Measuring Entropy in Molecular Recognition by Proteins

NMR and Single-Molecule FRET Insights into Fast Protein Motions and Their Relation to Function

Proteins often undergo large-scale conformational transitions, in which secondary and tertiary structure elements (loops, helices, and domains) change their structures or their positions with respect to each other. Simple considerations suggest that such dynamics should be relatively fast, but the functional cycles of many proteins are often relatively slow. Sophisticated experimental methods are starting to tackle this dichotomy and shed light on the contribution of large-scale conformational dynamics to protein function. In this review, we focus on the contribution of single-molecule Förster resonance energy transfer and nuclear magnetic resonance (NMR) spectroscopies to the study of conformational dynamics. We briefly describe the state of the art in each of these techniques and then point out their similarities and differences, as well as the relative strengths and weaknesses of each. Several case studies, in which the connection between fast conformational dynamics and slower function has been demonstrated, are then introduced and discussed. These examples include both enzymes and large protein machines, some of which have been studied by both NMR and fluorescence spectroscopies.

Deuterium spin relaxation of fractionally deuterated ribonuclease H using paired 475 and 950 MHz NMR spectrometers

Deuterium (2H) spin relaxation of 13CH2D methyl groups has been widely applied to investigate picosecond-to-nanosecond conformational dynamics in proteins by solution-state NMR spectroscopy. The B0 dependence of the 2H spin relaxation rates is represented by a linear relationship between the spectral density function at three discrete frequencies J(0), J(ωD) and J(2ωD). In this study, the linear relation between 2H relaxation rates at B0 fields separated by a factor of two and the interpolation of rates at intermediate frequencies are combined for a more robust approach for spectral density mapping. The general usefulness of the approach is demonstrated on a fractionally deuterated (55%) and alternate 13C-12C labeled sample of E. coli RNase H. Deuterium relaxation rate constants (R1, R1ρ, RQ, RAP) were measured for 57 well-resolved 13CH2D moieties in RNase H at 1H frequencies of 475 MHz, 500 MHz, 900 MHz, and 950 MHz. The spectral density mapping of the 475/950 MHz data combination was performed independently and jointly to validate the expected relationship between data recorded at B0 fields separated by a factor of two. The final analysis was performed by jointly analyzing 475/950 MHz rates with 700 MHz rates interpolated from 500/900 MHz data to yield six J(ωD) values for each methyl peak. The J(ω) profile for each peak was fit to the original (τM, Sf2, τf) or extended model-free function (τM, Sf2, Ss2, τf, τs) to obtain optimized dynamic parameters.

Decreasing Intracellular Entropy by Increasing Mitochondrial Efficiency and Reducing ROS Formation-The Effect on the Ageing Process and Age-Related Damage

A hypothesis is presented to explain how the ageing process might be influenced by optimizing mitochondrial efficiency to reduce intracellular entropy. Research-based quantifications of entropy are scarce. Non-equilibrium metabolic reactions and compartmentalization were found to contribute most to lowering entropy in the cells. Like the cells, mitochondria are thermodynamically open systems exchanging matter and energy with their surroundings-the rest of the cell. Based on the calculations from cancer cells, glycolysis was reported to produce less entropy than mitochondrial oxidative phosphorylation. However, these estimations depended on the CO2 concentration so that at slightly increased CO2, it was oxidative phosphorylation that produced less entropy. Also, the thermodynamic efficiency of mitochondrial respiratory complexes varies depending on the respiratory state and oxidant/antioxidant balance. Therefore, in spite of long-standing theoretical and practical efforts, more measurements, also in isolated mitochondria, with intact and suboptimal respiration, are needed to resolve the issue. Entropy increases in ageing while mitochondrial efficiency of energy conversion, quality control, and turnover mechanisms deteriorate. Optimally functioning mitochondria are necessary to meet energy demands for cellular defence and repair processes to attenuate ageing. The intuitive approach of simply supplying more metabolic fuels (more nutrients) often has the opposite effect, namely a decrease in energy production in the case of nutrient overload. Excessive nutrient intake and obesity accelerate ageing, while calorie restriction without malnutrition can prolong life. Balanced nutrient intake adapted to needs/activity-based high ATP requirement increases mitochondrial respiratory efficiency and leads to multiple alterations in gene expression and metabolic adaptations. Therefore, rather than overfeeding, it is necessary to fine-tune energy production by optimizing mitochondrial function and reducing oxidative stress; the evidence is discussed in this paper.

Breaking down walls: Continuous potential models for internal motions in NMR spin relaxation

Simple physical models for restricted diffusion in a potential, which provide important insights for NMR spin relaxation, usually are based on free diffusion within rigid boundaries or diffusion in relatively simple continuous potential energy surfaces. The diffusion-in-a-cone model is an example of the former and diffusion in an N-fold cosine potential is an example of the latter. The present work models restricted diffusion for arbitrary potential energy functions on the surface of a cone or a sphere, by expanding the potentials in Fourier or spherical harmonic series, respectively. The results exhibit simple relationships between generalized order parameters and effective correlation times, critical for analysis of experimental spin relaxation data, and illustrate the transition from diffusive-like to jump-like behavior in multi-well potentials.

Computational Approaches to Predict Protein-Protein Interactions in Crowded Cellular Environments

Investigating protein-protein interactions is crucial for understanding cellular biological processes because proteins often function within molecular complexes rather than in isolation. While experimental and computational methods have provided valuable insights into these interactions, they often overlook a critical factor: the crowded cellular environment. This environment significantly impacts protein behavior, including structural stability, diffusion, and ultimately the nature of binding. In this review, we discuss theoretical and computational approaches that allow the modeling of biological systems to guide and complement experiments and can thus significantly advance the investigation, and possibly the predictions, of protein-protein interactions in the crowded environment of cell cytoplasm. We explore topics such as statistical mechanics for lattice simulations, hydrodynamic interactions, diffusion processes in high-viscosity environments, and several methods based on molecular dynamics simulations. By synergistically leveraging methods from biophysics and computational biology, we review the state of the art of computational methods to study the impact of molecular crowding on protein-protein interactions and discuss its potential revolutionizing effects on the characterization of the human interactome.

Gauging Dynamics-driven Allostery Using a New Computational Tool: A CAP Case Study

In this study, we utilize Protein Residue Networks (PRNs), constructed using Local Spatial Pattern (LSP) alignment, to explore the dynamic behavior of Catabolite Activator Protein (CAP) upon the sequential binding of cAMP. We employed the Degree Centrality of these PRNs to investigate protein dynamics on a sub-nanosecond time scale, hypothesizing that it would reflect changes in CAP's entropy related to its thermal motions. We show that the binding of the first cAMP led to an increase in stability in the Cyclic-Nucleotide Binding Domain A (CNBD-A) and destabilization in CNBD-B, agreeing with previous reports explaining the negative cooperativity of cAMP binding in terms of an entropy-driven allostery. LSP-based PRNs also allow for the study of Betweenness Centrality, another graph-theoretical characteristic of PRNs, providing insights into global residue connectivity within CAP. Using this approach, we were able to correctly identify amino acids that were shown to be critical in mediating allosteric interactions in CAP. The agreement between our studies and previous experimental reports validates our method, particularly with respect to the reliability of Degree Centrality as a proxy for entropy related to protein thermal dynamics. Because LSP-based PRNs can be easily extended to include dynamics of small organic molecules, polynucleotides, or other allosteric proteins, the methods presented here mark a significant advancement in the field, positioning them as vital tools for a fast, cost-effective, and accurate analysis of entropy-driven allostery and identification of allosteric hotspots.

