Robert S. Manning

A continuum rod model of sequence-dependent DNA structure

Experimentally motivated parameters from a base‐pair‐level discrete DNA model are averaged to yield parameters for a continuum elastic rod with a curved unstressed shape reflecting the local DNA geometry. The continuum model permits computations with discretization lengths longer than the intrinsic discretization of the base‐pair model, and, for this and other reasons, yields an efficient computational formulation. Obtaining continuum stiffnesses is straightforward, but obtaining a continuum unstressed shape is hindered by the ‘‘noisy’’ small‐scale structure and rapid helix twist of the discrete unstressed shape. Filtering of the discrete data and an analytic transformation from the true normal‐vector field to a natural (untwisted) frame allows a stable continuum fit. Equilibrium energies of closed rings predicted by the continuum model are found to match the energies of the underlying discrete model to within 0.5%. The model is applied to a set of 11 short DNA molecules (≊ 150 bp) and properly distinguis...

Isoperimetric conjugate points with application to the stability of DNA minicircles

A conjugate point test determining an index of the constrained second variation in one–dimensional isoperimetric calculus of variations problems is described. The test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of DNA minicircles.

DNA Rings with Multiple Energy Minima

Within the context of DNA rings, we analyze the relationship between intrinsic shape and the existence of multiple stable equilibria, either nicked or cyclized with the same link. A simple test, based on a perturbation expansion of symmetry breaking within a continuum elastic rod model, provides good predictions of the occurrence of such multiple equilibria. The reliability of these predictions is verified by direct computation of nicked and cyclized equilibria for several thousand DNA minicircles with lengths of 200 and 900 bp. Furthermore, our computations of equilibria for nicked rings predict properties of the equilibrium distribution of link, as calculated by much more computationally intensive Monte Carlo simulations.

Conjugate Points Revisited and Neumann-Neumann Problems

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on the classic result that “stability requires the lack of conjugate points.” Furthermore, we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann-Neumann boundary conditions.

Stability of an elastic rod buckling into a soft wall

The conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field. We show that the operator approach presented by Manning (Manning et al. 1998 Proc. R. Soc. A 454, 3047–3074) can be used when the second-variation operator is an integrodifferential operator, rather than a differential operator as in the classical case. The external field is chosen to model two parallel ‘soft’ walls. We consider the examples of two-dimensional buckling under both pinned–pinned and clamped–clamped boundary conditions, as well as the three-dimensional clamped–clamped problem, where we consider the importance of the rod cross-section shape as it ranges from circular to extreme elliptical. For each of these problems, we find that in the appropriate limit, the soft-wall solutions approach a ‘hard-wall’ limit, and thus we make conjectures about these hard-wall contact equilibria and their stability. In the two-dimensional pinned–pinned case, this allows us to assign stability to the configurations reported by Holmes (Holmes et al. 1999 Comput. Methods Appl. Mech. Eng. 170, 175–207) and reconsider the experimental results discussed therein.

Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values

Human embryonic, fetal, and adult hemoglobins have different subunit interface strengths. Correlation with lifespan in the red cell

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Stability of n-covered Circles for Elastic Rods with Constant Planar Intrinsic Curvature

at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and

An Extended Conjugate Point Theory with Application to the Stability of Planar Buckling of an Elastic Rod Subject to a Repulsive Self-Potential

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Symmetry Breaking and the Twisted Elastic Ring

.Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter