Environ. Sci. Technol. 1992,26,763-771
Chiou, C. T.; Lee, J.-F., Boyd, S. A. Environ. Sci. Technol.
Valsaraj, K. T.; Thibodeaux, L. J. J. Hazard. Mater. 1988,
1990, 24, 1196.
19, 79.
Jurinak, J. J.; Volman, D. H. J. Phys. Chem. 1961,65,1853. Johnston, C. T.; Tipton, T.; Trabue, S. L.; Erickson, C.; Stone, D. A. Environ. Sci. Technol. 1992, 26, 382. Kiselev, A. V. Q.Rev. Chem. SOC.1961, 15, 116. Gammage, R. B.; Gregg, S. J. J. Colloid Interface Sci. 1972,
Blank, M.; Ottewill, R. H. J . Phys. Chem. 1964,68, 2206. Cutting, C. L.; Jones, D. C. J. Chem. SOC.1955, 4067. Hauxwell, F.; Ottewill, R. H. J. Colloid Interface Sci. 1968, 28, 514.
Karger, B. L.; Castells, R. C.; Sewell, P. A.; Hartkopf, A. J. J . Phys. Chem. 1971, 75, 3870. Karger, B. L.; Sewell, P. A.; Castells, R. C.; Hartkopf, A. J. J . Colloid Interface Sci. 1971, 35, 328. Reference Soil Test Methods for the Southern Region of the United States; Donohoe, S.J., Ed.; Southern Coop-
38, 118.
erative Services Bulletin 289; University of Georgia: Athens, GA, 1983. Handbook of Chemistry and Physics, 67th ed.; Weast, R. E., Ed.; CRC Press: Cleveland, OH, 1987. Pinal, R.; Rao, P. S. C.; Lee, L. S.; Cline, P. V. Environ. Sci. Technol. 1990, 24, 639. Harkins, W. D.; Brown, F. E. J. Am. Chem. SOC.1919,41,
Dorris, G. M.; Gray, D. G. J . Phys. Chem. 1981,85,3628. Okamura, J. P.; Sawyer, D. T. Anal. Chem. 1973,45,80. Lyman, W. J.; Reehl, W. F.; Rosenblatt, D. H. Handbook of Chemical Property Estimation Methods. McGraw-Hilk New York, 1982; Chapter 15. Brusseau, M. L.; Jessup, R. E.; Rao, P. S. C. Environ. Sci. Technol. 1991,25, 134. Gamerdinger, A. P.; Van Rees, K. C. J.; Rao, P. S. C. Submitted for publication in Environ. Sci. Technol. Karickhoff, S. W. J . Hydraul. Eng. 1984, 110, 707. Baumer, D.; Findenegg, G. H. J. Colloid Interface Sci. 1982,
499.
85, 118.
Sing, K. S.W.; Everett, D. H.; Haul, R. A. W.; Moscou, L.; Pierotti, R. A.; Rouquerol, J.; Siemieniewska,T. Pure Appl. Chem. 1981, 57, 603. Karnaukhov, A. P. J. Colloid Interface Sci. 1985,103,311. McClellan, A. L.; Harnsbarger, H. F. J . Colloid Interface Sci. 1967, 23, 577. Carter, D. L.; Mortland, M. M.; Kemper, W. D. In Methods of Soil Analysis, Part 1. Physical and Mineralogical Methods; Agronomy Monograph No. 9; American Society of Agronomy Soil Science Society of America: Madison, WI, 1986; Chapter 16, pp 413-423.
(36) (37) (38) (39) (40)
Jho, C.; Nealon, D.; Shogbola, S.; King, A. D., Jr. J. Colloid Interface Sci. 1978, 65, 141. Vidal-Madjar, C.; Gulochon, G.; Karger, B. L. J. Phys. Chem. 1976,80, 394. Fowkes, F. M. J . Colloid Interface Sci. 1968, 28, 493. Baehr, A. L. Water Resour. Res. 1987,23, 1926. Jury, W. A.; Russo, D.; Streile, G.; El Abd, H. Water Resour. Res. 1990, 26, 13. Ong, S. K.; Lion, L. W. Water Res. 1991, 25, 29.
Received for review August 7,1991. Accepted December 10,1991.
Comparison of Semianalytical Methods To Analyze Complexation with Heterogeneous Ligands Maarten M. Nederlof,*st Willem H. Van RiemsdiJk,+and Luuk K. Koopalz Department of Soil Science and Plant Nutrition, Dreijenplein 10, and Department of Physical and Colloid Chemistry, Dreijenplein 6 , Wageningen Agricultural University, 6703 HB Wageningen, The Netherlands
rn The binding of ions to natural ligands is influenced by the chemical heterogeneity of these ligands. Two approaches to calculate the affinity distribution are compared in terms of theory and practicality. The local isotherm approximation (LIA) methods use an approximation of the local binding function in order to solve the integral adsorption equation for the distribution function. The differential equilibrium function (DEF) method is based on the law of mass action, but it is shown that the method can also be interpreted within the LIA concept. The results obtained with the LOGA method, on the basis of accurate synthetic data on binding, are fairly good, but in general the distribution is somewhat flattened. The DEF distributions are asymmetrically distorted, but the position of a small peak at the high-affinity end of a distribution is clearly discernable.
