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Free Energy Decomposition of Protein-Protein Interactions

Free Energy Decomposition of Protein-Protein Interactions

Biophysical Journal Volume 81 August 2001 737–750 737 Free Energy Decomposition of Protein-Protein Interactions Sergey Yu. Noskov* and Carmay Lim...

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Biophysical Journal

Volume 81

August 2001



Free Energy Decomposition of Protein-Protein Interactions Sergey Yu. Noskov* and Carmay Lim*† *Institute of Biomedical Sciences, Academia Sinica, 11529 Taipei, and †Department of Chemistry, National Tsing-Hua University, 300 Hsinchu, Taiwan

ABSTRACT A free energy decomposition scheme has been developed and tested on antibody–antigen and protease– inhibitor binding for which accurate experimental structures were available for both free and bound proteins. Using the x-ray coordinates of the free and bound proteins, the absolute binding free energy was computed assuming additivity of three well-defined, physical processes: desolvation of the x-ray structures, isomerization of the x-ray conformation to a nearby local minimum in the gas-phase, and subsequent noncovalent complex formation in the gas phase. This free energy scheme, together with the Generalized Born model for computing the electrostatic solvation free energy, yielded binding free energies in remarkable agreement with experimental data. Two assumptions commonly used in theoretical treatments; viz., the rigid-binding approximation (which assumes no conformational change upon complexation) and the neglect of vdW interactions, were found to yield large errors in the binding free energy. Protein–protein vdW and electrostatic interactions between complementary surfaces over a relatively large area (1400 –1700 Å2) were found to drive antibody–antigen and protease– inhibitor binding.

INTRODUCTION To understand molecular recognition of protein surfaces, it is important to obtain the quantitative contributions of the individual forces governing binding affinity and specificity. Consequently, it is useful to develop reliable methods for computing absolute or relative free energies that can dissect the relative magnitudes of different thermodynamics effects. Many such methods have been developed and they fall generally into three classes. The first class includes free energy simulation methods (Kollman, 1996; Tembe and McCammon, 1984) that treat solute and solvent atoms explicitly with an atomic force-field description. These simulation methods have a rigorous statistical mechanics basis and can be used to compute the difference between the binding free energies of two closely related medium-sized compounds (e.g., ligands differing only in a single residue) in solution fairly accurately (Jorgensen and Tirado-Rives, 1995). However, they are less reliable for ligand–protein or protein–protein association due mainly to the problem of sampling the relevant configurations of the solute and solvent (McCammon, 1998). The second class encompasses empirical approaches that use parameters obtained from a “training set” of known interactions (Andrews et al., 1984; Bohm, 1994; Horton and Lewis, 1992; Weng et al., 1997; Williams et al., 1993). The binding free energy is taken to be a sum of different terms associated with hydrophobic contacts, hydrogen bonds or salt bridges, and conformational entropy loss. The various terms are usually not derived from theoretical backgrounds Received for publication 22 January 2001 and in final form 7 May 2001. Address reprint requests to Carmay Lim, Institute of Biomedical Sciences, Academia Sinica, 11529 Taipei, Taiwan. Tel.: 886-2-2652-3031; Fax: 886-2-2788-7641. E-mail: [email protected] Dr. Noskov’s permanent address is Institute of Solution Chemistry, Russian Academy of Sciences, Akademicheskaya str. 1, 153045 Ivanovo, Russia. © 2001 by the Biophysical Society 0006-3495/01/08/737/14 $2.00

(e.g., statistical thermodynamics or molecular mechanics), but from available binding data in a statistical manner. The key advantage of empirical methods is their speed. However, these phenomenological scoring functions rely critically on the quality of the training database. The third class includes methods that treat the solute atoms explicitly and the solvent as a continuum dielectric medium (Kollman et al., 2000). The electrostatic contribution to the solvation free energy is computed using finite difference Poisson–Boltzmann (PB) (Honig and Nicholls, 1995) methods or the Generalized Born (GB) approximation (Still et al., 1990; Qiu et al., 1997), whereas the nonelectrostatic contribution is treated as an empirical function of the solvent-accessible surface area (SASA) (Still et al., 1990). These methods appear promising for certain systems because they represent a good balance between speed (by avoiding sampling of solvent configurations) and accuracy (by incorporating long-range electrostatics, ionic strength, and polarization effects). Although the computed absolute/relative binding free energies for various systems have been reported to be in close agreement with the respective experimental values, this does not necessarily mean that the underlying theory or scheme for computing the binding free energy is accurate. This is because the agreement may be fortuitous and errors due to experimental issues may mask the errors inherent in computing the various free energy components. For example, in many cases the x-ray structure of the complex, but not its unbound counterparts, is known. In such cases, the binding free energy is often estimated from the atomic coordinates of the complex alone assuming no conformational change upon complexation (rigid-binding approximation) (Novotny et al., 1989; Williams et al., 1991; Horton and Lewis, 1992; Jackson and Sternberg, 1995; Friedman and Honig, 1995; Froloff et al., 1997; Novotny et al., 1997; Olson, 1999). In cases where the atomic coordinates of both free and bound proteins are available, the protein may have


been crystallized at a pH that is significantly different from the pH at which the binding free energy was measured. Furthermore, NMR structures are often solved at low pH. However, the free energy for protein association that is mediated by ionizable residues can be very sensitive to pH (Gibas, 1997; Xavier and Wilson, 1998). Discrepancy between the computed and experimental binding free energies has been attributed to the use of low pH structures rather than shortcomings in the free energy function/calculations, like the neglect of van der Waals (vdW) interactions (Krystek et al., 1993). Partly due to the above issues, the various theoretical studies have not reached a consensus regarding the major force driving the binding process. Pauling concluded that the cooperation of weak vdW and hydrogen-bonding interactions between complementary surfaces over an area sufficiently large to overcome the disrupting influence of thermal agitation govern protein–protein recognition (Pauling, 1974; Pauling and Pressman, 1945). In contrast, Chothia and Janin suggested that vdW and polar interactions contribute little to complex stability, and it is hydrophobicity (the release of interfacial water) that is the major factor stabilizing protein–protein associations (Chothia and Janin, 1975; Kauzmann, 1959). This view is also implicitly assumed in studies that fit the experimental binding free energy to buried surface areas (Horton and Lewis, 1992). Honig and coworkers and other groups found that nonpolar interactions, represented by a free energy–surface area relationship that is assumed to account for the hydrophobic effect and enhanced interfacial packing, provide the major driving force for MHC class I protein–peptide, trypsininhibitor and ␭cl repressor protein–DNA association (Froloff et al., 1997; Jayaram et al., 1999; Misra et al., 1998). Note that Chothia and Janin as well as Honig and coworkers did not take vdW interactions into account explicitly in computing the absolute binding free energy. In this work, Scheme 1 (see next section), in conjunction with continuum dielectric models, has been implemented to compute the absolute free energy of protein–protein association. Our initial goals are twofold: to assess the accuracy of our scheme for computing absolute binding free energies, and to assess the validity of the various assumptions in previous theoretical treatments like the rigid-binding approximation or the neglect of vdW interactions and vibrational entropy (Novotny et al., 1989; Williams et al., 1991; Horton and Lewis, 1992; Vajda et al., 1994; Jackson and Sternberg, 1995; Novotny et al., 1997; Froloff et al., 1997; Olson, 1999). To address our first goal, we chose systems for which experimental binding free energies are available for comparison with the computed values, and accurate structures are available to minimize errors due to experimental issues. In this way, any observed discrepancy between the calculated and experimental results would reflect primarily the assumptions or deficiencies in our methodology rather than inaccuracies in the experimental structures. Specifically, we chose systems for which the crystal strucBiophysical Journal 81(2) 737–750

Noskov and Lim

tures of both the complex and its components have been solved to ⱕ1.9-Å resolution to eliminate errors arising from poor quality structures or absence of free protein structures. Having the x-ray structures of the unbound and bound proteins also enable us to account for conformational changes that accompany binding (especially at the interaction surfaces). Also, we verified that the pH at which the free and complex structures were solved is close to the pH at which the corresponding binding free energy was measured to minimize errors due to structural changes at different pH values. To address our second goal, the systems chosen satisfied not only the above criteria, but have also been studied using various approximations. Only two systems were found to satisfy all of the aforementioned criteria. They are trypsin complexed with bovine pancreatic trypsin inhibitor (BPTI), and the fragment variable (Fv) region of mouse monoclonal antibody, D1.3, bound to hen egg-white lysozyme (HEL). Their experimental binding free energies, Protein Data Bank (PDB) entries (Bernstein et al., 1977) and various properties are listed in Table 1. The interface area is defined as the SASA difference between the complex and its components. Note that both systems correspond to association of positively charged proteins to form a ⫹12e complex with only small changes in conformation, as evidenced by root-mean-square deviation (RMSD) of the backbone atoms in the free x-ray structure from that in the corresponding complex crystal structure of ⬍1 Å (see Table 1). The excellent agreement found between the computed and experimental binding free energies (see Results) allowed us to identify the key forces governing protease-inhibitor and antibody–antigen complexation with interface areas in the range of 1400 –1700 Å2. Finally, we compare our findings with results obtained in previous studies for the same or homologous systems as well as for other protein–protein complexes (see Discussion). METHODS Theory The binding free energy calculations were based on the thermodynamic cycle ⌬Ggas




O ¡


关R 䡠 L兴gas 2⌬Gsolv共R䡠L兲

1⫺⌬Gsolv共l兲 ⫹


关共R 䡠 L兲min兴gas 2⌬Gisom共R䡠L兲





O ¡

关R 䡠 L兴sln


In Scheme 1, lower case r and l denote the unbound receptor and ligand conformations, whereas upper case R and L denote the bound receptor and ligand conformations. Note that [x]sln and [x]gas, where x ⫽ r, l, or R 䡠 L,

Protein–Protein Free Energy Decomposition TABLE 1


Properties of protein–protein complexes studied in this work

Property ⌬Gexp (kcal/mol)* (pH of binding) PDB entry Resolution (Å) pH of crystallization Number of atoms Net charge SASA (Å2)§ Conformational Change (Å)¶