Conformational entropy in molecular recognition of intrinsically disordered proteins

Broad conformational ensembles make intrinsically disordered proteins or regions entropically intriguing. Although methodologically challenging and understudied, emerging studies into their changes in conformational entropy (ΔS°conf) upon complex formation have provided both quantitative and qualitative insight. Recent work based on thermodynamics from isothermal titration calorimetry and NMR spectroscopy uncovers an expanded repertoire of regulatory mechanisms, where ΔS°conf plays roles in partner selection, state behavior, functional buffering, allosteric regulation, and drug design. We highlight these mechanisms to display the large entropic reservoir of IDPs for the regulation of molecular communication. We call upon the field to make efforts to contribute to this insight as more studies are needed for forwarding mechanistic decoding of intrinsically disordered proteins and their complexes.

Side-chain dynamics of the α1B -adrenergic receptor determined by NMR via methyl relaxation

G protein-coupled receptors (GPCRs) are medically important membrane proteins that sample inactive, intermediate, and active conformational states characterized by relatively slow interconversions (~μs-ms). On a faster timescale (~ps-ns), the conformational landscape of GPCRs is governed by the rapid dynamics of amino acid side chains. Such dynamics are essential for protein functions such as ligand recognition and allostery. Unfortunately, technical challenges have almost entirely precluded the study of side-chain dynamics for GPCRs. Here, we investigate the rapid side-chain dynamics of a thermostabilized α1B -adrenergic receptor (α1B -AR) as probed by methyl relaxation. We determined order parameters for Ile, Leu, and Val methyl groups in the presence of inverse agonists that bind orthosterically (prazosin, tamsulosin) or allosterically (conopeptide ρ-TIA). Despite the differences in the ligands, the receptor's overall side-chain dynamics are very similar, including those of the apo form. However, ρ-TIA increases the flexibility of Ile1764×56 and possibly of Ile2145×49 , adjacent to Pro2155×50 of the highly conserved P5×50 I3×40 F6×44 motif crucial for receptor activation, suggesting differences in the mechanisms for orthosteric and allosteric receptor inactivation. Overall, increased Ile side-chain rigidity was found for residues closer to the center of the membrane bilayer, correlating with denser packing and lower protein surface exposure. In contrast to two microbial membrane proteins, in α1B -AR Leu exhibited higher flexibility than Ile side chains on average, correlating with the presence of Leu in less densely packed areas and with higher protein-surface exposure than Ile. Our findings demonstrate the feasibility of studying receptor-wide side-chain dynamics in GPCRs to gain functional insights.

Temperature as a modulator of allosteric motions and crosstalk in mesophilic and thermophilic enzymes

Mesophilic and thermophilic enzyme counterparts are often studied to understand how proteins function under harsh conditions. To function well outside of standard temperature ranges, thermophiles often tightly regulate their structural ensemble through intra-protein communication (via allostery) and altered interactions with ligands. It has also become apparent in recent years that the enhancement or diminution of allosteric crosstalk can be temperature-dependent and distinguish thermophilic enzymes from their mesophilic paralogs. Since most studies of allostery utilize chemical modifications from pH, mutations, or ligands, the impact of temperature on allosteric function is comparatively understudied. Here, we discuss the biophysical methods, as well as critical case studies, that dissect temperature-dependent function of mesophilic-thermophilic enzyme pairs and their allosteric regulation across a range of temperatures.

Tsallis Entropy and Mutability to Characterize Seismic Sequences: The Case of 2007-2014 Northern Chile Earthquakes

Seismic data have improved in quality and quantity over the past few decades, enabling better statistical analysis. Statistical physics has proposed new ways to deal with these data to focus the attention on specific matters. The present paper combines these two progressions to find indicators that can help in the definition of areas where seismic risk is developing. Our data comes from the IPOC catalog for 2007 to 2014. It covers the intense seismic activity near Iquique in Northern Chile during March/April 2014. Centered in these hypocenters we concentrate on the rectangle Lat-22-18 and Lon-68-72 and deepness between 5 and 70 km, where the major earthquakes originate. The analysis was performed using two complementary techniques: Tsallis entropy and mutability (dynamical entropy). Two possible forecasting indicators emerge: (1) Tsallis entropy (mutability) increases (decreases) broadly about two years before the main MW8.1 earthquake. (2) Tsallis entropy (mutability) sharply decreases (increases) a few weeks before the MW8.1 earthquake. The first one is about energy accumulation, and the second one is because of energy relaxation in the parallelepiped of interest. We discuss the implications of these behaviors and project them for possible future studies.

Physical pictures of rotation mechanisms of F1- and V1-ATPases: Leading roles of translational, configurational entropy of water

We aim to develop a theory based on a concept other than the chemo-mechanical coupling (transduction of chemical free energy of ATP to mechanical work) for an ATP-driven protein complex. Experimental results conflicting with the chemo-mechanical coupling have recently emerged. We claim that the system comprises not only the protein complex but also the aqueous solution in which the protein complex is immersed and the system performs essentially no mechanical work. We perform statistical-mechanical analyses on V₁-ATPase (the A₃B₃DF complex) for which crystal structures in more different states are experimentally known than for F₁-ATPase (the α₃β₃γ complex). Molecular and atomistic models are employed for water and the structure of V₁-ATPase, respectively. The entropy originating from the translational displacement of water molecules in the system is treated as a pivotal factor. We find that the packing structure of the catalytic dwell state of V₁-ATPase is constructed by the interplay of ATP bindings to two of the A subunits and incorporation of the DF subunit. The packing structure represents the nonuniformity with respect to the closeness of packing of the atoms in constituent proteins and protein interfaces. The physical picture of rotation mechanism of F₁-ATPase recently constructed by Kinoshita is examined, and common points and differences between F₁- and V₁-ATPases are revealed. An ATP hydrolysis cycle comprises binding of ATP to the protein complex, hydrolysis of ATP into ADP and Pi in it, and dissociation of ADP and Pi from it. During each cycle, the chemical compounds bound to the three A or β subunits and the packing structure of the A₃B₃ or α₃β₃ complex are sequentially changed, which induces the unidirectional rotation of the central shaft for retaining the packing structure of the A₃B₃DF or α₃β₃γ complex stabilized for almost maximizing the water entropy. The torque driving the rotation is generated by water with no input of chemical free energy. The presence of ATP is indispensable as a trigger of the torque generation. The ATP hydrolysis or synthesis reaction is tightly coupled to the rotation of the central shaft in the normal or inverse direction through the water-entropy effect.

