mail:[email protected]
2 MikeSteel
theory’.Thistheoryincludesanalgorithmtodeterminewhethersuchnetworks
existswithinalargersystem,andforclassifyingthesenetworks;moreover,the
theory allows us to calculate the probability of the formation of such systems
withinnetworksbasedontheligationandcleavageofpolymers,andarandom
pattern of catalysis.
However, this theory relies heavily on the system being closed and finite.
In certain settings, it is useful to consider polymers of arbitrary length being
formed(e.g.ingeneratingthemembraneforaprotocell[4]).Intheseandother
unbounded chemical systems, interesting complications arise for RAF theory,
particularlywherethecatalysisofcertainreactionsispossibleonlybymolecule
types that are of greater complexity/length than the reactants or product of
the reactions in question. In this paper, we extend earlier RAF theory to deal
with unbounded chemical reaction systems. As in some of our earlier work,
our analysis ignores the dynamical aspects, which are dealt with in other
frameworks, such as ‘chemical organisation theory’ [1]; here we concentrate
instead on just the pattern of catalysis and the availability of reactants.
1.1 Preliminaries and definitions
In this paper, a chemical reaction system (CRS) consists of (i) a set X of
molecule types, (ii) a set of reactions, (iii) a pattern of catalysis C that
R
describeswhichmolecule(s)catalyseswhichreactions,and(iv)adistinguished
subset F of X called the food set.
We will denote a CRS as a quadruple = (X, ,C,F), and encode the
Q R
pattern of catalysis C by specifying a subset of X so that (x,r) C
×R ∈
preciselyifmoleculetypexcatalysesreactionr.SeeFig.1forasimpleexample
(from [11]).
In certain applications, X often consist of – or at least contain – a set of
polymers(sequences)oversomefinitealphabet (i.e.chainsx x x ,r 1,
1 2 r
A ··· ≥
where x ), as in Fig. 1; such polymer systems are particularly relevant
i
∈ A
to RNA or amino-acid sequence models of early life. Reactions involving such
polymers typically involve cleavage and ligation (i.e. cutting and/or joining
polymers), or adding or deleting a letter to an existing chain. Notice that if
no bound is put on the maximal length of the polymers, then both X and
R
are infinite for such networks, even when =1.
|A|
InthispaperwedonotnecessarilyassumethatX consistsofpolymers,or
that the reactions are of any particular type. Thus, a reaction can be viewed
formallyasanorderedpair(A,B)consistingofamulti-setAofelementsfrom
X (the reactants of r) and a multi-set B of elements of X (the products of
r); but we will mostly use the equivalent and more conventional notation of
writing a reaction in the form:
r =(a +a + +a b +b + +b ),
1 2 k 1 2 l
··· → ···
where the a ’s (reactants of r) and b ’s (products of r) are elements of X, and
i j
k,l 1 (e.g. x y,x+x y and y x+x are reactions).
(cid:48)
≥ → → →
Self-sustainingautocatalyticnetworkswithinopen-endedreactionsystems 3
001 01100 1000011 0011101
r7 r6 r5 r4
01 100 0011
101
r1 r2 r3
0 1 00 11
Fig. 1 A simple CRS based on polymers over a two-letter alphabet (0,1), with a food set
F ={0,1,00,11} and seven reactions. Dashed arrows indicate catalysis; solid arrows show
reactantsenteringareactionandproductsleaving.InthisCRSthereareexactlyfourRAFs
(definedbelow),namely{r1,r2},{r3},{r1,r2,r3},and{r1,r2,r3,r5}.
In this paper, we extend our earlier analysis of RAFs to the general (finite
or infinite) case and find that certain subtleties arise that are absent in the
finitecase.Wewillmostlyassumethefollowingconditions(A1)and(A2),and
sometimes also (A3).
(A1) F is finite;
(A2) each reaction r has a finite set of reactants, denoted ρ(r), and a finite
∈R
set of products, denoted π(r);
(A3) for any given finite set Y of molecule types, there are only finitely many
reactions r with ρ(r)=Y.
Givenasubset of ,wesaythatasubsetW X ofmoleculetypesisclosed
(cid:48)
R R ⊆
relative to if W satisfies the property r and ρ(r) W π(r) W.
