Torsional rigidity of DNA and length dependence of the free energy of DNA supercoiling

Direct observation of subunit rotation during DNA strand exchange by serine recombinases

Serine recombinases are proposed to catalyse site-specific recombination by a unique mechanism called subunit rotation. Cutting and rejoining DNA occurs within an intermediate synaptic complex comprising a recombinase tetramer bound to two DNA sites. After double-strand cleavage at both sites, one half of the complex rotates 180° relative to the other, before re-ligation of the DNA ends. We used single-molecule FRET (smFRET) methods to provide compelling direct physical evidence for subunit rotation by recombinases Tn3 resolvase and Sin. Synaptic complexes containing fluorescently labelled DNA show FRET fluctuations consistent with the subunit rotation model. FRET changes were associated with the rotation steps, on a timescale of 0.4-1.1s - 1 , as well as opening and closing of the gap between the scissile phosphates during cleavage and ligation. Multiple rounds of recombination were observed within the ~25 s observation period, including frequent consecutive rotation events in the cleaved-DNA state without evidence of intermediate ligation.

Bending DNA increases its helical repeat

In all biological systems, DNA is under high mechanical stress from bending and twisting. For example, DNA is tightly bent in nucleosome complexes, virus capsids, bacterial chromosomes, or complexes with transcription factors that regulate gene expression. A structurally and mechanically accurate model of DNA is therefore necessary to understand some of the most fundamental molecular mechanisms in biology including DNA packaging, replication, transcription and gene regulation. An iconic feature of DNA is its double helical nature with an average repeath 0 of ~10.45 base pairs per turn, which is commonly believed to be independent of curvature. We developed a ligation assay on nicked DNA circles of variable curvature that reveals a strong unwinding of DNA to over 11 bp/turn for radii around 3-4 nm. Our work constitutes a major modification of the standard mechanical model of DNA and requires reassessing the molecular mechanisms and energetics of all processes involving tightly bent DNA.

Designed architectural proteins that tune DNA looping in bacteria

Architectural proteins alter the shape of DNA. Some distort the double helix by introducing sharp kinks. This can serve to relieve strain in tightly-bent DNA structures. Here, we design and test artificial architectural proteins based on a sequence-specific Transcription Activator-like Effector (TALE) protein, either alone or fused to a eukaryotic high mobility group B (HMGB) DNA-bending domain. We hypothesized that TALE protein binding would stiffen DNA to bending and twisting, acting as an architectural protein that antagonizes the formation of small DNA loops. In contrast, fusion to an HMGB domain was hypothesized to generate a targeted DNA-bending architectural protein that facilitates DNA looping. We provide evidence from Escherichia coli Lac repressor gene regulatory loops supporting these hypotheses in living bacteria. Both data fitting to a thermodynamic DNA looping model and sophisticated molecular modeling support the interpretation of these results. We find that TALE protein binding inhibits looping by stiffening DNA to bending and twisting, while the Nhp6A domain enhances looping by bending DNA without introducing twisting flexibility. Our work illustrates artificial approaches to sculpt DNA geometry with functional consequences. Similar approaches may be applicable to tune the stability of small DNA loops in eukaryotes.

Surprising Twists in Nucleosomal DNA with Implication for Higher-order Folding

While nucleosomes are dynamic entities that must undergo structural deformations to perform their functions, the general view from available high-resolution structures is a largely static one. Even though numerous examples of twist defects have been documented, the DNA wrapped around the histone core is generally thought to be overtwisted. Analysis of available high-resolution structures from the Protein Data Bank reveals a heterogeneous distribution of twist along the nucleosomal DNA, with clear patterns that are consistent with the literature, and a significant fraction of structures that are undertwisted. The subtle differences in nucleosomal DNA folding, which extend beyond twist, have implications for nucleosome disassembly and modeled higher-order structures. Simulations of oligonucleosome arrays built with undertwisted models behave very differently from those constructed from overtwisted models, in terms of compaction and inter-nucleosome contacts, introducing configurational changes equivalent to those associated with 2-3 base-pair changes in nucleosome spacing. Differences in the nucleosomal DNA pathway, which underlie the way that DNA enters and exits the nucleosome, give rise to different nucleosome-decorated minicircles and affect the topological mix of configurational states.

A quantitative model of a cooperative two-state equilibrium in DNA: experimental tests, insights, and predictions

Quantitative parameters for a two-state cooperative transition in duplex DNAs were finally obtained during the last 5 years. After a brief discussion of observations pertaining to the existence of the two-state equilibrium per se, the lengths, torsion, and bending elastic constants of the two states involved and the cooperativity parameter of the model are simply stated. Experimental tests of model predictions for the responses of DNA to small applied stretching, twisting, and bending stresses, and changes in temperature, ionic conditions, and sequence are described. The mechanism and significance of the large cooperativity, which enables significant DNA responses to such small perturbations, are also noted. The capacity of the model to resolve a number of long-standing and sometimes interconnected puzzles in the extant literature, including the origin of the broad pre-melting transition studied by numerous workers in the 1960s and 1970s, is demonstrated. Under certain conditions, the model predicts significant long-range attractive or repulsive interactions between hypothetical proteins with strong preferences for one or the other state that are bound to well-separated sites on the same DNA. A scenario is proposed for the activation of the ilvPG promoter on a supercoiled DNA by integration host factor.

