The entropic cost of protein-protein association: a case study on acetylcholinesterase binding to fasciculin-2

Computational Assessment of Protein-Protein Binding Specificity within a Family of Synaptic Surface Receptors

Atomic-level information is essential to explain the formation of specific protein complexes in terms of structure and dynamics. The set of Dpr and DIP proteins, which play a key role in the neuromorphogenesis in the nervous system of Drosophila melanogaster, offer a rich paradigm to learn about protein-protein recognition. Many members of the DIP subfamily cross-react with several members of the Dpr family and vice versa. While there exists a total of 231 possible Dpr-DIP heterodimer complexes from the 21 Dpr and 11 DIP proteins, only 57 "cognate" pairs have been detected by surface plasmon resonance (SPR) experiments, suggesting that the remaining 174 pairs have low or unreliable binding affinity. Our goal is to assess the performance of computational approaches to characterize the global set of interactions between Dpr and DIP proteins and identify the specificity of binding between each DIP with their corresponding Dpr binding partners. In addition, we aim to characterize how mutations influence the specificity of the binding interaction. In this work, a wide range of knowledge-based and physics-based approaches are utilized, including mutual information, linear discriminant analysis, homology modeling, molecular dynamics simulations, Poisson-Boltzmann continuum electrostatics calculations, and alchemical free energy perturbation to decipher the origin of binding specificity of the Dpr-DIP complexes examined. Ultimately, the results show that those two broad strategies are complementary, with different strengths and limitations. Biological inter-relations are more clearly revealed through knowledge-based approaches combining evolutionary and structural features, the molecular determinants controlling binding specificity can be predicted accurately with physics-based approaches based on atomic models.

Developing end-point methods for absolute binding free energy calculation using the Boltzmann-quasiharmonic model

Understanding the physical forces underlying receptor-ligand binding requires robust methods for analyzing the binding thermodynamics. In end-point binding free energy methods the binding free energy is naturally decomposable into physically intuitive contributions such as the solvation free energy and configurational entropy that can provide insights. Here we present a new end-point method called EE-BQH (Effective Energy-Boltzmann-Quasiharmonic) which combines the Boltzmann-Quasiharmonic model for configurational entropy with different solvation free energy methods, such as the continuum solvent PBSA model and the integral equation-based 3D-RISM, to estimate the absolute binding free energy. We compare EE-BQH with other treatments of configurational entropy such as Quasiharmonic models in internal coordinates (QHIC) and in Cartesian coordinates (QHCC), and Normal Mode analysis (NMA), by testing them on the octa acids host-guest complexes from the SAMPL8 blind challenge. The accuracies in the calculated absolute binding free energies strongly depend on the configurational entropy and solvation free energy methods used. QHIC and BQH yield the best agreements with the established potential of mean force (PMF) estimates, with R² of ∼0.7 and mean unsigned error of ∼1.7 kcal mol^-1. These results from the end-point calculations are also in similar agreement with experiments. While 3D-RISM in combination with QHIC or BQH lead to reasonable correlations with the PMF results and experiments, the calculated absolute binding free energies are underestimated by ∼5 kcal mol^-1. While the binding is accompanied by a significant reduction in the ligand translational/rotational entropy, the change in the torsional entropy in these host-guest systems is slightly positive. Compared with BQH, QHIC underestimates the reduction of configurational entropy because of the non-Gaussian probability distributions in the ligand rotation and a small number of torsions. The study highlights the crucial role of configurational entropy in determining binding and demonstrates the potential of using the new end-point method to provide insights in more complex protein-ligand systems.

Quantification of Conformational Entropy Unravels Effect of Disordered Flanking Region in Coupled Folding and Binding

Intrinsic disorder (ID) constitutes a new dimension to the protein structure-function relationship. The ability to undergo conformational changes upon binding is a key property of intrinsically disordered proteins and remains challenging to study using conventional methods. A 1994 paper by R. S. Spolar and M. T. Record presented a thermodynamic approach for estimating changes in conformational entropy based on heat capacity changes, allowing quantification of residues folding upon binding. Here, we adapt the method for studies of intrinsically disordered proteins. We integrate additional data to provide a broader experimental foundation for the underlying relations and, based on >500 protein-protein complexes involving disordered proteins, reassess a key relation between polar and nonpolar surface area changes, previously determined using globular protein folding. We demonstrate the improved suitability of the adapted method to studies of the folded αα-hub domain RST from radical-induced cell death 1, whose interactome is characterized by ID. From extensive thermodynamic data, quantifying the conformational entropy changes upon binding, and comparison to the NMR structure, the adapted method improves accuracy for ID-based studies. Furthermore, we apply the method, in conjunction with NMR, to reveal hitherto undetected effects of interaction-motif context. Thus, inclusion of the disordered context of the DREB2A RST-binding motif induces structuring of the binding motif, resulting in major enthalpy-entropy compensation in the interaction interface. This study, also evaluating additional interactions, demonstrates the strength of the ID-adapted Spolar-Record thermodynamic approach for dissection of structural features of ID-based interactions, easily overlooked in traditional studies, and for translation of these into mechanistic knowledge.

