Multi‐equilibrium property of metabolic networks: Exclusion of multi‐stability for SSN metabolic modules

Abstract

It is a fundamental and important problem whether or not a metabolic network can admit multiple equilibria in a living organism. 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 viewpoints. In this paper, a structure-oriented modularization research framework is proposed to analyze the multi-stability of metabolic networks. We first decompose a metabolic network into four types of basic building blocks (called metabolic modules) according to the particularity of its structure, and then focus on one type of these basic building blocks—the single substrate and single product with no inhibition (SSN) module, by deriving a nonlinear ordinary differential equation (ODE) model based on the Hill kinetics. We show that the injectivity of the vector field of the ODE model is equivalent to the nonsingularity of its Jacobian matrix, which enables us equivalently to convert an unverifiable sufficient condition for the absence of multiple equilibria of an SSN module into a verifiable one. Moreover, we prove that this sufficient condition holds for the SSN module in a living organism. Such a theoretical result not only provides a general framework for modeling metabolic networks, but also shows that the SSN module in a living organism cannot be multi-stable. Copyright © 2011 John Wiley & Sons, Ltd.

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
Int. J. Robust. Nonlinear Control (2011)
Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/rnc.1718
Multi-equilibrium property of metabolic networks: Exclusion of
multi-stability for SSN metabolic modules
Hong-Bo Lei1, Ji-Feng Zhang1,∗,†and Luonan Chen2
1Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
2Key Laboratory of Systems Biology, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences,
Shanghai 200031, People’s Republic of China
SUMMARY
It is a fundamental and important problem whether or not a metabolic network can admit multiple equilibria
in a living organism. 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 viewpoints. In this paper, a structure-oriented
modularization research framework is proposed to analyze the multi-stability of metabolic networks. We
first decompose a metabolic network into four types of basic building blocks (called metabolic modules)
according to the particularity of its structure, and then focus on one type of these basic building blocks—
the single substrate and single product with no inhibition (SSN) module, by deriving a nonlinear ordinary
differential equation (ODE) model based on the Hill kinetics. We show that the injectivity of the vector
field of the ODE model is equivalent to the nonsingularity of its Jacobian matrix, which enables us
equivalently to convert an unverifiable sufficient condition for the absence of multiple equilibria of an
SSN module into a verifiable one. Moreover, we prove that this sufficient condition holds for the SSN
module in a living organism. Such a theoretical result not only provides a general framework for modeling
metabolic networks, but also shows that the SSN module in a living organism cannot be multi-stable.
Copyright ? 2011 John Wiley & Sons, Ltd.
Received 30 August 2010; Revised 28 January 2011; Accepted 2 February 2011
KEY WORDS:metabolic network; multiple equilibria; multi-stability; SSN module
1. INTRODUCTION
Metabolic networks are one type of the most important molecular networks in living organisms.
As nonlinear dynamical systems, they can generally admit three dynamic features: mono-stability
(only one stable steady state), bi-stability or multi-stability (more than one stable steady states), and
oscillation (periodic or non-periodic orbits) [1–4]. Mono-stability is very common in biological
systems. With the recent development of biotechnology, bi-stability [5,6] and oscillation [7,8]
were also observed at molecular level in various living organisms.
A question naturally arises on whether or not a given metabolic network is multi-stable in
all its feasible regions. For its great biological significance, this question attracts much attention
from both biologists and mathematicians. Take the photosynthetic carbon metabolic network in
the mesophyll cells of plants as an example. If it has two or more steady states, one of them will
∗Correspondence to: Ji-Feng Zhang, Key Laboratory of Systems and Control, Academy of Mathematics and Systems
Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China.
†E-mail: jif@iss.ac.cn
Copyright ? 2011 John Wiley & Sons, Ltd.

