Computational Methods to Study the Structure and Dynamics of Biomolecules and Biomolecular Processes

Highlights of the evolutionary development of the physical approach to biology during the last 80 years are traced in this chapter. The historical sequence of events that led to the introduction of modern simulation methods to treat biological processes is described in detail.

The knowledge of the three-dimensional structure of proteins is crucial for understanding many important biological processes. Most biologically important proteins are too large to handle for the classical simulation tools. In such cases, coarse-grained (CG) models nowadays offer various opportunities for efficient conformational sampling and thus prediction of the three-dimensional structure. A variety of CG models have been proposed, each based on a similar framework consisting of a set of conceptual components such as protein representation, force field, sampling, etc. In this chapter we discuss these components, highlighting ideas which have proven to be the most successful. As CG methods are usually part of multistage procedures, we also describe approaches used for the incorporation of homology data and all-atom reconstruction methods.

Simulations of protein dynamics may work on different levels of molecular detail. The levels of simplification (coarse-graining) can range from very low to atomic resolution and may concern different simulation aspects (including protein representation, interaction schemes or models of molecular motion). So-called coarse-grained (CG) models offer many advantages, unreachable by classical simulation tools, as demonstrated in numerous studies of protein dynamics. Followed by a brief introduction, we present example applications of CG models for efficient predictions of biophysical mechanisms. We discuss the following topics: mechanisms of chaperonin action, mechanical properties of proteins, membrane proteins, protein-protein interactions and intrinsically unfolded proteins. Presently, these areas represent emerging application fields of CG simulation models.

Work on a development of a new model of side-chain – side-chain interactions of amino acids, to be used in the UNRES force-field and in other largescale simulations, has been described in this chapter. In the presented model a polar/charged side chain consists of two interaction sites, ie., nonpolar and polar. General expressions for the effective energy of interaction between amino acids are given depending on the kind of interacting pair. The results of the studies on the influence of particle size on the free-energy profile of hydrophobic interactions, and the temperature dependence of the potential of mean force for side chain – side chain interactions are also presented.

Coarse-grained models and force fields have become useful in the studies of the dynamics and physicochemical properties of nucleic acids. Reduced representations of DNA or RNA allow saving computational cost of a few orders of magnitude in comparison with full-atomistic simulations. In this chapter we describe a few coarse-grained models of nucleic acids in which one nucleotide is represented as either one, two, or three beads.We selected the examples of the models designed to investigate the internal dynamics and temperature-dependent denaturation of nucleic acids, as well as created to predict the tertiary structure of RNA or used for large ribonucleoprotein complexes. We describe how the purpose of the model affects the design of the potential energy function and the choice of the simulation method.We also address the limitations of these models.

Electrostatic energy and forces are primary important factors defining macromolecular interactions and its self-organization in an aqueous solution. The unique property of electrostatic forces is its long-range character. Therefore, an accurate modeling of the long-range electrostatic interactions and related energy of macromolecule in an aqueous solvent at given temperature, salt, and hydrogen ion concentration is the long-standing problem. One of the most advanced solutions of macromolecular electrostatics is a single-molecule approach with an implicit solvent electrostatic model for macromolecular simulations in water proton bath is considered here. The fundamental quantity that implicit electrostatic models approximate is the solute potential of mean force, which is obtained by averaging over solvent degrees of freedom. The implicit solvent models suggest practical ways to calculate free energies of macromolecular conformations taking into account equilibrium interactions with water solvent and proton bath, while the explicit solvent approach is unable to do that due to the need to account for a large number of solvent degrees of freedom and long-range nature of the electrostatic interactions. The most advanced realizations of the implicit continuum electrostatic models by different research groups are discussed, their accuracy is examined and some applications of the implicit solvent electrostatic models to macromolecular modeling, such as protein free energy calculations, protein folding, ionization equilibria, and pK_a’s of ionizable groups and constant pH molecular dynamics are highlighted.

In this chapter we review our works on force fields for molecular simulations of protein systems.We first discuss the functional forms of the force fields and present some extensions of the conventional ones.We then present various methods for force-field parameter optimizations. Finally, some examples of our applications of these parameter optimization methods are given and they are compared with the results from the existing force fields.

The use of computer simulations as “virtual microscopes” is limited by sampling difficulties that arise fromthe large dimensionality and the complex energy landscapes of biological systems leading to poor convergences already in folding simulations of single proteins. In this chapter, we discuss a few strategies to enhance sampling in bimolecular simulations, and present some recent applications.

