J. theor. Biol. (1976) 58, 253-268

An Analysis of the Binding of Related Ligands H. P. RAPPAPORT

Department of Biology, Temple University, Philadelphia, Pennsylvania, 19122, U.S.A. (Received 1 February 1975) The thermodynamic relationship between the binding of the ligand, A-B, and analogs of its groups, A-X and B-Y, to a protein is derived using a thermodynamic cycle. Using the formalism, thermodynamic data, and simple models the following conclusions are reached: (1) A significant free energy barrier exists against a ligand and a protein coming together. The free energy barrier is essential for understanding the specificity of ligand binding. (2) In general the free energy of binding A-B is not equal to the sum of the free energies of binding A-H and B-H when A and B contribute independently to the binding of A-B. (3) It is physically possible for a methyl group to make a contribution greater than 2.3 kcal/mol to the binding of a ligand without an autosteric mechanism. (4) It is physically possible for tyrosine to bind at least 3.9 kcal/mol more strongly than phenylalanine if the hydroxyl group of tyrosine displaces a bound water molecule. (5) The co-operative interactions due to the joint presence of the protonated amino and carboxylate groups in the zwitter ion of an amino acid can make significant contributions tb binding.

1. Introduction The remarkable specificity displayed by proteins for the binding of ligands has been recognized for many years. F r o m a biological point of view the specificity may be considered a requirement placed on the design o f the chemical machinery of the cell in order that indiscriminate inhibition of the orderly chemical events o f the cell will not occur. The chemical severity o f the requirement o f specificity appears extreme because there are only a limited number of chemical groups used to make up biological molecules. Fischer (1894) recognized that steric interference provides a method for the discrimination against molecules that are larger than the normal ligand. F o r example, the bulk of the methyl group of alanine prevents the correct close T.B.

253

17

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H.

P. RAPPAPORT

fit of alanine in a glycine binding site. The complementary question is: What principle is used to discriminate against the binding of a smaller moiecule (glycine) in the site of a larger molecule (alanine) ? In the context of enzymic catalysis, Koshland & Neet (1968) have suggested the principle of an autosteric mechanism. The principle is that a part of a ligand induces a conformational change in the enzyme such that a stronger interaction between another part of the ligand and the enzyme occurs. A related problem arises when we attempt to understand the binding of a ligand in terms of the binding of its groups. What is the relationship between the binding of A-H, B-H and A-B to a protein ? Some investigators have assumed that if the free energy of association of A-B is equal to the sum of the free energies of association of A - H and B - H then the groups contribute essentially independently to the binding of A-B (Holier, Rarney, Orme, Bennett & Calvin, 1973). In order to understand how smaller molecules are prevented from binding to the sites of larger molecules and to be able to decide when the binding of a molecule has been explained in terms of the binding of its groups a rigorous thermodynamic formulation is required. Mulivor & Rappaport (1973) derived an approximate thermodynamic expression and used very rough approximations to interpret their data. In this paper the relationship between the binding of A-X, B-Y and A-B is obtained by using a thermodynamic cycle and the important free energy terms in the cycle are estimated. The analysis shows that a major factor for the understanding of the stronger binding of a normal ligand compared to a smaller analog is the rotational and translational free energy barrier in aqueous solutions for the binding reaction. The analysis further shows that the free energy of association of A-B in general is not the sum of the free energies of association of A - H and B-H when the contributions to the binding of A-B by A and B are essentially independent of each other.

