J . Phys. Chem. 1986, 90, 4104-41 10
4104
librium is maintained in the surface region throughout the desorption process. Analysis of experimental situations for which either of these assumptions is not valid could involve serious additional complexity. One may well ask about compensation in more general situations, such as an example given by GalweyZ5of a series of cracking reactions of hydrocarbons on Ni surfaces. In cases like these it is unclear what, if any, real or imagined equilibrium between the different hydrocarbons would permit us to set their chemical potentials equal, as we must if our model is to apply. Explaining the compensation that is observed in such sets of reactions remains an interesting challenge. It seems unlikely that the compensation
in sets of reactions like these can be explained by thermodynamic arguments and there may be no alternative to a microscopic treatment of the dynamics, perhaps along the lines proposed by Peacock-Lopez and S ~ h l . * ~ * ~ ’ Acknowledgment. We have benefited from stimulating discussions with M. W. Cole and S. c. Ying. We are grateful for the support of this work by the National Science Foundation through the Materials Research Laboratory of Brown University (P.J.E.) and by the Division of Chemical Sciences (Office of Basic Energy Sciences) of the U S . Department of Energy (E.F.G.). (26) Peacock-Lopex, E.; Suhl, H. Phys. Rev. B. 1982, 26, 3774. (27) Erickson, J. W.; Estrup, P. J. Surf. Sci., in press.
(25) Reference 1, Figure 2.
STATISTICAL MECHANICS AND THERMODYNAMICS Group Constant: A Measure of Submolecular Basicity BQa Noszdl Department of Inorganic and Analytical Chemistry, L. Eotvos University, Budapest H - 1443, P.O.B. 123, Hungary (Received: August 28, 1985; In Final Form: February 28, 1986)
Microscopic protonation constants belonging to the same group of multidentate ligands (in particular peptides and other biopolymers) are unifed in group constants. Chemical preconditions of this simplification are studied in detail. The mathematical relationships between the group constants and the traditional macroconstants are deduced, and the computational techniques are discussed. Group constants are useful for the characterization of individual basicity of independent groups and H-bonded group pairs in macromolecular systems. They can also be applied to estimate microspecies concentrations, including minor protonation isomers. Unexpectedly high and low group constant values indicate the existence of strong intramolecular interactions (e.g., H bond) in molecules of unknown fine structure.
Introduction Peptides, proteins, nucleic acids, and their constituents are molecules of low symmetry and of two or more (in many cases hundreds) functional groups, which are able to associate with (dissociate from) protons. Due to the two common fundamental characteristics above, these multidentate bioligands exist in a great number of microspecies and thus in a great number of fine structures. Microspecies are a distinct stage of protonation, when both the number and site(s) of bound proton(s) are definite. The physiological role (e.g., enzyme effect) of these polyfunctional biomolecules is strictly structure-dependent. The determination of the microspecies concentrations is a significant step toward a profound understanding of vitally important affector-receptor interactions. For reasons discussed in detail in the theoretical section, microspecies concentrations can never be measured directly. Previously the only way for their elucidation was the calculation in the knowledge of pH, total concentration, and microconstants. Although to date general methods for microconstant or microspecies determination have not appeared, microequilibria of certain small molecules can be nicely treated.14 If, however, protonation occurs simultaneously at more than three (1) Martin, R. B. J . Phys. Chem. 1971, 75, 2657. (2) Rabenstein, D. L.; Sayer, T.L.Anal. Chem. 1976, 48, 1141. (3) Tanokura, M. Biochim. Biophys. Acta 1983, 742, 576. (4) Kiss, T.; Tbth, B. Talanta 1982, 29, 539.
groups, the fully detailed microconstant determination is not feasible, because of the exponentially increasing number of microspecies and microconstants and the limited accuracy of the measurements. This paper presents a method that avoids these difficulties and makes possible the determination of microspecies concentrations for most biomacromolecules in cases when introduction and utilization of group constants is chemically allowed. Theory Equilibria between the various microspecies of a given molecule can be formulated in terms of a set of microscopic protonation constants.s-6 The microscopic protonation schemes belonging to molecules of two and three functional groups (denoted as A and B and A, B, and C ) are represented in Figures 1 and 2, respectively. The superscript on microconstant k denotes the functional group protonating in a given process, while the subscript(s) (if any) denotes (denote) the group(s) that is (are) attached to protons (5) Bjerrum, N. Z . Phys. Chem., Stoechiom. Venoandtschaftsl. 1923,106, 209. ( 6 ) Martin, R. B. Introduction to Biophysical Chemistry: McGraw-Hill: New York, 1964. (7) Edsall, J. T.; Wyman, J. Biophysics/ Chemistry; Academic: New York, 1958.
