Methods for determining stability in continuum elastic-rod models of DNA

When Is a Helix Stable?

We determine which helical equilibria of an isotropic Kirchhoff elastic rod with clamped ends are stable and which are unstable. Although the set of all helical equilibria is parametrized by four variables, with an additional fifth parameter determined by the rod's material, we show that only three of these five parameters are needed to distinguish between stable and unstable equilibria. We also show that the closure of the set of stable equilibria is star convex. With these results, we are able to compute and visualize the boundary between stable and unstable helices for the first time.

Infinitely long isotropic Kirchhoff rods with helical centerlines cannot be stable

It has long been known that every configuration of a planar elastic rod with clamped ends satisfies the property that if its centerline has constant nonzero curvature, then it is in stable equilibrium regardless of its length. In this paper, we show that for a certain class of nonplanar elastic rods, no configuration satisfies this property. In particular, using results from optimal control theory, we show that every configuration of an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with clamped ends that has a helical centerline with constant nonzero curvature becomes unstable at a finite length. We also derive coordinates for computing this critical length that are independent of the rod's bending and torsional stiffness. Finally, we derive a scaling relationship between the length at which a helical rod becomes unstable and the rod's curvature, torsion, and twist. In a companion paper, these results are used to compute the set of all stable rods with helical centerlines.

Linearized Bayesian inference for Young's modulus parameter field in an elastic model of slender structures

The deformations of several slender structures at nano-scale are conceivably sensitive to their non-homogenous elasticity. Owing to their small scale, it is not feasible to discern their elasticity parameter fields accurately using observations from physical experiments. Molecular dynamics simulations can provide an alternative or additional source of data. However, the challenges still lie in developing computationally efficient and robust methods to solve inverse problems to infer the elasticity parameter field from the deformations. In this paper, we formulate an inverse problem governed by a linear elastic model in a Bayesian inference framework. To make the problem tractable, we use a Gaussian approximation of the posterior probability distribution that results from the Bayesian solution of the inverse problem of inferring Young's modulus parameter fields from available data. The performance of the computational framework is demonstrated using two representative loading scenarios, one involving cantilever bending and the other involving stretching of a helical rod (an intrinsically curved structure). The results show that smoothly varying parameter fields can be reconstructed satisfactorily from noisy data. We also quantify the uncertainty in the inferred parameters and discuss the effect of the quality of the data on the reconstructions.

Quasi-static manipulation of a Kirchhoff elastic rod based on a geometric analysis of equilibrium configurations

Consider a thin, flexible wire of fixed length that is held at each end by a robotic gripper. Any curve traced by this wire when in static equilibrium is a local solution to a geometric optimal control problem, with boundary conditions that vary with the position and orientation of each gripper. We prove that the set of all local solutions to this problem over all possible boundary conditions is a smooth manifold of finite dimension that can be parameterized by a single chart. We show that this chart makes it easy to implement a sampling-based algorithm for quasi-static manipulation planning. We characterize the performance of such an algorithm with experiments in simulation.

An introduction to the mechanics of DNA

This article gives an overview of recent research on the mechanical properties and spatial deformations of the DNA molecule. Globally the molecule behaves like a uniform elastic rod, and its twisting and writhing govern its compaction and packaging within a cell. Meanwhile high mechanical stresses can induce structural transitions of DNA giving, for example, a phase diagram in the space of the applied tension and torque. Locally, the mechanical properties vary according to the local sequence organization. These variations play a vital role in the biological functioning of the molecule.