2. Formalism and Interpretation The relationship between the binding of the ligand A-B and its groups A and B to a protein M can be obtained by a series of steps introduced by Mulivor & Rappaport (1973). They are: (1) A-B is broken into A and B in solution; (2) A binds to M; (3) B binds to M(A); (4) A and B are joined together on M to give M(A-B). The four steps are equivalent to the binding of A-B to M. The experimental situation differs from this scheme because stable molecules are used instead of the radicals A and B. The stable analogs of A and B will be A - X and B-Y, where X and Y are atoms or groups. The physical characteristics of X and Y may be dictated by solubility

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requirements of the experiment or stereochemical considerations. To incorporate the use of A - X and B-Y instead of A and B into the scheme an enlarged series of steps is required. The thermodynamic cycle is: (1) (2) (3) (4) (5) (6) (7)

M(A-B) ~ A - B + M A-B + X + Y ~ A-X + B-Y M + A-X -~ M(A-X) M + B-Y --+ M(B-Y) M(A-X) --+ M(A) + X M(B-Y) -~ M(B)+Y M(B) -- M + B

(8) M(A)+B ~ M(A, B) (9) M(A, B) ~ M(A-B)

-At(A-B) AFd(A-B) - F(X) - F(Y) At(A-X) At(B-Y) AFt(A-X) +F(X) AF~(B-Y)+F(Y) -AF(B) AF(B)+AF(A, B) AFo(A, B).

F(X) and F(Y) are the partial molal free energies of the X and Y groups and At(A, B) is the change in the free energy of association of B with M when A is present on the protein. At(A, B) includes the contribution due to the autosteric effect. Adding the free energies At(A-B) = At(A-X) + At(B-Y) + AFa(A-B) + AFt(A-X) +AFc03-Y)+AF(A, B)+AFo(A, B).

(1)

It is permissible in the temperature range of biological interest to separate each free energy change into two parts: (1) The free energy change due to the electronic and vibrational variables of the molecules in the equivalent gas phase reaction, At(e, v). (2) The excess free energy change over At(e, v), AF(ex). In many cases the excess free energy change will be equal to the free energy change due to the rotational and translational variables of the solutes and solvent. Since equation (1) is derived from a cycle, it is satisfied by either class of free energy changes. It is convenient to write equation (1) as: At(A-B) = At(A-X) + At(B-Y) + AF(ex) + At(e, v)

(2)

where AF(ex) and At(e, v) are the free energy changes for the sum AFd(A-B) + AFt(A-X) + AFt(B-Y) + At(A, B) + AFa(A, a). The excess free energy change of AFd(A-B) has a simple and interesting interpretation under certain conditions. We assume that the molecular variables (electronic, vibrational, rotational and translational) that specify the excess free energy can be separated into an internal and an external group. The internal group specifies the microscopic internal states of the protein and the complex. The external group specifies the microscopic states

256

H. P. R A P P A P O R T

of the protein and the complex with respect to the solvent. The latter includes the center of mass translational variables and when appropriate the over all rotational variables. Writing the part of AF(A-B), AF(A-X) and AF(B-Y) due to the excess free energy in the form AFi(ex)+ AFe(ex): AFe(A-B; ex) = F,(M(A-B); e x ) - F e ( M ; e x ) - F ( A - B ; ex),

(3)

AF,(A-B; ex) = Fi(M(A-B); e x ) - F t ( M ; ex)

(4)

and assuming that AF(A, B; ex), AFt(A-X; ex), AFt(B-Y; ex) and AFa(A, B; ex) depend only on the internal variables AFe(A-B; ex) = AFt(A-X; ex)+AF,(B-Y; ex)+AF.,(A-B; ex).

(5)