0022-3654/86/2090-4104$01.50/0 0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4105
Group Constant: A Measure of Submolecular Basicity
1. Due to the high proton exchange rate, their lifetime is usually less than the “observation period” of the fastest spectroscopic techniques, thus all their spectra are composite ones. 2. There is continuous interconversion between all the different microspecies; consequently, they are inseparable. 3. Protonation isomers have a constant concentration ratio, which depends on neither the pH nor the total concentration. Figure 1. Protonation scheme of a molecule containing two functional
groups.
-yrH
. 0
‘t
4. Since protonation isomers occur exclusively in the presence of the other(s), their individual intensive characteristics (e.g., molar absorptivity) cannot be measured directly. 5. As the individual intensive characteristics are unknown, the individual concentrations are also unknown, and these two types of quantities are multiplied by each other (lack of linear independence). Absorption in the simplest case becomes
K
Figure 2. Protonation scheme of a molecule containing three functional
groups. during the process. In Figures 1 and 2, for example, symbols of microspecies protonated at group B and free of proton at group(s) Y
O
L
O
tH
6. The number of different protonation isomers enormously increases with the increase of the number of functional groups in the molecule. If the number of the groups is n and that of the protons attached to the molecule is i , the number of distinct protonation isomers in the solution is
Accordingly, the number of microspecies is
A (and C) are shown. Microconstants are defined by the same example as
n
Nrmp =
E
i=O
191 = 2”
while
(2)
Relations between the protonation micro- and macroconstants have been known since Bjerrum’s pioneer worka5 For a molecule of two groups the relations are
PI = kA + kB
(3)
P2 = kAk: = kBkk
(4)
For a molecule of three groups the relations-are
PI = kA + kB + kC
(5)
p2 = kAki + kAkz + kBk$ = k B k t + kCkP + kckt (6) p3 = kAkikzB= kBk$kgB= ...
(7)
The significance of these equations is that, in principle, macroconstants (with the exception of molecules of one single group) must not be assigned to any individual group: they belong to the whole molecule. Furthermore, their values are influenced not only by pure chemical factors but also by statistical factors. Microspecies of the same composition but different site(s) of protonation are protonation isomers (situated below each other in Figures 1 and 2). In view of the analytical determination, these species have the following peculiarities:
microconstants are needed to characterize the whole equilibrium systeme6This means that, e.g., in the solution of insulin (16 mobile protons within the normal pH range) there are 12870 nonidentical 8-proton-containing protonation isomers, 65536 microspecies, and 524288 microconstants. The microspecies concentration changes with pH (only the concentration ratio of the protonation isomers is pH-independent). Thus, in such a system as many as 65 536 equilibrium concentrations must be determined. The obvious difficulties involved in the peculiarities above can be eased by the use of group constants. Group constants can be deduced from microconstants, taking into consideration two principal effects of protonation: (1) the possibility of group coupling by H-bond formation and other out-of-chain interactions, and (2) electron attraction and repulsion along the covalent bonds of the molecular backbone. 1 . Influence of H-Bond-Forming Protonation on the Number of Microspecies and the Value of Constants. The usual microscopic protonation models (Figures 1 and 2) and the relevant equations (3, 4, and 5-7) are characteristic of protonation equilibria when no intramolecular H bond is formed between the protonation sites and the surrounding groups influence the protonation processes by inductive and mesomeric effects or possibly by hydrophobic interaction. However, if H-bond formation is energetically favorable for the system, the functional groups do not attach to the proton individually, the attached proton belongs to both of the bridgehead groups. This means that for molecules of two functional groups the number of singly protonated species in the system reduces to one. (This is the case of o-tyrosine?) Schemes in which H-bond formation occurs between groups A and B are illustrated in Figures 3 and 4. The H-bond formation modifies the relations between the macro- and microconstants as well: for a molecule of two groups
4106
The Journal of Physical Chemistry, Vol. 90, No. 17, I986
with H bond between A and B
B
pi = k f p2 = k'kfd
(10) (1 1)
for a molecule of three groups with H bond between B and C
8, = k f + k"
(12)
f12 = k f k t + kAkAf= kfk$ + kikiA
(13)
fl3
Noszgl
= k f k t k f d A= kAkAik,fd =
...
'
pH k,d
L
A o
Figure 3. Protonation scheme of a molecule containing two functional groups coupled by a H bond.
