Báo cáo khoa học: An extension to the metabolic control theory taking into account correlations between enzyme concentrations potx

In particular, negative correlations cause the flux to have a maximum value for a defined distribution of enzyme concentrations.. Redistribution coefficients of enzyme concentrations allowed

An extension to the metabolic control theory taking into account correlations between enzyme concentrations

Se´bastien Lion1,*, Fre´de´ric Gabriel1, Bruno Bost2, Julie Fie´vet1, Christine Dillmann1and

Dominique de Vienne1

1

Microbiologie, CNRS UMR 8621, Universite´ Paris Sud, Orsay Cedex, France

The classical metabolic control theory [Kacser, H &

Burns, J.A (1973) Symp Soc Exp Biol 27, 65–104;

Heinrich, R & Rapoport, T (1974) Eur J Biochem 42,

89–95.] does not take into account experimental evidence

for correlations between enzyme concentrations in the

cell We investigated the implications of two causes of

linear correlations: competition between enzymes, which

is a mere physical adaptation of the cell to the limitation

of resources and space, and regulatory correlations,

which result from the existence of regulatory networks

These correlations generate redistribution of enzyme

concentrations when the concentration of an enzyme

varies; this may dramatically alter the flux and metabolite concentration curves In particular, negative correlations cause the flux to have a maximum value for a defined distribution of enzyme concentrations Redistribution coefficients of enzyme concentrations allowed us to cal-culate the Ôcombined response coefficientÕ that quantifies the response of flux or metabolite concentration to a perturbation of enzyme concentration

Keywords: biochemical modelling; cellular constraint; flux; metabolite; response coefficient

The introduction of the metabolic control theory by Kacser

& Burns [1] and Heinrich & Rapoport [2] was a great

improvement in our understanding of the control of

metabolism (for a review see [3]) Numerous extensions to

the classical theory have been proposed to get rid of some

restrictive hypotheses of the initial theory Extensions exist,

for example, for nonproportionality of the rates of reaction

to enzyme concentration [4], enzyme–enzyme interaction

[5,6], time-varying systems [7,8], or supply–demand analysis

[9] Nevertheless, most studies have neglected the

correla-tions that exist between enzyme amounts in the cell

Concentration is a key parameter for enzyme activity

Changes in expression of enzyme genes play a central role in

the physiology of the cell, and dramatic modifications of the

cell proteome are consistently observed over development

and differentiation, or in response to environmental changes

(see http://us.expasy.org for examples in various species) In

addition, genetic studies have revealed natural variability for

enzyme concentration, for instance for alcohol

dehydro-genase in Drosophila [10] or lactate dehydrodehydro-genase in

Fundulus heteroclitus [11] Other examples can also be

found [12,13] Quantitative proteomic approaches have

confirmed that a majority of proteins/enzymes can display

a large range of variation within species [14–19] Those

physiological or genetic variations are expected to be interdependent There is evidence for cellular constraints that induce a variation of concentration of some enzymes

in response to a variation of others These correlations between enzyme concentrations undoubtedly have an impact on the behaviour of metabolic systems, and hence

on their evolution Two kinds of correlations will be studied

in this paper The first one will be referred to as competition

It is a mere physical adaptation of the cell to energetic or steric constraints The second one results from regulatory networks It will be referred to as regulation

Competitive constraints on the variation of enzyme concentrations have already been pointed out Such constraints have the effect of avoiding macromolecular crowding, which can result in a modification of catalytic and/or thermodynamic properties of enzymes [20], in a limitation of solubility leading to partial protein crystalliza-tion or aggregacrystalliza-tion [21,22], or a decrease in the diffusion of essential metabolites ([23], for a review see [24]) Other arguments include the limitation of resources, the energetic cost of maintaining the cellular concentrations of enzymes [25–27], and the availability of amino acids or elements of the transcription and translation machinery, which has been shown to be a limiting factor of protein synthesis in Escherichia coli [28] and Saccharomyces cerevisiae [29] Kacser & Beeby [30] were among the first to suggest that the hyperbolic flux–activity relationship must ultimately decline, for no more profound reason than that the cell or organism must eventually reach a point at which the cost of producing excess enzyme outweighs the benefit in fitness that can be derived from possessing the excess [31,32]

It is clear that such competitive constraints imply that variations of enzyme concentrations are negatively correla-ted: an increase in the concentration of some enzymes

Correspondence to D de Vienne, UMR de Ge´ne´tique Ve´ge´tale,

INRA/UPS/CNRS/INAPG, Ferme du Moulon, 91190

Gif-sur-Yvette, France Fax: +33 1 69 33 23 40, Tel.: +33 1 69 33 23 60,

E-mail: devienne@moulon.inra.fr

*Present address: Laboratoire d’e´cologie, E´cole normale supe´rieure,

46, rue d’Ulm, 75005 Paris, France.