Assessing the applicability of 19F labeled tryptophan residues to quantify protein dynamics

Nuclear magnetic resonance (NMR) spectroscopy is uniquely suited to study the dynamics of biomolecules in solution. Most NMR studies exploit the spins of proton, carbon and nitrogen isotopes, as these atoms are highly abundant in proteins and nucleic acids. As an alternative and complementary approach, fluorine atoms can be introduced into biomolecules at specific sites of interest. These labels can then be used as sensitive probes for biomolecular structure, dynamics or interactions. Here, we address if the replacement of tryptophan with 5-fluorotryptophan residues has an effect on the overall dynamics of proteins and if the introduced fluorine probe is able to accurately report on global exchange processes. For the four different model proteins (KIX, Dcp1, Dcp2 and DcpS) that we examined, we established that ¹⁵N CPMG relaxation dispersion or EXSY profiles are not affected by the 5-fluorotryptophan, indicating that this replacement of a proton with a fluorine has no effect on the protein motions. However, we found that the motions that the 5-fluorotryptophan reports on can be significantly faster than the backbone motions. This implies that care needs to be taken when interpreting fluorine relaxation data in terms of global protein motions. In summary, our results underscore the great potential of fluorine NMR methods, but also highlight potential pitfalls that need to be considered.

An NMR portrait of functional and dysfunctional allosteric cooperativity in cAMP-dependent protein kinase A

The cAMP-dependent protein kinase A (PKA) is the archetypical eukaryotic kinase. The catalytic subunit (PKA-C) structure is highly conserved among the AGC-kinase family. PKA-C is a bilobal enzyme with a dynamic N-lobe, harbouring the Adenosine-5'-triphosphate (ATP) binding site and a more rigid helical C-lobe. The substrate-binding groove resides at the interface of the two lobes. A distinct feature of PKA-C is the positive binding cooperativity between nucleotide and substrate. Several PKA-C mutations lead to the development of adenocarcinomas, myxomas, and other rare forms of liver tumours. Nuclear magnetic resonance (NMR) spectroscopy shows that these mutations disrupt the allosteric communication between the two lobes, causing a drastic decrease in binding cooperativity. The loss of cooperativity correlates with changes in substrate fidelity and reduced kinase affinity for the endogenous protein kinase inhibitor (PKI). The similarity between PKI and the inhibitory sequence of the kinase regulatory subunits suggests that the overall mechanism of regulation of the kinase may be disrupted. We surmise that a reduced or obliterated cooperativity may constitute a common trait for both orthosteric and allosteric mutations of PKA-C that may lead to dysregulation and disease.

Dissecting the Thermodynamics of ATP Binding to GroEL One Nucleotide at a Time

Variable-temperature electrospray ionization (vT-ESI) native mass spectrometry (nMS) is used to determine the thermodynamics for stepwise binding of up to 14 ATP molecules to the 801 kDa GroEL tetradecamer chaperonin complex. Detailed analysis reveals strong enthalpy-entropy compensation (EEC) for the ATP binding events leading to formation of GroEL-ATP₇ and GroEL-ATP₁₄ complexes. The observed variations in EEC and stepwise free energy changes of specific ATP binding are consistent with the well-established nested cooperativity model describing GroEL-ATP interactions, viz., intraring positive cooperativity and inter-ring negative cooperativity (Dyachenko A.; Proc. Natl. Acad. Sci. U.S.A.2013, 110, 7235-7239). Entropy-driven ATP binding is to be expected for ligand-induced conformational changes of the GroEL tetradecamer, though the magnitude of the entropy change suggests that reorganization of GroEL-hydrating water molecules and/or expulsion of water from the GroEL cavity may also play key roles. The capability for determining complete thermodynamic signatures (ΔG, ΔH, and -TΔS) for individual ligand binding reactions for the large, nearly megadalton GroEL complex expands our fundamental view of chaperonin functional chemistry. Moreover, this work and related studies of protein-ligand interactions illustrate important new capabilities of vT-ESI-nMS for thermodynamic studies of protein interactions with ligands and other molecules such as proteins and drugs.

4mCBERT: A computing tool for the identification of DNA N4-methylcytosine sites by sequence- and chemical-derived information based on ensemble learning strategies

N4-methylcytosine (4mC) is an important DNA chemical modification pattern which is a new methylation modification discovered in recent years and plays critical roles in gene expression regulation, defense against invading genetic elements, genomic imprinting, and so on. Identifying 4mC site from DNA sequence segment contributes to discovering more novel modification patterns. In this paper, we present a model called 4mCBERT that encodes DNA sequence segments by sequence characteristics including one-hot, electron-ion interaction pseudopotential, nucleotide chemical property, word2vec and chemical information containing physicochemical properties (PCP), chemical bidirectional encoder representations from transformers (chemical BERT) and employs ensemble learning framework to develop a prediction model. PCP and chemical BERT features are firstly constructed and applied to predict 4mC sites and show positive contributions to identifying 4mC. For the Matthew's Correlation Coefficient, 4mCBERT significantly outperformed other state-of-the-art models on six independent benchmark datasets including A. thaliana, C. elegans, D. melanogaster, E. coli, G. Pickering, and G. subterraneous by 4.32 % to 24.39 %, 2.52 % to 31.65 %, 2 % to 16.49 %, 6.63 % to 35.15, 8.59 % to 61.85 %, and 8.45 % to 34.45 %. Moreover, 4mCBERT is designed to allow users to predict 4mC sites and retrain 4mC prediction models. In brief, 4mCBERT shows higher performance on six benchmark datasets by incorporating sequence- and chemical-driven information and is available at http://cczubio.top/4mCBERT and https://github.com/abcair/4mCBERT.

Deep mining of the protein energy landscape

For over half a century, it has been known that protein molecules naturally undergo extensive structural fluctuations, and that these internal motions are intimately related to their functional properties. The energy landscape view has provided a powerful framework for describing the various physical states that proteins visit during their lifetimes. This Perspective focuses on the commonly neglected and often disparaged axis of the protein energy landscape: entropy. Initially seen largely as a barrier to functionally relevant states of protein molecules, it has recently become clear that proteins retain considerable conformational entropy in the "native" state, and that this entropy can and often does contribute significantly to the free energy of fundamental protein properties, processes, and functions. NMR spectroscopy, molecular dynamics simulations, and emerging crystallographic views have matured in parallel to illuminate dynamic disorder of the "ground state" of proteins and their importance in not only transiting between biologically interesting structures but also greatly influencing their stability, cooperativity, and contribution to critical properties such as allostery.

