DNA rings with multiple energy minima

DNA superhelicity

Closing each strand of a DNA duplex upon itself fixes its linking number L. This topological condition couples together the secondary and tertiary structures of the resulting ccDNA topoisomer, a constraint that is not present in otherwise identical nicked or linear DNAs. Fixing L has a range of structural, energetic and functional consequences. Here we consider how L having different integer values (that is, different superhelicities) affects ccDNA molecules. The approaches used are primarily theoretical, and are developed from a historical perspective. In brief, processes that either relax or increase superhelicity, or repartition what is there, may either release or require free energy. The energies involved can be substantial, sufficient to influence many events, directly or indirectly. Here two examples are developed. The changes of unconstrained superhelicity that occur during nucleosome attachment and release are examined. And a simple theoretical model of superhelically driven DNA structural transitions is described that calculates equilibrium distributions for populations of identical topoisomers. This model is used to examine how these distributions change with superhelicity and other factors, and applied to analyze several situations of biological interest.

Inverse design of an isotropic suspended Kirchhoff rod: theoretical and numerical results on the uniqueness of the natural shape

Solving the equations for Kirchhoff elastic rods has been widely explored for decades in mathematics, physics and computer science, with significant applications in the modelling of thin flexible structures such as DNA, hair or climbing plants. As demonstrated in previous experimental and theoretical studies, the natural curvature plays an important role in the equilibrium shape of a Kirchhoff rod, even in the simple case where the rod is isotropic and suspended under gravity. In this paper, we investigate the reverse problem: can we characterize the natural curvature of a suspended isotropic rod, given an equilibrium curve? We prove that although there exists an infinite number of natural curvatures that are compatible with the prescribed equilibrium, they are all equivalent in the sense that they correspond to a unique natural shape for the rod. This natural shape can be computed efficiently by solving in sequence three linear initial value problems, starting from any framing of the input curve. We provide several numerical experiments to illustrate this uniqueness result, and finally discuss its potential impact on non-invasive parameter estimation and inverse design of thin elastic rods.

Effect of spontaneous twist on DNA minicircles

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

Looping probabilities of elastic chains: a path integral approach

We consider an elastic chain at thermodynamic equilibrium with a heat bath, and derive an approximation to the probability density function, or pdf, governing the relative location and orientation of the two ends of the chain. Our motivation is to exploit continuum mechanics models for the computation of DNA looping probabilities, but here we focus on explaining the novel analytical aspects in the derivation of our approximation formula. Accordingly, and for simplicity, the current presentation is limited to the illustrative case of planar configurations. A path integral formalism is adopted, and, in the standard way, the first approximation to the looping pdf is obtained from a minimal energy configuration satisfying prescribed end conditions. Then we compute an additional factor in the pdf which encompasses the contributions of quadratic fluctuations about the minimum energy configuration along with a simultaneous evaluation of the partition function. The original aspects of our analysis are twofold. First, the quadratic Lagrangian describing the fluctuations has cross-terms that are linear in first derivatives. This, seemingly small, deviation from the structure of standard path integral examples complicates the necessary analysis significantly. Nevertheless, after a nonlinear change of variable of Riccati type, we show that the correction factor to the pdf can still be evaluated in terms of the solution to an initial value problem for the linear system of Jacobi ordinary differential equations associated with the second variation. The second novel aspect of our analysis is that we show that the Hamiltonian form of these linear Jacobi equations still provides the appropriate correction term in the inextensible, unshearable limit that is commonly adopted in polymer physics models of, e.g. DNA. Prior analyses of the inextensible case have had to introduce nonlinear and nonlocal integral constraints to express conditions on the relative displacement of the end points. Our approximation formula for the looping pdf is of quite general applicability as, in contrast to most prior approaches, no assumption is made of either uniformity of the elastic chain, nor of a straight intrinsic shape. If the chain is uniform the Jacobi system evaluated at certain minimum energy configurations has constant coefficients. In such cases our approximate pdf can be evaluated in an entirely explicit, closed form. We illustrate our analysis with a planar example of this type and compute an approximate probability of cyclization, i.e., of forming a closed loop, from a uniform elastic chain whose intrinsic shape is an open circular arc.

