Modeling DNA loops using the theory of elasticity

emDNA - A Tool for Modeling Protein-decorated DNA Loops and Minicircles at the Base-pair Step Level

Computational modeling of nucleic acids plays an important role in molecular biology, enhancing our general understanding of the relationship between structure and function. Biophysical studies have provided a wealth of information on how double-helical DNA responds to proteins and other molecules in its local environment but far less understanding of the larger scale structural responses found in protein-decorated loops and minicircles. Current computational models of DNA range from detailed all-atom molecular dynamics studies, which produce rich time and spatially dependent depictions of small DNA fragments, to coarse-grained simulations, which sacrifice detailed physical and chemical information to treat larger scale systems. The treatment of DNA used here, at the base-pair step level with rigid-body parameters, allows one to develop quality models hundreds of base pairs long from local, sequence-specific features found from experiment. The emDNA software takes advantage of this framework, producing elastically optimized models of DNA at thermal equilibrium with built-in or user-generated elastic models. This versatile program, in combination with case studies included in this article, allows users of any skill level to develop and investigate mesoscale models of their own design. The functionality of emDNA includes a tool to incorporate experiment-specific configurations, e.g. protein-bound and/or melted DNA from known high-resolution structures, within higher-order 3D models by fixing the orientation and position of user-specified base pairs. The software provides a new avenue into multiscale genetic modeling, giving a wide range of users a deeper understanding of DNA mesoscale organization and the opportunity to pose new questions in genetic research. The publicly available emDNA software, including build instructions and usage information, is available on GitHub (https://nicocvn.github.io/emDNA/).

Dual high-stroke and high-work capacity artificial muscles inspired by DNA supercoiling

Powering miniature robots using actuating materials that mimic skeletal muscle is attractive because conventional mechanical drive systems cannot be readily downsized. However, muscle is not the only mechanically active system in nature, and the thousandfold contraction of eukaryotic DNA into the cell nucleus suggests an alternative mechanism for high-stroke artificial muscles. Our analysis reveals that the compaction of DNA generates a mass-normalized mechanical work output exceeding that of skeletal muscle, and this result inspired the development of composite double-helix fibers that reversibly convert twist to DNA-like plectonemic or solenoidal supercoils by simple swelling and deswelling. Our modeling-optimized twisted fibers give contraction strokes as high as 90% with a maximum gravimetric work 36 times higher than skeletal muscle. We found that our supercoiling coiled fibers simultaneously provide high stroke and high work capacity, which is rare in other artificial muscles.

Three-dimensional nonlinear dynamics of prestressed active filaments: Flapping, swirling, and flipping

Initially straight slender elastic filaments or rods with constrained ends buckle and form stable two-dimensional shapes when prestressed by bringing the ends together. Beyond a critical value of this prestress, rods can also deform off plane and form twisted three-dimensional equilibrium shapes. Here, we analyze the three-dimensional instabilities and dynamics of such deformed filaments subject to nonconservative active follower forces and fluid drag. We find that softly constrained filaments that are clamped at one end and pinned at the other exhibit stable two-dimensional planar flapping oscillations when active forces are directed toward the clamped end. Reversing the directionality of the forces quenches the instability. For strongly constrained filaments with both ends clamped, computations reveal an instability arising from the twist-bend-activity coupling. Planar oscillations are destabilized by off-planar perturbations resulting in twisted three-dimensional swirling patterns interspersed with periodic flipping or reversal of the swirling direction. These striking swirl-flip transitions are characterized by two distinct timescales: the time period for a swirl (rotation) and the time between flipping events. We interpret these reversals as relaxation oscillation events driven by accumulation of torsional energy. Each cycle is initiated by a fast jump in torsional deformation with a subsequent slow decrease in net torsion until the next cycle. Our work reveals the rich tapestry of spatiotemporal patterns when weakly inertial strongly damped rods are deformed by nonconservative active forces. Taken together, our results suggest avenues by which prestress, elasticity, and activity may be used to design synthetic macroscale pumps or mixers.