DNA base order parameter determination without influence of chemical exchange

Nuclear magnetic resonance (NMR) spectroscopy is a versatile tool used to investigate the dynamic properties of biological macromolecules and their complexes. NMR relaxation data can provide order parameters S2, which represent the mobility of bond vectors reorienting within a molecular frame. Determination of S2 parameters typically involves the use of transverse NMR relaxation rates. However, the accuracy in S2 determination can be diminished by elevation of the transverse relaxation rates through conformational or chemical exchange involving protonation/deprotonation or non-Watson-Crick base-pair states of nucleic acids. Here, we propose an approach for determination of S2 parameters without the influence of exchange processes. This approach utilizes transverse and longitudinal 13C chemical shift anisotropy (CSA) - dipole-dipole (DD) cross-correlation rates instead of 13C transverse relaxation rates. Anisotropy in rotational diffusion is taken into consideration. An application of this approach to nucleotide base CH groups of a uniformly 13C/15N-labeled DNA duplex is demonstrated.

Locating dynamic contributions to allostery via determining rates of vibrational energy transfer

Determining rates of energy transfer across non-covalent contacts for different states of a protein can provide information about dynamic and associated entropy changes during transitions between states. We investigate the relationship between rates of energy transfer across polar and nonpolar contacts and contact dynamics for the β₂-adrenergic receptor, a rhodopsin-like G-protein coupled receptor, in an antagonist-bound inactive state and agonist-bound active state. From structures sampled during molecular dynamics (MD) simulations, we find the active state to have, on average, a lower packing density, corresponding to generally more flexibility and greater entropy than the inactive state. Energy exchange networks (EENs) are computed for the inactive and active states from the results of the MD simulations. From the EENs, changes in the rates of energy transfer across polar and nonpolar contacts are found for contacts that remain largely intact during activation. Change in dynamics of the contact, and entropy associated with the dynamics, can be estimated from the change in rates of energy transfer across the contacts. Measurement of change in the rates of energy transfer before and after the transition between states thereby provides information about dynamic contributions to activation and allostery.

Pressure, motion, and conformational entropy in molecular recognition by proteins

The thermodynamics of molecular recognition by proteins is a central determinant of complex biochemistry. For over a half-century, detailed cryogenic structures have provided deep insight into the energetic contributions to ligand binding by proteins. More recently, a dynamical proxy based on NMR-relaxation methods has revealed an unexpected richness in the contributions of conformational entropy to the thermodynamics of ligand binding. Here, we report the pressure dependence of fast internal motion within the ribonuclease barnase and its complex with the protein barstar. In what we believe is a first example, we find that protein dynamics are conserved along the pressure-binding thermodynamic cycle. The femtomolar affinity of the barnase-barstar complex exists despite a penalty by -TΔS_conf of +11.7 kJ/mol at ambient pressure. At high pressure, however, the overall change in side-chain dynamics is zero, and binding occurs with no conformational entropy penalty, suggesting an important role of conformational dynamics in the adaptation of protein function to extreme environments. Distinctive clustering of the pressure sensitivity is observed in response to both pressure and binding, indicating the presence of conformational heterogeneity involving less efficiently packed alternative conformation(s). The structural segregation of dynamics observed in barnase is striking and shows how changes in both the magnitude and the sign of regional contributions of conformational entropy to the thermodynamics of protein function are possible.

Binding and Functional Folding (BFF): A Physiological Framework for Studying Biomolecular Interactions and Allostery

EF-hand Ca2+-binding proteins (CBPs), such as S100 proteins (S100s) and calmodulin (CaM), are signaling proteins that undergo conformational changes upon increasing intracellular Ca2+. Upon binding Ca2+, S100 proteins and CaM interact with protein targets and induce important biological responses. The Ca2+-binding affinity of CaM and most S100s in the absence of target is weak (CaKD > 1 μM). However, upon effector protein binding, the Ca2+ affinity of these proteins increases via heterotropic allostery (CaKD < 1 μM). Because of the high number and micromolar concentrations of EF-hand CBPs in a cell, at any given time, allostery is required physiologically, allowing for (i) proper Ca2+ homeostasis and (ii) strict maintenance of Ca2+-signaling within a narrow dynamic range of free Ca2+ ion concentrations, [Ca2+]free. In this review, mechanisms of allostery are coalesced into an empirical "binding and functional folding (BFF)" physiological framework. At the molecular level, folding (F), binding and folding (BF), and BFF events include all atoms in the biomolecular complex under study. The BFF framework is introduced with two straightforward BFF types for proteins (type 1, concerted; type 2, stepwise) and considers how homologous and nonhomologous amino acid residues of CBPs and their effector protein(s) evolved to provide allosteric tightening of Ca2+ and simultaneously determine how specific and relatively promiscuous CBP-target complexes form as both are needed for proper cellular function.

Toward Reliable and Insightful Entropy Calculations on Flexible Molecules

The absolute entropy of a flexible molecule can be approximated by the sum of a rigid-rotor-harmonic-oscillator (RRHO) entropy and a Gibbs-Shannon entropy associated to the Boltzmann distribution for the occupation of the conformational energy levels. Herein, we show that such partitioning, which has received renewed interest, leads to accurate entropies of single molecules of increasing size provided that the conformational part is estimated by means of a set of discretization and expansion techniques that are able to capture the significant correlation effects among the torsional motions. To ensure a reliable entropy estimation, we rely on extensive sampling as that produced by classical molecular dynamics simulations on the microsecond time scale, which is currently affordable for small- and medium-sized molecules. According to test calculations, the gas-phase entropy of simple organic molecules is predicted with a mean unsigned error of 0.9 cal/(mol K) when the RRHO entropies are computed at the B3LYP-D3/cc-pVTZ level. Remarkably, the same protocol gives small errors [<1 cal/(mol K)] for the extremely flexible linear alkane molecules (C_nH_2n+2, n = 14, 16, and 18). Similarly, we obtain well-converged entropies for a more challenging test of drug molecules, which exhibit more pronounced correlation effects. We also perform equivalent entropy calculations on a 76 amino acid protein, ubiquitin, by taking advantage of the cutoff-dependent formulation of an expansion technique (correlation-consistent multibody local approximation, CC-MLA), which incorporates genuine correlation effects among the neighboring dihedral angles. Moreover, we show that insightful descriptors of the coupled torsional motions can be obtained with the CC-MLA approach.

How does it really move? Recent progress in the investigation of protein nanosecond dynamics by NMR and simulation

Nuclear magnetic resonance (NMR) spin relaxation experiments currently probe molecular motions on timescales from picoseconds to nanoseconds. The detailed interpretation of these motions in atomic detail benefits from complementarity with the results from molecular dynamics (MD) simulations. In this mini-review, we describe the recent developments in experimental techniques to study the backbone dynamics from ¹⁵N relaxation and side-chain dynamics from ¹³C relaxation, discuss the different analysis approaches from model-free to dynamics detectors, and highlight the many ways that NMR relaxation experiments and MD simulations can be used together to improve the interpretation and gain insights into protein dynamics.