On Restraints in End-Point Protein-Ligand Binding Free Energy Calculations

The impact of harmonic restraints on protein heavy atoms and ligand atoms on end-point free energy calculations is systematically characterized for 54 protein-ligand complexes. We observe that stronger restraints reduce the equilibration time and statistical inefficiency, suppress conformational sampling, influence correlation with experiment, and monotonically decrease the estimated loss of entropy upon binding, leading to stronger estimated binding free energies in most systems. A statistical estimator that reweights for the biasing potential and includes data prior to the estimated equilibration time has the highest correlation with experiment. A spring constant of 20 cal mol-1 Å-2 maintains a near-native energy landscape and suppresses artifactual energy minima while minimally limiting thermal fluctuations about the crystal structure. © 2019 Wiley Periodicals, Inc.

Towards a thermodynamic mechanistic model for the effect of temperature on arthropod vector competence for transmission of arboviruses

Arboviruses such as West Nile virus (WNV), bluetongue virus (BTV), dengue virus (DENV) and chikungunya virus (CHIKV) infect their arthropod vectors over a range of average temperatures depending on the ambient temperature. How the transmission efficiency of an arbovirus (i.e. vector competence) varies with temperature influences not only the short term risk of arbovirus outbreaks in humans and livestock but also the long term impact of climate change on the geographical range of the virus. The strength of the interaction between viral surface (glyco)protein (GP) and the host cell receptor (Cr) on binding of virus to host cell is defined by the thermodynamic dissociation constant K_{d_receptor} which is assumed to equal 10^-3 M (at 37 °C) for binding of a sialic acid (SA) on the arthropod midgut epithelial cell surface to a SA-binding site on the surface of BTV, for example. Here virus binding affinity is modelled with increasing number of GP/Cr contacts at temperatures from 10 °C to 35 °C taking into account the change in entropy on immobilization of the whole virus on binding (ΔS_{a_immob}). Based on published data, three thermodynamic GP/Cr binding scenarios, namely enthalpy-driven, entropy-assisted and entropy-driven, are shown to affect the temperature sensitivity of virus binding in different ways. Thus for enthalpy-driven GP/Cr binding, viruses bind host cells much more strongly at 10 °C than 35 °C. A mechanistic model is developed for the number of arthropod midgut cells with bound virus and by building in a kinetic component for the rate of arbovirus replication and subsequent spread to the arthropod salivary glands, a model for the effect of temperature on vector competence is developed. The model separates the opposing effects of temperature on midgut cell binding affinity from the kinetic component of virogenesis. It successfully accommodates both increases in vector competence with temperature as for DENV and WNV in mosquitoes and decreases as for the CHIKV 2010-1909 strain in various populations of Aedes albopictus mosquitoes. Enhanced cell binding at lower temperatures through enthalpy-driven GP/Cr binding compensates for the lower replication rate to some degree such that some transmission can still occur at lower temperatures. In contrast, the strength of entropy-driven GP/Cr binding diminishes at low temperatures although there is no minimum temperature threshold for transmission efficiency. The magnitude of ΔS_{a_immob} is an important data gap. It is concluded that thermodynamic and kinetic data obtained at the molecular level will prove important in modelling vector competence with temperature.

Simple Entropy Terms for End-Point Binding Free Energy Calculations

We introduce a number of computationally inexpensive modifications to the MM/PBSA and MM/GBSA estimators for binding free energies, which are based on average receptor-ligand interaction energies in simulations of a noncovalent complex, to improve the treatment of entropy: second- and higher-order terms in a cumulant expansion and a confining potential on ligand external degrees of freedom. We also consider a filter for snapshots where ligands have drifted from the initial binding pose. The variations were tested on six sets of systems for which binding modes and free energies have previously been experimentally determined. For some data sets, none of the tested estimators led to results significantly correlated with measured free energies. In data sets with nontrivial correlation, a ligand RMSD cutoff of 3 Å and a second-order truncation of the cumulant expansion was found to be comparable or better than the average interaction energy by several statistical metrics.

Host-guest interactions between sildenafil and cyclodextrins: Spectrofluorometric study and molecular dynamic modeling

Sildenafil (SF) was included in cyclodextrins (CD) to enhance its solubility. Spectrofluorometry was used to confirm the complexation constant (stability constant). The orientations of SF inside the β-CDs and γ-CDs were fully illustrated. Molecular dynamics simulations were performed on two inclusion complexes (β-CD/SF and γ-CD/SF) in the aqueous system. The polar methylpiperazine group was found to locate inside the β-CD cavity, both in the wide and narrow side and was positioned 2°A from the center. In contrast, the methylpiperazine group did not fit well within the γ-CD cavity. Moreover, these results also confirmed hydrogen bonding that the highest number of bonding formed between the polar methylpiperazine sulfonyl structure and the hydroxyl group of β-CD. The simulated binding free energy of the methylpiperazine-β-CD inclusion complex (-6.01kcal/mol), featured a large contribution from electrostatic and van der Waals forces, which was the most stable complex. The association constant of β-CD/SF (12.3) was higher than γ-CD/SF (3.3) and confirmed with in silico measurements of binding free energy. In summary, SF forms a stable complex with β-CD.

