Search

The electric solar wind sail (E-sail) has been proposed as a novel propulsion system that, if successful, promises to facilitate missions that would be inconvenient or impractical with traditional chemical propellants. The most common E-sail concept includes a central hub with several long positively charged tethers extending radially outward that harvest momentum from protons in the solar wind. Before the technology is ready for a demonstration mission, we must prove E-sail dynamics can be controlled throughout all phases of the mission. The objective of this paper is to present an analytical model for the linear vibrations of hub and spoke E-sails to facilitate the development of necessary control schemes. Although there is no limit to the number of spokes in our model, we demonstrate its capability by representing a system with 3 spokes. The resultant natural frequencies and mode shapes provide insight into the dynamics and stability of E-sails and lay the foundation for development of control schemes. In addition, our work reveals several simple calculations that accurately estimate many of the natural frequencies and could be used to facilitate the design of future E-sails.

DNA supercoiling plays an important role in a variety of cellular processes, including transcription, replication, and DNA compaction. To fully understand these processes, we must uncover and characterize the dynamics of supercoiled DNA. However, supercoil dynamics are difficult to access because of the wide range of relevant length and timescales. In this work, we present an algorithm to reconstruct the arrangement of identical fluorescent particles distributed around a circular DNA molecule, given their three-dimensional trajectories through time. We find that this curve reconstruction problem is analogous to solving the traveling salesman problem. We demonstrate that our approach converges to the correct arrangement with a sufficiently long observation time. In addition, we show that the time required to accurately reconstruct the fluorophore arrangement is reduced by increasing the fluorophore density or reducing the level of supercoiling. This curve reconstruction algorithm, when paired with next-generation super-resolution imaging systems, could be used to access and thereby advance our understanding of supercoil dynamics.

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.

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

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.

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.

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.

Several recent single-molecule experiments observe the response of supercoiled DNA to nicking endonucleases and topoisomerases. Typically in these experiments, indirect measurements of supercoil relaxation are obtained by observing the motion of a large micron-sized bead. The bead, which also serves to manipulate DNA, experiences significant drag and thereby obscures supercoil dynamics. Here we employ our discrete wormlike chain model to bypass experimental limitations and simulate the dynamic response of supercoiled DNA to a single strand nick. From our simulations, we make three major observations. First, extension is a poor dynamic measure of supercoil relaxation; in fact, the linking number relaxes so fast that it cannot have much impact on extension. Second, the rate of linking number relaxation depends upon its initial partitioning into twist and writhe as determined by tension. Third, the extensional response strongly depends upon the initial position of plectonemes.

Induced torsional stresses lead to an increase in the magnitude of torque sustained by double stranded DNA. There exists a critical magnitude of torque at which the DNA buckles signaling the beginning of the formation of a plectonemically supercoiled domain in the DNA. Further torsional deformation leads to an increase in the size of the supercoiled domain while the torque sustained by the DNA remains constant. The formation of the supercoiled domain also leads to a reduction in the end-to-end extension of the DNA starting with an abrupt reduction at the onset of buckling. Experiments have shown that this reduction in extension follows a linear trend. We investigate, by means of Brownian dynamics simulations, the extensional and torsional response of dsDNA to induced torsional stresses.Copyright © 2015 by ASME

A new end-effector was designed and built for microscribing processes using principles of compliant mechanisms. Initial testing of a chemomechanical microscribing process showed that low forces produce lines that are significantly smoother, more uniform, and exhibit less material chipping. From the testing it is apparent that there is a need for a specialized precision end-effector that (1) is passively controlled, (2) has low axial stiffness, and (3) has high lateral stiffness. To meet these needs a three segment folded-beam compliant mechanism was chosen. This design is passively controlled, has a high axial/lateral stiffness ratio, is ideal for clean sensitive applications, and can be designed for a range of low forces. The axial stiffness of the end-effector was modeled using both the compliant mechanism pseudo-rigid body model and linear-elastic beam theory. The lateral stiffness was modeled using FEA techniques. Ratios of lateral stiffness to axial stiffness were found to be nearly 1000:1. A fatigue analysis was also performed and it was determined that the mechanism could reach approximately 1 billion cycles before failure. The compliant end-effector design is a significant improvement over existing scribing alternatives and can produce smooth and uniform scribed lines that exhibit less material chipping.