Introduction The binding of metal ions and other chemical species to naturally occurring polyfunctional ligands affects the bioavailability and transport of those chemical species present in natural systems. The ligands may be present as small dissolved molecules or as reactive sites at the surface of mobile or immobile colloids. Natural colloids 'Department of Soil Science and Plant Nutrition. t Department of Physical and Colloid Chemistry. 0013-936X/92/0926-0763$03.00/0
such as fulvic and humic acids possess different types of reactive groups; that is to say, they are chemically heterogeneous. When the binding behavior of metal ions and other chemical species on natural ligands is studied, heterogeneity analysis is a valuable tool and may be helpful when a realistic model to describe the binding is to be chosen. Early work on the heterogeneity of polyfunctional ligands was done by researchers such as Simms (11, Scatchard et al. (2), Tanford (3), and Klotz and Hunston ( 4 ) , who treated the ligand system as a discrete series of site types, and Posner (5)and Gamble (6),who considered a continuous distribution of the heterogeneity. Hunston (7) applied the work of Ninomiya and Ferry (a),who worked on the viscoelastic behavior of materials, to the binding of small molecules to macromolecules. In the last decade, interest in the topic has increased (4-20). In the field of physical chemistry, the effect of the heterogeneity of a surface on the adsorption of small molecules has also attracted much attention; see the reviews by House (21) and Jaroniec and Brauer (22). Both purely numerical and semianalytical methods of heterogeneity analysis have been published. In this paper we will discuss the semianalytical methods for heterogeneity analysis based on continuous distributions. This critical comparison of the numerous methods of this type of analysis may help future users to select a method. As we recently discussed the local isotherm approximation (LIA)
0 1992 American Chemical Society
Environ. Sci. Technol., Vol. 26, No. 4, 1992 763
methods (18) and their relation to the affinity spectrum (AS) methods (7-9), in this paper we will focus on the comparison between the LIA methods and the differential equilibrium function (DEF) method (6, 13-1 7). We begin by discussing the fundamentals of the two methods. In the second part of the paper we will compare the methods on the basis of accurate, synthetic data on binding for systems with known distribution functions. Before we turn to the heterogeneity analysis as such, some attention will be paid to the equations describing the binding of a species to a polyfunctional ligand system. Binding Equations for Heterogeneous Systems The binding of a metal species or protons to a heterogeneous ligand system may be considered as the sum of the contributions of the different site types present. For one such site type i the binding of species M can be written as Si + M s SiM (1) where Siis the free site and SiM the complexed site. By application of the law of mass action to this equilibrium under the assumption of ideality (both in the solution and at the ligand system), the affinity constant Kimay be defined as
where [MI is the concentration of the species M in solution and (S,M)and (Si) are the site concentrations of complexed and free sites, respectively. When M is the only species reacting with (Si),the fraction of sites of type i occupied by M, ei, equals
ei =
{SiMJ (Si)+ (SiW
tesimal, eq 5 can be written as an integral equation: 8, = s eA( K , [ M I ) f(log K ) d log K
(7)
where 0(K,[M]) is the local binding function, f(1og K ) is the distribution function, and A indicates the range of log K values present. The distribution f(log K ) can be interpreted as a probability density function; that is, f(1og K) indicates the probability of finding sites with an affinity in the range log K + d log K. The product f(1og K ) d log K is equivalent to the fraction fi in the discrete case. Note that in eq 7 the subscript i is left out because individual site types can no longer be considered separately. Equations 5 and 7 are equivalent expressions for the generally used ideal overall binding function of a heterogeneous ligand system and provide a basis for obtaining the distribution function. In principle, 8, can be determined experimentally (see eq 6), and for 8(K,[M]),a theoretical relation can be used, such as eq 4. Inversion of the integral equation will provide f(1og K ) . Above it has been assumed that the total concentration of sites of the system (S,) = C(SjM)+ C(Si)is known. But in some cases it is difficult to determine the total site concentration experimentally. It follows from eq 6 that (St) is essentially a normalization constant. If (St)is unknown, a nonnormalized distribution f(1og K)(St)can still be obtained. An alternative approach to the heterogeneity analysis which has received much attention in the literature (6, 13-17) starts with the application of the mass action law to the overall ligand system. In this case the metal binding to the ligand system is represented as L+M*LM (8) and for each value of [MI the law of mass action is applied:
(3)
It follows from eqs 2 and 3 that
K is called a weighted average equilibrium function (6,13).
accessible when the total number of sites is known and can be expressed as {LMI &MI (6) ” = (LM) + (L) (s,)
is not a constant but a function of [MI or the metal loading of the ligand system. A plot of K versus, for instance, (LM)provides the course of this weighted average affinity as a function of the metal complexation. Gamble et al. (6,13,14) and later Bqffle et al. (15-17) elaborated this concept and analyzed K or quantities derived from it in great detail. Also, in this approach, the main goal is to quantify the binding heterogeneity. In order to satisfy the assumption of ideality in eq 5 and eq 7, the concentration (or activity) of species M is that near the binding sites. When the ligand system is charged, as many natural adsorbents are, the concentration near the binding sites differ from that in the bulk by a Boltzmann factor incorporating the electric potential near the binding sites. Neglecting this effect by simply using the bulk concentration is most common and leads to an “apparent” affinity distribution in which the electrostatic effects are incorporated (19). Only in favorable cases (e.g., with proton adsorption) can the electrostatic interactions be accounted for by explicitly assuming that the electrostatic potential is the same for all binding sites (23). In this case the affinity distribution reflects the chemical or “intrinsic” heterogeneity.