Electrostatic bending response of a charged helix

We explore the electrostatic bending response of a chain of charged particles confined on a finite helical filament. We analyze how the energy difference ΔE between the bent and the unbent helical chain scales with the length of the helical segment and the radius of curvature and identify features that are not captured by the standard notion of the bending rigidity, normally used as a measure of bending tendency in the linear response regime. Using ΔE to characterize the bending response of the helical chain we identify two regimes with qualitatively different bending behaviors for the ground state configuration: the regime of small and the regime of large radius-to-pitch ratio, respectively. Within the former regime, ΔE changes smoothly with the variation of the system parameters. Of particular interest are its oscillations with the number of charged particles encountered for commensurate fillings which yield length-dependent oscillations in the preferred bending direction of the helical chain. We show that the origin of these oscillations is the nonuniformity of the charge distribution caused by the long-range character of the Coulomb interactions and the finite length of the helix. In the second regime of large values of the radius-to-pitch ratio, sudden changes in the ground state structure of the charges occur as the system parameters vary, leading to complex and discontinuous variations in the ground state bending response ΔE.

Micro-mechanical, continuum-mechanical, and AFM-based descriptions of elasticity in open cylindrical micellar filaments

We present theoretical and experimental descriptions of the elasticity of cylindrical micellar filaments using micro-mechanical and continuum theories, and atomic force microscopy. Following our micro-mechanical elasticity model for micellar filaments [M. Asgari, Eur. Phys. J. E: Soft Matter Biol. Phys., 2015, 38(9), 1-16], the elastic bending energy of hemispherical end caps is found. The continuum description of the elastic bending energy of a cylindrical micellar filament is also derived using constrained Cosserat rod theory. While the continuum approach provides macroscopic description of the strain energy of the micellar filament, the micro-mechanical approach has a microscopic view of the filament, and provides expressions for kinetic variables based on a selected interaction potential between the molecules comprising the filament. Our model predicts the dependence of the elastic modulus of the micellar filaments on their diameter, which agrees with previous experimental observations. Atomic force microscopy is applied to estimate the elastic modulus of the filaments using force volume analysis. The obtained values of elastic modulus yield the persistence length of micellar filaments on the same order of the previously reported values. Consistent with previous studies, our results indicate that semi-flexible linear micelles have a relatively large local strain energy at their end points, which explains their tendency to fuse to minimize the number of end caps at relatively low total surfactant volume fractions. Also, the elastic modulus of micellar filaments was found to increase when the indentation frequency increases, a finding which agrees with previous rheological observations on the bulk shear modulus of micellar solutions.

Anisotropy of B-DNA Groove Bending

DNA bending is critical for DNA packaging, recognition, and repair, and occurs toward either the major or the minor groove. The anisotropy of B-DNA groove bending was quantified for eight DNA sequences by free energy simulations employing a novel reaction coordinate. The simulations show that bending toward the major groove is preferred for non-A-tracts while the A-tract has a high tendency of bending toward the minor groove. Persistence lengths were generally larger for bending toward the minor groove, which is thought to originate from differences in groove hydration. While this difference in stiffness is one of the factors determining the overall preference of bending direction, the dominant contribution is shown to be a free energy offset between major and minor groove bending. The data suggests that, for the A-tract, this offset is largely determined by inherent structural properties, while differences in groove hydration play a large role for non-A-tracts. By quantifying the energetics of DNA groove bending and rationalizing the origins of the anisotropy, the calculations provide important new insights into a key biological process.

Coarse-grained modeling of RNA 3D structure

Functional RNA molecules depend on three-dimensional (3D) structures to carry out their tasks within the cell. Understanding how these molecules interact to carry out their biological roles requires a detailed knowledge of RNA 3D structure and dynamics as well as thermodynamics, which strongly governs the folding of RNA and RNA-RNA interactions as well as a host of other interactions within the cellular environment. Experimental determination of these properties is difficult, and various computational methods have been developed to model the folding of RNA 3D structures and their interactions with other molecules. However, computational methods also have their limitations, especially when the biological effects demand computation of the dynamics beyond a few hundred nanoseconds. For the researcher confronted with such challenges, a more amenable approach is to resort to coarse-grained modeling to reduce the number of data points and computational demand to a more tractable size, while sacrificing as little critical information as possible. This review presents an introduction to the topic of coarse-grained modeling of RNA 3D structures and dynamics, covering both high- and low-resolution strategies. We discuss how physics-based approaches compare with knowledge based methods that rely on databases of information. In the course of this review, we discuss important aspects in the reasoning process behind building different models and the goals and pitfalls that can result.

