Báo cáo khoa học: Modular metabolic control analysis of large responses in branched systems – application to aspartate metabolism potx

Three properties determine the pattern for large changes in the variables: the values of infin-itesimal control coefficients, the effect of large rate changes on the control coefficients an

Modular metabolic control analysis of large responses in branched systems – application to aspartate metabolism

Fernando Ortega1and Luis Acerenza2

1 Computational Biology, Advanced Science and Technology Laboratory, AstraZeneca, Macclesfield, UK

2 Systems Biology Laboratory, Faculty of Sciences, University of the Republic, Montevideo, Uruguay

Introduction

Living organisms have complex cellular machineries

that have the ability to sense and adapt their metabolic

states to changes in external conditions The transition

between two metabolic states depends on the existence

of molecular mechanisms that, most often, produce

changes in many variable concentrations and fluxes

The design principles behind these responses and the

constraints limiting the patterns that could be achieved

have been studied within the framework of metabolic control analysis (MCA) [1–5]

Metabolic networks of organisms show thousands of variable concentrations and fluxes, so full application

of traditional MCA to intact metabolic systems is impracticable Therefore, to overcome this problem, module-based approaches, such as modular MCA, were introduced [6–9] Modular MCA conceptually

Keywords

Asp metabolism; large metabolic responses;

metabolic control analysis; metabolic

modeling; modular analysis

Correspondence

L Acerenza, Igua´ 4225, Montevideo 11400,

Uruguay

Fax: +598 2 5258629

Tel: +598 2 5258618/Ext.139

E-mail: aceren@fcien.edu.uy

(Received 17 February 2011, revised 16

April 2011, accepted 17 May 2011)

doi:10.1111/j.1742-4658.2011.08184.x

Organisms subject to changing environmental conditions or experimental protocols show complex patterns of responses The design principles behind these patterns are still poorly understood Here, modular metabolic control analysis is developed to deal with large changes in branched pathways Modular aggregation of the system dramatically reduces the number of explicit variables and modulation sites Thus, the resulting number of con-trol coefficients, which describe system responses, is small Three properties determine the pattern for large changes in the variables: the values of infin-itesimal control coefficients, the effect of large rate changes on the control coefficients and the range of rate changes preserving feasible intermediate concentrations Importantly, this pattern gives information about the possi-bility of obtaining large variable changes by changing parameters inside the module, without the need to perform any parameter modulations The framework is applied to a detailed model of Asp metabolism The system

is aggregated in one supply module, producing Thr from Asp (SM1), and two demand modules, incorporating Thr (DM2) and Ile (DM3) into pro-tein Their fluxes are: J1, J2, and J3, respectively The analysis shows simi-lar high infinitesimal control coefficients of J2 by the rates of SM1 and DM2 (Cv1J2¼ 0:6 and Cv2J2¼ 0:7, respectively) In addition, these coefficients present only moderate decreases when the rates of the corresponding mod-ules are increased However, the range of feasible rate changes in SM1 is narrow Therefore, for large increases in J2 to be obtained, DM2 must be modulated Of the rich network of allosteric interactions present, only two groups of inhibitions generate the control pattern for large responses

Abbreviations

AdoMet, S-adenosylmethionine; AK, aspartate kinase; DHDPS, dihydrodipicolinate synthase; HSDH, homoserine dehydrogenase;

MCA, metabolic control analysis; TD, threonine deaminase; TS, threonine synthase.

divides the system into modules, grouping together all

that is irrelevant to the question of interest, including

all that we ignore and knowledge of which is not

required to obtain the answer On the other hand, the

relevant metabolic variables, module exchange fluxes

and linking intermediate concentrations remain explicit

to perform a MCA on them MCA and modular

MCA have been applied to analyze the control of

met-abolic pathways [10–14] and intact cells [15–17]

One important limitation of traditional MCA and

modular MCA is that they have been mainly developed

for small, strictly speaking infinitesimal, changes Thus,

in general, the power of MCA to forecast, for example,

the flux change resulting from a change in enzyme

activ-ity is confined to cases where the enzymatic

perturba-tion is small However, many regulatory processes

in vivo and experiments that involve perturbations, as

well as biotechnological process of interest, require

large metabolic changes A modular MCA suitable for

the analysis of large responses in complex systems has

started to be developed The general theory for a system

divided into two modules and one linking intermediate

was obtained [18–21] Another type of sensitivity

ana-lysis for large changes is global sensitivity anaana-lysis

[22–24] This studies the effect that large changes in the

parameters have on the relevant outputs of the system,

using random sampling of the parameter space This

tool is mainly used for model characterization and

vali-dation It was not developed to analyze large responses

of complex experimental systems, where detailed

infor-mation of the structure and the types of rate laws

gov-erning many processes is not known For this purpose,

modular approaches could be used

Here, the general theory of modular MCA for large

responses is developed to include the analysis of

branched systems This formalism enables the analysis

of systems with three modules and one explicit

intermediate It may be used, for example, to predict

in what region or regions of the metabolic network an

effector would have to operate in order to produce a

large change in a particular metabolic concentration or

flux This is relevant for studying where a physiological

activator or inhibitor would act to regulate a cellular

process, or where the site of action of a drug would

have to be to compensate for the deviation of a

metabolic variable in a pathological condition The

application of the new method is illustrated using a

model of Asp metabolism [25]

