Anal. Chem. 2001, 73, 857-863
Concentration Jump Experiments for the Precise Determination of Rate Constants of Reverse Reactions in the Millisecond Time Range P. Buet, E. Lewitzki, and E. Grell*
Max-Planck-Institut fu¨r Biophysik, Kennedyallee 70, D-60596 Frankfurt, Germany A.-M. Albrecht-Gary
Laboratoire de Physico-Chimie Bioinorganique, Universite´ Louis Pasteur, F-67000 Strasbourg, France K. J. Wannowius, F. Mass, and H. Elias
Institut fu¨r Anorganische Chemie, Technische Universita¨t Darmstadt, D-64287 Darmstadt, Germany A. A. Mundt
Bio-Logic-Science Instrument S.A., F-38600 Claix, France Y. Dupont
CNRSsUMR 5090, Commissariat a` l'e´ nergie atomique, F-38054 Grenoble, France
Low dissociation or reverse rate constants of single-step or multistep complex formation equilibria are usually obtained with reduced precision from standard stoppedflow binding experiments by determination of the intercept of the concentration dependence of kobs. Large and fast concentration jumps, based on two different step-motordriven mixing setups, are performed with 60-300-fold dilutions that allow the precise, convenient, and independent determination of dissociation rate constants in the range of approximately 0.1-100 s-1 in a single stoppedflow dissociation experiment. A theoretical basis is developed for the design and for the evaluation of such dilution experiments by considering the rebinding occurring during dissociation. The kinetics of three chemical systems are investigated, the binding of Mg2+ to 8-hydroxyquinoline as well as of Ca2+ and K+ to the cryptand [2.2.2], by carrying out standard stopped-flow binding as well as dissociation experiments employing various dilution factors. The advantage of the dilution method for investigating chemical and biological systems is emphasized. To elucidate reaction mechanisms in the field of chemistry and biochemistry, it is common to carry out kinetic studies. In the case of fast reactions, the majority of the studies are done by employing the stopped-flow technique, with mixing times (or socalled dead times) around 1 ms or higher. For a single-step binding reaction (Figure 1a) under pseudo-first-order conditions, the time dependence of reactant or product concentration can then * Corresponding author: (tel) +4969 6303 290; (fax) +4969 6303 346; (email)
[email protected]. 10.1021/ac0007229 CCC: $20.00 Published on Web 02/02/2001
© 2001 American Chemical Society
Figure 1. Kinetic reaction schemes.
be described by an exponential, where the experimentally observed reciprocal decay time τ, often denoted kobs, is linearly dependent on the total concentration of the reactant applied in excess. The dissociation rate constant (koff) is obtained from the intercept of this linear dependence. In the case of a multistep reaction (as shown in Figure 1b), investigated again under pseudofirst-order conditions and where the first reaction step is not timeresolvable, the concentration dependence of kobs of the subsequent, slower second reaction step (kobs2) can exhibit saturation behavior. Here, the saturation value of kobs corresponds to the sum of k23 and k32, whereas the intercept yields the rate-limiting dissociation rate constant k32. If the value of koff (single-step equilibrium; Figure 1a) is several orders of magnitude lower than that of the formation rate constant kon, it is usually not possible to determine the rate constant of the reverse reaction with high precision. The same is true if k32 (multistep equilibrium; Figure 1b) is several orders of magnitude lower than that of k23. Another disadvantage due to the usually unprecise determination of the dissociation rate can be expected for a system where a fast reaction step precedes the Analytical Chemistry, Vol. 73, No. 5, March 1, 2001 857
rate-limiting dissociation step: A shift of the equilibrium between C’ and C (Figure 1c and d) upon change in reaction conditions, such as pH, can affect the value of the dissociation rate constant but may remain undiscovered in the case of insufficient precision. If the rate constant of the rate-limiting reverse reaction is very low, for example, smaller than 0.05 s-1, it can be determined precisely on the basis of a simple concentration jump experiment, for example, by applying conventional dilution techniques in the cuvette of a spectrophotometer. In addition, other techniques are applicable for the determination of the rate constant of a ratelimiting dissociation process, provided suitable competitors are available. In this case, the dissociation rate constant can be directly determined by carrying out standard stopped-flow experiments, provided the dissociation via a ternary complex does not occur faster. In principle, the dilution or concentration jump technique would also be applicable for the precise determination of faster dissociation steps with rate constants higher than 0.05 s-1 if an efficient and fast concentration jump experiment could be carried out in the millisecond time range, for example, in a stopped-flow instrument. The precise determination of dissociation rate constants is of great importance for characterizing the true stability (or lifetime) of a particular complex or if temperature-dependent studies for the