Coarse-Grained Protein Dynamics Studies Using Elastic Network Models

Modeling Conformational Transitions of Biomolecules from Atomic Force Microscopy Images using Normal Mode Analysis

Observing a single biomolecule performing its function is fundamental in biophysics as it provides important information for elucidating the mechanism. High-speed atomic force microscopy (HS-AFM) is a unique and powerful technique that allows the observation of biomolecular motion in a near-native environment. However, the spatial resolution of HS-AFM is limited by the physical size of the cantilever tip, which restricts the ability to obtain atomic details of molecules. In this study, we propose a novel computational algorithm designed to derive atomistic models of conformational dynamics from AFM images. Our method uses normal-mode analysis to describe the expected motions of the molecule, allowing these motions to be represented with a limited number of coordinates. This approach mitigates the problem of overinterpretation inherent in the analysis of AFM images with limited resolution. We demonstrate the effectiveness of our algorithm, NMFF-AFM, using synthetic data sets for three proteins that undergo significant conformational changes. NMFF-AFM is a fast and user-friendly program that requires minimal setup and has the potential to be a valuable tool for biophysical studies using HS-AFM.

Revolutionizing Molecular Design for Innovative Therapeutic Applications through Artificial Intelligence

The field of computational protein engineering has been transformed by recent advancements in machine learning, artificial intelligence, and molecular modeling, enabling the design of proteins with unprecedented precision and functionality. Computational methods now play a crucial role in enhancing the stability, activity, and specificity of proteins for diverse applications in biotechnology and medicine. Techniques such as deep learning, reinforcement learning, and transfer learning have dramatically improved protein structure prediction, optimization of binding affinities, and enzyme design. These innovations have streamlined the process of protein engineering by allowing the rapid generation of targeted libraries, reducing experimental sampling, and enabling the rational design of proteins with tailored properties. Furthermore, the integration of computational approaches with high-throughput experimental techniques has facilitated the development of multifunctional proteins and novel therapeutics. However, challenges remain in bridging the gap between computational predictions and experimental validation and in addressing ethical concerns related to AI-driven protein design. This review provides a comprehensive overview of the current state and future directions of computational methods in protein engineering, emphasizing their transformative potential in creating next-generation biologics and advancing synthetic biology.

Reused Protein Segments Linked to Functional Dynamics

Protein space is characterized by extensive recurrence, or "reuse," of parts, suggesting that new proteins and domains can evolve by mixing-and-matching of existing segments. From an evolutionary perspective, for a given combination to persist, the protein segments should presumably not only match geometrically but also dynamically communicate with each other to allow concerted motions that are key to function. Evidence from protein space supports the premise that domains indeed combine in this manner; we explore whether a similar phenomenon can be observed at the sub-domain level. To this end, we use Gaussian Network Models (GNMs) to calculate the so-called soft modes, or low-frequency modes of motion for a dataset of 150 protein domains. Modes of motion can be used to decompose a domain into segments of consecutive amino acids that we call "dynamic elements", each of which belongs to one of two parts that move in opposite senses. We find that, in many cases, the dynamic elements, detected based on GNM analysis, correspond to established "themes": Sub-domain-level segments that have been shown to recur in protein space, and which were detected in previous research using sequence similarity alone (i.e. completely independently of the GNM analysis). This statistically significant correlation hints at the importance of dynamics in evolution. Overall, the results are consistent with an evolutionary scenario where proteins have emerged from themes that need to match each other both geometrically and dynamically, e.g. to facilitate allosteric regulation.

