Entropy in molecular recognition by proteins

Significance Molecular recognition by proteins is a key element of biology. Appreciation of the underlying thermodynamics has been incomplete because of uncertainty in several contributions to the entropy. Here, we demonstrate a way to measure changes in protein conformational entropy using a dynamical proxy provided by NMR relaxation methods. We find that conformational entropy can contribute significantly and variably to the thermodynamics of binding. In addition, we determine the contribution of rotational-translational entropy loss upon forming a high-affinity complex involving a protein. The contribution of solvent entropy is also recalibrated. Thus, a more complete view of entropy in binding has been established and shows that inclusion of conformational entropy is necessary to understanding the origins of high-affinity interactions involving proteins. Molecular recognition by proteins is fundamental to molecular biology. Dissection of the thermodynamic energy terms governing protein–ligand interactions has proven difficult, with determination of entropic contributions being particularly elusive. NMR relaxation measurements have suggested that changes in protein conformational entropy can be quantitatively obtained through a dynamical proxy, but the generality of this relationship has not been shown. Twenty-eight protein–ligand complexes are used to show a quantitative relationship between measures of fast side-chain motion and the underlying conformational entropy. We find that the contribution of conformational entropy can range from favorable to unfavorable, which demonstrates the potential of this thermodynamic variable to modulate protein–ligand interactions. For about one-quarter of these complexes, the absence of conformational entropy would render the resulting affinity biologically meaningless. The dynamical proxy for conformational entropy or “entropy meter” also allows for refinement of the contributions of solvent entropy and the loss in rotational-translational entropy accompanying formation of high-affinity complexes. Furthermore, structure-based application of the approach can also provide insight into long-lived specific water–protein interactions that escape the generic treatments of solvent entropy based simply on changes in accessible surface area. These results provide a comprehensive and unified view of the general role of entropy in high-affinity molecular recognition by proteins.

Molecular recognition by proteins is fundamental to molecular biology. Dissection of the thermodynamic energy terms governing protein-ligand interactions has proven difficult, with determination of entropic contributions being particularly elusive. NMR relaxation measurements have suggested that changes in protein conformational entropy can be quantitatively obtained through a dynamical proxy, but the generality of this relationship has not been shown. Twenty-eight protein-ligand complexes are used to show a quantitative relationship between measures of fast side-chain motion and the underlying conformational entropy. We find that the contribution of conformational entropy can range from favorable to unfavorable, which demonstrates the potential of this thermodynamic variable to modulate protein-ligand interactions. For about one-quarter of these complexes, the absence of conformational entropy would render the resulting affinity biologically meaningless. The dynamical proxy for conformational entropy or "entropy meter" also allows for refinement of the contributions of solvent entropy and the loss in rotational-translational entropy accompanying formation of high-affinity complexes. Furthermore, structure-based application of the approach can also provide insight into long-lived specific water-protein interactions that escape the generic treatments of solvent entropy based simply on changes in accessible surface area. These results provide a comprehensive and unified view of the general role of entropy in high-affinity molecular recognition by proteins. entropy | molecular recognition | binding thermodynamics | protein dynamics | NMR relaxation M ost biological processes are controlled using molecular recognition by proteins. Protein-ligand interactions regulate enzyme function, signaling pathways, and cellular events essential to life, particularly through allosteric mechanisms (1). Indeed, efforts at pharmaceutical intervention in disease have largely centered on the manipulation of molecular recognition by proteins. The physical origin of high-affinity interactions involving proteins has been the subject of intense investigation for decades. Structural analysis at atomic resolution has helped illuminate in great detail the role played by enthalpy. On the other hand, the role of entropy in modulating the free energy of association of a protein with a ligand has remained elusive. Of prime interest here is the conformational entropy of the protein, which is defined by the distribution of conformational states populated by the protein. A binding event can result in a redistribution of populated states, and such differences are effectively invisible in a static, monolithic view of proteins. It remains a challenge for structure-based efforts, agnostic to the conformational landscape available to the protein, to relate the interactions of a few amino acids at a specific interface to the global, experimentally measured free energies (2)(3)(4). The potential role for conformational entropy in protein function has been speculated about and simulated for some time (4-9), but a comprehensive, quantitative, and experimental evaluation of the extent and variation of its importance has been lacking.
