Báo cáo khoa học: A kinetic study of a ternary cycle between adenine nucleotides

A kinetic study of a ternary cycle between adenine nucleotides Edelmira Valero1, Ramo´ n Varo´ n1 and Francisco Garcı´a-Carmona2

1 Departamento de Quı´mica-Fı´sica, Escuela Polite´ cnica Superior de Albacete, Universidad de Castilla-La Mancha, Albacete, Spain 2 Departamento de Bioquı´mica y Biologı´a Molecular A, Facultad de Biologı´a, Universidad de Murcia, Spain

Keywords enzymatic cycling; enzyme kinetics; moiety- conserved cycle; pyruvate kinase; S-acetyl coenzyme A synthetase ⁄ adenylate kinase

Correspondence E. Valero, Departamento de Quı´mica-Fı´sica, Escuela Polite´ cnica Superior de Albacete, Universidad de Castilla-La Mancha, Campus Universitario, E-02071-Albacete, Spain Fax: +34 967 599224 Tel: +34 967 599200 E-mail: Edelmira.Valero@uclm.es

The mathematical model described here has been submitted to the Online Cellular Systems Modelling Database and can be accessed at http://jjj.biochem.sun.ac.za/ database/valero/index.html free of charge

(Received 19 April 2006, revised 22 May 2006, accepted 8 June 2006)

doi:10.1111/j.1742-4658.2006.05366.x

In the present paper, a kinetic study is made of the behavior of a moiety- conserved ternary cycle between the adenine nucleotides. The system con- tains the enzymes S-acetyl coenzyme A synthetase, adenylate kinase and pyruvate kinase, and converts ATP into AMP, then into ADP and finally back to ATP. l-Lactate dehydrogenase is added to the system to enable continuous monitoring of the progress of the reaction. The cycle cannot work when the only recycling substrate in the reaction medium is AMP. A mathematical model is proposed whose kinetic behavior has been analyzed both numerically by integration of the nonlinear differential equations describing the kinetics of the reactions involved, and analytically under steady-state conditions, with good agreement with the experimental results being obtained. The data obtained showed that there is a threshold value of the S-acetyl coenzyme A synthetase ⁄ adenylate kinase ratio, above which the cycle stops because all the recycling substrate has been accumulated as AMP, never reaching the steady state. In addition, the concept of adenylate energy charge has been applied to the system, obtaining the enabled values of the rate constants for a fixed adenylate energy charge value and vice versa.

of amplifying a metabolic response against a signal, such as a change in a metabolic concentration. This phenomenon, called amplification or ultrasensitivity, has been experimentally proven to occur in binary closed cycles [6–9].

Abbreviations ACS, S-acetyl coenzyme A synthetase; AEC, adenylate energy charge; AK, adenylate kinase; LDH, L-lactate dehydrogenase; PEP, phosphoenolpyruvate; PK, pyruvate kinase; Pyr, pyruvate; ST, total adenylate substrate concentration.

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An important feature of intermediary metabolism is the existence of moiety-conserved cycles interconvert- ing different forms of a chemical moiety, while the sum of these forms remains constant [1,2]. The two best known groups of metabolites participating in such cycles are ATP–ADP–AMP (the moiety being the adenylate group) and NAD(P)–NAD(P)H (the oxidized and reduced forms of nicotinamide adenine dinucleo- tide). As in the case of substrate cycles [3,4], the occur- rence of cycling in closed (moiety-conserved) cycles [5] generally leads to an expenditure of energy, whereas there can be no changes in the total concentration of the converted substrates. The physiological role of this wasteful cycling has been proposed to be mainly a way The great sensitivity shown by cycles in metabolism has been applied in the laboratory to the quantitative determination of low levels of a metabolite or to the amplification of an enzymatic activity by coupling two bisubstrate enzyme-catalyzed reactions acting in oppos- ite directions [10,11] and in enzyme-linked immuno- assays [12–14]. Numerous kinetic studies about this reaction scheme have been performed [15–19] and even equations have been obtained for calculating enzyme

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quantities that minimize the cost of assays [20]. Sensi- tivity of the system can be further increased in several ways, such as using a 2 : 1 stoichiometry for the recyc- ling substrates [21] and double-cycling [11,22]. has previously been used by other authors to measure ACS activity in a continuous way [30–32], although the kinetic behavior shown by this multienzymatic sys- tem has not been studied.

The mathematical model described here has been sub- mitted to the Online Cellular Systems Modelling Data- base and can be accessed at http://jjj.biochem.sun.ac.za/ database/valero/index.html free of charge.

Kinetic analysis

The present paper addresses our investigation of the kinetic study of the behavior of a larger cycle involving three enzymes (a ternary cycle). Numerous metabolic loops involve at least three enzymes, including the tri- glyceride ⁄ fatty acids ⁄ fatty acyl-CoA cycle, the pyru- vate (Pyr) ⁄ oxaloacetate ⁄ phosphoenolpyruvate (PEP; or malate) cycle, the AMP ⁄ IMP ⁄ adenyl succinate cycle [23,24], the acetoacetyl-CoA ⁄ HMG-CoA ⁄ acetoacetate cycle [25] and the UTP ⁄ UDP ⁄ UDP glucose cycle con- nected to the glycogenn ⁄ glycogenn+1 cycle through gly- cogen synthase [3,26]. However, there are few reports in the literature dealing with the kinetic behavior of ternary cyclic systems [3,26–29] and, except one for one that includes an experimental illustration of the UTP ⁄ UDP glucose ⁄ UDP cycle [26], they are mainly devoted to theoretical considerations.

