fluorescence detection of the chemical relaxation of the reaction

THE JOURNAL OF BIOLOQIC~L CHEMISTRY Vol. 239, No. 3, March 1964

Printed in U.S. A.

Fluorescence Detection of the Chemical Relaxation of the

Reaction of Lactate Dehydrogenase with Reduced Nicotinamide Adenine Dinucleotide*

GEORG H. CZERLINSKI AND GISELA SCHRECK

From the Johnson Foundation for Medical Physics, University of Pennsylvania, Philadelphia 4, Pennsylvania

(Received for publication, May 20, 1963)

The ternary complex of liver alcohol dehydrogenase with reduced nicotinamide adenine dinucleotide and imidaeole exhibits a pronounced chemical relaxation in the millisecond range (2). This observation led to the investigation of other ternary complexes of dehydrogenases. Ternary complexes of lactate dehydrogenase appeared to be quite suitable (3). Preliminary experiments with A. Winer on the heart muscle enzyme did not give any measurable chemical relaxation. Later, careful investigation of the results indicated a small chemical relaxation effect in a system containing dehydrogenase and reduced nicotinamide adenine dinucleotide. After further improvement of the signal- to-noise ratio of the apparatus (4), we were able to investigate this relaxation effect in greater detail. Since the fluorescence of reduced nicotinamide adenine dinucleotide increases when it is bound to the dehydrogenase (5), we employed fluorescence detection.

EXPERIMENTAL PROCEDURE

Instrumentation-The temperature jump method with observation of changes in absorption was originally used on metal indi- cators (6). Recently, this method was applied to an amino trans- ferase system (7). The instrument used in the present investigation and its general performance has recently been described in detail (4). The angle between fluorescence excitation and observation of its emission was 0”; with fluorescence excitation via Kodak filter 18A, and emission via Kodak filter 2B. The temperature within the observation chamber was kept constant at 20” by cooling the electrodes with methanol. A temperature rise of 5’ was produced by capacitor discharge (determined with the pH indicator 4-methyl-umbelliferone in conjunction with phosphate and glycylglycine as buffer, as described earlier (8)).

Enzymes-Lactate dehydrogenase from rabbit skeletal muscle was purchased from C. F. Boehringer und Soehne GmbH, Mannheim, Germany, over a period of several months in quantities of 100 mg each to ascertain the experimental results by several repetitions. Every sample was dialyzed several times against K&O4 of ionic strength 0.09 with phosphate buffer, pH 7.0, ionic strength 0.01. The activity of the enzyme was determined kinetically by conventional means (9) (for 2-mm light path) and by fluorescence titration similar to a method introduced recently for liver alcohol dehydrogenase (10). For the fluorimetric

* This investigation was supported by Grants 19813 and GB 237 from the National Science Foundation. A preliminary report has been presented (1).

titrations, an Eppendorf photometer (Netheler und Hinz GmbH, Hamburg, Germany) was used, 366 rnp excitation, 410 rnp emission. A known quantity of enzyme solution was incubated with 0.03 M potassium oxalate-0.001 M EDTA-0.0073 M triethanol- amine-HCI, pH 6, ionic strength 0.1. Small quantities of NADH solution were added, and the concentration of NADH at the in- flection point of the titration curve was taken as equivalence point for the number of enzymatically active sites.

/?-Nicotinamide Adenine Dinucleofide-NADH was Sigma Grade from Sigma Chemical Company (stored dry at 4”). Its concentration was determined spectrophotometrically by meas- uring its absorption at 340 rnE.c in a l-cm cuvette with 6220 M-I cm-l as extinction coefficient at this wave length (11).

Other Chemical Components-Ordinary distilled water was re- distilled in quartz under nitrogen with addition of alkaline per- manganate. Ion migration out of the flask via the surface of the quartz was prevented by heating about 12 inches of quartz tube considerably above 100”. The water vapor had to pass this tube before being condensed. The purity of the water was tested conductometrically (a bridge circuit with lo-kc generator). The ionic strength of the solution was always adjusted to 0.1 by the addition of potassium sulfate, Baker Analyzed Reagent, Lot 3278, purchased from Arthur H. Thomas Company. Most of the experiments were conducted in potassium phosphate ionic strength 0.01, pH 7.0, Baker Analyzed Reagent, Lots 3246 and 3252, also from Arthur H. Thomas Company. Control experiments at pH 7 and all experiments at pH 8 and 9 were conducted in glycylglycine (Nutritional Biochemicals Corporation, Lot 1973) at 0.02 M. The pH was checked with a Radiometer pH meter, model 22.

