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arXiv:1001.1944v3 [q-bio.MN] 28 Jul 2010

On the Stability of Metabolic Cycles

Ed Reznika,1,2, Daniel Segr` ea,1,2,3,∗

aBioinformatics Program, 24 Cummington St, Boston MA

Abstract

We investigate the stability properties of two different classes of metabolic cycles using a combination

of analytical and computational methods. Using principles from structural kinetic modeling (SKM),

we show that the stability of metabolic networks with certain structural regularities can be studied

using a combination of analytical and computational techniques. We then apply these techniques

to a class of single input, single output metabolic cycles, and find that the cycles are stable under

all conditions tested. Next, we extend our analysis to a small autocatalytic cycle, and determine

parameter regimes within which the cycle is very likely to be stable. We demonstrate that analytical

methods can be used to understand the relationship between kinetic parameters and stability, and that

results from these analytical methods can be confirmed with computational experiments. In addition,

our results suggest that elevated metabolite concentrations and certain crucial saturation parameters

can strongly affect the stability of the entire metabolic cycle. We discuss our results in light of the

possibility that evolutionary forces may select for metabolic network topologies with a high intrinsic

probability of being stable. Furthermore, our conclusions support the hypothesis that certain types

of metabolic cycles may have played a role in the development of primitive metabolism despite the

absence of regulatory mechanisms.

1. Introduction

Cycles are at the heart of the metabolic networks of organisms spanning the entire tree of life

[1, 2, 3]. For example, the tricarboxylic acid (TCA) cycle sits at the core of energy production for many

species and additionally plays the role of regenerating essential cellular nutrients and components. A

great deal of research has been devoted to understanding the stability properties of such cycles, which

have often been represented as chemical reaction networks (CRNs). Many of these studies have drawn

conclusions about the steady state of a CRN without regard to details about the rate constants or

chemical concentrations of the particular CRN (for instance, see [4, 5, 6]). Instead, these approaches

rely on the topology of the network and commonly assume mass-action kinetics laws to limit (and

sometimes determine) the possible behaviors the system may exhibit at steady state. Can one hope

to extend these results to general, potentially nonlinear, rate laws?

In this work, we use a generalized modeling framework known as structural kinetic modeling (SKM)

to extend previous results on two classes of metabolic cycles. The first class has a special but relatively

flexible structure: it contains a single input, a single output, and unlimited length. This relatively

organized structure makes it amenable to analytical methods of study. For cycles belonging to this first

class, we prove that the cycle can only lose dynamical stability in an oscillatory manner, regardless of

absolute metabolite concentrations, flux values, kinetic rate constants, and form of monotonic kinetic

∗Corresponding Author

Email addresses: ereznik@bu.edu (Ed Reznik), dsegre@bu.edu (Daniel Segr` e)

1Boston University Department of Biomedical Engineering

2Boston University Center for Biodynamics

3Boston University Department of Biology

Preprint submitted to ElsevierJuly 29, 2010

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rate law. We further investigate the cycle computationally, and find that it remains completely stable

under all conditions tested, leading us to make a conjecture about its stability under all conditions.

We then apply the same SKM approach to a second class of cycles, autocatalytic cycles, and

demonstrate that these are not stable under all combinations of kinetic parameters and metabolic

conditions. Instead, we use analytical calculations to derive relationships between key parameters

which can enhance or attenuate the stability properties of the cycle. These relationships are then

computationally tested and verified.

Nearly all prior published work using SKM [7, 8] employs computational experiments and statistical

methods in order to draw conclusions about the system of interest. In this work, we show that metabolic

systems with relatively well-organized structures are not only analytically tractable but furthermore

that their analysis can actually lead to interesting and novel conclusions. Finally, we discuss the

implications of our results on the evolution of primitive metabolic networks. The SKM technique

used throughout the paper highlights essential parameters which directly impact the stability of the

cycle under study. Particular combinations of these parameters may have conferred an evolutionary

advantage to their respective metabolic networks by making them robust to small environmental

perturbations.

2. Background

2.1. Structural Kinetic Modeling (SKM)

The methods developed in this paper are based on the SKM framework[9]. SKM is a specific appli-

cation of generalized modeling [10] in which normalized parameters replace conventional parameters

(such as Vmax or KM in the modeling of metabolic networks). The normalized parameters have a

direct connection to the original kinetic parameters, but are much easier to work with. As will be

shown below, these parameters usually have well-defined and limited ranges (e.g. [0,1]), and sampling

them across this range effectively samples all possible values of the original kinetic parameters.

