Multi-equilibrium property of metabolic networks: SSI module.

Abstract

Revealing the multi-equilibrium property of a metabolic network is a fundamental and important topic in systems biology. Due to the complexity of the metabolic network, it is generally a difficult task to study the problem as a whole from both analytical and numerical viewpoint. On the other hand, the structure-oriented modularization idea is a good choice to overcome such a difficulty, i.e. decomposing the network into several basic building blocks and then studying the whole network through investigating the dynamical characteristics of the basic building blocks and their interactions. Single substrate and single product with inhibition (SSI) metabolic module is one type of the basic building blocks of metabolic networks, and its multi-equilibrium property has important influence on that of the whole metabolic networks.
In this paper, we describe what the SSI metabolic module is, characterize the rates of the metabolic reactions by Hill kinetics and give a unified model for SSI modules by using a set of nonlinear ordinary differential equations with multi-variables. Specifically, a sufficient and necessary condition is first given to describe the injectivity of a class of nonlinear systems, and then, the sufficient condition is used to study the multi-equilibrium property of SSI modules. As a main theoretical result, for the SSI modules in which each reaction has no more than one inhibitor, a sufficient condition is derived to rule out multiple equilibria, i.e. the Jacobian matrix of its rate function is nonsingular everywhere.
In summary, we describe SSI modules and give a general modeling framework based on Hill kinetics, and provide a sufficient condition for ruling out multiple equilibria of a key type of SSI module.