Since AFd(A-B; ex) is negative (see section 3), it represents the saving in the external excess free energy of binding when the groups A and B are bound as one molecule, A-B, instead of two molecules, A - X and B-Y. Some investigators have asserted that if the standard free energies based on a 1 molar standard state satisfy the relationship AF(A-B) = AF(A-H) + AF(B-H) then the groups A and B contribute independently to the binding o f A-B (Holler et aL, 1973). Several definitions of independence will be investigated to decide whether the assertion is supported. If the definition of independence is that AFo(ex) and AF(e, o) are equal to zero, then equation (2) shows the relationship is not obtained. The definition in principle is really inappropriate because the dependence of AFo(ex) on the mass and moments of inertia of A-B cannot be written as the sum of functions depending only on the mass and moments of inertia of A and B. To avoid this difficulty the definition can be taken to be either that the configurational integral part of AF,(A-B; ex) is equal to the coniigurational integral part of AFt(A; ex)+ AFt(B; ex) or that the average force exerted on A-B in the bound state is equal to the sum of the average forces exerted on A and B in their respective complexes. For the case where no change in the chemical composition of the ligands occurs on binding, the part of AFo(ex)+ AF~(A-H; ex) + AF,(B-H; ex) that depends on the mass and moments of inertia of the ligands is cancelled by the equivalent part of AFa(ex). However, with both definitions the configurational integral part of AF(ex), equation (2), is not zero. There does not appear to be a simple definition of independence that leads to the proposed relationship. Step (2) may involve the separation of net charges, and then steps (8) and (9) will result in the juxtaposition of the charges on the protein. It is informative in these cases to obtain the contribution to binding made by the joint presence of the groups carrying the charges. For concreteness the

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calculation is performed for the zwitter ion form of an amino acid. The free energy due to the joint presence of the protonated amino group and the carboxylate group on NH+-CRH-COO -, (A+-B-), is defined as the difference between the free energy changes for the two reactions: (10) A+-B --, A+-B - + H + (11) A-B --, A-B- + H +. The same definition is used for the free energy due to the joint presence of the groups on the protein-ligand complex. The net change in free energy due to the joint presence of the groups, AF(ch), is

AF(ch)= AF(A+-B-)-AF(A-B-)+AF(A-B)-AF(A+-B)

(6) where the AF's are the free energies of association of the ligands with the protein. If AF(ch) is removed from AFa(A+-B-)+ AF(A+, B-) + AFo(A+, B-), then from the definitions the sum can be replaced by AFa(A-B -) + AF(A, B -) + AFa(A, B-) + AF(exch). AF(exch) is the sum of the free energies of the two reactions of proton exchange: (12) A + - B + A - X ~ A - B + A + - X (13) M(A-B) + M(A ÷) ~ M(A+-B) + M(A). 3. Evaluation of AF d

(ex)

The method for the calculation of the excess free energy change for the reactions of category 1, Table 1, was: (1) The rotational and translational free energy of a molecule Was computed from canonical partition functions for an ideal gas at a concentration of 1 molar, 298 K. Rotational constants were taken from Herzberg (1966) or the moments of inertia were computed from crystallographic results (Pearson, 1953-1964). The free energies of internal rotation, column 2, were computed using the tabulation of the potential barriers by Lowe (1968) and the tables of Pitzer & Brewer (1961). (2) The free energies of transfer from the gas phase to the aqueous phase were obtained from Wauchope & Hague (1972), Nelson & de Ligny (1968) and Arnett et al. (1972). The standard state for the solutes is 1 molal, 298K, 1 atmosphere. The uncertainty is +0.5 kcal/mol. The values listed for the category 2 reactions are upper bounds. The method was to compute the total free energy change and to subtract a lower bound for the free energy change due to the electronic and vibrational variables in the gas phase. The general procedure was: (1) The free energies

258

~. P. RAPPAPORT TABLE 1

ExCess free energy change over the gas phase electronic and vibrational free energy change. The standard states are: gas phase 1 molar, 298 K, ideal gas; aqueous phase 1 molal, 298 K, and water as the pure liquid. Free energies due to rotational motion for the gas phase used rotational constants from Herzberg (1966) or moments of inertia calculated from crystallographic data (Pearson, 1953-1964). The free energies of internal rotation were calculated from the tabulation of potential energy functions by Lowe (1968) and the tables of Pitzer & Brewer (1961). The free energies of transfer from the gas to the aqueous phase were obtained from Wauchope & Hague (1972), Nelson & de Ligny (1968) and Arnett et al. (1972). The free energies for categories (2) and (3) are upper bounds. An example of the detailed calculations is presented in the text Difference AF of b e t w e e n AFn(ex), internal AF(ex)for aqueous gas phase rotation aqueous and phase (kcal/mol) 0ccal/mol) gas phase (kcal/mol) (kcal/rnol)

