Method for Determination of Association and Dissociation Rate

Anal. Chem. 2007, 79, 2972-2978

Method for Determination of Association and Dissociation Rate Constants of Reversible Bimolecular Reactions by Isothermal Titration Calorimeters Tsuyoshi Egawa,*,†,‡,§ Antonio Tsuneshige,†,| Makoto Suematsu,‡ and Takashi Yonetani†

Department of Biochemistry and Biophysics and the Johnson Research Foundation, University of Pennsylvania Medical Center, Philadelphia, Pennsylvania 19104-6059, and Department of Biochemistry and Integrated Medical Biology, School of Medicine, Keio University, Shinanomachi, Shinjuku-ku, Tokyo 160-8582, Japan

The rate law equation for reversible bimolecular reactions, which are describable by association and dissociation rate constants (k1 and k-1), is not solvable to a plain formula under stoichiometric reaction conditions. Therefore, it is a general technique to observe such reactions under pseudo first-order conditions, which make the reactions a single-exponential process, and enable us to determine k1 and k-1 without any complicated iterative computations needed to analyze the same reactions under stoichiometric reaction conditions. However, the accelerated reaction rates under pseudo first-order conditions are not always favorable to the physicochemical tools employing a slow or medium response time, such as thermal analysis instruments. In this study, we have developed a simple non-iterative analytical method to determine k1 and k-1 of reversible bimolecular reactions under stoichiometric conditions on the basis of experimental data of isothermal titration calorimetry (ITC), which is generally used to determine thermodynamic parameters rather than kinetic constants. Our method is principally based on the general principle of chemical bindings caused along with the titration processes, that is, the chemical relaxation kinetics, which had been hitherto considered in the analysis on the ITC data. Thermal analysis is a general method to study chemical reactions, because nearly all chemical reactions are accompanied by heat changes. Among them, isothermal titration calorimetry (ITC) is a technique to study binding thermodynamics of molecules in liquid phase.1 On the other hand, thermal analysis for chemical kinetic parameters is usually performed by reaction calorimetry instruments, or by specially designed laboratory-made calorimeters.2 * To whom correspondence should be addressed. Phone: 718-430-2533. Fax: 718-430-4230. E-mail: [email protected]. † University of Pennsylvania. ‡ Keio University. § Present address: Department of Physiology and Biophysics, Albert Einstein College of Medicine of Yeshiva University, 1300 Morris Park Avenue, Bronx, NY 10461. | Present address: Department of Frontier Bioscience, Faculty of Engineering, Hosei University, Tokyo.

2972 Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

The simplest applications of the thermal techniques to kinetic parameter determinations must be those to the irreversible firstorder reaction, which is represented by Scheme 1 and by its differential (eq 1) and integrated (eq 2) rate law equations,

d[P] ) k([A]0 - [P]) dt

(1)

[P] ) [A]0{1 - exp(-kt)}

(2)

where [A]0 stands for the concentration of A at t ) 0. The heat generation (or absorption), which can be observed by thermal analysis instruments, takes place upon the concentration change of the reaction; therefore, the heat flow at reaction time t is given as follows,

d[P] ) k[A]0 exp(-kt) dt

q[t] ) ∆H

(3)

where ∆H is the molar enthalpy of the reaction. Accordingly, the rate constant k can be obtained by a simple single-exponential fit to the experimental q[t] data, if the reaction is satisfactorily slower than the response of the instruments. On the other hand, the reversible bimolecular reaction (Scheme 2), which involves the association (k1) and dissociation (k-1) rate constants, brings some complexities in the kinetic analysis. The differential rate law equation (eq 4, where [M]0 and [L]0 stand for the initial concentrations of M and L, respectively) for this reaction cannot be solved into its integrated form that corresponds to eq 2 of the single-exponential reaction. (1) (a) Cooper, A.; Johnson, C. M. Methods Mol. Biol. 1994, 22, 109-124. (b) O’Brien, R.; Ladbury, J. E.; Chowdhry, B. Z. In Protein-Ligand Interactions: Hydrodynamics and Calorimetry; Harding, S. E., Chowdhry, B. Z., Eds.; Oxford University Press: Oxford, UK, 2001; pp 263-286. (c) Leavitt, S.; Freire, E. Curr. Opin. Struct. Biol. 2001, 11, 560-566. (d) Tellinghuisen, J. Methods Enzymol. 2004, 383, 245-282. (e) Lewis, E. A.; Murphy, K. P. Methods Mol. Biol. 2005, 305, 1-16. (2) (a) Hora´k, J.; Sˇilha´nek, J. Thermochim. Acta 1985, 92, 269-272. (b) Landau, R. N.; Williams, L. R. Chem. Eng. Prog. 1991, 87, 65-69. (c) Willson, R. J.; Beezer, A. E.; Mitchell, J. C.; Loh, W. J. Phys. Chem. 1995, 99, 71087113. 10.1021/ac062183z CCC: $37.00