Experimental insight into binding entropy often begins from a calorimetric perspective where the heat or enthalpy (ΔH total ) and free energy (ΔG total ) of binding are measured and the total binding entropy (ΔS total ) is determined by: [1] Here and throughout the main text, the thermodynamic values refer to a standard 1 M state of concentration. The challenge is to quantify the various microscopic contributions to the free energy of binding. Detailed atomic resolution structural models provide great insight into the origins of the enthalpy of binding. Much less certain are the various contributions to the total binding entropy. In principle, several types of entropy are potentially important (right side of Eq. 1). Historically, entropy has most often entered the discussion in terms of the changes in the entropy of solvent water (ΔS solvent ) and framed in terms of the so-called "hydrophobic effect" (10,11). ΔS solvent has, with some success, been related empirically to changes in accessible surface area (ΔASA) of the protein and ligand upon complexation (12). Changes in the conformational entropy (ΔS conf ) and the rotational-translation entropy (ΔS r-t ) (13) of the interacting species have received far less attention, presumably because they have resisted experimental measurement. Contributions to binding entropy from unrecognized sources (ΔS other ) are also included in Eq. 1 and discussed below.

Significance
Molecular recognition by proteins is a key element of biology. Appreciation of the underlying thermodynamics has been incomplete because of uncertainty in several contributions to the entropy. Here, we demonstrate a way to measure changes in protein conformational entropy using a dynamical proxy provided by NMR relaxation methods. We find that conformational entropy can contribute significantly and variably to the thermodynamics of binding. In addition, we determine the contribution of rotational-translational entropy loss upon forming a high-affinity complex involving a protein. The contribution of solvent entropy is also recalibrated. Thus, a more complete view of entropy in binding has been established and shows that inclusion of conformational entropy is necessary to understanding the origins of high-affinity interactions involving proteins.
Some time ago, it was realized that fast subnanosecond time scale motion between conformational states might provide access to various thermodynamic features (14), especially conformational entropy (15,16). Application of this idea has been thwarted by several technical limitations (17,18), but has nevertheless led to the strong suggestion that dynamical proxies made available by NMR relaxation measurements could provide access to measures of conformational entropy (19). More recently, efforts have been taken to overcome these technical barriers and limitations and to render this approach to protein entropy quantitative (20,21). The resulting NMR-based dynamical proxy for conformational entropy or "entropy meter" takes a simple form that requires a few assumptions, particularly about the precise nature of the underlying motion (21): [2] The first two terms contain the dynamical proxy, where O 2 axis is a measure of the degree of spatial restriction of the methyl group symmetry axis and ranges between 0, which represents complete isotropic disorder, and 1, which corresponds to no internal motion within the molecular frame (22), and is measured in various ways (17). Numerous models and simulations indicate that changes in entropy will be linearly related to changes in the relevant O 2 , varying between 0.1 and 0.9 (14-17, 21, 23). This linearity is the basis for the simple form of the entropy meter (20,21). Measurement of fast internal side-chain motion in proteins is largely restricted to the methyl group, which, due to its fast rotation, can be used as a relaxation probe in even very large proteins (17). The change in the average methyl group symmetry axis order parameter (Δ<O 2 axis >) upon binding a ligand is used as a dynamical proxy, and s d is the sought-after relationship (conversion) between these measures of fast internal motion and conformational entropy (20,21). To avoid issues associated with statements about absolute entropy, we restrict this treatment to changes in entropy upon a perturbation (e.g., binding of a ligand). In principle, all internal degrees of freedom of the protein contribute to ΔS conf , namely, bond lengths, angles, and torsions. In practice, empirical and computational studies show that changes upon binding are largely restricted to the softer, torsional modes (21,24). Eq. 2 also depends on the total number of torsion angles in the molecule (Nχ), because the motion of methyl-bearing amino acid side chains reports on side-chain motions across the entire protein molecule. Linear scaling with Nχ is expected when weak coupling of side-chain motion with neighbors is present (25). ΔS solvent is calculated from the structures of the free and complexed states by scaling changes in apolar and polar accessible surface area with empirically determined solvation entropy coefficients (12).