determination of the free activation energy of the rate-limiting dissociation step are anticipated. Recently, much interest has been devoted to concentration jump experiments for the investigation of the kinetics and the mechanism of fast protein refolding, such as in ref 1, for example. Refolding is induced by diluting the unfolded protein in the presence of a denaturing agent with buffer below the agent’s critical concentration. To evaluate a dilution or concentration jump experiment of equilibrium binding systems as those shown in Figure 1a or b in a simple and straightforward manner, at least a 103-fold dilution is required, so that the adjusted equilibrium concentration of C after the dilution is very small or even neglectable, compared to the initial one. This would imply that the fraction of C, which is re-formed from A and B while the dissociation of C is in progress, can be neglected. Under these circumstances, the observed kobs value would correspond to that of koff or k32, respectively. Since it appears rather unrealistic to perform such high dilution jumps for many reasons, a procedure is developed here to precisely determine rate constants of reverse reactions occurring in the millisecond or slower time range upon dilutions up to a factor of 300 by employing commercially available stopped-flow instruments with up to three mixing chambers. The choice of the concentrations for such experiments is of major importance for achieving a still measurable concentration shift under conditions of low material requirement. Three representative examples are presented here where the fast dilution technique is applied. THEORY The theoretical expression of the time dependence of the concentrations changes occurring during the reequilibration of a binding system must be available for the determination of the dissociation or reverse reaction rate constant. Because large concentration changes are anticipated as a consequence of large (1) Blond-Elguindi, S.; Friguet, B.; Goldberg, M. E. FEBS Lett. 1988, 241, 251256.
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dilutions, rebinding of the reaction partners has to be considered during the dissociation process. Concentration Shift upon Dilution. A stock solution consisttot tot and [B]stock and the ing of the total concentrations [A]stock equilibrium concentrations [A]stock, [B]stock, and [C]stock, characterized by the apparent stability constant K of a single-step equilibrium system (Figure 1a), where K ) [C]stock/([A]stock[B]stock), is mixed with a suitable reference solution so that a dilution by the factor V is reached. If the dilution is instantaneous, the resulting concentrations after the dilution at the nominal time t ) 0 are [A]0 ) [A]stock/V, [B]0 ) [B]stock/V, and [C]0 ) [C]stock/V. After equilibration under the new conditions at the assumed time t ) ∞, the concentrations [A]∞, [B]∞, and [C]∞ of the adjusted equilibrium are characterized by the same stability constant K, provided activity coefficients are unchanged. The equilibrium shift of the system is characterized on the basis of a change in the degree of complexation ∆[C]/[C]0, with tot tot R ) [A]stock + [B]stock + K-1, -1 ∆[C] K (1 - V) + VxΓ2 - xΓ1 ) [C]0 R - xΓ
(1)
1
and tot Γ1 ) R2 - 4[A]tot stock[B]stock,
(
Γ2 ) K-1 +
)
R - K-1 V
2
tot [A]tot stock[B]stock
-4
V2
(2)
Details are given in part A of the Supporting Information. Time Dependence of the Concentration Shift. Single-Step Equilibrium. Upon a V-fold dilution of a single-step equilibrium system (Figure 1a), the time dependence of the reequilibration is given by
e-konxΓ2t [C]t ) [C]∞ + xΓ2 e-konxΓ2t + γ with
2xΓ2
γ) K
-1
xΓ1 1 - 1 + xΓ2 V V
(
-1
(3)
)
Details are characterized in part A of the Supporting Information. Even, if the system is studied only on the basis of one single dilution, the value of koff can be determined in a straightforward manner with high precision, because the product kon(Γ2)1/2 consists of the dominant term konK-1, which is equal to koff. The larger the dilution, the smaller is the contribution resulting from rebinding of A and B occurring during the dissociation process. If indefinitely large dilutions were chosen, so that [C]∞ can be neglected compared to [C]0, γ would approach zero (cf. eq 3) and an exponential decay would result for the time dependence of [C]:
[C]t ) [C]∞ + [C]0e-kofft
(4)
Another approximation, also characterized by an exponential tot tot is much larger than [A]stock decay, is reached when [B]stock (pseudo-first-order conditions):
[C]t ) [C]∞ + ([C]0 - [C]∞) e-
(
)
tot [B]stock + koff t kon V
(5)
Under these circumstances, the following simple relation holds for the dependence of the reciprocal decay time τ, denoted kobs, for the dilution factor V
[B]tot 1 stock ) kobs ) kon + koff τ V
(6)
which can be applied for the direct determination of koff. Equation 6 corresponds to the expression derived for chemical relaxation (small concentration perturbations) of a one-step equilibrium system2 under conditions where the equilibrium concentration of B is also much larger than that of A. Multistep Equilibrium. If the system under investigation cannot be described by a single-step equilibrium reaction, a two-step process can be taken into consideration (Figure 1b). Provided the two expected equilibration processes are well separated on the time axis, a saturating concentration dependence is expected for the reciprocal decay time (τ2-1 ) kobs2) if the interconversion is slow compared to that of the preceding binding step. A comparatively simple dependence of kobs2 on the dilution factor V is then obtained, provided one reaction partner, for example, B, is applied in excess:
k23 1 + k32 ) kobs2 ) τ2 1 + (V/K1[B]tot stock)
(7)
If the faster process, characterized by the decay time τ1, is also observable under such conditions, an evaluation related to the one-step equilibrium procedure given above is possible. The considerations given above allow one to recognize conditions under which it is comparatively simple to determine dissociation rate constants of single- and two-step binding systems. How to choose suitable concentrations for the preparation of the stock solutions to perform experiments under optimal conditions is discussed in part B of the Supporting Information. In addition, graphical aids are also included in this part (Supporting Information, Figures 1 and 2). MATERIALS AND METHODS Materials, Solutions, and Spectra. Information is given in part C of the Supporting Information. Stopped-Flow Instrumentation. Multimixing, Step-MotorDriven Dilution Stopped-Flow Instrument. Variable multistep dilution stopped-flow studies are carried out using a Bio-Logic SFM(2) Eigen, M.; De Maeyer, L. C. M. In Techniques of Organic Chemistry; Friess, S. L., Lewis, E. S., Weissberger, A., Eds.; Wiley-Interscience: New York, 1963; Vol. III, Part II, p 895.
4/S stopped-flow instrument (cf. Supporting Information, Figure 3). The SFM-4/S is a four-syringe (20 or 5 mL) instrument in which all step-motor-driven syringes (S1, S2, S3, S4; cf. Supporting Information, Figure 3) can be operated independently to carry out single- to triple-mixing (mixing chambers MA, MB, MC; cf. Supporting Information, Figure 3). Dilutions up to 200-fold are reached. The resolution of the step motors is 35 nL/step for the 5-mL and 140 nL/step for the 20-mL syringe. Details and information about applied light sources, detection units, and signal amplification and evaluation are given in part D of the Supporting Information, together with the results of extensive dead time determinations (flow 8 mL s-1) employing the 2,6-dichloro-4-[(4hydroxyphenol)imino]-2,4-cyclohexadien-1-one (DICP)/ascorbic acid system similar to ref 3 (absorption) and the N-bromosuccinimide (NBS)/N-acetyl-L-tryptophanamide system according to ref 4 (fluorescence). The experimentally determined values correspond to the calculated ones, as described in detail in part D of the Supporting Information. Single-Mixing, Microstep-Motor-Driven Dilution Stopped-Flow Instrument. Single-step dilution stopped-flow experiments are performed with the Bio-Logic µSFM-20 instrument by employing the cell head and detection facilities of the SFM-4/S setup. In contrast to the SFM-4/S version, the µSFM-20 instrument consists of a single mixing chamber (only MA in Supporting Information, Figure 3) and two syringes (10 and 1 mL), S1 and S2 (cf. Supporting Information, Figure 3) for dilution experiments up to 300-fold. The instrument is equipped with refined step motors with a step size of 104 nL/step for the 10 and 10.4 nL/step for the 1-mL syringe. Further details, characterization of dead time and mixing linearity are given in part D of the Supporting Information. Pneumatic Stopped-Flow Instrument. Standard binding studies, based on the usual 1:1 dilution, are carried out with a SF17MV stopped-flow instrument (Applied Photophysics), controlled by an on-line PC and equipped with a thermostat. Details and properties of the setup as well as data evaluation are specified in part D of the Supporting Information. Kinetic Studies. Mg2+ Complex of 8-Hydroxyquinoline (HQn). For the study of the dynamics of the dissociation of the 8-oxoquinoline Mg2+ complex, dilution experiments are carried out by using the Bio-Logic SFM-4/S instrument equipped with the FC15 fluorescence cell (λex ) 330 nm and λemi > 495 nm; 25.0 ( 0.3 °C). For all experiments, a total flow of 8 mL s-1 is applied. Because Mg2+ dissociation is fast, a short dead time of the setup (∼6 ms) is chosen (only single mixer MA is applied; cf. Supporting Information, Figure 3). Thus, only 2-20-fold dilutions are used. Syringe S1 contains 50 mM triethanolamine hydrochloride, pH 8.0, adjusted with KCl to a total ionic strength of 0.2 M; S2 is filled with the same adjusted buffer containing 1.5 mM 8-hydroxyquinoline (dilution from a 120 mM methanolic stock solution) and 15 mM MgCl2 (final methanol content 1%). Standard stopped-flow binding experiments are performed with the SF17MV instrument accordingly by mixing 300 µM 8-hydroxyquinoline with different concentrations of MgCl2 (2-50 mM). Ca2+ Complex of 4,7,13,16,21,24-Hexaoxa-1,10-diazabicyclo[8.8.8]hexacosane ([2.2.2]). For the investigation of the kinetics of the dissociation of the Ca2+ complex of [2.2.2], dilution (3) Tonomura, B.; Nakatani, H.; Ohnishi, M.; Yamaguchi-Ito, J.; Hiromi, K. Anal. Biochem. 1978, 84, 370-383. (4) Peterman, B. F. Anal. Biochem. 1979, 93, 442-444.
