Physical Properties Governing Ligand Receptor Binding

4.09.3.1 Steric

As the lock-and-key model suggests, shape complementarity is very important for ligand-receptor binding and specificity. The overlap of atoms of the ligand and the receptor is prohibited by electron-electron repulsion. Clashes draw a large energetic penalty. In studies of receptor-ligand interactions, this penalty is usually modeled by the first term in the Lennard-Jones energy function8'9:

LJ r12 r6

The sum is over all ligand-receptor atom pairs and is a function of the distance, r, between the atoms in each pair, which have van der Waals' radii R and Rr.

and C is given by the Slater-Kirkwood formula and is dependent on atomic polarizability and the number of effective electrons per atom.

The functional form of the repulsive part is purely empirical and was chosen by Lennard-Jones to fit experimental data for rare gases and because of its computational convenience. In some studies of ligand-receptor interactions, it is useful to make the repulsion less abrupt by using a lower power of distance, e.g., r_ 8. This has the advantage that the repulsive energy is less sensitive to the atomic positions and thus more robust to errors in atomic coordinates or to the effects of atomic motions.

The attractive term in the Lennard-Jones function describes van der Waals' interactions due to dispersive attractions. These induced dipole-induced dipole interactions have an r_ 6 distance dependence and, while less strongly distance dependent than the repulsive interactions, are short range interactions.

Ligand-receptor binding involves the replacement of ligand-water and receptor-water interactions by ligand-receptor and water-water interactions. Whether van der Waals' interactions contribute favorably to ligand-receptor affinity or not depends on the strength of the ligand-receptor van der Waals' interactions compared to the interactions with water. Owing to this balance, any favorable contribution of van der Waals' interactions to binding affinity is typically small.

Figure 2 Water in receptor-ligand complexation. (a) Major urinary protein-I (MUP-I) with bound ligand (PDB code 1I06). Two waters (violet spheres) in hydrophobic environment of ligand binding pocket (surface representation) make possible sites for interactions with polar groups present on the lipophilic pheromones (stick representation).159 (b) The complex of the periplasmic oligopeptide binding protein OppA with peptide (LysGluLys) (PDB code 1JEU). Only active center residues of the protein are shown (stick representation). Water molecules (violet spheres) mediate interaction between OppA (periplasmic oligopeptide binding protein) and its peptide ligands.160 (c) The protein kinase C interacting protein (PKCI) dimer (PDB code 1KPA) has a dry interface with most crystallographically observed water molecules (violet spheres) forming a ring around the interface.161

Figure 2 Water in receptor-ligand complexation. (a) Major urinary protein-I (MUP-I) with bound ligand (PDB code 1I06). Two waters (violet spheres) in hydrophobic environment of ligand binding pocket (surface representation) make possible sites for interactions with polar groups present on the lipophilic pheromones (stick representation).159 (b) The complex of the periplasmic oligopeptide binding protein OppA with peptide (LysGluLys) (PDB code 1JEU). Only active center residues of the protein are shown (stick representation). Water molecules (violet spheres) mediate interaction between OppA (periplasmic oligopeptide binding protein) and its peptide ligands.160 (c) The protein kinase C interacting protein (PKCI) dimer (PDB code 1KPA) has a dry interface with most crystallographically observed water molecules (violet spheres) forming a ring around the interface.161

Nevertheless, consideration of steric complementarity is critical for ligand design. Empty vacuum cavities at the interface between ligand and receptor are energetically unfavorable: Nature abhors a vacuum. Interfacial cavities that are large enough to accommodate one or more water molecules may be hydrated and, if suitably filled, be energetically favorable. This depends not only on the size of the cavity but also on its polarity and ability to make hydrogen bonds with water molecules. Small nonpolar cavities with no hydrogen-bonding groups will be unoccupied whereas nonpolar cavities able to accommodate c. four or more water molecules may be hydrated if the water molecules themselves can make a hydrogen bond network.