Methods for Classical-Mechanical Molecular Simulation in Chemistry: Achievements, Limitations, Perspectives

More than a half century ago it became feasible to simulate, using classical-mechanical equations of motion, the dynamics of molecular systems on a computer. Since then classical-physical molecular simulation has become an integral part of chemical research. It is widely applied in a variety of branches of chemistry and has significantly contributed to the development of chemical knowledge. It offers understanding and interpretation of experimental results, semiquantitative predictions for measurable and nonmeasurable properties of substances, and allows the calculation of properties of molecular systems under conditions that are experimentally inaccessible. Yet, molecular simulation is built on a number of assumptions, approximations, and simplifications which limit its range of applicability and its accuracy. These concern the potential-energy function used, adequate sampling of the vast statistical-mechanical configurational space of a molecular system and the methods used to compute particular properties of chemical systems from statistical-mechanical ensembles. During the past half century various methodological ideas to improve the efficiency and accuracy of classical-physical molecular simulation have been proposed, investigated, evaluated, implemented in general simulation software or were abandoned. The latter because of fundamental flaws or, while being physically sound, computational inefficiency. Some of these methodological ideas are briefly reviewed and the most effective methods are highlighted. Limitations of classical-physical simulation are discussed and perspectives are sketched.

Flat-Bottom Elastic Network Model for Generating Improved Plausible Reaction Paths

Rapid generation of a plausible reaction path connecting a given reactant and product in advance is crucial for the efficient computation of precise reaction paths or transition states. We propose a computationally efficient potential energy based on the molecular structure to generate such paths. This potential energy has a flat bottom consisting of structures without atomic collisions while preserving nonreactive chemical bonds, bond angles, and partial planar structures. By combining this potential energy with the direct MaxFlux method, a recently developed reaction-path/transition-state search method, we can find the shortest plausible path passing within the bottom. Numerical results show that this combination yields lower energy paths compared to the paths obtained by the well-known image-dependent pair potential. We also theoretically investigate the differences between these two potential energies. The proposed potential energy and path generation routine are implemented in our Python version of the direct MaxFlux method, available on GitHub.

Pragmatic Coarse-Graining of Proteins: Models and Applications

The molecular details involved in the folding, dynamics, organization, and interaction of proteins with other molecules are often difficult to assess by experimental techniques. Consequently, computational models play an ever-increasing role in the field. However, biological processes involving large-scale protein assemblies or long time scale dynamics are still computationally expensive to study in atomistic detail. For these applications, employing coarse-grained (CG) modeling approaches has become a key strategy. In this Review, we provide an overview of what we call pragmatic CG protein models, which are strategies combining, at least in part, a physics-based implementation and a top-down experimental approach to their parametrization. In particular, we focus on CG models in which most protein residues are represented by at least two beads, allowing these models to retain some degree of chemical specificity. A description of the main modern pragmatic protein CG models is provided, including a review of the most recent applications and an outlook on future perspectives in the field.

Statistical Learning of Protein Elastic Network from Positional Covariance Matrix

Positional fluctuation and covariance during protein dynamics are key observables for understanding the molecular origin of biological functions. A frequently employed potential energy function for describing protein structural variation at the coarse-gained level is elastic network model (ENM). A long-standing issue in biomolecular simulation is thus the parametrization of ENM spring constants from the components of positional covariance matrix (PCM). Based on sensitivity analysis of PCM, the direct-coupling statistics of each spring, which is a specific combination of position fluctuation and covariance, is found to exhibit prominent signal of parameter dependence. This finding provides the basis for devising the objective function and the scheme of running through the effective one-dimensional optimization of every spring by self-consistent iteration. Formal derivation of the positional covariance statistical learning (PCSL) method also motivates the necessary data regularization for stable calculations. Robust convergence of PCSL is achieved in taking an all-atom molecular dynamics trajectory or an ensemble of homologous structures as input data. The PCSL framework can also be generalized with mixed objective functions to capture specific property such as the residue flexibility profile. Such physical chemistry-based statistical learning thus provides a useful platform for integrating the mechanical information encoded in various experimental or computational data.

The "violin model": Looking at community networks for dynamic allostery

Allosteric regulation of proteins continues to be an engaging research topic for the scientific community. Models describing allosteric communication have evolved from focusing on conformation-based descriptors of protein structural changes to appreciating the role of internal protein dynamics as a mediator of allostery. Here, we explain a "violin model" for allostery as a contemporary method for approaching the Cooper-Dryden model based on redistribution of protein thermal fluctuations. Based on graph theory, the violin model makes use of community network analysis to functionally cluster correlated protein motions obtained from molecular dynamics simulations. This Review provides the theory and workflow of the methodology and explains the application of violin model to unravel the workings of protein kinase A.

