J. Phys. Chem. 1992,96, 3 135-3 138
3135
Ionic Strength Effects on the Binding Constant of Calcium Chelators: Experiment and Theory Bo Svensson, Bo Jonsson,* Makoto Fushiki, and Sara Linse Physical Chemistry 2, Chemical Centre, University of Lund, POB 124, S-22100 Lund, Sweden (Received: April 30, 1991; In Final Form: August 20, 1991)
The effect of a KC1 electrolyte on the Ca2+binding properties of 5,S-Br,BAPTA and quin-2 has been investigated. The experimental determination of the binding constants was based on the absorbance difference between the free and calcium-loaded form of the chelator. The binding constant was measured in the range from 2 mM to 1.0 M KCl. Over the concentration range studied here the binding constants of the chelators change by 2 orders of magnitude. The theoretical calculations were based on a simple dielectric continuum model with a uniform dielectric permittivity, where only the charged species were explicitly considered. Binding constant shifts obtained from Monte Carlo simulations within this model are in excellent agreement with the experimental shifts. An equally good agreement is found from solutionsto the Poisson-Boltzmann equation. More surprising is that its linearized form gives virtually identical results over the whole concentration range.
Introduction It is well-known that the binding properties of ionic species may change by addition of an electrolyte. In some cases even small amounts of electrolyte may induce large changes, and it is of fundamental interest to understand and to be able to predict these effects.’** The Debye-Huckel theory has long since been the starting point for theoretical approaches to this p r ~ b l e m . The ~ limitations of Debye-Hiickel (DH) theory are well-known as well as its concepts. In practical applications, one can extend the validity of the D H theory by simple modification^.^ These extensions can sometimes be given theoretical justification,s but empirically adjusted parameters in the original D H expressions are commona6 The accuracy of what one might refer to as extended D H theories has been tested for simple electrolytes by comparison to results from more refined and computationally intensive calculations7 and also to Monte Carlo simulation^.^^^ These investigations have focused on monovalent salts, and the general conclusion emerging from the comparisons is that the extended D H theories are applicable up to concentrations of a few hundred millimolar. Asymmetric electrolytes are less frequently studied, and there does not seem to exist a coherent picture of the applicability of different theoretical approaches. Small ion binding chelators or even proteins, which may be highly charged polyions, represent a particularly interesting class of asymmetric electrolytes. For example, a well-known chelator like 5,5‘-Br2BAPTA has a net charge of -4 at neutral pH, and other common chelators normally carry a significant net charge. One would expect the binding of mono- or divalent ions in these systems to be strongly dependent on salt concentration. The interpretation of experimental binding constant shifts can sometimes be obscured by the fact that also nonelectrostatic interactions may change upon addition of salt. For example, if the chelator is flexible, it will certainly change its configuration when extra salt is added and screen electrostatic interactions. However, if the chelator structure is sufficiently rigid, one may assume that structural rearrangements play a minor role. Calcium binding proteins are in this respect particularly interesting to study, (1) Conway, 8. E. In Comprehensive Treatise of Electrochemistry; Conway, B. E., et al., Eds.; Plenum: New York, 1983; Vol. 5. (2) Lilley, T. H. Electrochemistry; Specialist Periodical Report; Alden Press: Oxford, 1973 and 1975; Vols. 3 and 5. (3) Friedman, H.L. Ionic Solution Theory; Interscience Publishers: New York, 1962. (4) Pitzer, K. S.J. Phys. Chem. 1973, 77, 268. (5) Nordholm, S. Chem. Phys. Lett. 1984, 105, 302. (6) Chan, K.-Y. J . Phys. Chem. 1990, 94, 8472. (7) Rasaiah, J. C.; Friedman, H. L. J . Chem. Phys. 1968, 48, 2742. (8) Card, D. N.; Valleau, J. P. J . Chem. Phys. 1970, 52, 6232. (9) Megen, W. Van; Snook, I. K. Mol. Phys. 1980, 39, 1043.
since they are known to be structurally very stable under different conditions.lOJ1 Mast of these usually bind more than one calcium ion; hence, the electrostatic contribution can be expected to dominate over other interaction terms. Recently, we have been able to quantitatively reproduce the salt effects in the calcium binding protein calbindin, using a remarkably simple model of the protein and the surrounding electrolyte s01ution’~J~ together with a new Monte Carlo (MC) technique for calculating individual chemical potentials. l 4 Originally, the method was developed for biochemical systems, but it should of course be equally applicable to simpler systems. To verify this, we have chosen to study ionic strength effects of two small Ca2+ binding chelators rather than proteins. The binding constant shifts seen for the small chelators are of course significantly smaller than those in calbindin, but on the other hand one avoids several of the complications involved in handling the biomolecular solutions, where the number of components is often large, some of which may interact with the protein in an unknown way. In addition to the MC simulations, we have also used the Poisson-Boltzmann (PB) equation and its linearized version to calculate the binding constant shifts. This gives us a possibility to investigate the validity of the approximations involved in the PB theory. The linearized PB or Debye-Huckel equation can be solved analytically and is from this point of view particularly interesting.
Experiment The two chelators we will investigate are 5,5’-Br2BAPTA and quin-2; see Figure 1. The Ca2+ binding constants of these chelators have been measured in 2 mM Tris/HCl buffer a t pH 7.5 at various concentrations of KCl. The tetrapotassium salts of the chelators have been used throughout. The chelators bind Ca2+ strongly, and at low ionic strength the binding constants of quin-2 and 5,5’-Br2BAPTA are around 2 X lo8 and 1 X lo7 M-I, re~pectively.’~These values were obtained from titrations monitoring the competition for Ca2+between the chelator and EDTA, (10) Wendt, B.; Hofmann, T.; Martin, S. R.; Bayley, P.; Brodin, P.; Grundstrom, T.; Thulin, E.; Linse, S.;Forsen, S. Eur. J . Biochem. 1988, 175, 439. (11) Brzeska, H.; Venyaminov, S. V.; Grabarek, Z.; Drabikowski, W. FEES Lett. 1983, 153, 169. (12) Svensson, B.; Jonsson, B.; Woodward, C. Biophys. Chem. 1990,38, 179. ..
(13) Svensson, B.; Jonsson, B.; Woodward, C.; Linse, S. Biochemistry
1991, 30, 5209.
(14) Svensson, B.; Woodward, C. Mol. Phys. 1988,64, 247. (1 5) Linse, S.;Johansson, C.; Brodin, P.; Grundstrom, T.; Drakenberg, T.; Forsen, S. Biochemistry 1991, 30, 154.
0022-3654/92/2096-3 135%03.00/0 0 1992 American Chemical Society
Svensson et al.
3136 The Journal of Physical Chemistry, Vol, 96, No. 7, 1992 5 , 5 * -B r2 BAPTA
coo. coo.
ApK = pK - p F C f= @(AGE’- AGE1*E3/(ln10)
quin-2
The electrostatic free energy contribution can be expressed in terms of the excess chemical potentials, p , of the bound and free Ca2+
coo. coo-
AGE’ = p ( ~ -) p ( ~ )
Br
cy
Br
Figure 1. Schematic structures of 5,5’-Br,BAPTA and quin-2.
at different pH’s close to 7.5. The binding constants of the chelators a t low ionic strength thus depend on the value used for the Ca2+binding constant of EDTA. We used 1.8 X 10” M-I for the fully deprotonated form of EDTA a t low ionic strength (an average of published values) and pK, values of the carboxylate groups according to standard textbooks.I6 At 10 mM to 1.O M KC1 the Ca2+ binding constants of the chelators were determined by titrating a solution of the chelator with CaZ+(in the absence of EDTA). The absorbance at 263 nm of the chelator solution was followed during the titration. Since there is a dramatic decrease in the absorbance at this wavelength when the chelators bind Ca2+, the binding constant could be obtained from a least-squares fit to the absorbance as a function of the total Ca2+ concentration. The volumes of the chelator solution (usually 2.5 or 3.0 mL) and Ca2+aliquots (5.0 pL) were calibrated against each other by weight. Ca2+ stock solutions (3 and 10 mM) were prepared in 2 mM Tris/HCl at pH 7.5 at the same values of KCl concentration as the chelator solutions, and the Ca2+ concentration was determined by atomic absorption spectroscopy. The concentration of the chelator, approximately 30 pM, was determined from the absorbance at 239.5 nm in the presence of excess Ca2+,using c239,5values of 4.2 X lo4 and 1.6 X lo4 L mol-’ cm-I for quin-2 and 5,5’-Br2BAPTA, respectively. The residual Ca2+concentration before starting the titration was determined by comparing the absorbance at 263 nm with the values obtained in the presence of excess EDTA and Ca2+,respectively. The accuracy of the determined values is highest for 53’Br2BAPTA at high ionic strength but considerably lower for 5,5’-Br,BAFTA at 10 and 25 mM KCl and for quin-2 at all values of the KC1 concentration. The values for 5,5‘-Br2BAPTA a t 10 and 25 mM KC1 and for both chelators at 50,100, and 150 mM KCl have been published e l ~ e w h e r e . l ~ , ~ ~
Binding Model The stoichiometric calcium binding constant, K, to the chelator, Q, can be written as
where the brackets denote molar concentration. The binding constant is simply related to the Gibbs free energy
K = exp[-B(AGE’
(3)
+ AGO)]
(2)
where 0 equals l/kBT, kBbeing Boltzmann’s constant and T the temperature. The first term in the exponential is the change in electrostatic free energy, which is dominated by long-range interactions, while the second term contains all other contributions. This includes contributions from structural changes of the chelator and changes in solvation of the chelator and the calcium ion upon binding as well as trivial terms from the choice of standard state. These are essential for determining the absolute value of the binding constant, which is outside the scope of this work. We now make the assumption that AGO is independent of salt concentration. Relative to a reference salt concentration, we then have for the shift in the logarithm of the binding constant (16) Skwg, D. A.; West, D.M. Fundamentals of Analytical Chemistry; Holt-Saunders: New York, 1976. (17) Linse, S.; Helmersson, A,; ForsBn, S . J . Biol. Chem. 1991, 266, 8050.
(4)
Model and Methods The primitive model is used to describe the electrolyte solution? which means that the ions are treated as hard spheres with a diameter CT and a charge qe. The diameter is the same, 4 A, for all ions, and q is the valency and e the unit charge. The solvent is replaced by a structureless continuum, which enters the calculations only through its dielectric constant cr assumed to be uniform throughout the solution. The temperature and dielectric constant were chosen to be 298 K and 78.7, respectively, to correspond to water at room temperature. The chelator was modeled as a hard sphere with a diameter of 14 A and a charge of -4e, since both quin-2 and 5,5’-BrzBAFTA are fully protolyzed at the experimental pH. The charge has been placed either in the center of the chelator (model I) or 2 A from its boundary (model 11). Within this model the interaction energy between two ions i and j a distance rV apart is given by ulJ = q,q,e2/(4.eoerr,,) =
‘1,
rlJ
r,J 2
(a, +
0,)/2
< (a! +
(54 (5b)
where to is the permittivity of free space. Simulations. The simulations were performed with the Metropolis algorithm’* in the canonical ensemble, where the number of particles, temperature, and volume are constant. A spherical simulation cell was used. This contained one chelator, counterions, and salt ions. The chelator was kept fixed at the center of the cell during the simulation. In dilute salt solution with 2 mM salt or less, the cell volume was determined from the chelator concentration. Above this concentration the cell contained 50 salt pairs, and the cell volume was determined by the salt concentration. Additional simulations including 100 salt pairs confirmed that the simulated averages were converged. The system was allowed to equilibrate for a few thousand configurations per ion, and the averages were collected from at least loo00 codigurations per ion. The excess chemical potential of free and bound Ca2+ was obtained by a modified Widom technique.I4 The traditional Widomlg procedure is to insert a fictitious test particle at a point r and calculate the change in energy AU(r) of the system P 4 r ) = -In (exp(-BAU(r)))
(6)
where the angular brackets denote an ensemble average. A straightforward application of this formula to ionic systems introduces unacceptable finite size effects. It has been shown that a simple charge scaling procedure can correct for much of this error in both uniform and nonuniform systems.14 The binding site has been taken as the position of the chelator charge. At salt concentrations above 2 mM, p(F) was obtained from separate simulations of uniform salt solutions. The Poisson-Boltzmann Equation. The comparison between Poisson-Boltzmann and Monte Carlo results will be limited to model I where the binding site is assumed to coincide with the center of the chelator. It is straightforward to extend the comparison to model I1 or to other nonspherical systems as has been done for proteins.*O The ion distribution outside the chelator is then obtained by the spherically symmetric PB equation d2p/dr2 + 2 / r dp/dr = -p(r)/totr P(T)
= Cqaena(r) = Cqo,eno,oexp(-qaecp(r)/kT) a
(7) (8)
a
~~
~
(18) Metropolis, N.; Rosenbluth, A. W.; Roscnbluth, M. N.; Teller, A. H.; Teller, E. J . Chem. Phys. 1953, 21, 1087. (19) Widom, B. J . Chem. Phys. 1963, 39, 2808.
(20) Fushiki, M.; Svensson, B.; Jonsson, B.; Woodward, C. Biopolymers 1991, 31, 1149.
The Journal of Physical Chemistry, Vol. 96, No. 7, 1992 3137
Ionic Strength Effects on Calcium Chelators
TABLE I: Excess Cbemical Potential, &I, of Bound and Free Calcium in chelator' CKCl MC PB DH 0.29 0.60 1.15 1.59 1.96 2.35 2.57 2.98 3.25 3.47 3.67
Ob
2 10 25 50 100 150 300 500 700 1000
"Model I.
0.59 1.14 1.58 1.95 2.35 2.58 2.97 3.24 3.40 3.56
0.56 1.08 1.51 1.89 2.29 2.54 2.94 3.21 3.38 3.55
free calcium PB
DH
-0.18 -0.26 -0.44 -0.65 -0.86 -1.10 -1.26 -1.52 -1.73 -1.84 -1.94
-0.21 -0.44 -0.65 -0.85 -1.10 -1.25 -1.54 -1.76 -1.90 -2.05
-0.20 -0.41 -0.61 -0.80 -1.03 -1.18 -1.49 -1.72 -1.87 -2.02
the chelator counterions contribute.
where na(r) and n, are the number density of the ath ionic species in the spherical cell and salt reservoir, respectively. Equation 7 is complemented by two boundary conditions: one at the spherical cell boundary
dR,)= 0,
R, = cell radius
(9)
and the other a t the surface of the chelator (dq/dr),=, = -qoe/4neoera2
(10)
where a and q,,e are the radius and charge of the chelator, respectively. Equation 7 is solved by the RungeKutta method. The solution for a free calcium ion is obtained in the same way. The excess chemical potential, p(B), is obtained as the difference in free energy between the bound and unbound state of the chelator p(B)
= G(Ca2+) - G(0)
(11)
where G(0) is the free energy without calcium and G(Ca2+)with one calcium ion bound to the chelator. Since the system is in equilibrium with a salt reservoir, the free energy can be written as20 GPB
MC
=
qoedO)/2kT - 2 7 r I p d d r - kT47r
E I [ n a ( r ) - nao19 d r a
If we expand the exponential term in eq 8 and keep only the first term, we get the linearized Poisson-Boltzmann or Debye-Hiickel equation d29/d9
+ 2/r
dp/dr = K2&)
(13)
where K = [E(qg)2nd/%erkT]1/2 is the Debye screening constant. The linearized equation can be solved in the same way as the nonlinear one, and the free energy expression simply becomes
GDH = qoedO)/2kT
(14)
with chemical potentials obtained from eq 11.
Results and Discussion Table 1 shows the excess chemical potentials of bound and free calcium ions from MC simulations of a chelator, where the trivial contribution to p(B) from the chelator charge has been subtracted off. One can note that the chemical potentials for both the chelator complex and a free calcium ion are surprisingly accurately determined by the approximate theories. This is true not only at low ionic strength but also in 1.O M KC1. A similar comparison for the chemical potential of a monovalent ion in a 1:l salt solution would look different both on a relative and on an absolute scale; see below. It is only the shifts relative to a reference that are meaningful to discuss in relation to experiments, and we have chosen the chelator in 2 mM KCl as a reference, which should be close to the experimental conditions a t the lowest ionic strength, where at least 2 mM Tris/HCl buffer is present. Table 11 summarizes the binding constant shifts obtained from experiments on the two chelators as well as those from the different theories and models.
TABLE 11: ApK for the Chelators Relative to 2 mM KCP cKCl 0" Oc
2 10 25 50 100 150 300 500 700 1000
MC"
MCb
PB
DH
-0.37 -0.17 0.00 0.32 0.60 0.85 1.12 1.29 1.58 1.79 1.93 2.05
-0.41 -0.17 0.00 0.34 0.68 0.97 1.30 1.49 1.86 2.12 2.30 2.48
0.00 0.34 0.62 0.87 1.15 1.32 1.61 1.82 1.95 2.09
0.00 0.32 0.59 0.84 1.11 1.29 1.59 1.81 1.95 2.09
exP BAPTA quin-2 0.00 0.26 0.64 0.89 1.20 1.36 1.58 1.77 1.86 1.97
0.00 0.88 1.18 1.35 1.60 1.72 1.82 1.94
'Model I. Model 11. Only the chelator counterions contribute. "Chelator in infinite dilution. The experimental shifts for the two chelators are within the experimental uncertainties identical, which indicate that from an electrostatic point of view they behave very similarly. Figure 1 shows that their covalent structures differ to some extent, but this is obviously of minor importance. These results are in agreement with Pethig et a1.,21 who compiled salt shifts in the Mg2+binding of a dozen different BAPTA analogues, which all showed very similar salt behavior. The theoretical and experimental shifts for model I are in excellent agreement, despite the simplicity of the model. Even at the highest concentration the difference between theoretical and experimental data is only a few percent. Model 11, where the binding site is displaced from the center of the sphere, gives a quantitative agreement with experiment up to 0.15 M KCl, but at higher salt concentrations the comparison becomes qualitative. One could argue that this model is unrealistic, in placing the binding site as close as 2 A from the surface of the chelator. Already an increase of this distance to 3 A would improve the comparison with experiment significantly. However, the position of the binding site is a parameter in the model and as such it will have an influence on the final results. Table I1 also shows that it is important to treat the reference system as realistic as possible. Neglecting the 2 mM salt in the reference system would shift all the simulated numbers by 0.2 pK units. Similarly, the four K+ ions acting as counterions to the chelator are responsible for a shift of 0.2 pK units in a salt-free system, relative to an infinitely dilute chelator (see the first row in Table 11). This contribution is automatically incorporated in the M C simulations, although it is less straightforward to include in the PB calculations, particularly in a nonspherical model. The excellent agreement between experiment and theory depends of course on the choice of the different model parameters as discussed above in the comparison between models I and 11. However, the results are surprisingly insensitive to the choice of radii for the small ions, the calcium ion, and the chelator. For example, ApK between 2 and 150 mM KCl changes from 1.29 (21) Pethig, R.; Kuhn, M.; Payne, R.; Adler, E.; Chen, T.-H.; Jaffe, L. F. Cell Calcium 1989, 10, 491.
Svensson et al.
3138 The Journal of Physical Chemistry, Vol. 96, No. 7, 1992
to 1.14 if the chelator diameter is increased from 14 to 20 A. Decreasing the diameter to 10 A increases the shift to 1.48 pK units. Changing the chelator diameter may also to some degree be compensated by a change in the size of the calcium ion. Keeping the chelator diameter at 14 A and increasing the diameter of CaZ+to an unrealistic 8 A gives a shift of 1.15 pK units. Similarly, a change of the salt ion radii would hardly be noticeable, u n ! they ~ were chosen to be very small or very large. One reason for the insensitivity is that the results are dominated by the long-range electrostatic interactions, and the hard-core or excluded-vo!ume effects play only minor roles. This is also likely to be the reason for the nearly identical salt dependence of 5,5'-Br,BAPTA and quin-2 seen in the experiment. It should be pointed out that the chemical potentials and pK shifts become more model dependent at increased salt concentration. As an illustration to this, consider the excess chemical potential of CaZ+ in a 1 M KCl solution, where K+ and C1- have a diameter of 4 A. With a the diameter of Ca2+equal to 2,4, and 8 A, we obtain P.u(F) 2s -2.61, -1.94, and -0.55, respectively. In the above calculations we have assumed a uniform dielectric permittivity equal to that of pure water. One may argue that the chelator sphere should be represented by a somewhat lower permittivity. We have investigated what effect a low dielectric permittivity, E, = 2 inside the chelator, would have on the shift. In this case the electrostatic interactions become more complicated