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H.-B. LEI, J.-F. ZHANG AND L. CHEN
correspond to the higher or highest photosynthesis rate, which clearly can be used to increase the
grain yields by driving the system into this steady state; otherwise, the only thing one can do is
to improve the photosynthesis at the existing steady state [9–11]. Thus, it is crucial to accurately
judge whether or not this molecular network is able to admit multiple equilibria so that a correct
strategy can be adopted.
It is not only expensive but also difficult, if not impossible, to answer such a question via
biological experiments. Hence, a system modeling approach to tackle this problem is strongly
demanded. However, in the traditional theoretical analysis, necessary information about model
parameters is always required. Due to the limitation of measurement tools and measurement errors,
most of the model parameters are either unavailable or uncertain. This makes it difficult to analyze
the model. Moreover, the biological variability, which makes the model parameters vary drastically
from different individuals, limits the applications of the theoretical results based on a model with
fixed parameter values. Thus, to detect multi-stability, another good way is to analyze the network
from a structure perspective, which needs only the topological structure information but has a wide
range suitability to various individuals or systems.
There are some pioneering works in the area of detecting multiple equilibria of some specific
systems. In 1981, a conjecture of Thomas asserts that a necessary condition for multiple equilibria
is the presence of a positive circuit in the interaction graph, i.e. a circuit in which the product of the
signs of the edges is positive [12]. Since then, it took more than two decades for people to prove
the Thomas conjecture from a particular case to a general case [13–17]. Based on the Thomas
conjecture and other three open conjectures [18–20], Kaufman et al. gave another necessary
condition for multiple equilibria of a dynamical system described by a set of ordinary differential
equations (ODEs) [21]. Craciun and Feinberg [22–24] analyzed the chemical network constituted
by the reactions in an isothermal homogeneous continuous flow stirred tank reactor (CFSTR [25]).
They described the rate of each chemical reaction in the CFSTR by the law of mass-action and
gave sufficient conditions for the absence and presence of multiple positive equilibria, respectively
[22]. Moreover, by regarding a cell as a CFSTR, they extended the work in [22,23] to an enzyme-
driven reaction network [24]. Chesi proposed a recursive algorithm to compute equilibrium point
of genetic regulatory network [26] and analyzed their global asymptotic stability [27].
Generally, a metabolic network in a living organism is a large-scale system. Hence, it is difficult
to directly study it as a whole, especially when the only available information is the network
structure. To overcome such a difficulty, we resort to the modularization idea, which is widely used
in the area of systems control [28–30], and view a metabolic network as an assembly of several
basic building blocks. Specifically, by exploiting the topological structures, we first decompose a
general metabolic network into four types of basic building blocks, which are called metabolic
modules, and then investigate the multi-stability of the original network through studying the
dynamical characteristics of the basic modules and their interactions. Such a strategy can not
only reduce the difficulty in studying a complex metabolic network, but also make full use of its
structural information. After getting a profound insight in the basic building blocks, we can use
them to reconstruct a metabolic network.
To practice such a modularization idea, we model a general metabolic network in a mathematical
manner. Generally, an enzyme-catalyzed metabolicreaction contains several intermediate reactions,
such as thebindingof substrateand enzymeand thedegradingof theintermediateenzyme–substrate
complex. If the details of all those intermediate reactions are known, one can model the network
based on the law of mass action and use the theory of Craciun et al. [22–24] to investigate its
multi-stability. Unfortunately, the detailed intermediate process of a metabolic reaction is unknown
for many cases. Compared with the law of mass action, less information about the intermediate
reactions is required when the Michaelis–Menten kinetics [31] or Hill kinetics [32] is chosen to
describe the rate of a metabolic reaction. Moreover, the parameters in the Hill or Michaelis–Menten
kinetics have clear biological meaning and can be regulated by experimental techniques. Thus, we
model the enzyme-driven metabolic network based on the Hill kinetics in this work, and investigate
whether or not it can admit multiple equilibria from a structure perspective.
In this paper, we show that a metabolic network can be constructed by four basic modules,
and specifically focus on one key type of the basic modules, i.e. the single substrate and single
Copyright ? 2011 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control (2011)
DOI: 10.1002/rnc