Structural Basis for Elastic Mechanical Properties of the DNA Double Helix

In this article, we investigate the principal structural features of the DNA double helix and their effects on its elastic mechanical properties. We develop, in the pursuit of this purpose, a helical continuum model consisting of a soft helical core and two stiff ribbons wrapping around it. The proposed model can reproduce the negative twist-stretch coupling of the helix successfully as well as its global stretching, bending, and torsional rigidities measured experimentally. Our parametric study of the model using the finite element method further reveals that the stiffness of phosphate backbones is a crucial factor for the counterintuitive overwinding behavior of the duplex and its extraordinarily high torsional rigidity, the major-minor grooves augment the twist-stretch coupling, and the change of the helicity might be responsible for the transition from a negative to a positive twist-stretching coupling when a tensile force is applied to the duplex.

Exploring LacI-DNA dynamics by multiscale simulations using the SIRAH force field

The lac repressor protein (LacI) together with its target regulatory sequence are a common model for studying DNA looping and its implications on transcriptional control in bacteria. Owing to the molecular size of this system, standard all-atom (AA) simulations are prohibitive for achieving relevant biological time scales. As an alternative, multiscale models, which combine AA descriptions at particular regions with coarse-grained (CG) representations of the remaining components, were used to address this computational challenge while preserving the relevant details of the system. In this work, we implement a new multiscale approach based on the SIRAH force field to gain deeper insights into the dynamics of the LacI-DNA system. Our methodology allows for a dual resolution treatment of the solute and solvent, explicitly representing the protein, DNA, and solvent environment without compromising the AA region. Starting from the P1 loop configuration in an undertwisted conformation, we were able to observe the transition to the more stable overtwisted state. Additionally, a detailed characterization of the conformational space sampled by the DNA loop was done. In agreement with experimental and theoretical evidence, we observed the transient formation of kinks at the loop, which were stabilized by the presence of counterions at the minor groove. We also show that the loop's intrinsic flexibility can account for reported FRET measurements and bent conformations required to bind the CAP transcription factor.

A molecular model for the free energy, bending elasticity, and persistence length of wormlike micelles

An expression for the elastic free-energy density of a wormlike micelle is derived taking into account interactions between its constituent molecules. The resulting expression is quadratic in the curvature and torsion of the centerline of micelle and thus resembles free-energy density functions for polymer chains and helical filaments such as DNA. The model is applied on a wormlike micelle in the shape of a circular arc, open or closed. Conditions under which linear chains in dilute systems transform into toroidal rings are analyzed. Two concrete anisotropic soft-core interaction potentials are used to calculate the elastic moduli present in the derived model, in terms of the density of the molecules and their dimensions. Expressions for the persistence length of the wormlike micelle are found based on the flexural rigidities so obtained. Similar to previous observations, our results indicate that the persistence length of a wormlike micelle increases as the aspect ratio of its constituent molecules increases. A detailed application of the model on wormlike micelles of toroidal geometry, along with employing statistical-thermodynamical concepts of self-assembly is performed, and the results are found to be well consistent with the literature. Steps to obtain the material parameters through possible experiments are discussed.

