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Optimizing Enzymatic Cycling Assays: Spectrophotometric Determination of Low Levels of Pyruvate andL -Lactate
ANALYTICAL BIOCHEMISTRY ARTICLE NO.
239, 47–52 (1996)
0289
Optimizing Enzymatic Cycling Assays: Spectrophotometric Determination of Low Levels of Pyruvate and L-Lactate Edelmira Valero* and Francisco Garcı´a-Carmona†,1 *Departamento de Quı´mica-Fı´sica, E.U. Polite´cnica, Universidad de Castilla–La Mancha, E-02071 Albacete, Spain; and †Departamento de Bioquı´mica y Biologı´a Molecular (A), Facultad de Biologı´a, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Received October 13, 1995
A kinetic analysis of enzymatic cycling systems covering the whole course of the reaction, i.e., the transient phase and the steady state, is presented. The cost of enzymatic cycling assays is minimalized by using equations to calculate the smallest amount of enzymes which should be used to obtain a given rate constant of the cycle. The model chemical system chosen for illustration purposes involved the determination of pyruvate and/or L-lactate via the coupling of L-lactate dehydrogenase and L-lactate oxidase cycling and NADH mediation. The method is simple although thorough and can be applied to any technique that uses enzymatic cycling. q 1996 Academic Press, Inc.
Enzyme assays are extensively used in biochemical analysis because of the extremely high catalytic power and specificity of enzymes which may quite often be detectable with great ease. The simplest method to determine the concentration of a known metabolite S in a sample is to follow the characteristic enzymatic reaction S r P, by measuring either the accumulation of P or the disappearance of S. This may be performed continuously or when the reaction is practically complete (endpoint methods) (1). When the decrease in S or the increase in P cannot be established, it is a common practice to couple one or more secondary enzymes to generate a measurable product. The kinetics of these systems have been studied in depth and a number of solutions for systems containing one or more coupling enzymes have been reported (2–5), together with equations that allow the calculation of the amount of coupling enzyme that must be added to give any given coupling time, thus minimizing the cost of assay (6, 7).
However, it is not always possible to use these kinetic methods to determine directly very small quantities of a metabolite, such as those which frequently occur in biochemistry, since they do not allow amplification of the signal. Several systems of enzyme amplification have been developed and applied; these include enzyme cascades (8, 9), release of enzymes from liposomes (10), prosthetogenic amplification systems (11, 12), and enzymatic cycling (13, 14). This last amplification system is based on the regeneration in situ of the target metabolite by means of two enzymatic reactions occurring in the opposite direction, so that its concentration remains constant, while at the same time other products of the enzymatic reactions are accumulated in the medium. The amount of product accumulated at a fixed time is directly proportional to the original amount of metabolite in the sample when certain conditions are fulfilled (15). The sensitivity of the assay may be increased by using longer incubation times or larger amounts of the two enzymes, the upper limit being imposed by the availability of free substrate for cycling (16). The rate of such cycling assays can be much faster than conventional assays, although the increase in sensitivity increases costs substantially. This fact, together with the absence of time-based equations that would make it possible to follow the process in a continuous form, led us to solve the corresponding set of differential equations and to develop a simple and convenient method to minimize the cost of assays and increase confidence in the results obtained. This procedure was applied to an assay to determine pyruvate and/or L-lactate involving the enzymes L-lactate dehydrogenase and L-lactate oxidase (Scheme I), that has been recently implemented in a dual bioreactor with the immobilized enzymes to amplify the signal (17). THEORY
1
To whom correspondence should be addressed. Fax: (Spain) 68 36 41 47.
The enzymatic cycling system shown in Scheme I may be described as 47
0003-2697/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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d[P2]/dt Å k2[B],
SCHEME I. Cyclic enzymatic system for the assay of pyruvate and/or L-lactate involving the enzymes LDH and LOD.