Methods

Large parameter changes produce changes in the

meta-bolic concentrations and fluxes The effect that a change

in the parameter p has on the steady-state value of a variable w (metabolite concentration, S, or flux, J) is quantified by the response coefficient, Cw

p, representing the relative change in the variable divided by the rela-tive change in the parameter (see definitions of the coef-ficients in Doc S1 and [21]) The number of response coefficients in cellular metabolism (number of parame-ters· number of variables) is very large, and measuring all of them is not practically feasible To overcome this problem, we can conceptually divide the system into a small number of modules, leaving explicit only the vari-able concentrations and fluxes relevant to the analysis that we want to perform [6–9] For example, in Fig 1,

we represent a system divided into three modules and one linking intermediate Each module can therefore be considered as a ‘super-reaction’, consisting of many enzyme-catalyzed reactions Modularization drastically reduces the number of explicit variables, but there are still a large number of response coefficients, because of the large number of parameters involved

Parameter changes affect the explicit metabolite concentrations and fluxes through the effect on the rates of the modules to which the parameters belong Therefore, the effect that a parameter change has on a variable can be decomposed into two parts, namely, the effect that the parameter has on the rate of the module, and the effect that the resulting rate change

J1

J3

J2

Fig 1 Branched modular system The metabolic system is conceptually aggregated into one input and two output modules connected by one linking intermediate, S J 1 , J 2 and J 3 are the fluxes of modules 1, 2 and 3, respectively.

has on the variable It is possible to obtain identical

rate changes modifying different parameters or

combi-nations of parameters, operating on the same rate, by

different amounts Thus, we can quantify the effect

that changing the rate of a module has on a variable

without specifying the parameter responsible for the

change For this purpose, we use the control coefficient

for large changes, Cw

v, representing the relative change

in the variable divided by the relative change in the

rate that produced the variable change In the modular

representation of the system, there is a small number

of explicit variables and of module rates, resulting in a

small number of control coefficients This set of

con-trol coefficients quantifies the concon-trol properties of the

modular representation of the system, and can be

experimentally determined (see below)

In the definition of flux control coefficient,

CJ

v¼ ðrJ 1Þ=ðr  1Þ, rJ is the factor by which the flux

Jhas changed, and r is the factor by which the rate, v,

that originated the flux change was modified Note that

v represents both the rate equation governing the rate

of the step and the value that this rate equation takes

By definition, the steady-state flux, J, is the value taken

by the rate equation when embedded in a metabolic

system that reaches steady state However, there is an

important difference between flux change and rate

change, which will be analyzed next For the sake of

clarity, let us first focus on a reaction step in a

metabolic system governed by a rate law where the

rate is proportional to the enzyme concentration:

vab= g(Sa)Eb g(S) is an arbitrary function of the

intermediate concentration, S, and E is the enzyme

con-centration We will assume that when the parameter E

is changed, the system goes from the reference state to a

final state The superscripts a and b indicate the state at

which S and E are evaluated, respectively J is the

steady-state flux carried by the step Initially, the system

is at the reference state, o, where the quantities involved

take the values: Eo, So, voo, and Jo (with Jo= voo)

The enzyme is changed to the final steady state, Ef, the

final values taken by the other quantities being: Sf, vff,

and Jf(with Jf= vff) The flux change is:

rJ= Jf⁄ Jo= vff⁄ voo

We will call r the factor by which the enzyme

concen-tration is changed (r = Ef⁄ Eo); r is also the factor by

which the rate was changed, because the rate is

propor-tional to the enzyme concentration As we will see next,

rcan also be calculated from rate values The following

equalities hold:

vff= g(Sf)Ef= g(Sf)rEo = rvfo

By solving this equation, r can be obtained:

r= vff⁄ vfo The results obtained for the flux change and the rate change remain valid if we consider a module including many reaction steps governed by a rate law that is not proportional to the parameter or group of parameters that are changed: vab= v(Sa,pb) In this general case,

ris the factor by which the initial rate is effectively mul-tiplied when one or more parameters in the module (nonproportional to the rate) are changed In general, the difference between flux change (rJ= Jf⁄ Jo= vff⁄ voo) and rate change (r = vff⁄ vfo) is that, whereas the rate change is a local change, obtained by changing one or more parameters at constant intermediate concentra-tion, flux change is a systemic change, involving simul-taneous changes in the parameters and intermediate concentration r < 1, r = 1 and r > 1 correspond to rate decrease, rate unchanged (or changed infinitesi-mally) and rate increase, respectively In the analysis and plots given below, r =1 corresponds to the values

