- Email: [email protected]
Control, regulation and thermodynamics of free-energy transduction
877
Biochimie 71 (1989) 877-886 © Socirt6 de Chimie biologique/Elsevicr, Paris
Control, regulation and thermodynamics of free-energy transduction Hans V. WESTERHOFF
Section on Membrane Enzymology, Laboratory of Cell Biology, National Heart, Lung, and Blood Institute, National Institutes of Health, USA; and Netherlands Cancer Institute, H5, Division of Molecular Biology, Plesmanlaan 121, 1066 CX Amsterdam, The Netherlands (Received 14-4-1989, accepted 16-5-1989)
Summary n The quantitative formalism called Metabolic Control Theory makes it possible to be precise in discussions of metabolic control. To illustrate this, I will mention 2 experimental systems where free energy is converted from one form to another, i.e., bacteriorhodopsin liposomes and mitochondriai oxidative phosphorylation. More specifically I shall discuss how the distribution of the control of fluxes, concentrations and potentials, among the various enzymes (catalysts) in these systems has been measured and how this distribution can be understood in terms of the enzyme properties. From the outset, Metabolic Control Theory was valid for branched metabolic pathways with nonlinear kinetics. Yet, it seemed to be limited to metabolic pathways without e n z y m e - e n z y m e interactions and to steady states. It is now clear that these limitations were apparent only and recent extensions to Metabolic Control Theory deal explicitly with enzyme-enzyme interaction and with transient-time analysis. Other limitations are inherent. For instance, Metabolic Control Theory pays for its clarity and exactness by being limited to small modulations. Mosaic Non Equilibrium Thermodynamics and Biochemical System Analysis are formalisms that deal with larger changes, at the cost of accuracy and exactness. metabolic control theory / rate-limiting step / elasticity coefficient / non-equilibriumthermodynamics/ coupling / sensitivity analysis
Principles of metabolic control One of the advances brought by metabolic control theory was the more precise definition of what is meant with the statement that a certain enzyme controls a flux through a pathway. Higgins [I] proposed to define what we now [2, 31 call the flux-control coeftlcient as the effect on the pathway flux of a modulation of the enzyme activity. To eliminate the problem of dimensior '.e defined changes in relative terms, i.e., the change in flux normalized by the flux was to b ' compared to the modulation in enzyme activity normalized by the enzyme activity. He also stressed that the modulation should be small. In Metabolic Control Theory (MCT) the limit is taken to infinitely small modulations [1-27; reviewed in 3, 9, 27]. Although this limit is not
achievable experimentally, it ensures that the unperturbed system is considered, and that the control coefficient becomes independent of considerations such as the magnitude of the perturbation. An additional advantage is that the mathematics for infinitely small changes is well developed (differential calculus). Finally, the definition has the practical advantage that the flux control coefficient is the slope in the double logarithmic plot of flux (J) versus enzyme activity (e): CJe = lim ( r J / J / ( r e / e ) = ( d J / J ) / ( d e / e ) = t~e----~o
din IJI/din(e) Similarly one can define a control coefficient for the control of any variable by any parameter. Here we encounter a second point stressed by Metabolic Control Theory: it is important to
878
H. V. Westerhoff
have a clear conception of what controls what [4, 3, 11, 16]. Parameters are properties that have been set and will only be changed by actions from the outside. In "metabolic systems" (which here, by definition, have constant gene expression), examples are enzyme concentrations, Km values, Vmax values, temperature, pressure. By contrast, variables are not set from the outside but are, at steady state, functions of the parameters. Also, the variables change between steady states. Examples of variables are metabolite concentrations, membrane potentials (except in cases where the membrane potential is clamped), reaction rates and metabolic fluxes. Parameters are controllers and the variables are the controlled. A corollary is that a question such as to what extent the intracellular glycerol phosphate concentration controls lactate production is illegal: because intracellular phosphate is itself a variable, it cannot be modulated from the outside so as sensibly to ask the question what happens if we set its concentration at a slightly different value (and, importantly, keep it there). It may be clear that control theory allows many more control coefficients to be defined than the ones we have recently become used to. For instance, one may define a coefficient relating to the control of cellular volume by temperature, or another coefficient relating to the control of thermodynamic efficiency by the proton permeability coefficient of the energy coupling membrane. Why are control coefficients so useful? Well, this differs depending on how one is interested in metabolic control. One reason is perhaps that they offer a standard definition of the extent to which an enzyme is important for the magnitude of a flux or concentration. Thus, rather than having to describe in each discussion what we precisely mean by "enzyme k controls flux J to extent x", we can now simply say: "the flux control coefficient C~. is 0.37". That the meaning of this statement has its limitations (e.g., precisely because it is defined for very small changes, it does not necessarily say everything about larger modulations) is to be well understood by everyone, and not a problem of the definition per se. A second charm of the control-coefficient definition is perhaps that it is so closely related to experiment. The recipe of measurement is virtually enclosed in its definition: modulate the enzyme activity and measure the effect on the flux, trying to make the enzyme modulation as small as consistent with experimental accuracy.
The modulation of enzyme activity can obviously be achieved by adding inhibitors or activators of the enzyme [9], or by altering the concentration of the enzyme (either through addition, or by genetic means) [5, 8, 9, 27-32]. Of course, not in all experimental systems any of these methods is simple, but this is a problem that cannot and should not be solved by concepts of metabolic control. For those that like to be systematic, it is perhaps attractive that there is an end to determining flux control coefficients; there is only 100% of the control to be distributed and indeed, ~he flux control coefficients must add up to 1. Consequently, one may exercise one's bookkeeping as one determines flux control coefficients. A caveat is that flux control coefficients can be negative, such that sums of flux control coefficients can exceed 1 if negatives are omitted. Enzymes catalyzing branches competing with the main pathway under consideration tend to have negative control coefficients [10, 29]. For concentration control coefficients it is more common to be negative (enzymes for which the metabolite in question is the substrate will have a negative concentration control coefficient with respect to that metabolite) and indeed, they add up to zero rather than one [6]. Perhaps the most important advantage of the control coefficients is that their definition allows one to use additio~al laws in Metabolic Control Theory that relate c(mtroi of fiuxes~ __m_etabo!ite concentrations, etc. to the properties of the enzymes in the system. That is, one can explain the behaviour of the system as a whole on the basis (i) of the properties of its components and (ii) of the way in which the components are arranged. Metabolic Control Theory shows that the sole enzyme property that is relevant to the distribution of control is not the extent to which the reaction catalyzed by the enzyme is displaced from equilibrium, but the responsiveness (called "elasticity") of the enzyme with respect to changes in metabolite concentrations. The "elasticity coefficient" is defined as the measure of this responsiveness. It corresponds to the relative change in rate (v) of the enzyme-catalyzed reaction induced by a small standard relative change in concentration of a metabolite (M): EM--= lim ( S v / v ) / ( ~ [ M ] / [ M ] ) = . 8v---)O (o~v/ v) / (O[M] / [M]) = O(lnlvl) / aln[M] In this case, changes in rate are considered at constant concentrations of all other factors that
Comrol, regulation and thermodynamics of free-energy transduction may influence the rate (this is stressed by using the curved 0s of partial derivatives). In contrast to the definition of control coefficients, no evolution to a new steady state is considered. What is important of the way in which enzymes are arranged is their sequence in the metabolic pathway, at which points in the pathway there are branches, and which part of the flux flows through each branch. That these are indeed the properties that determine the distribution of control can be deduced from a method that can be used to calculate the control coefficients for fluxes, concentrations and flux ratios [3, 12, 15-18, 25]. We shall illustrate this for the pathway of Figure 1. The first step is to construct an n by n matrix of all ones, where n (5 in the case of Fig. 1) is the number of enzymes in the system. The second step is to assign subsequent columns of the matrix to subsequent enzymes in the pathway and (skipping the first line), subsequent lines to subsequent metabolites. At each position that corresponds to an enzyme j (column wise) and to a metabolite (row wise) k one inserts the corresponding elasticity coefficient E~,. In branched pathways the number of metabolites is less than the number of enzymes minus 1. Consequently, the bottom lines of the matrix will still contain ones. To these lines information about the branches in the metabolic pathway is now added. For all enzymes prior to the branch, a zero is inserted; for all enzymes after a branch, the part of the flux that does not flow though that branch is inserted. Thus the matrix C -~ is obtained, which is the inverse of the matrix that contains all the control coefficients:
C-1=
l1
1
k
4 4
\~
1
1
1
4)
~4
~I
e4
~4
0
Jb/J
Jb/J
Ja/J
Here ei is the elasticity coefficient of enzyme j with respect to metabolite (or metabolic variable [3] Xk. Jb is the steady-state flux through enzyme 5, whereas J~ is the flux through the upper branch.
S -
•
X2 -,
X3 -
879
X/,-
P
,
II
Fig. 1. Sample metabolic pathway. The concentrations of S and P are constant. Any metabolite Xk may influence the rate of any reaction; such an influence is reflected in a nonzero elasticity coefficient. J, J~,, and Jb are the steady state fluxes through el (and e~), e3 (and e4), and e5 respectively.