Dissecting the Influence of Protein Flexibility on the Location and Thermodynamic Profile of Explicit Water Molecules in Protein-Ligand Binding

Explicit water molecules in the binding site of proteins play a crucial role for protein-ligand association. Recent advances in computer-aided drug discovery methodology allow for an accurate prediction of the localized position and thermodynamic profile of water molecules (i.e., hydration sites) in the binding site. The underlying calculations are based on MD simulations of explicit water molecules in a restrained protein structure. However, the ligand-binding process is typically associated with protein conformational change that influences the position and thermodynamic properties of the hydration site. In this manuscript, we present the developments of two methods to incorporate the influence of protein conformational change on hydration sites either by following the conformational transition step-by-step (method I) or to match the hydration sites of the two transition end states using local coordinate systems (method II). Using these methods, we highlight the difference in the estimated protein desolvation free energy with and without inclusion of protein flexibility. To the best of our knowledge, this is the first study that explicitly studies the influence of protein conformational change on the position and thermodynamic profiles of water molecules and provides methodology to incorporate protein flexibility into the estimation of the desolvation free energy.

New Computational Approach for External Entropy in Protein-Protein Binding

Molecular recognition through the noncovalent association of biomolecules is of central importance in biology and pharmacology. Developing reliable computational methods for estimating binding thermodynamic parameters is therefore of great practical value. However, considerable uncertainty remains regarding the external entropy that is associated with the reduction in the external (positional and orientational) degrees of freedom upon complex formation. Here, we present a novel statistical mechanical method for computing the external entropy by extending the energetic approach we have developed for unimolecular processes to association processes. We find that, in contrary to what is postulated in most of the previous methods, intrinsic couplings between the internal and external degrees of freedom of bound complex cannot in general be neglected in the determination of the external entropy. Nevertheless, there exists the best choice of the external coordinates with which those couplings are minimized. With such a choice of the external coordinates, the lowest upper bound of the external entropy is obtained from a tractable expression, which serves as an estimate of the external entropy. Our method can be implemented in a straightforward manner with molecular dynamics simulations, and its applicability is demonstrated through the application to the barnase-barstar complex.

Evaluations of the Absolute and Relative Free Energies for Antidepressant Binding to the Amino Acid Membrane Transporter LeuT with Free Energy Simulations

The binding of ligands to protein receptors with high affinity and specificity is central to many cellular processes. The quest for the development of computational models capable of accurately evaluating binding affinity remains one of the main goals of modern computational biophysics. In this work, free energy perturbation/molecular dynamics simulations were used to evaluate absolute and relative binding affinity for three different antidepressants to a sodium-dependent membrane transporter, LeuT, a bacterial homologue of human serotonin and dopamine transporters. Dysfunction of these membrane transporters in mammals has been implicated in multiple diseases of the nervous system, including bipolar disorder and depression. Furthermore, these proteins are key targets for antidepressants including fluoxetine (aka Prozac) and tricyclic antidepressants known to block transport activity. In addition to being clinically relevant, this system, where multiple crystal structures are readily available, represents an ideal testing ground for methods used to study the molecular mechanisms of ligand binding to membrane proteins. We discuss possible pitfalls and different levels of approximation required to evaluate binding affinity, such as the dependence of the computed affinities on the strength of constraints and the sensitivity of the computed affinities to the particular partial charges derived from restrained electrostatic potential fitting of quantum mechanics electrostatic potential. Finally, we compare the effects of different constraint schemes on the absolute and relative binding affinities obtained from free energy simulations.

Distance-Based Configurational Entropy of Proteins from Molecular Dynamics Simulations

Estimation of configurational entropy from molecular dynamics trajectories is a difficult task which is often performed using quasi-harmonic or histogram analysis. An entirely different approach, proposed recently, estimates local density distribution around each conformational sample by measuring the distance from its nearest neighbors. In this work we show this theoretically well grounded the method can be easily applied to estimate the entropy from conformational sampling. We consider a set of systems that are representative of important biomolecular processes. In particular: reference entropies for amino acids in unfolded proteins are obtained from a database of residues not participating in secondary structure elements;the conformational entropy of folding of β2-microglobulin is computed from molecular dynamics simulations using reference entropies for the unfolded state;backbone conformational entropy is computed from molecular dynamics simulations of four different states of the EPAC protein and compared with order parameters (often used as a measure of entropy);the conformational and rototranslational entropy of binding is computed from simulations of 20 tripeptides bound to the peptide binding protein OppA and of β2-microglobulin bound to a citrate coated gold surface. This work shows the potential of the method in the most representative biological processes involving proteins, and provides a valuable alternative, principally in the shown cases, where other approaches are problematic.

