Glycine: A Simple Zwitterion Analysis of Its Proton-Binding Isotherm Harry A. Saroff Laboratory of Biochemical Pharmacology, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Building 8,Room 227,Bethesda, MD 20892 Glycine, H2N-CH&OOH, with its amino and carboxyl groups is a good example for developing the principles of binding a ligand to different interacting sites. If we start with the completely unprotonated form
the first step in the binding of protons yields primarily the zwitterion. (+)
(-)
H3N-CHrCOO
The strong electrostatic attraction between the positively charged amino group and the negative carboxyl group is the predominant ligand-dependent interaction or perturbation in this binding process. Evaluation of interactions between binding sites may be used to reveal important structural details in the binding molecule. One of the early attempts was that of Bjermm who calculated the distance between the charged carboxyl groups using the experimental constants of malonic acid (7 .?i ~-,-,. This paper will analyze the experimental data for the binding of protons to glycine using data from chemically modified derivatives and data from individual-site isotherms. An appendix is included that contains a brief development of the general binding equations for multiple sites and a detailed description of ligand-dependent interactions in a two-site system. The student is advised to consult the appendix when some point seems obscure.
Figure 1. Binding isotherms for glycine. The combined isotherm is designated by a + b. Cuwes a and bare the individual-site isotherms for the amino snd carboxyl groups. Dashed curves are the unperturbed binding isotherms. where CH is the concentration of free protons. We wish to translate these two experimental constants, Kl and K2,into quantities that reflect the intrinsic properties of the amino and carboxyl groups and their interactions on protonation. In order to proceed with such an analysis, some assumptions are necessary The followingis the first.
Analysis of the Data for Glyclne Binding of protons to glycine may be formulated schematically as follows.
m a b
-
U
KZ
H+cH~-co~-) ==K, ==
(+)
H
630 U ~
+
~
Assumption: The ligand-dependentinteractions derive only from electrostatic forces of attraction and repulsion between the ionic groups of glycine. The binding of prw tons to one of the sites, though inducing changes at that site, has no effect on the other site except through eoulomhic forces between ~ the ~ resulting ions. ~ ~ ~ ~
This assumption is applied in analyzing three of the microscopic reactions in eq 1. (1)
The reference state is taken as the unliganded species -
H2N-CH2-COO Figure 1illustrates the experimental binding isotherm for glycine, labelled (a + b). Macroscopic or experimental constants are KI = 109.78 M-' and Kz = M-' obtained by fitting this isotherm with the following general binding equation for two sites (see the appendix).
Protonation of the Carboxyl Group The followingreaction for the protonation of the carboxyl group is described with the microscopic constant Kb. (See appendix for explanation of microscopic constants.) a b a b
U
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637
The reference state has one negative charge, and the mono~rotonatedstate in ea 3 has no chame. Conseauentlv. there is no change in el&rostatic force-between the tw'o sites on ~mtonationof the carboxvl m u v . Therefore. the binding of a pmton to the carboxyi &ougis not perturbed in this step of the reaction. Thus, it follows that Kb = kb, which is the intrinsic constant for the binding of a proton to the carboxyl group, and the relative wncentration of species H2N-CHrCOOH is simply kbc~.
-
Protonatwn of the Amino Group Similarly, in the microscopic reaction a b
OLP
is kdzac$, where k, is the intrinsic constant for the pmtonation of the amino group. A Perturbed Binding There are no perturbations in the binding of protons resulting from site-site interactions in the two micmscopic reactions just described.
However, there is a significant change in the s i t e site interactions for the microswpic reaction
(kaalabc~)
Chemical Modification Data added to this problem come fmm analyses of the binding isotherms for the methyl and ethyl esters of glycine Ha-CHrCOOCH3 and H2N-CH,COOCzH, The average of the binding constants for these com~ o u n d sis lo7" M-' (4). If we maintain the ~recedine Hsumption and add the assumption that a-C& i o r - ~ ~ ~ ; j ~ the eauivalent of a hvdroeen .m . o u is " ., atom. then the measured'associnti~nconstant for the esters equals that for the un~enurbrdarninorrrou~.Conseauentlv. k.= 10' hl '.AddiGonal support forvthis'assumption c& from the data on the association constants for the hindine of ~ m t o n to s polymers ofglyc~nr.'ihe'constants the aminofloups of!.t? are lo7"' 11 ' and 10 ' ' M for the te~minalamino rrrou~i of triglycine and tetraglycine (4). This solves the problem. From eq 7, A
(6)
The microscopic constant describing eq 6 is
where slab is defined as the ligand-dependent interaction factor. The ligand-dependent free energy of interaction is thus -RT In al.b, and the relative wncentration of the species Journal of Chemical Education
Equation 7 is the same as eq 32 in the appendix with U1ba = = 1. Using the preceding reasoning, we defme three parameters, k,, kb, and al,b, but there are only two experimental valuesKl and Kz. Some additional data or assumptions are still required. Two ~otentialsources from which additional data miy be applid are chemical modification of one of the sites and measurement of the individual-site isotherms.