Entropies Derived from the Packing Geometries within a Single Protein Structure

A fast, simple, yet robust method to calculate protein entropy from a single protein structure is presented here. The focus is on the atomic packing details, which are calculated by combining Voronoi diagrams and Delaunay tessellations. Even though the method is simple, the entropies computed exhibit an extremely high correlation with the entropies previously derived by other methods based on quasi-harmonic motions, quantum mechanics, and molecular dynamics simulations. These packing-based entropies account directly for the local freedom and provide entropy for any individual protein structure that could be used to compute free energies directly during simulations for the generation of more reliable trajectories and also for better evaluations of modeled protein structures. Physico-chemical properties of amino acids are compared with these packing entropies to uncover the relationships with the entropies of different residue types. A public packing entropy web server is provided at packing-entropy.bb.iastate.edu, and the application programing interface is available within the PACKMAN (https://github.com/Pranavkhade/PACKMAN) package.

Kinematic Vibrational Entropy Assessment and Analysis of SARS CoV-2 Main Protease

The three-dimensional conformations of a protein influence its function and select for the ligands it can interact with. The total free energy change during protein-ligand complex formation includes enthalphic and entropic components, which together report on the binding affinity and conformational states of the complex. However, determining the entropic contribution is computationally burdensome. Here, we apply kinematic flexibility analysis (KFA) to efficiently estimate vibrational frequencies from static protein and protein-ligand structures. The vibrational frequencies, in turn, determine the vibrational entropies of the structures and their complexes. Our estimates of the vibrational entropy change caused by ligand binding compare favorably to values obtained from a dynamic Normal Mode Analysis (NMA). Higher correlation factors can be achieved by increasing the distance cutoff in the potential energy model. Furthermore, we apply our new method to analyze the entropy changes of the SARS CoV-2 main protease when binding with different ligand inhibitors, which is relevant for the design of potential drugs.

Ligand binding remodels protein side-chain conformational heterogeneity

While protein conformational heterogeneity plays an important role in many aspects of biological function, including ligand binding, its impact has been difficult to quantify. Macromolecular X-ray diffraction is commonly interpreted with a static structure, but it can provide information on both the anharmonic and harmonic contributions to conformational heterogeneity. Here, through multiconformer modeling of time- and space-averaged electron density, we measure conformational heterogeneity of 743 stringently matched pairs of crystallographic datasets that reflect unbound/apo and ligand-bound/holo states. When comparing the conformational heterogeneity of side chains, we observe that when binding site residues become more rigid upon ligand binding, distant residues tend to become more flexible, especially in non-solvent-exposed regions. Among ligand properties, we observe increased protein flexibility as the number of hydrogen bonds decreases and relative hydrophobicity increases. Across a series of 13 inhibitor-bound structures of CDK2, we find that conformational heterogeneity is correlated with inhibitor features and identify how conformational changes propagate differences in conformational heterogeneity away from the binding site. Collectively, our findings agree with models emerging from nuclear magnetic resonance studies suggesting that residual side-chain entropy can modulate affinity and point to the need to integrate both static conformational changes and conformational heterogeneity in models of ligand binding.

Electrostatic Interactions between CSTF2 and pre-mRNA Drive Cleavage and Polyadenylation

Nascent pre-mRNA 3'-end cleavage and polyadenylation (C/P) involves numerous proteins that recognize multiple RNA elements. Human CSTF2 binds to a downstream U- or G/U-rich sequence through its RNA recognition motif (RRM) regulating C/P. We previously reported the only known disease-related CSTF2 RRM mutant (CSTF2^D50A) and showed that it changed the on-rate of RNA binding, leading to alternative polyadenylation in brains of mice carrying the same mutation. In this study, we further investigated the role of electrostatic interactions in the thermodynamics and kinetics of RNA binding for the CSTF2 RRM and the downstream consequences for regulation of C/P. By combining mutagenesis with NMR spectroscopy and biophysical assays, we confirmed that electrostatic attraction is the dominant factor in RRM binding to a naturally occurring U-rich RNA sequence. Moreover, we demonstrate that RNA binding is accompanied by an enthalpy-entropy compensation mechanism that is supported by changes in pico-to-nanosecond timescale RRM protein dynamics. We suggest that the dynamic binding of the RRM to U-rich RNA supports the diversity of sequences it encounters in the nucleus. Lastly, in vivo C/P assays demonstrate a competition between fast, high affinity RNA binding and efficient, correct C/P. These results highlight the importance of the surface charge of the RRM in RNA binding and the balance between nascent mRNA binding and C/P in vivo.

Complementary approaches to obtaining thermodynamic parameters from protein ligand systems-challenges and opportunities

Protein ligand interactions play an important role in biology. Increasingly the aim is to understand and influence protein ligand binding. The binding process is heavily influenced by its thermodynamic parameters. In order to understand how the whole system thermodynamics work it is important to characterise the individual contribution of each of the systems components. While the change in conformational entropy of the protein can be determined using QENS complementary methods are necessary in order to characterise all components. This paper will describe the challenges that can occur when combining the different methods, as well as how they can be overcome.

Mapping Enzyme Landscapes by Time-Resolved Crystallography with Synchrotron and X-Ray Free Electron Laser Light

Directly observing enzyme catalysis in real time at the molecular level has been a long-standing goal of structural enzymology. Time-resolved serial crystallography methods at synchrotron and X-ray free electron laser (XFEL) sources have enabled researchers to follow enzyme catalysis and other nonequilibrium events at ambient conditions with unprecedented time resolution. X-ray crystallography provides detailed information about conformational heterogeneity and protein dynamics, which is enhanced when time-resolved approaches are used. This review outlines the ways in which information about the underlying energy landscape of a protein can be extracted from X-ray crystallographic data, with an emphasis on new developments in XFEL and synchrotron time-resolved crystallography. The emerging view of enzyme catalysis afforded by these techniques can be interpreted as enzymes moving on a time-dependent energy landscape. Some consequences of this view are discussed, including the proposal that irreversible enzymes or enzymes that use covalent catalytic mechanisms may commonly exhibit catalysis-activated motions. Expected final online publication date for the Annual Review of Biophysics, Volume 51 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Tuning Protein Dynamics to Sense Rapid Endoplasmic-Reticulum Calcium Dynamics

Multi-scale calcium (Ca²⁺ ) dynamics, exhibiting wide-ranging temporal kinetics, constitutes a ubiquitous mode of signal transduction. We report a novel endoplasmic-reticulum (ER)-targeted Ca²⁺ indicator, R-CatchER, which showed superior kinetics in vitro (k_off ≥2×10³  s^-1 , k_on ≥7×10⁶  M^-1  s^-1 ) and in multiple cell types. R-CatchER captured spatiotemporal ER Ca²⁺ dynamics in neurons and hotspots at dendritic branchpoints, enabled the first report of ER Ca²⁺ oscillations mediated by calcium sensing receptors (CaSRs), and revealed ER Ca²⁺ -based functional cooperativity of CaSR. We elucidate the mechanism of R-CatchER and propose a principle to rationally design genetically encoded Ca²⁺ indicators with a single Ca²⁺ -binding site and fast kinetics by tuning rapid fluorescent-protein dynamics and the electrostatic potential around the chromophore. The design principle is supported by the development of G-CatchER2, an upgrade of our previous (G-)CatchER with improved dynamic range. Our work may facilitate protein design, visualizing Ca²⁺ dynamics, and drug discovery.

