how Reversible is Irreversible?

This page contains the text (but not the illustrations) of the following article:
A. Cornish-Bowden and M. L. Cárdenas (2000)
Irreversible Reactions in Metabolic Simulations: how Reversible is Irreversible?

in
Animating the Cellular Map
(ed. J.-H. S. Hofmeyr, J. H. Rohwer and J. L. Snoep)
Stellenbosch University Press, Stellenbosch, pp. 65–71. A fully formatted PDF file including
illustrations is
also available.

Mathematically and thermodynamically there are no irreversible reactions in chemistry,
and all enzymes catalyse reversible reactions. All reactions have finite standard Gibbs energies,
which is just another way of saying that they all have finite equilibrium constants. Nonetheless,
for an enzyme like pyruvate kinase, with an equilibrium constant of the order of 10^{5}, one may
feel that this is just a mathematical nicety and that the thermodynamic necessity for the reverse
reaction to be possible has no practical consequences for cell physiology, and in practice many
metabolic reactions are conventionally regarded as irreversible. This typically means that they
have equilibrium constants of the order of 1000 or more, though when the mass action ratio in
the cell also strongly favours the forward direction a reaction with a substantially smaller
equilibrium constant than this can be regarded as irreversible. For example, the hexokinase
reaction has an equilibrium constant that is less than 250 at pH 6.5 (Gregoriou
et al., 1981),
but the reaction is still considered irreversible because in normal physiological
conditions the concentration of glucose is always much greater than that of glucose 6-phosphate
nd the concentration of ATP is usually greater than
that of ADP.

In metabolic simulations the difficulty and inconvenience of using complete reversible rate equations for all the enzymes is aggravated by the fact that for nearly all "irreversible" enzymes virtually no kinetic measurements of the rates of the reverse reactions, or even of the effects of products on the rates of the forward reactions, have ever been reported. In the absence of any data, therefore, one is forced to choose between ignoring the reverse reaction altogether or guessing what its kinetic parameters might be on the basis of the values of the equilibrium constant and of the kinetic parameters for the forward direction. It would hardly be surprising if many simulators faced with such an unappealing choice preferred the easier option of ignoring reverse reactions, apparently safe and no worse than just guessing the parameter values.

Our view is that it is always best to use reversible equations in metabolic simulations for all
processes apart from exit fluxes, and similar sentiments are expressed in the documentation for
the simulation program Gepasi (Mendes, 1993). Even so, we were surprised how large the effect
of allowing for the reverse reaction of pyruvate kinase in a model of glycolysis in *Trypanosoma
brucei* proved to be (Eisenthal & Cornish-Bowden, 1998; Cornish-Bowden & Eisenthal, 2000).
With this step treated as strictly irreversible transport of pyruvate from the cell had a flux
control coefficient of zero, as expected for a step following an irreversible step. What was not
expected, however, was that allowing for the reverse reaction of pyruvate kinase by assuming
reasonable parameter values consistent with the equilibrium constant of about 10^{5}
would perceptibly change all of the flux control coefficients in the system and that export of
pyruvate would go from having no flux control at all to having the second largest flux control
coefficient in the system.

The model of glycolysis in *Trypanosoma brucei*, developed originally by Bakker et al. (1997), is relatively complex, with 20 steps in four compartments; many of the steps
have multiple substrates and products, and the metabolite concentrations as a whole are constrained
by four different conservation relationships, one of them highly complicated and unintuitive. All of
this makes it a rather unsuitable model for analysing the effect of allowing for the reversibility of
pyruvate kinase, and one may suspect that the unexpected behaviour observed was related in some
way to the complexity of the model. We therefore decided to investigate the question in the context
of a model simple enough for the results to be fully understood.