(Received 19 July 2004, revised 20 September 2004,

accepted 22 September 2004)

causes a decrease in the concentration of other enzymes,

which can lead to important metabolic perturbations, i.e

to the so-called protein burden effect [33] For instance,

overexpression of b-galactosidase in E coli was found to

reduce the synthesis of the other proteins [34] and

over-expression of glycolysis enzymes in Zymomonas mobilis has

been shown to reduce glycolytic flux [35]: therefore, for large

enzyme concentrations, the classical hyperbolic shape of the

flux curve, as predicted by the metabolic control theory,

does not describe in a satisfactory way the behaviour of the

metabolic pathway Flux can be expected to decrease when

enzyme concentration becomes too high, and it may be

interesting to model such behaviour

Regulatory correlations can be positive or negative The

production and degradation of enzymes, which determines

their concentration, is related to the structure of the genetic

regulatory network [36] The lactose operon in E coli [37] is

a well known example of a regulatory system that induces

correlations between the concentrations of the enzymes

involved in lactose metabolism Several experimental and

theoretical studies have been devoted to the understanding

of the mechanisms of regulatory networks [38–41]

Meta-bolic engineering makes an important use of regulation of

metabolic pathways to achieve overexpression of the

products of interest For instance, Prati et al [42] reported

a way to achieve simultaneous inhibition and activation of

two glycosyltransferases of the O-glycosylation pathway in

Chinese hamster ovary cells In Lactococcus lactis, several

genes of glycolysis have been shown to be expressed at

higher levels on glucose than on galactose [43] The authors

interpreted this as a result of two different regulatory

networks With the growing use of quantitative proteomics

methods, we can expect to find many more examples of

correlations between enzymes, even if we still lack the tools

to determine whether regulatory networks actually underlie

these correlations

The existence of these competitive and regulatory

corre-lations between enzymes is assumed to constrain the

response of the metabolic systems Here, we present an

extension of the metabolic control theory in which response

coefficients allow us to quantify the change of a metabolic

variable (flux or metabolite concentration) in response to a

perturbation of a parameter (enzyme concentration) and to

the variations of other parameters resulting from that

perturbation We apply the general concept of a Ôcombined

response coefficientÕ to a linear model of redistribution of

enzyme concentrations in order to study the systemic

consequences of enzyme correlations

Control of metabolic pathways and

redistribution of enzyme concentrations

Control of metabolic variables

Let us consider a metabolic pathway with n enzymes

E1, E2,…, Encatalyzing reversible reactions between

sub-strates S1,…, Sm(m metabolites)

To quantify the response of a systemic variable y, such as

the flux in the pathway or the concentration of a metabolite, to

an infinitesimal change in the activity (concentration) of

enzyme Ek, Kacser & Burns [1] and Heinrich & Rapoport [2]

introduced the control coefficient In the revised

nomen-clature for metabolic control analysis, the control coefficient

Cykis defined as the steady-state response in y to a change in the local rate of step k, vk, with no reference to enzyme con-centration (http://www.sun.ac.za/biochem/mcanom.html)

In particular, the control coefficient of flux J with respect

to step k is:

CJk¼vk J

@J

@vk and the control coefficient of metabolite concentration Si with respect to step k is:

CSi

k ¼vk

Si

@Si

@vk Summation theorems can be derived for metabolite and flux control coefficients Summing over all reactions, we have [1]:

Xn j¼1

CJj ¼ 1 and Xn

j¼1

CSi

These relationships show that the control of flux (or metabolite concentration) is shared among all enzymes in the pathway

Control coefficients are systemic properties We can also define local properties such as the elasticity, which quantifies the effect of any parameter p that affects the local rate of an individual (isolated) step The elasticity coefficient ekp for step k is written as [1]:

ekp¼ p

vk

@vk

@p

Introducing correlations between enzyme concentrations The classical form of metabolic control theory implicitly considers that enzyme concentration can increase towards infinity, which is biologically inconsistent Competitive and regulatory constraints on enzyme concentrations exist, that can be described with a model of redistribution of enzyme concentrations

We considered a system starting in a state defined by the concentrations

E0¼ ðE01; E02; :::; E0k; :::; E0nÞ

of the n enzymes, and supposed that a variation of the concentration of a target enzyme Ekresults in a variation of the concentrations of other enzymes

Redistribution coefficient In order to quantify the impact

of variation of enzyme Ekon enzyme Ej, we defined the redistribution coefficient (akj) as the ratio of an infinitesimal change in the concentration Ejto an infinitesimal change in the concentration Ek:

akj¼@Ej

@Ek

ð1Þ

In this framework, the enzyme concentrations become interdependent parameters

Combined response coefficient of the flux If an effector p acts on the flux through its effect on enzyme j, the response

coefficient RJis the product of the flux response coefficient

with respect to enzyme j and the elasticity of enzyme j with

respect to p [1]:

RJp¼ CJ

jejp Let us now assume that the effector p acts on more than

one enzyme in a metabolic pathway We can define the

overall, multisite response obtained from the n enzymes of

the system as [44,45]:

RJp¼Xn j¼1

CJjejp

This is only true for very small changes in p because the

response coefficient is defined as a first order

approxima-tion For a large change in p, we should add correction

terms to account for nonlinearities

Considering an effector p causing the redistribution of

enzyme concentrations through the modification of

con-centration Ekof the enzyme Ek(e.g p is a mutation causing

an increase of Ekand consequently modification of other

enzyme concentrations), we can write, replacing p by Ek:

RJEk¼Xn j¼1

CJjejk

Assuming that the response of an isolated reaction is

directly proportional to change in enzyme concentration, we

have:

ejk¼Ek

Ej

akj

so that

RJEk¼ Ek

Xn j¼1

CJjakj

Ej

ð2Þ

We call RJE

k the combined response coefficient [46] We

will show later (in the case of a linear metabolic pathway)

that the combined response coefficient can be equivalently

written as:

RJ

E k¼Ek J

@J

@Ek where the partial derivative is taken on a set of enzyme

concentrations that describes the correlations between

enzymes

Biologically speaking, this means that the combined

response coefficient contains information about the

corre-lations between the enzyme concentrations, hence the term

ÔcombinedÕ We can see the combined response coefficient

as a Ôresponse coefficient under constraintÕ We can split

Eqn 2 into two terms:

RJ

E k¼ CJkakkþ Ek

X j6¼k

CJjakj

Ej Note that akk¼ 1 (Eqn 1), so that

RJE

k ¼ CJ

kþ Ek X j6¼k

CJjakj

Ej

ð3Þ

The effect of a variation of enzyme Ekon the flux appears

then to be dependent on two factors: (a) the control exerted by

enzyme Ekon the flux, and (b) the effect of enzyme Ekon the others, through the redistribution rules, which is modulated

by the control exerted by those enzymes on the flux Thus, even if enzyme Ekhas a high control coefficient on the flux, an increase of Ekshould cause a decrease of the flux if Ekis negatively correlated with the concentrations of other enzymes We can also note that the response coefficient of enzyme Ekwill be higher than its control coefficient in cases where enzyme Ekis positively correlated with at least one other enzyme of the pathway, and not correlated to the others Thus, we have given a general expression for the combined response coefficient of the flux, valid for a network of any complexity, with no assumption on the rules

of redistribution of enzyme concentrations In the next paragraph, we will present the theoretical framework that allowed us to describe the linear correlations between enzyme concentrations, and in the second part of the paper we will analyse in detail the particular case of a linear pathway of enzymes far from saturation, considering the response of both flux and metabolite concentrations

Linear models of redistribution of enzyme concentrations

We assumed linear redistribution, which means that akjis considered to be constant

Figure 1 shows how enzyme concentrations are redis-tributed due to their correlations Figure 1A corresponds to the case of independent enzyme concentrations that was studied in the founding papers of the metabolic control theory [1,2] Figure 1B–G corresponds to various con-straints that result in a redistribution of enzyme concentra-tions over the variation of a particular enzyme

Let us examine these constraints, the mathematical expressions of which are summarized in Table 1 We focus

on a linear model of redistribution of enzyme concentra-tions but other models are possible Let us further introduce the normalized concentration ejdefined as:

ej¼ Ej

Etot where

Etot¼Xn j¼1

Ej

Competitive correlations In order to take into account the fact that enzyme concentrations are likely to be bounded, Heinrich et al [47–49], and de Vienne et al [46], have proposed to put a constraint on the total concentration Etot

of the enzymes in the pathway In this paper, this is designed

as competition and we limit the study to the quite rigid constraint where Etotis a constant We have:

Xn j¼1

Ej¼ Etot¼ const

Using the normalized concentration ej, the competitive constraint on the metabolic pathway reads

Xn j¼1

In systems with only competitive constraints, the

con-centrations of enzymes Ej ("j „ k) decrease when the

concentration of enzyme E increases and the proportions

of enzymes Ej remain constant So we can define the competition coefficient ckjbetween enzymes Ekand Ej(i.e the constant proportion between these enzymes) as

G

Fig 1 Redistribution of enzyme concentrations when the concentration of the target enzyme changes We considered a six-enzyme pathway E 3 is the concentration of the target enzyme The y-axis shows the concentrations of enzymes E 1 , E 2 , E 4 , E 5 and E 6 Unless otherwise stated, the starting distribution of enzyme concentrations is the vector E 0 ¼ (0.04,0.02,0.04,0.37,0.44,0.09), indicated with dots on the figures The concentration of the target enzyme varies either between 0 and E tot ¼ 1, or between E 3,min and E 3,max , depending on the constraints imposed on the system (A) Independence between enzyme concentrations (B) Pure regulation with positive correlations The redistribution coefficients are a 3 ¼ b 3 ¼ (0.99,0.63,1,0.94,0.43,0.29) (C) Pure regulation with one enzyme being negatively correlated a 34 ¼ )0.94, the other redistribution coefficients being the same as in (B) (D) Competition when the starting distribution of enzyme concentrations is the optimal one, which maximizes the flux (E) Competition when the starting distribution of enzyme concentrations is E 0 (F) Regulation with competition The starting distribution of enzyme concentrations is E 1 ¼ (0.13,0.13,0.31,0.04,0.02,0.37) and the redistribution coefficients are a 3 ¼ (0.05,0.05,1,0.5,0.5,-2.1) (G) Regulation with competition when the starting distribution of enzyme concentrations is E 0 and coregulation coefficients b (Eqn 6) are as in (B).