(cid:48) (cid:48)
R ∈ R ⊆ ⇒ ⊆
Inotherwords,asetofmoleculetypesisclosedrelativeto ifeverymolecule
(cid:48)
R
that can be produced from W using reactions in is already present in W.
(cid:48)
R
Notice that the full set X is itself closed. The global closure of F relative to
, denoted here as gcl (F), is the intersection of all closed sets that contain
(cid:48) (cid:48)
RF (since X is closed, tRhis intersection is well defined). Thus gcl (F) is the
(cid:48)
unique minimal set of molecule types containing F that is closedRrelative to
.
(cid:48)
R
We can also consider a constructive closure of F relative to , denoted
(cid:48)
R
here as ccl (cid:48)(F), which is union of the set F and the set of molecule types x
R
that can be obtained from F by carrying out any finite sequence of reactions
from where,foreachreactionr inthesequence,eachreactantofr iseither
(cid:48)
R
an elements of F or a product of a reaction occurring earlier in the sequence,
and x is a product of the last reaction in the sequence.
4 MikeSteel
Note that gcl (cid:48)(F) always contains ccl (cid:48)(F) (and these two sets coincide
when the CRS isRfinite) but, for an infinRite CRS, ccl (cid:48)(F) can be a strict
R
subset of gcl (F), even when (A1) holds. To see this, consider the system
(cid:48)
(X, ) whereRX = x ,x ,x ,... , F = f , where = r ,r ,r ,r ,... is
0 1 2 (cid:48) 0 1 2 3
R { } { } R { }
defined as follows:
r =(f x );
1 1
→
r =(f +x x ), for all j 1;
j j j+1
→ ≥
r =(x +x + x ).
0 1 2 0
···→
Then x0 gcl (cid:48)(F) ccl (cid:48)(F). In this example, notice that r0 has infinitely
many rea∈ctantRs, whic−h vioRlates (A2). By contrast, when (A2) holds, we have
the following result.
Lemma 1 Suppose that (A2) holds. Then ccl (cid:48)(F) = gcl (cid:48)(F). Moreover,
under (A1) and (A2), if is countable, thenRthis (commRon) closure of F
(cid:48)
R
relative to is countable also.
(cid:48)
R
Proof SupposetheconditionofLemma1holdsbutthatccl (cid:48)(F)isnotclosed;
R
we will derive a contradiction. Lack of closure means there is a molecule x
in X ccl (cid:48)(F) which is the product of some reaction r (cid:48) that has all
− R ∈ R
its reactants in ccl (cid:48)(F). By (A2), the set of reactants of r is finite, so we
R
may list them as x1,x2,...,xk, and, by the definition of ccl (cid:48)(F), for each
R
i 1,...,k , either x F or there is a finite sequence S of reactions
i i
∈ { } ∈
from that generates x starting from reactants entirely in F and using just
(cid:48) i
R
elementsofF orproductsofreactionsappearingearlierinthesequenceS .By
i
concatenating these sequences (in any order) and appending r at the end, we
obtainafinitesequenceofreactionsthatgeneratexfromF,whichcontradicts
the assumption that ccl (cid:48)(F) is not closed. If follows that ccl (cid:48)(F) is closed
R R
relative to , and since it is clearly a minimal set containing F that is closed
(cid:48)
R
relativeto (cid:48),itfollowsthatccl (cid:48)(F)=gcl (cid:48)(F).Thatccl (cid:48)(F)iscountable
under (A1)Rand (A2) follows froRm the fact tRhat any countaRble union of finite
sets is countable. (cid:50)
In view of Lemma 1, whenever (A2) holds, we will henceforth denote the
(common) closure of F relative to (cid:48) as cl (cid:48)(F).
R R
Definition [RAF, and related concepts] Suppose we have a CRS =
Q
(X, ,C,F), which satisfies condition (A2). An RAF for is a non-empty
R Q
subset of for which
(cid:48)
R R
(i) for each r (cid:48), ρ(r) cl (cid:48)(F); and
∈R ⊆ R
(ii) for each r (cid:48), at least one molecule type in cl (cid:48)(F) catalyses r.