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EXCLUSION OF MULTI-STABILITY FOR SSN METABOLIC MODULES
product without inhibition (SSN) module, and prove that it cannot be multi-stable. By our results,
although an SSN module satisfies the necessary condition for multiple equilibria given in the
Thomas conjecture [12], there is still no more than one equilibrium. This implies that the necessary
condition is not sufficient.
This paper is organized as follows. Section 2 introduces the main idea of the modularization
decomposition of a metabolic network, defines the SSN metabolic module and provides a general
framework to model it. Section 3 proves the SSN metabolic modules in living organisms cannot
be multi-stable. Section 4 gives a biological example to illustrate the application of the theoretical
results. Section 5 provides some concluding remarks and some topics worth investigating.
2. MODELING SSN MODULE
A metabolic network is generally composed of a large number of enzyme-catalyzed reactions.
Some of them contain only one substrate and one product, and the others do not. Furthermore,
there may exist inhibitors for some metabolic reaction, which acts to decrease the reaction rate.
Due to the special feature of enzyme-catalyzed reactions, we can classify metabolic reactions into
four classes according to the number of substrates and products and the existence of inhibitors, as
follows.
Definition 2.1
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) metabolic 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.
Due to the particularity of activation, we will discuss it in detail as future work and do not
consider it in thisarticle. And in this sense, thefour classes ofreactions definedin Definition2.1, i.e.
SSI, MMI, SSN, and MMN reactions, can cover all metabolic reactions. Then we can decompose
a metabolic network into four types of basic building blocks, and call them SSN, SSI, MMN, and
MMI metabolic modules, respectively. In fact, initially taking each reaction as a module and then
extending it to the maximum subnetwork through adding its neighbor nodes is a nice and efficient
way to do the modularization decomposition. We aim to study a metabolic network with the
modularization idea, regard it as a system consisting of these basic modules, and then investigate
the original network through studying the characteristics of the basic modules and their interactions.
There are several steps for such a process, from the optimal modularization decomposition to the
characteristics of each module, to the interactions among these modules, and to the integration of
the four types of basic modules. As a first step in this line, we will discuss the SSN module in
this paper.
2.1. Definitions
In this section, we will give the definition for SSN module.
Definition 2.2 (Reaction graph)
For a group of SSN metabolic reactions, take each metabolite as a node. If two nodes appear in
a same reaction, link them with a directed edge from the substrate to the product. Then we get a
graph, called reaction graph of the group of SSN reactions.
Remark 2.1
We regard a reversible reaction as two reactions. For example, take A
E
− ?
? −B as the forward reaction
A
E
− →B and the reverse reaction B
E
− → A.
Copyright ? 2011 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control (2011)
DOI: 10.1002/rnc

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H.-B. LEI, J.-F. ZHANG AND L. CHEN
Remark 2.2
Any node in a reaction graph must be a substrate or a product of some reaction. Hence, it will
have at least one connecting edge.
Definition 2.3
In the reaction graph of a group of SSN metabolic reactions, a node is called an input node, if the
direction of each edge connecting it points to other node; a node is called an output node, if the
direction of each edge connecting it points to itself. 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 2.4
A reaction is said to be relevant to a metabolite S, if S is the reactant, product or inhibitor of this
reaction.
Definition 2.5 (SSN module)
For a given metabolic network, denote?
are called the state node set and the reaction set of the SSN module, respectively, if the following
conditions are satisfied:
M the set of all the metabolites, and? R the set of all the
reactions. The pair S×R is called an SSN module within the metabolic network, and S and R
(i) S is a nonempty subset of?
metabolites in S.
(iii) The reactions in R are all SSN reactions.
(iv) If there exist both input and output nodes, then for any S∈S, there exists one directed
path‡from some input node to some output node passing S in the reaction graph of R.
(v) The undirected graph constructed as follows is a connected graph: remove all the input and
output nodes (if there exist) and the edges connected with them in the reaction graph of
R, and replace each directed edge by an undirected one.
M.
(ii) R, a nonempty subset of? R, is constituted of all the reactions which are relevant to the
Remark 2.3
The first three conditions of Definition 2.5 are the elementary requirements. The condition (iv)
is from biological systems. In a living organism, any metabolite must be synthesized from other
metabolites and be converted into an output. The condition (v) is essential for the modularization
decomposition of a metabolic network.
Remark 2.4
Although each reaction in an SSN module is with single substrate and single product, an SSN
module could be with multiple inputs and multiple outputs, i.e. multiple input and output nodes.
Similarly, we can define the SSI, MMI, and MMN metabolic modules, which will be studied
in detail in our future works. We now give an example to elucidate the concepts in the above
definitions.
Suppose that there is a group of metabolic reactions, H
S2
− →S3, S2
where H, Si, and Pkare metabolites (i.e. substrates and products) and Ejare enzymes. They are
all SSN reactions, and their reaction graph is shown in Figure 1. By Definition 2.3, H is an input
node, P1and P2are output nodes, S1is a head node, S3is both a head and an end node, and
S4and S5are end nodes. Denote S={S1,S2,S3,S4,S5} and R the set of those reactions. Then
S×R is an SSN module.
A metabolic network in a living organism is usually very complex and is generally composed
of various types of metabolic reactions, including SSN, SSI, MMN, and MMI reactions. Thus, the
E1
− →S1, H
− − → P1, S4
E2
− →S3, S1
E11
− − →P2, S5
E3
− →S2, S1
E12
− − →S1, S5
E4
− →S3,
E13
− − → P1,
E5
E6
− →S4, S2
E7
− →S5, S3
E8
− →S4, S3
E9
− → P2, S4
E10
‡Here, a path in a directed graph is a sequence of nodes such that from each of its nodes there is a directed edge
to the next node in the sequence.
Copyright ? 2011 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control (2011)
DOI: 10.1002/rnc

