Structural compliance: A new metric for protein flexibility

Proteins are the active players in performing essential molecular activities throughout biology, and their dynamics has been broadly demonstrated to relate to their mechanisms. The intrinsic fluctuations have often been used to represent their dynamics and then compared to the experimental B-factors. However, proteins do not move in a vacuum and their motions are modulated by solvent that can impose forces on the structure. In this paper, we introduce a new structural concept, which has been called the structural compliance, for the evaluation of the global and local deformability of the protein structure in response to intramolecular and solvent forces. Based on the application of pairwise pulling forces to a protein elastic network, this structural quantity has been computed and sometimes is even found to yield an improved correlation with the experimental B-factors, meaning that it may serve as a better metric for protein flexibility. The inverse of structural compliance, namely the structural stiffness, has also been defined, which shows a clear anticorrelation with the experimental data. Although the present applications are made to proteins, this approach can also be applied to other biomolecular structures such as RNA. This present study considers only elastic network models, but the approach could be applied further to conventional atomic molecular dynamics. Compliance is found to have a slightly better agreement with the experimental B-factors, perhaps reflecting its bias toward the effects of local perturbations, in contrast to mean square fluctuations. The code for calculating protein compliance and stiffness is freely accessible at https://jerniganlab.github.io/Software/PACKMAN/Tutorials/compliance .     

Selection of optimal variants of Gō-like models of proteins through studies of stretching

The Gō-like models of proteins are constructed based on the knowledge of the native conformation. However, there are many possible choices of a Hamiltonian for which the ground state coincides with the native state. Here, we propose to use experimental data on protein stretching to determine what choices are most adequate physically. This criterion is motivated by the fact that stretching processes usually start with the native structure, in the vicinity of which the Gō-like models should work the best. Our selection procedure is applied to 62 different versions of the Gō model and is based on 28 proteins. We consider different potentials, contact maps, local stiffness energies, and energy scales—uniform and nonuniform. In the latter case, the strength of the nonuniformity was governed either by specificity or by properties related to positioning of the side groups. Among them is the simplest variant: uniform couplings with no i, i + 2 contacts. This choice also leads to good folding properties in most cases. We elucidate relationship between the local stiffness described by a potential which involves local chirality and the one which involves dihedral and bond angles. The latter stiffness improves folding but there is little difference between them when it comes to stretching.     

Improved Gō-like models demonstrate the robustness of protein folding mechanisms towards non-native interactions

The use of simple theoretical models has provided a considerable contribution to our present understanding of the means by which proteins adopt their native fold from the plethora of available unfolded states. A common assumption in building computationally tractable models has been the neglect of stabilizing non-native interactions in the class of models described as "Gō-like." The focus of this study is the characterization of the folding of a number of proteins via a Gō-like model, which aims to map a maximal amount of information reflecting the protein sequence onto a "minimalist" skeleton. This model is shown to contain sufficient information to reproduce the folding transition states of a number of proteins, including topologically analogous proteins that fold via different transition states. Remarkably, these models also demonstrate consistency with the general features of folding transition states thought to be stabilized by non-native interactions. This suggests that native interactions are the primary determinant of most protein folding transition states, and that non-native interactions lead only to local structural perturbations. A prediction is also included for an asymmetrical folding transition state of bacteriophage lambda protein W, which has yet to be subjected to experimental characterization.     

Temperature‐induced unfolding behavior of proteins studied by tensorial elastic network model

Motivated by single molecule experiments and recent molecular dynamics (MD) studies, we propose a simple and computationally efficient method based on a tensorial elastic network model to investigate the unfolding pathways of proteins under temperature variation. The tensorial elastic network model, which relies on the native state topology of a protein, combines the anisotropic network model, the bond bending elasticity, and the backbone twist elasticity to successfully predicts both the isotropic and anisotropic fluctuations in a manner similar to the Gaussian network model and anisotropic network model. The unfolding process is modeled by breaking the native contacts between residues one by one, and by assuming a threshold value for strain fluctuation. Using this method, we simulated the unfolding processes of four well‐characterized proteins: chymotrypsin inhibitor, barnase, ubiquitein, and adenalyate kinase. We found that this step‐wise process leads to two or more cooperative, first‐order‐like transitions between partial denaturation states. The sequence of unfolding events obtained using this method is consistent with experimental and MD studies. The results also imply that the native topology of proteins “encrypts” information regarding their unfolding process. Proteins 2016; 84:1767–1775. © 2016 Wiley Periodicals, Inc.     