Observation of Sub-Microsecond Protein Methyl-Side Chain Dynamics by Nanoparticle-Assisted NMR Spin Relaxation

Amino-acid side-chain properties in proteins are key determinants of protein function. NMR spin relaxation of side chains is an important source of information about local protein dynamics and flexibility. However, traditional solution NMR relaxation methods are most sensitive to sub-nanosecond dynamics lacking information on slower ns-μs time-scale motions. Nanoparticle-assisted NMR spin relaxation (NASR) of methyl-side chains is introduced here as a window into these ns-μs dynamics. NASR utilizes the transient and nonspecific interactions between folded proteins and slowly tumbling spherical nanoparticles (NPs), whereby the increase of the relaxation rates reflects motions on time scales from ps all the way to the overall tumbling correlation time of the NPs ranging from hundreds of ns to μs. The observed motional amplitude of each methyl group can then be expressed by a model-free NASR S² order parameter. The method is demonstrated for ²H-relaxation of CH₂D methyl moieties and cross-correlated relaxation of CH₃ groups for proteins Im7 and ubiquitin in the presence of anionic silica-nanoparticles. Both types of relaxation experiments, dominated by either quadrupolar or dipolar interactions, yield highly consistent results. Im7 shows additional dynamics on the intermediate time scales taking place in a functionally important loop, whereas ubiquitin visits the majority of its conformational substates on the sub-ns time scale. These experimental observations are in good agreement with 4-10 μs all-atom molecular dynamics trajectories. NASR probes side-chain dynamics on a much wider range of motional time scales than previously possible, thereby providing new insights into the interplay between protein structure, dynamics, and molecular interactions that govern protein function.

Transparent window 2D IR spectroscopy of proteins

Proteins are complex, heterogeneous macromolecules that exist as ensembles of interconverting states on a complex energy landscape. A complete, molecular-level understanding of their function requires experimental tools to characterize them with high spatial and temporal precision. Infrared (IR) spectroscopy has an inherently fast time scale that can capture all states and their dynamics with, in principle, bond-specific spatial resolution. Two-dimensional (2D) IR methods that provide richer information are becoming more routine but remain challenging to apply to proteins. Spectral congestion typically prevents selective investigation of native vibrations; however, the problem can be overcome by site-specific introduction of amino acid side chains that have vibrational groups with frequencies in the "transparent window" of protein spectra. This Perspective provides an overview of the history and recent progress in the development of transparent window 2D IR of proteins.

Understanding the Binding Transition State After the Conformational Selection Step: The Second Half of the Molecular Recognition Process Between NS1 of the 1918 Influenza Virus and Host p85β

Biomolecular recognition often involves conformational changes as a prerequisite for binding (i.e., conformational selection) or concurrently with binding (i.e., induced-fit). Recent advances in structural and kinetic approaches have enabled the detailed characterization of protein motions at atomic resolution. However, to fully understand the role of the conformational dynamics in molecular recognition, studies on the binding transition state are needed. Here, we investigate the binding transition state between nonstructural protein 1 (NS1) of the pandemic 1918 influenza A virus and the human p85β subunit of PI3K. 1918 NS1 binds to p85β via conformational selection. We present the free-energy mapping of the transition and bound states of the 1918 NS1:p85β interaction using linear free energy relationship and ϕ-value analyses. We find that the binding transition state of 1918 NS1 and p85β is structurally similar to the bound state with well-defined binding orientation and hydrophobic interactions. Our finding provides a detailed view of how protein motion contributes to the development of intermolecular interactions along the binding reaction coordinate.

THE IMS PARADOX: A PERSPECTIVE ON STRUCTURAL ION MOBILITY-MASS SPECTROMETRY

Studies of large proteins, protein complexes, and membrane protein complexes pose new challenges, most notably the need for increased ion mobility (IM) and mass spectrometry (MS) resolution. This review covers evolutionary developments in IM-MS in the authors' and key collaborators' laboratories with specific focus on developments that enhance the utility of IM-MS for structural analysis. IM-MS measurements are performed on gas phase ions, thus "structural IM-MS" appears paradoxical-do gas phase ions retain their solution phase structure? There is growing evidence to support the notion that solution phase structure(s) can be retained by the gas phase ions. It should not go unnoticed that we use "structures" in this statement because an important feature of IM-MS is the ability to deal with conformationally heterogeneous systems, thus providing a direct measure of conformational entropy. The extension of this work to large proteins and protein complexes has motivated our development of Fourier-transform IM-MS instruments, a strategy first described by Hill and coworkers in 1985 (Anal Chem, 1985, 57, pp. 402-406) that has proved to be a game-changer in our quest to merge drift tube (DT) and ion mobility and the high mass resolution orbitrap MS instruments. DT-IMS is the only method that allows first-principles determinations of rotationally averaged collision cross sections (CSS), which is essential for studies of biomolecules where the conformational diversities of the molecule precludes the use of CCS calibration approaches. The Fourier transform-IM-orbitrap instrument described here also incorporates the full suite of native MS/IM-MS capabilities that are currently employed in the most advanced native MS/IM-MS instruments. © 2020 John Wiley & Sons Ltd. Mass Spec Rev.

On Catalytic Kinetics of Enzymes

Classical enzyme kinetic theories are summarized and linked with modern discoveries here. The sequential catalytic events along time axis by enzyme are analyzed at the molecular level, and by using master equations, this writing tries to connect the microscopic molecular behavior of enzyme to kinetic data (like velocity and catalytic coefficient k) obtained in experiment: 1/k = t equals to the sum of the times taken by the constituent individual steps. The relationships between catalytic coefficient k, catalytic rate or velocity, the amount of time taken by each step and physical or biochemical conditions of the system are discussed, and the perspective and hypothetic equations proposed here regarding diffusion, conformational change, chemical conversion, product release steps and the whole catalytic cycle provide an interpretation of previous experimental observations and can be testified by future experiments.