∈R R
Inwords,anon-emptyset ofreactionsformsanRAFfor if,forevery
(cid:48)
R Q
reaction r in , each reactant of r and at least one catalyst of r is either
(cid:48)
R
present in F or able to be constructed from F by using just reactions from
within the set .
(cid:48)
R
Self-sustainingautocatalyticnetworkswithinopen-endedreactionsystems 5
An RAF for is said to be a finite RAF or an infinite RAF depending on
(cid:48)
R Q
whether or not is finite or infinite. The concept of an RAF is a formalisa-
(cid:48)
|R|
tionofa‘collectivelyautocatalyticset’,pioneeredbyStuartKauffman[8]and
[9]. Since the union of any collection of RAFs is also an RAF, any CRS that
contains an RAF necessarily contains a unique maximal RAF. An irrRAF is
an(infiniteorfinite)RAFthatisminimal–i.e.itcontainsnoRAFasastrict
subset. In contrast to the uniqueness of the maximal RAF, a finite CRS can
have exponentially many irrRAFs [7].
The RAF concept needs to be distinguished from the stronger notion of
a constructively autocatalytic and F-generated (CAF) set [10] which requires
that can be ordered r ,r ,...,r so that all the reactants and at least
(cid:48) 1 2 N
R
one catalyst of r are present in cl (F) for all i 1,...,N (in the
i {r1,...,ri−1} ∈ { }
initialcasewherei=1,wetakecl (F)=F).Thisconditionessentiallymeans
∅
that in a CAF, a reaction can only proceed if one of its catalysts is already
available, whereas an RAF could become established by allowing one or more
reactionsr toproceeduncatalysed(presumablyatamuchslowerrate)sothat
later, in some chain of reactions, a catalyst for r is generated, allowing the
whole system to ‘speed up’. Notice that although the CRS in Fig. 1 has four
RAFs it has no CAF.
f1 f4
p4
f5
p3 p1
RAF

f3 f2 f1
p2
f1 f4 p4
p4
f5
p3 p1 p3 p1
f3 f2 f3 f2
p2 p2
CAF pseudo-RAF

Fig.2 ExamplesofafiniteRAF(thatisnotaCAF),afiniteCAFandafinitepseudo-RAF
(thatisnotanRAF).Intheseexamples,themoleculetypesareround nodes(thefoodset
isdenotedf1,f2,...,andp1,p2,...areproducts),reactionsarehollowsquares,anddashed
arrowsindicatecatalysis.
6 MikeSteel
TheRAFconceptalsoneedstobedistinguishedfromtheweakernotionof
a pseudo-RAF [12], which replaces condition (ii) with the relaxed condition:
(ii): for all r , there exists x F or x π(r) for some r such
(cid:48) (cid:48) (cid:48)
∈R ∈ ∈ ∈R
that (x,r) C.
∈
In other words, a pseudo-RAF that fails to be an RAF is an autocatalytic
system that could continue to persist once it exists, but it can never form
from just the food set F, since it is not F-generated.
ThesetwoalternativesnotionstoRAFsareillustrated(inthefinitesetting)
in Fig. 2. Notice that every CAF is an RAF and every RAF is a pseudo-RAF,
but these containments are strict, as Fig. 2 shows.
While the notion of a CAF may seem reasonable, it is arguably too con-
servative in comparison to an RAF, since a reaction can still proceed if no
catalyst is present, albeit it at a much slower rate, allowing the required cat-
alyst to eventually be produced. However relaxing the RAF definition further
toapseudo-RAFisproblematic(sinceareactioncannotproceedatall,unless
all its reactants are present, and so such a system cannot arise spontaneously
just from F). This, along with other desirable properties of RAFs (their for-
mationrequiresonlylowlevelsofcatalysisincontrasttoCAFs[10]),suggests
that RAFs are a reasonable candidate for capturing the minimal necessary
condition for self-sustaining autocatalysis, particularly in models of the origin
of metabolism.
