19148
J. Phys. Chem. 1996, 100, 19148-19152
Bistability in Coupled Open Substrate Cycles: Numerical and Experimental Approaches Emilie Simonet, Christian Bourdillon, and Jean-Franc¸ ois Hervagault* Laboratoire de Technologie Enzymatique, URA 1442 du CNRS, UniVersite´ de Compie` gne, BP 529, 60205 Compie` gne Cedex, France
Michel Gervais Centre de Ge´ ne´ tique Mole´ culaire, UPR 2420 du CNRS, 91190 Gif sur YVette, France ReceiVed: June 13, 1996; In Final Form: October 2, 1996X
The dynamic and steady-state behaviors of two open substrate cycles sharing a common interconversion enzyme are investigated in a homogeneous flow-through reactor. Lactate dehydrogenase (LDH) converts pyruvate and NADH into lactate and NAD, respectively. In turn, NAD (+ formate) is recycled into NADH (+ CO2) by formate dehydrogenase (FDH), and in the presence of the oxidized form of 2-(hydroxymethyl)6-methoxy-1,4-benzoquinone (Q), lactate is reoxidized into pyruvate (+ Qred) by flavocytochrome b2 (FCytb2). When operating under thermodynamically open conditions by a continuous supply of pyruvate, quinone, NADH, and formate, this multienzyme system can exhibit multiple steady states under the form of dynamic hysteresis when using, among others, the pyruvate input concentration as the control parameter. This nonlinear behavior results from the strong inhibition of LDH exerted by its substrate pyruvate. The numerical predictions of a simple mathematical model, taking into account the coupling between the actual enzyme rate equations and mass transfers, agree both quantitatively and qualitatively with the observed experiments.
Introduction Substrate cycles are topological enzyme organizations ubiquitously distributed throughout metabolic pathways. They take place in many instances such as equivalent reduction and oxidation, synthesis and degradation, and in the generation of energy. The covalent modification of proteins by interconverting enzymes such as kinases and phosphatases is also an important feature of cycles involved in signalling pathways. The roles played by cycles (futile or not) in the regulation of metabolism is still a matter of controversy. However, from experimental data and theoretical considerations, it is likely that substrate cycling is physiologically involved in thermogenesis, in the orientation of fluxes, in the control of concentrations, and in the amplification of sensitivity.1 When dealing with minimum cycles where a reaction intermediate either triggers its own synthesis (product activation) or inhibits its own degradation (substrate inhibition), more sophisticated steadystate and dynamic behaviors such as multistability (alternative steady states), sustained temporal oscillations, and chaotic motion may be expected from theoretical considerations.2 The more familiar feature related to bistability is the emergence of reversible hysteretic transitions between two functionally different branches of steady states in response to a change in a control parameter value: these abrupt transitions may occur for two different threshold values of the parameter considered (limit points, LPs). Thus, in addition to their ability to store information (short-term memory), such bistable substrate cycles can be regarded as powerful regulatory devices with dual capacities by either supporting the maintainance of metabolite levels and fluxes (homeostasis) or amplifying sensitivity (switching effect) in response to a regulatory signal.3 Moreover, under realistic conditions, these non-self-releasing triggers may lose their ability to switch reversibly from one stable steady state to the other. If one of the limit points either disappears * To whom all correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01734-0 CCC: $12.00
or is no longer physically accessible, then a transition between the two branches of steady states is only possible one time, and from that moment on, the system is unable to trace back its previous history even when the constraints are released and selecting thus one mode of operation in an irreversible manner.4 Such reversible and irreversible hysteretic transitions were observed experimentally in isolated model reconstituted binary and ternary substrate cycles.5 When dealing with cellular metabolism, glycolysis can be viewed as a prototype of nonlinear dynamic pathways. Hysteretic transitions, damped and sustained oscillatory motions, and irreversible transitions of the glycolytic intermediates were observed experimentally in yeast and muscle whole cells, in cell-free extracts, and in reconstituted (homogeneous) enzyme systems.6 Surprisingly, if these potential nonlinear behaviors, in general, and multiplicity of steady states, in particular, exhibited by enzymes and multienzyme complexes are still seldom studied and investigated by biochemists, some chemical reactions, such as the Belousov-Zhabotinsky, BriggsRauscher, and peroxidase reactions, aroused a great deal of interest and are as yet well documented.7 In the present report, the dynamic and steady-state behaviors of two open substrate cycles sharing a common interconverting enzyme are investigated. The system operates under thermodynamically open conditions by continuous in- and outfluxes of substrates and products (homogeneous flow-through reactor). Materials and Methods Chemicals and Enzymes. NADH, L-lactate, pyruvic and formic acids, phenylhydrazine, and L-lactic dehydrogenase [from rabbit heart, isoenzyme H4 (EC 1.1.1.27)] were purchased from Sigma. 