Full text

Page 1

REPORTOpen Access
Multi-equilibrium property of metabolic networks:
SSI module
Hong-Bo Lei1, Ji-Feng Zhang1*, Luonan Chen2,3*
From The 4th International Conference on Computational Systems Biology (ISB 2010)
Suzhou, P. R. China. 9-11 September 2010
Abstract
Background: Revealing the multi-equilibrium property of a metabolic network is a fundamental and important
topic in systems biology. Due to the complexity of the metabolic network, it is generally a difficult task to study
the problem as a whole from both analytical and numerical viewpoint. On the other hand, the structure-oriented
modularization idea is a good choice to overcome such a difficulty, i.e. decomposing the network into several
basic building blocks and then studying the whole network through investigating the dynamical characteristics of
the basic building blocks and their interactions. Single substrate and single product with inhibition (SSI) metabolic
module is one type of the basic building blocks of metabolic networks, and its multi-equilibrium property has
important influence on that of the whole metabolic networks.
Results: In this paper, we describe what the SSI metabolic module is, characterize the rates of the metabolic
reactions by Hill kinetics and give a unified model for SSI modules by using a set of nonlinear ordinary differential
equations with multi-variables. Specifically, a sufficient and necessary condition is first given to describe the
injectivity of a class of nonlinear systems, and then, the sufficient condition is used to study the multi-equilibrium
property of SSI modules. As a main theoretical result, for the SSI modules in which each reaction has no more than
one inhibitor, a sufficient condition is derived to rule out multiple equilibria, i.e. the Jacobian matrix of its rate
function is nonsingular everywhere.
Conclusions: In summary, we describe SSI modules and give a general modeling framework based on Hill kinetics,
and provide a sufficient condition for ruling out multiple equilibria of a key type of SSI module.
Background
Revealing the multi-equilibrium property of a metabolic
network is a fundamental and important topic in sys-
tems biology [1-5]. Generally, it is not only expensive
but also difficult, if not impossible, to solve this problem
via biological experiments. Hence, a systematical model-
ing approach is strongly demanded [6-8]. However, in
the traditional theoretical analysis, necessary information
on model parameters is always required. Due to the lim-
itation of measurement tools, measurement errors and
biological variability, most of the model parameters are
either unavailable or uncertain. This not only makes it
difficult to analyze the model, but also limits the appli-
cations of the theoretical results based on a model with
fixed parameter values. In contrast to detailed model
parameters, the topological structure of a metabolic net-
work is relatively easier to be obtained and is invariant
for many cases. Hence, a structure-oriented analysis
should be much more useful on understanding qualita-
tive dynamics of metabolic networks, since it can not
only overcome the difficulty due to the lack of para-
meter information, but also provide a deep insight into
the essential design principles.
There are some pioneering works in structure-
oriented study on multiple equilibria of networks
[3,9-17], which have recently been surveyed in [5]. A
metabolic network in a living cell is a large-scale
* Correspondence: jif@iss.ac.cn; lnchen@sibs.ac.cn
1Key Laboratory of Systems and Control, Academy of Mathematics and
Systems Science, Chinese Academy of Sciences, Beijing 100190, China
2Key Laboratory of Systems Biology, Shanghai Institutes for Biological
Sciences, Chinese Academy of Sciences, Shanghai 200031, China
Full list of author information is available at the end of the article
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© 2011 Lei et al; licensee BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
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molecular network and contains a great number of
metabolites and reactions, and thus, is generally difficult
to be theoretically analyzed as a whole, especially when
there is no parameters but only structure information
available.
To overcome such a difficulty, we proposed a struc-
ture-oriented modularization framework in [5]: using
the modularization idea commonly used in the area of
control theory [18,19], viewing a metabolic network as
an assembly of basic building blocks (called metabolic
modules) with specific structures, and investigating the
multi-equilibrium property of the original network by
studying the characteristics of these basic modules and
their interactions. Such an idea not only reduces the dif-
ficulty in investigating a complex metabolic network,
but also makes full use of the structure information,
thereby overcomes the limitation of the methods based
on models with fixed parameter values. After getting a
deep insight of the basic building blocks, people can use
them to reconstruct new metabolic networks.
In particular, in [5] we showed that a metabolic net-
work can be decomposed into four types of basic mod-
ules according to the topological structure, and proved
that one type of those modules, i.e. the single substrate
and single product with no inhibition (SSN) modules,
cannot admit multiple equilibria. Here we will focus on
another important type of those basic modules, i.e. the
single substrate and single product with inhibition (SSI)
modules, and investigate their multi-equilibrium
property.
Comparing with SSN modules, an SSI module
contains metabolic reactions which are inhibited by
other metabolites. Hence, the topological structure of an
SSI module is much more complex from theoretical
