PARENT: A Parallel Software Suite for the Calculation of Configurational Entropy in Biomolecular Systems

Lasso Peptides: Exploring the Folding Landscape of Nature's Smallest Interlocked Motifs

Lasso peptides make up a class of natural products characterized by a threaded structure. Given their small size and stability, chemical synthesis would offer tremendous potential for the development of novel therapeutics. However, the accessibility of the pre-folded lasso architecture has limited this advance. To better understand the folding process de novo, simulations are used herein to characterize the folding propensity of microcin J25 (MccJ25), a lasso peptide known for its antimicrobial properties. New algorithms are developed to unambiguously distinguish threaded from nonthreaded precursors and determine handedness, a key feature in natural lasso peptides. We find that MccJ25 indeed forms right-handed pre-lassos, in contrast to past predictions but consistent with all natural lasso peptides. Additionally, the native pre-lasso structure is shown to be metastable prior to ring formation but to readily transition to entropically favored unfolded and nonthreaded structures, suggesting that de novo lasso folding is rare. However, by altering the ring forming residues and appending thiol and thioester functionalities, we are able to increase the stability of pre-lasso conformations. Furthermore, conditions leading to protonation of a histidine imidazole side chain further stabilize the modified pre-lasso ensemble. This work highlights the use of computational methods to characterize lasso folding and demonstrates that de novo access to lasso structures can be facilitated by optimizing sequence, unnatural modifications, and reaction conditions like pH.

Influence of metal binding on the conformational landscape of neurofilament peptides

In order to understand the preferred modes of chelation in metal-binding peptides, quantum mechanical calculations can be used to compute energies, resulting in a hierarchy of binding affinities. These calculations often produce increasing stabilization energies the higher the coordination of the complex. However, as the coordination of a metal increases, the conformational freedom of the polypeptide chain is inevitably reduced, resulting in an entropic penalty. Estimating the magnitude of this penalty from the many different degrees of freedom of biomolecular systems is very challenging, and as a result this contribution to the free energy is often ignored. Here we explore this problem focusing on a family of phosphorylated neuropeptides that bind to aluminum. We find that there is a general negative correlation between both stabilization energy and entropy. Our results suggest that a subtle interplay between enthalpic and entropic forces will determine the population of the most favourable species. Additionally, we discuss the requirements for a possible "Metal Ion Hypothesis" based on our findings.

In Silico Simulations Reveal Molecular Mechanism of Uranyl Ion Toxicity towards DNA-Binding Domain of PARP-1 Protein

The molecular toxicity of the uranyl ion (UO22+) in living cells is primarily determined by its high affinity to both native and potential metal-binding sites that commonly occur in the structure of biomolecules. Recent advances in computational and experimental research have shed light on the structural properties and functional impacts of uranyl binding to proteins, organic ligands, nucleic acids, and their complexes. In the present work, we report the results of the computational investigation of the uranyl-mediated loss of DNA-binding activity of PARP-1, a eukaryotic enzyme that participates in DNA repair, cell differentiation, and the induction of inflammation. The latest experimental studies have shown that the uranyl ion directly interacts with its DNA-binding subdomains, zinc fingers Zn1 and Zn2, and alters their tertiary structure. Here, we propose an atomistic mechanism underlying this process and compute the free energy change along the suggested pathway. Our Quantum Mechanics/Molecular Mechanics (QM/MM) simulations of the Zn2-UO22+ complex indicate that the uranyl ion replaces zinc in its native binding site. However, the resulting state is destroyed due to the spontaneous internal hydrolysis of the U-Cys162 coordination bond. Despite the enthalpy of hydrolysis being +2.8 kcal/mol, the overall reaction free energy change is -0.6 kcal/mol, which is attributed to the loss of domain's native tertiary structure originally maintained by a zinc ion. The subsequent reorganization of the binding site includes the association of the uranyl ion with the Glu190/Asp191 acidic cluster and significant perturbations in the domain's tertiary structure driven by a further decrease in the free energy by 6.8 kcal/mol. The disruption of the DNA-binding interface revealed in our study is consistent with previous experimental findings and explains the loss of PARP-like zinc fingers' affinity for nucleic acids.