2-(Hydroxymethyl)-6-methoxy-1,4-benzoquinone was obtained from Aldrich and formate dehydrogenase (EC 1.2.1.2) from Boehringer Mannheim. Flavocytochrome b2 (L-lactate cytochrome c oxidoreductase, EC 1.1.2.3) from Hansunela anomala was purified according to methods previously described.8 All solutions were freshly prepared in 0.1 M phosphate buffer, pH 7.0. © 1996 American Chemical Society
Bistability in Coupled Open Substrate Cycles
J. Phys. Chem., Vol. 100, No. 49, 1996 19149 mental conditions), NAD is reduced in turn into NADH by formate dehydrogenase, FDH, with the production of CO2. The oxidized form of the quinone, Qox, and lactate are reoxidized and reduced into pyruvate and Qred, respectively, by flavocytochrome b2, Fcytb2. Therefore, the evolution of the various metabolite concentrations in the reactor can be described by a set of three differential equations, set to zero at steady state, and taking into account the actual reaction rates, as determined under our experimental conditions, and the various metabolite fluxes throughout the reactor.
Figure 1. Model bicyclic multienzyme system (see text for details).
Kinetic Assay. All kinetic measurements were performed on a Hewlett Packard 8452 diode-array spectrophotometer thermostated at 25 °C. L-Lactate dehydrogenase (LDH) and formate dehydrogenase (FDH) activities were determined by following the change in the NADH concentration at 340 nm (340,NADH ) 6230 M-1 cm-1). The flavocytochrome b2 (FCytb2) activity was measured by following the absorption decrease of 2-(hydroxymethyl)-6-methoxy-1,4-benzoquinone9 at 374 nm (374 ) 974 M-1 cm-1). Reactor. Experimental data were obtained using a thermostated stirred flow-through reactor (3 mL) thermostated at 25 °C and maintained under a continuous N2 atmosphere to facilitate the removal of CO2 produced by FDH. The separate inputs of enzymes and substrates was achieved by four different channels (enzymes, NADH, quinone, pyruvate + formate) with equal flow-rate controlled by peristaltic pumps (30 µL min-1, i.e. residence time ) 100 min). The outflux solution was passed continuously through a spectrophotometer cuvette to measure the absorbances at 340 and 408 nm for further determination of the NADH and quinone concentrations from the following relationships, with 340,Q ) 681 and 408,Q ) 681 M-1 cm-1:
[NADH] ) (A340 - A408)/340,NADH and [Q] ) A408/408,Q The pyruvate concentration was also determined (end point method) from absorbance measurements at 374 nm after 1 min incubation with 0.2 M phenylhydrazine. Preliminary controls have confirmed that there are no detectable cross reactions between phenylhydrazine, quinone, formate, and lactate. Numerical Procedures. The bifurcation diagrams (detection and continuation of singularities) were computed by using the software package AUTO9 complemented with a Rosenbrock method of integration.10 Results and Discussion Description of the Model. The bicyclic multienzyme system under study is depicted and detailed in Figure 1. All reactions take place in a flow-through reaction chamber maintained at a constant volume, i.e. with equal in- and outflux flow-rates. The inlet solution contains only pyruvate (Prv), NADH, and quinone (Qox, oxidized form) with fixed concentrations ([Prv]0, [NADH]0, and [Q]0) along with the enzymes. The outlet solution contains all the metabolites (and enzymes). Within the reaction chamber, pyruvate and NADH are oxidized into lactate (Lac) and NAD, respectively, by lactate dehydrogenase, LDH. Pyruvate inhibits its own degradation by LDH (minus sign). In the presence of formate at saturating concentrations (>10Km, under our experi-
{
d[Prv] ) -VLDH + VFCytb2 + R([Prv]0 - [Prv]) dt d[NADH] ) -VLDH - Vi + VFDH + dt R([NADH]0 - [NADH]) d[Q] ) -VFCytb2 - Vi + R([Q]0 - [Q]) dt
(1)
Provided that there is no influx of lactate, NAD, and Qred within the reactor, equation system (1) is complemented with the following mass conservation equations,
{
[Prv]0 ) [Prv] + [Lac] [NADH]0 ) [NADH] + [NAD] [Q]0 ) [Q] + [Q]red
(2)
where R stands for the residence time within the reactor [)(in and out) flow-rate/volume] and subscript 0 refers to the influx concentrations. Derivation of the Individual Enzyme Rate Equations. L-Lactic dehydrogenase catalyses reversibly11 the reaction
