Multiscale modeling of double-helical DNA and RNA: a unification through Lie groups

A Study on the Bending Stiffness of a New DNA Origami Nano-Joint

The present article aims to investigate the mechanical properties of a new DNA origami nano-joint using the steered molecular dynamics (SMD) simulation. Since the analysis of mechanical properties is of great importance in bending conditions for a nano-joint, the forces are selected to achieve angular changes in the joint by the resultant torque. In this study, the nano-joint is considered as a beam in order to use mechanical equations to extract the mechanical properties of the designed nano-joint. In addition, the bending stiffness of the beam is investigated in different modes of deflection using the Euler-Bernoulli beam theory. The results revealed that the value of bending stiffness increases with increasing deflection, and the changes in the bending stiffness relative to the deflection is linear. The proposed DNA origami nano-joint can be used as a joint in nanorobots and can be effectively applied in nanorobotic systems to move different components.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker-Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Elasticity as the Basis of Allostery in DNA

Allosteric interactions in DNA are crucial for various biological processes. These interactions are quantified by measuring the change in free energy as a function of the distance between the binding sites for two ligands. Here, we show that trends in the interaction energy of ligands binding to DNA can be explained within an elastic birod model, which accounts for the deformation of each strand as well as the change in stacking energy due to perturbations in position and orientation of the bases caused by the binding of ligands. The strain fields produced by the ligands decay with distance from the binding site. The interaction energy of two ligands decays exponentially with the distance between them and oscillates with the periodicity of the double helix in quantitative agreement with experimental measurements. The trend in the computed interaction energy is similar to that in the perturbation of groove width produced by the binding of a single ligand, which is consistent with molecular simulations. Our analysis provides a new framework to understand allosteric interactions in DNA and can be extended to other rod-like macromolecules whose elasticity plays a role in biological functions.

Theory of wavelet-based coarse-graining hierarchies for molecular dynamics

We present a multiresolution approach to compressing the degrees of freedom and potentials associated with molecular dynamics, such as the bond potentials. The approach suggests a systematic way to accelerate large-scale molecular simulations with more than two levels of coarse graining, particularly applications of polymeric materials. In particular, we derive explicit models for (arbitrarily large) linear (homo)polymers and iterative methods to compute large-scale wavelet decompositions from fragment solutions. This approach does not require explicit preparation of atomistic-to-coarse-grained mappings, but instead uses the theory of diffusion wavelets for graph Laplacians to develop system-specific mappings. Our methodology leads to a hierarchy of system-specific coarse-grained degrees of freedom that provides a conceptually clear and mathematically rigorous framework for modeling chemical systems at relevant model scales. The approach is capable of automatically generating as many coarse-grained model scales as necessary, that is, to go beyond the two scales in conventional coarse-grained strategies; furthermore, the wavelet-based coarse-grained models explicitly link time and length scales. Furthermore, a straightforward method for the reintroduction of omitted degrees of freedom is presented, which plays a major role in maintaining model fidelity in long-time simulations and in capturing emergent behaviors.

Influence of model resolution on geometric simulations of antibody aggregation

It is estimated that allergies afflict up to 40% of the world's population. A primary mediator for allergies is the aggregation of antigens and IgE antibodies bound to cell-surface receptors, FcεRI. Antibody/antigen aggregate formation causes stimulation of mast cells and basophils, initiating cellular degranulation and releasing immune mediators which produce an allergic or anaphylactic response. Understanding the shape and structure of these aggregates can provide critical insights into the allergic response. We have previously developed methods to geometrically model, simulate and analyze antibody aggregation inspired by rigid body robotic motion simulations. Our technique handles the large size and number of molecules involved in aggregation, providing an advantage over traditional simulations such as molecular dynamics (MD) and coarse-grained energetic models. In this paper, we study the impact of model resolution on simulations of geometric structures using both our previously developed Monte Carlo simulation and a novel application of rule-based modeling. These methods complement each other, the former providing explicit geometric detail and the latter providing a generic representation where multiple resolutions can be captured. Our exploration is focused on two antigens, a man-made antigen with three binding sites, DF3, and a common shrimp allergen (antigen), Pen a 1. We find that impact of resolution is minimal for DF3, a small globular antigen, but has a larger impact on Pen a 1, a rod-shaped molecule. The volume reduction caused by the loss in resolution allows more binding site accessibility, which can be quantified using a rule-based model with implicit geometric input. Clustering analysis of our simulation shows good correlation when compared with available experimental results. Moreover, collisions in all-atom reconstructions are negligible, at around 0.2% at 90% reduction.

