Methods for modeling cytoskeletal and DNA filaments

This review summarizes the models that researchers use to represent the conformations and dynamics of cytoskeletal and DNA filaments. It focuses on models that address individual filaments in continuous space. Conformation models include the freely jointed, Gaussian, angle-biased chain (ABC), and wormlike chain (WLC) models, of which the first three bend at discrete joints and the last bends continuously. Predictions from the WLC model generally agree well with experiment. Dynamics models include the Rouse, Zimm, stiff rod, dynamic WLC, and reptation models, of which the first four apply to isolated filaments and the last to entangled filaments. Experiments show that the dynamic WLC and reptation models are most accurate. They also show that biological filaments typically experience strong hydrodynamic coupling and/or constrained motion. Computer simulation methods that address filament dynamics typically compute filament segment velocities from local forces using the Langevin equation and then integrate these velocities with explicit or implicit methods; the former are more versatile and the latter are more efficient. Much remains to be discovered in biological filament modeling. In particular, filament dynamics in living cells are not well understood, and current computational methods are too slow and not sufficiently versatile. Although primarily a review, this paper also presents new statistical calculations for the ABC and WLC models. Additionally, it corrects several discrepancies in the literature about bending and torsional persistence length definitions, and their relations to flexural and torsional rigidities.

Diffusion, crowding & protein stability in a dynamic molecular model of the bacterial cytoplasm

A longstanding question in molecular biology is the extent to which the behavior of macromolecules observed in vitro accurately reflects their behavior in vivo. A number of sophisticated experimental techniques now allow the behavior of individual types of macromolecule to be studied directly in vivo; none, however, allow a wide range of molecule types to be observed simultaneously. In order to tackle this issue we have adopted a computational perspective, and, having selected the model prokaryote Escherichia coli as a test system, have assembled an atomically detailed model of its cytoplasmic environment that includes 50 of the most abundant types of macromolecules at experimentally measured concentrations. Brownian dynamics (BD) simulations of the cytoplasm model have been calibrated to reproduce the translational diffusion coefficients of Green Fluorescent Protein (GFP) observed in vivo, and “snapshots” of the simulation trajectories have been used to compute the cytoplasm's effects on the thermodynamics of protein folding, association and aggregation events. The simulation model successfully describes the relative thermodynamic stabilities of proteins measured in E. coli, and shows that effects additional to the commonly cited “crowding” effect must be included in attempts to understand macromolecular behavior in vivo.

The interior of a typical bacterial cell is a highly crowded place in which molecules must jostle and compete with each other in order to carry out their biological functions. The conditions under which such molecules are typically studied in vitro, however, are usually quite different: one or a few different types of molecules are studied as they freely diffuse in a dilute, aqueous solution. There is therefore a significant disconnect between the conditions under which molecules can be most usefully studied and the conditions under which such molecules usually “live”, and developing ways to bridge this gap is likely to be important for properly understanding molecular behavior in vivo. Toward this end, we show in this work that computer simulations can be used to model the interior of bacterial cells at a near atomic level of detail: the rates of diffusion of proteins are matched to known experimental values, and their thermodynamic stabilities are found to be in good agreement with the few measurements that have so far been performed in vivo. While the simulation approach is certainly not free of assumptions, it offers a potentially important complement to experimental techniques and provides a vivid illustration of molecular behavior inside a biological cell that is likely to be of significant educational value.

Modeling 2D and 3D diffusion

Modeling obstructed diffusion is essential to the understanding of diffusion-mediated processes in the crowded cellular environment. Simple Monte Carlo techniques for modeling obstructed random walks are explained and related to Brownian dynamics and more complicated Monte Carlo methods. Random number generation is reviewed in the context of random walk simulations. Programming techniques and event-driven algorithms are discussed as ways to speed simulations.

Two-angle model and phase diagram for chromatin

We have studied the phase diagram for chromatin within the framework of the two-angle model. Only a rough estimation of the forbidden surface of the phase diagram for chromatin was given in a previous work of Schiessel. We revealed the fine structure of this excluded-volume borderline numerically and analytically. Furthermore, we investigated the Coulomb repulsion of the DNA linkers to compare it with the previous results.

