Nonlinear enzymatic cycling systems: the exponential cycling system. I. Mathematical models.

Abstract A previously undescribed class of enzymatic cycling systems, the nonlinear cycling systems, is presented and mathematical models are derived for three examples of the simplest type of nonlinear cycling system, the exponential cycling system. In this system, the concentrations of the cycling intermediates and by-products increase exponentially with time, and, at any given time, are linear functions of the initial concentrations of the cycling intermediates. The systems may be useful as biochemical amplifiers in analytical techniques, and a simple modification of the systems gives systems that are potentially operative in vivo as components of some types of biochemical regulatory systems.

In an accompanying paper (1) we demonstrated the theoretical existence of a previously undescribed type of enzymatic cycling system, the exponential cycling system. Mathematical models for three such systems were derived and some of the properties of the systems were discussed.
The present communication presents a detailed investigation comparing some of the predictions of the model of the myokinasepyruvate kinase exponential cycling system (shown in Fig. 1) with the behavior of the corresponding experimental cycling system.

MATERIALS AND METHODS
The following reagents were purchased from Sigma: ADP, ATP, P-enolpyruvate, NADH, and crystalline rabbit skeletal muscle enzymes; myokinase (Grade III, 600 to 1000 units per mg), pyruvate kinase (type II, 300 to 500 units per mg), and lactate dehydrogenase (type II, 400 units per mg). The myokinase and pyruvate kinase preparations were further purified to reduce adenine nucleotide contamination.
Escherichia coli alkaline phosphatase (salt fractionated, 25 units per mg) was obtained from Worthington.
Crystalline bovine plasma albumin was a product of Armour Inc. AMP was obtained from P-L Riochemicals, and was further purified to remove ADP and ATP * This work was supported by a grant from the United States Public Health Service NS 06034-06. contamination.
Tris(hydroxymethyl)aminomethane was the Ultrapure grade of Mann Biochemicals. Triethylamine was a product of Matheson, Coleman, and Bell and was redistilled before use. Bio-Gel P-150 (100-200 mesh) was obtained from Calbiochem, and Bio-Gel TE-2 anion exchange resin was a gift of Mr. Anthony Ross (Brown University).
Purification of Pyruvate K&use---Pyruvate kinase was purified by the following modification of the Turtle and Kipnis method (2) which was originally used for the purification of myokinase.
One milliliter of a 3.2 M ammonium sulfate suspension of pyruvate kinase, containing 10 mg of protein, was adjusted to 0.9 to 0.95 saturation in ammonium sulfate by the addition of the solid salt. The precipitate was collected by centrifugation and redissolved in 200 ~1 of M Tris (Cl), pH 8.0, containing 50 Kg of E.
coli alkaline phosphatase. After incubation for 1 hour at 30", the mixture was clarified by centrifugation and applied to a Bio-Gel P-150 column (9 mm x 750 mm) equilibrated at room temperature with 50 mM Tris (Cl), pH 8.0, 100 mM KCI, 1 mM EDTA plus 5% (v/v) glycerol.
The column was developed at room temperature with the same buffer. Fractions (0.5 ml) were collected and assayed for pyruvate kinase and alkaline phosphatase activities.
The pyruvate kinase fractions containing no detectable phosphatase activity were used in the timed point cycling procedure (Procedure B, below).
Aliquots (10 ~1) of each fraction diluted with 50 mM Tris (Cl) pH 8.0 were added to 100-~1 portions of the assay mixture, and the rate of reaction was determined from the recorded rate of change in absorbancy at 340 nm.
Alkaline phosphatase activity was estimated with P-enolpyruvate as the substrate.
After incubation, the reaction velocity was determined from the recorded rate of change in absorbancy at 340 nm. The incubation period served to increase the sensitivity and reproducibility of the phosphatase assay.
Pur$cation of Myokinase-Myokinase was purified by essen-Issue of June 10, 1972 tially the same procedure used for the purification of pyruvate kinase. Approximately 4 mg of myokinase in an ammonium sulfate suspension were collected by centrifugation and redissolved in 200 ~1 of M Tris (Cl), pH 8.0, containing 50 pg of E. coli alkaline phosphatase.
