Chapter 6
Extreme Currents,
Elementary Flux Modes and
Extreme Pathways
Clarke [2] defined the concept of extreme current for networks with irreversible
reactions only. Schuster and coworkers (e.g. [7, 8]) introduced the concept of
elementary flux modes to cover the case of networks with reversible reactions.
Roughly at the same time Palsson and coworkers [6] introduced the concept of
extreme pathways for the same purpose. A draw-back of the latter is, that it can
be applied only under particular assumptions on the structure of the reaction
network, which will be explained later.
In this chapter our main focus is on explaining the intimate and subtile relationship between extreme currents and elementary flux modes.
6.1
Extreme Currents of an irreversible metabolic
network
Assume that the metabolic network N has irreversible reactions only. We call
such a network simply an ‘irreversible metabolic network’. Then the admissable flux cone Γ, hence Γ∗ , is a pointed cone. Recall that Γ∗ is polyhedral
(Proposition 5.3.2).
Definition 6.1.1. The extreme currents (ECs) of an irreversible metabolic
network are the extreme rays of the associated steady state flux cone Γ∗ , with
for each 0 removed.
According to Theorem B.2.5 and Theorem B.2.6, Γ∗ is generated by its extreme
rays, of which it has finitely many. Note that this set of extreme rays is uniquely
determined by the geometry. Thus we have
21
22CHAPTER 6. EXTREME CURRENTS, ELEMENTARY FLUX MODES AND EXTREME PATH
Corollary 6.1.2. An irreversible metabolic network N has a unique and finite
set of extreme currents that generates the steady state flux cone Γ∗ .
Proposition 6.1.3. Any representative of an extreme current is indecomposable (or simple).
Proof. Let p1 , . . . , pR ∈ Γ∗ be a complete set of representatives for the extreme
currents of N . Assume that pj0 is decomposable. Then there exist vi ∈ Γ∗ ,
vi 6= 0 such that v1 6∈ ray(v2 ) (i.e. they define different flux modes) and λi > 0
such that
p j 0 = λ1 v 1 + λ 2 v 2
and null(pj0 ) is properly contained in null(vi ), i = 1, 2. The collection P =
(i)
{p1 , . . . , pR } generates Γ∗ , so there exist µj ≥ 0 such that
vi =
R
X
(i)
µ j pj .
j=1
Hence
pj 0 =
R
X
(1)
(2) λ1 µj + λ2 µj
j=1
pj
The set P is systemically independent according to Proposition B.3.2, so
(
0, if j 6= j0 ,
(1)
(2)
λ1 µj + λ 2 µj =
1, if j = j0 .
(1)
Because λi > 0, we must have µj
(i)
µj 0 p j 0
6.2
(2)
= 0 = µj
if j 6= j0 . Consequently vi =
and we arrive at a contradiction. So each pj is indecomposable.
Elementary Flux Modes
Schuster and coworkers (e.g. [original ref(s)], [7, 8]) started to consider reaction networks with reversible reactions. They introduced the concept of elementary flux mode (EFM) as important tool in the analysis of such networks.
Definition 6.2.1. An elementary flux mode is a flux mode associated to a
simple flux distribution v ∗ ∈ Γ∗ .
According to Proposition 5.6.3, equivalently one can say that an EFM is represented by an indecomposable steady state flux distribution.
The following issues need to be addressed concerning EFMs:
EFMs exist;
There are only finitely many EFMs;
The set of EFMs generate Γ∗ ;
6.3. RELATIONSHIP BETWEEN ECS AND EFMS
23
The collection of EFMs is uniquely determined by the network N ;
What is the relationship between ECs and EFMs?
EFMs have a useful biological interpretation.
The latter point is addressed in Chapter 7. We start with the point dealing
with the relationship between ECs and EFMs, as it will turn out that these
are closely related. Moreover, the results on ECs and understanding of this
relationship results in the answer to the first four points.
6.3
6.3.1
Relationship between ECs and EFMs
Case of irreversible metabolic networks
In Proposition 6.1.3 we showed that if the network has irreversible reactions
only, then each EC is simple, hence is an EFM. The reverse also holds:
Proposition 6.3.1. An elementary fluxmode of a steady state flux cone Γ∗ ⊂
Rr+ is an extreme ray (with 0 removed). In particular, an EFM of an irreversible
metabolic network is an EC.
Proof. Let 0 6= v ∗ ∈ Γ∗ be a flux distribution that represents an elementary flux
mode EFM1 : EFM1 = {λv ∗ | λ > 0}. So v ∗ is indecomposable. Suppose that
EM1 is not a one-dimensional face of Γ∗ . Since Γ∗ ⊂ Rr+ , a set of representatives
P = {p1 , . . . , pR } for the extreme rays (or one-dimensional faces) of Γ∗ generate
this cone. Moreover P is systemically independent. Thus,
∗
v =
R
X
µj pj ,
λj ≥ 0,
j=1
while at least two of the µj are strictly positive (otherwise v ∗ were an extreme
ray...). Say µj1 > 0. Put v1 := pj1 and λ1 := µj1 . Let
X
v2 :=
µj p j .
j6=j1 :µj >0
Then v1 and v2 define different flux modes, because of systemic independence
of P , v ∗ = λ1 v1 + v2 , while null(v ∗ ) is a proper subset of null(vi ), i = 1, 2,
because null(pj ) cannot be contained in null(pj1 ) by systemic independence.
Therefore v ∗ is decomposable and we arrive at a contradiction. So EFM1 is a
one-dimensional face, with 0 removed.
6.3.2
The reconfigured reaction network
Let N be a reaction network that may contain reversible reactions. We start
by defining an associated augmented or reconfigured reaction network N ′ (as it
is called in [3]), as follows.
24CHAPTER 6. EXTREME CURRENTS, ELEMENTARY FLUX MODES AND EXTREME PATH
The set of metabolites for N ′ is the same as for N . In N ′ each reversible
reaction j ∈ Rev(N ) is split into a pair of irreversible reactions labelled by
(j, +) and (j, −), where (j, +) refers to the original reaction when the reaction
flux Φj > 0, while (j, −) refers to the reaction with negative flux, Φj < 0. We
call the set {(j, +), (j, −)} a reversible reaction pair, since the corresponding
reaction j in N is a reversible reaction.
So
R(N ′ ) = Irr(N ′ ) := Irr(N ) ∪ Rev(N ) × {±}.
The admissible flux cone associated to N ′ now equals
R(N ′ )
Γ′ := R+
Irr(N )
= R+
2Rev(N )
× R+
.
Define the reduction map Ψ : Γ′ → Γ that identifies reversible reaction pairs in
N ′ and defines the flux through the corresponding reversible reaction in N as
the net reaction flux. That is, for v ′ ∈ Γ′ ,
(
vj′ ,
if j ∈ Irr(N ),
Ψ(v ′ )j :=
(6.1)
′
′
v(j,+) − v(j,−) ,
if j ∈ Rev(N ).
The map Ψ is as ‘linear’ as it can be:
Proposition 6.3.2. Ψ is a surjective homomorphism of convex cones, i.e. it
maps onto Γ and satisfies:
(i ) Ψ(v ′ + w′ ) = Ψ(v ′ ) + Ψ(w′ ) for all v ′ , w′ ∈ Γ′ (additivity),
(ii ) Ψ(λv ′ ) = λΨ(v ′ ) for all λ ≥ 0 and v ′ ∈ Γ′ (positive homogeneity).
Ψ will not be injective generally, when Rev(N ) 6= ∅.
One can however define a ‘right-inverse’ Ψ† : Γ → Γ′ , that we call standard
splitting or standard reconfiguration: if v ∈ Γ, then


vi ,
if j = i, i ∈ Irr(N ),



v,
if j = (i, +) and vi ≥ 0,
i
Ψ† (v)j :=
(6.2)

−vi ,
if j = (i, −) and vi ≤ 0,



 0,
otherwise.
That is, Ψ† does not change the fluxes through irreversible reactions, while for
reversible reactions the net flux through such a reaction is completely allocated
to the reaction in the reversible reaction pair that has the same sign as the net
flux of the reversible reaction. It is easily checked that
Ψ ◦ Ψ† = IdΓ .
(6.3)
Note that generally, Ψ† ◦ Ψ 6= IdΓ′ .
The stoichiometric matrix S ′ of the reconfigured network N ′ is defined as
S ′ v ′ := SΨ(v ′ )
(6.4)
6.3. RELATIONSHIP BETWEEN ECS AND EFMS
25
for all v ′ ∈ Γ′ . Definition (6.4) is such that the rate of change of metabolite
numbers (or concentrations) of metabolites that are involved as substrate or
product in a reversible reaction in N are determined by the net reaction flux,
as it should. Denote the steady state flux cone in Γ′ determined by S ′ by (Γ′ )∗ .
Combining (6.4) and (6.3) yields
S ′ Ψ† (v) = S ′ ΨΨ† (v) = Sv,
for all v ∈ Γ.
(6.5)
Consequently,
Proposition 6.3.3. Γ∗ = Ψ (Γ′ )∗ .