/q C=
q q
-c? -c? -q -q -q -q -q -q -q
c~ c!/ c!l
c~
-c4
-c~4 -c4
c!/
-c~ -c~ -q
ci/
\c~
Here C~ measures the control exerted by enzyme k on the main flux J. C ~ m e a s u r e s the control exerted by enzyme k on metabolite Xm. C~ measures the control exerted by enzyme k on the branch ratio j = J a / J b . The basis for this method lies in the so-called connectivity relations between control coefficients and elasticity coefficients [4, 11, 17]. In II|dlly
~ d ~
tll~
~aLlivvay
UI
lllt~lK.al.
|~ ~LL[II~|VlILI~
simple to employ these connectivity relations themselves, rather than the matrix inversion procedure. The law that connects flux control coefficients to elasticity coefficients was discovered by Kacser ar d Burns [4]" (for any metabolic variable X) J. 1 + J. 2 g xCJ-e i i = 0 = C IE x C 2E x The one that relates concentration control coefficients was discovered by Burns (Kacser, personal communication) and rediscovered by Westerho'ff and Chen [11]:
.cX-i~% = -aj~; C x., l E:~ + C x.~= 2 ex
-1
where .~)k equals 0 if X i and Xk are different metabohtes and 1 if they are identical. On the right-hand side in these equations, these laws have been written for a pathway of only 2 enzymes, e~ and e2 and the metabolite X in between. This helps to grasp their implication. The former implies that the ratio of the flux control coefficients in this case equals the inverse
H. V. Westerhoff
880
ratio of the elasticity coefficients, i.e., the more elastic enzyme exerts less of the control on the pathway flux. Taken together with the summation theorem, the latter implies that: : - c
x =
+
That is, in absolute terms the control of the enzymes over the concentration of the intermediary metabolite, diminishes if their elasticities increase.
Control of free-energy transduction Various groups have applied Metabolic Control Theory to mitochondrial oxidative phosphorylation, one of the prime examples of biological free-energy transduction. Before Metabolic Control Theory was applied, discussions of what controlled oxidative phosphorylation tended to be confusing. With respect to the role of the adenine nucleotide translocator, some would stress that titration of mitochondrial respiration with an inhibitor of that process did not lead to direct inhibition; in the titration curve there was an initial plateau phase. Consequently, the adenine nucleotide translocator could not be the ratelimiting step. Confusingly, when titrations were carded out with different inhibitors, or under different conditions, the lengths of the plateaus in the titration curves varied. Some authors pointed at the obvious control exerted by extramitochondrial A D P on the respiratory rate and suggested that the translocator, being the first step, must control respiration. Others pointed at the fact ~hat the only reaction in oxidative phosphorylation that is, in practice, irreversible is the reacaon catalyzed by cytochrome oxidase and that therefore the adenine nucleotide translocator could not be the rate-limiting step. The application of Metabolic Control Theory did a great deal to resolve these issues. First, it rationalized that titrations with different inhibitors should yield different results, depending on the type of interaction the inhibitor has with its target enzyme [9]. Second, actual determination of the control coefficients for the control of mitochondrial respiration showed that the control of the process was distributed among the participating enzymes, with, in isolated rat-liver mitochondria, ~ 0.3 of the control residing in the adenine nucleotide translocator [28, 33-36]. That the latter number was neither 0.0 nor 1.0 may serve to illustrate that, without quantitative help of the sort offered by Metabolic Control
Theory, the discussion concerning the control of oxidative phosphorylation could have gone on forever: if one can only discuss in terms of "the rate-limiting step" or "not the rate limiting step", then it will remain difficult to decide whether 0.3 is 1.0 (the former case), or 0.0 (the latter case). It was also shown that the control distribution among the enzymes varied with the work load (in terms of the amount of hexokinase) imposed on the mitochondria, with concentrations of metabolites (such as phosphate [37]) and with organism of origin of the mitochondria [37], another reason why without precise analysis, agreement would never have been obtained. The question was also asked as to why the control of respiration was distributed in the way it was. The dominant role of elasticity coefficient over distance from equilibrium in the above equations, already shows that arguments about the cytochrome oxidase reaction being far from equilibrium and therefore being the rate-limiting step, were faulty. Indeed, the control of mitochondrial respiration by citochrome oxidase at most active respiration was < 0.6 ([28, 33, 37]; cf. [38] and its discussion [3]). By measuring the relative magnitudes of the elasticity coefficients of respiration and the adenine nucleotide translocator with respect to the electrochemical potential difference for protons across the inner mitochondrial membrane, if was found that the ratio ko .~. ,. ., .~. ,. . the 2 elasticity coefficients was high enough to explain that much of the flux control resides in the adenine nucleotide translocator rather than in cytochrome oxidase [39]. In a different study it was confirmed experimentally that the degree of control by the transIocator depends strongly on the elasticity coefficient of the extramitochondrial ATP consumer with respect to the [ A T P ] / [ A D P ] ratio [40].
Regulation
versus
control
In metabolic control analyses it is important not to confuse control with regulation. If, along the lines summarized above, one has determined how the control over the flux through a metabolic pathway is distributed among its enzymes, one still does not know how the pathway flux is actually regulated [8, 20]. Flux control by an enzyme indicates to what extent the flux would change if the activity of that enzyme were
Control, regulation and thermodynamics or free-energy transduction changed. It does not by itself indicate whether or not the organism actually modulates the activ ity of that enzyme to regulate the flux. Tc discuss regulation of a metabolic flux, one needs to be precise about which question one discusses, For instance, the question "How is glycolytic flux regulated?" is too vague. A better question would read "How is glycolytic flux regulated in the Pasteur effect?" After asking a good question, one should identify possible routes of regulation, e.g., through citrate or ATP and their action on phosphofructokinase (PFK) and pyruvate kinase. One may then specify the question as "which of these routes of regulation is quantitatively the most important in the Pasteur effect?". The answer should com,: from the comparison of the control coefficients of each of these regulators for each route. The control coefficient of ATP through PFK equals the arithmetic product of the flux control coefficient of PFK and the elasticity coefficient of PFK for ATP [8]. That is, one calculates to what extent ATP can modulate the activity of PFK and multiplies this by the effect of such a modulation on the pathway flux. This control coefficient then has to be multiplied by the actual change in ATP concentration produced by the mitochondria during the Pasteur effect: (dJ /J)due l o
ATP change, along PFK regulatory route =
C'J .~PFK ~PFK -
=
_.dIATPI/IATP] ATe -t .... J"
t ....
J
This should then be calculated for every regulatory route. The largest value then refers to the most important regulatory route [16]. The determination of all these control coefficients may be quite some work. However, it is in principle straightforward and its complexity reflects the complexity of the problem of metabolic regulation itself; any simpler answer may to be too simple.
Enzyme-enzyme interaction and metabolite channeling In Metabolic Control Theory, the enzyme activities are assumed to be independent parameters [6]. That is, one can change the activity of enzyme number I without immediately affecting the activities of the other enzymes in the system (except through subsequent changes in metabolite concentrations). As a simplification, we have often discussed Metabolic Control Theory in terms of enzyme concentrations rather than
881
enzyme activities. There are important cases in which this simplification breaks down. One is where an enzyme exists in a monomer-dimer equilibrium, with the dimer having an enzyme activity that is not equal to twice the activity of the monomer (at any concentration of the relevant metabolites). A second is where 2 enzymes interact physically and influence each other's activity. Metabolic channelling is an extreme variant of this. Metabolic Control Theory formulated in terms of enzyme activities remains valid in these cases [6]: but Metabolic Control Theory formulated in terms of enzyme concentrations loses its validity. On the basis of this, Kacser et al. (personal communication) have recently developed a method that essentially formulates Metabolic Control Theory in terms of enzyme concentrations, and corrects for the effects of enzyme-enzyme interactions. This approach is useful because Metabolic Control Theory in terms of enzyme concentrations is closer to experimental control analysis than Metabolic Control Theory in terms of activities.
Larger changes It has been pointed out quite rightly that Metabolic Control Theory is limited because it does not address large modulations of metabolism, whereas in many cases, modulations are substantial [20]. indeed, the definitions iv Metabolic Control Theory are such that they only address infinitely small changes (like differential calculus does). Of course, experience is that the differential of a property tends to be virtually equal to the ratio of small changes, but the ratio of larger changes may deviate from this. Thus, one could define a primed flux control coefficient as: C' = ( ~ J / J ) / ( ~ e / e ) This flux control coefficient would be a function of ~e, i.e., the extent of modulation of e. It is important that, just like in the case of the usual control coefficient, the modulation in e is made at constant activities of the other enzymes in the system. Otherwise, one would not be sure if the change in flux is the effect of the change in the concentration of the enzyme under study, or an effect of changes in the activities of the other enzymes. Could one develop a Metabolic Control Theory for finite changes, for instance by fixing the change in enzyme activity to 50%? Would the summation and connectivity laws be valid for
882
H. V. Westerhoff
control coefficients redefined in this way? The answer is: no. At least, the proof for the summation law cannot be simply reformulated for the case of 50%-control coefficients. Precisely because in the definition only the activity of one enzyme is supposed to be modulated, it is not generally so that the response to a simultaneous 50% increase in the activities of all enzymes is given by: 0.50 --- 81nJ = ~ C'i.81ne ----Z.C'-0.50 (This relationship only holds in the limit of 8lne to 0). Hence, such ("50%"-) flux control coefficients would not generally add up to 1. For similar reasons, the connectivity theorems lose their proof if control and elasticity coefficients are redefined in terms of large changes. 1
Integral approaches "~nen changes are not extremely large, one may utilize Metabolic Control Theory to get a first order approximation to the effects of a modulation. Alternatively, one could attempt to determine the precise rate equations of all enzymes in the system and then perform a numerical solution for the steady state equations [41]. Some methods address larger changes without going all the way of determining the details of all the kinetic relations. One of these is Biochemical Systems Analysis (BSA) [42-44]. In this approach one effectively assumes the elasticity coefficients to be constant over larger variations of met~bolite concentrations. Integration then leads to a power law dependence of reaction rates on metabolite concentrations. A difference with Metabolic Control Theory used to be that Biochemical Systems Analysis described kinetic systems in terms of unidirectional reaction rates, whereas Metabolic Control Theory discusses in terms of net reaction rates. The elasticity coefficients of Metabolic Control Theory do not always correspond to the exponents in the power laws of Biochemical Systems Analysis. The power laws of Biochemical Systems Analysis have the advantage that they linearize through a logarithmic coordinate transformation. Systems of such linear equations can be readily solved and system behaviour studied. It is not clear how accurately the power laws describe actual biochemical rate equations [see however, 44]. Especially with respect to the known saturation of enzyme catalyzed reactions at high substrate concentrations, this seems problematic.