1.2 Properties of RAFs in an infinite CRS
As in the finite CRS setting, the union of all RAFs is an RAF, so any CRS
that contains an RAF has a unique maximal one. It is easily seen that an
infiniteCRSthatcontainsanRAFneednothaveamaximalfiniteRAF,even
under (A1)–(A3), but in this case, the CRS would necessarily also contain an
infinite RAF (the union of all the finite RAFs).
A natural question is the following: if an infinite CRS contains an infinite
RAF, does it also contain a finite one? It is easily seen that even under condi-
tions (A1) and (A2), the answer to this last question is ‘no’. We provide three
examplestoillustratedifferentwaysinwhichthiscanoccur.Thisisincontrast
to CAFs, for which exactly the opposite holds: if a CRS contains an infinite
CAF, then it necessarily contains a sequence of finite ones. Moreover, two of
the infinite RAFs in the following example contain no irrRAFs (in contrast to
the finite case, where every RAF contains at least one irrRAF).
Example1:LetX = f,x ,...,x ,... ,F = f and = r ,r ,...,r ,... .
1 n 1 2 n
{ } { } R { }
Let r = (f x ). We will specify particular CRS’s by describing r ,r ,...,
1 1 2 3
→
and the pattern of catalysis as follows.
– has a reaction r = (f +x x ) for each i > 1 and r is catalysed
1 i i 1 i i
Q − →
by x for each i 1.
i+1
≥
Self-sustainingautocatalyticnetworkswithinopen-endedreactionsystems 7
– has a reaction r = (f +f + +f[i times] x ) for each i > 1 and
2 i i
Q ··· →
r is catalysed by x for each i 1.
i i+1
≥
– hasthesamereactionsas butr isnowcatalysedbyeveryx :j >i.
3 2 i j
Q Q
Fig. 3 illustrates the three CRS’s.
f f f
x1 x2 x3 x1 x2 x3 x1 x2 x3
Q1 Q2 Q3
Fig. 3 ThreesimpleexamplesofinfiniteCRSsthathaveinfiniteRAFsbutnofiniteRAF.
Each of , , satisfy (A1) and (A2), but only satisfies (A3). All
1 2 3 1
Q Q Q Q
three CRSs contain infinite RAFs, but no finite RAF, and no CAF. More
precisely:
– has as its unique RAF (which is therefore an irrRAF).
1
Q R
– The RAFs of consist precisely of all subsets of r ,r ,..., for some
2 j j+1
Q { }
j. Thus has a countably infinite number of RAFs but no irrRAF.
2
Q
– TheRAFsof consistpreciselyofallinfinitesubsetsof .Thus,theset
3
Q R
of RAFs for in uncountably infinite, and it contains no irrRAF.
3
Q
2 Determining whether or not a CRS contains an RAF
In this section, we assume that both (A1) and (A2) hold. Given a CRS =
Q
(X, ,C,F), consider the following nested decreasing sequence of reactions:
R
, ,..., defined by = and for each i>1:
1 2 1
R R R R
= r :ρ(r) cl (F), and x cl (F):(x,r) C . (1)
Ri+1 { ∈Ri ⊆ Ri ∃ ∈ Ri ∈ }
Thus, is obtained from by removing any reaction that fails to have
i+1 i
R R
either all its reactants or at least one catalyst in the closure of F relative to
(cid:84)
. Let µ( )= . It is easily shown that any RAF present in is
Ri Q i 1Ri R(cid:48) Q
necessarily a subset≥of µ( ) (since for all i 1 by induction on i).
(cid:48) i
Q R ⊆ R ≥
Thus if µ( ) = then does not have an RAF. In the finite case there is
Q ∅ Q
a strong converse – if µ( ) = then has an RAF, and µ( ) is the unique
Q (cid:54) ∅ Q Q
maximal RAF for (this is the basis for the ‘RAF algorithm’ [5] and [6]).
Q
However, in contrast, this result can fail for an infinite CRS, as we now show
with a simple example, which also satisfies (A1)–(A3).