E1
S1 / A r P1 / B
[1]
E2
S2 / B r P2 / A,
[2]
where S1 , P1 , S2 , and P2 are b-NADH, b-NAD/, O2 , and H2O2 , respectively, A and B are pyruvate and Llactate, respectively, and E1 and E2 are the enzymes 2 L-lactate dehydrogenase (LDH) and L-lactate oxidase (LOD), respectively. To derive analytical expressions for the time dependence of the species involved in Scheme I, the following assumptions were made: (1) S1 and S2 concentrations (b-NADH and O2) in the reaction medium must be high enough to be saturating or remain constant during the reaction time. Under these conditions and with relatively high activities of both enzymes, E1 and E2 , a small quantity of A or B can ‘‘catalyze’’ the formation of large quantities of the products P1 and P2 . When either P1 or P2 is measured, the system acts as a chemical amplifier in the determination of A and B concentrations. (2) During the cycling, the concentration of A and B must be clearly lower than their respective Michaelis– Menten constants (KMA and KMB toward E1 and E2 , respectively), so that the reaction rates of steps (1) and (2) remain proportional to their respective concentrations. Taking into account these assumptions, the differential equation system that describes the mechanism shown by Eqs. [1] and [2] is d[A]/dt Å 0k1[A] / k2[B]
[3]
d[B]/dt Å k1[A] 0 k2[B]
[4]
d[P1]/dt Å k1[A]
[5]
where k1 and k2 are apparent first-order rate constants, k1 Å Vm1/KM1 (Vm1 is the reaction rate of step (1) at A and S1 saturating concentrations; KM1 is a function of the Michaelis–Menten constants of A and S1 toward E1 and of the initial concentration of S1 , with KM1 á KMA if S1 is saturating), k2 Å Vm2/KM2 (Vm2 is the reaction rate of step (2) at B and S2 saturating concentrations; KM2 is a function of the Michaelis–Menten constants of B and S2 toward E2 and of the initial concentration of S2 , with KM2 á KMB if S2 is saturating). In this case, k1 and k2 have units of min01, although if Vm is expressed as specific activity, ki (i Å 1, 2) for each enzyme would be expressed as min01mg01 (the notation used in this latter case will be k*i (i Å 1, 2); then ki Å k*i Mi , where Mi is the amount of enzyme Ei per assay, expressed as milligrams). The solution to the differential Eqs. [3] and [4], whose initial conditions are at t Å 0, [A] Å A0 , and [B] Å B0 , is [A] Å
k1A 0 0 k2B0 0l t k2(A 0 / B0) e / l l
[B] Å 0
k1A 0 0 k2B0 0l t k1(A 0 / B0) e / , l l
[P1] Å
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[8]
k1(k1A 0 0 k2B0) k1k2(A 0 / B0) (1 0 e0l t) / t 2 l l [9] k2(k1A 0 0 k2B0) k1k2(A 0 / B0) (1 0 e0lt) / t 2 l l
[P2] Å 0
[10] being V1 Å
k1(k1A 0 0 k2B0) 0l t k1k2(A 0 / B0) e / l l
V2 Å 0
k2(k1A 0 0 k2B0) 0l t k1k2(A 0 / B0) e / l l
[11] [12]
the accumulation rates of products P1 and P2 , respectively. The corresponding initial rates are:
Abbreviations used: LDH, L-lactate dehydrogenase; LOD, L-lactate oxidase; BSA, bovine serum albumin.
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where l Å k1 / k2 . Inserting Eqs. [7] and [8] into Eqs. [5] and [6], respectively, and integrating again, taking into account that at t Å 0, [P1] Å [P2] Å 0, we obtain:
2
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[6]
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V1,0 Å k1A 0
[13]
V2,0 Å k2B0 .
[14]
49
OPTIMIZING ENZYMATIC CYCLING ASSAYS
In the steady state (t r `), exponential terms in Eqs. [9] and [10] are negligible and, reordering terms: [P1]ss Å
k1(k1A 0 0 k2B0) k1k2(A 0 / B0) / t l2 l
[P2]ss Å 0
[15]
k2(k1A 0 0 k2B0) k1k2(A 0 / B0) / t. [16] l2 l
Equations [15] and [16] indicate that both products of the cycle, P1 and P2 , are accumulated in linear form with time when the steady state is reached. The rate of the reaction in the steady state will be: Vss Å k(A 0 / B0)
[17]
with kÅ
k1k2 k1 / k2
[18]
the general rate constant for the overall reaction (15). Equation [18] may also be written as: kÅ
k*1M1k*2 M2 k*1 M1 / k*2 M2
.
[19]
FIG. 1. Graphical determination of the optimum conditions for an enzymatic cycling assay using the optimized straight line.
zero, one can solve for the levels of M2 and M1 (and therefore for k2 and k1) which give the desired rate constant k but minimize cost:
F S DG F S DG
k2 Å k 1 {
Minimum Cost The efficiency of a recycling process when working at the steady state is measured by the rate constant k, which is the number of cycles per minute that can be achieved before a metabolite molecule is destroyed (i.e., the total number of moles of product formed per mole of metabolite during the course of a complete reaction). Taking k1 from Eq. [18], we obtain (it may be done the same taking k2): k1 Å
k2k . k2 0 k
1{
Cost Å C1M1 / C2 M2 ,
[21]
where M1 and M2 are the amounts of enzymes E1 and E2 , respectively, expressed as milligrams per assay. (Note that in this case the final volume was 1 ml). Taking M1 from Eq. [19], substituting it into Eq. [21], taking partial derivatives of the cost with respect to the remaining M2 value, and setting the derivative equal to
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[22]
C2k*1
1/2
C2k*1
C1k*2
.
[23]
The representation of k1 vs k2 for different k values is a straight line which goes through the origin and which contains all points for the pairs of enzyme concentrations, with the minimum cost for each k value being the slope (a) of this line of the minimum cost:
[20]
Thus, a graphical representation of k1 vs k2 will be a hyperbola whose points are all the possible combinations of k1 and k2 allowing to obtain a given k (Fig. 1). C1 and C2 being the cost per milligram for the enzymes 1 and 2, respectively, the cost per assay will be
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k1 Å k 1 {
1/2
C1k*2
aÅ 1{
S D S D C2k*1
1/2
C1k*2
C1k*2
1/2
.
[24]
C2k*1
Therefore, the point of intersection of the hyperbola obtained by plotting Eq. [20] after fixing k, with the optimized straight line will give the coordinates of the desired optimum point (ko1 and ko2) (Fig. 1). MATERIALS AND METHODS
b-NADH, sodium pyruvate, and L-lactate dehydrogenase (EC 1.1.1.27) from rabbit muscle (960 U mg01) were obtained from Sigma. L-Lactate oxidase (EC
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VALERO AND GARCI´A-CARMONA
50
Data fitting was performed by linear regression using the SigmaPlot Scientific Graphing System. RESULTS
The theory here presented was applied to a representative practical case, the determination of pyruvate and/or L-lactate by enzymatic cycling by measuring in continuous form in the steady state with the enzymes LDH and LOD. Sensitivity of the Assay
FIG. 2. Standard cycling assay calibration straight line for pyruvate. Conditions as indicated under Materials and Methods.