The experimental system chosen for the present study was a closed (moiety-conserved) ternary cycle in which ATP, ADP and AMP are the interconverted substrates. The converting enzymes involved in the sys- tem are adenylate kinase (AK; EC 2.7.4.3), pyruvate kinase (PK; EC 2.7.1.40) and S-acetyl coenzyme A synthetase (ACS; EC 6.2.1.1). The indicator reaction that enables the progress of the cyclic process to be followed is the coupling of l-lactate dehydrogenase (LDH; EC 1.1.1.27), where NADH consumption is measured with time (Scheme 1). This reaction scheme

The experimental system being studied here is depicted in Scheme 1. In this system, ATP is transformed into AMP in the presence of sufficiently high concentrations of acetate ions and coenzyme A by the catalytic action of the enzyme ACS. In the next step, two molecules of ADP are generated at each turn of the cycle from one AMP and one ATP catalyzed by AK. The cycle is closed by the conversion of one molecule of ADP to one mole- cule of ATP in the presence of a sufficient amount of PEP in a reaction catalyzed by the enzyme PK. LDH and NADH are added to the reaction medium to con- tinuously monitor the reaction. The reaction turns clockwise in the presence of a sufficient amount of acet- ate ions, coenzyme A, PEP and NADH. Note that the system cannot work when the only recycling substrate present in the reaction medium is AMP, as was experi- mentally and theoretically checked. Note also that this is a moiety-conserved ternary cycle [5], as the sum of [ATP], [ADP] and [AMP] remains constant during the whole course of the reaction (provided that adenylate levels bound to the enzymes involved in the cycle may be considered negligible against free adenine nucleotides concentration), i.e.

Scheme 1. Schematic representation of the ternary cycle under investigation interconverting the moiety ATP ⁄ ADP ⁄ AMP catalyzed by the enzymes ACS, AK and PK. Pyruvate is converted into L-lac- tate by the enzyme LDH, thus preventing the reversibility of the AK reaction and allowing the progress of the reaction to be continu- ously monitored. AcÆ, acetate ion; CoA, coenzyme A; AcCoA, acetyl coenzyme A; PPi, pyrophosphate; PEP, phosphoenolpyruvate; Pyr, pyruvate; Lac, L-lactate.

ð1Þ ST ¼ ½ATP(cid:1) þ ½ADP(cid:1) þ ½AMP(cid:1)

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To study the kinetics of the proposed reaction scheme, the following two assumptions, which can be easily implemented in the experimental conditions, were made: (a) Non-recycling substrates concentrations (acetate ions, coenzyme A and PEP) are sufficiently high to be saturating or remain constant during the reaction time. The same holds for NADH concentration. This assumption is common practice in enzyme kinetics, where to derive approximate analytical solutions corres- ponding either to the transient phase or to the steady state of an enzyme reaction, it is usually assumed that the substrate concentration remains approximately con- stant [21,33–35] and therefore the results obtained are only valid under these conditions. Taking into account that nonrecycling substrates and also NADH (the chromogenic substrate) are continuously consumed in the reaction medium from the outset of the reaction,

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the equations obtained here will be less accurate as the reaction time progresses. (b) During cycling,

system will reach a steady state. In this situation, the concentration of recycling substrates, [ATP]ss, [ADP]ss and [AMP]ss will be a constant value. So, making Eqns (2)–(4) equal to zero and taking into account condition the following expressions are obtained for the (1), concentration of adenine nucleotides attained in the steady state:

m;2 ½ATP(cid:1)ssÞ ½ATP(cid:1)ssðVm;2(cid:2)Vmapp;1Þ(cid:2)Vmapp;1K ATP

m;2 þVm;2Kmapp;1 ð8Þ

Vmapp;1ðKþK AMP the concentration of Pyr is clearly lower than its Michaelis–Menten constants towards the enzyme LDH, so that the reaction rate of the chromogenic step remains of the first order with respect to its concentration. This assumption is com- monly used in coupled enzyme assays when the rate of the chromogenic step is sufficiently high [36,37]. ½AMP(cid:1)ss¼

Under these conditions, the evolution of ATP, ADP and AMP concentrations with time is described by the following set of three differential equations: ð9Þ ½ADP(cid:1)ss ¼ ð2Þ d[ATP]/dt ¼ (cid:2)m1 (cid:2) m2 þ m3 2m1;ssKmapp;3 Vmapp;3 (cid:2) 2m1;ss

ð3Þ d[AMP]/dt ¼ m1 (cid:2) m2 ð10Þ ½ATP(cid:1)SS ¼ ST (cid:2) ½ADP(cid:1)SS (cid:2) ½AMP(cid:1)SS ð4Þ d[ADP]/dt ¼ 2m2 (cid:2) m3 being

ð11Þ m1;ss ¼ where v1, v2 and v3 are the velocities of the reactions catalyzed by ACS, AK and PK, respectively, being: Vmapp;1½ATP(cid:1)ss Kmapp;1 þ ½ATP(cid:1)ss

ð5Þ m1 ¼ Vmapp;1½ATP(cid:1)=Kmapp;1 þ ½ATP(cid:1) and

ð6Þ m3 ¼ Vmapp;3½ADP(cid:1)=Kmapp;3 þ ½ADP(cid:1)

Note that it is not possible to find AXP (X ¼ T,D,M) levels attained in the steady state as a function of the kinetic parameters of the system and initial conditions. Nevertheless, AMP concentration must be a finite pos- itive value so that the system can reach a steady state. From Eqn (8), it is easy to obtain the following condi- tion to attain a stationary situation:

> ð12Þ Vm;2 Vmapp;1 ½ATP(cid:1)ss þ K ATP m;2 ½ATP(cid:1)ss þ Kmapp;1

where Vmapp,i and Kmapp,i (i ¼ 1,3) are apparent con- stants for a fixed nonrecycling substrates concentra- tion, i.e. for a fixed concentration of acetate ions and coenzyme A in the case of ACS, and for a fixed con- centration of PEP in the case of PK. Vmapp,1 ¼ Vm,1 the reaction catalyzed by (the maximal velocity of ACS at the concentration used) and Kmapp;1 ¼ K ATP m;1 towards ACS if acetate ions and coenzyme A concen- trations are saturating, and Vmapp,3 ¼ Vm,3 (the max- imal velocity of the reaction catalyzed by PK at the concentration used) and Kmapp;3 ¼ K ADP m;3 towards PK if PEP concentration is saturating.

