Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

Key Laboratory of Systems Biology, SIBS-Novo Nordisk Translational Research Centre for PreDiabetes, Shanghai Institutes for Biological Sciences, Chinese Academy of Sciences, Shanghai 200031, China

Collaborative Research Center for Innovative Mathematical Modelling, Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan

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 systems biology

There are some pioneering works in structure-oriented study on multiple equilibria of networks

To overcome such a difficulty, we proposed a structure-oriented modularization framework in

In particular, in

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 negative 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 substrates, the inhibitions and their interactions in SSI modules.

Owing to these inherent characteristics, both the modeling 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 metabolite 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 modules, 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 mechanisms 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 modules, and is a polynomial that is increasing in any of its variables in the work

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 necessary 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 reaction 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 investigate a reaction with inhibition from the viewpoints of both experiment and theory, and special analysis methods is required. Hence, to investigate a metabolic network, it may be necessary and feasible to divide the metabolic reactions into two groups, one is with inhibition and the other is with no inhibition. In real metabolic 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 metabolic reactions into four classes according to the number of substrates and products and the existence of inhibition

**Definition 1.**

**Remark 1.**

Before giving the definition of the SSI module, we need the following concepts.

**Definition 2.**

Now, we give an example to show how to get a reaction graph. Suppose that there are two SS reactions:

A reaction graph

**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.

**Definition 3.**

**Definition 4.**

**Definition 5.**

Now we can define the SSI module.

**Definition 6** (SSI module).

(

**Remark 2.**

**Remark 3.**

An SSI module

**An SSI module** The SSI module (a) can be decomposed into the SSN module (b) and the smaller SSI module (c).

An SSI module

**An SSI module** The SSI module (a) cannot be decomposed into an SSN module and a new SSI module.

Modeling SSI metabolic modules

We will give an appropriate expression to describe the rate of each metabolic reaction in an SSI module before modeling it in a mathematical manner, especially for the reactions with inhibition.

Two broad classes of enzyme inhibitions, i.e. irreversible and reversible, are generally recognized

where

where _{S}_{I}_{max}_{M}_{C}

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

In this case, the rate of the reaction

where _{U}

An noncompetitive inhibitor can combine with both free enzyme and enzyme-substrate complexes. Enzyme is inactivated when such an inhibitor is bound, and cannot catalyze the conversion from substrate into product. In the following reactions, the metabolite

In this case, the rate of the reaction

Although the above three types of reversible inhibitions were observed in experiments, from the theoretical viewpoint, (3) is a general expression of (1) and (2) with appropriate parameter values. Hence, we will take (3) to describe the rate of reaction

Let (

_{1},⋯, _{n}

_{1} → _{1},⋯, _{m}_{m}

Assume that all the reactions in _{j}_{j}_{j}

to describe the reaction rate; and if there is no inhibitor, then we take

where _{Aj}_{Ij}_{j}_{j}_{maxj}_{Mj}_{j}_{j}_{j}_{Mj}^{nj}_{j}

Let _{i}_{Si}_{i}_{1},⋯, _{n}^{τ}_{i}_{i}_{i}

where _{j}

where _{i}

**Remark 4.**_{j} is an input node, then its concentration C_{Aj} in

**Definition 7.**
_{0}
_{0}
_{0}
_{0}

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 analyzing. So we convert it to another equivalent form first.

Define

We can show that ℝ^{ℒ}_{1},⋯, _{n}^{ℒ}^{ℒ}

_{i}^{τ}

whose entries are all zero except the

where

and the column vectors ^{n}_{j}_{j}^{ℒ}

Now, we can go on the model analysis based on the new equivalent model (8).

**Definition 8.**^{n}^{n}_{1} ≠ _{2} ∈ ℝ^{n}_{1}) = _{2}).

**Lemma 1.**^{n}^{n}^{n}

**D**, then the system cannot admit multiple equilibria in ** D**, i.e. the equations F** D**.

Lemma 1 provides a sufficient condition for the absence of multiple equilibria of a general system, but such a condition is difficult to be verified. Hence, we need to convert it into an equivalent one which is relatively easy to be verified. For some simple cases, for example, ^{2}, but the determinant of its Jacobian matrix is ^{2}, which is zero on line

**Lemma 2.**** D** ⊂ ℝ

_{1},⋯, _{n}^{τ}^{n}, _{k}^{n} → ℝ (_{i}

**Lemma 3.**^{n}_{1}, _{2}, _{3}, _{4}} _{k}_{3} ∪ _{4}, _{k}_{k}_{2} ∪ _{4}, _{k}_{k}_{k}

_{k}_{k}_{k}_{k}_{k}_{k}

_{1},⋯,_{m}** P**.

**Thorem 1.**

Discussion

The above result provides a sufficient condition for the absence of multiple equilibria of a type of SSI modules. But this condition cannot be satisfied by all such SSI modules. In other words, some SSI modules can actually admit multiple equilibria. We will give such an example. The SSI module is shown in Figure _{0}, _{1}, _{2} and _{3} represent the concentration of the metabolites _{1}, _{2}, _{3})^{τ}

An SSI module having multiple equilibria

**An SSI module having multiple equilibria** This SSI module can admit multiple equilibria.

Then we can get the model,

_{1}/_{1} – _{2} – _{3} (13a)

_{2}/_{2} – _{4} (13b)

_{3}/_{3} – _{5}, (13c)

where

Denote

Then the Jacobian matrix of the rate function

Consequently,

Note that that

_{1} = 0.35000 _{2} = 0.72835 _{3} = 2.44647, (14)