of the infinitesimal control coefficients

According to the definition of Cw

v, experimental deter-mination of these coefficients requires change of param-eters in the different modules in order to determine, for each module, the rates voo, vff, and vfo This experimen-tal approach has some drawbacks On the one hand, it

is a laborious approach, because of the relatively high number of parameter modulations and measurements required On the other hand, in a large system, control

is normally distributed, and most of the parameters have a relatively low effect on the fluxes, the errors involved in the determination of the coefficients being high An important result of the theory of modular MCA for large responses is that the Cw

v coefficients can

be calculated in an alternative way, using data obtained

by modulating S and measuring the resulting rates In this approach, values of voo and vfo are sufficient to perform the calculations, measurement of vffnot being required Modulation of S may be performed by using

an auxiliary reaction, in which case manipulation of parameters of the system is not necessary This alterna-tive method, based on the theory of modular MCA for large responses, does not have the drawbacks mentioned above We will describe this alternative method in the case of a metabolic system grouped into three modules and one linking intermediate (Fig 1)

Results

Relationships between system responses and component responses in branched systems The responses of the rates of the isolated modules to changes in the intermediate concentration, i.e the

component responses, are represented by the e-elasticity

coefficients In the scheme of Fig 1, there are three

e-elasticity coefficients for large changes: evi

S(i = 1, 2, 3) These may be directly calculated replacing the data,

module rates versus S, in the definition (see definitions

of the coefficients in Doc S1 and [21]) With the values

of the e-elasticity coefficients, evi

S, and rS= Sf⁄ So, the factors by which the rates are changed, ri, and the

sys-temic responses, Cw

vi(w = S, Ji and i = 1, 2, 3), are calculated from the equations of Tables 1 and 2,

respec-tively Note that all the relationships involving system

properties and component properties given in Tables 1

and 2 reduce to the well-known expressions of

tradi-tional MCA and modular MCA when infinitesimal

changes are considered (i.e when rS= 1) Derivation

of the equations given in Tables 1 and 2 is given in

Doc S1 The formalism developed here is based on

control coefficients with respect to rates, and therefore

remains valid if the rates of the reaction steps are not

pro-portional to the corresponding enzyme concentrations

Concentration and flux control coefficients for large

changes satisfy summation relationships For the

branched metabolic network, the summation

relation-ships are:

CS

v1 þ CS

v2 þ CS

v3 ¼ 1  rS

CJ1

v1 þ CJ1

v2 þ CJ1

v3 ¼ 1

CJ2

v 1 þ CJ 2

v 2 þ CJ 2

v 3 ¼ 1

CJ3

v1 þ CJ3

v2 þ CJ3

v3 ¼ 1:

In addition, the flux control coefficients are

con-strained by flux conservation relationships:

CJ1

v1 ¼ a CJ2

v1 þ 1  að Þ CJ3

v1

CJ1

v 2 ¼ a CJ2

v 2 þ 1  að Þ CJ3

v 2

CJ1

v 3 ¼ a CJ2

v 3 þ 1  að Þ CJ3

v 3

where: a ¼ Jo

2



Jo

1and 1  a ¼ Jo

3



Jo

1 With the r-factors and control coefficients given in Tables 1 and 2, the variable changes produced by a rate change may be calculated from the following expression:

wf

wo¼ 1 þ Cw

vi ðri  1Þ

It is important to note that, in this general expression,

Cw

vi is a function of ri If, for a module i, the value of the control coefficient is close to 0, or only r-factors close to unity can be achieved, significant changes in the variable cannot be obtained by modulating this module This is a strong result, because it implies that the impossibility of changing the variable does not depend on which parameter or combination of param-eters of the module is changed So, if we want to change a variable, it is necessary to change a parame-ter or set of parameparame-ters in a module with a control coefficient and r-factor substantially different from 0 and 1, respectively

Usefulness of module control coefficients in branched systems

In the scheme of Fig 1, with one supply module (module 1) and two demand modules (modules 2 and

Table 1 r-factors versus e-elasticity coefficients Expressions used

to calculate the rate changes (r i ) from the component responses

(e vi

S ), the change in the intermediate concentration (rS) and the initial

flux distribution (a ¼ J o 

J o ).

r 1 ¼

1 þ a e v 2

S þ ð 1  a Þ e v 3

S

r S  1

1 þ e v 1

S ð r S  1 Þ

r 2 ¼

a þ e v 1

S  1  a ð Þ e v 3

S

rS  1

a 1 þ e v 2

S ð rS  1 Þ

r3 ¼

1  a

ð Þ þ ev1

S  a ev2

S

r S  1

1  a

ð Þ 1 þ ev3

S ð r S  1 Þ

Table 2 Control coefficients versus e-elasticity coefficients Expressions used to calculate the system responses (C w

v i ) from the component responses (e vi

S ), the change in the intermediate concen-tration (rS), and the initial flux distribution (a ¼ J o 

J o ).