1.1.3.x; x to be determined (18)) from Pediococcus species (33 U mg01) was obtained from Boehringer Manheim. Stock enzyme solutions (70 and 132 U/ml, respectively) were prepared in 50 mM Tris/HCl buffer, pH 7.6, containing 1 mg/ml BSA, and further dialyzed at 47C against this buffer (in the absence of BSA) in order to reduce the presence of pyruvate and L-lactate as contaminants. All other buffer and reagents were of analytical grade and were used without further purification. Spectrophotometric readings were obtained on a Uvikon 940 spectrophotometer from Kontron Instruments. Cycling Assay Unless otherwise stated, the cycling reagent consisted of 50 mM Tris/HCl buffers, pH 7.6, 256 mM bNADH, 1.8 mg LDH, 60 mg LOD, and different volumes of a stock solution of 10 mM sodium pyruvate. The reaction was started by the addition of LDH, the final volume being 1 ml. The time course of the reaction was followed by measuring the disappearance of b-NADH at 340 nm (e Å 6270 M01 cm01), at 377C. If the reaction is started by the addition of the substrate (pyruvate or L-lactate) instead of the enzyme, spontaneous oxidation of b-NADH is observed during preincubation. This fact is indicative of the presence of pyruvate and/or L-lactate in one or both enzyme solutions. The sum of pyruvate plus L-lactate as contaminants present at each enzyme concentration was evaluated by extrapolation of velocity line vs pyruvate concentration to the abscissa axis (Fig. 2).
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Figure 2 shows the sensitivity and linearity of this cycling system for the measurement of low levels of pyruvate (it would be the same for the measurement of L-lactate). Using cycling enzymes at the mentioned concentrations, good linearity was obtained over a range of 0.25 to 2 nmol of pyruvate. The rate of the reaction was constant over a period of at least 30 min (data not shown). The presence of pyruvate and/or Llactate as contaminants can be observed, their concentration being evaluated by extrapolating to velocity zero, according to Eq. [17]. A value of 0.75 nmol of pyruvate plus L-lactate was obtained under these conditions, which is about the lowest level that this type of contamination may be reduced (16). Evaluation of Kinetic Parameters The general rate constant of the cycle (k) was evaluated from data from Fig. 2, according to Eq. [17], obtaining a value of 1.8 min01 under the present conditions. The apparent first-order rate constants, k1 and k2 , may be evaluated by determining their corresponding kinetic parameters, Vm and KM , in a classical way. However, it is not always possible to perform this determination under the same experimental conditions and it is not necessary to know the individual Vm and KM . These constants may be directly determined from the cycle by varying the concentration of one enzyme while maintaining the concentration of the other fixed, at a determined concentration of pyruvate plus L-lactate. Taking into account that 1/k Å 1/k1 / 1/k2 (from Eq. [18]), a double reciprocal plot of k vs LOD concentration would be a straight line with the ordinate value of 1/k1 , and by analogy, a double reciprocal plot of k vs LDH concentration, would be a straight line with the ordinate value of 1/k2 (14). However, in the present case, the presence of pyruvate plus L-lactate as contaminants in the enzyme solutions does not allow the direct determination of k from the velocity data thus obtained, being necessary to perform these experiments at several substrate concentrations. The results obtained are shown in Fig. 3, LDH being the contaminated enzyme as can be seen. According to Eq. [17] the slope of each
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OPTIMIZING ENZYMATIC CYCLING ASSAYS
51
FIG. 3. Variation of steady-state rate with the concentration of the enzymes LDH (A) and LOD (B). The following concentrations of enzymes were used: A, (s) 0.5, (l) 0.9, (,) 1.4, (.) 1.8, (h) 2.5, (j) 3.3, and (n) 4 mg/ml LDH; the concentration of LOD used was 60 mg/ml. B, (s) 20, (l) 40, (,) 60, (.) 80, and (h) 100 mg/ml LOD; the concentration of LDH used was 1.8 mg/ml.
