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Discussion Forum

Should irreversible thermodynamics be applied to metabolic systems ?

D a v i d F. W i l s o n - No t in m o s t cases

David F. Wilson received a Ph.D. in Biochemistry with Dr Tsoo E. King at Oregon State University in 1964. He went to the University o f Pennsylvania for a post-doctoral position with Dr Britton Chance and stayed to become professor in the Department of Bio- chemistry and Biophysics. In 1971 he received the Eli Lilly Award in Bio- Chemistry from the American ( hemical Society.

Precise descriptions of both the energy relationships among the metabolic reactants and the flux through metabolic pathways are important goals of biochemistry. Historically, classical ther- modynamics has been used to describe the energy relationships while enzyme kinetics has been used to describe the metabolic flux. Recently, some workers I a have argued that the formalism of irreversible thermodynamics (also given the misnomer 'non- equilibrium thermodynamics') is more appropriate for this pur- pose. Discussion must be based on the relative effectiveness of the two approaches and the limitations of each.

Classical thermodynamics and its limitations In classical thermodynamics the transition of a chemical sys-

tem from one state to another at constant temperature and pressure involves changes in the heat content (enthalpy or H), the free energy content (G) and the entropy (order or S) of the system. The changes in these properties are related by the expression: AH = AG-TAS where T is the absolute temperature. The change in enthalpy, AH, is dependent only on the molecular rearrangements in the chemicals involved and is independent of the pathway by which the reaction occurs. The Gibbs free energy change, AG, is the maximum work (fully reversible conditions) which the chem- ical reaction can do on its environment minus that done if there is an increase in volume of the reactants (PAV). The entropy change, AS, is the change in molecular 'order' of the chemical reactants. When the reaction occurs under conditions for which it is not fully reversible, the work done on its environment will be less than the maximum value. Measurement of the actual work done (W) allows the efficiency (W/AG) to be calculated and the loss in free energy (AG-W) appears as an increase in heat content of the environment. The calculated values are in- dependent of the rate at which the chemical reaction occurs Since the values calculated by classical thermodynamics do not contain the units of time, the classical view is that thermodynamic parameters can not be used to predict the rate at which reactions will occur. This is well exemplified by the observation that a 2:1 mixture of hydrogen and oxygen gases at atmospheric pressure and 25°C shows no measurable reaction, but introduction of a trace of platinum dust causes an immediate explosion.

Hans V. Westerhoff- Yes. Kinetics alone are impracticable

Hans V. Westerhoff ~tudied chemistry (including biochemistry, non- equilibrium thermodynamics and oncology) at the University of Amster- dam and the Antonie van Leeuwenhoek huis. In 1977, he started in the group o f and in intense collaboration with Karel van Dam at the University of Amster- dam on the experimental and theoretical application of system kinetic approaches ;~ to mitochondrial oxidative phosphoryla- tion. Bacteriorhodopsin liposomes, 'soluble metabolism', metabolic control and microbial growth have now and then 7wapped places with oxidative phosphorylation.

I shall begin by briefly sketching the type of problems for which enzyme kinetics alone fails and in which non-equilibrium ther- modynamics (NET)* may well be one of the possible approaches. Then, after comparing various theoretical techniques in their application to that type of problems, I shall discuss which combinations of methods and problems seem optimal. Finally, I shall respond to Dr Wilson's criticism.

NET's subject: quantitative understanding of metabolism Enzyme kinetics is a very powerful method; it usually provides

a complete description of the dependence of the reaction rate of a single enzyme on the concentrations of substrates and modifiers. However, metabolism involves many enzymes, which influence the rates of each other through the concentrations of common metabolites. A quantitative description of metabolism on the basis of enzyme kinetics alone would involve the determination of the complete kinetics of every enzyme in the system, followed by mathematical solution of the equations arising from the steady state conditions. (Alternatively, the equations could be treated as differential equations and integrated. ) In general, it is impossible to provide such (analytical) mathematical solutions for systems consisting of more than two enzymes. Consequently, histori- cally, enzyme kinetics alone has n o t proved suitable for the description of metabolic fluxes in systems of any complexity.