where (LM)= C{SiM)is the total concentration of metal bound to the ligand system, (L) = C(Si)is the total concentration of free sites of the ligand system, and (S,) is the total concentration of sites. Equation 5 applies to a heterogeneous system with a discrete number of site types. When the differences between the K ior rather the log Kivalues become infini-
Heterogeneity Analysis In general, the methods for heterogeneity analysis can be divided into (a) a group of methods that starts with the assumption of a discrete distribution of heterogeneity and (b) a group that assumes the presence of a continuous distribution. In environmental science, the Scatchard
(4)
Equation 4 is called the local isotherm or local binding function and is equivalent to the well-known Langmuir equation. It applies both to a solution with a large number of equal single-site ligands and to one with homogeneous polyfunctional ligands each containing a number of equal sites. This is a consequence of the assumed ideality. In the ideal case the overall binding function is simply obtained by a weighted summation of the local contributions, i.e. et = Cfiei (5) 1
where 0, is the total proportion of sites occupied by M and f i is the proportion of site type i with respect to the total concentration of sites in the system. 0, is experimentally
764
Envlron. Sci. Technol., Vol. 26, No. 4, 1992
graphical analysis method (2) is a classic example of the first group. In this method, a limited number of affinity constants and their fractions can be determined when the binding function is known (4). Recently, Brassard et al. (20) developed a numerical method that is able to determine a large set of discrete affinity constants together with their proportions. In the case of a continuous distribution, a series of methods is proposed in which, for known values of et and a given local binding function 8,f(1og K ) is obtained by solving eq 7 without making a priori assumptions about the type of distribution function. Equation 7 is a Fredholm integral equation of the first kind, which is difficult to solve numerically for f(1og K ) . Good results are obtained with numerical methods such as regularization (24) and singular value decomposition (25) for very accurate data that cover the whole range of concentrations. So far, none of these methods has been applied to ion binding data. By making some approximations it is also possible to solve eq 7 analytically for f(1og K ) . The affinity spectrum method (7-9) is a well-known example of this group of methods. Similar types of methods from the literature on adsorption are the condensation approximation (CA) (26), the asymptotically correct approximation (ACA) (27), the Rudzinsky-Jagiello (RJ)method (28,29), and the LOGA method (18). The CA, ACA, and LOGA methods are based on various approximations of the local binding function, that is the binding function for a homogeneous subset of sites. Nederlof et al. (18) called these the local isotherm approximation methods. They also showed that the firstand second-order affinity spectra are closely related to the CA and LOGA distributions, respectively. Gamble (6) and later Buffle (15) developed the semianalytical differential equilibrium function method for the heterogeneity analysis. They assumed a continuous distribution function. The DEF approach is based on applying the law of mass action to a heterogeneous system, an ideal originally proposed by Simms (1). In this paper we concentrate on a comparison of the LIA methods (18, 26-29) with the differential equilibrium function method (13-17). Local Isotherm Approximation (LIA) Methods. In the LIA methods, the kernel 8 (the equation for the local binding function) of the integral equation (eq 7) is replaced by another function which approximates the local isotherm in such a way that the integral equation can be solved analytically for the distribution function f(1og K ) . Deviations of the “LIA function” from the true local binding function will, in principle, result in deviations of the calculated distribution function from the true distribution function. The better the LIA function approximates the local binding function, the more the calculated distribution function will resemble the true distribution function. For a review of the older LIA functions and some results obtained for a new LIA function, see ref 18. Here we will discuss the condensation approximation, the asymptotically correct approximation, and the new LIA function named LOGA. (a) Condensation Approximation (CA). The condensation approximation (26) is a classic and very illustrative method to solve eq 7 for f(1og K ) . In this method the local isotherm function is replaced by a step function. This is equivalent to assuming that the different site types are occupied sequentially, i.e., the sites with the highest affinity first, then those with the second highest affinity, and so on. The position of the step depends on the local binding function and is determined by a best fit criterion. In the case of eq 4 as local isotherm, the best fit is obtained
wwl Flgure 1. CA, LOGA-1 (p = 0.7),and LOGA-AS (p = 1.0) iocai isotherm approximations in comparison with the Langmuir local binding function.
when the step function intersects the Langmuir isotherm at 8 = 0.5; see Figure 1. According to eq 4, at 8 = 0.5, K[M] = 1, so that in the CA the affinity KCAcan be replaced by 1/[M] and the approximation of the local binding function becomes 8CA = 0 for [MI < 1/&A (104 8CA = 1 for [MI 2 1 / & A (lob) where O CA is used to indicate that eq 10 is the CA approximation of the true local binding function (eq 4). The CA distribution function is obtained by substituting eq 10 into the integral equation and then inverting the integral equation (18, 26):
log KCA = -log [MI
(Ilb)
(b) Asymptotically Correct Approximation (ACA). The ACA method, also called the asymptotically correct condensation approximation (ACCA),was developed by Cerofolini (27) as a first improvement of the CA method. We mention the method here because of the similarity with the local isotherm derived for the DEF method (see DEF section). In the ACA method the local isotherm is now replaced by e A C A = K[M] for [MI < ~ / K A C A(1%) 8ACA