C S

v 1 ¼ 1 þ ev1

S ð r S  1 Þ

den

C S

v 2 ¼  a 1 þ e v 2

S ð r S  1 Þ

den

C S

v 3 ¼  1  a ð Þ 1 þ e v 3

S ð rS  1 Þ

den

CJ1

v 1 ¼ a ev2

S þ ð 1  a Þ e v 3

S

1 þ e v 1

S ð r S  1 Þ

den

CJ1

v 2 ¼  a e v 1

S 1 þ e v 2

S ð r S  1 Þ

den

CJ1

v 3 ¼  1  a ð Þ ev1

S 1 þ ev3

S ð r S  1 Þ

den

C J 2

v 1 ¼ e v 2

S 1 þ e v 1

S ð r S  1 Þ

den

C J 2

v 2 ¼  e v 1

S þ ð 1  a Þ e v 3

S

1 þ e v 2

S ð rS  1 Þ

den

CJ2

v 3 ¼  1  a ð Þ e v 2

S 1 þ e v 3

S ð r S  1 Þ

den

CJ3

v 1 ¼ e v 3

S 1 þ e v 1

S ð r S  1 Þ

den

CJ3

v 2 ¼  a ev3

S 1 þ ev2

S ð r S  1 Þ

den

CJ3

v 3 ¼  e v 1

S þ a e v 2

S

1 þ ev3

S ð r S  1 Þ

den den ¼  e v 1

S þ a e v 2

S þ 1  a ð Þe v 3

S

3), there are three concentration and nine flux control

coefficients for large responses (Table 2) These

quan-tify how the variables are affected by changes in the

supply or demand rates For example, CJ2 and CJ3

represent how a change in supply rate affects the

fluxes J2 and J3, producing Y and Z If CJ2v1 and Cv1J3

have similar values, those fluxes are similarly affected,

in relative terms, by the supply rate change On the

other hand, CJ2v2 quantifies the effect that a change in

the rate of the demand of S for Y synthesis has on

the flux that produces Y A high value of this flux

control coefficient, in a wide range of values of

rate change r2, indicates that large flux changes could

be achieved by changing the rate of the process

carry-ing the flux Similar considerations apply to CJ3v3

Finally, CJ2v3 and Cv2J3 quantify how one demand flux

is affected by changing the rate of the competing

branch

Normally, CJ2, CJ3, CJ2 and CJ3 are positive, and CJ2

and CJ3 are negative, indicating increase and decrease

of the flux with rate increase, respectively In the case

that J2 and J3 produce molecules essential for cell

functioning (e.g for protein synthesis), one would

expect them to show a relatively high response to the

demand rate of the corresponding modules, and

there-fore CJ2v2 and CJ3v3 to have relatively high values (the

control exerted by supply being smaller) In addition,

the change in one of these demand rates should not

significantly affect the flux of the competing branch,

the values of CJ2v3 and Cv2J3 being relatively small in

absolute terms

Control pattern

In traditional modular MCA, the control pattern is the

set of values that the infinitesimal control coefficients

take at the reference state For example, in Fig 1 the

control pattern comprises the values of the 12

infinites-imal control coefficients

In the framework of modular MCA for large

responses, apart from the values of the infinitesimal

control coefficients at the reference state, the control

pattern includes two important additional properties

One property is how the values of the control

coeffi-cients change when the rate is changed by a large

(non-infinitesimal) amount The normal behavior of

flux control coefficients of the type CJiviis that their

val-ues decrease when the rate of the corresponding

mod-ule increases; that is, CviJi decreases when ri increases

When CJivi stays approximately constant or increases,

the control pattern is called sustained or paradoxical,

respectively [26,27] Note that, in modular MCA for

large responses, the values of the infinitesimal control

coefficients are also relevant to the control pattern obtained Normally, CJivi decreases when the rate increases If the initial infinitesimal control coefficient

is small, then, as the rate is increased, it will become even smaller, with little effect on the flux Therefore,

to obtain a substantial increase in the flux, a relatively large infinitesimal control coefficient is usually required The other important property is the range of values of ri that can be achieved It is, in principle, possible to obtain any value of ri manipulating the parameters in module i However, some ri values would result in unrealistic values of the concentration

of the linking intermediate (S), e.g either too high or too low to be compatible with the physical chemistry

or the physiology of the cell The range of values of S (Smin, Smax) determines a range of values of ri (rimin,

rimax) that can be achieved (Table 1)

In summary, the three relevant properties character-izing the control pattern are: the value of the infinitesi-mal control coefficients at the reference state, the effect that non-infinitesimal rate changes have on the values

of the control coefficients for large responses, and the range of rate changes that can be achieved, maintain-ing the concentration of the linkmaintain-ing intermediate at feasible values Next, we will illustrate how these three properties constrain the range of values that a flux can take

