Looping dynamics of linear DNA molecules and the effect of DNA curvature: a study by Brownian dynamics simulation

Dissecting the general mechanisms of protein cage self-assembly by coarse-grained simulations

The development of artificial protein cages has recently gained massive attention due to their promising application prospect as novel delivery vehicles for therapeutics. These nanoparticles are formed through a process called self-assembly, in which individual subunits spontaneously arrange into highly ordered patterns via non-covalent but specific interactions. Therefore, the first step toward the design of novel engineered protein cages is to understand the general mechanisms of their self-assembling dynamics. Here we have developed a new computational method to tackle this problem. Our method is based on a coarse-grained model and a diffusion-reaction simulation algorithm. Using a tetrahedral cage as test model, we showed that self-assembly of protein cage requires of a seeding process in which specific configurations of kinetic intermediate states are identified. We further found that there is a critical concentration to trigger self-assembly of protein cages. This critical concentration allows that cages can only be successfully assembled under a persistently high concentration. Additionally, phase diagram of self-assembly has been constructed by systematically testing the model across a wide range of binding parameters. Finally, our simulations demonstrated the importance of protein's structural flexibility in regulating the dynamics of cage assembly. In summary, this study throws lights on the general principles underlying self-assembly of large cage-like protein complexes and thus provides insights to design new nanomaterials.

Classification of protein-protein association rates based on biophysical informatics

BACKGROUND

Proteins form various complexes to carry out their versatile functions in cells. The dynamic properties of protein complex formation are mainly characterized by the association rates which measures how fast these complexes can be formed. It was experimentally observed that the association rates span an extremely wide range with over ten orders of magnitudes. Identification of association rates within this spectrum for specific protein complexes is therefore essential for us to understand their functional roles.

RESULTS

To tackle this problem, we integrate physics-based coarse-grained simulations into a neural-network-based classification model to estimate the range of association rates for protein complexes in a large-scale benchmark set. The cross-validation results show that, when an optimal threshold was selected, we can reach the best performance with specificity, precision, sensitivity and overall accuracy all higher than 70%. The quality of our cross-validation data has also been testified by further statistical analysis. Additionally, given an independent testing set, we can successfully predict the group of association rates for eight protein complexes out of ten. Finally, the analysis of failed cases suggests the future implementation of conformational dynamics into simulation can further improve model.

CONCLUSIONS

In summary, this study demonstrated that a new modeling framework that combines biophysical simulations with bioinformatics approaches is able to identify protein-protein interactions with low association rates from those with higher association rates. This method thereby can serve as a useful addition to a collection of existing experimental approaches that measure biomolecular recognition.

Understanding the Impacts of Conformational Dynamics on the Regulation of Protein-Protein Association by a Multiscale Simulation Method

Complexes formed among diverse proteins carry out versatile functions in nearly all physiological processes. Association rates which measure how fast proteins form various complexes are of fundamental importance to characterize their functions. The association rates are not only determined by the energetic features at binding interfaces of a protein complex but also influenced by the intrinsic conformational dynamics of each protein in the complex. Unfortunately, how this conformational effect regulates protein association has never been calibrated on a systematic level. To tackle this problem, we developed a multiscale strategy to incorporate the information on protein conformational variations from Langevin dynamic simulations into a kinetic Monte Carlo algorithm of protein-protein association. By systematically testing this approach against a large-scale benchmark set, we found the association of a protein complex with a relatively rigid structure tends to be reduced by its conformational fluctuations. With specific examples, we further show that higher degrees of structural flexibility in various protein complexes can facilitate the searching and formation of intermolecular interactions and thereby accelerate their associations. In general, the integration of conformational dynamics can improve the correlation between experimentally measured association rates and computationally derived association probabilities. Finally, we analyzed the statistical distributions of different secondary structural types on protein-protein binding interfaces and their preference to the change of association rates. Our study, to the best of our knowledge, is the first computational method that systematically estimates the impacts of protein conformational dynamics on protein-protein association. It throws lights on the molecular mechanisms of how protein-protein recognition is kinetically modulated.

Using Coarse-Grained Simulations to Characterize the Mechanisms of Protein-Protein Association

The formation of functionally versatile protein complexes underlies almost every biological process. The estimation of how fast these complexes can be formed has broad implications for unravelling the mechanism of biomolecular recognition. This kinetic property is traditionally quantified by association rates, which can be measured through various experimental techniques. To complement these time-consuming and labor-intensive approaches, we developed a coarse-grained simulation approach to study the physical processes of protein–protein association. We systematically calibrated our simulation method against a large-scale benchmark set. By combining a physics-based force field with a statistically-derived potential in the simulation, we found that the association rates of more than 80% of protein complexes can be correctly predicted within one order of magnitude relative to their experimental measurements. We further showed that a mixture of force fields derived from complementary sources was able to describe the process of protein–protein association with mechanistic details. For instance, we show that association of a protein complex contains multiple steps in which proteins continuously search their local binding orientations and form non-native-like intermediates through repeated dissociation and re-association. Moreover, with an ensemble of loosely bound encounter complexes observed around their native conformation, we suggest that the transition states of protein–protein association could be highly diverse on the structural level. Our study also supports the idea in which the association of a protein complex is driven by a “funnel-like” energy landscape. In summary, these results shed light on our understanding of how protein–protein recognition is kinetically modulated, and our coarse-grained simulation approach can serve as a useful addition to the existing experimental approaches that measure protein–protein association rates.