viewpoint. The metabolites interconnect with each other
via reactions without inhibitions in SSN modules, while
via reactions with inhibitions in SSI modules. Inhibitions
make the metabolites (state variables) couple with each
other in SSI modules, which are actually a kind of nega-
tive feedbacks. Moreover, the reaction mechanisms are
much more complicated in SSI modules than those in
SSN modules. For instance, when the other conditions
(such as temperature, pH, the concentration and activity
of the enzymes) are unchanged, the reaction rates
depend mainly on the substrate concentrations in SSN
modules but are simultaneously affected by the sub-
strates, the inhibitions and their interactions in SSI
modules.
Owing to these inherent characteristics, both the mod-
eling procedure and theoretical analysis for SSI modules
are much more difficult than those for SSN modules.
Specifically, first, the intricate topological structure
makes the modeling procedure for SSI modules much
complicated. It is relatively easy to describe the rate of a
metabolic reaction based on Hill kinetics if its inhibitors
are known. But in a general SSI module, each reaction
may be inhibited by other metabolites, and each meta-
bolite may act as an inhibitor for other reactions.
Hence, it is difficult to construct a unified model for SSI
modules. Second, the strong coupling in SSI modules
makes the model difficult to analysis. The metabolites
mutually restrain each other via inhibitions in SSI mod-
ules, which may result in a loop or other complex
structure.
Therefore, we have to consider all the metabolites
simultaneously, which makes the dimension reduction
of the system useless. Third, the complicated mechan-
isms of the reactions in SSI modules make the reaction
rate equations more complex. In fact, the reaction rate
is an increasing function of one variable in SSN mod-
ules, and is a polynomial that is increasing in any of its
variables in the work [3,16,17] of Craciun et al.. But, in
SSI modules, the reaction rate involves two or more
variables, and is increasing in the concentration of sub-
strate and decreasing in the concentration of inhibitor,
which is also the essential difference between this work
and that of Craciun et al..
The above characteristics of SSI modules makes the
analytical skills developed for the SSN module cases no
longer applicable. To overcome these difficulties, we
first construct a special vector space, and represent the
unified model of SSI modules via a system of nonlinear
ordinary differential equations in a vector form. And
then, we investigate the multi-equilibrium property of
SSI modules through analyzing a sufficient and neces-
sary condition of the injectivity of a particular nonlinear
system. For the SSI modules in which each reaction has
at most one inhibitor, we derive a sufficient condition
for the absence of multiple equilibria, i.e. the Jacobian
matrix of the rate function is nonsingular everywhere.
Results and Discussion
SSI metabolic module
If a metabolite can bind the enzyme of a metabolic reac-
tion to repress its activity and decrease the reaction rate,
then it is generally called an inhibitor of the enzyme or
the reaction. This process is called the inhibition of the
enzyme or the reaction. Generally, it is difficult to inves-
tigate a reaction with inhibition from the viewpoints of
both experiment and theory, and special analysis meth-
ods is required. Hence, to investigate a metabolic net-
work, it may be necessary and feasible to divide the
metabolic reactions into two groups, one is with inhibi-
tion and the other is with no inhibition. In real meta-
bolic networks, many reactions are with only one
substrate and one product. Compared with other type of
reactions, such reactions have particular properties, and
is worth investigating first. Hence, we classified the
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metabolic reactions into four classes according to the
number of substrates and products and the existence of
inhibition [5].
Definition 1.([5]) A metabolic reaction is called a
single substrate and single product (SS) reaction, if it
contains only one substrate and one product; otherwise,
called a multiple substrates or multiple products (MM)
reaction. An SS (or MM) reaction is called an SS (or
MM) reaction with inhibition, SSI (or MMI) for short, if
there exist some inhibitors of the reaction; otherwise,
called an SS (or MM) reaction with no inhibition, SSN
(or MMN) for short.
Remark 1.A reversible reaction will be viewed as two
reactions. For example, takeA
reactionAB
⎯→⎯
and the reverse reactionB
Before giving the definition of the SSI module, we
need the following concepts.
Definition 2. For a group of SS metabolic reactions (
including SSN and SSI reactions), take each metabolite as a
node. If two nodes appear in a same reaction, link them
with a directed edge (arrow) from the substrate to the pro-
duct, and such an edge is called reaction edge. If a metabo-
lite can inhibit some reaction, link it and the reaction edge
with a line that contains a bar at the end near the reaction
edge, and such an edge is called inhibition edge. Then we get
a graph, called reaction graph of the group of SS reactions.
Now, we give an example to show how to get a reaction
graph. Suppose that there are two SS reactions: A ® B, C
® D, and the metabolite D is an inhibitor of the first reac-
tion. The corresponding reaction graph is shown in Figure 1.
Definition 3. In the reaction graph of a group of SS
metabolic reactions, a node is called an input node, if
the direction of each reaction edge connecting it points to
other node; a node is called an output node, if the direc-
tion of each reaction edge connecting it points to itself.
B
E