Effect of the Neck Linker on Processive Stepping of Kinesin Motor

Kinesin motor protein, which is composed of two catalytic domains connected together by a long coiled-coil stalk via two flexible neck linkers (NLs), can step processively on a microtubule towards the plus end by hydrolyzing adenosine triphosphate (ATP) molecules. To understand what the role is that the NL plays in the processive stepping, the dynamics of the kinesin motor are studied theoretically here by considering the mutation or deletion of an N-terminal cover strand that contributes to the docking of the NL in kinesin-1, the extension of the NL in kinesin-1, the mutation of the NL in kinesin-1, the swapping of the NL of kinesin-2 with that of kinesin-1, the joining of the stalk and neck of Ncd that moves towards the minus end of MT to the catalytic domain of kinesin-1, the replacement of catalytic domain of kinesin-1 with that of Ncd, and so on. The theoretical results give a consistent and quantitative explanation of various available experimental results about the effects of these mutations on motor dynamics and, moreover, provide predicted results. Additionally, the processive motility of kinesin-6 MKLP2 without NL docking is also explained. The available experimental data about the effect of NL mutations on the dynamics of the bi-directional kinesin-5 Cin8 are also explained. The studies are critically implicative to the mechanism of the stepping of the kinesin motor.

Protein Function Analysis through Machine Learning

Machine learning (ML) has been an important arsenal in computational biology used to elucidate protein function for decades. With the recent burgeoning of novel ML methods and applications, new ML approaches have been incorporated into many areas of computational biology dealing with protein function. We examine how ML has been integrated into a wide range of computational models to improve prediction accuracy and gain a better understanding of protein function. The applications discussed are protein structure prediction, protein engineering using sequence modifications to achieve stability and druggability characteristics, molecular docking in terms of protein-ligand binding, including allosteric effects, protein-protein interactions and protein-centric drug discovery. To quantify the mechanisms underlying protein function, a holistic approach that takes structure, flexibility, stability, and dynamics into account is required, as these aspects become inseparable through their interdependence. Another key component of protein function is conformational dynamics, which often manifest as protein kinetics. Computational methods that use ML to generate representative conformational ensembles and quantify differences in conformational ensembles important for function are included in this review. Future opportunities are highlighted for each of these topics.

Optimising Elastic Network Models for Protein Dynamics and Allostery: Spatial and Modal Cut-offs and Backbone Stiffness

The family of coarse-grained models for protein dynamics known as Elastic Network Models (ENMs) require careful choice of parameters to represent well experimental measurements or fully-atomistic simulations. The most basic ENM that represents each protein residue by a node at the position of its C-alpha atom, all connected by springs of equal stiffness, up to a cut-off in distance. Even at this level a choice is required of the optimum cut-off distance and the upper limit of elastic normal modes taken in any sum for physical properties, such as dynamic correlation or allosteric effects on binding. Additionally, backbone-enhanced ENM (BENM) may improve the model by allocating a higher stiffness to springs that connect along the protein backbone. This work reports on the effect of varying these three parameters (distance and mode cutoffs, backbone stiffness) on the dynamical structure of three proteins, Catabolite Activator Protein (CAP), Glutathione S-transferase (GST), and the SARS-CoV-2 Main Protease (M pro ). Our main results are: (1) balancing B-factor and dispersion-relation predictions, a near-universal optimal value of 8.5 Å is advisable for ENMs; (2) inhomogeneity in elasticity brings the first mode containing spatial structure not well-resolved by the ENM typically within the first 20; (3) the BENM only affects modes in the upper third of the distribution, and, additionally to the ENM, is only able to model the dispersion curve better in this vicinity; (4) BENM does not typically affect fluctuation-allostery, which also requires careful treatment of the effector binding to the host protein to capture.

