Báo cáo khoa học: Modular metabolic control analysis of large responses The general case for two modules and one linking intermediate docx

We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a func-tion of the linking intermediate there is no need to change para

Modular metabolic control analysis of large responses

The general case for two modules and one linking intermediate

Luis Acerenza1and Fernando Ortega2

1 Laboratorio de Biologı´a de Sistemas, Facultad de Ciencias, Universidad de la Repu´blica, Igua´, Montevideo, Uruguay

2 School of Biosciences, The University of Birmingham, UK

The quantitative study of metabolic responses in intact

cells is essential in research programs that require

understanding of the differences in physiological and

pathological cellular functioning or predicting the

phenotypic consequences of genetic manipulations To

perform this type of studies, a systemic approach

called metabolic control analysis (MCA) was

deve-loped [1–5] One of its central goals is to determine

how the responses of system variables, quantified by

control coefficients, depend on the properties of the

component reactions, described by elasticity

coeffi-cients

Predicting the responses of intact cellular systems to environmental and genetic changes has not been an easy task This could explain the lack of success in many biotechnological and biomedical applications that require changing metabolic variables in a pre-established way [6,7] Two of the major challenges to understanding metabolic responses are the structural complexity of the molecular networks sustaining cellu-lar functioning and the nonlinearity inherent in the interaction and kinetic laws involved In the develop-ment of MCA, some strategies have been devised to deal with these difficulties

Keywords

metabolic control analysis; metabolic control

design; metabolic responses; modular

control analysis; top-down control analysis

Correspondence

L Acerenza, Laboratorio de Biologı´a de

Sistemas, Facultad de Ciencias, Universidad

de la Repu´blica, 4225, Montevideo 11400,

Uruguay

Fax: +598 2 525 8629

Tel: +598 2 525 8618–23, Ext 139

E-mail: aceren@fcien.edu.uy

Note

Dedicated to the memory of Reinhart

Heinrich, one of the fathers of Metabolic

Control Theory

(Received 5 October 2006, accepted

7 November 2006)

doi:10.1111/j.1742-4658.2006.05575.x

Deciphering the laws that govern metabolic responses of complex systems

is essential to understand physiological functioning, pathological conditions and the outcome of experimental manipulations of intact cells To this aim,

a theoretical and experimental sensitivity analysis, called modular meta-bolic control analysis (MMCA), was proposed This field was previously developed under the assumptions of infinitesimal changes and⁄ or propor-tionality between parameters and rates, which are usually not fulfilled

in vivo Here we develop a general MMCA for two modules, not relying on those assumptions Control coefficients and elasticity coefficients for large changes are defined These are subject to constraints: summation and response theorems, and relationships that allow calculating control from elasticity coefficients We show how to determine the coefficients from top-down experiments, measuring the rates of the isolated modules as a func-tion of the linking intermediate (there is no need to change parameters inside the modules) The novel formalism is applied to data of two experi-mental studies from the literature In one of these, 40% increase in the activity of the supply module results in less than 4% increase in flux, while infinitesimal MMCA predicts more than 30% increase in flux In addition,

it is not possible to increase the flux by manipulating the activity of demand The impossibility of increasing the flux by changing the activity of

a single module is due to an abrupt decrease of the control of the modules when their corresponding activities are increased In these cases, the infini-tesimal approach can give highly erroneous predictions

Abbreviations

ANT, adenine nucleotide translocator; MCA, metabolic control analysis; MMCA, modular metabolic control analysis.

Regarding network complexity, top-down or

modu-lar strategies have been proposed [8–10] These

strat-egies abstractly divide the system into modules,

lumping together irrelevant (and unknown)

compo-nents and representing explicitly only the processes

that we are interested in describing The aim is to

sti-mulate and measure the responses using the intact

sys-tem, so that we are certain that the analysis performed

and the conclusions obtained apply to this system

To deal with nonlinearity two assumptions have been

made The first is that metabolic perturbations and

responses are small, so that they can be described using

a first order infinitesimal treatment The second

assumption is that in vivo enzyme catalysed reaction

rates are proportional to the corresponding enzyme

concentrations, as is normally the case when measured

in diluted in vitro conditions It is important to note

that, to our knowledge, all the developments in

steady-state MCA have included at least one of these two

assumptions [11,12] However, many, if not most, of

the responses exhibited by metabolic systems subject to

environmental changes or genetic manipulations

involve large changes in metabolic variables Moreover,

the assertion that in vivo rates are proportional to

enzyme concentration is difficult to justify The

cyto-plasm of cells is far from being diluted, showing a very

crowded state where the validity of the proportionality

found in vitro has still not been demonstrated [13]

Attempts to extend infinitesimal control analysis to

large changes in the variables have been reviewed in

previous publications [5,12] Our previous

contribu-tions to extend infinitesimal modular metabolic control

analysis (MMCA) consisted on the following steps

First, control coefficients for large changes were

defined and summation theorems, in terms of enzyme

concentrations, derived [14] Expressions to calculate

these control coefficients in terms of the elasticity

coef-ficients for large changes were obtained [12,15]

