The chemical and biological sciences face unprecedented opportunities in the 21st century. A confluence of factors from parallel universes – advances in experimental techniques in biomolecular structure determination, progress in theoretical modeling and simulation for large biological systems, and breakthroughs in computer technology – has opened new avenues of opportunity as never before. Now, experimental data can be interpreted and further analysed by modeling, and predictions from any approach can be tested and advanced through companion methodologies and technologies. This two volume set describes innovations in biomolecular modeling and simulation, in both the algorithmic and application fronts. With contributions from experts in the field, the books describe progress and innovation in areas including: simulation algorithms for dynamics and enhanced configurational sampling, force field development, implicit solvation models, coarse-grained models, quantum-mechanical simulations, protein folding, DNA polymerase mechanisms, nucleic acid complexes and simulations, RNA structure analysis and design and other important topics in structural biology modeling. The books are aimed at graduate students and experts in structural biology and chemistry and the emphasis is on reporting innovative new approaches rather than providing comprehensive reviews on each subject.

Innovations in Biomolecular Modeling and Simulations Volume 2

By Tamar Schlick

The Royal Society of Chemistry

Copyright © 2012 Royal Society of Chemistry
All rights reserved.
ISBN: 978-1-84973-462-2

Contents

Volume 1,
Beginnings,
Chapter 1 Personal Perspective Harold A. Scheraga, 3,
Chapter 2 Fashioning NAMD, a History of Risk and Reward: Klaus Schulten Reminisces Lisa Pollack, 8,
Force Fields and Electrostatics,
Chapter 3 Towards Biomolecular Simulations with Explicit Inclusion of Polarizability: Development of a CHARMM Polarizable Force Field based on the Classical Drude Oscillator Model C. M. Baker, E. Darian and A. D. MacKerell Jr, 23,
Chapter 4 Integral Equation Theory of Biomolecules and Electrolytes Tyler Luchko, In Suk Joung and David A. Case, 51,
Chapter 5 Molecular Simulation in the Energy Biosciences Xiaolin Cheng, Jerry M. Parks, Loukas Petridis, Benjamin Lindner, Roland Schulz, Hao-Bo Guo, Goundla Srinivas and Jeremy C. Smith, 87,
Sampling and Rates,
Chapter 6 Enhancing the Capacity of Molecular Dynamics Simulations with Trajectory Fragments Alfredo E. Cardenas and Ron Elber, 117,
Chapter 7 Computing Reaction Rates in Bio-molecular Systems Using Discrete Macro-states Eric Darve and Ernest Ryu, 138,
Chapter 8 Challenges in Applying Monte Carlo Sampling to Biomolecular Systems M. Mezei, 207,
Coarse Graining and Multiscale Models,
Chapter 9 Coarse-grain Protein Models N. Ceres and R. Lavery, 219,
Chapter 10 Generalised Multi-level Coarse-grained Molecular Simulation and its Application to Myosin-V Movement William R. Taylor and Zoe Katsimitsoulia, 249,
Chapter 11 Top-down Mesoscale Models and Free Energy Calculations of Multivalent Protein-Protein and Protein-Membrane Interactions in Nanocarrier Adhesion and Receptor Trafficking Jin Liu, Neeraj J. Agrawal, David M. Eckmann, Portonovo S. Ayyaswamy and Ravi Radhakrishnan, 272,
Chapter 12 Studying Proteins and Peptides at Material Surfaces Jun Feng, Gillian C. Lynch and B. Montgomery Pettitt, 293,
Chapter 13 Multiscale Design: From Theory to Practice J. Fish, V. Filonova and Z. Yuan, 321,
Subject Index, 345,
Volume 2,
Atomistic Simulations of Nucleic Acids and Nucleic Acid Complexes,
Chapter 1 Modelling Nucleic Acid Structure and Flexibility: From Atomic to Mesoscopic Scale Filip Lankas, 3,
Chapter 2 Molecular Dynamics and Force Field Based Methods for Studying Quadruplex Nucleic Acids Shozeb M Haider and Stephen Neidle, 33,
Chapter 3 Opposites Attract: Shape and Electrostatic Complementarity in Protein-DNA Complexes Robert C. Harris, Travis Mackoy, Ana Carolina Dantas Machado, Darui Xu, Remo Rohs and Marcia Oliveira Fenley, 53,
Chapter 4 Intrinsic Motions of DNA Polymerases Underlie Their Remarkable Specificity and Selectivity and Suggest a Hybrid Substrate Binding Mechanism Meredith C. Foley, Karunesh Arora and Tamar Schlick, 81,
Chapter 5 Molecular Dynamics Structure Prediction of a Novel Protein–DNA Complex: Two HU Proteins with a DNA Four-way Junction Elizabeth G. Wheatley, Susan N. Pieniazek, Iulia Vitoc, Ishita Mukerji and D.L. Beveridge, 111,
Chapter 6 Molecular Dynamics Simulations of RNA Molecules J. Šponer, M. Otyepka, P. Banáš, K. Réblová and N. G. Walter, 129,
Chapter 7 The Structure and Folding of Helical Junctions in RNA David M. J. Lilley, 156,
DNA Folding, Knotting, Sliding and Hopping,
Chapter 8 Structure and Dynamics of Supercoiled DNA Knots and Catenanes Guillaume Witz and Andrzej Stasiak, 179,
Chapter 9 Monte Carlo Simulations of Nucleosome Chains to Identify Factors that Control DNA Compaction and Access Karsten Rippe, Rene Stehr and Gero Wedemann, 198,
Chapter 10 Sliding Dynamics Along DNA: A Molecular Perspective Amir Marcovitz and Yaakov Levy, 236,
Drug Design,
Chapter 11 Structure-based Design Technology CONTOUR and its Application to Drug Discovery Zhijie Liu, Peter Lindblom, David A. Claremon and Suresh B. Singh, 265,
Chapter 12 Molecular Simulation in Computer-aided Drug Design: Algorithms and Applications Robert V. Swift and Rommie E. Amaro, 281,
Chapter 13 Computer-aided Drug Discovery: Two Antiviral Drugs for HIV/AIDS J. Andrew McCammon, 316,
Subject Index, 320,