Solvation effects and driving forces for protein thermodynamic and kinetic cooperativity: how adequate is native-centric topological modeling?

What energetic and solvation effects underlie the remarkable two-state thermodynamics and folding/unfolding kinetics of small single-domain proteins? To address this question, we investigate the folding and unfolding of a hierarchy of continuum Langevin dynamics models of chymotrypsin inhibitor 2. We find that residue-based additive Gō-like contact energies, although native-centric, are by themselves insufficient for protein-like calorimetric two-state cooperativity. Further native biases by local conformational preferences are necessary for protein-like thermodynamics. Kinetically, however, even models with both contact and local native-centric energies do not produce simple two-state chevron plots. Thus a model protein's thermodynamic cooperativity is not sufficient for simple two-state kinetics. The models tested appear to have increasing internal friction with increasing native stability, leading to chevron rollovers that typify kinetics that are commonly referred to as non-two-state. The free energy profiles of these models are found to be sensitive to the choice of native contacts and the presumed spatial ranges of the contact interactions. Motivated by explicit-water considerations, we explore recent treatments of solvent granularity that incorporate desolvation free energy barriers into effective implicit-solvent intraprotein interactions. This additional feature reduces both folding and unfolding rates vis-à-vis that of the corresponding models without desolvation barriers, but the kinetics remain non-two-state. Taken together, our observations suggest that interaction mechanisms more intricate than simple Gō-like constructs and pairwise additive solvation-like contributions are needed to rationalize some of the most basic generic protein properties. Therefore, as experimental constraints on protein chain models, requiring a consistent account of protein-like thermodynamic and kinetic cooperativity can be more stringent and productive for some applications than simply requiring a model heteropolymer to fold to a target structure.     

Probing possible downhill folding: native contact topology likely places a significant constraint on the folding cooperativity of proteins with approximately 40 residues

Experiments point to appreciable variations in folding cooperativity among natural proteins with approximately 40 residues, indicating that the behaviors of these proteins are valuable for delineating the contributing factors to cooperative folding. To explore the role of native topology in a protein's propensity to fold cooperatively and how native topology might constrain the degree of cooperativity achievable by a given set of physical interactions, we compared folding/unfolding kinetics simulated using three classes of native-centric C^α chain models with different interaction schemes. The approach was applied to two homologous 45-residue fragments from the peripheral subunit-binding domain family and a 39-residue fragment of the N-terminal domain of ribosomal protein L9. Free-energy profiles as functions of native contact number were computed to assess the heights of thermodynamic barriers to folding. In addition, chevron plots of folding/unfolding rates were constructed as functions of native stability to facilitate comparison with available experimental data. Although common Gō-like models with pairwise Lennard-Jones-type interactions generally fold less cooperatively than real proteins, the rank ordering of cooperativity predicted by these models is consistent with experiment for the proteins investigated, showing increasing folding cooperativity with increasing nonlocality of a protein's native contacts. Models that account for water-expulsion (desolvation) barriers and models with many-body (nonadditive) interactions generally entail higher degrees of folding cooperativity indicated by more linear model chevron plots, but the rank ordering of cooperativity remains unchanged. A robust, experimentally valid rank ordering of model folding cooperativity independent of the multiple native-centric interaction schemes tested here argues that native topology places significant constraints on how cooperatively a protein can fold.     

The Limited Role of Nonnative Contacts in the Folding Pathways of a Lattice Protein