'
The intrinsic wnstant k, for the binding of protons to the amino group is perturbed in this reaction. The perturbation derives fmm the change in an electrostatic force of zero in the reference state to an attractive force in the monoprotonated species
638
with and
there is no change in electrostatic force between the two sites in this step of the pmtonation. Consequently, the relative concentration of the species
(11
+HaN-CHrCOO- i s kaalabcn.It should be emphasized that the intrinsic constants, k, and kb, are the binding constants under conditions where the ligand-dependent interaction factors are unity. We may now expand eq 2 to the following.
and slab= lozo8or -2.8 kcal. yielding kb= The dashed curves of Figure 1, labelled k = and k = 104.43,illustrate the binding isotherms for the unperturbed amino and carboxyl groups. The concentration for 112 saturation of the amino group (in glycine) is decreased because the value of slab is greater than unity, whereas the concentration for 112 saturation of the carboxyl group is increased. This may be quantified using the macroscopic constants Kl and Kz.
The intrinsic or unperturbed constant k, is increased approximately by the factor slab. whereas the unperturbed constant kb is decreased approximately by the same amount.
Indivdual-SiteData
-
Figure 1, in addition to illustrating the combined isotherm a + b, illustrates the binding isotherms generated by the individual interacting sites, a and b. Individual-site isothcnns are derived from data collcctcd on the binding of a ligand at one site while the other site binds the same ligand. Methods used are Raman spectroscopy (5),nuclear magnetic resonance (61,ultraviolet absorption spectmscopy (71,and selective enzyme action (8). Nuclear magnetic resonance data have been reported for the binding of protons to glycine (9). The species in which the amino group is pmtonated are '+'H3N-CH,COO'-' and '+'H8N4!H&OOH, yielding the following equation for the number of protons hound to the amino group. -
k.%bc~
I U C I I21~s I 1 k,,al,h -- K I ~ Cs -HK ~ 2 kb K,I(.H~A, T K I K 2 ( ~ ~ s-~1, . j - K 2 ( c ~ b ) ~-Kz(c~s)o.5 .~ + Kz(cH.)o~ - K~(cm)os+ 1
]
It appears, particularly from eq 12, that the individualsite isotherm for the carboxyl group (site b) may provide the necessary added experimental value. To test whether this appearance is real, let us assume the following.
+ kakbci
The experimental data illustrated in Figure 1 derive from and , alsb= 102.08,and the constants k, = lo7.', kb = 1 0 ~ " ~ these constants are without error. The experimental data are collected with a precision to give five significantfigures.
.
Similarly, the number of protons bound to the carboxyl group is vm =
Consequently, the values derived from the experimental isotherms would be
kbcH+ k,kbc& 1+ (ka%b
+ k b h + k.k&
(9)
. =~ ~ ~ K, = 1 0 ~ K, -2.3500 -9.7800 (cH.)o.~ = 10 (cada.s=lo
Data from experimental individual-site isotherms provide values for the concentration at lJ2 saturation, (c~.)o,s and (c&~, for each of the sites. The following relationships may be derived (10)from eqs 2,8,and 9.
Applying eq 12, we get kb =
Table 1. Calculated Data for lndividual8ite Curves with the Following Parameters in eqs 8 and 9
Log ka 10.7799965 6.7799968 7.7000000 IV 6.7800266 V 5.7802983 VI 4.7830980 V H is~ lisied for log cx = -12.00 to -7.35.