How-ever, the interpretation of the results of all these

previous contributions to MMCA for large changes

requires that the rates of the steps are proportional to

the corresponding enzyme concentrations

In the present contribution, we develop an MMCA

that applies to steady-state responses of any extent and

that does not assume proportionality between reaction

rates and parameters Therefore, it applies to any

parameter (enzyme concentration, external effector,

etc.), irrespective of its functional relationship with the

reaction rate To achieve this, rate control coefficients

(where parameters are not specified) and p-elasticity

coefficients for large changes were defined Combining

these two types of newly defined coefficients, we derive

response theorems, which are essential to study the

response of metabolic variables to external activators

or inhibitors We also show, in the framework of large changes, that rate control coefficients verify the same constraints (summation theorems, etc.) as those satis-fied, when rates are proportional to enzyme concentra-tions, by enzyme response coefficients Another central result is that the rate control coefficients can be used

to determine the flux and intermediate changes that would be obtained by changing the rates of the isola-ted modules by large factors These relationships are useful to analyse where to modulate the system in order to change a variable in a desirable way, or to speculate about possible sites at which cell physiology operates to modify the variables, when adapting to dif-ferent conditions All the quantities and relationships developed here may be applied to data obtained from top-down experiments Notably, this type of experi-ment may be performed by direct modulation of the intermediate, without changing parameters inside the modules The way the formalism is applied and the type of conclusions that can be drawn are illustrated with two studies, taken from the literature, performed using top-down experiments: the control of glycolytic flux and biomass production in Lactococcus lactis [16] and the control of oxidative phosphorylation in isola-ted rat liver mitochondria [17]

Results

The modular approach to large metabolic responses

A central issue to solving many biotechnological and biomedical problems is to assess how to modulate a metabolic system in order to obtain a pre-established change in the concentration of an intermediate or a flux Within the framework of reductionist approaches, the studies to solve this type of problem are performed

on isolated component reactions, reconstructed small portions of the network or extracts As a consequence, the results obtained may not be extrapolated with con-fidence to the in vivo system because, in the reduction process, it is more likely that relevant interactions are lost In contrast, modular approaches study the intact system and therefore the conclusions obtained apply to this system

Let us consider a metabolic network with any num-ber of intermediates and reactions In the modular approach, we focus on an intermediate S which divides the system into two parts or modules (Scheme 1) The system has three variables: the concentration of the linking intermediate (S), the rate at which the interme-diate is produced by the supply module (v1) and the

rate at which it is consumed by the demand module

(v2) [18] In this strategy, it is assumed that the only

interactions between modules are linking intermediates

[10] A module could be an enzyme catalysed reaction,

a metabolic pathway, a large portion of metabolism

(i.e., carbohydrate metabolism), an organelle (e.g.,

mitochondria) or a cell

The rate v1 depends on S and on all the parameters

belonging to the supply module Similarly, v2 depends

on S and on the parameters belonging to the demand

module Examples of parameters could be the

concen-trations of external substrates, products or effectors

and the concentrations of enzymes The functional

dependence of v1 and v2 on S and on the parameters

could be very complex because, in the case that

mod-ules are large portions of metabolism, many enzyme

catalysed reactions and metabolites are involved But,

for our purposes, we only need to consider explicitly

one parameter for each module: p1 for the supply

module and p2for the demand module In this context,

the functional dependence of the rates could be

expressed as follows: v1 ¼ v1(S, p1) and v2¼ v2(S, p2)

Note that proportionality between rates and

parame-ters is not assumed in this treatment At steady state,

both rates are equal to the flux, J (v1¼ v2 J)

There are three types of metabolic changes relevant

to the modular control analysis that we shall develop

below In the first, p1(or p2) is changed and the

pertur-bation propagates throughout the system, with S and J

settling to new steady-state values In the second type,

Sis kept at a constant value by some external means so

that when p1 (or p2) is changed, the perturbation will

not be able to propagate to the other module, resulting

in different final values of v1 and v2 In the third type,

one changes S without changing any parameter of the

system (for example, adding an auxiliary reaction

which consumes S), also resulting in different changes

in the rates These three types of metabolic changes are

the basis for the definitions of response (and control),

p-elasticity and e-elasticity coefficients for large

chan-ges, respectively, given below

Quantification of metabolic responses

The sensitivity of response of a steady-state variable, w

(usually metabolite concentration, S, or flux, J) to a

large change in a parameter, pi, from an initial state o

to a final state f, is quantified by the mean-response coefficient (or mean-sensitivity coefficient) [14]:

Rw¼ w

f

wo 1

pfi

po i

 1

! ,

ð1Þ

It represents the relative change in the variable divided

by the relative change in the parameter that originated the variable change This sensitivity coefficient is a sys-temic property because the effect of the parameter change propagates through all the system Because, in Scheme 1, we have two system variables, S and J, and two parameters, p1 and p2, we will consider four of these coefficients: RS

p1, RS p2, RJp1and RJp2 Next, the parameter, pi, is changed, keeping S at a fixed value For the sake of convenience, S is kept at the value Sf, i.e., the value of the final state that S would reach if the parameter was changed without keeping S fixed (definition of response coefficient given above) We shall quantify the sensitivity of the rate, vi,

to a large change in pi, from an initial value po

i to a final value pfi, by the mean p-elasticity coefficient:

pvi¼ v

ff i

vfoi  1

!

pfi

po i

 1

! ,

ð2Þ

vab

i ¼ viðSa;pb

iÞ Having two rates and two parameters there are four mean p-elasticity coefficient: pv1

p1, pv1 p2, pv2 p1 and pv2

p2 Because v1 is independent of p2 and v2 inde-pendent of p1it follows that: pv1

p2¼ pv2 p1¼ 0 p-Elasticity coefficients represent the sensitivities of the rates of the isolated component modules to changes in the parame-ters

Finally, we consider that the concentration, S, is changed by some external means, without changing the parameters pi The sensitivity of the rate, vi, to a large change in S, from an initial value Soto a final value Sf,

is quantified by the mean e-elasticity coefficient [12]:

evi

fo i

voo i

 1

!

Sf

So 1



ð3Þ

Here we have also used the notation: vab

i ¼ viðSa;pb

iÞ Having two rates and one intermediate there are two e-elasticity coefficients: ev1

S and ev2

S These e-elasticity coefficients represent the sensitivity of the rate of the supply module to changes in the concentration of its product and the sensitivity of the rate of the demand module to changes in the concentration of its sub-strate, respectively

In the case of mean elasticity coefficients, pi and S both play the role of parameters But note that while

S

v 1 v 2

supply demand

Scheme 1 Metabolic system constituted by a supply module (1)

and a demand module (2) linked by one intermediate S.

in the definition of mean p-elasticity coefficients the

change in the rate with pi is performed keeping S at

the final value, in the definition of mean e-elasticity

coefficients the change in the rate with S is performed

keeping piat the initial value

Parameter changes affect S or J through the effects

on the rates to which the parameters belong Control

coefficients can, therefore be defined in terms of rates,

i.e., as relative change in the variable divided by the

relative change in rate that produced the variable

change [11,19,20] More specifically, a parameter pi is

changed from the initial value po

i to a final value pfi, at fixed S, producing a change in the rate vi As in the

definition of mean p-elasticity coefficients [Eqn (2)],

the rate change is evaluated at S¼ Sf.To quantify the

sensitivity of response of the steady-state variable w to

a large change in the rate viwe define the mean-control

coefficient:

Cw¼ w

f

wo 1

vffi

vfoi  1

! ,

ð4Þ

Remember that: vab

i ¼ viðSa;pb

iÞ The value taken by this coefficient is a system property, because the effect

of the rate change propagates throughout There are

four of these coefficients: CS

v1, CS v2, Cv1J and CJv2

It can be easily shown, using Eqns (1), (2), and (4),

that the two types of control coefficients defined above

[Eqns (1) and (4)] are related by the response theorem:

w stands for S or J and i¼ 1,2 This theorem states

that the effect that a change in a parameter has on a

metabolic variable depends on two factors: the local

effect that the parameter has on the isolated rate

through which it operates and the systemic effect that

a change in rate has on the metabolic variable If the

parameter piis an enzyme concentration or other

inter-nal parameter its initial value, po

i, is not zero and its relative change ðpfi=po

i 1Þ has a finite value In this case, the coefficients Rw and pvi are well defined But,

if piis an external effector (inhibitor, activator or new

enzyme activity), po

i will normally be zero and the coef-ficients would tend to infinity This could easily be

solved by replacing in the definitions of Rw and pvi

relative changes in pi by the corresponding absolute

changes, i.e., replacing ðpfi=po

i  1Þ by ðpfi po

iÞ ¼ pfi The rates of the supply and demand modules, vi, are

non zero and therefore the coefficients Cw are always

well defined

One of the central aims of the present work is to

show how the coefficients Cw can be calculated using

data obtained from top-down experiments In this type

of experiment only the rates of the modules for differ-ent values of the intermediate concdiffer-entration are deter-mined, the measurement of parameter values not being necessary However, to derive the equations that calcu-late the values of Cw from measurements of v1, v2 and

S, the effect that particular changes in the parameter values would have on the variables will be analysed These particular parameter changes and their conse-quences on the values of the variables are the subject matter below

Parameter changes

We shall assume that the system starts at a reference state o, where the parameters, rates and variables take the values: po