DNA buckling in bacteriophage cavities as a mechanism to aid virus assembly

While relatively simple biologically, bacteriophages are sophisticated biochemical machines that execute a precise sequence of events during virus assembly, DNA packaging, and ejection. These stages of the viral life cycle require intricate coordination of viral components whose structures are being revealed by single molecule experiments and high resolution (cryo-electron microscopy) reconstructions. For example, during packaging, bacteriophages employ some of the strongest known molecular motors to package DNA against increasing pressure within the viral capsid shell. Located upstream of the motor is an elaborate portal system through which DNA is threaded. A high resolution reconstruction of the portal system for bacteriophage ϕ29 reveals that DNA buckles inside a small cavity under large compressive forces. In this study, we demonstrate that DNA can also buckle in other bacteriophages including T7 and P22. Using a computational rod model for DNA, we demonstrate that a DNA buckle can initiate and grow within the small confines of a cavity under biologically-attainable force levels. The forces of DNA-cavity contact and DNA-DNA electrostatic repulsion ultimately limit cavity filling. Despite conforming to very different cavity geometries, the buckled DNA within T7 and P22 exhibits near equal volumetric energy density (∼1kT/nm(3)) and energetic cost of packaging (∼22kT). We hypothesize that a DNA buckle creates large forces on the cavity interior to signal the conformational changes to end packaging. In addition, a DNA buckle may help retain the genome prior to tail assembly through significantly increased contact area with the portal.

Conformational Modeling of Continuum Structures in Robotics and Structural Biology: A Review

Hyper-redundant (or snakelike) manipulators have many more degrees of freedom than are required to position and orient an object in space. They have been employed in a variety of applications ranging from search-and-rescue to minimally invasive surgical procedures, and recently they even have been proposed as solutions to problems in maintaining civil infrastructure and the repair of satellites. The kinematic and dynamic properties of snakelike robots are captured naturally using a continuum backbone curve equipped with a naturally evolving set of reference frames, stiffness properties, and mass density. When the snakelike robot has a continuum architecture, the backbone curve corresponds with the physical device itself. Interestingly, these same modeling ideas can be used to describe conformational shapes of DNA molecules and filamentous protein structures in solution and in cells. This paper reviews several classes of snakelike robots: (1) hyper-redundant manipulators guided by backbone curves; (2) flexible steerable needles; and (3) concentric tube continuum robots. It is then shown how the same mathematical modeling methods used in these robotics contexts can be used to model molecules such as DNA. All of these problems are treated in the context of a common mathematical framework based on the differential geometry of curves, continuum mechanics, and variational calculus. Both coordinate-dependent Euler-Lagrange formulations and coordinate-free Euler-Poincaré approaches are reviewed.

DNA topology confers sequence specificity to nonspecific architectural proteins

Topological constraints placed on short fragments of DNA change the disorder found in chain molecules randomly decorated by nonspecific, architectural proteins into tightly organized 3D structures. The bacterial heat-unstable (HU) protein builds up, counter to expectations, in greater quantities and at particular sites along simulated DNA minicircles and loops. Moreover, the placement of HU along loops with the "wild-type" spacing found in the Escherichia coli lactose (lac) and galactose (gal) operons precludes access to key recognition elements on DNA. The HU protein introduces a unique spatial pathway in the DNA upon closure. The many ways in which the protein induces nearly the same closed circular configuration point to the statistical advantage of its nonspecificity. The rotational settings imposed on DNA by the repressor proteins, by contrast, introduce sequential specificity in HU placement, with the nonspecific protein accumulating at particular loci on the constrained duplex. Thus, an architectural protein with no discernible DNA sequence-recognizing features becomes site-specific and potentially assumes a functional role upon loop formation. The locations of HU on the closed DNA reflect long-range mechanical correlations. The protein responds to DNA shape and deformability—the stiff, naturally straight double-helical structure—rather than to the unique features of the constituent base pairs. The structures of the simulated loops suggest that HU architecture, like nucleosomal architecture, which modulates the ability of regulatory proteins to recognize their binding sites in the context of chromatin, may influence repressor-operator interactions in the context of the bacterial nucleoid.

Structural ensemble and dynamics of toroidal-like DNA shapes in bacteriophage ϕ29 exit cavity

In the bacteriophage ϕ29, DNA is packed into a preassembled capsid from which it ejects under high pressure. A recent cryo-EM reconstruction of ϕ29 revealed a compact toroidal DNA structure (30-40 basepairs) lodged within the exit cavity formed by the connector-lower collar protein complex. Using multiscale models, we compute a detailed structural ensemble of intriguing DNA toroids of various lengths, all highly compatible with experimental observations. In particular, coarse-grained (elastic rod) and atomistic (molecular dynamics) models predict the formation of DNA toroids under significant compression, a largely unexplored state of DNA. Model predictions confirm that a biologically attainable compressive force of 25 pN sustains the toroid and yields DNA electron density maps highly consistent with the experimental reconstruction. The subsequent simulation of dynamic toroid ejection reveals large reactions on the connector that may signal genome release.