1.3500035 3.3500032 4.4300000 5.3499729 6.3496971 7.3468696
-12.000 -11.700 -10.600 -10.000 -9.750 -9.350 -8.750 -7.350
1
and the value 102.35 kb=- 2
-1.000 1.000 2.060 3.000 4.000 5.0W
is recovered. Clearly this is incorrect. The following table shows calculated values for the expected experimental results from the constants, k. = lo7', kb = 104.43,and slab = 102.08.Results are given to twelve significant figures. log Ki log Kz log ( c ~ ~ k . 5
V H is~ listed for log cx = -4.65 to 0.00 Log Cx
(fl+ 102.3500 - 10~12.130-2.3500)
Log a i a b
Log kb
I II 111
(12)
Application of Equation 12
(8)
-
(11)
I
II
111
IV
V
VI
0.0059895 0.011880 0.13146 0.37600 0.51726 0.72911 0.91464 0.99630
0.0059621 0.011826 0.13093 0.37492 0.51611 0.72820 0.91428 0.99628
0.0059895 0.011880 0.13146 0.37600 0.51726 0.72911 0.91464 0.99629
0.0059899 0.011881 0.13147 0.37601 0.51727 0.72910 0.91461 0.99626
0.0059936 0.011888 0.13153 0.37611 0.51733 0.72905 0.91438 0.99593
o.0060320 0.011963 0.13221 0.37715 0.51806 0.72857 0.91214 0.99270
log (CH~)O.S
'
9.78000193992 2.34999806008 -9.77999807624 -2.35000192319
Six significant figures in the experimental ~ ~ ~the l kb= ~ 102'049, ~ andy seveni significant figures yield the value kb=104.M5. Consequently, seven significant figures in the experimental determination are required to give meaningful results, me &f1culty in using individual-site data from the isotherms of glycine derives from the large value of the ratio k.al.dkb. This value is 105-35.
Application of Equations 8 and 9 Simultaneous use of eqs 8 and 9, in fitting the two individual-site isotherms, also fails to give reliable results. Alarge range of values will give satisfactoryfits to the data with the product k,alab = An extensive set of calculations is given in Table 1. 0.000
0.99555
0.99555
0.99555
0.99555
0.99555
0.99552
Volume 71 Number 8 August 1994
639
~
~
Evaluation of ligand-dependent interactions requires explicit assumptions regarding the nature of the interactious and some means of defining the system under conditions in which no lieand-denendent interactions occur. In the case of glycine treated & this paper these two assumptions are the following. T h e sole source OF the logand-drpmdent mternetmns is the eleetrovtatrc attractmn or repulsmn between the mnrzed amino and carboxyl gmups. The system with no ligand-dependentinteractions is defined by the methyl ester of glydne. These assumptions provide a reasonable basis for describing the binding system far glycine.
The interaction between the occupied amino and the unokupied carboxyl is the predominak ligand-dependent interaction in glycine. Acombination of the as.ymmetry in the occupied-u&ccupied interactions and the large ratio k,alablkb thwarted the use of individual-site isotherms in the analysis for glycine. However, when the occupied-unoccupied interactions are equal (symmetrical) and when the ratio k.lkb is not too large, analysis of individual-site isotherms may provide reasonable values for the intrinsic constants and ligand-dependent interactions (10). Appendix-General Equations for Multiple Sies Macroscopic Constants, Experimental Constants, Apparent Constants, Intrinsic Constants, and lndependent Sites
Macroscopic Constants Let us assume one of the following.
-
The binding of a ligand X to the multiple sites of a molecule P may be represented as a series of stepwise reactions. The value of each constant does not change with the extent of binding.
Then the binding of X may be represented by the followingseriesofreach&s in which thespecies in square brackets represent their activities in the equilibrium mixture.
where ox is the concentration offree ligandX. Equations 13 snd 14 are general. The reactions may occur with or without interactions between sites that may or may not be identical. Independent Sites and the Intrinsic Constant If no ligaud-dependent interactious occur on binding then we classify the sites as independent. This statement does not imply the absence of interactions between the sites. We assume that the interactions that do occur between the sites remain unchanged on binding of ligand. Also, this statement does not imply that the binding site or ligand remain unchanged. All the changes that do occur are restricted to the individual site and its ligand. Under these conditions the binding constant is defined as the intrinsic or unperturbed constant. If a further restriction is imposed, requiring that all sites he identical as well as independent, then the relationship between the set ofK; values and the intrinsic or unperturbed constant k is
Under these conditions only combinatorial terms in addition to the value of the intrinsic constant determine the macroscopic or experimental constant. This general equation will be derived below for two sites. When the unliganded molecule Pis designated as a reference state, its relative concentration consequently becomes unity, and these eqs 13-15 lead to the followingcommonly used binding polynomial (also known as the grand partition function or sum of the relative concentrations of the species).