8 MikeSteel
Example 2: Consider the following infinite CRS, = (X, ,C,F), where
4
Q R
F = f , and X = f,s,t, x ,x ,x ,... , where
1 2 3
{ } { }∪{ }∪F
= f,ff,fff, ,f(i),
F { ··· ···}
(this set can be thought of as all polymers of f). The reaction set is =
R
r ,r ,r ,... r ,r ,... , where, for all i 1:
{ 1 2 3 }∪{ 2(cid:48) 3(cid:48) } ≥
r =(f +f(i) f(i+1));
i
→
r =(f(i) x +s).
i(cid:48) → i
The pattern of catalysis is defined as follows: s catalyses r and t catalyses
1
r , and for all i > 1 f(i) catalyses r and x catalyses r . This CRS is
2(cid:48) i i i(cid:48)+1
illustrated in Fig 4. Notice that satisfies (A1), (A2) and (A3). However, if
4
Q
f
f r1 ff r2 fff r3 ffff
r20 r30 r40
t
x2 x3 x4
s
Fig. 4 AninfiniteCRSQ4 whichhasnoRAFeventhoughµ(Q4)isnon-empty(equalto
{r1,r2,...}).ThisCRSsatisfies(A1)–(A3)and(A5),butnot(A4).
we construct the sequence described above, then as the sole catalyst (t) of
i
R
r is neither in the food set, nor generated by any other reaction, it follows
2(cid:48)
that r will be absent from , and so r will also be absent from (since
2(cid:48) R2 3(cid:48) R3
the only catalyst of r is produced by r ). Continuing in this way, we obtain
3(cid:48) 2(cid:48)
µ( ) = r ,r ,r ,... , but this set is not an RAF, since the sole catalyst
4 1 2 3
Q { }
s of r does not lie lie in the closure of F relative to r ,r ,r ,... – it was
1 1 2 3
{ }
produced by the r reactions and in these have all disappeared in the limit;
j(cid:48)
moreover it is clear that no subset of is an RAF. (cid:50)
4
Q
Thus,werequireslightlystrongerhypothesesthanjust(A1)–(A3)inorder
toensurethat hasanRAFwhenµ( )= .This,isprovidedbythefollowing
Q Q (cid:54) ∅
result.
Self-sustainingautocatalyticnetworkswithinopen-endedreactionsystems 9
Proposition 1 Let = (X, ,C,F) satisfy (A1) and (A2). The following
Q R
then hold:
(i) µ( ) contains every RAF for ; in particular, if µ( )= , then has no
Q Q Q ∅ Q
RAF.
(ii) Suppose that satisfies both of the following further conditions:
(cid:84) Q
(A4) cl (F) cl (F), for the sequence defined in (1).
(A5) Eai≥ch1 reRaiction⊆r µ(Q)is catalysed by only finRiteily many molecule types.
∈R
Then contains an RAF if and only if µ( ) is non-empty (in which case,
Q Q
µ( ) is the maximal RAF for ).
Q Q
Before proving this result, we pause to make some comments and obser-
vations concerning the new conditions (A4) and (A5). Regarding Condition
(A4),containmentintheoppositedirectionisautomatic(byvirtueofthefact
that f( Y ) f(Y ) for any function f and sets Y ), so (A4) amounts to
i i i i
∩ ⊆ ∩
saying that the two sets described are equal.
Notice also that in Example 2 (Fig. 4) satisfies (A5) but it violates
4
Q
(A4), as it must, since does not have an RAF. To see how violates
4 4
Q (cid:84) Q
(A4), notice that cl (F)= , while cl (F)= s .
µ(Q4) F i≥1 Ri F ∪{ }
Condition (A5) is quite strong, but Proposition 1 is no longer true if it is
removed. To see why, consider the following modification of in which
Q(cid:48)4 Q4
the only product of r (for i > 1) is x , and x catalyses r for all i > 1 (in
i(cid:48) i i 1
(cid:84)
asodd(iAti4o)nhtooldrsi(cid:48);+h1)o,waesvesrhoµw(n i)n=Figr. 5,r. T,.h.e.n wclhµ(iQch(cid:48)4,)(aFs)b=eforei,≥i1scnlRoti(aFn)R=AFF
Q(cid:48)4 { 1 2 }
for since there is no catalyst of r in cl (F). Notice that (A5) fails
Q(cid:48)4 1 {r1,r2,...}
for since r has infinitely many catalysts. Nevertheless, it is possible to
Q(cid:48)4 1
obtain a result that dispenses with (A5) at the expense of a strengthening
(A4), which we will do shortly in Proposition 2.
f
f r1 ff r2 fff r3 ffff
r20 r30 r40
t
x2 x3 x4
Fig. 5 AninfiniteCRSQ(cid:48) whichhasnoRAFeventhoughµ(Q(cid:48))isnon-empty(equalto
4 4
{r1,r2,...}).ThisCRSsatisfies(A1)–(A3)and(A4),butnot(A5),nor(A4)(cid:48).