(14)
where kf is the microconstant characteristic of the H-bond-forming protonation and k t is the microconstant referring to the uptake of the second proton when the H bond decomposes. As Figures 3 and 4 and eq 10-14 show, the number of the and the microprotonation isomers, Npi,the microspecies, Nmsp, constants all decrease if H bonding occurs in the molecule. If the numbers of H bonds and doubly protonated (ruptured) H . . bonds are h and j , respectively
__
Figure 4. Protonation scheme of a molecule containing three functional groups. Groups B and C are coupled by an H bond.
groups) that it does not exceed the error of the microconstant determination, all the microconstants of the same group are practically equal. This means that all of them can be unified in one parameter; we call it the group constant. Chemical preconditions of this simplification will be discussed later. The unification of microconstants can be represented as
while the number of microspecies is
and
microconstants are needed to describe all the equilibria definition if i - 2 j < 0, then
[
BY
n-h-j li-2j
The numerical dependence of Nmpand N,, upon n and h is shown in Table I to illustrate that H-bond forming moieties (group pairs) reduce the number of both the microspecies and the microconstants. Another important consequence of E-bond-forming protonation appears in the value of equilibrium Lonstants. If the H-bondforming protonation establishing linkage between groups B and C is more favorable for the system than any of the solitary protonations of B or C, the first proton association takes place at a pH higher than expected on the basis of the individual basicities. On the other hand, the second protonation, which causes the rupture of the stable H bond, requires a low pH relative to the individual protonations of either B or C. Accordingly, if unusually high and low constants are determined for a system, the molecule certainly contains H-bond-coupled groups. 2. Effect of Protonation on the Basicity of the Adjacent Groups. The most usual reason (in rigid molecules, the only one) why microconstants of the same group (e.g., kB,k:, k:, and k i C for group B in Figure 2) differ from each other is the electronattracting effect, which occurs when the proximate group(s) (group A or C or both in this example) protonate. This causes more or less decrease in basicity of the group in question. The effect is significant if the adjacent group (A or C) is separated by an adequately low number of intervening carbon (or other) atoms from the given group (B), Le., the static inductive effect actually reaches the group under consideration. The more groups hold protons in an intramolecular chemical environment, the stronger the electron-attracting effect, the less the basicity of the group in question. Thus, among microconstants belonging to the same group the greatest one, kB,refers to the proton association when no other sites are protonated. The smallest one, kACB,on the other hand, quantifies the proton association ability when all other sites hold protons. If the difference between these two extreme microconstants is so small (due to the relative remoteness of the other
kA = k$ = k$ = k& = kA
(15)
kB = k i = kg = kBAC - k B
(16)
kC = k: = kg = k z B = kc
(17)
where constants on the rightmost side with subscripts only are the group constants. This type of simplification can analogously be formulated for H-bond-containing molecules; see the example of Figure 4.
kf = kAf = k f
(18)
k A = k f A = kidA = k A
(19)
kfd = kfAd = kd
(20)
Equations 18-20 show that the lack of perceptible molecularbackbone-mediated interaction between the sites does not indicate a lack of group-coupling interactions between spatially close groups. Substituting eq 15-17 into eq 5-7 yields
pi = kA + k ,
+ kc f l 2 = kAke + kAkc + kBkc D3
=k ~ k ~ k c
(21) (22) (23)
From this system of equations group constants cannot be expressed explicitly. Nevertheless, the whole system of equations can be compressed into a polynom of the third degree: p3
- &k
+ @lk2- k3 = 0
(24)
No subscript is necessary on the group constant because starting with any of the three group constants will lead to the same formula. Equation 24 can be generalized for an arbitrary number of functional groups, n: x ( - l ) i @ n - i k i= 0
i=O
(25)
To consider H-bond-containing systems eq 18-20 are substituted into eq 12-14: (26) P I = k f + kA P2
= kfkA + k f k d
(27)
03 = k f k A k d
(28)
From eq 26-28 a simple and concise formula, such as (24) or (25),
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4107
Group Constant: A Measure of Submolecular Basicity
TABLE I: Number of Microspecies (Nmp)and Microconstants (N,) as They Depend on the Number of Groups ( n ) and H Bonds ( h ) Linking the Groups
n 2
1
0 1 2 3
2
1
4 3
3 4 2
8 6
4 12 7
5
16 12 9
32 20 12
6 8
TABLE 11: Protonation Constants of near 25 O C and Ionic Strength = 0.1 no. of intervening compd atoms oxalic acid 0 malonic acid 1 succinic acid 2 glutaric acid 3 adipic acid 4 pimelic acid 5 mucic acid 4 3,3-dithiopropjonic acid 6 1,2-bis(carboxy6 methy1thio)benzene