DNA is a long flexible polymer and is involved in several fundamental cellular processes such as transcription, replication and chromosome packaging. These processes induce forces and torques in the DNA which deform it. These deformations in turn affect the structure and function of DNA. However, understanding of the dynamic response of DNA to the various forces that act on it is still far from complete. Several experiments have been carried out to study these responses most of which use a micron sized magnetic bead attached to the DNA molecule to both manipulate it and to observe its dynamics. One limitation of this approach is that the dynamics of the DNA molecule has mostly been characterized “indirectly” by observing the dynamics of the magnetic bead. It is also reasonable to expect that, because of the size of the bead relative to that of the DNA, the magnetic bead dynamics will obscure that of the DNA. We adapt existing coarse-grained Brownian dynamics models of DNA to develop a model capable of representing the dynamics of DNA without any of the artifacts inherent to the experiments. This model accounts for bending, torsion, extension, electrostatics, hydrodynamics and the random thermal forces acting on DNA in an electrolyte solution. We then carry out Brownian dynamics simulations with our model to benchmark with well established theoretical results of a stretched polymer in solution. Finally, we employ our model to predict the relaxation time scale for single molecule experiments which sets the framework for future studies in which we plan to further shed light on the dynamics of DNA over long length and time scales.

DNA supercoiling is an often overlooked participant in essential cellular processes including transcription, replication and recombination. Supercoils are both a byproduct of and a significant input to these processes. For example, RNA polymerase prefers binding negatively supercoiled DNA and produces positive supercoiling ahead of (and negative supercoiling behind) the transcription site. Subject to thermal fluctuations and DNA-protein interactions, supercoils are dynamic structures which accumulate, rearrange, translocate, and dissipate. Recent single molecule studies have observed the dynamic formation, diffusion, hopping, and dissipation of supercoils. Here, we employ coarse-grain Brownian dynamics simulations, paralleling recent experiments, to build understanding beyond the capabilities of current experimental techniques. Our computational model incorporates hydrodynamic interactions, thermal fluctuations, bending, torsion, extension, and electrostatics. We perform many simulations including trajectories of 21 kilo-basepair lengths of supercoiled DNA spanning up to about 100 ms. We observe several quantities describing the dynamics of supercoils, including: the average number of supercoils, their lifetime, first juxtaposition time, and diffusion constants.

The stresses induced in DNA during cell processes such as replication and transcription lead to the formation of plectonemic supercoils. Supercoiling in turn affects these processes. As a result of this relationship between supercoiling and these processes, the degree of supercoiling in DNA needs to be controlled closely in order to optimize these cell processes. This control is facilitated by enzymes such as type 1b topoisomerases and nicking endonucleases which relax supercoiling in DNA.The dynamics of DNA supercoil relaxation have been studied in recent experiments by means of single-molecule magnetic and/or optical tweezer experiments. Novel as these experiments are, they do not permit a direct observation of the structural changes that occur in DNA during supercoil relaxation. They depend on components (e.g. a paramagnetic bead) attached to the DNA to indirectly obtain information about these dynamics.We studied the dynamics of supercoil relaxation by means of Brownian Dynamics simulations of a discrete wormlike-chain (dWLC) model of DNA. These simulations parallel the single-molecule experiments in which a single DNA molecule is held under constant tension so that its end-to-end extension increases as supercoils are relaxed by a nicking endonuclease. The dWLC model accounts for elasticity, electrostatics and entropic forces as well as for hydrodynamic interactions.From the simulation results, we more directly extract the rate at which supercoiling in DNA is relaxed by nicking endonucleases. We also determine the dependence of the relaxation timescales on the tension applied to the DNA molecule.