NADH + pyruvate T NAD + lactate with an equilibrium constant equal to 1.8 × 10-6 at pH 7. Under our experimental conditions, the enzyme operates irreversibly toward the right-hand side (oxidation and reduction of NADH and Prv, respectively). Indeed, the further oxidation and reduction of the products Lac and NAD, respectively, proceed through irreversible enzymic steps (see below). The enzyme follows a sequential ordered mechanism with substrate inhibition by pyruvate:
VM,LDH
VLDH ) 1+
KmNADH [NADH]
+
KmPrv [Prv]
+
KiaKmPrv [NADH][Prv]
+
[Prv] KIPrv
(3)
where VM,LDH is the maximal activity proportional to enzyme concentration, KmNADH and KmPrv (13 and 29 mM, respectively) are the Michaelis constants for NADH and pyruvate, respectively, and KIPrv (2 mM) is the inhibition constant for pyruvate. Kia (83 µM) is a complex kinetic constant. FlaVocytochrome b2 oxidizes lactate (Lac) into pyruvate in the presence of 2-(hydroxymethyl)-6-methoxy-1-4-benzoquinone (Q, oxidized form) and is inhibited noncompetitively by pyruvate:
VFCytb2 )
VM,FCytb2 KmLac KmQ KmLacKmQ [Prv] + + 1+ 1+ KIPrv [Lac] [Q] [Lac][Q]
(
)(
)
(4)
where VM,FCytb2 is the maximal activity of the enzyme; Kmlac and KmQ (0.25 mM and 60 µM, respectively) are the Michaelis constants for lactate and quinone, respectively.
19150 J. Phys. Chem., Vol. 100, No. 49, 1996
Simonet et al.
Figure 2. Continuation of limit points in the VLDH/[Prv]0 parametric plane [equation system (1)]. The loci of limit points delineate the bistability domain (shaded area). VLDH ) 0.33 mM min-1, VFCytb2 ) 5 µM min-1, VFDH ) 0.45 mM min-1, and R ) 10-2 min-1.
Formate dehydrogenase catalyzes12 the reaction
formate + NAD f CO2 + NADH Under our experimental conditions, where the flux of N2 through the reactor is enough to ensure a complete removal of the CO2 produced by the reaction, this latter enzyme operates irreversibly toward the production of NADH. In addition, the enzyme is inhibited competitively by NADH. In the following rate expression, the concentration of formate is not taken explicitely into account since its influx concentration is much larger than its Michaelis constant with respect to the enzyme:
VFDH )
VM,FDH KmNAD
(
)
[NADH] + 1+ 1+ KINADH [NAD]
(5)
where VM,FDH is the maximal activity of the enzyme; KmNAD and KINADH (0.2 mM and 54 µM, respectively) are the Michaelis and the inhibition constants for NAD and NADH, respectively. In (1), Vi accounts for the nonenzymatic interaction between NADH and quinone (Q). This “interfering” reaction13 is first order with respect to both substrates and can be written as
Vi ) Kr[NADH][Q]
(6)
with Kr ) 18 µM-1 min-1. Steady-State Behavior of the System: Numerical Predictions. Basically, numerous parameters may control the emergence of alternative stable steady states in such a cyclically coupled multienzyme system, including the influx metabolite and moiety-conserved concentrations, the flow-rates, and the interconverting enzyme maximal activities (or ratios of). In the present report, we will restrict our study to the pyruvate influx concentration, [Prv]0, and the lactic dehydrogenase maximal activity, VM,LDH. The continuation of limit points in the [Prv]0/VM,LDH plane (Figure 2) shows that the system may exhibit multistability in a large domain of the parameter values. The two loci of limit points coalesce at VM,LDH and [Prv]0 equal to 50 µM min-1 and 4 mM, respectively. The domain of bistability with respect to [Prv]0 increases with increasing VM,LDH up to 0.75 mM min-1 and then remain almost constant. Indeed, for higher LDH activities, this latter enzyme is in large excess with respect to the other ones. Also noticeable is that the domain shrinks with decreasing VFDH activity from 0.45 to 0.15 mM min-1 and disappears (monostability domain) for still lower values and that it is not significantly altered with changes in the VFCytb2 activity
Figure 3. Time evolution of pyruvate, NADH (b), and quinone ([) concentrations as calculated from the absorbance measurements in the spectrophotometer cuvette. The full square at t ) 350 min represents the steady-state concentration of pyruvate calculated from end-point measurements. The influx and initial concentrations (within the reactor) are identical for each substrate, i.e. [Prv]0 ) [Prv]t)0 ) 7.5 mM, [NADH]0 ) [NADH]t)0 ) [Q]0 ) [Q]t)0 ) 0.5 mM. In these experiments and the following ones (Figures 4 and 5), the first measurements of NADH and quinone concentrations are made at t ) 10 min (vertical dashed line). This constraint is imposed by the time delay generated by the fluidic circuit between the inside of the reactor and the measurement cuvette. The solid lines through the data points were calculated by integration of (1) with VLDH ) 0.33 mM min-1; VFCytb2 ) 5 µM min-1, VFDH ) 0.45 mM min-1, and R ) 10-2 min-1.