DNA Manipulation and Single-Molecule Imaging

DNA replication, repair, and recombination in the cell play a significant role in the regulation of the inheritance, maintenance, and transfer of genetic information. To elucidate the biomolecular mechanism in the cell, some molecular models of DNA replication, repair, and recombination have been proposed. These biological studies have been conducted using bulk assays, such as gel electrophoresis. Because in bulk assays, several millions of biomolecules are subjected to analysis, the results of the biological analysis only reveal the average behavior of a large number of biomolecules. Therefore, revealing the elementary biological processes of a protein acting on DNA (e.g., the binding of protein to DNA, DNA synthesis, the pause of DNA synthesis, and the release of protein from DNA) is difficult. Single-molecule imaging allows the analysis of the dynamic behaviors of individual biomolecules that are hidden during bulk experiments. Thus, the methods for single-molecule imaging have provided new insights into almost all of the aspects of the elementary processes of DNA replication, repair, and recombination. However, in an aqueous solution, DNA molecules are in a randomly coiled state. Thus, the manipulation of the physical form of the single DNA molecules is important. In this review, we provide an overview of the unique studies on DNA manipulation and single-molecule imaging to analyze the dynamic interaction between DNA and protein.

Overtwisting induces polygonal shapes in bent DNA

By combining analytical results and simulations of various coarse-grained models, we investigate the minimal energy shape of DNA minicircles which are torsionally constrained by an imposed over or undertwist. We show that twist-bend coupling, a cross interaction term discussed in the recent DNA literature, induces minimal energy shapes with a periodic alternation of parts with high and low curvature resembling rounded polygons. We briefly discuss the possible experimental relevance of these findings. We finally show that the twist and bending energies of minicircles are governed by renormalized stiffness constants, rather than the bare ones. This has important consequences for the analysis of experiments involving circular DNA meant to determine DNA elastic constants.

Probing hyper-negatively supercoiled mini-circles with nucleases and DNA binding proteins

It is well accepted that the introduction of negative supercoils locally unwinds the DNA double helix, influencing thus the activity of proteins. Despite the use of recent methods of molecular dynamics simulations to model the DNA supercoiling-induced DNA deformation, the precise extent and location of unpaired bases induced by the negative supercoiling have never been investigated at the nucleotide level. Our goals in this study were to use radiolabeled double-stranded DNA mini-circles (dsMCs) to locate the unpaired bases on dsMCs whose topology ranged from relaxed to hyper-negatively supercoiled states, and to characterize the binding of proteins involved in the DNA metabolism. Our results show that the Nuclease SI is nearly ten times more active on hyper-negatively supercoiled than relaxed DNA. The structural changes responsible for this stimulation of activity were mapped for the first time with a base pair resolution and shown to be subtle and distributed along the entire sequence. As divalent cations modify the DNA topology, our binding studies were conducted with or without magnesium. Without magnesium, the dsMCs topoisomers mostly differ by their twist. Under these conditions, the Escherichia coli topoisomerase I weakly binds relaxed dsMCs and exhibits a stronger binding on negatively and hyper-negatively supercoiled dsMCs than relaxed dsMCs, with no significant difference in the binding activity among the supercoiled topoisomers. For the human replication protein A (hRPA), the more negatively supercoiled is the DNA, the better the binding, illustrating the twist-dependent binding activity for this protein. The presence of magnesium permits the dsMCs to writhe upon introduction of negative supercoiling and greatly modifies the binding properties of the hRPA and Escherichia coli SSB on dsMCs, indicating a magnesium-dependent DNA binding behavior. Finally, our experiments that probe the topology of the DNA in the hRPA-dsMC complexes show that naked and hRPA-bound dsMCs have the same topology.

Twist-stretch profiles of DNA chains

Helical molecules change their twist number under the effect of a mechanical load. We study the twist-stretch relation for a set of short DNA molecules modeled by a mesoscopic Hamiltonian. Finite temperature path integral techniques are applied to generate a large ensemble of possible configurations for the base pairs of the sequence. The model also accounts for the bending and twisting fluctuations between adjacent base pairs along the molecules stack. Simulating a broad range of twisting conformation, we compute the helix structural parameters by averaging over the ensemble of base pairs configurations. The method selects, for any applied force, the average twist angle which minimizes the molecule's free energy. It is found that the chains generally over-twist under an applied stretching and the over-twisting is physically associated to the contraction of the average helix diameter, i.e. to the damping of the base pair fluctuations. Instead, assuming that the maximum amplitude of the bending fluctuations may decrease against the external load, the DNA molecule first over-twists for weak applied forces and then untwists above a characteristic force value. Our results are discussed in relation to available experimental information albeit for kilo-base long molecules.

Direct single-molecule observations of DNA unwinding by SV40 large tumor antigen under a negative DNA supercoil state

Superhelices, which are induced by the twisting and coiling of double-helical DNA in chromosomes, are thought to affect transcription, replication, and other DNA metabolic processes. In this study, we report the effects of negative supercoiling on the unwinding activity of simian virus 40 large tumor antigen (SV40 TAg) at a single-molecular level. The supercoiling density of linear DNA templates was controlled using magnetic tweezers and monitored using a fluorescent microscope in a flow cell. SV40 TAg-mediated DNA unwinding under relaxed and negative supercoil states was analyzed by the direct observation of both single- and double-stranded regions of single DNA molecules. Increased negative superhelicity stimulated SV40 TAg-mediated DNA unwinding more strongly than a relaxed state; furthermore, negative superhelicity was associated with an increased probability of SV40 TAg-mediated DNA unwinding. These results suggest that negative superhelicity helps to regulate the initiation of DNA replication.