8j 6¼ k ckj¼
ej

1
ek

ð5Þ

Thus partial derivation of Eqn 5 with respect to ekleads to

@ej

@ek

¼ ckj

and we have ckk¼ 1 As by definition akj¼@ej

@e k, we have for pure competitive systems, a simple relationship between

competition and redistribution coefficients (Table 1,

Appendix A):

8j 6¼ k akj¼ ckj¼
ej

1
ek

akk¼ ckk¼ 1 which can also be derived from summation of Eqn 5

Regulatory correlations When redistribution of enzyme

concentrations is only due to regulatory mechanisms, total

enzyme content has no upper limit, but enzyme

concen-trations are correlated Variation of the concentration of

enzyme Ekfrom Ekto Ek00drives the system to a new state

E100; :::; E00j; :::; E00n, where

8j E00j ¼ Ejþ bkjðE00k
EkÞ ð6Þ

where Ej is the concentration of enzyme Ej before the

variation of enzyme Ek, and bkj is the coregulation

coefficient between enzymes Ej and Ek The coefficients

can be positive, negative or null, but at least one is different

from 0 It is worth noting that bkk¼ 1

In systems with only regulatory constraints, the

coregu-lation coefficient corresponds to the redistribution

coeffi-cient, i.e akj¼ bkj, as shown in Appendix A (also Table 1)

Redistribution coefficients in competitive-regulatory

path-ways When both competition and regulation are present in

a pathway, it is interesting to note that a simple relationship

exists between the redistribution coefficient akj and the coregulation coefficients bkj(Appendix A):

akj¼bkj
ejBk

1
ekBk

ð7Þ

where

Bk¼Xn j¼1

bkj:

This relationship does not involve explicitly the competi-tion coefficient ckj But when there is no coregulation in the system, we have"j „ k bkj¼ 0 and Bk¼ bkk¼ 1, so that:

akj¼
ej

1
ek

¼ ckj

Application: the case of a linear pathway

of enzymes

We applied our model of redistribution of enzyme con-centrations to the linear pathway of enzymes far from saturation studied by Kacser & Burns [1]

Flux and metabolite concentrations in a linear pathway Let us consider a linear metabolic pathway, with n enzymes

E1, E2,…, Enconverting a substrate X0into a final product

Xnby a series of unimolecular reversible reactions:

X0¢

E1

S1¢

E2

S2¢

E3 ¢En
2Sn
2 ¢

En
1

Sn
1¢

E n

Xn The enzymes are supposed to be Michaelian and far from saturation The steady-state flux through the pathway is [1,2]:

Pn j¼1

1

V j

M jK0;j
1

ð8Þ

and the steady-state concentration of metabolite Siis

Si¼J

XK0;i X0

X j>i

1

V j

M jK0;j
1

þ Xn

K0;n

X ji

1

V j

M jK0;j
1

0

@

1

A ð9Þ

where X0and Xnare the concentrations of substrate X0and product Xn, respectively, and X¼ X0) Xn/K0,n X0and Xn are considered as fixed parameters of the systems, while the intermediate metabolite concentrations Si (1£ i £ n) 1) are variables Vkis the maximum velocity of enzyme Ek, Mk

is its Michaelis constant, and K0;k¼Qk

j¼1Kj
1;j is the product of the equilibrium constants of reactions 1, 2,…, k

To make apparent the concentration of enzymes, Ek, in Eqns 8 and 9, we used the relationship:

Vk¼ kcat;kEk where kcat,kis the turnover number of enzyme Ek

We can then define the activity parameter Akof enzyme

Ekby:

Ak¼kcat;k

Mk

K0;k
1

with K ¼ 1 by convention

Table 1 Mathematical expressions of the redistribution coefficients of

enzyme concentrations a kj when introducing competitive and/or

regula-tory constraints a kj is the ratio of a change in the concentration E j to a

change in the concentration E k Note that values of a kj are only true for

j „ k because a kk is always equal to unity The subscript k refers in this

table to the enzyme whose concentration we want to vary, for instance

through experimental or genetic means (see Appendices A to D for

more details).

