Writhe formulas and antipodal points in plectonemic DNA configurations

Torsional Stiffness of Extended and Plectonemic DNA

DNA torsional elastic properties play a crucial role in DNA structure, topology, and the regulation of motor protein progression. However, direct measurements of these parameters are experimentally challenging. Here, we present a constant-extension method integrated into an angular optical trap to directly measure torque during DNA supercoiling. We measured the twist persistence length of extended DNA to be 22 nm under an extremely low force (∼0.02  pN) and the twist persistence length of plectonemic DNA to be 24 nm. In addition, we implemented a rigorous data analysis scheme that bridged our measurements with existing theoretical models of DNA torsional behavior. This comprehensive set of torsional parameters demonstrates that at least 20% of DNA supercoiling is partitioned into twist for both extended DNA and plectonemic DNA. This work provides a new experimental methodology, as well as an analytical and interpretational framework, which will enable, expand, and enhance future studies of DNA torsional properties.

WASP: a software package for correctly characterizing the topological development of ribbon structures

We introduce the Writhe Application Software Package (WASP) which can be used to characterisze the topology of ribbon structures, the underlying mathematical model of DNA, Biopolymers, superfluid vorticies, elastic ropes and magnetic flux ropes. This characterization is achieved by the general twist–writhe decomposition of both open and closed ribbons, in particular through a quantity termed the polar writhe. We demonstrate how this decomposition is far more natural and straightforward than artificial closure methods commonly utilized in DNA modelling. In particular, we demonstrate how the decomposition of the polar writhe into local and non-local components distinctly characterizes the local helical structure and knotting/linking of the ribbon. This decomposition provides additional information not given by alternative approaches. As example applications, the WASP routines are used to characterise the evolving topology (writhe) of DNA minicircle and open ended plectoneme formation magnetic/optical tweezer simulations, and it is shown that the decomponsition into local and non-local components is particularly important for the detection of plectonemes. Finally it is demonstrated that a number of well known alternative writhe expressions are actually simplifications of the polar writhe measure.

Transfer-matrix calculations of DNA polymer micromechanics under tension and torque constraints

Recent development of single-molecule manipulation technologies has made it possible to exert constant force and torque on individual DNA biopolymers to probe their elastic characteristics and structural stability. It has been previously shown that depending on the nature of applied mechanical constraints, DNA can exist in several forms including B-, L-, and P-DNA. However, there is still a lack of understanding of how structural heterogeneity of DNA, which may naturally arise due to sequence-dependent DNA properties, protein binding, or DNA damage, influences local stability of the above DNA states. To provide a more complete and detailed description of the DNA mechanics, we developed a theoretical framework based on transfer-matrix calculations and demonstrated how it can be used to predict the DNA behavior upon application of a wide range of force and torque constraints. The resulting phase diagram shows DNA structural transitions that are in good agreement with previous experimental and theoretical studies. We further discuss how the constructed formalism can be extended to include local inhomogeneities in the DNA physical properties, thus making it possible to investigate the effect of DNA sequence as well as protein binding on DNA structural stability.

Thermodynamics of long supercoiled molecules: insights from highly efficient Monte Carlo simulations

Supercoiled DNA polymer models for which the torsional energy depends on the total twist of molecules (Tw) are a priori well suited for thermodynamic analysis of long molecules. So far, nevertheless, the exact determination of Tw in these models has been based on a computation of the writhe of the molecules (Wr) by exploiting the conservation of the linking number, Lk=Tw+Wr, which reflects topological constraints coming from the helical nature of DNA. Because Wr is equal to the number of times the main axis of a DNA molecule winds around itself, current Monte Carlo algorithms have a quadratic time complexity, O(L(2)), with respect to the contour length (L) of the molecules. Here, we present an efficient method to compute Tw exactly, leading in principle to algorithms with a linear complexity, which in practice is O(L(1.2)). Specifically, we use a discrete wormlike chain that includes the explicit double-helix structure of DNA and where the linking number is conserved by continuously preventing the generation of twist between any two consecutive cylinders of the discretized chain. As an application, we show that long (up to 21 kbp) linear molecules stretched by mechanical forces akin to magnetic tweezers contain, in the buckling regime, multiple and branched plectonemes that often coexist with curls and helices, and whose length and number are in good agreement with experiments. By attaching the ends of the molecules to a reservoir of twists with which these can exchange helix turns, we also show how to compute the torques in these models. As an example, we report values that are in good agreement with experiments and that concern the longest molecules that have been studied so far (16 kbp).

