Elastic rod model incorporating shear and extension, generalized nonlinear Schrödinger equations, and novel closed‐form solutions for supercoiled DNA

Dynamics of certain Euler-Bernoulli rods and rings from a minimal coupling quantum isomorphism

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in one dimension is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in three dimensions with ratio of bending modulus to linear density (ℏ/2m)^{2}. For m=m_{e}, this quantity is comparable to that of a microtubule. Axial forces and torques applied to the rod play the role of scalar and vector potentials, respectively, and rod inextensibility plays the role of normalization. We show how an uncertainty principle ΔxΔp_{x}≳ℏ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of ℏ or ℏ/2 (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wave numbers, a "twist quantum" appears that is somewhat analogous to the magnetic flux quantum. These and other results are obtained from a purely classical treatment of the rod, i.e., without quantizing any classical fields.

Comment on "Modulating DNA configuration by interfacial traction: an elastic rod model to characterize DNA folding and unfolding"

In this comment, we point out that the tractions induced by interfacial energy, which are referred to as the tractions on the central axis curve of the DNA elastic rod presented by Huang (J. Biol. Phys. 37(1), 79-90, 2011), are incorrect. The correct tractions are provided in this literature. Further, with the use of the correct tractions, we present new numerical results, which for the values given by Zaixing Huang do not give rise to the physical behavior observed for DNA by the author.

An elastic rod model to evaluate effects of ionic concentration on equilibrium configuration of DNA in salt solution

As a coarse-gained model, a super-thin elastic rod subjected to interfacial interactions is used to investigate the condensation of DNA in a multivalent salt solution. The interfacial traction between the rod and the solution environment is determined in terms of the Young-Laplace equation. Kirchhoff's theory of elastic rod is used to analyze the equilibrium configuration of a DNA chain under the action of the interfacial traction. Two models are established to characterize the change of the interfacial traction and elastic modulus of DNA with the ionic concentration of the salt solution, respectively. The influences of the ionic concentration on the equilibrium configuration of DNA are discussed. The results show that the condensation of DNA is mainly determined by competition between the interfacial energy and elastic strain energy of the DNA itself, and the interfacial traction is one of forces that drive DNA condensation. With the change of concentration, the DNA segments will undergo a series of alteration from the original configuration to the condensed configuration, and the spiral-shape appearing in the condensed configuration of DNA is independent of the original configuration.

Molecular Dynamics Simulations of a Nucleosome and Free DNA

All atom molecular dynamics simulations (10ns) of a nucleosome and of its 146 basepairs of DNA free in solution have been conducted. DNA helical parameters (Roll, Tilt, Twist, Shift, Slide, Rise) were extracted from each trajectory to compare the conformation, effective force constants, persistence length measures, and fluctuations of nucleosomal DNA to free DNA. The conformation of DNA in the nucleosome, as determined by helical parameters, is found to be largely within the range of thermally accessible values obtained for free DNA. DNA is found to be less flexible on the nucleosome than when free in solution, however such measures are length scale dependent. A method for disassembling and reconstructing the conformation and dynamics of the nucleosome using Fourier analysis is presented. Long length variations in the conformation of nucleosomal DNA are identified other than those associated with helix repeat. These variations are required to create a proposed tetrasome conformation or to qualitatively reconstruct the 1.75 turns of the nucleosome's superhelix. Reconstruction of free DNA using selected long wavelength variations in conformation can produce either a left-handed or a right-handed superhelix. The long wavelength variations suggest 146 basepairs is a natural length of DNA to wrap around the histone core.

Modulating DNA configuration by interfacial traction: an elastic rod model to characterize DNA folding and unfolding

As a continuum model of DNA, a thin elastic rod subjected to interfacial interactions is used to investigate the equilibrium configuration of DNA in intracellular solution. The interfacial traction between the rod and the solution environment is derived in detail. Kirchhoff's theory of elastic rods is used to analyze the equilibrium configuration of a DNA segment under the action of the interfacial traction. The influences of the interfacial energy factor and bending stiffness on the toroidal spool formation of the DNA segment are discussed. The results show that the equilibrium configuration of DNA is mainly determined by competition between the interfacial energy and elastic strain energy of the DNA itself, and the interfacial traction is one of the forces that drives DNA folding and unfolding.

