Molecularand CellularBiochem&try115: 43-48, 1992. © 1992KluwerAcademic Publishers. Printedin the Netherlands.

A kinetic description of sequential, reversible, Michaelis-Menten reactions: practical application of theory to metabolic pathways Stephen P.J. Brooks and Kenneth B. Storey Departments of Biology and Chemistry, Carleton University, Ottawa, Ontario, K1S 5B6, Canada Received 26 November 1991; accepted10 March 1992

Abstract Equations are presented which describe a linear coupled system of reactions that utilize a single substrate and convert it to product by way of several intermediate enzyme catalysed steps. The present analysis extends previous results by assuming that the enzymes obey reversible Michaelis-Menten kinetics. In order for the system to reach steady state one must assume that the initial substrate concentration and the final product concentration are buffered to a constant value. Using the present analysis it can be shown that the system will not enter a steady state if the maximal velocity of any forward reaction is less than the steady state flux through the system. This condition represents a practical test for determining if a system will enter steady state but is valid only when the rate of the primary enzyme is not affected allosterically be intermediates in the pathway. The equations are used to analyse a portion of the rat liver glycogenic pathway that catalyses the conversion of glucose to fructose 1,6-bisphosphate. (Mol Cell Biochem 115: 43-48, 1992) Key words: coupled enzymes, steady state, transition times, metabolism

Introduction Analysis of linear systems of enzymes, which catalyse monomolecular reactions (with respect to the intermediates in the pathway) has so far been confined to either (a) pseudo-first-order kinetic reactions which may or may not be reversible [1, 2], or (b) irreversible (unidirectional with respect to flux) sequences of MichaelisMenten enzymes [3, 4]. The reasons for this are twofold: firstly, if the primary reaction is not irreversible the equations describing the intermediates are complex, and secondly, most uses of coupling enzyme theory are confined to minimization of lag times or transition times in order to measure kinetic parameters of a primary enzyme [5]. Unfortunately, the equations re-

suiting from these analyses are not useful for studying a series of metabolic enzymes where the concentrations of coupling enzymes (enzymes subsequent to the primary enzyme) are not in large kinetic excess with respect to the rate of the primary enzyme in the sequence. Often, application of a kinetic theory to a linear enzyme system requires that the system is in steady state before one measures overall flux through the pathway. A good example of this is the use of metabolic control theory to study rate controlling enzymes; here the system must be at steady state prior to measurement of flux-control parameters and elasticity coefficients [6]. Presently, no practical test exists to determine if a linear

Address for offprints: K.B. Storey,Departmentof Biology,CarletonUniversity,Ottawa, Ontario, KIS 5B6, Canada

44 system of non-first order enzymes is in steady state. This is especially important when metabolic pathways are kinetically isolated in vitro by use of specific substrates and coupling enzymes because it is important that all enzymes participating in the pathway be clearly identified. The present work outlines methods to ensure that a steady state has been achieved and provides equations to completely describe the kinetic parameters of linear enzyme systems. The resulting equations should be evaluated prior to application of any kinetic theory to a series of enzymes measured under in vitro conditions.

Theory

minimal amount of S is used during the time course of the experiment. If the reaction velocity is measured sufficiently early during the time course, [P] = 0 so that a constant steady state flux can be maintained. Alternately, the production of P can be coupled to coupling enzymes which are themselves irreversible to ensure this condition. At steady state the concentrations of the intermediates are constant and so: 0 = d[SsJ/dt = d[f1,~J/dt . . . . . d[In,ss]/dt

(3)

and

In general, a linear series of n enzyme reactions that obey Michaelis-Menten kinetics can be represented by Scheme 1. Scheme 1: Linear sequence of enzymes with back reactions. Jo Ja J2 J(j~) Jj J(ns,.D Jn S --~ I1-~---I2 ~ . . . . Ij . . . . . Ir, ~-~--P Where the net flux through each individual enzyme locus, Jj, is the difference between the forward and backward reactions.

J~ = J0 = J1 . . . . .