Disentangling DNA molecules

The widespread circular form of DNA molecules inside cells creates very serious topological problems during replication. Due to the helical structure of the double helix the parental strands of circular DNA form a link of very high order, and yet they have to be unlinked before the cell division. DNA topoisomerases, the enzymes that catalyze passing of one DNA segment through another, solve this problem in principle. However, it is very difficult to remove all entanglements between the replicated DNA molecules due to huge length of DNA comparing to the cell size. One strategy that nature uses to overcome this problem is to create the topoisomerases that can dramatically reduce the fraction of linked circular DNA molecules relative to the corresponding fraction at thermodynamic equilibrium. This striking property of the enzymes means that the enzymes that interact with DNA only locally can access their topology, a global property of circular DNA molecules. This review considers the experimental studies of the phenomenon and analyzes the theoretical models that have been suggested in attempts to explain it. We describe here how various models of enzyme action can be investigated computationally. There is no doubt at the moment that we understand basic principles governing enzyme action. Still, there are essential quantitative discrepancies between the experimental data and the theoretical predictions. We consider how these discrepancies can be overcome.

The role of structural parameters in DNA cyclization

BACKGROUND

The intrinsic bendability of DNA plays an important role with relevance for myriad of essential cellular mechanisms. The flexibility of a DNA fragment can be experimentally and computationally examined by its propensity for cyclization, quantified by the Jacobson-Stockmayer J factor. In this study, we use a well-established coarse-grained three-dimensional model of DNA and seven distinct sets of experimentally and computationally derived conformational parameters of the double helix to evaluate the role of structural parameters in calculating DNA cyclization.

RESULTS

We calculate the cyclization rates of 86 DNA sequences with previously measured J factors and lengths between 57 and 325 bp as well as of 20,000 randomly generated DNA sequences with lengths between 350 and 4000 bp. Our comparison with experimental data is complemented with analysis of simulated data.

CONCLUSIONS

Our data demonstrate that all sets of parameters yield very similar results for longer DNA fragments, regardless of the nucleotide sequence, which are in agreement with experimental measurements. However, for DNA fragments shorter than 100 bp, all sets of parameters performed poorly yielding results with several orders of magnitude difference from the experimental measurements. Our data show that DNA cyclization rates calculated using conformational parameters based on nucleosome packaging data are most similar to the experimental measurements. Overall, our study provides a comprehensive large-scale assessment of the role of structural parameters in calculating DNA cyclization rates.

Elastic Energy Partitioning in DNA Deformation and Binding to Proteins

We study the elasticity of DNA based on local principal axes of bending identified from over 0.9-μs all-atom molecular dynamics simulations of DNA oligos. The calculated order parameters describe motion of DNA as an elastic rod. In 10 possible dinucleotide steps, bending about the two principal axes is anisotropic yet linearly elastic. Twist about the centroid axis is largely decoupled from bending, but DNA tends to overtwist for unbending beyond the typical range of thermal motion, which is consistent with experimentally observed twist-stretch coupling. The calculated elastic stiffness of dinucleotide steps yield sequence-dependent persistence lengths consistent with previous single-molecule experiments, which is further analyzed by performing coarse-grained simulations of DNA. Flexibility maps of oligos constructed from simulation also match with those from the precalculated stiffness of dinucleotide steps. These support the premise that base pair interaction at the dinucleotide-level is mainly responsible for the elasticity of DNA. Furthermore, we analyze 1381 crystal structures of protein-DNA complexes. In most structures, DNAs are mildly deformed and twist takes the highest portion of the total elastic energy. By contrast, in structures with the elastic energy per dinucleotide step greater than about 4.16 kBT (kBT: thermal energy), the major bending becomes dominant. The extensional energy of dinucleotide steps takes at most 35% of the total elastic energy except for structures containing highly deformed DNAs where linear elasticity breaks down. Such partitioning between different deformational modes provides quantitative insights into the conformational dynamics of DNA as well as its interaction with other molecules and surfaces.

Normal-Mode Analysis of Circular DNA at the Base-Pair Level. 1. Comparison of Computed Motions with the Predicted Behavior of an Ideal Elastic Rod

We have extended a newly developed approach to study the low-frequency normal modes of mesoscopic fragments of linear DNA in order to investigate the dynamics of closed circular molecules of comparable size, i.e., a few hundred base pairs. We have added restraint energy terms and a global minimization step to treat the more complicated, spatially constrained duplex in terms of the intrinsic conformation and flexibility of the constituent base-pair "step" parameters. Initial application of the methodology to the normal modes of an ideal closed circular DNA molecule [Formula: see text] which is naturally straight in its relaxed open linear state, inextensible, and capable of isotropic bending and independent twisting at the base-pair level [Formula: see text] matches theoretical predictions of elastic rod dynamics. The energy-optimized closed circular states and the types of low frequency motions follow expected behavior, with (1) uniform twist density and uniform energy density in the minimum energy state; (2) a near-zero frequency torsional mode with "free" rotation about the global helical axis; (3) higher-order torsional modes accompanied by global rocking motions and pure in-plane and out-of-plane bending motions in the torsionally relaxed circle; and (4) mixed modes of bending when the chain is supercoiled (over- or undertwisted). Furthermore, the computed changes in normal-mode frequencies with imposed supercoiling or with variation of chain length are virtually identical to theoretically predicted values.

