Appendix 1: enzyme kinetics
In this Appendix the rate equations, experimental kinetic data and an overview of the biochemical knowledge concerning the glycolytic enzymes are given. Derivations follow established practice [32,59]. One substrate, one product reversible Michaelis–Menten kinetics was used to describe the enzymes PGI, PGM and ENO:
where a and p represent the concentrations of the corresponding substrate and product, respectively. Γ is the mass-action ratio, p/a, Keq is the equilibrium constant, peq/aeq. Ka and Kp are the Michaelis–Menten constants for a and p. Reversible Michaelis–Menten kinetics for two noncompeting substrate-product couples was used for HK, GraPDH, PGK and PYK:
where a and b represent the concentrations of the substrates and p and q the concentrations of the products. For the other enzymes specific rate equations apply, as described in the following sections.
Transport of glucose: HXT
The transport of glucose across the cell membrane occurs via facilitated diffusion [56,60–62]. We have used a symmetrical carrier model:
in which [Glcout] and [Glcin] are the concentrations of extracellular and intracellular glucose, respectively. The ‘interactive constant’Ki depends on the relative mobility of the unbound and bound carrier [60,63], and was calculated in . Despite the large number of transporters, glucose transport in derepressed yeast cells can be kinetically characterized by one high affinity component with a Km of 1–2 mm. This was also the case for the yeast used in this study: zero trans-influx of glucose occurred with a Km of 1.2 mm (Table 2).
In the literature it has been proposed that glucose 6-phosphate inhibits glucose transport [64,65]. Such a feedback regulation has been incorporated in most of the glycolytic models [26,28–30,66]. The experimental basis for this feedback is controversial . We have shown experimentally that, for high-affinity transport, the simplest model for a facilitated diffusion transporter (a symmetrical carrier whose rate depends on extra- and intracellular glucose only) can account for the difference between initial uptake rates and steady-state glucose consumption . This removed the necessity of postulating inhibition by glucose 6-phosphate. This model has therefore also been used here.
In yeast three enzymes can phosphorylate glucose: hexokinase PI, hexokinase PII, and glucokinase . In our glucose-derepressed compressed yeast, hexokinase PI appears to dominate: we found that KmFru/KmGlc = 1.5 and VmaxFru/VmaxGlu = 2  (results not shown). Our Km values for glucose and ATP agree well with published ones [47,68–70]. The kinetics of hexokinase appear more complicated than the simple Michaelis–Menten kinetics employed in this study. For example, binding of glucose enhances the affinity for ATP [69,71]. Such kinetics have not yet been incorporated. They may have implications for the role of glucose signaling by the hexokinases . The impact of the more complex kinetics of hexokinase cannot be assessed yet, as data on the internal glucose concentration are yet lacking, and because product inhibition of the enzyme is not fully understood. Rather than being sensitive to its direct product, G6P (the Ki was estimated to be 40 mm), hexokinase appears negatively regulated, in an as yet unknown way, by trehalose 6-phosphate synthase . Competitive inhibition by Tps1's metabolic product trehalose 6-phosphate may offer a mechanism . In our model, as in all other published ones, no Tps1-mediated feedback mechanism was incorporated, as the feedback has not been characterized kinetically (see main text for the consequences).
Phosphoglucose isomerase: PGI
The kinetics of PGI have been measured in both directions and a reversible Michaelis–Menten equation was used to describe the kinetics of this enzyme. Our Km values are in agreement with those found in the literature . The kinetics fit reasonably well with the equilibrium constant (Keq,PGI = 0.31) according to the Haldane relationship: we found an equilibrium constant of 0.26.
Aldolase is generally assumed to follow an ordered uni-bi mechanism, with GraP binding after glycerone phosphate [27,74]. The equation is:
where a represents [F1,6bP2], p represents [glycerone phosphate] and q represents [GraP]. The kinetic parameters are scarce, but all agree on the Km for F1,6bP2: around 0.3 mm[47–49]. Not many studies have been performed on the backward reaction, the only numbers found were from . Unknown is the inhibitory constant for GraP: it was estimated by Richter  and used by others (including us), but experimental data is lacking.