Protein compactness and interaction valency define the architecture of a biomolecular condensate across scales

Non-membrane-bound biomolecular condensates have been proposed to represent an important mode of subcellular organization in diverse biological settings. However, the fundamental principles governing the spatial organization and dynamics of condensates at the atomistic level remain unclear. The Saccharomyces cerevisiae Lge1 protein is required for histone H2B ubiquitination and its N-terminal intrinsically disordered fragment (Lge11-80) undergoes robust phase separation. This study connects single- and multi-chain all-atom molecular dynamics simulations of Lge11-80 with the in vitro behavior of Lge11-80 condensates. Analysis of modeled protein-protein interactions elucidates the key determinants of Lge11-80 condensate formation and links configurational entropy, valency, and compactness of proteins inside the condensates. A newly derived analytical formalism, related to colloid fractal cluster formation, describes condensate architecture across length scales as a function of protein valency and compactness. In particular, the formalism provides an atomistically resolved model of Lge11-80 condensates on the scale of hundreds of nanometers starting from individual protein conformers captured in simulations. The simulation-derived fractal dimensions of condensates of Lge11-80 and its mutants agree with their in vitro morphologies. The presented framework enables a multiscale description of biomolecular condensates and embeds their study in a wider context of colloid self-organization.

On a mechanistic impact of transmembrane tetramerization in the pathological activation of RTKs

Constitutive activation of receptor tyrosine kinases (RTKs) via different mutations has a strong impact on the development of severe human disorders, including cancer. Here we propose a putative activation scenario of RTKs, whereby transmembrane (TM) mutations can also promote higher-order oligomerization of the receptors that leads to the subsequent ligand-free activation. We illustrate this scenario using a computational modelling framework comprising sequence-based structure prediction and all-atom 1 µs molecular dynamics (MD) simulations in a lipid membrane for a previously characterised oncogenic TM mutation V536E in platelet-derived growth factor receptor alpha (PDGFRA). We show that in the course of MD simulations the mutant TM tetramer retains stable and compact configuration strengthened by tight protein-protein interactions, while the wild type TM tetramer demonstrates looser packing and a tendency to dissociate. Moreover, the mutation affects the characteristic motions of mutated TM helical segments by introducing additional non-covalent crosslinks in the middle of the TM tetramer, which operate as mechanical hinges. This leads to dynamic decoupling of the C-termini from the rigidified N-terminal parts and facilitates more pronounced possible displacement between the C-termini of the mutant TM helical regions that can provide more freedom for mutual rearrangement of the kinase domains located downstream. Our results for the V536E mutation in the context of PDGFRA TM tetramer allow for the possibility that the effect of oncogenic TM mutations can go beyond alternating the structure and dynamics of TM dimeric states and might also promote the formation of higher-order oligomers directly contributing to ligand-independent signalling effectuated by PDGFRA and other RTKs.

Computational Strategies for Entropy Modeling in Chemical Processes

Computational simulations of entropy are important in understanding the thermodynamic forces that drive chemical reactions on a molecular scale. In recent years, various algorithms have been developed and applied in conjunction with molecular modeling techniques to evaluate the change of entropy in solvation, hydrophobic interactions, and chemical reactions. The aim of this review is to highlight four specific computational entropy calculation methods: normal mode analysis, free volume theory, two-phase thermodynamics, and configurational entropy modeling. The technical aspects, applications, and limitations of each method will be discussed in detail.

Quasiclassical Trajectory Simulation as a Protocol to Build Locally Accurate Machine Learning Potentials

Direct trajectory calculations have become increasingly popular in recent computational chemistry investigations. However, the exorbitant computational cost of ab initio trajectory calculations usually limits its application in mechanistic explorations. Recently, machine learning-based potential energy surface (ML-PES) provides a powerful strategy to circumvent the heavy computational cost and meanwhile maintain the required accuracy. Despite the appealing potential, constructing a robust ML-PES is still challenging since the training set of the PES should cover a broad enough configuration space. In this work, we demonstrate that when the concerned properties could be collected by the localized sampling of the configuration space, quasiclassical trajectory (QCT) calculations can be invoked to efficiently obtain locally accurate ML-PESs. We prove our concept with two model reactions: methyl migration of i-pentane cation and dimerization of cyclopentadiene. We found that the locally accurate ML-PESs are sufficiently robust for reproducing the static and dynamic features of the reactions, including the time-resolved free energy and entropy changes, and time gaps.