1vfb 1.80 7.1 5,384 12 15,049

⫺11.4 ⫾ 0.1† (7.1) 1vfa 1.80 7.1 3,424 4 10,027 1.07 (0.87)


1dpx 1.65 8.0 1,960 8 6,470 1.27 (0.88)




2ptc 1.90 7.0 4,112 12 11,441

⫺18.1 ⫾ 0.1‡ (8.0) 2ptn 1.55 7.0 3,220 6 9,095 0.83 (0.61)

1bpi 1.10 8.2 892 6 4,013 1.23 (0.93)

*Experimental binding free energy, ⌬Gexp, in kcal/mol, at 298 K and the pH of the ⌬Gexp measurement in brackets. From Verhoeyen et al. (1988). ‡ From Vincent & Lazdunski (1972). § SASA values are based on the x-ray structures. ¶ The RMSD of the heavy atoms (or backbone atoms) of the bound protein from the free one. †

correspond to the same structure; viz., the relaxed “all-hydrogen” x-ray structure (see next section). The latter was energy minimized in the gas phase to yield [xmin]gas, which corresponds to a local energy minimum that is required to compute gas-phase vibrational frequencies and thus the vibrational entropy change accompanying binding (see Gas-phase Binding Free Energy). The standard free energy change (⌬G0) for the noncovalent association of two molecules in solution according to Scheme 1 is given by

⌬G⬚ ⫽ ⌬⌬Gsolv ⫹ ⌬⌬Gisom ⫹ ⌬Ggas,


⌬⌬Gsolv ⫽ ⌬Gsolv共R 䡠 L兲 ⫺ ⌬Gsolv共r兲 ⫺ ⌬Gsolv共l兲




⌬⌬Gisom ⫽ ⌬Gisom共R 䡠 L兲 ⫺ ⌬Gisom共r兲 ⫺ ⌬Gisom共l兲. (3) The desolvation free energy, ⫺⌬Gsolv, corresponds to the work of transferring the molecule in its solution conformation to the same conformation in the gas phase at 298 K. The isomerization free energy, ⫺⌬Gisom, is the work of transforming the molecule in its x-ray conformation to a nearby local minimum structure in the gas phase. ⌬Ggas is the standard free energy change per mole for the noncovalent association of the molecules in the gas phase at 298 K. The calculations of ⌬Gsolv, ⌬Gisom, and ⌬Ggas are described below.

Structures In computing the absolute free energy according to Scheme 1, x-ray structures of the free proteins and their respective complex were used (see Table 1). The ionizable groups in the free proteins and respective complex were protonated or deprotonated according to available experimental pKa values and the pH of crystallization. All aspartic acid and glutamic acid


residues as well as COOH-terminal groups were deprotonated, whereas arginine, lysine, cysteines, and NH2-terminal groups were protonated. Histidine residues were protonated only if their side-chain nitrogen atoms were within 3.5 Å of a hydrogen acceptor in the crystal structures. However, for the proteins studied here, the histidine side chains were found not to be involved in hydrogen bonding. Thus, they were treated as neutral by protonating at N⑀2 or N␦1 according to their local environment in the crystal structure. In trypsin, His40 and His91 were protonated at N⑀2, but His57 was protonated at N␦1 (Jackson and Sternberg, 1995). BPTI has no histidine residues. For the Fv region of mouse monoclonal D1.3 antibody and lysozyme, all the histidines were protonated at N⑀2 (Tanokura, 1983). Only one histidine is located in the interface region in the complex crystal structures; viz., His57 in the trypsin/BPTI complex. First, hydrogen atoms were added to the crystal structure using the HBUILD module of the CHARMM program (Brooks et al., 1983). The resulting structure was subjected to several steps of minimization using steepest descent followed by adopted-basis Newton–Raphson with strong (10 kcal/mol/Å2) harmonic constraints on all heavy atoms. This relieved close contacts in the protein without disrupting its overall conformation (Philippopoulos and Lim, 1995), as evidenced by RMSDs from the respective crystal structure of ⱕ0.064 Å (see Table 2). The relaxed all-hydrogen x-ray structures, denoted by [r]sln, [lsln, and [R 䡠 L]sln, were used to compute the solvation free energies (see below). For the isomerization step, each all-hydrogen x-ray structure was energy minimized (initially with constraints, then without) using a dielectric constant of one for ⱖ2500 steps of steepest descent, followed by ⬃7500 steps of adopted-basis Newton– Raphson until the average force was ⬍2 ⫻ 10⫺6 kcal/mol/Å. The fully minimized [rmin]gas, [lmin]gas, and [(R 䡠 L)min]gas structures, which remain close to their respective crystal structure (see Table 2), were used to obtain the gas-phase complexation energies and entropies. All energy minimizations were carried out using the CHARMM program (Brooks et al., 1983) and the version 22 all-hydrogen forcefield (MacKerell et al., 1998). In computing ⌬G0 using the rigid-binding approximation, the starting conformations of the unbound r and l proteins were obtained from the

RMSD of heavy atoms from crystal structures*

Complex Structure

Free Structures







[(R 䡠 L)]sln [(R 䡠 L)]sln [(R 䡠 L)]sln [(R 䡠 L)min]gas

[r/l]sln [R/L]sln [R/Lmin⫹cons]sln [r/lmin]gas

0.06 0.06 0.06 1.23

0.06 1.05 1.05 1.15

0.04 1.29 1.25 1.48

0.05 0.05 0.05 1.37

0.06 0.97 0.97 1.23

0.05 1.20 1.18 1.35

*The RMSD of the heavy atoms in the free or bound protein structure from the respective x-ray structure (see Methods). Biophysical Journal 81(2) 737–750


Noskov and Lim

relaxed all-hydrogen x-ray structure of the R 䡠 L complex; i.e., [r]sln ⫽ [R]sln and [l]sln ⫽ [L]sln. These rigid free structures exhibit heavy atom RMSDs from the respective crystal structures that range from 0.97 to 1.29 Å (see Table 2). The latter value is greater than the motion of protein crystal atoms, 0.3– 0.5 Å, derived from their B-factors (Froloff et al., 1997; Luzzati, 1952). Thus, in the rigid-binding approximation, ⌬G0 is computed according to ⌬Ggas



关R兴gas 1

关L兴gas 1

关共R 䡠 L兲min兴gas 2⌬Gisom共R䡠L兲




O ¡


关R 䡠 L]gas 2⌬Gsolv共R䡠L兲


O ¡


关R 䡠 L兴sln

Honig, 1995) derived from the partition coefficients of linear alkanes between water and the gas phase using AMBER radii to compute the SASA. ␥vdW was set to ⫺38.8 cal/mol/Å2 (Jayaram et al., 1999), the value obtained from experimental vaporization enthalpies of hydrocarbons (Ohtaki and Fukushima, 1992). The difference between ␥ and ␥vdW yields ␥cav ⫽ 46.0 cal/mol/Å2 (Sharp et al., 1991). The SASA of the molecule was computed using the GEPOL program (Pascual-Ahuir and Silla, 1990) and CHARMM vdW radii (MacKerell et al., 1998). Hence, ⌬⌬Gnonel solv is proportional to the loss in SASA at the protein–protein interface. The electrostatic solvation free energy, ⌬Gelec solv, was computed using two continuum solvent approaches: the analytical GB model (Qiu et al., 1997; Still et al., 1990) and numerical PB methods (Honig and Nicholls, elec 1995). The former is denoted by ⌬Gsolv,GB , whereas the latter by elec elec ⌬Gsolv,PB . In the GB model (Still et al., 1990), ⌬Gsolv,GB is expressed as a sum of Coulombic interactions between each pair of charges (qi, qj) in a solvent of dielectric constant ⑀ and the Born self solvation energy of each individual charge,


elec ⌬Gsolv,GB ⫽ ⫺ 166 1 ⫺


The ⌬G was also evaluated using free receptor and free ligand structures that have been modeled from the respective complex by minimizing the receptor R and ligand L conformations using CHARMM with a distance-dependent dielectric constant and constraints on all heavy atoms (denoted by min ⫹ cons in Scheme 3). In this case, [r]sln ⫽ [Rmin⫹cons]sln and [l]sln ⫽ [Lmin⫹cons]sln. Their heavy-atom RMSDs from the respective crystal structures are similar to those of the rigid structures (see Table 2). ⌬Ggas


O ¡ 关共R 䡠 L兲min兴gas

⫹ 关Lmin兴gas

1⫺⌬Gisom共R 关Rmin⫹cons兴gas






2⌬Gisom共R䡠L兲 关R 䡠 L]gas



1⫺⌬Gisom共L ⫹ 关Lmin⫹cons兴gas 1⫺⌬Gsolv共L

⫹ 关Lmin⫹cons兴sln

O ¡ 关R 䡠 L兴sln ⌬ G⬚

fGB ⫽

2 ij

r ⫹ ␣i␣jexp

␣i ⫽ ⫺

Solvation free energy ⌬Gsolv

cav vdW elec ⌬Gsolv ⫽ ⌬Gsolv ⫹ ⌬Gsolv ⫹ ⌬Gsolv


The first two terms in Eq. 4 constitute the nonelectrostatic contribution (⌬Gnonel solv ) to the solvation free energy due to the work required to create the solute cavity in solution (⌬Gcav solv) and to vdW interactions between the solute and solvent (⌬GvdW solv ). The last term in Eq. 4 gives the electrostatic contribution (⌬Gelec solv) to the solvation free energy. The nonelectrostatic solvation free energy, ⌬Gnonel solv , was approximated by a linear function of the SASA (Lee and Richards, 1971), nonel vdW cav ⌬Gsolv ⫽ ⌬Gsolv ⫹ ⌬Gsolv

⬇ 共␥vdW ⫹ ␥cav兲 ⫻ SASA⫽␥ ⫻ SASA. 2

␥ was set to 7.2 cal/mol/Å . This value in Eq. 5 and the GB model (Eq. 6 below) could reproduce the experimental hydration free energies of a wide range of neutral small molecules (Still et al., 1990) and the experimental free energies of dihydrofolate reductase and trypsin binding (Zou et al., 1999). It is also in accord with that (6.4 ⫾ 1.7 cal/mol/Å2) (Friedman and Biophysical Journal 81(2) 737–750




i j


i⫽1 j⫽1

rij2 ⫺ . 4 ␣ i␣ j


166 , Gpol,i


冉 冊冋 冉 冊 冉 冊

Gpol,i ⫽ 1 ⫺

1 ␧

1 166 166 ⫺ ⫹ P1 2 ␭ RvdW,i RvdW,i

冘 Pr V ⫹ 冘 Pr V ⫹ 冘


2 j 4 ij



CCF ⫽ 1.0, (5)