Symmetric Dicarboxylic Acids
log
K, 3.81 5.32 5.24 5.01 5.00 5.04 3.63 4.47 3.79
log K2
log K2 -
1.37
2.44 2.66 1.30 0.87 0.72 0.74 0.55 0.59 0.69
2.66 3.94 4.14 4.28 4.30 3.08 3.88 3.10
log
K,
ref 17, 18 19 19 20
20 21, 22 23 24 25
cannot be deduced. However, it is a mutual important meaning of systems 21-23 and 26-28 that the number of unknown group constants is equal to the number of macroconstants. Thus, group constants can be calculated from microconstants. Data Referring to the Chemical Preconditions Related with the Introduction of Group Constants. Group constants (or the same constants called otherwise) cannot be found in monographs devoted to chemical equilibria.'-" Yet the notion is not new. Relations between "titration" and dissociation constants of polybasic acids containing independent or even equivalent and independent groups have been analyzed in a few p a p e r ~ . ~These J~ works restrict their considerations to ideal systems, free of interactions, so the real circumstances when the application is allowed have not been studied. On the other hand, chemical conclusions have also been drawn by the use of group-constant-like quantities,I4 but due to the neglect of significant interactions these systems had to be reana1y~ed.l~This is why it is a crucial question to determine'the cases when group constants can be used. As stated above, microconstants may be treated as group constants if the effect of protonation on the adjacent group decreases to a negligible level along the intervening bonds. What is the number and character of atoms or bonds between the groups when this negligibility is virtually perfect? Monitoring the protonation shifts for methylene protons near basic sites by IH N M R , has shownI3 that amine deprotonation downfield shifts are still sensitive at the third (y) methylene moiety, but not at any farther one. In the case of carboxylate groups the effect disappears at the protons of the third carbon atom. 13C (8) Rossotti, F. J. C.; Rossotti, H. S. The Determination of Stability Constanfs: McGraw-Hill: New York, 1961. (9) Butler, J. N. Ionic Equilibrium. A Mathematical Approach: Addisson-Wesley: Reading, MA, 1964. (10) Beck, M. T. Chemistry of Complex Equilibria; Akademiai: Budapest, 1970. (1 1) fncztdy, J. Analytical Application of Complex Equilibria; Adadhiai: Budapest, 1977. (12) Simms, H.S. J . Am. Chem. SOC.1926,48, 1239. von Muralt, A. J . Am. Chem. SOC.1930,52, 3518. (13) Sarneski, J. E.; Reilley, C. N. 'The Determination of Proton Binding Sites by NMR-Titrations", in Essays on Analytical Chemistry; Wiinninen, E., Ed.; Pergamon: Oxford, 1977; p 35. (14) Jameson, R. F. J . Chem. SOC.,Dalton Trans. 1978, 43. (15) Jameson, R. F.; Hunter, G . ; Kiss, T. J . Chem. SOC.1978, 768.
32 24 18
6 80 52 33
64 48 36 27
13 192 128 84 54
16
8192 6144 4608 3456
53248 37888 26880 19008
65536 49152 36864 27648
524288 376832 270336 193536
1458
6561
11664
69984
6561
34992
N M R spectroscopy has led to similar conclusion^.^^ This statement is supported by protonation macroconstants of symmetrical dicarboxylic acids, where 2-2 of the four microconstants are identical; see Table I (data suggested by Critical Stability Constand6). In every compound of Table I1 the intramolecular environment of the functional groups is identical. This is why in the absence of interaction between the groups, according to statistical considerations for equivalent coordination sites,5~26-27 the log K,- log K2 difference has to be log 4 = 0.602. The last three compounds of hindered rotation (including mucic acid with only four intervening atoms) really show this difference (concerning the accuracy of the data). In the homologous series from oxalic to pimelic acid, electron attraction along the carbon chain and (partial) electrostatic or H-bond-forming effect can be observed. The former decreases as the number of intervening bonds increases, the latter do not fully disappear even at the higher homologues due to the flexibility of the molecules. In this way the perfect independence of the groups (and thus A Ig K = 0.602) cannot be accomplished. Despite this, the value and the difference of the constants for adipic and pimelic acids is practically equal. This also supports the spectroscopic conclusion that, regardless of conjugated systems, influencing of proximate group's basicity along the chain takes place if more more than three atoms are situated between the protonation sites. At peptides this occurs in two cases: in dipeptides or in higher peptides if the C-terminal end is aspartic or glutamic acid; at the ornithine N-terminal the number of connecting atoms is four, while at the nearest non-terminal-peptide bond separated case (-Asp-Asp- sequence) this number is six. The backbone-mediated independence is visually exemplified by IH and 13C N M R protonation shifts of glutathione (y-Lglutamyl-L-cysteinylglycine). In this molecule the glycyl carboxylate is separated by five connecting atoms from the -SH group and by nine connecting atoms from both the carboxylate and ammonium groups of the glutamyl residue. The NMR-pH profile clearly show^**^^^ that the a and 6 carbon and methylene proton resonances of the cysteinyl residue are totally undisturbed by the glycyl and glutamyl carboxylic acid ionization (pH 2-3). The reverse is also true: chemical shifts of the glycyl moiety do not (16) Critical Stability Constants; Martell, A. E., Smith, R. M., Ed.; Plenum: New York, 1977. (17) McAulay, A.; Nancollas, G. H. Trans. Faraday SOC.1960,56, 1165. (18) Schwarzenbach, G.; Fisher, A. Helu. Chim. Acta 1960, 43, 1365. (19) Campi, E. Ann. Chim. (Rome) 1963, 53, 96. (20) Yasada, M.; Yamasaki, K.; Ohtaki, H. Bull. Chem. SOC.Jpn 1960, 33, 1067. (21) Christensen, J. J.; Izatt, R. M.; Hansen, L. D. J . Am. Chem. SOC. 1968, 89, 213. (22) Laing, D. K.; Pettit, L. D. J . Chem. SOC.,Dalton Trans. 