Multiscale modeling of nucleic acids: insights into DNA flexibility

The elastic rod theory is used together with all-atom normal mode analysis in implicit solvent to characterize the mechanical flexibility of duplex DNA. The bending, twisting, stretching rigidities extracted from all-atom simulations (on linear duplexes from 60 to 150 base pairs in length and from 94-bp minicircles) are in reasonable agreement with experimental results. We focus on salt concentration and sequence effects on the overall flexibility. Bending persistence lengths are about 20% higher than most experimental estimates, but the transition from low-salt to high-salt behavior is reproduced well, as is the dependence of the stretching modulus on salt (which is opposite to that of bending). CTG and CGG trinucleotide repeats, responsible for several degenerative disorders, are found to be more flexible than random DNA, in agreement with several recent studies, whereas poly(dA).poly(dT) is the stiffest sequence we have encountered. The results suggest that current all-atom potentials, which were parameterized on small molecules and short oligonucleotides, also provide a useful description of duplex DNA at much longer length scales.

3D reconstruction and comparison of shapes of DNA minicircles observed by cryo-electron microscopy

We use cryo-electron microscopy to compare 3D shapes of 158 bp long DNA minicircles that differ only in the sequence within an 18 bp block containing either a TATA box or a catabolite activator protein binding site. We present a sorting algorithm that correlates the reconstructed shapes and groups them into distinct categories. We conclude that the presence of the TATA box sequence, which is believed to be easily bent, does not significantly affect the observed shapes.

Modeling DNA loops using the theory of elasticity

An elastic rod model of a protein-bound DNA loop is adapted for application in multi-scale simulations of protein-DNA complexes. The classical Kirchhoff system of equations which describes the equilibrium structure of the elastic loop is modified to account for the intrinsic twist and curvature, anisotropic bending properties, and electrostatic charge of DNA. The effects of bending anisotropy and electrostatics are studied for the DNA loop clamped by the lac repressor protein. For two possible lengths of the loop, several topologically different conformations are predicted and extensively analyzed over the broad range of model parameters describing DNA bending and electrostatic properties. The scope and applications of the model in already accomplished and in future multi-scale studies of protein-DNA complexes are discussed.

DNA Mechanics

We review the history of DNA mechanics and its analysis. We evaluate several methods to analyze the structures of superhelical DNA molecules, each predicated on the assumption that DNA can be modeled with reasonable accuracy as an extended, linearly elastic polymer. Three main approaches are considered: mechanical equilibrium methods, which seek to compute minimum energy conformations of topologically constrained molecules; statistical mechanical methods, which seek to compute the Boltzmann distribution of equilibrium conformations that arise in a finite temperature environment; and dynamic methods, which seek to compute deterministic trajectories of the helix axis by solving equations of motion. When these methods include forces of self-contact, which prevent strand passage and preserve the topological constraint, each predicts plectonemically interwound structures. On the other hand, the extent to which these mechanical methods reliably predict energetic and thermodynamic properties of superhelical molecules is limited, in part because of their inability to account explicitly for interactions involving solvent. Monte Carlo methods predict the entropy associated with supercoiling to be negative, in conflict with a body of experimental evidence that finds it is large and positive, as would be the case if superhelical deformations significantly disrupt the ordering of ambient solvent molecules. This suggests that the large-scale conformational properties predicted by elastomechanical models are not the only ones determining the energetics and thermodynamics of supercoiling. Moreover, because all such models that preserve the topological constraint correctly predict plectonemic interwinding, despite these and other limitations, this constraint evidently dominates energetic and thermodynamic factors in determining supercoil geometry. Therefore, agreement between predicted structures and structures obtained experimentally, for example, by electron microscopy, does not in itself provide evidence for the correctness or completeness of any given model of DNA mechanics.

Finite-element analysis of the displacement of closed DNA loops under torsional stress