Models of protein energetics that neglect interactions between amino acids that are not adjacent in the native state, such as the Gō model, encode or underlie many influential ideas on protein folding. Implicit in this simplification is a crucial assumption that has never been critically evaluated in a broad context: Detailed mechanisms of protein folding are not biased by nonnative contacts, typically argued to be a consequence of sequence design and/or topology. Here we present, using computer simulations of a well-studied lattice heteropolymer model, the first systematic test of this oft-assumed correspondence over the statistically significant range of hundreds of thousands of amino acid sequences that fold to the same native structure. Contrary to previous conjectures, we find a multiplicity of folding mechanisms, suggesting that Gō-like models cannot be justified by considerations of topology alone. Instead, we find that the crucial factor in discriminating among topological pathways is the heterogeneity of native contact energies: The order in which native contacts accumulate is profoundly insensitive to omission of nonnative interactions, provided that native contact heterogeneity is retained. This robustness holds over a surprisingly wide range of folding rates for our designed sequences. Mirroring predictions based on the principle of minimum frustration, fast-folding sequences match their Gō-like counterparts in both topological mechanism and transit times. Less optimized sequences dwell much longer in the unfolded state and/or off-pathway intermediates than do Gō-like models. For dynamics that bridge unfolded and unfolded states, however, even slow folders exhibit topological mechanisms and transit times nearly identical with those of their Gō-like counterparts. Our results do not imply a direct correspondence between folding trajectories of Gō-like models and those of real proteins, but they do help to clarify key topological and energetic assumptions that are commonly used to justify such caricatures.     

Chevron behavior and isostable enthalpic barriers in protein folding: successes and limitations of simple Gō-like modeling

It has been demonstrated that a "near-Levinthal" cooperative mechanism, whereby the common Gō interaction scheme is augmented by an extra favorability for the native state as a whole, can lead to apparent two-state folding/unfolding kinetics over a broad range of native stabilities in lattice models of proteins. Here such a mechanism is shown to be generalizable to a simplified continuum (off-lattice) Langevin dynamics model with a Calpha protein chain representation, with the resulting chevron plots exhibiting an extended quasilinear regime reminiscent of that of apparent two-state real proteins. Similarly high degrees of cooperativity are possible in Gō-like continuum models with rudimentary pairwise desolvation barriers as well. In these models, cooperativity increases with increasing desolvation barrier height, suggesting strongly that two-state-like folding/unfolding kinetics would be achievable when the pairwise desolvation barrier becomes sufficiently high. Besides cooperativity, another generic folding property of interest that has emerged from published experiments on several apparent two-state proteins is that their folding relaxation under constant native stability (isostability) conditions is essentially Arrhenius, entailing high intrinsic enthalpic folding barriers of approximately 17-30 kcal/mol. Based on a new analysis of published data on barnase, here we propose that a similar property should also apply to a certain class of non-two-state proteins that fold with chevron rollovers. However, several continuum Gō-like constructs considered here fail to predict any significant intrinsic enthalpic folding barrier under isostability conditions; thus the physical origin of such barriers in real proteins remains to be elucidated.     

Toward a molecular understanding of the anisotropic response of proteins to external forces: insights from elastic network models

With recent advances in single-molecule manipulation techniques, it is now possible to measure the mechanical resistance of proteins to external pulling forces applied at specific positions. Remarkably, such recent studies demonstrated that the pulling/stretching forces required to initiate unfolding vary considerably depending on the location of the application of the forces, unraveling residue/position-specific response of proteins to uniaxial tension. Here we show that coarse-grained elastic network models based on the topology of interresidue contacts in the native state can satisfactory explain the relative sizes of such stretching forces exerted on different residue pairs. Despite their simplicity, such models presumably capture a fundamental property that dominates the observed behavior: deformations that can be accommodated by the relatively lower frequency modes of motions intrinsically favored by the structure require weaker forces and vice versa. The mechanical response of proteins to external stress is therefore shown to correlate with the anisotropic fluctuation dynamics intrinsically accessible in the folded state. The dependence on the overall fold implies that evolutionarily related proteins sharing common structural features tend to possess similar mechanical properties. However, the theory cannot explain the differences observed in a number of structurally similar but sequentially distant domains, such as the fibronectin domains.     

Optimization and evaluation of a coarse-grained model of protein motion using x-ray crystal data

Simple coarse-grained models, such as the Gaussian network model, have been shown to capture some of the features of equilibrium protein dynamics. We extend this model by using atomic contacts to define residue interactions and introducing more than one interaction parameter between residues. We use B-factors from 98 ultra-high resolution (相似文献   

The mechanical unfolding of ubiquitin through all-atom Monte Carlo simulation with a Go-type potential

The mechanical unfolding of proteins under a stretching force has an important role in living systems and is a logical extension of the more general protein folding problem. Recent advances in experimental methodology have allowed the stretching of single molecules, thus rendering this process ripe for computational study. We use all-atom Monte Carlo simulation with a Gō-type potential to study the mechanical unfolding pathway of ubiquitin. A detailed, robust, well-defined pathway is found, confirming existing results in this vein though using a different model. Additionally, we identify the protein's fundamental stabilizing secondary structure interactions in the presence of a stretching force and show that this fundamental stabilizing role does not persist in the absence of mechanical stress. The apparent success of simulation methods in studying ubiquitin's mechanical unfolding pathway indicates their potential usefulness for future study of the stretching of other proteins and the relationship between protein structure and the response to mechanical deformation.     