AFa(ex),

(1) C~H6-~ 2CH4 CHaNH2 ~ CH4 q- NHa CeHsCHa -+ C6H6 + CH, CHaCOOH -~ HCOOH q- CH4 CHaNHa+ "-~ CH4 q- NH4 + CHaCH2NHa+ -~ CHa -~ CH3NHa+ (2) ala ~ gly + CH4 ilu ~ val q- CH~ tyr -+ pile q- H20 phe ~ gly d- CTHa (3) phe ~ gly -k CTHTOH

--6'4 --6"6 --6"8 --6"6 --6"8 --7"0

0"18 0"22 0"38 0"66 0"20 0"4

2"2 2"2 1"8 1"5 --5"4 --1 "4

--4"2 --4"4 --5"0 --5"1 --12"2 --8"4 --6"7 --6"6 --8"9 --13"1 --21

o f formation were used to calculate the total free energy change for the aqueous reaction H 2 + A - B ~ A - H + B - H . The standard state of the solutes is 1 molal, 298 K, 1 atmosphere. The standard state of water is the pure liquid, 298 K, 1 atmosphere. (2) The electronic and vibrational enthalpy change, AH(C-D), was computed for the gas phase reaction H 2 + C - D ~ C - H + D - H . The compound C - D was selected so that its broken bond represented the broken bond of A-B. The actual choice insured that A H ( A - B ) / > AH(C-D). The total enthalpy change is the sum of the enthalpy changes due to electronic and vibrational variables, internal rotation variables, and the over-all rotational and translational variables. The enthalpy change due to the over-all rotational and translational variables

BXNDING OF RELATED LmANDS

259

is 0"3 kcal/mol. The enthalpy change for the loss of an internal rotation will not exceed - 0 . 6 kcal/mol. Because the sum does not exceed - 0 . 3 kcal/mol, the total enthalpy change was not corrected. (3) For the calculation of AF(e, v), AS(e, v) is required. The ~ompounds considered are in their electronic ground states at 298 K, so AS(e) is zero. Spectral data is not in general available to calculate AS(v). The frequencies of the vibrations of the hydrogen nuclei added in the reaction, > 1000 cm -1, are so high that they make an insignificant contribution to AS(v). The contribution to the carbon skeleton vibrations of the bond between A and B may not be insignificant, and hence AS(v) can be negative. Thus

AH(e, v; C-D) < AH(e, v; A-B) < AF(e, v; A-B). (4) The excess free energy change, AF(ex), is

A F - AF(e, v) < A F - AH(e, v; C-D). (5) The standard excess molal free energy of H2 in water was used to obtain the upper bound for AFa(ex). The details of the calculation for ala ~ gly + CH# are presented below: H2

4.2

-b ala --, gly + CH4 --89.13 -90.15 -8.25 AFf~9s(1M, aq) AF ° = -- 13"47 kcal/mol.

(7)

The free energies of formation of crystalline alanine, crystalline glycine and gaseous methane were obtained from Greenstein & Wintz (1961) and Stull, Westrum & Sinke (1969). The free energies of transfer to 1 molal, aqueous phase, zwitter ion form, were taken from Greenstein & Winitz (1961). The free energies of transfer of hydrbgen and methane were taken from Wauchope & Hague (1972). The reaction for the lower bound of the enthalpy change was the gas phase reaction H2 + CH3CH2NH~- --* CH4 + CH3NH~,

AH = -- 9-9 kcal/mol.