as the forward
E
A
E
⎯→⎯
.
The other nodes are called state nodes. A state node that
directly connects with an input (or output) node is called
a head (or an end) node.
Definition 4. ([5]) A reaction is said to be relevant to
a metabolite S, if S is a reactant, a product or an inhibi-
tor of this reaction.
Definition 5. A path is a sequence of nodes such that
from each of its nodes there is a directed reaction edge to
the next node in the sequence.
Now we can define the SSI module.
Definition 6 (SSI module). For a given metabolic net-
work, denoteM the set of all the metabolites, andR
the set of all the reactions. The triple (L, ℛ, ℐ) is called
an SSI module within the metabolic network, L, ℛ and
ℐ are called the state node set, the reaction set and the
inhibition set of the SSI module, respectively, if the fol-
lowing conditions are satisfied:
(i)LM
⊂ is nonempty.
(ii)RR
⊂ is nonempty and constituted of all the
reactions which are relevant to the metabolites in L.
(iii) The reactions in ℛ are all SS (including SSN and
SSI) reactions.
(iv) ℐ ⊂ L × ℛis nonempty, and its element (I, A ®
B) means the metabolite I is an inhibitor of the reaction
A ® B.
(v) If there exist both input and output nodes, then for
any S Î L, there exist a directed path from some input
node to some output node passing S in the reaction
graph of ℛ.
(vi) The undirected graph constructed as follows is
connected: remove all the input and output nodes, the
inhibition edges, and the reaction edges connected with
the input or output nodes in the reaction graph of ℛ;
replace each directed reaction edge by an undirected one.
Remark 2.Not all reactions in an SSI module are SSI
reactions.
Remark 3. Although each reaction in an SSI module has
single substrate and single product, an SSI module could
be with multiple inputs and multiple outputs, i.e. having
multiple input and output nodes. Furthermore, an SSI
module could contain an SSN module or be decomposed
into an SSN module and a smaller SSI module. For exam-
ple, the SSI module shown in Figure 2(a)can be decom-
posed into the SSN module shown in Figure 2(b)and the
SSI module shown inFigure 2(c). But not all SSI modules
can be decomposed in this way. For example, the SSI mod-
ule ofFigure 3(a) contains the SSN module shownFigure 3
(b), but cannot be decomposed any more. This question
comes into the modularization decomposition of a meta-
bolic network, which is beyond the scope of this paper.
Modeling SSI metabolic modules
We will give an appropriate expression to describe the
rate of each metabolic reaction in an SSI module before
CD
AB
Figure 1 A reaction graph Each node means a metabolite. An
arrow represents a reaction, and the bar at the end of a line
denotes an inhibitor.
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modeling it in a mathematical manner, especially for the
reactions with inhibition.
Two broad classes of enzyme inhibitions, i.e. irreversi-
ble and reversible, are generally recognized [20-23]. In
an irreversible inhibition, the inhibitor combines with or
destroys a functional group on the enzyme that is essen-
tial for its activity. The irreversible inhibitor dissociates
very slowly from its target enzyme because it tightly
binds to its active site. Such a process is always irrever-
sible, and we do not consider it here. In contrast, in a
reversible inhibition, the inhibitor dissociates very
rapidly from its target enzyme because it becomes very
loosely bound with the enzyme. Three types of reversi-
ble inhibitions are observed: competitive, uncompetitive
and noncompetitive. Next we will introduce those rever-
sible inhibitions [20-26]. A competitive inhibitor can
combine reversibly with the active site of the enzyme
and compete with the substrate. If the active site is
occupied by the inhibitor, then it is unavailable for the
binding of the substrate, which decreases the reaction
rate. In the following reactions, the metabolite I is acting
as a competitive inhibitor of the reaction S ® P,
S + E ⇌ SE ® P + E
I + E ⇌ EI,
where S, E, P and I are substrate, enzyme, product and
inhibitor, respectively. Based on the Michaelis-Menten
kinetics with the conservation condition on E, the rate
of the reaction S ® P can be described as
v
V
1
C
KC
max
+
S
)
M
C
K
S
I
C
=
+
(
,
(1)
where CSand CIrepresent the concentrations of the
substrate S and the inhibitor I, respectively; Vmaxmeans
the maximum rate of the reaction, KMis the Michaelis-
Menten constant, and KCis the competitive inhibition
constant with respect to I.
An uncompetitive inhibitor cannot combine with a
free enzyme, but only with an enzyme-substrate
complex, and precludes the complex from converting
into product. In the following reactions, the metabolite
I is acting as a uncompetitive inhibitor of the reaction
S ® P,
S + E ⇌ SE ® P + E
I + SE ⇌ SEI.
In this case, the rate of the reaction S ® P can be
described as
v
V
+
C
(1
KC
maxS
+
MS
C
K
I
U
=
)
,
(2)
where KUis the uncompetitive inhibition constant
with respect to I.
An noncompetitive inhibitor can combine with both
free enzyme and enzyme-substrate complexes. Enzyme
is inactivated when such an inhibitor is bound, and can-
not catalyze the conversion from substrate into product.
In the following reactions, the metabolite I is acting as a
noncompetitive inhibitor of the reaction S ® P,
S + E ⇌ SE ® P + E
I + E ⇌ EI
I + SE ⇌ SEI.
In this case, the rate of the reaction S ® P can be
described as
v
V
C
K
C
C
+
K
max
)
S
M
S
C
K
I
C
I
U
=
++
(()
.
11
(3)
Although the above three types of reversible inhibitions
were observed in experiments, from the theoretical view-
point, (3) is a general expression of (1) and (2) with appropri-
ate parameter values. Hence, we will take (3) to describe the
rate of reaction S ® P when I is known to be an inhibitor.
Let (L, ℛ, ℐ) be an SSI metabolic module containing
n state nodes and m reactions, and denote
L = {S1,..., Sn},
AB
C
D
E
F
G
H
I
J
(a)
AB
D
E
F
G
H
(b)
C
D
E
F
G
H
I
J
(c)
Figure 2 An SSI module The SSI module (a) can be decomposed
into the SSN module (b) and the smaller SSI module (c).
AB
C
D
EF
(a)
B
C
D
E
(b)
Figure 3 An SSI module The SSI module (a) cannot be
decomposed into an SSN module and a new SSI module.
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ℛ = {A1® B1,..., Am® Bm}.
Assume that all the reactions in ℛ obey the Hill
kinetics or the Michaelis-Menten kinetics. Then, for the
reaction Aj® Bj, if Ijis an inhibitor, then we use
v
VC
KC
VC
j
maxj
C
K
Cj
A
n
M
n
A
n
C
K
maxj
j
j
j
j
Ij
j
j
Ij
Uj
=
+++
()
() ()() ()
(
11

A A
n
j
C
K
A
n
C
K
j
j
Ij
Cj
j
j
Ij
Uj
KC
)
()() ()11
+++
(4)
to describe the reaction rate; and if there is no inhibi-
tor, then we take
v
VC
KC
VC
KC
j
maxjA
n
M
n
A
n
maxjA
n
jA
n
j
j
j
j
j
j
j
j
j
j
=
++
()
()()
()
()
,

(5)
where CAjand CIjare the concentration of the
metabolite Ajand the inhibitor Ij, Vmaxjrepresents the
maximum rate of the reaction, KMjis the Michaelis-
Menten constant of the substrate Aj, njis the Hill coeffi-
cient, Kj= (KMj)nj. If nj= 1, (4) and (5) are also called
Michaelis-Menten equations.
Let Ci ≜ CSirepresent the concentration of the
metabolite Si, and C = (C1,..., Cn)τ, where τ means the
transpose of a matrix. Note that the rate of change of
the concentration of Si is given by the difference
between the rate(s) of the reaction(s) generating Siand
the rate(s) of the reaction(s) consuming Si. Then
∑
R
,
dC
dt
vv
i
j
ABBS
j
ABAS
jjjijjji
=−
→∈=→∈=
∑
R
,
,
(6)
where vjis given by (4) if there exists inhibitor of the
reaction and by (5) if there is no inhibitor. Then, we can
get a model of the SSI module (L, ℛ, ℐ),
dC
dt
R C P
1
( ; )
R C P
n
( ; )
R C
( ; ; ),
dC
dt

dC
dt
n
=
⎛
⎜
⎜
⎜⎜
⎝
⎞
⎟
⎟
⎟⎟
⎠
=
⎛
⎜
⎜⎜
⎝
⎞
⎟
⎟⎟
⎠
1

P
(7)
where Ri(C; P) is given by the right hand side of (6),
P is vector-valued model parameter. R(C; P) is called the
rate function of the model.
Remark 4.If node Ajis an input node, then its concen-
tration CAjin (4) or (5) is not a variable of the model (7)
but a parameter.
Definition 7.For a fixed parameter P0, an equilibrium
of ( 7 ) is a state C that satisfiesdC
dt= 0 , i.e. a solution
of the algebraic equations R(C; P0) = 0. System (7) or the
SSI module (L, ℛ, ℐ) is said to have the capability of
multiple equilibria, if there exists a parameter P0such
that the algebraic equations R(C;P0) = 0 have more than
one positive solutions.
Theoretical results
In this section, we will derive a sufficient condition for
the absence of multiple equilibria of a common type of
SSI modules. But the system (7) is not effective for ana-
lyzing. So we convert it to another equivalent form first.
Define

L=∈ = −∞ ∞
(
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
=∑z S
i
1
z
i ii
n
:, ) .
We can show that ℝLis a vector space spanned by
L = {S1,..., Sn}. For any reaction A®B Î ℛ, if A (or B)
is a state node in L, we view it as a vector in ℝL; if A
(or B) is an input (or output) node, we make a conven-
tion viewing it as the zero vector in ℝL. Denote
εi= (0,..., 1,..., 0)τ,
whose entries are all zero except the ith position.
Then
CCCC
ni i
e
i
n
==
=∑
(,,),
1
1

t
dC
dt
dC
dt
dC
dt
dC
dt
ni
i
i
n
=⎛
⎝⎜
⎧
⎨⎪
⎩⎪
⎞
⎠⎟=
=∑
1
1
,,

t
e
=−
⎫
⎬⎪
⎭⎪
→∈=→∈==
∑
j
R,
∑
B
j
∑
i
vv
jj
ABASABS
n
i
jjijji
R,
1
e
=−
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
===∑
1
∑
j
∑
i
dde
B S
j i
j
j
m
A Sj
mn
i
j i
vv
11
=−
⎧
⎨⎪
⎩⎪
⎫
⎬⎪
⎭⎪
===∑
1
∑
i
∑
j
dede
B S
j i
j
i
n
A Si
nm
j
j i
v
11

() ,
v
ba
jjj
j
m
−
=∑
1
where
dMS
i
i
MS
=
=
⎧
⎨
⎩
1
0
,
,
,if
otherwise,
and the column vectors
∑
1
tors Bjand Ajin ℝLwith respect to the basis L, respec-
tively. Thus, we get the equivalent model,
bde
j B S
j i
i
i
n
=
=
∑
1
and
ade
jA S
i
n
i
j i
=
=
in ℝnare the coordinates of the vec-
dC
dt
R C P
( ; )
v
jjj
j
m
==−
=∑
() .
ba
1
(8)
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