1, po

2;voo

1;voo

2; Soand Jo(Table 1) We shall consider six different ways of modifying the initial state, o, which give the final states: xsp, ysp, xp, yp, xs and ys In two of them, one parameter is changed (p1

or p2) and the variables (S and J) freely adjust to the final steady state If p1 is changed the final state is xsp and if p2is changed the final state is ysp(Table 1) The second two ways of modifying the system is to change

a parameter, keeping S at a fixed value In this case, if

p1 is changed the final state is xp, S being kept at the constant value Sx, and if p2is changed the final state is

yp, S being kept at Sy (Table 1) Finally, the third two ways of modifying the system are to change S by some external means, without changing any parameter; S will be changed from So to Sx and from So to Sy, being the final states xsand ys, respectively (Table 1)

We call r1 the factor by which the rate v1 changes when we go from state xs to state xsp, i.e., when p1 is changed from po

1 to px, keeping S fixed at Sx.Similarly,

we call r2the factor by which the rate v2changes when

we go from state ys to state ysp, i.e., when p2 is chan-ged from po

2 to py2, keeping S fixed at Sy As was men-tioned above, to develop the theory for a MMCA for large changes we need to consider particular changes

in the parameters These particular parameter changes

Table 1 Different ways of modifying the reference state Details given in text [note that v ab

i ¼ v i ðS a ; p b

i Þ].

1 v 00

1 v x0

1 v yy

1 v x0

1 v x0

are those resulting in r2 equal to the reciprocal of r1.

In equations, we have (Table 1):

r1Bv1xx

vxo

1

and r2Bv

yy 2

v2yo with r2¼

1

r1

ð6Þ

If p1 and p2 are changed so that Eqn (6) is fulfilled,

the values of the variables satisfy the following

rela-tionships (see Appendix for proof):

Sx¼ Sy and Jx¼ r1Jy ð7Þ

As a consequence of the steady state condition, and

Eqns (6) and (7), eight equalities between the rates are

fulfilled:

Jo¼ voo

1 ¼ voo 2

Jx¼ vxx

1 ¼ vxo

2 ¼ vyo2

Jy¼ vyy2 ¼ vyo1 ¼ vxo1

ð8Þ

Therefore, experimental determination of three rates,

Jo, vxo

1 and vyo2, allows the calculation of the 11 rates

involved (Table 1) In Fig 1, we give a graph (similar

to the graph of combined rate characteristics used by

Hofmeyr and Cornish-Bowden [18]) representing the

effects on the rates of two sets of parameter changes,

one fulfilling and the other not fulfilling the condition given in Eqn (6)

Next, we will derive useful relationships involving the mean control coefficients [defined in Eqn (4)] and the mean e-elasticity coefficients [defined in Eqn (3)]

Relationships between system properties and module properties

The fundamental relationships of MMCA for large changes, in the case of two modules, are the following:

CJv1¼ ev1

SCS v1þ ev1

SðrS 1Þ þ 1

CJv2¼ ev1

SCS v2

CJv1¼ ev2

SCS v1

CJv2¼ ev2

SCS v2þ ev2

SðrS 1Þ þ 1

ð9Þ

where rs¼ Sx⁄ So¼ Sy⁄ So These four equations are the starting point to derive all the other relationships and theorems for large changes given below Their validity can be tested using Eqns (3) (4), (6), (7) and (8), and Table 1

Equation (9) can be solved to obtain the mean control coefficients in terms of the mean e-elasticity coefficients and rs The result is:

CJv1¼e

v2

Sðev1

SðrS 1Þ þ 1Þ

ev2

S  ev1 S

CJv2¼e

v1

Sðev2

SðrS 1Þ þ 1Þ

ev2

S  ev1 S

CS v1¼e

v1

SðrS 1Þ þ 1

ev2

S  ev1 S

CS v2¼ðe

v2

S ðrS 1Þ þ 1Þ

ev2

S  ev1 S

ð10Þ

From these equations it is easily shown that mean con-trol coefficients fulfil the following summation theo-rems:

Cv1J þ CJ

CS v1þ CS

The factors r1and r2can also be calculated in terms of the mean e-elasticity coefficients and rs:

r1¼ 1

r2

¼e

v2

S ðrS 1Þ þ 1

ev1

This relationship was obtained using Eqns (4), (6), (7) and (10)

Fig 1 Rates versus S Schematic representations when condition

Eqn 6 (A) is not fulfilled and (B) is fulfilled.