No competition

(E tot is not constant)

Competition (E tot is constant)

No regulation 8j 6¼ k a kj ¼ 0

a kk ¼ 1

8j 6¼ k a kj ¼ c kj ¼
ej

1
e k

a kk ¼ c kk ¼ 1

X n j¼1

c kj ¼ 0 Regulation 8j 6¼ k a kj ¼ b kj

bkk¼ 1 8j 6¼ k

a kj ¼bkj
ejBk

1
e k B k

a kk ¼ 1

X n j¼1 a kj ¼ 0

The steady-state flux through the pathway is thus

Pn j¼1

1

A j E j

ð10Þ

and the steady-state concentration of metabolite Siis

Si¼ J

XK0;i X0

X j>i

1

AjEj

þ Xn

K0;n

X ji

1

AjEj

! ð11Þ

Below, we will consider the catalytic component Ak is

constant and only consider variations of enzyme

concen-trations, in order to study how biological constraints on

these concentrations can modify the behaviour of metabolic

pathways

Variation of flux and metabolite concentrations

in unconstrained pathways

When enzyme concentrations are not correlated, i.e when

there are no competitive or regulatory constraints, both flux

and metabolite concentrations reach a plateau when the

concentration of a particular enzyme Ekincreases (Fig 2)

Considering the concentrations of the other enzymes as constants, the maximum flux value is (Eqn 10 and a general theoretical background in Appendix B):

Jmax¼PX j6¼k

1

A j E j

The concentration of a metabolite located downstream of the variable enzyme increases until it reaches a plateau (Appendix C):

Sdowni ¼PK0;i j6¼k

1

A j E j

X0 X j>i

1

AjEj

þ Xn

K0;n

X ji

j6¼k

1

AjEj

0 B

1 C

The concentration of a metabolite located upstream of the variable enzyme decreases until it reaches a minimum value:

Supi ¼PK0;i j6¼k

1

A j E j

X0 X j>i

j6¼k

1

AjEj

þ Xn

K0;n

X ji

1

AjEj

0 B

1 C

It is clear that the level of the asymptotes depends on the concentrations of the other enzymes

Consequences of enzyme redistribution on the flux Figure 3 describes the change in flux that results from different types of correlations and corresponds to various constraints that result in a redistribution of enzyme concentrations over the variation of a particular enzyme

Competitive-regulatory pathways Introducing both com-petitive and regulatory correlations in the system will alter the flux curve with respect to enzyme concentration These constraints result in a limited range of variation for enzyme concentrations, and in the variation of concentration of

an enzyme being limited by that of the others If the concentration of a given enzyme becomes high, the concentration of another is likely to vanish, therefore bringing the value of the flux to zero Therefore, in this model, each enzyme has a range of variation [emin,emax] in which the flux is positive

Over the range of variation of the concentration of

an enzyme, the flux increases to a maximal value, then decreases when the concentration is lower or higher This is due to the fact that at least one redistribution coefficient must be negative when competition is introduced

Moreover, everything else being equal, each set of redistribution coefficients results in a particular flux curve

As will be mentioned later, all the possible curves are restricted by an envelope curve

Relationship between redistribution and combined res-ponse coefficients In order to analyse the response of the flux to the variation of enzyme concentration in constrained pathways, we used the combined response coefficient we have defined previously All the results in this section depend

on the assumption that the pathway is linear with mass-action kinetics, which ensures analytical tractability Replacing CJby its expression in Eqn 2, we easily find an analytical expression for the combined response coefficient:

0.0

A

B

0.2 0.4 0.6 0.8 1.0 Concentration of enzyme 3

0.0 0.2 0.4 0.6 0.8 1.0

Concentration of enzyme 3

Fig 2 Variation of metabolic variables in a linear unconstrained

path-way with respect to concentration of one enzyme Unless stated

other-wise, all plots describe a six-enzyme pathway with activities A 1 ¼ 0.32,

A 2 ¼ 0.83, A 3 ¼ 0.72, A 4 ¼ 0.04, A 5 ¼ 0.40, A 6 ¼ 0.16 The x-axis

is the concentration of enzyme 3 The concentrations of enzyme are

(0.04,0.02,E 3 ,0.37,0.44,0.09) Furthermore, we choose X 0 ¼ 2 and

X n /K 0,n ¼ 1 (therefore, X ¼ 1) (A) Variation of flux in a linear

unconstrained pathway with respect to concentration of one enzyme.

The figure shows the classical hyperbolic flux curve (B) Variation of

metabolite concentration in a linear unconstrained pathway with

respect to concentration of one enzyme The figure shows the variation

of concentration of a metabolite upstream (solid line) and a metabolite

downstream (dashed line) the variable enzyme.

e k ¼ J

XEtot

ekXn j¼1

akj

Aje2 j

ð12Þ

In Appendix D we present another derivation using the

fact that RJe

k¼ek

J

@J

@e k Unlike the control coefficient, the flux combined response

coefficient can become negative, as a consequence of

competition and/or negative coregulation (Fig 4) Limits

of flux combined response coefficient are +1 for the

minimal value of ekand –1 for the maximal value (Appendix

D) Flux reaches an absolute maximal value for a vector of

enzyme concentrations (e1,…, en) defined by [47,50] such that

ek¼

1ffiffiffiffiA

k

p

Pn j¼1

1ffiffiffiffiA

j

p

ð13Þ

and the flux combined response coefficient is null when

ek¼ e

k

The envelope curve For each value of ek, we can determine

a maximum value for the flux (Appendix E) and consider

the curve that passes through all these points This curve will

be called the envelope curve

Does this envelope curve correspond to a peculiar

redistribution system or to a mere mathematical

construc-tion? In Appendix E, we used the optimization method

proposed by Heinrich et al [47,50] to show that the envelope

curve corresponds to a pure competitive model It passes through the absolute maximum of flux under the constraint

of Eqn 4, which is reached for a vector of concentrations (e1; :::; en) as defined in Eqn 13 For all values of ek, we have:

8i; j 6¼ k ei

ej¼

ffiffiffiffiffi

Aj

Ai r

Fig 3 Relationship between flux and enzyme concentration for different models of enzyme redistribution We consider a six-enzyme linear pathway with activities as in Fig 2 The other parameters are the same as in Fig 1 The vertical dashed line indicates the point corresponding to reference distribution E 0 (A) Pure regulation with positive or null correlations Solid line: all the a i ’s are positive Dashed line: a 36 ¼ 0 Dotted line: a 36 ¼ 0 and a 35 ¼ 0 Dashed-dotted line: all the a i ’s are zero (B) Pure regulation with one negative correlation Dashed line: with one enzyme being negatively correlated Solid line: pure regulation with positive correlations [compare with (A)] (C) Competition Solid line corresponds to the redistribution coefficients in Fig 1D, and dashed line to those in Fig 1E (D) Regulation with competition Solid line is competition as shown in (C); dashed and dotted lines describe regulation with competition (dotted line corresponds to the redistribution coefficients in Fig 1F, dashed line to those in Fig 1G).

Concentration of enzyme 3 0.2 0.3 0.4 0.5 0.6

Fig 4 Variation of flux combined response coefficient with respect

to the concentration of one enzyme, under the constraint of Eqn 4 (competition) Flux combined response coefficient is positive for

e < e*, null for e ¼ e* and negative for e > e*, where e* is the concentration leading to the optimal value of flux, as defined by [47] (Appendix E).

Moreover, we show that the redistribution coefficients of

the envelope curve are given by:

akj¼
e

 j

1
e k Therefore, whatever the redistribution rules, for a given

set of fixed activities, all flux curves will be bounded by the

envelope curve we have defined

Pure regulatory pathways When only regulatory

con-straints are present, two subcases of interest should be

mentioned: positive and negative correlations

Positive correlation is presented in Fig 3A

When all coregulation coefficients are positive, the flux

asymptotically tends towards a straight line, as the

concen-tration of enzyme Ekincreases The coefficients of this line

are given in Appendix B If, starting from the situation

where all the enzymes are positively coregulated, we choose

to set p coregulation coefficients to zero, the flux curve will

reach a plateau, which is characterized by the following flux

value (Appendix B):

JðpÞmax¼PX j2I p

1

A j E j0

with i2 Ipif aki¼ 0 (i.e enzymes Eiare independent from

enzyme Ek)

The higher the number of coregulation coefficients (i.e

the less the enzymes are coregulated), the lower the plateau,

and when no enzymes are coregulated with enzyme Ek, the

maximum flux value Jmaxðn
1Þ is the one found by Kacser &

Burns [1]:

Jðn
1Þmax ¼PX

j6¼k

1

A j E j0

Negative correlation is presented in Fig 3B

When at least one redistribution coefficient is negative,

the flux curve reaches a maximum beyond which it declines

towards zero (Appendix B)

Pure competitive pathways Here, we consider that the

total concentration of the enzymes is constant (Eqn 4) and

that only proportional redistribution occurs between the

enzymes of the pathway (akj¼ ckj¼
ej

1
e k) This means that an increase in the concentration of a given enzyme Ek

causes a decrease in the concentrations of the others, in

proportions remaining constant Mathematically speaking,

this is equivalent to setting"j„ k, bkj¼ 0 in the expression

of akj(Eqn 7)

Therefore, in this particular model the flux response

coefficient reads (Appendix D):

RJ

e k¼ ek

1
ek

J

XEtot

1

Ake2
1

for the enzyme Ekthat causes the redistribution of the other

concentrations, and:

8j6¼ k RJej¼ 1
J

XEtot

1

Ake2 k for the other enzymes

Thus it must be stressed that because of the absence of coregulation, the limit of the combined response coeffi-cients for ek¼ 0 is not +1 anymore, but is now 1 (Appendix D)

As in the competitive-regulatory model, the shape of the flux-activity curve is altered and points out that enzymes should possess an optimal concentration beyond which the flux decreases (Fig 3C), as it had been predicted previously [30,49,51]

When concentration of enzyme Ekvaries from 0 to Etot,

we only need to know the proportional redistribution coefficients in order to determine the concentrations of the other enzymes Therefore, we can draw several flux curves, each one determined by a set of proportional redistribution rules When ek¼ 0 or ek¼ 1, the flux is null The maximum value of the flux curve depends on the redistribution coefficients for this curve These different curves correspond to unoptimized distributions of the concentrations, i.e distributions where akj6¼
e

j

 ð1
e

kÞ,

ek being defined by Eqn 13 (the optimum distribution) As the optimum distribution corresponds to the envelope curve (see above), all the curves corresponding to unop-timized distributions have flux values less than those of the envelope curve