The remainder of the purification was identical to that for pyruvate kinase except that the buffer used to equilibrate the column contained 0.1 mM dithiothreitol, and the collected fractions were assayed for myokinase and alkaline phosphatase activities.
The myokinase fractions containing no detectable phosphatase activity were used in the timed point cycling procedure (Procedure B).
Myokinase activity was assayed spectrophotometrically in a coupled assay. The assay mixture contained 100 mM Tris (Cl), pH 8.0, 100 m&f KCI, 4 mM MgSO+ 2 mM P-enolpyruvate, 2.5 mM ATP, 4 mM AMP, 0.15 mM NADH, 10 pg per ml of pyruvate kinase and 10 pg per ml of lactate dehydrogenase. Samples (10 ~1) of each fraction, previously diluted with 50 mM Tris (Cl) pH 8.0 containing 1 mg per ml of bovine plasma albumin, were added to lOO-~1 portions of the assay mixture.
The rate of reaction was determined from the recorded rate of change in absorbancy at 340 nm.
Alkaline phosphatase activity was assayed as in the purification of pyruvate kinase.
Puri$cattin of AMP-A 12 mm x 120 mm column was packed with Bio-Gel TE-2 anion exchange resin, and washed successively with loo-ml portions of water, 0.1 M NaOH, water, 0.1 M and 0.01 M triethylamine buffer (Buffer A) saturated with carbon dioxide.
AMP (200 mg of the free acid) was dissolved in 10 ml of distilled water with the aid of concentrated ammonium hydroxide.
The pH of the solution was 8.0 when it was applied to the column.
The AMP was eluted with a linear gradient of Buffer A from 0.01 to 0.4 M over 500 ml. The fractions containing AMP were pooled and concentrated to dryness by flash evaporation.
Dynamic Cycling Procedure, Procedure A-The dynamic cycling procedure was developed to test the agreement between some of the kinetic predictions of the model cycling system and the behavior of the experimental cycling system. The procedure used lactate dehydrogenase as an indicator enzyme and monitored the behavior of the system by the changing absorbancy of the reaction mixture at 340 nm.
Cycling reaction mixtures were prepared containing 50 mM Tris (Cl), pH 8.0, 100 mM KCl, 2 mM MgS04, 0.50 mu P-enolpyruvate, 1 mM AMP, 50 to 60 pM NADH, appropriate amounts of myokinase and pyruvate kinase and lactate dehydrogenase to give a pseudo-first order rate constant k~ of at least 15 per min with respect to pyruvate.
The reaction was initiated by the addition of the AMP.
It was not necessary to purify the myokinase and AMP preparations used in this procedure; however, the commercial pyruvate kinase preparation contained an inhibitor which was removed by dialysis for 2 hours at room temperature against 50 mM Tris (Cl), pH 7.5, 100 mM KCI, and 1 mM EDTA.
The amount of lactate dehydrogenase required to give the desired rate constant was calculated from the relation kL = Vmax/K m where Vmax and K, were determined from a Lineweaver-Burk plot under conditions similar to the cycling reaction conditions except that pyruvate kinase and myokinase were omitted and AMP and P-enolpyruvate (PEP) are present at relatively high concentrations.
The cycling intermediates, ATP and ADP are initially present at very low concentrations, and increase in concentration exponentially with time. Pyruvate, a by-product of the system, also increases its concentration exponentially with time.
appropriate concentrations of pyruvate were added to the reaction mixtures.
The apparent K, for pyruvate under those conditions was 1 mM.
Timed Point Cycling Procedure, Procedure B-The timed point cycling procedure was developed to test the prediction that the concentration of pyruvate at any given time in the cycling system is a linear function of the initial concentration of ATP in the system. Cycling reaction mixtures were prepared containing appropriate concentrations of ATP and the cycling reaction was allowed to proceed for a constant period of time.
The reaction was terminated by the addition of EDTA and the pyruvate generated by the cycling reaction was quantitated by its conversion to lactate with lactate dehydrogenase.
The NAD produced in the conversion was quantitated fluorometrically by a modification of the procedure by Lowry et al. (3).