Trajectory fragments algorithms are a set of methods that partition the relevant trajectory space between reactant and products into smaller regions of phase space. Many short trajectories are launched to evaluate transition probabilities between these regions. Each of the methods processes this short trajectory data with different kinetic models and as a result long-time kinetic and thermodynamic information for the overall molecular event can be extracted. This chapter focuses on Milestoning, providing detailed analysis of the approximations involved in the algorithm and its computational implementation. Two other trajectory fragments methods (Partial Path Transition Interface Sampling and Markov State Models) are briefly discussed as well. Finally, two recent applications of trajectory fragments methods are described.

Recent advances in nanotechnology have provided new experimental tools to study biological processes at the molecular level [1]. Instead of monitoring biochemical reactions involving macroscopic numbers of molecules one can now observe behavior of individual molecules by techniques of single molecule optical and force spectroscopies. The optical spectroscopy has been used primarily for identification of stages in protein folding [2, 3, 4, 5]. On the other hand, the force spectroscopy has been usually applied to establish a degree of mechanical stability through stretching either at constant speed or at constant force to induce unfolding. However, monitoring of the subsequent refolding events in a mechanically controlled environment has also been accomplished [9, 10].

Biomembranes are omnipresent in the living world. Each cell is bounded by a cell (plasma) membrane. Also, sub-cellular structures (organelles and nucleus) are enclosed in internal membranes (Fig. 1). Biomembranes are thin lamellar structures of an enormous number of molecules of several chemical types, among them proteins, peptides, phospholipids, glycolipids, sterols, terpenoids.

The membrane proteins are still the “Wild West” of structural biology. Although more and more membrane proteins structures are determined, their functioning is still difficult to investigate because they are fully functional only in the membranous environments. Several specific methodologies were developed to investigate various aspects of their cellular life but still they are challenging for computational methods. In this chapter we summarize the efforts made on elucidation the structural and dynamical properties of different types of membrane proteins emphasizing on those computational methods which were designed and employed particularly to study membrane proteins including their interactions in complex membranous systems.

The ability to aggregate into β-sheet-rich fibrillar structures is a common property shared by many proteins. However, the propensity to aggregate and the precise mechanisms involved vary from protein to protein. Two currently intensely studied proteins are the Alzheimer’s-related amyloid β-peptide (Aβ) and the Parkinson’s-related α-synuclein (αS), both of which are disordered as free monomers and form fibrils. Here, we present studies of Aβ monomers and dimers and monomeric αS, based on an implicit solvent all-atom Monte Carlo (MC) approach. Somewhat unexpectedly, in the αS study, two distinct phases are observed. As a result, in the simulations, disordered αS has to overcome a rather large free-energy barrier in order to acquire a fibril-like fold. No corresponding barrier is observed in the Aβ simulations. Recently, the same computational model was used to study the folding of the Top7 protein, with > 90 residues and a mixed α + β fold. This chapter provides a summary of these Aβ , αS and Top7 studies.

Molecular dynamics simulations, coupled with experimental investigations, could improve our understanding of the protein aggregation and fibrillization process of amyloidogenic proteins. Computational tools are being applied to solve the protein aggregation and fibrillization problem, providing insight into amyloid structures and aggregation mechanisms. Experimental studies of the nature of protein aggregation are unfortunately limited by the structure of aggregates and their insolubility in water. The difficulties have stimulated the development of new experimental methods, and intensive efforts to match computational results with the results of experimental investigations. The number of papers published on simulations of amyloidogenic proteins has increased rapidly during the last decade. The simulation systems covered a range from simple peptides (Alzheimer Aβ peptides or peptides being fragments of amyloidogenic proteins), to large proteins (transthyretin, prion protein, cystatin C, β2-microglobulin etc.). In studies of aggregation, very important is the integration of experimental and computational methods. Computational simulations constitute an “analytical tool” for obtaining and processing biological information and to make useful explanations of the physicochemical principles of amyloidogenesis, as well as to understand the role of amino acid sequences in amyloidogenic proteins. Very efficient theoretical models for prediction of protein aggregation propensities from primary structures have been proposed. At a minimal computational cost, some of these models can determine putative, aggregationprone regions (“hot-spots”) within a protein sequence. The in silico simulations increase our understanding of the protein aggregation process. In this chapter, the molecular studies of amyloidogenic proteins like prion protein, transthyretin, and human cystatin C are presented. The MD studies of these proteins show the first steps during amyloids formation. In addition, in this chapter, the MD studies of protein fibrils are presented. Based on MD simulations of fibril models it is possible to interpret some experimental results and suggest a mechanism of elongation for the fibril protofilament formation.