straight line thus obtained is the value of k corresponding to the indicated LDH and LOD concentrations. When these values were plotted reciprocally against enzyme concentrations as previously indicated (data not shown), the values of k*1 and k*2 thus obtained were 1.6 min01/mg of enzyme and 0.0675 min01/mg of enzyme, respectively. Economical Optimization Using data from Fig. 1, a value of k Å 1.8 min01 was found, as has been mentioned above. The corresponding hyperbola, where all value pairs for k1 and k2 yielding this k value lie, was drawn (Fig. 1). After the calculation of a (Eq. [24]) from k*1 of LDH and k*2 of LOD, and the price relation of both enzymes, which is 5.14, the corresponding straight line can also be plotted. From the coordinates of the point of intersection of the hyperbola with the optimized straight line, ko1 Å 20.5 min01 and ko2 Å 1.9 min01 are obtained. These values allowed us to calculate the optimum concentrations of both cycling enzymes to obtain a cycling rate of 1.8 min01. These values are 12.8 mg/ml of LDH and 27.4 mg/ml of LOD. The enzymatic assay carried out under these conditions gives this cycling rate, after subtracting the blank (data not shown). DISCUSSION
When designing enzymatic assays, cost is an important aspect to bear in mind, since the high cost of enzymes means that there is a natural tendency to try to reduce their concentration. However, this may lead to serious problems such as incomplete reaction, the
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nonattainment of steady state, and nonlinearity with time. The procedure that is usually employed in optimizing cyclic enzyme systems consists of searching for a maximum cycling rate by using various ratios of the cycling enzymes, while maintaining as constant the total amount of protein (19, 20). This procedure requires the realization of a great number of assays and would be feasible from an economical point of view only if the price relation for the two enzymes was 1. The optimization procedure here described is useful for any cycling assay and has several advantages over the above method. (a) It is a more thorough method based on the kinetics of the system. This allows a more exact and a more economical approximation than is available at present, such optimizing conditions being attained with smaller quantities of cycling enzymes. Thus, for example, Eq. [20] reveals that neither k1 or k2 can be smaller than k. If, for example, k1 is smaller than k, the rate constant k observed would be smaller meaning that more of enzyme 2 had been used than was strictly necessary. (b) The cycling constant k is fixed in advance, which makes it possible to select the cycling rate in accordance with the cycling time and sensitivity needed. (c) Knowledge of the apparent first-order rate constants, k1 and k2 , leads to greater kinetic control of the cycling reaction, which may help in detecting problems. This method has been illustrated by the measurement of low levels of pyruvate (or L-lactate) with the cycling system LDH/LOD. With the cycling rate here used (1.8 cycles/min) the assay is about 100 times more
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VALERO AND GARCI´A-CARMONA
52
sensitive than the widely used colorimetric assay of LDH activity which measures the disappearance or appearance of b-NADH for pyruvate and L-lactate, respectively (16, 21, 22). Sensitivity may be increased in several ways such as reducing the cycling volume, using longer cycling times, increasing the enzymes concentrations, immobilizing enzymes (17), or using other physical techniques. The limit of sensitivity is governed by assay reproducibility and the blank. An improvement in the detection limit would entail further purification of LDH to reduce the background signal. ACKNOWLEDGMENT This paper has been partially supported by a grant from the Comisio´n Interministerial de Ciencia y Tecnologı´a (CICYT), Project No. ALI 93/0573.
REFERENCES 1. Bergmeyer, H. U. (1978) in Principles of Enzymatic Analysis (Bergmeyer, H. U., Ed.), Verlag Chemie, Weinheim/New York. 2. McClure, W. R. (1969) Biochemistry 8, 2782–2786. 3. Coward, J. K., and Wu, F. Y. (1973) Anal. Biochem. 55, 406– 410. 4. Storer, A. C., and Cornish-Bowden, A. (1974) Biochem. J. 141, 205–209. 5. Easterby, J. S. (1984) Biochem. J. 219, 843–847. 6. Cleland, W. W. (1979) Anal. Biochem. 99, 142–145.
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7. Garcı´a-Carmona, F., Garcı´a-Ca´novas, F., and Lozano, J. A. (1981) Anal. Biochem. 113, 286–291. 8. Chock, P. B., and Stadtman, E. R. (1977) Proc. Natl. Acad. Sci. USA 74, 2766–2770. 9. Varo´n, R., Molina, M., Garcı´a-Moreno, M., Garcı´a-Sevilla, F., and Valero, E. (1994) Biol. Chem. Hoppe-Seyler 375, 365–371. 10. Litchfield, W., Freytag, J. W., and Adamich, M. (1984) Clin. Chem. 30, 1441–1445. 11. Harbron, S., Eggelte, H. J., and Rabin, B. R. (1991) Anal. Biochem. 198, 47–51. 12. Harbron, S., Fisher, M., and Rabin, B. R. (1993) Biochim. Biophys. Acta 1161, 311–316. 13. Lowry, O. H., and Passonneau, J. V., Schulz, D. W., and Rock, M. K. (1961) J. Biol. Chem. 236, 2746–2755. 14. Valero, E., Varo´n, R., and Garcı´a-Carmona, F. (1995) Biochem. J. 309, 181–185. 15. Passonneau, J. V., and Lowry, O. H. (1978) in Principles of Enzymatic Analysis (Bergmeyer, H. U., Ed.), Verlag Chemie, Weinheim/New York. 16. Passonneau, J. V., and Lowry, O. H. (1993) in Enzymatic Analysis: A Practical Guide (Passonneau, J. V., and Lowry, O. H., Eds.), Humana Press, Totowa, NJ. 17. Raba, J., and Mottola, H. (1994) Anal. Biochem. 220, 297–302. 18. Naka, K. (1993) Clin. Chem. 39, 2351. 19. Cha, S., and Cha, C.-J. M. (1965) Mol. Pharmacol. 1, 178–179. 20. Stephon, R. L., Niedbala, R. S., Schray, K. J., and Heindel, N. D. (1992) Anal. Biochem. 202, 6–9. 21. Noll, F. (1966) Biochem. Z. 346, 41–49. 22. Tietz, N. W. (1976) in Fundamentals of Clinical Chemistry (N. W. Tietz, Ed.), p. 937, Saunders, Philadelphia, PA.