A Multiscale Computational Model for Simulating the Kinetics of Protein Complex Assembly

Proteins fulfill versatile biological functions by interacting with each other and forming high-order complexes. Although the order in which protein subunits assemble is important for the biological function of their final complex, this kinetic information has received comparatively little attention in recent years. Here we describe a multiscale framework that can be used to simulate the kinetics of protein complex assembly. There are two levels of models in the framework. The structural details of a protein complex are reflected by the residue-based model, while a lower-resolution model uses a rigid-body (RB) representation to simulate the process of complex assembly. These two levels of models are integrated together, so that we are able to provide the kinetic information about complex assembly with both structural details and computational efficiency.

Jörg Langowski: his scientific legacy and the future it promises

Background

With the passing of Jörg Langowski 6 May 2017 in a sailplane accident, the scientific community was deprived of a strident and effective voice for DNA and chromatin molecular and computational biophysics, for open access publishing and for the creation of effective scientific research networks.

Methods

Here, after reviewing some of Jörg's key research contributions and ideas, we offer through the personal remembrance of his closest collaborators, a deep analysis of the major results of his research and the future directions they have engendered.

Conclusions

The legacy of Jörg Langowski has been to propel a way of viewing biological function that considers living systems as dynamic and in three dimensions. This physical view of biology that he pioneered is now, finally, becoming established also because of his great effort.

Out of the Randomness: Correlating Noise in Biological Systems

The study of the dynamics of biological systems requires one to follow relaxation processes in time with micron-size spatial resolution. This need has led to the development of different fluorescence correlation techniques with high spatial resolution and a tremendous (from nanoseconds to seconds) temporal dynamic range. Spatiotemporal information can be obtained even on complex dynamic processes whose time evolution is not forecast by simple Brownian diffusion. Our discussion of the most recent applications of image correlation spectroscopy to the study of anomalous sub- or superdiffusion suggests that this field still requires the development of multidimensional image analyses based on analytical models or numerical simulations. We focus in particular on the framework of spatiotemporal image correlation spectroscopy and examine the critical steps in getting information on anomalous diffusive processes from the correlation maps. We point out how a dual space-time correlative analysis, in both the direct and the Fourier space, can provide quantitative information on superdiffusional processes when these are analyzed through an empirical model based on intermittent active dynamics. We believe that this dual space-time analysis, potentially amenable to mathematical treatment and to the exact fit of experimental data, could be extended to include the rich phenomenology of subdiffusive processes, thereby quantifying relevant parameters for the various motivating biological problems of interest.

Analytical expressions for the closure probability of a stiff wormlike chain for finite capture radius

Estimating the probability that two monomers of the same polymer chain are close together is a key ingredient to characterize intramolecular reactions and polymer looping. In the case of stiff wormlike polymers (rigid fluctuating elastic rods), for which end-to-end encounters are rare events, we derive an explicit analytical formula for the probability η(r_{c}) that the distance between the chain extremities is smaller than some capture radius r_{c}. The formula is asymptotically exact in the limit of stiff chains, and it leads to the identification of two distinct scaling regimes for the closure factor, originating from a strong variation of the fluctuations of the chain orientation at closure. Our theory is compatible with existing analytical results from the literature that cover the cases of a vanishing capture radius and of nearly fully extended chains.

Interplay of Protein Binding Interactions, DNA Mechanics, and Entropy in DNA Looping Kinetics

DNA looping plays a key role in many fundamental biological processes, including gene regulation, recombination, and chromosomal organization. The looping of DNA is often mediated by proteins whose structural features and physical interactions can alter the length scale at which the looping occurs. Looping and unlooping processes are controlled by thermodynamic contributions associated with mechanical deformation of the DNA strand and entropy arising from thermal fluctuations of the conformation. To determine how these confounding effects influence DNA looping and unlooping kinetics, we present a theoretical model that incorporates the role of the protein interactions, DNA mechanics, and conformational entropy. We show that for shorter DNA strands the interaction distance affects the transition state, resulting in a complex relationship between the looped and unlooped state lifetimes and the physical properties of the looped DNA. We explore the range of behaviors that arise with varying interaction distance and DNA length. These results demonstrate how DNA deformation and entropy dictate the scaling of the looping and unlooping kinetics versus the J-factor, establishing the connection between kinetic and equilibrium behaviors. Our results show how the twist-and-bend elasticity of the DNA chain modulates the kinetics and how the influence of the interaction distance fades away at intermediate to longer chain lengths, in agreement with previous scaling predictions.

Sequence Affects the Cyclization of DNA Minicircles

Understanding how the sequence of a DNA molecule affects its dynamic properties is a central problem affecting biochemistry and biotechnology. The process of cyclizing short DNA, as a critical step in molecular cloning, lacks a comprehensive picture of the kinetic process containing sequence information. We have elucidated this process by using coarse-grained simulations, enhanced sampling methods, and recent theoretical advances. We are able to identify the types and positions of structural defects during the looping process at a base-pair level. Correlations along a DNA molecule dictate critical sequence positions that can affect the looping rate. Structural defects change the bending elasticity of the DNA molecule from a harmonic to subharmonic potential with respect to bending angles. We explore the subelastic chain as a possible model in loop formation kinetics. A sequence-dependent model is developed to qualitatively predict the relative loop formation time as a function of DNA sequence.

Establishing epigenetic domains via chromatin-bound histone modifiers

The eukaryotic nucleus harbors the DNA genome, which associates with histones and other chromosomal proteins into a complex referred to as chromatin. It provides an additional layer of so-called epigenetic information via histone modifications and DNA methylation on top of the DNA sequence that determines the cell's active gene expression program. The nucleus is devoid of internal organelles separated by membranes. Thus, free diffusive transport of proteins and RNA can occur throughout the space accessible for a given macromolecule. At the same time, chromatin is partitioned into different specialized structures such as nucleoli, chromosome territories, and heterochromatin domains that serve distinct functions. Here, we address the question of how the activity of chromatin-modifying enzymes is confined to chromatin subcompartments. We discuss mechanisms for establishing activity gradients of diffusive chromatin-modifying enzymes that could give rise to distinct chromatin domains within the cell nucleus. Interestingly, such gradients might directly result from immobilization of the enzymes on the flexible chromatin chain. Thus, locus-specific tethering of these enzymes to chromatin could have the potential to establish, maintain, or modulate epigenetic patterns of characteristic domain size.