© 2007 American Chemical Society Published on Web 02/21/2007

Scheme 1

Scheme 2

d[ML] ) k1[ML]2 - {k1([M]0 + [L]0) + k-1}[ML] + dt k1[M]0[L]0 (4) Therefore, for this type of reaction, the concentration term [ML] remains in its q[t] equation, which is d[ML]/dt multiplied by the binding enthalpy. This means that the determination of k1 and k-1 on the basis of the experimental q[t] requires certain iterative computation steps to assume reliable [ML] values on the basis of the q[t] changes in the entire reaction period. The well-known way, which has been mostly combined with spectroscopic methods, to overcome the problem of the unsolvable differential rate law equation (eq 4) is to observe the reversible bimolecular reactions under pseudo first-order conditions, for example, L . M for Scheme 2.3 Under the pseudo first-order condition, eq 4 is degenerated into eq 5, where kapp is given by eq 6. This yields a simple integrated rate law equation (eq 7).

d[ML] ) -kapp[ML] + k1[M]0[L]0 dt

(5)

kapp ) k1[L]0 + k-1

(6)

[ML] ) exp(-kappt) +

k1 kapp

[M]0[L]0

(7)

Equation 7 indicates that a simple single-exponential fit is enough to determine kapp, and then k1 and k-1 are obtained from a plot of kapp versus [L]0 according to eq 6; the slope and intercept of the plot are k1 and k-1, respectively.3 Examples of such pseudo first-order/spectroscopic applications are given in the Supporting Information (Figure S1); we examined valency hybrid hemoglobins and azide anion (N3-) reactions, which are the test reactions used in this study to evaluate the validity of the present thermal analysis method. The pseudo first-order method is, in principle, applicable to the thermal analysis. If so, the determination of kinetic parameters of the reversible bimolecular reactions can be also straightforwardly done for colorless compounds, which are difficult to study by spectrophotometric methods. However, the pseudo first-order conditions inevitably accelerate the overall rate of reactions, causing difficulties in the observations of binding processes of most chemical reactions with ordinary calorimeters, which employ a response time (∼s) much slower than that of optical detectors. The purpose of this study is to develop a thermal analysis method, which enables us to determine rate constant values for (3) Antonini, E.; Brunori, M. In Hemoglobin and Myoglobin in Their Reactions with Ligands; Neuberger, A., Tatum, E. L., Eds.; North-Holland Publishing Co.: Amsterdam, 1971; Vol. 21, pp 193-199.