EVALUATION OF RESULTS

Fig. 1 shows an oscilloscope trace obtained with a solution of 6 fin NADH with an equivalent amount of enzyme. Only the shot noise-carrying trace is on the original photograph. The initial equilibrium value is above the screen and determined at less vertical amplification. The “instantaneous” temperature rise causes an instantaneous change in the fluorescence. The beginning of this step (see also Fig. 2) is “time zero” and indicated in Fig. 1 by the short vertical lines. The short horizontal line gives the final equilibrium value. The crossing point of the short horizontal line is the origin of a two-dimensional coordinate system. A set of theoretical curves with this same coordinate system has been prepared on transparent film before the ex-

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914 Complexes of Lactate Dehydrogenase with NADH Vol. 239, No. 3

177-6C -

FIG. 1. Evaluation of a trace with smallest error, shown in Fig. 4 by the straight line at ci = 6 PM. The thin lines are drawn in from theoretical curves, prepared separately on transparent film. The long vertical and horizontal lines are 1 cm apart. The centimeter deflections have to be converted into the real observables by the vertical and horizontal sensitivities, which are for the above trace: 5 mv per centimeter and 1 msec per centimet.er, respectively.

FIG. 2. Idealized curves of temperature jump experiments of NADH with lactate dehydrogenase, introducing the various equilibrium parameters, as they appear in the evaluation (in the experiments, Aso > Age).

periments. Each photograph of theoretical curves contains the function

As, = A& exp( - t/T)

with constant 7 in centimeters of trace and A& in centimeters as parameter.’ Various photographs of theoretical curves with 7 increasing from 0.5 to 4 cm are kept for evaluations.

In the determination of the observables A& and 7, calculated curves were placed on the observed trace, beginning with small 7 values. As T of the calculated curves is increased, one reaches a “limiting fit from the small side.” This curve is drawn into Fig. 1 for demonstration, Q- = 1.2 cm, A& = 1.0 cm. Then the same procedure is repeated, beginning with sample curves with large 7 values. As T is now decreased from one sample set to the next one, one reaches a “limiting fit from the large side.” This curve is also drawn into Fig. 1 for demonstration, 7 = 1.3 cm, A& = 0.9 cm. The two values of r and of A& are then used as limits in the plots of these observables (or some related parameters) as a function of concentration. One such series of one day is shown in Figs. 4 and 5 as a set of vertical straight lines. The dashed lines (extending through the straight lines) contain the limits of at least three different experiments on different days.

A general theory of the kinetics of chemical relaxation was given a few years ago (12). Here, only a brief derivation of the specific equations necessary for the full evaluation of the results will be given. In Appendix I, statics2 of chemical relaxation will be derived, in Appendix II, its kinetics,2 in Appendix III, its

I Index 2 in ASZ refers to a special kinetic condition in the later evaluation, see Equation 48.

Z The introduction of the terms “statics,” “kinetics,” and “thermodynamics of chemical relaxat’ion” is certainly associated

thermodynamics,2 and in Appendix IV, the change in the fluorescence yield with temperature. The final results were reached by successively fitting theoretical curves to the various experimental data, until a “best fit” to all data was obtained. To facilitate the reading of the text, a glossary of symbols follows immediately.

GLOSSARY

A bar over a symbol represents its equilibrium value. Sym- bols with a bar will therefore not be listed senaratelv. Y

z

SO

ST

XY

AST

ASY

Aso

Generalized component in a biochemical”system, here Y = E, R, ER, EX, which are generally ions, but the charges have been omitted for convenience. E = active site of the enzyme; R = NADH; ER and EX are binary complexes of E and R. Used like Y when more than one generalized individual component has to be considered. Residual signal (in millivolts) obtained without NADH being present, “background fluorescence,” with the conditions mentioned in the section on “In- strumentation” and a load resistor at the photomultiplier anode of 100 kQ, this signal was near 500 mv. Total fluorescence signal (in millivolts) of the biochemical system before the temperature jump, thus including X0. Fluorescence signal (in millivolts) of the individual component Y under the conditions of the individual experiment,. Momentary change in the total fluorescence signal (in millivolts), which thus includes all components of the system. Reference-zero is at the elevated temperature (here 25”). Momentary change in the fluorescence signal (in millivolts) of component Y. Momentary total change in the fluorescence signal (in millivolts) because of t.he change in fluorescence yield of all components on raising the temperature from 20-25”, counting from the upper temperature. With j 2 1, momentary total change in the fluorescence signal (in millivolts) because of the change in the concentrations of all components caused by the jth relaxation process (reference-zero at elevated temperature). Also without index, especially if there is only one relaxation process, or as long as several steps have not been distinguished, (chemical) relaxation time (constant) (mostly in milliseconds (msec) ; also in microsec- onds (psec) or seconds (set)). j = 0, refers to the heating time constant; j = 1, to the faster; j = 2, to the second fastest relaxation process, etc., wit’h the highest j indicating the slowest chemical step.