Modeling DNA loops using the theory of elasticity

An elastic rod model of a protein-bound DNA loop is adapted for application in multi-scale simulations of protein-DNA complexes. The classical Kirchhoff system of equations which describes the equilibrium structure of the elastic loop is modified to account for the intrinsic twist and curvature, anisotropic bending properties, and electrostatic charge of DNA. The effects of bending anisotropy and electrostatics are studied for the DNA loop clamped by the lac repressor protein. For two possible lengths of the loop, several topologically different conformations are predicted and extensively analyzed over the broad range of model parameters describing DNA bending and electrostatic properties. The scope and applications of the model in already accomplished and in future multi-scale studies of protein-DNA complexes are discussed.

Mesoscopic modeling for nucleic acid chain dynamics

To gain a deeper insight into cellular processes such as transcription and translation, one needs to uncover the mechanisms controlling the configurational changes of nucleic acids. As a step toward this aim, we present here a mesoscopic-level computational model that provides a new window into nucleic acid dynamics. We model a single-stranded nucleic as a polymer chain whose monomers are the nucleosides. Each monomer comprises a bead representing the sugar molecule and a pin representing the base. The bead-pin complex can rotate about the backbone of the chain. We consider pairwise stacking and hydrogen-bonding interactions. We use a modified Monte Carlo dynamics that splits the dynamics into translational bead motion and rotational pin motion. By performing a number of tests, we first show that our model is physically sound. We then focus on a study of the kinetics of a DNA hairpin--a single-stranded molecule comprising two complementary segments joined by a noncomplementary loop--studied experimentally. We find that results from our simulations agree with experimental observations, demonstrating that our model is a suitable tool for the investigation of the hybridization of single strands.

Deformations of promoter DNA bound to carcinogens help interpret effects on TATA-element structure and activity

The TATA-box binding protein (TBP) is required by eukaryotic RNA polymerases for correct transcription initiation. TBP binds to the minor groove of an 8 base pair (bp) DNA-promoter element known as the TATA box and severely bends the TATA box. The promoter-DNA substrate can be damaged by components present in the cell or the environment to produce covalent carcinogen-DNA adducts. These may lead to transcription blockage or unfaithful transcription. Benzo[a]pyrene (BP) is a widespread environmental chemical carcinogen which can be metabolically converted to DNA-reactive enantiomeric (+) and (-)-anti-benzo[a]pyrene diol epoxides (BPDEs). Recent experimental studies of a pair of stereoisomeric adenine adducts, derived from (+) and (-)-anti-BPDEs, have revealed how these lesions influence the complexation of TBP with the TATA box. Depending on the adduct's location in the TATA box and its stereochemistry, the stability of monomeric TATA-TBP complexes was found to increase or decrease relative to the unmodified DNA. We report here analyses of molecular-dynamics simulations to interpret these findings. Structural analyses of 12 DNA-protein systems representing different combinations of adduct stereoisomer type and placement within the promoter reveal that the location of the adduct within the TATA octamer determines whether the stability of TATA-TBP complexes is increased or decreased. The effect on binding stability can be interpreted in terms of conformational freedom and major-groove space available to BP due to the hydrogen bonds and inserted phenylalanines of the TATA-TBP complex; that is, depending on the position of the adenine to which BP is covalently bound, BP can be accommodated in an intercalated or major-groove orientation with ease or with difficulty (due to interference with TATA-TBP interactions). The unravelled structures and interactions thus reveal the effect of different adduct locations on TATA-TBP complex formation and suggest how transcription initiation may be affected by the presence of a bulky BP.

Constructing irregular surfaces to enclose macromolecular complexes for mesoscale modeling using the discrete surface charge optimization (DISCO) algorithm