the reversible bimolecular reactions under stoichiometric conditions, but without any complicated iterative computations to obtain numerical solutions of the rate law equation (eq 4). To this end, we have examined general aspects of ITC data, which comprise time-dependent signals upon chemical injections in a titration experiment, and we also studied details of the kinetic event at each titration step on the basis of the principle of chemical relaxation kinetics. Our main findings are: (1) ITC data from a single titration experiment in principle give certain exponential injection signals even under stoichiometric conditions; (2) the apparent first-order rate constant of such single-exponential processes holds a linear correlation with the concentration term of the titration, giving a kapp versus [chemicals] plot, which is analogous to that obtained by optical method under the pseudo first-order conditions; and (3) the plot gives k1 and k-1. According to the present findings, we could determine k1 and k-1 values for the test reactions with only simple single-exponential and linear least-squares fits. EXPERIMENTAL PROCEDURES Materials. The preparation of the 2,3-diphosphoglycerate-free form of Hb was carried out according to a method described elsewhere.4 Ferric and carbonmonoxy forms of isolated R and β chains of Hb were prepared according to Geraci et al.5 with a slight modification. Hybrid Hb tetramer sample stoichiometrically consisting of ferric R and carbomonoxy β subunits (R3+/β2+-CO), and that consisting of ferric β and carbomonoxy R subunits (β3+/R2+CO), were prepared by mixing the appropriate R and β subunits, respectively. ITC Experiments: General. Isothermal calorimetric experiments were carried out using a VP-ITC MicroCalorimeter (MicroCal, Northampton, MA). In all titration experiments, each injection step introduced 10 µL of solution of chemicals into the 1.4381 mL reaction cell. Buffer systems used for the experiments were 100 mM sodium phosphate (pH 7.0) and 100 mM HEPES (pH 7.0). ITC Experiments: Instrument Response Function. The response function of our ITC instrument was estimated by measuring the dilution heat of methanol/buffer system. We investigated ITC signals upon injecting methanol/buffer solution (0.125-4.5%) into a pure buffer in the reaction cell. This concentration range was chosen so as to confirm that the signal shape does not depend on the signal intensity in a wide range (-0.15 to -60 µJ s-1 of the intensity), which fully covers the intensity range (-0.42 to -25 µJ s-1) of ITC signals observed in other experiments on the test reactions. RESULTS AND DISCUSSION Test Reactions: Binding of Azide Anion to the Ferric Hemes in Valency Hybrid Hemoglobins. As test reactions, we observed the bindings between ferric hemes of valency hybrid Hbs and azide anion (N3-). The valency hybrid Hbs used were ferric R and carbomonoxy ferrous β (R3+/β2+-CO) and ferric β and carbomonoxy ferrous R (β3+/R2+-CO) tetramer (R2β2) proteins. Because only the ferric hemes bind with N3-, the valency (4) (a) Tsuneshige, A.; Park, S.; Yonetani, T. Biophys. Chem. 2002, 98, 4963. (b) Yonetani, T.; Park, S.; Tsuneshige, A.; Imai, K.; Kanaori, K. J. Biol. Chem. 2002, 277, 34508-34520. (5) Geraci, G.; Parkhurst, L. J.; Gibson, Q. H. J. Biol. Chem. 1969, 244, 46644667.

Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

2973

Figure 1. Calorimetric isothermal titration for the binding of N3- to the ferric hemes of R3+/β2+-CO (a and b) and β3+/R2+-CO (c and d) hemoglobins. The upper panels (a and c) show heat flow profiles upon 28 successive injections of 10 µL of 4 mM NaN3 dissolved in a phosphate buffer (100 mM NaPi, pH 7.0) into a 1.4381 mL isothermal cell containing 0.2 mM of R3+/β2+-CO (a) or β3+/R2+-CO (c) Hb samples in the same phosphate buffer. The negative peaks in the heat flow indicate heat evolution in the isothermal cell. The dilution heat of NaN3 was measured in a separate control experiment and was subtracted. Baseline of the profile was shifted appropriately so as to place at ∼0. Experiments were carried out at 15 °C. The lower panels (b and d) show the binding isotherm (O) created by integrating each peak area of heat flow profiles in (a) and (c) and by normalizing the integer by the molar amount of N3- injected. The molar ratio of N3- to the ferric heme of the samples was calculated as described in Appendix I of the Supporting Information. Solid lines represent theoretical binding curves employing association constant (K) values at 2.2 × 105 (b) and 8.2 × 105 (d) M-1.

hybrid Hbs enabled us to observe the bindings to N3- at the ferric hemes in the R and β subunits separately. Using the pseudo firstorder/spectroscopic method, we determined k1 values of the N3binding to be 8.4 × 101 and 2.5 × 102 M-1 s-1 at 15 °C and pH 7.0 for the R and β hemes, respectively (Figure S1 in the Supporting Information). These data were in reasonable agreement with previous report6 in which k1 values were determined for R3+/β3+ tetramer Hb (R, 6.9 × 101; β, 3.9 × 102 M-1 s-1 at 20 °C and pH 6.91) or isolated R and β chains (R, 9.0 × 101; β, 4.0 × 102 M-1 s-1 at 20 °C and pH 7.0). The k-1 values were 3.8 × 10-4 and 3.0 × 10-4 s-1 for the R and β hemes, respectively. The practical goal of this study is to reproduce these rate constant values by using the present thermal analysis method. General Aspects of ITC Data. Isothermal titration calorimetry measures heat evolution or absorption in liquid phase upon injecting aliquots of one reactant into the other in a sample cell.1 Upon such heat evolution or absorption, the ITC instrument automatically adjusts the flow of heat between the sample cell and outer spaces so as to keep the cell temperature at a constant value and records the time profile of the heat flow (heat flow profile). Figure 1a and c shows heat flow profiles observed by using our ITC instrument for the Hb-azide reactions, which are the same reactions as those introduced before, although the azide concentrations used here were stoichiometric (NaN3/heme ratio was 2-2.5 at the final). (6) Gibson, Q. H.; Parkhurst, L. J.; Geraci, G. J. Biol. Chem. 1969, 244, 46684676.