(8)

The enthalpy change was computed from the gas phase reactions (i) CH3CH2NH~ --* H + + CH3CH2NH2

11"8 kcal/mol

(ii) H2 + CH3CH2NH2 ~ CH4 +CH3NH2

-12-39 kcal/mol

(iii) CH3NH2 + H + ~ CH3NH~

- 9 . 3 kcal/mol

The enthalpy changes for reactions (i) and (iii) are taken from Arnett et aL (1972), using NH,~ ~ N H 3 + H + as the base of comparison. The enthalpy change for reaction (ii) was computed from the data of Stull, Westrum &

260

H.P.

RAPPAPORT

Sinke (1969). Because gas phase enthalpies of ionization for carboxylic acids have not been reported, an indirect estimate of the effect of the carboxylate ion was necessary. The following two reactions show that the presence of a carboxyl group will make the enthalpy change less negative: (iv) H2 +CH3CH3 --' 2CH, (v) H2 + CHaCH2COOH -* CH4 + CHaCOOH

- 15.54 kcal/mol - 13-1 kcal/mol

The removal of the proton from the carboxyl group to give the carboxylate ion would be expected to make the enthalpy change for reaction (v) less negative. The only available gas phase reactions that are at all analogous are for the ammonium ion. (vi) H2 + CH3CH2CH2NH~ -~ C H , + CH3CH2NH~- - 12-79 kcaI/mol (vii) H2 + CH3CH2CH2NH2 ~ CH4 + CHaCH2CH2NH2 - 11.59 kcal/mol The computed enthalpy change of reaction (vi) used the data of Aue, Webb & Bowers (1972). The comparison of the enthalpy changes for the reactions (iv) to (vii) indicates that the enthalpy change of reaction (7) is a lower bound. From the rotational and translational free energy of hydrogen gas and the free energy of transfer to the aqueous phase, the standard state excess partial molal free energy of hydrogen in water is - 3 . 1 2 kcal/mol. Thus

AFd(ex) < AFf;9a-AH(e, v)+F(H2, 1 M) =-13"47+9.9-3.12=-6.69.

(9)

The enthalpy changes for the other reactions in their order of appearance in Table 1 were estimated from: H2 + 3-methyl pentane ~ pentane + methane; H2 + p-cresol ~ toluene + water; H2 + propylbenzene -. toluene + ethane + 5.6. The 5.6 kcal/mol is an estimate of the effect of the NH~ group. It is obtained from the enthalpy differences between the reactions H2+C2H6 ~ 2CH 4 and H 2 + C H a C H 2 N H ~" ~ C H 4 + C H a N H ~ . In the case of the p-cresol reaction, which does not involve the breaking of a carbon carbon bond, AS(v) is - 5 . 5 e.u. The uncertainty in the calculated values is about 4-1.5 kcal/mol. The enthalpy calculation for category (3) used the gas phase reaction H 2 0 + CHaCHxNH~j ~ CH3OH + CHaNH~. The values of AF~(ex) for the small neutral molecules in category (1) are close to the values predicted by Koshland's lattice model (Koshland, 1962; Mulivor & Rappaport, 1973). When a charge is introduced into a group that is next to the broken bond or one bond removed, the AFd(ex) values are substantially larger. The substitution of a hydrogen atom for a larger group allows a greater interaction with water, and in the case of NH~ more

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hydrogen bonding with the water. A comparison of the last reaction in category (2) and the reaction of category 3 illustrates the very significant changes that occur when analogs differ by a group that interacts strongly with water.