with some arbitrariness. But these terms describe the particular treatment in a very short and clear way: statics deals with the static conditions (ST as a function of the various Kr,z and c$) immediately before imposing the heat surge and in the temperature jump apparatus; kinetics gives the relationships between the experimentally measured relaxation times 7f and the kinetic parameters of the system (the various L) ; thermodynamics relates the observed (“chemical”) change in the fluorescence signal (caused by the change in temperature) to the thermodynamic

parameters of the system (ASj = f(Affy.~)).

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March 1964 G. H. Cxerlinski and G. Xchreck 915

vy Molar fluorescence coefficient (in millivolts per PM)

of component Y.

A%= Change in the molar fluorescence coefficient of Y (in millivolts per PM per heat surge (that is, 5”)) because of the change in fluorescence efficiency of Y upon heating by 5” (to 25”).

knl Velocity constant of an individual reaction step for that direction, which in the symbolism of chemical reaction steps is indicated by the arrow next to i&, ; see Equation 1.

K m, m+ Equilibrium constant of an individual reaction step with m and m + 1 indicating the numbering of the associated velocity constants; Equations 2 and 3.

KO Complex dissociation constant, introduced for the simplification of some expressions; see Equation 7.

AKu.z Change in the dissociation constant Ky,Z caused by the temperature jump of 5”.

AHY,z Change in enthalpy of a reaction step (in kilocalories per mole), which is described by Ky,Z for 23”.

0 CY Analytical concentration of component Y (in micromo-

lar or molar) ; see Equation 4.

CY Momentary concentration of component Y (in micromolar or molar) ; see Equation 14.

ACY Momentary deviation in the concentration of component Y from the equilibrium value after the temperature jump of 5” for all relaxation processes, that is, at 25”; see Equation 17.

(Acu)~ Momentary deviation in the concentration of component Y from the equilibrium value & after the temperature jump of 5” up to and including the jth relaxation process. Two of such consecutive equilibrium

deviations give the Aij, as is shown in Equation 49. The index j has thus here a slightly different meaning than in Asj and ri.

Aii Equilibrium concentration coefficient (in micromolar or in molar), referring to that component Y which was selected as the ith concentration variable in the system of differential equations and which is associated with thejth relaxation time.

aii Coefficient of time constants (in msec-i or in set-l), referring with i to that component Y which was selected as the ith concentration variable in the system of differential equations (like i in AiJ. The second index, j, refers to any one component, the concentration of which was selected as variable in the system of differential equations (in contrast to j in Aij).

b Dimensionless parameter used in the solution of differential equations (see Equation 34).

RESULTS

The results of fluorimetric titrations of active sites are reported in Table I. The number of active sites fluctuated considerably from one stock solution to the other. The same was found for the optical density of loo-fold diluted stock solutions. They fluctuated in a different way than the molarities of active sites, indicating that some inactive protein is present in varying amounts. This certainly contributes to the error in the experiments. The method of titration should be improved as it is still connected with an error of about 15 %.

In an actual experimental series, the background signal, X0, of only the buffer is determined initially from the Tektronix 545

TABLE I Comparison between optical density of enzyme solution

and concentration cf active enzyme sites

Lactate dehydrogenase was incubated u-ith 0.031 M potassium oxalate and phosphate, pH 7.0, ionic strength 0.1. The concentration of active sites was then determined by fluorimetric titration with NADH (analogous to the method of Theorell and McKinley-McKee (IO)).

Enzyme sample Active sites in stock solution*

O.D. of loo-fold diluted enzyme stock at 280 III,,

1 2 3

5

PM

360 420 490 600 550

0.128 0.147 0.119 0.183 0.220

* Determined after 40.fold dilution.

oscilloscope screen (Tektronix, Inc., Beaverton, Oregon). This measurement is repeated at frequent intervals between the various solutions investigated. It generally remains constant within 5% during a series of experiments. In the later evaluations o the different signals, that value at 80 is always taken which is closest in time to the time at which XT was measured. ST is the total signal obtained from the solution with enzyme and coen- zyme.

A calibration curve was taken for the fluorescence of NADH. The fluorescence signal of NADH was plotted as a function of its concentration, and the slope of this curve results in the molar fluorescence coefficient VR. Temperature jumps are also performed on these solutions and the height of the signal change, A&, is plotted as a function of the concentration of NADH. The slope of the straight line gives AVE. Both of these coefficients are important for later descriptions of the experimental results.