=1
for
[MI 1~ / K A C A (12b)
The ACA distribution function is
log KACA = -log [MI (13b) The coefficient 0.43 occurs because log K is used instead of In K. The ACA distribution results in an asymmetrical deformation of the true distribution function (18),which is a serious disadvantage of the method. (c) The Logarithmic Symmetrical Approximation (LOGA). The CA and ACA local isotherms are rather poor approximations of the Langmuir function. A much closer approximation is obtained with the LOGA function (18) which is given by 8LOGA
= LU(K[MI)~ for
= 1 - a(K[M])-fl
[MI
< ~ / K L O G A (14a)
[MI 1 ~ / K L O G A (14b) where (11and P have to be optimized to fit the Langmuir @LOGA
for
Environ. Sci. Technol., Vol. 26, No. 4, 1992 765
isotherm. Like the CA, the LOGA function intersects the Langmuir isotherm at 0 L O G A = 0.5, which leads to the relation log[M] = -log KLOFA. The approximated local binding function (eq 14) is only continuous at ~ L O G A= 0.5 for CY = 0.5, and we will restrict ourselves to this situation. Insertion of eq 14 with CY = 0.5 in the integral equation, followed by inversion of the integral, leads to the LOGA distribution function (18):
The coefficient 0.189 (=0.432)results from log K being used instead of In K. Fitting eq 14 with CY = 0.5 to the Langmuir function using a least sum of squares method results in a value of 0 = 0.7. In Figure 1 this approximation is indicated by LOGA-1. A reasonable good fit is obtained for 0 = 1 too; the approximate isotherm now intersects the Langmuir function in 0 = 0.5 only. This property of the approximation has the advantage that the resulting distribution never becomes negative. In Figure 1 this approximation is indicated by LOGA-AS, because it is essentially equivalent to Hunston’s second-order affinity spectrum (7-9,181. For /3 = 0.79, eq 15 corresponds with the result found by Hsu et al. (28)and Rudzinski et al. (29). Differential Equilibrium Function (DEF) Method. At first sight the DEF method seems completely different from the LIA methods even though the main goal is the same, namely, to quantify the binding heterogeneity of a ligand system. The DEF method is based on the approach using the law of mass action (6,13-17) and starts with eq 9 instead of the general binding equation (eq 7). However, in the derivation of the DEF functions, the basic relations eqs 2-7 are also applied. After normalization of the concentration of bound and free sites by the total site number (S,),eq 9 can also be written as
al. (15-17) extended the DEF method with a distribution function by stating that 0Jog KDW) can be seen as a “good estimate” or “analogue” of the true cumulative distribution function. The distribution density becomes
In the last paragraph of the DEF section we will interpret the DEF method within the LIA concept. For ease of understanding, the derivation of Gamble et al. is discussed first. (a) Gamble’s Derivation of K D E p For the derivation of &EF, Gamble et al. (6,13) started with the definition of K (eq 16) and introduced the relation between the overall binding and the contributions of the individual sites (eq 5), so that eq 16 can be written as
Cfiei i K(1- 0,) = -
[MI Then ei in the right-hand side of eq 20 is replaced by 0i = K;[M](l - 0;) (21) which follows directly from the Langmuir equation (eq 4). The result is
If(1 - 0,) = CKifi(1 - 0;) i
(22)
Next Gamble et al. (6, 13) replaced eq 22 by
which means that the summation over all the site types i is replaced by a summation over changes in coverage A(1 - Ogp Because of the assumption involved, Ki is replaced by its DEF approximation KDEFj. The assumption is a simplification of reality and not in accordance with eq 21, which essentially states that at any change in concentration [MI the coverages of all site types change. Without the assumption it follows from eq 22 that A(R(1- 0,)l = CK;fiA(l- 0;) (24) i
For a homogeneous ideal ligand system, eq 16 is equivalent to eq 2 and the correct affinity constant is obtained. For a heterogeneous system, K is a function of the concentrations of species M, or the extent of binding e,, and represents a special, weighted, average affinity. AccoLding to Gamble et al. (6,13,14), two disadvantages of log K are that for its calculation the total number of sites _of the ligand system (S,) must be known and that log K as a function of 0, gives a poor representation of the underlying heterogeneity. In order to overcome these problems, Gamble introduced the “differential equilibrium function”, KDEF, which is intended to be more closely related to Ki and is defined as (6, 13) W(1- et)) (17) KDEF = d ( l - 0,) Substitution of eq 16 in eq 17 gives d(et/[Ml) = - d(&MJ/[MI) KDEF= det d(LM1
(18)
from which it follows that (S,] is not required for the calculation of KDEF. The reasoning that led Gamble and co-workers to the definition of KDm will be analyzed below. Gamble et al. (6, 13,14) accepted a plot of log KDEF versus the concentration of complexed sites {LM)(or e,) as a sufficient characterization of the heterogeneity. Buffle et 766
Environ. Sci. Technol., Vol. 26, No. 4, 1992
whereas from eq 23 it follows that A(R(1- 0,))= KDEFJA(~ - et),
(25)
When eq 24 is compared with eq 25 it is apparent that both equations are equivalent only when the site type i changes from completely occupied to completely empty at a decrease in concentration corresponding to (1- A0,) .. Thus A(1 - ei) = 1 and A(0J = -1; this means that the sites empty sequentially. In a later paper, Gamble and Langford (14) derived KDEF using eq 25. In the original derivation of Gamble et al. it was assumed that very many site types are present; then the summation in eq 23 can be replaced by an integral (6, 13):
R(l - 0,) = JYKDEF d(1 - 0,)
(26)