Jj = J + j - J_j

(1)

When the reactions are reversible Michaelis-Menten enzymes, the flux through each step is given by [7]:

V_jK+jI(j+I )

JJ= K_jK+j+ I0+~)K+j+ IiK_ j

(2)

In Equation 2, V+j and K+j represent the maximal velocity and Michaelis constant respectively for the forward direction, and V_j and K_j represent the corresponding values for the backward direction. Fulfillment of the steady state condition for Scheme 1 requires that the substrate and product concentrations are fixed at constant values. This condition is necessary so that, at steady state, d[S]/dt = d[P]/dt = 0. Note that, if both [S] and [P] are fixed at finite, non-zero concentrations, the flux at steady state will reflect the overall equilibrium value of the [S] ~ [P] reaction. If [P] = 0, the final reaction is treated as irreversible. In vitro, S may be buffered to remain constant over the time course of the experiment or the concentration of S may be much greater than the flux through the pathway so that a

Jj . . . . .

J.

(4)

Combining Equations 2 and 4 permits one to define the steady state concentrations of the intermediates in terms of the kinetic constants of the reactions and the overall flux through the system:

[Iss]j = ~j + [Pss]" I] (ti + ~'~ {]~i" iSI1 (Xk} i=j i=j+l k=j where: cti = K + i ( V _ i +

(5)

Jss)/(K_i'(V+i- Jss))

[~i = Jss'K+i/(V+in

V+jK_jIj --

d[Ii,sj/dt . . . . .

Jss)

= total number of intermediates (excluding initial substrate and product).

Equation 5 presents a quick method for calculating the concentrations of the intermediates when the steady state flux and kinetic constants of the individual enzymes are known. Alternately, if the concentration of any intermediate is known, Equation 5 can be solved for the overall flux by a numerical method. Although Equation 5 was developed for reversible MichaelisMenten reactions it can be used to predict steady state concentrations for systems that include mixtures of reversible and irreversible reactions. In this case, one simply sets ai = 0 for the irreversible reactions and proceeds to solve for the intermediates. Inspection of Equation 5 shows that unless the inequality of the Equation 6 is fulfilled,

V+i > Jss

(6)

the concentrations of the intermediates are not defined (they are negative numbers) and a steady state is not achieved. In fact, fulfillment of Equation 6 provides a

45 quick and easy test to determine if a chain of sequential enzymes will achieve steady state. This test is especially useful if one is studying control of metabolic pathways in crude cell isolates (see Results and discussion section). Equation 5 shows that the capability of the system to achieve steady state depends on both (a) the maximal velocities of each enzyme in the sequence and (b) the K m value for each enzyme in the sequence. This result extends the previous analysis of Easterby [2] who defined transition times for a linear system of reversible first-order reactions. In the case of first-order reactions, the equations predict that a steady state could be defined for any value of the first-order rate constants; no kinetic constraints were placed on the steady state when the reactions were first-order. In the case of reversible Michaelis-Menten enzymes, the present analysis shows that a steady state may not be achieved without fulfillment of Equation 6. Another useful test for defining a linear sequence of enzymes involves calculating the transition time for the system [1-4]: •=

~ ~i = ~ [I~li/J~s

i=l

i=1

(7)

and comparing calculated values with those determined experimentally. Differences in the calculated and experimental x values are evidence for either (i) substrate channelling (which must be confirmed by other tests) or for (ii) the presence of enzymes in the sequence which participate in the overall reaction but have not yet been accounted for in the mathematical analysis.