Simulation Assisted Analysis of the Intrinsic Stiffness for Short DNA Molecules Imaged with Scanning Atomic Force Microscopy

Studying the mechanical properties of short segments of dsDNA can provide insight into various biophysical phenomena, from DNA looping to the organization of nucleosomes. Scanning atomic force microscopy (AFM) is able to acquire images of single DNA molecules with near-basepair resolution. From many images, one may use equilibrium statistical mechanics to quantify the intrinsic stiffness (or persistence length) of the DNA. However, this approach is highly dependent upon both the correct microscopic polymer model and a correct image analysis of DNA contours. These complications have led to significant debate over the flexibility of dsDNA at short length scales. We first show how to extract accurate measures of DNA contour lengths by calibrating to DNA traces of simulated AFM data. After this calibration, we show that DNA adsorbed on an aminopropyl-mica surface behaves as a worm-like chain (WLC) for contour lengths as small as ~20 nm. We also show that a DNA binding protein can modify the mechanics of the DNA from that of a WLC.

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

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

DNA origami compliant nanostructures with tunable mechanical properties

DNA origami enables fabrication of precise nanostructures by programming the self-assembly of DNA. While this approach has been used to make a variety of complex 2D and 3D objects, the mechanical functionality of these structures is limited due to their rigid nature. We explore the fabrication of deformable, or compliant, objects to establish a framework for mechanically functional nanostructures. This compliant design approach is used in macroscopic engineering to make devices including sensors, actuators, and robots. We build compliant nanostructures by utilizing the entropic elasticity of single-stranded DNA (ssDNA) to locally bend bundles of double-stranded DNA into bent geometries whose curvature and mechanical properties can be tuned by controlling the length of ssDNA strands. We demonstrate an ability to achieve a wide range of geometries by adjusting a few strands in the nanostructure design. We further developed a mechanical model to predict both geometry and mechanical properties of our compliant nanostructures that agrees well with experiments. Our results provide a basis for the design of mechanically functional DNA origami devices and materials.

Destabilization of the MutSα's protein-protein interface due to binding to the DNA adduct induced by anticancer agent carboplatin via molecular dynamics simulations

DNA mismatch repair (MMR) proteins maintain genetic integrity in all organisms by recognizing and repairing DNA errors. Such alteration of hereditary information can lead to various diseases, including cancer. Besides their role in DNA repair, MMR proteins detect and initiate cellular responses to certain type of DNA damage. Its response to the damaged DNA has made the human MMR pathway a useful target for anticancer agents such as carboplatin. This study indicates that strong, specific interactions at the interface of MutSα in response to the mismatched DNA recognition are replaced by weak, non-specific interactions in response to the damaged DNA recognition. Data suggest a severe impairment of the dimerization of MutSα in response to the damaged DNA recognition. While the core of MutSα is preserved in response to the damaged DNA recognition, the loss of contact surface and the rearrangement of contacts at the protein interface suggest a different packing in response to the damaged DNA recognition. Coupled in response to the mismatched DNA recognition, interaction energies, hydrogen bonds, salt bridges, and solvent accessible surface areas at the interface of MutSα and within the subunits are uncoupled or asynchronously coupled in response to the damaged DNA recognition. These pieces of evidence suggest that the loss of a synchronous mode of response in the MutSα's surveillance for DNA errors would possibly be one of the mechanism(s) of signaling the MMR-dependent programed cell death much wanted in anticancer therapies. The analysis was drawn from dynamics simulations.

Non-specificity and synergy at the binding site of the carboplatin-induced DNA adduct via molecular dynamics simulations of the MutSα-DNA recognition complex

MutSα is the most abundant mismatch-binding factor of human DNA mismatch repair (MMR) proteins. MMR maintains genetic stability by recognizing and repairing DNA defects. Failure to accomplish their function may lead to cancer. In addition, MutSα recognizes at least some types of DNA damage making it a target for anticancer agents. Here, complementing scarce experimental data, we report unique hydrogen-bonding motifs associated with the recognition of the carboplatin induced DNA damage by MutSα. These data predict that carboplatin and cisplatin induced damaging DNA adducts are recognized by MutSα in a similar manner. Our simulations also indicate that loss of base pairing at the damage site results in (1) non-specific binding and (2) changes in the atomic flexibility at the lesion site and beyond. To further quantify alterations at MutSα-DNA interface in response to damage recognition, non-bonding interactions and salt bridges were investigated. These data indicate (1) possible different packing and (2) disruption of the salt bridges at the MutSα-DNA interface in the damaged complex. These findings (1) underscore the general observation of disruptions at the MutSα-DNA interface and (2) highlight the nature of the anticancer effect of the carboplatin agent. The analysis was carried out from atomistic simulations.

Framed curves and knotted DNA

The present mini-review covers the local and global geometry of framed curves and the computation of twist and writhe in knotted DNA circles. Classical inequalities relating the total amount of bending of a closed space curve and associated knot parameters are also explained.