where y = 1- 0e is the coverage at a certain concentration [MI = [MI.*. Finally, the definition of KDEF (eq 17) is obtained by differentiation of eq 26 with respect to (18,). (b) Interpretation of the DEF Method in Terms of the LIA Concept. KDEF can also be interpreted in terms of the LIA concept. When it is assumed that the DEF distribution fDEF(1og KDEF) presented by eq 19 is an approximation of the true distribution, a DEF local isotherm, 0DEF, that is an approximation of the true local isotherm, can be sought under the condition that
0, =
pDEFfI)EF(log
KDEF)d 1% KDEF
(27)
In order to derive an expression for 0DEF we return to the analysis of Gamble and co-workers. The assumption that Oi goes from zero to 1 instantaneously at an infinitesimal increase in coverage means that sites fill sequentially. Due to this simplification an approximation of K iis found, defined as KDEF. One would expect that, due to the sequential filling, the DEF distribution would resemble the CA distribution and that the DEF local isotherm would resemble the CA isotherm. However, this is not the case, because the sequential filling concept used in the derivation of Gamble is inconsistent with the use of eq 21. Note that in eq 21 appears at both the left- and right-hand sides. Gamble et al. (6,13)replaced Bi only at the righthand side by a step function and obtained a new expression for eiat the left-hand side. So two types of local isotherms which mutually conflict are used in the same equation. The resulting Eli at the left-hand side, which we will denote as 0DEF, now becomes linearly dependent on [MI. For each site with affinity KDEF one can write ~DEF = KDEF[M] for [MI 5 [MI# (284 0DEF
=0
for
[MI
0
> [MI#
where [MI#is the concentration corresponding to KDEF as obtained from eq 17. It follows from eq 28 that not only is (KDEF[M])M# the point where the isotherm e D E F is split into two parts (the breakpoint), but it is also the maximum value that 0 D E F can have. Normally, 0 D E F should have a maximum value of 1,that is, at KDEF[M]= 1. However, this only happens if KDEF = 1/ [MI, which is generally not the case (in fact it occurs only in one point of the isotherm). This means that, for KDEF < 1/[M], e D E F does not reach the value 1 (the site is not completely filled) and that, for KDEF> l/[M], 0 D E F becomes larger than 1 (the site becomes oversaturated). The breakpoint value (KD~[M])M# follows from eq 18 by first working out the differentiation and then multiplying the result by [MI:
Equation 29 shows that the “breakpoint” value of QDEF is not a fixed value but depends on [MI and 0,. In the previously discussed LIA approximations, the breakpoint = 1, value at any concentration [MI# occurs at KLIAIM]# independent of the overall isotherm. This means that the LIA local isotherm for each site type is the same on a K[M] scale. In the DEF case, (KDEF[M])M# is not constant and the maximum of 0DEF varies with the affinity KDEF. Another artifact is that eDm drops to zero after its maximum value has been reached: the site empties when the concentration increases; this is physically unrealistic. A graphical representation of the true and approximated local isotherms is shown in Figure 2. In Figure 2a the local isotherm is plotted as 0,(K[M]),which clearly shows the linear relationship for the initial part of the isotherm. Figure 2b shows the Bi(log K[M]) plot, which can be compared directly with the plots shown in Figure 1. At low [MI the DEF isotherm is equivalent to the ACA isotherm; however, at the breakpoint 0 A C A reaches 0 = 1, whereas the DEF function reaches a maximum not equal to 1 and it drops to zero after the breakpoint. As e D E F is not a logical approximation of eq 4, the distribution function obtained is hard to interpret in terms of a weighted true distribution, as can be done with the CA and LOGA methods (28). The equation for the DEF distribution function was already suggested by Buffle, but it can
Flgure 2. L I A interpretation of the DEF method, &, with the Langmuir local blnding function.
in comparison
now be derived by introducing eq 28 in the overall adsorption equation (eq 27) for any fixed value [MI#:
where y = log a E F is the value of log KDEF at concentration [MI#as obtained by eq 17. Equation 30 is essentially the sum of all the contributions 0DEF(M#) for each site type with affinity KDEF weighted by the relative occurrence of the sites. As eq 30 applies for each value of [MI#, the distribution of log KDEF can be obtained by taking the first derivative of €),/[MI with respect to log
KDEF.
Substituting eq 18 for KDEF in eq 31 results finally in eq 19. The fact that this result is obtained shows that KDEF and fDEF(1og KDEF) can indeed be interpreted within the LIA concept. Similarly, as with fCA(1Og KCA)and fLOGA(l0g KLOGA), fDEF(1og KDEF) is an approximation of the true distribution function f(1og K ) . Dzombak et al. (30)derived a different expression for the distribution of KDEF. They started by rewriting eq 20 for a continuous distribution: K ‘(l et) = f ( K )dK (32) 1 + K[M]
-
The distribution is derived by equating the right-hand side of eq 26 with the right-hand side of eq 32; the result is
At first sight this may seem correct. However, by equating the right-hand side of eq 26 to that of eq 32, it has to be assumed that K in eq 32 has the same meaning as KDEF in eq 26. As we have shown in the analysis (compare, for Environ. Sci. Technol., Vol. 26, No. 4, 1992
767
12
I a
1 CA
:
2
0
-
'
'
1
'
1
0
2
logK
4
6
7
-
1
2
! 18
logK
logK
I
I
1
I
TRUE
i\ 0
2
4
6
I
8
1 0
2
logK
4
6
8
logK
IogK
Figure 3. Cumulative and differential distributions obtained with the DEF, CA, and the LOGA methods for the first example. For details of the true distribution, see Table I . In a-c, the cumulative distributions are presented in combination with the true cumulative distribution. I n d-f, the differential distributionsobtained with the DEF, CA, and LOGA methods, respectively, are presented in combination with the true differential distribution.
example, eqs 24 and 25), these K quantities have a different meaning: KDEF is a weighted function of the true K values. Therefore we reject the distribution equation developed by Dzombak et al.