冊 冘 冘 qf q ,

fGB is an effective distance function that depends on the interatomic distances (rij) and the effective Born radii of atoms i and j (␣i, ␣j). In computing the electrostatic solvation free energies, ⑀ was set to 80, the dielectric constant for bulk water, and the atomic charges were taken from the CHARMM version 22 forcefield (MacKerell et al., 1998). The effective Born radii were computed using an empirical scheme first proposed by Still and co-workers (Qiu et al., 1997) and modified for protein solvation calculations by Dominy and Brooks (1999). The effective Born radius of an atom depends on the atom’s specific molecular environment (Babu and Lim, 1999, 2001). It is defined as the atomic radius that would give the Born solvation free energy (Born, 1920) of the atom with unit charge, if all other atoms in the molecule were uncharged and served only to displace the solvent dielectric medium; i.e.,

Calculations The standard solvation free energy, ⌬Gsolv, can be expressed as a sum of 3 terms,

1 ␧

再 冋




3 j 4 ij


rij 共RvdW,i ⫹ RvdW,j兲

CCF ⫽ 0.5 1.0 ⫺ cos





P 4V j CCF rij4


1 , P5

rij 共RvdW,i ⫹ RvdW,j兲


冊 冊册冎 冊 2


P 5␲

2 rij 1 ⱕ 共RvdW,i ⫹ RvdW,j兲 P5


Protein–Protein Free Energy Decomposition


In Eq. 8, rij is the distance from the point charge of interest, i, to an uncharged atom j in the molecule; RvdW,i is the vdW radius of atom i, and 3 Vj ⫽ 4⁄3␲RvdW,j is the volume of atom j. The vdW radii were taken from the CHARMM version 22 all-hydrogen forcefield with the polar hydrogen radii set to 0.8 Å following previous works (Dominy and Brooks, 1999; Qiu et al., 1997). P1, P2, P3, P4, P5 and ␭ are fitting parameters. They have been optimized by Dominy and Brooks (1999) for a combined protein and nucleic acid structure database and the CHARMM version 22 forcefield. The parameters used in this work are P1 ⫽ 0.448, P2 ⫽ 0.173, P3 ⫽ 0.013, P4 ⫽ 9.015, P5 ⫽ 0.9, and ␭ ⫽ 0.705 (Dominy and Brooks, 1999). The reader is referred to the original works for the derivation of Eq. 8 (Qiu et al., 1997) and the fitting procedure to obtain the parameters (Dominy and Brooks, 1999). The ⌬Gelec solv energies have also been evaluated by solving numerically the linearized Poisson–Boltzmann equation using the DelPhi program (Gilson and Honig, 1988). The protein was treated as a low dielectric medium (⑀int ⫽ 1) surrounded by a high dielectric solvent (⑀out ⫽ 80 for water). The ionic strength was set to zero, but increasing it to 0.1 M had little effect on the magnitude of the solvation free energy, as found in previous works (Froloff et al., 1997; Jackson and Sternberg, 1995). The low-dielectric region of the protein was defined as the region inaccessible to contact by a 1.4-Å sphere rolling over the molecular surface, defined by the atomic coordinates and vdW radii. The vdW radii and atomic charges used in the DelPhi calculations were taken from the CHARMM version 22 forcefield. The electrostatic potentials were calculated on a cubic grid with a spacing of 0.5 Å and a 90% grid fill to avoid grid boundary artifacts. The difference between the electrostatic potential calculated in solution (⑀out ⫽ 80) and in the gas phase (⑀out ⫽ 1) yielded the electrostatic contribution to the solvation free energy.

Isomerization free energy ⌬Gisom The isomerization free energy for each molecule was estimated from the energy difference between the relaxed all-hydrogen structure and the respective fully minimized structure (see above); i.e.,

⌬Gisom ⬇ ⌬Eisom ⫽ E共X兲 ⫺ E共Xmin兲,


The energy E was calculated using the CHARMM program (Brooks et al., 1983) with the version 22 all-hydrogen forcefield (MacKerell et al., 1998) with ⑀ ⫽ 1 and an atom-based force-shifting function, which shifts the nonbonded forces to zero at 12 Å.

Gas-phase binding free energy, ⌬Ggas The standard gas-phase free energy change is given by

⌬Ggas ⫽ ⌬Egas ⫹ p⌬V ⫺ T共⌬Strans ⫹ ⌬Srot ⫹ ⌬Svib ⫹ ⌬Sconf兲 (12) where the intramolecular energy change, ⌬Egas is computed from Eqs. 11 and 13,

⌬Egas ⫽ E共共R 䡠 L兲min兲 ⫺ E共rmin兲 ⫺ E共lmin兲

TSrot ⫽


and p⌬V ⫽ ⫺RT ⫽ ⫺0.6 kcal.mol at 298 K. The gas-phase translational and rotational entropies for the free proteins and their complex were calculated from ideal gas partition functions (Qtrans, Qrot) using classical statistical mechanics (McQuarrie, 1976).

5 3 2␲mkBT ⫺ ln共␳兲 kBT, ⫹ ln 2 2 h2


3 1 8 ␲ k BT 3 ⫺ ln共␴兲 kBT. ⫹ ln共␲IaIbIc兲 ⫹ ln 2 2 2 h2 2

In Eq. 14, m is the total mass of the molecule, kB is Boltzmann’s constant, T is the temperature (298 K), h is Planck’s constant, ␳ is the number density (1 M per liter), IA, IB, IC are the 3 principal moments of inertia, and ␴ is the symmetry number (which is equal to 1 for nonsymmetric molecules). The gas-phase vibrational entropies for the free proteins and the respective complex were calculated from normal mode frequencies ␯j based on the expression (McQuarrie, 1976):



TSvib ⫽


hvj ⫺ kBT ln共1 ⫺ e⫺h␯j/kBT兲. (15) ⫺1


The normal mode frequencies ␯j were computed using the CHARMM program (Brooks et al., 1983) with the version 22 all-hydrogen forcefield (MacKerell et al., 1998) using an iterative diagonalization scheme (Janezic and Brooks, 1995), a dielectric constant of one, and a cutoff of 12 Å to truncate the nonbonded forces. For each structure, there were six zerofrequency modes corresponding to overall translational and rotational motions and no imaginary frequencies. The conformational entropy change, ⌬Sconf, was approximated by the loss of side-chain conformational entropy upon binding, which, in turn, was estimated using the empirical scale of Pickett and Sternberg (1993, 1994). This model assumes that solvent-accessible side chains (with relative SASA ⬎ 60%) populate different rotamers, whereas buried side chains (with relative SASA ⱕ 60%) are restricted to one rotamer. The conformation entropy of a given side chain r (Sconf,r) is given by,

冘 p ln共p 兲, N

Sconf,r ⫽ ⫺ R


where X ⫽ r, l, or R 䡠 L, and

E ⬇ EvdW ⫹ Eelec

TStrans ⫽





where pi is the probability of the side chain in rotameric state i. The values of pi were obtained from the observed distribution of side-chain rotamers in 50 nonhomologous protein crystal structures, taking into account the effects of symmetry and free rotation (Pickett and Sternberg, 1993).

RESULTS In computing ⌬G0 according to Schemes 1–3, two assumptions were made: the free energy components in Eq. 1 were assumed to be additive; and they were based on a single time-averaged x-ray structure rather than ensemble averaged. Hence, the present scheme is best suited for the noncovalent binding of two proteins that undergo only slight conformational changes to form a high affinity complex, as in the case here (see Table 1). However, for the binding of flexible peptides to proteins, free energy estimates based on a single conformation of the unbound and bound species may not be appropriate. Tables 3 and 4 summarize the results for D1.3䡠Hel and trypsin䡠BPTI binding, respectively. In each case, the binding free energy was computed using three different sets of structures for the free proteins (see Methods): the relaxed all-hydrogen x-ray structures (Scheme 1); rigid [r]sln ⫽ Biophysical Journal 81(2) 737–750


Noskov and Lim

TABLE 3 Free energy and its components for the binding of the Fv region of mouse monoclonal antibody D1.3 to hen egg-white lysozyme (Hel)* This Work §


Interface area¶ Total binding free energy㛳,** ⌬G° Solvation contributions ⌬⌬Gcav solv ⌬⌬GvdW solv ⌬⌬Gelec solv** ⌬⌬Gsolv Gas phase contributions vdW†† ⌬⌬Eisom elec †† ⌬⌬Eisom vdW ⌬Egas elec ⌬Egas ⫺T⌬Strans ⫺T⌬Srot ⫺T⌬Svib‡‡ ⫺T⌬Sconf ⌬Ggasⴙisom§§ Net component contributions ⌬⌬Gnonel¶¶ solv ⌬EvdW㛳㛳 elec ⌬G *** ⫺T⌬Strans⫹rot⫹vib†††

Scheme 1 1448

Scheme 2 1479

Scheme 3 1470

ⴚ11.4 (ⴚ9.4)

38.4 (44.2)

5.4 (7.4)

⫺66.6 56.2 98.1 (100.1) 87.7 (89.7)

⫺68.0 57.4 118.7 (124.5) 108.1 (113.9)

⫺67.6 57.0 118.1 (120.2) 107.6 (109.6)