Bullied no more: when and how DNA shoves proteins around

The predominant protein-centric perspective in protein-DNA-binding studies assumes that the protein drives the interaction. Research focuses on protein structural motifs, electrostatic surfaces and contact potentials, while DNA is often ignored as a passive polymer to be manipulated. Recent studies of DNA topology, the supercoiling, knotting, and linking of the helices, have shown that DNA has the capability to be an active participant in its transactions. DNA topology-induced structural and geometric changes can drive, or at least strongly influence, the interactions between protein and DNA. Deformations of the B-form structure arise from both the considerable elastic energy arising from supercoiling and from the electrostatic energy. Here, we discuss how these energies are harnessed for topology-driven, sequence-specific deformations that can allow DNA to direct its own metabolism.

Characterization of the geometry and topology of DNA pictured as a discrete collection of atoms

The structural and physical properties of DNA are closely related to its geometry and topology. The classical mathematical treatment of DNA geometry and topology in terms of ideal smooth space curves was not designed to characterize the spatial arrangements of atoms found in high-resolution and simulated double-helical structures. We present here new and rigorous numerical methods for the rapid and accurate assessment of the geometry and topology of double-helical DNA structures in terms of the constituent atoms. These methods are well designed for large DNA datasets obtained in detailed numerical simulations or determined experimentally at high-resolution. We illustrate the usefulness of our methodology by applying it to the analysis of three canonical double-helical DNA chains, a 65-bp minicircle obtained in recent molecular dynamics simulations, and a crystallographic array of protein-bound DNA duplexes. Although we focus on fully base-paired DNA structures, our methods can be extended to treat the geometry and topology of melted DNA structures as well as to characterize the folding of arbitrary molecules such as RNA and cyclic peptides.

Building enhancers from the ground up: a synthetic biology approach

A challenge of the synthetic biology approach is to use our understanding of a system to recreate a biological function with specific properties. We have applied this framework to bacterial enhancers, combining a driver, transcription factor binding sites, and a poised polymerase to create synthetic modular enhancers. Our findings suggest that enhancer-based transcriptional control depends critically and quantitatively on DNA looping, leading to complex regulatory effects when the enhancer cassettes contain additional transcription factor binding sites for TetR, a bacterial transcription factor. We show through a systematic interplay of experiment and thermodynamic modeling that the level of gene expression can be modulated to convert a variable inducer concentration input into discrete or step-like output expression levels. Finally, using a different DNA-binding protein (TraR), we show that the regulatory output is not a particular feature of the specific DNA-binding protein used for the enhancer but a general property of synthetic bacterial enhancers.

Twist propagation in dinucleosome arrays

We present a Monte Carlo simulation study of the distribution and propagation of twist from one DNA linker to another for a two-nucleosome array subjected to externally applied twist. A mesoscopic model of the array that incorporates nucleosome geometry along with the bending and twisting mechanics of the linkers is employed and external twist is applied in stepwise increments to mimic quasistatic twisting of chromatin fibers. Simulation results reveal that the magnitude and sign of the imposed and induced twist on contiguous linkers depend strongly on their relative orientation. Remarkably, the relative direction of the induced and applied twist can become inverted for a subset of linker orientations-a phenomenon we refer to as "twist inversion". We characterize the twist inversion, as a function of relative linker orientation, in a phase diagram and explain its key features using a simple model based on the geometry of the nucleosome/linker complex. In addition to twist inversion, our simulations reveal "nucleosome flipping", whereby nucleosomes may undergo sudden flipping in response to applied twist, causing a rapid bending of the linker and a significant change in the overall twist and writhe of the array. Our findings shed light on the underlying mechanisms by which torsional stresses impact chromatin organization.

Filamentous biopolymers on surfaces: atomic force microscopy images compared with Brownian dynamics simulation of filament deposition

Nanomechanical properties of filamentous biopolymers, such as the persistence length, may be determined from two-dimensional images of molecules immobilized on surfaces. For a single filament in solution, two principal adsorption scenarios are possible. Both scenarios depend primarly on the interaction strength between the filament and the support: i) For interactions in the range of the thermal energy, the filament can freely equilibrate on the surface during adsorption; ii) For interactions much stronger than the thermal energy, the filament will be captured by the surface without having equilibrated. Such a ‘trapping’ mechanism leads to more condensed filament images and hence to a smaller value for the apparent persistence length. To understand the capture mechanism in more detail we have performed Brownian dynamics simulations of relatively short filaments by taking the two extreme scenarios into account. We then compared these ‘ideal’ adsorption scenarios with observed images of immobilized vimentin intermediate filaments on different surfaces. We found a good agreement between the contours of the deposited vimentin filaments on mica (‘ideal’ trapping) and on glass (‘ideal’ equilibrated) with our simulations. Based on these data, we have developed a strategy to reliably extract the persistence length of short worm-like chain fragments or network forming filaments with unknown polymer-surface interactions.