Lactate dehydrogenase and NADH are then mixed in buffer in equivalent amounts and investigated. Before each temperature jump experiment, the total fluorescence signal, XT, is measured at the output of the photomultiplier with a Tektronix 545 oscilloscope and 53/54D plug-in unit. The signal is then plotted as a function of the analytical concentration of NADH, c”,. The plot of XT = f(ci) should give a rising line with very slight curvature (see Fig. 6). Any deviations from this smooth curve would indicate some error for the deviating point in the particular experiment (initiating a repetition). The theoretical descrip- tion of this curve is not possible until after the evaluation of the kinetic experiments.

Temperature jumps are then performed on the various solutions. The samples are changed after each individual jump experiment in order to avoid disturbances by the electrolysis prod- ucts in the solution and by photolytic degeneration caused by the high light level. Fig. 3 shows a photograph of a temperature jump experiment at the upper concentration limit (beyond which evaluation is no longer possible).

Fig. 2 shows the general oscillographic appearance of temperature jump experiments at medium vertical amplification. The initial horizontal section of the trace indicates the equilibrium position before the temperature jump. At the “instant of the heat surge” (instantaneous with respect to the horizontal deflec- tion used to observe the chemical relaxation) the signal changes by Ai, because of the dependence of the fluorescence yield on

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Complexes of Lactate Dehydrogenase with NADH Vol. 239, No. 3

175-3c

C;= CR= 58/m&M

FIG. 3. Oscilloscope trace of a temperature jump experiment with a mixture of equivalent amounts of lactate dehydrogenase and NADH at the analytical concentration CL = 58 FM. The total fluorescence signal is 440 mv with 65 mv as signal from the buffer, SO (for a load resistor of 10 kQ at the photomultiplier anode). Phosphate buffer of ionic strength 0.01, pH 7, ionic strength increased to 0.1 by addition of K&04.

temperature. The chemical effect causes a total change A%. Unfortunately, for the system under investigation, A&, >> A& (up to 10 times). This relationship requires considerably increased vertical amplification and an offset of the initial horizontal section of the trace such that the amplified relaxation process appears on the oscilloscope screen, as was done for Figs. 1 and 3. The amount of offset is observed on an outside meter connected to a cathode follower within Tektronix 545. The relaxation times are evaluated from several traces of the same solution, as mentioned previously.

The reciprocal of the chemical relasation time is plotted as a function of c”,. Figs. 4 and 5 show such graphical representa- tions. The height of the individual lines indicates the error in the determination of reciprocal relaxation times. They comprise the limits derived from many experimental curves from various enzyme stock solutions.

Fig. 4 gives 7-2 as ordinate in order to investigate the validity of Equation 39, which represents a one-step mechanism. Two limiting straight lines are drawn into the graph, leading to a monomolecular ks of 600 se@ and a bimolecular kl of about 107 ~-1 see-1, while kz/kl = 6 X 10-b M. This dissociation constant is much larger than the one reported by Zewe and Fromm (13) (see under “Discussion”). The most probable values (at the centers of the dashed lines) are also far off any straight line. This outcome suggests the investigation of the next simplest reaction scheme:3

h kt E+R-ER-EX

kz k4 (1)

The velocity constants, L , are connected to the equilibrium constants of the individual steps by

- - K2,’ = $ a ‘e = Kr;2 (2)

1

k4 ZER & = - = - = K;*;

ks EEX (3)

The index refers to the symbol in Equation 1, while the overbar on the concentration symbol indicates equilibrium values.

Critical for comparison are Equations 39 and 40 of Appendix

3 R is NADH; E is the active site of lactate dehydrogenase; see also the “Glossary.”

II. The square root relationship between l/7 and e; in both of them requires one to cover at least two orders of magnitude in c”, (around 2cR = K2J. To show the broad concentration range better, the results are given in Fig. 5 on a semilog plot. The theoretical curve in Fig. 5 was drawn with Equation 40 and the parameters kq = 300 set-I, k3 = 1000 set-I, and K2,l = 6 KM. The error on these parameters, comparing all experiments, is estimated to be 30 %. Such a high error is unfortunate but the nevitable noise in the experiment (shot noise at photocathode)

$7” m<’

2.0 T /’

/

/

w IO 20 30 40 50 60

55.29 Ci in pM

FIG. 4. Linear plot of the square of the reciprocal relaxation time as a function of the analytical concentration of NADH, CL, with lactate dehydrogenase added in equivalent amounts. Phos- phate buffer, pH 7.0, of ionic strength 0.01 -+ 0.1 by K&04. The length of the individual line represents the width of the error in the determination of the relaxation times, see “Evaluation of Results.” (The load resistor at the anode of RCA tube 2020 was 100 kn throughout).