The kinetic description of s u c h systems, which are too com- plex for enzyme kinetics alone, is the subject of NET.

Possible approaches to the quantitative description of metabolic systems

The closest approach to the (impossible) analytical solution, uses a computer to solve the rate equations (e.g. Ref. 1). The potential accuracy is mostly limited by the uncertainties in the kinetic constants that must be plugged into the calculations. A

* As the common meaning of'irreversible' is 'that cannot be reyersed or revoked'. 'irreversible' thermodynamics might be supposed to be the thermodynamics of processes that cannot mn backward. In fact,'irreversible' thermodynamics studies all processes that are not in equilibrium, independently of whether they can (in practice) run backward or not. 1 therefore prefer to use the term 'non-equilibrium' thermodynamics.

Elsevier Biomedical Press 1982 B376 5067,82/0000 ~888)/$()1 (8)

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Irreversible thermodynamics: A basis Irreversible thermodynamics was developed in an attempt to

bridge the gap between classical thermodynamics and kinetics. There are several rationales for the development of the existing formalism of irreversible thermodynamics 4. It is useful to examine one of the systems which permit a clear statement of the assumptions inherent in the formalism as it applies to chemical systems. For the bimolecular reaction:

kl A + B r - ~ - C + D (1)

the forward reaction rate (v,) is equal tokl[A][B] and the reverse reaction rate (vr) is equal tok. ~[C][D]. The free energy change is expressed:

AG = AG °' + RTIn [CllD] (2) [A][B]

Multiplying and dividing the last term by k- ~/k~ we may write

AG AG° '+ RTIn k ~[C][D].k, = (3) k~ [AI[B] k ,

Making use of the relationship AG °' = RTIn k~.t this equation simplifies to: k~

AG = RTIn v--L (4) VI

Although this derivation is for a simple bimolecular reaction it can be shown to be applicable to all reactions under selected con- ditions. Taylor series expansion of eqn 4 shows that linearity be- tween the net flux (vt-w) and AG is approximated only when AG is between _+ 0.8 kJ/mole 4 ~. The formalism of irreversible ther- modynamics makes use of eqn 4. However, different symbolism

is used, with J = LA (5)

replacing v, constantx ( A G ) (6)

In eqn 5 where J is the flux through the reaction, A is called the chemical affinity (usually set equal to AG) and L is an empiri- cally determined constant required to fit eqn 5 to the available data (phenomenological coefficient). In the region near equilibrium this is a linear relationship and can be solved readily while for AG >/0.8 k J/mole the relationship is usually non-linear and not pre- dictable.

Irreversible thermodynamics: Applicability to biological metabolic systems

The applicability of irreversible thermodynamics to biological metabolic systems is dependent on both the correctness of the required assumptions and the accuracy with which eqn 5 predicts the behavior of the system outside the region for which measure- ments have been made. When an enzyme catalysed reaction is being considered, in addition to the requirement for near equilib- rium (AG ~ 0.8 kJ/mole) the activity of the catalyst must not change, i.e. there must be no change in any effectors of the enzymatic activity excepting substrate and product, and then only as they enter into the equilibrium expression. This means there can be: (a) no change in the concentration of noncovalent activators or inhibitors of the enzyme; (b) no change in the activ- ity of the enzyme through activation or inhibition of the enzyme by covalent modification such as phosphorylation; (c) no change in the other effectors of enzyme activity such as association and dissociation of subunits etc. The need for these restrictions is evi- dent since any change in catalytic activity implies a proportional increase or decrease in both u~ and ~t, resulting in an increase or decrease in the net flux (w-w) with no change in AG. In eqn 5 the catalytic activity is incorporated in the constant L and changes in

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further disadvantage is that every result corresponds to only one specific set of values of the kinetic parameters. An analytical solu- tion would lead to a continuous picture.