In Fig 2A (inset), we represent curves of CJvi versus

ri for a hypothetical system The a-curve corresponds

to a starting system, and the b-curve to the system after several parameters have been changed Black circles are the values of the infinitesimal control coeffi-cients at the reference state In this example, the value

of the infinitesimal control coefficient at the reference state in the a-curve is twice the value in the b-curve However, the control coefficient in a drastically decreases when riis increased, while the control coeffi-cient in b remains constant (i.e shows sustained con-trol) In Fig 2A, we show the flux as a function of ri calculated form the a-curve and b-curve appearing in the inset of the figure The b-curve, showing the lowest infinitesimal control coefficient in the reference state, results in greater increases in flux for moderate or high values of ri increase Note that, in an analysis based

on the infinitesimal control coefficients at the reference state only, the opposite, erroneous conclusion, would have been reached

In the previous example, we assumed, that in both systems, ricould be changed in identical ranges Let us consider another hypothetical example, represented in Fig 2B The value of the infinitesimal control coeffi-cient at the reference state in the a-curve is also twice the value in the b-curve In this case, however, the

a-curve shows sustained control, whereas in the b-curve,

the control drops smoothly as the rate is increased

Importantly, in this case, the maximum ri that can be

achieved in the a-curve (rimax= 1.10) is lower than

that in in the b-curve (rimax= 1.40) As a consequence

of this property, the b-curve would result in greater

increases in flux for high values of rate increase

Con-sidering the infinitesimal control coefficients at the

ref-erence state only, once again, the opposite, erroneous,

conclusion would have been obtained

Determination of the control coefficients from top-down experiments

Here, we will show how to calculate the coefficients for large responses from top-down experiments in a branched system with three modules and one linking intermediate

First, the fluxes and concentration of the linking intermediate, S, at the reference state are determined Second, S is changed by addition of an auxiliary reaction For each value of S, the rates of the three modules are measured Changing S by parameter mod-ulation is also possible, but in this case only the rates where the parameter was not modulated are computed From the table of the rates versus S, the three e-elas-ticity coefficients for large changes are calculated in terms of S Introducing the e-elasticity coefficients in the equations of Tables 1 and 2 renders the values of the factors r1, r2 and r3 and of the control coefficients

as a function of S Finally, the parametric plot of the control coefficients for large changes as a function of ri (i = 1–3) may be constructed

In an experimental system, a complete modular MCA for large responses ideally requires both upmod-ulation and downmodupmod-ulation of the linking intermedi-ate concentration in the range of feasible values, and measurement of this concentration and the correspond-ing values of the rates To our knowledge, there is no set of data in the literature that allows performing a complete modular MCA for large responses in branched systems The analysis of incomplete datasets requires undesirable extrapolations to be made outside the experimental range Therefore, we decided to illus-trate the new method with a mathematical model based on measured kinetic parameters, to avoid this type of extrapolation The model is manipulated in exactly the same way as an experimental system, as is described in the next section

Application of modular MCA of large responses

to Asp metabolism

We will analyze a detailed kinetic model of Asp metabolism in Arabidopsis constructed on the basis of measured kinetic parameters [25] The structure of the metabolic system shows several branch points regulated

by a relatively complex network of allosteric interac-tions The model was originally built for analyzing the function of these allosteric interactions in the branched pathway For this purpose, the authors performed numerical simulations and traditional MCA

Here, we will apply to the model the new modular MCA framework, to study the control pattern for

0.7

0.8

0.9

1.0

1.1

1.2

r i

J r i

J o

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.0

0.4

0.8

C v

i

J

0.7

0.8

0.9

1.0

1.1

1.2

r i

J r i

J o

0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 0.0

0.4

0.8

C v

i

J

A

B

b

a

a b

b a

a b

Fig 2 Flux control pattern and flux changes The flux control

coefficients with respect to the rate (C J

vi ) and the flux relative to the reference state value (Jri⁄ J o

) are plotted against the factor by which the rate of the module is changed (ri), for two hypothetical

situations (A, B) In both cases, the infinitesimal flux control

coefficient at the reference state (•) in the a-curve is twice that in

the b-curve (A) The control coefficient in the a-curve decreases

with module rate, whereas that in the b-curve remains constant.

As a consequence, higher increases in flux may be obtained in the

b-system (B) The control coefficient in the a-curve is constant and

that in the b-curve decreases smoothly However, the b-system

can achieve larger flux changes, because the feasible range of

rates in the a-curve is smaller In both cases, using only

infinitesi-mal control coefficients to predict large flux changes leads to

erroneous conclusions.

large responses of Thr and Ile incorporation into

pro-tein To this end, a modular aggregation of the model

in three modules and one linking intermediate, Thr,

was made (Fig 3) Module 1 is the ‘supply’ module

producing Thr from Asp Module 2 is a ‘demand’

module consuming Thr for protein synthesis Module 3

is another ‘demand’ module, producing Ile from Thr,

used for protein synthesis

It is important to emphasize, before beginning our

analysis, that there are two key differences between the

MCA applied in Curien et al [25] and the modular MCA that we will perform next The first is that, whereas the MCA used by Curien et al applies to small (strictly speaking, infinitesimal) changes around the reference state only, our modular MCA is also valid for large changes To study the effect of large modulations, Curien et al used numerical simulation

to obtain the effects that large changes in particular parameters have on selected variables