Glyceraldehyde-3-phosphate dehydrogenase: GraPDH
The enzyme GraPDH is an enigma. It is one of the most abundant enzymes in yeast . In S. cerevisiae, it has been found in the cytosol and in the nucleus, and in some related yeast species it appears also located at the cell wall . The enzyme appears to bind specific tRNAs .
The kinetics of the enzyme are renowned for their complexity. The enzyme forms monomers, dimers and tetramers that have different activities . Cooperative binding of NAD and NADH has been documented well [53,78,79]. However, the cooperativity is only obvious at pH > 7.5 . A complete data set of Km values under conditions very similar to ours exists in the literature ; the authors did not observe cooperativity either. We have therefore not included cooperative binding in the rate equation, but rather used reversible two-substrates, two-products Michaelis–Menten kinetics. We have measured the Vmax in either direction. Inhibition by the adenine nucleotides has been reported  and incorporated in some models [28,29], but the effects were very small with the inhibition constants and concentrations of the adenine nucleotides used in the model (result not shown). Therefore this inhibition has been left out.
When the Haldane relationship and our maximal rates and the affinities of Lambeir et al.  were used, the calculated equilibrium constant differed fivefold from the published equilibrium constant . The most suspect parameter in the rate equation of GraPDH is its affinity for 1,3GriP2. 1,3GriP2 was varied indirectly by in situ generation of 1,3GriP2 by PGK, using excess 3GriP and variable amounts of ATP. It was assumed that the ATP concentration added equaled the 1,3GriP2 concentration at the excess of auxiliary enzyme added . We have changed the uncertain Km of 1,3GriP2 (fivefold) so as to arrive at the known equilibrium constant for GraPDH.
Phosphoglycerate kinase: PGK
PGK is difficult to assay in the forward direction because of the instability of its substrate, 1,3GriP2. Many studies have been performed on the reverse reaction, however, with Km values for 3GriP in the range 0.2–0.5 mm[47,82–85], and for ATP between 0.1 and 0.5 mm[47,82–84,86]. Our values of 0.53 mm for 3GriP and 0.3 mm for ATP agree with those. The Km for 1,3GriP2 was estimated by the Ki as measured by . The Km for ADP was taken from .
Phosphoglycerate mutase: PGM
PGM activity in yeast depends on 2,3-diphosphoglycerate. We have not taken 2,3GriP2 into account, assuming that the enzyme is saturated with 2,3GriP2. The Km for 2,3GriP2 is in the low micromolar range , whereas the concentration of 2,3GriP2 was estimated to be 0.1 mm (result not shown). The Km of PGM for its substrate 3GriP differs considerably between studies. Km values from 2 mm to 0.2 mm have been reported [47,88–90]. We measured and implemented a Km value of 1.2 mm. For the product 2GriP, a Km value of 0.08 mm was used .
Two isoenzymes of enolase are present in yeast. Class I enolase is expressed under glucose derepressed conditions; class II enolase is expressed during growth on glucose . We measured a Km value for 2GriP that agrees well with that of class I enolase , in line with the derepressed state of the compressed yeast. Subsequently the Km of the product phosphoenolpyruvate was taken from class I enolase .
Pyruvate kinase: PYK
The most striking feature of pyruvate kinase is its strong activation by F1,6bP2. At the low F1,6bP2 concentrations prevailing during gluconeogenic conditions, the enzyme showed cooperativity with respect to phosphoenolpyruvate. At high F1,6bP2 concentrations, however, the enzyme exhibited hyperbolic Michaelis–Menten kinetics with increased affinity for phosphoenolpyruvate, in line with earlier findings [92,93]. We found that the activation is maximal at 0.5 mm of F1,6bP2, which is more than 10 times lower than the concentration that we have measured. We have therefore used Michaelis–Menten kinetics. Our Km values for phosphoenolpyruvate and ADP fit well with the values found in numerous studies [47,92–99]. The affinities of the products are much less studied: we have equated them to the dissociation constants measured by Macfarlane and Ainsworth .