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.

Affinity of disordered protein complexes is modulated by entropy-energy reinforcement

Intrinsically disordered proteins (IDPs), which are very common and essential to many biological activities, sometimes function via interaction with another IDP and form a fuzzy complex, which can be highly stable. It is unclear what the biophysical forces are that govern their thermodynamics and specificity, which are essential for de novo fuzzy complex design. Here, we explored the fuzzy complex formed between ProTα and H1, which are oppositely charged IDPs, by swapping the charges between them, generating variants that have either greater polyampholytic or polyelectrolytic nature as well as different charge patterns. Charge swapping and shuffling dramatically change the affinity of the fuzzy complex, which is contributed to by both enthalpy and entropy, where the latter is dominated by counterion release.

The association between two intrinsically disordered proteins (IDPs) may produce a fuzzy complex characterized by a high binding affinity, similar to that found in the ultrastable complexes formed between two well-structured proteins. Here, using coarse-grained simulations, we quantified the biophysical forces driving the formation of such fuzzy complexes. We found that the high-affinity complex formed between the highly and oppositely charged H1 and ProTα proteins is sensitive to electrostatic interactions. We investigated 52 variants of the complex by swapping charges between the two oppositely charged proteins to produce sequences whose negatively or positively charged residue content was more homogeneous or heterogenous (i.e., polyelectrolytic or polyampholytic, having higher or lower absolute net charges, respectively) than the wild type. We also changed the distributions of oppositely charged residues within each participating sequence to produce variants in which the charges were segregated or well mixed. Both types of changes significantly affect binding affinity in fuzzy complexes, which is governed by both enthalpy and entropy. The formation of H1–ProTa is supported by an increase in configurational entropy and by entropy due to counterion release. The latter can be twice as large as the former, illustrating the dominance of counterion entropy in modulating the binding thermodynamics. Complexes formed between proteins with greater absolute net charges are more stable, both enthalpically and entropically, indicating that enthalpy and entropy have a mutually reinforcing effect. The sensitivity of the thermodynamics of the complex to net charge and the charge pattern within each of the binding constituents may provide a means to achieve binding specificity between IDPs.

Computational Modeling of Conformer Stability in Benenodin-1, a Thermally Actuated Lasso Peptide Switch

Benenodin-1 is a thermally actuated lasso peptide rotaxane switch with two primary translational isomers that differ in the relative position of the residue Gln15. The conversion from one conformer to the other involves substantial enthalpy-entropy compensation: one conformer is energetically favored and the other is entropically favored. Here, we take a multi-scale quantum mechanical (QM) and classical molecular dynamic (MD) approach to reveal residue-specific sources of these differences in stability. QM reveals that the two benenodin-1 conformers involve distinct hydrogen bonding networks, with the enthalpically favored conformer having more intra-peptide hydrogen bonds between the Gln15 side chain and nearby residues. The evaluation of configurational entropy over the MD-sampled geometries reveals that the entropically favored conformer has enhanced conformational flexibility. By computing the by-residue-sum entropies, we identify the role of Gln15 and neighboring Glu14 in mediating the entropic variation during the switching process. These computational insights help explain the effects of Glu14Ala and Gln15Ala mutations on the conformational population of benenodin-1 observed experimentally.

Decoupled coordinates for machine learning-based molecular fragment linking

Recent developments in machine learning-based molecular fragment linking have demonstrated the importance of informing the generation process with structural information specifying the relative orientation of the fragments to be linked. However, such structural information has so far not been provided in the form of a complete relative coordinate system. We present a decoupled coordinate system consisting of bond lengths, bond angles and torsion angles, and show that it is complete. By incorporating this set of coordinates in a linker generation framework, we show that it has a significant impact on the quality of the generated linkers. To elucidate the advantages of such a coordinate system, we investigate the amount of reliable information within the different types of degrees of freedom using both detailed ablation studies and an information-theoretical analysis. The presented benefits suggest the application of a complete and decoupled relative coordinate system as a standard good practice in linker design.