10 MikeSteel
Proof of Proposition 1: Suppose is any RAF for . Induction on i 1
(cid:48)
R Q ≥
shows that for all i, so that µ( ); in particular, if µ( ) = ,
(cid:48) i (cid:48)
R ⊆ R R ⊆ Q Q ∅
then has no RAF. The proof of part (ii) of Proposition 1 relies on a simple
Q
lemma.
Lemma 2 Suppose that (A ,i 1) is any nested decreasing sequence of sub-
i
≥
setsandB isafinitesetforwhichA B = foralli 1.Thensomeelement
i
∩ (cid:54) ∅ ≥
of B is present in every set A .
i
Proof of lemma: Suppose, to the contrary, that for every element b B, there
∈
is some set A in the sequence that fails to contain b (we will show this is
i(b)
not possible by deriving a contradiction). Let I = max i(b) : b B . Since
{ ∈ }
B is a finite set, I is a finite integer, and since the sequence (A ,i 1) is a
i
nested decreasing sequence, it follows that A B = , a contradicti≥on. (cid:50)
I
∩ ∅
Returning to the proof of Part (ii), suppose that µ( )= ; we will show that
Q (cid:54) ∅
µ( ) is an RAF for (and so, by Part (i), the unique maximal RAF for ).
Q Q Q
Forr µ( ),ρ(r) cl (F)foreachi(otherwiserwouldnotbeanelementof
∈ Q ⊆ Ri (cid:84)
and thereby fail to lie in µ( )). Thus ρ(r) cl (F) cl (F)
Rbyi+(1A4). It remains to show thatQr is catalysed⊆by ia≥t1leaRsit one⊆elemµ(eQn)t of
cl (F). Let B = x X : (x,r) C . By (A5), B is finite. Moreover,
µ( ) r r
Q { ∈ ∈ }
for each i 1, B cl (F) = (otherwise r would fail to be in and
≥ r ∩ Ri (cid:54) ∅ Ri+1
thereby not lie in µ( )). By Lemma 2, there is a molecule type x B that
r
(cid:84) Q ∈
lies in cl (F) and this latter set is contained in cl (F) by (A4). In
summaryi,≥e1verRyireactioninµ( )hasallitsreactantsandµa(tQl)eastonecatalyst
Q
present in cl (F) and so µ( ) is an RAF for , as claimed.
µ( )
Q Q Q (cid:50)
Suppose we now remove Condition (A5) in Proposition 1. In this case, by
a slight strengthening of (A4), we obtain a positive result (Proposition 2). To
describe this, we first require a further definition. Recall that C is the set of
pairs (x,r) where molecule type x catalyses reaction r. Given a subset C of
(cid:48)
C, let
[C ]= r :(x,r) C for some x X .
(cid:48) (cid:48)
R { ∈R ∈ ∈ }
Define a nested decreasing sequence of subsets C ,C ,..., by C =C and for
1 2 1
each i 1,
≥
C = (x,r) C : x ρ(r) cl (F) , (2)
i+1 { ∈ i { }∪ ⊆ R[Ci] }
(cid:84)
and let C = C .
i 1 i
∞ ≥
Proposition 2 Let satisfy(A1)and(A2),aswellasthefollowingproperty:
Q
(cid:92)
(A4) cl (F) cl (F), for the sequence C defined in (2).
(cid:48) R[Ci] ⊆ R[C∞] i
i 1
≥
Then hasanRAFifandonlyifC = ,inwhichcase [C ]isamaximal
RAF fQor . ∞ (cid:54) ∅ R ∞
Q