The second difference is that Curien et al deter-mined the control coefficients of all the variable metabolite concentrations and fluxes with respect to the rate of all the steps, and we will use a modular aggregation of the model, applying modular MCA to one supply and two demand modules connected by Thr Therefore, our conclusions will not be referred to the control by individual steps, but to the control by regions of the network relevant to the particular meta-bolic processes that we aim to understand

Our analysis will consist of two stages First, we will calculate the control pattern of large responses of the modular system, as was described in ‘Control pattern’ The analysis of this pattern will show, for example, how the system responds to large changes in supply of and demand for Thr for protein synthesis In this first stage, the modules will be treated as ‘black boxes’ The perturbations and determination of the responses in the model will follow the same steps described

in ‘Determination of control coefficients from top-down experiments’ It is important to emphasize that none of the conclusions obtained at this stage require know-ledge about the processes taking place inside the modules In the second stage, we will look inside the modules and study how the control pattern, deter-mined in the first stage, is affected by eliminating the allosteric interactions operating in the system This study will allow investigation of which allosteric inter-actions are relevant for establishing the control pattern

of Thr and Ile incorporation into protein and which are not The rate equations and parameter values used are given in Doc S1

To start the analysis, the model was manipulated following the same general procedure that would be performed on an experimental system The Thr con-centration was changed up and down, and the corre-sponding rates of the three modules were computed With these quantities, all of the coefficients and factors were calculated

The elasticity coefficients for large responses ( ev1

S , ev2

S and ev3

S ) were obtained in a range of Thr concentration between 60 and 3000 lm, i.e approximately between

1⁄ 5 and 10 times the reference steady-state value, Thro= 303 lm (see Discussion below) This range of

Fig 3 Modular aggregation of Asp metabolism The model of Asp

metabolism is aggregated into one input and two output modules,

Thr being the linking intermediate Modules 1, 2 and 3 have fluxes

J1, J2, and J3, respectively, as in Fig 1 AdoMet, Asp and Cys are

external species, their concentrations remaining constant Aspartate

semialdehyde (ASA), aspartyl phosphate (Asp-P), homoserine

(HSer), Ile, Lys, phosphohomoserine (PHSer) and Thr are internal

variable species AK, aspartate semialdehyde dehydrogenase

(ASADH), cystathionine-c-synthase (CGS), DHDPS, HSDH,

homo-serine kinase (HSK), TD and TS represent enzyme activities The

five groups of allosteric interactions, four inhibitions (G-I to G-IV)

and one activation (G-V), are indicated by dashed lines For

addi-tional information, see Doc S1 and [25].

values of Thr concentration is the one used in all of the

calculations, and will determine the ranges of values of

ri that could be achieved The product elasticity

coeffi-cient, ev1

S , is negative and the substrate elasticity

coeffi-cients, ev2

S and ev3

S , are positive in of all the range of values of Thr concentration When the concentration

of Thr increases, the three coefficients decrease, in

absolute terms These decreases correspond to increases

in saturation of the processes (Fig S1)

In Fig 4, the concentration control coefficients for

large responses (CThr

v1 , CThr v2 and CThr

v3 ) are represented as

a function of the factors by which the rates of the

modules were changed (r1, r2, and r3) The signs of

these control coefficients are those normally expected:

CThr

v1 , quantifying control with respect to ‘supply’, is

positive, and CThr

v2 and CThr

v3 , quantifying control with respect to ‘demand’, are negative The absolute values

taken at the reference state are close to the minimum

values attained in all of the range of plausible rates

Moderate increases in r1 or decreases in r2 and r3

result in relatively high changes in the concentration

control coefficients The ranges of ri values, (rimin,

rimax) i = 1–3, that could be achieved without

produc-ing unfeasible concentrations of Thr, are: (r1min,

r1max) = (0.36, 1.69), (r2min, r2max) = (0.23, 5.2), and

(r3min, r3max) = (0.14, 4.0) Importantly, these are the

ranges of rate changes that can be achieved when

the fluxes adapt to changing external conditions or

when the system is manipulated for biotechnological

purposes

In Fig 5, we represent the flux control coefficients for large responses, Cv1J1, Cv1J2, CJ3v1, CJ1v2, CJ2v2, CJ3v2, CJ1v3, Cv3J2 and CJ3v3, and the steady-state fluxes, J1, J2and J3, as a function of the corresponding factors r1, r2, and r3 The flux control coefficients with respect to v1 are fairly constant, in most of the range of r1, and show reasonably high values However, the range of r1 values, maintaining the concentration of the linking intermediate at plausible values, is relatively narrow: (r1min, r1max) = (0.36, 1.69) As a consequence, the maximum increases in the output fluxes, J2 and J3, that can be achieved by increasing the rate of the sup-ply module are modest: 29% and 16%, respectively (Fig 5A) On the other hand, the ranges of rate changes of the demand modules are much wider: (r2min, r2max) = (0.23, 5.2) and (r3min, r3max) = (0.14, 4.0) In addition, if we look in the insets of Fig 5B,C, the flux control coefficients CJ2and CJ3 at the reference state are high, and show only moderate decreases when rate is increased This is why the fluxes J2 and J3 can achieve increases of 160% and 182%, changing the rates of the corresponding modules (Fig 5B,C) Cv2J3 shows, in all the range of rates, low values in absolute terms (Fig 5B) As a consequence, J3 suffers only minor perturbations if the rate of the competing mod-ule is changed Cv3J2 shows higher absolute values than