Using biochemistry and biophysics to extinguish androgen receptor signaling in prostate cancer

Castration resistant prostate cancer (CRPC) continues to be androgen receptor (AR) driven. Inhibition of AR signaling in CRPC could be advanced using state-of-the-art biophysical and biochemical techniques. Structural characterization of AR and its complexes by cryo-electron microscopy would advance the development of N-terminal domain (NTD) and ligand-binding domain (LBD) antagonists. The structural basis of AR function is unlikely to be determined by any single structure due to the intrinsic disorder of its NTD, which not only interacts with coregulators but likely accounts for the constitutive activity of AR-splice variants (SV), which lack the LBD and emerge in CRPC. Using different AR constructs lacking the LBD, their effects on protein folding, DNA binding, and transcriptional activity could reveal how interdomain coupling explains the activity of AR-SVs. The AR also interacts with coregulators that promote chromatin looping. Elucidating the mechanisms involved can identify vulnerabilities to treat CRPC, which do not involve targeting the AR. Phosphorylation of the AR coactivator MED-1 by CDK7 is one mechanism that can be blocked by the use of CDK7 inhibitors. CRPC gains resistance to AR signaling inhibitors (ARSI). Drug resistance may involve AR-SVs, but their role requires their reliable quantification by SILAC-mass spectrometry during disease progression. ARSI drug resistance also occurs by intratumoral androgen biosynthesis catalyzed by AKR1C3 (type 5 17β-hydroxysteroid dehydrogenase), which is unique in that its acts as a coactivator of AR. Novel bifunctional inhibitors that competitively inhibit AKR1C3 and block its coactivator function could be developed using reverse-micelle NMR and fragment-based drug discovery.

Exploring the Thermodynamics of 7-Amino Actinomycin D-Induced Single-Stranded DNA Hairpin by Spectroscopic Techniques and Computational Simulations

NMR studies have indicated that the anti-tumor therapeutic agent actinomycin D (ACTD) can induce seemingly single-stranded DNA (ssDNA) oligomer 5'-CCGTT₃GTGG-3' to form a hairpin structure with tandem GT mismatches at the stem region next to a loop of three stacked thymine bases. In an effort to uncover the preference of binding sequence and to elucidate the thermodynamics properties of the binding, a combination of spectroscopic techniques and computational simulation studies was performed with d(CCGTT_nGTGG) and d(CCGAA_nGAGG) (denoted as GTT_n and GAA_n, respectively; n = 3, 5, and 7) sequences. In the presence of 7-amino actinomycin D (7AACTD), all the six oligomers formed stable hairpin structures. The GTT₅-7AACTD/GAA₅-7AACTD hairpin structure was more stable than the corresponding GTT_n-7AACTD and GAA_n-7AACTD (n = 3, 7). No significant ΔG difference was observed between GTT_n-7AACTD and GAA_n-7AACTD complexes with the same loop length. In agreement with the 7AACTD-induced hairpin stability results, the binding affinity of GTT_n and GAA_n with 7AACTD increased from n = 3 to n = 5 and then decreased when n is 7. Moreover, GTT_n and GAA_n with the same loop length showed comparable binding affinities to 7AACTD. Furthermore, molecular dynamics simulations found that van der Waals interactions between GTT_n/GAA_n and 7AACTD were the primary attractive forces for 7AACTD binding, and the electrostatic interactions between the carbonyl groups of 7AACTD and bases in the hairpin were the major unfavorable forces. These findings furthered our understanding that 7AACTD is sensitive to the loop size and sequence as well as tandem GT/GA mismatches of their deoxyribonucleic acid (DNA) targets. A deep understanding of the thermodynamics and the molecular recognition mechanism of 7AACTD with ssDNAs would further the development of ACTD-like antitumor agents.

The role of NMR in leveraging dynamics and entropy in drug design

Nuclear magnetic resonance (NMR) spectroscopy has contributed to structure-based drug development (SBDD) in a unique way compared to the other biophysical methods. The potency of a ligand binding to a protein is dictated by the binding free energy, which is an intricate interplay between entropy and enthalpy. In addition to providing the atomic resolution structural information, NMR can help to identify protein-ligand interactions that potentially contribute to the enthalpic component of the free energy. NMR can also illuminate dynamic aspects of the interaction, which correspond to the entropic term of the free energy. The ability of NMR to access both terms in the free energy equation stems from the suite of experiments developed to shed light on various aspects that contribute to both entropy and enthalpy, deepening our understanding of the biological function of macromolecules and assisting to target them in physiological conditions. Here we provide a brief account of the contribution of NMR to SBDD, highlighting hallmark examples and discussing the challenges that demand further method development. In the era of integrated biology, the unique ability of NMR to directly ascertain structural and dynamical aspects of macromolecule and monitor changes in these properties upon engaging a ligand can be combined with computational and other structural and biophysical methods to provide a more complete picture of the energetics of drug engagement with the target. Such efforts can be used to engineer better drugs.

Conformational Entropy from Mobile Bond Vectors in Proteins: A Viewpoint that Unifies NMR Relaxation Theory and Molecular Dynamics Simulation Approaches

A new method for determining conformational entropy in proteins is reported. Proteins prevail as conformational ensembles, p ∝ exp(-u). By selecting a bond vector (e.g., N-H) as a conformation representative, molecular dynamics simulations can provide (relative to a reference structure) p as approximate Boltzmann probability density and u as N-H potential of mean force (POMF). The latter is as accurate as implied by the force field but statistical in character; this limits the insights it can provide and its utilization. Conformational entropy is given exclusively by u. Deriving it from POMFs renders it accurate but statistical in character. Previously, we devised explicit (i.e., analytical but not exact) potentials made of Wigner functions, D_0K^L, with L ≤ 4, which closely resemble the corresponding POMFs in form; hence, they also approach the latter in accuracy. Such potentials can be beneficially characterized/compared in terms of composition, symmetry, and associated order parameters. In this study, we develop a method for deriving conformational entropy from them, which also features the benefits specified above. The method developed is applied to the dimerization of the Rho GTPase-binding domain of plexin-B1. Insights into local ordering, entropy compensation, and features of allostery are gained. In previous work, we developed the slowly relaxing local structure (SRLS) approach for the analysis of NMR relaxation from restricted bond vector motion in proteins. SRLS comprises explicit (restricting) potentials of the kind developed here. It also comprises diffusion tensors describing the local motion and related features of local geometry. The complete model fits experimental data. In future work, the explicit potentials developed here will be inserted unchanged in SRLS-based data fitting, thereby improving the picture of structural dynamics. Given that SRLS is unique in featuring potentials that can closely approach the corresponding POMFs in accuracy, the present study is an important step toward generally improving protein dynamics by NMR relaxation.

Protein conformational entropy is not slaved to water

Conformational entropy can be an important element of the thermodynamics of protein functions such as the binding of ligands. The observed role for conformational entropy in modulating molecular recognition by proteins is in opposition to an often-invoked theory for the interaction of protein molecules with solvent water. The "solvent slaving" model predicts that protein motion is strongly coupled to various aspects of water such as bulk solvent viscosity and local hydration shell dynamics. Changes in conformational entropy are manifested in alterations of fast internal side chain motion that is detectable by NMR relaxation. We show here that the fast-internal side chain dynamics of several proteins are unaffected by changes to the hydration layer and bulk water. These observations indicate that the participation of conformational entropy in protein function is not dictated by the interaction of protein molecules and solvent water under the range of conditions normally encountered.