Fig. 1. A model pathway with (a,b) and without (c,d) feedback inhibition of the first step
by the end product, and taking account (a,c) or not (b,d) of a very small degree of reversibility in
the fourth step. The rate of the reaction catalysed by E_{1} was
10X_{0}(1 – X_{0}/25S_{1})(X_{0} + S_{1})/[1 + (X_{0} + S_{1})2 + S_{5}2];
when there was no feedback inhibition the term in S_{5} was omitted from the denominator.
Step 2 was treated as an equilibrium with equilibrium constant of 1. E_{3} and E_{5}
followed reversible Michaelis–Menten kinetics with forward and reverse limiting rates of 8 and 0.5
respectively for E_{3} and 12 and 2 respectively for E_{5}, and forward and reverse
Michaelis constants of 2 and 3 respectively for E_{2} and 0.5 and 2 respectively for
E_{5}. In models (a) and (c) E_{4} also followed reversible Michaelis–Menten
kinetics, with forward and reverse limiting rates of 10 and 0.001 respectively , and forward and
reverse Michaelis constants of 0.5 and 25 respectively; in models (b) and (d) the ordinary ireversible
Michaelis–Menten equation was used with limiting rate 10 and Michaelis constant 0.5 (effectively setting
the reverse limiting rate to 0 and the reverse Michaelis constant to infinity). The exit flux catalysed by
E_{6} followed Michaelis–Menten kinetics with Michaelis constant 0.5 and a limiting rate V6
that was varied to simulate changes in demand. The concentration of X_{0} was fixed at 10. All simulations
were done with Gepasi (Mendes, 1993), version 3.21, which was downloaded from an address that
no longer exists. The current address is
http://www.gepasi.org/.

We have therefore compared the results of simulating the metabolic behaviour of the four models
illustrated in Fig. 1. The most complete version of this model is shown at the top-left as Fig. 1a. In this
version there is a feedback loop from the end-product S_{5} to the first step, catalysed by
E_{1}, which follows the reversible Hill equation (Hofmeyr & Cornish-Bowden, 1997) with a
Hill coefficient of 2; in addition, E_{4} is represented by a reversible rate equation even
though the equilibrium constant of 5 x 10^{5} might be regarded as large enough for the
reverse reaction to be neglected. In two of the variant versions of the model, 1c and 1d, the feedback
effect of S_{5} on E_{1} is absent, and in models 1b and 1d the reaction catalysed
by E_{4} is made strictly irreversible.

The four versions of the model thus include all possibilities necessary for deciding whether taking proper account of feedback inhibition is more or less important than allowing for the small degree of reversibility in a reaction with a very large equilibrium constant. If allowing for this small degree of reversibility is trivial, whereas ignoring a feedback loop is crucial, we should expect the results from models 1a and 1b to resemble one another and to differ from those from models 1c and 1d. However, if it proves essential to allow for even a small degree of reversibility, but unimportant to take full account of a feedback loop, then we should expect models 1a and 1c to resemble one another and to differ from models 1b and 1d.

Fig. 2. Variation of the distribution of flux control as a function of demand for the four models illustrated in Fig. 1. The layout is the same as in Fig. 1, and the letters (a–d) refer to the same four cases. The variation of the flux J with the demand is shown in the inset to each plot.

So far as the flux and the flux control coefficients are concerned it turns out that neither of these expectations is fulfilled. As illustrated in Fig. 2, three models give qualitatively very similar behaviour but the fourth, from model 1d, is very different. In other words one can ignore either feedback loops or reversibility with relatively little effect on the flux properties, but one cannot ignore both. Although at first sight perhaps surprising, both aspects of this result are consistent with what was known before. The contribution of feedback inhibition to metabolic regulation has been previously analysed in several models without any internal irreversible steps but of varying degrees of complexity (Hofmeyr & Cornish-Bowden, 1991; Cornish-Bowden et al., 1994, 1995), with consistent results: in all cases the importance of feedback inhibition (and the degree of cooperativity of such inhibition) has been found to lie not in more effective flux regulation but in allowing metabolite concentrations to remain relatively little changed when the flux changes. Similarly, the finding that a step that is isolated from the rest of the system by an irreversible step cannot exert any flux control is a classical result in control analysis.