2974 Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

These heat flow profiles were obtained upon 28 successive injections of NaN3 solution into the R3+/β2+-CO (Figure 1a) or β3+/R2+-CO (Figure 1c) samples. In the profiles, the negative signals represent heat evolution in the sample cell. During the measurement, known amounts (10 µL of 12-14 µM) of N3- were introduced at every injection into the sample cell, which contained the R3+/β2+-CO or β3+/R2+-CO samples (170-200 µM) (for details about the concentrations of the reactants, see Appendix I in the Supporting Information). Upon every injection, a certain proportion of the injected amount of N3- bound with the ferric R or β heme to cause the signal, and the strength of the signal progressively became weaker (Figure 1a and c) because the ferric heme of the R or β subunit became saturated with N3-. Because each ITC heat flow signal displays a characteristic time profile after the injection, the integrated area of each signal with respect to time (t) gives total heat changes upon each injection.1 When the integer is normalized by the molar amount of the applied injectant, it gives an apparent enthalpy change (∆Happ).1 In Figure 1b and d, ∆Happ values of the above ITC experiments are plotted against the molar ratio of N3- to the ferric heme of R3+/β2+-CO (b) or β3+/R2+-CO (d) (O). From such a kind of ∆Happ plot (isothermal titration plot), the association constant (K) between injectant and sample molecules can be generally determined.1 For the present data, the K value between N3- and the ferric heme of R3+/β2+-CO was calculated to be 2.2 × 105 M-1 at 15 °C and pH 7.0 (Figure 1b, theoretical titration curve was indicated by solid curve), while K of the binding between N3-

changes in the heat flow signal were also found for the data of β3+/R2+-CO (Figure 2b), although fwhm values were significantly smaller than those found for the R3+/β2+-CO reaction. These observations can be fully explained on the basis of the principle of relaxation kinetics as described below. Relaxation Kinetics and Implications to the ITC Data. Scheme 2 is a general scheme for reversible binding between chemicals L and M. When the equilibrium of the reaction system of Scheme 2 is perturbed by adding a small amount of the ligand L, the system undergoes an adjustment, so-called relaxation, until the concentrations of the species (M, L, and ML) have the values required by the new equilibrium configuration.7 Kinetics of such relaxation processes have been fully established.7 Here, we define a variable x as

cML - cML[t] ) cMf[t] - cMf ) cLf[t] - cLf ≡ x

Figure 2. Comparison of the heat flow signals of the N3- binding to R3+/β2+-CO and β3+/R2+-CO. The shapes of the signals that appeared in the ITC heat flow data of Figure 1a and b are compared in (a) and (b), respectively. The instrument function of the ITC machine (see text) is also shown in (c).

and the ferric heme of β3+/R2+-CO was obtained to be 8.2 × 105 M-1 at the same condition (Figure 1d). Details of the Heat Flow Profile. In the conventional application of the ITC technique, the ITC heat flow profile is simply used to determine ∆Happ values, which are integers of heat flow signals. The ITC heat flow signal is, however, expected to carry kinetic information too, because the signal is a time profile initiated by the injection of a reactant, although the quality of kinetic information must depend on how the kinetic events taking place in the reaction cell are slower than the response of the instrument. In Figure 2a, the 2nd, 12th, 16th, and 21st heat flow signals shown by Figure 1a are redrawn, where the intensity of each signal was appropriately normalized, while the starting point of each signal was set at t ) 0 for the purpose of comparisons of signal shapes. Some ITC signals of the reaction of β3+/Ra+-CO from Figure 1c were also redrawn in Figure 2b in the same way. As seen, the full width at half-maximum (fwhm) values of the signals in Figure 2a and b are significantly larger than the fwhm of the instrument response function (∼20 s) (Figure 2c), demonstrating that the chemical kinetic processes of the Hbs and N3reactions significantly contributed to the shapes of the ITC signals under the present experimental conditions. It should be emphasized that such kinetic information was available because the reaction was observed under stoichiometric biding conditions that slowed the overall reaction rate. Interestingly, the fwhm was increased from 100 s in the 2nd signal of R3+/β2+-CO reaction (Figure 2a, thick solid curve) to 170 s in the 12th one (thick broken curve), and progressively decreased when the injection advanced from the 12th to subsequent ones (16th, thin broken curve, 21st, thin solid curve). Similar (7) Connors, K. A. Chemical Kinetics. The Study of Reaction Rates in Solution; VCH: New York, 1990; pp 136-139.