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239, 47–52 (1996)
0289
Optimizing Enzymatic Cycling Assays: Spectrophotometric Determination of Low Levels of Pyruvate and L-Lactate Edelmira Valero* and Francisco Garcı´a-Carmona†,1 *Departamento de Quı´mica-Fı´sica, E.U. Polite´cnica, Universidad de Castilla–La Mancha, E-02071 Albacete, Spain; and †Departamento de Bioquı´mica y Biologı´a Molecular (A), Facultad de Biologı´a, Universidad de Murcia, E-30100 Espinardo, Murcia, Spain
Received October 13, 1995
A kinetic analysis of enzymatic cycling systems covering the whole course of the reaction, i.e., the transient phase and the steady state, is presented. The cost of enzymatic cycling assays is minimalized by using equations to calculate the smallest amount of enzymes which should be used to obtain a given rate constant of the cycle. The model chemical system chosen for illustration purposes involved the determination of pyruvate and/or L-lactate via the coupling of L-lactate dehydrogenase and L-lactate oxidase cycling and NADH mediation. The method is simple although thorough and can be applied to any technique that uses enzymatic cycling. q 1996 Academic Press, Inc.
Enzyme assays are extensively used in biochemical analysis because of the extremely high catalytic power and specificity of enzymes which may quite often be detectable with great ease. The simplest method to determine the concentration of a known metabolite S in a sample is to follow the characteristic enzymatic reaction S r P, by measuring either the accumulation of P or the disappearance of S. This may be performed continuously or when the reaction is practically complete (endpoint methods) (1). When the decrease in S or the increase in P cannot be established, it is a common practice to couple one or more secondary enzymes to generate a measurable product. The kinetics of these systems have been studied in depth and a number of solutions for systems containing one or more coupling enzymes have been reported (2–5), together with equations that allow the calculation of the amount of coupling enzyme that must be added to give any given coupling time, thus minimizing the cost of assay (6, 7).
However, it is not always possible to use these kinetic methods to determine directly very small quantities of a metabolite, such as those which frequently occur in biochemistry, since they do not allow amplification of the signal. Several systems of enzyme amplification have been developed and applied; these include enzyme cascades (8, 9), release of enzymes from liposomes (10), prosthetogenic amplification systems (11, 12), and enzymatic cycling (13, 14). This last amplification system is based on the regeneration in situ of the target metabolite by means of two enzymatic reactions occurring in the opposite direction, so that its concentration remains constant, while at the same time other products of the enzymatic reactions are accumulated in the medium. The amount of product accumulated at a fixed time is directly proportional to the original amount of metabolite in the sample when certain conditions are fulfilled (15). The sensitivity of the assay may be increased by using longer incubation times or larger amounts of the two enzymes, the upper limit being imposed by the availability of free substrate for cycling (16). The rate of such cycling assays can be much faster than conventional assays, although the increase in sensitivity increases costs substantially. This fact, together with the absence of time-based equations that would make it possible to follow the process in a continuous form, led us to solve the corresponding set of differential equations and to develop a simple and convenient method to minimize the cost of assays and increase confidence in the results obtained. This procedure was applied to an assay to determine pyruvate and/or L-lactate involving the enzymes L-lactate dehydrogenase and L-lactate oxidase (Scheme I), that has been recently implemented in a dual bioreactor with the immobilized enzymes to amplify the signal (17). THEORY
1
To whom correspondence should be addressed. Fax: (Spain) 68 36 41 47.
The enzymatic cycling system shown in Scheme I may be described as 47
0003-2697/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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VALERO AND GARCI´A-CARMONA
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d[P2]/dt Å k2[B],
SCHEME I. Cyclic enzymatic system for the assay of pyruvate and/or L-lactate involving the enzymes LDH and LOD.