The packing and the hydration patterns at ligand-receptor interfaces have been studied in crystal structures. Some interfaces are well packed with water molecules totally excluded from them (see, e.g., Figure 2c). Ordered interfacial water sites are often observed at positions where water molecules can mediate hydrogen bonds between receptor and ligand (Figure 2b). Particularly ordered water sites are sometimes considered to be an intrinsic part of the solute molecule. Interfacial water sites can alter the specificity of a binding site due to their ability to accept and donate up to four hydrogen bonds. As shown in the example in Figure 2a, interfacial water molecules may permit a rather hydrophobic receptor site to interact with a ligand and accommodate its hydrogen-bonding ability.

To optimize steric complementarity in ligand design, a means to detect interfacial cavities and crevices is necessary. A solvent accessible surface or a molecular (Connolly) surface can be computed by rolling a spherical probe with a radius of 1.4 A, corresponding to the size of a water molecule, over the solute molecules.11 The size of the probe can be varied to represent different molecules, ions, or molecular fragments, and a range of analytical and numerical algorithms can be used to compute surfaces and identify cavities. A useful tool, which employs alpha shape theory to identify pockets and cavities in protein structures, is the CASTp webserver.12'13'171

Design of a ligand to fill a crevice or an interfacial space requires not only knowing the size and shape of the space but also its physicochemical properties. Suitable functional groups to add to a molecule can be identified by computing molecular interaction fields for probes with hydrogen-bonding and electrostatic properties as well as steric ones. In the GRID approach,14,172 an empirical energy function is used to describe the interaction between probe and target, and energetically favorable binding sites for a range of chemical probes, including a water probe, can be mapped out. There are many examples of successful modification of ligands to fill interfacial cavities. The modifications not only need to fill space but also have the appropriate chemical properties. They may also need to displace interfacial water molecules. This was, for example, the approach taken for inhibitors of scytalone dehydratase, which were designed to displace a water molecule from the active site.15 Further examples are given in the next sections.

4.09.3.2 Electrostatic

Electrostatic interactions play a significant role in ligand-receptor binding. They are long-range, varying with distance as r_ 1, and therefore can be particularly important for molecular recognition. Each molecule can be considered to have not only a net charge but also a charge distribution over the whole molecule. In modeling studies of ligand-receptor binding, partial charges are usually assigned to atom centers on the basis of ab initio or semiempirical quantum mechanical calculations or simpler approaches such as electronegativity equalization. Like charges repel and unlike charges attract. Consequently, many positive and negative terms contribute to the coulombic electrostatic energy of a ligand-receptor complex, which is given by:

4re£o£ r where 80 is the permittivity of free space, 8 the relative dielectric constant of the surrounding medium, and q/ and qr are partial atomic point charges in the ligand and receptor, respectively.

Coulomb's law applies for a homogeneous dielectric medium. If all atoms of the system are modeled explicitly, including all water molecules and ions in the solvent, and the system is subjected to molecular dynamics simulation, 8 = lis usually used. Often, however, the water molecules and ions are treated implicitly and an aqueous solvent is modeled as a continuum having 8e80. The ions are assumed to have a Boltzmann distribution. In this case, the dielectric constant of the solvent medium differs from that of the molecular solutes. The electrostatic potential of this heterogeneous system is described by the Poisson-Boltzmann equation.

The Poisson-Boltzmann equation (or its linearized form) may be solved numerically for biomacromolecules. Finite difference and multigrid methods on grid(s) superimposed on the target molecule are the most commonly used methods to solve the Poisson-Boltzmann equation. Details of solution of the Poisson-Boltzmann equation for biomacromolecules are given in references.16-19 To solve the Poisson-Boltzmann equation, a suitable value of the dielectric constant for the molecular interior and the definition of the dielectric boundary must be chosen. These are adjustable parameters to be fitted in the context of the complete energy function, the treatment of molecular flexibility, and the properties to be computed.20'21 The dielectric boundary can be chosen as the van der Waals' surface, the solvent accessible molecular surface (mapped by a solvent probe surface) or the solvent accessible surface (mapped by a solvent probe center). The choice of dielectric boundary definition can significantly impact the magnitude of electrostatic binding free energies.22,23

In the Poisson-Boltzmann model, the electrostatic binding free energy consists of ligand charge-receptor charge interaction terms (of similar functional form to coulombic interactions) and ligand and receptor charge desolvation terms, which depend on the square of each partial charge. The charge interaction energy is the first and the charge desolvation energies are the second and third terms on the right-hand side of this equation:

Eel-PB = Pl®r + TjP l (®bound, l - Ffree, l) + TjPr (®bound, r - Ffree, r)

p/ and pr are the charge distributions over ligand and receptor, respectively, and F/ and Fr the electrostatic potentials of the ligand and the receptor, respectively, and are computed in the presence (bound) and absence (free) of the other molecule's dielectric cavity. The potential of a charge q, in homogeneous dielectric e at distance r is:

4ne0e r

Charge desolvation due to the transfer of a charge from a high to a low dielectric medium disfavors binding. Thus, the overall energetic contribution of electrostatic interactions to the binding affinity of a complex may be small or even unfavorable, while at the same time being very important for specificity. Binding affinity can be improved by systematically optimizing the charge distribution of a ligand considering the balance between charge interaction and charge desolvation terms; an example is the design of improved inhibitors of HIV-1 cell entry.24

The computational demands of solution of the Poisson-Boltzmann equation can be prohibitive for some studies of ligand-receptor interactions, so simpler electrostatic models are often used. The effective charge model25 permits calculation of electrostatic forces between two molecules based on Poisson-Boltzmann calculations for the individual molecules, thus permitting their electrostatic interactions to be computed quickly for many different molecular arrangements. It reproduces the Poisson-Boltzmann forces well at intermolecular separations greater than one water molecule. Other methods to account for the dielectrically discontinuous boundary between molecules and the implicit solvent are the method of images (e.g., assuming an infinite planar boundary14) or the generalized Born model.26'27 A simple and computationally efficient but rather rough approximation is to use Coulomb's law with e = nr (n is an integer), i.e., dependent on interatomic distance, r. Most treatments of electrostatics in drug design neglect charge desolvation effects and thus provide an incomplete description of electrostatic interactions.

Specific polarization effects, beyond those modeled by a continuum dielectric model and the movement of atoms, are usually neglected. Many-body effects are also neglected by use of a pair-wise additive energy function. Polarizable forcefields are, however, becoming more common in the molecular mechanics force fields used for molecular dynamics simulations. Furthermore, polarization effects can be considered directly by treating the ligand, and sometimes the surrounding protein, quantum mechanically. The full ab initio quantum mechanical treatment of a protein for computing ligand-receptor binding affinities was recently achieved for a study of ligand binding to the human estrogen receptor alpha.28

The strength of electrostatic interactions in a ligand-receptor complex can be investigated experimentally by measuring the ionic strength dependence and pH dependence of binding properties. The strength of electrostatic interactions varies widely. Electrostatic interactions are of clear importance for the recognition and binding properties of highly charged molecules such as nucleic acids. However, all types of bound ligand-receptor complexes can show electrostatic complementarity at the interface. It is not so much the net charge of a molecule as its charge distribution that is important. For example, in many protein-protein complexes, the two proteins have net charges of the same sign but bind with electrostatically complementary interfaces. Importantly, the electrostatic potentials at the interface are complementary, not necessarily the adjacent charges at the interface.29 This is due to the long-range nature of the electrostatic interactions as well as the effects of the heterogeneous dielectric constant. Complementary electrostatic interactions are shown in Figure 3a and d for inhibitor-enzyme and peptide-signaling domain complexes, respectively. Solvation effects may make electrostatic interactions quite complex as shown by the examples in Figure 3b and c for the urokinase-type plasminogen activator with two different inhibitors bound at its active site.30 In Figure 3c the interactions of the inhibitor with the whole active site are such that the carboxylate group of Asp189 and the inhibitor amidine group are too far apart to make a hydrogen-bonded salt link. Their electrostatically favorable interactions are

Figure 3 Electrostatic interactions. (a) An inhibitor, Aeruginosin 98-B (stick representation), bound to a negatively charged pocket of bovine trypsin (surface representation) (PDB code 1AQ7).162 The electrostatic potential of the protein was calculated with UHBD,173 mapped onto the protein surface and color coded (negative: red; positive: blue). (b) Water (violet sphere) mediated short hydrogen bond network in urokinase-type plasminogen activator (grey sticks) complexed with an inhibitor (green sticks) (PDB code 1GI8). Short hydrogen bonds (dotted lines) are presumed ionic over a large range of pH, with a negative charge distributed among Ophenoh Ooxy, and OSer195 and a positive charge on His57.30 (c) Water (violet sphere)-mediated salt bridge between urokinase-type plasminogen activator Asp189 (grey sticks) and inhibitor (PDB code 1GI8).30 (d) Phosphoserine-proline containing peptide (stick representation, phosphate groups in ball-and-stick representation) bound to a group IV WW domain (surface representation) face with positive electrostatic potential (PDB code 1F8A) (electrostatic potential computed and mapped as in part a).163