From adjacent activation in Escherichia coli and DNA cyclization to eukaryotic enhancers: the elements of a puzzle

Deoxyribonucleic acid cyclization, Escherichia coli lac repressor binding to two spaced lac operators and repression enhancement can be successfully used for a better understanding of the conditions required for interaction between eukaryotic enhancers and the machinery of transcription initiation. Chronologically, the DNA looping model has first accounted for the properties initially defining enhancers, i.e., independence of action with distance or orientation with respect to the start of transcription. It has also predicted enhancer activity or its disruption at short distance (site orientation, alignment between promoter and enhancer sites), with high-order complexes of protein, or with transcription factor concentrations close or different from the wild-type situation. In another step, histones have been introduced into the model to further adapt it to eukaryotes. They in fact favor DNA cyclization in vitro. The resulting DNA compaction might explain the difference counted in base pairs in the distance of action between eukaryotic transcription enhancers and prokaryotic repression enhancers. The lac looping system provides a potential tool for analysis of this discrepancy and of chromatin state directly in situ. Furthermore, as predicted by the model, the contribution of operators O2 and O3 to repression of the lac operon clearly depends on the lac repressor level in the cell and is prevented in strains overproducing lac repressor. By extension, gene regulation especially that linked to cell fate, should also depend on transcription factor levels, providing a potential tool for cellular therapy. In parallel, a new function of the O1–O3 loop completes the picture of lac repression. The O1–O3 loop would at the same time ensure high efficiency of repression, inducibility through the low-affinity sites and limitation of the level of repressor through self-repression of the lac repressor. Last, the DNA looping model can be successfully adapted to the enhancer auxiliary elements known as insulators.

What controls DNA looping?

The looping of DNA provides a means of communication between sequentially distant genomic sites that operate in tandem to express, copy, and repair the information encoded in the DNA base sequence. The short loops implicated in the expression of bacterial genes suggest that molecular factors other than the naturally stiff double helix are involved in bringing the interacting sites into close spatial proximity. New computational techniques that take direct account of the three-dimensional structures and fluctuations of protein and DNA allow us to examine the likely means of enhancing such communication. Here, we describe the application of these approaches to the looping of a 92 base-pair DNA segment between the headpieces of the tetrameric Escherichia coli Lac repressor protein. The distortions of the double helix induced by a second protein--the nonspecific nucleoid protein HU--increase the computed likelihood of looping by several orders of magnitude over that of DNA alone. Large-scale deformations of the repressor, sequence-dependent features in the DNA loop, and deformability of the DNA operators also enhance looping, although to lesser degrees. The correspondence between the predicted looping propensities and the ease of looping derived from gene-expression and single-molecule measurements lends credence to the derived structural picture.

Blind predictions of DNA and RNA tweezers experiments with force and torque

Single-molecule tweezers measurements of double-stranded nucleic acids (dsDNA and dsRNA) provide unprecedented opportunities to dissect how these fundamental molecules respond to forces and torques analogous to those applied by topoisomerases, viral capsids, and other biological partners. However, tweezers data are still most commonly interpreted post facto in the framework of simple analytical models. Testing falsifiable predictions of state-of-the-art nucleic acid models would be more illuminating but has not been performed. Here we describe a blind challenge in which numerical predictions of nucleic acid mechanical properties were compared to experimental data obtained recently for dsRNA under applied force and torque. The predictions were enabled by the HelixMC package, first presented in this paper. HelixMC advances crystallography-derived base-pair level models (BPLMs) to simulate kilobase-length dsDNAs and dsRNAs under external forces and torques, including their global linking numbers. These calculations recovered the experimental bending persistence length of dsRNA within the error of the simulations and accurately predicted that dsRNA's “spring-like” conformation would give a two-fold decrease of stretch modulus relative to dsDNA. Further blind predictions of helix torsional properties, however, exposed inaccuracies in current BPLM theory, including three-fold discrepancies in torsional persistence length at the high force limit and the incorrect sign of dsRNA link-extension (twist-stretch) coupling. Beyond these experiments, HelixMC predicted that ‘nucleosome-excluding’ poly(A)/poly(T) is at least two-fold stiffer than random-sequence dsDNA in bending, stretching, and torsional behaviors; Z-DNA to be at least three-fold stiffer than random-sequence dsDNA, with a near-zero link-extension coupling; and non-negligible effects from base pair step correlations. We propose that experimentally testing these predictions should be powerful next steps for understanding the flexibility of dsDNA and dsRNA in sequence contexts and under mechanical stresses relevant to their biology.

DNA and RNA are fundamental molecules in the central dogma of molecular biology. Many biological behaviors of double-stranded DNA and RNA – including transcription/translation by proteins and packaging into compact structures – depend on their ability to flex and twist. Single-molecule tweezers now provide accurate mechanical measurements of DNA and RNA helices under force and torque but have not been used to rigorously falsify and thereby advance computational models. Here we present the first such blind challenge, involving recent dsRNA tweezers data that were kept hidden from modelers and a new HelixMC toolkit that resolves challenges in simulating long double helices from base-pair level models. The predictions gave excellent agreement with bending and stretching measurements of dsRNA but failed to recover twisting properties, pinpointing a critical area of future investigation. HelixMC also predicted that poly(A)/poly(T) and Z-DNA–biologically important variants whose elastic responses have not been studied with tweezers–will have distinct mechanical properties. These results open a route to iteratively falsifying and refining computational models of long nucleic acid helices, as is necessary for attaining a predictive understanding of their biological behaviors.