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experiments are carried out by using all available syringes and mixing chambers of the Bio-Logic SFM-4/S setup, equipped with the TC100/15 absorption cell (detection at 580 nm; 25.0 ( 0.3 °C). Dilutions up to ∼1:10 per single mixing unit are chosen, leading to total dilutions of 5-100-fold, flow times between 125 and 620 ms, volumes between 300 and 2350 µL in S1 (cf. Supporting Information, Figure 3) and S2, 200 and 50 µL in S3, and 200 and 250 µL in S4, with a total flow of 8.0 mL s-1 (dead time ∼12 ms). S1, S2, and S4 (each 20 mL) contain 5 mM N-ethylmorpholine hydrochloride, pH 8.5, including 20 µM Cresol Red, adjusted with choline chloride to a total ionic strength of 0.2 M; S3 (volume 5 mL) contains 10 mM [2.2.2] and 100 mM CaCl2 in the same ionic strength adjusted indicator/buffer system. In addition, a single dilution experiment is carried out in the same medium under the conditions of equal [2.2.2] and CaCl2 concentration (for details, cf. legend of Figure 5a). Standard binding experiments are performed with the SF17MV setup by mixing 153 µM [2.2.2] with different concentrations of CaCl2 in the same medium. K+ Complex of [2.2.2]. For the characterization of the dissociation of the K+ complex of [2.2.2], dilution experiments are carried out by using the Bio-Logic µSFM-20 instrument equipped with the TC100/10 absorption cell (detection at 580 nm; 25.0 ( 0.3 °C). The stock solution in the 1-mL syringe contains 60 mM [2.2.2] and 600 mM KCl in the same indicator/buffer system used for the Ca2+ dissociation experiments. The 10-mL syringe is filled with the same ionic strength-adjusted indicator/buffer system. The 200fold dilutions are obtained by combining 10 µL of the smaller syringe with 1990 µL of the larger syringe (flow time 350 ms; dead time ∼4 ms). Standard stopped-flow binding studies are also carried out with the SF17MV instrument (absorption detection at 520 nm) by mixing 150 µM [2.2.2] with different concentrations of KCl (0.3-12.7 mM) in the same medium. APPLICATION OF THE MULTISTEP DILUTION STOPPED FLOW FOR THE DETERMINATION OF RATE CONSTANTS OF REVERSE REACTIONS General Considerations. Four important factors limit the applicability of the concentration jump method. One factor concerns the dead time of the setup, which should be short compared to the decay time of the system under investigation. The dead time depends on the mode of dilution, cell dimensions, and flow rate. The second aspect is related to the shift of the chemical equilibrium upon dilution. The maximum dilution factor that can be reached experimentally with step-motor-driven setups is currently ∼300. Depending on the binding affinity of the investigated system, the concentrations of the involved species have to be selected appropriately so that a realistic detection of the dissociation process after dilution is possible. Besides these aspects, the absolute signal of the chemical system after dilution represents a severe limitation. Details concerning optimization, instrumentation, and dead times as well as general recommendations to perform dilution experiments are given in the Supporting Information. In the following, the dilution stopped-flow technique is applied for the characterization of known chemical systems. Mg2+ Binding to 8-Hydroxyquinoline. This system (cf. Figure 2a) has been previously analyzed on the basis of a singlestep binding model.5,6 A recorded kinetic binding experiment in the millisecond time range, performed with excess MgCl2 (pseudo860 Analytical Chemistry, Vol. 73, No. 5, March 1, 2001
Figure 2. (a) Schematic illustration of Mg2+ binding to 8-hydroxyquinoline (HQn) and (b) of cation binding (Mn+: Ca2+, K+) to the cryptand [2.2.2], coupled to an acid/base indicator system.