Deoxyribonucleic acid (DNA) is a long and flexible biopolymer that contains genetic information. Building upon the discovery of the iconic double helix over 50 years ago, subsequent studies have emphasized how its biological function is related to the mechanical properties of the molecule. A remarkable system which highlights the role of DNA bending and twisting is the packing and ejection of DNA into and from viral capsids. A recent 3D reconstruction of bacteriophage φ29 reveals a novel toroidal structure of highly bent/twisted DNA contained in a small cavity below the viral capsid. Here, we extend an elastic rod model for DNA to enable simulation of the toroid as it is compacted and subsequently ejected from a small volume. We compute biologically-relevant forces required to form the toroid and predict ejection times of several nanoseconds.

DNA looping mediated by the Lac repressor is an archetypal test case for modeling protein and DNA flexibility. Understanding looping is fundamental to quantitative descriptions of gene expression. Systematic analysis of LacI•DNA looping was carried out using a landscape of DNA constructs with lac operators bracketing an A-tract bend, produced by varying helical phasings between operators and the bend. Fluorophores positioned on either side of both operators allowed direct Forster resonance energy transfer (FRET) detection of parallel (P1) and antiparallel (A1, A2) DNA looping topologies anchored by V-shaped LacI. Combining fluorophore position variant landscapes allows calculation of the P1, A1 and A2 populations from FRET efficiencies and also reveals extended low-FRET loops proposed to form via LacI opening. The addition of isopropyl-β-d-thio-galactoside (IPTG) destabilizes but does not eliminate the loops, and IPTG does not redistribute loops among high-FRET topologies. In some cases, subsequent addition of excess LacI does not reduce FRET further, suggesting that IPTG stabilizes extended or other low-FRET loops. The data align well with rod mechanics models for the energetics of DNA looping topologies. At the peaks of the predicted energy landscape for V-shaped loops, the proposed extended loops are more stable and are observed instead, showing that future models must consider protein flexibility.

DNA supercoils arise from processes such as replication, transcription and the nearly 10000-fold compaction of long DNA molecules into the cell nucleus. Because of their potential to enhance or repress cellular processes (e.g. replication, transcription, and recombination), supercoils must be carefully regulated. In fact, chemotherapy drugs have been designed to impede topoisomerase enzymes that are responsible for the regulation of supercoils. Motivated by the biological significance of supercoils, several experimental and computational studies have focused on probing their structure and dynamics. For example, recent single molecule experiments have uncovered a dynamic buckling transition while interrogating the torsion-extension relationship for supercoils. As another example, Brownian Dynamics (BD) simulations of kilo base-pair lengths of circular DNA have explored the sensitivity of site juxtaposition times to supercoiling. Unfortunately, however, there remain many fundamental questions concerning the structure and dynamics of supercoils. In this study, our objective is to quantify the diffusion constant of plectonemic supercoils along stretched DNA. To this end, we employ BD simulations of kilo base-pair lengths of DNA with biologically relevant levels of supercoiling and extension. From these simulations, we quantify the diffusion constant for plectonemes and further compute site juxtaposition and plectoneme nucleation timescales.