Cycling reaction mixtures were prepared with 25-~1 samples of known ATP concentrations, 25 ~1 of 2 mM AMP and 50 ~1 of a solution containing 100 mM Tris (Cl), pH 8.0, 200 mM KCl, 4 mM MgSO+ 1 mM P-enolpyruvate, 300 pg per ml of bovine plasma albumin, and myokinase and pyruvate kinase to give pseudofirst order rate constants of 7.5 per min and 0.47 per min for ATP and ADP, respectively.
The cycling reaction was initiated at timed intervals by the addition of the AMP.
After incubation at 30" for the appropriate cycling time, the reactions were stopped by the addition of 25 ~1 of 50 mM EDTA (adjusted to pH 8.0 with sodium hydroxide) at the same time intervals used to initiate the reaction.
Each reaction then received 25 ~1 of a mixture containing 25 mM EDTA, pH 8.0, 10 pg per ml of lactate dehydrogenase and approximately 150 pM NADH (made NAD free by the method of Lowry et al.) (3). After incubation for at least 10 min at room temperature, the NAD concentration of the mixtures was determined by a slight modification of the Lowry procedure (3). Hydrochloric acid (25 ~1 of 1.2 N HCI) was added to each reaction mixture followed by 150 ~1 of 9 N sodium hydroxide after at least 5 min. The mixtures were heated at 95" for 10 min, cooled to room temperature, and diluted with 1 ml of distilled water.
The fluorescence of the mixtures was determined with a Turner filter fluorometer equipped with a 7-60 primary filter and 2A plus 48 filters as secondary filters. The change in fluorescence was calibrated using the same procedure with known pyruvate concentrations, and was found to be reproducible from day to day. Vol. 247,No. 11 RESULTS

AND DISCUSSION
The mathematical description of the model myokinase-pyruvate kinase cycling system has been given previously (1). For comparison of the model cycling system and its experimental counterpart, only a portion of the original set of equations is necessary: where lcl and kz are first order rate constants for the reactions catalyzed by myokinase and pyruvate kinase, respectively. P, D, and T are concentrations of pyruvate, ADP and ATP in the cycling system, respectively.
The subscript zero denotes the initial concentrations of these substances. The prediction from Equation 1 that the amount of pyruvate generated by the system increases exponentially with time was tested by use of the dynamic cycling procedure (Procedure A). This procedure uses lactate dehydrogenase to convert pyruvate to lactate, simultaneously giving a stoichiometrically equal conversion of NADH to NAD.
The use of lactate dehydrogenase introduces an additional kinetic parameter, kL, the pseudo-first order rate constant with respect to pyruvate, which must be considered in the interpretation of the data. A model Line a shows the crude data for a typical cycling reaction by the dynamic cycling procedure.
Line b is a replot of the same data after the addition of 0.0014 absorbancy unit to the initial absorbancy of the reaction mixture.
The value of ~1 as calculated from the slope of line b is 0.801 per min. drogenase reaction is derived in Appendix I and given by the where N is the NADH concentration in the cycling system and -XI? z kL. No is N when t is equal to zero. Equation 6 which is used in the interpretation of the experimental data was obtained from Equation 5 by assuming that the terms involving eXz t and epkL 2 were negligible during the period of data collection. 2klk2kLTo The theoretical validity of these assumptions may be checked by direct calculation from Equation 5. In calculations used for determining the point in time at which the negligibility assump. tions became valid, it was assumed that a term was negligible if its magnitude was less than 1% of the magnitude of the term involving exit. A typical example of calculations of this type indicated that, for a value of kL 15 times larger than the largest value of X1 in a set of cycling reactions, the assumptions were valid for Xlt greater than or equal to approximately 1.0, regardless of the values of k, and kp. For the dynamic cycling reported in this paper, direct calculations with Equation 5 indicated that Equation 6 could be used in the interpretation of the data after the 1st min of cycling.
In practice, the data from the 1st min of the cycling reaction in most cases was below the limits of resolution of the spectrophotometric determinations used in the cycling procedure.