(2) The equilibrium thermodynamics approach assumes the entire system to be so close to equilibrium that the powerful and mathematically simple equilibrium thermodynamical relation- ships apply. However. this approach is hardly applicable to metabolic systems which, almost by definition, house fluxes and are therefore out of equilibrium. Whenever flux occurs it is pref- erable to treat the system with near-equilibrium-NET (see (4)), which encompasses equilibrium thermodynamics and adds powerful laws to it.

(3) Another approach (of. Ref. 2) treats all processes but one with equilibrium thermodynamics and the remaining 'rate- limiting step' with single-enzyme kinetics. The mathematics are simple, but as the processes are assumed to be in equilibrium no allowance can be made for fluxes (contrast Ref. 2). Anyhow NET would be preferable for those steps.

(4) In the accompanying discussion Dr Wilson implicitly focusses on the approach that is most properly called near- equilibrium-NET (NNET) ".~. Like NET in general, this approach treats the reaction rates ('flows') as being caused by Gibbs free energy (G) gradients ('forces') rather than by concen- trations. NNET uses proportionalt and symmetrical relationships between flows and forces, proof for which is limited, though much less limited than Dr Wilson suggests (see below). Conse- quently, the application of NNET is absolutely safe only when all 'elemental' processes are close to equilibrium, or when propor- tionality and symmetry have been verified experimentally :*'4. NNET brings powerful theorems 5. an impressive simplification of the mathematics (proportionality and symmetry):', as well as an easily accessible theoretical and practical understanding of the system under study 4.

Of the two variants of NNET, the most common one, phenomenological NNET (e.g. Refs 3,5) describes the system as a black box; the proportionality constants in the flow-force rela- tionships do not bear upon the structure of the black box and the properties of the enzymes within it. This variant is useful for deriving the general properties of energy transducing systems (e.g. Ref. 5). In the second variant, mechanistic NNET (e.g. Refs 4,6,7,8), the proportionality constants, and thereby the sys- tem behaviour, are explicit functions oftbe structure of the system and the properties of the enzymes.

(5) Many metabolic systems contain reactions that are signifi- cantly displaced from equilibrium and for which the proportional- ity and symmetry of the flow-force relationships cannot (yet) be verified experimentally. Some of those systems may be treated by assuming (or rather verifying) that all reactions but one can be treated with NNET, the remaining reaction being treated with enzyme kinetics, or rather a thermodynamic translation thereof (Ref. 9).

(6) In Mosaic non-equilibrium thermodynamics ('MNET') 8 ~" the reaction rate of every individual ('elemental') reaction is written as a function of the (electro)chemical poten- tials (~ = zF~0 + RT In(c)) of the reactants. If the relationship between the chemical potential and the rate is not known for an enzyme then prototype relationships derived" TM from enzyme kinetics are used~'; this is just like assuming Michaelis-Menten kinetics (with adjustable Km and V) in the first approach of yet uncharacterized single enzymes.

Next, these rate-chemical potential relationships are approx- imated" by relationships that may be slightly less accurate than the

~ Here the term 'proportional' rather than 'linear' is ttsed to distinguish this case ol v :L.( AG) fromthe generally linear case v L-( AG) ,~ constant.

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catalytic activity make L a variable and invalidate the equation. In general biochemical experience with metabolic pathways, net flux through each pathway is determined by enzyme(s) which are strongly displaced from equilibrium (AG more negative than approximately - 4 M/mole) and subject to regulation by activators and inhibitors. Application of irreversible thermo- dynamics denies the existence of such regulation or at least its importance in determining metabolic flux.

There are also reasons for doubting that the rate of enzymatic reactions can be accurately described by linearized equations of the type shown. The rates of enzyme reactions can be approxi- mated by such linear equations only over a limited range of sub- strate concentrations 6.7 and even in this region systematic devia- tion from linearity is pmsenP. In such a case extrapolation of the equation from the region in which the experimental data was fit- ted to outside of the measurement region is clearly unjustified.