CJ3v2, and J2decreases to a greater extent than J3, when the rate of the competing module is increased (Fig 5C), although the effect is not dramatic

In summary, with the control pattern found, the fluxes J2 and J3 show a large response to the demand

of the corresponding modules, the effect of changing supply being much smaller In addition, the change in one of the demand rates does not severely affect the flux of the competing branch These properties are those to be expected when the products of the demand modules are essential, simultaneously, for cell function-ing, as is the case in the system under study If a mod-ular analysis based on infinitesimal control coefficients only were performed, some of these conclusions would have been different For instance, as the infinitesimal module control coefficients CJ2v1 and CJ2v2 take similar values (CJ2v1¼ 0:6 and CJ2v2¼ 0:7), this information alone would suggest that the increases in the flux J2 that could be achieved by changing, independently, the rate

of supply and the rate of demand are quantitatively similar As we have seen, the analysis for large responses shows that this conclusion based on infinites-imal module control coefficients only is erroneous

In the work of Curien et al [25], where MCA was applied for infinitesimal changes and without module aggregation, the authors found a rather high level

of control of protein-forming fluxes in the reaction

1 2

3

0

2

4

6

8

10

r i

Thr ri

Thr o

1 2 3

0 1 2 3 4 5 –10

–5 0 5 10

C vi Thr

Fig 4 Concentration control pattern The concentration control

coefficients with respect to the rates (C Thr

vi ) and Thr concentration relative to the reference state value (Thr ri ⁄ Thr o ) are plotted against

the factor by which the rate of module i (r i , i = 1, 2, 3) is changed,

for the system in Fig 3 Starting at the reference state (•),

moder-ate increases in r 1 or decreases in r 2 and r 3 result in relatively high

changes in the control coefficients (r 1min , r 1max ) = (0.36, 1.69),

(r2min, r2max) = (0.23, 5.2) and (r3min, r3max) = (0.14, 4.0) are the

ranges of r i values that could be achieved without producing

unfea-sible Thr concentrations.

catalyzed by isoform AK1 of aspartate kinase (AK), located in the supply region, suggesting that important increases in the fluxes could be achieved by modulating supply However, if the maximum velocity of AK1 at the reference state (AK1 = 0.25) is multiplied by a fac-tor 2.26, r1 reaches its maximum feasible value (1.69) (Fig 5A) As was discussed above, the parameter increase could produce, at most, a 29% in J2 and a 16% increase in J3 Therefore, increases in supply rate may produce only moderate increases in protein-form-ing fluxes Note that the upper bounds to output fluxes increases are independent of which parameter or com-bination of parameters of the supply module are chan-ged and to what extent, as was discussed above The modular MCA performed for large responses and the conclusions obtained up to now did not require knowledge of details from inside the modules Now we will look inside the modules to analyze the effect of eliminating the allosteric interactions on the control pattern for large responses (Fig 3) In mod-ule 1, there are several groups of allosteric interactions: inhibition of both activities of bifunctional AK-HSDH (two isoforms: AKI-HSDHI and AKII-HSDHII) by Thr (G-II), inhibition of the isoforms of monofunc-tional AK (AK1 and AK2) by Lys (G-III), inhibition

of the isoforms of DHDPS (DHDPS1 and DHDPS2)

by Lys (G-IV), and activation of TS (TS1) by AdoMet (G-V) In module 3, there is only one protein subject

to allosteric regulation, namely, TD, which is inhibited

by Ile (G-I) and, in module 2, there is no allosteric interaction (for a full description of the allosteric regu-lations in the model, see [25] and references therein) Next, we will study the control pattern after elimina-tion of the five groups of interacelimina-tions (G-I to G-V), one at a time, to assess the relative importance that these groups have in determining the type of control pattern for large responses exhibited by the system

It is important to note that this type of modification will also produce the undesirable effect of affecting the reference values of the variable metabolite concentra-tions and fluxes, i.e the reference steady state of the system To avoid this simultaneous effect on state and control, we have modified the maximal rates in the rate equations where the allosteric interaction is elimi-nated in such a way that the starting values of the con-centrations and fluxes remain unaltered

In Fig 6, we represent the effect of eliminating the feedback inhibition of TD by Ile (G-I in Fig 3) The main differences from the original control pattern are

as follows The range of r1 increases, the modified sys-tem showing large increases in control by supply of J3