The conformational entropy meter approach has thus far been applied to only two protein-ligand systems: calcium-activated calmodulin-binding peptides representing the domains of regulated proteins (20) and a series of mutants of the catabolite activator protein binding DNA (26). It is unknown whether this relationship holds generally. We address this issue by examining 28 protein-ligand complexes that span a broad range of binding affinities (K d = 10 −4 to 10 −10 M) and ligand types (nucleic acids, enzyme substrates and cofactors, carbohydrates, and polypeptides) (SI Appendix, Table S1).

Results and Discussion
We set out to test the hypothesis that the entropy meter is universally applicable [i.e., that the scaling (s d ) between changes in fast internal motion and changes in conformational entropy is constant]. There are now roughly two dozen published studies of the change in methyl-bearing side-chain dynamics that are sufficiently complete to use to test this idea (SI Appendix, Table S1). Criteria for selection of these complexes included the availability of comprehensive assignments and relaxation data in both free and complexed protein states, as well as a largely dry interface in the complex. We have augmented many of these NMR relaxation studies by measuring the binding thermodynamics using isothermal titration calorimetry (ITC) under corresponding NMR solution conditions (i.e., buffer, pH, temperature). Completely new examples were also examined. The curated dataset is summarized in SI Appendix, Tables S1-S4.
Previously, the empirically determined coefficients relating changes in accessible polar and apolar surface area to the solvation entropies of these two types of surface had to be derived subject to various assumptions about the nature of conformational entropy (12). The unprecedented amount of dynamical data used here now allows us to determine the area coefficients directly. The data on the 28 proteins used here were obtained at somewhat different temperatures (298-308 K). Significant heat capacity changes accompany hydration of apolar and polar groups. Fortunately, the corresponding temperature variation of the desolvation entropy is well described experimentally by the relations ΔS apolar = ΔCp apolar ·ln(T/385K)·ΔASA apolar (27) and ΔS polar = ΔCp polar ·ln(T/176K)·ΔASA polar (28), where ΔCp apolar and ΔCp polar are the changes in hydration heat capacities per unit area of apolar (ΔASA apolar ) and polar (ΔASA polar ) surface, respectively, and 385 K and 176 K are reference temperatures at which the apolar and polar solvation entropies extrapolate to 0 (27,28). The intercept of Eq. 2 contains the loss in rotationaltranslational entropy upon formation of the complex. We therefore recast Eq. 2 as: where the heat capacity terms are evaluated at the experimental temperature of each complex. When combined with the known changes in apolar and polar accessible surface area, this formulation gives the contributions to solvent entropy. Entropies are given here with respect to a standard state of 1 M. Should unaccounted contributions to the binding entropy (i.e., ΔS other ) be insignificant, then ΔS r-t will dominate the ordinate intercept. Violations of the few assumptions used to construct the calibration line for the entropy meter will result in deviation from linearity. It is assumed that the methyl groups are sufficiently numerous, are well distributed, and are adequately coupled to non-methyl-bearing amino acids such that their motions provide comprehensive coverage of internal motion in the protein.
These assumptions are strongly supported by simulation (21).
The fit of Eq. 3 to the dynamical data indicates a strong linear correlation (R of −0.85, R 2 = 0.72, P < 10 −7 ) (Fig. 1). To address the stability of the fit, we carried out a jackknife-type analysis, where individual members of the dataset were randomly sampled. A minimum of only five points is required to define s d with reasonable precision, which smoothly converges as additional points are added (SI Appendix, Fig. S1). The determined values of s d , ΔS r-t , ΔCp apolar , and ΔCp polar are listed in Table 1, and further statistics are summarized in SI Appendix, Fig. 2 and Table S5. The parameter of central interest (s d ) is well determined and provides a robust and apparently general means to obtain the change in conformational entropy upon protein-ligand association experimentally. It is somewhat smaller than the parameter obtained using only the two protein systems, catabolite activator protein and calmodulin, used previously for calibration (21) (Fig. 1, green and blue circles, respectively). Although it is likely that s d will vary in detail between different proteins, the precision captured by Fig. 1 indicates that this variability is limited.
In this treatment, we have ignored the contribution to the binding entropy from the backbone of the protein. Recent simulations suggest that the binding of ligands by structured proteins will involve little contribution from the backbone of the polypeptide chain (25). Unfortunately, only six of the complexes used have sufficiently characterized dynamics to allow backbone motion to be fully included in our analysis. However, when analyzed in a similar fashion, this subset indicates that the contribution by the backbone to the binding entropy is indeed small (<5%). Using the determined value of s d , we can now establish quantitatively that the contribution of conformational entropy to molecular recognition by proteins is quite variable. Conformational entropy can highly disfavor, have no effect on, or strongly favor association, and it is often a large determinant of the thermodynamics of binding (Fig. 2). Indeed, for approximately one-quarter of the complexes used in the calibration of the entropy, the absence of the contribution of conformational entropy to the binding thermodynamics would result in biologically ineffective affinities.