55.33 4 IN ph4

FIG. 5. Semilog plot of the reciprocal chemical relaxation time as a function of the concentration of NADH, ci. The conditions are like tbose for Fig. 4.

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March 1964 G. H. Czerlinski and G. Schreck

prevented the attainment of better precision. It is immediately apparent that the theoretical curve comes much closer to the most probable values. The reaction sequence of Equation 1 is therefore correspondingly more probable. The results were

mV ST-SO

0.5 LO 2 5 IO 20 50 100 55.45

CN OS I” plb4

FIG. 6. Graphic representation of the fluorescence due to chemical components, ST - SO, as a function of the total concentration of NADH, ci, which is equal to the concentration of enzymatically active sites of lactate dehydrogenase. Phosphate buffer, pH 7.0, etc., as for Fig. 4.

.OI t I I I I ! I c

OJ 0.2 0.5 I 2 5 IO 55.4 7

2Ea

K 2.1

FIG. 7. Relative change of the signal, $z, as a function of the relative concentration 2~JK2.1 for the evaluation of the thermodynamics of chemical relaxation from temperature jumps. ER is evaluated according to Equation 9; conditions as for Fig. 4.

4 mV A I,

500-

200-

IOO-

50-

20-

917

05 1.0 2 5 IO 20 50 100

55.51 CR IN ,LLM

FIG. 8. A plot of the measured change in the yield of fluorescence emission due to the temperature change, A&,, as a function of the concentration of free NADH. The same conditions prevail as in Fig. 4.

verified in 1W2 M glycine as buffer, pH 7, 8, and 9, with ionic strength always 0.1 by addition of KzS04. Neither change in buffer nor change in pH caused a deviation from the results reported beyond the 30 To error. Thus, one can state that the constants reported above are independent of buffer and pH within the specified range.

Fig. 6 shows a log plot of 8, - & as a function of ci. The rising line is given by Equation 13 of Appendix I in conjunction with Equation 9 and the above values for the kinetic and equilibrium constants; furthermore, VR = 50 mv per pM and qEX = 30 mv per pM. The agreement between experimental points and theoretical curve is quite satisfactory.

Fig. 7 represents the plot for the relative change of the signal from EX as a function of the parameter 2z~JK2,~. The relationship between the experimentally determined A& and the theoretical value A?fBx is given by Equation 52 of Appendix III. With values for the efficiencies given previously, the numerical value for Equation 52 is 3/8. CR and the equilibrium constants Kz,r and K4,3 are then obtainable from Fig. 7 with the help of Equation 45. The curve inserted in Fig. 7 is valid for the fol- lowing values: AK~,I:KQ,I = 0.19, AK~,z:KQ = 0.04. The error in the determination is expected to be slightly more than in the kinetic evaluations. Employing Equation 41 with Equation 42, T = 297” K and AT = 5” C, one obtains AHz,r = 6.8 kcal per mole, AH4,3 = 1.3 kcal per mole. The enthalpies of these reactions are determined rather crudely but have definitely the same sign and their ratio is close to 5.

Fig. 8 shows the dependence of changes in fluorescence yield with temperature on the concentration of free NADH. The theory for this dependence is briefly treated in Appendix IV.

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The theoretical curve was obtained with Equation 55 and the values Aqx = 4.8 mv per PM and AqEx = 7.6 mv per PM. The latter value was obtained from the relationship AvEx = AqRqEx/qR , but the insertion of the theoretical curve shows that the points are scattered closely around it. Thus, there seems to be no necessity at this time to improve the value for AqEx further.

DISCUSSION

The results derived independently above will now be compared with the literature. The most elaborate investigations on rabbit skeletal muscle lactate dehydrogenase appear to have been performed recently by Zewe and Fromm (13). Kinetic and equilibrium parameters were evaluated from over-all reaction kinetics. But purification difficulties prevented the determination of the enzyme concentration, so that the kinetic constants could not be computed. Only a dissociation constant for ER was given in Table I of Zewe and Fromm (13). Zewe and Fromm report KNADH = 1.08 X 1O-6 M.4 It would correspond to our Ko. KO was defined according to Equation 7 which results in terms of concentrations in

Ko = FEFR

CER + ZEX (56)

The numerical value with the above results is 1.4 x low6 M. I f one calculates the equilibrium constant from kinetic (kin) data, one obtains the sequence of relationships