Biomolecular Structure and Modeling: Problem and Application Perspective

The experimental progress described in the previous chapter has been accompanied by an increasing desire to relate the complex three-dimensional (3D) shapes of biomolecules to their biological functions and interactions with other molecular systems. Structural biology, computational biology, genomics, proteomics, bioinformatics, chemoinformatics, and others are natural partner disciplines in such endeavors.

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.

Stability results for constrained calculus of variations problems: an analysis of the twisted elastic loop

Problems with a variational structure are ubiquitous throughout the physical sciences and have a distinguished scientific history. Constrained variational problems have been much less studied, particularly the theory of stability, which determines which solutions are physically realizable. In this paper, we develop stability exchange results appropriate for parameter-dependent calculus of variations problems with two particular features: either the parameter appears in the boundary conditions, or there are isoperimetric constraints. In particular, we identify an associated distinguished bifurcation diagram, which encodes the direction of stability exchange at folds. We apply the theory to a twisted elastic loop, which can naturally be formulated as a calculus of variations problem with both isoperimetric constraints and parameter-dependent boundary conditions. In combination with a perturbation expansion that classifies certain pitchfork bifurcations as sub- or super-critical, the distinguished diagram for the twisted loop provides a classification of the stability properties of all equilibria. In particular, an unanticipated sensitive dependence of stability properties on the ratio of twisting to bending stiffness is revealed.

Structural basis for cooperative DNA binding by CAP and lac repressor

Catabolite gene activator protein (CAP) and lac repressor (LR) are celebrated transcription-regulating proteins that bind to DNA cooperatively forming a ternary complex with the promoter loop. Here we present a multiscale model of the ternary complex derived from crystal structures of the proteins and a continuous structure of the DNA loop built using the theory of elasticity. We predict that the loop is underwound in the binary complex with the LR, whereas in the ternary complex with the LR and CAP, the loop is overwound and extended due to an upstream relocation of a DNA binding hand of LR. The computed relocation distance matches the experimental observations and the energy balance of the system explains the cooperativity effect. Using the multiscale approach, we build an all-atom model of the ternary complex that suggests a series of further experimental investigations.

Mechanical model of the nucleosome and chromatin

A theoretical framework for evaluating the approximate energy and dynamic properties associated with the folding of DNA into nucleosomes and chromatin is presented. Experimentally determined elastic constants of linear DNA and a simple fold geometry are assumed in order to derive elastic constants for extended and condensed chromatin. The model predicts the Young s modulus of extended and condensed chromatin to within an order of magnitude of experimentally determined values. Thus we demonstrate that the elastic properties of DNA are a primary determinant of the elastic properties of the higher order folded states. The derived elastic constants are used to predict the speed of propagation of small amplitude waves that excite an extension(sound), twist, bend or shear motion in each folded state. Taken together the results demonstrate that folding creates a hierarchy of time, length and energy scales.

Calculations of nucleic acid conformations

The present computational power and sophistication of theoretical approaches to nucleic acid structural investigation are sufficient for the realization of static and dynamic models that correlate accurately with current crystallographic, NMR and solution-probing structural data, and consequently are able to provide valuable insights and predictions for a variety of nucleic acid conformational families. In molecular dynamics simulations, the year 1995 was marked by the foray of fast Ewald methods, an accomplishment resulting from several years' work in the search for an adequate treatment of the electrostatic long-range forces so primordial in nucleic acid behavior. In very large systems, and particularly in the RNA-folding field, techniques originating from artificial intelligence research, like constraint satisfaction programming or genetic algorithms, have established their utility and potential.

Simulating DNA at low resolution

The past year has witnessed the development of several new mathematical approaches to analyzing the structure of double-helical DNA and to incorporating the sequence-dependent features of the chain in computer simulations of long polymers. Of special interest in this respect are the local and global structural changes induced by the binding of various proteins to DNA, ranging from subtle bending, untwisting and sliding motions at the base-pair level to the apparent organization of supercoiled structure in chains that are thousands residues long. The computational effort has also included both new ways to incorporate the polyelectrolyte character of DNA and other environmental forces in simulations of long chains and new methods to keep track of the multitude of configurations so generated. The collective advances are pointing to ways that will soon connect the sequences of base pairs in large genomes to folded three-dimensional structures based on natural bending, twisting and translational tendencies and in response to deformations produced by the binding of different proteins.