(8)

where cMf[t] (or cLf[t]) and cMf (or cLf) are the concentration of free M (or L) at t ) t and at the new equilibrium, respectively, while cML[t] and cML represent the concentration of ML at t ) t and at the new equilibrium, respectively. The rate equation for x is then given as follows.7

dx - ) k1x2 + (k1cMf + k1cLf + k-1)x dt

(9)

If the perturbation added to the system is sufficiently small, the term k1x2 in the above equation is negligible, and then the rate equation can be approximated to

dx ) kappx dt

-

(10)

where

kapp ) k1(cMf + cLf) + k-1

(11)

Hence, x is given by

x ) x[0] exp(-kappt)

(12)

where x[0] is the x value at t ) 0. The above equation indicates that the relaxation undergoes a first-order process if the approximation given by eq 10 is reasonable, that is, the perturbation to the reaction system is sufficiently small, and then the heat changes of the reaction in such a case are given as follows.

dx q[t] ) ∆H ) -∆Hkappx[0] exp(-kappt) dt

(13)

When we apply eq 13 to the practical cases, we need to evaluate whether the approximation of eq 10 is indeed acceptable. The criterion for judging this will be found when eq 9, which is the original rate law equation, is rewritten as the following formula:7

dx ) kappx(1 + r) dt

(14)

Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

2975

-

Figure 3. Changes in the concentration term and criterion parameter in the ITC experiments on the binding of N3- to the ferric hemes of R3+/β2+-CO (a and b) and β3+/R2+-CO (c and d). Based on the K values, the concentration term cMf + cLf and criterion parameter r′ were calculated and plotted against the injection number.

where

r ) (cMf + cLf + K-1)-1x

(15)

K ) k1/k-1

(16)

and

Equation 14 approaches to the approximated equation eq 10 if r is sufficiently smaller than 1. In this study, we introduce another practical parameter r′ given by:

r′ ) [cMf(n) + cLf(n) + K-1]-1[cML(n) - cML(n - 1)] (17)

where the variable n indicates the injection number in the titration; for example, cML(n) is the equilibrium concentration of ML after the nth injection in the ITC experiments. Because the x value is at most cML(n) - cML(n - 1) for the chemical event upon the nth injection, r is smaller than r′. Thus, the criterion for accepting the approximation eq 10 is r′ , 1. Among the parameters in the above equations, the binding association constant K is systematically obtained by ITC experiments; it is the original function of the ITC instruments to determine K from the isothermal titration plot. Based on the experimental K value thus determined, the concentration terms cMf, cLf, and cML are also obtainable in a straightforward manner (see Appendix I in the Supporting Information). We then calculated numerical values of the parameter cMf + cLf at every injection of the present ITC experiments for R3+/β2+-CO (Figure 3a) and β3+/R2+-CO (Figure 3c) reactions. 2976

Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

In both cases, cMf + cLf first decreases with the injection number, and then increases. Such a behavior of the parameter cMf + cLf, which is the sum of the concentrations of free molecules, is not surprising; in early steps of the titration, almost all of the injected ligand binds to M, decreasing cMf; on the other hand, the binding becomes saturated in the later steps, and most of the injected ligand remains as free form, increasing cLf. According to eq 17, the criterion parameter r′ decreases as increasing cMf + cLf, and it also decreases as decreasing cML(n) cML(n - 1). Among the latter two parameters, how cMf + cLf changes during the titration was described already (Figure 3a and c). On the other hand, the parameter cML(n) - cML(n - 1) represents how much the concentration of complex ML increases at the nth injection. Therefore, this parameter decreases with saturating the binding. Because of these properties of the parameters cMf + cLf and cML(n) - cML(n - 1), the criterion parameter r′ generally shows an asymmetric parabola curve as a function of the titration injection number (Figure 3b and d for the R3+/β2+-CO and β3+/R2+-CO reactions, respectively), and it becomes satisfactorily small (r′ , 1) when the titration approaches to the end point. In other words, the binding titration for the reversible bimolecular reaction always involves a certain end phase, in which the kinetics is describable as a single-exponential process even under the stoichiometric binding conditions. Determination of the Kinetic Constants. According to the results shown here and discussions in the previous sections, now we find a systematic way to determine the kinetic constants of reversible bimolecular reactions on the basis of data from a single ITC experiment as follows: (1) determine K and the concentration terms cMf, etc., from the isothermal titration plot; (2) according to eq 17 and the determined values of K and the concentration terms, calculate the criterion parameter r′; (3) based on the r′