The goal of SKM is to capture the local stability properties of a biochemical system. In this sense,

it bridges the gap between genome-scale steady state modeling [11] and explicit kinetic modeling of a

metabolic process. To study stability, one usually determines the Jacobian J of the system of interest

and evaluates it at the system’s steady states. Assuming knowledge of the form of kinetic rate laws

for a CRN, it is quite possible to write down the corresponding J. However, in many cases J will be

unnecessarily complicated and quite difficult to work with. By performing a change of variables, SKM

actually simplifies the mathematical form of each entry in J. The new entries are in almost all cases

easier to work with. The specific method by which the change of variables occurs is briefly described

below and more completely in Appendix A. Much of this material is based on [9] and we refer readers

to that reference and its supplementary materials for more information.

If we let S be the m-dimensional vector of metabolite concentrations, N be the m×r stoichiometric

matrix, and v be the r-dimensional vector of reaction rates, then we can describe the dynamics of the

system with the equation

dS

dt= Nv(S,k)(2.1)

where v(S,k) denotes that the reaction rates are dependent on both metabolite concentrations S

and kinetic parameters (such as Michaelis-Menten constants) k. If we assume that a non-negative

steady state S0exists, then we can redefine our system using the definitions

xi=Si(t)

S0

i

,Λij= Nijvj(S0)

S0

i

,µj(x) =

vj(S)

vj(S0)

(2.2)

where i = 1...m and j = 1...r, x is a vector of metabolite concentrations normalized with respect

to their steady state concentrations and µ represents flux normalized with respect to steady state flux

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values. The matrix Λ represents the stoichiometric matrix normalized with respect to steady state

fluxes and steady state metabolite concentrations.

The Jacobian, evaluated at x0= 1 (which, because of the way x is defined, is actually the equilib-

rium of the system), can be written as

Jx= Λθµ

x

(2.3)

Note that the equations above were derived without regard to the actual form of the kinetic

equations that determined the ODE system. The matrix θµ

degree of saturation of normalized flux µj with respect to normalized substrate concentration xi.

In terms of derivatives, each element of θ represents the degree of change in a flux as a particular

metabolite is incrementally increased. This is analogous to the concept of elasticity in metabolic

control analysis [12].

What does the θ matrix look like? Its columns correspond to each metabolite, and its rows to each

flux. A non-zero element θi

jin the matrix represents the effect a small change in a metabolite j has on

flux i. In the case of Michaelis-Menten kinetics, this element in the matrix may take values ranging

from [0,1]. In the case of standard competitive inhibition (e.g. allosteric inhibition by a product), the

element takes values in [-1,0].

To usefully illustrate the meaning of a single θ element, consider an equation following Michaelis-

Menten kinetics shown in (2.4). First, we write µ(x), which we recall is the flux normalized by the

flux at the steady state. Here, S0is the concentration of the substrate S at steady state. Then, we

manipulate the equation by substituting xS0for S, where x is the normalized steady state concentration

of substrate S. The result is shown in (2.5).

xcontains elements which represent the

V =

k2E0S

KM+ S

(2.4)

µ(x) =

k2E0S

KM+S

k2E0S0

KM+S0

= xKM+ S0

KM+ xS0

(2.5)

Finally, we take a derivative with respect to x and evaluate it at x = 1 to obtain θ in (2.6). Notice

that θ can only take values between 0 and 1 for any positive value of S0.

θ =

1

1 +

S0

KM

(2.6)

The power of SKM comes from the parametrization illustrated above. Each element in θ has a

precise correspondence to some combination of kinetic parameters in the original model. However, it

is far more tractable to study a system using θ parameters rather than the original kinetic parameters.

To see this more clearly, recall that in most cases, biochemical kinetic constants are poorly estimated.

In order to build precise ODE models for biochemical systems, it is usually necessary to actually choose

values for these constants. While the chosen values may be estimated from experimental measurements,

hidden dependencies in the model may actually result in non-obvious correlations between parameters

that can strongly affect the output of the model.

On the other hand, if one is only interested in the stability of a characterized steady-state, it may

not be necessary to actually have knowledge of kinetic parameters. Experimental measurements can

provide data on absolute metabolite concentrations and flux values, and the stoichiometry is often

known a priori. Then, one can parameterize the system as shown above and sample many possible

combinations of θ parameters. For each unique set of θ parameters, the stability of the Jacobian is

determined and recorded. Analysis of the results can lead to several important conclusions such as

which θ parameters have the strongest correlation with stability of the entire metabolic system. Thus,

the benefit of employing SKM over other techniques is that it provides the means to analyze and make

sense of a large number of possible cases of a metabolic network, rather than a single instance.

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At first glance, it is unclear whether this will make things easier or harder; values for fluxes and

concentrations are almost never known. However, our work to follow shows that the absence of this

knowledge does not make the problem of stability intractable. Instead, we operate with the following

idea: we assume that the steady state flux vector v and the metabolite concentration vector X are

variables, and we do not specify them. Instead, we simply determine the stability of the system in

terms of these and other (θ) variables. Then, we can find trends in stability as the flux and metabolite

concentrations change: it may be that as a concentration goes up, the stability of the system tends to

fall. Isolating these trends is the motivation of our analysis.