Nucleosome disassembly intermediates characterized by single-molecule FRET

The nucleosome has a central role in the compaction of genomic DNA and the control of DNA accessibility for transcription and replication. To help understanding the mechanism of nucleosome opening and closing in these processes, we studied the disassembly of mononucleosomes by quantitative single-molecule FRET with high spatial resolution, using the SELEX-generated "Widom 601" positioning sequence labeled with donor and acceptor fluorophores. Reversible dissociation was induced by increasing NaCl concentration. At least 3 species with different FRET were identified and assigned to structures: (i) the most stable high-FRET species corresponding to the intact nucleosome, (ii) a less stable mid-FRET species that we attribute to a first intermediate with a partially unwrapped DNA and less histones, and (iii) a low-FRET species characterized by a very broad FRET distribution, representing highly unwrapped structures and free DNA formed at the expense of the other 2 species. Selective FCS analysis indicates that even in the low-FRET state, some histones are still bound to the DNA. The interdye distance of 54.0 A measured for the high-FRET species corresponds to a compact conformation close to the known crystallographic structure. The coexistence and interconversion of these species is first demonstrated under non-invasive conditions. A geometric model of the DNA unwinding predicts the presence of the observed FRET species. The different structures of these species in the disassembly pathway map the energy landscape indicating major barriers for 10-bp and minor ones for 5-bp DNA unwinding steps.

First-principles calculation of DNA looping in tethered particle experiments

We calculate the probability of DNA loop formation mediated by regulatory proteins such as Lac repressor (LacI), using a mathematical model of DNA elasticity. Our model is adapted to calculating quantities directly observable in tethered particle motion (TPM) experiments, and it accounts for all the entropic forces present in such experiments. Our model has no free parameters; it characterizes DNA elasticity using information obtained in other kinds of experiments. It assumes a harmonic elastic energy function (or wormlike chain type elasticity), but our Monte Carlo calculation scheme is flexible enough to accommodate arbitrary elastic energy functions. We show how to compute both the 'looping J factor' (or equivalently, the looping free energy) for various DNA construct geometries and LacI concentrations, as well as the detailed probability density function of bead excursions. We also show how to extract the same quantities from recent experimental data on TPM, and then compare to our model's predictions. In particular, we present a new method to correct observed data for finite camera shutter time and other experimental effects. Although the currently available experimental data give large uncertainties, our first-principles predictions for the looping free energy change are confirmed to within about 1 k(B)T, for loops of length around 300 basepairs. More significantly, our model successfully reproduces the detailed distributions of bead excursion, including their surprising three-peak structure, without any fit parameters and without invoking any alternative conformation of the LacI tetramer. Indeed, the model qualitatively reproduces the observed dependence of these distributions on tether length (e.g., phasing) and on LacI concentration (titration). However, for short DNA loops (around 95 basepairs) the experiments show more looping than is predicted by the harmonic-elasticity model, echoing other recent experimental results. Because the experiments we study are done in vitro, this anomalously high looping cannot be rationalized as resulting from the presence of DNA-bending proteins or other cellular machinery. We also show that it is unlikely to be the result of a hypothetical 'open' conformation of the LacI tetramer.

Geometry of mediating protein affects the probability of loop formation in DNA

Recent single molecule experiments have determined the probability of loop formation in DNA as a function of the DNA contour length for different types of looping proteins. The optimal contour length for loop formation as well as the probability density functions have been found to be strongly dependent on the type of looping protein used. We show, using Monte Carlo simulations and analytical calculations, that these observations can be replicated using the wormlike-chain model for double-stranded DNA if we account for the nonzero size of the looping protein. The simulations have been performed in two dimensions so that bending is the only mode of deformation available to the DNA while the geometry of the looping protein enters through a single variable which is representative of its size. We observe two important effects that seem to directly depend on the size of the enzyme: 1), the overall propensity of loop formation at any given value of the DNA contour length increases with the size of the enzyme; and 2), the contour length corresponding to the first peak as well as the first well in the probability density functions increases with the size of the enzyme. Additionally, the eigenmodes of the fluctuating shape of the looped DNA calculated from simulations and theory are in excellent agreement, and reveal that most of the fluctuations in the DNA occur in regions of low curvature.

Brownian dynamics of double-stranded DNA in periodic systems with discrete salt

Numerical models of mesoscale DNA dynamics relevant to in vivo scenarios require methods that incorporate important features of the intracellular environment, while maintaining computational tractability. Because the explicit inclusion of ions leads to electrostatic calculations that scale as the square of the number of charged particles, such models typically handle these calculations using low-potential, mean-field approaches, rather than by considering the discrete interactions of ions. This allows approximation of the long-range, screened self-repulsion of DNA, but is unable to capture detailed electrostatic phenomena, such as short-range attractions mediated by ion-ion correlations. Here, we develop a dynamical model of explicitly double-stranded, sequence-specific DNA in a bulk environment consisting of other polyions and explicitly represented counterions and coions. DNA is represented as two interwound chains of charged Stokes spheres, and ions as free, monovalently charged Stokes spheres. Brownian dynamics simulations performed at salt concentrations of 0.1, 1, 10, and 100 mM demonstrate this model captures anticipated behaviors of the system, including increasing compaction of the polyion by the ionic atmosphere with increasing ionic strength. The decay of the distance dependence of the ion concentrations as one moves away from the polyion approaches their equilibrium values in quantitative agreement with predictions of Poisson-Boltzmann theory. The simulation results also demonstrate quantitative agreement with experimental measurements of the persistence length of B-DNA, which increases significantly at low ionic strengths. The model also captures behaviors intimating the importance of explicitly representing ionic and polyionic structure. These include penetration of the polyion interior by both coions and counterions, and counterion-mediated accumulation of coions near the surface of the polyion. Such phenomena are likely to play an important role in the formation of alternative DNA secondary structures, suggesting the present methods will prove valuable to dynamic models of superhelical stress-induced DNA structural transitions.