Salt-mediated electrostatics interactions play an essential role in biomolecular structures and dynamics. Because macromolecular systems modeled at atomic resolution contain thousands of solute atoms, the electrostatic computations constitute an expensive part of the force and energy calculations. Implicit solvent models are one way to simplify the model and associated calculations, but they are generally used in combination with standard atomic models for the solute. To approximate electrostatics interactions in models on the polymer level (e.g., supercoiled DNA) that are simulated over long times (e.g., milliseconds) using Brownian dynamics, Beard and Schlick have developed the DiSCO (Discrete Surface Charge Optimization) algorithm. DiSCO represents a macromolecular complex by a few hundred discrete charges on a surface enclosing the system modeled by the Debye-Hückel (screened Coulombic) approximation to the Poisson-Boltzmann equation, and treats the salt solution as continuum solvation. DiSCO can represent the nucleosome core particle (>12,000 atoms), for example, by 353 discrete surface charges distributed on the surfaces of a large disk for the nucleosome core particle and a slender cylinder for the histone tail; the charges are optimized with respect to the Poisson-Boltzmann solution for the electric field, yielding a approximately 5.5% residual. Because regular surfaces enclosing macromolecules are not sufficiently general and may be suboptimal for certain systems, we develop a general method to construct irregular models tailored to the geometry of macromolecules. We also compare charge optimization based on both the electric field and electrostatic potential refinement. Results indicate that irregular surfaces can lead to a more accurate approximation (lower residuals), and the refinement in terms of the electric field is more robust. We also show that surface smoothing for irregular models is important, that the charge optimization (by the TNPACK minimizer) is efficient and does not depend on the initial assigned values, and that the residual is acceptable when the distance to the model surface is close to, or larger than, the Debye length. We illustrate applications of DiSCO's model-building procedure to chromatin folding and supercoiled DNA bound to Hin and Fis proteins. DiSCO is generally applicable to other interesting macromolecular systems for which mesoscale models are appropriate, to yield a resolution between the all-atom representative and the polymer level.

Atomic force microscope investigation of large-circle DNA molecules

A circular bacterial artificial chromosome of 148.9kbp on human chromosome 3 has been extended and fixed on bare mica substrates using a developed fluid capillary flow method in evaporating liquid drops. Extended circular DNA molecules were imaged with an atomic force microscope (AFM) under ambient conditions. The measured total lengths of the whole DNA molecules were in agreement with sequencing analysis data with an error range of +/-3.6%. This work is important groundwork for probing single nucleotide polymorphisms in the human genome, mapping genomic DNA, manipulating biomolecular nanotechnology, and studying the interaction of DNA-protein complexes investigated by AFM.

Effect of DNA superhelicity and bound proteins on mechanistic aspects of the Hin-mediated and Fis-enhanced inversion

Using a recently developed inhomogeneous, macroscopic model for long DNA bound to proteins, we examine topological and geometric aspects of DNA/protein structures and dynamics on various stages of the Hin inversion pathway. This biological reaction involves exchange of DNA in a synaptic complex that brings together several DNA sites bound to Hin dimers as well as Fis enhancers. Brownian dynamics simulations in the millisecond timescale allow us to follow and analyze the DNA/protein dynamics trajectories and to examine the effects of DNA superhelicity and protein binding on various reaction steps. Analysis of the generated kinetic pathways helps explain mechanistic aspects regarding the process by which two or three protein-bound DNA sites come to close spatial proximity and show that how topological selectivity (two trapped supercoils), enhancer binding, and properties of supercoiled DNA play critical roles in regulating the inversion reaction. Specifically, a critical amount of DNA superhelicity (e.g., |sigma| > 0.02) leads to an optimal interplay for the first reaction step-two-site juxtaposition-between large-scale random rearrangements of Hin-bound DNA and local slithering within branches of plectonemes. The three-site juxtaposition, the second step, is significantly accelerated by the presence of an enhancer protein that, due to severe local bending, also alters juxtaposition mechanisms, especially for superhelical density magnitude greater than around 0.04.

Long time dynamics of complex systems

Molecular dynamics trajectories of large biological molecules are restricted to nanoseconds. We describe a computational method, based on optimization of a functional, to extend the time of molecular simulations by orders of magnitude. Variants of our technique have already produced microsecond and millisecond trajectories. The large steps enable feasible computations of atomically detailed approximate trajectories. Numerical examples are provided: (i) a conformational change in blocked glycine peptide and (ii) helix formation of an alanine-rich peptide.

Biomolecular diffusional association

The primary method for the computational study of biomolecular diffusional association is Brownian dynamics. Recent work has seen advances in the efficiency of computing association rates and in the accuracy of simulation models. New areas to which Brownian dynamics has been applied include protein polymerisation and protein adsorption to a surface. There has recently been particularly intense study of protein-protein association, and Brownian dynamics, together with other theoretical and experimental approaches, has led to new insights into the determinants of protein-protein binding kinetics.

Computational modeling predicts the structure and dynamics of chromatin fiber

BACKGROUND

The compact form of the chromatin fiber is a critical regulator of fundamental processes such as transcription and replication. These reactions can occur only when the fiber is unraveled and the DNA strands contained within are exposed to interact with nuclear proteins. While progress on identifying the biochemical mechanisms that control localized folding and hence govern access to genetic information continues, the internal structure of the chromatin fiber, let alone the structural pathways for folding and unfolding, remain unknown.