Figure 3 Electrostatic interactions. (a) An inhibitor, Aeruginosin 98-B (stick representation), bound to a negatively charged pocket of bovine trypsin (surface representation) (PDB code 1AQ7).162 The electrostatic potential of the protein was calculated with UHBD,173 mapped onto the protein surface and color coded (negative: red; positive: blue). (b) Water (violet sphere) mediated short hydrogen bond network in urokinase-type plasminogen activator (grey sticks) complexed with an inhibitor (green sticks) (PDB code 1GI8). Short hydrogen bonds (dotted lines) are presumed ionic over a large range of pH, with a negative charge distributed among Ophenoh Ooxy, and OSer195 and a positive charge on His57.30 (c) Water (violet sphere)-mediated salt bridge between urokinase-type plasminogen activator Asp189 (grey sticks) and inhibitor (PDB code 1GI8).30 (d) Phosphoserine-proline containing peptide (stick representation, phosphate groups in ball-and-stick representation) bound to a group IV WW domain (surface representation) face with positive electrostatic potential (PDB code 1F8A) (electrostatic potential computed and mapped as in part a).163

instead mediated by a water molecule, whose presence reduces the charge desolvation penalty usually associated with salt-link formation. In Figure 3b inhibitor-enzyme interactions are again mediated by a water molecule, but this time the water molecule helps to stabilize very short hydrogen bonds.

Electrostatic interactions can be exploited in drug design. An example is provided by the anti-influenza drug Relenza. This compound is an inhibitor of the neuraminidase coat protein of the influenza virus. The design strategy (see 4.24.3 Structure-Based Drug Design - The Use of Protein Structure in Drug Discovery and for review, see 31) was to modify a transition state analog on the basis of the crystal structure of the enzyme so as to optimize binding affinity. By probing the protein with different probes with the GRID program,14 an interfacial pocket was identified where a positively charged amino or guanidine group could favorably be added as a substituent to the transition state analog.32 Compounds with these positively charged substituents turned out to be very potent and one of them, Relenza, is used in the clinic. It is delivered to its site of action as a nasal spray: its charged functional groups make it too polar for oral delivery, which requires the compound to be able to pass through membranes.

Subsequent modification to improve inhibitor-protein interactions, the stability of the inhibitor and its bioavailability, resulted in Tamiflu, an anti-influenza drug that is taken orally.33 It is delivered in an uncharged proform and then converted in the body to the active charged form. This shows how charged moieties in ligands can be optimized for binding to the macromolecular target but also illustrates the need to try to consider simultaneously absorption, distribution, metabolism, excretion, and toxicity (ADMET) properties.

4.09.3.3 Hydrogen Bonds

Hydrogen bonds are specific, short-range, directional nonbonded interactions. They are predominantly electrostatic in character, although charge transfer also contributes to their strength. In molecular mechanics force fields, they are usually treated as resulting from the sum of coulombic terms, and this is possible if polar hydrogen atoms are modeled explicitly.

In some force fields, it is also considered necessary to model the positions and point charges of lone pairs of electrons on hydrogen bond acceptor atoms, e.g., sulfur atoms and carboxylate oxygen atoms, in order to model the appropriate hydrogen-bonding geometry. In other energy functions, there is a hydrogen-bonding term in addition to the electrostatic and Lennard-Jones terms. This hydrogen-bonding term models the distance and angular dependence of hydrogen bonds. The GRID energy function, for example, has been specifically designed to reproduce observed hydrogen-bonding geometries in crystal structures of small molecules and of proteins. It has a hydrogen-bonding energy describing the interaction between a probe (p) (a small ligand or fragment of a ligand) and a target (t) receptor atom that is the product of three terms:

Er is dependent on the separation between target and probe nonhydrogen atoms participating in the hydrogen bond. It has the form:

rm fti where M and N depend on the chemical nature of the hydrogen-bonding atoms. Possible values of the m and n parameters are m = 6, n = 4; m = 8, n = 6 (as in GRID); m = 12, and n = 10. The angular terms take different functional forms depending on the chemical types of the hydrogen bonding atoms and whether they are in the probe or the target receptor. The angular dependence differs for the same atom type in the probe and in the target. This is because the target includes the interactions of the hydrogen-bonding atom's neighbors whereas these are absent for the probe, which is able to rotate to an orientation that results in an optimal hydrogen bond energy. When multiple hydrogen bonds are possible, the best combination is found by systematic search or analytically.

Hydrogen bonds are critical for the structure and interactions of biological macromolecules. In proteins, hydrogen-bonding patterns are key signatures of secondary structure elements. The same hydrogen-bonding patterns that occur between backbone atoms in adjacent strands in b-sheets in a single protein chain can serve to 'zip' together two proteins that interact via extended strands - see the example in Figure 4d. The hydrogen bonds between ligands and receptors can of course also adopt many other arrangements (see examples in Figure 4). Hydrogen bonds are important contributors to the specificity of receptor-ligand interactions. The specificity is achieved not only through favorable short-range directionally specific interactions but also because ligand-receptor arrangements that leave hydrogen-bonding capacity unsatisfied are disfavored. It is unfavorable to place the carbonyl oxygen of a ligand so that it is buried pointing into a hydrophobic pocket of a receptor because it cannot make any hydrogen bonds in this location and loses the hydrogen bonds it could make to water molecules in the unbound state.

Figure 4 Various intermolecular hydrogen bond interactions. (a) The catalytically important residue Arg55 (grey sticks) of cyclophilin A (CypA) forms a hydrogen bond with its substrate, the succinyl-Ala-Ala-Pro-Phe-p-nitroanilide (AAPF) peptide (green sticks) (PDB code 1RMH).164 (b) Two GTP analogs (stick representation) in the Ffh/FtsY complex interact with each other via hydrogen bonds between the ribose O3' hydroxyl of one and the g-phosphate oxygen of the other.165 (c) Ligand S-1153 (green sticks) forms two direct hydrogen bonds with the HIV-1 reverse transcriptase (PDB code 1EP4) main-chain carbonyl of Pro236 and nitrogen of Lys-103. A water (violet sphere)-mediated hydrogen bond is formed with the carbonyl oxygen of Lys101.166 (d) Hydrogen bonds between adjacent parallel strands in the kinesin dimer (PDB code 2KIN).

Figure 4 Various intermolecular hydrogen bond interactions. (a) The catalytically important residue Arg55 (grey sticks) of cyclophilin A (CypA) forms a hydrogen bond with its substrate, the succinyl-Ala-Ala-Pro-Phe-p-nitroanilide (AAPF) peptide (green sticks) (PDB code 1RMH).164 (b) Two GTP analogs (stick representation) in the Ffh/FtsY complex interact with each other via hydrogen bonds between the ribose O3' hydroxyl of one and the g-phosphate oxygen of the other.165 (c) Ligand S-1153 (green sticks) forms two direct hydrogen bonds with the HIV-1 reverse transcriptase (PDB code 1EP4) main-chain carbonyl of Pro236 and nitrogen of Lys-103. A water (violet sphere)-mediated hydrogen bond is formed with the carbonyl oxygen of Lys101.166 (d) Hydrogen bonds between adjacent parallel strands in the kinesin dimer (PDB code 2KIN).

Charged residues in proteins and charged moieties of ligands can engage in hydrogen bonds to other charged groups (salt links or salt bridges) or to polar groups without a net charge (see Figure 4a and b). While the burying of a charged group on a ligand or protein in a protein has a large charge desolvation penalty, this can be compensated by hydrogen bonds from surrounding polar groups. Indeed, favorable local hydrogen bonds can even overcome an unfavorable electrostatic surface potential at the binding site of a charged ligand. For example, the buried binding sites for phosphate and sulfate ions in bacterial ion active transport receptor proteins have a negative surface potential but nevertheless the anions bind very specifically due to the optimal ligand-receptor hydrogen-bonding arrangement, which leaves no hydrogen bond donor or acceptor unpaired.34