Torsional behavior of chromatin is modulated by rotational phasing of nucleosomes

Torsionally stressed DNA plays a critical role in genome organization and regulation. While the effects of torsional stresses on naked DNA have been well studied, little is known about how these stresses propagate within chromatin and affect its organization. Here we investigate the torsional behavior of nucleosome arrays by means of Brownian dynamics simulations of a coarse-grained model of chromatin. Our simulations reveal a strong dependence of the torsional response on the rotational phase angle Ψ₀ between adjacent nucleosomes. Extreme values of Ψ₀ lead to asymmetric, bell-shaped extension-rotation profiles with sharp maxima shifted toward positive or negative rotations, depending on the sign of Ψ₀, and to fast, irregular propagation of DNA twist. In contrast, moderate Ψ₀ yield more symmetric profiles with broad maxima and slow, uniform propagation of twist. The observed behavior is shown to arise from an interplay between nucleosomal transitions into states with crossed and open linker DNAs and global supercoiling of arrays into left- and right-handed coils, where Ψ₀ serves to modulate the energy landscape of nucleosomal states. Our results also explain the torsional resilience of chromatin, reconcile differences between experimentally measured extension-rotation profiles, and suggest a role of torsional stresses in regulating chromatin assembly and organization.

Chromatin regulates DNA torsional energy via topoisomerase II-mediated relaxation of positive supercoils

Eukaryotic topoisomerases I (topo I) and II (topo II) relax the positive (+) and negative (-) DNA torsional stress (TS) generated ahead and behind the transcription machinery. It is unknown how this DNA relaxation activity is regulated and whether (+) and (-)TS are reduced at similar rates. Here, we used yeast circular minichromosomes to conduct the first comparative analysis of topo I and topo II activities in relaxing chromatin under (+) and (-)TS. We observed that, while topo I relaxed (+) and (-)TS with similar efficiency, topo II was more proficient and relaxed (+)TS more quickly than (-)TS. Accordingly, we found that the relaxation rate of (+)TS by endogenous topoisomerases largely surpassed that of (-)TS. We propose a model of how distinct conformations of chromatin under (+) and (-)TS may produce this unbalanced relaxation of DNA. We postulate that, while quick relaxation of (+)TS may facilitate the progression of RNA and DNA polymerases, slow relaxation of (-)TS may serve to favor DNA unwinding and other structural transitions at specific regions often required for genomic transactions.

Surface charge effects on the 2D conformation of supercoiled DNA

We have adsorbed plasmid pUc19 DNA on a supported bilayer. By varying the fraction of cationic lipids in the membrane, we have tuned the surface charge. Plasmid conformations were imaged by Atomic Force Microscopy (AFM). We performed two sets of experiments: deposition from salt free solution on charged bilayers and deposition from salty solutions on neutral bilayers. Both sets show similar trends: at low surface charge density or low bulk salt concentration, the internal electrostatic repulsion forces plasmids to adopt completely opened structures, while at high surface charge density or higher bulk salt concentration, usual supercoiled plectonemes are observed. We experimentally demonstrate the equivalence of surface screening by mobile interfacial charges and bulk screening from salt ions. At low to medium screening, the electrostatic repulsion at plasmid crossings is predominant, leading to a number of crossovers decreasing linearly with the characteristic screening length. We compare our data with an analytical 2D-equilibrated model developed recently for the system and extract the DNA effective charge density when strands are adsorbed at the surface.

Effect of the boundary conditions and influence of the rotational inertia on the vibrational modes of an elastic ring

We present the small-amplitude vibrations of a circular elastic ring with periodic and clamped boundary conditions. We model the rod as an inextensible, isotropic, naturally straight Kirchhoff elastic rod and obtain the vibrational modes of the ring analytically for periodic boundary conditions and numerically for clamped boundary conditions. Of particular interest are the dependence of the vibrational modes on the torsional stress in the ring and the influence of the rotational inertia of the rod on the mode frequencies and amplitudes. In rescaling the Kirchhoff equations, we introduce a parameter inversely proportional to the aspect ratio of the rod. This parameter makes it possible to capture the influence of the rotational inertia of the rod. We find that the rotational inertia has a minor influence on the vibrational modes with the exception of a specific category of modes corresponding to high-frequency twisting deformations in the ring. Moreover, some of the vibrational modes over or undertwist the elastic rod depending on the imposed torsional stress in the ring.

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.

DNA loop domain organization as revealed by single-cell gel electrophoresis

At higher order levels chromatin is organized into loops. This looping, which plays an important role in transcription regulation and other processes, remains poorly understood. We investigated the kinetics of DNA loop migration during single cell gel electrophoresis (the comet assay). The migration of a part of the loops was shown to be reversible after switching off electrophoresis and to be sensitive to intercalation-induced changes in supercoiling. Another group of the loops migrates rapidly, the rate being insensitive to the supercoiling level. The largest part of the loops cannot migrate at all, presumably because of their large size. The loop ends can be detached in the presence of high concentrations of intercalators or protein denaturants, thus increasing the fraction of DNA that cannot migrate in the gel. The distribution of the loop length up to 100kilobases appears to be consistent with the fractal globule organization.

The thermodynamics of DNA loop formation, from J to Z

The formation of DNA loops is a ubiquitous theme in biological processes, including DNA replication, recombination and repair, and gene regulation. These loops are mediated by proteins bound at specific sites along the contour of a single DNA molecule, in some cases many thousands of base pairs apart. Loop formation incurs a thermodynamic cost that is a sensitive function of the length of looped DNA as well as the geometry and elastic properties of the DNA-bound protein. The free energy of DNA looping is logarithmically related to a generalization of the Jacobson-Stockmayer factor for DNA cyclization, termed the J factor. In the present article, we review the thermodynamic origins of this quantity, discuss how it is measured experimentally and connect the macroscopic interpretation of the J factor with a statistical-mechanical description of DNA looping and cyclization.