Folding with downhill behavior and low cooperativity of proteins

The downhill folding observed experimentally for a small protein BBL is studied using off-lattice Gō-like model. Our simulations show that the downhill folding has low cooperativity and is barrierless, which is consistent with the experimental findings. As an example of comparison in detail, the two-state folding behavior of proteins, for example, protein CI2, is also simulated. By observing the formation of contacts between the residues for these two proteins, it is found that the physical origin of the downhill folding is due to the deficiency of nonlocal contacts which determine the folding cooperatively. From a statistics on contacts of the native structures of 17 well-studied proteins and the calculation of their cooperativity factors kappa2 based on folding simulations, a strong correlation between the number of nonlocal contacts per residue NN and the factors kappa2 is obtained. Protein BBL with a value of NN = 0.73 has the lowest cooperativity factor kappa2 = 0.34 among all 17 proteins. A crossover around NNc approximately 0.9 could be defined to separate the two-state folders and the downhill folder roughly. A protein would behave downhill folding when its NN = NNc. For proteins with their NN values are about (or slightly larger than) NNc, the folding behaves with low cooperativity and the barriers are small, showing a weak two-state behavior or a downhill-like behavior. Furthermore, simulations on mutants of a two-state folder show that a mutant becomes a downhill folder when its NN is reduced to a value smaller than NNc. These could enable us to identify the downhill folding or the cooperative two-state folding behavior solely from the native structures of proteins.     

基于弹性网络模型的蛋白质变构路径与关键残基识别研究

目的 变构效应在蛋白质生物学功能执行过程中发挥着重要的调控作用，如何基于蛋白质空间结构，有效识别变构信号的传播路径和关键的残基位点是蛋白质结构-功能关系研究领域的热点科学问题。方法 本研究利用基于弹性网络模型（elastic network model，ENM）的力分布计算方法，通过分析蛋白质对外力的响应过程，来识别体系的变构路径以及变构过程中的关键残基。在该方法中，对蛋白质的关键变构位点施加外力，通过对体系形变以及内力分布情况的分析，有效识别与外力承载区域形变相耦合的关键残基，从而得到力信号在蛋白质结构内的传播路径。结果 利用该方法研究了人类磷酸甘油酸激酶（human phosphoglycerate kinase，hPGK）和蛋白质酪氨酸磷酸酶（protein tyrosine phosphatase，PTP）PDZ2结构域的变构调控路径和关键残基。对于hPGK，识别出从底物结合位点到铰链区的两条变构信号传导路径。对于PTP PDZ2，也成功识别出从配体结合位点传递到蛋白质远端的两条长程变构调控路径。计算结果与实验和分子动力学（molecular dynamics，MD）模拟得到的结果一致。结论 本研究为蛋白质体系关键残基识别及变构路径研究提供了有效的分析方法。     

Sequence of events in folding mechanism: beyond the Gō model

Simplified Gō models, where only native contacts interact favorably, have proven useful to characterize some aspects of the folding of small proteins. The success of these models is limited by the fact that all residues interact in the same way so that the folding features of a protein are determined only by the geometry of its native conformation. We present an extended version of a Calpha-based Gō model where different residues interact with different energies. The model is used to calculate the thermodynamics of three small proteins (Protein G, Src-SH3, and CI2) and the effect of mutations (DeltaDeltaGU-N, DeltaDeltaGdouble dagger-N, DeltaDeltaGdouble dagger-U, and phi-values) on the wild-type sequence. The model allows us to investigate some of the most controversial areas in protein folding, such as its earliest stages and the nature of the unfolded state, subjects that have lately received particular attention.     