2.2. Previous results on stability of metabolic cycles

Our work is preceded by a great deal of work aimed at understanding the stability of metabolic

networks[3, 13, 14, 15, 16]. One flavor of such work is referred to as chemical reaction network theory

(CRNT). CRNT is an elegant and powerful framework for understanding how the structure of chemical

networks dictates their equilibrium and stability properties. By applying CRNT to the cycles studied

in this paper, one can (in some cases) prove the stability of their steady states. This result can be

easily obtained, for example, by feeding the stoichiometric matrices of the corresponding networks

into the ERNEST MATLAB Toolbox [17]. Notably, these results cannot be easily extended beyond

the case of mass-action kinetics (where the rate of reaction is proportional to the concentration of its

reactants). The SKM method presented in this article extends the stability results for these cycles to

general monotonic kinetics.

Dynamical systems approaches have also been used in understanding the stability properties of

autocatalytic cycles (the second type of cycle presented in the paper). Previous research by King [14]

(focusing on dynamics) and by Eigen [3] (focusing on steady-state stability properties) have studied

autocatalytic cycles assuming first-order (non-saturating) mass-action kinetic rate laws. Extending

such work, our analysis takes into account all forms of saturating rate laws up to order one, and

derives simple expressions for the stability of metabolic cycles given adequate information about rate

constants and steady-state concentrations. To put this in context, the prior work on autocatalytic

cycles represents a single point (at which all θ parameters equal 1) in the large space of θ parameters

within which the actual metabolic network resides. Our approach searches over this entire space, and

identifies trends in stability across it.

3. A Simple Example

To illustrate the main concepts behind analytically characterizing the stability of a metabolic cycle

using SKM, we will start with a simple example. Consider the cycle shown in Figure 1. To analyze

the conditions under which it is stable, we will write the Λ and θ matrices, multiply them together to

obtain the Jacobian J, and then find the eigenvalues of J by writing out its characteristic equation.

The cycle contains 3 metabolites and 6 reactions. We impose that the reactions can only proceed

in the forward direction. Furthermore, the 6 reactions are constrained by mass balance, and we

note that two linearly independent rate vectors will characterize the fluxes of all 6 reactions in the

network. Assume that the steady states metabolite concentrations (A0,B0,C0,O0

are (1,1,1,1,1). Furthermore, assume that a flux of magnitude αF enters the cycle through reaction

v1, and (1 − α)F flux returns through reaction v5(where 0 < α < 1).

Finally, we impose that there is no complicated activation or inhibition present in the system. This

means that v2uses only metabolite A and cofactor O1as its substrates, v3uses only metabolite B, v4

and v5use only metabolite C, v6uses only O2, and v1is constant and uses none of the metabolites

as substrates. Notice that this constrains all θ parameters to be greater than or equal to zero. Con-

straining θ to be positive ensures that any kinetic rate laws we consider in our analysis are monotonic,

i.e. that an increase in the concentration of the substrate of a reaction will never decrease the rate of

a reaction. This will be essential in the analysis to follow. Now, we can write our system in terms of

SKM variables:

1,O0

2) of the system

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Figure 1: A simple metabolic cycle

Λ =

αF

0

0

0

0

−F

F

0

0

0

00

0

(1 − α)F

0

−(1 − α)F

0

0

0

0

0

F

−F

F

−F

F

−αF

0

0−F

(3.1)

θ =

0

θ1

0

0

0

0

0

0

θ2

0

0

0

0

0

0

θ3

θ4

0

0

0

θ5

0

0

0

0

0

0

0

0

θ6

(3.2)

Because the cofactors O1and O2are conserved, we can remove them from the two matrices above

(for details and justification, see the Supplementary Material of [9]). Doing this, we omit the last row

of Λ, and θ becomes

Then, to obtain the Jacobian we compute the product Λθ. In order to determine the stability of

the system, we find the eigenvalues of the Jacobian and determine whether any of them are positive.

We find the eigenvalues by imposing |J − λI| = 0, i.e. by setting to zero the following determinant:

θ =

0

θ1

0

0

0

0

0

0

θ2

0

0

0

0

0

0

θ3

θ4

0

0

0

θ5

0

0

−θ6

(3.3)

????????

−λ − Fθ1

Fθ1

0

0

0(1 − α)Fθ4

0

0

−λ − Fθ2

Fθ2

−θ2

−θ5

θ5

−λ − αFθ3− (1 − α)Fθ4

0−θ5− θ6

????????

(3.4)

This corresponds to the equation

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