The calibrated entropy meter allows the determination of the contribution of conformational entropy to a binding process using only the dynamical information; that is, for a heterodimer (AB): [4] Importantly, other entropic contributions to protein-ligand associations are also made accessible by the approach summarized in Fig. 1 and Eq. 3. Eq. 3 allows for other unknown sources of entropy. These sources might include, for example, (de)protonation or (de)solvation of charge associated with binding (29). Clearly, if ΔS other is both significant and varies between complexes, then the linearity of Eq. 3 will be degraded. The observed linear correlation strongly suggests that such is not the case. Thus, if ΔS other is small relative to ΔS conf and ΔS solvent , then the ordinate intercept represents the loss in rotational-translational entropy upon formation of high-affinity complexes. ΔS r-t has been the subject of extensive theoretical debate but, like conformational entropy, has resisted experimental definition. The apparent ΔS r-t is −0.10 ± 0.01 kJ·mol −1 ·K −1 (standard state of 1 M), which compares reasonably well with the apparent ΔS r-t obtained through molecular dynamics simulations (30).
The contribution of solvent entropy to processes involving proteins is generally derived from changes in solvent accessible surface area. Before this work, such approaches have required strong assumptions regarding conformational entropy (12), which are not required here because all four parameters of the entropy terms of Eq. 3 can be directly determined. We find that burial of both apolar and polar surfaces upon binding produces a positive (favorable) change in entropy. This positive change occurs because hydration of polar groups, like hydration of apolar groups, has negative entropy of hydration, in agreement with a wide range of thermodynamic data on solute, ion, and protein hydration (28,31,32). We determined the surface area coefficients for apolar (a apolar ) and polar (a polar ) desolvation entropy at 298 K to be −0.096 ± 0.029 J·mol −1 ·K −1 ·Å −2 and −0.027 ± 0.005 J·K −1 ·Å −2 , respectively ( Table 1). The corresponding hydration heat capacity coefficients are also listed in Table 1. Burial of the hydrophobic area stabilizes the complex through the Fig. 1. Calibration of the dynamical proxy for protein conformational entropy. Fitting of Eq. 3 to data provided by 28 protein-ligand associations (blue, green, and magenta circles) is shown. The difference in the measured total binding entropy and calculated solvent entropy is plotted against the change in the dynamical proxy upon binding of ligand. The dynamical proxy is the difference of the average Lipari-Szabo squared generalized order parameters of methyl group symmetry axes (Δ<O 2 axis >) scaled by the number of total torsion angles (Nχ) in the protein. Horizontal error bars include the SD of <O 2 axis >, which is less than ±0.01 for our data. For complexes involving small peptides or oligonucleotides, the uncertainty of the ligand contributes to the abscissa error bars (SI Appendix). The ordinate error bars are from the propagated quadrature errors of experimental ΔS total and the precision of the fitted coefficients used to determine ΔS solvent , and ΔS r-t . In many cases, the ordinate error is less than the size of the symbol. The fitted slope (s d ) of −0.0048 ± 0.0005 kJ·mol −1 ·K −1 allows for the conversion of measured changes in methyl-bearing side-chain motion and the associated contribution to the total conformational entropy. Other parameters determined for Eq. 3 are summarized in Table 1. Also shown are the values for the barnase/dCGAC complex without addition of a net contribution from long-lived water with the protein (red diamond) and with a net contribution from water rigidly associated only in the complex added (gray diamond). The latter is calculated using the entropy of fusion for the immobilization of nine water molecules in the complex only (9 × −22 J·mol −1 ·K −1 ). Further details are provided in the main text. The orange square represents the HBP(D24R)histamine binary complex binding to serotonin. The CaM and CAP data subsets are shown in green and blue, respectively.
