Entropy Meter Research Articles

Characterization of Internal Protein Dynamics and Conformational Entropy by NMR Relaxation.

Recent studies suggest that the fast timescale motion of methyl-bearing side chains may play an important role in mediating protein activity. These motions have been shown to encapsulate the residual conformational entropy of the folded state that can potentially contribute to the energetics of protein function. Here, we provide an overview of how to characterize these motions using nuclear magnetic resonance (NMR) spin relaxation methods. The strengths and limitations of several techniques are highlighted in order to assist with experimental design. Particular emphasis is placed on the practical aspects of sample preparation, data collection, data fitting, and statistical analysis. Additionally, discussion of the recently refined "entropy meter" is presented and its use in converting NMR observables to conformational entropy is illustrated. Taken together, these methods should yield new insights into the complex interplay between structure and dynamics in protein function.

Entropy in 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 in Molecular Recognition by Proteins

At a fundamental level, biological processes are most often controlled using molecular recognition by proteins. Protein-ligand interactions impact critical events ranging from the catalytic action of enzymes, the assembly of macromolecular structures, complex signaling and allostery, transport phenomena, force generation and so on. The physical origin of high affinity interactions involving proteins continues to be the subject of intense investigation. Conformational entropy represents perhaps the last piece of the thermodynamic puzzle that governs protein structure, stability, dynamics and function. The presence and importance of internal conformational entropy in proteins has been debated for decades but has resisted experimental quantification. Over the past few years we have introduced, developed and validated an NMR-based approach that uses a dynamical proxy to determine changes in conformational entropy. This new approach, which we term the NMR “entropy meter,” requires few assumptions, is empirically calibrated and is apparently robust and universal. Using this “entropy meter,” it can now be quantitatively shown that proteins retain considerable conformational entropy in their native functional states and that this conformational entropy can play a decisive role in the thermodynamics of molecular recognition by proteins. Recent results show that changes conformational entropy of a protein upon binding a high affinity ligand is highly system specific and can vary from strongly inhibiting to even strongly promoting binding and everything in between. Thus one cannot possibly understand comprehensively how proteins work without knowledge of the breadth and underlying principles of the role of conformational entropy in protein function. This approach also allows for the refinement of empirical coefficients that relate changes in accessible surface area to changes in the entropy of water and the determination of the loss of rotational-translational entropy in high affinity protein complexes. Supported by the NIH and the Mathers Foundation.

Role of Conformational Entropy in Extremely High Affinity Protein Interactions

Interactions of extreme affinity (Kd ∼ fM) underlie many biochemical processes necessary to life. The physical determinants of such large binding energies are not well understood. Specific interactions at the interface (ΔHbinding) and the release of solvating water (TΔSsolvation) are usually assumed to dominate the binding energetics. The role of conformational entropy (TΔSconf) in determining binding affinity has remained elusive, in part due to the difficulties in measuring such changes in entropy experimentally. Recent developments in the Wand laboratory have bridged this gap by using solution NMR measurements of dynamics to empirically calibrate a “conformational entropy meter.” It has enabled quantitative measurements of the change in conformational entropy in protein-ligand binding. The toxin-antitoxin system studied here, barnase-barstar, forms a complex with fM affinity (ΔGbinding ∼ 19 kcal/mol) without undergoing any major structural changes and retaining a hydrated interface. To explore the role of conformational entropy, the fast (ps-ns time-scale) motions of backbone and side chains of the two proteins were measured in both the free (unbound) and the complexed (bound) states using NMR spectroscopy. Furthermore, hydration dynamics were measured in water and in the confined space of a reverse micelle. The dynamic response observed leads to an unfavorable change in TΔSconf, with a more rigid, still hydrated interface. This comprehensive study of both protein and “ligand” (in this case, another protein) and the measured site-specific changes in dynamics and hydration sheds light on the thermodynamic contributions that enable fM affinities. Supported by grants from the NIH, The Mathers Foundation and NSF.