Intrinsic curvature of DNA influences LacR-mediated looping

Protein-mediated DNA looping is a common mechanism for regulating gene expression. Loops occur when a protein binds to two operators on the same DNA molecule. The probability of looping is controlled, in part, by the basepair sequence of inter-operator DNA, which influences its structural properties. One structural property is the intrinsic or stress-free curvature. In this article, we explore the influence of sequence-dependent intrinsic curvature by exercising a computational rod model for the inter-operator DNA as applied to looping of the LacR-DNA complex. Starting with known sequences for the inter-operator DNA, we first compute the intrinsic curvature of the helical axis as input to the rod model. The crystal structure of the LacR (with bound operators) then defines the requisite boundary conditions needed for the dynamic rod model that predicts the energetics and topology of the intervening DNA loop. A major contribution of this model is its ability to predict a broad range of published experimental data for highly bent (designed) sequences. The model successfully predicts the loop topologies known from fluorescence resonance energy transfer measurements, the linking number distribution known from cyclization assays with the LacR-DNA complex, the relative loop stability known from competition assays, and the relative loop size known from gel mobility assays. In addition, the computations reveal that highly curved sequences tend to lower the energetic cost of loop formation, widen the energy distribution among stable and meta-stable looped states, and substantially alter loop topology. The inclusion of sequence-dependent intrinsic curvature also leads to nonuniform twist and necessitates consideration of eight distinct binding topologies from the known crystal structure of the LacR-DNA complex.

Computational modeling of the chromatin fiber

The packing of the genomic DNA in the living cell is essential for its biological function. While individual aspects of the genome architecture, such as DNA and nucleosome structure or the arrangement of chromosome territories are well studied, much information is missing for a unified description of cellular DNA at all its structural levels. Computer modeling can contribute to such a description. We present here some typical approaches to models of the chromatin fiber, including different amounts of detail in the description of the local nucleosome structure. The main results from our simulations are that the physical properties of the chromatin fiber can be well described by a simplified model consisting of cylinder-like nucleosomes connected by flexible DNA segments, with a geometry determined by the bending and twisting angles between nucleosomes. Randomness in the local geometry - such as random absence of linker histone H1 - leads to a dramatic increase in the chromatin fiber flexibility. Furthermore, we show that chromatin is much more flexible to bending than to stretching, and that the structure of the chromatin fiber favors the formation of sharp bends.

Kinetics of interior loop formation in semiflexible chains

Loop formation between monomers in the interior of semiflexible chains describes elementary events in biomolecular folding and DNA bending. We calculate analytically the interior distance distribution function for semiflexible chains using a mean field approach. Using the potential of mean force derived from the distance distribution function we present a simple expression for the kinetics of interior looping by adopting Kramers theory. For the parameters, that are appropriate for DNA, the theoretical predictions in comparison with the case are in excellent agreement with explicit Brownian dynamics simulations of wormlike chain (WLC) model. The interior looping times (tauIC) can be greatly altered in the cases when the stiffness of the loop differs from that of the dangling ends. If the dangling end is stiffer than the loop then tauIC increases for the case of the WLC with uniform persistence length. In contrast, attachment of flexible dangling ends enhances rate of interior loop formation. The theory also shows that if the monomers are charged and interact via screened Coulomb potential then both the cyclization (tauc) and interior looping (tauIC) times greatly increase at low ionic concentration. Because both tauc and tauIC are determined essentially by the effective persistence length [lp(R)] we computed lp(R) by varying the range of the repulsive interaction between the monomers. For short range interactions lp(R) nearly coincides with the bare persistence length which is determined largely by the backbone chain connectivity. This finding rationalizes the efficacy of describing a number of experimental observations (response of biopolymers to force and cyclization kinetics) in biomolecules using WLC model with an effective persistence length.

Gradual melting of a replication origin (Schizosaccharomyces pombe ars1): in situ atomic force microscopy (AFM) analysis

Local DNA melting is integral to fundamental processes such as replication or transcription. In vivo, these two processes do not occur on molecules free in solution but, instead, involve DNA molecules which are organized into DNA/proteins complexes. Atomic force microscopy imaging offers a possibility to look at individual molecules. It allowed us to follow the progress of local denaturation in liquid, but with the added constraints of DNA lying on a surface. We present a kinetic analysis of the mapping of the temperature-driven melting seen at a replication origin (Schizosaccharomyces pombe ars1). The results indicate an expected base composition dependency, but also a strong extremity effect. Noteworthy, a "structural" effect is clearly occurring - which is shown by the greater susceptibility of the strongly curved region present in the sequence to unwind. DNA melting, at this place, is seen to occur after an increase in the curvature amplitude and a simultaneous shift of the nucleotide sequence positioned at the apex. Because this may determine the position of the Replication Initiation (R.I.) site, the result suggests that eukaryotic replication origins, although described as possessing no consensus sequences, may well have their mechanics sustained by the properties of common structural features. Our analysis may, therefore, provide new information that will give genuine insights on how DNA molecules behave when organized into primosomes, replisomes, promoter initiation complexes, etc. and thus, be essential to better understanding the way genes function.