The directions of the largest thermal fluctuations of the structure of a protein in its native state are the directions of its low-frequency modes (below 1 THZ), named acoustical modes by analogy with the acoustical phonons of a material. The acoustical modes of a protein assist its conformational changes and are related to its biological functions. Low-frequency modes are difficult to detect experimentally. A survey of experimental data of low-frequency modes of proteins is presented. Theoretical approaches, based on normal mode analysis, are of first interest to understand the role of the low-frequency modes in proteins. In this chapter, the fundamentals of normal mode analysis using all-atom models and coarse-grained elastic models are reviewed. They are applied to proteins intimately related to human diseases: ubiquitin and the 70kDa Heat-Shock Protein (Hsp70). The ubiquitin protein is a single domain protein which is a benchmark for biomolecular Nuclear Magnetic Resonance spectroscopy. Present all-atom calculations predict a “boson peak” near 20 cm^− 1 in the inelastic neutron scattering spectra of this protein. The molecular chaperone Hsp70 is an exemplary model to illustrate the different properties of the low-frequency modes of a multi-domain protein, which occurs in two well distinct structural states (open and closed states). The role of the low-frequency modes in the transition between the two states of Hsp70 is analyzed in detail. It is shown that the low-frequency modes provide an easy means of communication between protein domains separated by a large distance.

Intrinsically unstructured/disordered proteins (IUPs/IDPs) exist as highly flexible conformational ensembles without adopting a stable three-dimensional structure. Experimental and bioinformatical studies in the past two decades have shown that these proteins play a central role in various signaling and regulatory processes. Accordingly, their frequency in higher eukaryotes reaches high proportions and their malfunction can be connected to a wide variety of diseases. Recognizing the biological importance of these proteins motivated researchers to understand various aspects of disordered proteins and protein segments from the viewpoint of biochemistry, molecular biology and pharmacology. In general, IDPs are difficult to study experimentally because of the lack of a unique structure in the isolated form. Nevertheless, various bioinformatics tools developed over the last few years enable their identification and characterization using only the amino acid sequence. In this chapter — after a brief introduction to IDPs in general — we present a small survey of current methods aimed at identifying disordered proteins or protein segments, focusing on those that are publicly available as web servers. We also discuss in more detail approaches that predict disordered regions and specific regions involved in protein binding by modeling the physical background of protein disorder. Furthermore, we argue that the heterogeneity of disordered segments needs to be taken into account for a better understanding of protein disorder and the correct use and interpretation of the output of disorder prediction algorithms.

Computing alignments of proteins based on their structure is one of the fundamental tasks of bioinformatics. It is crucial in all kinds of comparative analysis as well as in performing evolutionary and functional classification. Whereas determination of sequence relationships is well founded in statistical models, there is still considerable uncertainty over how to describe geometric relationships between proteins. Continuous growth of structural databases calls for fast and reliable algorithmic methods, enabling one to effectively compute alignments of pairs and larger sets of protein molecules. Although such methodologies have been developed over the past two decades, there exist so-called “difficult similarities” which may include repeats, insertions or deletions, permutations, and conformational changes. A brief overview of existing methodologies with emphasis on different approaches to decomposition of structures into smaller fragments is followed by a presentation of a formalism of local descriptors of protein structures. A formal definition of the problem of computing optimal alignments accommodating aforementioned difficulties is presented along with an analysis of the computational complexity of its important variants. Examples of “difficult similarities” and practical aspects of protein structure comparison are discussed.

This chapter introduces a novel protein folding simulation model which involves several stages. In particular, it distinguishes the so-called early stage (ES) and late state (LS) intermediates, though it can also account for a greater number of intermediates or — alternatively — using ES as the sole intermediate. The early stage intermediate is generated by geometric modeling of polypeptide bond chains, expressed as pairs of their relative binding angles (V-angle) and radii of curvature (R - which is dependent on V). Results of this process point to a limited conformational subspace, providing a convenient set of starting structures for subsequent free internal energy optimization algorithms. The late stage folding model acknowledges the influence of water on the folding process, with hydrophobic residues directed toward the center of the emerging protein body and hydrophilic residues exposed on its surface. Overall, the structure of the protein’s hydrophobic core can be modeled with a 3D Gauss function (hence the “fuzzy oil drop” designation). The presented algorithm reflects the influence of the aqueous environment on the protein’s structure as addition to the optimization of its internal free energy components.