Solving Eqn (13) for (rs) 1) and replacing the

resulting expression into Eqn (10) gives:

CJv1¼ e

v2 S

ev2

S  r1ev1 S

CJv2¼ r1e

v1 S

ev2

S  r1ev1 S

CS

ev2

S  r1ev1 S

CS

ev2

S  r1ev1 S

ð14Þ

These expressions constitute a different way to

calcu-late the mean control coefficients in terms of the mean

e-elasticity coefficients, to the one given in Eqn (10)

Finally, expressions to calculate the mean e-elasticity

coefficients from the mean control coefficients, i.e., the

metabolic control design equations for large changes,

can be readily obtained from Eqn (14)

ev1

J v2

CS v2

ev2

J v1

CS v1

ð15Þ

Equations (9) to (13) are valid independently of the

functional relationship between the rates v1 and v2,

and the corresponding parameters p1 and p2 They

were previously derived under the restrictive

assump-tion that the rates are proporassump-tional to the

correspond-ing enzyme concentrations [12,14,15] It is easy to

show that when the changes of the parameters and

rates are small (r1 and rs tend to one) they reduce to

the well-known relationships of traditional MCA,

based on infinitesimal changes [1–5,21–23]

Up to this point, the analysis performed did not

require the measurement of parameter values In fact,

to calculate CS

v1, CS

v2, Cv1J and Cv2J , only measurements

of So, Sx, Jo, vxo

1 and vyo2 are needed Nevertheless, if

we want to determine RS

p1, RS p2, RJp1 and RJp2, the initial and final values of the parameter, po

1, px, po

2 and py2, and the corresponding rates have to be known, in

order to calculate the mean p-elasticity coefficients

(Eqn 2)

With these, the mean response coefficients (Eqn 1),

are obtained introducing Eqn (2) and (10) into the

response theorems (Eqn 5)

The relationships that we have derived show that

the control coefficients for large changes are subject to

constraints, which condition the responses of the

meta-bolic variables to parameter changes As a

conse-quence, an important issue in MCA is to determine

how a variable (w) would respond if a parameter or a rate of the system is modulated with a large change The mean control coefficients can be used to perform this calculation, employing the following equation, derived from Eqn (4)

wf

wo ¼ 1 þ Cwðri 1Þ with i ¼ 1; 2 ð16Þ where w0 and wfare the initial and final values of the variable (intermediate or flux), respectively, Cw is the mean control coefficient (Eqn 10), and ri is the factor

by which the rate of the isolated module i has been changed (Eqn 13) If Cw and (ri – 1) have the same sign the variable increases and if they have opposite signs the variable decreases Rate changes are pro-duced by parameter changes The change in the vari-able that results from the change in a particular parameter, pi, can be calculated with an analogous equation to Eqn (16):

wf=wo¼ 1 þ Cwpvipfi=poi 1

with i¼ 1; 2:

Calculation of systemic responses from top-down experiments

Next, we shall show how the mean control coefficients may be calculated from top-down experiments using the relationships derived in the previous section Adding to Scheme 1 an auxiliary reaction, it is poss-ible to modulate the concentration of the intermediate,

S, and measure the rates of the supply and demand modules, v1 and v2 Applying fitting procedures to the table of experimental values v1, v2 and S, continuous functions, represented by v1(S) and v2(S), can be obtained These two functions are the basis for all the calculations

In the reference state, o, the auxiliary rate is zero:

S¼ So, v1¼ voo

1 ¼ v1ðSoÞ and v2¼ voo

2 ¼ v2ðSoÞ When the auxiliary rate is gradually changed, the values taken by intermediate and rates are: S¼ Sx¼ Sy,

v1¼ vxo

1 ¼ v1ðSÞ and v2¼ vyo1 ¼ v2ðSÞ The mean e-elas-ticity coefficients (Eqn 3), expressed in terms of the fitting functions, are given by:

ev1

S ¼ v1ðSÞ

v1ðSoÞ 1

S

So 1



ev2

S ¼ v2ðSÞ

v2ðSoÞ 1

S

So 1

Introducing these functions and rs¼ S ⁄ So into Eqns (10) and (13) we obtain CS

v1, CS v2, Cv1J , CJv2, r1and

r2as a function of S With these functions several plots can be built We can represent CS

v1, CS v2, CJv1, Cv2J ,

v1þ CS

v2 and Cv1J þ CJ

v2 as a function of S⁄ So These plots show how the overall control, given by the

sum-mation theorems (Eqns 11 and 12), is distributed

among the blocks On the other hand, we can

repre-sent CS

v1 and CJv1 as a function of r1, and CS

v2 and CJv2

as a function of r2 These are useful to analyse how

the control of each module varies as its activity

chan-ges In the case of the flux the control normally drops

when the activity is increased

The procedure of analysis that we have described

does not require the measurement of parameter values

But, as was mentioned above, to calculate the mean

p-elasticity coefficient, pv1

p1 and pv2

p2, and the mean response coefficients, RS

p1, RS p2, RJp1 and RJp2, the param-eter values, po

1, px, po

2 and py2, and the rates for these parameter values must be measured The calculations

for the case of parameters acting, say, on the rate v1

are performed as follows The increase in the

param-eter from po

1 to px, results in a new steady state in the

intermediate, Sx pv1

p1, CS v1 and Cv1J are evaluated at Sx Introducing these values in the response theorems