Entropic Path Sampling: Computational Protocol to Evaluate Entropic Profile along a Reaction Path

Fleeting intermediates constitute dynamically stepwise mechanisms. They have been characterized in molecular dynamics trajectories, but whether these intermediates form a free energy minimum to become entropic intermediates remains elusively defined. We developed a computational protocol known as entropic path sampling to evaluate the entropic variation of reacting species along a reaction path based on an ensemble of trajectories. Using cyclopentadiene dimerization as a model reaction, we observed an entropy maximum along the reaction path which originates from an enhanced conformational flexibility as the reacting species enter into a flat energy region. As the reacting species further approach product formation, unfavorable entropic restriction fails to offset the potential energy drop, resulting in no free energy minimum along the post-TS pathway. Our results show that cyclopentadiene dimerization involves an entropy maximum that leads to dynamic intermediates with elongated lifetimes, but the reaction does not involve entropic intermediates.

Phospholipids dock SARS-CoV-2 spike protein via hydrophobic interactions: a minimal in-silico study of lecithin nasal spray therapy

Understanding the physical and chemical properties of viral infections at molecular scales is a major challenge for the scientific community more so with the outbreak of global pandemics. There is currently a lot of effort being placed in identifying molecules that could act as putative drugs or blockers of viral molecules. In this work, we computationally explore the importance in antiviral activity of a less studied class of molecules, namely surfactants. We employ all-atoms molecular dynamics simulations to study the interaction between the receptor-binding domain of the SARS-CoV-2 spike protein and the phospholipid lecithin (POPC), in water. Our microsecond simulations show a preferential binding of lecithin to the receptor-binding motif of SARS-CoV-2 with binding free energies significantly larger than [Formula: see text]. Furthermore, hydrophobic interactions involving lecithin non-polar tails dominate these binding events, which are also accompanied by dewetting of the receptor binding motif. Through an analysis of fluctuations in the radius of gyration of the receptor-binding domain, its contact maps with lecithin molecules, and distributions of water molecules near the binding region, we elucidate molecular interactions that may play an important role in interactions involving surfactant-type molecules and viruses. We discuss our minimal computational model in the context of lecithin-based liposomal nasal sprays as putative mitigating therapies for COVID-19.

Calculating Transfer Entropy from Variance-Covariance Matrices Provides Insight into Allosteric Communication in ERK2

We develop an approach by which reliable estimates of the transfer entropy can be obtained from the variance-covariance matrix of atomic fluctuations, which converges quickly and retains sensitivity to the full chemical profile of the biomolecular system. We validate our method on ERK2, a well-studied kinase involved in the MAPK signaling cascade for which considerable computational, experimental, and mutation data are available. We present the results of transfer entropy analysis on data obtained from molecular dynamics simulations of wild-type active and inactive ERK2, along with mutants Q103A, I84A, L73P, and G83A. We show that our method is systematically consistent within the context of other approaches for calculating transfer entropy, and we provide a method for interpreting networks of interconnected residues in the protein from a perspective of allosteric coupling. We introduce new insights about possible allosteric activity of the extreme N-terminal region of the kinase, and we describe evidence that suggests that activation may occur by different paths or routes in different mutants. Our results highlight systematic advantages and disadvantages of each method for calculating transfer entropy and show the important role of transfer entropy analysis for understanding allosteric behavior in biomolecular systems.

Cyclic peptides: backbone rigidification and capability of mimicking motifs at protein-protein interfaces