Equation 6 may be transformed to give a more usable form of the equation: 2klk2kLTo In (No -N + 2To) = Alt + ldAl(hl + ,Q~' The equation implies that a semilog plot of change in concentration of NADH, corrected for the initial ATP concentration, versus time should give a straight line with its slope equal to X1 (Fig. 2). Fig. 2 shows a typical example of data obtained by the dynamic cycling reaction.
It was transformed to give a line with slope X1, by addition of a small constant absorbancy to the initial absorbancy of the reaction mixture, as indicated in Equation 7. The magnitude of the correction is proportional to the initial ATP present in the system. Since the average correction for all of the results obtained by the dynamic cycling procedure was between 0.001 and 0.002 absorbancy units, the apparent initial ATP concentration for Equation 7 was between 0.08 and 0.16 PM. It is evident that the experimental data shown in the figure are consistent with the predicted behavior of the cycling system over a substantial portion of the cycling reaction.
Eventually, the crude experimental cycling data depart from the straight line instead of continuing to approach it, presumably because the The theoretical curve was calculated from Equation 2 with kz (pyruvate kinase activity) set equal to 0.855 per min as estimated from Fig. 4. The exnerimental values of ~1 were calculated from data obtained by the' dynamic cycling procedure, as described.
The experimental values of kl were calculated from the dilutions of the stock myokinase solution required to prepare the reaction mixtures. The activitv of the myokinase solution was estimated by the dynamic cycling procedure and the iterative approximation procedure, as described in Appendix II. The line labeled Limit is equal to kz (0.855 per min).
concentrations of ATP and ADP in the system have become large enough to invalidate the first order kinetic assumptions used in the derivation of the mathematical model. Thus, over a substantial portion of the measurable cycling reaction, pyruvate is generated in exponentially increasing amounts with time. The simple method for transforming the crude data from the dynamic cycling procedure to give a straight line with its slope equal to X1 provides a way of testing the relationship of X1 to kr and kz, as predicted by Equation 2. Qualitatively, as one of the reaction rate constants (e.g. k,) becomes much larger than the other (kz), the value of X1 asymptotically approaches the value of the rate limiting rate constant (kz) (Fig. 3).
The asymptotic behavior of X1 as one of the reaction rate constants becomes much larger than the other gives a method of quantitating the activities of myokinase and pyruvate kinase in terms of their apparent pseudo-first order rate constants in the cycling reaction; the value of the rate constant for the rate limiting enzyme is approximately equal to XX when the other enzyme is present in large excess. The method of quantitating the reaction rate constants, kl and kz, gives a way of testing the quantitative aspects of Equation 2. In its original form, the equation makes the problem of demonstrating the quantitative aspects of the relation unnecessarily complex.
The equation may be algebraically rearranged to give two equivalent equations emphasizing different aspects of the same relation: These equations predict that for a set of cycling reactions in which one of the reaction rate constants is held constant at L, while the other, k,, is varied, a plot of X1(X1 + k,) versus k, -X1 should give a straight line intercepting the origin and having its slope equal to k, (Figs. 4 and 5). Taking both sets of data together, Equation 2 quantitatively describes the relationship of X1 to kl and leg over more than four orders of magnitude change in the ratio of kl to kz. The prediction that Equation 3 quantitatively describes the relation between XZ and k1 and kz over the same interval is consistent with this result.
The apparent validity of Equation 2 gives a method of refining the data somewhat.
For example, a preliminary plot of X1(X1 + kl) versus kI -Xi gives a better estimate of kz than a single asymptotic determination of kz. The refined estimate of kz can be in turn used to refine the original estimate of kl. The refined estimates of k1 and kz generally differ from the original asymptotic estimates by less than lo'%, so that the results are not materially changed except for a slightly better fit by the experimental data to the predicted relation.
Equation 2 also gives a simple iterative procedure (described in Appendix II) for estimating the reaction rate constants for preparations of myokinase and pyruvate kinase without resorting to asymptotic conditions in the assays. For the data shown Values of XI were calculated from data obtained by the dynamic cycling procedure, as described.
Values of kz were calculated from the dilutions of the stock pyruvate kinase solution required to prepare the cycling reaction mixtures.