Competition between B-Z and B-L transitions in a single DNA molecule: Computational studies

Under negative torsion, DNA adopts left-handed helical forms, such as Z-DNA and L-DNA. Using the random copolymer model developed for a wormlike chain, we represent a single DNA molecule with structural heterogeneity as a helical chain consisting of monomers which can be characterized by different helical senses and pitches. By Monte Carlo simulation, where we take into account bending and twist fluctuations explicitly, we study sequence dependence of B-Z transitions under torsional stress and tension focusing on the interaction with B-L transitions. We consider core sequences, (GC)_{n} repeats or (TG)_{n} repeats, which can interconvert between the right-handed B form and the left-handed Z form, imbedded in a random sequence, which can convert to left-handed L form with different (tension dependent) helical pitch. We show that Z-DNA formation from the (GC)_{n} sequence is always supported by unwinding torsional stress but Z-DNA formation from the (TG)_{n} sequence, which are more costly to convert but numerous, can be strongly influenced by the quenched disorder in the surrounding random sequence.

Biophysics of protein-DNA interactions and chromosome organization

The function of DNA in cells depends on its interactions with protein molecules, which recognize and act on base sequence patterns along the double helix. These notes aim to introduce basic polymer physics of DNA molecules, biophysics of protein-DNA interactions and their study in single-DNA experiments, and some aspects of large-scale chromosome structure. Mechanisms for control of chromosome topology will also be discussed.

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

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

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

The dependence of DNA supercoiling on solution electrostatics

We develop an elastic-isotropic rod model for twisted DNA in the plectonemic regime. We account for DNA elasticity, electrostatic interactions and entropic effects due to thermal fluctuations. We apply our model to single-molecule experiments on a DNA molecule attached to a substrate at one end, while subjected to a tensile force and twisted by a given number of turns at the other end. The free energy of the DNA molecule is minimized subject to the imposed end rotations. We compute values of the torsional stress, radius, helical angle and key features of the rotation-extension curves. We also include in our model the end loop energetic contributions and obtain estimates for the jumps in the external torque and extension of the DNA molecule seen in experiments. We find that, while the general trends seen in experiments are captured simply by rod mechanics, the details can be accounted for only with the proper choice of electrostatic and entropic interactions. We perform calculations with different ionic concentrations and show that our model yields excellent fits to mechanical data from a large number of experiments. Our methods also allow us to consider scenarios where we have multiple plectonemes or a series of loops forming in the DNA instead of plectonemes. For a given choice of electrostatic and entropic interactions, we find there is a range of forces in which the two regimes can coexist due to thermal motion.

Biopolymer elasticity: Mechanics and thermal fluctuations

We present an analytical study of the role of thermal fluctuations in shaping molecular elastic properties of semiflexible polymers. Our study interpolates between mechanics and statistical mechanics in a controlled way and shows how thermal fluctuations modify the elastic properties of biopolymers. We present a study of the minimum-energy configurations with explicit expressions for their energy and writhe and plots of the extension versus link for these configurations and a study of fluctuations around the local minima of energy and approximate analytical formulas for the free energy of stretched twisted polymers. The central result of our study is a closed-form expression for the leading thermal fluctuation correction to the free energy around the nonperturbative writhing family solution for the configuration of a biopolymer. From the derived formulas, the predictions of the wormlike chain model for molecular elasticity can be worked out for a comparison against numerical simulations and experiments.