Modeling spatial correlation of DNA deformation: DNA allostery in protein binding

We report a study of DNA deformations using a coarse-grained mechanical model and quantitatively interpret the allosteric effects in protein-DNA binding affinity. A recent single-molecule study (Kim et al. Science 2013, 339, 816) showed that when a DNA molecule is deformed by specific binding of a protein, the binding affinity of a second protein separated from the first protein is altered. Experimental observations together with molecular dynamics simulations suggested that the origin of the DNA allostery is related to the observed deformation of DNA's structure, in particular, the major groove width. To unveil and quantify the underlying mechanism for the observed major groove deformation behavior related to the DNA allostery, here we provide a simple but effective analytical model where DNA deformations upon protein binding are analyzed and spatial correlations of local deformations along the DNA are examined. The deformation of the DNA base orientations, which directly affect the major groove width, is found in both an analytical derivation and coarse-grained Monte Carlo simulations. This deformation oscillates with a period of 10 base pairs with an amplitude decaying exponentially from the binding site with a decay length lD ≈10 base pairs as a result of the balance between two competing terms in DNA base-stacking energy. This length scale is in agreement with that reported from the single-molecule experiment. Our model can be reduced to the worm-like chain form at length scales larger than lP but is able to explain DNA's mechanical properties on shorter length scales, in particular, the DNA allostery of protein-DNA interactions.

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 conformation and energy in nucleosome core: a theoretical approach

DNA conformation in complex with proteins is far from its canonical B-form. The affinity of complex formation and structure of DNA depend on its attachment configuration and sequence. In this article, we develop a mechanical model to address the problem of DNA structure and energy under deformation. DNA in nucleosome core particle is described as an example. The structure and energy of nucleosomal DNA is calculated based on its sequence and positioning state. The inferred structure has remarkable similarity with X-ray data. Although there is no sequence-specific interaction of bases and the histone core, we found considerable sequence dependency for the nucleosomal DNA positioning. The affinity of nucleosome formation for several sequences is examined and the differences are compatible with observations. We argue that structural energy determines the natural state of nucleosomal DNA and is the main reason for affinity differences in vitro. This theory can be utilized for the DNA structure and energy determination in protein-DNA complexes in general. An animated Interactive 3D Complement (I3DC) is available in Proteopedia at http://proteopedia.org/w/Journal:JBSD:17.

DNA modeling reveals an extended lac repressor conformation in classic in vitro binding assays

Protein-mediated DNA looping, such as that induced by the lactose repressor (LacI) of Escherichia coli, is a well-known gene regulation mechanism. Although researchers have given considerable attention to DNA looping by LacI, many unanswered questions about this mechanism, including the role of protein flexibility, remain. Recent single-molecule observations suggest that the two DNA-binding domains of LacI are capable of splaying open about the tetramerization domain into an extended conformation. We hypothesized that if recent experiments were able to reveal the extended conformation, it is possible that such structures occurred in previous studies as well. In this study, we tested our hypothesis by reevaluating two classic in vitro binding assays using a computational rod model of DNA. The experiments and computations evaluate the looping of both linear DNA and supercoiled DNA minicircles over a broad range of DNA interoperator lengths. The computed energetic minima align well with the experimentally observed interoperator length for optimal loop stability. Of equal importance, the model reveals that the most stable loops for linear DNA occur when LacI adopts the extended conformation. In contrast, for DNA minicircles, optimal stability may arise from either the closed or the extended protein conformation depending on the degree of supercoiling and the interoperator length.