(Eqn 5), RS

p1 and RJp1 are obtained An analogous

pro-cedure can be followed to calculate RS

p2and RJp2 Finally, using Eqn (16), the mean control coefficients

can be used to calculate the change in the system

vari-able (w¼ J or S) that could be obtained with a large

change in the rate of the isolated module by a factor r

For this purpose, the ratios Jf⁄ Joand Sf⁄ Soare plotted

as a function of r, for each one of the modules These

plots show where and in what extent the system has to

be modulated in order to obtain a desirable change in

a variable

Below, we will apply this analysis to data

deter-mined with top-down experiments obtained from the

literature

Analysis of experimental cases

Here, we shall apply the formalism developed in two

studies, performed using top-down experiments The

first analyses the control of glycolytic flux and biomass

production of L lactis [16] and the other studies the

control of oxidative phosphorylation in isolated rat liver

mitochondria [17] The choice of these cases was not

based on the particular interest of the systems studied,

but on the appropriateness of the examples to illustrate

the application of the analysis developed in this work

In the study of Koebmann and colleagues [16],

energy metabolism of L lactis was split into a supply

module, that produces ATP (glycolytic module or

module 1), and a demand module, that consumes ATP

(biomass production module or module 2) The

inter-mediate is the ratio of concentrations ATP⁄ ADP

(Scheme 1 with S¼ ATP ⁄ ADP) Top-down experi-ments consisted of varying the ATP⁄ ADP ratio and measuring the supply and demand rates independently The decrease in the ATP⁄ ADP ratio was achieved by overexpressing the hydrolytic part of the F1 domain of the (F1 F2) H+-ATPase, that increases ATP consump-tion To perform an infinitesimal top-down control analysis at the reference state, the authors obtained fit-ting functions for the experimental values of v1 and v2 versus S These functions are adequate for their pur-pose, but they are not sufficiently good for points away from the reference state, which should be consi-dered when performing a top-down control analysis for large changes Here, the values of v1 and v2 versus

S were fitted to the following functions: v1(S)¼ 82.14 S0.4⁄ (0.8574 + 0.2107 S0.75) and v2(S)¼ 2.325

S3.5⁄ (2.253 + 0.02219 S3.5) (Scheme 1) As mentioned above, these two functions are the basis for all our cal-culations The parameters of the fitting functions do not have units, because S (i.e., ATP⁄ ADP) is dimen-sionless and the values of the rates are expressed as a percentage of the rate at the reference state The refer-ence state is So¼ 9.7 and the ratios S ⁄ So, studied experimentally, are in the interval (0.49, 1) The mean e-elasticity coefficients, ev1

S and ev2

S, are calculated replacing the fitting curves given above, v1(S) and

v2(S), in Eqn (17) ev2

S is always positive This is the sign normally expected because a substrate is an acti-vator of the reaction rate, its increase normally result-ing in an increase in rate ev1

S, a product elasticity, exhibits the normal (negative) sign around the refer-ence state (So¼ 9.7) However, at approximately S ¼ 6.24 the elasticity vanishes, taking a positive sign under this value This behaviour represents ‘product activa-tion’ of S on the rate of module 1 Finally, CS

v1, CS v2,

CJv1, Cv2J , r1and r2are obtained, introducing the expres-sions for the mean e-elasticity coefficients and rs¼

S⁄ So into Eqn (10) and (13) At the reference state (when S tends to So), these expressions give the values

of the infinitesimal control coefficients: Cv1J ¼ 0:80,

CJv2¼ 0:20, CS

v1¼ 6:55 and CS

v2¼ 6:55 In Fig 2 we represent CS

v1, CS v2, CJv1, Cv2J , CS

v1þ CS v2 and Cv1J þ CJv2 as

a function of S⁄ So

CJv1þ CJv2 is always one, according to what it states

in the flux summation relationship for large changes (Eqn 11) In the region of S⁄ So values between 0.49 and 0.65, Cv1J >1 and CJv2<0 This is due to the posit-ive sign of the product elasticity, ev1

S, in this region In addition, the values CJv1 and CJv2 are quantitatively very different from those obtained with infinitesimal chan-ges (Fig 2A) The concentration summation relation-ship (Eqn 12) states that in the case of large changes

CS v1þ CS v2 is not equal to zero Because in all the

experimental range S£ So, the sum of the coefficients

is positive In this case, CS

v2 is negative and slightly smaller in absolute value than CS

v1, which is positive

Only at the reference state, both coefficients take the

same absolute value, i.e., when the changes are

infini-tesimal (Fig 2B)