4. Evaluation of AF. (ex) Detailed data is not available for the calculation of AF,(ex) for specific cases. A variety of changes may accompany the process M(A, B) ~ M(A-B). The variations could range from a process having no significant conformational change of the protein, no change of the interaction with the solvent, and no change of the average positions of the ligands on the protein, to a process with significant changes in all the categories. Because of the possible variations, it is appropriate to define an "ideal process" so that deviations from it may be recognized. The ideal process is defined to have the following characteristics: (1) No conformational change of the protein occurs. (2) No change of the interaction of the protein ligand complex with the solvent occurs. (3) The average positions of A and B on the protein do not change. (4) The rotation about the bond joining A and B is not constrained to the top of the rotational potential energy curve, resulting in a large positive free energy of rotation. The only change in an ideal process is the conversion of 12 degrees of rotational and translational freedom, six for each ligand, into seven degrees of freedom for the final ligand, one for internal rotation and 6 for rotation and translation. Since the free energy associated with the internal rotation is small, it will be neglected. The simplest model to represent the jiggling motions of the ligands on the protein is harmonic oscillators with frequencies found for vibrations in the solid and liquid states due to intermolecular motions. Frequencies corresponding to the experimental Debye temperatures of crystals provide an estimate for the frequency that a single molecule would have if an equivalent interaction occurred with a massive object. Organic crystals have Debye temperatures between 100 and 150 K (Westrum & McCullough, 1963), which corresponds to a frequency range of 66 cm - t to 100 cm -1. Benzene for example has a Debye temperature of 150 K for both rotational and translational motion (Lord, Ahlberg & Andrew, 1937). Herzberg & Litovitz (1959) have calculated the average collision times of simple organic molecules in the liquid state from the velocities of ultrasound. The corresponding frequencies of vibration are equal to or greater than 200 cm-1. Infra red and Raman data from ice and liquid water show frequencies due to intermolecular motions from 60 to 800 cm - t (Ockman, 1958). The peaks at 229 cm-1 and 800 cm-1 may be due to single molecule hindered translation

262

H.P.

RAPPAPORT

and hindered rotation. The frequency of vibration of A-B on M for a particular degree of freedom will lie between the equivalent frequencies of vibration of A and B for an ideal process. An upper bound for AFo(ex) is established by calculating the free energy change due to the loss o f the six degrees of freedom with the lowest frequencies. The free energy changes for the loss of six degrees of freedom with frequencies of 66 cm- 1, 100 cm- 1 and 200 cm-1 are 4-3 kcaI/moI, 2-7 kcal/mol and 0-02 kcal/mol. 5. AFc(A-X) and AFt(B-Y) It is permissible to imagine that sufficient energy is added in these processes to the electronic and vibrational degrees of freedom so that the stereochemistry of A and B is not changed by the removal of X and Y. The energy will be recovered when A-B is formed and no change in the overall AF(e, v) will occur. 6. Analysis The point of view developed in the previous sections will be used to-analyze the binding of selected amino acids to amino acyl-tRNA synthetases. The binding of amino acids has been chosen because experimental data is available for the binding of smaller analogs and because bounds for the required AFd(ex)'s can be calculated. Although considerable experimental data is available for the binding of other ligands, the required free energies of formation are unknown. Except for the calculations of AF(ch), the following conditions are assumed: (1) An ideal process takes place on the protein for the reaction M(A, B) --, M(A-B). (2) AF(e, v) is zero. (3) AF(A, B) is zero. Condition (1) limits the conclusions to one physical possibility. Condition (2) is reasonable because the bonds that are broken in the analysis are not involved in the assumed chemical mechanism of the reaction during catalysis. Condition (3) is required when no autosteric mechanism operates. Santi & Danenberg (1971) and Mulivor & Rappaport (1973) have measured the binding of phenylalanine and smaller analogs to phenylalanyl-tRNA synthetases. Both groups found that the removal of the protonated amino group resulted in a very large loss of binding. The standard free energies of binding phenylalanine, hydrocinnamic acid, benzyI alcohol and glycine were - 7 kcal/mol, - 3 . 5 kcal/mol, - 4 . 5 kcal/mol and > 0 kcal/mol (Mulivor & Rappaport, 1973). Since the removal of the glycine group has a substantial effect on binding, why doesn't glycine have a measurable binding? Can we conclude from the results that an autosteric mechanism i~ required? The difference between the binding of phenylalanine and

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263

benzyl alcohol, --2-5 kcal/mol, attributable to the glycine group, is not the free energy of association of glycine. It is the free energy contribution of glycine already present on the protein. To calculate the free energy of association, the free energy of bringing glycine and the protein together must be taken into account. Using equation (2) with the bound for AFa(ex) from Table 1, category (3), AF(gly) > 21 - AFa(ex) - AFc(ex; bz-OH) - AFc(ex; gly-H).