Study of Allosteric Transitions of Human P-Glycoprotein by Using the Two-State Anisotropic Network Model

Human P-glycoprotein (P-gp) is a kind of ATP-binding cassette (ABC) transporters. Once human P-gp is overexpressed in tumor cells, which can lead to tumor multidrug resistance (MDR). However, the present experimental methods are difficult to obtain the large-scale conformational transition process of human P-gp. In this work, we explored the allosteric pathway of human P-gp from the inward-facing (IF) to the outward-facing (OF) state in the substrate transport process with the two-state anisotropic network model (tANM). These results suggest that the allosteric transitions proceed in a coupled way. The conformational changes of nucleotide-binding domains (NBDs) finally make the transmembrane domains (TMDs) to the OF state via the role of the allosteric propagation of the intracellular helices IH1 and IH2. Additionally, this allosteric pathway is advantageous in energy compared with other methods. This study reveals the conformational transition of P-gp, which contributes to an understanding of the allosteric mechanism of ABC exporters.

A Comparative Evaluation of the Structural and Dynamic Properties of Insect Odorant Binding Proteins

Insects devote a major part of their metabolic resources to the production of odorant binding proteins (OBPs). Although initially, these proteins were implicated in the solubilisation, binding and transport of semiochemicals to olfactory receptors, it is now recognised that they may play diverse, as yet uncharacterised, roles in insect physiology. The structures of these OBPs, the majority of which are known as “classical” OBPs, have shed some light on their potential functional roles. However, the dynamic properties of these proteins have received little attention despite their functional importance. Structural dynamics are encoded in the native protein fold and enable the adaptation of proteins to substrate binding. This paper provides a comparative review of the structural and dynamic properties of OBPs, making use of sequence/structure analysis, statistical and theoretical physics-based methods. It provides a new layer of information and additional methodological tools useful in unravelling the relationship between structure, dynamics and function of insect OBPs. The dynamic properties of OBPs, studied by means of elastic network models, reflect the similarities/dissimilarities observed in their respective structures and provides insights regarding protein motions that may have important implications for ligand recognition and binding. Furthermore, it was shown that the OBPs studied in this paper share conserved structural ‘core’ that may be of evolutionary and functional importance.

In silico investigation and surmounting of Lipopolysaccharide barrier in Gram-Negative Bacteria: How far has molecular dynamics Come?

Lipopolysaccharide (LPS), a main component of the outer membrane of Gram-negative bacteria, has crucial implications on both antibiotic resistance and the overstimulation of the host innate immune system. Fighting against these global concerns calls for the molecular understanding of the barrier function and immunostimulatory ability of LPS. Molecular dynamics (MD) simulations have become an invaluable tool for uncovering important findings in LPS research. While the reach of MD simulations for investigating the immunostimulatory ability of LPS has been already outlined, little attention has been paid to the role of MD simulations for exploring its barrier function and synthesis. Herein, we give an overview about the impact of MD simulations on gaining insight into the shield role and synthesis pathway of LPS, which have attracted considerable attention to discover molecules able to surmount antibiotic resistance, either circumventing LPS defenses or disrupting its synthesis. We specifically focus on the enhanced sampling and free energy calculation methods that have been combined with MD simulations to address such research. We also highlight the use of special-purpose MD supercomputers, the importance of appropriate LPS and ions parameterization to obtain reliable results, and the complementary views that MD and wet-lab experiments provide. Thereby, this work, which covers the last five years of research, apart from outlining the phenomena and strategies that are being explored, evidences the valuable insights that are gained by MD, which may be useful to advance antibiotic design, and what the prospects of this in silico method could be in LPS research.