FLEXIBILITY AND CONTROL OF PROTEIN–DNA LOOPS

Protein–DNA loops are essential for efficient transcriptional repression and activation. The geometry and stability of the archetypal Lac repressor tetramer (LacI)–DNA loop were investigated using designed hyperstable loops containing lac operators bracketing a sequence-directed bend. Electrophoretic mobility shift assays, DNA cyclization, and bulk and single-molecule fluorescence resonance energy transfer (FRET) demonstrate that the DNA sequence controls whether the LacI–DNA loop forms a compact loop with positive writhe or an open loop with little writhe. Monte Carlo methods for simulation of DNA ring closure were extended to DNA loops, including treatment of variable protein hinge angles. The observed distribution of topoisomer products upon cyclization provides a strong constraint on possible models. The experiments and modeling imply that LacI–DNA can adopt a wide range of geometries but has a strong intrinsic preference for an open form. The flexibility of LacI helps explain in vivo observations that DNA looping is less sensitive to DNA length and shape than that expected from the physical properties of DNA. While DNA cyclization suggests two pools of precursor loops for the 9C14 construct, single-molecule FRET demonstrates a single population. This discrepancy suggests that the LacI–DNA structure is strongly influenced by flanking DNA.

A multiscale dynamic model of DNA supercoil relaxation by topoisomerase IB

In this study, we report what we believe to be the first multiscale simulation of the dynamic relaxation of DNA supercoils by human topoisomerase IB (topo IB). We leverage our previous molecular dynamics calculations of the free energy landscape describing the interaction between a short DNA fragment and topo IB. Herein, this landscape is used to prescribe boundary conditions for a computational, elastodynamic continuum rod model of a long length of supercoiled DNA. The rod model, which accounts for the nonlinear bending, twisting, and electrostatic interaction of the (negatively charged) DNA backbone, is extended to include the hydrodynamic drag induced by the surrounding physiological buffer. Simulations for a 200-bp-long DNA supercoil in complex with topo IB reveal a relaxation timescale of ∼0.1-1.0 μs. The relaxation follows a sequence of cascading reductions in the supercoil linking number (Lk), twist (Tw), and writhe (Wr) that follow companion cascading reductions in the supercoil elastic and electrostatic energies. The novel (to our knowledge) multiscale modeling method may enable simulations of the entire experimental setup that measures DNA supercoiling and relaxation via single molecule magnetic trapping.

Torque-induced deformations of charged elastic DNA rods: thin helices, loops, and precursors of DNA supercoiling

We study the deformations of charged elastic rods under applied end forces and torques. For neutral filaments, we analyze the energetics of initial helical deformations and loop formation. We supplement this elastic approach with electrostatic energies of bent filaments and find critical conditions for buckling depending on the ionic strength of the solution. We also study force-induced loop opening, for parameters relevant for DNA. Finally, some applications of this nano-mechanical DNA model to salt-dependent onset of the DNA supercoiling are discussed.

Genome3D: a viewer-model framework for integrating and visualizing multi-scale epigenomic information within a three-dimensional genome

Background

New technologies are enabling the measurement of many types of genomic and epigenomic information at scales ranging from the atomic to nuclear. Much of this new data is increasingly structural in nature, and is often difficult to coordinate with other data sets. There is a legitimate need for integrating and visualizing these disparate data sets to reveal structural relationships not apparent when looking at these data in isolation.

Results

We have applied object-oriented technology to develop a downloadable visualization tool, Genome3D, for integrating and displaying epigenomic data within a prescribed three-dimensional physical model of the human genome. In order to integrate and visualize large volume of data, novel statistical and mathematical approaches have been developed to reduce the size of the data. To our knowledge, this is the first such tool developed that can visualize human genome in three-dimension. We describe here the major features of Genome3D and discuss our multi-scale data framework using a representative basic physical model. We then demonstrate many of the issues and benefits of multi-resolution data integration.

Conclusions

Genome3D is a software visualization tool that explores a wide range of structural genomic and epigenetic data. Data from various sources of differing scales can be integrated within a hierarchical framework that is easily adapted to new developments concerning the structure of the physical genome. In addition, our tool has a simple annotation mechanism to incorporate non-structural information. Genome3D is unique is its ability to manipulate large amounts of multi-resolution data from diverse sources to uncover complex and new structural relationships within the genome.