Next, we represent CS

v1 and CJv1 as a function of r1, and CS

v2 and CJv2 as a function of r2 in two parametric

plots: CJv1 and CJv2 in Fig 3A and CS

v1 and CS

v2 in Fig 3B These are useful plots to analyse how the flux

and concentration control of each module changes as

the activity of the corresponding module is increased

For the flux control, we obtain the normal behaviour,

i.e., the control of both modules diminishes as their

activity is increased (Fig 3A) In addition, CJv1 is

greater than CJv2 in all the range of r factors studied

(0.72, 1.38), but they both fall dramatically in this

rather small range CJv1 decreases from 1.04 to 0.09 and

CJv2 from 0.91 to )0.04 For the concentration control,

the control of the supply module increases and the control of the demand module decreases, in absolute terms, when the corresponding activity is increased (Fig 3B) In the range studied (0.72, 1.38), CS

v1 increa-ses from 1.9 to 15.9 and CS

v2 decreases from 22.0 to 1.4 At r¼ 0.72, CS

v2 is more than 11 times greater than CS

v1 and, at r¼ 1.38, CS

v1 is more than 11 times greater than CS

v2 At r¼ 1, where the mean coeffi-cients coincide with the infinitesimal coefficoeffi-cients, CS

v1 andCS

v2 are equal

Finally, we determine the changes in the flux and intermediate that could be obtained by changing the rates of the modules This calculation is performed using Eqn (16) and is represented in Fig 4 Figure 4A shows that it is not possible to increase the flux signifi-cantly, which is due to the abrupt decrease in CJv1 and

CJv2 with r1 and r2, respectively In this respect, a 40% increase in the activity of the supply module (mod-ule 1) results in less that 4% increase in flux and, in practice, increasing the activity of the demand module

Fig 2 Mean control coefficients versus S ⁄ S o in L lactis (A) Flux

mean control coefficients and their sum and (B) intermediate mean

control coefficients and their sum The reference state is indicated

by d at S ⁄ S o ¼ 1 The range of S ⁄ S o represented corresponds to

the experimental range reported in [16].

Fig 3 Mean control coefficients versus module activity, r, in

L lactis (A) Flux mean control coefficients and (B) intermediate mean control coefficients Solid lines represent values in the experi-mental range and dashed lines give values extrapolated outside this range The reference state is indicated by d at S ⁄ S o ¼ 1.

(module 2) the flux decreases (there is a very small

increase in the flux when increasing the activity of

module 2 for r between 1 and 1.1, which is, for all

practical purposes, irrelevant) In contrast, decreasing

both rates, independently, produces significant and

similar decreases of the flux In this region, flux and

rate are approximately proportional for both modules,

the decrease in rate of module 1 producing a slightly

bigger decrease of the flux Note that, in this example,

there is no way to obtain significant increases in the

flux by changing the activity of a single module

Regarding the intermediate, Fig 4B shows that

decreasing the supply rate or increasing the demand

rate produces moderate decreases (less than 50%),

while increasing the supply rate or decreasing the

demand rate produces increases by a large factor (up

to more than seven times)

Let us now analyse the second experimental case,

concerning the control of oxidative phosphorylation in

isolated rat liver mitochondria [17] Oxidative

phos-phorylation was divided into two modules linked

by the fraction of mitochondrial matrix ATP

(ATP-consuming module or module 2) is the adenine nucleotide translocator (ANT) and the supply module (ATP-producing module or module 1) is the rest of mitochondrial oxidative phosphorylation, including respiratory chain, ATP synthesis and the associated transport processes Membrane potential (Dw) is an intermediate included inside module 1 In the following analysis, we shall assume that the direct effect of this intermediate on module 2 can be neglected, existing only an indirect effect through S Experimental evi-dence for this assumption was reported by Ciapaite

et al [24] Under these conditions, the analysis remains valid even if large changes in Dw take place when the system is modulated with effectors One of these effec-tors is palmitoyl-CoA, an inhibitor of module 2 (ANT) that has no direct effect on module 1 To apply the top-down control analysis developed in the present work to this case, we fitted the experimental points reported in Fig 5 of [17] to continuous functions

Fig 4 Fluxes (A) and intermediate concentrations (B) produced by

independent modulations in the activity of the supply or demand in

L lactis Solid lines represent values in the experimental range and

dashed lines give values extrapolated outside this range The

refer-ence state is indicated by d at S ⁄ S o ¼ 1.

Fig 5 Mean control coefficients versus S ⁄ S o in isolated rat liver mitochondria (A) Flux mean control coefficients and their sum and (B) intermediate mean control coefficients and their sum The refer-ence state is indicated by d at S ⁄ S o ¼ 1 The range of S ⁄ S o repre-sented corresponds to the experimental range reported in [17].