Dynamics of Bacteriorhodopsin in the Dark-Adapted State from Solution Nuclear Magnetic Resonance Spectroscopy

To achieve efficient proton pumping in the light-driven proton pump bacteriorhodopsin (bR), the protein must be tightly coupled to the retinal to rapidly convert retinal isomerization into protein structural rearrangements. Methyl group dynamics of bR embedded in lipid nanodiscs were determined in the dark-adapted state, and were found to be mostly well ordered at the cytosolic side. Methyl groups in the M145A mutant of bR, which displays only 10 % residual proton pumping activity, are less well ordered, suggesting a link between side-chain dynamics on the cytosolic side of the bR cavity and proton pumping activity. In addition, slow conformational exchange, attributed to low frequency motions of aromatic rings, was indirectly observed for residues on the extracellular side of the bR cavity. This may be related to reorganization of the water network. These observations provide a detailed picture of previously undescribed equilibrium dynamics on different time scales for ground-state bR.

Disulfide Reduction Allosterically Destabilizes the β-Ladder Subdomain Assembly within the NS1 Dimer of ZIKV

The Zika virus (ZIKV) was responsible for a recent debilitating epidemic that till date has no cure. A potential way to reduce ZIKV virulence is to limit the action of the nonstructural proteins involved in its viral replication. One such protein, NS1, encoded as a monomer by the viral genome, plays a major role via symmetric oligomerization. We examine the homodimeric structure of the dominant β-ladder segment of NS1 with extensive all atom molecular dynamics. We find it stably bounded by two spatially separated interaction clusters (C1 and C2) with significant differences in the nature of their interactions. Four pairs of distal, intramonomeric disulfide bonds are found to be coupled to the stability, local structure, and wettability of the interfacial region. Symmetric reduction of the intramonomeric disulfides triggers marked dynamical heterogeneity, interfacial wettability, and asymmetric salt-bridging propensity. Harnessing the model-free Lipari-Szabo based formalism for estimation of conformational entropy (S_conf), we find clear signatures of heterogeneity in the monomeric conformational entropies. The observed asymmetry, very small in the unperturbed state, expands significantly in the reduced states. This allosteric effect is most noticeable in the electrostatically bound C2 cluster that underlies the greatest stability in the unperturbed state. Allosteric induction of conformational and thermodynamic asymmetry is expected to affect the pathways leading to symmetric higher-ordered oligomerization, and thereby affect crucial replication pathways.

Membrane Proteins Have Distinct Fast Internal Motion and Residual Conformational Entropy

The internal motions of integral membrane proteins have largely eluded comprehensive experimental characterization. Here the fast side-chain dynamics of the α-helical sensory rhodopsin II and the β-barrel outer membrane protein W have been investigated in lipid bilayers and detergent micelles by solution NMR relaxation techniques. Despite their differing topologies, both proteins have a similar distribution of methyl-bearing side-chain motion that is largely independent of membrane mimetic. The methyl-bearing side chains of both proteins are, on average, more dynamic in the ps-ns timescale than any soluble protein characterized to date. Accordingly, both proteins retain an extraordinary residual conformational entropy in the folded state, which provides a counterbalance to the absence of the hydrophobic effect. Furthermore, the high conformational entropy could greatly influence the thermodynamics underlying membrane-protein functions, including ligand binding, allostery, and signaling.

Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation

Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by S_k ≡ S/k = -∫(P_eq ln P_eq)dΩ - ln∫dΩ. The quantity P_eq = exp(-u)/Z is the orientational probability density, where Z is the partition function and u is the spatially restricting potential exerted by the immediate internal protein surroundings at the site of the motion of the bond-vector. It is appropriate to expand the potential, u, which restricts local rotational reorientation, in the basis set of the real combinations of the Wigner rotation matrix elements, D_0K^L. For small molecules dissolved in anisotropic media, one typically keeps the lowest even L, L = 2, nonpolar potential in axial or rhombic form. For bond-vectors anchored at the protein, the lowest odd L, L = 1, polar potential is to be used in axial or rhombic form. Here, we investigate the effect of the symmetry and polarity of these potentials on S_k. For L = 1 (L = 2), S_k is the same (differs) for parallel and perpendicular ordering. The plots of S_k as a function of the coefficients of the rhombic L = 1 (L = 2) potential exhibit high-symmetry (specific low-symmetry) patterns with parameter-range-dependent sensitivity. Similar statements apply to analogous plots of the potential minima. S_k is also examined as a function of the order parameters defined in terms of u. Graphs displaying these correlations, and applications illustrating their usage, are provided. The features delineated above are generally useful for devising orienting potentials that best suit given physical circumstances. They are particularly useful for bond-vectors acting as NMR relaxation probes in proteins, when their restricted local motion is analyzed with stochastic models featuring Wigner-function-made potentials. The relaxation probes could also be molecules adsorbed at surfaces, inserted into membranes, or interlocked within metal-organic frameworks.

Entropy Hotspots for the Binding of Intrinsically Disordered Ligands to a Receptor Domain

Proline-rich motifs (PRMs) are widely used for mediating protein-protein interactions with weak binding affinities. Because they are intrinsically disordered when unbound, conformational entropy plays a significant role for the binding. However, residue-level differences of the entropic contribution in the binding of different ligands remain not well understood. We use all-atom molecular dynamics simulation and the maximal information spanning tree formalism to analyze conformational entropy associated with the binding of two PRMs, one from the Abl kinase and the other from the nonstructural protein 1 of the 1918 Spanish influenza A virus, to the N-terminal SH3 (nSH3) domain of the CrkII protein. Side chains of the stably folded nSH3 experience more entropy change upon ligand binding than the backbone, whereas PRMs involve comparable but heterogeneous entropy changes among the backbone and side chains. In nSH3, two conserved nonpolar residues forming contacts with the PRM experience the largest side-chain entropy loss. In contrast, the C-terminal charged residues of PRMs that form polar contacts with nSH3 experience the greatest side-chain entropy loss, although their "fuzzy" nature is attributable to the backbone that remains relatively flexible. Thus, residues that form high-occupancy contacts between nSH3 and PRM do not reciprocally contribute to entropy loss. Furthermore, certain surface residues of nSH3 distal to the interface with PRMs gain entropy, indicating a nonlocal effect of ligand binding. Comparing between the PRMs from cAbl and nonstructural protein 1, the latter involves a larger side-chain entropy loss and forms more contacts with nSH3. Consistent with experiments, this indicates stronger binding of the viral ligand at the expense of losing the flexibility of side chains, whereas the backbone experiences less entropy loss. The entropy "hotspots" as identified in this study will be important for tuning the binding affinity of various ligands to a receptor.