E1
S1 / A r P1 / B
[1]
E2
S2 / B r P2 / A,
[2]
where S1 , P1 , S2 , and P2 are b-NADH, b-NAD/, O2 , and H2O2 , respectively, A and B are pyruvate and Llactate, respectively, and E1 and E2 are the enzymes 2 L-lactate dehydrogenase (LDH) and L-lactate oxidase (LOD), respectively. To derive analytical expressions for the time dependence of the species involved in Scheme I, the following assumptions were made: (1) S1 and S2 concentrations (b-NADH and O2) in the reaction medium must be high enough to be saturating or remain constant during the reaction time. Under these conditions and with relatively high activities of both enzymes, E1 and E2 , a small quantity of A or B can ‘‘catalyze’’ the formation of large quantities of the products P1 and P2 . When either P1 or P2 is measured, the system acts as a chemical amplifier in the determination of A and B concentrations. (2) During the cycling, the concentration of A and B must be clearly lower than their respective Michaelis– Menten constants (KMA and KMB toward E1 and E2 , respectively), so that the reaction rates of steps (1) and (2) remain proportional to their respective concentrations. Taking into account these assumptions, the differential equation system that describes the mechanism shown by Eqs. [1] and [2] is d[A]/dt Å 0k1[A] / k2[B]
[3]
d[B]/dt Å k1[A] 0 k2[B]
[4]
d[P1]/dt Å k1[A]
[5]
where k1 and k2 are apparent first-order rate constants, k1 Å Vm1/KM1 (Vm1 is the reaction rate of step (1) at A and S1 saturating concentrations; KM1 is a function of the Michaelis–Menten constants of A and S1 toward E1 and of the initial concentration of S1 , with KM1 á KMA if S1 is saturating), k2 Å Vm2/KM2 (Vm2 is the reaction rate of step (2) at B and S2 saturating concentrations; KM2 is a function of the Michaelis–Menten constants of B and S2 toward E2 and of the initial concentration of S2 , with KM2 á KMB if S2 is saturating). In this case, k1 and k2 have units of min01, although if Vm is expressed as specific activity, ki (i Å 1, 2) for each enzyme would be expressed as min01mg01 (the notation used in this latter case will be k*i (i Å 1, 2); then ki Å k*i Mi , where Mi is the amount of enzyme Ei per assay, expressed as milligrams). The solution to the differential Eqs. [3] and [4], whose initial conditions are at t Å 0, [A] Å A0 , and [B] Å B0 , is [A] Å
k1A 0 0 k2B0 0l t k2(A 0 / B0) e / l l
[B] Å 0
k1A 0 0 k2B0 0l t k1(A 0 / B0) e / , l l
[P1] Å
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[8]
k1(k1A 0 0 k2B0) k1k2(A 0 / B0) (1 0 e0l t) / t 2 l l [9] k2(k1A 0 0 k2B0) k1k2(A 0 / B0) (1 0 e0lt) / t 2 l l
[P2] Å 0
[10] being V1 Å
k1(k1A 0 0 k2B0) 0l t k1k2(A 0 / B0) e / l l
V2 Å 0
k2(k1A 0 0 k2B0) 0l t k1k2(A 0 / B0) e / l l
[11] [12]
the accumulation rates of products P1 and P2 , respectively. The corresponding initial rates are:
Abbreviations used: LDH, L-lactate dehydrogenase; LOD, L-lactate oxidase; BSA, bovine serum albumin.
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where l Å k1 / k2 . Inserting Eqs. [7] and [8] into Eqs. [5] and [6], respectively, and integrating again, taking into account that at t Å 0, [P1] Å [P2] Å 0, we obtain:
2
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[6]
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V1,0 Å k1A 0
[13]
V2,0 Å k2B0 .
[14]
49
OPTIMIZING ENZYMATIC CYCLING ASSAYS
In the steady state (t r `), exponential terms in Eqs. [9] and [10] are negligible and, reordering terms: [P1]ss Å
k1(k1A 0 0 k2B0) k1k2(A 0 / B0) / t l2 l
[P2]ss Å 0
[15]
k2(k1A 0 0 k2B0) k1k2(A 0 / B0) / t. [16] l2 l
Equations [15] and [16] indicate that both products of the cycle, P1 and P2 , are accumulated in linear form with time when the steady state is reached. The rate of the reaction in the steady state will be: Vss Å k(A 0 / B0)
[17]
with kÅ
k1k2 k1 / k2
[18]
the general rate constant for the overall reaction (15). Equation [18] may also be written as: kÅ
k*1M1k*2 M2 k*1 M1 / k*2 M2
.
[19]
FIG. 1. Graphical determination of the optimum conditions for an enzymatic cycling assay using the optimized straight line.
zero, one can solve for the levels of M2 and M1 (and therefore for k2 and k1) which give the desired rate constant k but minimize cost:
F S DG F S DG
k2 Å k 1 {
Minimum Cost The efficiency of a recycling process when working at the steady state is measured by the rate constant k, which is the number of cycles per minute that can be achieved before a metabolite molecule is destroyed (i.e., the total number of moles of product formed per mole of metabolite during the course of a complete reaction). Taking k1 from Eq. [18], we obtain (it may be done the same taking k2): k1 Å
k2k . k2 0 k
1{
Cost Å C1M1 / C2 M2 ,
[21]
where M1 and M2 are the amounts of enzymes E1 and E2 , respectively, expressed as milligrams per assay. (Note that in this case the final volume was 1 ml). Taking M1 from Eq. [19], substituting it into Eq. [21], taking partial derivatives of the cost with respect to the remaining M2 value, and setting the derivative equal to
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[22]
C2k*1
1/2
C2k*1
C1k*2
.
[23]
The representation of k1 vs k2 for different k values is a straight line which goes through the origin and which contains all points for the pairs of enzyme concentrations, with the minimum cost for each k value being the slope (a) of this line of the minimum cost:
[20]
Thus, a graphical representation of k1 vs k2 will be a hyperbola whose points are all the possible combinations of k1 and k2 allowing to obtain a given k (Fig. 1). C1 and C2 being the cost per milligram for the enzymes 1 and 2, respectively, the cost per assay will be
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k1 Å k 1 {
1/2
C1k*2
aÅ 1{
S D S D C2k*1
1/2
C1k*2
C1k*2
1/2
.
[24]
C2k*1
Therefore, the point of intersection of the hyperbola obtained by plotting Eq. [20] after fixing k, with the optimized straight line will give the coordinates of the desired optimum point (ko1 and ko2) (Fig. 1). MATERIALS AND METHODS
b-NADH, sodium pyruvate, and L-lactate dehydrogenase (EC 1.1.1.27) from rabbit muscle (960 U mg01) were obtained from Sigma. L-Lactate oxidase (EC
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VALERO AND GARCI´A-CARMONA
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Data fitting was performed by linear regression using the SigmaPlot Scientific Graphing System. RESULTS
The theory here presented was applied to a representative practical case, the determination of pyruvate and/or L-lactate by enzymatic cycling by measuring in continuous form in the steady state with the enzymes LDH and LOD. Sensitivity of the Assay
FIG. 2. Standard cycling assay calibration straight line for pyruvate. Conditions as indicated under Materials and Methods.