Within cells, supercoiled DNA is inherently subjected to various degrees of bending and twisting, and many DNA-binding proteins are now known to be sensitive to these mechanical features of DNA. Under significant levels of bending and/or torsional stress, DNA has been shown to depart from native B-form structure and adopt a number of more energetically favorable alternate conformations. However, very little is known about the structural details of these stress-induced structures. Recently, the DNA nuclease BAL 31 was used to assay for helical destabilizations in small DNA minicircles sustaining fixed amounts of bending and torsional stresses (Nucl. Acids Res., 36(4), 2008). Here, we seek to determine if the helical destabilizations that accompany elevated levels of negative torsional stress in tightly looped minicircles produce localized structures that confer enhanced bending flexibility to the DNA structure, such as would occur in kinked DNA. Towards this end, we synthesized untwisted, 100-bp minicircles that are recognized and digested by BAL 31 and overtwisted 106-bp minicircles that are resistant to degradation by BAL 31. Using cryo-electron microscopy, we then generated three-dimensional image reconstructions of these two minicircle species. From these quantitative descriptions of the minicircle geometries, we observe no evidence of DNA kinking in either the BAL 31-sensitive, 100-bp minicircles or the BAL 31-insensitive, 106-bp minicircles. Since the torsional destabilizations that are recognized by BAL 31 were not observed to confer significant enhancements in bending flexibility, we propose that the torsional destabilizations appearing with the 100-bp minicircle relieve negative torsional stress through the formation of a left-handed dinucleotide stack with no interruption in base-stacking at the right-to-left transition, thus minimally affecting the local bending stiffness. Our observations are consistent with the formation of Z(WC)-DNA, which has been previously theorized to be the predominate form of left-handed DNA appearing in nature.

The well-recognized Lac repressor protein (LacI) regulates transcription by bending DNA into a loop. In addition to the known role of DNA flexibility, there is accumulating evidence suggesting that the flexibility of LacI also plays a role in this gene regulation. Here we extend our elastic rod model for DNA (previously used to model DNA only) to represent LacI. Specifically, we represent sites of concentrated flexibility in the protein with flexible elastic rod domains; and we represent relatively rigid domains of the protein with stiff elastic rod domains. Our analysis shows the sensitivity of looping energetics to the degree of flexibility within the protein over a large range of DNA lengths. In addition, we show that the predicted energetically dominant binding topology (A ) remains upon introducing protein flexibility.Copyright © 2011 by ASME

A single DNA molecule is a long and flexible biopolymer that contains the genetic code. Building upon the discovery of the iconic double helix over 50 years ago, subsequent studies have emphasized how its biological function is related to the mechanical properties of the molecule. A remarkable system which high-lights the role of DNA bending and twisting is the packing and ejection of DNA into and from viral capsids. A recent 3D reconstruction of bacteriophage φ29 reveals a novel toroidal structure thought to be 30–40 bp of highly bent/twisted DNA contained in a small cavity below the capsid. Here, we extend an elastic rod model for DNA to enable simulation of the toroid as it is compacted and subsequently ejected from a small volume. We compute biologically-realistic forces required to form the toroid and predict ejection times of several nanoseconds.

Essential cellular processes such as replication and transcription induce torsional stresses in DNA which lead to an accumulation of supercoils. Excessive, as well as insufficient, levels of supercoiling can impede the progress of these processes. Therefore, topoisomerases are necessary to maintain an optimal degree of DNA supercoiling. Recent experiments have considered the relaxation dynamics of DNA supercoils by both Topo1B and nicking endonucleases. In these experiments, nicking endonucleases or topoisomerases act on individual torsionally constrained DNA molecules which are held at constant tension. Unfortunately, supercoil relaxation dynamics occur faster than can be measured by observing merely the end-to-end extension of the DNA, as in magnetic or optical trap assays. Our objective is to simulate the relaxation of DNA supercoils to better understand the dynamics and quantify the relaxation timescales. We represent DNA with a coarse-grained Brownian dynamics model which accounts for DNA bending, torsion, electrostatics, and hydrodynamics. Our simulations are designed to parallel experiments and permit direct comparisons to the experiments.

Deoxyribonucleic acid (DNA) is an essential molecule that enables the storage and retrieval of genetic information. In its role during cellular processes, this long flexible molecule is significantly bent and twisted. Previously, we developed an elastodynamic rod approximation to study DNA deformed into a loop by a gene regulatory protein (lac repressor) and predicted the energetics and topology of the loops. Although adequate for DNA looping, our model neglected electrostatic interactions, which are essential when considering processes that result in highly supercoiled DNA including plectonemes. Herein, we extend the rod approximation to account for electrostatic interactions and present strategies that improve computational efficiency. Our calculations for the stability for a circularly bent rod and for an initially straight rod compare favorably to existing equilibrium models. With this new capability, we are now well-positioned to study the dynamics of transcription and other dynamic processes that result in DNA supercoiling.