Dynamics of single DNA looping and cleavage by Sau3AI and effect of tension applied to the DNA

Looping and cleavage of single DNA molecules by the two-site restriction endonuclease Sau3AI were measured with optical tweezers. A DNA template containing many recognition sites was used, permitting loop sizes from approximately 10 to 10,000 basepairs. At high enzyme concentration, cleavage events were detected within 5 s and nearly all molecules were cleaved within 5 min. Activity decreased approximately 10-fold as the DNA tension was increased from 0.03 to 0.7 pN. Substituting Ca(2+) for Mg(2+) blocked cleavage, permitting measurement of stable loops. At low tension, the initial rates of cleavage and looping were similar (approximately 0.025 s(-1) at 0.1 pN), suggesting that looping is rate limiting. Short loops formed more rapidly than long loops. The optimum size decreased from approximately 250 to 45 basepairs and the average number of loops (in 1 min) from 4.2 to 0.75 as tension was increased from 0.03 to 0.7 pN. No looping was detected at 5 pN. These findings are in qualitative agreement with recent theoretical predictions considering only DNA mechanics, but we observed weaker suppression with tension and smaller loop sizes. Our results suggest that the span and elasticity of the protein complex, nesting of loops, and protein-induced DNA bending and wrapping play an important role.

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.

Tension-dependent DNA cleavage by restriction endonucleases: two-site enzymes are "switched off" at low force

DNA looping occurs in many important protein-DNA interactions, including those regulating replication, transcription, and recombination. Recent theoretical studies predict that tension of only a few piconewtons acting on DNA would almost completely inhibit DNA looping. Here, we study restriction endonucleases that require interaction at two separated sites for efficient cleavage. Using optical tweezers we measured the dependence of cleavage activity on DNA tension with 15 known or suspected two-site enzymes (BfiI, BpmI, BsgI, BspMI, Cfr9I, Cfr10I, Eco57I, EcoRII, FokI, HpaII, MboII, NarI, SacII, Sau3AI, and SgrAI) and six one-site enzymes (BamHI, EcoRI, EcoRV, HaeIII, HindIII, and DNaseI). All of the one-site enzymes were virtually unaffected by 5 pN of tension, whereas all of the two-site enzymes were completely inhibited. These enzymes thus constitute a remarkable example of a tension sensing "molecular switch." A detailed study of one enzyme, Sau3AI, indicated that the activity decreased exponentially with tension and the decrease was approximately 10-fold at 0.7 pN. At higher forces (approximately 20-40 pN) cleavage by the one-site enzymes EcoRV and HaeIII was partly inhibited and cleavage by HindIII was enhanced, whereas BamHI, EcoRI, and DNaseI were largely unaffected. These findings correlate with structural data showing that EcoRV bends DNA sharply, whereas BamHI, EcoRI, and DNaseI do not. Thus, DNA-directed enzyme activity involving either DNA looping or bending can be modulated by tension, a mechanism that could facilitate mechanosensory transduction in vivo.

Lac repressor hinge flexibility and DNA looping: single molecule kinetics by tethered particle motion

The tethered particle motion (TPM) allows the direct detection of activity of a variety of biomolecules at the single molecule level. First pioneered for RNA polymerase, it has recently been applied also to other enzymes. In this work we employ TPM for a systematic investigation of the kinetics of DNA looping by wild-type Lac repressor (wt-LacI) and by hinge mutants Q60G and Q60 + 1. We implement a novel method for TPM data analysis to reliably measure the kinetics of loop formation and disruption and to quantify the effects of the protein hinge flexibility and of DNA loop strain on such kinetics. We demonstrate that the flexibility of the protein hinge has a profound effect on the lifetime of the looped state. Our measurements also show that the DNA bending energy plays a minor role on loop disruption kinetics, while a strong effect is seen on the kinetics of loop formation. These observations substantiate the growing number of theoretical studies aimed at characterizing the effects of DNA flexibility, tension and torsion on the kinetics of protein binding and dissociation, strengthening the idea that these mechanical factors in vivo may play an important role in the modulation of gene expression regulation.

DNA looping by two-site restriction endonucleases: heterogeneous probability distributions for loop size and unbinding force

Proteins interacting at multiple sites on DNA via looping play an important role in many fundamental biochemical processes. Restriction endonucleases that must bind at two recognition sites for efficient activity are a useful model system for studying such interactions. Here we used single DNA manipulation to study sixteen known or suspected two-site endonucleases. In eleven cases (BpmI, BsgI, BspMI, Cfr10I, Eco57I, EcoRII, FokI, HpaII, NarI, Sau3AI and SgrAI) we found that substitution of Ca2+ for Mg2+ blocked cleavage and enabled us to observe stable DNA looping. Forced disruption of these loops allowed us to measure the frequency of looping and probability distributions for loop size and unbinding force for each enzyme. In four cases we observed bimodal unbinding force distributions, indicating conformational heterogeneity and/or complex binding energy landscapes. Measured unlooping events ranged in size from 7 to 7500 bp and the most probable size ranged from less than 75 bp to nearly 500 bp, depending on the enzyme. In most cases the size distributions were in much closer agreement with theoretical models that postulate sharp DNA kinking than with classical models of DNA elasticity. Our findings indicate that DNA looping is highly variable depending on the specific protein and does not depend solely on the mechanical properties of DNA.

Modeling a self-avoiding chromatin loop: relation to the packing problem, action-at-a-distance, and nuclear context

There is now convincing evidence that genomes are organized into loops, and that looping brings distant genes together so that they can bind to local concentrations of polymerases in "factories" or "hubs." As there remains no systematic analysis of how looping affects the probability that a gene can access binding sites in such factories/hubs, we used an algorithm that we devised and Monte Carlo methods to model a DNA or chromatin loop as a semiflexible (self-avoiding) tube attached to a sphere; we examine how loop thickness, rigidity, and contour length affect where particular segments of the loop lie relative to binding sites on the sphere. Results are compared with those obtained with the traditional model of an (infinitely thin) freely jointed chain. They provide insights into the packing problem (how long genomes are packed into small nuclei), and action-at-a-distance (how firing of one origin or gene can prevent firing of an adjacent one).