1.1.3.x; x to be determined (18)) from Pediococcus species (33 U mg01) was obtained from Boehringer Manheim. Stock enzyme solutions (70 and 132 U/ml, respectively) were prepared in 50 mM Tris/HCl buffer, pH 7.6, containing 1 mg/ml BSA, and further dialyzed at 47C against this buffer (in the absence of BSA) in order to reduce the presence of pyruvate and L-lactate as contaminants. All other buffer and reagents were of analytical grade and were used without further purification. Spectrophotometric readings were obtained on a Uvikon 940 spectrophotometer from Kontron Instruments. Cycling Assay Unless otherwise stated, the cycling reagent consisted of 50 mM Tris/HCl buffers, pH 7.6, 256 mM bNADH, 1.8 mg LDH, 60 mg LOD, and different volumes of a stock solution of 10 mM sodium pyruvate. The reaction was started by the addition of LDH, the final volume being 1 ml. The time course of the reaction was followed by measuring the disappearance of b-NADH at 340 nm (e Å 6270 M01 cm01), at 377C. If the reaction is started by the addition of the substrate (pyruvate or L-lactate) instead of the enzyme, spontaneous oxidation of b-NADH is observed during preincubation. This fact is indicative of the presence of pyruvate and/or L-lactate in one or both enzyme solutions. The sum of pyruvate plus L-lactate as contaminants present at each enzyme concentration was evaluated by extrapolation of velocity line vs pyruvate concentration to the abscissa axis (Fig. 2).
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Figure 2 shows the sensitivity and linearity of this cycling system for the measurement of low levels of pyruvate (it would be the same for the measurement of L-lactate). Using cycling enzymes at the mentioned concentrations, good linearity was obtained over a range of 0.25 to 2 nmol of pyruvate. The rate of the reaction was constant over a period of at least 30 min (data not shown). The presence of pyruvate and/or Llactate as contaminants can be observed, their concentration being evaluated by extrapolating to velocity zero, according to Eq. [17]. A value of 0.75 nmol of pyruvate plus L-lactate was obtained under these conditions, which is about the lowest level that this type of contamination may be reduced (16). Evaluation of Kinetic Parameters The general rate constant of the cycle (k) was evaluated from data from Fig. 2, according to Eq. [17], obtaining a value of 1.8 min01 under the present conditions. The apparent first-order rate constants, k1 and k2 , may be evaluated by determining their corresponding kinetic parameters, Vm and KM , in a classical way. However, it is not always possible to perform this determination under the same experimental conditions and it is not necessary to know the individual Vm and KM . These constants may be directly determined from the cycle by varying the concentration of one enzyme while maintaining the concentration of the other fixed, at a determined concentration of pyruvate plus L-lactate. Taking into account that 1/k Å 1/k1 / 1/k2 (from Eq. [18]), a double reciprocal plot of k vs LOD concentration would be a straight line with the ordinate value of 1/k1 , and by analogy, a double reciprocal plot of k vs LDH concentration, would be a straight line with the ordinate value of 1/k2 (14). However, in the present case, the presence of pyruvate plus L-lactate as contaminants in the enzyme solutions does not allow the direct determination of k from the velocity data thus obtained, being necessary to perform these experiments at several substrate concentrations. The results obtained are shown in Fig. 3, LDH being the contaminated enzyme as can be seen. According to Eq. [17] the slope of each
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OPTIMIZING ENZYMATIC CYCLING ASSAYS
51
FIG. 3. Variation of steady-state rate with the concentration of the enzymes LDH (A) and LOD (B). The following concentrations of enzymes were used: A, (s) 0.5, (l) 0.9, (,) 1.4, (.) 1.8, (h) 2.5, (j) 3.3, and (n) 4 mg/ml LDH; the concentration of LOD used was 60 mg/ml. B, (s) 20, (l) 40, (,) 60, (.) 80, and (h) 100 mg/ml LOD; the concentration of LDH used was 1.8 mg/ml.
