article published online: 11 april 2010 | doi: 10.1038/nchembio.347

The role of conformational entropy in molecular recognition by calmodulin

Michael S Marlow1–3, Jakob Dogan1–3, Kendra K Frederick1,2, Kathleen G Valentine1,2 & A Joshua Wand1,2*

The physical basis for high-affinity interactions involving proteins is complex and potentially involves a range of energetic ­contributions. Among these are changes in protein conformational entropy, which cannot yet be reliably computed from ­molecular structures. We have recently used changes in conformational dynamics as a proxy for changes in conformational entropy of calmodulin upon association with domains from regulated proteins. The apparent change in conformational entropy was linearly related to the overall binding entropy. This view warrants a more quantitative foundation. Here we calibrate an ‘entropy meter’ using an experimental dynamical proxy based on NMR relaxation and show that changes in the conformational entropy of calmodulin are a significant component of the energetics of binding. Furthermore, the distribution of motion at the interface between the target domain and calmodulin is surprisingly noncomplementary. These observations promote modifica- tion of our understanding of the energetics of protein-ligand interactions.

he formation of protein complexes involves a complicated with high affinity15. Using NMR relaxation methods, we have manifold of interactions that often includes dozens of amino shown that calcium-saturated calmodulin (CaM) is an unusually acids and thousands of square Ångstroms of contact area1. dynamic protein and is characterized by a broad distribution of T 10 The origins of high-affinity interactions are quite diverse and com- the amplitudes of fast side chain dynamics . The binding of target plex and are reflected in the difficulty of computing the energet- domains causes a redistribution of these motions in calmodulin10,11. ics of interactions between proteins using molecular structure Interpretation of the changes in motion using a simple harmonic alone2. Indeed, structure-based design of pharmaceuticals has been oscillator model13 yielded a linear correlation between the appar- impeded by this barrier3. Of interest here is the role of protein con- ent change in conformational entropy of CaM and the total bind- formational entropy in modulating the free energy of the associa- ing entropy11. Taken at face value, this suggested a considerable tion of a protein with a ligand. Decomposition of the total binding contribution from conformational entropy to the total binding free energy emphasizes that the entropy of binding is comprised of entropy and indicated a role for conformational entropy in ‘tun- contributions from the protein, the ligand and the solvent: ing’ calmodulin’s high-affinity interactions. However, the analytical strategy used suffers from several 11,12 © 2010 Nature America, Inc. All rights reserved. All rights Inc. America, Nature © 2010 ∆∆GHtott=−ot TS∆∆totb=−HTindp( ∆∆ SS rotein ++ ligand ∆ S so lven t ) (1) potential limitations . That approach effectively takes an inven- tory of the change in motion at a limited number of sites and inter- Historically, the contributions by solvent entropy have taken center prets this within the context of a simple physical model such as stage and are usually framed in terms of the so-called hydrophobic the harmonic oscillator13. This raises several issues including the effect4. Hydrophobic solvation by water continues to be the subject effects of correlated motion, the operation of a more complex of extensive analysis5,6. In principle, the entropic contributions of a potential energy function, the completeness of the oscillator count, 12 structured protein to the binding of a ligand (∆Sprotein) include both and so on . Although the presence of a linear correlation between changes in its internal conformational entropy (∆Sconf) and changes the apparent change in conformational entropy and total binding 7 in rotational and translational entropy (∆SRT) . Equation (1) empha- entropy is a compelling indication of the importance of the former, sizes that the measurement of the entropy of binding does not resolve the quantitative scale of the interpreted ∆Sconf is potentially suspect. contributions from internal protein conformational entropy. Despite Here we resolve this issue by taking a different approach based on the suggestion that the conformational entropy of structured pro- an empirical calibration of the dynamical proxy for conformational teins is significant8,9, it is only recently that experimental evidence entropy. Rather than attempt a model-dependent interpretation has become available to indicate that it is sufficiently responsive to of an inventory of changes in local dynamics, we use experimen- influence the thermodynamics of protein association10,11. tal measures of local dynamics as a proxy for local disorder and