(10)

Even if AFo(ex) is assigned our largest estimate, 4-3 kcal/mol, and the removal of the hydroxyl group from benzyl alcohol resulted in the loss of two hydrogen bonds, AFc(ex; bz-OH) equal to 10 kcal/mol, the free energy of the binding of glycine would be > 6.7 kcal/mol. The result demonstrates the importance of the free energy barrier in the binding reaction. Although there is a significant interaction between glycine and the enzyme, it is not sufficient to overcome the free energy barrier of coming together. Holler, Rarney, Orme, Bennett & Calvin (1973) have determined the standard free energy of association of isoleucine ( - 6 . 9 kcal/mol) and valine ( - 4 - 7 kcal/mol) with an isoleucyl-RNA synthetase. The difference between the standard free energy of transfer from water to an organic solvent of small hydrocarbons differing by one methyl group is not greater than - 1 . 2 kcal/mol (Kauzmann, 1959). Since the difference in the binding of isoleucine and valine is - 2 . 2 kcal/mol it might seem reasonable to conclude that an autosteric mechanism is required. However, the reasoning is mis1ending. The transfer of a molecule into a liquid can be viewed as a two-step process. First, a cavity is formed in the liquid for the molecule which requires energy. Second, the molecule is placed into the cavity and .;nteracts with the surrounding molecules. If the cavity already existed, as some kind of incipient cavity probably does in the enzyme, then the free energy of transfer would be more negative. In addition, the density of atoms may be higher in the protein than the liquid resulting in stronger dispersion forces. Wishnia (1969) has measured the free energy of association of butane with 13-1actoglobin and hemoglobin. Converting his results to the molal concentration scale, the free energies are - 4 . 1 kcat/moI and - 3 - 8 kcal/moI. The binding sites are in the interior of the proteins. Water molecules are not displaced in hemoglobin and probably not in 13-1actoglobin (Wishnia, 1969). The following calculation assumes no autosteric mechanism for the binding of butane. To obtain an expression for the free energy of association of methane from Wishnia's data a thermodynamic cycle equivalent to the one in section 2 is used. Butane is converted to four methanes. The methanes bind to the four sites that compose the site of the binding of butane in 13-1actoglobin. AFj(ex) for the aqueous phase reaction C4Hz0 + 4 H ~ 4CH4,

264

H . P . RAPPAPORT

AF = AFd(ex)-4F(H), can be calculated from the free energies of transfer from the gas to the aqueous phase (Wauchope & Hague, 1972), and the rotational and translational free energies in the gas phase (Pitzer, 1937). It is -13.3 kcal/mol. The result of the calculation is that the average free energy of association of methane with the four sites of ~-lactoglobin, AF(CH4), is AF(CH4) = 3"28

AF'(ex) 4

AF~(CH,; ex) 4

(11)

AF'(ex) is the excess free energy change for the reaction M(CHa, C a 2, C H 2, C H 3 ) ~ M(C4Hlo). AF~(CH4; ex) is the change in the free energy when the hydrogen atoms are removed from the methane molecules with the symmetry number change already taken into account. It is positive. Assuming that the binding site of the 5-methyl group of isoleucine is equivalent to the average site in [3-1actoglobin AF'(ex) AF~(CH4; ex) AF(ilu)-AF(val) = 3.28 - + AF~(val, CH3; ex) + 4 4 + AF~(CHa-H; ex) + AFd(val-CH a; ex) + AFc(val-H; ex). (12) AF~(CHa-H; ex) is AFc(CHa-H; ex) corrected for the symmetry number change. AF~(val-H; ex) is essentially zero. For equivalent sites in the two cases AF'~(ex) ~- 3AFo(val, CH3; ex) and AF'(CH4; ex) > 6AF~(CH3-H; ex). The inequality is due to the greater effect on the mass and moments of inertia when two hydrogens are removed from two of the methanes instead of one hydrogen. Taking the bound for AFd(val-CHa; ex) from Table 1, 6.6 kcal/mol, -