- 7 Kkin = 2 = K~,,K~J = E (57)

1 3

The difference between the two is immediately apparent, and one would expect to find

Kkin 2 Ko (58)

Kkin of our investigations would result in 1.8 pM which fulfills Equation 58, but KC,’ $KNA~H of Zewe and Fromm (better with the revised value of 2.5 PM~). A discrepancy between thermodynamically and kinetically determined equilibrium constants was reported for the horse liver alcohol dehydrogenase system (14). Condition 58 is valid only if all reactions “beyond” EX (that is, involving substrates) are much faster than the one under consideration. Any deviation could be caused by the relative size of some of the monomolecular constants and would in- validate Condition 58.

There is one other point which could lead to a considerable difference between results derived from steady state kinetics and from chemical relaxation. kS was evaluated as lo3 see-l. Then kz should be at least lo4 see-I. As Kz,l = 6 MM, it is from (2)

k, = IO4 set-l/(6 X lC@ M) = 1.7 X lo9 M-I see-l

ki, to be diffusion controlled, could still be somewhat higher (de- pending on effective charges and geometry), which simultaneoulsy would increase kz further. But there is no reason why kt could not be smaller than given above; it actually could approach ka (but kf + k3 5 1000 se@). Under such conditions, K4,3 would certainly deviate from 0.3 and Kz,l would also change. It would then be necessary to solve Equation 33 as such without the simplification of Equation 36. The presently available precision in

4 H. J. Fromm obtained recently from fluorescence titrations a dissociation constant of 2.5 pM (personal communication), which is close to our value of KO = 1.4 PM.

the determination of the data does not permit a more detailed evaluation. The condition kz >> k3 was assumed therefore.

One may desire to know the structural differences in the two binary complexes ER and EX. It was already mentioned earlier5 that ER may be considered as the diffusion complex formed upon approach of the component molecules E and R. Three possibilities are presently considered for the differences between ER and EX. First, the slow monomolecular step might be caused by the removal of a water molecule between part of NADH and the zinc ion bound to the enzyme. Second, a re- folding process of NADH on the surface of the enzyme could be rate-limiting (such as a separation between the adenine ring and the nicotinamide ring). Third, some rearrangement in the helix configuration of the enzyme could be the slowest process. At this time, no final conclusions can be drawn, as all three conver- sions are expected to be involved in the binding process of E to R.

In general, it can be said that chemical relaxation methods are a powerful tool for the investigation of t,he detailed kinetics of chemical reactions. The results for the lactate dehydrogenase system have been shown to be in reasonably good agreement with previous results from the literature. They go beyond previous results in so far as only a two-step mechanism gives good agreement with the most probable experimental points and with the latest value for the over-all dissociation constant. To show irrevocably the presence of a two-step mechanism, the experiments would have to be repeated with improved instrumentation and more highly purified enzymes.

APPENDIX I

For the system of Equation 1, one can distinguish two analytical concentrations, indicated by the exponent 0

c; = FE + e ER + EER + EEX (4)

and

Ci = CR + FER + FEX (5)

Then, one has Equations 2 to 5, with four unknowns which may be solved via a quadratic equation. One finds for

CR = f(Ko + C”, - C”,, I[ 4K& ' + (K,, + & _ &z (6)

with

(7)

I f CEX is identical to zero, then Ko would become K~J.

6 The equality in Equation 12 is an assumption but a reasonable one if one considers ER as diffusion complex of E and R. With respect to enzyme systems, it might be necessary to broaden the concept of the diffusion complex somewhat. Only the slow step to the highly fluorescing EX shows up in the experiments, evi- dently associated with some electronic rearrangement (shift in the peak of fluorescence emission). But the complex to which the kinetic constants kt and kd are connected at the left side, could be the last member in a chain of fast monomolecular interconversions. All the bimolecular components would then comprise ER. The structural complexity of the enzyme and of NADH make such a reaction sequence probable. No differences in the yield of fluorescence emission should then exist before the (slow) electronic rearrangement to EX occurs. The assumption is experimentally verified by using Equation 13 on the various signals obtained (see main text).