Results and discussion The above equations were developed to characterize a linear system of equations that metabolize glucose to fructose 1,6-bisphosphate in vitro. This system has been previously analyzed by metabolic control theory to determine control strengths of the individual enzymes in the sequence [8-10]. Metabolic control theory highlights the sensitivity of overall pathway flux to very small changes in the concentrations of individual metabolites and to very small changes in the activities of individual enzymes of the pathway. This can be expressed mathematically as: CJEj = (dJ/dvj. vj/J)ss = (dlnJ/dlnvj)s~

(s)

where J is the overall flux through the system, vj is the activity of enzyme Ej and CJEj is the flux-control coefficient [11]. By definition, summation of the flux-control coefficients for any particular pathway containing n enzymes gives a value of 1.0 [12], or CJEi = 1

(9)

i-1

For closed systems at steady state, the experimentally determined sum of flux-control coefficients is approximately 1.0 lending credence to Equations 8 and 9. Fluxcontrol coefficients are thus a measure of how much each individual enzyme contributes to limiting the overall flux through the pathway [11-14]. Application of Equations 8 and 9 to in vitro linear enzyme systems gives

Jss =

Cjl"Ej/(Cj2-t- Ej)

(10)

where Cjl and Cj2 are complex expressions including enzyme parameters and equilibrium constants for each individual enzyme Ej [6]. Combination of Equations 8 and 10 gives a simple expression for determining fluxcontrol coefficients of individual enzymes CJzj = ( C i l - J)/Cjl

(11)

with Cjl values obtained from double reciprocal plots of flux versus E1 from Equation 15 [8-10]. In a recent study [10] Equations 10 and 11 were used to calculateflux-control coefficients for enzymes in a rat liver extract in vitro. The enzyme steps were kinetically isolated into the three step sequence as shown in Scheme 2. In Scheme 2, the three primary enzymes of glycolysis, glucokinase (GK), phosphoglucoisomerase (PGI) and phosphofructokinase (PFK), catalyse the conversion of glucose into fructose 1,6-bisphosphate. These enzymes were isolated kinetically by (a) adding glucose directly to enzyme-containing homogenates, and (b) adding a large excess of aldolase, triose phosphate isomerase and cx-glycerophosphate dehydrogenase. Scheme 2: Three enzyme system of Torres et al. [10]. J0 J1 J2 Glucose ~ G6P ~-- F6P ~ FDP Where J0 represents the overall flux through glucokinase, Jx represents the flux through PGI and J2 repre-

46 sents the flux through PFK. In the original analysis [10] Scheme 2 was restricted by adding i mM ATP, 2.5 mM creatine phosphate and 5 IU/ml of creatine kinase to maintain a constant ATP concentration [10]. Under these conditions, J-0 (see Equation 1) is negligible (reverse glucokinase reaction does not occur because ADP is limiting) so that J~ = J+0 = forward rate of GK. Limiting ADP concentrations and the presence of a large excess of coupling enzymes (aldolase, triosephosphate isomerase and a-glycerophosphate dehydrogenase) means that the final reaction is irreversible (reverse PFK reaction does not occur) so that J2-- J+2 (J-2 = 0). Scheme 2 can thus be reduced to a simple linear coupled enzyme system. Application of Equation 5 to Scheme 2 under the conditions of Torres et al. [10], and assuming that the velocity of PFK can be adequately described by a Michaelis-Menten equation, gives [G6el~s =

I1 =

81 "~- [Pssl'Gtl'~2 +

= 81 q- 82"~1 [F6P]ss = I 2 =

82 +

[P~]'a2 =

82"0[1 (12)

82

(13)

Note that, under the conditions of the reaction, [Pss] = 0 and % = 0. The Vm,x and K m values for the individual reactions, used to calculate individual forward and backward fluxes, are presented in Table 1. In the experiments of Torres et al. [10], glucose concentrations were varied (5, 20, and 100 mM) to produce different values of Js~. In the present analysis we calculated Jss using the values of V+0 and K+0 found in Table 1 and assumed that J+0 was constant during the time course of the experiment. Torres et al. [10] also titrated the system of Scheme 2 with increasing concentrations of hexokinase, PGI, or PFK and measured the observed velocity after 2 to 3 min. The observed velocity was then plotted against activity of added enzyme according to Equation 15 to obtain estimates of Cjl and Cj2. The value of Cil was then used to calculate CJEi values for fixed initial fluxes. Using the maximal enzyme activities determined for the individual enzymes and the literature Km values it was possible to obtain estimates of the steady state concentrations of F6P and G6P at increasing concentrations of glucose (see Equations 12 and 13). These values were subsequently used to calculate the transition time for the overall reaction (Equation 7). At steady state, the overall flux (Jss) is equal to the rate of the GK reaction so that Jss is directly proportional to the concentration of glucose in the assay. A plot of • versus