1

2

3

0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.4

0.6

0.8

1.0

1.2

r 1

J i r1

2 3

0.4 0.6 0.8 1.0 1.2 1.4 1.6 0.0

0.4

0.8

C v1

1 2

3

0.5

1.0

1.5

2.0

2.5

3.0

3.5

r 2

J i r2

J i o

1 2

3

–0.4 0.0 0.4 0.8

C v2

1

2 3

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r 3

J i r3

J i o

1

2 3

–0.4 0.0 0.4 0.8

C v3

A

B

C

Fig 5 Flux control pattern The flux control coefficients (CJi

v 1 , CJi

v 2

and C J i

v 3 ) and the flux values, relative to the reference state value

(J r1

i



J o

i ,J r 2

i



J o

i and J r 3

i



J o

i ) are plotted against the factor by which the rate of the module is changed (r 1 , r 2 and r 3 , respectively), for

the system in Fig 3 The three numbered curves in each plot

cor-respond to the three fluxes Ji(i = 1, 2, 3) We can see that there

is a relatively high infinitesimal control of the output fluxes, J 2

and J 3 , by the rate of the supply module [i.e CJ2

v 1 and CJ3

v 1 , repre-sented in the inset of (A), are relatively high in all the feasible

range of r 1 ], but, owing to the narrow range of feasible supply

rates, substantial increases of these fluxes require increases in

the rates of demand modules, r2 and r3 [see J r2

2



J o in (B) and

J r3

3



J o in (C)].

(the flux of the module where the feedback was elimi-nated) but not J2 (Fig 6A) CJ3v2 increases in absolute terms, producing an undesirably large change in J3 when the rate of the competing branch is increased (Fig 6B) The range of r3 increases But, since the absolute values of the control coefficients with respect

to v3 are reduced, the maximum effects on the fluxes with r3 remain approximately unchanged (Fig 6C) However, the system with the feedback inhibition has the advantage of requiring a smaller r3 to achieve the same J3

In Fig 7, we represent the effect of eliminating the feedback inhibition of bifunctional AKI-HSDHI and AKII-HSDHII by Thr (G-II in Fig 3) In contrast to what was observed for the inhibition of TD, the ranges

of values of r1, r2 and r3 decrease After removal of the inhibition, the maximum values of the fluxes that can be obtained by changing r1 are almost the same (Fig 7A), but the changes in r1 required are smaller, which strengthens the control by supply In addition, the maximum effects on J2 of changing r2 and on J3

of changing r3 are drastically reduced (Fig 7B,C), impinging on the potential of the system to control the output fluxes by demand Finally, the maximum reduc-tion of J2 by r3 and of J3by r2 resulting from branch competition remains unchanged after elimination of the inhibition, but is achieved with smaller changes in rate, what is another disadvantage for the independent regulation of the branches

Eliminating the feedback inhibition of AK1 and AK2 by Lys (G-III in Fig 3) has minor effects on the control pattern, the flux changes that can be achieved being similar to those of the original system (Fig S2) The inhibition of DHDPS1 and DHDPS2 by Lys (G-IV in Fig 3) also has minor effects on the control pattern Finally, because, in the model, AdoMet is treated as a parameter, eliminating the activation of TS1 by AdoMet (G-V in Fig 3) and compensating by changing the maximal rate has no effect on the control pattern (data not shown)

In summary, only elimination of G-I and elimina-tion of G-II (Fig 3) produce important changes in the control pattern Moreover, the resulting changes in the control pattern impair the regulatory responses of the system: weakening the control by demand, which is needed, strengthening the unwanted control by supply, reducing the desirable independence between compet-ing branches, or a combination of these Therefore, these two groups of allosteric inhibitions appear to be essential for establishing the adequate control pattern

A natural question is whether the main factor responsible for generating the control pattern in the inhibition of AKI-HSDHI and AKII-HSDHII by Thr

1

2 3

1 2 3

0

2

4

6

8

10

12

r 1

J i r1

J i o

1

2

3

1

2

3

0 1 2 3 4 5 6 7 0.0

0.4 0.8 1.2

C v1

1 2

3

1 2

3

0

1

2

3

4

r 2

J i r2

J i o

1

2

3

1 2

3

0 1 2 3 4 5 6

–0.8 –0.4 0.0 0.4 0.8

C v2

1

2

3

1

2 3

0

1

2

3

4

5

r 3

J i r3

J i o

1

3

2

3

0 2 4 6 8 10 12 14

–0.4 0.0 0.4 0.8

C v3

A

B

C

Fig 6 Effect of eliminating G-I allosteric interactions The flux

con-trol coefficients and the flux values, relative to the reference state

value, are plotted against the factor by which the rate of the

mod-ule is changed, for the system in Fig 3 (solid lines) and the system

modified by eliminating G-I allosteric interactions (dashed lines).

The main effects on the control pattern of eliminating G-I are as

follows: (A) the feasible range of supply rates and the control of J3

by supply increase dramatically; (B) a large increase in J 2 can still

be achieved by increasing r2, but there is a concomitant large

decrease in the flux of the competing branch, J3; and (C) by

increasing r 3 , the same maximum value of J 3 can be achieved, but

requires much higher increases in rate Therefore, eliminating G-I

interactions produces undesirable effects on the control pattern.