than eq 5 due to the reaction field contribution from the low dielectric sphere. Details about simulations of electrolytes in such systems have been given e l s e ~ h e r e . ~Although * ~ ~ ~ a rather low value of the interior dielectric constant was chosen, the effect on the shift was found to be very small. For a chelator with a diameter of 14 A in 0.15 M KCl, we find a shift of 1.27 relative to the 2. mM reference solution, which should be compared to 1.29 for a uniform dielectric permittivity. So far we have mainly discussed the comparison of simulated pK shifts with the experimental values. Turning to the shifts obtained with the Poisson-Boltzmann equation we find an equally good agreement. The agreement does not originate from fortuitous cancellation in eq 4, since the chemical potentials of free and bound calcium as noted in Table I are each in excellent agreement with simulation data. Moreover, the linearized PB equation gives surprisingly enough equally good results. In the linearized theory the chemical potentials can be obtained analytically from the simple formula
+
= - ~ q ~ e ~ / 8 r c ~ t , (K lC )
/.l
(15)
where u is the average ionic diameter. The inclusion of (1 + K U ) in the denominator of eq 15 leads to a substantial improvement over the DebyeHiickel limiting law, valid when KU 0. Figure 2a,b shows a comparison of the excess chemical potential obtained from MC simulations and from eq 15. It is possible to distinguish the hard-core and electrostatic contributions in a simulation, and these two terms are plotted separately in Figure 2. The extended DH equation, despite the correction term KU in the denominator of eq 15, still lacks a significant part of the hard-core contribution, which can be seen from Figure 2. This term is substantial for a monovalent ion in a 1:l salt solution, while it plays a minor role in the excess chemical potential of a divalent ion in the same salt solution. This hard-core contribution is the same for a monovalent and a divalent ion, but in the latter case it is partially canceled by the too large estimate of the electrostatic contribution in the PB or DH theory. This cancellation of hard-core and electrostatic correlation terms is probably responsible for the good agreement seen between MC and PB results.
-
(22) Friedman, H. L. Mol. Phys. 1975, 29, 1533 (23) Linse, P.J . Phys. Chem. 1986, 90,6821.
I\
, total
'1
J
...............................hard
i...................................
core
20 30 40 (mM)]'I2 Figure 2. Excess chemical potential for a monovalent ion (a, top) and a divalent ion (b, bottom) in a 1: 1 salt solution. The solid lines are Monte Carlo results, which are decomposed into hard-core (dotted lines) and electrostatic (dash-dottedlines) contributions. The dashed lines are from 0
10
[c
eq 15.
The electrostatic contribution to the excess chemical potential will become more important with higher valency. Thus, we expect the approximate theory to be relatively more accurate for multivalent ions. The neglect of ion-ion correlations in the PB theory would always lead to a too high electrostatic contribution to the chemical potentials, thereby partially compensating for the neglect of the hard-core term. Within which limits this is true is of course difficult to define, but both the PB and D H theories work very well for a protein with a charge of -8e in 100 mM 1:l or a +1:-2 salt.zo
Conclusion We have shown that two calcium binding chelators, 53'Br2BAPTA and quin-2, have an identical salt dependence over a KCl concentration range from 2 mM to 1.0 M. A dielectric continuum model with the simplest possible model of the chelators is able to give a quantitative agreement with experimentally observed shifts in the binding constant. Furthermore, simulated shifts are rather insensitive to parameters of the model, which is an effect of the long-ranged nature of the Coulomb interaction. We have also shown that the Poisson-Boltzmann equation and even the linearized Debye-Huckel theory are capable of reproducing the experimental shifts with a high accuracy. The two approximate theories seem to work well due to a fortuitous cancellation of hard-core and correlation effects. It seems as if the dielectric interior of the chelator plays a negligible role, and it is justified to assume a uniformly high permittivity throughout the aqueous solution. We expect the type of calculations presented here to be applicable to a wide range of systems, where the shifts are dominated by electrostatic interactions.