Cyclization is commonly employed in efforts to improve the target binding affinity of peptide-based probes and therapeutics. Many structural motifs have been identified at protein-protein interfaces and provide promising targets for inhibitor design using cyclic peptides. Cyclized peptides are generally assumed to be rigidified relative to their linear counterparts. This rigidification potentially pre-organizes the molecules to interact properly with their targets. However, the actual impact of cyclization on, for example, peptide configurational entropy, is currently poorly understood in terms of both its magnitude and molecular-level origins. Moreover, even with thousands of desired structural motifs at hand, it is currently not possible to a priori identify the ones that are most promising to mimic using cyclic peptides nor to select the ideal linker length. Instead, labor-intensive chemical synthesis and experimental characterization of various cyclic peptide designs are required, in hopes of finding one with improved target affinity. Herein, using molecular dynamics simulations of polyglycines, we elucidated how head-to-tail cyclization impacts peptide backbone dihedral entropy and developed a simple strategy to rapidly screen for structures that can be reliably mimicked by preorganized cyclic peptides. As expected, cyclization generally led to a reduction in backbone dihedral entropy; notably, however, this effect was minimal when the length of polyglycines was >9 residues. We also found that the reduction in backbone dihedral entropy upon cyclization of small polyglycine peptides does not result from more restricted distributions of the dihedrals; rather, it was the correlations between specific dihedrals that caused the decrease in configurational entropy in the cyclic peptides. Using our comprehensive cyclo-Gn structural ensembles, we obtained a holistic picture of what conformations are accessible to cyclic peptides. Using "hot loops" recently identified at protein-protein interfaces as an example, we provide clear guidelines for choosing the "easiest" hot loops for cyclic peptides to mimic and for identifying appropriate cyclic peptide lengths. In conclusion, our results provide an understanding of the thermodynamics and structures of this interesting class of molecules. This information should prove particularly useful for designing cyclic peptide inhibitors of protein-protein interactions.

Application and Comprehensive Analysis of Neighbor Approximated Information Theoretic Configurational Entropy Methods to Protein-Ligand Binding Cases

The binding entropy is an important thermodynamic quantity which has numerous applications in studies of the biophysical process, and configurational entropy is often one of the major contributors in it. Therefore, its accurate estimation is important, though it is challenging mostly due to sampling limitations, anharmonicity, and multimodality of atomic fluctuations. The present work reports a Neighbor Approximated Maximum Information Spanning Tree (A-MIST) method for conformational entropy and presents its performance and computational advantage over conventional Mutual Information Expansion (MIE) and Maximum Information Spanning Tree (MIST) for two protein-ligand binding cases: indirubin-5-sulfonate to Plasmodium falciparum Protein Kinase 5 (PfPK5) and P. falciparum RON2-peptide to P. falciparum Apical Membrane Antigen 1 (PfAMA1). Important structural regions considering binding configurational entropy are identified, and physical origins for such are discussed. A thorough performance evaluation is done of a set of four entropy estimators (Maximum Likelihood (ML), Miller-Madow (MM), Chao-Shen (CS), and James and Stein shrinkage (JS)) with known varying degrees of sensitivity of the entropy estimate on the extent of sampling, each with two schemes for discretization of fluctuation data of Degrees of Freedom (DFs) to estimate Probability Density Functions (PDFs). Our comprehensive evaluation of influences of variations of parameters shows Neighbor Approximated MIE (A-MIE) outperforms MIE in terms of convergence and computational efficiency. In the case of A-MIE/MIE, results are sensitive to the choice of root atoms, graph search algorithm used for the Bond-Angle-Torsion (BAT) conversion, and entropy estimator, while A-MIST/MIST are not. A-MIST yields binding entropy within 0.5 kcal/mol of MIST with only 20-30% computation. Moreover, all these methods have been implemented in an OpenMP/MPI hybrid parallel C++11 code, and also a python package for data preprocessing and entropy contribution analysis is developed and made available. A comparative analysis of features of current implementation and existing tools is also presented.

Entropy of Proteins Using Multiscale Cell Correlation

A new multiscale method is presented to calculate the entropy of proteins from molecular dynamics simulations. Termed Multiscale Cell Correlation (MCC), the method decomposes the protein into sets of rigid-body units based on their covalent-bond connectivity at three levels of hierarchy: molecule, residue, and united atom. It evaluates the vibrational and topographical entropy from forces, torques, and dihedrals at each level, taking into account correlations between sets of constituent units that together make up a larger unit at the coarser length scale. MCC gives entropies in close agreement with normal-mode analysis and smaller than those using quasiharmonic analysis as well as providing much faster convergence. Moreover, MCC provides an insightful decomposition of entropy at each length scale and for each type of amino acid according to their solvent exposure and whether they are terminal residues. While the residue entropy depends weakly on solvent exposure, there is greater variation in entropy components for larger, more polar amino acids, which have increased conformational entropy but reduced vibrational entropy with greater solvent exposure.