Low-Frequency Harmonic Perturbations Drive Protein Conformational Changes

Protein dynamics has been investigated since almost half a century, as it is believed to constitute the fundamental connection between structure and function. Elastic network models (ENMs) have been widely used to predict protein dynamics, flexibility and the biological mechanism, from which remarkable results have been found regarding the prediction of protein conformational changes. Starting from the knowledge of the reference structure only, these conformational changes have been usually predicted either by looking at the individual mode shapes of vibrations (i.e., by considering the free vibrations of the ENM) or by applying static perturbations to the protein network (i.e., by considering a linear response theory). In this paper, we put together the two previous approaches and evaluate the complete protein response under the application of dynamic perturbations. Harmonic forces with random directions are applied to the protein ENM, which are meant to simulate the single frequency-dependent components of the collisions of the surrounding particles, and the protein response is computed by solving the dynamic equations in the underdamped regime, where mass, viscous damping and elastic stiffness contributions are explicitly taken into account. The obtained motion is investigated both in the coordinate space and in the sub-space of principal components (PCs). The results show that the application of perturbations in the low-frequency range is able to drive the protein conformational change, leading to remarkably high values of direction similarity. Eventually, this suggests that protein conformational change might be triggered by external collisions and favored by the inherent low-frequency dynamics of the protein structure.

Parameterizing elastic network models to capture the dynamics of proteins

Coarse-grained normal mode analyses of protein dynamics rely on the idea that the geometry of a protein structure contains enough information for computing its fluctuations around its equilibrium conformation. This geometry is captured in the form of an elastic network (EN), namely a network of edges between its residues. The normal modes of a protein are then identified with the normal modes of its EN. Different approaches have been proposed to construct ENs, focusing on the choice of the edges that they are comprised of, and on their parameterizations by the force constants associated with those edges. Here we propose new tools to guide choices on these two facets of EN. We study first different geometric models for ENs. We compare cutoff-based ENs, whose edges have lengths that are smaller than a cutoff distance, with Delaunay-based ENs and find that the latter provide better representations of the geometry of protein structures. We then derive an analytical method for the parameterization of the EN such that its dynamics leads to atomic fluctuations that agree with experimental B-factors. To limit overfitting, we attach a parameter referred to as flexibility constant to each atom instead of to each edge in the EN. The parameterization is expressed as a non-linear optimization problem whose parameters describe both rigid-body and internal motions. We show that this parameterization leads to improved ENs, whose dynamics mimic MD simulations better than ENs with uniform force constants, and reduces the number of normal modes needed to reproduce functional conformational changes.

Dynamics-Evolution Correspondence in Protein Structures

The genotype-phenotype mapping of proteins is a fundamental question in structural biology. In this Letter, with the analysis of a large dataset of proteins from hundreds of protein families, we quantitatively demonstrate the correlations between the noise-induced protein dynamics and mutation-induced variations of native structures, indicating the dynamics-evolution correspondence of proteins. Based on the investigations of the linear responses of native proteins, the origin of such a correspondence is elucidated. It is essential that the noise- and mutation-induced deformations of the proteins are restricted on a common low-dimensional subspace, as confirmed from the data. These results suggest an evolutionary mechanism of the proteins gaining both dynamical flexibility and evolutionary structural variability.

From System Modeling to System Analysis: The Impact of Resolution Level and Resolution Distribution in the Computer-Aided Investigation of Biomolecules

The ever increasing computer power, together with the improved accuracy of atomistic force fields, enables researchers to investigate biological systems at the molecular level with remarkable detail. However, the relevant length and time scales of many processes of interest are still hardly within reach even for state-of-the-art hardware, thus leaving important questions often unanswered. The computer-aided investigation of many biological physics problems thus largely benefits from the usage of coarse-grained models, that is, simplified representations of a molecule at a level of resolution that is lower than atomistic. A plethora of coarse-grained models have been developed, which differ most notably in their granularity; this latter aspect determines one of the crucial open issues in the field, i.e. the identification of an optimal degree of coarsening, which enables the greatest simplification at the expenses of the smallest information loss. In this review, we present the problem of coarse-grained modeling in biophysics from the viewpoint of system representation and information content. In particular, we discuss two distinct yet complementary aspects of protein modeling: on the one hand, the relationship between the resolution of a model and its capacity of accurately reproducing the properties of interest; on the other hand, the possibility of employing a lower resolution description of a detailed model to extract simple, useful, and intelligible information from the latter.