⫺39.4 71.0 ⫺67.8 ⫺106.1 13.4 14.1 7.0 9.3 ⴚ99.1 ⫺10.4 ⫺107.2 63.0 (65.0) 34.5

⫺26.3 88.7 ⫺52.0 ⫺163.0 13.4 14.1 14.2 9.3 ⴚ102.2

31.1 148.7 ⫺62.1 ⫺242.1 13.4 14.1 18.5 9.3 ⴚ69.7 ⫺10.6 ⫺31.0 25.3 (31.1) 46.0

⫺10.6 ⫺78.3 43.9 (45.9) 41.7



1640 ⴚ9 ⴞ 3



42 ⫾ 3 ⫺41 ⫺21 9


*All energies are in kcal/mol; a blank means that the value is not available. From Novotny et al. (1989). The authors also computed a cratic entropy term, ⫺T⌬Scratic, which was set to 2 kcal/mol. ‡ Data for homologous HyHel10-HEL system from Novotny et al. (1997). Note that an internal dielectric constant ␧int ⫽ 4 was used to solve the PB equation, whereas an ␧int ⫽ 1 was used in this work. § See Methods and Table 2; “rigid” implies that the rigid-binding approximation was used, see text. ¶ The difference between the solvent-accessible surface areas of bound and free proteins in Å2. 㛳 ⌬G⬚ ⫽ ⌬⌬Gsolv,PB/GB ⫹ ⌬Ggas⫹isom; see Methods. **The numbers without and with parentheses were computed using the GB model and finite difference PB methods, respectively. †† ⌬Eisom is the energy required to isomerize the local energy minimum in the gas phase to the respective relaxed all-hydrogen X-ray structure. ‡‡ The TS values of the individual species are taken from the last column of Table 6. §§ ⌬Ggas⫹isom ⫽ ⌬⌬Eisom ⫹ ⌬Ggas using Eqs. 10 –12. ¶¶ cav vdW ⌬⌬Gnonel solv ⫽ ⌬⌬Gsolv ⫹ ⌬⌬Gsolv using Eq. 5. 㛳㛳 vdW vdW vdW ⌬E ⫽ ⌬⌬Eisom ⫹ ⌬Egas . elec elec ***⌬Gelec ⫽ ⌬⌬Gelec solv ⫹ ⌬⌬Eisom ⫹ ⌬Egas . ††† trans⫹rot⫹vib trans rot T⌬S ⫽ T⌬S ⫹ T⌬S ⫹ T⌬Svib. †

[R]sln and [l]sln ⫽ [L]sln structures (Scheme 2); and [r]sln ⫽ [Rmin⫹cons]sln and [l]sln ⫽ [Lmin⫹cons]sln (Scheme 3). In what follows, we will first compare the free energies computed using the relaxed all-hydrogen x-ray structures with experiment. Agreement with experiment then allows us to assess the relative contributions from protein–protein versus protein–solvent versus solvent–solvent interactions and electrostatic versus vdW forces to the binding free energy. We then evaluate the accuracy of binding free energies computed using rigid as well as [Rmin⫹cons]sln and [Lmin⫹cons]sln free protein structures. Comparison between computed and experimental ⌬Gexp: GB versus PB 0 Tables 3 and 4 show that the ⌬GGB (⫺11.4 kcal/mol for D1.3䡠Hel and ⫺18.6 kcal/mol for trypsin䡠BPTI) based on

Biophysical Journal 81(2) 737–750

[r]sln, [l]sln, and [R 䡠 L]sln relaxed x-ray structures and elec ⌬⌬Gsolv,GB are in excellent agreement with the experimental values (⫺11.4 and ⫺18.1 kcal/mol). In contrast, the 0 based on [r]sln, [l]sln, and [R 䡠 L]sln relaxed x-ray ⌬GPB elec structures and ⌬⌬Gsolv,PB exhibit larger deviations from the 0 0 experimental numbers than ⌬GGB , whereas the ⌬GGB and 0 ⌬GPB differ by roughly 20%. The remarkable agreement 0 between ⌬GGB and experiment indicates that systematic errors involved in computing free energies/energies in the right and left legs of Scheme 1 have largely cancelled. elec Because the ⌬Gsolv,GB based on x-ray structures are more elec negative than the respective ⌬Gsolv,PB by ⱕ7.6%, but the elec CPU time needed to compute ⌬Gsolv,GB is 5 to 6 times less elec than that for ⌬Gsolv,PB , the GB formulation together with Scheme 1 appears to be an efficient and reliable way of computing binding free energies and obtaining trends (see below).

Protein–Protein Free Energy Decomposition TABLE 4


Free energy and its components for the binding of trypsin to BPTI* This Work

Free Structures Interface area Total binding free energy ⌬G° Solvation contributions ⌬⌬Gcav solv ⌬⌬GvdW solv ⌬⌬Gelec solv ⌬⌬Gsolv Gas phase contributions vdW ⌬⌬Eisom elec ⌬⌬Eisom vdW ⌬Egas elec ⌬Egas ⫺T⌬Strans ⫺T⌬Srot ⫺T⌬Svib ⫺T⌬Sconf ⌬Ggasⴙisom Net component contributions ⌬⌬Gnonel solv ⌬EvdW ⌬Gelec ⫺T⌬Strans⫹rot⫹vib

Scheme 1 1667







ⴚ7.5 (ⴚ4.4)

ⴚ34.1 (ⴚ31.1)




⫺70.9 59.8 198.0 (201.1) 186.9 (190.0)

⫺70.7 59.7 180.5 (183.5) 169.4 (172.4)









Scheme 3


ⴚ18.6 (ⴚ23.1) ⫺76.7 64.7 84.6 (80.1) 72.6 (68.1) 44.3 166.1 ⫺105.8 ⫺228.3 12.6 13.6 ⫺5.0 11.9 ⴚ91.3


Scheme 2

42.1 132.0 ⫺86.0 ⫺314.7 12.6 13.4 ⫺2.7 9.4 ⴚ194.4

⫺12.0 ⫺61.6 22.5 (18.0) 21.2

⫺11.1 ⫺43.8 15.3 (18.4) 23.4

21.7 41.1 ⫺80.4 ⫺213.1 12.6 13.4 ⫺7.6 9.4 ⴚ203.5 ⫺11.1 ⫺58.7 8.5 (11.5) 18.5


⫺37.5 ⫺30.0 9.0

*See footnotes under Table 3. From Krystek et al. (1993). ‡ From Jackson and Sternberg (1995). § From Polticelli et al. (1999). †

Solvation versus gas-phase contributions to binding affinity Scheme 1 enables the net binding free energy to be dissected into solvation (protein–solvent and solvent–solvent) versus gas-phase (protein–protein) contributions. The latter, which is a sum of ⌬Ggas and ⌬⌬Gisom terms, is negative, thus favoring complexation, whereas ⌬⌬Gsolv is positive and opposes binding (Tables 3 and 4). For both systems, gas-phase vdW and electrostatic protein–protein interacelec tions favor binding (⌬EvdW gas and ⌬Egas negative). Note that elec ⌬Egas is negative even though the receptor and ligand have net positive charges (Table 1). The solvent–solvent cavitation term (Eq. 5) also favors protein–protein complexation (⌬⌬Gcav solv negative) as less work is required to create the complex cavity than the free protein cavities in solution. In contrast to the favorable protein–protein and solvent–solvent interactions, vdW and electrostatic protein–solvent inelec teractions generally oppose binding (⌬⌬GvdW solv and ⌬⌬Gsolv positive) due to the cost of desolvating the unbound proteins. Decomposition of ⌬G0 into component energies Tables 3 and 4 show that, for both systems, ⌬⌬Gnonel solv (⫺10.4 kcal/mol for D1.3䡠Hel and ⫺12.0 kcal/mol for trypsin䡠BPTI) roughly cancels the gas-phase conformational

entropy term, ⫺T⌬Sconf (9.3 kcal/mol for D1.3䡠Hel and 11.9 kcal/mol for trypsin䡠BPTI). This finding can be rationalized nonel if ⌬⌬Gnonel solv ⬇ ⫺T⌬⌬Ssolvent, based on the fact that ⌬⌬Gsolv is related to the hydrophobic effect, which, at room temperature, is dominated by the solvent entropy term. The observed cancellation can then be attributed to the increase in the solvent entropy upon complexation (as protein side chains at the interface release water molecules), and the concomitant decrease in the solute entropy (as protein side chains at the interface lose torsional degrees of freedom upon interacting with other residues) (Novotny et al., 1989). To verify whether the observed cancellation of ⌬⌬Gnonel solv and T⌬Sconf is general, these two quantities were also computed for other systems for which x-ray structures are available for both the free and bound proteins (although the structures for these systems may not be as accurate because they do not satisfy all the criteria specified in the Introducconf are not very sensitive to the tion, ⌬⌬Gnonel solv and T⌬S structures because they depend on the SASA, see Tables 3 and 4). The results in Table 5 show that ⌬⌬Gnonel solv and ⫺T⌬Sconf oppose each other and their difference is less than conf 1.5 kcal/mol, indicating that ⌬⌬Gnonel gensolv and ⫺T⌬S erally offset one another. It should be emphasized that the conf compensation rests on the observed ⌬⌬Gnonel solv and T⌬S magnitude of the coefficient ␥, 7.2 cal/mol/Å2, used in this work (see below). Biophysical Journal 81(2) 737–750


Noskov and Lim

nonel TABLE 5 Changes in the side-chain conformational entropy (T⌬Sconf) and nonelectrostatic solvation free energy (⌬⌬Gsolv ) in various protein–protein associations


Ligand* Complex Interface area† ⌬⌬Gnonel‡ solv ⫺T⌬Sconf‡ ⌬⌬Gnonel solv ⫺T⌬Sconf

Proteinase B



(1sgb) OMTKY3 (1omt) 3sgb 1447 ⫺10.4 8.9 ⫺1.5

(1vfa) HEL (1dpx) 1vfb 1448 ⫺10.4 9.3 ⫺1.1

(2cha) OMTKY3 (1omt) 1cho 1632 ⫺11.8 10.4 ⫺1.4

Subtilisin Carlsberg (1sbc) Eglin C (1egl) 1cse 1640 ⫺11.8 10.7 ⫺1.1



(2ptn) BPTI (1bpi) 2ptc 1667 ⫺12.0 11.9 ⫺0.1

(1thm) Eglin C (1egl) 1tec 1719 ⫺12.3 11.2 ⫺1.1

Subtilisin Novo (1sup) CI-2 (2ci2) 2sni 1795 ⫺12.9 12.1 ⫺0.8

*PDB entry in brackets. † The difference between the solvent-accessible surface areas of bound and free proteins in Å2. ‡ In kcal/mol.

conf Because the ⌬⌬Gnonel terms offset one solv and ⫺T⌬S another in Tables 3 and 4, the remaining components of the binding affinity can be partitioned into net gas-phase protein–protein vdW effects (⌬Evdw), net protein–protein and protein–solvent electrostatic effects (⌬Gelec), and the sum of translational, rotational, and vibrational entropy changes (⫺T⌬Strans⫹rot⫹vib). The magnitudes of ⌬EvdW, ⌬Gelec, and T⌬Strans⫹rot⫹vib in Tables 3 and 4 are sizable, indicating that all three components contribute to the overall binding affinity, so that

⌬G⬚ ⬃ ⌬Evdw ⫹ ⌬Gelec ⫺ T⌬Strans⫹rot⫹vib.