A computational framework for mechanical response of macromolecules: application to the salt concentration dependence of DNA bendability

A computational framework is presented for studying the mechanical response of macromolecules. The method combines a continuum mechanics (CM) model for the mechanical properties of the macromolecule with a continuum electrostatic (CE) treatment of solvation. The molecules are represented by their shape and key physicochemical characteristics such as the distribution of materials properties and charge. As a test case, we apply the model to the effect of added salt on the bending of DNA. With a simple representation of DNA, the CM/CE framework using a Debye-Hückel model leads to results that are in good agreement with both analytical theories and recent experiments, including a modified Odijk-Skolnick-Fixman theory that takes the finite length of DNA into consideration. Calculations using a more sophisticated CE model (Poisson-Boltzmann), however, suffer from convergence problems, highlighting the importance of balancing numerical accuracy in the CM and CE models when dealing with very large systems, particularly those with a high degree of symmetry.

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.

Concentration and length dependence of DNA looping in transcriptional regulation

In many cases, transcriptional regulation involves the binding of transcription factors at sites on the DNA that are not immediately adjacent to the promoter of interest. This action at a distance is often mediated by the formation of DNA loops: Binding at two or more sites on the DNA results in the formation of a loop, which can bring the transcription factor into the immediate neighborhood of the relevant promoter. These processes are important in settings ranging from the historic bacterial examples (bacterial metabolism and the lytic-lysogeny decision in bacteriophage), to the modern concept of gene regulation to regulatory processes central to pattern formation during development of multicellular organisms. Though there have been a variety of insights into the combinatorial aspects of transcriptional control, the mechanism of DNA looping as an agent of combinatorial control in both prokaryotes and eukaryotes remains unclear. We use single-molecule techniques to dissect DNA looping in the lac operon. In particular, we measure the propensity for DNA looping by the Lac repressor as a function of the concentration of repressor protein and as a function of the distance between repressor binding sites. As with earlier single-molecule studies, we find (at least) two distinct looped states and demonstrate that the presence of these two states depends both upon the concentration of repressor protein and the distance between the two repressor binding sites. We find that loops form even at interoperator spacings considerably shorter than the DNA persistence length, without the intervention of any other proteins to prebend the DNA. The concentration measurements also permit us to use a simple statistical mechanical model of DNA loop formation to determine the free energy of DNA looping, or equivalently, the for looping.

Molecular mechanics of the alpha-actinin rod domain: bending, torsional, and extensional behavior

α-Actinin is an actin crosslinking molecule that can serve as a scaffold and maintain dynamic actin filament networks. As a crosslinker in the stressed cytoskeleton, α-actinin can retain conformation, function, and strength. α-Actinin has an actin binding domain and a calmodulin homology domain separated by a long rod domain. Using molecular dynamics and normal mode analysis, we suggest that the α-actinin rod domain has flexible terminal regions which can twist and extend under mechanical stress, yet has a highly rigid interior region stabilized by aromatic packing within each spectrin repeat, by electrostatic interactions between the spectrin repeats, and by strong salt bridges between its two anti-parallel monomers. By exploring the natural vibrations of the α-actinin rod domain and by conducting bending molecular dynamics simulations we also predict that bending of the rod domain is possible with minimal force. We introduce computational methods for analyzing the torsional strain of molecules using rotating constraints. Molecular dynamics extension of the α-actinin rod is also performed, demonstrating transduction of the unfolding forces across salt bridges to the associated monomer of the α-actinin rod domain.

The cell interacts with its environment in both biochemical and mechanical ways. In this study we explore one of the ways in which the cell interacts mechanically with its environment. α-Actinin is a cytoskeletal crosslinker: it functions to scaffold the cytoskeletal actin filaments that provide mechanical reinforcement to the cell. In its functional environment α-actinin is exposed to a multitude of mechanical stresses as it attaches itself to a dynamic network of actin filaments. The actin filaments extend, rotate, and bend the α-actinin crosslinkers. In this study we employ molecular dynamics techniques to understand the structural characteristics of α-actinin that underlie its ability to provide a scaffold in such a stressed environment. We analyzed the natural frequencies of α-actinin and simulated force-induced bending, extension, and twisting. Our results suggest that α-actinin has structural flexibility facilitating crosslinking in a dynamic environment and also structural rigidity stabilizing the linkage in the stressed environment. We have discovered novel natural bending movements of the rod domain that enhance its function as a crosslinker. We have also demonstrated the specific structural characteristics of α-actinin that give it the previously suggested property of having partial flexibility. Our results enhance the understanding of structural mechanics in the cytoskeletal molecules.