Diffusion and partitioning of macromolecules in casein microgels: evidence for size-dependent attractive interactions in a dense protein system

Understanding the mechanisms that determine the diffusion and interaction of macromolecules (such as proteins and polysaccharides) that disperse through dense media is an important fundamental issue in the development of innovative technological and medical applications. In the current work, the partitioning and diffusion of macromolecules of different sizes (from 4 to 10 nm in diameter) and shapes (linear or spherical) within dispersions of casein micelles (a protein microgel) is studied. The coefficients for diffusion and partition are measured using FRAP (fluorescence recovery after photobleaching) and analyzed with respect to the structural characteristics of the microgel determined by the use of TEM (transmission electron microscopy) tomography. The results show that the casein microgel displays a nonspecific attractive interaction for all macromolecules studied. When the macromolecular probes are spherical, this affinity is clearly size-dependent, with stronger attraction for the larger probes. The current data show that electrostatic effects cannot account for such an attraction. Rather, nonspecific hydration molecular forces appear to explain these results. These findings show how weak nonspecific forces affect the diffusion and partitioning of proteins and polysaccharides in a dense protein environment. These results could be useful to better understand the mechanisms of diffusion and partitioning in other media such as cells and tissues. Furthermore, there arises the possibility of using the casein micelle as a size-selective molecular device.

Inferring the microscopic surface energy of protein-protein interfaces from mutation data

Mutations at protein-protein recognition sites alter binding strength by altering the chemical nature of the interacting surfaces. We present a simple surface energy model, parameterized with empirical ΔΔG values, yielding mean energies of -48 cal mol(-1) Å(-2) for interactions between hydrophobic surfaces, -51 to -80 cal mol(-1) Å(-2) for surfaces of complementary charge, and 66-83 cal mol(-1) Å(-2) for electrostatically repelling surfaces, relative to the aqueous phase. This places the mean energy of hydrophobic surface burial at -24 cal mol(-1) Å(-2) . Despite neglecting configurational entropy and intramolecular changes, the model correlates with empirical binding free energies of a functionally diverse set of rigid-body interactions (r = 0.66). When used to rerank docking poses, it can place near-native solutions in the top 10 for 37% of the complexes evaluated, and 82% in the top 100. The method shows that hydrophobic burial is the driving force for protein association, accounting for 50-95% of the cohesive energy. The model is available open-source from http://life.bsc.es/pid/web/surface_energy/ and via the CCharpPPI web server http://life.bsc.es/pid/ccharppi/.

Analysis of factors influencing hydration site prediction based on molecular dynamics simulations

Water contributes significantly to the binding of small molecules to proteins in biochemical systems. Molecular dynamics (MD) simulation based programs such as WaterMap and WATsite have been used to probe the locations and thermodynamic properties of hydration sites at the surface or in the binding site of proteins generating important information for structure-based drug design. However, questions associated with the influence of the simulation protocol on hydration site analysis remain. In this study, we use WATsite to investigate the influence of factors such as simulation length and variations in initial protein conformations on hydration site prediction. We find that 4 ns MD simulation is appropriate to obtain a reliable prediction of the locations and thermodynamic properties of hydration sites. In addition, hydration site prediction can be largely affected by the initial protein conformations used for MD simulations. Here, we provide a first quantification of this effect and further indicate that similar conformations of binding site residues (RMSD < 0.5 Å) are required to obtain consistent hydration site predictions.

WATsite: hydration site prediction program with PyMOL interface

Water molecules that mediate protein-ligand interactions or are released from the binding site on ligand binding can contribute both enthalpically and entropically to the free energy of ligand binding. To elucidate the thermodynamic profile of individual water molecules and their potential contribution to ligand binding, a hydration site analysis program WATsite was developed together with an easy-to-use graphical user interface based on PyMOL. WATsite identifies hydration sites from a molecular dynamics simulation trajectory with explicit water molecules. The free energy profile of each hydration site is estimated by computing the enthalpy and entropy of the water molecule occupying a hydration site throughout the simulation. The results of the hydration site analysis can be displayed in PyMOL. A key feature of WATsite is that it is able to estimate the protein desolvation free energy for any user specified ligand. The WATsite program and its PyMOL plugin are available free of charge from http://people.pnhs.purdue.edu/~mlill/software.

Molecular dynamics simulations of the mammalian glutamate transporter EAAT3

Excitatory amino acid transporters (EAATs) are membrane proteins that enable sodium-coupled uptake of glutamate and other amino acids into neurons. Crystal structures of the archaeal homolog GltPh have been recently determined both in the inward- and outward-facing conformations. Here we construct homology models for the mammalian glutamate transporter EAAT3 in both conformations and perform molecular dynamics simulations to investigate its similarities and differences from GltPh. In particular, we study the coordination of the different ligands, the gating mechanism and the location of the proton and potassium binding sites in EAAT3. We show that the protonation of the E374 residue is essential for binding of glutamate to EAAT3, otherwise glutamate becomes unstable in the binding site. The gating mechanism in the inward-facing state of EAAT3 is found to be different from that of GltPh, which is traced to the relocation of an arginine residue from the HP1 segment in GltPh to the TM8 segment in EAAT3. Finally, we perform free energy calculations to locate the potassium binding site in EAAT3, and find a high-affinity site that overlaps with the Na1 and Na3 sites in GltPh.