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EXCLUSION OF MULTI-STABILITY FOR SSN METABOLIC MODULES
Figure 1. The reaction graph of the reaction set R.
one with only SSN modules is a special case, but SSN metabolic module exists in many metabolic
networks. Take the reversible reaction R5P
← →Ru5P in the photosynthetic carbon metabolism
as an example, where R5P, Ru5P, and Rpi represent, respectively, the Ribose 5-phosphate,
Ribulose 5-phosphate and Ribose 5-phosphate isomerase. Let S={R5P,Ru5P}, and R be the
set of R5P
− − →Ru5P, Ru5P
S×R is an SSN module of the photosynthetic carbon metabolism.
Rpi
Rpi Rpi
− − →R5P and the other reactions relevant to R5P and Ru5P. Then
2.2. Modeling SSN module
Let S×R be an SSN metabolic module containing n state nodes and m reactions, and denote
S={S1,...,Sn},
R={A1→B1,..., Am→Bm}.
Assume that all the reactions in R obey the Hill kinetics. Then, the Hill function
Vmaxj(CAj)nj
(KMj)nj+(CAj)nj?Vmaxj(CAj)nj
vj=
Kj+(CAj)nj
(1)
can be used to describe the rate of the enzyme-catalyzed reaction Aj
concentration of the metabolite Aj, Vmaxjrepresents the maximum rate of the reaction, KMjis the
Michaelis–Menten constant of the substrate Aj, njis the Hill coefficient, Kj=(KMj)nj. If nj=1,
(1) is also called Michaelis–Menten equation. The Hill kinetics or the Michaelis–Menten kinetics
is a good choice to describe the rate of a biochemical reaction, which has been widely accepted
by both biologists and mathematicians. Moreover, an enzyme-catalyzed metabolic reaction can be
formulated by these kinetics without the detailed information on its intermediate process.
Let Ci?CSirepresent the concentration of the metabolite Si, and C=(C1,...,Cn)T, where T
means the transpose of a matrix. Note that the rate of change of the concentration of Siis given by
the difference between the rate(s) of the reaction(s) generating Siand the rate(s) of the reaction(s)
consuming Si. Then
Ej
− →Bj, where CAjis the
dCi
dt
=
?
Aj→Bj∈R,
Bj=Si
vj−
?
Aj→Bj∈R,
Aj=Si
vj,
(2)
where vjis given in (1). Then, we can get a model of the SSN module S×R,
⎛
⎜
⎜
dt
where Ri(C; P) is given by the right-hand side of (2), P is vector-valued model parameter. R(C; P)
is called the rate function of the model.
dC
dt=
⎜
⎜
⎜
⎝
⎜
dC1
dt
...
dCn
⎞
⎟
⎟
⎟
⎟
⎟
⎠
⎟
=
⎛
⎜
⎜
⎜
⎝
R1(C; P)
...
Rn(C; P)
⎞
⎟
⎟
⎟
⎠?R(C; P),
(3)
Copyright ? 2011 John Wiley & Sons, Ltd.
Int. J. Robust. Nonlinear Control (2011)
DOI: 10.1002/rnc