1975, 2291. (23) Bottari, E. Monatsh. Chem. 1968, 99, 176. (24) Hawkins, C. J.; Perrin, D. D. Inorg. Chem. 1963, 2, 843. (25) Ford, G. J.; Pettit, L. D.; Sherringbon, C. J . Inorg. Nucl. Chem. 1971, 33, 4119. (26) Adams, E. Q. J. Am. Chem. SOC.1916, 38, 1503. (27) Wegscheider, R. Monatsh. Chem. 1895, 16, 153. (28) Rabenstein, D. L. J . Am. Chem. SOC.1973, 95, 2797. (29) Rabenstein, D. L.; Greenber, M. S.; Evans, C. A. Biochemistry 1977, 16, 977. (30) Rabenstein, D. L.; Guevremont, R.; Evans, C. A. "Glutathione and its Metal Complexes", in Metal Ions in Biological Systems; Sigel, H., Ed.; Marcell Dekker: New York, 1979; Vol. 9.
4108
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986
change when the glutamyl ammonium and cysteinyl sulfhydryl deprotonate (pH 8-9). Since microconstants of different value have been published for this system,28~30*31 the question inevitably arises, how can the nine-atoms-apart glutamyl carboxylate (slightly) influence the basicity of its glycyl counterpart if neither the also nine-atomdistant ammonium nor the relatively nearby sulfhydryl cannot deflect the chemical shifts of the glycyl moiety? Efforts to answer this problem of glutathione protonation are in progress.32 Nevertheless, calculating the group constants on the basis of the published macroconstants allows the microconstants and group constants and their characteristics to be directly compared. The microconstants (original symbols are used first and then, for uniformity, our symbols) of the carboxylate protonation region are log k 2 , = log kA = 2.33 f 0.01 log k , = log k$ = 2.09 f 0.05 log k 1 2= log kB = 3.36 f 0.10 log k2 = log kf: = 3.12 f 0.05 where A and B stand for glutamyl and glycyl carboxylates, respectively. The difference between the microconstants of the same group is 0.24 in log k units. This is much less than the corresponding values for tartaric, malonic, and oxalic acids, which indicates a lack of backbone-mediated interactions. From the data above it can be calculated that 91.4% of the protonations take place in the processes assigned to two constants (log kB = 3.36 f 0.10 and log kBA = 2.09 f 0.05). By the use of eq 25 and the macroconstants, the group constants can be calculated: log kB = 3.38
log
kA
= 2.07
From a comparison of these values with the relevant microconstants, it can be stated that the group constants furnish data on the main pathway of the protonation process. Their values are very near the appropriate microconstants (see the error of microconstants). Analogous comparisons with similar results can be done for other di- and tripeptide systems where microconstants are available.29 The following are two more examples of the applicability of group constants: In solutions of synthetic macromolecules Barbucci et al.34 have stated that in the protonation processes of polymers built up of basic rigid monomers the repeating moieties behave like independent units, due to the lack of intramolecular interactions. In a thymopoietin pentapeptide fragment, where overlapping microconstants of the tyrosyl and the lysyl residues could be determined spectrophotometrically, both the two phenolic as well as the amino microconstants proved to be identical in every state of the other Limiting Cases for the Relations of Macro- and Microequilibria. Group constants can also be approached by defining the limiting cases for macro- and microequilibria. In this aspect dibasic acids (and divalent bases) have the advantage of simplicity in treatment . All the relations below can equally be regarded as association (protonation) constants of divalent bases or dissociation constants of dibasic acids. In order to maintain their validity, the only restriction is that in the same set of equations all the constants should be association or dissociation ones. It can be seen from eq 3 and 4 that there are at least three microconstants expressed by only two independent macroconstants. Unfortunately, the frequently used relation
(31) Martin, R. B.; Edsall, J. T. Bull. SOC.Chim. Bioi. 1958, 40, 1763. (32) Noszil, B.; Osztis, E., unpublished. (33) Nosdl, B.; Scheller-Krattiger, V.;Martin, R. B. J . Am. Chem. SOC. 1982, 104, 1078. (34) Barbucci, R.; Ferruti, P.; Micheloni, M.; Delfini, N.; Segre, A. L.; Conti, F. Polymer 1978, 19, 1329. (35) S6vig6, I.; Kiss, T.; Gergely, A. Inorg. Chim.1984, 93(4), L53.