Configurational Entropy Components and Their Contribution to Biomolecular Complex Formation

Configurational entropy change is a central constituent of the free energy change in noncovalent interactions between biomolecules. Due to both experimental and computational limitations, however, the impact of individual contributions to configurational entropy change remains underexplored. Here, we develop a novel, fully analytical framework to dissect the configurational entropy change of binding into contributions coming from molecular internal and external degrees of freedom. Importantly, this framework accounts for all coupled and uncoupled contributions in the absence of an external field. We employ our parallel implementation of the maximum information spanning tree algorithm to provide a comprehensive numerical analysis of the importance of the individual contributions to configurational entropy change on an extensive set of molecular dynamics simulations of protein binding processes. Contrary to commonly accepted assumptions, we show that different coupling terms contribute significantly to the overall configurational entropy change. Finally, while the magnitude of individual terms may be largely unpredictable a priori, the total configurational entropy change can be well approximated by rescaling the sum of uncoupled contributions from internal degrees of freedom only, providing support for NMR-based approaches for configurational entropy change estimation.

β-Branched Amino Acids Stabilize Specific Conformations of Cyclic Hexapeptides

Cyclic peptides (CPs) are a promising class of molecules for drug development, particularly as inhibitors of protein-protein interactions. Predicting low-energy structures and global structural ensembles of individual CPs is critical for the design of bioactive molecules, but these are challenging to predict and difficult to verify experimentally. In our previous work, we used explicit-solvent molecular dynamics simulations with enhanced sampling methods to predict the global structural ensembles of cyclic hexapeptides containing different permutations of glycine, alanine, and valine. One peptide, cyclo-(VVGGVG) or P7, was predicted to be unusually well structured. In this work, we synthesized P7, along with a less well-structured control peptide, cyclo-(VVGVGG) or P6, and characterized their global structural ensembles in water using NMR spectroscopy. The NMR data revealed a structural ensemble similar to the prediction for P7 and showed that P6 was indeed much less well-structured than P7. We then simulated and experimentally characterized the global structural ensembles of several P7 analogs and discovered that β-branching at one critical position within P7 is important for overall structural stability. The simulations allowed deconvolution of thermodynamic factors that underlie this structural stabilization. Overall, the excellent correlation between simulation and experimental data indicates that our simulation platform will be a promising approach for designing well-structured CPs and also for understanding the complex interactions that control the conformations of constrained peptides and other macrocycles.

How phosphorylation influences E1 subunit pyruvate dehydrogenase: A computational study

Pyruvate (PYR) dehydrogenase complex (PDC) is an enzymatic system that plays a crucial role in cellular metabolism as it controls the entry of carbon into the Krebs cycle. From a structural point of view, PDC is formed by three different subunits (E1, E2 and E3) capable of catalyzing the three reaction steps necessary for the full conversion of pyruvate to acetyl-CoA. Recent investigations pointed out the crucial role of this enzyme in the replication and survival of specific cancer cell lines, renewing the interest of the scientific community. Here, we report the results of our molecular dynamics studies on the mechanism by which posttranslational modifications, in particular the phosphorylation of three serine residues (Ser-264-α, Ser-271-α, and Ser-203-α), influence the enzymatic function of the protein. Our results support the hypothesis that the phosphorylation of Ser-264-α and Ser-271-α leads to (1) a perturbation of the catalytic site structure and dynamics and, especially in the case of Ser-264-α, to (2) a reduction in the affinity of E1 for the substrate. Additionally, an analysis of the channels connecting the external environment with the catalytic site indicates that the inhibitory effect should not be due to the occlusion of the access/egress pathways to/from the active site.