Proof. Let v ′ ∈ (Γ′ )∗ . Then SΨ(v ′ ) = S ′ v ′ = 0, so Ψ(v ′ ) ∈ Γ∗ . To show that
we get all of Γ∗ , let v ∗ ∈ Γ∗ and put v ′ := Ψ† (v ∗ ) ∈ Γ′ . Then
S ′ v ′ = S ′ Ψ† (v ∗ ) = Sv ∗ = 0,
so indeed, v ′ ∈ (Γ′ )∗ .
(Γ′ )∗ is a pointed polyhedral cone, because N ′ contains irreversible reactions
only. So it is generated by its extreme rays of which there are finitely many. In
particular, (Γ′ )∗ is generated by its extreme currents, say EC1 , . . . , ECR . Let
e1 , . . . , eR be a set of representative flux distributions these ECs. ej corresponds
to ECj . An immediate consequence of Proposition 6.3.3 is
Corollary 6.3.4. Γ∗ is generated by
{vj∗ | vj∗ = Ψ(ej ), vj∗ 6= 0, j = 1, . . . , R}.
Definition 6.3.5. The flux mode associated to a steady-state flux distribution
v ′ ∈ (Γ′ )∗ such that supp(v ′ ) is a single reversible reaction pair and such that
′
′
for (j, ±) ∈ supp(v ′ )) is called a futile cycle.
= v(j,−)
Ψ(v ′ ) = 0 (i.e. v(j,+)
It is clear that a flux distribution that represents a futile cycle is simple. So,
futile cycles constitute a particular class of extreme currents of N ′ . In view of
Corollary 6.3.4 futile cycles are irrelevant to the generation of Γ∗ .
6.3.3
Case of general reaction networks
Let N now be a reaction network with (some) reversible reactions. The crucial
and fundmental theorem on elementary fluxmodes is the following (cf. [3] Theorem 11 ), that answers almost all questions raised in Section 6.2. The question of
biological interpretation will be considered after algorithms for the computation
of ECs and EFMs have been discussed.
1
In the cited theorem the relationship to ECs is implicitly present, through our Propositions
6.1.3 and 6.3.1 that exhibit the equivalence of the concept of EFM and EC for irreversible
reaction networks.
26CHAPTER 6. EXTREME CURRENTS, ELEMENTARY FLUX MODES AND EXTREME PATH
Theorem 6.3.6 (Relationship ECs and EFMs). The following holds:
(i ) The standard reconfiguration of an EFM of N through Ψ† is an EC of
N ′ , consequently any EFM of N is the reduction of an EC of N ′ under
Ψ.
(ii ) An EC of N ′ is either the standard reconfiguration of an EFM of N , or
it is a futile cycle.
Thus, the EFMs of Γ∗ are in one-to-one correspondence with the non-zero reductions of ECs of N ′ . In particular, the set of EFMs is finite, generates Γ∗
and is uniquely determined by N .
Proof. (i ): Let v ∗ ∈ Γ∗ generate an EFM of N . Put e∗ := Ψ† (v ∗ ). Then
e∗ ∈ (Γ′ )∗ according to (6.5) and v ∗ = Ψ(e∗ ), using (6.3). Suppose that e∗
does not represent an EC of N ′ . Since N ′ is an irreversible metabolic network
by construction, e∗ cannot be simple (Proposition 6.3.1). Thus there exists
v ′ ∈ (Γ′ )∗ , v ′ 6= 0 such that supp(v ′ ) is a proper subset of supp(e∗ ). Put
v := Ψ(v ′ ). So v ∈ Γ∗ . We make the following claims:
Claim 1:
v 6= 0,
Claim 2:
supp(v) is properly contained in supp(v ∗ )2 .
Once we have proven the two claims, we immediately arrive at a contradiction
with the simplicity of v ∗ . So e∗ must be representing an EC of N ′ . Then (6.3)
yields the second statement of part (i ).
′
Proof of Claim 1: If v were zero, then supp(v ′ ) ⊂ Rev(N ) × {±} and v(j,+)
=
′
′
v(j,−) for all (j, ±) ∈ supp(v ). Thus
(j, +) ∈ supp(v ′ )
iff
(j, −) ∈ supp(v ′ ).
(6.6)
However, by definition of Ψ† and because e∗ = Ψ† (v ∗ ), if (j, ±) ∈ supp(e∗ ),
then (j, ∓) 6∈ supp(e∗ ). Since supp(v ′ ) ⊂ supp(e∗ ), (6.6) cannot hold, yielding
a contradiction.
Proof of Claim 2: Because e∗ = Ψ† (v ∗ ), either (j, +) ∈ supp(e∗ ) or (j, −) ∈
supp(e∗ ), but not both. So (j, α) ∈ supp(e∗ ) implies that j ∈ supp(Ψ(e∗ )).