Motion and Conformational Entropy in Protein Function: Creation of an NMR-Based Entropy Meter

Conformational entropy is a potentially important thermodynamic parameter contributing to protein function. Quantitative measures of conformational entropy are necessary for an understanding of its role but have been difficult to obtain. We have recently introduced empirical method that utilizes changes in conformational dynamics as a proxy for changes in conformational entropy. We have now used molecular dynamics simulations to probe the microscopic origins of the link between conformational dynamics and conformational entropy. Simulation of seven proteins gave an excellent correlation with measures of side-chain motion derived from NMR relaxation. The simulations show that the motion of methyl-bearing side-chains are sufficiently coupled to that of other side chains to serve as excellent reporters of the overall side-chain conformational entropy. These results tend to validate the use of experimentally accessible measures of methyl motion - the NMR-derived generalized order parameters - as a proxy from which to derive changes in protein conformational entropy due to a perturbation such as the binding of a ligand. A slightly modified weighting scheme to project the change in dynamics of experimental methyl dynamics into conformational entropy is presented. Originally based on data from the calmodulin system, we will describe experimental results from other systems that indicate that the “entropy meter” approach is both robust and general and that the involved conformational entropies are often large and cannot be ignored. Supported by the NIH and the Mathers Foundation.

Probing the Role of Conformational Entropy in Protein-Inhibitor Binding

In silico drug discovery has long held the promise to revolutionize the way drugs are designed, but has largely failed to deliver. Historically inhibitors were designed based mainly on static structures that emphasized interaction energetics but ignored potential contributions from conformational entropy. This issue has been difficult to address experimentally. Recently, we have developed an NMR-based approach that employs measures of motion as a proxy for the conformational entropy. Using an empirical calibration, we have shown that the resulting “entropy meter” is both robust and general. We hypothesize that conformational entropy represents a critical “missing piece” in rational drug design. Using the “entropy meter,” we examine the effect of conformational entropy on protein-inhibitor interactions using the drug target and mitogen-activated protein kinase (MAPK) p38α as a model system. Given the relatively large size of this protein, all studies were performed in a perdeuterated background. Using standard methods, we have obtained nearly complete ILV methyl assignments of this protein. 15N and 13C NMR spin relaxation experiments were used to determine the sub-ns dynamics of the protein both in the apo state and bound to either a competitive and a non-competitive (allosteric) inhibitor. In both cases, we observe the a large unfavorable entropic effect of binding. We find that the conformational entropy change upon binding represents a significant contribution to the overall binding free energy. These results promote further studies to understand how classes of inhibitors may modulate the dynamics and corresponding internal entropy of protein targets, and how this information can be used to design better lead compounds.This work was supported by NIH Grant GM100910

Resolving Internal Motional Correlations to Complete the Conformational Entropy Meter.

Conformational entropy (SΩ) has long been used to theoretically characterize the dynamics of proteins, DNA, and other polymers. Though recent advances enabled its calculation also from simulations and nuclear magnetic resonance (NMR) relaxation experiments, correlated molecular motion has hitherto greatly hindered both numerical and experimental determination, requiring demanding empirical and computational calibrations. Herein, we show that these motional correlations can be estimated directly from the temperature-dependent SΩ series that reveal effective persistence lengths of the polymers, which we demonstrate by measuring SΩ of amphiphilic molecules in model lipid systems by spin-labeling electron paramagnetic resonance (EPR) spectroscopy. We validate our correlation-corrected SΩ meter against the basic biophysical interactions underlying biomembrane formation and stability, against the changes in enthalpy and diffusion coefficients upon phase transitions, and against the energetics of fatty acid dissociation. As the method can be directly applied to conformational analysis of proteins and other polymers, as well as adapted to NMR or polarized fluorescence techniques, we believe that the approach can greatly enrich the scope of experimentally available statistical thermodynamics, offering new physical insights into the behavior of biomolecules.

Entropy meters and the entropy of non-extensive systems.

In our derivation of the second law of thermodynamics from the relation of adiabatic accessibility of equilibrium states, we stressed the importance of being able to scale a system's size without changing its intrinsic properties. This leaves open the question of defining the entropy of macroscopic, but unscalable systems, such as gravitating bodies or systems where surface effects are important. We show here how the problem can be overcome, in principle, with the aid of an 'entropy meter'. An entropy meter can also be used to determine entropy functions for non-equilibrium states and mesoscopic systems.

Measuring Entropy in Molecular Recognition by Proteins.

Molecular recognition by proteins is fundamental to the molecular basis of biology. Dissection of the thermodynamic landscape governing protein-ligand interactions has proven difficult because determination of various entropic contributions is quite challenging. Nuclear magnetic resonance relaxation measurements, theory, and simulations suggest that conformational entropy can be accessed through a dynamical proxy. Here, we review the relationship between measures of fast side-chain motion and the underlying conformational entropy. The dynamical proxy reveals that the contribution of conformational entropy can range from highly favorable to highly unfavorable and demonstrates the potential of this key thermodynamic variable to modulate protein-ligand interactions. The dynamical so-called entropy meter also refines the role of solvent entropy and directly determines the loss in rotational-translational entropy that occurs upon formation of high-affinity complexes. The ability to quantify the roles of entropy through an entropy meter based on measurable dynamical properties promises to highlight its role in protein function.