CHAPTER 1

Modelling Nucleic Acid Structure and Flexibility: From Atomic to Mesoscopic Scale

FILIP LANKAS

Centre for Complex Molecular Systems and Biomolecules, Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo nám. 2, 166 10 Praha 6, Czech Republic

Email: filip.lankas@uochb.cas.cz

1.1 Introduction

Atomic-resolution models provide detailed information about nucleic acid structure, dynamics and flexibility. However, they are rather limited in the time and length scales they can cover. For instance, present-day all-atom molecular dynamics (MD) simulations with explicit representation of water and ions can model nucleic acid molecules up to roughly 100 nucleotides in size for simulation times of about 100–1000 ns, which is still behind relevant scales of many important processes in nucleic acid biology and in nanotechnology applications. Moreover, all-atom MD is computationally intensive, a production of one typical MD trajectory requiring several weeks on a standard multiprocessor machine. For these reasons, researches have focused on the development of coarse-grained models, using groups of atoms as elementary units of the model. Coarse-grained models have proved useful in problems involving longer time and length scales and where detailed atomistic information is not required.

In this chapter we provide a short survey of several recently published coarse-grained models used to study nucleic acid structure and flexibility. Section 1.2 is devoted to pseudoatom models, in which groups of atoms are typically modelled by effective, spherical interacting particles. In section 1.3 we provide a more detailed account of models representing each base or base pair as a general rigid body. The position and orientation of the body is fully characterized by giving a reference point and a right-handed, orthonormal frame attached to the body, the relative rotation and displacement between the bodies are captured by suitably chosen internal coordinates. In section 1.3.1 we describe a standard construction of the reference point and frame attached to a base. Section 1.3.2 concerns internal coordinate definition. Internal coordinates implemented in two popular conformational analysis programs, 3DNA and Curves+, are presented, and their similarities and differences are discussed. In section 1.3.3, we describe rigid base and basepair models with nonlocal, quadratic interaction potentials recently proposed by Lankas, Gonzalez et al. Application to a DNA oligomer containing A-tract is presented in section 1.3.4. We infer model parameters for the oligomer from all-atom MD simulations using both 3DNA and Curves+ coordinates, and compare the values. Section 1.3.5 briefly discusses anharmonic behaviour related to the presence of conformational substates or to high loads. Section 1.4 is devoted to coarser models with elementary units comprising more than one or two nucleotides, and to alternative approaches.

1.2 Models Based on Pseudoatoms

Many researches use coarse-grained models in which several atoms are represented as one effective atom (pseudoatom). The way real atoms are assigned to pseudoatoms, and the form of the effective potential energy describing the pseudoatom interaction, vary substantially among the models.