An approach that also approximates the actual rate equations, but then by piecewise linear equations, is Mosaic Non Equilibrium Thermodynamics (MNET) [3, 45]. Because of its thermodynamic origin, this approach has ~,dinlv been applied to biological free-energy transduction. The basis for the linear assumption is that most biuc~iemic~! r~act'.'.on rates are bounded by a negative and a positive maximum rate. As a consequence there must be an inflection point in the relationship between rate and free-energy difference of reaction. Around this inflection point, a linear approximation should be good. Indeed, in enzyme kinetic calculations it was shown that with a tolerance of 15% error, > 75% of the range of the reaction rate is described. Much of the rest of the rate can be described by additional linear relationships, i.e., by assuming a Vmax(or Vmi,) has been attained. The applications and theory of MNET have recently been reviewed [3], and here it may suffice to discuss 2 issues where MNET comes into close contact with Metabolic Control Theory. The first address~,s the relationships between reaction rates and free-energy differences of reaction. Until recently, the relationship between mitochondrial respiration rate and extramitochondrial phosphorylation potential was found to be linear except at low phesphorylation potentials. The enzyme kinetic considerations mentioned above suggested that the linearity of that relationship might be disrupted at high phosphorylation potentials [3, 46]. By increasing the accuracy of the determination of phosphorylation potentials, Wanders et al. were able to detect sigmoidicity in the relationship between respiration rate and phosphorylation potential at high magnitudes of the latter [47]. The question arose as to whether this sigmoidicity lies in the relationship between mitochondrial respiration and A/2n or in the flow-force relationship of the adenine nucleotide translocator. This question was addressed by measuring the variation of respiration with phosphorylation potential both in a titration with uncoupler and in a titration with hexokinase (in the presence of glucose) [48]. This method is illustrated by Figure 2. The comparison of 2 points, one out of either titration, that have the same respiratory rate (and hence, by the chemiosmotic paradigm, the same membrane potential), yields one rate of respiration (Jo) for a difference in free energy (6(8G)) across the adenine nucleotide translocator. By making these comparisons for all respira-
Control, regulation and thermodynamics of free-energy transduction tory rates, the flux through the translocator was evaluated as it varied with the free energy difference across the translocator (assuming that in the uncoupler titration the free energy difference across the translocator was essentially constant; for a more complete interpretation see [48]). The flow-force relationship exhibited sigmoidicity at high phosphorylation potentials
[481.
MNET has shown that, away from equilibrium, the flow-force relationship for an enzyme is not necessarily unique; it may depend on the way in which the force is varied [3]. In the method in Figure 2, all 3 components of the free energy difference across the translocator, i.e., the membrane potential, thc intramitochondrial phosphorylation potential and the extramitochondrial phosphorylation potential varied
Iog([ATP]/IADP])out Fig.
2. Method to determine a flow-force relationship of the adenine nucleotide translocator. Two relationships between respiration rate (J~) and extramitochondrial phosphorylation potential (which is linearly related to Iog([ATP / ADP])o,t) are shown. The full line is obtained in a titration with hexokinase, the dashed line in a titration with a protonophore. For every respiration rate one can find a difference (8(~G)) in extramitochondrial log([ATP/ADP]) between the 2 lines. Assuming that in the protonophore titration respiration varies linearly with the (small) freeenergy difference across the adenine nucleotide translocator, and because phosphorylation rate varies linearly with respiration rate in the hexokinase titration [48], curvature in the plot of respiration rate versus this B(t$G) must originate in a nonlinear flow-force relationship of the adenine nucleotide translocator in the titration with hexokinase.
883
simultaneously. To see if the sigmoidicity of the flow-force relationship of the adenine nucleotide translocator would also occur if the force was varied differently, we devised a way of varying mainly the extramitochondriai phosphorylation potential [48]. This is illustrated in Figure 3. The principle is to inhibit only some of the adenine nucleotide translocators by adding the tight binding inhibitor carboxyatractyloside and t h e n returning the respiratory rate to the rate in the absence of carboxyatractyloside by adding hexokinase. This gives one data point with a rate of respiration and an extramitochondrial phosphorylation potential. Additional data points are obtained by adding va~ing amounts of carboxyatractyloside and hexolonase, but such that the respiratory rate comes out the same. The data points (the circles in Fig. 3) then
tog(IATPIIIADPI)o t Fig. 3. Method to determine the flow-force relationship across the adenine nucteotide translocator at constant membrane potential and virtually constant intramitochondrial phosphorylation potential. Starting with a certain amount of hexokinase, hence at a certain respiration rate, carboxyatractyloside (full arrow) and hexokinase (dashed arrow) are added such that the respiration rate remains the same. This leads to the data points indicated by the circles. From the amount of carboxyatractyloside added, one can calculate the fraction of adenine nucleotide transiocators that are still active. By dividing the phosphorylation rate by that fraction and plotting the results as a function of Iog([ATP / ADP])o,t one obtains a flow-force relationship of the adenine nucleotide transloeator at constant membrane potential (because respiration rate is constant) and virtually constant intramitochondrial phosphorylation potential (to the extent that the H+-ATPase remains close to static head).
$84
H. V. Westerhoff
have the same respiratory rate and hence, by the paradigm of the chemiosmotic coupling hypothesis, the same membrane potential. However, they differ in extramitochondrial phosphorylation potential and in turnover rate of the uninhibited adenine nucleotide translocators. The latter is obtained by dividing the total transport rate of ATP (which is equated to the rate of oxidative phosphorylation) by the fraction of uninhibited translocators, 1-1/ Imax (lma x is defined as the amount of inhibitor necessary to completely inhibit all translocators supposing that binding is irreversible; it is equal to the total amount of adenine nucleotide translocators). Plotting this turnover rate of uninhibited translocators versus the extramitochondrial phosphorylation potential, we again observed a sigmoidal flow-force relationship for the adenine nucleotide translocator [48]. We concluded that the flow-force relationship of the adenine nucleotide translocator can exhibit sigmoidicity (nonlinearity) at high extramitochondrial phosphorylation potentials.This occurred when the membrane potential was held constant (the experiment along the lines of Fig. 3), and in the actual variation of the force in a case of varying work load (the experiment along the lines of Fig. 2). Because in a titration with 2,4-dinitrophenol (a protonophore) in the absence of hexokinase, the relationship between mitochondriai respiration and extramitochondrial phosphorylation potential lacked sigmoidicity, this suggests that the sigmoidicity in the relationship between respiration and extramitoehondrial phosphorylation potential under conditions of varying extramitochondrial ATP consumption is due to the sigmoidicity in the flow-force relationship of the adenine nucleotide translocator. As we discussed [48], this conclusion is somewhat, though not quite, dependent on the validity of the delocalized chemiosmotic coupling concept [cf., 49 l. The finding of this sigmoid flow-force relationship not only underlined the importance of using MNET rather than the older types of NET, it also provided an explanation for why at intermediary work loads some of the flux control is still in extramitochondrial ATP consuming processes. The sigmoidicity of the flow-force relationship of the translocator increases the (absolute) elasticity coefficient of the latter at intermediary work loads and hence shifts its flux control to the ATP consuming processes [3]. To develop and check both MNET and Metabolic Control Theory, we have experimented
m', "h in a reasonably well defined reconstituted system of biological free energy transduction, i.e., bacteriorhodopsin liposomes [3, 29]. Here the input free energy, contained in photons of high intensity, is converted to an electrochemical potential difference for protons by the photondriven proton-pump bacteriorhodopsin. The membrane potential across the liposomal membrane can be made to drive K + extrusion by the addition of the K+-specific ionophore valinomycin. For electroneutrality reasons net K + extrusion is equal to net proton uptake, which can be assayed with a pH electrode. We measured the control of net proton uptake (just after the light was switched on, i.e., at zero pH gradient) and found it to be distributed among bacteriorhodopsin and valinomycin. Surprisingly, the flux control of either was close to one, leading to a sum well exceeding 1. That this was not inconsistent with the summation theorem for flux control coefficients, which prescribes a sum of 1, was shown when the flux control coefficients of the proton permeability of the liposomal membrane was found to be close to - 1. Earlier we had developed a complete MNET description of free energy transduction in bacteriorhodopsin liposomes [45, 50]. An MNET description has the advantage that it is valid, to a good approximation, for large modulations of enzyme activities. Metabolic Control Theory is strictly valid, but only for infinitely small changes. However, in the limit to infinitely small changes, also MNET becomes strictly valid. We demonstrated this for the case of bacteriorhodopsin liposomes by deriving expressions for the control coefficients of net proton uptake flux by bacteriorhodopsin, proton permeability and K ÷ permeability, both by using Metabolic Control Theory and by using MNET (which leads to an integral expression for the proton flux in terms of these three activities; the partial derivatives of this expression with respect to those three activities give the flux control coefficients). The two routes led to the same results for the flux controls with respect to the activity of bacteriorhodopsin, the non-proton ion permeability and the proton permeability respectively [3]:
C~ =(Li + L~)/(Li + L~ + n~L.) C.I = (I ,~' + n~L~)/(L i + LHI + n~L~) C~ = - L ~ / ( L i + LH~ + n2L.)