Computation of standard binding free energies of polar and charged ligands to the glutamate receptor GluA2

Accurate calculation of the binding affinity of small molecules to proteins has the potential to become an important tool in rational drug design. In this study, we use the free energy perturbation (FEP) method with restraints to calculate the standard binding free energy of five ligands (ACPA, AMPA, CNQX, DNQX, and glutamate) to the glutamate receptor GluA2, which plays an essential role in synaptic transmission. To deal with the convergence problem in FEP calculations with charged ligands, we use a protocol where the ligand is coupled in the binding site while it is decoupled in bulk solution simultaneously. The contributions from the conformational, rotational, and translational entropies to the standard binding free energy are determined by applying/releasing respective restraints to the ligand in bulk/binding site. We also employ the confine-and-release approach, which helps to resolve convergence problems in FEP calculations. Our results are in good agreement with the experimental values for all five ligands, including the charged ones which are often problematic in FEP calculations. We also analyze the different contributions to the binding free energy of each ligand to GluA2 and discuss the nature of these interactions.

Protein pharmacophore selection using hydration-site analysis

Virtual screening using pharmacophore models is an efficient method to identify potential lead compounds for target proteins. Pharmacophore models based on protein structures are advantageous because a priori knowledge of active ligands is not required and the models are not biased by the chemical space of previously identified actives. However, in order to capture most potential interactions between all potentially binding ligands and the protein, the size of the pharmacophore model, i.e. number of pharmacophore elements, is typically quite large and therefore reduces the efficiency of pharmacophore based screening. We have developed a new method to select important pharmacophore elements using hydration-site information. The basic premise is that ligand functional groups that replace water molecules in the apo protein contribute strongly to the overall binding affinity of the ligand, due to the additional free energy gained from releasing the water molecule into the bulk solvent. We computed the free energy of water released from the binding site for each hydration site using thermodynamic analysis of molecular dynamics (MD) simulations. Pharmacophores which are colocalized with hydration sites with estimated favorable contributions to the free energy of binding are selected to generate a reduced pharmacophore model. We constructed reduced pharmacophore models for three protein systems and demonstrated good enrichment quality combined with high efficiency. The reduction in pharmacophore model size reduces the required screening time by a factor of 200-500 compared to using all protein pharmacophore elements. We also describe a training process using a small set of known actives to reliably select the optimal set of criteria for pharmacophore selection for each protein system.

Free energy simulations of ligand binding to the aspartate transporter Glt(Ph)

Glutamate/Aspartate transporters cotransport three Na(+) and one H(+) ions with the substrate and countertransport one K(+) ion. The binding sites for the substrate and two Na(+) ions have been observed in the crystal structure of the archeal homolog Glt(Ph), while the binding site for the third Na(+) ion has been proposed from computational studies and confirmed by experiments. Here we perform detailed free energy simulations of Glt(Ph), giving a comprehensive characterization of the substrate and ion binding sites, and calculating their binding free energies in various configurations. Our results show unequivocally that the substrate binds after the binding of two Na(+) ions. They also shed light into Asp/Glu selectivity of Glt(Ph), which is not observed in eukaryotic glutamate transporters.

Estimating absolute configurational entropies of macromolecules: the minimally coupled subspace approach

We develop a general minimally coupled subspace approach (MCSA) to compute absolute entropies of macromolecules, such as proteins, from computer generated canonical ensembles. Our approach overcomes limitations of current estimates such as the quasi-harmonic approximation which neglects non-linear and higher-order correlations as well as multi-minima characteristics of protein energy landscapes. Here, Full Correlation Analysis, adaptive kernel density estimation, and mutual information expansions are combined and high accuracy is demonstrated for a number of test systems ranging from alkanes to a 14 residue peptide. We further computed the configurational entropy for the full 67-residue cofactor of the TATA box binding protein illustrating that MCSA yields improved results also for large macromolecular systems.

In silico relationship between configurational entropy and soft degrees of freedom in proteins and peptides

Configurational entropies from temperature-dependent free energy calculations of proteins and peptides are compared with entropies determined as the sum of the Shannon entropies of a kernel density estimate of the dihedral angle probability distributions. We find that the entropy changes can be separated into independent contributions from hard and soft degrees of freedom. Furthermore, because the effects of motional correlations between dihedral angles on entropy differences are small, the functional role of configurational entropy changes can be interpreted in a spatially resolved manner.

The ruggedness of protein-protein energy landscape and the cutoff for 1/r(n) potentials

MOTIVATION

Computational studies of the energetics of protein association are important for revealing the underlying fundamental principles and for designing better tools to model protein complexes. The interaction cutoff contribution to the ruggedness of protein-protein energy landscape is studied in terms of relative energy fluctuations for 1/r(n) potentials based on a simplistic model of a protein complex. This artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase.