straight line thus obtained is the value of k corresponding to the indicated LDH and LOD concentrations. When these values were plotted reciprocally against enzyme concentrations as previously indicated (data not shown), the values of k*1 and k*2 thus obtained were 1.6 min01/mg of enzyme and 0.0675 min01/mg of enzyme, respectively. Economical Optimization Using data from Fig. 1, a value of k Å 1.8 min01 was found, as has been mentioned above. The corresponding hyperbola, where all value pairs for k1 and k2 yielding this k value lie, was drawn (Fig. 1). After the calculation of a (Eq. [24]) from k*1 of LDH and k*2 of LOD, and the price relation of both enzymes, which is 5.14, the corresponding straight line can also be plotted. From the coordinates of the point of intersection of the hyperbola with the optimized straight line, ko1 Å 20.5 min01 and ko2 Å 1.9 min01 are obtained. These values allowed us to calculate the optimum concentrations of both cycling enzymes to obtain a cycling rate of 1.8 min01. These values are 12.8 mg/ml of LDH and 27.4 mg/ml of LOD. The enzymatic assay carried out under these conditions gives this cycling rate, after subtracting the blank (data not shown). DISCUSSION
When designing enzymatic assays, cost is an important aspect to bear in mind, since the high cost of enzymes means that there is a natural tendency to try to reduce their concentration. However, this may lead to serious problems such as incomplete reaction, the
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nonattainment of steady state, and nonlinearity with time. The procedure that is usually employed in optimizing cyclic enzyme systems consists of searching for a maximum cycling rate by using various ratios of the cycling enzymes, while maintaining as constant the total amount of protein (19, 20). This procedure requires the realization of a great number of assays and would be feasible from an economical point of view only if the price relation for the two enzymes was 1. The optimization procedure here described is useful for any cycling assay and has several advantages over the above method. (a) It is a more thorough method based on the kinetics of the system. This allows a more exact and a more economical approximation than is available at present, such optimizing conditions being attained with smaller quantities of cycling enzymes. Thus, for example, Eq. [20] reveals that neither k1 or k2 can be smaller than k. If, for example, k1 is smaller than k, the rate constant k observed would be smaller meaning that more of enzyme 2 had been used than was strictly necessary. (b) The cycling constant k is fixed in advance, which makes it possible to select the cycling rate in accordance with the cycling time and sensitivity needed. (c) Knowledge of the apparent first-order rate constants, k1 and k2 , leads to greater kinetic control of the cycling reaction, which may help in detecting problems. This method has been illustrated by the measurement of low levels of pyruvate (or L-lactate) with the cycling system LDH/LOD. With the cycling rate here used (1.8 cycles/min) the assay is about 100 times more
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sensitive than the widely used colorimetric assay of LDH activity which measures the disappearance or appearance of b-NADH for pyruvate and L-lactate, respectively (16, 21, 22). Sensitivity may be increased in several ways such as reducing the cycling volume, using longer cycling times, increasing the enzymes concentrations, immobilizing enzymes (17), or using other physical techniques. The limit of sensitivity is governed by assay reproducibility and the blank. An improvement in the detection limit would entail further purification of LDH to reduce the background signal. ACKNOWLEDGMENT This paper has been partially supported by a grant from the Comisio´n Interministerial de Ciencia y Tecnologı´a (CICYT), Project No. ALI 93/0573.
REFERENCES 1. Bergmeyer, H. U. (1978) in Principles of Enzymatic Analysis (Bergmeyer, H. U., Ed.), Verlag Chemie, Weinheim/New York. 2. McClure, W. R. (1969) Biochemistry 8, 2782–2786. 3. Coward, J. K., and Wu, F. Y. (1973) Anal. Biochem. 55, 406– 410. 4. Storer, A. C., and Cornish-Bowden, A. (1974) Biochem. J. 141, 205–209. 5. Easterby, J. S. (1984) Biochem. J. 219, 843–847. 6. Cleland, W. W. (1979) Anal. Biochem. 99, 142–145.
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7. Garcı´a-Carmona, F., Garcı´a-Ca´novas, F., and Lozano, J. A. (1981) Anal. Biochem. 113, 286–291. 8. Chock, P. B., and Stadtman, E. R. (1977) Proc. Natl. Acad. Sci. USA 74, 2766–2770. 9. Varo´n, R., Molina, M., Garcı´a-Moreno, M., Garcı´a-Sevilla, F., and Valero, E. (1994) Biol. Chem. Hoppe-Seyler 375, 365–371. 10. Litchfield, W., Freytag, J. W., and Adamich, M. (1984) Clin. Chem. 30, 1441–1445. 11. Harbron, S., Eggelte, H. J., and Rabin, B. R. (1991) Anal. Biochem. 198, 47–51. 12. Harbron, S., Fisher, M., and Rabin, B. R. (1993) Biochim. Biophys. Acta 1161, 311–316. 13. Lowry, O. H., and Passonneau, J. V., Schulz, D. W., and Rock, M. K. (1961) J. Biol. Chem. 236, 2746–2755. 14. Valero, E., Varo´n, R., and Garcı´a-Carmona, F. (1995) Biochem. J. 309, 181–185. 15. Passonneau, J. V., and Lowry, O. H. (1978) in Principles of Enzymatic Analysis (Bergmeyer, H. U., Ed.), Verlag Chemie, Weinheim/New York. 16. Passonneau, J. V., and Lowry, O. H. (1993) in Enzymatic Analysis: A Practical Guide (Passonneau, J. V., and Lowry, O. H., Eds.), Humana Press, Totowa, NJ. 17. Raba, J., and Mottola, H. (1994) Anal. Biochem. 220, 297–302. 18. Naka, K. (1993) Clin. Chem. 39, 2351. 19. Cha, S., and Cha, C.-J. M. (1965) Mol. Pharmacol. 1, 178–179. 20. Stephon, R. L., Niedbala, R. S., Schray, K. J., and Heindel, N. D. (1992) Anal. Biochem. 202, 6–9. 21. Noll, F. (1966) Biochem. Z. 346, 41–49. 22. Tietz, N. W. (1976) in Fundamentals of Clinical Chemistry (N. W. Tietz, Ed.), p. 937, Saunders, Philadelphia, PA.
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