Rationally designed coiled-coil DNA looping peptides control DNA topology

Artificial DNA looping peptides were engineered to study the roles of protein and DNA flexibility in controlling the geometry and stability of protein-mediated DNA loops. These LZD (leucine zipper dual-binding) peptides were derived by fusing a second, C-terminal, DNA-binding region onto the GCN4 bZip peptide. Two variants with different coiled-coil lengths were designed to control the relative orientations of DNA bound at each end. Electrophoretic mobility shift assays verified formation of a sandwich complex containing two DNAs and one peptide. Ring closure experiments demonstrated that looping requires a DNA-binding site separation of 310 bp, much longer than the length needed for natural loops. Systematic variation of binding site separation over a series of 10 constructs that cyclize to form 862-bp minicircles yielded positive and negative topoisomers because of two possible writhed geometries. Periodic variation in topoisomer abundance could be modeled using canonical DNA persistence length and torsional modulus values. The results confirm that the LZD peptides are stiffer than natural DNA looping proteins, and they suggest that formation of short DNA loops requires protein flexibility, not unusual DNA bendability. Small, stable, tunable looping peptides may be useful as synthetic transcriptional regulators or components of protein–DNA nanostructures.

Strong bending of the DNA double helix

During the past decade, the issue of strong bending of the double helix has attracted a lot of attention. Here, we overview the major experimental and theoretical developments in the field sorting out reliably established facts from speculations and unsubstantiated claims. Theoretical analysis shows that sharp bends or kinks have to facilitate strong bending of the double helix. It remains to be determined what is the critical curvature of DNA that prompts the appearance of the kinks. Different experimental and computational approaches to the problem are analyzed. We conclude that there is no reliable evidence that any anomalous behavior of the double helix happens when DNA fragments in the range of 100 bp are circularized without torsional stress. The anomaly starts at the fragment length of about 70 bp when sharp bends or kinks emerge in essentially every molecule. Experimental data and theoretical analysis suggest that kinks may represent openings of isolated base pairs, which had been experimentally detected in linear DNA molecules. The calculation suggests that although the probability of these openings in unstressed DNA is close to 10⁻⁵, it increases sharply in small DNA circles reaching 1 open bp per circle of 70 bp.

Structural ensemble and microscopic elasticity of freely diffusing DNA by direct measurement of fluctuations

Precisely measuring the ensemble of conformers that a macromolecule populates in solution is highly challenging. Thus, it has been difficult to confirm or falsify the predictions of nanometer-scale dynamical modeling. Here, we apply an X-ray interferometry technique to probe the solution structure and fluctuations of B-form DNA on a length scale comparable to a protein-binding site. We determine an extensive set of intrahelix distance distributions between pairs of probes placed at distinct points on the surface of the DNA duplex. The distributions of measured distances reveal the nature and extent of the thermally driven mechanical deformations of the helix. We describe these deformations in terms of elastic constants, as is common for DNA and other polymers. The average solution structure and microscopic elasticity measured by X-ray interferometry are in striking agreement with values derived from DNA-protein crystal structures and measured by force spectroscopy, with one exception. The observed microscopic torsional rigidity of DNA is much lower than is measured by single-molecule twisting experiments, suggesting that torsional rigidity increases when DNA is stretched. Looking forward, molecular-level interferometry can provide a general tool for characterizing solution-phase structural ensembles.

Small DNA circles as probes of DNA topology

Small DNA circles can occur in Nature, for example as protein-constrained loops, and can be synthesized by a number of methods. Such small circles provide tractable systems for the study of the structure, thermodynamics and molecular dynamics of closed-circular DNA. In the present article, we review the occurrence and synthesis of small DNA circles, and examine their utility in studying the properties of DNA and DNA–protein interactions. In particular, we highlight the analysis of small circles using atomistic simulations.

Interplay of protein and DNA structure revealed in simulations of the lac operon

The E. coli Lac repressor is the classic textbook example of a protein that attaches to widely spaced sites along a genome and forces the intervening DNA into a loop. The short loops implicated in the regulation of the lac operon suggest the involvement of factors other than DNA and repressor in gene control. The molecular simulations presented here examine two likely structural contributions to the in-vivo looping of bacterial DNA: the distortions of the double helix introduced upon association of the highly abundant, nonspecific nucleoid protein HU and the large-scale deformations of the repressor detected in low-resolution experiments. The computations take account of the three-dimensional arrangements of nucleotides and amino acids found in crystal structures of DNA with the two proteins, the natural rest state and deformational properties of protein-free DNA, and the constraints on looping imposed by the conformation of the repressor and the orientation of bound DNA. The predicted looping propensities capture the complex, chain-length-dependent variation in repression efficacy extracted from gene expression studies and in vitro experiments and reveal unexpected chain-length-dependent variations in the uptake of HU, the deformation of repressor, and the folding of DNA. Both the opening of repressor and the presence of HU, at levels approximating those found in vivo, enhance the probability of loop formation. HU affects the global organization of the repressor and the opening of repressor influences the levels of HU binding to DNA. The length of the loop determines whether the DNA adopts antiparallel or parallel orientations on the repressor, whether the repressor is opened or closed, and how many HU molecules bind to the loop. The collective behavior of proteins and DNA is greater than the sum of the parts and hints of ways in which multiple proteins may coordinate the packaging and processing of genetic information.