In summary: regulated metabolic pathways do not in general fulfil the assumptions essential to the formalism of irreversible thermodynamics. Thus, application of the equations developed for this formalism is an empirical curve-fitting procedure under conditions for which the implied relationships can not be expected to hold. Although empirical curve fitting can be a valid and important descriptive tool, linearizations such as that given in eqn 5, even for ideal conditions, do not accurately describe the kinetics of enzyme catalysed reactions. In contrast, analysis using classical thermodynamics coupled with enzyme kinetics is not subject to limits other than the accuracy of the experimental data and the ability of the investigator to correctly describe the kinetic behavior of the reactions involved. Moreover, the final equations contain real biochemical proposals concerning mechanism, reg- ulatory parameters etc. which can be subjected to experimental test. It is my opinion that the formalism of irreversible ther- modynamics should not be applied to regulated metabolic path- ways, not only because of its inherent limitations, but also because better approaches are available.

Reply to H. V. Westerhoff Dr Westerhoff has claimed impressive powers for analysis

using irreversible thermodynamics. Most of these perceived powers are fictitious but extensive further discussion is not feas- ible at this time. However, 1 wish to ask the readers to consider two points.

(I) Irreversible thermodynamics has been applied to mitochondrial oxidative phosphorylation by Dr Westerhoff and others. The overall reaction is:

H' + NADH + 3 ADP + 3 P~ + 1 / 2 0 ~ NAD ~ + 3 ATP + I-I20 (7)

Irreversible thermodynamics predicts that the reaction rate is related to the 'force' or reaction affinity:

: L [ NADI-II ( IADPI[P~I ):' [O~1'/~ [W l (8) J [NAD ~] ' , ~

Eqn 8 is immediately recognizable as incorrect because the rate of respiration by suspensions of isolated mitochondria is essentially independent of oxygen concentrations above approximately 20 p~m. Similar observations for other metabolic pathways such as glycolysis (flux is essentially independent of the concentrations of lactate and pyruvate and usually that of glucose) provide experimental proof that metabolic fluxes can not be related to the reaction affinity in the manner claimed by Dr Westerhoff.

(2) Almost any equation with a suitable number of variables will fit a given set of experimental data. Thus, the ability of Dr Westerhoff and others to 'fit' the equations of irreversible ther-

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correct ones, but are mathematically just simple enough to allow an analytical solution of the steady state equations. Calculations based on enzyme kinetics" 1:, and experimental evidence z" ,1.1:, suggest the form of such approximations. The important differ- ence between MNET and (mechanistic) NNET is that these rela- tionships are neither limited to near-equilibrium, nor necessarily proportional, nor necessarily symmetrical. Yet, when processes within the system turn out to be near-equilibrium, the MNET rela- tionships automatically reduce to NNET relationships.

The third step in MNET is ~.1° to use the boundary conditions arising from steady state and physical constraints and transform the assembly of elemental rate equations into a much smaller number of equations describing the steady state flows as functions of both the forces (free energy differences of substrates such as ATP and ADP plus phosphate in oxidative phosphorylation) imposed on the system from outside and the activities, reaction stoichiometries and regulatory properties of the enzymes within the system. Also, the 'structure' of the system (e.g. 'chemiosmo- tic' vs. 'chemico-conformational') bears on the equations ", 'L

(7) Network thermodynamics ~4 is similar to MNET, except that the approximations are omitted. A computer is used to obtain numerical solutions. (Dis)advantages are similar to those of approach 1.

When to use which approach? No single approach is the best for all metabolic systems.

Approach 1 is probably the best for systems consisting of a small number of enzymes where the kinetic constants are fairly well known'. For more complex systems in which more than one enzyme operates far from equilibrium, MNET ~,1° and network thermodynamics 14 seem to be most suitable. When no enzyme or only one is far from equilibrium, it seems advisable to use approaches 4 (Refs 3-7) and 5 (Ref. 9) respectively. The scope and mathematical advantages of approaches 2 and 3 are limited. Only rarely are these approaches not inferior to approaches 4 and 5 respectively.