Binding free energy analysis of protein-protein docking model structures by evERdock

To aid the evaluation of protein-protein complex model structures generated by protein docking prediction (decoys), we previously developed a method to calculate the binding free energies for complexes. The method combines a short (2 ns) all-atom molecular dynamics simulation with explicit solvent and solution theory in the energy representation (ER). We showed that this method successfully selected structures similar to the native complex structure (near-native decoys) as the lowest binding free energy structures. In our current work, we applied this method (evERdock) to 100 or 300 model structures of four protein-protein complexes. The crystal structures and the near-native decoys showed the lowest binding free energy of all the examined structures, indicating that evERdock can successfully evaluate decoys. Several decoys that show low interface root-mean-square distance but relatively high binding free energy were also identified. Analysis of the fraction of native contacts, hydrogen bonds, and salt bridges at the protein-protein interface indicated that these decoys were insufficiently optimized at the interface. After optimizing the interactions around the interface by including interfacial water molecules, the binding free energies of these decoys were improved. We also investigated the effect of solute entropy on binding free energy and found that consideration of the entropy term does not necessarily improve the evaluations of decoys using the normal model analysis for entropy calculation.

Testing the mutual information expansion of entropy with multivariate Gaussian distributions

The mutual information expansion (MIE) represents an approximation of the configurational entropy in terms of low-dimensional integrals. It is frequently employed to compute entropies from simulation data of large systems, such as macromolecules, for which brute-force evaluation of the full configurational integral is intractable. Here, we test the validity of MIE for systems consisting of more than m = 100 degrees of freedom (dofs). The dofs are distributed according to multivariate Gaussian distributions which were generated from protein structures using a variant of the anisotropic network model. For the Gaussian distributions, we have semi-analytical access to the configurational entropy as well as to all contributions of MIE. This allows us to accurately assess the validity of MIE for different situations. We find that MIE diverges for systems containing long-range correlations which means that the error of consecutive MIE approximations grows with the truncation order n for all tractable n ≪ m. This fact implies severe limitations on the applicability of MIE, which are discussed in the article. For systems with correlations that decay exponentially with distance, MIE represents an asymptotic expansion of entropy, where the first successive MIE approximations approach the exact entropy, while MIE also diverges for larger orders. In this case, MIE serves as a useful entropy expansion when truncated up to a specific truncation order which depends on the correlation length of the system.

Self-Consistent Framework Connecting Experimental Proxies of Protein Dynamics with Configurational Entropy

The recently developed NMR techniques enable estimation of protein configurational entropy change from the change in the average methyl order parameters. This experimental observable, however, does not directly measure the contribution of intramolecular couplings, protein main-chain motions, or angular dynamics. Here, we carry out a self-consistent computational analysis of the impact of these missing contributions on an extensive set of molecular dynamics simulations of different proteins undergoing binding. Specifically, we compare the configurational entropy change in protein complex formation as obtained by the maximum information spanning tree approximation (MIST), which treats the above entropy contributions directly, and the change in the average NMR methyl and NH order parameters. Our parallel implementation of MIST allows us to treat hard angular degrees of freedom as well as couplings up to full pairwise order explicitly, while still involving a high degree of sampling and tackling molecules of biologically relevant sizes. First, we demonstrate a remarkably strong linear relationship between the total configurational entropy change and the average change in both methyl and backbone-NH order parameters. Second, in contrast to canonical assumptions, we show that the main-chain and angular terms contribute significantly to the overall configurational entropy change and also scale linearly with it. Consequently, linear models starting from the average methyl order parameters are able to capture the contribution of main-chain and angular terms well. After applying the quantum-mechanical harmonic oscillator entropy formalism, we establish a similarly strong linear relationship for X-ray crystallographic B-factors. Finally, we demonstrate that the observed linear relationships remain robust against drastic undersampling and argue that they reflect an intrinsic property of compact proteins. Despite their remarkable strength, however, the above linear relationships yield estimates of configurational entropy change whose accuracy appears to be sufficient for qualitative applications only.