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March 1964 G. H. Czerlinski and G. Schreck 919

One may now introduce the experimental condition

c”, = c;

Then Equation 6 results in

in which

(8) all = -[kl@E + h2) + k21 (25)

a12 = -kz (26)

i[ 1 l/Z FR = +Ko 1,g -1 (9)

a21 = -ks

a22 = -(kg + k4)

(27)

(28)

I f Sy is the signal of any component Y, it is related to concentra- The general solution of the differential Equations 23 and 24 is

tion cy by given by (as observed from the final equilibrium condition)

SY = 17YFY (10) ACR = AH exp(-t/d + Al2 exp (-t/n) (29)

As SE << SR, the difference in fluorescence signals is ACBX = AH exp(--t/TX) + AH exp(-tt/~~) (30)

ST - So = SR + SER + SEX (11) The concentration coefficients Aij will be considered later in Ap-

The condition pendix III, where this third equation is also needed

?R = ?ER < ‘,EX

is now assumed.5 Then Equation 11 becomes

c”, + &JR 6

ST - SO = ?R

CR - &

1 + KM + VEX ~

1 + Kn,,

(12) ACER = AH exp(-t/d + AU exp(-f/72) (31)

The roots -7;’ and -T,’ are obtained from the characteristic equation of the determinant

(13) A = an a12

I I (32)

a21 a22

APPENDIX II One then obtains

The kinetics of the two reaction steps in Equation 1 may be described by the two differential equations

1 _ a* [l * (1 - b)l/2] (33) 71.2

(14) with

bx - = k&R - kacBX

dt (15)

Here none of the concentration parameters carries a bar, as gen- It is now set theoretically6

era1 nonequilibrium conditions are considered. At chemical -all >> -a22 (35) equilibrium

dFy Such relaxation phenomena will only be considered here. One

- =o dt

(16) realizes from Equations 25 to 28 that all > a12 and az2 > as1. Then one obtains from Equations 34 and 35

One may now substitute for each concentration

cy = Fy + Acy (17)

O<b=~[a~~--a~l~]<1 (36)

and maintain experimenfally One may now expand the square root, omit terms higher than linear ones, and reintroduce Equations 25 to 28. Then the two

AC y << E y. (18) relaxation times are given by

Equation 18 then allows one to neglect the A2 terms. The dif- 1 ferential Equations 14 and 15 therefore become (37)

d&R -=

clt -kl(FEACR f FRACE) + kzAf2z.q (19)

(38)

dACEx - = dt

k&m - k&Ex (20) If only 1 /rl would be present, K. in Equation 9 would have to be replaced by KQ,~. Equation 37 would then become (with Con-

From stoichiometry and conservation laws, one obtains dition 8, Equation 2, and squaring)

ACR = ACE (21) a

71 = k; + 4k,k& (39)

which result in

ACER = -ACR - ACEX (22) 6 This relationship means that the first (bimolecular) reaction step proceeds much faster than the second (monomolecular) one. Although this is set here as condition for simplifying the eoua-

dAcR -T = UIIACR -I- alzAcEa

dt

dAcEx - = UzdCR + aztAcEx

dt

tions, ii is a quite practical definition, as one Ean-generally only (23) distinguish two relaxation times from each other experimenlallu.

when ihey are sufficiently apart. Although there are other pas: sible derivations for 71 and 72, the one with Equation 35 is selected,

(24) since it permits, via Equation 33, any precision in the calculations, if 71 and 72 are not very far apart.

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Equation 38 on the other hand becomes with Equation 9 and its conditions

1 1 - = ” l

+ KM -1 - + k4 (40) 7-2 1 _ (1 + 4&K;‘)‘/” 1

Both 71 and Q are written as a function of the concentration parameter & only.

APPENDIX III

In the temperature jump method, the change in the signal height of an individual reaction is determined by the enthalpy of this reaction, AH. The relationship between an equilibrium constant K and its AH is given by the well known van% Hoff equation

dInK AH -=- dT RT=

The equilibrium constant K may be expressed in terms of the various concentrations &. The differential of the natural log of an individual concentration & leads, for the purpose of experimental utilization, to

dlnEy 1 AEy -=-- dT AT Ey

(42)

Applying this relationship together with the temperature dependence of the equilibrium constant to Equations 2 and 3 results, after simplification, in

A&a AEER AEEX _---

K4.8 EER EEX

(43)

These equations may be simplified with stoichiometric relationships which agree with those of Equations 21 and 22, except that c carries a bar. One may then solve for any of the concentration changes. It is here solved for A&x/c~x. One finally obtains

ACEX -= EEX

(45)

Experimentally, one measures as tot.al change Aa, with

AST = ABo + A.91 + AS2 (46)

The change A& will be considered in Appendix IV. It is

A& = qRAlt + qsxA21 + ?1ER A 31 (47)

As EX is separated from the fast reaction process by a slow step, API = 0. Further, An = (A& and ASI = (AC&, which may be computed from Equation 43 only. But as - (A&), = (ACER)~, one obtains with Equation 12: A& = 0. This explains why As1 does not appear in Fig. 2.