glucose concentration, presented in Fig. 1, shows that the reaction entered steady state very slowly even at very low initial glucose concentrations (x = 17.1min). Furthermore, as glucose concentrations increased above 30 mM (x = 90 min), T increased rapidly to infinity. Figure 1 also shows that at 50mM glucose, ~ is undefined (infinitely large) because the condition of Equation 9 is no longer fulfilled; at this point log(V+2/ Jss) = 0. These calculations indicate that rate measurements at times less than 1 h (for [glucose]0 = 5 mM) or less than 4hr (for [glucose]0 = 20mM) would not give correct steady state velocities [8]. At glucose concentrations -> 50 mM the system would not enter steady state. These calculations were confirmed by numerical integration (4th order Runga-Kutta; [15]) which showed that the observed velocity increased for i h (5 mM) or 4 hr (20 raM) after the initiation of the reaction (data not shown). Inspection of Equation 6 and numerical analysis of Scheme 2 also revealed that, when the primary reaction is irreversible, the flux-control coefficient for the initial reaction must be equal to 1.0 and the CJEivalues for all subsequent (coupling) enzymes in the sequence must be equal to 0 by definition (Equation 11). This is true because, for the system of Scheme 2 to achieve steady state, the primary enzyme m u s t be rate limiting (Jss < V+j for all values of j). This analysis applies only to systems in which (a) the primary enzyme is an irreversible step, (b) there is no feed-back inhibition, and (c) Table 1. Maximal activities and kinetic parameters of enzymes from rat liver homogenates

Enzyme

Activity/ml 1

K m value 2

GK PGI forward reverse PFK

V+o= 0.105mM/min

K+o = 20mM

V < = 0.405mM/min V 1 = 0.454mM/min V+_, = 0.075 mM/min

K+I = K 1= K+2 =

0.03mM 0.01mM 0.2raM

1Values were taken from Torres et al. [8]. The original assay conditions, in 2ml of assay, were: 50mM HEPES, 10mM sodium phosphate (pH 7.4), 100raM KCI, 10raM MgC12, 1raM ATP, 0.28raM N A D H , 2.5 m M phosphocreatine, 15/zM fructose 2,6-bisphosphate, 5 IU creatine kinase/ml 1 IU aldolase/ml, 5 IU tfiosephosphate isomerase/ml, and 3 IU ~x-glycerolphosphate dehydrogenase/ml [8]. Activities were measured at 35 ° C and are based on 1.5 mg/ml protein in the reaction cuvette [10]. Maximal activities measured by Tortes et al. [8] were: 0.07+ 0.002IU/mg (GK), 0.27+ 0.008IU/mg (PGI), and 0.05 + 0.015 IU/mg (PFK). Maximal enzyme activities are expressed in concentration units ( l I U / m l = 1 mM/min). 2Values taken from Barman [18] and Dunaway [19]. The value for V_ 1 was calculated based on the Haldane relationship: Keq = V1.K_J(V_I.K1).

47

'

500

i



i

,

i

'

i

'

i

2.0

i

i

[

i

2.6

2 O

m

400

1.5

0

1.0

+ ,,a

2.4 c-

r"

3OO

i

E v

2O0

E

+

NI

2.2

f__ U~ UI

r__ UI

100 0 0

10

20

,.30

0.5

2.0

0.0

1.8

40

[gJ cos ] (raM) Fig 1. Kinetic parameters of Scheme 2 as a function of glucose concentration. The transition time and the ratio of the maximal velocity of the terminal reaction to the overall flux (V+JJss) were calculated as a function of increasing glucose concentrations. The x value was obtained using Equations 7, 12, and 13 and the kinetic constants of Table 1.