Computational analysis of looping of a large family of highly bent DNA by LacI

Sequence-dependent intrinsic curvature of DNA influences looping by regulatory proteins such as LacI and NtrC. Curvature can enhance stability and control shape, as observed in LacI loops formed with three designed sequences with operators bracketing an A-tract bend. We explore geometric, topological, and energetic effects of curvature with an analysis of a family of highly bent sequences, using the elastic rod model from previous work. A unifying straight-helical-straight representation uses two phasing parameters to describe sequences composed of two straight segments that flank a common helically supercoiled segment. We exercise the rod model over this two-dimensional space of phasing parameters to evaluate looping behaviors. This design space is found to comprise two subspaces that prefer parallel versus anti-parallel binding topologies. The energetic cost of looping varies from 4 to 12 kT. Molecules can be designed to yield distinct binding topologies as well as hyperstable or hypostable loops and potentially loops that can switch conformations. Loop switching could be a mechanism for control of gene expression. Model predictions for linking numbers and sizes of LacI-DNA loops can be tested using multiple experimental approaches, which coupled with theory could address whether proteins or DNA provide the observed flexibility of protein-DNA loops.

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.

Effect of supercoiling on formation of protein-mediated DNA loops

DNA loop formation is one of several mechanisms used by organisms to regulate genes. The free energy of forming a loop is an important factor in determining whether the associated gene is switched on or off. In this paper we use an elastic rod model of DNA to determine the free energy of forming short (50-100 basepair), protein mediated DNA loops. Superhelical stress in the DNA of living cells is a critical factor determining the energetics of loop formation, and we explicitly account for it in our calculations. The repressor protein itself is regarded as a rigid coupler; its geometry enters the problem through the boundary conditions it applies on the DNA. We show that a theory with these ingredients is sufficient to explain certain features observed in modulation of in vivo gene activity as a function of the distance between operator sites for the lac repressor. We also use our theory to make quantitative predictions for the dependence of looping on superhelical stress, which may be testable both in vivo and in single-molecule experiments such as the tethered particle assay and the magnetic bead assay.

DNA confined in nanochannels: Hairpin tightening by entropic depletion

A theory is presented of the elongation of double-stranded DNA confined in a nanochannel based on a study of the formation of hairpins. A hairpin becomes constrained as it approaches the wall of a channel which leads to an entropic force causing the hairpin to tighten. The DNA in the hairpin remains double-stranded. The free energy of the hairpin is significantly larger than what one would expect if this entropic effect were unimportant. As a result, the distance between hairpins or the global persistence length is often tens of micrometer long and may even reach millimeter sizes for 10 nm thin channels. The hairpin shape and size and the DNA elongation are computed for nanoslits and circular and square nanochannels. A comparison with experiment is given.

Molecular mechanisms of cellular mechanics

Mechanical forces play an essential role in cellular processes as input, output, and signals. Various protein complexes in the cell are designed to handle, transform and use such forces. For instance, proteins of muscle and the extracellular matrix can withstand considerable stretching forces, hearing-related and mechanosensory proteins can transform weak mechanical stimuli into electrical signals, and regulatory proteins are suited to forcing DNA into loops to control gene expression. Here we review the structure-function relationship of four protein complexes with well defined and representative mechanical functions. The first example is titin, a protein that confers passive elasticity on muscle. The second system is the elastic extracellular matrix protein, fibronectin, and its cellular receptor integrin. The third protein system is the transduction apparatus in hearing and other mechanical senses, likely containing cadherin and ankyrin repeats. The last system is the lac repressor protein, which regulates gene expression by looping DNA. This review focuses on atomic level descriptions of the physical mechanisms underlying the various mechanical functions of the stated proteins.