DNA looping in prokaryotes: experimental and theoretical approaches

Transcriptional regulation is at the heart of biological functions such as adaptation to a changing environment or to new carbon sources. One of the mechanisms which has been found to modulate transcription, either positively (activation) or negatively (repression), involves the formation of DNA loops. A DNA loop occurs when a protein or a complex of proteins simultaneously binds to two different sites on DNA with looping out of the intervening DNA. This simple mechanism is central to the regulation of several operons in the genome of the bacterium Escherichia coli, like the lac operon, one of the paradigms of genetic regulation. The aim of this review is to gather and discuss concepts and ideas from experimental biology and theoretical physics concerning DNA looping in genetic regulation. We first describe experimental techniques designed to show the formation of a DNA loop. We then present the benefits that can or could be derived from a mechanism involving DNA looping. Some of these are already experimentally proven, but others are theoretical predictions and merit experimental investigation. Then, we try to identify other genetic systems that could be regulated by a DNA looping mechanism in the genome of Escherichia coli. We found many operons that, according to our set of criteria, have a good chance to be regulated with a DNA loop. Finally, we discuss the proposition recently made by both biologists and physicists that this mechanism could also act at the genomic scale and play a crucial role in the spatial organization of genomes.

Plasmid DNA topology assayed by two-dimensional agarose gel electrophoresis

Two-dimensional (2D) agarose gel electrophoresis is nowadays one of the best methods available to analyze DNA molecules with different masses and shapes. The possibility to use nicking enzymes and intercalating agents to change the twist of DNA during only one or in both runs, improves the capacity of 2D gels to discern molecules that apparently may look alike. Here we present protocols where 2D gels are used to understand the structure of DNA molecules and its dynamics in living cells. This knowledge is essential to comprehend how DNA topology affects and is affected by all the essential functions that DNA is involved in: replication, transcription, repair and recombination.

Supercoiled Minivector DNA resists shear forces associated with gene therapy delivery

Supercoiled DNAs varying from 281 to 5302 bp were subjected to shear forces generated by aerosolization or sonication. DNA shearing strongly correlated with length. Typical sized plasmids (≥ 3000 bp) degraded rapidly. DNAs 2000-3000 bp persisted ~10 min. Even in the absence of condensing agents, supercoiled DNA <1200 bp survived nebulization, and increased forces of sonication were necessary to shear it. Circular vectors were considerably more resistant to shearing than linear vectors of the same length. DNA supercoiling afforded additional protection. These results show the potential of shear-resistant Minivector DNAs to overcome one of the major challenges associated with gene therapy delivery.

Unlinking of supercoiled DNA catenanes by type IIA topoisomerases

It was found recently that DNA catenanes, formed during replication of circular plasmids, become positively (+) supercoiled, and the unlinking of such catenanes by type IIA topoisomerases proceeds much more efficiently than the unlinking of negatively (-) supercoiled catenanes. In an attempt to explain this striking finding we studied, by computer simulation, conformational properties of supercoiled DNA catenanes. Although the simulation showed that conformational properties of (+) and (-) supercoiled replication catenanes are very different, these properties per se do not give any advantage to (+) supercoiled over (-) supercoiled DNA catenanes for unlinking. An advantage became evident, however, when we took into account the established features of the enzymatic reaction catalyzed by the topoisomerases. The enzymes create a sharp DNA bend in the first bound DNA segment and allow for the transport of the second segment only from inside the bend to its outside. We showed that in (-) supercoiled DNA catenanes this protein-bound bent segment becomes nearly inaccessible for segments of the other linked DNA molecule, inhibiting the unlinking.

Cooperative kinking at distant sites in mechanically stressed DNA

In cells, DNA is routinely subjected to significant levels of bending and twisting. In some cases, such as under physiological levels of supercoiling, DNA can be so highly strained, that it transitions into non-canonical structural conformations that are capable of relieving mechanical stress within the template. DNA minicircles offer a robust model system to study stress-induced DNA structures. Using DNA minicircles on the order of 100 bp in size, we have been able to control the bending and torsional stresses within a looped DNA construct. Through a combination of cryo-EM image reconstructions, Bal31 sensitivity assays and Brownian dynamics simulations, we have been able to analyze the effects of biologically relevant underwinding-induced kinks in DNA on the overall shape of DNA minicircles. Our results indicate that strongly underwound DNA minicircles, which mimic the physical behavior of small regulatory DNA loops, minimize their free energy by undergoing sequential, cooperative kinking at two sites that are located about 180° apart along the periphery of the minicircle. This novel form of structural cooperativity in DNA demonstrates that bending strain can localize hyperflexible kinks within the DNA template, which in turn reduces the energetic cost to tightly loop DNA.

Effect of spontaneous twist on DNA minicircles

Monte Carlo simulations are used to study the effect of spontaneous (intrinsic) twist on the conformation of topologically equilibrated minicircles of dsDNA. The twist, writhe, and radius of gyration distributions and their moments are calculated for different spontaneous twist angles and DNA lengths. The average writhe and twist deviate in an oscillatory fashion (with the period of the double helix) from their spontaneous values, as one spans the range between two neighboring integer values of intrinsic twist. Such deviations vanish in the limit of long DNA plasmids.

Temperature dependence of DNA persistence length

We have determined the temperature dependence of DNA persistence length, a, using two different methods. The first approach was based on measuring the j-factors of short DNA fragments at various temperatures. Fitting the measured j-factors by the theoretical equation allowed us to obtain the values of a for temperatures between 5°C and 42°C. The second approach was based on measuring the equilibrium distribution of the linking number between the strands of circular DNA at different temperatures. The major contribution into the distribution variance comes from the fluctuations of DNA writhe in the nicked circular molecules which are specified by the value of a. The computation-based analysis of the measured variances was used to obtain the values of a for temperatures up to 60°C. We found a good agreement between the results obtained by these two methods. Our data show that DNA persistence length strongly depends on temperature and accounting for this dependence is important in quantitative comparison between experimental results obtained at different temperatures.