Experimental measurement of ∆Sconf has been difficult. Motion seek an empirical scaling between them. The aim is to effectively between different microscopic structural states has been devel- solve for each term of equation (1). An essential component of oped as an indirect measure of or proxy for conformational this approach is knowledge of the dynamics of the target domains, entropy12. Motion expressed on the subnanosecond timescale cor- which are determined here. The availability of these data allows responds to significant entropy13, and NMR relaxation methods for the quantitative calibration of the dynamical proxy for confor- are particularly well suited for its characterization12. Previously, we mational entropy. We found that the conformational entropy of used calmodulin as a model system to investigate the role of con- calmodulin not only contributes significantly to the free energy formational entropy in the high-affinity association of proteins11. of binding of target domains but also appears to be the dominant Calmodulin is central to calcium-mediated signal transduction factor in tuning the affinity of calmodulin for the various target pathways of eukaryotes14 and interacts with hundreds of proteins domains examined.

1Johnson Research Foundation and 2Department of Biochemistry and Biophysics, University of Pennsylvania, Philadelphia, Pennsylvania, USA. 3These authors contributed equally to this work. *e-mail: [email protected]

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10 2 smMLCK(p) are heterogeneously distributed, with O axis values ­ranging from 0.05 to 0.95 (Fig. 1). The distribution is non-uniform and reminis- CaMKI(p) cent of the multimodal distributions of the calmodulin component eNOS(p) of these complexes11. 8 CaMKKα(p) PDE(p) Variable and counterintuitive motion at the interface nNOS(p) The methyl-bearing side chains of the target domains are distri­ buted throughout the CaM-peptide interface, thereby providing an 6 excellent system to examine the intricacies of structure-­dynamics relationships. The structures of all but the complex with PDE(p) are known. CaM-target complexes have ‘anchor’ residues that

Occurrences 4 localize to hydrophobic pockets formed by the N-terminal and C-terminal domains of CaM. Typically, one anchor resi- due is ­aromatic (tryptophan or phenylalanine) and the other is aliphatic. Anchor residues are believed to be essential for com- 2 plex formation21. The aliphatic side chain anchors of the bound target domains were localized to the N-terminal domain of CaM (Fig. 2). Surprisingly, most aliphatic peptide side chains histori- cally identified as anchor residues were more dynamic than one 0 2 might expect. Specifically, the O axis values of the eNOS(p) and 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 nNOS(p) leucine δ methyls and the CaMKI(p) methionine ε 2 O axis methyl were at or below the residue-specific averages for the CaM complexes (Supplementary Table 3). With Leu19 of smMLCK(p) Figure 1 | Distribution of methyl symmetry axis generalized order providing the lone exception, the binding within the pocket need parameters for the target domains bound to calcium-saturated wild- not significantly confine the motion of the anchor residues of the type calmodulin. We determined the parameters using deuterium NMR bound target domain. relaxation (see Methods). Complex formation results in a pattern of the dynamics of the CaM methyl-bearing residues that form the binding pocket (Ile27, Leu32, Met51, Ile52, Val55, Ile63 and Met71). For example, in every RESULTS complex, Ile27δ and Ile63δ exhibit relatively restricted motion, 2 Dynamics of the target domains in complex with CaM with an average O axis of 0.69 (n = 12), which is 0.19 greater than Using deuterium NMR relaxation methods16, we examined fast the residue average for isoleucine δ methyls in the CaM complexes. 2 motion of the methyl-bearing side chains of the target domains in In contrast, the O axis for the δ methyl of Ile52 averages 0.36 (n = 6), the six CaM complexes examined previously11. These include com- which is 0.14 less than the residue average in the CaM complexes. 2 plexes between calmodulin and the calmodulin-binding domains The average O axis for the δ methyls of Leu32 of 0.50 (n = 12) is of the endothelial and neuronal nitric oxide synthases (eNOS and also substantially smaller than the residue average of 0.60. We

© 2010 Nature America, Inc. All rights reserved. All rights Inc. America, Nature © 2010 nNOS, respectively), calmodulin kinase kinase-α (CaMKKα), calm- observed a similar pattern in the residues found in the C-terminal odulin kinase I (CaMKI), phosphodiesterase (PDE) and the smooth pocket that bind aromatic peptide anchor residues. Some residues, muscle myosin light chain kinase (smMLCK). The ­calmodulin- such as Leu105 and Val136, have methyl group dynamics at their binding domains are represented here as peptides, and we will use residue-specific averages in the CaM complexes. Others are highly the nomenclature eNOS(p), for example, to emphasize this fact. All constrained, such as Ile100δ (n = 5) and Ala128 (n = 5), which show 2 of the peptide domains have basic amphiphilic sequences typical of average O axis values that are 0.3 and 0.23 larger than the residue CaM binding domains (Supplementary Table 1). These domains averages, respectively. Clearly, binding does not induce a uniform have been found by isothermal titration calori­metry at 308 K to reduction in side chain motion within the hydrophobic pockets. We bind with roughly the same affinity to CaM but with widely also observed more nuanced responses. This may be a feature of different thermodynamic parameters defining the free energy of proteins that have evolved to bind numerous targets. Such responses association11,17,18. In the case of the CaMKKα(p) and smMLCK(p) suggest that the view of hot spot interactions22 should be extended domains, binding is driven by a large favorable change in total bind- to include resolution of dynamical (entropic) from specific enthal- ing enthalpy overcoming a large unfavorable change in total binding pic contributions. entropy. At the other extreme, nNOS(p) binding is driven by a favorable change in total enthalpy accompanied by a small favorable Calibration of the entropy meter change in entropy. The other domains represent intermediate cases. We now turn to the insights into the thermodynamic origins of The entropy of binding of these domains varies by 90 kJ mol−1 and binding offered by the characterization of internal motion in the changes sign (Supplementary Table 2)11. calmodulin complexes. One main goal is to empirically calibrate the All bound target domain methyl resonances are well resolved dynamical proxy of conformational entropy for the calmodulin sys- in 13C NMR spectra, and deuterium relaxation parameters could tem. We first decompose the entropy of binding in terms of contri- be measured with high precision (Supplementary Fig. 1). The butions from calmodulin, the target domains and solvent: degree of spatial restriction of each motional probe was assigned a CaM target number between 0, corresponding to complete isotropic disorder, ∆∆SStot =+conf ∆∆SSconf ++sol ∆SRT (2) and 1, corresponding to a fixed orientation within the molecular frame. This parameter is the so-called model-free squared gen- Contributions from rotational and translational entropy of CaM 19 eralized order parameter as it applies to the methyl symmetry and the peptide (∆SRT) have been grouped. The similarity in peptide 2 19 axis (O axis). A recent re-evaluation of the model-free treatment lengths, the structures of the complexes, and the binding affinities reinforces confidence in its robustness with respect to highly asym- suggest that ∆SRT is essentially constant across the complexes. We 20 2 metric side chain motion . The 80 methyl O axis para­meters from 53 further postulate that the contribution of changes in the conforma- residues of the target domains in the 6 wild-type CaM complexes tional entropy of CaM and the target domains are linearly related to

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local disorder represented by the squared generalized order para­ Supplementary Table 1 for further explanation). Equation (3)

meters determined by NMR relaxation in methyl groups: also includes potential contributions (∆Sother) from other sources of entropy of the protein that are not sensed (directly or indirectly) by  CaM 2 CaM target 2 target  ∆∆Smconf =+nOres iiaxis nOres ∆ axis  +∆Sother (3) the deuterium NMR relaxation probes used here. This includes, for    example, vibrational entropy involving motion that does not average The parameter m defines the scaling between average changes in the angle of the methyl symmetry axis with the magnetic field and residue-weighted generalized order parameters for calmodulin and motion slower than overall tumbling of the macromolecule12. We