.0 a apolar (298 K)/10 −5 kJ·mol −1 ·K −1 ·Å −2 −9.6 ± 2.9 a polar (298 K)/10 −5 kJ·mol −1 ·K −1 ·Å −2 −2.7 ± 0.5 Calibration of the entropy meter is derived from a global fit for the parameters ΔCp apolar , ΔCp polar , ΔS r-t , and s d of Eq. 3 using the experimental binding entropy changes, order parameter changes, and known experimental temperatures for the 28 complexes summarized in Fig. 1 (R = −0.85, R 2 = 0.72, P < 10 −7 ). Uncertainties were determined by Monte Carlo sampling. SDs are shown. The average covariance matrix is provided in SI Appendix, Table  S6. Values for the hydration entropy coefficients per unit area are also tabulated at the standard temperature of 298 K. Calibration is performed with reference to a standard state concentration of 1 M. *Entropic contribution of the backbone is not directly included (as discussed in the main text). hydrophobic effect. Concomitantly, burial of the polar area also stabilizes the complex via release of its hydrating water into the bulky, less ordered state (31). The coefficients are smaller than obtained previously (12,33) and, in part, reflect the impact of inadequately assessing the contribution of residual side-chain entropy in the folded state of proteins in prior work. It is also interesting to note that the smaller magnitude likely arises from a difference in the nature of solvation. Prior solvation entropies have been estimated from group transfers between water and organic solvents or global unfolding of globular proteins (e.g., refs. 12,27,28,[33][34][35][36], which involves complete solvation of side chains. In contrast, the studies here focus on the (de)solvation of more extended protein surfaces and may reflect differences in length scale, detailed geometry, and inherent dynamics of protein surfaces (e.g., refs. [37][38][39]. This issue remains to be explored further. Several fundamental properties of protein molecules in solution are related to their heat capacity, which, in turn, is related to the underlying entropy. The most pertinent definition of C p here is the derivative of the entropy with respect to the natural logarithm of the temperature. Thus, the determined s d parameter of the entropy meter, along with suitable temperature dependence data, allows the protein conformational contribution to C p changes to be determined. The relative importance of conformational and solvation contributions to ΔC p of binding, as for ΔS, has been the subject of considerable debate (40), because, previously, there was no experimental way to isolate the different contributions. The temperature dependence of fast methyl-bearing side-chain motion has been examined for only a few proteins: for example, the drkN (41) and α-spectrin (42) SH3 domains, ubiquitin (43,44), and a calmodulin-peptide complex (45). Early NMR studies suggested that the backbone contributed less to the heat capacity of folded proteins than side chains (41). Using the entropy meter, we find that the amino acid side chains contribute only a small fraction (∼5-6%) to the total heat capacity measured by differential scanning calorimetry of the latter two proteins (SI Appendix, Table S6). This finding reinforces the proposal that most of the heat capacity comes from solvent-protein interactions (10,40,46).
In addition to exploring the role of conformational entropy in protein function, the entropy meter can be used to illuminate other important manifestations of entropy. For example, the complexes used for calibration of the entropy meter largely have dry interfaces. The presence of retained "structural" water at the buried surface comprising the interface would affect the solvent entropy but would be masked by the standard method to calculate it based on accessible surface area. Following calibration (Fig. 1), the entropy associated with the organization of water within a protein complex can be explored. To illustrate this view, we examined the complex between the extracellular ribonuclease barnase of Bacillus amyloliquefaciens and an oligonucleotide dCGAC model substrate. Barnase is a 110-residue ribonuclease and is inhibited by the 89-residue barstar to suppress its potentially lethal activity inside the cell. The barnase/dCGAC complex has nine fully buried and crystallographically well-defined structural water molecules at the interface (47) (Fig. 3). Determination of the binding thermodynamics by direct titration monitored by NMR spectroscopy and ITC indicates that the free energy of binding of dCGAC to barnase is essentially provided by a gain in entropy (SI Appendix, Fig. S3 and Table S2). Dynamic analysis indicates a corresponding increase in side-chain motion upon binding of the oligonucleotide that corresponds to a favorable contribution to the binding entropy (SI Appendix, Fig. S3 and Table S4). An inventory of the entropy contributions as outlined by Eq. 3 results in good agreement, which is in apparent contradiction to the implications of the structure of the complex noted above. This contradiction arises because rigidification of nine water molecules is not accommodated by the solvation term of Eq. 3. Using the entropy of fusion of water (−22 J·mol −1 ·K −1 ) (48), the rigidification of these waters in the complex is be predicted to result in a distinct deviation from the calibration line (Fig. 1, gray and blue diamonds). Thus, either the structural waters retain considerable S r-t in the complex and/or they are rigidified in free barnase. To answer this question, we turned to an approach introduced some time ago by Wüthrich and coworkers (49), where the nuclear Overhauser effect (NOE), and its rotating frame counterpart (ROE), between hydrogens of the protein and hydrogens of long-lived associated water can provide information about protein-water dynamics. An unprecedented number of NOE contacts consistent with the long-lived attachment of the nine waters buried at the interface in the complex are seen in free barnase (4). It is noted that artifacts can potentially occur when using this approach in bulk solution (50). However, encapsulation of proteins within the protective water core of a reverse micelle suppresses hydrogen exchange and other processes that can corrupt the NOE and ROE (51). Dipolar contacts consistent with nine of the buried waters of the crystallographic structure are also observed from encapsulated barnase (Fig. 3). The average NOE/ROE ratio of these waterprotein contacts in bulk solution is approximately −0.15, indicating that the waters are relatively rigidly held to the protein. Thus, the rigidly held water by free barnase appears to nullify the entropic cost of using water as a structural element in the complex.