The basic kinetic pattern for AK has been reported to be random Bi Bi [38,39] so, assuming rapid equilib- rium for all binding and dissociation steps, the equa- tion corresponding to v2 will be:

m;2 ½AMP(cid:1) þ K AMP

m;2 ½ATP(cid:1) þ ½ATP(cid:1)½AMP(cid:1)

Vm;2½ATP(cid:1)½AMP(cid:1) ð7Þ m2 ¼ K þ K ATP This equation indicates that the system only will reach a steady state when the relationship between the enzymes AK and ACS is such that condition (12) is fulfilled. In all other cases, the system will operate until all the recycling substrate is accumulated as AMP, at which point the reaction will stop, never reaching the steady state, as was experimentally and theoretically checked. In addition, taking into account that [ADP]ss cannot take infinite (ST) or negative values by the own dynamics of the cycle, Eqn (9) indicates that Vmapp,3 must always be greater than 2v1,ss, as was checked by numerical integration.

where Vm,2 is the maximal velocity of the reaction cat- alyzed by AK at the concentration used and K is a constant value. As the catalytic activity of the cycle has been deter- mined using LDH as indicator enzyme, the differential the time-dependence of Pyr equation which gives concentration will be: The set of differential Eqns (2)–(4)

ð13Þ d[Pyr]/dt ¼ m3 (cid:2) m4

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where v4 is the velocity of the reaction catalyzed by LDH and, taking into account assumptions (a) and (b), it is given by: is nonlinear owing to the expression corresponding to v2 (Eqn 7), thus it cannot be analytically solved. This means that the kinetic behavior of the system must be studied by means of particular solutions obtained numerically. However, under certain experimental conditions, the

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ð14Þ m4 ¼ k4½Pyr(cid:1)

at the integration ([NADH]NI)

where k4 is an apparent first-order rate constant, with k4 ¼ Vmapp,4 ⁄ Kmapp,4, and the same explanation given above for Vmapp,i and Kmapp,i is valid. Making Eqn (13) equal to zero, the following expression is obtained for the steady-state rate of the cycle:

ð15Þ V ss ¼ V mapp;3½ADP(cid:1)ss K mapp;3 þ ½ADP(cid:1)ss

Taking into account that the reaction is followed by measuring the amount of NADH present in the reac- tion medium, the differential equation which gives the NADH consumption is:

ð16Þ d[NADH]=dt ¼ (cid:2)m4

initial

condition Integrating Eqn (16) with the [NADH] ¼ NADH0,s (the NADH concentration value when the steady state is reached) at t ¼ 0 (the start of the steady state) gives:

t ð17Þ ½NADH(cid:1)ss ¼ NADH0;s (cid:2) V mapp;3½ADP(cid:1)ss K mapp;3 þ ½ADP(cid:1)ss

errors of NADH values predicted from Eqn (18) ([NADH]) with regard to those obtained from the numerical times corresponding to a given depletion of NADH (44%, approximately the same that has been measured experimentally) at different ST concentrations. It can be seen that relative error increases as the ST-value increa- ses, and that the errors obtained for the steady-state rates were very small. It must be noted that final steady- state concentration values of adenine nucleotides were independent of their initial concentration values used at the same ST-value (except in the case when [AMP] ¼ ST at t ¼ 0, as has been mentioned above). However, at high initial concentrations of ADP it was necessary to increase the NADH0 value to reach the steady state owing to the coupling of LDH with the enzyme PK. As these conditions do not allow the experimental monitor- ing of the reaction progress in the spectrophotometer, we have preferred the input of ST-values as ATP0. It was also checked that Pyr levels attained in the steady state (data not shown) were clearly below its corres- towards LDH ponding Michaelis–Menten constant (30 lm with 100 lm NADH [11]), indicating that the assumptions made can be considered valid during the reaction time used in each case.

If the difference between NADH0 (the initial concentra- tion of NADH at the start of the reaction) and NADH0,s can be considered negligible, Eqn (17) becomes: Particular cases of the model

Case (a): [ATP]ss (cid:3)ST t ð18Þ ½NADH(cid:1)ss ¼ NADH0 (cid:2) V mapp;3½ADP(cid:1)ss K mapp;3 þ ½ADP(cid:1)ss

Table 1. Relative error of NADH concentration values and steady-state rates predicted from Eqns (18) and (15), respectively, with regard to the values predicted by numerical integration (Eqns A1–A5).

Time for 44% depletion of NADH (s)

Relative error (%)

Relative error (%)

ST (lM)

[NADH]NI (lM)

[NADH] (lM)

a Vss (lMÆs)1)

6.5 15 25 35 50 60 100 250 500 750 1000 1500

3480.5 1327.5 777.5 554.8 392.4 330.3 207.8 99.6 64.0 52.2 46.4 40.6

143.36 143.37 143.40 143.38 143.37 143.37 143.38 143.36 143.42 143.43 143.38 143.38

141.17 140.56 140.23 139.91 139.51 139.26 138.36 135.59 132.49 130.37 128.79 126.75

1.53 1.96 2.21 2.42 2.69 2.86 3.50 5.42 7.62 9.10 10.17 11.59

3.29 · 10)2 8.69 · 10)2 1.49 · 10)1 2.09 · 10)1 2.97 · 10)1 3.53 · 10)1 5.66 · 10)1 1.21 1.93 2.40 2.74 3.18

4.67 · 10)4 1.34 · 10)3 6.10 · 10)5 3.48 · 10)4 3.06 · 10)3 1.87 · 10)3 2.04 · 10)3 1.06 · 10)3 1.56 · 10)3 5.36 · 10)3 5.30 · 10)3 1.94 · 10)2

aVss values obtained from Eqn (15) and from numerical integration are the same as the significant numbers shown. The values of the rate constants used and initial concentration of NADH were as indicated in Fig. 1(A). [ADP]ss values for Eqns (15) and (18) were calculated using Eqn (9), inserting [ATP]ss values obtained from numerical integration. ST ¼ [ATP]0.