In general, thermodynamic approaches have the advantage that insight in energy transduction (e.g. efficiency) of the system is obtained automatically ~,15.1'~. Moreover, thermodynamic para- meters such as electric potentials, electrochemical potential dif- ferences across a membrane (e.g. protonmotive force) and redox potentials '' are most easily treated in terms of NET. Violation of the first and second law of thermodynamics are avoided automat- ically in NET treatments, in kinetic treatments these points require special attention.

Reply to D. F. Wilson Dr Wilson's criticism essentially consists of three theses:

( 1 ) The relationship between the rate of a reaction and the Gibbs free energy difference of the reaction is non-proportional and even not predictable for I~GI ~ 0.8 kJ/mol.

Only one of the above-mentioned variants of NET, i.e. NNET, depends on the proportionality of such relationships. Theretbre, this criticism comes down to the criticism that some authors have too lightly assumed that in the metabolic systems they were treat- ing. they could safely use this simplest of all NET approaches rather than MNET, or network thermodynamics. Reviewing the literature on this point, it is remarkable (though only in view of Dr Wilson's criticism) how often authors using NNET did show experimentally that relationships between flows and Gibbs tree energy differences in their system were linear ~.~' H.,:, see even ~7. The range of the experimentally observed linearity exceeded three [e.g?'.'"":~], seven [e.g.:"~7], or even ten ~. ~' kJ/mol, which suggested that most experimental systems do not correspond to

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~,llsott

modynamics to selected sets of experimental data does not provide scientific evidence that the equations are applicable or that the author's interpretations of the assigned variables and con- stants are correct.

Acknowledgements The author thanks Drs Joseph Higgins and Maria Erecifiska for

their suggestions and criticisms.

T I B S - Augus t 1982

References 1 Rottenherg, H. (1979)Biochim. Biophys. Acta 549, 225-253 2 Walz, D. (1979)Biochim. Biophys. Acta 505, 27%353 3 Stucki, J. W. (1980) Eur. J. Biochem. 109, 269-283 4 Prigogine, I. (1967) Introduction to Thermodynamics of Irreversible

Processes. John Wiley and Sons Ltd, New York 5 Heinz, E, (1973) Mechanics and Energetics o f Biological Transport in

Molecular Biology, Biochemistry and Biophysics (Kleinzeller, A. and Wittman, H. G,, eds), Vol. 12, Spfinger-Verlag, New York

6 Wilson, D. F. (1980) Biochim Biophys. Acta 616,371-380 7 Rottenberg, H. (1973)Biophys. J. 13. 503-511

Pe e~terhoJ] the condition tbr which Dr Wilson derives that linearity is limited to Gibbs free energy ranges of twice 0.8 kJ/mol (1.5 kJ/mol tbr <15% error in rate description). Dr Wilson treats the linearity approximation as based on just neglecting all second and higher order terms in the Taylor series expansion, or, to put it in graphi- cal terms, as based on the use of the tangent to the curve rather than the actual curve itself. On the basis of enzyme kinetics, Rottenberg ~, see also :~, ~2. ~:~, plotted rates as a function of the free energy of reaction and noted that such curves generally :~,H TM have inflection points, i.e. points where the second-order term in the Taylor series disappears. Such non-trivial linearity leads to linear approximations of the mathematically correct relationship that predict the rate with less than 15% error for a free energy range of 7.4 kJ/mol (equivalent to 75% of the total velocity range)H'x:': much wider ranges of linearity than the 1.5 kJ/mol derived by Dr Wilson are expected and found 5.a H":~'~L

On the other hand, experimental limitations have often pre- cluded the experimental demonstration that the linear flow-force relationships were proportional and symmetrical as well. In such cases, the use of NNET should still be considered as a somewhat uncertain, first approximation.

Neither MNET "~'a°, nor network thermodynamics ~4 rely on proportionality or symmetry. They merely rely on the demonstra- tions (e.g. Refs 11-13) that relationships between v and - AG are as predictable as the relationships between v and IS] used in enzyme kinetics.

(2) In v L . ( - A G ) ( J -L-A) the catalytic activity is incor- porated in L, so that L is a variable; therefore changes in cataly- tic activity invalidate the equation.