Group theory and biomolecular conformation: I. Mathematical and computational models

Biological macromolecules, and the complexes that they form, can be described in a variety of ways ranging from quantum mechanical and atomic chemical models, to coarser grained models of secondary structure and domains, to continuum models. At each of these levels, group theory can be used to describe both geometric symmetries and conformational motion. In this survey, a detailed account is provided of how group theory has been applied across computational structural biology to analyze the conformational shape and motion of macromolecules and complexes.

Dynamics of an electrostatically charged elastic rod in fluid

We investigate the effects of electrostatic and steric repulsion on the dynamics of a pre-twisted charged elastic rod immersed in a viscous incompressible fluid. Equations of motion of the rod include the fluid–structure interaction, rod elasticity and a combination of two interactions that prevent self-contact, namely the electrostatic interaction and hard-core repulsion. The governing equations are solved using the generalized immersed-boundary method. We find that after perturbation, a pre-twisted minicircle collapses into a compact supercoiled configuration. The collapse proceeds along a complex trajectory that may pass near several unstable equilibrium configurations, before it settles in a locally stable equilibrium. The dwell time near an unstable equilibrium can be up to several microseconds. Both the final configuration and the transition path are sensitive to the initial excess link, ionic strength of the solvent and the initial perturbation.

DNA supercoiling helps to unlink sister duplexes after replication

DNA supercoiling is one of the mechanisms that can help unlinking of newly replicated DNA molecules. Although DNA topoisomerases, which catalyze the strand passing of DNA segments through one another, make the unlinking problem solvable in principle, it remains difficult to complete the process that enables the separation of the sister duplexes. A few different mechanisms were developed by nature to solve the problem. Some of the mechanisms are very intuitive while the others, like topology simplification by type II DNA topoisomerases and DNA supercoiling, are not so evident. A computer simulation and analysis of linked sister plasmids formed in Escherichia coli cells with suppressed topoisomerase IV suggests an insight into the latter mechanism.

How stiff is DNA?

The natural stiffness of DNA, which contributes to the interactions of the many proteins involved in its biological processing and packaging, also plays an important role in modern nanotechnology. Here we report new Monte-Carlo simulations of deformable DNA molecules of potential utility in understanding the behavior of the long, double-helical polymer in the tight confines of a cell and in the design of novel nanomaterials and molecular devices. We directly determine the fluctuations in end-to-end extension associated with the conventional elastic-rod representation of DNA and with more realistic models that take account of the precise deformability of the constituent base-pair steps. Notably, the variance of end-to-end distance shows a quadratic increase with chain length in short chains of both types. We also consider the contributions to chain extension from the chemical linkages used to attach small molecular probes to DNA. The distribution of computed distances is sensitive to the intrinsic structure and allowed deformations of the tether. Surprisingly, the enhancement in end-to-end variance associated with the presence of the probe depends upon chain length, even when the probe is rigidly connected to DNA. We find that the elastic rod model of DNA in combination with a slightly fluctuating tether accounts satisfactorily for the distributions of end-to-end distances extracted from the small-angle X-ray scattering of gold nanocrystals covalently linked to the ends of short DNAs. There is no need to introduce additional structural fluctuations to reproduce the measured uptake in end-to-end fluctuations with chain length.

Simulation of DNA catenanes

DNA catenanes are important objects in biology, foremost as they appear during replication of circular DNA molecules. In this review we analyze how conformational properties of DNA catenanes can be studied by computer simulation. We consider classification of catenanes, their topological invariants and the methods of calculation of these invariants. We briefly analyze the DNA model and the simulation procedure used to sample the equilibrium conformational ensemble of catenanes with a particular topology. We consider how to avoid direct simulation of many DNA molecules when we need to account for the linking-unlinking process. The simulation methods and their comparisons with experiments are illustrated by some examples. We also describe an approach that allows simulating the steady state fraction of DNA catenanes created by type II topoisomerases.

Temperature dependence of circular DNA topological states

Circular double-stranded DNA has different topological states which are defined by their linking numbers. Equilibrium distribution of linking numbers can be obtained by closing a linear DNA into a circle by ligase. Using Monte Carlo simulation, we predict the temperature dependence of the linking number distribution of small circular DNAs. Our predictions are based on flexible defect excitations that resulted from local melting or unstacking of DNA base pairs. We found that the reduced bending rigidity alone can lead to measurable changes of the variance of linking number distribution of short circular DNAs. If the defect is accompanied by local unwinding, the effect becomes much more prominent. The predictions can be easily investigated in experiments, providing a new method to study the micromechanics of sharply bent DNAs and the thermal stability of specific DNA sequences. Furthermore, the predictions are directly applicable to the studies of binding of DNA-distorting proteins that can locally reduce DNA rigidity, form DNA kinks, or introduce local unwinding.

Effect of the matrix on DNA electrophoretic mobility

DNA electrophoretic mobilities are highly dependent on the nature of the matrix in which the separation takes place. This review describes the effect of the matrix on DNA separations in agarose gels, polyacrylamide gels and solutions containing entangled linear polymers, correlating the electrophoretic mobilities with information obtained from other types of studies. DNA mobilities in various sieving media are determined by the interplay of three factors: the relative size of the DNA molecule with respect to the effective pore size of the matrix, the effect of the electric field on the matrix, and specific interactions of DNA with the matrix during electrophoresis.