Noszil is a derivative of eq 3 and 4 , so it does not hold any new information. For these reasons microconstants normally cannot be calculated from macroconstants only. However, there may be some special cases: 1. The molecule is symmetrical, like oxalic, malonic, succinic, ..., etc. acid, or oxalate, malonate, succinate, ..., etc. bases. For these compounds k A = kB and k t = k i . This is a new relation by which k A = kB = @ , / 2 and kk = 2K2. It should be noted, however, that there is no definite relation between kA and k t or kBand k i and so between K , and K2. It depends chemically on the extent of interaction (distance, etc.) of the groups. 2. The molecule is symmetrical and the groups are well se k A = kB = kA -k arated. This is the “statistical B - A’ Therefore K , / K 2 = 4. This perfect independence hardly occurs in practice; rigid, symmetrical molecules with remote groups are necessary. 3 . The molecule is not symmetrical, but the groups are separated by several intervening atoms. This is the case of group constants, which can be divided into two subcategories. a . There is no out-of-chain interaction between the groups. kA = k e = kA kB = k f : = k B
t
b. The groups are separated by many intervening atoms, but they are coupled by the accepted proton. There is only one single type of monoprotonated species and equilibrium parameter. p, = K , = k f K2 = kd p2 = K , K 2 = kfkd It has to be emphasized that group-constant treatment does not exclude the interaction between any two groups. In fact, the group-constant values are indicators of group-group couplings. Typical polypeptide molecules are composites of solitary and coupled group-containing loci. Computation of Group Constants
Group constants have definite mathematical relations with the conventional equilibrium macroconstants and of course with the experimental data. It is worth surveying the possibilities of their determination from both types of relationships, because there are well-known sophisticaed methods for the determination of mac r o c ~ n s t a n t s .On ~ ~ the ~ ~ other hand, in principle, the less conversions are made on the experimental data, the more precisely the constants can be determined. Determination of Group Constants When H-Bond-Forming Protonation Does Not Occur. Equation 25 can be used as basis for a fast and simple method utilizing macroconstants. x(-i)i@,,-iki= 0 i=O
In the knowledge of the previously elucidated macroconstants a suitable method (e.g., Newton-Raphson iteration) can furnish all the roots of eq 25, and so all the group constants can be determined. However, usually the possibility of H-bond formation at ligands of unknown structure in solution may not be excluded. This procedure is more adequate for producing initial data for methods that can take into account the effect of the H-bond formation. Determination of Group Constants When the Protonation Takes Place with H-Bond Formation of Unknown Number. If the H-bond formation of a functional group is favorable with one other group, the possible number of intramolecular H-bonds ( h ) in a molecule of n groups may be h = 0, 1, ..., [n/2] (where [ n / 2 ] is the integer part of n/2). Accordingly, the protonation processes may take place in [ n / 2 ] + 1 types of schemes, meaning [ n / 2 ] + (36) SillCn, L. G. Acta Chem. Scand. 1962, 16, 159. (37) Ingri, N.; SillCn, L. G. Acfa Chem. Scand. 1962, 16, 173. (38) Perrin, D. D.; Sayce, I. G. Talanta 1968, 14, 833. (39) Sayce, I. G. Talanta 1968, 15, 1397. (40) Nagypil, I. Mugy. Kem. Foly. 1974, 80, 49. (41) Barcza, L. Magy. Kem. Foly. 1969, 75, 513.