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The error committed by Eqn (18) will be greater as the rate of the cycle increases. Table 1 shows the relative In those cases in which ATP levels attained in the steady state are near to adenylate total concentration (ST-value), it is possible to know in an approximate

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way the adenine nucleotides concentrations attained in the steady state, since Eqns (8) and (11) can be rewrit- ten as follows:

m;2 STÞ STðVm;2 (cid:2) Vmapp;1Þ (cid:2) Vmapp;1K ATP

m;2 þ Vm;2Kmapp;1 ð19Þ

Vmapp;1ðK þ K AMP stants towards the corresponding enzyme, so that the reaction rates of the three steps of the cycle remain of the first order with respect to their respective concen- trations, Eqns (5) and (6) can be simplified to a rate law for first-order kinetics: ½AMP(cid:1)ss ¼ ð22Þ m1 ¼ k1½ATP(cid:1)

ð23Þ m3 ¼ k3½ADP(cid:1) In addition,

ð20Þ m1;ss ¼ where ki (i ¼ 1,3) are apparent first-order rate con- stants, with ki ¼ Vmapp,i ⁄ Kmapp,i. Vmapp;1ST Kmapp;1 þ ST Eqn (7) can also be simplified to the following expression: Eqn (12) becomes:

ð24Þ m2 ¼ k2½ATP(cid:1)½AMP(cid:1) > ð21Þ Vm;2 Vmapp;1 ST þ K ATP m;2 ST þ Kmapp;1 where k2 is an apparent second-order rate constant, being:

ð25Þ k2 ¼ Vm;2 K

if the following condition is fulfilled:

m;2 ½AMP(cid:1) þ K AMP

m;2 ½ATP(cid:1) þ ½ATP(cid:1)½AMP(cid:1)

K >> K ATP ð26Þ Eqn (21) allows the experimental determination of lev- els of the enzymes AK and ACS necessary to reach the steady state in these cases. It is a very important equa- tion to be taken into account when measuring ACS activity in the presence of a sufficient initial amount of ATP using this cycle as the coupled system [30–32].

If Eqns (2)–(4) are now made equal to zero, and taking into account condition (1), the following expressions are obtained for the concentration of adenine nucleo- tides attained in the steady state when the cycle oper- ates under first-order kinetics: relative errors for [ATP]ss,

ð27Þ ½ATP(cid:1)ss ¼ k3ðk2ST (cid:2) k1Þ k2ð2k1 þ k3Þ Table 2 shows the relative errors of [ATP]ss, [ADP]ss, [AMP]ss, Vss and [NADH]ss values predicted from equa- tions corresponding to this particular case with regard to those obtained from the numerical integration at dif- ferent relatively high initial ATP concentrations. It can [ADP]ss, be seen that [AMP]ss and Vss decrease as [ATP]0 increases, indicating that this approach can be used at relatively high initial concentrations of ATP, under these conditions.

ð28Þ ½ADP(cid:1)ss ¼ 2k1ðk2ST (cid:2) k1Þ k2ð2k1 þ k3Þ

Case (b): First-order kinetics with respect to adenine nucleotides concentration ð29Þ ½AMP(cid:1)SS ¼ k2=k1

[ADP]ss,

[ADP] and [AMP], [ATP], In those cases in which the concentration of the recyc- ling substrates, is clearly lower than their respective Michaelis–Menten con-

Table 2. Relative errors of [AMP]ss, Vss and [ATP]ss, [NADH]ss values predicted from equations corresponding to case (a) with regard to those obtained from the numerical integration (Eqns A1–A5) at different relatively high initial ATP concentrations. Conditions are as indicated in Fig. 1A. [ATP]0 ¼ ST.

These equations clearly indicate that, under these con- ditions, the cycle will only reach a steady state when k1 ⁄ k2 < ST. In all other cases, the system will operate until all the recycling substrate is accumulated as AMP, at which point the reaction will stop, never reaching the steady state, as was experimentally and theoretically checked. Eqn (18) now becomes:

Relative error (%)

ð30Þ ½NADH(cid:1)SS ¼ NADH0 (cid:2) k3½ADP(cid:1)SS (cid:4) t

[ATP]0 (lM)

[ATP]ss

[ADP]ss

[AMP]ss

Vss

[NADH]ss

and

100 250 500 750 1000 1500

3.44 1.99 1.17 0.79 0.57 0.35

3.41 1.96 1.14 0.77 0.55 0.34

6.30 7.06 8.60 9.77 10.66 11.89

8.41 · 10)2 4.16 · 10)2 1.96 · 10)2 1.11 · 10)2 6.88 · 10)3 3.28 · 10)3

4.02 · 10)2 4.89 · 10)2 4.50 · 10)2 3.73 · 10)2 3.04 · 10)2 2.08 · 10)2

ð31Þ VSS ¼ k3½ADP(cid:1)SS

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Table 3 shows the relative errors of [ATP]ss, [ADP]ss, [AMP]ss, Vss and [NADH]ss values predicted from equations corresponding to this particular case with regard to those obtained from the numerical integra- tion at different relatively low initial ATP concentra-

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[ADP]ss,

m

[AMP]ss, Vss and [ATP]ss, Table 3. Relative errors of [NADH]ss values predicted from Eqns (27)–(31) with regard to those obtained from the numerical integration (Eqns A1–A5) at different relatively low initial ATP concentrations. Conditions as indicated in Fig. 1A. [ATP]0 ¼ ST.

Relative error (%)

[ATP]0 (lM)

[ATP]ss

[ADP]ss

[AMP]ss

Vss

[NADH]ss

6.5 15 25 35 50 60 100

0.67 1.73 2.98 4.24 6.13 7.38 12.43

0.11 0.23 0.36 0.48 0.67 0.78 1.22

0.72 1.86 3.22 4.58 6.62 7.97 13.41

2.11 3.46 4.81 6.13 8.07 9.36 14.51

1.49 · 10)2 2.21 · 10)2 5.72 · 10)2 9.03 · 10)2 1.38 · 10)1 1.68 · 10)1 2.83 · 10)1

tions. There is good concordance between the analyt- ical and numerical solutions at relatively low ST-val- ues, because relative errors calculated at the times corresponding to a consumption of NADH as high as 44% are relatively small, decreasing at lower ST-values.