Do femtonewton forces affect genetic function? A review

Protein-Mediated DNA looping is intricately related to gene expression. Therefore any mechanical constraint that disrupts loop formation can play a significant role in gene regulation. Polymer physics models predict that less than a piconewton of force may be sufficient to prevent the formation of DNA loops. Thus, it appears that tension can act as a molecular switch that controls the much larger forces associated with the processive motion of RNA polymerase. Since RNAP can exert forces over 20 pN before it stalls, a 'substrate tension switch' could offer a force advantage of two orders of magnitude. Evidence for such a mechanism is seen in recent in vitro micromanipulation experiments. In this article we provide new perspective on existing theory and experimental data on DNA looping in vitro and in vivo. We elaborate on the connection between tension and a variety of other intracellular mechanical constraints including sequence specific curvature and supercoiling. In the process, we emphasize that the richness and versatility of DNA mechanics opens up a whole new paradigm of gene regulation to explore.

Loops in DNA: an overview of experimental and theoretical approaches

DNA loop formation plays a central role in many cellular processes. The aim of this paper is to present the state of the art and open problems regarding the experimental and theoretical approaches to DNA looping. A particular attention is devoted to the effects of the protein bridge size and of protein induced sharp DNA bending on DNA loop formation enhancement.

Protein-mediated DNA loops: effects of protein bridge size and kinks

This paper focuses on the probability that a portion of DNA closes on itself through thermal fluctuations. We investigate the dependence of this probability upon the size of a protein bridge and/or the presence of a kink at half DNA length. The DNA is modeled by the wormlike chain model, and the probability of loop formation is calculated in two ways: exact numerical evaluation of the constrained path integral and the extension of the Shimada and Yamakawa saddle point approximation. For example, we find that the looping free energy of a 100-base-pairs DNA decreases from 24 kBT to 13 kBT when the loop is closed by a protein of r=10 length. It further decreases 5 kBT to when the loop has a kink of 120 degrees at half-length.

Modeling of intramolecular reactions of polymers: An efficient method based on Brownian dynamics simulations

By the traditional approach to the Brownian dynamics simulations of intrachain reactions of polymers, the initial chain conformation is sampled from the equilibrium distribution. A dynamic trajectory is carried out until a "collision" of the reactive groups takes place, i.e., the distance between their centers becomes less that a certain reaction radius. The average length of the trajectory is equal to the mean time tauF of a diffusion-controlled reaction. In this work we propose another computational scheme. The trajectory begins at the instant of collision and is carried out until the chain is relaxed. The length of the trajectory has the order of the relaxation time taurel of the distance between the reactive groups. For polymer systems with taurel << tauF, this scheme allows the computation of tauF with considerable gain in computational time. Using the present approach, we calculated the mean time of DNA cyclization for the molecule length in the range from 50 to 500 nm.

Modelling DNA loops using continuum and statistical mechanics

The classical Kirchhoff elastic-rod model applied to DNA is extended to account for sequence-dependent intrinsic twist and curvature, anisotropic bending rigidity, electrostatic force interactions, and overdamped Brownian motion in a solvent. The zero-temperature equilibrium rod model is then applied to study the structural basis of the function of the lac repressor protein in the lac operon of Escherichia coli. The structure of a DNA loop induced by the clamping of two distant DNA operator sites by lac repressor is investigated and the optimal geometries for the loop of length 76 bp are predicted. Further, the mimicked binding of catabolite gene activator protein (CAP) inside the loop provides solutions that might explain the experimentally observed synergy in DNA binding between the two proteins. Finally, a combined Monte Carlo and Brownian dynamics solver for a worm-like chain model is described and a preliminary analysis of DNA loop-formation kinetics is presented.

Spontaneous sharp bending of double-stranded DNA

Sharply bent DNA is essential for gene regulation in prokaryotes and is a major feature of eukaryotic nucleosomes and viruses. The explanation normally given for these phenomena is that specific proteins sharply bend DNA by application of large forces, while the DNA follows despite its intrinsic inflexibility. Here we show that DNAs that are 94 bp in length-comparable to sharply looped DNAs in vivo-spontaneously bend into circles. Proteins can enhance the stability of such loops, but the loops occur spontaneously even in naked DNA. Random DNA sequences cyclize 10(2)-10(4) times more easily than predicted from current theories of DNA bending, while DNA sequences that position nucleosomes cyclize up to 10(5) times more easily. These unexpected results establish DNA as an active participant in the formation of looped regulatory complexes in vivo, and they point to a need for new theories of DNA bending.

Dynamics of DNA loop capture by the SfiI restriction endonuclease on supercoiled and relaxed DNA

The SfiI endonuclease is a prototype for DNA looping. It binds two copies of its recognition sequence and, if Mg(2+) is present, cuts both concertedly. Looping was examined here on supercoiled and relaxed forms of a 5.5 kb plasmid with three SfiI sites: sites 1 and 2 were separated by 0.4 kb, and sites 2 and 3 by 2.0 kb. SfiI converted this plasmid directly to the products cut at all three sites, though DNA species cleaved at one or two sites were formed transiently during a burst phase. The burst revealed three sets of doubly cut products, corresponding to the three possible pairings of sites. The equilibrium distribution between the different loops was evaluated from the burst phases of reactions initiated by adding MgCl(2) to SfiI bound to the plasmid. The short loop was favored over the longer loops, particularly on supercoiled DNA. The relative rates for loop capture were assessed after adding SfiI to solutions containing the plasmid and MgCl(2). On both supercoiled and relaxed DNA, the rate of loop capture across 0.4 kb was only marginally faster than over 2.0 kb or 2.4 kb. The relative strengths and rates of looping were compared to computer simulations of conformational fluctuations in DNA. The simulations concurred broadly with the experimental data, though they predicted that increasing site separations should cause a shallower decline in the equilibrium constants than was observed but a slightly steeper decline in the rates for loop capture. Possible reasons for these discrepancies are discussed.