the target domain upon binding—that is, assume that these other contributions (∆Sother) are the same for all ∆OO22=−()complexedfO2 ()ree (4) target peptides and complexes, which is not unreasonable given that axis axis axis the peptides all have roughly the same number of degrees of free- CMa target nres and nres are the number of residues in CaM and a given dom. This leads to the prediction of a linear relationship between target domain, respectively, used in the entropy calibration (see the difference of the total binding entropy and the solvent entropy and the apparent change in conformational entropy measured by NMR relaxation: 180°  CaM 22CaM target target   ()∆∆SStots− ol =+mn res ••∆∆Onaxis res Oaxis  ++∆∆SSRT other (5)    Some of the assumptions leading to equation (5) may appear drastic. However, should any of them be violated, a significant devi- ation from linearity should be observed. As suggested by equation (5), to compare dynamics in the various Trp complexes, we used a normalization procedure to account for varia- CaMKKα(p) tion in the number of methyl sites in CaM whose motion could be quantified and to account for the fact that, although fully resolved, the number of residues in the target domains vary. We used a simple average (see equation (5) and Supplementary Table 4). The appar- ent change in conformational entropy due to fast motion was then calculated without explicit consideration of the classical entropy due to rotamer interconversion11,23. This entropy is contained within Leu the calibrated dynamical proxy (that is, within the scalar m). As the free target states cannot be assessed using the model-free formal- smMLCK(p) 1.0 ism19, we assume that the dynamics of the free unstructured target 2 domains are uniform and correspond to an O axis of 0.05. This lim- iting value is seen, for example, in Val3 of CaMKKα(p) bound to CaM, which is completely solvent exposed and hence provides an internal reference for unrestricted motion that meets the criteria of

© 2010 Nature America, Inc. All rights reserved. All rights Inc. America, Nature © 2010 the model-free treatment. 2 The O axis parameter only detects motion on timescales signifi- cantly shorter than the macromolecular tumbling, which for these Met complexes is on the order of 8.5 ns11. States that interconvert on the CaMKI(p) microsecond to millisecond timescale are illuminated by chemi- 24 0.0 cal shift averaging effects . Methyl dispersion experiments of the O 2 CaM-smMLCK(p) and CaM-nNOS(p) complexes indicate that axis these complexes are silent in this time regime. Minor states that are not averaged on the chemical shift timescale result in weak peaks in the NMR spectrum. A small amount of microheterogeneity was observed in CaM in some of the complexes11. This is a small con- Leu tribution that we have ignored. There was no microheterogeneity of eNOS(p) side chain conformations of the bound target domains evident in their 13C heteronuclear single quantum coherence (HSQC) spectra, and the corresponding contribution to the conformational entropy was taken to be zero. To solve equation (5), we used the binding entropies obtained by isothermal titration calorimetry11 and calculated the change in solvent entropy using the known structures of free CaM25 and the five complexes for which high-resolution structures are avail- Leu 26–29 nNOS(p) able . We used the empirical relationship between changes in accessible surface area and the entropy of solvent30, and we calcu- lated this relationship assuming a fully solvated structure for the Figure 2 | Dynamical character of the hydrophobic anchor in the dissociated target domains (see Methods and Supplementary N-terminal domain of CaM. Shown are the heavy atom surface Table 5). We did not include changes in solvent entropy due to representations of CaM residues 6–146 and the associated target domains. electro­striction of water by solvation of explicit charge. To further test The target domains were flipped 180° and translated to the right to show this approach, we also examined the complex of a mutant CaM with the binding interface. Methyl groups are color coded according to their the smMLCK(p) domain. Of several studied previously, we chose mobility. The arrows point to the so-called anchor residues. The figure to ­examine the CaM(E84K)-smMLCK(p) complex as it showed was generated using PyMOL50. the most ­varied dynamical response compared to the wild-type

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–0.40 Excluding the CaMKKα(p) complex, the quantitative linearity of the remaining points strongly suggests that the assumptions under- lying equation (5) are largely valid and that a self-consistent view –0.50 CaM-nNOS(p) of the origin of the thermodynamics of binding in the calmodulin CaM-eNOS(p) system has been established. Most important is the apparent validity ) –0.60 of using measures of motion as a quantitative proxy for conforma- –1

K tional entropy. Furthermore, the quantitative consistency also sug- –1 CaM(E84K)-smMLCK(p) gests that the contribution of vibrational entropy, which is largely –0.70 contained in the constant intercept in Figure 3, is not variable across