Control, regulation and thermodynamics o f free-energy transduction
Here L,, L i and L~t correspond to the activity of bacteriorhodopsin, valinomycin, and to proton permeability respectively, n to the number of protons pumped per absorbed photon by bacterio-rhodopsin. 3' measures the back-pressure sensitivity of bacteriorhodopsin to the electrochemical potential difference for protons. 3/is zero if bacteriorhodopsin is not sensitive to back pressure, and 1 if its completely sensitive [3]. That in the limit to infinitely small changes, MNET and Metabolic Control Theory lead to the same result, may be interpreted as that Metabolic Control Theory is a subset of MNET. However, this is not a fair representation of the situation. The fact is that the summation and connectivity theorems are not naturally present in MNET and that Metabolic Control Theory is indeed a much more efficient theory to attain the description of the effects of infinitely small changes. For the suggestion that Metabolic Control Theory may be a subset of Biochemical Systems Analysis, the latter arguments also hold. In this case however, Metabolic Control Theory is certainly not a subset of Biochemical Systems Analysis. For, Biochemical Systems Analysis describes systems in terms of unidirectional rather than net fluxes (which is the way MNET and Metabolic Control Theory describe systems). Only recently it has been shown that Biochemical Systems Analysis can also be formulated in terms of net fluxes [51]. Concluding remarks When discussing control and regulation it is important to rigidly define what one is talking about: - C o n t r o l tends to be distributed and condition dependent; -Elasticity of enzymes rather than their distance from equilibrium determines the distribution of the control; -Metabolic Control Theory is not limited to linear pathways; -Metabolic Control Theory is tailored to small changes; - A l s o metabolic regulation can be studied by Metabolic Control Analysis. Acknowledgments 1 thank Jos Arents, Bert Groen and Ron Wanders for the collaborations on this topic, and Drs. C. and
885
E. Kohen for discussions. Ron Wanders devised the method illustrated in Figure 2 [48]. References 1 Higgins J.J. (1965) in: ControlofEnergy Metabolism (B. Chance, R.W. Estabrook and J.R. Williamson, eds.), Academic Press, New York, pp. 13-46 2 Burns J.A., Cornish-Bowden A., Groen A.K., Heinrich R., Kacser H., Porteous J.W., Rapoport S.M., Rapoport T.A., Stueki J.W., Tager J.M., Wanders R.J.A. & Westerhoff H.V. (1985) Trends Biochem. Sci. 10, 16 3 Westerhoff H.V. & Van Dam K. (1987) "tllermodynamics and Control of Biological Free-Energy Transduction. Elsevier, Amsterdam 4 Kacser H. & Burns J. (1973) in: Rate Control of Biological Processes (D.D. Davies, ed.), Cambridge University Press, London, pp. 65-104 5 Rapoport T.A. & Heinrich R. (1975) BioSystems 7,120-129 6 Heinrich R., Rapoport S.M. & Rapoport T.A. (1977) Progr. Biophys. Mol. Biol. 32, 1-83 7 Kohn M.C., Whitley L.M. & Garfinkel D. (1979) J. Theor. Biol. 76, 437-452 8 Kacser H. & Burns J.A. (1979) Biochem. Soc. Trans. 7, 1149-1161 9 Groen A.K., Van der Meet R., Westerhoff H.V., Wanders R.J.A., Akerboom T.P.M. & Tager J.M. (1982) in: Metabolic Compartmentation (H. Sies, ed.), Academic Press, New York, pp. 9-37 10 Kacser H. (1983) Biochem. Soc. Trans. 11, 35 -40 11 Westerhoff H.V. & Chen Y. (1984) Eur. J. Biochem. 142,425-430 12 Fell. D.A. & Sauro H.M. (1985) Eur. J. Biochem. 148, 555-561 13 }tofmcyr J.-lI.S., Kacscr H. & Van der Merwe K.J. (1986) Ear. J. Biochem. 155,631-641 14 Sorribas A. & Bartrons R. (1986) Eur. J. Biochem. 158, 107-115 15 Hofmeyr J.-H. (1c~86) Studies in steaay state modelling and co/itrol analysis of metabol:c systems. Ph.D. Thesis, University of Stellenbosch 16 Kell D.B. & Westerhoff H.V. (1986) FEMS Microbiol. Rev. 39, 305-320 17 Westerhoff H.V. & Keli D.B. (1987) Biotechnol. Bioengin. 30, 101-107 18 Sauro H.M., Small J.R. & Fell D.A. (1987) Eur. J. Biochem. 165,215-221 19 Westerhoff H.V., Plomp P.J.A.M., Groen A.K. & Wanders R.J.A. (1987) Cell Biophys. 10, 239-267 20 Crabtree B. & Newsholme E.A. (1987) Trends Biochem. Sci. 12, 4-12 21 Crabtree B. & Newsholme E.A. (1987) Biochem. J. 247, 113-120 22 Giersch C. (1988) Eur. J. Biochem. 174,509-513
886
H. V. Westerhoff
23 Giersch C. (1988) Eur. J. Biochem. 174,515-517 24 Reder C. (1988)J. Theor. Biol. 135, 175-202 25 Small J.R. & Fell D.A. (1989) J. Theor. Biol. 136, 181-197 26 Giersch C. (1988) J. Theor. Biol. 134, 451-462 27 Westerhoff H.V. (1989) (ed.) Principles of Metabolic Control. Plenum Press, New York, (in press) 28 Groen A.K., Wanders R.J.A., WesterhoffH.V., Van der Meer R. & Tager J.M. (1982) J. Biol. Chem. 257, 2754-2757 29 Westerhoff H V . & Arents J.C. (1984) Biosci. Rep. 4, 23-31 30 Torres N.V., Mateo F., Melendez-Hevia E. & Kacser H. (1986) Biochem. J. 234, 169-174 31 Rapoport T.A., Heinrich R. & Rapoport S.M. (1974) Eur. J. Biochem. 42,107-120 32 Nimmo H.G. & Cohen P.T.W. (1987) 247, 1 - 3 33 Gellerich F.N., Bohnensack R. & Kunz W. (1983) Biochim. Biophys. Acta 722, 381-391 34 Duszynski J., Groen A.K., Wanders R.J.A., Vervoorn R.C. & Tager J.M. (1982) FEBS Lett. 146, 263-266 35 Holness M., Crespo-Armas A. & Mowbray J. (1984) FEBS Lett. 177, 231-235 36 Moreno-Sanchez R. (1985) J. Biol. Chem. 260, 12554-12560 37 Mazat J.-P., Bart E.J.-B., Rigoulet M. & GuErin B. (1986) Biochim. Biophys. Acta 849, 7 - 1 5 38 Forman N. & Wilson D.F. (1983) J. Biol. Chem. 258, 8649-8655 39 Westerhoff H.V., Plomp P.J.A.M., Groen A.K., Wanders R.J.A., Bode J.A. & Van Dam K . (1987) Arch. Biochem. Biophys. 257, 154-169 40 Wander~ R I A , ~,rn,-n A v Van ~ - mund C.W.T. & Tager J.M. (1984) Eur. J. Biochem. 142, 417-424 41 Kohn M.C. & Gartinkel D. (1983) Ann. Biomed. Engin. 11, 511 42 Savageau M.A. (1972) Curt. Top. Cell. Reg. 6, 63-130 .... 43 Savageau M.C. (1976) Biochemical systems analysis; a study of function and design, in." Molecular Biology. Addison-Wesley, Reading, MA 44 Voit E.O. & Savageau M.A. (1987) Biochemistry 26, 6869-6880 45 Westerhoff H.V., Scholte B.J. & Hellingwerf K.J. (1979) Biochim. Biophys. Acta 547, 544-560
46 Van der Meer R., Westerhoff H.V. & Van Dam K. (1980) Biochim. Biophys. Acta 591, 488-493 4"7 Wanders R.J.A., Groen A.K., Meijer A.J. & Tager J.M. (1981) FEBS Lett. 132,201-206 48 Wanders R.J.A. & Westerhoff H.V. (1988) Biochemistry 27, 7832-7840 49 Westerhoff H.V., Kell D.B., Kamp F. & Van Dam K. (1988) in: Microcompartmentation (D.P. Jones, ed.), CRC Press, Boca Raton, Florida, USA, pp. 116-154 50 Westerhoff H.V., Hellingwerf K.J., Arents J.C., Scholte B.J. & Van Dam K. (1981) Proc. Natl. Acad. Sci. USA 78, 3554-3558 51 Cornish-Bowden A. (ed.) (1989) Control of Metabolic Processes. Plenum Press, New York (in press)
Appendix A. Summation theorems for control coefficients How can one predict whether for a given variable the summation theorem should amount to 0, 1, or perhaps to some other value? There is a rather general approach to this problem [3, 22], which is similar to the use of Euler's theorem in equilibrium thermodynamics_ For t h i ~ , n n p n , ~ , ~ r l c * . o . . o ; n . . . . . k ~ . L ~ when the system is at steady state and simultaneously all enzyme activities are increased by a factor f, the system remains at steady state. If so, and the magnitude of the property under consideration increases by the factor fn, then the sum of all control coefficients by all the enzymes on that property must equal n. For metabolite concentrations and potentials n equals 0, for fluxes n equals 1. The proof is found by taking the derivative of the logarithm of the following equation with respect to the logarithm of f, and then taking the limit of f to 1 [3, 22]: X(f'el,f'% ..... f'em) = fn'X(et, e2, %) For transient times, n equals - 1.