Consequences of enzyme redistribution on metabolite concentrations

When the metabolite concentrations are considered as systemic variables, similar treatment applies because the results on redistribution, competition and coregulation coefficients between enzyme concentrations are still valid Metabolite concentrations are always bounded For a linear pathway with a positive flux, the metabolite concen-tration will have a lower limit corresponding to the weighted concentration of the input substrate of the system (X0K0,i) and an upper limit corresponding to XnK0,i/K0,n(Eqn 11) The range of variation of the concentration of metabolite Si

is therefore equal to K0,iG, where G is the equilibrium ratio

X0K0,n/Xn This means that the further the system is from equilibrium, the more the metabolites are free to vary Interestingly, this also implies that the variation of metabo-lite concentrations is constrained by environmental param-eters, independently of the catalytic properties of the enzymes It is important to note that this feature only results from the particular expression of Si in a linear pathway and not from the introduction of redistribution rules

As for the flux, we summarize in Fig 5 the change in metabolite concentrations (actually Si/K0,iand not Si) as a result of various correlations between enzyme concentra-tions

Relationship between redistribution and metabolite com-bined response coefficients As in the case of the flux, we can define a combined response coefficient for the concen-tration of metabolite Si, with respect to enzyme Ek:

RSi

e k ¼Xn j¼1

CSi

keje

k ¼Xn j¼1

CSi

kakj

ek

ej

We can show that it is equivalent to calculate:

e k¼ek

Si

@Si

@ek

on a suitable set of enzyme concentrations describing the

constraints

In competitive-regulatory systems, we have the following

relationship between the metabolite combined response

coefficient and the redistribution coefficients (Appendix F):

RSi

e k¼ek

Si

J2K0;i

XE2

tot

X ji

akj

Aje2 j

X j>i

1

Ajej

j<i

akj

Aje2 j

X ji

1

Ajej

!

ð14Þ

It is easy to show that the value of metabolite combined

response coefficient is 0 for ek¼ emin and that it can be

positive or negative, but is always bounded

It is worth noting that in the absence of any regulatory

constraint in the system, we get two relationships between

metabolite combined response coefficient and redistribution

coefficients, depending on the position of the variable

enzyme Ekwith respect to the metabolite Si(downstream or

upstream), as shown in Appendix F:

k i RSi

e k¼ J

2

XE2 tot

K0;i

Si

X j>i

1

Ajej

! 1

Akek

1

1
ek

k>i RSi

e k ¼
J

2

XE2 tot

K0;i

Si

X ji

1

Ajej

! 1

Akek

1

1
ek

8

>

>

>

>

Competitive-regulatory pathways The competitive and

regulatory constraints also change the pattern of variation

of metabolites concentrations when enzyme concentration

becomes too high The global behaviour of metabolite concentration with respect to enzyme concentration is dramatically altered for large enzyme concentrations

As we take into account competition in this section, there

is at least one negative redistribution coefficient, which means that the concentration of at least one enzyme ‘ vanishes for ek¼ emax Therefore, metabolite concentration will be minimal when ek¼ emaxif ‘ > k and maximal if

‘£ k (Appendix C)

The behaviour of the system is fully determined by the sign and the magnitude of the redistribution coefficients between enzymes Therefore, three kinds of behaviour can

be distinguished in this system (Fig 5D): (a) a ÔU-shapedÕ variation whereby the upstream metabolite decreases from

XnK0,i/K0,nto a minimum value and then increases until the maximal concentration is reached again; (b) a monotonous variation allowing the metabolite concentration to describe the whole range of variation; (c) a Ôhump-shapedÕ variation whereby the downstream metabolite concentration increa-ses from X0K0,i to a maximum value, and then decreases towards X0K0,i

Hence, the model can account for a variety of behaviours

of the metabolite concentrations We can say that the behaviour of the system depends on the position in the pathway of the enzyme whose concentration becomes 0 when the target enzyme reaches its maximal value The key point is to know whether the enzyme is located upstream or downstream the metabolite (Appendix C) Thus, an increase

in the concentration of an enzyme can induce either an increase or a decrease in the concentration of a metabolite This can be related to what is observed in many human metabolic diseases, which can be caused either by an excess

Fig 5 Relationship between metabolite concentrations and enzyme concentration for different models of enzyme redistribution The parameters are the same as in Fig 1 Note that plots are S i /K 0,i and not S i alone (A) Independent enzymes (B) Pure regulation with positive correlations (C) Competition (D) Regulation with competition The two upper curves represent metabolites upstream of the variable enzyme; the three other are downstream metabolites.