RESULTS

The critical values of the cutoff were calculated for each of 11 popular power-type potentials with n=0/9, 12 and for two thresholds of 5% and 10%. The artificial ruggedness decreases to tolerable thresholds for cutoffs larger than the critical ones. The results showed that for both thresholds the critical cutoff is a non-monotonic function of the potential power n. The functions reach the maximum at n=3/4 and then decrease with the increase of the potential power. The difference between two cutoffs for 5% and 10% artificial ruggedness becomes negligible for potentials decreasing faster than 1/r(12). The analytical results obtained for the simple model of protein complexes agree with the analysis of artificial ruggedness in a dataset of 62 protein-protein complexes, with different parameterizations of soft Lennard-Jones potential and two types of protein representations: all-atom and coarse-grained. The results suggest that cutoffs larger than the critical ones can be recommended for protein-protein potentials.

Extracting Kinetic and Stationary Distribution Information from Short MD Trajectories via a Collection of Surrogate Diffusion Models

Low-dimensional stochastic models can summarize dynamical information and make long time predictions associated with observables of complex atomistic systems. Maximum likelihood based techniques for estimating low-dimensional surrogate diffusion models from relatively short time series are presented. It is found that a heterogeneous population of slowly evolving conformational degrees of freedom modulates the dynamics. This underlying heterogeneity results in a collection of estimated low-dimensional diffusion models. Numerical techniques for exploiting this finding to approximate skewed histograms associated with the simulation are presented. In addition, statistical tests are also used to assess the validity of the models and determine physically relevant sampling information, e.g. the maximum sampling frequency at which one can discretely sample from an atomistic time series and have a surrogate diffusion model pass goodness-of-fit tests. The information extracted from such analyses can possibly be used to assist umbrella sampling computations as well as help in approximating effective diffusion coefficients. The techniques are demonstrated on simulations of Adenylate Kinase.

E9-Im9 colicin DNase-immunity protein biomolecular association in water: a multiple-copy and accelerated molecular dynamics simulation study

Protein-protein transient and dynamic interactions underlie all biological processes. The molecular dynamics (MD) of the E9 colicin DNase protein, its Im9 inhibitor protein, and their E9-Im9 recognition complex are investigated by combining multiple-copy (MC) MD and accelerated MD (aMD) explicit-solvent simulation approaches, after validation with crystalline-phase and solution experiments. Im9 shows higher flexibility than its E9 counterpart. Im9 displays a significant reduction of backbone flexibility and a remarkable increase in motional correlation upon E9 association. Im9 loops 23-31 and 54-64 open with respect to the E9-Im9 X-ray structure and show high conformational diversity. Upon association a large fraction (approximately 20 nm2) of E9 and Im9 protein surfaces become inaccessible to water. Numerous salt bridges transiently occurring throughout our six 50 ns long MC-MD simulations are not present in the X-ray model. Among these Im9 Glu31-E9 Arg96 and Im9 Glu41-Lys89 involve interface interactions. Through the use of 10 ns of Im9 aMD simulation, we reconcile the largest thermodynamic impact measured for Asp51Ala mutation with Im9 structure and dynamics. Lys57 acts as an essential molecular switch to shift Im9 surface loop towards an ideal configuration for E9 inhibition. This is achieved by switching Asp60-Lys57 and Asp62-Lys57 hydrogen bonds to Asp51-Lys57 salt bridge. E9-Im9 recognition involves shifts of conformational distributions, reorganization of intramolecular hydrogen bond patterns, and formation of new inter- and intramolecular interactions. The description of key transient biological interactions can be significantly enriched by the dynamic and atomic-level information provided by computer simulations.

Poisson-Boltzmann model analysis of binding mRNA cap analogues to the translation initiation factor eIF4E

The electrostatic free energy of binding of two analogues of the 5'-mRNA cap, differing in size and electric charge, to the wild type and mutated eukaryotic initiation factor eIF4E was computed using the finite difference solutions to the Poisson-Boltzmann equation. Two definitions of the solute-solvent dielectric boundary were used: van der Waals model, solvent exclusion (SE) model. The computed electrostatic energies were supplemented by estimations of the non polar and entropic contributions. A comparison with experimental data for the investigated systems was done. It appears that the SE model with additional contribution fits experimental findings better than the van der Waals model does.

The size of the intermolecular energy funnel in protein-protein interactions

Revealing the fundamental principles of protein interactions is essential for the basic knowledge of molecular processes and designing better predictive tools. Protein docking procedures allow systematic sampling of intermolecular energy landscapes, revealing the distribution of energy basins and their characteristics. A systematic search docking procedure GRAMM-X was applied to a comprehensive nonredundant database of nonobligate protein-protein complexes to determine the size of the intermolecular energy funnel. The unbound structures were simulated using rotamer library. The procedure generated grid-based matches, based on a smoothed Lennard-Jones potential, and minimized them off the grid with the same potential. The minimization generated a distribution of distances, based on a variety of metrics, between the grid-based and the minimized matches. The metric selected for the analysis, ligand interface RMSD, provided three independent estimates of the funnel size: based on the distribution amplitude for the near-native matches, deviation from random, and correlation with the energy values. The three methods converge to similar estimates of approximately 6-8 A ligand interface RMSD. The results indicated dependence of the funnel size on the type of the complex (smaller for antigen-antibody, medium for enzyme-inhibitor, and larger for the rest of the complexes) and the funnel size correlation with the size of the interface. Guidelines for the optimal sampling of docking coordinates, based on the funnel size estimates, were explored.