Two Identical and Independent Sites
The following represents the stepwise reaction of a lieand X bindine to a molecule with two inde~endentand identical sites ihown schematicallyas UU. A
[P&I = IPIKIK&. ..K,[XI" (13)
The various K,'s are the macroscopic constants (also characterized as experimental, apparent, or Adair constants (11).If an additional assumntion is made that the activity coefficients are unlty, then i h number ~ of moles of lieand X bound to the total number of moles of the bindine molecule P i s
-
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Journal of Chemical Education
In the reactions above each of the schematic symbols represents the concentration of that species in the equilibrium mixture. The symbol P of eq 13 above is represented by UU. PXl of eq 13 equals the sum of the two microspecies XU and UX, and PX2 is represented by XX. The macroscopic constants K l and K2 may be evaluated as follows.
and
These values of K;could be derived fmm eq 15. The total concentration of the molecule UU in all of its forms is
The number of moles of ligand X bound to the total number of moles ofthe binding molecule is, according to eq 14,
The fractional binding is
The first part of eq 25 may be obtained from eq 22 by clearingfractions. The properties of eq 25 are illustrated in Figure 2 with the values k.= 108and kb = lo6 M-'. Two Sites with Ligand-Dependent Interactions
In this section we explore the binding isotherms under conditions in which the sites are no longer independent as defined above. Ligand binding to a given site is now considered to perturb a neighboring site. Again, we arbitrarily choose the unliganded species UU as the reference state. Because the two sites exist on the same molecule.. they. interact in a myriad of ways that we need not define. It i s sufficient to identifv these interactions in the reference state with the factor,-&, and a free energy of interaction equal to -RT in &. We are interested in the change of this free energy upon the binding of ligand X. These changes are defmed as the ligand-dependent free energies of interaction. Identical and Symmetrical Sites
The same reasoning used in deriving eq 21 will demonstrate that this equation applies to any number of identical and independent sites. Equation 21 is called by many names, usually the Langmuir isotherm and, in logarithmic form, the Henderson-Hasselbalch equation.
The following reaction illustrates only the interactions occurring between two identical and symmetrical sites on binding of a ligand X.
Two Different Independent Sites
When the ligand X binds to two different and independent sites a and b, the binding isotherm may be formulated, using eq 21, as follows.
The following schematic representation of the reaction is written in a form that emphasizes the macmscopic constants K, and Kz.
The interaction in the reference species UU with its two unoccupied sites is identified as the unoccupied-unoccupied interaction. Both of the species, XU and UX, have one occupied site and one unoccupied site. The free energy of the unoccupied-occupied interaction is -RT In a;. The interaction of the species XX is identified as the occupied-occupied interaction with free energy of interaction -RT In 4. Changes in free energy of interaction on binding the ligand X are (-RTln a;)- (-RT in 6)
This schematic representation differs only symbolically from that in eq 17. The expressions below each of the species quantify the concentrations of each ofthe species relative to that of the unliganded species. The sum of the relative concentrations of the species (grand partition function) is
with
The free energies of interaction and binding are combined in Table 2 to give free energies relative to the reference state. These free energy terms yield the statistical weights or relative concentrations of the four species from which the following grand partition function and binding equation are derived. The ligand-dependent interaction factor is
and K,=k.+kb
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641
Table 2. Free Energies and Relative Concentrations for =I 25
State Free Energy Free Energy Relative Relative of of Binding Free Energy Concentration lnteracction (Statistical Weight) UU XU
-RTln& -RTln ai
0 -RTln k
0 kai
-RT 1 x 1 7
-RTlnai
-~~lnk'
-RTln a5
-RTln k2
m 0.75 3 ?
-2.. C
kZai k a m -RT lnkai -RTln7
o
.-0
0.25
U.
ah
XX
-c
8
0
ao
-10
/
/'/
(a+b
k=10 /
,/ k = 1 0
/ - ,
-, /
k2azc.$
.--
1 L 8
/
*m 0.5 -
1 ka~cx
uo
UX
1
/
-9
-7
-8
-6
-5
-4
Figure 2. Binding isothermsfortwosites.The solid line represents the isotherm or the combined sites a and b generated with eqs 25,28,31 and 34. See Table 3 for values of the constants. Dashed lines iilustrate the unperturbed binding isotherms for the independent sites of eq 25.
with 2ka1cx + 2k2a& Klcx + ~K~K.& vx = 1+ 2kal% + k2a& = 1+Klcx + K~K~C; (28)
Ligand-dependent 6ee energies of interaction are thus -RT in a,. Two requirements for evaluating these important (and elusive) quantities are designation of a reference state and estimation of the intrinsic constant k. The binding isotherm of Figure 2, which was generated by two different and independent sites, may also be generated by aninfinity of two identical and interacting sites. Table 3 lists a small number of values fork, a l , and Q that fit the binding curve of Figure 2. Calculations for fmding these values are the following.