or by a lack in a given metabolite By extending our model

to nonlinear correlations and pathways, we can expect

to observe similar patterns with nonbounded metabolite

concentrations

Pure regulatory pathways In this case, the total enzyme

concentration is not constant but at least one regulation

coefficient is non-zero As for flux, two subcases of interest

should be mentioned

If there are positive correlations (Fig 5B), i.e all the

redistribution coefficients are positive, the metabolite

con-centration curve reaches a plateau whatever the position of

the metabolite in the pathway with respect to the enzyme:

downstream or upstream The level of the plateau is

different from that of the unconstrained case and is given in

Appendix C

With negative correlations, i.e when at least one

redis-tribution coefficient is negative, the concentration of

meta-bolite Siis seen to increase or decrease towards the upper or

the lower limits of metabolite concentrations (Appendix C)

This is due to the fact that, when two enzymes are

negat-ively correlated, an increase in the first one ultimately causes

the second one to vanish

Pure competitive pathways In a pure competitive pathway

(Fig 5C), the behaviour of metabolite concentrations is not

affected by the introduction of proportional redistribution

and the general behaviour of the system is the same as the

one predicted by the classical metabolic control theory

(Fig 5A) Upstream metabolites are found to decrease until

a plateau is reached (when all enzyme concentrations are

null except the varying enzyme), whereas downstream

metabolites are found to increase until a plateau is reached

The values of the plateau for both an upstream and a

downstream enzyme are different from those of an

unconstrained pathway (Appendix C)

The pure competitive model shows therefore that taking

into account proportional redistribution between enzymes

can dramatically modify the flux through the pathway

without altering the qualitative behaviour of metabolite

concentrations

Discussion

We developed an extension to the metabolic control theory

that takes into account the existence of correlations

between enzyme concentrations in the cell In our model,

enzyme concentrations are linked by so-called

redistribu-tion coefficients, which account for the effect of the

variation of one enzyme concentration onto the

concen-tration of other enzymes We have distinguished two kinds

of correlations: competition and regulation This

distinc-tion is not a mere artifice In the literature, there are

multiple examples of correlations due to regulatory

mech-anisms, at the transcriptional and/or (post) translational

levels Competition is less popular, but is also documented

For instance Snoep et al [35] showed experimentally that

overexpression of plasmid-encoded protein in Z mobilis

could lead to the dilution of other enzymes and therefore

cause a reduction in the glycolytic flux This protein burden

effect is likely to be more critical in organisms like

Z mobilis, where 50% of the cytoplasmic proteins are

reserved for glycolytic enzymes [52], than in E coli where these enzymes are present at low concentration In the same line, Parsch et al [53] showed in Drosophila that the deletion of a conserved regulatory element in the Adh gene resulted in increased ADH overexpression and activity, but delayed development

As enzyme concentrations are not independent, the control of flux or metabolite concentrations cannot be quantified with the classical control coefficient anymore Using the concept of response coefficients, we have showed how correlations between enzyme concentrations can affect flux and metabolite concentrations in a pathway, and how this effect can be quantitatively measured For the flux, we gave a general expression (Eqn 3) showing how the interplay of the redistribution of enzyme concentrations and of the control of enzymes on the flux determine the response of any metabolic pathway to a variation of enzyme concentration A similar treatment can be applied to metabolite concentrations The combined response coeffi-cient can take any positive or negative value, while the control coefficient varies between 0 and 1 (at least in simple linear pathways) As a major and general conclusion of the model, we showed that, if the concentration of an enzyme is negatively correlated with the concentrations of other enzymes, increasing the concentration of that enzyme will cause the flux to have a maximum, even if the control of this enzyme on the flux is strong

The influence of the redistribution coefficients on the combined response coefficient means that the correlations between enzyme concentrations modify the control distri-bution pattern within the pathway However the combined response coefficients do not exhibit a simple summation property, unlike the classical control coefficients Thus, it would be hazardous to use control coefficient summation property in top-down control analysis to estimate the control of steps that have not been studied through modulation of enzyme efficiency, especially in cases where competition and/or regulation are likely to be present Another approach to study the distribution of control has been developed by Westerhoff’s group [54,55], which applies

to multilevel networks These networks are divided into modules where reactions are linked by mass transfer, whereas modules can interact with each other only by regulatory effects This approach allows the determination

of the role of enzyme level in metabolic control, by considering that nonmetabolic modules can have a share

of the metabolic control

For the sake of analytical tractability, we chose to study in detail a simple linear metabolic pathway We showed that introducing correlations between enzyme concentrations alters the shape of the flux and metabolite curves For the flux, there is indeed a maximum value for any redistribution rule, provided that at least one coefficient is negative (due to competition or negative regulation): the enzymes have an optimal concentration beyond which the flux decreases, as already predicted or demonstrated [30,47,50,51] in the case of competition alone The only case where there is no maximum flux value is when all enzymes are positively correlated In metabolic engineering, the only way to have high fluxes is to increase all enzyme concentrations simultaneously As it can

be technically difficult, it should be more practicable to optimize the distribution of enzyme amounts in the system