Hydrogen bonds between a ligand and its receptor may be mediated by water molecules (see, e.g., Figure 4c). The ability of water molecules to donate and accept, as well as their freedom to move, means that the water molecules can serve to weaken the geometric restrictions on hydrogen-bonding interactions and extend their reach. This ability means they can act as a lubricant during binding processes and functional motions, i.e., the water molecules can serve to facilitate conformational changes that maintain or improve hydrogen-bonding network. Such a role has, for example, been proposed for water molecules in the active site gorge of acetylcholinesterase.35

Hydrogen bonds are exploited in drug design, particularly to obtain specificity. Docking programs generally model hydrogen bond interactions better than hydrophobic ones (see below). The number of hydrogen bonds in a drug molecule may be limited by requirements on polarity for absorption and permeation. The Lipinski rule-of-five,36 for example, suggests that compounds with more than five hydrogen bond donors or more than 10 hydrogen bond acceptors are more likely to have poor absorption or permeation characteristics.

4.09.3.4 Hydrophobic Effect

The hydrophobic effect describes the energetic preference of nonpolar molecular surfaces to interact with other nonpolar molecular surfaces and thereby to displace water molecules from the interacting surfaces. The hydrophobic effect is due to both enthalpic and entropic effects. Hydrophobic interactions are short-range attractive interactions that make an important contribution to ligand-receptor binding affinities. They also contribute to specificity but in a less geometrically constrained way than hydrogen-bonding interactions.

In simulations made with an explicit solvent model, there is no need for a special term in the force field to describe the hydrophobic effect. It occurs as a natural consequence of the Lennard-Jones and electrostatic interactions between water molecules and the atoms in the ligand and receptor molecules. However, when the solvent model is an implicit, continuum model or only includes a few particularly ordered water molecules, then hydrophobic interactions should be modeled by an additional term in the energy function. One way to do this is by employing an empirical term dependent on the surface area buried upon binding:

The buried surface area may be defined from the molecular (Connolly) surface or the solvent-accessible surface. The coefficients (ct) are dependent on the surface area definition and on the atom type. The coefficients can be assigned according to the polarity of the atoms with polar atoms having opposite sign coefficients to nonpolar atoms.37

To identify hydrophobic patches on receptors, the GRID program provides a 'DRY probe. It can be considered to be like an 'inverse' water probe. It makes a Lennard-Jones interaction in the same way as a water probe. It is also neutral like a water probe and has no electrostatic interaction term. The hydrogen bond energy term is however inverted to reflect the fact that polar parts of the target that are able to make hydrogen bonds will not be energetically favored next to a hydrophobic probe. Hydrophobic patches on proteins detected by the 'DRY probe and exploited by the hydrophobic sides of ligands are shown in Figure 5b and c (see 4.11 Characterization of Protein-Binding Sites and Ligands Using Molecular Interaction Fields).

4.09.3.5 Entropy

Ligand-receptor binding is accompanied by several types of entropic changes (see 38 for review).

The transformation of two mobile molecules into one mobile complex results in the loss of translational and rotational entropy. Assuming that the ligand is completely immobilized with respect to the receptor, this entropy loss can be estimated from analytical expressions (Sackur-Tetrode equation) for the translational and rotational entropy of molecules in the gas phase. Generally, the entropy loss is less because the ligand still has translational and rotational freedom in the receptor binding site. The contribution of entropy loss to binding affinity is however far from negligible: Murray and Verdonk,39 estimated that the barrier to binding of small molecules to proteins due to loss of rigid body entropy is

Figure 5 Hydrophobic interactions. (a) Proposed fatty acid ligand C23H48O2 in the active site of cholesterol esterase (PDB code 1LLF). The long alkyl chain of the ligand makes interactions with the hydrophobic environment of the gorge below the reactive site of the enzyme.167 (b) Two tryptophan residues (grey sticks), in the carbohydrate-binding module of xylanase 10A, provide planar hydrophobic stacking interactions for a glucose disaccharide (green sticks) (PDB code 1I82).168 Yellow patches indicate favorable regions for the DRY probe as calculated with GRID. (c) Structure of a cross-linked helical peptide, C14linkmid, bound to IQN17, a soluble peptide that contains the HIV-1 gp41 hydrophobic pocket (surface representation) (PDB code 1GZl).169 Yellow patches indicate favorable regions for the DRY probe as calculated with GRID.