Practical Comparison On the basis of the fundamentals of the methods it is clear that in the absence of experimental error the LOGA method will, in general, result in a better approximation of the true distribution than the CA method. From the derivation of the DEF method it follows that this method results in a perfect approximation for a homogeneous system, but gives an asymmetrically deformed distribution function for heterogeneous systems. But we also need to understand in more detail the quality of the methods when applied to highly accurate data. To test this quality we will use overall binding functions that are calculated using known affinity distributions. Only in this way can the outcome of a certain method be compared with the true distribution function. As the selection of examples for this comparison is arbitrary we will include two examples that were previously selected by Gamble and Langford (14) and Buffle et al. (17) to illustrate the DEF method. The first example, taken from Gamble and Langford (14), describes a system consisting of a discrete mixture of four monofunctional ligands; see Table I. The Alog K between two successive types of ligand ranges from 0.6 to 0.9 log K unit, which is a rather narrow interval. To compare the results obtained with the true distribution function, f(1og K ) should be plotted. However, both the distribution function obtained with the DEF method and the CA (or LOGA) method are continuous distribution functions, whereas the chosen example is based on a discrete distribution. Since the ligand proportions, fi, and the affinity density, f(1og K ) , are different quantities, it is impossible to compare the resulting distribution function with the true distribution in an unarbitrary way in one figure: there will always be an arbitrary scaling factor involved. This problem can be overcome by calculating the cumulative distribution function F(1og K ) for the true (discrete) distribution as well as for the distribution functions resulting from a LIA or DEF method. The cu768
Environ. Sci. Technol., Vol. 26, No. 4, 1992
Table I. Discrete Ligand Mixture" i
compound
proportion
log K
1
malonate carbohydrazide glycine ethyl ester tartrate
0.32
5.55
2 3 4
0.22
4.92
0.16 0.30
4.14
3.20
Example taken from ref 26.
mulative LIA and DEF distributions can be obtained from their distribution function by integration. The value of F(1og K ) for a certain value of log K corresponds with the area under the distribution function up to this log K value. In the case of a normalized distribution function, the maximum value of F(1og K ) is the surface area under the entire distribution function, which equals 1. For CA and DEF the cumulative distributions can be obtained directly by plotting 8,versus log KCAor log KDEF. Note that this plot for log KDEFis essentially equivalent to the affinity plot drawn by Gamble et al. (6, 13). In Figure 3a-c, the true cumulative distribution is compared with the results obtained using the DEF, CA, and LOGA methods, respectively. All these methods result in smooth curves, which is in contrast with the stepped true cumulative distribution. The DEF method (Figure 3a) only approximates the true distribution closely for the highest log K value. Both the CA (Figure 3b) and the LOGA (Figure 3c) follow the true distribution pattern over the whole range of the distribution, although the steps are smoothed to a continuous function. The LOGA method (Figure 3c) follows the true distribution most closely, as is to be expected. Although the cumulative distribution allows the best comparison of results for this example, we also prefer to show the differential distributions. The results are presented in Figure 3d-f together with the true (discrete) distribution. As stated before, the vertical scales belonging to the discrete and the continuous representation cannot be matched. They axis on the left-hand side corresponds to f(1og K ) , whereas the proportions f, can be read from the y axis on the right-hand side. The true log K values
I
L W '
Flgure 4. Cumulatlve and differential distributlons obtained with the DEF, CA, and the LOGA methods for the second example. The true distribution is a bl-Gaussian distributlon with a dlfference of 1.5 log K units in peak positions and a width of 1. The results are presented in the same order as in Figure 3.
of the ligand mixture and their relative occurrence are directly evident from, respectively, the position and the length of the spikes. Figure 3d gives the resulting DEF distribution. The distribution shows only two large peaks at the extremes of the log KDEF interval. The high-affinity peak provides a good estimate of the log K value of the highest affinity sites. However, an extremely high value (10.7) is obtained for the density at log KDEF = 5.5. In this case the DEF distribution gives a poor indication of the true distribution. The CA distribution, given in Figure 3e, is a rather broad featureless distribution and is also not capable of indicating the presence of the four ligands. The LOGA method (Figure 3f) results in two clearly separated peaks instead of four individual peaks. The high-affinity peak corresponds with the two high-affinity sites, the low-affinity peak with the lowest affinity site. In general it may be concluded that all three methods are incapable of resolving the affinity distribution if the peak positions of the individual ligands making up the system are closer than 1log K unit. Nevertheless, the best image of the distribution is obtained with the LOGA method. The second example, taken from Nederlof et al. ( l a ) ,is a combination of two Gaussian distributions with a difference in peak position of 1.5 log K units. The log K values are scaled by an arbitrary scaling parameter K (see the caption of Figure 4 for details). In Figure 4a-c is given the true cumulative distribution, together with the results obtained with the DEF, CA, and LOGA methods, respectively. For this case, the results obtained with the DEF method deviates strongly from the true cumulative distribution over almost the entire log K interval. The CA cumulative distribution deviates strongly at the extremes of the distribution. The LOGA distribution follows the true distribution closely, but it is not as pronounced as the true distribution. In Figure 4d-f, the results are replotted as differential distributions. It follows that the DEF method (Figure 4d) does not retain the symmetry of the distribution: the high-affinity peak is much larger than the low-affinity