first-order conditions) and characterized by a time-dependent fluorescence emission increase, is shown in Figure 5 of the Supporting Information (cf. part E). As expected, a linear dependence of the experimentally determined kobs values on MgCl2 concentration is observed (cf. Figure 6 in part E of the Supporting Information). The resulting kinetic parameters are 18 700 ( 500 M-1 s-1 for kon and 47 ( 6 s-1 for k′off, determined from the slope and intercept, respectively. Here, k′off is treated as an apparent rate constant and corresponds to the product of koff[H+], which is constant in the buffered medium (cf. ref 6). The results are consistent with earlier data, obtained under slightly different experimental conditions5,6 and allow the calculation of the pHdependent stability constant K′ of the complex, the log K′ value being 2.59. An independent determination by spectrofluorometric titration provides a similar value of 2.5 ( 0.1. For the independent determination of the dissociation rate constant of this chemical system, dilution stopped-flow experiments are carried out with a single mixing chamber (3.7 ms dead time). Experimental details are given in Materials and Methods as well as in parts D and E of the Supporting Information. The stock solution containing 8-hydroxyquinoline in the presence of excess MgCl2 at pH 8.5 is diluted 2-20-fold under conditions where the ionic strength is maintained. These moderate dilutions imply that no very large degree of dissociation can be achieved; the calculated values are 16, 37, and 78% for 2-, 4-, and 20-fold dilutions, respectively. The recorded kinetic phase of a 4-fold dilution experiment, characterized now by a fluorescence decrease, is shown in Figure 5 of the Supporting Information. Because an excess of MgCl2 is applied, the time-resolved process is evaluated according to eqs 5 and 6, predicting a monoexponential function. From the dependence of the kobs values on the dilution factor, shown in Figure 3, a dissociation rate constant k′off of 48 ( 0.5 s-1 is obtained. This value agrees with that obtained from the intercept of the stopped-flow binding experiment, mentioned above. It may be interesting to note that the experi(5) Brissette, P.; Ballou, D. P.; Massey, V. Anal. Biochem. 1989, 181, 234238. (6) Bugnon, P.; Laurenczy, G.; Ducommun, Y.; Sauvageat, P. Y.; Merbach, A. E.; Ith, R.; Tschanz, R.; Doludda, M.; Bergbauer, R.; Grell, E. Anal. Chem. 1996, 68, 3045-3049.
Figure 3. Concentration jump studies by carrying out stopped-flow dissociation experiments of the 8-oxoquinoline-Mg2+ complex at pH 8.0 under conditions of excess MgCl2 by employing different dilution factors (for detailed experimental conditions, see Materials and Methods). Plot of kobs values versus dilution factor. The solid line represents the theoretical dependence of kobs versus dilution for a one-step binding equilibrium according to a fit with eq 6, which provides the parameters K ) 280 M-1, k′off ) 48 s-1. The dotted line represents the level of the k′off value. Further details are given in the text.
mentally observed kobs value resulting for the 20-fold dilution is still ∼20% higher than that of the k′off value, due to the contribution of the kon-dependent term. Ca2+ Binding to [2.2.2]. The kinetics of Ca2+ binding to the cryptand [2.2.2] has been reported earlier at pH 11.57 (ionic strength 0.1 M, 25.0 ( 0.3 °C). Formation and dissociation rate constants of 7.3 × 103 M-1 s-1 and 0.26 s-1, respectively, based on a single-step binding model was obtained. A precise determination of the dissociation rate constant of this order of magnitude from the intercept of the linear concentration dependence of kobs was not possible. Under our conditions, at pH 8.5, the ligand preferentially exists in its monoprotonated state. Therefore, one H+/ligand molecule is assumed to be released upon cation binding. This can be followed spectrophotometrically, even in weakly buffered solutions, by employing a fast-responding pH indicator such as Cresol Red (cf. Figure 2b; Mn+ ) Ca2+). In the presence of 20 mM CaCl2, cation binding under our conditions leads to a maximum pH change of ∼0.07. Before carrying out concentration jump experiments, a kinetic binding study is performed (experimental details are given under Materials and Methods). In Figure 4a, the time-resolved absorption change of a representative Ca2+ binding experiment is shown. The observed CaCl2 concentration dependence of the apparent rate constant kobs is illustrated in Figure 4b. Only the kobs values of the five lowest CaCl2 concentrations appear to exhibit the expected linear dependence of kobs, typical for a single-step binding system. A kon value of 230 ( 12 M-1 s-1 and, according to the intercept, of 0.4 ( 0.15 s-1 for k′off are obtained. As indicated before, the apparent rate constant k′off corresponds here to koff[H+]. If the complete experimental data are considered in Figure 4b, the saturation behavior, as expected, for example, for a two-step binding model (scheme b in Figure 1), can no longer be overlooked. This implies that the time-resolved process corresponds to a slow equilibration (7) Loyola, V. M.; Pizer, R.; Wilkins, R. G. J. Am. Chem. Soc. 1977, 99, 71857188.