"Dividing and Conquering" and "Caching" in Molecular Modeling

Molecular modeling is widely utilized in subjects including but not limited to physics, chemistry, biology, materials science and engineering. Impressive progress has been made in development of theories, algorithms and software packages. To divide and conquer, and to cache intermediate results have been long standing principles in development of algorithms. Not surprisingly, most important methodological advancements in more than half century of molecular modeling are various implementations of these two fundamental principles. In the mainstream classical computational molecular science, tremendous efforts have been invested on two lines of algorithm development. The first is coarse graining, which is to represent multiple basic particles in higher resolution modeling as a single larger and softer particle in lower resolution counterpart, with resulting force fields of partial transferability at the expense of some information loss. The second is enhanced sampling, which realizes "dividing and conquering" and/or "caching" in configurational space with focus either on reaction coordinates and collective variables as in metadynamics and related algorithms, or on the transition matrix and state discretization as in Markov state models. For this line of algorithms, spatial resolution is maintained but results are not transferable. Deep learning has been utilized to realize more efficient and accurate ways of "dividing and conquering" and "caching" along these two lines of algorithmic research. We proposed and demonstrated the local free energy landscape approach, a new framework for classical computational molecular science. This framework is based on a third class of algorithm that facilitates molecular modeling through partially transferable in resolution "caching" of distributions for local clusters of molecular degrees of freedom. Differences, connections and potential interactions among these three algorithmic directions are discussed, with the hope to stimulate development of more elegant, efficient and reliable formulations and algorithms for "dividing and conquering" and "caching" in complex molecular systems.

Coarse-grained elastic network modelling: A fast and stable numerical tool to characterize mesenchymal stem cells subjected to AFM nanoindentation measurements

The knowledge of the mechanical properties is the starting point to study the mechanobiology of mesenchymal stem cells and to understand the relationships linking biophysical stimuli to the cellular differentiation process. In experimental biology, Atomic Force Microscopy (AFM) is a common technique for measuring these mechanical properties. In this paper we present an alternative approach for extracting common mechanical parameters, such as the Young's modulus of cell components, starting from AFM nanoindentation measurements conducted on human mesenchymal stem cells. In a virtual environment, a geometrical model of a stem cell was converted in a highly deformable Coarse-Grained Elastic Network Model (CG-ENM) to reproduce the real AFM experiment and retrieve the related force-indentation curve. An ad-hoc optimization algorithm perturbed the local stiffness values of the springs, subdivided in several functional regions, until the computed force-indentation curve replicated the experimental one. After this curve matching, the extraction of global Young's moduli was performed for different stem cell samples. The algorithm was capable to distinguish the material properties of different subcellular components such as the cell cortex and the cytoskeleton. The numerical results predicted with the elastic network model were then compared to those obtained from hertzian contact theory and Finite Element Method (FEM) for the same case studies, showing an optimal agreement and a highly reduced computational cost. The proposed simulation flow seems to be an accurate, fast and stable method for understanding the mechanical behavior of soft biological materials, even for subcellular levels of detail. Moreover, the elastic network modelling allows shortening the computational times to approximately 33% of the time required by a traditional FEM simulation performed using elements with size comparable to that of springs.

Guardians of the Cell: State-of-the-Art of Membrane Proteins from a Computational Point-of-View

Membrane proteins (MPs) encompass a large family of proteins with distinct cellular functions, and although representing over 50% of existing pharmaceutical drug targets, their structural and functional information is still very scarce. Over the last years, in silico analysis and algorithm development were essential to characterize MPs and overcome some limitations of experimental approaches. The optimization and improvement of these methods remain an ongoing process, with key advances in MPs' structure, folding, and interface prediction being continuously tackled. Herein, we discuss the latest trends in computational methods toward a deeper understanding of the atomistic and mechanistic details of MPs.