assumption (a) in the Kinetic analysis, curves were registered in all cases up to an absorbance value of 0.9 (143.5 lm NADH), a concentration much higher than towards LDH ((cid:3)1 lm with 3 lm the apparent K NADH Pyr [11]). An experimental progress curve in which the system cannot reach the steady state is shown in Fig. 1D, with an excess of ACS. The inset plots show the results obtained by HPLC analysis of the reaction medium before the start of the reaction, in the absence of ACS (chromatogram a) and at the end of the reac- tion (chromatogram b). The first chromatogram reveals the presence of the chemicals added to the reac- tion medium, PEP, ATP and NADH (retention time of coenzyme A was longer, so it was eluted in the cleaning of the column), and the presence of an small amount of ADP and AMP due to a contamination of ATP and NADH standard solutions (data not shown). It can be seen that at the end of the reaction, peaks corresponding to ATP and ADP have disappeared and the adenylate substrate has been accumulated as AMP. All of these results are in agreement with theo- retically predicted data, supporting the validity of the proposed model for the multienzymatic system under study.

Results and Discussion

Time course of the cycle Steady-state behavior of the system

Figure 1A,B shows the time progress curves obtained by numerical integration of the nonlinear set of differ- ential equations shown in the Appendix (which takes into account the depletion of NADH, but not the depletion of the nonrecycling substrates since their ini- tial concentrations were higher), using the rate con- stants set experimentally evaluated (see Experimental procedures) (Fig. 1A), and a set of rate constants pre- dicting that all recycling substrate will be accumulated as AMP in the steady state (Fig. 1B), under initial con- ditions similar to those used experimentally. It can be seen that in the first case, nucleotide concentration rea- ches a specific steady-state value after a small transient phase, with the disappearance rate of NADH varying in parallel. In contrast, the system cannot reach a steady state in the second case (Fig. 1B), as the reac- tion is stopped after all the recycling adenylate sub- strate has been accumulated as AMP.

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Figure 1C shows a selection of experimental pro- gress curves obtained at several different initial concen- trations of ATP under the conditions described in the Experimental procedures. It can be appreciated that the system reaches a steady state in which the NADH consumption rate is constant after a small transient phase, whose duration diminishes when increasing into account concentration. Taking ATP initial Figure 2A shows the experimental dependence of steady-state rates of the cycle obtained at different ini- tial concentrations of ATP. A hyperbolic dependence can be seen, in agreement with theoretically obtained data (Fig. 2C). Adenine nucleotides concentrations attained in the steady state under these experimental conditions are shown in Fig. 2B. It can be seen that ATP levels attained in the steady state increased line- arly when increasing ATP initial concentrations in the reaction medium (the inset plot). It can also be seen in this plot that under the experimental conditions used, at higher initial ATP levels, ATP concentrations attained in the steady state were near to ATP initial concentrations, with much lower ADP and AMP lev- els being attained in the steady state [particular case (a)]. Dependence of ADP levels attained in the steady state fit well to an hyperbolic equation, and AMP lev- els attained in the steady state lightly increased with ATP initial concentrations, in agreement with data obtained by computer simulation (Fig. 2D). We also checked by experiment at low initial concentrations of ADP as ST (in this case the reaction was started by the addition of ADP; data not shown), that final steady-state adenine nucleotides concentrations were the same than values obtained at the same ST-value when ST ¼ [ATP]0.

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A

B

C

D

m;2 ¼ 2.5 · 101 lM, K AMP

Fig. 1. (A) Simulated progress curves corresponding to the species involved in the reaction scheme shown in Scheme 1. The values of the 2, K ATP rate constants used were: Km app,1 ¼ 7.0 · 102 lM, Km app,3 ¼ 2.6 · 102 lM, K ¼ 7.1 · 104 lM m;2 ¼ 1.1 · 102 )1Æs)1. The initial concentrations values used lM, Vmapp,1 ¼ 2.3 lMÆs)1, Vm,2 ¼ 1.7 · 102 lMÆs)1, Vmapp,3 ¼ 6.5 · 101 lMÆs)1 and k 0 4 ¼ 5 lM were: [NADH]0 ¼ 256 lM and [ATP]0 ¼ 16.3 lM. (B) Simulated progress curves obtained using Vmapp,1 ¼ 3.3 · 101 lMÆs)1. The remaining conditions are as described in Fig. 1A. (C) Experimental progress curves of b-NADH consumption obtained for the ternary cycle under study. Conditions are as indicated in the Experimental procedures. The following initial concentrations of ATP were used for curves 1–8, respect- ively: 16.3, 26.1, 32.6, 39.1, 65.3, 200, 500 lM and 1 mM. (D) Experimental progress curve of NADH consumption obtained using a final con- centration of ACS ¼ 0.34 units and [ATP]0 ¼ 16.3 lM. The rest of conditions are as described in Fig. 1C. Insets: Chromatogram a, the reaction mixture was injected before the start of the reaction, in the absence of ACS; chromatogram b, the reaction mixture was injected at the end of the reaction.

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Figure 3A shows the experimental variation of the steady-state rate obtained when the cycle is run at dif- ferent ACS concentrations. We chose a low ST-value for the performance of the next set of experiments, due the nonrecycling substrates to less consumption of [assumption (a)] and greater simplicity at the theoret- [particular case (b)], although the results ical level obtained at higher ST-values would be similar when- ever in the assumptions performed were fulfilled, agreement with the equations obtained. It could be observed that the cycling rate in the steady state increases as the ACS concentration increases at relat- ively low levels of this enzyme. However, if we contin- ued to increase the ACS activity in the reaction

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Kinetics of a ternary cycle

A

B

C

D

Fig. 2. (A) Experimental steady-state rates obtained at different ATP initial concentrations in the reaction medium. Conditions as indicated in Experimental procedures. (B) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of ATP initial concentration in the reaction medium. Experimental conditions are as described in Fig. 2A. The points represent experimental data (they are the mean of three assays), the error bars represent SD and the lines correspond to regression analysis plot. (C) Theoretical steady-state rates obtained by numerical integration of differential Eqns (A1)–(A5) in the Appendix at different ATP initial concentration values. Conditions as indicated in Fig. 1A. (D) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of ATP initial concentration values. Conditions are as described in Fig. 2C.

which cannot be recovered by PK because there is not enough ATP for the enzyme AK.