0 The ⌬GGB computed using Eq. 17 are ⫺9.7 kcal/mol for D1.3䡠Hel and ⫺17.9 kcal/mol for trypsin䡠BPTI, which underestimate the experimental numbers by 1.7 and 0.2 kcal/ mol, respectively. Of the three terms in Eq. 17, only the gas-phase protein–protein vdW interactions favor binding (⌬Evdw negative). Furthermore, the magnitude of ⌬Evdw is larger than that of ⌬Gelec or T⌬Strans⫹rot⫹vib for both D1.3䡠Hel and trypsin䡠BPTI complexation. These findings are consistent with the experimental observation that enthalpy drives D1.3䡠Hel binding from measurements of enthalpy and entropy changes by titration calorimetry (Bhat et al., 1994). Hence, close packing and shape complementarity play important roles in noncovalent protein association. The net electrostatic interactions generally oppose formation of complexes with an interface area between 1400 and 1700 Å2 (⌬Gelec positive). This finding does not mean that electrostatic interactions are unimportant for protein binding, but shows that the gain in hydrogen bonding and charge– charge protein–protein interactions across the interface is offset by the loss of favorable electrostatic protein–solvent interactions. Although translational and rotational entropy loss inevitably oppose complexation (⌬Strans⫹rot negative), vibrational entropy can favor or disfavor protein–protein association: ⌬Svib is negative for D1.3䡠Hel complexation, but is positive for trypsin–BPTI binding (see also Discussion).

Biophysical Journal 81(2) 737–750

Because the magnitude of the T⌬Strans⫹rot term is similar for two ligands of roughly equal volumes, l and l⬘, binding to a common receptor r, their binding free energy difference will be given by

⌬⌬G⬚ ⬃ ⌬Evdw ⫺ ⌬E⬘vdw ⫹ ⌬Gelec ⫺ ⌬G⬘elec ⫹ T⌬Svib ⫺ T⌬S⬘vib,


where the prime denotes the thermodynamic change for binding of l⬘. Hence, the discrimination between two similarly shaped ligands, which defines specificity, is governed by close packing of the protein surfaces so that the proper hydrogen bonds can be formed (McCammon, 1998). Eq 18 shows that the affinity of a drug ligand l⬘ for its receptor protein with an interface area between 1400 and 1700 Å2 can be improved by mutations that increase the magnitude of ⌬E⬘vdw, but decrease the magnitude of ⌬G⬘elec. Evaluation of the rigid-binding approximation The RMSDs of the heavy atoms in the free x-ray structure from the corresponding complex crystal structure is ⬍1.3 Å for both systems studied (Table 1). Therefore, the structural changes upon complexation are relatively small and it seems reasonable, to a first approximation, to neglect any conformational changes upon protein–protein association; i.e., to assume [r]sln ⫽ [R]sln and [l]sln ⫽ [L]sln. In fact, previous studies used this rigid-binding approximation to compute the free energy for D1.3䡠Hel and trypsin䡠BPTI binding (Tables 3 and 4). The binding free energy computed using Scheme 2 does not agree with the experimental value (Tables 3 and 4, column 3), and is even quite positive for the binding of D1.3 antibody to Hel. The discrepancy between theory and experiment arises from both protein–protein as well as protein–solvent electrostatic interactions. The rigid free receptor and free ligand conformations isomerize to local minima, which have less favorable electrostatic interactions than the local minima derived from the relaxed x-ray structures of the free proteins. Consequently,

Protein–Protein Free Energy Decomposition

the ⌬Eelec gas values computed using the rigid free receptor and free ligand structures (⫺242 and ⫺315 kcal/mol) are more negative than the respective numbers in the second column of Tables 3 and 4 (⫺106 and ⫺228 kcal/mol). In contrast, the ⌬⌬Gelec solv values computed using the rigid free structures (119 and 198 kcal/mol) are more positive than the respective numbers in column 2 of Tables 3 and 4 (98 and 85 kcal/mol). In the case of D1.3 antibody binding to Hel, the relative component contributions to the binding free energy computed using Scheme 2 do not agree with those obtained using Scheme 1. In particular, Scheme 2 predicts that the dominant contribution to the net binding free energy for D1.3䡠Hel complexation is not ⌬EvdW, but the entropy term, T⌬Strans⫹rot⫹vib, in contrast to Scheme 1 and experimental observations (see above). These results show that, even if the structural changes upon complexation are known to be small, the rigid-binding approximation (Scheme 2) will generally not yield accurate binding free energy, and may also not reveal the dominant contribution to the binding free energy.

Evaluation of free Rminⴙcons receptor and free Lminⴙcons ligand structures In cases where the free receptor or free ligand structures have not been experimentally solved but their complex crystal structure is known, the former can be predicted from the latter by minimizing the receptor and ligand conformations in the complex (see Structures section above) provided that binding results in minimal conformational changes. Like the rigid structures, the [r]sln ⫽ [Rmin⫹cons]sln and [l]sln ⫽ [Lmin⫹cons]sln structures predict a positive binding free energy for D1.3䡠Hel complexation, whereas they severely overestimate the magnitude of the free energy for trypsin䡠BPTI binding. However, they yield trends in the relative component contributions to the binding free energy that are similar to those based on the relaxed x-ray structures (Tables 3 and 4). The magnitude of the component contributions based on the [Rmin⫹cons]sln and [Lmin⫹cons]sln structures are in-between the respective numbers based on the x-ray and rigid structures (Tables 3 and 4). This suggests that approximating [r]sln and [l]sln by [Rmin⫹cons]sln and [Lmin⫹cons]sln, respectively, in cases where slight conformational rearrangements accompany complexation, as in the cases studied here, can reveal qualitative features in the binding free energies.

Dependence of component free energies on free and complex structures The results using Schemes 1–3 show that accurate structures are needed to obtain reliable absolute binding free energies (Sharp, 1998). This sensitivity of the atomic coordinates is expected because electrostatic and vdW interactions depend

745 TABLE 6 Lowest four nonzero vibrational frequencies (cmⴚ1) and vibrational entropies (TS in kcal/mol) from normal mode calculations for the fully minimized structures* Protein D1.3 D1.3 D1.3 HEL

(X-ray) (rigid) (predicted) (X-ray)

HEL (rigid) HEL (predicted) D1.3䡠HEL Trypsin (X-ray) Trypsin (rigid) Trypsin (predicted) BPTI (X-ray) BPTI (rigid) BPTI (predicted) BPTI䡠Trypsin

Mode 1

Mode 2

Mode 3

Mode 4


3.95 3.42 3.65 4.35 3.72† 3.48 3.96 2.17 4.44 6.21 6.15 6.31 6.19‡ 7.87 7.86 2.40

4.70 4.77 4.82 5.01 5.29† 5.01 4.96 4.62 5.91 6.83 6.52 8.24 7.70‡ 8.64 9.05 3.61

5.48 5.41 5.43 5.36 6.17† 5.12 5.89 4.97 6.39 7.14 6.81 8.58 8.70‡ 9.98 10.67 4.93

5.87 6.17 6.09 5.97 6.41† 6.39 6.46 5.44 7.14 7.62 7.49 9.80 9.56‡ 10.78 11.95 5.45

1569.2 1562.1 1564.2 718.9 737.4 731.2 2281.1 1476.0 1478.3 1473.9 401.2 401.1 400.7 1882.1

*The notations X-ray, rigid, and predicted refer to the structures in the first three rows of Table 2, which have been fully minimized as described in Methods; none of the structures resulted in an imaginary frequency. † Gibrat and Go (1990). ‡ Janezic and Brooks (1995).

on the interatomic distances. Hence, the ⌬Evdw and ⌬Gelec terms computed using different free structures vary significantly (see Tables 3 and 4). The vibrational entropy term is also sensitive to the local minimum structure used to compute the frequencies (see Table 6). In contrast, the ⌬⌬Gnonel solv and T⌬Sconf terms computed using different free structures do not vary as much as they depend on the solvent-accessible solvent surface area. These findings are in accord with previous works (Froloff et al., 1997; Jackson and Sternberg, 1995; Novotny et al., 1997). DISCUSSION Below, we compare our above findings with results obtained in previous studies for the same or homologous systems and for other protein–protein complexes. Comparison of the various free energy decomposition schemes The scheme used here to compute the free energy for protein–protein complexation is closest in spirit to that used by Jayaram et al. (1999) for protein–DNA complexation. The key difference lies in the isomerization step in Scheme 1, which was introduced to obtain local-minimum structures (with no imaginary frequencies) for normal mode frequency calculations. In contrast, Jayaram et al. computed first the isomerization/adaptation of the protein structure in the free state to the conformation in the complex state (the adaptation energy), and, subsequently, their desolvation (see Scheme 4). As a result, the gas-phase complexation step Biophysical Journal 81(2) 737–750


Noskov and Lim

corresponds to species that are not in energy minima, thus their normal mode frequencies cannot be computed. ⌬Ggas


O ¡ 关共R 䡠 L兲]gas

⫹ 关L兴gas

1⫺⌬Gsolv共r兲 1⫺⌬Gsolv共l兲 关R兴sln ⫹ 关L]sin 1




2⌬Gsolv共R 䡠 L兲


O ¡ 关R 䡠 L]sln

⫹ 关l]sln


In computing the absolute free energies for D1.3䡠HEL (Novotny et al., 1989) and trypsin䡠BPTI binding (Krystek et al., 1993) in Tables 3 and 4, Novotny and coworkers used rigid [r]sln ⫽ [R]sln and [l]sln ⫽ [L]sln structures and neglected all vdW interactions, protein–solvent electrostatic interactions, and the vibrational entropy change. Furthermore, they computed the free energy components directly in solution via the expression,

nonel elec elec ⌬G⬚ ⬇ ⌬⌬Gsolv ⫹ ⌬⌬Gsolv,PB ⫹ ⌬Esln ⫺ T⌬Sconf.