_{1} = 0.35438 _{2} = 2.48512 _{3} = 0.78621, (15)

_{1} = 0.34243 _{2} = 1.11200 _{3} = 1.34743. (16)

Parameter values

parameter

value

parameter

value

parameter

value

_{max}_{1}

2.6

_{1}

0.14

_{1}

1

_{max}_{2}

3.7

_{2}

0.23

_{2}

2

_{max}_{3}

4.3

_{3}

0.23

_{3}

2

_{max}_{4}

1.3

_{4}

0.29

_{4}

1

_{max}_{5}

1.5

_{5}

0.27

_{5}

1

_{C}_{2}

5.4

_{u}_{2}

9.8

_{0}

1

_{C}_{3}

6.2

_{u}_{3}

8.7

The parameter values used in the simulation of the numeric example.

The Jacobian matrix of the rate function

Its eigenvalues are

_{1} = –9.1666, _{2} = –0.1730, _{3} = –0.0267.

They are all negative numbers, which implies that the system (13) is asymptotically stable at the equilibrium (14).

The Jacobian matrix of the rate function

And its eigenvalues are

_{1} = –8.8773, _{2} = –0.3005, _{3} = 0.0209.

The first two are negative and the last one is positive. This means the system (13) is not stable at the equilibrium (15). Figure _{1} = 0.2, _{2} = 0.728 and _{3} = 2.519. If we take a small change on the initial value for _{1} = 0.2, _{2} = 0.728 and _{3} = 2.520, then the trajectory would be diverged, see Figure _{1} = 0.2, _{2} = 0.729 and _{3} = 2.520, see Figure

Simulation of the example

**Simulation of the example** The dynamic behaviors of the system (13) starting from four different initial values. The model parameters are listed in Table _{1} = 0.2, _{2} = 0.728, _{3} = 2.519. (b) _{1} = 0.2, _{2} = 0.728, _{3} = 2.520. (c) _{1} = 0.2, _{2} = 0.723, _{3} = 2.519. (d) _{1} = 0.2, _{2} = 0.729, _{3} = 2.520.

Conclusions

The multi-equilibrium property of metabolic networks is of great practical significance and difficult to be investigated biologically or theoretically. To study it, we proposed a structure-oriented modularization framework: viewing a metabolic network as an assembly of basic building blocks with particular structures, and investigating the multi-equilibrium property of the original network by studying the characteristics of the basic modules and their interactions. The SSI module is one of the four types of basic building blocks, whose multi-equilibrium property was studied in this paper.

Due to the complexity of its topological structure, the strong coupling between each metabolite and the intricacy of the reaction mechanism, it is a difficult task to analyze the dynamic properties of SSI modules. In particular, comparing with SSN modules, there exists negative feedbacks in SSI modules caused by inhibitions, which makes the module structure and the reaction mechanism much more complicated. This paper mainly discussed one common type of SSI modules in which each reaction has no more than one inhibitor, which is considered as the first step towards elucidating the design principle of metabolic networks in living organisms. In the near future, we will further discuss the SSI modules in which there are reactions with more than one inhibitor. In addition, the main idea of this work can be extended to study the problem of networkomics (or netomics) which covers all stable forms of biomolecular networks

Methods

Proof of Lemma 2

Now, we will prove (i). Assume, to arrive a contradiction, that ** D** for some

This implies that the Jacobian matrix ** D** for all

(ii) can be proved similarly.

Proof of Lemma 3

_{k}_{i}

where

Let

where

Denote

Now we will first show the necessity. By Lemma 2(i), it to sufficient to show that for any

that is, (10) holds.

_{k}_{2}, the parameters _{3}, and the parameters _{4}, and does not contain the parameter _{k}_{2}, the parameters _{3}, the parameters _{4}, and does not contain the parameter

Then _{k}_{k}_{k}_{k}_{k}_{k}_{k}

Case _{1}_{k}_{k}

Case _{2}: Now, _{qk}_{k}_{k}_{k}_{k}_{k}_{k}_{k}

Case _{3}: Note that _{rk}_{2} we can show that _{k}_{k}

Case _{4}

If _{rk}_{qk}_{k}_{k}_{k}

If _{k}_{k}_{k}_{k}

If

where

Since _{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}_{k}

Next, we will show the sufficiency. By Lemma 2 (ii) and the discussion in the proof of the necessity, it is sufficient to show for any _{k}_{k}

Take

Then _{k}_{k}

Case _{1}_{k}_{k}

Case _{2} ∪ _{3}: Similar to the proof of necessity for the cases _{2} and _{3}, respectively, we can show that _{k}_{k}

Case _{4}_{k}_{k}_{k}_{k}

When _{k}_{k}_{k}

which implies _{k}_{k}_{k}_{k}_{k}

Proof of Theorem 1

_{1} = {_{j}_{j}_{j}

_{2} = {_{j}_{j}_{j}

_{3} = {_{j}_{j}_{j}

_{4} = {_{j}_{j}_{j}

When each reaction in _{j}_{k}_{k}

Authors contributions

HBL, JFZ and LC proposed the main idea. HBL developed theoretical results and drafted the manuscript. JFZ and LC gave valuable suggestions. All authors wrote and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (under grant 60821091, 91029301 and 61072149), by CAS KSCX2-EW-R-01, and by the Chief Scientist Program of Shanghai Institutes for Biological Sciences, CAS (under grant 2009CSP002). This work was also partially supported by JSPS FIRST Program, Japan.

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