In a rather detailed model of nucleic acid bases developed by Maciejczyk et al., each base is represented by a rigid body with three to five interaction centres. The van der Waals component of the interactions is modelled by Lennard-Jones spheres (beads), the charge distribution is approximated by a set of electric dipole moments located at the centres of the spheres. The Lennard-Jones parameters and the position of the beads are determined by fitting them to the all-atom AMBER van der Waals energy, the dipoles are fitted to quantum mechanical electrostatic potential. Such a detailed base model, when incorporated into a coarse-grained model of the whole nucleic acid, should be suitable for simulation of protein-DNA complexes.

A more coarse-grained approach has been proposed by Knotts et al. Their model, which follows up earlier work by Drukker et al., represents a nucleotide by three interaction sites, one each for the phosphate, sugar and base. The interactions are captured by bond, angle and dihedral terms, complemented by stacking (6/12) and base-pairing (10/12) potentials and a Debye-Hückel electrostatic term. Interestingly, the cutoff for stacking interactions had to be chosen to include not only nearest-neighbour, but also next nearest-neighbour base-base interactions along the strand. Although the model includes many adjustable parameters, only three of them are found to play an important role: the dihedral force constant and the stacking and base pairing energy depths. The model is parameterized using thermal denaturation experimental data and is able to predict salt-dependent melting, bubble formation and rehybridization. It can also qualitatively capture the dependence of the persistence length on salt concentration. The persistence length itself is smaller than the experimental value roughly by a factor of two.

The model of Knotts et al. inspired a lot of subsequent research activity. Sambirski et al. included an effective dielectric constant dependent on temperature and salt concentration, as well as a novel term meant to represent many-body effects associated with water rearrangement during denaturation. Moreover, they reparameterized the model to improve the simulated persistence length, which now falls between 45 nm and 56 nm, close to the consensus value of 50 nm. In a separate study, the authors used the model to explore reaction pathways of DNA hybridization. In their melting study of a small-molecule- DNA hybrid relevant for nanotechnology applications, Prytkova et al. 10 modified the Sambirski et al. model by including explicit sodium ions rather than using the implicit Debye-Hückel approximation, and found that the explicit counterions contribute to the sharp melting transition observed experimentally.

DeMille et al. combined their coarse-grained water and ion model with the DNA model of Knotts et al. They optimized the coarse-grained interactions between DNA and solvent using atomistic simulations to reproduce the solvent structure around DNA. Each water molecule or ion is represented by a chargeless particle interacting through a short-range potential. A key feature is the use of a combination of two-body and three-body potentials that encourage the tetrahedral configuration of water. The model reproduces rather quantitatively the distribution and relative residence times of water and ions around DNA, but the simulated melting temperature is unrealistically high. The authors note that the model cannot simultaneously represent the melting temperature and the solvation, but they conjecture that improvement could be obtained through a finer resolution of the bases. A step in this direction was made by Pantano and coworkers: Dans et al. proposed a model which maps each nucleotide onto six pseudoatoms, Darre et al. developed a solvent model in which ca. 11 water molecules are represented by four tetrahedrally interconnected beads and solvated ions are modelled as charged van der Waals spherical particles. Darre et al. found that, when combined with the DNA coarse-grained description of Dans et al., the model reproduces reasonably well a number of structural features of DNA and its solvation. The model Hamiltonian is close in its functional form to the Hamiltonians employed in atomistic molecular dynamics, so that standard atomistic simulation codes can in principle be used to perform the simulation.

Niewieczerzal and Cieplak used representations differing in the number of pseudoatoms (between 2 and 5) per nucleotide in their dynamical models of DNA micromanipulations such as stretching, twisting and unzipping. Instead of using multi-bead structures, Morris-Andrews et al. modelled the DNA bases as rigid ellipsoids to capture their anisotropic properties. Interaction potentials (a modification of Gay-Berne potential is used for the ellipsoids) are estimated systematically from all-atom simulations. Prévost and co-workers developed a model with 5–6 beads per nucleotide specifically designed for modelling protein-DNA complexes. Recently, they used the model to study the early stage of DNA sequence recognition within RecA filaments.

Ouldridge et al. model a DNA strand as a string of rigid nucleotides with one interaction site for the backbone and three for the base (stacking site, hydrogen-bonding site, and base repulsion site). An additional vector indicates the plane of the base. Backbone sites are connected via finitely extensible nonlinear elastic spring. A key role is played by the stacking interactions, which directly imply the helicity of the model. They are modulated according to the relative alignment of the normal vectors and the alignment of the normals with the base-base intersite vector. Analogous directional dependence is introduced also for the hydrogen-bonding potential. The specificity of Watson-Crick pairing is taken into account but any other sequence dependence is neglected. The model reproduces the transition of ssDNA from an ordered, helical form at low temperature to a disordered form at high temperature. It also quantitatively captures the temperature and transition width of duplex formation, and the temperature of hairpin formation, including its dependence on loop and stem length. In addition, the model yields a pitch of 10.4 base pairs per turn and a bending persistence length of 154 base pairs, values close to reported experimental results. Twist fluctuations and single strand persistence length are very well reproduced, too. The authors applied their model to simulate the operation of DNA tweezers, molecular machines driven by hybridization and strand displacement. In a subsequent study, they explored in detail the hybridization behaviour and mechanical properties of the model, as well as the representation of more complex structural motifs. The authors anticipate many potential applications both in nanotechnology and in modelling biologically relevant structural transitions.