-
peak. Moreover, the position of the low-affinity peak is shifted to higher affinities, which makes the entire distribution much too narrow. The shift in the low-affinity peak results from the fact that each log KDEF value is weighted by the distribution itself. The first peak area contributes substantially in the weighting of log KDEF to obtain the second part of the distribution. The distribution obtained with the CA method (Figure 4e) is not very informative; it results in one flat broad peak. The LOGA method (Figure 4f) is capable of clearly distinguishing the two separate peaks, although they are smoother and broader than the true peaks. The positions of the two maximums and the minimum in the LOGA distribution correspond exactly with those of the true distribution, and the symmetry of the distribution is retained. The last example is comparable to one of the examples used by Buffle et al. (17). The example is a continuous Sips distribution (semi-Gaussian) (31)with three discrete site types superposed on the high-affinity tail of the continuous distribution (see the caption of Figure 5 for details). According to Buffle et al. (17),this type of highaffinity site is important for the binding of heavy metal ions. Buffle called these sites the dominant minor sites; the Sips distribution represents the major sites. In Figure 5a-c are given the resulting cumulative distributions of log together with the true disKDEF, log KCA,and log KLoCA tribution. In the high-affinity range the F(1og KDEF) and the true distribution are fairly similar. Around log K = 3 (the position of the largest discrete peak), the true and the DEF distribution begin to deviate markedly. For the lower affinities, the DEF result is systematically shifted toward too high affinities. The CA Cumulative distribution (Figure 5b) is much closer to the true distribution than the DEF cumulative distribution (Figure 5a). The LOGA method (Figure 5c) follows the true cumulative distribution fairly close over the whole affinity range, with a smooth step for the largest minor site. The two small minor sites cannot be discerned in any of the cumulative distributions. In Figure 5d-f, the calculated differential distributions are compared with the true distribution. Note again that Environ. Sci. Technol., Vol. 26, No. 4, 1992 769
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LOGA )
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the height of the discrete spikes cannot be compared directly with the height of a continuous peak at the same position of the affinity axis. The proportions can be read on the y axis on the right-hand side of Figure 5f; the densities can be read on the left-hand side. The DEF approximation (Figure 5d) shifts the broad peak toward higher affinities compared to the true distribution. The first discrete ligand (with the highest affinity) shows up very pronounced at about the correct position, whereas the other two discrete peaks appear as weak shoulders in the distribution. As expected, the CA method (Figure 5e) results in broadened peaks. Nevertheless, a clear indication of the largest minor site peak and its position is obtained. The small discrete peaks are not seen. The LOGA (Figure 5f) method follows the main broad peak almost perfectly, and the three discrete peaks are also discernible. In order to show the small peaks in the distribution clearly, Buffle usually plotted the distribution on a log frequency scale. In addition, he was mainly interested in the high-affinity part (16). In that case, the small peaks do indeed become more prominent for both the LOGA and DEF methods. However, the question is whether the derivatives can be determined accurately enough to allow this enlargement in practice. At low values for metal binding, small errors in the data can easily be enlarged in the distribution to unrealistic peaks.
Discussion and Conclusions So far the comparison of the different methods has been based on exact data. In each method, at least the first derivative of a binding function is needed. Experimental errors may easily give rise to spurious peaks in the calculated distribution functions (32-34), because of the difficulties inherent in the determination of derivatives of experimental data. The higher the derivatives needed, the more difficulties are expected. The CA method needs only a first derivative and is thus the least prone to errors. Although the LOGA method leads in principle to better results than the CA method, in practice this may not be the case because a reliable third derivative of an experimental function is hard to obtain. 770
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According to eq 17 log KDEF is based on a first derivative, and for the distribution function a second differentiation is needed (eq 19). Since the CA needs only one differentiation, it follows that the DEF distribution is more prone to errors than the CA distribution. The DEF method is probably somewhat less sensitive than the LOGA method. In general, the error problem can be tackled by applying sophisticated smoothing spline techniques to the binding data (35,36). In these methods the extent of smoothing can be related to the magnitude of the experimental error in the data. A smoothing spline technique adapted to the present problem will be presented in a forthcoming paper. Finally, it has been stated (14) that the “Gamble plot”, log K D E F ( e t ) , is sufficient to describe the heterogeneity of a ligand system and that therefore only such a Gamble plot need be stored after an experiment. As the LOGA method gives at least as much information as the DEF method does, we think that the original et values as a function of log [MI should be stored so that different methods can be applied. In addition, it is advisable to store the original data for the