Microscopic Insights into the NMR Relaxation-Based Protein Conformational Entropy Meter

Conformational entropy is a potentially important thermodynamic parameter contributing to protein function. Quantitative measures of conformational entropy are necessary for an understanding of its role but have been difficult to obtain. An empirical method that utilizes changes in conformational dynamics as a proxy for changes in conformational entropy has recently been introduced. Here we probe the microscopic origins of the link between conformational dynamics and conformational entropy using molecular dynamics simulations. Simulation of seven proteins gave an excellent correlation with measures of side-chain motion derived from NMR relaxation. The simulations show that the motion of methyl-bearing side chains are sufficiently coupled to that of other side chains to serve as excellent reporters of the overall side-chain conformational entropy. These results tend to validate the use of experimentally accessible measures of methyl motion--the NMR-derived generalized order parameters--as a proxy from which to derive changes in protein conformational entropy.

Molecular Dynamics Simulations Support the use of Methyl Generalized Order Parameters in the “Entropy Meter”

Conformational entropy is a potentially important thermodynamic parameter contributing to protein function. Unfortunately, conformational entropy has been difficult to measure experimentally. We have been working towards using measures of fast internal motion as a proxy for conformational entropy. Implementation of this approach has been impeded by the apparent need to employ a model-dependent interpretation of the obtained generalized order parameters. We have recently proposed that the quantitative relationship between generalized order parameters and conformational entropy could be determined empirically and effectively create an ‘entropy meter’ (Marlow et al. Nat. Chem. Biol. 6, 353). This approach has initially employed methyl symmetry axis order parameters that can be obtained relatively easily. The approach then rests on the coupling between motion of the methyl bearing side chains and the rest of the amino acids being sufficient to report on the whole protein. Here, we examine the generality of this assumption using molecular dynamics simulations of a number of proteins. The MD simulations performed with the NAMD suite showed excellent agreement between the calculated and experimental side chain order parameters for all the proteins, a marked improvement from previous studies. We find a general correlation between the internal dynamics of protein and the protein conformational entropy. The calculated protein conformational entropy showed excellent linear correlation to both the calculated NMR order parameter of methyl groups as well as the experimentally measured NMR methyl order parameters. Supported by NIH grant GM102447 and a grant from the Mathers Foundation.

The role of conformational entropy in molecular recognition by calmodulin

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 employed 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” employing 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 are surprisingly non-complementary. These observations promote modification of our understanding of the energetics of protein-ligand interactions.

Conformational entropy in molecular recognition by proteins

Molecular recognition by proteins is fundamental to almost every biological process, particularly the protein associations underlying cellular signal transduction. Understanding the basis for protein‐protein interactions requires the full characterization of the thermodynamics of their association. Historically it has been virtually impossible to experimentally assess changes in residual protein entropy, a potentially important component of the change in the binding free energy. NMR spectroscopy has recently emerged as a powerful tool for characterizing the dynamics of proteins and has thereby gained access to their conformational entropy. We have previously shown that the dynamics of calmodulin varies significantly upon binding a variety of target domains. Surprisingly, the apparent change in the corresponding entropy is linearly related to the change in the overall binding entropy. We have completed a complimentary study of the dynamics of the bound calmodulin‐binding domains, which allows for the calibration of the dynamical “entropy meter.” The results indicate that changes in residual protein conformational entropy can indeed contribute significantly to the free energy of protein‐ligand association. This is not a generally held view and has significant implications for the maturation of high affinity interactions either by biological evolution or by human intervention.

An entropy meter based on the thermoelectric potential of a nonisothermal solid-electrolyte cell

The theory, design, and performance of a solid electrolyte twin thermocell for the direct determination of the partial molar entropy of oxygen in a single-phase or multiphase mixture are described. The difference between the Seebeck coefficients of the concentric thermocells is directly related to the difference in the partial molar entropy of oxygen in the electrodes of each thermocell. The measured potentials are sensitive to small deviations from equilibrium at the electrodes. Small electric disturbances caused by simultaneous potential measurements or oxygen fluxes caused by large oxygen potential gradients between the electrodes also disturb the thermoelectric potential. An accuracy of ±0.5 cal th K −1 mol −1 has been obtained by this method for the entropies of formation of NiO and NiAl 2O 4. This “entropy meter” may be used for the measurement of the entropies of formation of simple or complex oxides with significant residual contributions which cannot be detected by heat-capacity measurements.