A second example of the further insights that can be garnered from the entropy meter centers on the idea that the loss of rotation-translational entropy will diminish for weaker interactions (i.e., K d > 100 μM) due to significant residual motion in the complex. To explore this idea, we examined the interaction of Contribution of protein conformational entropy to the free energy of ligand binding to proteins. The broad range of contributions available to proteins for high-affinity binding of ligands is illustrated by the proteinligand complexes used to calibrate the parameters of Eq. 3. The 28 proteinligand complexes are arranged in descending order of the contribution of conformational entropy (red bars) to the total free energy of binding (blue bars). Conformational entropy contributed by the response of amino acid side chains to the binding of a ligand can vary from highly unfavorable, to negligible, to highly favorable. In some cases, conformational entropy is essential for high-affinity binding. The structural origins of the variable utilization of conformational entropy in molecular recognition are unknown. In most cases, the change in solvent entropy remains a dominant contribution. Note that −TΔS r-t , ΔS ligand , and ΔS solvent are not shown here. The thermodynamics of each complex are summarized in SI Appendix, Tables  S2 and S4. histamine and serotonin with the histamine-binding protein (HBP) from the Rhipicephalus appendiculatus tick. HBP is a member of the lipocalin family of proteins, which have been shown to bind to histamine and serotonin (52). These heterocyclic molecules serve as primary mediators of the inflammatory response upon tissue damage (53). HBP is a 171-residue, β-barrel protein that, unlike most other lipocalin family proteins, has evolved to possess two histamine-binding sites, with one having high affinity (H-site) and one having low affinity (L-site) (52). The tick secretes HBP to interfere with the defensive inflammatory response of the host. For our studies, we have used the D24R mutation located in the L-site that largely abolishes the binding of the second histamine molecule while retaining the primary high-affinity histamine binding (H-site) (54). ITC established that histamine binds with very high affinity to HBP(D24R) [K d = 3.2 ± 0.7 nM, consistent with earlier measurements using trace radiolabeling (52)] and is accompanied by a large favorable change in enthalpy and an unfavorable total binding entropy (SI Appendix, Table S2). NMR relaxation analysis indicates a heterogeneous response of the protein to the binding of histamine (SI Appendix, Fig. S4). These results give a datum very close to the consensus fitted line of the entropy meter. Titration of the HBP(D24R)•histamine complex with serotonin reveals a weak binding site largely centered on residues Y30, V49, V51, A53, F67, E82, which are near the L-site (SI Appendix, Fig. S5). This finding could be indicative of residual binding activity at the L-site. ITC reveals no detectable heat, indicating that the modest binding free energy (−13.7 kJ/mol) is driven entirely by entropy. NMR relaxation analysis indicates an increase in side-chain dynamics that corresponds to a very modest ΔS conf of 3 ± 1 J·mol −1 ·K −1 . Interestingly, when completing the inventory of entropy, one finds only a small deviation from the entropy predicted (Fig. 1). This deviation is likely traced to an uncertainty in the solvation parameters for this complex and perhaps to a small residual S r-t of the serotonin ligand in the complex. Thus, ΔS r-t estimated in Fig. 1 is likely valid down to relatively low affinities corresponding to dissociation constants in the millimolar range.