nonel elec conf The ⌬⌬Gcav terms in solv, ⌬⌬Gsolv , ⌬⌬Gsolv,PB, and T⌬S Eqs. 21 and 22 were computed as in this work, except that Jackson and Sternberg (1995) used ␥cav ⫽ 42.0 cal/mol/Å2, whereas Polticelli et al. (1999) used ␥ ⫽ 58.2 cal/mol/Å2 with curvature-corrected accessible area. Furthermore, both groups used atomic charges and radii from the PARSE parameter set (Sitkoff et al., 1994), and a dielectric constant elec of two for the protein in computing ⌬⌬Gsolv,PB , but a dielectric constant of one for the protein was used in this work because the CHARMM vdW radii and atomic charges have been parameterized using ⑀ ⫽ 1. Note that the ⌬Eelec sln in Eqs. 20 and 21 corresponds to the electrostatic interactions between the two proteins embedded in a medium of dielectric constant two. In general, the internal dielectric constant of a protein is ambiguous in computing free energy components directly in solution using a continuum dielectric model. However, it can be unambiguously set to unity in computing gas-phase ⌬Eelec using Schemes 1–3.

nonel ⌬G⬚ ⬇ ⌬⌬Gsolv ⫹ ⌬Gelec

⫺ T⌬Sconf ⫺ T⌬Strans⫹rot ⫺ T⌬Scratic.


⌬⌬Gnonel solv was computed using Eq. 5 with ␥ set to 25 cal/mol/Å2, whereas ⌬Gelec was computed from a solventscreened Coulomb potential with ⑀ ⫽ 4rij,

冘 冘 16q␲qr .


⌬Gelec ⫽


i⫽1 j⫽i⫹1

i j

2 ij


The ⫺T⌬Sconf term was estimated by 0.6N, where N is the number of immobilized side-chain torsions, and the ⫺T⌬Strans⫹rot and ⫺T⌬Scratic terms were set to 9 and 2 kcal/mol, respectively. Like Novotny and coworkers, Jackson and Sternberg (1995) and Polticelli et al. (1999) used the rigid-binding approximation and computed the free energy components for trypsin䡠BPTI binding directly in solution. Both groups omitted the entropy terms (T⌬Strans⫹rot⫹vib) because their focus was on relative rather than absolute binding affinities. In addition, Jackson and Sternberg (1995) neglected protein–protein and protein–solvent vdW interactions assuming that atoms making interactions at the interface will also make contacts with water in the unbound state. Hence, they assumed that ⫺T⌬Sconf ⬇ ⌬Evdw ⫹ ⌬⌬GvdW solv and estimated the binding free energy by cav elec elec ⌬G⬚ ⬇ ⌬⌬Gsolv ⫹ ⌬⌬Gsolv,PB ⫹ ⌬Esln .


Polticelli et al. (1999) also did not consider protein–protein and protein–solvent vdW interactions explicitly, but assumed that any difference will be implicitly taken into account by ␥ (see below) (Honig et al., 1993), Biophysical Journal 81(2) 737–750

nonel ⌬⌬Gsolv and T⌬Sconf compensation conf Using Scheme 1, ⌬⌬Gnonel for solv was found to offset ⫺T⌬S seven systems for which x-ray structures are available for both the free and bound proteins (see Table 5). Novotny and coworkers also found ⌬⌬Gnonel solv (using ␥ in Eq. 5 equal to 25 2 cal/mol/Å , see above) to be roughly equal to T⌬Sconf for both D1.3䡠HEL (Novotny et al., 1989) and trypsin䡠BPTI binding (Krystek et al., 1993) (see Tables 3 and 4) as well as for the homologous system, HyHEL-5䡠HEL, and McPC603䡠phosphocholine (Novotny et al., 1989). Likewise, in computing the change in free energy upon binding of HEL to antibody HyHEL-10 arising from point mutations in HEL, Sharp (1998) found that ⌬⌬Gnonel solv (using ␥ ⫽ 25 cal/mol/Å2) and T⌬Sconf contributions compensate to some extent. If this feature is indeed a general one, then the hydrophobic effect and the loss in side-chain conformational entropy do not dictate high binding affinity. Support for this comes from the observed poor correlation between the change in free energy on mutating a side chain to alanine and the change in total (or hydrophobic) side-chain SASA on complex formation (Bogan and Thorn, 1998). Further corroboration comes from the large and favorable (negative) enthalpy changes observed for several protein–protein association reactions (Bhat et al., 1994; Pearce et al., 1996). For example, the binding between barnase and barstar is enthalpically driven with negligible entropy change (Frisch et al., 1997). In contrast, previous works have attributed hydrophobicity to be the major factor stabilizing protein– protein association (Kauzmann, 1959; Chothia and Janin, 1975; Horton and Lewis, 1992; Froloff et al., 1997).

Protein–Protein Free Energy Decomposition


vdW interactions (⌬Evdw)

Vibrational entropy (⌬Svib)

Tables 3 and 4 show that the single largest attractive contribution to the binding free energy in the two systems studied here is ⌬EvdW. This is in accord with the free energy perturbation results of Miyamoto and Kollman (1993) for the highaffinity interaction between biotin and streptavidin, whose binding free energy (⌬Gexp ⫽ ⫺18.3 cal/mol) is similar to that for trypsin and BPTI. In contrast, Novotny and coworkers neglected ⌬EvdW, assuming that this term contributes little to the overall free energy for D1.3䡠HEL and trypsin䡠BPTI binding. As well, Jackson and Sternberg (1995) neglected protein– protein and protein–solvent vdW interactions, assuming that conf their sum, ⌬EvdW ⫹ ⌬⌬GvdW (see above). Howsolv ⬇ ⫺T⌬S vdW vdW ever, using Scheme 1, ⌬E ⫹ ⌬⌬Gsolv for D1.3䡠HEL binding (⫺51 kcal/mol) opposes, but does not cancel, the corresponding ⫺T⌬Sconf term (9 kcal/mol), and ⌬EvdW ⫹ ⌬⌬GvdW solv for trypsin䡠BPTI complexation is positive (3 kcal/mol), and does not even oppose the respective ⫺T⌬Sconf term, which is also positive. In other studies (Horton and Lewis, 1992; Nicholls et al., 1991; Shen, 1997; Vajda et al., 1994; Williams et al., 1991; Wilson et al., 1991) protein–protein vdW interactions at the interface were thought to be balanced by protein– water vdW interactions in the unbound state. Although this may seem to be the case for trypsin䡠BPTI binding because ⌬EvdW is largely offset by ⌬⌬GvdW solv (⫺62 versus 65 kcal/mol, Table 4), the former is almost twice the latter for D1.3䡠HEL complexation (⫺107 versus 56 kcal/mol, Table 3).

The four lowest nonzero vibrational frequencies obtained here for HEL and BPTI are in good agreement with those reported in previous studies (Gibrat and Go, 1990; Janezic and Brooks, 1995) (see Table 6). Furthermore, the mean square fluctuations of the backbone atoms obtained from normal mode analyses of HEL and BPTI were found to be in good agreement with those deduced from B-factors (Gibrat and Go, 1990; Janezic and Brooks, 1995). The four lowest frequencies in the D1.3䡠HEL and trypsin䡠BPTI complexes were found to be significantly lower than those in the free structures. However, the overall vibrational entropy change opposes D1.3䡠HEL binding but favors trypsin䡠BPTI complexation. The finding that vibrational entropy can make opposite contributions to protein–protein association is consistent with previous normal mode analysis studies, which found that vibrational entropy opposes subtilisin-eglin C complexation (Ishida et al., 1998), but favors insulin dimerization (Tidor and Karplus, 1994). Furthermore, the ⫺T⌬Svib values for D1.3䡠HEL (7.0 kcal/mol) and trypsin䡠BPTI (⫺5.0) binding are similar in magnitude to those for subtilisineglin C complexation (9.0 kcal/mol) (Ishida et al., 1998) and insulin dimerization (⫺7.2 kcal/mol), respectively. Errors in the normal mode vibrational frequencies due to truncation and anharmonic effects are present in both the free and bound states and are expected to cancel, thus they are unlikely to change the key findings of this work. Net electrostatic interactions (⌬Gelec)

Translational and rotational entropy (⌬Stransⴙrot) Using Scheme 1, the loss of translational and rotational degrees of freedom (⫺T⌬Strans⫹rot) was found to oppose D1.3䡠HEL and trypsin䡠BPTI binding by 26 to 28 kcal/mol. The difference between this value and that (9 kcal/mol, see above) used by Novotny and coworkers (Krystek et al., 1993) is because the latter is an estimate in solution (Page and Jencks, 1971), whereas the T⌬Strans⫹rot obtained in this work was computed from Eq. 14 in the gas phase. However, the T⌬Strans⫹rot obtained for D1.3䡠HEL and trypsin䡠BPTI binding is similar to that computed for insulin dimerization (⫺27 kcal/mol) (Tidor and Karplus, 1994) and for DNAEcoRi complexation (⫺32 kcal/mol) (Jayaram et al., 1999), which are also based on ideal gas partition functions. The similarity of the magnitude of T⌬Strans⫹rot computed for the noncovalent association of molecules with varying masses suggests a near cancellation of the mass-dependent terms in the translational and rotational entropy (Eq. 14). Hence, Scheme 1 leads to a binding free energy that is nearly independent of molecular mass (Gilson et al., 1997). Because the magnitude of T⌬Strans⫹rot is comparable to that of ⌬EvdW and ⌬Gelec, it should not be neglected in computing absolute binding free energies.