aF(ilu) - aF(val)
- 1 . 4 kcal/mol), and 2-methyl-l-butanol ( - 2 . 5 kcal/mol). Using 2-methyl-l-butylamine to represent A+-B, 3-methyl-pentanoic acid to represent A - B - and 2-methyl-l-butanol to represent A-B,

AF(ch) < - 6"9 -- 2.5 + 2-8 + 1.4 = - 5.2 kcal/mol.

(17)

In this example note that the lack of an amino group on 2-methyl-l-butanol which represents (A-B) is compensated by the lack of an amino group on 3-methyl-pentanoic acid which represents A - B - . The lack of a carboxyl group on 2-methyl-l-butylamine, representing A÷-B, is not compensated by its absence on 2-methyl-l-butanol, which has a hydroxyl group. Mulivor & Rappaport (1973) measured the standard free energies of association of phenylalanine ( - 7 kcal/mol), phenylalaninol (-5.3-kcal/mol), hydrocinnamic acid ( - 3 - 5 kcal/mol) and benzyl alcohol ( - 4 . 5 kcal/mol). Using hydrocinnamic acid to represent A - B - , phenylalaninol to represent A+-B and benzyl alcohol to represent A-B, AF(ch) is --2-8 kcal/mol. The choice of the analogs in this case has allowed a compensation for the lack of a carboxyl and an amino group in the analog of A-B, benzyl alcohol. The presence of the hydroxyl group in benzyl alcohol is compensated by its presence in phenylalaninol so far as its contribution to the free energies of the ligands in solution is concerned. In both of these examples the strength of the binding of the respective analogs is A+-B > A-B > A - B - . The order suggests that the protonated amino group of the ligand is attracted to a carboxylate group on the protein and that the carboxylate group of the ligand is repelled by the carboxylate group of the protein. The. autosteric mechanism of this model is: The presence of only the protonated amino group distorts the average position of the carboxylate group in the protein towards it. The presence of only the carboxylate group in the ligand distorts the position of the carboxylate group in the protein away from it, and towards the position the protonated amino group would occupy. When both charges are present in the ligand, the average position of the negative charge of the protein will be even closer to the protonated amino group and even farther away from the carboxylate group of the ligand than when they are present singly. The total value of AF(ch) need not represent an autosteric effect. Part of AF(ch) could be due

BINDING OF RELATED LIGANDS

267

to the increased interaction between the protonated amino group and the carboxylate group when the amino acid is transferred from the aqueous phase to the protein surface. The major conclusions from the calculations are: (1) A significant free energy barrier exists against a ligand and protein coming together and the barrier is important in the specificity of binding. (2) It is physically possible for a methyl group to make a contribution to binding that exceeds - 2 . 3 kcal/mol without an autosteric mechanism. (3) It is physically possible for the free energy of association of tyrosine to be 3.9 kcal/mol lower than phenylalanine if the hydroxyl group of tyrosine displaces a bound water molecule. (4) The joint presence of the protonated amino group and the carboxylate group on an amino acid can make a significant contribution to binding. Finally, I wish to emphasize that some of the conclusions refer to what is physically possible. They do not prove that individual proteins use these methods. The calculations do show that considerable caution should be used before concluding that binding results require an autosteric mechanism. Research supported in part by American Cancer Society Grant p-615.

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