In summary, the coefficients relating changes in fast protein side-chain motion to changes in conformational entropy, changes in accessible surface area to changes in solvent entropy, and the loss of rotational-translational entropy in high-affinity complexes have been determined. The range of ligand types used here demonstrates that the relationship between fast internal side-chain motion and the underlying conformational entropy is universal and represents a fundamental property of soluble proteins. It is demonstrated that conformational entropy has a highly variable role in the formation of complexes involving proteins: It can favor, disfavor, or have no impact on the free energy of binding. There are no obvious structural correlates apparent for this behavior. The connection between structure, the enthalpy that it represents, conformational entropy, and internal motion presents an immediate challenge to our current understanding of protein thermodynamics and function. In this vein, the view emerging from crystallographic analysis of minor conformers in proteins (55)(56)(57), combined with the dynamical proxy validated here, provides a means to quantify the role of conformational entropy in protein structure and function.

Methods
Various proteins and their complexes (47,52,(58)(59)(60)(61) were prepared as described in detail in SI Appendix. Some published NMR relaxation and thermodynamic studies were used without further analysis (SI Appendix, Table  S1). Macromolecular rotational correlation times and backbone O 2 NH were determined (62) from 15 N relaxation obtained at two magnetic fields. Model-free parameters (22) were determined using a grid search approach (63) using a version of Relxn2A (64) implemented in the C ++ /AMP language. The 15 N-relaxation analysis used an effective N-H bond length of 1.04 Å (65) and a general 15 N tensor breadth of 170 ppm (66). Tumbling models were identified through standard statistical analysis (67). Methyl group O 2 axis parameters were determined from measured (68) deuterium T 1 and T 1ρ relaxation. A quadrupolar coupling constant of 167 kHz was used (69). Analysis of NMR relaxation in these systems is summarized in SI Appendix, Tables S3, S4, and S7. Changes in polar and apolar accessible surface area were calculated using AREAIMOL (70) as described previously (20) (SI Appendix, Table  S4). BioMagResBank accession numbers are summarized in SI Appendix, Table S8. Studies of protein hydration dynamics used 3D 15 N-resolved 1 H-1 H NOE spectroscopy and ROE spectroscopy (7.2-kHz spin lock field) experiments (51,71). Perdeuterated barnase prepared in 25 mM imidazole (pH 6.2) and 10 mM KCl was encapsulated by defined volume injection (72) into 75 mM deuterated hexadecyltrimethylammonium bromide (CTAB) and 190 mM deuterated hexanol in deuterated pentane to a molar ratio of water to CTAB (W 0 or water loading) of 20. Structural fidelity of encapsulated barnase was confirmed by comparison of 15 N-HSQC spectra. These experiments were performed at 25°C and 500 MHz ( 1 H).
ACKNOWLEDGMENTS. We thank Professor Mark Greene for generous access to the isothermal titration calorimeter. We thank Veronica Moorman and Kendra Frederick for preliminary calorimetry experiments and initial curation of published studies. This work was supported by the Mathers   3. Identification of rigidly held water in free barnase. Expansion of the 2D slice at the water resonance of 3D 15 N-resolved NOE spectroscopy (NOESY) spectra of barnase in bulk aqueous solution (A, 50-ms NOE mixing time) and barnase encapsulated in CTAB/hexanol reverse micelles prepared in pentane (B, 40-ms NOE mixing time) is illustrated. These crosspeaks correspond to NOEs between amide hydrogens of barnase and local hydration water. The detection of hydration water in bulk aqueous solution is potentially corrupted by various mechanisms, most notably hydrogen exchange. These artifacts are suppressed in the reverse micelle. The correspondence of cross-peaks indicated by annotated assignments confirms those cross-peaks in the aqueous spectrum as genuine NOEs. Comparison of NOE and ROE cross-peak intensities indicates that these waters are rigidly held in free barnase. Additional NOEs in the reverse micelle spectrum result from the general slowing of water that brings motion of additional waters in the hydration layer into a detectable time regime. (C) Structural mapping of NOE cross-peaks between the protein and water in free barnase in bulk solution. Sites with long-lived hydration interactions are shown as green spheres. Crystallographic waters of free barnase in the binding site are shown as blue spheres.