Assigning the pKa's of Polyprotic Acids

Assigning the pKa 's ol Polyprotic Acids While teaching а Ьiochemistry lab course in which stu­ dents measured the pKa 's of amino acids, the author was asked about the assignment of the pKa 's of cysteine, which was reported in the textbook as follows. (8.33) SH 1 (cysteine) СН2 1 H3N+-СН-СО2- (1.71) (10.78) Не Ыithely answered that this assignment made sense, and picked up а сору of the "Handbook of Biochemistry" 1 to look up data that might prove its validity to the student. Не started Ьу noting that H2S (7.00) was more acidic than the NH4 + ion (9.24) and that the difference between the pKa 's of H2S and NH4 + (2.24) was roughly equivalent to the differ­ ence between the second and third pK8 's of cysteine (2.45). Не then naively suggested that the simplest way to test this assignment would Ье to see which рК8 disappeared when either the -NH3+ or -SH group of cysteine was deriva­ tized. Не was therefore quite pleased to find an entry for N,N,N-trimethyl cysteine, (8.65) SH 1 (N,N,N-trimethyl cysteine) СН2 1 (CHз)Hз№-CH-C02which showed the disappearance of the рК8 at 10.78, in accord with the textbook assignment. Unfortunately, the student found an entry for S-methyl cysteine which showed that derivatization of the sulfhydryl group also leads to the disappearance of the рК8 at 10.78. SСНз 1 СН2 1 Нз№-СН-СО2(8.75) (S-methyl cysteine) When we then turned to "Dissociation Constants of Organic Bases in Aqueous Solution" 2, the student noted that the рК8 data for both cysteine and cysteic acid were also in agreement with an assignment of рК2 and рК3 for cysteine that is the opposite of what appears in most textbooks. (8.71) Нз№-СН-СО2J СН2 1 СН2 (cystine) 1 Нз№-СН-СО2(8.02) The answer to why derivatization of cysteine at either the -NH3+ or -SH groups leads to the loss of рК3 is well known to Ьiochemists but not as well known to chemists who are used to working with polyprotic acids such as H2S, Н2СО3, and Н3РО4, or simple amino acids such as glycine. Classical treatments of polyprotic acids are based on the assumption that the difference between the pKa 's is largf. enough to assume stepwise dissociation. нnм + Н2О Hn-lм- + Нзо+ Hn-lм- + Н2О = нn-2м2- + Нзо + = If the dissociation is in fact stepwise, most of the Н3О + ion comes from the loss of the first proton, and the equilibrium concentrations of the Н3О+ and Hn-iM- ions are more or less equal. Substituting this approximation into the equilibrium ex­ pression for the first step in the dissociation allows us to solve for the Н3О + or Hn-iM- ion concentration. [H3Q+] ""' [Н п_ 1 М-] = y'К1 [HnMJ Substituting the same approximation into the expression for the second step suggests that the concentration of the Н n_ 2м 2- ion is roughly equal to the value of К2 for the acid. [Н30 +] [Нп_ 2м2-] =К -----2 [Hn_ 1M-] The Henderson-Hasselbach Equatlon When there is а large difference between the pK8 's of an asymmetric polyprotic acid such as glycine, we can assign these pK8 's to individual titrataЫe groups. For example, H3 N+-СН2-СО2(2.35) (9.78) We can then use the Henderson-Hasselbach equation, base] рН=рК. + log [ [acid] to calculate the percentage of each titrataЬ!e group in its acid or conjugate base form at a given рН. At a рН of 7.00, for example, only 1 in 600 of the amino groups in glycine is present as the conjugate base (-NH2 ), and only 1 in 45,000 of the a-carboxylic acid groups is present as the conjugate acid (-СО2Н). Polyprotlc Aclds for whlch the DIHerence between к;s 1s Small 1 s 1 s Polyprotlc Aclds for whlch the DIHerence between K8 's 1s Large Cysteine is а classic example of а polyprotic acid for which SОзН 1 СН2 1 Нз№-СН-СО2(8.7) (cysteic acid) 1 Sober, Н. А., Ed., "Handbook of Biochemistry", 2nd ed.; Chemi­ cal Rubber Company: Cleveland, ОН, 1970. 