) (kJ mol CaM-CaMKI(p) the calmodulin complexes. In this respect, it is noteworthy that the sol S –0.80 CaM-smMLCK(p) ∆ corresponding ordinate intercept is nominally positive even though

– the loss of rotational and translational entropy would result in a neg- tot S

∆ ative contribution to the ordinate intercept (see equation (2)). This

( –0.90 apparent discrepancy is easily explained by recognizing that the for- CaM-CaMKKα(p) mation of each of the complexes results in the burial of six charged –1.00 side chains through the formation of ion pairs. The removal of charge from bulk water will result in a significant increase in solvent entropy33. The degree of electrostriction in the free state can be esti- –1.10 15 20 25 30 35 mated from the pressure dependence of the formation of the CaM- CaM 2 CaM target 2 target smMLCK(p) complex, which has been measured using hydrogen n ∆O + n 〈〉∆O 34 res ⋅ 〈 axis 〉 res ⋅ axis exchange–based methods . Comparison to solvent entropy values for model charged species33 suggests that this effect is comparable Figure 3 | Calibration of the dynamical proxy for protein conformational to the predicted positive contribution to the free energy of binding entropy. Simple considerations lead to the prediction of a quantitative by ∆SRT. Furthermore, vibrational entropy has also been suggested linear relationship between the total binding entropy and the entropy of to provide a favorable contribution to the binding of ligands to pro- solvent to the conformational entropy by NMR relaxation parameters teins in some cases; dihydrofolate reductase is one such example35. derived from methyl-bearing amino acids (see equation (5)). The error It is possible that an increase in vibrational entropy upon formation 2 bars represent the average s.d. of the difference of the average O axis of the calmodulin complexes also diminishes the counterbalancing parameters between free CaM and the complex with each target (see text). contribution of the loss of rotational and translational entropy to 2 Each O axis parameter was derived from T1 and T1ρ values obtained at two the ordinate intercept of Figure 3. Regardless, the linear regression magnetic field strengths. The average dynamics of wild-type and E84K statistics indicate that these contributions are constant across the CaM were based on 52 resolved methyl sites. The average dynamics of the five well-behaved complexes. six complexes shown were based on 73 to 88 resolved methyl sites (see The possible exception of the CaM-CaMKKα(p) complex to the Supplementary Table 7 for further details). The lower CaM-CaMKKα(p) linear relationship defined by the other five complexes (Fig. 3) has datum is a clear outlier (Jackknife distance 8.8, all others < 2.3). The upper several implications. As noted above, this domain is expected to CaM-CaMKKα(p) point results from a simple correction to the solvent form a collapsed hydrophobic cluster in the free state. A simple cal-