Transcription-driven twin supercoiling of a DNA loop: A Brownian dynamics study

The torque generated by RNA polymerase as it tracks along double-stranded DNA can potentially induce long-range structural deformations integral to mechanisms of biological significance in both prokaryotes and eukaryotes. In this paper, we introduce a dynamic computer model for investigating this phenomenon. Duplex DNA is represented as a chain of hydrodynamic beads interacting through potentials of linearly elastic stretching, bending, and twisting, as well as excluded volume. The chain, linear when relaxed, is looped to form two open but topologically constrained subdomains. This permits the dynamic introduction of torsional stress via a centrally applied torque. We simulate by Brownian dynamics the 100 micros response of a 477-base pair B-DNA template to the localized torque generated by the prokaryotic transcription ensemble. Following a sharp rise at early times, the distributed twist assumes a nearly constant value in both subdomains, and a succession of supercoiling deformations occurs as superhelical stress is increasingly partitioned to writhe. The magnitude of writhe surpasses that of twist before also leveling off when the structure reaches mechanical equilibrium with the torsional load. Superhelicity is simultaneously right handed in one subdomain and left handed in the other, as predicted by the "transcription-induced twin-supercoiled-domain" model [L. F. Liu and J. C. Wang, Proc. Natl. Acad. Sci. U.S.A. 84, 7024 (1987)]. The properties of the chain at the onset of writhing agree well with predictions from theory, and the generated stress is ample for driving secondary structural transitions in physiological DNA.

Entropy loss in long-distance DNA looping

The entropy loss due to the formation of one or multiple loops in circular and linear DNA chains is calculated from a scaling approach in the limit of long chain segments. The analytical results allow us to obtain a fast estimate for the entropy loss for a given configuration. Numerical values obtained for some examples suggest that the entropy loss encountered in loop closure in typical genetic switches may become a relevant factor in comparison to both k(B)T and typical bond energies in biopolymers, which has to be overcome by the released bond energy between the looping contact sites.

Making contacts on a nucleic acid polymer

The interaction of proteins bound at distant sites on a nucleic acid chain plays an important role in many molecular biological processes. Contact between the proteins is established by looping of the intervening polymer, which can comprise either double- or single-stranded DNA or RNA, or interphase or metaphase chromatin. The effectiveness of this process, as well as the optimal separation distance, is highly dependent on the flexibility and conformation of the linker. This article reviews how the probability of looping-mediated interactions is calculated for different nucleic acid polymers. In addition, the application of the equations to the analysis of experimental data is illustrated.

Atomic force microscopy analysis of intermediates in cobalt hexammine-induced DNA condensation

The packaging pathway of cobalt hexammine-induced DNA condensation on the surface of mica was examined by varying the concentration of Co(NH3)6(3+) in a dilute DNA solution and visualizing the condensates by atomic force microscopy (AFM). Images reveal that cobalt hexammine-induced DNA condensation on mica involves well-defined structures. At 30 microM Co(NH3)6(3+), prolate ellipsoid condensates composed of relatively shorter rods with linkages between them are formed. At 80 microM Co(NH3)6(3+), the condensed features include toroids with average diameter of approximately 240 nm as well as U-shaped and rod-like condensates with nodular appearances. The results imply that the condensates, whether toroids, U-shaped or rod-like structures have similar intermediate state which includes relatively shorter rod-like segments. The average size of the condensed toroids after incubated at room temperature for 5 h (approximately 240 nm) is much larger than that incubated for 0.5 h (approximately 100 nm). The results indicate that the condensation of DNA by Co(NH3)6(3+) is a kinetic-controlled process.

The effect of the DNA conformation on the rate of NtrC activated transcription of Escherichia coli RNA polymerase.sigma(54) holoenzyme

The transcription activator protein NtrC (nitrogen regulatory protein C) can catalyze the transition of Escherichia coli RNA polymerase complexed with the sigma 54 factor (RNAP.sigma(54)) from the closed complex (RNAP.sigma(54) bound at the promoter) to the open complex (melting of the promoter DNA). This process involves phosphorylation of NtrC (NtrC-P), assembly of an octameric NtrC-P complex at the enhancer sequence, interaction of this complex with promoter-bound RNAP.sigma(54) via DNA looping, and hydrolysis of ATP. We have used this system to study the influence of the DNA conformation on the transcription activation rate in single-round transcription experiments with superhelical plasmids as well as linearized templates. Most of the templates had an intrinsically curved DNA sequence between the enhancer and the promoter and differed with respect to the location of the curvature and the distance between the two DNA sites. The following results were obtained: (i) a ten- to 60-fold higher activation rate was observed with the superhelical templates as compared to the linearized conformation; (ii) the presence of an intrinsically curved DNA sequence increased the activation rate of linear templates about five times; (iii) no systematic effect for the presence and/or location of the inserted curved sequence was observed for the superhelical templates. However, the transcription activation rate varied up to a factor of 10 between some of the constructs. (iv) Differences in the distance between enhancer and promoter had little effect for the superhelical templates studied. The results were compared with theoretical calculations for the dependence of the contact probability between enhancer and promoter expressed as the molar local concentration j(M). A correlation of j(M) with the transcription activation rate was observed for values of 10(-8) M<j(M)<10(-6) M and a kinetic model for NtrC-P-catalyzed open complex formation was developed.