Because supp(v ′ ) ⊂ supp(e∗ ), a similar argument applies to v ′ . Since Ψ does
not affect the irreversible reactions, the claim follows from supp(v ′ ) ⊂ supp(e∗ ).
(ii ): We already observed that futile cycles are ECs of N ′ . So suppose that there
exists an EC of N ′ that is neither a futile cycle, nor a standard reconfiguration
of an EFM of N . Let e∗ be a flux distribution in this EC.
Claim 3: supp(e∗ ) cannot contain a reversible reaction pair, i.e. both (j, +)
and (j, −). Consequently, Ψ† (Ψ(e∗ )) = e∗ ,
2
In general, if v ′ , w′ ∈ (Γ′ )∗ such that supp(v ′ ) ⊂ supp(w′ ), then it need not be in general
that supp(Ψ(v ′ )) ⊂ supp(Ψ(w′ )) In fact, if (i, α) ∈ supp(v ′ ) (or supp(w′ )...), then whether or
′
′
not i is in supp(Ψ(v ′ )) is determined by the values of v(j,α)
and v(j,−α)
. So there is something
to prove here.
6.3. RELATIONSHIP BETWEEN ECS AND EFMS
27
According to our assumption, v ∗ := Ψ(e∗ ) cannot generate an EFM of N . In
particular, v ∗ is not simple and one can find v ∈ Γ∗ , v 6= 0, such that supp(v) is
a proper subset of supp(v ∗ )3 . We cannot conclude that supp(Ψ† (v)) is a proper
subset of supp(Ψ† (v ∗ )), because it may be that the sign of vj is different from
that of vj∗ for some j ∈ Rev(N ) ∩ supp(v)4 . However, following a construction
in the proof of [8], Lemma 2, p.160, one can obtain
Claim 4: There exists v̂ ∈ Γ∗ , v̂ 6= 0, such that supp(v̂) is a proper subset of
supp(v ∗ ), while supp(Ψ† (v̂)) is a proper subset of supp(Ψ† (v ∗ )) too.
Then we arrive at a contradiction with the simplicity of e∗ = Ψ† (v ∗ ) (Claim
3) and obtain a proof of Part (ii ). The last statements follow by combining
parts (i ) and (ii ) and Corollaries 6.1.2 and 6.3.4. We complete the proof by
providing the arguments yielding Claims 3 and 4.
Proof of Claim 3: If supp(e∗ ) does contain a reversible reaction pair, say for
reaction i ∈ Rev(N ), then define a futile cycle represented by e∗1 ∈ (Γ′ )∗ through
e∗1,j
 ∗

 e(i,+) ,
=
e∗(i,−) ,


0,
if j = (i, ±) and e∗(i,+) ≤ e∗(i,−) ,
if j = (i, ±) and e∗(i,+) > e∗(i,−) ,
otherwise.
Then e∗ = e∗1 + e∗2 with e∗2 ∈ (Γ′ )∗ , e2 6= 0, while null(e∗ ) ⊂ null(e∗i ) for i = 1, 2.
That is, e∗ is decomposable, contrary to our assumption of simplicity of e∗
(Proposition 6.1.3).
Proof of Claim 4: If j ∈ Rev(N ) ∩ supp(v) is such that sgn(vj ) = −sgn(vj∗ )
(note that both vj and vj∗ must be non-zero), then define
v̂ (j) := |vj∗ |v + |vj |v ∗ .
(6.7)
Then v̂ (j) ∈ Γ∗ . Because supp(v) is a proper subset of supp(v ∗ ), v̂ (j) 6= 0,
(j)
v̂j = 0, and consequently supp(v̂ (j) ) is a proper subset of supp(v ∗ ). Note that
(6.7) is such that if the components with the same index of the flux distributions
v and v ∗ have the same sign, this remains so for the resulting flux distribution.
Thus, by repeatedly applying the reconfiguration (6.7) (first with v̂ (j) replacing
v, then using the newly obtained distribution, etc.), we obtain in finitely many
steps (due to the finite number of reactions) a flux distribution v̂ ∈ Γ∗ , v̂ 6= 0,
such that supp(v̂) is a proper subset of supp(v ∗ ) while all reaction fluxes that
are non-zero for both v̂ and v ∗ have the same sign. In this case, one does have
supp(Ψ† (v̂)) ⊂ supp(Ψ† (v ∗ )).
3
The reasoning in [3] concerning the proof of Theorem 1 in the Methods section is very
brief concerning what follows.
4
In that case, (j, α) ∈ supp(Ψ† (v)), while (j, −α) ∈ supp(Ψ† (v ∗ ) and (j, α) is not.