Water-membrane partition thermodynamics of an amphiphilic lipopeptide: an enthalpy-driven hydrophobic effect

To shed light on the driving force for the hydrophobic effect that partitions amphiphilic lipoproteins between water and membrane, we carried out an atomically detailed thermodynamic analysis of a triply lipid modified H-ras heptapeptide anchor (ANCH) in water and in a DMPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) bilayer. Combining molecular mechanical and continuum solvent approaches with an improved technique for solute entropy calculation, we obtained an overall transfer free energy of approximately -13 kcal mol(-1). This value is in qualitative agreement with free energy changes derived from a potential of mean force calculation and indirect experimental observations. Changes in free energies of solvation and ANCH conformational reorganization are unfavorable, whereas ANCH-DMPC interactions-especially van der Waals-favor insertion. These results are consistent with an enthalpy-driven hydrophobic effect, in accord with earlier calorimetric data on the membrane partition of other amphiphiles. Furthermore, structural and entropic analysis of molecular dynamics-generated ensembles suggests that conformational selection may play a hitherto unappreciated role in membrane insertion of lipid-modified peptides and proteins.

Chasing funnels on protein-protein energy landscapes at different resolutions

Studies of intermolecular energy landscapes are important for understanding protein association and adequate modeling of protein interactions. Landscape representation at different resolutions can be used for the refinement of docking predictions and detection of macro characteristics, like the binding funnel. A representative set of protein-protein complexes was used to systematically map the intermolecular landscape by grid-based docking. The change of the resolution was achieved by varying the range of the potential, according to the variable resolution GRAMM methodology. A formalism was developed to consistently parameterize the potential and describe essential characteristics of the landscape. The results of gradual landscape smoothing, from high to low resolution, indicate that i), the number of energy basins, the landscape ruggedness, and the slope decrease accordingly; ii), the number of near-native matches, defined as those inside the funnel, increases until the trend breaks down at critical resolution; the rate of the increase and the critical resolution are specific to the type of a complex (enzyme inhibitor, antigen-antibody, and other), reflect known underlying recognition factors, and correlate with earlier determined estimates of the funnel size; iii), the native/nonnative energy gap, a major characteristic of the energy minima hierarchy, remains constant; and iv), the putative funnel (defined as the deepest basin) has the largest average depth-related ruggedness and slope, at all resolutions. The results facilitate better understanding of the binding landscapes and suggest directions for implementation in practical docking protocols.

Interaction cutoff effect on ruggedness of protein-protein energy landscape

The concept of the energy landscape is important for better understanding of protein-protein interactions and for designing adequate docking procedures. The intermolecular landscape has a rugged terrain that impedes search procedures. Its inherent ruggedness is related to the conformational characteristics of the molecules and to the form of the potential function--more rugged for short-range potentials and less rugged for "soft," typically long-range potentials. Our study determined that the landscape ruggedness is further substantially exacerbated by truncation of the potentials. This additional ruggedness appears below certain critical interaction ranges that depend on the form of the potential. The theoretical model describing the cutoff effect on the landscape ruggedness is confirmed by the energy calculation on a dataset of protein-protein complexes. The negative effect of the potentials cutoff is well known. However, revealing its physical basis in terms of the energy landscape is important for better understanding of intermolecular interactions.

Accelerated entropy estimates with accelerated dynamics

Accelerated dynamics is applied to entropy calculations on a set of toy and molecular systems and is found to enhance the rate of convergence.

Comparative MD analysis of the stability of transthyretin providing insight into the fibrillation mechanism

Proteins can misfold and aggregate, which is believed to be the cause of a variety of diseases, affecting very diverse organs in the body. Many questions about the nature of aggregation and the proteins that are involved in these events are still left unanswered. One of the proteins that is known to form amyloids is transthyretin (TTR), the secondary transporter of thyroxine, and transporter of retinol-binding protein. Several experimental results have helped to explain this aberrant behavior of TTR; however, structural insights of the amyloidgenic process are still lacking. Therefore, we have used all-atom MD simulation and free energy calculations to study the initial phase of this process. We have calculated the free energy changes of the initial tetramer dissociation under different conditions and in the presence of thyroxine. We show that tetramer formation is indeed only thermodynamically favorable in neutral pH conditions. We find that binding of two thyroxine molecules stabilizes the complex, and that this occurs with negative cooperativity. In addition to the energetic calculations, we have also investigated the dominant motions of the TTR and found that only the dimeric form of the protein could undergo the initial fibril formation.