The slow change is similarly described by

A.Sz = ~11-412 + ~EXAZZ + 11EaA32 (48)

I f (AC,), is the concentration change of any component Y, it

must be computed from both Equations 43 and 44. Then it is

any

Aiz = (AcY)~ - (AcY); (49)

for the properly selected component Y. It becomes thus with the previous consideration

Ah = ?R&R)z -I r)~x@&x)z -I VER@EER)I (50)

Use of Equation 12 and the equilibrium-analogue of Equation 22 gives

A.92 = (T,TX - e&%x (51)

Equation 45 is then related to experimental parameters by

AS2 7EX - 1)R -=~

ALEX (52)

?BX

and SEX, which is identical with the second term on the right side of Equation 13. At this point, it might be advisable to refer again to the two basic assumptions. They are given by the equality in Equation 12 and Condition 35.

APPENDIX IV

The yield of fluorescence emission is dependent on temperature, so that one obtains a very fast change in fluorescence concurrent with the change in temperature. This is caused by the competi- tion between radiative emission and vibrational losses, the latter increasing with temperature. This very fast “physical” change is, with Aq, as change in molar fluorescence coefficient of Y

ASo = AVRCR -I- AV,$ER -I- AVBXFEX (53)

I f one assumes7

&R = AVER <AVEX (54)

and uses Equations 5 and 9 for the solution of Equation 53, one obtains

0

ASO = AVR c", + &.~ER

+ AVEX CR - CR

1 + K3,4 1 + Kasa

(55)

SUMMARY

The combination of lactate dehydrogenase from rabbit skeletal muscle with reduced nicotinamide adenine dinucleotide was investigated fluorimetrically with the temperature jump method. Best agreement with the experimental results and the data of Zewe and Fromm is obtained by assuming a fast bimolecular step which is followed by a slow monomolecular interconversion. The dissociation constant of the former is 6 pM the equilibrium constant of the latter is 0.3, resulting in an over-all dissociation constant of 1.4 PM. The bimolecular velocity constant is estimated near log M-’ set-I; the consecutive monomolecular constant is evaluated at 1000 se@. The enthalpy of the bimolecular reaction step is near 7 kcal per mole; that of the monomolecular one is nearly one-fifth of the former. The signal height of the chemical change is generally only 10% of the total change, the re- mainder being caused by the change of fluorescence yield with temperature.

Acknowledgments-We would like to express our high apprecia-

7 The reasoning for assuming this condition is quite similar to the reasoning outlined in footnote 5.

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March 1964 G. H. Cxerlinski and G. Xchreck 921

tion to Dr. Britton Chance for his continuous stimulating interest in the progress of this investigation. We are highly indebted to the National Science Foundation for their extensive support of this research project.

REFERENCES 1. CZERLINSKI, G., Federation Proc., 22, 2984 (1963). 2. CZERLINSKI; G.; Biochim. et Biophys. Acta, 64, 199 (1962). 3. WINER, A. D., AND SCHWERT, G. W., J. Biol. Chem., 234,1155

(1959). 4. CZERLINSKI, G., Rev. Sci. Znstr., 33,1184 (1962). 5. WINER. A. D.. SCHWERT. G. W.. AND MILLAR. D. P. S.. J. Biol.

Chem., 234,‘1149 (1959). ’

6. CZERLINSKI, G., AND EIGEN, M., 2. Elektrochem., 63, 652 (1959).

7. HAMMES, G. G., AND FASELLA, P., J. Am. Chem. Sot., 64,4644 (1962).

8. CZERLINSKI, G., Dissertation, Gottingen (1958). 9. BUCHER, T., 2. Naturjorsch., 8b, 555 (1953).

10. THEORELL, H., MCKINLEY-MCKEE, J. S., Acta Chem. Scud, 16, 1811 (1961).

11. KORNBERG, A., in E. E. SNELL (Editor), Biochemical prer)uru- tions, Voi. S,.John Wiley &So&, Inc.jNew York, 1953, b. 20.

12. EIGEN. M.. 2. Elektrochem.. 64. 115 119601. 13. ZEWE,‘~., XND FROMM, H. j., j. Bioi Chem., 237, 1668 (1962). 14. THEORELL, H., AND MCKINLEY-MCKEE, J. S., Acta Chem.

Scud, 16, 1803 (1961).

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Georg H. Czerlinski and Gisela SchreckDehydrogenase with Reduced Nicotinamide Adenine Dinucleotide

Fluorescence Detection of the Chemical Relaxation of the Reaction of Lactate

1964, 239:913-921.J. Biol. Chem.

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