In many respects the parameter L in this equation is comparable to the parameters V or Km in the Michaelis-Menten equation. Such parameters are sometimes called 'constants' only to indicate their independence of the explicit variable in the equation (i.e. [S], or AG). The equation v = L . ( - A G ) separates the effects on the reaction rate of the substrate and product (in AG) from the effects of the properties of the enzyme and invariant boundary conditions (in L) H- ~:~. The fact that L's (and accompanying para- meters ~.~°) carry information concerning the properties of the individual enzymes, is one of the powerful properties of the NET approaches: when made expliciP ,o.~.~,~, this property takes away yet another bias against NET, i.e. that NET, like equilib- rium thermodynamics can not be used to obtain mechanistic information about systems.

(3) Analyses using classical thermodynamics coupled with enzyme kinetics are not subject to limits other than the accuracy of the experimental data and the ability of the investigator to cor- rectly describe the kinetic behaviour of the reactions involved. Moreover, the final equations contain real biochemical propos- als concerning mechanism, regulatory parameters etc., which can be subjected to experimental tests.

The former statement provides the opportunity to warn that the application of classical ( : equilibrium) thermodynamics to reac- tions through which flux flows (e.g. sites I and I1 of oxidative

phosphorylation ~) is tbrmally wrong and certainly limited (in contrast to Dr Wilson's statement). For those reactions applica- tion of NET and even its near-equilibrium subform is evidently more accurate, more informative and preferable.

As to the second statement most variants of NET (notably (mechanistic) NNET 4,'~ ~.1,~, MNET~.~o and network ther- modynamics TM) do contain biochemical proposals concerning mechanisms, regulatory properties and reaction stoichiometries of the individual enzymes in the system, as well as proposals con- cerning the structure of the system (e.g. chemiosmotic versus chemico-conformational). Phenomenological proportionality constants were used only in studies that focussed on the general energy transduction aspects of a system and were not intended to obtain information about mechanisms. Perhaps quite signifi- cantly, the application of the proposedly (see Dr Wilson's con- tribution) better approach to oxidative phosphorylation left out the mechanistically important protonsL NET descriptions of oxidative phosphorylation have explicitly considered the role of protons :~.4,",1~.~. These have, for instance, led to the determina- tion of the H ~/O and H ' /ATP stoichiometries under conditions of high protonmotive force *'' ~ and to further insight into the question whether the proton motive force is the only 'high energy inter- mediate' in oxidative phosphorylation, e.g. :',1H.

Reaction to Dr Wilson's reply My co-workers and I do not use Dr Wilson's equation, but

equations that fully account for substrate concentrations exceed- ing their K,,? I;'. '" 21. in fact this is one of the characteristics of MNET.

For instance: the data in Ref. 9 could not have been 'fitted' with H ' /ATP = 4 and H ' /O -- 5, but could be 'fitted' with the values 2 . 6 ( + / - 0 . 4 ) and 3.6(+/-0.4)respect ively. Those in Refs 2l and 22 could be fitted with an H~/e of 1.6(+/ 0,2) only. By what Dr Wilson seems to disapprove of as being mere 'fitting', my co-workers determined proton stoichiometries of proton pumps,~.22, the extent to which the energy coupling membrane is permeable to protons ~,~° the degree of which specific enzymes control metabolic fluxes (Westerhoff, H. V., Groen, A. K., Wanders, R. J. A. and Bode, J. A., submitted for publication), and the (proton-) current, voltage (A/2,) characteristics of a light driven proton pump TM, like others determine K~ and V for isolated enzymes. Those who think that such information (obtained under conditions in which the systems are normally operating) is irrelevant may be convinced by Dr Wilson's point 2.

Conclusions (1) For the quantitative description of metabolic sys tems ,

enzyme kinetics p e r se is impracticable. (2) None of the existing methods is ideal for that purpose, but

numerical computer methods and MNET are most often the best. (3) For enzymes through which flux flows, near-equilibrium

assumptions are always less erroneous than /n-equilibrium assumptions.