Many cellular processes involving DNA, including replication and transcription, result in significant superhelical stresses. During transcription, for example, RNA polymerase locally untwists about a helical turn of the DNA double helix. Then to elongate the RNA transcript, it proceeds along the template strand of the DNA and thereby induces supercoiling. DNA topoisomerases play an important role in relieving these stresses. Here we focus on understanding the action of human DNA topoisomerase I (Topo1) which operates in three basic steps: (i) cleaving a single strand of the DNA double helix, (ii) allowing the DNA superhelical stresses to relax, and (iii) religating the DNA. Recently, the Dekker lab, at Delft University of Technology, performed single molecule experiments to probe the relaxation of supercoils by topo1. A significant molecular dynamics (MD) effort (>100 cpu years) by the Andricioaei lab, at the University of California Irvine, characterized the energetics and topological changes of topo1 in complex with only a short fragment of DNA (∼20 bp). Including a longer length of DNA to represent a biologically relevant length-scale (greater than a persistence length), is computationally prohibitive for MD and was necessarily neglected. Here we introduce an elasto-dynamic rod model as a first approximation to provide a dynamic description of the DNA as it relaxes. The rod model describes bending and torsion of the DNA helical axis, electrostatic and self-contact interactions, and approximates the hydrodynamic drag on the molecule. For our simulations, we provide as initial conditions, a plectomemic supercoil. The MD simulations serve to provide boundary conditions to the rod model by characterizing the torque applied to the DNA by topo1 as it rotates. Here we present preliminary results for the relaxation rates of supercoils.

DNA is a life-sustaining molecule that enables the storage and retrieval of genetic information. In its role during essential cellular processes, this long flexible molecule is significantly bent and twisted. Previously, we developed an elasto-dynamic rod approximation to study DNA deformed into a loop by a gene regulatory protein (lac repressor) and predicted the energetics and topology of the loops. Although adequate for DNA looping, our model neglected electrostatic interactions which are essential when considering processes that result in highly super-coiled DNA including plectonemes. Herein we extend the rod approximation to account for electrostatic interactions and present strategies that improve computational efficiency. Our calculations for the stability for a circularly bent rod and for an initially straight rod compare favorably to existing equilibrium models. With this new capability, we are now well-positioned to study the dynamics of transcription and other dynamic processes that result in DNA supercoiling.

Protein-mediated DNA loop formation is an important biological process that regulates key functions such as transcription. We present a mechanical model for these DNA-protein complexes that can take effects of the DNA sequence such induced curvature into account. This model provides the equilibrium shape and elastic energy of the DNA loop, using boundary conditions from the protein crystal structure. We then construct a Hamiltonian for small perturbations of the DNA around the equilibrium shape, which in turn allows us to calculate the eigenmodes and the entropic contributions of the thermal fluctuations to the free energy of the DNA loop. Here we present computations related to the short wild-type lactose repressor loop of Escheria coli (E. coli), and find that the entropic contributions are significant and amount to up to 3.9 k B T of the free energy. We also show that this entropic contribution from the stiffening of the DNA loop depends strongly on the phase angle between the two operator sites, which adds to the known phasing effect of the elastic energy of the loop.