Figure 4. Kinetics of Ca2+ binding to [2.2.2] at pH 8.5 (experimental details are given in Materials and Methods). (a) Standard stoppedflow binding experiment (absorption detection) due to mixing (1:1 ratio) with 20 mM CaCl2. The solid line corresponds to a monoexponential fit leading to a kobs value of 5.05 s-1. (b) Plot of kobs versus values of total CaCl2 concentration after mixing. Evaluation of the kobs values obtained at low CaCl2 concentrations (filled circles, solid line expresses a linear concentration dependence), based on a one-step binding equilibrium (scheme a in Figure 1), leads to a kon value of 232 M-1 s-1 and of 0.39 s-1 for koff. Evaluation of the complete experimental data (filled and open circles; dashed line expresses saturation behavior) based on a two-step binding equilibrium (scheme b in Figure 1) according to eq 8 (with Mn+ corresponding to Ca2+) provides values for K1 of 17 M-1, for k23 of 20 s-1, and for k32 of 0.2 s-1.
(characterized by kobs2) between an intermediate and a final state of the complex, coupled to a fast, preceding binding process (preequilibration), which is not time-resolvable under our conditions. The evaluation of the concentration dependence of kobs2 can be performed according to the following equation,8 with Mn+ ) Ca2+ and k23 and k32 representing the corresponding rate constants of interconversion:
kobs2 )
k23K1[Mn+] 1 + K1[Mn+]
+ k32
(8)
Here, it is assumed that ligand deprotonation takes place within the fast preequilibration step, which appears to represent the most likely possibility. Consequently, k32 represents, in contrast to k′off, a true and thus pH-independent rate constant which characterizes (8) Doludda, M.; Kastenholz, F.; Lewitzki, E.; Grell, E. J. Fluoresc. 1996, 6, 159-163.
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the rate-limiting step of Ca2+ dissociation. Up to now, such a kinetic saturation behavior for alkaline earth cation binding to cryptands has not been reported. The evaluation according to the two-step model leads to values for the preequilibrium constant K1 of 17 ( 3 M-1, for k23 of 20 ( 5 M-1 s-1, and for k32 of 0.2 ( 0.1 s-1. These data suggest that the intermediate state (corresponding to AB in Figure 1b) is only weakly populated. The fact that a very similar value for the Ca2+ dissociation rate constant at pH 11.5 has been determined,7 confirms our assumption with regard to the assignment of the deprotonation step. Dissociation Experiments with Evaluation According to a SingleStep Equilibrium. If the chemical system is treated as a singlestep binding equilibrium (at least for the lower [CaCl2]), there are two different possibilities for carrying out and evaluating dilution experiments: either by applying similar reactant concentrations to achieve a maximum complex dissociation upon dilution and using eq 3 for the evaluation or by choosing the concentration of one reactant in excess and evaluating the observed data on the basis of eq 5. Both possibilities provide an independent determination of the apparent dissociation rate constant k′off. Based on the upper diagram of Figure 1 in the Supporting Information (part B), a degree of complex dissociation of ∼75% can be expected for a 100-fold dilution at pH 8.5 for a stock solution containing 30 mM [2.2.2] and 30 mM CaCl2, based on a stability constant of 1.75 × 103 M-1 (calculated from data given in refs 9 and 10). The kinetic response to a single concentration jump experiment for a 70-fold dilution is shown in Figure 5a). In contrast to the Ca2+ binding experiment, an increase of the indicator absorption is observed here and a small deviation of the experimental curve from a monoexponential decay is only visible in the time range 0-2 s. Evaluation according to eq 3 (single-step equilibrium) provides a k′off value of 0.15 ( 0.03) s-1. In addition to this experiment, a set of dilution studies is carried out under conditions of excess CaCl2 in the stock solution (for details, see Materials and Methods). If the kinetic phases obtained for different dilutions are evaluated on the basis of a monoexponential decay for the purpose of a coarse estimation, the kobs values are expected to decrease with increasing degree of dilution and appear to reach a constant value of ∼0.26 s-1 for very high dilutions (cf. Figure 5b). Under such circumstances, this value could be attributed to k′off. However, as expected according to the theoretical considerations discussed before, these kinetic phases still contain a nonnegligible contribution from the rebinding process. The evaluation according to eqs 5 and 6, still related to a singlestep binding model, leads to similar k′off values for all dilutions (cf. Figure 5c). The resulting mean k′off value is 0.15 ( 0.02 s-1. For the rate-limiting step of Ca2+ dissociation, which is assumed to be pH independent, a good correspondence is found between the values of the dissociation rate constant obtained from the stopped-flow binding and dissociation experiments performed at pH 8.5, although the latter study provides a more precise value. Dissociation Experiments with Evaluation According to a TwoStep Equilibrium. If, however, the dissociation experiments (10100-fold dilutions), performed with excess CaCl2 in the stock solution, are evaluated on the basis of a two-step binding model (fit with monoexponential function), eq 7 can be applied for the (9) Lehn, J. M.; Sauvage, J. P. Chem. Commun. 1971, 440. (10) Lehn, J. M.; Sauvage, J. P. J. Am. Chem. Soc. 1975, 97, 6700-6707.