The rates of module 1 and 2 are given by:

v1(S)¼ 14.04 ⁄ (0.03625 + S10.19) and v2(S)¼ 1259S ⁄

(1.136 + S) (Scheme 1) When 5 lmolÆL)1 of

palmi-toyl-CoA (I¼ 5) was added, the rate v1 was described

by the same function [v1(S, I¼ 5) ¼ v1(S)] and the rate

v2 changed, being described by: v2(S, I¼ 5) ¼

378.3S⁄ (0.4796 + S) The reference states, without and

with 5 lmolÆL)1of palmitoyl-CoA, were So¼ 0.49 and

So

I ¼ 0:70, respectively The mean e-elasticity

coeffi-cients, ev1

S and ev2

S, are calculated replacing the fitting curves, v1(S) and v2(S), into Eqn (17) For the entire

range of S studied, ev2

S is positive and ev1

S is negative as would normally be expected Introducing ev1

S, ev2

rs¼ S ⁄ So

into Eqn (10) and (13) CS

v1, CS v2, CJv1, CJv2, r1 and r2 are obtained Note that, v1 and v2 were

meas-ured for different ranges of values of S (see Fig 5 of

[17]) As a consequence, all values of mean-control

coefficients calculated by this analysis involve values of

mean-elasticity coefficients extrapolated outside the

experimental range Accordingly, in the figures that we

will present next, no distinction between experimental

and extrapolated range will be made (in contrast to

Figs 2–4) In Fig 5, we plot CS

v1, CS v2, CJv1, CJv2,

CS

v1þ CS

v2 and CJv1þ Cv2J as a function of S⁄ So

Cv1J þ CJv2 is always one (Eqn 11) and, in this case,

0 < Cv1J <1 and 0 < Cv2J <1 because ev1

S and ev2

normal signs (Fig 5A) CS

v1>0 and CS

v2<0 in all the range of S⁄ So values (Fig 5B) For S⁄ So < 1,

CS

v1> CS

v2



  and CS

v1þ CS v2>0 (total concentration con-trol dominated by supply), while for S⁄ So > 1,

CS

v1< CS

v2



  and CS

v1þ CS v2<0 (total concentration con-trol dominated by demand) CS

v1þ CS v2¼ 0 at the refer-ence state only (Eqn 12)

Finally, we have quantified the effect of

palmytoil-CoA (I, specific inhibitor of module 2) on the

interme-diate, S, and the flux, J, using the corresponding mean

response coefficients, RS

I and RJI Here, definitions involving absolute changes in I are used because the

initial value of I is zero [RS

I ¼ ðSf=So 1Þ=ðIf IoÞ and

RJI ¼ ðJf=Jo 1Þ=ðIf  IoÞ] These coefficients are

calcu-lated using the response theorems for large changes

(Eqn 5), i.e., RS

I ¼ CS

v2 pv2

I and RJI ¼ CJv2 pv2

I , where pv2

I

is the mean p-elasticity coefficient, defined in terms of

absolute changes in I ½pv2

I ¼ ðvff2=vfo2  1Þ=ðIf IoÞ ¼

v2ðS; I ¼ 5Þ=v2ðSÞ  1=ð5  0Þ In Fig 6, we represent

RS

I, RJI and pv2

I as a function of S=So

I In the range of values analysed, pv2

I varies between, approximately, )0.07 and )0.1 Therefore, its effect, in the response

theorem is, roughly speaking, to lower by a tenth the

absolute value of the mean control coefficients, CS

v2 and CJv2, and to change their sign (compare Figs 5 and

6) Another interesting representation would be to plot

RS

I, RJI and pv2

I as a function of the concentration of

inhibitor I This was not possible for this example because the data available was determined at one inhibitor concentration only

Discussion

In MCA, elasticity analysis is the procedure that allows calculation of the control coefficients in terms

of elasticity coefficients In this contribution, we develop a completely general modular elasticity analy-sis of large metabolic responses, for the case of two modules and one intermediate, which also constitutes

an extension of the infinitesimal supply demand analy-sis developed by Hofmeyr and Cornish-Bowden [18] to large changes The stages to achieving this goal were the following: In the first elasticity analysis of large metabolic responses that we previously developed [12], the equations obtained were valid for variable elasticity coefficients and could be applied to analyse model si-mulations involving this type of coefficient However, they could not be applied to analyse top-down experi-ments that result in variable elasticity coefficients, because the relationship between the factor r and the elasticity coefficients had not been deduced In this context, we applied the analysis to an experimental case where the elasticity coefficients were reported to

be approximately constant [12] In a recent contribu-tion [15], the relacontribu-tionship between r and the elasticity coefficients was established and applied to an experi-mental case with variable elasticity coefficients These two preceding formalisms still relied, for the interpret-ation of the results obtained, on the assumption that all the reaction rates are proportional to the corres-ponding enzyme concentrations In addition, they

Fig 6 Mean response coefficients versus S=S o

I in isolated rat liver mitochondria The mean p-elasticity coefficient of the demand block with respect to the inhibitor (I, palmitoyl-CoA), p v 2

I , is also represen-ted The reference state (with 5 lmolÆL)1of palmitoyl-CoA) is indi-cated by d at S=S o

I ¼ 1.