DNA is a long flexible biopolymer containing genetic information. Proteins often take advantage of DNA’s inherent flexibility to perform their cellular functions. Here we present selected results from our computational studies of the mechanical looping of DNA by the Lactose repressor protein. The Lactose repressor resides in the bacterium E. coli and deforms DNA into a loop as a means of controlling the production of enzymes necessary for digesting lactose. We examine this looping process using a computational rod model [1–3] to understand the strain energy and geometry for the resultant DNA loops. Our model captures the multiple looped conformations of the molecule arising from both multiple boundary conditions and geometric nonlinearities. In addition, the model captures the periodic variation of strain energy with base-pair length as suggested by repression experiments (see, for example, [4, 5]).Copyright © 2007 by ASME

The dynamics of DNA supercoils are intimately connected to several fundamental cellular processes. Transcription is a prime example. While the rate of transcription is sensitive to the level of supercoiling, transcription produces positive (and negative) supercoils ahead of (and behind) RNA polymerase. Subject to thermal fluctuations and DNA-protein interactions, supercoils are dynamic structures which accumulate, rearrange, translocate, and dissipate. Recent single molecule studies have observed the dynamic formation, diffusion, hopping, and dissipation of supercoils. Here, we employ Brownian dynamics simulations of a discrete worm-like chain to build understanding beyond that provided by recent experimental efforts. Our computational model accounts for hydrodynamic interactions, thermal fluctuations, bending, torsion, extension, and electrostatics in stretched DNA. We perform many simulations including trajectories representing 21 kilobasepairs of supercoiled DNA over the course of about 500 ms. We observe several metrics describing the dynamics of supercoils, including: the average number of supercoils, their lifetime, first juxtaposition time, and diffusion constants. In addition, we explore the sensitivity of these quantities to DNA extension as well as sequence dependence (through the introduction of sites with elevated DNA flexibility).

Tailed bacteriophages are highly efficient machines for infecting a host cell. A single virion must attach to the host surface, penetrate the cell wall, then release its double-stranded DNA genome. In these remarkable DNA delivery machines, controlling how and when the genome is released is a task of paramount importance. Recently, a three-dimensional cryo-electron microscopy (cryo-EM) reconstruction of mature bacteriophage φ29 revealed an intriguing toroidal DNA structure contained in a small cavity below the viral capsid. This highly bent toroidal DNA supercoil is thought to be 30-40 basepairs of dsDNA and its function remains unknown.In this study, we employ an elastic rod model to simulate highly strained DNA as it is compressed within the protein cavity. The model provides estimates of force and energy required to form the toroid as well as its equilibrium conformation. Results reveal that a toroid can indeed form under the biologically relevant forces for the viral packing motor (up to 100 pN). To understand the effects of the highly stressed toroid on DNA structure, we then employ molecular dynamics (MD) simulations starting with the rod model-computed equilibrium as the initial condition. Following equilibration in MD, we construct an approximate density map using the final predicted toroid to compare with the recently published cryo-EM data. The computed density map correctly predicts the major dimensions of the toroid as well as regions of high and low density. Finally, we simulate the start of the ejection process by integrating the rod model forward in time upon sudden removal of the protein “tail knob.” The resulting fast dynamic collapse of the toroid suggests that its function could be to prime the ejection of the remaining viral genome.

Interwound DNA structures (plectonemes) arise from processes such as replication, transcription and the nearly 10000-fold compaction of long DNA molecules into the cell nucleus. These supercoils can enhance or repress cellular processes such as replication, transcription, and recombination. In fact, an increased frequency of DNA-DNA juxtapositions within plectonemic supercoils facilitates DNA looping. Although plectonemes have been studied with recent single molecule techniques, little is known about their dynamic formation, diffusion, rearrangement, and dissolution. In fact, single molecule assays often are confounded by the relatively slow dynamics of a large magnetic or optical bead used to manipulate the DNA. To characterize plectoneme dynamics, we employ Brownian dynamics simulations of stretched supercoiled DNA. Here we consider a range of system parameters describing length, extension, and superhelical density. Our model incorporates viscous drag, thermal fluctuations, bending, torsion, extension, and electrostatics. Additionally, we employ periodic boundary conditions to prevent plectonemes from diffusing off the ends of linear DNA. Our simulations reveal that both the number and size of plectonemes vary with time. Interestingly, this precludes the characterization of plectoneme motion with a diffusion constant. To quantify plectoneme dynamics, we define τ as the median time for first juxtaposition of two sites on a DNA molecule separated by a prescribed distance. Here we show that τ depends on system parameters as well as separation distance and ranges from milliseconds to seconds. Our simulations shed light on processes requiring juxtaposition (e.g. DNA looping) as well as processes in which supercoils are transmitted along DNA (e.g. transcription).