Pyruvate decarboxylase: PDC
Pyruvate decarboxylase exhibits cooperative kinetics with respect to its substrate pyruvate [43,101]. Irreversible Hill kinetics were used to describe this cooperativity:
The cooperativity and affinity for pyruvate are phosphate-dependent [43,101]. We have adopted the K0.5 of 4.3 mm and the Hill coefficient of 1.9 as measured by Boiteux and Hess  at 25 mm phosphate. This K0.5-value is in line with the 4 mm found by  (see also ).
Alcohol dehydrogenase: ADH
ADH follows ordered bi-bi kinetics, with the cofactor binding first :
where a is [ethanol], b is [NAD], p is [acetaldehyde] and q is [NADH].
Of the five isoenzymes of ADH present in yeast, ADH-I and ADH-II have clearly established metabolic functions. We estimated the contribution of ADH-I and ADH-II to the total ADH activity by varying the ethanol concentration and measuring the initial rate of NAD reduction in cell-free extracts. As the two enzymes differ in their affinity for ethanol by one order of magnitude (ADH-II having the higher affinity ), the two components were expected to be visible in this way. Indeed, Eadie–Hofstee plots of ADH activity vs. ethanol showed curved kinetics, indicative of more than one component. When two components were fitted, Km values of 45 ± 10 and 0.7 ± 2 mm were found, with corresponding Vmax values of 2.8 ± 0.2 and 0.3 ± 0.2 U·(mg total protein)−1. Although the errors are substantial, especially in the high affinity component (corresponding to ADH-II), it can be concluded that the low-affinity component (corresponding to ADH-I) was the dominant isoenzyme under our conditions. It was checked that also the in vitro activity of ADH-I was tenfold higher than that of ADH-II when the measured metabolite levels (Table 1) were substituted in the rate equation for ADH. We have therefore used the kinetics of ADH-I only.
Kinetic parameters for ADH can be found in [102–104] We have chosen the data from Ganzhorn et al. as for each isoenzyme a complete data set including inhibition constants was given, under conditions very similar to ours . The affinities for ethanol found by us are in reasonable agreement with the ones reported by Ganzhorn et al. ( 17 and 0.8 mm for ADH-I and ADH-II, respectively).
For glycolysis to proceed, the net ATP produced by PGK and PYK should be consumed by ATP consuming processes. These processes are lumped into one general ATPase, whose activity is set to be dependent on ATP, according to experimental observations . A linear relation was used:
Branches of glycolysis
For simplicity, the fluxes to trehalose and glycogen were introduced as constants at the experimentally determined values. This is allowed because we here analyze one steady state. For further analysis, the substrate dependencies of these two branches will be required. For glycerol metabolism, it was assumed that glycerol 3-phosphate dehydrogenase completely controlled the flux through that pathway, as experimental evidence has shown that its control should be high [106–108]. Reversible Michaelis–Menten kinetics (Eqn. A3) were used, with glycerone phosphate and NADH as substrates and glycerol 3-phosphate and NAD as products. The glycerol 3-phosphate concentration was fixed at the measured value of 0.15 mm (Table 1).
The kinetics of G3PDH have not been studied extensively. We measured the affinity of G3PDH for glycerone phosphate, which was in agreement with that measured by Albertyn et al. . Kinetic constants for NAD and NADH were from . No data were found on the Km of G3PDH to glycerol 3-phosphate. We have adopted a Km value of 1 mm. The Vmax of G3PDH turned out to be very sensitive to the salt and protein concentration in the assay, and no reliable Vmax could be measured. Instead, we have adjusted the G3PDH Vmax values to the measured glycerol flux.