Designing Well-Structured Cyclic Pentapeptides Based on Sequence-Structure Relationships

Cyclic peptides are a promising class of molecules for unique applications. Unfortunately, cyclic peptide design is severely limited by the difficulty in predicting the conformations they will adopt in solution. In this work, we use explicit-solvent molecular dynamics simulations to design well-structured cyclic peptides by studying their sequence-structure relationships. Critical to our approach is an enhanced sampling method that exploits the essential transitional motions of cyclic peptides to efficiently sample their conformational space. We simulated a range of cyclic pentapeptides from all-glycine to a library of cyclo-(X1X2AAA) peptides to map their conformational space and determine cooperative effects of neighboring residues. By combining the results from all cyclo-(X1X2AAA) peptides, we developed a scoring function to predict the structural preferences for X1-X2 residues within cyclic pentapeptides. Using this scoring function, we designed a cyclic pentapeptide, cyclo-(GNSRV), predicted to be well structured in aqueous solution. Subsequent circular dichroism and NMR spectroscopy revealed that this cyclic pentapeptide is indeed well structured in water, with a nuclear Overhauser effect and J-coupling values consistent with the predicted structure.

Mapping the sequence-structure relationships of simple cyclic hexapeptides

Cyclic peptides are promising protein-protein interaction modulators with high binding affinities and specificities, as well as enhanced stabilities and oral availabilities over linear analogs. Despite their relatively small size and cyclic architecture, it is currently difficult to predict the favored conformation(s) of most classes of cyclic peptides. An improved understanding of the sequence-structure relationships for cyclic peptides will offer an avenue for the rational design of cyclic peptides as possible therapeutics. In this work, we systematically explored the sequence-structure relationships for two cyclic hexapeptide systems using molecular dynamics simulation techniques. Starting with an all-glycine cyclic hexapeptide, cyclo-G₆, we systematically replaced glycine residues with alanines and characterized the structural ensembles of different variants. The same process was repeated with valines to investigate the effects of larger side chains. An analysis of the origin of structure preferences was performed using thermodynamics decomposition and several general observations are reported.

Toward accurately modeling N-methylated cyclic peptides

Cyclic peptides have unique properties and can target protein surfaces specifically and potently. N-Methylation provides a promising way to further optimize the pharmacokinetic and structural profiles of cyclic peptides. The capability to accurately model structures adopted by N-methylated cyclic peptides would facilitate rational design of this interesting and useful class of molecules. We apply molecular dynamics simulations with advanced enhanced sampling methods to efficiently characterize the structural ensembles of N-methylated cyclic peptides, while simultaneously evaluating the overall performance of several simulation force fields. We find that one of the residue-specific force fields, RSFF2, is able to recapitulate experimental structures of the N-methylated cyclic peptide benchmarks tested here when the correct amide isomers are used as initial configurations and enforced during the simulations. Thus, using our simulation approach, it is possible to accurately and efficiently predict the structures of N-methylated cyclic peptides if sufficient information is available to determine the correct amide cis/trans configuration. Moreover, our results suggest that, upon further optimization of RSFF2 to more reliably predict cis/trans isomers, molecular dynamics simulations will be able to de novo predict N-methylated cyclic peptides in the near future, strongly motivating such continued optimization.

Entropy Transfer between Residue Pairs and Allostery in Proteins: Quantifying Allosteric Communication in Ubiquitin

It has recently been proposed by Gunasakaran et al. that allostery may be an intrinsic property of all proteins. Here, we develop a computational method that can determine and quantify allosteric activity in any given protein. Based on Schreiber's transfer entropy formulation, our approach leads to an information transfer landscape for the protein that shows the presence of entropy sinks and sources and explains how pairs of residues communicate with each other using entropy transfer. The model can identify the residues that drive the fluctuations of others. We apply the model to Ubiquitin, whose allosteric activity has not been emphasized until recently, and show that there are indeed systematic pathways of entropy and information transfer between residues that correlate well with the activities of the protein. We use 600 nanosecond molecular dynamics trajectories for Ubiquitin and its complex with human polymerase iota and evaluate entropy transfer between all pairs of residues of Ubiquitin and quantify the binding susceptibility changes upon complex formation. We explain the complex formation propensities of Ubiquitin in terms of entropy transfer. Important residues taking part in allosteric communication in Ubiquitin predicted by our approach are in agreement with results of NMR relaxation dispersion experiments. Finally, we show that time delayed correlation of fluctuations of two interacting residues possesses an intrinsic causality that tells which residue controls the interaction and which one is controlled. Our work shows that time delayed correlations, entropy transfer and causality are the required new concepts for explaining allosteric communication in proteins.