(not shown). As it affects the dynamic behavior of the whole system in a similar way as does the influx pyruvate concentration, the flow-rate might also have been chosen as the control parameter. However, from a purely experimental point of view, its value ranging around some tens of microliters per min, reliable determination and reproducibility cannot be fairly granted. Experimental Observations Numerous generic experiments were performed under various conditions (external and initial metabolite concentrations) in order to test both the robustness of the numerical method and, more importantly, the validity and reliability of the model. Figure 3 illustrates one such experiment, where the system is expected to evolve toward a unique stable steady state. During the first 100 min, the evolution of the various metabolite concentrations is mainly controlled by the activity of LDH, which consumes pyruvate and NADH to produce lactate and NAD. Due to the low activity of FCytb2 in addition to its poor affinity with respect to lactate, the recycling yield of pyruvate is maintained at a very low rate. At the same time, the decrease in the pyruvate concentration has the effect of removing gradually the inhibition of LDH. These two processes act synergetically to speed up the depletion in NADH (and pyruvate). When the concentration of pyruvate comes close to zero after about 100 min, the NADH concentration increases again. In that second step, the final steady-state NADH and quinone concentrations depend mainly on the FDH and FCytb2 activities, respectively. The occurrence of two successive limiting steps during the evolution of the metabolite concentrations within the reactor (VLDH, for t < 100 min, and VFDH, for t > 100 min) is confirmed by the numerical integration of (1). It is also shown that the lower the VLDH rate, the longer the first step. Existence of Two Stable Steady States (Time Evolution). The experiment shown in Figure 4 demonstrates that given a set of parameter values granting multistability, the (two) alternative steady-state regimes can be actually observed. All
Bistability in Coupled Open Substrate Cycles
Figure 4. Experimental evidence for multistability: time evolution of the NADH (b) and quinone (3) concentrations from two differnt initial pyruvate concentrations within the reactor (internal perturbation). In the two experiments, the influx and initial concentrations for NADH and quinone are identical as well as the influx pyruvate concentration, i.e. [NADH]0 ) [NADH]t)0 ) [Q]0 ) [Q]t)0 ) 0.5 mM and [Prv]0 ) 15 mM. The final steady-state concentration for each metabolite depends upon the initial pyruvate concentration within the reactor: [NADH] ≈ [Q] ) 0.15 mM and [Prv] ) 14 mM for [Prv]t)0 ) 0 (A) and [NADH] ) 0.3 mM, [Q] ) 0.1 mM, and [Prv] ) 0 for [Prv]t)0 ) 15 mM (B). The solid lines through the data points were calculated by integration of the equation system (1) with VLDH ) 0.33 mM min-1, VFCytb2 ) 5 µM min-1, VFDH ) 0.45 mM min-1, and R ) 10-2 min-1.
influx metabolite concentrations (external parameters) being kept constant, two experiments are made with different initial concentrations of a selected metabolite (internal variable). Thus, under a constant influx pyruvate concentration ([Pyr]0 ) 15 mM), its initial concentration within the reactor will determine which final steady state is reached by the different metabolites. For [Prv]t)0 ) 0 (Figure 4A) the lower branch of steady states with respect to each substrate is attained, i.e. low concentrations of quinone (≈0.12 mM), NADH (≈0.12 mM), and pyruvate (≈0). On the other hand, for [Prv]t)0 ) 15 mM (Figure 4B) the higher branch is attained with high concentrations of NADH (≈0.3 mM) and pyruvate (≈14 mM). Transition between Two Branches of Steady States (Time Evolution). Figure 5 illustrates a transition between the two alternative branches of stable steady states when changing the influx pyruvate concentration from 0 to 20 mM (see legend for details). Steady-State Behavior. The experimental determinations of pyruvate, NADH, and quinone steady-state values as a function of the influx pyruvate concentration (0 < [Pyr]0 < 30 mM) are summarized in Figure 6. These data were obtained from experiments carried out under the conditions as shown in Figure 4 (internal perturbations) and fit fairly well the numerically computed S-shaped continuation curves.