Closed DNA loops that contain intrinsic curvature occur in biologically important structures that are formed by bringing together proteins attached at distinct sites. Such loops constitute topological domains that are characterized by a linking number Delta Lk. We calculate, using finite-element analysis, the structural changes induced by small changes in this linking number, Delta Lk. Because of the intrinsic curvature, the slightest change in linking number induces writhe and the loop begins to fold in space. We previously studied the case in which the initial curvature is uniformly distributed along the DNA rod. We found that there are two different folding modes, depending on the amount of intrinsic curvature and the Poisson ratio, a quantity that measures the ratio of bending stiffness to torsional rigidity. For combinations of the Poisson ratio and curvature that lie below a critical curve, called the Fickel curve, the folding is monotonic in the sense that the writhe uniformly increases as Delta Lk increases, until self-contact occurs. For combinations below this curve, the folding is non-monotonic in the sense that as Delta Lk increases the writhe first increases, then decreases back to essentially zero, and then increases uniformly until self-contact occurs. The folding behaviour and the self-contact points in the two folding modes are completely different. In this paper we first review this previous work. We then extend those results to more-complex situations in which the curvature is initially distributed non-uniformly along the DNA rod. We show that the location of the Fickel curve depends upon both the extent of the initial curvature and upon its distribution along the rod. We also show that two DNAs with the same total intrinsic curvature will fold differently depending upon the distribution of that curvature along the DNA axis, and upon the point of the loop at which the applied rotation or change in Delta Lk is introduced.

Implications of the dependence of the elastic properties of DNA on nucleotide sequence

Recent advances in structural biochemistry have provided evidence that not only the geometric properties but also the elastic moduli of duplex DNA are strongly dependent on nucleotide sequence in a way that is not accounted for by classical rod models of the Kirchhoff type. A theory of sequence-dependent DNA elasticity is employed here to calculate the dependence of the equilibrium configurations of circular DNA on the binding of ligands that can induce changes in intrinsic twist at a single base-pair step. Calculations are presented of the influence on configurations of the assumed values and distribution along the DNA of intrinsic roll and twist and a modulus coupling roll to twist. Among the results obtained are the following. For minicircles formed from intrinsically straight DNA, the distribution of roll-twist coupling strongly affects the dependence of the total elastic energy Psi on the amount alpha of imposed untwisting, and that dependence can be far from quadratic. (In fact, for a periodic distribution of roll-twist coupling with a period equal to the intrinsic helical repeat length, Psi can be essentially independent of alpha for -90 degrees < alpha <90 degrees.) When the minicircle is homogeneous and without roll-twist coupling, but with uniform positive intrinsic roll, the point at which Psi attains its minimum value shifts towards negative values of alpha. It is remarked that there are cases in which one can relate graphs of Psi versus alpha to the 'effective values' of bending and twisting moduli and helical repeat length obtained from measurements of equilibrium distributions of topoisomers and probabilities of ring closure. For a minicircle formed from DNA that has an 'S' shape when stress-free, the graphs of Psi versus alpha have maxima at alpha = 0. As the binding of a twisting agent to such a minicircle results in a net decrease in Psi, the affinity of the twisting agent for binding to the minicircle is greater than its affinity for binding to unconstrained DNA with the same sequence.

Link, twist, energy, and the stability of DNA minicircles

We describe how the stability properties of DNA minicircles can be directly read from plots of various biologically intuitive quantities along families of equilibrium configurations. Our conclusions follow from extensions of the mathematical theory of distinguished bifurcation diagrams that are applied within the specific context of an elastic rod model of minicircles. Families of equilibria arise as a twisting angle alpha is varied. This angle is intimately related to the continuously varying linking number Lk for nicked DNA configurations that is defined as the sum of Twist and Writhe. We present several examples of such distinguished bifurcation diagrams involving plots of the energy E, linking number Lk, and a twist moment m3, along families of cyclized equilibria of both intrinsically straight and intrinsically curved DNA fragments.

Statistical mechanics of sequence-dependent circular DNA and its application for DNA cyclization

DNA cyclization is potentially the most powerful approach for systematic quantitation of sequence-dependent DNA bending and flexibility. We extend the statistical mechanics of the homogeneous DNA circle to a model that considers discrete basepairs, thus allowing for inhomogeneity, and apply the model to analysis of DNA cyclization. The theory starts from an iterative search for the minimum energy configuration of circular DNA. Thermodynamic quantities such as the J factor, which is essentially the ratio of the partition functions of circular and linear forms, are evaluated by integrating the thermal fluctuations around the configuration under harmonic approximation. Accurate analytic expressions are obtained for equilibrium configurations of homogeneous circular DNA with and without bending anisotropy. J factors for both homogeneous and inhomogeneous DNA are evaluated. Effects of curvature, helical repeat, and bending and torsional flexibility in DNA cyclization are analyzed in detail, revealing that DNA cyclization can detect as little as one degree of curvature and a few percent change in flexibility. J factors calculated by our new approach are well consistent with Monte Carlo simulations, whereas the new theory has much greater efficiency in computations. Simulation of experimental results has been demonstrated.