Biochimie 71 (1989) 877-886 © Socirt6 de Chimie biologique/Elsevicr, Paris
Control, regulation and thermodynamics of free-energy transduction Hans V. WESTERHOFF
Section on Membrane Enzymology, Laboratory of Cell Biology, National Heart, Lung, and Blood Institute, National Institutes of Health, USA; and Netherlands Cancer Institute, H5, Division of Molecular Biology, Plesmanlaan 121, 1066 CX Amsterdam, The Netherlands (Received 14-4-1989, accepted 16-5-1989)
Summary n The quantitative formalism called Metabolic Control Theory makes it possible to be precise in discussions of metabolic control. To illustrate this, I will mention 2 experimental systems where free energy is converted from one form to another, i.e., bacteriorhodopsin liposomes and mitochondriai oxidative phosphorylation. More specifically I shall discuss how the distribution of the control of fluxes, concentrations and potentials, among the various enzymes (catalysts) in these systems has been measured and how this distribution can be understood in terms of the enzyme properties. From the outset, Metabolic Control Theory was valid for branched metabolic pathways with nonlinear kinetics. Yet, it seemed to be limited to metabolic pathways without e n z y m e - e n z y m e interactions and to steady states. It is now clear that these limitations were apparent only and recent extensions to Metabolic Control Theory deal explicitly with enzyme-enzyme interaction and with transient-time analysis. Other limitations are inherent. For instance, Metabolic Control Theory pays for its clarity and exactness by being limited to small modulations. Mosaic Non Equilibrium Thermodynamics and Biochemical System Analysis are formalisms that deal with larger changes, at the cost of accuracy and exactness. metabolic control theory / rate-limiting step / elasticity coefficient / non-equilibriumthermodynamics/ coupling / sensitivity analysis
Principles of metabolic control One of the advances brought by metabolic control theory was the more precise definition of what is meant with the statement that a certain enzyme controls a flux through a pathway. Higgins [I] proposed to define what we now [2, 31 call the flux-control coeftlcient as the effect on the pathway flux of a modulation of the enzyme activity. To eliminate the problem of dimensior '.e defined changes in relative terms, i.e., the change in flux normalized by the flux was to b ' compared to the modulation in enzyme activity normalized by the enzyme activity. He also stressed that the modulation should be small. In Metabolic Control Theory (MCT) the limit is taken to infinitely small modulations [1-27; reviewed in 3, 9, 27]. Although this limit is not
achievable experimentally, it ensures that the unperturbed system is considered, and that the control coefficient becomes independent of considerations such as the magnitude of the perturbation. An additional advantage is that the mathematics for infinitely small changes is well developed (differential calculus). Finally, the definition has the practical advantage that the flux control coefficient is the slope in the double logarithmic plot of flux (J) versus enzyme activity (e): CJe = lim ( r J / J / ( r e / e ) = ( d J / J ) / ( d e / e ) = t~e----~o
din IJI/din(e) Similarly one can define a control coefficient for the control of any variable by any parameter. Here we encounter a second point stressed by Metabolic Control Theory: it is important to
878
H. V. Westerhoff
have a clear conception of what controls what [4, 3, 11, 16]. Parameters are properties that have been set and will only be changed by actions from the outside. In "metabolic systems" (which here, by definition, have constant gene expression), examples are enzyme concentrations, Km values, Vmax values, temperature, pressure. By contrast, variables are not set from the outside but are, at steady state, functions of the parameters. Also, the variables change between steady states. Examples of variables are metabolite concentrations, membrane potentials (except in cases where the membrane potential is clamped), reaction rates and metabolic fluxes. Parameters are controllers and the variables are the controlled. A corollary is that a question such as to what extent the intracellular glycerol phosphate concentration controls lactate production is illegal: because intracellular phosphate is itself a variable, it cannot be modulated from the outside so as sensibly to ask the question what happens if we set its concentration at a slightly different value (and, importantly, keep it there). It may be clear that control theory allows many more control coefficients to be defined than the ones we have recently become used to. For instance, one may define a coefficient relating to the control of cellular volume by temperature, or another coefficient relating to the control of thermodynamic efficiency by the proton permeability coefficient of the energy coupling membrane. Why are control coefficients so useful? Well, this differs depending on how one is interested in metabolic control. One reason is perhaps that they offer a standard definition of the extent to which an enzyme is important for the magnitude of a flux or concentration. Thus, rather than having to describe in each discussion what we precisely mean by "enzyme k controls flux J to extent x", we can now simply say: "the flux control coefficient C~. is 0.37". That the meaning of this statement has its limitations (e.g., precisely because it is defined for very small changes, it does not necessarily say everything about larger modulations) is to be well understood by everyone, and not a problem of the definition per se. A second charm of the control-coefficient definition is perhaps that it is so closely related to experiment. The recipe of measurement is virtually enclosed in its definition: modulate the enzyme activity and measure the effect on the flux, trying to make the enzyme modulation as small as consistent with experimental accuracy.
The modulation of enzyme activity can obviously be achieved by adding inhibitors or activators of the enzyme [9], or by altering the concentration of the enzyme (either through addition, or by genetic means) [5, 8, 9, 27-32]. Of course, not in all experimental systems any of these methods is simple, but this is a problem that cannot and should not be solved by concepts of metabolic control. For those that like to be systematic, it is perhaps attractive that there is an end to determining flux control coefficients; there is only 100% of the control to be distributed and indeed, ~he flux control coefficients must add up to 1. Consequently, one may exercise one's bookkeeping as one determines flux control coefficients. A caveat is that flux control coefficients can be negative, such that sums of flux control coefficients can exceed 1 if negatives are omitted. Enzymes catalyzing branches competing with the main pathway under consideration tend to have negative control coefficients [10, 29]. For concentration control coefficients it is more common to be negative (enzymes for which the metabolite in question is the substrate will have a negative concentration control coefficient with respect to that metabolite) and indeed, they add up to zero rather than one [6]. Perhaps the most important advantage of the control coefficients is that their definition allows one to use additio~al laws in Metabolic Control Theory that relate c(mtroi of fiuxes~ __m_etabo!ite concentrations, etc. to the properties of the enzymes in the system. That is, one can explain the behaviour of the system as a whole on the basis (i) of the properties of its components and (ii) of the way in which the components are arranged. Metabolic Control Theory shows that the sole enzyme property that is relevant to the distribution of control is not the extent to which the reaction catalyzed by the enzyme is displaced from equilibrium, but the responsiveness (called "elasticity") of the enzyme with respect to changes in metabolite concentrations. The "elasticity coefficient" is defined as the measure of this responsiveness. It corresponds to the relative change in rate (v) of the enzyme-catalyzed reaction induced by a small standard relative change in concentration of a metabolite (M): EM--= lim ( S v / v ) / ( ~ [ M ] / [ M ] ) = . 8v---)O (o~v/ v) / (O[M] / [M]) = O(lnlvl) / aln[M] In this case, changes in rate are considered at constant concentrations of all other factors that
Comrol, regulation and thermodynamics of free-energy transduction may influence the rate (this is stressed by using the curved 0s of partial derivatives). In contrast to the definition of control coefficients, no evolution to a new steady state is considered. What is important of the way in which enzymes are arranged is their sequence in the metabolic pathway, at which points in the pathway there are branches, and which part of the flux flows through each branch. That these are indeed the properties that determine the distribution of control can be deduced from a method that can be used to calculate the control coefficients for fluxes, concentrations and flux ratios [3, 12, 15-18, 25]. We shall illustrate this for the pathway of Figure 1. The first step is to construct an n by n matrix of all ones, where n (5 in the case of Fig. 1) is the number of enzymes in the system. The second step is to assign subsequent columns of the matrix to subsequent enzymes in the pathway and (skipping the first line), subsequent lines to subsequent metabolites. At each position that corresponds to an enzyme j (column wise) and to a metabolite (row wise) k one inserts the corresponding elasticity coefficient E~,. In branched pathways the number of metabolites is less than the number of enzymes minus 1. Consequently, the bottom lines of the matrix will still contain ones. To these lines information about the branches in the metabolic pathway is now added. For all enzymes prior to the branch, a zero is inserted; for all enzymes after a branch, the part of the flux that does not flow though that branch is inserted. Thus the matrix C -~ is obtained, which is the inverse of the matrix that contains all the control coefficients:
C-1=
l1
1
k
4 4
\~
1
1
1
4)
~4
~I
e4
~4
0
Jb/J
Jb/J
Ja/J
Here ei is the elasticity coefficient of enzyme j with respect to metabolite (or metabolic variable [3] Xk. Jb is the steady-state flux through enzyme 5, whereas J~ is the flux through the upper branch.
S -
•
X2 -,
X3 -
879
X/,-
P
,
II
Fig. 1. Sample metabolic pathway. The concentrations of S and P are constant. Any metabolite Xk may influence the rate of any reaction; such an influence is reflected in a nonzero elasticity coefficient. J, J~,, and Jb are the steady state fluxes through el (and e~), e3 (and e4), and e5 respectively.