The formation of glycerol leads to a redox imbalance in glycolysis, as NADH is oxidized in the process . We have measured production of pyruvate, acetate and succinate. These weak carboxylic acids can account for 95% of the glycerol formation, with the production of succinate accounting for 80% of the glycerol production. For simplicity, we have only included a branch to succinate to counterbalance the glycerol branch. The origin of succinate is not clear; it can either be formed in the glyoxylate cycle or by part of the Krebs' cycle. As the latter involves both pyruvate and acetaldehyde as precursors and the former only acetaldehyde, we have chosen the glyoxylate cycle as the origin of succinate, for simplicity reasons only. As no kinetic data were available, the parameters had to be fit to the succinate production rate, and subsequently only stochiometric differences between the two alternative synthesis routes exist. The branch from acetaldehyde towards succinate via the glyoxylate cycle comprises many steps and was necessarily simplified by:
Appendix 2: kinetics of phosphofructokinase
PFK may be an enzymologists favorite, but it is a modelers nightmare. The difficulty is the many regulatory interactions and the resulting combinatorial explosion. Simplification is therefore required, and many effectors are necessarily assumed to be constant in the time window of the model, such as ammonium, phosphate, protons and fructose 2,6-bisphosphate. The regulatory effects that have been used explicitly in the existing glycolytic models are the cooperative binding of F6P, the inhibition by ATP (in some), the activation by AMP (in all, [26–29,66]) and ADP (only in ). The role of ADP in the regulation of PFK is much less important than those of ATP and AMP, and has not been included in our rate equation for PFK.
Product inhibition by F1,6bP2 was not included in any of the existing models. The main inhibitory action of F1,6bP2 is a decrease in the activation by F2,6bP2[111–113]. Therefore, a minimal kinetic model of PFK should be a function of the concentrations of F6P, ATP, AMP, F2,6bP2 and F1,6bP2. We have measured the activity of PFK in partially purified enzyme preparations as a function of F6P and ATP, and then looked at the effects of AMP and F2,6bP2. The inhibitory effect of F1,6bP2 was incorporated on the basis of data from Otto et al. . To our knowledge, no model is available that describes all these effects at the same time. We have successfully tried to fit the same model as was used by Galazzo & Bailey , Schlosser et al.  and Cortassa & Aon . It is based on the Monod, Wymann, Changeux model for allosteric enzymes, as adapted by Hess and Plesser  to apply to enzymes with two substrates.
The effect of AMP, F2,6bP2 and F1,6bP2, and the inhibitory effect of ATP are assumed to be mediated by displacement of the equilibrium between the Tense state and the Relaxed state, i.e. they affect the equilibrium constant L. It was assumed that the Tense state did not bind F6P and thus, that this state is inactive. The rate equation used to fit the in vitro kinetic data was therefore:
The kinetic parameters for F6P and ATP were estimated by nonlinear regression of over our 200 experimental data points of the rate of PFK as a function of the concentrations of ATP and F6P. Kinetic parameters are shown in Table 2. They are quite different from those used in the other models, most notably L0. One reason may be that ATP inhibition was explicit in our equation, where it is implicit (and therefore ATP-independent) in the L0 of the equation used by the other groups. If physiological ATP concentrations were substituted in the right hand side of Eqn. A12 (i.e. in the second factor in which ATP is involved), L increased by more than three orders of magnitude. Hofmann's group, however, have also found low L0 values [115–117], in good agreement with ours.
Activation of AMP was measured at different F6P concentrations and at an ATP concentration of 1 mm. These data were used to fit the binding parameters of AMP for the tense and relaxed state. The same procedure was followed for F2,6bP2 activation . F1,6bP2 decreases the activation of F2,6bP2, most probably by competition for the binding site. This can explain activation of PFK by F1,6bP2 in the absence of F2,6bP2 (results not shown; see [112,118]). Otto et al. showed an inhibition of F2,6bP2 activation by F1,6bP2. Their data was used to fit a model describing competition of F1,6bP2 and F2,6bP2 for the same site . The same mechanism was used by Kessler et al. to describe the inhibitory effect of F1,6bP2 on the activation by F2,6bP2. A model of inhibition by F1,6bP2 via competition with F6P could not describe the observed inhibition curves .
To our knowledge this is a first attempt to capture the effects of so many metabolites in a single rate equation. Our rate equation was able satisfactorily to describe the effects of the substrates F6P and ATP, the inhibition by ATP, the activation by AMP and F2,6bP2 and the inhibitory effect of F1,6bP2 on the activation by the latter.