Table 3. A Sample of Constants for Fitting with eqs 25,28,31, and 34, the Binding Isotherm (a + b) of Figure 2
Sites Different, independent
Equation
(28)
Different,interacting, symmetrical
(31)
108.W432 al=-- 5.05 From KIKz = k2%,
IQ
lo8
1o6
1o7 lo6 lo5
10'
a2
-
-
(34)
lo9 lo9
lo5
5.05
o6 1o5 1o7 1o3
50.5
1
1o6
505.0 0.1 0.101
1.O 100.0 1o4 0.01 100.0
alab
alba
a2
100
91
lo3
Equation 29 is characterized ivith the following grand partition function and binding equations. = 1+ k p p x + kbalex + k,kba&
1014 %=-10' x 10' - LO
a1
(25)
Identical, interacting, symmetrical
1. Far the binding isotherm of Figure 2, k. = lo8 Different,interand kb = lo6 to give K, 108.W432,~d acting, asymmetrical ",v "2 v - ln14 I" 2. Take k = lo7 for usein eq 28. From K1 = 2kal,
k
2
= 1+ Klcx + KlKzex
(30)
and KI = (k, + kb)al
Similarly, when k = lo6, al = 50.5 and Q = 100.0. When k = lo5, a, = 505.0 and uz = lo4.
KIKZ=k&%
Different Sites with Symmetrical Interactions
Two different sites, a and b, with symmetrical ligand-dependent interactions may be described with the following reaction. ab XU fk.a~cxI
Because there are two different intrinsic constants and two ligand-dependent interactions, four parameters are required to describe the system. However, there are still only two experimental parameters, Kl and K 2 Again, an infinity of values of k., kb, a], and Q will fit the data of Figure 2 generated by two different and independent sites. A sample calculation for eq 29 follows. Take k, = lo9 and % = 1iT2foruse in eq 31. Then fromKlK2= k$2bazr
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Journal of Chemical Education
FromKl = (k, + kb)al,
Similarly, when k. = kb = lo3.
lo9 and @ =
symbol (and its microscopic constant) between the species UX and XX is redundant.
10'; al = 0.101 and
Different Sites with Asymmetrical Unoccupied-Occupied lntemctions Different sites that are asvmmetrical in their unoceupied-xcupied interactions require a distinction between the lig-and-dependentinteractions for the species
ab
UX
Equation 29 above is modified to include the factors al* and alb.in eq 32 below.
IkbalbaCX) (35) This brings us to the definition of the microscopic constant. Microscopic constants may be defined as those constants that describe the equilibrium between two individually specified (microscopic) species. In the reaction below the equilibrium between the species ab ab UU and XU
is described with the microscopic constant K., etc. ab
(32) Equation 32 is the general equation for describing the binding of protons to glycine. This reaction has already been described above. The grand partition function and binding equations now become (kbalbacxl
The four microscopic constants are thus evaluated as follows in terms of the intrinsic wnstants and the ligand-dependent interaction factors.
This asymmetry introduces an added parameter in the values for fitting the isotherm of F i r e 2. (This isotherm has two independent sites with k. = 10' and k b = 10%-'.) Some of these values are included in Table 3. Asample calculation follows. Take k, = lo5, kb = lo6, and slab = lo2 for use in eq 34. Then from Kl = k,ahb + kbalb,,
.. ab
ad'
The system characterized in eq 36 is wnsidered to be in equilibrium. Therefore any one of these four microscopic constants may be considered redundant. Three microscopic constants sre sufficient to describe the equilibriium. Thus, & = (KSJKb. It is important to notice that when the sites are independent (noninteractmg, all a, = unity) the micmscopic and intrinsic constants are identical. Literature Cited
Microscopic Constants Equation 32 above, and previous reactions, were written in a form to emphasize the macroswpic constants, Ki and K2. Equation 35 below is the equivalent of eq 32. The system is wnsidered to be in equilibrium. Consequently, the species will adjust to their proper relative concentrations regardless of the path. Introducing an added equilibrium
5. Asher, S. A.:Adams, M. L.:Sehusta,T M. Biochemislry 1881,20,33393346. 6. Bradbury. J.H.;Scheraga,H.A. J A m r Chcrn. Soc. 1966,88,42404246. 7 Benesch, R. E.; Beneseh, R. J.h e r Chem. Soc. 1956.77, S877-5881. 8. Galas.D.;Schmitz,A.NudAci&RDs.1978,5,31573170.
9. Horsley, W J.;Stemlkht, H.J A m r Chrn. Soc. 1988,90,313&3148.
10. SamK, H.A. J. Theor Bid. 1987,129,427438. 11. Ada", G.S. J Bid. C h m . 1826,63,529-545.
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