there is no feed-forward activation of the individual enzymes. Although the above analysis suggests that the system of Torres et al. [10] did not enter steady state at high glucose concentrations, and was not at steady state during their measurements at low glucose concentrations, their data suggest that a true steady state was achieved. We thus postulated that an unidentified reaction must be involved in the reactions in Scheme 2 which would enable the system to enter steady state. The best candidate for this enzyme is glucose 6-phosphatase (G6Pase) which dephosphorylates glucose 6-phosphate to give glucose and inorganic phosphate. G6Pase is abundant in mammalian liver with a maximal velocity of approximately 0.16IU/mg [16, 17] and a Km value of 0.42mM [18]. Including this reaction (see Scheme 3) immediately resolved the system into a steady state at all glucose concentrations as shown in Fig. 2; • was defined and V+2 > Jss for all glucose concentrations. S c h e m e 3." Proposed four enzyme system. J+l

J+0

Glucose

> G6P ~

~

J+2

F6P ~

FDP

'-'-1

J+3

glucose Note that Scheme 3 now contains a branch point at G6P so that application of Equation 5 to the system requires knowledge of the flux both before and after the branch point. Thus, the concentrations of the intermediates

I o g (V + 2 / d ss)

0

20

40

60

80

1 O0

Eglucose] (me) Fig. 2. Kinetic parameters of Scheme 3 as a function of glucose concentration. The transition time and the ratio of the maximal velocity of the terminal reaction to the overall flux (V+2/Jss) were determined for increasing concentrations of glucose. The ~ value was calculated using Equations 5 and 7 using Jbeforeinstead of Jss. The kinetic constants of the individual reactions are presented in Table 1.

were calculated by setting Jbefore = Jafter + J+3 and using either Jbefore o r Jaft, (in place of Jss in Equation 5) to calculate the intermediate concentration depending on its position relative to the branchpoint. The concentration of G6P was calculated using the Jb~foreflux value. The transition time value was calculated by substituting Jbefo~efor Jss in Equation 7. The inclusion of a fourth enzyme which competes directly with PGI for G6P shows that Equations 10 and 11 do not apply to Scheme 2 in rat liver homogenates: the kinetically isolated reaction sequence cannot be modeled by a linear set of reactions but should include a branch point to produce a 'dead end' product. In this system, glucose can be considered as a 'dead end' product because it does not affect the rate of glucokinase (glucose concentration is in large excess with respect to the amount of product produced during the time course of the experiment). The dependence of Jaft~ (the steady state flux through PFK) on the activities of the individual enzymes is illustrated in Fig. 3. Here the relative velocity is plotted as a function of relative enzyme concentration for each enzyme in the pathway. It is obvious from inspection of Fig. 3 that the flux is dependent on the rate of three enzymes: GK, PFK and G6Pase. The sensitivity of Scheme 3 to changes in enzyme concentrations can also be demonstrated by calculating flux control coefficients for the individual enzymes of our mathematical model. These values are simply the slopes of tangents to the curves of Fig. 3 and are: 1.030 for GK, - 0.865 for G6Pase, 0.019 for PGI, and 0.816 for PFK. The large dependence of the reaction velocity on G6Pase activity demonstrates that Equations 10 and 11

48

1.0 -%

0.5

i



i

.

'~

~

.

i

.

.........G6Pase

-Z 0.0

_ _P~

.

0"~

.

.

.

.

"

-0.5

-'"'..... "'.........,

PF'Kj "~'. _

--1.0 ~ I

-1.0

GK I

-0.5

I

0.0

[

0.5

I

1.0

I o g ( E / E o) Fig. 3. Relative flux through PFK (Jafter, s e e text) as a function of changing enzyme concentrations: numerical simulation. Jaftervalues were calculated at different concentrations of GK (solid line), PGI (long dashed line), PFK (short dashed line) or G6Pase (dotted line). Individual enzyme concentrations (E) were varied relative to the in vivo values of Table 1 (E0). The ratio of the flux at the new enzyme concentration (Jafter) to the flux at the original enzyme concentration (J0) is plotted as a function of the E/Eo ratio. Flux values were calculated by numerical integration of Equation 2.

do not adequately describe the overall flux through the primary sequence of glycolytic reactions in rat liver. The above calculations illustrate the importance of using Equations 5 and 7 prior to kinetic evaluation of isolated systems of linear reversible enzyme reactions. The usefulness of these equations was demonstrated by highlighting inconsistencies between predicted and experimentally observed results in studying kinetic patterns of rat liver cell free extracts. Application of these equations may also prove useful in verifying the linearity of other cell free extracts and become an integral part of an overall kinetic characterization of enzyme systems.