862 Analytical Chemistry, Vol. 73, No. 5, March 1, 2001
Figure 5. Stopped-flow dissociation experiments of the [2.2.2]Ca2+ complex at pH 8.5 by employing the three-step mixing setup: (a) 70-fold dilution of 30 mM [2.2.2] and 30 mM CaCl2 in 5 mM N-ethylmorpholine hydrochloride, pH 8.5 (containing 20 µM Cresol Red), adjusted to an ionic strength of 0.2 M with choline chloride, with the same ionic strength-adjusted indicator/buffer system at a flow rate of 8 mL/s (25.0 ( 0.3 °C). Evaluation based on a fit with a monoexponential function leads to a kobs value of 0.26 s-1 (dashed line) and, according to eq 3, to a Ca2+ dissociation rate constant k′off of 0.15 s-1 (solid line). (b) Variable dilution experiments under conditions of excess CaCl2, as described in Materials and Methods. Plot of kobs2 versus the dilution factor. Circles represent kobs2 values due to a two-step binding system, resulting from fits with a monoexponential function. The solid line represents the theoretical dependence according to the fit with eq 7 providing the parameters K1 ) 22 M-1, k23 ) 10.5 s-1, and k32 ) 0.17 s-1. The dashed line indicates the level of the mean k32 value for Ca2+ dissociation of 0.17 s-1. (c) Plot of k′off values obtained from evaluations according to eq 6, again related to a one-step binding model, versus dilution factor. The dashed line represents the mean k′off value for Ca2+ dissociation, calculated from the values of the seven largest dilutions, which is 0.15 s-1. Further details are given in the text.
determination of the k32 value from the experimentally obtained kobs2 data (cf. Figure 5b). The corresponding fit provides values for K1 of 22 ( 2 M-1, for k23 of 10.5 ( 0.4 s-1, and for k32 of 0.17 ( 0.02 s-1. The resulting k32 value is very close to that obtained for k′off according to the single-step binding model. K+ Binding to [2.2.2]. This system (Mn+ in Figure 2b corresponds to K+) has to be investigated in a shorter time range because K+ dissociation is faster than Ca2+ dissociation. The kinetic binding study of K+ and [2.2.2] under pseudo-first-order conditions is carried out in the same medium as that for Ca2+
(details are given in Materials and Methods). K+ binding is detected by employing the same indicator and is characterized by an absorption decrease. Plotting the observed, apparent rate constant versus the total KCl concentration leads to a more pronounced saturation behavior than observed for Ca2+ binding kinetics. Also here, a kinetic reaction model consisting of a slow interconversion step, coupled to a fast preceding binding step (cf. Scheme b in Figure 1) has to be assumed. Evaluation according to eq 8 (with Mn+ ) K+) yields values for the preequilibrium constant K1 of (4 ( 1) × 103 M-1, of 24 ( 6 s-1 for k23, and of 10 ( 8 s-1 for the rate-limiting dissociation rate constant k32. According to literature data,9,10 the stability constant K′ of the + K complex of [2.2.2] at pH 8.5 is calculated to be 7.1 × 103 M-1. On the basis of a single-step binding process, only a degree of complex dissociation around 3% can be expected according to eq 1 for a 100-fold dilution of a solution containing 60 mM [2.2.2] and 600 mM KCl, respectively, and of 5% for a 200-fold dilution. Even under such unfavorable conditions, the detection of the absorption change is possible in the case of a 200-fold dilution experiment. As in the case of Ca2+ dissociation, an absorption increase upon K+ release is observed, which is consistent with a pH decrease of the solution. Because KCl is applied in excess, an evaluation of the resolved kinetic phase on the basis of a monoexponential function is possible and leads to a value of 29 ( 3 s-1 for kobs2. Employing eq 7 and inserting the known values of K′1 and k23, resulting from the kinetic binding study, a value of 6 ( 2 s-1 is obtained for k32, the rate constant of K+ dissociation. This value lies within the error range of the less precise value of 10 ( 8 s-1, originating from conventional stopped-flow binding studies. DISCUSSION The precise and independent determination of the rate constant of the rate-limiting dissociation step of a complex is of importance, because it expresses the true stability of a particular state in terms of its lifetime. For this purpose, fast dissociation stopped-flow experiments leading to large concentration jumps are easier to perform and offer greater accuracy than conventional methods for the determination of dissociation rate constants ranging between 0.1 and 100 