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medium, the steady-state rate of the cyclic reaction would reach a maximum, after which it would decrease until it reached the point at which NADH consump- tion (and therefore the cyclic reaction) was abolished in agreement at relatively high ACS concentrations, with condition (12). This result was due to an excessive consumption of ATP at high ACS concentrations, Adenine nucleotide concentrations attained in the steady state as a function of ACS concentration levels in the reaction medium are shown in Fig. 3(B). It can be seen that as ACS activity was increased in the reac- tion medium, the levels of ATP attained in the steady

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Kinetics of a ternary cycle

A

B

D

C

Fig. 3. (A) Experimental steady-state rates obtained at different ACS concentrations in the reaction medium. Conditions as indicated in Experimental procedures, with [ATP]0 ¼ 16.3 lM. (B) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of ACS con- centration in the reaction medium. Experimental conditions are as described in Fig. 3A. The points represent experimental data (they are the mean of three assays) and the error bars represent SD. The straight line through [AMP]ss points corresponds to data obtained by linear regression analysis. (C) Theoretical steady-state rates obtained from Eqn (31) at different k1-values. Conditions are as indicated in Fig. 1A. (D) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of k1-values, obtained from Eqns (27)–(29). Conditions are as described in Fig. 3C.

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that steady-state rates are governed by the ADP con- centrations attained in the steady state. On the other hand, AMP concentrations in the steady state increase linearly with ACS activity until they approach the ST- value, i.e. all of the recycling substrate has been accu- mulated as AMP, at which point the cycle stops as there is no more ATP available for the enzyme AK. state decreased to reach a near zero-value at relatively high ACS levels, when the system is unable to reach the steady state. ADP levels attained at the steady state show an evolution parallel to the steady-state rates, with a maximum reached when the steady-state rate was maximum, and decreasing thereafter. This result is in agreement with Eqn (15), which predicts

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Kinetics of a ternary cycle

This result is in agreement with Eqn (29), which indi- cates that AMP concentrations attained in the steady state are directly proportional to the concentration of ACS in the reaction medium.

that there is a threshold value of k1

reaction reached a steady state even at low levels of PK activity. At very low levels of PK activity, this step becomes the limiting factor of the cycle, and ADP is accumulated in the reaction medium, although the cycle does not stop in this case, in agreement with Eqn (9). Steady-state rates increased as PK activity was increased until they reached a constant level, both experimentally (Fig. 5A) and theoretically (Fig. 5C). Figure 5B,D shows the steady-state concentrations of the three recycling substrates obtained both experi- mentally (Fig. 5B) and theoretically (Fig. 5D) as a function of PK activity in the reaction medium and k3-value, respectively. As can be observed, AMP final concentration was independent of k3, which agrees with Eqn (29), whereas ATP and ADP final concentra- tions increased and decreased, respectively, until they reached constant values. the three interconverted substrates

Adenylate energy charge Figure 3C shows the theoretical steady-state rates obtained when varying the k1-value. As can be seen, the steady-state rates were correctly predicted by the model, indicating that this is not an effect lying out- side the cycle. It is also possible to observe that when the ratio k1 ⁄ k2 ¼ ST-value (the last point in the plot), the system cannot reach the steady state and the reac- tion stops, which is in agreement with Eqn (29). This (or means Vmapp,1), above which the cycle cannot attain a steady state, this value being k1 ¼ k2 ST (when the cycle operates under first-order kinetics). Figure 3D shows the theoretical values of the steady-state concentra- tions of (Eqns (27)–(29)). The dependences obtained were parallel to those obtained experimentally.

the steady-state Adenine nucleotides constitute a well known group of metabolites participating in a moiety-conserved cycle in the intermediary metabolism. The role of these com- pounds in regulating metabolism has long been recog- nized and referred to as the adenylate energy charge (AEC) in the cell [40], which was defined through the following dimensionless parameter, varying between 0 and 1:

ð32Þ AEC ¼ ½ATP(cid:1) þ 0:5½ADP(cid:1) ½ATP(cid:1) þ ½ADP(cid:1) þ ½AMP(cid:1)

The AEC could be visualized in our model, as the three adenine nucleotides are involved in it. Thus, inserting Eqns (27)–(29) [we have used equations cor- responding to case (b) for greater simplicity; the AEC value for case (a) is (cid:3)1] into Eqn (32), the following expression is obtained for the AEC when the cyclic system being studied operates under steady-state condi- tions and under first-order kinetics:

ð33Þ AEC ¼ ðk1 þ k3Þðk2ST (cid:2) k1Þ k2STð2k1 þ k3Þ to a higher

Figure 4A shows the steady-state rates obtained experimentally when varying the AK concentration in the reaction medium. In this case, there was a threshold level of AK activity under which the sys- tem could not reach the steady state; this value was k2 ¼ k1 ⁄ ST, in agreement with Eqns (27)–(29). At higher AK concentrations rates increased to reach a constant value. ATP, ADP and AMP concentrations attained in the steady state are shown in Fig. 4B. It can be observed that when AK activity is not sufficient to reach a steady state, the recycling substrates accumulate as AMP, and when this occurs, the cyclic reaction is stopped (mathemat- ically this would be [AMP] ¼ ST when t fi ¥). At higher levels of AK activity, the AMP concentrations attained in the steady state decreased with increasing k2 (or Vm,2), while ATP and ADP final concentra- they reached a near constant tions increased until value, varying parallel in to the steady-state rate, agreement with Eqn (31). It was also experimentally (data not shown) and theoretically checked (Fig. 4C) that an increase in ACS activity in the reaction level of AK activity medium leads (k2-value) being necessary for the system to reach a steady state. Figure 4D shows the theoretical values of the steady-state concentrations of the three inter- converted substrates. It can be seen that dependences predicted by the model were very similar to those obtained experimentally. This equation also predicts that k2ST > k1 so that the system can attain a steady state. It is not possible to draw a plot of Eqn (33) as a function of k1, k2 and k3 to illustrate the dependence of AEC upon them. How- ever, for a fixed AEC value and total recycling sub- strate concentration, ST, the following expression is obtained, for example, for k2:

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ð34Þ k2 ¼ k2 1 þ k1k3 ST½k1 þ k3 (cid:2) AECð2k1 þ k3Þ(cid:1) Figure 5A shows the steady-state rates obtained experimentally when varying the PK concentration in the reaction medium. It can be seen that the response of the system was different, since in this case the

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Kinetics of a ternary cycle

B

A

c

C

D

b

a

Fig. 4. (A) Experimental steady-state rates obtained at different AK concentrations in the reaction medium. Conditions are as indicated in the Experimental procedures, with [ATP]0 ¼ 16.3 lM. (B) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of AK con- centration in the reaction medium. Experimental conditions are as described in Fig. 4A. The points represent experimental data (they are the mean of three assays) and the error bars represent SD. (C) Theoretical steady-state rates obtained from Eqn (31) at different k2-values. The values used for Vmapp,1 were as follows: curve a, 2.3, curve b, 4.7 and curve c, 10.5 lMÆs)1. The rest of the conditions are as indicated in Fig. 1A. (D) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of k2-values, obtained from Eqns (27)–(29). Conditions are as described in Fig. 4C, with Vmapp,1 ¼ 2.3 lMÆs)1.

AEC < ð35Þ k1 þ k3 2k1 þ k3

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Figure 6 shows a three-dimensional plot of Eqn (34) for a fixed ST-value and two different values of AEC, 0.8 (a) and 0.9 (b), which are normal values in a num- ber of tissues and organisms [41]. It can be seen that infinite or positive This function leads to negative, (the only situation with biological meaning) k2-values when the AEC value is higher, equal or smaller, respectively, than the relationship k1 + k3 ⁄ (2k1 + k3). This indicates that the AEC values allowed with given k1- and k3-values are those that fulfill the following condition:

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Kinetics of a ternary cycle

A

B

C

D

Fig. 5. (A) Experimental steady-state rates obtained at different PK concentrations in the reaction medium. Conditions are as indicated in the Experimental procedures, with [ATP]0 ¼ 16.3 lM. (B) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of PK con- centration in the reaction medium. Experimental conditions are as described in Fig. 5A. The points represent experimental data (they are the mean of three assays) and the error bars represent SD. The straight line through [AMP]ss points corresponds to data obtained by linear regression analysis. (C) Theoretical steady-state rates obtained from Eqn (31) at different k3-values. Conditions as indicated in Fig. 1A. (D) Steady-state ATP (d), ADP (s) and AMP (.) concentrations as a function of k3-values, obtained from Eqns (27)–(29). Conditions are as des- cribed in Fig. 5C.

there are pairs of k1 and k3 values (relatively small val- ues of k3) that are not allowed by fixed AEC values, leading to negative or infinite k2-values.The k3 ⁄ k1 rela- tionship enabled for a fixed AEC value can easily be derived from Eqn (35), with the following result:

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> ð36Þ 2AEC (cid:2) 1 1 (cid:2) AEC According to Eqn (36), when AEC 6 0.5, the relation- ship between the rates of formation and destruction of ATP (k3 and k1) can take any value; however, for AEC > 0.5, not all values of k1 and k3 are allowed, as can be deduced from Eqn (35). It is obvious that a minimum level of ATP regeneration with respect to ATP consumption is necessary to keep a high AEC value, i.e. the coupling between ATP-yielding and k3 k1

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Kinetics of a ternary cycle

Spectrophotometric readings were obtained on a Uvikon 940 spectrophotometer from Kontron Instruments (Zu¨ rich, Switzerland). The time course of the reaction was followed by measuring the disappearance of b-NADH at 340 nm (e340 ¼ 6270 m)1Æcm)1) at 37 (cid:1)C. The temperature was maintained using a Hetofrig Selecta (Barcelona, Spain) cir- culating bath with a heater ⁄ cooler and checked using a Cole-Parmer (Vernon Hill, IL, USA) digital thermometer with a precision of ± 0.1 (cid:1)C.

Methods

Fig. 6. Three-dimensional plot of k2 against k1 and k3 values (Eqn 34) for a fixed ST-value (16.3 lM) and the following AEC values: (a) 0.8 and (b) 0.9.

Individual enzyme activity measurements

The initial activity dependence of PK as a function of ADP was studied by using the Pyr assay coupled with the enzyme LDH. Measurements were performed in the presence of 256 lm NADH, 1.6 mm PEP, 75 mm KCl, 2 mm MgCl2, 10 mm sodium acetate, 0.05–1.1 mm ADP in 50 mm imidazole ⁄ HCl buffer and 5.3 U of LDH, (pH 7.6). The reaction was started by the addition of 0.03 U of PK, the final volume being 0.5 mL. Rate data versus initial ADP concentration thus obtained were fitted by nonlinear regression to the Michaelis–Menten equa- values of Kmapp,3 tion, providing the corresponding (0.26 mm) and Vmapp,3.

ATP-demanding processes is controlled by the AEC value.

m;2 (0.025 mm), K AMP m;2

Interconverting cycles are increasingly referred to as regulating structures of the metabolism. In the present paper, we have illustrated the particular kinetic behav- ior of a closed (moiety-conserved) ternary cycle between ATP, ADP and AMP. It has been shown that the ratio between the enzymatic activities involved in the cycle cannot take any value, but some conditions must be fulfilled to prevent accumulation of adenosine in the form of AMP. The results obtained herein may contribute to our knowledge of the behavior of cycles in metabolic regulation.

Experimental procedures

The initial activity dependence of AK as a function of ATP and AMP was studied by using the PK- and LDH- coupled assay. Measurements were performed in the pres- ence of 256 lm NADH, 1.6 mm PEP, 75 mm KCl, 2 mm MgCl2, 10 mm sodium acetate, 0.09–1.53 mm ATP, 0.11– 1.28 mm AMP and 5.3 U of LDH, in 50 mm imidaz- ole ⁄ HCl buffer (pH 7.6). The reaction was started by the addition of 4.1 U of PK and 0.04 U of AK (premixed), the final volume being 0.5 mL. The rate data versus initial sub- strate concentrations thus obtained were fitted by nonlinear regression to the Michaelis–Menten equation, obtaining from the secondary replots of these data the corresponding (0.11 mm) and Vm,2. The values of K ATP fulfillment of condition (26) was also checked at low ATP and AMP levels, giving a value for K of 0.071 mm2.