© 2010 Nature America, Inc. All rights reserved. All rights Inc. America, Nature © 2010 entropy arising from a postulated hydrophobic cluster in the free state culation suggests that such a dehydrated cluster could significantly of this target domain (see Supplementary Table 5). Excluding the CaM- reduce the apparent discrepancy (see Supplementary Table 5 and CaMKKα(p) points results in a linear regression statistic R of 0.95 Fig. 3). The resulting chain compaction would cause an opposing (P < 0.02). This regression line is shown. The slope (m = −0.037 ± 0.007 correction but in this situation is predicted to be relatively small36. It kJ K−1 mol−1 residue−1) allows for empirical calibration of the conversion of should be noted that the isothermal titration calorimetry profile of changes in side chain dynamics to a quantitative estimate of changes in the formation of this complex is simple and unremarkable and does conformational entropy. The ordinate intercept is 0.26 ± 0.18 kJ K−1 mol−1. not indicate the presence of a more complex equilibrium involving the disassembly of aggregates of the target, for example17. In addition, calmodulin interacts with this domain in reverse orientation from complex31. The entropy of binding was determined by isothermal the others, which may be reflected in the apparently anomalously titration calorimetry to be +41 ± 1 kJ mol−1 at 308 K, which is substan- higher motion of the bound domain. Despite these uncertainties tially less unfavorable than the wild-type complex. The binding free about the nature of the solvent and free target domain contributions energy and enthalpy were also determined be −43.6 ± 0.5 and −84.3 ± to the binding entropy of the formation of this complex, the contri- −1 2 0.8 kJ mol , respectively. The methyl O axis parameters were previously bution of CaM to the binding of the CaMKKα(p) target is consistent determined for the complex31. To use this complex for calibration, we with the other complexes, as we will now show below. had to measure the dynamics of the bound smMLCK(p) domain and characterize the dynamics of the free CaM(E84K) mutant. Insights into the role of protein entropy in binding Equation (5) requires a quantitative linear relationship and is We calibrated the dynamical proxy relating local measures of dis­ plotted for the six complexes (Fig. 3). Five give an excellent lin- order quantified by the squared generalized order parameters of the ear relationship (R = 0.95), a slope of −0.037 ± 0.007 kJ K−1 mol−1 symmetry axis of methyl groups to residual conformational entropy residue−1 and an ordinate intercept of 0.26 ± 0.18 kJ K−1 mol−1 (Fig. 3). This allowed us to determine the contributions of changes (Fig. 3). The CaMKKα(p) complex is a clear outlier (Fig. 3). This in conformational entropy to the total binding entropy (Fig. 4). is not surprising. The primary sequence is unusually hydrophobic The empirical calibration of measures of motion as a proxy for con- (Supplementary Table 1) and the peptide has limited solubility17. formational entropy circumvents the microscopic details that are dif- Hydrophobic cluster analysis32 illuminates a hydrophobic patch ficult to accommodate in a model-dependent calculation. In a sense, and suggests that the dissociated domain exists in a collapsed, we have simply created an entropy meter analogous to a simple ther- less hydrated state than is assumed for the solvent entropy mometer. It is important to note that in this view the motion of the calculations. A simple correction based on the size of the pre- methyl group is used to sense the disorder of its local surroundings. dicted hydrophobic cluster shows that this is not an unreasonable The apparent relationship between the change in the conforma- explanation for the apparent discrepancy (Fig. 3). tional entropy of CaM and the target domains and the total entropy

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smMLCK(p) eNOS(p) chain dynamics seen generally in proteins and the similarly hetero­ CaMKKα(p) CaMKI(p) PDE(p) nNOS(p) geneous response to ligand binding seen here raises the question of 0.80 how faithful the coupling is. Indeed, as exemplified by the variable behavior of the CaM-target domain interface (Fig. 2), the coupling 0.60 at specific sites may seem quite variable. However, the approach

) illustrated here relies on the average response of dozens of probes. –1

K Indeed, as the mutant CaM complex perhaps demonstrates best, the –1 0.40 empirical correlation of motion with conformational entropy seems to be surprisingly robust (Fig. 3). 0.20 This robustness presumably arises from several sources. The empirical calibration of the entropy meter developed here assumes 2 0.00 the existence of a linear relationship between motion (that is, O axis) and the local conformational entropy (disorder) sampled by the −0.20 NMR spin probe. In a packed protein, this local disorder reflects not only the spin probe itself but also that of its neighbors. In addition, a wide range of potential energy functions, including the highly asym- −0.40 metric step function, show a roughly linear correlation with the cor- Component binding entropy (kJ mol 2 23 2 responding O axis parameter . The O axis parameters of various CaM −0.60 complexes have a trimodal distribution11. We have termed these groupings or classes the J-, α- and ω-classes, in accordance with 12 2 −0.80 the character of the motions underlying them . The lowest O axis −0.30 −0.25 −0.20 −0.15 −0.10 −0.05 0 parameter J-class is accompanied by rotamer interconversion23 that Total binding entropy (kJ mol–1 K–1) is reflected by averaging of associated J-coupling constants. In prin- 2 ciple this distortion of the O axis parameter could introduce a non- Figure 4 | Decomposition of the entropy of binding of target domains to linear response. However, for energetic parameters consistent with calcium-saturated calmodulin. Decomposition is based on equation (5) rotamer populations in packed proteins, the correlation remains and the calibration of the dynamical proxy (see Fig. 3). Solid diamonds are very linear23. Furthermore, in most cases, the dynamical response the solvent entropies calculated from the changes in accessible surface of individual sites within calmodulin does not result in a change in area and include the correction resulting from the postulated hydrophobic motional class (see ref. 23 for an illustration). This tends to minimize cluster of the free CaMKα(p) target domain (see text and Supplementary the introduction of qualitatively distinct motion upon transition Table 5). The uncorrected value for CaMKα(p) is shown as an open from the free to bound state of the protein. These various factors, diamond. No structure is available for the CaM-PDE(p) complex, so the when averaged over a large number of probes distributed across corresponding solvent entropy cannot be calculated. Solid circles and the entire molecule, would tend to produce a robustness to local triangles are the contributions to the binding entropy by the conformational exceptions to the assumptions underlying the use of a dynamical entropy of CaM and the target domains, respectively. Solid squares are proxy for conformational entropy. the contributions to the binding entropy not reflected in the measured The analysis presented here is somewhat limited by knowl-