Acknowledgements

The authors wish to acknowledge an NSERC (Canada) operating grant to K.B.S. and the editorial criticisms of J.M. Storey and Dr. M. Guppy.

References 1. Easterby JS: Coupled enzyme assays: A general expression for the transient. Biochim Biophys Acta 293: 552-558, 1973

2. Easterby JS: A generalized theory of the transition time for sequential enzyme reactions. Biochem J 199: 155-161, 1981 3. Brooks SPJ, Espinola T, Suelter CH: Theory and practical application of coupled enzyme reactions: One and two auxiliary enzymes. Can J Biochem Cell Biol 62: 945-955, 1984 4. Brooks SPJ, Espinola T, Suelter CH: Theory and practical application of coupled enzyme systems: One and two coupling enzymes with mutarotation of an intermediate. Can J Biochem Cell Biol 62: 956-963, 1984 5. Brooks SPJ, Suelter CH: Practical aspects of coupling enzyme theory. Anal Biochem 176: 1-14, 1989 6. Kascer H, Burns JA: Molecular Democracy: Who Shares the controls? Biochem Soc Trans 7: 1151-1160, 1979 7. Cornish-Bowden A: Fundamentals of Enzyme Kinetics. Butterworth & Co., London, 1979, pp 31 8. Torres NV, Mateo F, Melendez-Hevia E, Kacser H: Kinetics of metabolic pathways. A system in vitro to study the control of flux. Biochem J 234: 169-174, 1986 9. Tortes NV, Souto R, Melendez-Hevia E: Study of the flux and transition time control ceofficient profiles in a metabolic system in vitro and the effect of an external stimulator. Biochem J 260: 763-769, 1989 10. Torres NV, Mateo F, Rioi-Cimas JM, Melendez-Hevia E: Control of glycolysis in rat liver by glucokinase and phosphofructokinase: Influence of glucose concentration. Mol Cell Biochem 93: 21-26, 1990 11. Kascer H, Burns JA: The control of flux. Symp Soc Exp Bio127: 65-104, 1973 12. Porteous JW: Control analysis: A theory that works. In: A Cornish-Bowden, ML Cardenas (eds.) Control of Metabolic Processes. Plenum Press, New York, 1990, pp 51-67 13. Heinrich R, Rapoport TA: A linear steady-state treatment of enzymatic chains: General properties, control and effectorstrength. Eur J Biochem 42: 89-95, 1974 14. Groen AK, Westerhoff HV: Modern control theories: A consumers' test. In: A Cornish-Bowden, ML Cardenas (eds.) Control of Metabolic Processes. Plenum Press, New York, 1990, pp 101-118 15. Press WH, Flannery BP, Teukolsky SA, Vetterling WT: Numerical recipes in C. The art of scientific computing. Cambridge University Press, Cambridge, 1986, pp 547-563 16. Scrutton MC, Utter MF: The regulation of glycolysis and gluconeogenesis in animals tissues. Ann Rev Biochem 37: 249-302, 1986 17. Hers HG, Hue L: Regulation of glucolysis in animal tissue. Ann Rev Biochem 52: 617-653, 1983 18. Barman TE: Enzyme Handbook, Springer-Verlag, New York, 1969 19. Dunaway GA: A review of animal phosphofructokinase isozymes with an emphasis on their physiological role. Mol Cell Biochem 52: 75-91, 1983

A kinetic description of sequential, reversible, Michaelis-Menten reactions: practical application of theory to metabolic pathways.

Equations are presented which describe a linear coupled system of reactions that utilize a single substrate and convert it to product by way of severa...
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