/q C=
q q
-c? -c? -q -q -q -q -q -q -q
c~ c!/ c!l
c~
-c4
-c~4 -c4
c!/
-c~ -c~ -q
ci/
\c~
Here C~ measures the control exerted by enzyme k on the main flux J. C ~ m e a s u r e s the control exerted by enzyme k on metabolite Xm. C~ measures the control exerted by enzyme k on the branch ratio j = J a / J b . The basis for this method lies in the so-called connectivity relations between control coefficients and elasticity coefficients [4, 11, 17]. In II|dlly
~ d ~
tll~
~aLlivvay
UI
lllt~lK.al.
|~ ~LL[II~|VlILI~
simple to employ these connectivity relations themselves, rather than the matrix inversion procedure. The law that connects flux control coefficients to elasticity coefficients was discovered by Kacser ar d Burns [4]" (for any metabolic variable X) J. 1 + J. 2 g xCJ-e i i = 0 = C IE x C 2E x The one that relates concentration control coefficients was discovered by Burns (Kacser, personal communication) and rediscovered by Westerho'ff and Chen [11]:
.cX-i~% = -aj~; C x., l E:~ + C x.~= 2 ex
-1
where .~)k equals 0 if X i and Xk are different metabohtes and 1 if they are identical. On the right-hand side in these equations, these laws have been written for a pathway of only 2 enzymes, e~ and e2 and the metabolite X in between. This helps to grasp their implication. The former implies that the ratio of the flux control coefficients in this case equals the inverse
H. V. Westerhoff
880
ratio of the elasticity coefficients, i.e., the more elastic enzyme exerts less of the control on the pathway flux. Taken together with the summation theorem, the latter implies that: : - c
x =
+
That is, in absolute terms the control of the enzymes over the concentration of the intermediary metabolite, diminishes if their elasticities increase.
Control of free-energy transduction Various groups have applied Metabolic Control Theory to mitochondrial oxidative phosphorylation, one of the prime examples of biological free-energy transduction. Before Metabolic Control Theory was applied, discussions of what controlled oxidative phosphorylation tended to be confusing. With respect to the role of the adenine nucleotide translocator, some would stress that titration of mitochondrial respiration with an inhibitor of that process did not lead to direct inhibition; in the titration curve there was an initial plateau phase. Consequently, the adenine nucleotide translocator could not be the ratelimiting step. Confusingly, when titrations were carded out with different inhibitors, or under different conditions, the lengths of the plateaus in the titration curves varied. Some authors pointed at the obvious control exerted by extramitochondrial A D P on the respiratory rate and suggested that the translocator, being the first step, must control respiration. Others pointed at the fact ~hat the only reaction in oxidative phosphorylation that is, in practice, irreversible is the reacaon catalyzed by cytochrome oxidase and that therefore the adenine nucleotide translocator could not be the rate-limiting step. The application of Metabolic Control Theory did a great deal to resolve these issues. First, it rationalized that titrations with different inhibitors should yield different results, depending on the type of interaction the inhibitor has with its target enzyme [9]. Second, actual determination of the control coefficients for the control of mitochondrial respiration showed that the control of the process was distributed among the participating enzymes, with, in isolated rat-liver mitochondria, ~ 0.3 of the control residing in the adenine nucleotide translocator [28, 33-36]. That the latter number was neither 0.0 nor 1.0 may serve to illustrate that, without quantitative help of the sort offered by Metabolic Control
Theory, the discussion concerning the control of oxidative phosphorylation could have gone on forever: if one can only discuss in terms of "the rate-limiting step" or "not the rate limiting step", then it will remain difficult to decide whether 0.3 is 1.0 (the former case), or 0.0 (the latter case). It was also shown that the control distribution among the enzymes varied with the work load (in terms of the amount of hexokinase) imposed on the mitochondria, with concentrations of metabolites (such as phosphate [37]) and with organism of origin of the mitochondria [37], another reason why without precise analysis, agreement would never have been obtained. The question was also asked as to why the control of respiration was distributed in the way it was. The dominant role of elasticity coefficient over distance from equilibrium in the above equations, already shows that arguments about the cytochrome oxidase reaction being far from equilibrium and therefore being the rate-limiting step, were faulty. Indeed, the control of mitochondrial respiration by citochrome oxidase at most active respiration was < 0.6 ([28, 33, 37]; cf. [38] and its discussion [3]). By measuring the relative magnitudes of the elasticity coefficients of respiration and the adenine nucleotide translocator with respect to the electrochemical potential difference for protons across the inner mitochondrial membrane, if was found that the ratio ko .~. ,. ., .~. ,. . the 2 elasticity coefficients was high enough to explain that much of the flux control resides in the adenine nucleotide translocator rather than in cytochrome oxidase [39]. In a different study it was confirmed experimentally that the degree of control by the transIocator depends strongly on the elasticity coefficient of the extramitochondrial ATP consumer with respect to the [ A T P ] / [ A D P ] ratio [40].
Regulation
versus
control
In metabolic control analyses it is important not to confuse control with regulation. If, along the lines summarized above, one has determined how the control over the flux through a metabolic pathway is distributed among its enzymes, one still does not know how the pathway flux is actually regulated [8, 20]. Flux control by an enzyme indicates to what extent the flux would change if the activity of that enzyme were
Control, regulation and thermodynamics or free-energy transduction changed. It does not by itself indicate whether or not the organism actually modulates the activ ity of that enzyme to regulate the flux. Tc discuss regulation of a metabolic flux, one needs to be precise about which question one discusses, For instance, the question "How is glycolytic flux regulated?" is too vague. A better question would read "How is glycolytic flux regulated in the Pasteur effect?" After asking a good question, one should identify possible routes of regulation, e.g., through citrate or ATP and their action on phosphofructokinase (PFK) and pyruvate kinase. One may then specify the question as "which of these routes of regulation is quantitatively the most important in the Pasteur effect?". The answer should com,: from the comparison of the control coefficients of each of these regulators for each route. The control coefficient of ATP through PFK equals the arithmetic product of the flux control coefficient of PFK and the elasticity coefficient of PFK for ATP [8]. That is, one calculates to what extent ATP can modulate the activity of PFK and multiplies this by the effect of such a modulation on the pathway flux. This control coefficient then has to be multiplied by the actual change in ATP concentration produced by the mitochondria during the Pasteur effect: (dJ /J)due l o
ATP change, along PFK regulatory route =
C'J .~PFK ~PFK -
=
_.dIATPI/IATP] ATe -t .... J"
t ....
J
This should then be calculated for every regulatory route. The largest value then refers to the most important regulatory route [16]. The determination of all these control coefficients may be quite some work. However, it is in principle straightforward and its complexity reflects the complexity of the problem of metabolic regulation itself; any simpler answer may to be too simple.
Enzyme-enzyme interaction and metabolite channeling In Metabolic Control Theory, the enzyme activities are assumed to be independent parameters [6]. That is, one can change the activity of enzyme number I without immediately affecting the activities of the other enzymes in the system (except through subsequent changes in metabolite concentrations). As a simplification, we have often discussed Metabolic Control Theory in terms of enzyme concentrations rather than
881
enzyme activities. There are important cases in which this simplification breaks down. One is where an enzyme exists in a monomer-dimer equilibrium, with the dimer having an enzyme activity that is not equal to twice the activity of the monomer (at any concentration of the relevant metabolites). A second is where 2 enzymes interact physically and influence each other's activity. Metabolic channelling is an extreme variant of this. Metabolic Control Theory formulated in terms of enzyme activities remains valid in these cases [6]: but Metabolic Control Theory formulated in terms of enzyme concentrations loses its validity. On the basis of this, Kacser et al. (personal communication) have recently developed a method that essentially formulates Metabolic Control Theory in terms of enzyme concentrations, and corrects for the effects of enzyme-enzyme interactions. This approach is useful because Metabolic Control Theory in terms of enzyme concentrations is closer to experimental control analysis than Metabolic Control Theory in terms of activities.
Larger changes It has been pointed out quite rightly that Metabolic Control Theory is limited because it does not address large modulations of metabolism, whereas in many cases, modulations are substantial [20]. indeed, the definitions iv Metabolic Control Theory are such that they only address infinitely small changes (like differential calculus does). Of course, experience is that the differential of a property tends to be virtually equal to the ratio of small changes, but the ratio of larger changes may deviate from this. Thus, one could define a primed flux control coefficient as: C' = ( ~ J / J ) / ( ~ e / e ) This flux control coefficient would be a function of ~e, i.e., the extent of modulation of e. It is important that, just like in the case of the usual control coefficient, the modulation in e is made at constant activities of the other enzymes in the system. Otherwise, one would not be sure if the change in flux is the effect of the change in the concentration of the enzyme under study, or an effect of changes in the activities of the other enzymes. Could one develop a Metabolic Control Theory for finite changes, for instance by fixing the change in enzyme activity to 50%? Would the summation and connectivity laws be valid for
882
H. V. Westerhoff
control coefficients redefined in this way? The answer is: no. At least, the proof for the summation law cannot be simply reformulated for the case of 50%-control coefficients. Precisely because in the definition only the activity of one enzyme is supposed to be modulated, it is not generally so that the response to a simultaneous 50% increase in the activities of all enzymes is given by: 0.50 --- 81nJ = ~ C'i.81ne ----Z.C'-0.50 (This relationship only holds in the limit of 8lne to 0). Hence, such ("50%"-) flux control coefficients would not generally add up to 1. For similar reasons, the connectivity theorems lose their proof if control and elasticity coefficients are redefined in terms of large changes. 1
Integral approaches "~nen changes are not extremely large, one may utilize Metabolic Control Theory to get a first order approximation to the effects of a modulation. Alternatively, one could attempt to determine the precise rate equations of all enzymes in the system and then perform a numerical solution for the steady state equations [41]. Some methods address larger changes without going all the way of determining the details of all the kinetic relations. One of these is Biochemical Systems Analysis (BSA) [42-44]. In this approach one effectively assumes the elasticity coefficients to be constant over larger variations of met~bolite concentrations. Integration then leads to a power law dependence of reaction rates on metabolite concentrations. A difference with Metabolic Control Theory used to be that Biochemical Systems Analysis described kinetic systems in terms of unidirectional reaction rates, whereas Metabolic Control Theory discusses in terms of net reaction rates. The elasticity coefficients of Metabolic Control Theory do not always correspond to the exponents in the power laws of Biochemical Systems Analysis. The power laws of Biochemical Systems Analysis have the advantage that they linearize through a logarithmic coordinate transformation. Systems of such linear equations can be readily solved and system behaviour studied. It is not clear how accurately the power laws describe actual biochemical rate equations [see however, 44]. Especially with respect to the known saturation of enzyme catalyzed reactions at high substrate concentrations, this seems problematic.