Applying information theory concepts to the process of structural modeling justifies the need for a limited conformational subspace, comprising selected fragments of the Ramachandran map. Comparing the quantity of information present in the initial amino acid sequence with the quantity of information required to accurately describe the resulting 3D structure (pairs of Φ and Ψ angles) reveals a deficit and therefore calls for an additional source of information. We postulate that this information is contributed by the aqueous environment which triggers the generation of a hydrophobic core. In addition, deviations from the idealized “fuzzy oil drop” structure are found to correspond to active sites capable of binding ligands or forming protein complexes. Multicriteria optimization concept is applied to strike a balance between internal and external (environmental) optimization.

Despite the formidable progress in Nuclear Magnetic Resonance (NMR) spectroscopy, quality assessment of NMR-derived structures remains as an important problem. Thus, validation of protein structures is essential for the spectroscopists, since it could enable them to detect structural flaws and potentially guide their efforts in further refinement. Moreover, availability of accurate and efficient validation tools would help molecular biologists and computational chemists to evaluate quality of available experimental structures and to select a protein model which is the most suitable for a given scientific problem. The ¹³C ^α nuclei are ubiquitous in proteins, moreover, their shieldings are easily obtainable from NMR experiments and represent a rich source of encoded structural information that makes ¹³C ^α chemical shifts an attractive candidate for use in computational methods aimed at determination and validation of protein structures. In this chapter, the basis of a novel methodology of computing, at the quantum chemical level of theory, the ¹³C ^α shielding for the amino acid residues in proteins is described. We also identify and examine the main factors affecting the ¹³C ^α -shielding computation. Finally, we illustrate how the information encoded in the ¹³C chemical shifts can be used for a number of applications, viz., from protein structure prediction of both α-helical and β-sheet conformations, to determination of the fraction of the tautomeric forms of the imidazole ring of histidine in proteins as a function of pH or to accurate detection of structural flaws, at a residue-level, in NMR-determined protein models.

In biological processes, the charge distribution is modified moving electrons and positive holes, mostly protons and metal ions, within hydrated macromolecular assemblies. These events are crucial to transfer the energy of chemical bonds into electric currents and ionic gradients, representing, respectively, energy flow and storage in cells. The modeling of the forces behind these processes is challenging, involving different space and time scales, ranging, at least, from confined electrons to macromolecules in the liquid water environment. Thanks to theoretical advances in first-principles computer simulations and to high performance computers, movements of electrons and transferable cations can be combined into robust and detailed dynamical models. This is also of great help in understanding the role of metal cofactors in important biological processes, like photosynthesis and oxidative stress. This chapter summarizes, through simple examples, statistical applications of density-functional theory, one of the most promising modeling techniques available for this level of description. Particular emphasis is devoted to bridge coarse grained models (built at whatever empirical level) with a refined description of the “reactive” portion of the system involving water molecules.

This chapter is focused on recent advances in quantum chemical modeling of active sites in heme proteins and iron porphyrin complexes. After introducing the computational methods (density functional theory and correlated ab initio ones), several case studies are reviewed to show how these methods unravel the electronic structure of heme and heme-related systems; in particular, how they deal with description of: (a) spin state energetics in ferrous and ferric complexes; (b) binding properties of CO, NO, and O₂ ligands to heme; (c) electronic structure of P450 Cpd I and alike systems. Making conclusive calculations for the heme species requires a balanced treatment of electron correlation, which is a great challenge for the present computational methods. Further challenge is situated in a correct translation of the computational results into chemical terms. Achievements of modern ab initio methods on the two fronts are highlighted and disscussed in relation to DFT calculations.

This chapter is focused on applications of quantum chemical (QC) DFT methodology to study reaction mechanisms of metalloenzymes, emphasising new insights that could be obtained thanks to the computations and showing the limitations of the QC approach. Several case studies taken from authors’ research serve to explain and rationalize modelling protocols and to underline information provided by computations, which are not accessible from experiment. Case studies are assorted as to illustrate how the most likely mechanisms may be identified among mechanistic proposals. It is also highlighted how deliberate model constructing and probing various scenarios and/or electronic states help in identifying key factors ruling enzymatic reactions.We hope also to make it definitely clear that the credibility of theoretical modeling still relies heavily on chemical knowledge, intuition as well as on experience of the researcher.