Group Constant: A Measure of Submolecular Basicity
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4109
1 systems of equations in Figures 2 and 4. The difference originates from the different number of H bonds. In each one of the [n/2] 1 systems of equations the relationships between the complex products and the group constants are different. There are n - 2h group constants assigned to lonely groups, plus q u a l h (coupled) group constants, which belong either to the formation or to the decomposition of H bonds. In other words, the relations between the n group constants and the n macroconstants may be formulated in [n/2] 1 different ways, 1 different sets of constants. resulting in [n/2] In principle only one of these sets is correct (where the model counts with the correct number of coupled groups). However, in practice, results of some other models may fall near the correct one, due to the distorting effect of experimental errors. If the coupled groups are of significantly different basicity, the effect of coupling is sensitive only in the value of the group constants, but not in any of the error functions. These error functions, standard deviation, etc. are usually used to characterize the correctness of a model. A method that makes use of macroconstants for groupconstant determination has appeared in our previous paper.42 Here two other methods are described, where neither macroconstants nor number of H bonds need to be known. A. A possible basis for groupconstant evaluation is the classical complex formation function.43 For deprotonation processes its more practical derivative is
bK6
+
+
+
n
+ F ( n - i)j3,[H+Ii i= 1
R=
l"S51"9Kk lqK,logK2kgK,
I
I
9
12
Figure 5. log of successive macroconstant ( K ) and group-constant ( k ) values in the basic region for ACTHIwB2. Subscripts of group constants correspond to values in Table 111. Pattern of calc~lation:~~= K I= EEHkt$ 82 = C$H+lx.f;Hkikp 83 E ~ ~ l + l C , L = H + I C f i , H k i k ] k 8~6, = &,k,, K, = &/j3x+ where H + 1 = I , I + 1 = J , etc.
by successive approximation. After a few cycles their real values can be determined. B, In the past few decades computerized methods have become overwhelmingly predominant in the field of equilibrium constant determination.3641 These programs usually require knowledge of the total ligand (CTL)and acid (CTH)concentrations as well as the equilibrium concentration of one of the components, most often [H+]. If XI-o denotes the xth independent group and yl," the yth isolated not necessarily adjacent pair of groups the following relationships can be written:
n-Zh
(29)
i=o iBi[H+l'
CTH
= [H']
+ x= 1 [~l-o][H+]k, + h I v l 2 l [ H + l ( 1 + kyf + 2kYfkfiW+l) (35)
y= 1
where H is the mole of protons dissociated from one mole of ligand at a given pH and n is the number of functional groups in the molecule. If the number of intervening atoms between each of two functional groups exceeds three (precondition of microconstant conversion into group constant), eq 29 can be given as the sum of n - 2h, one-step, and h, two-step, partial proton dissociation functions h
n-2h
R = x=l C A + y=1 CB
(30)
where A = (1
+ k,[H+])-]
In the above functions k, denotes the group constant of the xth independent group, and kyf and kyd denote the group constants of the yth H-bonded pair of groups. From eq 30 any of the actual independent (ki) or H-bonded (kd and kgd)group constants can be expressed explicitly n-2h
i-1
h
- l)[W]-'
(31)
kfi = (2 - C)((C- 1)[H+] + k@[H+l2C)-'
(32)
kgd = (2 - C + (1 - C)[H+]kd)(kg.[H+]2C)-1
(33)
ki = ( ( R -
E
E A - CB)-'
x=i+l x=l
y=l
where x=1
y=g+l y-1
Knowing the set of R - [H+] value pairs and estimating initial groupconstant values allows all group constants to be calculated (42) Nosi3, B.;Burger, K. Acta Chim. Acad. Sci. Hung. 1979,1Do,275. (43) Bjcrrum, J. Metal Ammine Complex Formation in Aqueous Solution; Haase: Copenhagen, 1941.
Equations 34 and 35 are formally identical with relations of acid mixture concentrations expressed by macroconstants. In this way the computation of group constants can be transformed into methods elaborated for macroconstant determination.
Applications One of the main applications of group constants is the characterization of the individual basicity of groups or group pairs in complicated molecules, where several protonation processes overlap. Data on every protonating group of multifunctional polymeric molecules (especially polypeptides) over the entire pH range are hardly accessible by any other method. Some aspects of this utilization have already been p ~ b l i s h e d . ~ ~ , ~ ~ Group Constants of ACTH. Experimental Methods. The natural aH adrenocorticotropic hormone (ACTH) consists of 39 amino acids. We used its N-terminal 4, 14, and 32 amino acid containing fragments. The last one produces all the physiological effects of the hormone. The substances for titration were twice lyophilized perprotonated preparations. The following devices have been used for the pH-metric measurement: digital pH meter Radiometer pHM 64 (precision of display 0.1 mV or 0.001 pH unit); automatic burette Radiometer ABU 12 (precision of reading 0.001 cm3); glass electrode (Radiometer G 202 B); Ag/AgCl electrode of the second kind as reference, which has been connected to the measuring cell by a Wilhelm bridge. The ionic strength was kept constant at 0.3 mol dm-3 by N a N 0 3 as an auxiliary electrolyte. The titrant was carbonate-free NaOH in 0.01, 0.1, and 0.3 mol dm-3 concentrations. The peptide was dissolved in nitric acid solutions, the concentrations of which were identical with those of the titrants. The temperature was kept at 25.0 i 0.1 O C by an ultrathermostat. Results and Discussion In Figure 5 log of successive macroconstants (K, ... K6) and group constants (kM ... kH) for ACTH,-,, are represented on the same scale. Macroconstants, which are governed not only by chemical but also by statistical factors, fall nearly equidistantly, (44) Noszll, B.;Burger, K. Mum.Kern. Foly. 1981, 87, 175.