An approach that also approximates the actual rate equations, but then by piecewise linear equations, is Mosaic Non Equilibrium Thermodynamics (MNET) [3, 45]. Because of its thermodynamic origin, this approach has ~,dinlv been applied to biological free-energy transduction. The basis for the linear assumption is that most biuc~iemic~! r~act'.'.on rates are bounded by a negative and a positive maximum rate. As a consequence there must be an inflection point in the relationship between rate and free-energy difference of reaction. Around this inflection point, a linear approximation should be good. Indeed, in enzyme kinetic calculations it was shown that with a tolerance of 15% error, > 75% of the range of the reaction rate is described. Much of the rest of the rate can be described by additional linear relationships, i.e., by assuming a Vmax(or Vmi,) has been attained. The applications and theory of MNET have recently been reviewed [3], and here it may suffice to discuss 2 issues where MNET comes into close contact with Metabolic Control Theory. The first address~,s the relationships between reaction rates and free-energy differences of reaction. Until recently, the relationship between mitochondrial respiration rate and extramitochondrial phosphorylation potential was found to be linear except at low phesphorylation potentials. The enzyme kinetic considerations mentioned above suggested that the linearity of that relationship might be disrupted at high phosphorylation potentials [3, 46]. By increasing the accuracy of the determination of phosphorylation potentials, Wanders et al. were able to detect sigmoidicity in the relationship between respiration rate and phosphorylation potential at high magnitudes of the latter [47]. The question arose as to whether this sigmoidicity lies in the relationship between mitochondrial respiration and A/2n or in the flow-force relationship of the adenine nucleotide translocator. This question was addressed by measuring the variation of respiration with phosphorylation potential both in a titration with uncoupler and in a titration with hexokinase (in the presence of glucose) [48]. This method is illustrated by Figure 2. The comparison of 2 points, one out of either titration, that have the same respiratory rate (and hence, by the chemiosmotic paradigm, the same membrane potential), yields one rate of respiration (Jo) for a difference in free energy (6(8G)) across the adenine nucleotide translocator. By making these comparisons for all respira-
Control, regulation and thermodynamics of free-energy transduction tory rates, the flux through the translocator was evaluated as it varied with the free energy difference across the translocator (assuming that in the uncoupler titration the free energy difference across the translocator was essentially constant; for a more complete interpretation see [48]). The flow-force relationship exhibited sigmoidicity at high phosphorylation potentials
[481.
MNET has shown that, away from equilibrium, the flow-force relationship for an enzyme is not necessarily unique; it may depend on the way in which the force is varied [3]. In the method in Figure 2, all 3 components of the free energy difference across the translocator, i.e., the membrane potential, thc intramitochondrial phosphorylation potential and the extramitochondrial phosphorylation potential varied
Iog([ATP]/IADP])out Fig.
2. Method to determine a flow-force relationship of the adenine nucleotide translocator. Two relationships between respiration rate (J~) and extramitochondrial phosphorylation potential (which is linearly related to Iog([ATP / ADP])o,t) are shown. The full line is obtained in a titration with hexokinase, the dashed line in a titration with a protonophore. For every respiration rate one can find a difference (8(~G)) in extramitochondrial log([ATP/ADP]) between the 2 lines. Assuming that in the protonophore titration respiration varies linearly with the (small) freeenergy difference across the adenine nucleotide translocator, and because phosphorylation rate varies linearly with respiration rate in the hexokinase titration [48], curvature in the plot of respiration rate versus this B(t$G) must originate in a nonlinear flow-force relationship of the adenine nucleotide translocator in the titration with hexokinase.
883
simultaneously. To see if the sigmoidicity of the flow-force relationship of the adenine nucleotide translocator would also occur if the force was varied differently, we devised a way of varying mainly the extramitochondriai phosphorylation potential [48]. This is illustrated in Figure 3. The principle is to inhibit only some of the adenine nucleotide translocators by adding the tight binding inhibitor carboxyatractyloside and t h e n returning the respiratory rate to the rate in the absence of carboxyatractyloside by adding hexokinase. This gives one data point with a rate of respiration and an extramitochondrial phosphorylation potential. Additional data points are obtained by adding va~ing amounts of carboxyatractyloside and hexolonase, but such that the respiratory rate comes out the same. The data points (the circles in Fig. 3) then
tog(IATPIIIADPI)o t Fig. 3. Method to determine the flow-force relationship across the adenine nucteotide translocator at constant membrane potential and virtually constant intramitochondrial phosphorylation potential. Starting with a certain amount of hexokinase, hence at a certain respiration rate, carboxyatractyloside (full arrow) and hexokinase (dashed arrow) are added such that the respiration rate remains the same. This leads to the data points indicated by the circles. From the amount of carboxyatractyloside added, one can calculate the fraction of adenine nucleotide transiocators that are still active. By dividing the phosphorylation rate by that fraction and plotting the results as a function of Iog([ATP / ADP])o,t one obtains a flow-force relationship of the adenine nucleotide transloeator at constant membrane potential (because respiration rate is constant) and virtually constant intramitochondrial phosphorylation potential (to the extent that the H+-ATPase remains close to static head).
$84
H. V. Westerhoff
have the same respiratory rate and hence, by the paradigm of the chemiosmotic coupling hypothesis, the same membrane potential. However, they differ in extramitochondrial phosphorylation potential and in turnover rate of the uninhibited adenine nucleotide translocators. The latter is obtained by dividing the total transport rate of ATP (which is equated to the rate of oxidative phosphorylation) by the fraction of uninhibited translocators, 1-1/ Imax (lma x is defined as the amount of inhibitor necessary to completely inhibit all translocators supposing that binding is irreversible; it is equal to the total amount of adenine nucleotide translocators). Plotting this turnover rate of uninhibited translocators versus the extramitochondrial phosphorylation potential, we again observed a sigmoidal flow-force relationship for the adenine nucleotide translocator [48]. We concluded that the flow-force relationship of the adenine nucleotide translocator can exhibit sigmoidicity (nonlinearity) at high extramitochondrial phosphorylation potentials.This occurred when the membrane potential was held constant (the experiment along the lines of Fig. 3), and in the actual variation of the force in a case of varying work load (the experiment along the lines of Fig. 2). Because in a titration with 2,4-dinitrophenol (a protonophore) in the absence of hexokinase, the relationship between mitochondriai respiration and extramitochondrial phosphorylation potential lacked sigmoidicity, this suggests that the sigmoidicity in the relationship between respiration and extramitoehondrial phosphorylation potential under conditions of varying extramitochondrial ATP consumption is due to the sigmoidicity in the flow-force relationship of the adenine nucleotide translocator. As we discussed [48], this conclusion is somewhat, though not quite, dependent on the validity of the delocalized chemiosmotic coupling concept [cf., 49 l. The finding of this sigmoid flow-force relationship not only underlined the importance of using MNET rather than the older types of NET, it also provided an explanation for why at intermediary work loads some of the flux control is still in extramitochondrial ATP consuming processes. The sigmoidicity of the flow-force relationship of the translocator increases the (absolute) elasticity coefficient of the latter at intermediary work loads and hence shifts its flux control to the ATP consuming processes [3]. To develop and check both MNET and Metabolic Control Theory, we have experimented
m', "h in a reasonably well defined reconstituted system of biological free energy transduction, i.e., bacteriorhodopsin liposomes [3, 29]. Here the input free energy, contained in photons of high intensity, is converted to an electrochemical potential difference for protons by the photondriven proton-pump bacteriorhodopsin. The membrane potential across the liposomal membrane can be made to drive K + extrusion by the addition of the K+-specific ionophore valinomycin. For electroneutrality reasons net K + extrusion is equal to net proton uptake, which can be assayed with a pH electrode. We measured the control of net proton uptake (just after the light was switched on, i.e., at zero pH gradient) and found it to be distributed among bacteriorhodopsin and valinomycin. Surprisingly, the flux control of either was close to one, leading to a sum well exceeding 1. That this was not inconsistent with the summation theorem for flux control coefficients, which prescribes a sum of 1, was shown when the flux control coefficients of the proton permeability of the liposomal membrane was found to be close to - 1. Earlier we had developed a complete MNET description of free energy transduction in bacteriorhodopsin liposomes [45, 50]. An MNET description has the advantage that it is valid, to a good approximation, for large modulations of enzyme activities. Metabolic Control Theory is strictly valid, but only for infinitely small changes. However, in the limit to infinitely small changes, also MNET becomes strictly valid. We demonstrated this for the case of bacteriorhodopsin liposomes by deriving expressions for the control coefficients of net proton uptake flux by bacteriorhodopsin, proton permeability and K ÷ permeability, both by using Metabolic Control Theory and by using MNET (which leads to an integral expression for the proton flux in terms of these three activities; the partial derivatives of this expression with respect to those three activities give the flux control coefficients). The two routes led to the same results for the flux controls with respect to the activity of bacteriorhodopsin, the non-proton ion permeability and the proton permeability respectively [3]:
C~ =(Li + L~)/(Li + L~ + n~L.) C.I = (I ,~' + n~L~)/(L i + LHI + n~L~) C~ = - L ~ / ( L i + LH~ + n2L.)