The activity of the stock solution was determined by the dynamic cycling procedure and the iterative approximation procedure given in Appendix II. The data in the inset near the origin of a are shown plotted on an expanded scale in b. The slope of the line is 1.01 per min, and is theoretically equal to kr by Equation 9. in Figs. 3, 4, and 5, the initial estimates for kr and kz were made by the iterative procedure.
These estimates were left unchanged by the procedure described above for refining them.
Aside from the kinetic properties of the myokinase-pyruvate kinase cycling system, the most important remaining predicted characteristic of the system is that the concentrations of pyruvate, ATP, and ADP in the system after a constant cycling reaction period are a linear function of the initial concentrations of ATP and ADP.
This prediction was tested for the case of pyruvate as a function of the initial ATP concentration.
The results of two such tests are shown in Fig. 6. Taken together, the results indicate that the pyruvate concentration is a linear function of the initial ATP concentration for at least two orders Each ATP concentration was run in triplicate by the timed point cycling procedure, as described. The cvcline times used were 17.5 and 27 min for a and 6, respectively." Teifluorescence units are equivalent to approximately 0.68 PM pyruvate.
The vertical bars through the point are the standard deviations for those points. of magnitude variation in the initial ATP concentration.
The fact that the pyruvate increased with time in the cycling reaction mixtures prepared with no added ATP suggests that the reagents used to prepare the mixtures still contained traces of contaminating ATP or ADP.
The magnitude of the apparent contamination may be estimated from the ATP-axis intercept in Fig. 6 b as equivalent to approximately 1.2 nru ATP. The apparent contamination was much higher when the same reagents were used before purification.
In particular, the commercial pyruvate kinase and AMP contained significant ADP or ATP contamination Purification of the myokinase preparation had no effect on the apparent residual ATP contamination in the cycling system. One major source of the remaining contamination appears to be the bovine plasma albumin added to stabilize the myokinase activity.
In the absence of the albumin, the results showed much greater variability, but consistently indicated an apparent contamination equivalent to about 0.2 no ATP in the reaction mixture.
Models for two other exponential cycling systems have been derived (1). The first of these, the pyruvate carboxylase system, shown in Fig. 7, is based on the reactions catalyzed by pyruvate carboxylase (4, 5), pyruvate kinase and oxalacetate decarboxylase (6). Preliminary results with this system (5) indicate that the system is more complicated than the model Orthophosphate, a byproduct of the cycling reaction, also increases in concentration exponentially with time.
FIG. 8. The P-enolpyruvate synthase cycling system. The reactions designated kl, k2, and ka are catalyzed by P-enolpyruvate (PEP) synthase, myokinase, and pyruvate kinase, respectively. ATP and P-enolpyruvate are present at relatively high concentrations. The cycling intermediates, pyruvate, AMP, and ADP, are initially present at very low concentrations and their concentrations increase exponentially with time during the course of the cycling reaction.
Orthophosphate, a by-product of the cycling reaction, also increases in concentration exponentially with time.
suggests. The system is a nonlinear cycling system, but apparently cannot be described in terms of simple exponential functions.
However, the concentrations of the cycling intermediates at any given time appear to be linear functions of the initial concentrations of the cycling intermediate, as predicted. The second cycling system, shown in Fig. 8, is based on the reactions catalyzed by P-enolpyruvate synthase (7), myokinase, and pyruvate kinase. Preliminary results with this system (5) indicate that the agreement between the mathematical model and the experimental cycling system may be good. The increasing concentrations of the cycling intermediates at any given time appear to be linear functions of the initial concentrations of the intermediates.
Ac~rwwledgments-The authors are indebted to Drs. S. Cha and R. E. Parks, Jr. for their stimulating discussions of this work, and to Dr. G. Heppner for her helpful criticisms of the manuscript. where JCL is a first order rate constant, the rate of change in pyruvate concentration in a system in which pyruvate is being generated in time by Equation I-l, and simultaneously converted to lactate by Equation I-2 is given by Cycling System: Experimental System Vol. 247,No. 11 where N is NADH. Substitution of I-5 into I-6 and integration