Recent cryo-electron microscopy (cryo-EM) images of DNA minicircles, about 100 basepairs in length, provide a new perspective on the mechanics of DNA. In essence, the resultant 3-dimensional reconstructions capture the mechanically deformed state of the double helix at an instant in time. Such deformations may include sites with high bending and/or torsional flexibility that could result from the tight bending required to cyclize, superhelical stress, and thermal fluctuations. Unfortunately, these reconstructions resolve only the DNA helical axis and provide no information about (i) how to register their known basepair sequence with the reconstruction and (ii) how torsional deformations are distributed along the length of the minicircle. In addition, the experimental procedures are complicated and consequently limit the the number of reconstructed minicircles to about 20. Our objective is to understand the mechanics of these DNA minicircles, and specifically, to develop a method capable of detecting the presence of kinks or torsional destabilizations in their cryo-EM reconstructions. To this end, we developed a modal analysis approach to describe the mechanics of DNA minicircles. In our method, we use the thermal modes of a homogeneous elastic rod representation for circular DNA to assign modal amplitudes to each reconstruction. The distribution of modal amplitudes for a population of minicircles provides a unique ‘signature’ dependent upon several variables, including superhelical density and the presence of kinks or torsional destabilizations. To test our method and predict these signatures, we developed a statistical mechanics model to simulate ensembles of DNA minicircles. This model can represent sequence dependence (elasticity/curvature) and prescribed kinks or torsional destabilizations. Our preliminary analysis suggests that the observed signatures are inconsistent with a homogeneous elastic rod and thereby implicate the role of sequence dependence, kinks or torsional instabilities.

While the typical viral genome is several kilobases long, it is packed to near crystalline density within a viral capsid only tens of nanometers in diameter. In the case of bacteriophage φ29, this enormous compaction results from strong molecular motors that generate forces of approximately 100 pN. Recently, a three-dimensional cryo-electron microscopy reconstruction of mature φ29 was published. An intriguing feature in the reconstruction is a 60A diameter toroidal DNA supercoil, estimated to be only 30-40 base pairs in length, within the cavity formed by the connector and the lower collar. The function of this highly-bent DNA, remains unknown. In this study, we use an elastic rod model to simulate the DNA inside the cavity. We learn that as more DNA is pushed into the capsid, compressive forces build until a critical load is reached and the DNA ‘buckles’ to fill the cavity thereby forming the toroidal supercoil observed in the reconstruction. We estimate the energy, forces, and torques required to form this toroidal DNA inside the cavity. Based on these results, we propose possible biological functions of this intriguing structure.

Protein mediated DNA looping is a well known gene regulatory mechanism. A commonly studied system that controls gene expression is lactose repressor (LacI) induced DNA looping. In two in vitro studies, the Muller-Hill group investigates how the lac repressor protein in E. coli forms loops with linear and cyclized DNA. Their experiments analyzed LacI induced looping on linear DNA over a wide range of interoperator lengths (6-21 helical turns) and on supercoiled DNA minicircles of 452 base pairs. In these experiments, electron microscopy, non-denaturing polyacrylamide gel elecrophoresis, and DNase I protection experiments were used to detect loop formation, estimate loop size, quantify loop stability, and for supercoiled DNA to detect loop topology (ΔLk). In our study, we exercise our computational rod model to make side-by-side comparisons of our predictions with their experimental observations. By making comparisons, we look to understand the energetic cost of loop formation and the resulting topology of the looped complex.