Allosteric communication in molecular machines via information exchange: what can be learned from dynamical modeling

Allosteric regulation is crucial for the operation of protein machines and molecular motors. A major challenge is to characterize and quantify the information exchange underlying allosteric communication between remote functional sites in a protein, and to identify the involved relevant pathways. We review applications of two topical approaches of dynamical protein modeling, a kinetic-based single-molecule stochastic model, which employs information thermodynamics to quantify allosteric interactions, and structure-based coarse-grained modeling to characterize intra-molecular couplings in terms of conformational motions and propagating mechanical strain. Both descriptions resolve the directionality of allosteric responses within a protein, emphasizing the concept of causality as the principal hallmark of protein allostery. We discuss the application of techniques from information thermodynamics to dynamic protein elastic networks and evolutionary designed model structures, and the ramifications for protein allostery.

Intrinsic dynamics is evolutionarily optimized to enable allosteric behavior

Allosteric behavior is central to the function of many proteins, enabling molecular machinery, metabolism, signaling and regulation. Recent years have shown that the intrinsic dynamics of allosteric proteins defined by their 3-dimensional architecture or by the topology of inter-residue contacts favors cooperative motions that bear close similarity to structural changes they undergo during their allosteric actions. These conformational motions are usually driven by energetically favorable or soft modes at the low frequency end of the mode spectrum, and they are evolutionarily conserved among orthologs. These observations brought into light evolutionary adaptation mechanisms that help maintain, optimize or regulate allosteric behavior as the evolution from bacterial to higher organisms introduces sequential heterogeneities and structural complexities.

Can Conformational Changes of Proteins Be Represented in Torsion Angle Space? A Study with Rescaled Ridge Regression

Torsion angles are the natural degrees of freedom of protein structures. The ability to determine torsional variations corresponding to observed changes in Cartesian coordinates is highly valuable, notably to investigate the mechanisms of functional conformational changes or to develop computational models of protein dynamics. This issue is far from trivial in practice since the impact of modifying one torsion angle strongly depends on all other angles, and the compounding effects of small variations in bond lengths and valence angles can completely disrupt a protein fold. We demonstrate that naive strategies, such as directly comparing torsion angles between structures without correcting for variations in bond lengths and valence angles or fitting torsional variations without a proper regularization scheme, fail at producing an adequate representation of conformational changes in internal coordinates. In contrast, rescaled ridge regression, a method recently introduced to regularize multidimensional regressions with correlated explanatory variables, is shown to consistently identify a minimal set of torsion angles variations that closely reproduce changes in Cartesian coordinates. This torsional representation of conformational changes is shown to be robust to the choice of experimental structures. It also provides a better agreement with theoretical models of protein dynamics than the Cartesian representation, regarding notably the predominance of low-frequency normal modes in functional motions and the presence, in predicted equilibrium dynamics, of hints of natural selection for specific functional motions. The software is available at https://github.com/ugobas/tnm .

Analyzing Fluctuation Properties in Protein Elastic Networks with Sequence-Specific and Distance-Dependent Interactions

Simple protein elastic networks which neglect amino-acid information often yield reasonable predictions of conformational dynamics and are broadly used. Recently, model variants which incorporate sequence-specific and distance-dependent interactions of residue pairs have been constructed and demonstrated to improve agreement with experimental data. We have applied the new variants in a systematic study of protein fluctuation properties and compared their predictions with those of conventional anisotropic network models. We find that the quality of predictions is frequently linked to poor estimations in highly flexible protein regions. An analysis of a large set of protein structures shows that fluctuations of very weakly connected network residues are intrinsically prone to be significantly overestimated by all models. This problem persists in the new models and is not resolved by taking into account sequence information. The effect becomes even enhanced in the model variant which takes into account very soft long-ranged residue interactions. Beyond these shortcomings, we find that model predictions are largely insensitive to the integration of chemical information, at least regarding the fluctuation properties of individual residues. One can furthermore conclude that the inherent drawbacks may present a serious hindrance when improvement of elastic network models are attempted.