Figure 5 Hydrophobic interactions. (a) Proposed fatty acid ligand C23H48O2 in the active site of cholesterol esterase (PDB code 1LLF). The long alkyl chain of the ligand makes interactions with the hydrophobic environment of the gorge below the reactive site of the enzyme.167 (b) Two tryptophan residues (grey sticks), in the carbohydrate-binding module of xylanase 10A, provide planar hydrophobic stacking interactions for a glucose disaccharide (green sticks) (PDB code 1I82).168 Yellow patches indicate favorable regions for the DRY probe as calculated with GRID. (c) Structure of a cross-linked helical peptide, C14linkmid, bound to IQN17, a soluble peptide that contains the HIV-1 gp41 hydrophobic pocket (surface representation) (PDB code 1GZl).169 Yellow patches indicate favorable regions for the DRY probe as calculated with GRID.

15-20 kJmol_ 1 i.e., around 3 orders of magnitude in affinity at 298 K. There are several methods to compute rigid body entropy loss from the extent of conformational sampling of the molecules in the bound and unbound states; these usually assume harmonic or quasiharmonic motion in the bound state (see 40 and methods discussed therein).

Binding affects the internal molecular dynamics and this affects the vibrational entropy and the entropy of rotamers. When two molecules bind, it is most often reported that the interface region becomes more ordered upon binding. This results in an entropic penalty. However, it appears that this penalty is quite often neutralized by binding, inducing greater flexibility in the receptor at sites distant to the ligand binding face,41'42 which may result in allosteric effects at remote sites. Interestingly, regions of proteins that participate in binding are often quite mobile or disordered and become ordered upon binding. Although this is entropically unfavorable, the advantage is that the mobile region of the protein can fish out and adapt to its ligand by a 'fly casting mechanism' that can be kinetically advantageous.43

The change in vibrational entropy upon binding can be estimated from normal mode44 or quasiharmonic45 analysis of simulated structures. This permits changes in the entropy of harmonic vibrational motions to be computed. Entropic changes in nonharmonic motions, such as rotamer transitions (see Section 4.09.4.4.2), can be estimated by computing accessible rotamer states. Upon binding, the number of possible rotamers, n, that an interfacial side chain can adopt usually decreases and the entropic cost can be estimated as:

Such an approach has been used to analyze structures of protein-protein complexes by Cole and Waricker46 who found that, on the basis of computed side chain entropy, the interfaces were less flexible than the rest of the protein surface.

Water molecules are displaced upon protein-ligand binding and their environment is changed; this also has an entropic effect. The displacement of ordered water molecules from an interface into bulk solution makes a favorable entropic contribution. An upper limit for this entropic change is given by the entropy change on transferring a water molecule from ice to water. This was estimated by Dunitz as 28 J K 1 mol 1 (2 kcal mol 1 at 300 K). In general, the contribution is much smaller as the water around protein surfaces, as well as most of the water in protein interiors,48 has much greater freedom to move than water in ice.

In modeling and simulation, the treatment of solvent entropic contributions depends on the type of model. In an all-atom model with explicit solvent molecules simulated by molecular dynamics methods, the entropic contributions of the solvent arise from the model and no special entropy terms in the force field are necessary. If the solvent is modeled implicitly as a continuum, then a term to account for the change in solvent entropy upon binding should be included in expressions to compute binding affinity. For example, in the GRID method, the entropic cost of displacing a water molecule by a hydrophobic probe is assumed to be 0.848 kcal mol_ 1.14 In other approaches, a surface area dependent free energy term includes both entropic and enthalpic contributions resulting from burying molecular surface upon binding.

Entropic effects can be exploited along with enthalpic effects in drug design. To optimize entropic contributions, compounds are usually designed to be relatively rigid with few rotatable bonds (see Section 4.09.4.2). They often have hydrophobic patches that interact with the receptor and are entropically favorable. Compounds can be designed to bind to more rigid parts of a protein target. This is entropically advantageous. However, flexibility in parts of small molecules may be useful to allow adaptation to flexible targets and to variants of targets, such as HIV protease.49

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