4110 The Journal of Physical Chemistry, Vol. 90, No. 17, 1986
TABLE 111 Group-Constant Values in log k Units of the Functional Groups for Three Fragments in Aqueous Solution
site terminal carboxylate glutamic acid and aspartic acid carboxylate histidine imidazole terminal amino lysine amino and tyrosine phenolate
ACTH,-, 3.20
1.27 110.70
ACTHI-14 3.47
6.31 7.29 9.94 10.86
ACTHI-12 3.54 3.59 4.10 4.25 5.03 6.43 7.47 9.70 10.13 10.45 10.82 10.94 10.98
A B C
D E F G H
I J
K L M
while group constants reflect pure basicity. Table I11 shows group constant values for three corticotropin fragments (ACTHi4, ACTHl+ and and their most reasonable assignments to the functional groups protonating in the investigated pH range. Some important properties of the above data are 1. Constants to the C-terminal carboxylate groups (ACTHl4 methionine log k = 3.20, ACTHI-14 glycine log k = 3.47, alanine log k = 3.54) are approximately 1 log unit greater than the relevant values of the given monomeric amino acid showing the same order. Thus, their values can be well explained by taking into account the effect of the peptide bond and the side chain of the amino acid. 2. The group constant log k = 4.19 of ACTHI-,, obviously reflects the pure basicity of a side chain carboxylate, which is influenced by neither H-bond nor any other secondary effects. Two similar constants have been determined for ACTHlw3*,too. However, the other two carboxylate constants of this molecule are significantly different. As it has been stated in the theoretical section, group constants belonging to the formation (B + C H+ ~ r ?B - H - C+) and to the decomposition (B - H - C+ + H+ BH+ CH') of a hydrogen bond are supposed to be greater and smaller by the same value than the individual basicity of the pillar groups. This means that the also uninfluenced group constants (4.25 and 4.10) most presumably belong to the glutamic acid and aspartic acid side chains, located at positions 5 and 29, respectively. These are remote enough to be independent. The two deviant constants (5.03 and 3.59) certainly belong to the glutamyl carboxylates at positions 28 and 30. These are connected by H-bond and/or electrostatic field effects, reflected by these remarkably different group-constant values. 3. Tyrosine phenolate and lysyl e-amino groups are of very similar basicity. This is why there is no evident assignment in
+
+
*
Noszd the basic region. In principle, grougconstant values for the two tyrosine phenolate groups could be determined by spectrophotometric method. In practice, however, the ACTHl-,, and molecules contain tryptophane, phenylalanine, and histidine residues, disturbing seriously the selective change of absorbance due to deprotonation of the tyrosyl moiety. Therefore, more precise values and unambiguous assignments of phenolates could not be achieved even by combining spectrophotometry and potentiometry . 4. The most deviant constant relative to the third groups of trifunctional amino acids or peptide side chains is log k = 9.70 at ACTHi-32. Since there is a similar value at ACTH,-l4 too, the moiety responsible for this phenomenon is certainly part of both fragments. This is the e-amino group of lysine. This "decreased" constant does not have any "increased" counterpart (at least its deprotonation does not occur below pH 13). Consequently, the H-bond-forming (or electrostatically coupled) pair of this t-amino group is obviously such a site (serine alcoholic hydroxyl, arginine guanidinium, peptide -NH- etc.) that loses protons at extremely low hydrogen ion concentration. 5. Those five groups, which deprotonate at the most basic part of the studied pH range (e-amino and phenolate), are certainly interacting ones, as indicated by their scattered nonequivalence. An important conclusion of the examples above is that group constants cannot substitute microconstants in the cases of small molecules. However, in several systems (especially in polypeptide-containing ones), it can provide valuable information not only on the functional groups but also on their intramolecular environment. The other most important application of group constants is the determination of microspecies concentrations in systems where hundreds or thousands of differently protonated species exist in the presence of one another. Group constants can be used to estimate the concentration of even the least abundant minor microspecies. This becomes increasingly significant as kinetic studies reveal more and more reactions where the minor species is the reactive For these reasons this problem is discussed in a separate paper.45 Acknowledgment. The author is indebted to the G. Richter Chemical Works, Budapest, for making the ACTH substances available, and to Prof. K. Burger and A. LadPnyi for helpful discussions. (45) Noszll, B. J. Phys. Chem., in press. (46) Slight differences between macroconstants calculated so and in Table I11 originate from ignoring group constants A-G and N , 0, and P. Accurate values for constants N, 0, and P of arginine guanidino groups with log k between 12 and 1 3 are out of our reach.