Calculations of protein-ligand binding entropy of relative and overall molecular motions

In the context of virtual database screening, calculations of protein-ligand binding entropy of relative and overall molecular motions are challenging, owing to the inherent structural complexity of the ligand binding well in the energy landscape of protein-ligand interactions and computing time limitations. We describe a fast statistical thermodynamic method for estimation the binding entropy to address the challenges. The method is based on the integration of the configurational integral over clusters obtained from multiple docked positions. We apply the method in conjunction with 11 popular scoring functions (AutoDock, ChemScore, DrugScore, D-Score, F-Score, G-Score, LigScore, LUDI, PLP, PMF, X-Score) to evaluate the binding entropy of 100 protein-ligand complexes. The averaged values of binding entropy contribution vary from 6.2 to 9.1 kcal/mol, showing good agreement with literature. We calculate positional sizes and the angular volume of the native ligand wells. The averaged geometric mean of positional sizes in principal directions varies from 0.8 to 1.4 A. The calculated range of angular volumes is 3.3-11.8 rad(2). Then we demonstrate that the averaged six-dimensional volume of the native well is larger than the volume of the most populated non-native well in energy landscapes described by all of 11 scoring functions.

Inference of macromolecular assemblies from crystalline state

We discuss basic physical-chemical principles underlying the formation of stable macromolecular complexes, which in many cases are likely to be the biological units performing a certain physiological function. We also consider available theoretical approaches to the calculation of macromolecular affinity and entropy of complexation. The latter is shown to play an important role and make a major effect on complex size and symmetry. We develop a new method, based on chemical thermodynamics, for automatic detection of macromolecular assemblies in the Protein Data Bank (PDB) entries that are the results of X-ray diffraction experiments. As found, biological units may be recovered at 80-90% success rate, which makes X-ray crystallography an important source of experimental data on macromolecular complexes and protein-protein interactions. The method is implemented as a public WWW service.

Calculations of solute and solvent entropies from molecular dynamics simulations

The translational, rotational and conformational (vibrational) entropy contributions to ligand-receptor binding free energies are analyzed within the standard formulation of statistical thermodynamics. It is shown that the partitioning of the binding entropy into different components is to some extent arbitrary, but an appropriate method to calculate both translational and rotational entropy contributions to noncovalent association is by estimating the configurational volumes of the ligand in the bound and free states. Different approaches to calculating solute entropies using free energy perturbation calculations, configurational volumes based on root-mean-square fluctuations and covariance matrix based quasiharmonic analysis are illustrated for some simple molecular systems. Numerical examples for the different contributions demonstrate that theoretically derived results are well reproduced by the approximations. Calculation of solvent entropies, either using total potential energy averages or van't Hoff plots, are carried out for the case of ion solvation in water. Although convergence problems will persist for large and complex simulation systems, good agreement with experiment is obtained here for relative and absolute ion hydration entropies. We also outline how solvent and solute entropic contributions are taken into account in empirical binding free energy calculations using the linear interaction energy method. In particular it is shown that empirical scaling of the nonpolar intermolecular ligand interaction energy effectively takes into account size dependent contributions to the binding free energy.

Stochastic dynamics of macromolecular-assembly networks

The formation and regulation of macromolecular complexes provides the backbone of most cellular processes, including gene regulation and signal transduction. The inherent complexity of assembling macromolecular structures makes current computational methods strongly limited for understanding how the physical interactions between cellular components give rise to systemic properties of cells. Here, we present a stochastic approach to study the dynamics of networks formed by macromolecular complexes in terms of the molecular interactions of their components. Exploiting key thermodynamic concepts, this approach makes it possible to both estimate reaction rates and incorporate the resulting assembly dynamics into the stochastic kinetics of cellular networks. As prototype systems, we consider the lac operon and phage lambda induction switches, which rely on the formation of DNA loops by proteins and on the integration of these protein-DNA complexes into intracellular networks. This cross-scale approach offers an effective starting point to move forward from network diagrams, such as those of protein-protein and DNA-protein interaction networks, to the actual dynamics of cellular processes.

Flexibility and Conformational Entropy in Protein-Protein Binding

To better understand the interplay between protein-protein binding and protein dynamics, we analyzed molecular dynamics simulations of 17 protein-protein complexes and their unbound components. Complex formation does not restrict the conformational freedom of the partner proteins as a whole, but, rather, it leads to a redistribution of dynamics. We calculate the change in conformational entropy for seven complexes with quasiharmonic analysis. We see significant loss, but also increased or unchanged conformational entropy. Where comparison is possible, the results are consistent with experimental data. However, stringent error estimates based on multiple independent simulations reveal large uncertainties that are usually overlooked. We observe substantial gains of pseudo entropy in individual partner proteins, and we observe that all complexes retain residual stabilizing intermolecular motions. Consequently, protein flexibility has an important influence on the thermodynamics of binding and may disfavor as well as favor association. These results support a recently proposed unified model for flexible protein-protein association.