2 Perrin, D. D. "Dissociation Constants of Organic Bases in Aque­ ous Solution"; Butterworths: London, 1965. the stepwise dissociation assumption fails.3 When cysteine is titrated with base, а significant fraction of the -SH groups are deprotonated at more or less the same time as the -NH3+ ions. Because the presence or absence of а charge on the -SH group can influence the acidity of the -NH3+ group, and vice versa, it is possiЫe to define four microscopic dissociation constants for cysteine (k n, ks, kn, and k8,) that describe what happens when the -SH and -NH3 + groups are titrated. К3) dissociation constants measured when cysteine is titrat­ ed and the microscopic or molecular (k., kn , etc.) dissocia­ tion constants of the-NH3+ and-SH groups? Ву juggling the equilibrium expressions forК 2, Кз, ks, kn, k.,, and kn, we can see that, [H3N +--S-] + [H 2 N--SH] [H3O+ ][H 2N--S-] and, s- SH 1 СН 2 1 Нз№-СН-СО2- 1 СН 2 1 H2 N-CH-CO2 - /J,,, These microscopic dissociation constants are associated with the following equilibria. [Н O +][H N +--S-J 3 3 k• = -=-------=----+ The macroscopic dissociation constants (К2 and К3 ) mea­ sured when cysteine is titrated are therefore related to the acidity of both the-NH3+ and-SH groups, and it is а mistake to attempt to assign these macroscopic constants to either the -SH or -NНз + group of cysteine unless there is reason to believe that there is а significant difference in the magnitude of the microscopic constants ks and kп. Analysis of Microscoplc Dlssociatlon Constants for Cystelne An analysis of UV spectral data for cysteine derivatives enaЫed Benesch and Benesch4 to determine the microscop­ ic pka constants for cysteine. [H3N pk, = 8.53 (H 3O + j [H 2N--SН] pk n = 8.86 k " = ____::___ + [H 3N --SH] [H O+][H N--S-] [H 2N--SН] = 3 2 ------ k ", = [Н 3O ] [H2N--S-] -=-------=---+ k, • + [H 3N --s-] Because two products are formed when the Нз№ СН(СН2SН)СО2- ion loses а proton, H2N--SH and the Н3№--S- ion, the equilibrium concentration of the con­ jugate base in this dissociation is equal to the sum of the concentrations of these products. The equilibrium expres­ sion for the macroscopic К2 constant which measures the ease with which cysteine loses а second proton is therefore, [Н О +]([н N + ---s-1 + [H N---SН]) 3 3________ 2 К2=-=----__: _ [H3N+ --sнJ Both the H2N---SH and H3№---S- produced in the second dissociation of cysteine can go on to lose а third proton to form the H2N---S- ion. Because two acids, H2N---SH and H3№---s-, serve as sources of the H2N---S- conjugate base, the equilibrium expression for the macroscopic К3 dissocia­ tion constant for cysteine is + [H3O ][H2N--S-] Kз = ---'-----[H3N + --s-] + [H2N--SH] What is the relationship between the macroscopic (К2 and 3 Edsall, J. Т.; Wyman, J. "Biophysical Chemistry"; Academic: New York, 1958; Vol 1 рр 450-504. 4 Benesch, R. Е.; Benesch, R. J. Amer. Chem. Soc. 1955, 77, 5877. pk,, = 10.03 pkn, = 10.36 These data suggest that the microscopic or molecular disso­ ciation constants for the -SH and -NH3 + groups in cysteine are virtually identical. Furthermore, the loss of а proton Ьу either the -SH or -NH3 + groups leads to а decrease of about 1.5 рК units in the acidity of the other functional group, as might Ье expected. Analysls of рК8 Data The close similarity between the microscopic dissociation constants for the -NH3+ and -SH groups in cysteine might appear surprising at first glance, since the pKa's of H2S and NH4 + differ Ьу 2 рН units. However, the рК8 of а more reasonaЫe model compound, СНзСН2SН (10.60), is remark­ aЬly close to the рК8 of CH3NH3 + (10.66), and the results of this analysis of cysteine are therefore not particularly sur­ prising. Concluslon Because the microscopic pk8 's of the -SH and -NH3+ groups in cysteine are almost the same, derivatization of either group leads to the disappearance of the same macro­ scopic рК. The difference of roughly 2 рК units between рК2 and рК3 of cysteine is not а reflection of the relative acidities of the -SH and -NH3 + groups but the effect of the loss of а proton Ьу either the -SH or -NH3 + group on the acidity of the remaining functional group. Although it is tempting to assign the pKa 's of polyprotic acids to individual titrataЫe groups, this exercise should Ье reserved for those acids where the difference between the Ka 's is large enough that we can safely assume stepwise dissociation.