The initial activity dependence of ACS as a function of ATP was studied by using the AK-, PK- and LDH-coupled assay [30–32]. Measurements were performed in the pres- ence of 256 lm NADH, 1.6 mm PEP, 75 mm KCl, 2 mm MgCl2, 10 mm sodium acetate, 0.6 mm coenzyme A, 0.2– 1.65 mm ATP, 3.1 U of PK, 3.1 U of AK, 5.3 U of LDH and 3.8 · 10)3 U of ACS, in 50 mm imidazole ⁄ HCl buffer (pH 7.6). The reaction was started by the addition of ATP, the final volume being 0.5 mL. Rate data versus initial ATP concentration thus obtained were fitted by nonlinear regression to the Michaelis–Menten equation, obtaining the corresponding values of Kmapp,1 (0.75 mm) and Vmapp,1, tak- ing into account that two moles of NADH are oxidized per mole of substrate.

b-NADH, coenzyme A, AMP, ADP, ATP, sodium PEP, imidazole, PK (448 UÆmg)1) from rabbit muscle, AK (2020 UÆmg)1) from rabbit muscle, LDH (1120 UÆmg)1) from rabbit muscle, ACS (8.5 UÆmg)1) from baker’s yeast (Saccharomyces cerevisiae) and BSA were obtained from Sigma (Madrid, Spain). Stock solutions of enzymes (207.0, 210.1, 526.4 and 1.15 UÆmL)1, respectively) were prepared daily in 50 mm imidazole ⁄ HCl acid buffer (pH 7.6) contain- ing 2 mgÆmL)1 bovine serum albumin. All other reagents were of analytical grade and were used without further purification. All solutions were prepared in ultrapure deion- ized nonpyrogenic water (Milli Q; Millipore, Madrid, Spain).

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Reagents

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To obtain rate data, the experimental progress curves obtained in each case were fitted by linear regression to a first-order polynomial equation of the reaction time using the sigmaplot scientific graphing system version 8.02 (2002, SPSS Inc., Chicago, IL, USA).

Computer simulation

Simulated progress curves were obtained by numerical solu- tion of the nonlinear set of differential Eqns (A1)–(A5) des- cribed in the Appendix, which takes into account the depletion of NADH, using the experimentally obtained val- ues of the rate constants and the same initial concentration values. Numerical integration was performed by means of the fourth- and fifth-order Runge–Kutta–Fehlberg formula [42], using the ode45 function from matlab software version 6.5 (http://www.mathworks.com). The data thus obtained were plotted using the sigmaplot scientific graph- ing system for windows version 8.02 (http://www.spss.com).

Operation of the complete cycle

Acknowledgements

The kinetic studies dealing with the complete ternary cycle were carried out in a reaction medium containing 256 lm NADH, 1.6 mm PEP, 75 mm KCl, 2 mm MgCl2, 10 mm sodium acetate, 0.6 mm coenzyme A, ATP and ⁄ or ADP at the indicated concentrations, 3.1 U of PK, 3.1 U of AK, 5.3 U of LDH and 0.023 U of ACS in 50 mm imidaz- ole ⁄ HCl buffer (pH 7.6). The reaction was started by the addition of ACS; the final volume was 0.5 mL. Steady-state rate data were obtained by linear regression fitting to a first-order polynomial equation of the reaction time of the linear portion of experimental progress curves, using the software mentioned above.

The work described in this paper was supported by a grant from the Direccio´ n General de Investigacio´ n del Ministerio de Ciencia y Tecnologı´ a (Spain), Project num- ber BQU2002-01960, and by a grant from the Consejerı´ a de Educacio´ n y Ciencia de la Junta de Comunidades de Castilla-La Mancha, Project number PAI-05–036. HPLC nucleotides determination

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Appendix

m2 ½AMP(cid:1) þ K AMP m2

¼ (cid:2)

0

d½ATP(cid:1) dt

Vmapp;1½ATP(cid:1) Kmapp;1 þ ½ATP(cid:1)

¼ d½AMP(cid:1) dt Vmapp;1½ATP(cid:1) Kmapp;1 þ ½ATP(cid:1) Vm;2½ATP(cid:1)½AMP(cid:1) (cid:2) ½ATP(cid:1) þ ½ATP(cid:1)½AMP(cid:1) K þ K ATP The system of differential equations corresponding to the cyclic mechanism shown in Scheme 1, taking into account the depletion of NADH, is as follows: ðA3Þ

4½Pyr(cid:1)½NADH(cid:1)

Vm;2½ATP(cid:1)½AMP(cid:1)

(cid:2)

K þ K ATP

½ATP(cid:1) þ ½ATP(cid:1)½AMP(cid:1)

m2 ½AMP(cid:1) þ K AMP m2

0

¼ (cid:2) k ðA4Þ d½Pyr(cid:1) dt Vmapp;3½ADP(cid:1) Kmapp;3 þ ½ADP(cid:1)

4½Pyr(cid:1)½NADH(cid:1)

ðA1Þ

þ

Vmapp;3½ADP(cid:1) Kmapp;3 þ ½ADP(cid:1)

ðA5Þ ¼ (cid:2)k d½NADH(cid:1) dt

4½NADH(cid:1) corresponds to k4 in the text.

m2 ½AMP(cid:1) þ K AMP m2

where k0 2Vm;2½ATP(cid:1)½AMP(cid:1) ¼ d½ADP(cid:1) dt ½ATP(cid:1) þ ½ATP(cid:1)½AMP(cid:1) K þ K ATP

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(cid:2) ðA2Þ Vmapp;3½ADP(cid:1) Kmapp;3 þ ½ADP(cid:1)