Critical evaluation of simple network models of protein dynamics and their comparison with crystallographic B-factors

In recent years, elastic network models (ENM) have been widely used to describe low-frequency collective motions in proteins. These models are often validated and calibrated by fitting mean-square atomic displacements estimated from x-ray crystallography (B-factors). We show that a proper calibration procedure must account for the rigid-body motion and constraints imposed by the crystalline environment on the protein. These fundamental aspects of protein dynamics in crystals are often ignored in currently used ENMs, leading to potentially erroneous network parameters. Here we develop an ENM that properly takes the rigid-body motion and crystalline constraints into account. Its application to the crystallographic B-factors reveals that they are dominated by rigid-body motion and thus are poorly suited for the calibration of models for internal protein dynamics. Furthermore, the translation libration screw (TLS) model that treats proteins as rigid bodies is considerably more successful in interpreting the experimental B-factors than ENMs. This conclusion is reached on the basis of a comparative study of various models of protein dynamics. To evaluate their performance, we used a data set of 330 protein structures that combined the sets previously used in the literature to test and validate different models. We further propose an extended TLS model that treats the bulk of the protein as a rigid body while allowing for flexibility of chain ends. This model outperforms other simple models of protein dynamics in interpreting the crystallographic B-factors.     

Models with energy penalty on interresidue rotation address insufficiencies of conventional elastic network models

In this study, I present a new elastic network model, to our knowledge, that addresses insufficiencies of two conventional models—the Gaussian network model (GNM) and the anisotropic network model (ANM). It has been shown previously that the GNM is not rotation-invariant due to its energy, which penalizes rigid-body rotation (external rotation). As a result, GNM models are found contaminated with rigid-body rotation, especially in the most collective ones. A new model (EPIRM) is proposed to remove such external component in modes. The extracted internal motions result from a potential that penalizes interresidue stretching and rotation in a protein. The new model is shown to pertinently describe crystallographic temperature factors (B-factors) and protein open↔closed transitions. Also, the capability of separating internal and external motions in GNM slow modes permits reexamining important mechanochemical properties in enzyme active sites. The results suggest that catalytic residues stay closer to rigid-body rotation axes than their immediate backbone neighbors. I show that the cumulative density of states for EPIRM and ANM follow different power laws as functions of low-mode frequencies. When using a cutoff distance of 7.5 Å, The cumulative density of states of EPIRM scales faster than that of all-atom normal mode analysis and slower than that of simple lattices.     

Topological frustration and the folding of interleukin-1 beta

The cytokine, interleukin-1beta (IL-1beta), adopts a beta-trefoil fold. It is known to be much slower folding than similarly sized proteins, despite having a low contact order. Proteins are sufficiently well designed that their folding is not dominated by local energetic traps. Therefore, protein models that encode only the folded structure and are energetically unfrustrated (Gō-type), can capture the essentials of the folding routes. We investigate the folding thermodynamics of IL-1beta using such a model and molecular dynamics (MD) simulations. We develop an enhanced sampling technique (a modified multicanonical method) to overcome the sampling problem caused by the slow folding. We find that IL-1beta has a broad and high free energy barrier. In addition, the protein fold causes intermediate unfolding and refolding of some native contacts within the protein along the folding trajectory. This "backtracking" occurs around the barrier region. Complex folds like the beta-trefoil fold and functional loops like the beta-bulge of IL-1beta can make some of the configuration space unavailable to the protein and cause topological frustration.     

i‐Patch: Interprotein contact prediction using local network information

Biological processes are commonly controlled by precise protein‐protein interactions. These connections rely on specific amino acids at the binding interfaces. Here we predict the binding residues of such interprotein complexes. We have developed a suite of methods, i‐Patch, which predict the interprotein contact sites by considering the two proteins as a network, with residues as nodes and contacts as edges. i‐Patch starts with two proteins, A and B, which are assumed to interact, but for which the structure of the complex is not available. However, we assume that for each protein, we have a reference structure and a multiple sequence alignment of homologues. i‐Patch then uses the propensities of patches of residues to interact, to predict interprotein contact sites. i‐Patch outperforms several other tested algorithms for prediction of interprotein contact sites. It gives 59% precision with 20% recall on a blind test set of 31 protein pairs. Combining the i‐Patch scores with an existing correlated mutation algorithm, McBASC, using a logistic model gave little improvement. Results from a case study, on bacterial chemotaxis protein complexes, demonstrate that our predictions can identify contact residues, as well as suggesting unknown interfaces in multiprotein complexes. Proteins 2010. © 2010 Wiley‐Liss, Inc.