J. Phys. Chem., Vol. 100, No. 49, 1996 19151
Figure 5. Illustration of a transition between two branches of stable steady states when changing the influx pyruvate concentration, [Prv]0 (external perturbation). The influx and initial metabolite concentrations are as follows: [NADH]0 ) [NADH]t)0 ) [Q]0 ) [Q]t)0 ) 0.5 mM and [Prv]0 ()[Prv]t)0) ) 0. The various metabolites rapidly reach their steady-state concentration, i.e. [NADH] ≈ 0.45 mM, [Q] ) 0.15 mM, and [Prv] ) 0. At t ) 70 min (black arrows) the influx pyruvate concentration is increased to 20 mM. If the quinone concentration (3) remains almost constant to a final (steady-state) concentration of about 0.15 mM, NADH (b), on the other hand, decreases abruptly to 0.15 mM and increases again to a final concentration of 0.35 mM. The pyruvate steady-state concentration is close to its influx value, i.e. ≈15 mM (9, end-point measurement). The solid lines through the data points (A and B) were calculated by integration of equation system (1) with VLDH ) 0.33 mM min-1, VFCytb2 ) 5 µM min-1, VFDH ) 0.45 mM min-1, and R ) 10-2 min-1.
Discussion and Conclusion In this report we have studied at both numerical and experimental levels the dynamic and steady-state behaviors of two substrate cycles sharing a common interconversion enzyme operating in a homogeneous flow-through reactor. The mass transfer and individual enzyme rate equations were combined to derive a simple mathematical model. A thorough continuation analysis in parametric space did show that the system could exhibit bistability with respect to numerous parameters, including, among others, the influx pyruvate concentration. These predictions were confirmed experimentally, both for steady-state and evolution measurements. As already stressed in the Introduction, examples of experimental multistability in homgeneous cyclic systems are seldom seen in the biochemical literature. This work is then a contribution to the study and analysis of nonlinear properties and dynamic behaviors of complex substrate cycles, as theoretically predicted by many authors.6 One of the raisons d’eˆtre of a bistable system is its ability to exist in two alternative functional steady-state regimes, depending upon its short-term history (initial conditions). Such a property, if it is a powerful way to regulate the metabolite concentration levels (stabilization and/or orientation of fluxes), may also be a good device to use in the control of energetic needs. Indeed, the rather low pyruvate and NADH concentrations belonging to the lower steady-state branches of continuation (see Figure 6) result from a high LDH activity. On the other hand, the upper (steady-state) branches are mainly
19152 J. Phys. Chem., Vol. 100, No. 49, 1996
Simonet et al. others for by simple entrainment reasons. This remains true when one of the cycles exhibits a nonlinear dynamic behavior resulting from the introduction of a destabilizing factor (e.g. substrate inhibition). Recent results dealing with the introduction of an electrochemical recycling of the oxidized form of the quinone show that the amplitude of the bistability domains with respect to the influx pyruvate concentration is retained but shifted toward lower values of the parameter. More importantly is that irreversible transitions may also occur when varying the yield of the electrochemical recycling of the quinone (article in preparation). References and Notes
Figure 6. Pyruvate (A), NADH (B), and quinone (C) steady-state concentrations as a function of the influx pyruvate concentration. Each experimental steady-state solution (full and empty points) was obtained from experiments carried out under the conditions as shown in Figure 4. The full and dashed lines (stable and unstable solutions, respectively) through the experimental points were calculated from (1) with VLDH ) 0.33 mM min-1, VFCytb2 ) 5 µM min-1, VFDH ) 0.45 mM min-1, and R ) 10-2 min-1; [Q]0 ) [NADH]0 ) 0.5 mM.
controlled by the activity of FDH, with a low consumption of these substrates (or a favorable balance between their production and use). Therefore, under identical enzyme activities and metabolite influx, the whole system may “choose” to operate at low energy cost in response to emergency situations, for example. The bicyclic multienzyme system studied here is topologically similar to a set of geared wheels where the concentration of each metabolite (and moiety) is totally interdependent on the
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JP961734U