A coarser type of model has been proposed by Kenward and Dorfman. They represent a nucleotide as two Lennard-Jones spheres, one for the base and the other for the backbone. Covalent links are modelled by the finitely extensible nonlinear spring. Hydrogen bonding and stacking between the bases are captured by additional potentials of identical functional forms, with prefactors depending on the type of interaction (hydrogen bonding or stacking) and on the identity of the bases involved. The configurations are propagated in time using a Brownian dynamics simulation scheme. The authors applied the model to single-stranded DNA hairpins and found that it correctly captures the effect of base-base interactions and temperature on the thermodynamics and kinetics of hairpin formation and melting. They also used the model to study the 10-23 DNAzyme. DNAzymes, or deoxyribozymes, are single-stranded DNA molecules that catalyze nucleic acid reactions. The study identifies a transition state of the reaction, providing a possible microscopic interpretation of experimental observations.

Savelyev and Papoian designed a sequence-independent model in which each nucleotide is represented by a single bead. Apart from bond and angle potentials, the model also includes what the authors call a fan potential, through which a given base interacts with eleven nearest neighbouring bases in the opposite strand. The bond, angle and fan potentials are polynomials of up to 4th order. The electrostatic interactions are modelled by a Debye-Hückel potential with added short-range repulsive term. The adjustable model parameters are the coefficients of the polynomials and the prefactor of the repulsive electrostatic term. Thus, the part of the Hamiltonian to be parameterized is a sum of terms, each term being a product of an unknown constant parameter and a fluctuating quantity observable in the simulation. The parameters are chosen to minimize the differences between the mean observables computed from the coarse-grained simulation and those from a reference all-atom simulation. The authors proposed an iterative parameter optimization procedure which they call molecular renormalization group-coarse graining, or MRG-CG. Later, the authors extended the model by explicitly including mobile ions. The interionic potential involves a Coulombic term, a short- range repulsion term, and a sum of five Gaussians to account for hydration effects. The model leads to quantitative agreement (after a uniform rescaling) with the experimental ionic strength dependence of the persistence length, and predicts a structural transition of a torsionally stressed DNA minicircle upon increasing the ionic strength.

In their wrapped-around models for the Lac operon complex, La Penna and Perico adopted a model of DNA in which each base pair is modelled as a single isotropic, charged Lennard-Jones site. Models of nucleosomes and polynucleosome arrays have been proposed in which the beads representing the DNA comprise one nucleotide up to ten basepairs each. A more detailed discussion of these models is beyond the scope of the present chapter.

The models presented in this section mostly concern DNA. We refer the reader to the recent review by Trylska, which discusses bead models of complexes involving proteins and RNA and which focuses on a key protein-RNA complex, the ribosome.

Pseudoatom models vary a lot in their level of detail, underlying interaction potentials and the way solvent and ions are represented. Explicit inclusion of water and ions (in a coarse-grained form) allows one to capture aspects of the intimate relationship between nucleic acids and their solvation. Just as for atomic-resolution force fields, properly balanced interactions with the solvent are of primary importance. For instance, DeMille et al. found an inverse correlation between the ability of the coarse-grained model to describe the radial distribution functions and the degree of coarsening of the DNA moiety involved, and envisaged improvement for models where DNA and solvent would be represented at a comparable level of detail. Another feature that emerges is a possible compensation between the level of coarse-graining and the required range of interactions to be included. Even in the rather detailed model of Knotts et al., the stacking interactions are extended to next nearest-neighbour bases. The much coarser model of Savelyev and Papoian includes interactions over an entire helical turn. The Ouldridge et al. model is both rather coarse and with short-range interactions, but it includes a unique element: the effective normal vector capturing the base orientation on which the stacking and hydrogen bonding depend. Indeed, bases are in reality far from spherical – rather, they are anisotropic objects with stacking surfaces and hydrogen bonding edges. Thus, models representing bases as general rigid bodies may capture important properties of nucleic acids.

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