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

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

Competition between curls and plectonemes near the buckling transition of stretched supercoiled DNA

Recent single-molecule experiments have observed that formation of a plectonemically supercoiled region in a stretched, twisted DNA proceeds via abrupt formation of a small plectonemic "bubble." A detailed mesoscopic model is presented for the formation of plectonemic domains, including their positional entropy, and the influence of small chiral loops or "curls" along the extended DNA. Curls begin to appear just before plectoneme formation, and are more numerous at low salt concentrations (<20 mM univalent ions) and at low forces (<0.5 pN). However, plectonemic domains quickly become far more stable slightly beyond the transition to supercoiling at moderate forces and physiological salt conditions. At the supercoiling transition, for shorter DNAs (2 kb) only one supercoiled domain appears, but for longer DNAs at lower forces (<0.5 pN) positional entropy favors formation of more than one plectonemic domain; a similar effect occurs for low salt. Although they are not the prevalent mode of supercoiling, curls are a natural transition state for binding of DNA-loop-trapping enzymes; we show how addition of loop-trapping enzymes can modify the supercoiling transition. The behavior of DNA torque is also discussed, including the effect of the measurement apparatus torque stiffness, which can play a role in determining how large the torque "overshoot" is at the buckling transition.

Geometry of twist transport in a rotating elastic rod

An elastic rod rotating in a viscous fluid undergoes a shape transition from a twirling (axial spinning) to a whirling state (crankshafting motion) at a certain critical frequency [Wolgemuth et al., Phys. Rev. Lett. 84, 1623 (2000)]. The physical properties of such whirling rods are largely unknown, owing to their strongly nonlinear character. We analytically and numerically demonstrate that this dynamical transition occurs to reduce the viscous energy dissipation. A simple geometric interpretation underlying this observation is also given. These results provide a fundamental scenario for viscous twist transport in flexible filaments and are potentially important in the analysis of biopolymer dynamics such as DNA supercoiling during transcriptions.

Linker histones incorporation maintains chromatin fiber plasticity

Genomic DNA in eukaryotic cells is organized in supercoiled chromatin fibers, which undergo dynamic changes during such DNA metabolic processes as transcription or replication. Indeed, DNA-translocating enzymes like polymerases produce physical constraints in vivo. We used single-molecule micromanipulation by magnetic tweezers to study the response of chromatin to mechanical constraints in the same range as those encountered in vivo. We had previously shown that under positive torsional constraints, nucleosomes can undergo a reversible chiral transition toward a state of positive topology. We demonstrate here that chromatin fibers comprising linker histones present a torsional plasticity similar to that of naked nucleosome arrays. Chromatosomes can undergo a reversible chiral transition toward a state of positive torsion (reverse chromatosome) without loss of linker histones.

Comment on "Writhe formulas and antipodal points in plectonemic DNA configurations"

We point out that the disagreement between the paper by Neukirch and Starostin [S. Neukirch and E. L. Starostin, Phys. Rev. E 78, 041912 (2008)] and ours [J. Samuel, S. Sinha, and A. Ghosh, J. Phys.: Condens. Matter 18, S253 (2006)] is only apparent and stems from a difference in approach. Neukirch and Starostin are concerned with classical elasticity and individual curves while we focus on statistical averages over curves.

Elasticity and electrostatics of plectonemic DNA

We present a self-contained theory for the mechanical response of DNA in single molecule experiments. Our model is based on a one-dimensional continuum description of the DNA molecule and accounts both for its elasticity and for DNA-DNA electrostatic interactions. We consider the classical loading geometry used in experiments where one end of the molecule is attached to a substrate and the other one is pulled by a tensile force and twisted by a given number of turns. We focus on configurations relevant to the limit of a large number of turns, which are made up of two phases, one with linear DNA and the other one with superhelical DNA. The model takes into account thermal fluctuations in the linear phase and electrostatic interactions in the superhelical phase. The values of the torsional stress, of the supercoiling radius and angle, and key features of the experimental extension-rotation curves, namely the slope of the linear region and thermal buckling threshold, are predicted. They are found in good agreement with experimental data.