RESULTS

To offer structural insights into how this nucleoprotein complex might be organized, we present a macroscopic computer model describing the mechanics of the chromatin fiber on the polymer level. We treat the core particles as electrostatically charged disks linked via charged elastic DNA segments and surrounded by a microionic hydrodynamic solution. Each nucleosome unit is represented by several hundred charges optimized so that the effective Debye-Hückel electrostatic field matches the field predicted by the nonlinear Poisson-Boltzmann equation. On the basis of Brownian dynamics simulations, we show that oligonucleosomes condense and unfold in a salt-dependent manner analogous to the chromatin fiber.

CONCLUSIONS

Our predicted chromatin model shows good agreement with experimental diffusion coefficients and small-angle X-ray scattering data. A fiber of width 30 nm, organized in a compact helical zigzag pattern with about 4 nucleosomes per 10 nm, naturally emerges from a repeating nucleosome folding motif. This fiber has a cross-sectional radius of gyration of R(c) = 8.66 nm, in close agreement with corresponding values for rat thymus and chicken erythrocyte chromatin (8.82 and 8.5 nm, respectively).

Dynamics of site juxtaposition in supercoiled DNA

Juxtaposition kinetics between specific sites in supercoiled DNA is investigated at close to physiological ionic conditions by Brownian dynamics simulations. At such conditions, supercoiled DNA is interwound, and the probability of spatial site juxtaposition is much higher than in relaxed DNA. We find, however, that supercoiling does not correspondingly increase the rate of juxtaposition at these physiological conditions. An explanation to this unexpected finding emerges on analysis of the juxtaposition dynamics. We note that although a particular site i(1) in supercoiled DNA is often in close proximity (juxtaposed) to another site i(2), the change of i(2) occurs very slowly and depends largely on internal slithering of opposite segments of the DNA superhelix. Such slithering results in long correlations between successive values of i(2); these correlations increase the average time of juxtaposition between two DNA sites. Random collisions between sites located on different superhelix branches-although increasing in importance with DNA size-contribute less substantially to site juxtaposition at high salt than slithering for DNA up to 6 kb in length.

A-Tract bending: insights into experimental structures by computational models

While solution structures of adenine tract (A-tract) oligomers have indicated a unique bend direction equivalent to negative global roll (commonly termed "minor-groove bending"), crystallographic data have not unambiguously characterized the bend direction; nevertheless, many features are shared by all A-tract crystal and solution structures (e.g. propeller twisting, narrow minor grooves, and localized water spines). To examine the origin of bending and to relate findings to the crystallographic and solution data, we analyze molecular dynamics trajectories of two solvated A-tract dodecamers: 1D89, d(CGCGA(6)CG), and 1D98, d(CGCA(6)GCG), using a new general global bending framework for analyzing bent DNA and DNA/protein complexes. It is significant that the crystallographically-based initial structures are converted from dissimilar to similar bend directions equivalent to negative global roll, with the average helical-axis bend ranging from 10.5 degrees to 14.1 degrees. The largest bend occurs as positive roll of 12 degrees on the 5' side of the A-tracts (supporting a junction model) and is reinforced by gradual curvature at each A-tract base-pair (bp) step (supporting a wedge model). The precise magnitude of the bend is subtly sequence dependent (consistent with a curved general sequence model). The conversion to negative global roll only requires small local changes at each bp, accumulated over flexible moieties both outside and inside the A-tract. In contrast, the control sequence 1BNA, d(CGCGA(2)TTCGCG), bends marginally (only 6.9 degrees ) with no preferred direction. The molecular features that stabilize the bend direction in the A-tract dodecamers include propeller twisting of AT base-pairs, puckering differences between A and T deoxyriboses, a narrow minor groove, and a stable water spine (that extends slightly beyond the A-tract, with lifetimes approaching 0.2 ns). The sugar conformations, in particular, are proposed as important factors that support bent DNA. It is significant that all these curvature-stabilizing features are also observed in the crystallographic structures, but yield overall different bending paths, largely due to the effects of sequences outside the A-tract. These results merge structural details reported for A-tract structures by experiment and theory and lead to structural and dynamic insights into sequence-dependent DNA flexibility, as highlighted by the effect of an A-tract variant of a TATA-box element on bending and flexibility required for TBP binding.