Model 1b gives results that resemble those from the complete system much more than they
resemble those from model 1d, even though it interposes a completely irreversible step between
the last two enzymes and the rest of the pathway. The explanation is that even though E_{5} and
E_{6} follow an irreversible step they are not isolated by it from the rest of the pathway
because they can communicate with it through the feedback loop. So far as flux control is concerned,
therefore, what is important is that the end product can communicate with the early steps in a pathway;
it does not matter whether this communication is along the chain or via a feedback loop. A practical
illustration is provided by serine biosynthesis in bacteria and mammals (Fell & Snell, 1988): in bacteria
the end product serine acts as a feedback inhibitor, but in mammals it does not, acting only as an
ordinary product inhibitor; in both cases, however, it regulates its own synthesis quite effectively.
Moreover, in the case of an almost irreversible reaction it is not the near-zero term in the numerator
of the rate expression that is essential for communication with the earlier steps but the terms in product
concentration in the denominator. In other words it is not reversibility as such that is important but
the possibility of product inhibition.

Fig. 3. Variation of metabolite concentrations as a function of demand for the four models illustrated in Fig. 1. The layout is the same as in Figs. 1–2, and the letters (a–d) refer to the same four cases.

As in previous studies of feedback inhibition (Hofmeyr & Cornish-Bowden, 1991; Cornish-Bowden
et al., 1994, 1995), the crucial difference between models with and without such inhibition
is not in the flux behaviour but in the changes in metabolite concentration that accompany changes in flux.
In the present case, illustrated in Fig. 3, the concentration patterns for models with feedback inhibition
are virtually identical whether or not the reversibility of the fourth reaction is allowed for (Fig. 3ab).
The two cases without feedback inhibition (Fig. 3cd) are not identical but are still much more similar
than the corresponding patterns for the flux (Fig. 2cd) might lead one to expect. In the fully irreversible
case (Fig. 3d) lowering the demand to below the flux delivered by E_{1} causes the concentrations
of the intermediates that follow the irreversible step to rise to infinity, so that no steady state is possible;
in the almost-irreversible case (Fig. 3c) these concentrations also rise to very high values, albeit not to
infinity.

If flux regulation were the only consideration, one might consider Fig. 2c to represent the ideal, with
flux strictly equal to demand until the limit that the system can deliver. In contrast, in Fig. 2ab the
proportionality is only approximate and the flux is always a little less than the demand. However,
this imperfection is very slight, and the perfection

of Fig. 2c is bought at the very heavy price of
complete loss of concentration regulation (Fig. 3c).

Gaining insight into the design of metabolic regulation has been only part of our objective in this
investigation. We have also been concerned to determine when it is safe in a metabolic simulation to
treat reactions with very large equilibrium constants as strictly irreversible. The conclusion is that
no matter how large the equilibrium constant one must not allow a step to completely isolate one part
of a pathway from the rest. If there is a feedback loop that allows communication around an irreversible
step then the behaviour is virtually identical whether the reversibility of the nearly irreversible step is
allowed for or not. Similarly, if terms for product inhibition are included in the denominator of the rate
expression then it will make little or no difference whether the small negative term is included in the
numerator or not. Considered in these terms the initially puzzling observation in the model of glycolysis
in *Trypanosoma brucei* makes good sense and does not require any explanation in terms of obscure
properties arising from the complexity of the model. In the original model with pyruvate kinase treated
as strictly irreversible (Bakker et al., 1997) there was no feedback loop around this
enzyme, and no possibility of product inhibition was allowed for. Consequently pyruvate transport
was completely isolated from the rest of the pathway and any effects that it could have had on the
regulatory structure were concealed; they were only revealed when communication between pyruvate
and the earlier steps was permitted.

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