application of a proper smoothing technique. In general, it should be realized that in order to obtain analytical expressions for the distribution function, approximations have to be made. Therefore, the results obtained will deviate from the true distribution. Of the methods considered, the CA method provides a simple but rather approximate solution. A considerable improvement of the CA result can be achieved with the LOGA method. In a previous paper we showed that the affinity spectrum methods can be interpreted within the LIA framework. The AS1 and AS2 methods correspond closely to the CA and the LOGA methods, respectively. Here we have shown that the DEF distribution can also be analyzed within this concept. The DEF method is shown to be the least satisfactory: the local isotherm is approximated by a physically impossible “saw-tooth”type isotherm. Tests of the methods in practice show that the resolving power of the CA method is weak; the resulting distribution is always rather smooth. For fairly smooth continuous distributions with several well-separated peaks, the LOGA method gives a very good result. With the DEF method, peak positions
will be shifted to higher affinities and peak areas will not result in a correct representation of the site fractions. In the case of discrete site fractions superposed on a smooth continuous distribution, the LOGA method corresponds most closely with the true function. The DEF method works well for the high-affinity minor sites, but results in a distorted distribution at the lower affinity end. The CA method smoothes the distribution strongly and details are lost. In general, it may be concluded that for exact data the LOGA method is the best of the methods studied, if information is wanted on the entire affinity range. To detect high-affinity minor sites, the results of the LOGA method can be complemented with DEF results. Comparison of the results of the LOGA method with DEF results also gives a clear indication of the limits of the log K range: the LOGA distribution is always somewhat too wide; the DEF distribution is always too narrow. Acknowledgments
C. H. Langford is acknowledged for the pleasant and fruitful discussions we had in Wageningen about the DEF method. We idso thank H. P. Van Leeuwen, J. Buffle, and D. S. Gamble for their critical comments on a preliminary version of this paper. Literature Cited (1) Simms, 13. S. J. Am. Chem. SOC.1926, 48, 1239-1250. (2) Scatchard, G.; Scheinberg, I. H.; Armstrong, S. H., Jr. J. Am. Chem. SOC.1950, 72, 535-540. (3) Tanford, C. Physical Chemistry of Macromolecules; Wiley Interscience: New York, 1961. (4) Klotz, I. M.; Hunston, D. L. Biochemistry 1971, 10, 3065-3069. (5) Posner, A. M. Trans. Int. Congr. Soil Sci. 1967,8,161-174. (6) Gamble, D. S. Can. J. Chem. 1970, 48, 2662-2669. (7) Hunston, D. L. Anal. Biochem. 1975, 63, 99-109. (8) Ninomiya, K.; Ferry, J. D. J. Colloid Interface Sci. 1959, 14, 36-48. (9) Thakur, A. K.; Munson, P. J.; Hunston, D. L.; Rodbard, D. Anal. Biochem. 1980, 103, 240-254. (10) Shuman, M. S.; Collins, B. J.; Fitzgerald, P. J.; Olson, D. L. In Aquatic and Terrestrial Humic Materials;Christman, R. F., Gjessing, E. T., Eds.; Ann Arbor Science: An Arbor, MI, 1983; Chapter XVII, pp 349-370. (11) Perdue, E. M.; Lytle, C. R. In Aquatic and Terrestrial Humic Materials; Christman, R. F., Gjessing, E. T., Eds.; Ann Arbor Science: Ann Arbor, MI, 1983; Chapter XIV, pp 295-313. (12) Sposito, G. CRC Crit. Rev. Environ. Control 1986, 16, 193-229.
(13) Gamble, D. S.; Underdown, A. W.; Langford, C. H. Anal. Chem. 1980,52, 1901-1908. (14) Gamble, D. S.;Langford, C. H. Environ. Sci. Technol. 1988, 22, 1325-1336. (15) Buffle, J. In Circulation of Metals in the Environment; Sigel, H., Ed.; Metal Ions in Biological Systems 18; M. Dekker: New York, 1984; pp 165-221. (16) Altmann, R. S.; Buffle, J. Geochim. Cosmochim.Acta 1988, 52, 1505-1519. (17) Buffle, J.; Altmann, R. S.;Filella, M.; Tessier, A. Geochim. Cosmochim. Acta 1990, 54, 1535-1553. (18) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. J. Colloid Interface Sci. 1990, 135, 410-426. (19) Van Riemsdijk, W. H.; Bolt, G. H.; Koopal, L. K. In Interactions at the Soil Colloid-Soil Solution Interface;Bolt, G. H., De Boodt, M. F., Hayes, M. H. B., McBride, M. B., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1991; pp 81-114. (20) Brassard, P.; Kramer, J. R.; Collins, P. M. Environ. Sci. Technol. 1990,24, 195-201. (21) House, W. A. Specialist Periodical Report; Everett, D. H., Ed. Colloid Sci. 1983, 4, 1-58. (22) Jaroniec, M.; Brauer, P. Surf. Sci. Rep. 1986, 6, 65-117. (23) De Wit, J. C. M.; Van Riemsdijk, W. H.; Nederlof, M. M.; Kinniburgh, D. G.; Koopal, L. K. Anal. Chim. Acta 1990, 232, 189-207. (24) House, W. A. J. Colloid Interface Sci. 1978, 67, 166-180. (25) Vos, C. H.; Koopal, L. K. J. Colloid Interface Sci. 1985, 105, 183-196. (26) Harris, L. B. Surf. Sci. 1968, 10, 129-145. (27) Cerofolini, G. F. Thin Solid Films 1974, 23, 129-152. (28) Hsu, C. C.; Wojciechowski, B. W.; Rudzinski, W.; Narkiewicz, J. J. Colloid Interface Sci. 1978, 67, 292-303. (29) Rudzinski, W.; Jagiello, J.; Grillet, Y. J. Colloid Interface Sci. 1982, 87, 478-491. (30) Dzombak, D. A.; Fish, W.; Morel, F. M. M. Environ. Sci. Technol. 1986,20, 669-675. (31) Sips, R. J. Chem. Phys. 1950,18, 1024-1026. (32) Turner, D. R.; Varney, M. S.; Whitfield, M.; Mantoura, R. F. C.; Riley, J. P. Geochim. Cosmochim. Acta 1986, 50, 289-297. (33) Fish, W.; Dzombak, D. A.; Morel, F. M. M. Enuiron. Sci. Technol. 1986,20, 676-683. (34) Nederlof, M. M.; Van Riemsdijk, W. H.; Koopal, L. K. In Heavy Metals in the Environment;Vernet, J. P., Ed.; CEP Consultants Ltd.: Edinburgh, 1989; Vol. 2, pp 400-403. (35) Reinsch, C. H. Numer. Math. 1967, 10, 177-183. (36) Craven, P.; Wahba, G. Numer. Math. 1979, 31, 377-403.
Received for review July 9, 1991. Revised manuscript received November 4,1991. Accepted November 11,1991. This work was partially funded by the Netherlands Integrated Soil Research Programme under Contract PCBB 8948 and partially by the EC Environmental Research Programme on Soil Quality under Contract EV4 V-0100- NL(GDF).
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