We catalogue configurations that locally minimize energy for a planar elastic rod (extensible-shearable or inextensible-unshearable) subject to arbitrary Dirichlet boundary conditions in position and orientation. Via a combination of analysis and computation, we determine several bifurcation surfaces in the 3-parameter space of boundary conditions and explore how they depend on the rod material parameters, including in the inextensible limit. For each possible boundary condition, we find all stable equilibria with sufficiently low energy that they might be competitive within a Boltzmann distribution if the rod were used to model DNA with tens or hundreds of base pairs, the length-scale relevant for DNA looping. Depending on the boundary conditions, there are as many as three such equilibria.

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Coiled-coil protein structural motifs have proven amenable to the design of structurally well-defined biomaterials. Mesoscale structural properties can be fairly well predicted based on rules governing the chemical interactions between the helices that define this structural motif. We explore the role of the hydrophobic core residues on the selfassembly of a coiled-coil polymer through a mutational analysis coupled with a salting-out procedure. Because the resultant polymers remain in solution, a thermodynamic approach is applied to characterize the polymer assembly using conventional equations from polymer theory to extract nucleation and elongation parameters. The stabilities and lengths of the polymers are measured using circular dichroism spectropolarimetry, sizing methods including dynamic light scattering and analytical ultracentrifugation, and atomic force microscopy to assess mesoscale morphology. Upon mutating isoleucines at two core positions to serines, we find that polymer stability is decreased while the degree of polymerization is about the same. Differences in results from circular dichroism and dynamic light scattering experiments suggest the presence of a stable intermediate state, and a scheme is proposed for how this intermediate might relate to the monomer and polymer states.

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The bifurcation diagram for the buckling of a planar elastica under a load lambda is made up of a "trivial" branch of unbuckled configurations for all lambda and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions to determine how this diagram perturbs with the addition of a small intrinsic shape in the elastica, focusing in particular on the effect near the bifurcation points. We find that for almost all intrinsic shapes epsilon f(s), the difference between the buckled solution and the trivial solution is O(epsilon^(1/3)), but for some ``ineffective'' f, this difference is O(epsilon), and we find functions u_j(s) so that f is ineffective at bifurcation point number j when < f, u_j > = 0. These ineffective perturbations have important consequences in numerical simulations, in that the perturbed bifurcation diagram has sharper corners near the former bifurcation points, and there is a higher risk of a numerical simulation inadvertently hopping between branches near these corners.

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The theory of conjugate points in the classic calculus of variations allows, for a certain class of functionals, the characterization of a critical point as stable (i.e., a local minimum) or not. In this work, we generalize this theory to more general functionals, assuming certain generic properties of the second variation operator. The extended conjugate point theory is then applied to a two-dimensional elastic rod subject to pointwise self-repulsion. The critical points are computed by numerically solving first-order integro-differential equations using a finite difference scheme. The stability of each critical point is then computed by determining conjugate points of the second variation operator. In addition, the generalized theory requires the numerical evaluation of the crossing velocity of the zero eigenvalue of the second variation operator at each conjugate point, a feature not present in the classic case (where the crossing velocity can be shown to always be negative). Results demonstrate that the repulsive potential has a stabilizing influence on some branches of critical points.

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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 see the logic behind the classic result that "stability requires the lack of conjugate points". Furthermore, we show how the spectral perspective allows the easy extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann-Neumann boundary conditions.

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We describe a quantitative anlysis of

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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 m

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The conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field, using the operator approach presented in (Manning, Rogers, and Maddocks, Proc. Roy. Soc. A

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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 sigma <= 1 at which a second ODE (the Jacobi equation) has a solution vanishing at 0 and sigma.

Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter lambda, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of lambda of interest.

Here we present an alternative approach that takes advantage of the presence of a parameter lambda. Rather than search for conjugate points sigma <= 1 at a fixed value of lambda, we search for a set of special parameter values lambda

We apply this approach to the elastic strut, in which the the force appears linearly in S, and as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute 2-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE.

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A stability index is computed for the

The index is computed by framing the standard Euler-Cosserat equilibrium equations within a constrained variational principle with an isoperimetric constraint ensuring the ring closure. The fact that

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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 basepairs. Further, 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.

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Symmetry breaking is considered in the context of a parameter-dependent two-point boundary value problem describing a twisted elastic ring that arises as a model of DNA minicircles. We explicitly determine, via a perturbation expansion, those representatives of a manifold of solutions to a symmetric BVP that persist when the symmetry is broken. Conditions generating a generic splitting are identified, and some degenerate cases are studied. In both the generic and degenerate cases, the results of the perturbation expansion are used to improve the efficiency of existing algorithms for computing symmetry-broken bifurcation diagrams.

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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.

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Nonlinear problems arising in modelling applications are frequently parameter dependent, so that families of solutions are of interest. Such problems naturally lend themselves to interactive computation that exploits parameter continuation methods combined with visualization techniques. Visualization provides both understanding of the solution set and feedback for computational steering. We describe various issues that have arisen in our investigations of problems of this general type.

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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 (approximately 150 bp) and properly distinguishes their cyclization probabilities (J factors) when compared both to experimental cyclization rates and to Monte Carlo simulations. The continuum model does not include entropic contributions to the free energy. However, because of its rapid and accurate computation of internal energy, the continuum model should, when combined with further work on entropic effects, be a useful method for computing experimental DNA free energies.

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We apply recent methods for semiclassical time propagation involving non-Cartesian variables to the repulsive 1D potential V(x)=x

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We derive a regularized semiclassical radial propagator for the Coulomb potential, a case for which standard approaches run into well-known difficulties associated with a non-Cartesian radial coordinate and a potential singularity. Following Kleinert [

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Often it is important to consider the expansion of a quantum state

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The rheological behavior of sickle cells was evaluated with a constant flow filtration system at 37

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Return to my home page rmanning@haverford.edu