Control, regulation and thermodynamics o f free-energy transduction
Here L,, L i and L~t correspond to the activity of bacteriorhodopsin, valinomycin, and to proton permeability respectively, n to the number of protons pumped per absorbed photon by bacterio-rhodopsin. 3' measures the back-pressure sensitivity of bacteriorhodopsin to the electrochemical potential difference for protons. 3/is zero if bacteriorhodopsin is not sensitive to back pressure, and 1 if its completely sensitive [3]. That in the limit to infinitely small changes, MNET and Metabolic Control Theory lead to the same result, may be interpreted as that Metabolic Control Theory is a subset of MNET. However, this is not a fair representation of the situation. The fact is that the summation and connectivity theorems are not naturally present in MNET and that Metabolic Control Theory is indeed a much more efficient theory to attain the description of the effects of infinitely small changes. For the suggestion that Metabolic Control Theory may be a subset of Biochemical Systems Analysis, the latter arguments also hold. In this case however, Metabolic Control Theory is certainly not a subset of Biochemical Systems Analysis. For, Biochemical Systems Analysis describes systems in terms of unidirectional rather than net fluxes (which is the way MNET and Metabolic Control Theory describe systems). Only recently it has been shown that Biochemical Systems Analysis can also be formulated in terms of net fluxes [51]. Concluding remarks When discussing control and regulation it is important to rigidly define what one is talking about: - C o n t r o l tends to be distributed and condition dependent; -Elasticity of enzymes rather than their distance from equilibrium determines the distribution of the control; -Metabolic Control Theory is not limited to linear pathways; -Metabolic Control Theory is tailored to small changes; - A l s o metabolic regulation can be studied by Metabolic Control Analysis. Acknowledgments 1 thank Jos Arents, Bert Groen and Ron Wanders for the collaborations on this topic, and Drs. C. and
885
E. Kohen for discussions. Ron Wanders devised the method illustrated in Figure 2 [48]. References 1 Higgins J.J. (1965) in: ControlofEnergy Metabolism (B. Chance, R.W. Estabrook and J.R. Williamson, eds.), Academic Press, New York, pp. 13-46 2 Burns J.A., Cornish-Bowden A., Groen A.K., Heinrich R., Kacser H., Porteous J.W., Rapoport S.M., Rapoport T.A., Stueki J.W., Tager J.M., Wanders R.J.A. & Westerhoff H.V. (1985) Trends Biochem. Sci. 10, 16 3 Westerhoff H.V. & Van Dam K. (1987) "tllermodynamics and Control of Biological Free-Energy Transduction. Elsevier, Amsterdam 4 Kacser H. & Burns J. (1973) in: Rate Control of Biological Processes (D.D. Davies, ed.), Cambridge University Press, London, pp. 65-104 5 Rapoport T.A. & Heinrich R. (1975) BioSystems 7,120-129 6 Heinrich R., Rapoport S.M. & Rapoport T.A. (1977) Progr. Biophys. Mol. Biol. 32, 1-83 7 Kohn M.C., Whitley L.M. & Garfinkel D. (1979) J. Theor. Biol. 76, 437-452 8 Kacser H. & Burns J.A. (1979) Biochem. Soc. Trans. 7, 1149-1161 9 Groen A.K., Van der Meet R., Westerhoff H.V., Wanders R.J.A., Akerboom T.P.M. & Tager J.M. (1982) in: Metabolic Compartmentation (H. Sies, ed.), Academic Press, New York, pp. 9-37 10 Kacser H. (1983) Biochem. Soc. Trans. 11, 35 -40 11 Westerhoff H.V. & Chen Y. (1984) Eur. J. Biochem. 142,425-430 12 Fell. D.A. & Sauro H.M. (1985) Eur. J. Biochem. 148, 555-561 13 }tofmcyr J.-lI.S., Kacscr H. & Van der Merwe K.J. (1986) Ear. J. Biochem. 155,631-641 14 Sorribas A. & Bartrons R. (1986) Eur. J. Biochem. 158, 107-115 15 Hofmeyr J.-H. (1c~86) Studies in steaay state modelling and co/itrol analysis of metabol:c systems. Ph.D. Thesis, University of Stellenbosch 16 Kell D.B. & Westerhoff H.V. (1986) FEMS Microbiol. Rev. 39, 305-320 17 Westerhoff H.V. & Keli D.B. (1987) Biotechnol. Bioengin. 30, 101-107 18 Sauro H.M., Small J.R. & Fell D.A. (1987) Eur. J. Biochem. 165,215-221 19 Westerhoff H.V., Plomp P.J.A.M., Groen A.K. & Wanders R.J.A. (1987) Cell Biophys. 10, 239-267 20 Crabtree B. & Newsholme E.A. (1987) Trends Biochem. Sci. 12, 4-12 21 Crabtree B. & Newsholme E.A. (1987) Biochem. J. 247, 113-120 22 Giersch C. (1988) Eur. J. Biochem. 174,509-513
886
H. V. Westerhoff
23 Giersch C. (1988) Eur. J. Biochem. 174,515-517 24 Reder C. (1988)J. Theor. Biol. 135, 175-202 25 Small J.R. & Fell D.A. (1989) J. Theor. Biol. 136, 181-197 26 Giersch C. (1988) J. Theor. Biol. 134, 451-462 27 Westerhoff H.V. (1989) (ed.) Principles of Metabolic Control. Plenum Press, New York, (in press) 28 Groen A.K., Wanders R.J.A., WesterhoffH.V., Van der Meer R. & Tager J.M. (1982) J. Biol. Chem. 257, 2754-2757 29 Westerhoff H V . & Arents J.C. (1984) Biosci. Rep. 4, 23-31 30 Torres N.V., Mateo F., Melendez-Hevia E. & Kacser H. (1986) Biochem. J. 234, 169-174 31 Rapoport T.A., Heinrich R. & Rapoport S.M. (1974) Eur. J. Biochem. 42,107-120 32 Nimmo H.G. & Cohen P.T.W. (1987) 247, 1 - 3 33 Gellerich F.N., Bohnensack R. & Kunz W. (1983) Biochim. Biophys. Acta 722, 381-391 34 Duszynski J., Groen A.K., Wanders R.J.A., Vervoorn R.C. & Tager J.M. (1982) FEBS Lett. 146, 263-266 35 Holness M., Crespo-Armas A. & Mowbray J. (1984) FEBS Lett. 177, 231-235 36 Moreno-Sanchez R. (1985) J. Biol. Chem. 260, 12554-12560 37 Mazat J.-P., Bart E.J.-B., Rigoulet M. & GuErin B. (1986) Biochim. Biophys. Acta 849, 7 - 1 5 38 Forman N. & Wilson D.F. (1983) J. Biol. Chem. 258, 8649-8655 39 Westerhoff H.V., Plomp P.J.A.M., Groen A.K., Wanders R.J.A., Bode J.A. & Van Dam K . (1987) Arch. Biochem. Biophys. 257, 154-169 40 Wander~ R I A , ~,rn,-n A v Van ~ - mund C.W.T. & Tager J.M. (1984) Eur. J. Biochem. 142, 417-424 41 Kohn M.C. & Gartinkel D. (1983) Ann. Biomed. Engin. 11, 511 42 Savageau M.A. (1972) Curt. Top. Cell. Reg. 6, 63-130 .... 43 Savageau M.C. (1976) Biochemical systems analysis; a study of function and design, in." Molecular Biology. Addison-Wesley, Reading, MA 44 Voit E.O. & Savageau M.A. (1987) Biochemistry 26, 6869-6880 45 Westerhoff H.V., Scholte B.J. & Hellingwerf K.J. (1979) Biochim. Biophys. Acta 547, 544-560
46 Van der Meer R., Westerhoff H.V. & Van Dam K. (1980) Biochim. Biophys. Acta 591, 488-493 4"7 Wanders R.J.A., Groen A.K., Meijer A.J. & Tager J.M. (1981) FEBS Lett. 132,201-206 48 Wanders R.J.A. & Westerhoff H.V. (1988) Biochemistry 27, 7832-7840 49 Westerhoff H.V., Kell D.B., Kamp F. & Van Dam K. (1988) in: Microcompartmentation (D.P. Jones, ed.), CRC Press, Boca Raton, Florida, USA, pp. 116-154 50 Westerhoff H.V., Hellingwerf K.J., Arents J.C., Scholte B.J. & Van Dam K. (1981) Proc. Natl. Acad. Sci. USA 78, 3554-3558 51 Cornish-Bowden A. (ed.) (1989) Control of Metabolic Processes. Plenum Press, New York (in press)
Appendix A. Summation theorems for control coefficients How can one predict whether for a given variable the summation theorem should amount to 0, 1, or perhaps to some other value? There is a rather general approach to this problem [3, 22], which is similar to the use of Euler's theorem in equilibrium thermodynamics_ For t h i ~ , n n p n , ~ , ~ r l c * . o . . o ; n . . . . . k ~ . L ~ when the system is at steady state and simultaneously all enzyme activities are increased by a factor f, the system remains at steady state. If so, and the magnitude of the property under consideration increases by the factor fn, then the sum of all control coefficients by all the enzymes on that property must equal n. For metabolite concentrations and potentials n equals 0, for fluxes n equals 1. The proof is found by taking the derivative of the logarithm of the following equation with respect to the logarithm of f, and then taking the limit of f to 1 [3, 22]: X(f'el,f'% ..... f'em) = fn'X(et, e2, %) For transient times, n equals - 1.