We have developed a semi-analytic method for calculating the Stockmayer Jacobson J-factor for protein mediated DNA loops, as well as DNA ring cyclization. The formation of DNA loops on the order of a few persistence lengths is a key component in many gene regulatory functions. The binding of LacI protein within the Lac Operon of E.coli serves as the canonical example in which loop regulated transcription is understood. This fundamental looping motif consists of one protein simultaneously bound to two DNA operator binding sites. We explore as inputs the effect of sequence-dependent curvature and elasticity on the formation of DNA loops by constructing a Hamiltonian describing thermal fluctuations about the open and looped states. These fluctuations allow us to compute the entropic cost of loop formation, and thus allow a full computation of the free energy. Our work demonstrates that even for short sequences of the order one persistence length, entropic contributions are required to correctly compute the J factor.We determine the lowest energy shape of the inter-operator DNA loop using a non-linear mechanical rod model under prescribed binding topologies (e.g, parallel and anti-parallel binding). Expanding about this shape allows us to calculate the J factors associated with parallel and anti-parallel binding topologies within the Lac system, and thus how entropy influences the most energetically favorable topology. The J factor can be used to compare the relative loop lifetimes of various DNA sequences, making it a useful tool in sequence design. Our work also allows the computation of an effective torsional persistence length, which demonstrates how torsion bending coupling affects the conversion of writhe to twist.

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.

Many DNA-binding proteins interact with twisted or bent DNA. To characterize the activity of these proteins as a function of the torsional and bending stresses, we must first understand how these mechanical stresses affect the DNA tertiary structure (topology). To experimentally define this relationship on scales that are biologically relevant to DNA-binding proteins requires DNA molecules which stably maintain high degrees of stress and deformation on a length scale appreciably below the persistence length. DNA minicircles of ∼100bp in size offer a unique opportunity to achieve our required specifications. We have prepared circular DNA constructs (100bp, 106bp, and 108bp) sustaining comparable magnitudes of bending stress and varying degrees of torsional stress (which arises when linear DNA molecules of a non-integral number of helical turns are circularized). Using cryo-electron microscopy (cryo-EM) combined with 3-D image reconstruction, we have been able to quantitatively characterize the structural details at the molecular level of the topological effects of torsional stress within these minicircle constructs. We have observed the three species of minicircles under conditions of both weak and mild electrostatic repulsion, and measured the observed distributions of curvature (indicative of kink formation) and writhe (reflective of torsional stress). Despite the significant torsional stress sustained within the most highly stressed construct, all three are roughly planar, though the writhe and curvature distributions do depart significantly from theoretically predicted values. We are attempting to resolve the discrepancies between theoretical expectations and our observed experimental data using Brownian dynamics simulations of DNA minicircles sustaining varying degrees of torsional stress. We expect that this work will begin to define the behavior of highly stressed DNA at biologically relevant scales, and will broaden our understanding of how sub-persistence length DNA responds to mechanical stress.

The formation of protein mediated DNA loops are a key component in many biological regulatory functions. The binding of LacI protein within the Lac Operon of E.coli serves as the canonical example in which loop regulated transcription is understood. This fundamental looping motif consists of one protein simultaneously bound to two DNA operator binding sites. We calculate the free energy cost of loop formation using a Hamiltonian constructed about the looped state as well as the intrinsically open state of the DNA. The shape of the inter-operator DNA loop in mechanical equilibrium with the protein, is determined using a non-linear mechanical rod model. Our rod model captures the effect of sequence dependent curvature, sequence dependent persistence length, including any anisotropic bending. The equilibrium DNA-protein binding orientations are inferred from LacI protein crystallized with DNA operator segments.Our Hamiltonian describes the change in bending energy of the DNA due to linear perturbations about either the looped or open state. We now calculate the normal modes of the Hamiltonian in order to characterize thermal fluctuations of the loop. Comparing the change stiffness we then calculate the Stockmayer J-factor(looping probability), free energy as well as the entropic contributions, of loop formation. Our works shows that these entropic contributions can play a significant role in determining loop stability and formation.