Figure 4. Plot of kapp versus cMf + cLf for the binding of N3- to the ferric hemes of R3+/β2+-CO (a and c) and β3+/R2+-CO (b and d). The kapp values were obtained by least-squares fits of the data to eq 18 (a and b) and eq 19 (c and d). Insets in (a) and (d) are examples of least-squares results by the former and latter equations, respectively. The association rate constants k1 were obtained from the slopes of the plots, while the dissociation rate constants k-1 were obtained according to the equation K ) k1/k-1.

values, find ITC heat flow signals that can be satisfactorily regarded as a single-exponential process; (4) for each of these heat flow signals, carry out single-exponential fit to determine the time constant kapp; and (5) make a plot of kapp versus cMf + cLf, and determine k1 and k-1 as the slope and intercept of the plot (eq 11), respectively. For step 4, we may need to include the instrument response function in the least-squares calculations, depending on how the chemical processes are slower than the instrument response. This can be easily done by the standard deconvolution fit, which is given by the following equation:

f(t) )

∫ h(τ)g(t - τ) dτ t

0

(18)

where f(t), h(t), and g(t) are the experimental data, instrument response function, and single-exponential function having the constant kapp, respectively. An alternative least-squares way to obtain kapp is given by eq 19:

φ′(t) ) kapp{h(t) - φ(t)}

(19)

where φ(t) and φ′(t) are the normalized experimental data and its derivative (see Appendix II in the Supporting Information for details), for which numerical values can be easily calculated by most of the commercial mathematical software. Regarding step 3 described above, we selected the ITC heat flow signals at 20-25th and 19-24th injections (at which r′ is less than 0.01) from the experimental data of R3+/β2+-CO and β3+/ R2+-CO reactions, respectively. The kapp values calculated for the selected signals were plotted against cMf + cLf in Figure 4, where

kapp data obtained by least-squares fits of eqs 18 and 19 are shown in (a) and (b) (a, R3+/β2+-CO; b, β3+/R2+-CO) and (c) and (d) (c, R3+/β2+-CO; d, β3+/R2+-CO), respectively. Some examples of the results of least-squares fits with these equations are also shown in the insets of Figure 4a and d, demonstrating that the experimental data (broken curves) were well reproduced by the fitting results (solid curves). Other experimental signals and their numerical derivatives were also reproduced well by fitting results by eq 18, as well as by eq 19 (not shown). The data in each panel of Figure 4 well lie on a straight line, being consistent with what is expected from eq 11. The slopes of the data obtained with eqs 18 and 19 are consistent with each other. According to the slopes of the plots, k1 values were determined to be (8.8-10) × 101 and (2.0-2.1) × 102 M-1 s-1 for the binding of N3- to R3+/β2+-CO and β3+/R2+-CO, respectively. On the other hand, the intercept of each plot, which corresponds to k-1, was accompanied by a somewhat large error value (not shown). This is not surprising because the intercept position is given by extrapolating data to the zero position along the concentration axis. Thus, we calculated more reliable k-1 values according to the relation K ) k1/k-1 to be (4.0-4.5) × 10-4 and (2.4-2.6) × 10-4 (s-1) for the R and β hemes, respectively. The k1 and k-1 values shown here are in reasonable agreement with those obtained in this study by pseudo first-order/spectroscopic techniques; that is, k1 ) 8.4 × 101 (R heme) or 2.5 × 102 (β heme) M-1 s-1 and k-1 ) 3.8 × 10-4 (R heme) or 3.0 × 10-4 (β heme) s-1 (small differences between the data obtained by thermal and spectroscopic methods could be further explained in terms of the autoxidation reactions of hemes; see the Supporting Information). Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

2977

These consistencies demonstrate that the new kinetic analysis method developed in this study successfully determined the association and dissociation rate constants of the present bimolecular binding reactions on the basis of ITC experiment data. ACKNOWLEDGMENT This work was supported in part by the National Institutes of Health Grant HL 14508, and by a grant from the Japan-U.S. Cooperative Science Program.

2978

Analytical Chemistry, Vol. 79, No. 7, April 1, 2007

SUPPORTING INFORMATION AVAILABLE Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review November 19, 2006. Accepted January 16, 2007. AC062183Z