s-1. The application of step-motor-controlled mixing units appears to be advantageous to keep concentration fluctuations during the mixing low and to enable short dead times which, depending on the applied equipment, are between 1 and 8 ms. If a dissociation rate constant is smaller than ∼0.1 s-1, simpler mixing techniques can be applied. In principle, it is sufficient to carry out a single dilution experiment for the precise and independent determination of a dissociation rate constant as long as the formation rate constant is known and a 50-200-fold dilution can performed. Previous attempts to apply the concentration jump technique for the determination of dissociation rate constants11,12have been limited by performing only small concentration jumps. Such experiments have thus been treated as chemical relaxation studies, (11) Bernasconi, C. F. Relaxation Kinetics; Academic Press: New York, 1976. (12) Schelly, Z. A.; Farina, R. D.; Eyring, E. M. J. Phys. Chem. 1970, 74, 617620. (13) Wong, M. M.; Schelly, Z. A J. Phys. Chem. 1974, 78, 1891-1895. (14) Lang, J. In Chemical and Biological Applications of Relaxation Spectrometry; Wyn-Jones, E., Ed.; D. Reidel: Boston, 1975; pp 195-200.
where small perturbations of the chemical equilibrium are induced. Under these experimental conditions, a linearization of the rate equation was possible. The problem of the extent of equilibrium displacement and errors related to an analysis of the recorded processes on the basis of pure exponentials has been treated extensively by Bernasconi.11 As a general rule, the equilibrium displacement should not exceed 10% if the linearized rate equations of chemical relaxation are intended to be applied. Evidently, these conditions are violated if large concentration jumps are chosen as the result of corresponding dilution experiments, as recommended in this contribution. Several factors limit the applicability of this technique. The maximum desirable decrease of the concentration of a complex upon dilution depends on the spectral properties of the investigated system. If the dilution is too large, the detection of components is no longer possible. With regard to the sensitivity of optical detection methods, fluorescence offers marked advantages over absorption detection, although light scattering detection can sometimes be very sensitive, too. On the other hand, highly concentrated stock solutions of the complex often cannot be prepared due to too low solubility properties or the lack of sufficient amounts of biological material, for example. In addition, if a molecular interaction is extremely strong, even comparatively large dilutions within the detectable concentration range, may not lead to an appreciable equilibrium displacement. On the basis of a single-step equilibrium, contour line plots have been calculated for four typical equilibrium stability constants of complexes, which allow a fast and convenient selection of suitable conditions for a given case (cf. Figures 1 and 2 in part A of the Supporting Information). To carry out dissociation stopped-flow experiments with an excess of one reaction partner represents the simplest possibility to obtain a precise value of the dissociation rate constant. The experimentally obtained kinetic phase under these circumstances is monoexponential for single- and two-step binding reactions, provided the initial binding is considerably faster than the subsequent equilibration process. If in the case of a single-step binding process, the concentrations of both reactants are similar, which usually provides better detection possibilities, the response to a large dilution experiment is no longer exponential and eq 3 has to be applied for the corresponding evaluation. Our results, obtained on different chemical systems, clearly illustrate that dilution experiments represent a fast and convenient method for an independent determination of rate constants of reverse reactions. ACKNOWLEDGMENT The authors thank Drs. H. Ruf and E. Bombarda for helpful discussions. SUPPORTING INFORMATION AVAILABLE Details concerning theory, optimization of experimental conditions, instrumentation with dead time determinations, practical recommendations for performing concentration jump experiments, and additional results are given. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review June 21, 2000. Accepted November 30, 2000. AC0007229 Analytical Chemistry, Vol. 73, No. 5, March 1, 2001
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