Using Scheme 1, the net electrostatic effect is to oppose D1.3䡠HEL and trypsin䡠BPTI binding (⌬Gelec positive). In contrast, Novotny and coworkers found the net ⌬Gelec to be negative for D1.3䡠HEL (Novotny et al., 1989) and trypsin䡠BPTI binding (Krystek et al., 1993) (Tables 3 and 4). This is because they computed the ⌬Gelec term from a solvent-screened Coulomb potential (Eq. 20) that neglects charge solvation/desolvation effects. In later works, Novotny et al. (1997) computed ⌬Gelec using PB methods and an internal dielectric constant of 4, which yielded a positive value for the homologous HyHEL-10䡠HEL system (see Table 3, last column). The same (positive ⌬Gelec) trend was also found by Jackson and Sternberg (1995) and Polticelli et al. (1999) for trypsin䡠BPTI binding. This feature is observed in the binding of other proteins; e.g., the binding of a series of peptides to MHC class I compounds (Froloff et al., 1997), the binding of DNA to ␭cI repressor (Misra et al., 1998) and EcoRi endonuclease (Jayaram et al., 1999), and dimer formation by the capsids of three icosahedral viruses (Reddy et al., 1998) and by the GCN4 leucine zipper (Hendsch and Tidor, 1999). Thus, it seems that the net electrostatic interactions generally oppose formation of complexes with an interface area between 1400 and 1700 Å2, but it is not clear if this is the case for near neutral proteins that form smaller interface areas, which are likely to form weaker complexes. Biophysical Journal 81(2) 737–750


As mentioned above, electrostatic protein–solvent interactions are found to oppose binding (positive ⌬⌬Gelec solv), whereas hydrogen bonding or charge– charge protein–protein interactions drive complexation (negative ⌬Eelec gas ⫹ ⌬⌬Eelec ). This was also found by Novotny et al. (1997) for isom D1.3䡠HEL binding and by Jackson and Sternberg (1995) for trypsin䡠BPTI binding (see Tables 3 and 4). Xu et al. (1997) found that salt bridges across the binding interface can significantly stabilize complexes. In sharp contrast, Polticelli et al. (1999) found the opposite for trypsin䡠BPTI binding: the solvation term was found to favor binding (⌬⌬Gelec solv ⫽ ⫺101.9 kcal/mol), whereas the Coulomb energy was found to oppose binding (⌬Eelec gas ⫽ 149.7 kcal/mol). They attributed the latter to Coulomb repulsion between a ⫹8e trypsin and ⫹6e BPTI, and the former to the more favorable solvation of the ⫹14e complex compared to the free proteins. Because the aforementioned groups computed the solvation term using the same continuum dielectric method (PB), the observed discrepancy may be due to the higher net charge on trypsin of ⫹8e used by Polticelli et al. (1999) (as opposed to ⫹6e in this work, see Table 1), which probably resulted in significant conformational changes, as evidenced by the much smaller trypsin䡠BPTI interface area of 1231 Å2 (see Table 4). CONCLUSIONS 1. For proteins that form high-affinity complexes, relatively accurate absolute binding free energies can be obtained using Scheme 1 and the GB model to compute the electrostatic solvation free energy, provided accurate x-ray structures for the free proteins and respective complex are available. The latter is required, not only to account for conformational changes upon binding, but because ⌬Evdw , ⌬Gelec and ⌬Svib are sensitive to the 3-dimensional structure (see Tables 3 and 4). 2. There are three major advantages in using Scheme 1. First, each of the free energy components in Eq. 1 corresponds to a well-defined physical process. Second, by separating the gas-phase protein–protein interactions from protein–solvent and solvent–solvent interactions, the electrostatic components do not depend on the internal dielectric constant of the protein, whose value is uncertain in computing the electrostatic components of the binding free energy directly in solution. Third, the translational, rotational, and vibrational entropy changes can be computed in a straightforward manner (using Eqs. 14 and 15) in the gas-phase, but not in solution. 3. In cases where an accurate complex crystal structure is known but not its unbound counterparts, approximating [r]sln by [Rmin⫹cons]sln and [l]sln by [Lmin⫹cons]sln can yield trends in the relative contributions of the component terms to the binding free energy. 4. Neglecting vdW interactions from the free energy function can be a source of large errors because the gain in Biophysical Journal 81(2) 737–750

Noskov and Lim

protein–protein vdW interaction energy at the interface may not be totally offset by the loss of protein-water vdW energy in the unbound state. 5. The following findings apply to protease—inhibitor and antibody–antigen complexes with an interface area in the range of 1400 –1700 Å2. conf a. Due to the observed ⌬⌬Gnonel comsolv and ⫺T⌬S pensation, the magnitude of the binding affinity can be estimated from three components using Scheme 1 (see Eq. 17); viz., net gas-phase, protein–protein vdW effects (⌬Evdw); net protein–protein and protein–solvent electrostatic effects (⌬Gelec); and the sum of gas-phase translational, rotational, and vibrational entropy changes (⫺T⌬Strans⫹rot⫹vib). b. The forces driving antibody–antigen and protease– inhibitor noncovalent association stem from vdW and electrostatic (including salt bridges and hydrogen bonds) protein–protein interactions between complementary surfaces over a relatively large area (1400 – 1700 Å2) (Pauling, 1974; Pauling and Pressman, 1945). In contrast, vdW and electrostatic protein– solvent interactions oppose antibody–antigen and protease–inhibitor binding. These two opposite trends may be a general feature of highly charged complexes with an interface area in the range of 1400 –1700 Å2. c. As a consequence of b., the affinity of an antigen (inhibitor) for an antibody (protease) may be improved by increasing the close packing and hydrogen bonding (as opposed to charge– charge) interactions between the two proteins, and reducing the electrostatic cost of desolvating the free proteins (e.g., using water molecules to mediate charge– charge interactions at the protein interface) (Bhat et al., 1994).

We are grateful to Professor M. Karplus for the CHARMM program. S.N. is supported by a postdoctoral fellowship from Academia Sinica. This work is supported by the Institute of Biomedical Sciences at Academia Sinica, the National Center for High Performance Computing, and the National Science Council, Republic of China.

REFERENCES Andrews, P. R., D. J. Craik, and J. L. Martin. 1984. Functional group contributions to drug–receptor interactions. J. Med. Chem. 27: 1648 –1657. Babu, S., and C. Lim. 1999. Theory of ionic hydration: insights from molecular dynamics simulations and experiment. J. Phys. Chem. B. 103: 7958 –7968. Babu, S., and C. Lim. 2001. Incorporating nonlinear solvent response in continuum dielectric models using a two-sphere description of the born radius. J. Phys. Chem. A. 105:5030 –5036. Bernstein, F. C., T. F. Koetzle, G. J. B. Williams, E. F. Meyer, M. D. Brice, J. R. Rodgers, O. Kennard, T. Shimanouchi, and M. Tasumi. 1977. The Protein Data Bank: a computer-based archival file for macromolecular structures. J. Mol. Biol. 122:535–542.

Protein–Protein Free Energy Decomposition Bhat, N. T., G. A. Bentley, G. Boulot, M. I. Greene, D. Tello, W. Dall’acqua, H. Souchon, F. P. Schwarz, R. A. Mariuzza, R. J. and Poljak. 1994. Bound water molecules and conformational stabilization help mediate an antigen–antibody association. Proc. Natl. Acad. Sci. U.S.A. 91:1089 –1093. Bogan, A. A., and K. S. Thorn. 1998. Anatomy of hot spots in protein interfaces. J. Mol. Biol. 280:1–9. Bohm, H. J. 1994. The development of a simple empirical scoring function to estimate the binding constant for a protein–ligand complex of known three-dimensional structure. J. Comput. Aided Mol. Des. 8:243–256. Born, M. 1920. Z. Phys. 1:45– 48. Brooks, B. R., R. E. Bruccoleri, B. D. Olafson, D. J. States, S. Swaminathan, and M. Karplus. 1983. CHARMm: a program for macromolecular energy, minimization, and dynamics calculations. J. Comp. Chem. 4:187–217. Chothia, C., and J. Janin. 1975. Principles of protein–protein recognition. Nature. 256:705–708. Dominy, B. N., and C. L. Brooks, III. 1999. Development of a generalized Born model parametrization for proteins and nucleic acids. J. Phys. Chem. B. 103:3765–3773. Friedman, R. A., and B. Honig. 1995. A free energy analysis of nucleic acid base stacking in aqueous solution. Biophys. J. 69:1528 –1535. Frisch, C., G. Schreiber, J. C. M., and, F. A. R. 1997. Thermodynamics of the interaction of barnase and barstar: changes in the free energy vs. changes in enthalpy on mutation. J. Mol. Biol. 267:696 –706. Froloff, N., A. Windemuth, and B. Honig. 1997. On the calculation of binding free energies using continuum methods: application to MHC class I protein–peptide interactions. Prot. Sci. 6:1293–1301. Gibas, C. J. 1997. pH dependence of antibody/lysozyme complexation. Biochemistry. 36:15599 –15614. Gibrat, J.-F., and N. Go. 1990. Normal mode analysis of human lysozyme: study of the relative motion of the two domains and characterization of the harmonic motion. Proteins: Struct. Func. Genet. 8:258 –279. Gilson, M. K., J. A. Given, B. A. Bush, and J. A. McCammon. 1997. The statistical-thermodynamic basis for computation of binding affinities: a critical review. Biophys. J. 72:1047–1069. Gilson, M. K., and B. Honig. 1988. Calculation of the total electrostatic energy of macromolecular system: solvation energy, binding energies and conformational analysis. Proteins: Struct. Func. Genet. 4:7–18. Hendsch, Z., and B. Tidor. 1999. Electrostatic interaction in the GCN4 leucine zipper: substantial contribution arise from intramolecular interaction enhanced on binding. Protein Sci. 8:1381–1392. Honig, B., and A. Nicholls. 1995. Classical electrostatics in biology and chemistry. Science. 268:1144 –1149. Honig, B., K. Sharp, and A. Yang. 1993. Macroscopic models of aqueous solutions: biological and chemical application. J. Phys. Chem. 97: 1101–1109. Horton, N., and M. Lewis. 1992. Calculation of free energy of association for protein complexes. Protein Sci. 1:169 –181. Ishida, H., Y. Jochi, and A. Kidera. 1998. Dynamic structure of subtilisineglin c complex studied by normal mode analysis. Proteins: Struct. Func. Genet. 32:324 –333. Jackson, R. M., and M. J. E. Sternberg. 1995. A continuum model for protein–protein interactions: application to the docking problem. J. Mol. Biol. 250:258 –275. Janezic, D., and B. Brooks. 1995. Harmonic analysis of large systems. II. Comparison of different protein models. J. Comp. Chem. 16:1543–1553. Jayaram, B., K. J. McConnell, B. D. Surjit, and D. L. Beveridge. 1999. Free energy analysis of protein-DNA binding: The EcoRI EndonucleaseDNA complex. J. Comp. Phys. 151, 333–357. Jorgensen, W. L., and J. Tirado-Rives. 1995. Free energies of hydration for organic molecules from Monte Carlo simulations. In Perspectives in Drug Discovery and Design, Vol. 3, K. Muller (editor). ESCOM Science Publishers, Leiden, The Netherlands. 123–138. Kauzmann, W. 1959. Some factors in the interpretation of protein denaturation. Adv. Prot. Chem. 14:1– 63.