© 2010 Nature America, Inc. All rights reserved. All rights Inc. America, Nature © 2010 dynamics—that is, ∆Sother + ∆SRT (see equation (5), which is obtained from edge of the free state of the ligand. Calmodulin generally relieves linear regression (see Fig. 3)). Though not required, there are noteworthy auto-inhibition by competition for a pseudosubstrate sequence15. linear correlations between the total binding entropy and its components. Molecular recognition of the target sequence involves its refold- There is a significant (R = 0.94) but weakly dependent (slope = −0.25 ± 0.04) ing on the surface of CaM34,37 and is an example of “folding upon negative linear correlation between solvent entropy and total binding binding”38. Complexes involving structured proteins, where the entropy. In contrast, there is a significant (R = 0.91) and relatively strong NMR-based inventory of changes in conformational entropy can be positive dependence (slope = +1.0 ± 0.2) observed between the change of undertaken directly in both the free and complexed states, should conformational entropy of CaM and the total binding entropy. The apparent prove more tractable. Nevertheless, the observations made here negative correlation between target binding entropy and the total binding extend the database of site-resolved experimental measurements entropy is not statistically significant (R = 0.045; P > 0.10). of fast dynamics in proteins12, which in turn provides challenging test sets for molecular dynamics simulations39–42 and for structure- of binding can now be quantitatively revealed. The changes in con- based predictions of local motion43. The results reported here repre- formational entropy of the target domains and CaM are large rela- sent to our knowledge the first quantitative experimental measure tive to the free energy of binding and are the same magnitude as the of the role of conformational entropy in high-affinity interactions solvent entropy (Fig. 4 and Supplementary Table 6). involving proteins. As such, they should provide a benchmark for theoretical analyses based on statistical thermodynamic or molecu- DISCUSSION lar dynamics­ approaches44. The use of a dynamical proxy as an empirical entropy meter is more The conformational entropy of CaM is linearly correlated with simple and precise than the “oscillator inventory” attempted previ- the total binding entropy. There is no physical requirement for such ously11. Although the inventory approach provides a qualitatively a statistical relationship. Its existence suggests roots in the evolu- correct view, it seems to significantly underestimate the contribu- tion of the target calmodulin-binding domains and the need to tion of conformational entropy to the binding free energy. This is resolve a complex optimization of structural specificity (molecular perhaps not surprising. An inventory requires completeness in the recognition) and affinity. Notably, calcium-saturated calmodulin sampling of motion, whereas the method developed here has quite is an unusually dynamic protein, and this feature has clearly been a different view and uses local measures of motion as an indirect exploited and may have been the solution arrived at by evolution. In measure of local disorder. The latter approach does require good effect, variation of the conformational entropy of calmodulin ‘tunes’ sampling of the volume of the protein, which is met here by the ~90 the binding free energy. In contrast, the estimated increases in methyl groups in each of the calmodulin complexes. It also requires ­solvent entropy upon binding are large and favorable but only vary significant coupling of the motion of methyl groups to surround- a little with binding entropy (Fig. 4). Thus, although solvent entropy ing nonmethyl amino acid side chains. The heterogeneous side is a powerful favorable driving force for these binding interactions,

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