749 Kollman, P. A. 1996. Advances and continuing challenges in achieving realistic and predictive simulations of the properties of organic and biological molecules. Acc. Chem. Res. 29:461– 469. Kollman, P. A., I. Massova, C. Reyes, B. Kuhn, S. Huo, L. Chong, M. Lee, T. Lee, Y. Duan, W. Wang, O. Donini, P. Cieplak, J. Srinivasan, D. A. Case, and T. E. Cheatham, III. 2000. Calculating structures and free energies of complex molecules: combining molecular mechanics and continuum models. Acc. Chem. Res. 33:889 – 897. Krystek, K., T. Stouch, and J. Novotny. 1993. Affinity and specificity of serine endopeptidase–protein inhibitor interactions: empirical free energy calculations based on x-ray crystallographic structures. J. Mol. Biol. 234:661– 679. Lee, B., and F. M. Richards. 1971. The interpretation protein structures: estimation of static accessibility. J. Mol. Biol. 55:379 – 400. Luzzati, V. 1952. Traitement statistique des erreurs dans la determination des structures cristallines. Acta Cryst. 5:802– 810. MacKerell, J. A. D., D. Bashford, M. Bellott, R. Dunbrack, J. D. Evanseck, M. J. Field, S. Fischer, J. Gao, H. Guo, S. Ha, D. Joseph-McCarthy, L. Kuchnir, K. Kuczera, F. T. K. Lau, C. Mattos, S. Michnick, T. Ngo, D. T. Nguyen, B. Prodhom, W. E. I. Reiher, B. Roux, M. Schlenkrich, J. C. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz-Kuczera, D. Yin, and M. Karplus, M. 1998. All-hydrogen empirical potential for molecular modeling and dynamics studies of proteins using the CHARMM22 force field. J. Phys. Chem. B. 102:3586 –3616. McCammon, J. A. 1998. Theory of biomolecular recognition. Curr. Opin. Struct. Biol. 8:245–249. McQuarrie, D. A. 1976. Statistical Mechanics, Harper and Row, New York. Misra, V. K., J. L. Hecht, A.-S. Yang, and B. Honig. 1998. Electrostatic contributions to the binding free energy of the ␭cl repressor to DNA. Biophys. J. 75:2262–2273. Miyamoto, S., and P. A. Kollman. 1993. Absolute and relative binding free energy calculations of the interaction of biotin and its analogs with streptavidin using molecular dynamics/free energy perturbations approaches. Proteins: Struct. Funct. Genet. 16:226 –245. Nicholls, A., K. A. Sharp, and B. Honig. 1991. Protein folding and association: insights from the interfacial and thermodynamic properties of hydrocarbons. Proteins: Struct. Func. Genet. 11:281–296. Novotny, J., R. Bruccoleri, M. Davis, and K. A. Sharp. 1997. Empirical free energy calculations: a blind test and further improvements to the method. J. Mol. Biol. 268:401– 414. Novotny, J., R. Bruccoleri, and F. A. Saul. 1989. On the attribution of binding energy in antigen–antibody complexes McPC 603, D1.3, and HyHEL-5. Biochemistry. 28:4735– 4746. Ohtaki, H., and N. Fukushima. 1992. A structural study of saturated aqueous solutions of some alkali halides by x-ray diffraction. J. Sol. Chem. 21:23–25. Olson, M. A. 1999. Mean-field analysis of protein–protein interaction. Biophys. Chem. 75:115–128. Page, M. I., and W. P. Jencks. 1971. Entropic contributions to rate accelerations in enzymic and intramolecular reactions and the chelate effect. Proc. Natl. Acad. Sci. U.S.A. 68:1678 –1683. Pascual-Ahuir, J. L., and E. Silla. 1990. GEPOL: an improved description of molecular surfaces. I. Building the spherical surface set. J. Comp. Chem. 11:1047–1060. Pauling, L. 1974. Molecular basis of biological specificity. Nature. 248: 769 –771. Pauling, L., and D. Pressman. 1945. J. Am. Chem. Soc. 67:1003–1012. Pearce, K. H., M. H. Ultsch, K. R. H., V. A. d. M., and J. A. Wells. 1996. Structural and mutational analysis of affinity-inert contact residues at the growth hormone–receptor interface. Biochemistry. 35:10300 –10307. Philippopoulos, M., and C. Lim. 1995. Molecular dynamics simulation of E. Coli ribonuclease H1 in solution: correlation with NMR and x-ray data and insights into biological function. J. Mol. Biol. 254:771–792. Pickett, S. D., and M. J. Sternberg. 1994. Protein side-chain conformational entropy derived from fusion data— comparison with other empirical scales. Protein Eng. 7:149 –155. Biophysical Journal 81(2) 737–750

750 Pickett, S. D., and M. J. E. Sternberg. 1993. Empirical scale of side-chain conformational entropy in protein folding. J. Mol. Biol. 231:825– 839. Polticelli, F., P. Ascenzi, M. Bolognesi, and B. Honig. 1999. Structural determinants of trypsin affinity and specificity for cationic inhibitors. Prot. Sci. 8:2621–2629. Qiu, D., P. S. Shenkin, F. P. Hollinger, and W. C. Still. 1997. The GB/SA continuum model for solvation. A fast analytical method for the calculation of approximate Born radii. J. Phys. Chem. 101:3005–3014. Reddy, V. S., H. A. Giesing, R. Morton, A. Kumar, C. Post, C. L. I. Brooks, and J. E. Johnson. 1998. Energetics of quasiequivalence: computational analysis of protein–protein interactions in icosahedral viruses. Biophys. J. 74:546 –558. Sharp, K. A. 1998. Calculation of HyHel10-Lysozyme binding free energy changes: effect of ten point mutations. Proteins: Struct. Func. Genet. 33:39 – 48. Sharp, K. A., A. Nicholls, R. F. Fine, and B. Honig. 1991. Reconciling the magnitude of the macroscopic and microscopic hydrophobic effects. Science. 252:106 –109. Shen, J. 1997. A theoretical investigation of tight-binding thermolysin inhibitors. J. Med. Chem. 40:2953–2958. Sitkoff, D., K. A. Sharp, and B. Honig. 1994. Accurate calculation of hydration free energies using macroscopic solvent models. J. Phys. Chem. 98:1978 –1988. Still, W. C., A. Tempczyk, R. C. Hawley, and T. Hendrickson. 1990. Semianalytical treatment of solvation for molecular mechanics and dynamics. J. Am. Chem. Soc. 112:6127– 6129. Tanokura, M. 1983. 1H-NMR study on the tautomerism of the imidazole ring of histidine residues. I. Microscopic pK values and molar ratios of tautomers in histidine-containing epptides. Biochim. Biophys. Acta. 742: 576 –585. Tembe, B. L., and J. A. McCammon. 1984. Ligand–receptor interactions. Comput. Chem. 8:281–283.

Biophysical Journal 81(2) 737–750

Noskov and Lim Tidor, B., and M. Karplus. 1994. The contribution of vibrational entropy to molecular association. The dimerization of insulin. J. Mol. Biol. 238: 405– 414. Vajda, S., Z. Weng, R. Rosenfeld, and C. DeLisi. 1994. Effect of conformational flexibility and solvation on receptor–ligand binding free energies. Biochemistry. 33:13977–13988. Verhoeyen, M., C. Milstein, and G. Winter. 1988. Reshaping human antibodies: grafting an antilysozyme activity. Science. 239:1534 –1535. Vincent J. P., and M. Lazdunski. 1972. Trypsin-pancreatic trypsin inhibitor association. Dynamics of the interaction and role of disulfide bridges. Biochemistry. 11:2967–2979. Weng, Z., C. Delisi, and S. Vajda. 1997. Empirical free energy calculation: comparison to calorimetric data. Protein Sci. 6:1976 –1984. Williams, D. H., J. P. Cox, A. J. Doig, M. Gardner, I. A. Nicholls, C. J. Salter, and R. C. Mitchell. 1991. Toward the semiquantitative estimation of binding constants: guides for peptide–peptide binding in aqueous solution. J. Am. Chem. Soc. 113:7020 –7030. Williams, J. P., M. S. Searle, J. P. Mackay, U. Gerhard, and R. A. Maplestone. 1993. Toward an estimation of binding constants in aqueous solution. Proc. Natl. Acad. Sci. U.S.A. 90:1172–1178. Wilson, C., J. E. Mace, and D. A. Agard. 1991. Computational method for the design of enzymes with altered substrate specificity. J. Mol. Biol. 220:495–506. Xavier, K. A., and R. C. Wilson. 1998. Association and dissociation kinetics of anti-hen egg lysozyme monoclonal antibodies HyHEL-5 and HyHEL-10. Biophys. J. 74:2036 –2045. Xu, D., S. L. Lin, and R. Nussinov. 1997. Protein binding vs. protein folding: the role of hydrophilic bridges in protein associations. J. Mol. Biol. 265:68 – 84. Zou, X., Y. Sun, and I. D. Kuntz. 1999. Inclusion of solvation in ligand binding free energy calculations using the generalized-Born model. J. Am. Chem. Soc. 121:8033– 8043.