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 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.

A Monte Carlo code applied to the cgDNA coarse-grain rigid-base model of B-form double-stranded DNA is used to predict a sequence-averaged persistence length of l = 53.5 nm in the sense of Flory, and of l = 160 bp or 53.5 nm in the sense of apparent tangent-tangent correlation decay. These estimates are slightly higher than the consensus experimental values of 150 bp or 50 nm, but we believe the agreement to be good given that the cgDNA model is itself parametrized from molecular dynamics simulations of short fragments of length 10-20 bp, with no explicit fit to persistence length. Our Monte Carlo simulations further predict that there can be substantial dependence of persistence lengths on the specific sequence [Formula: see text] of a fragment. We propose, and confirm the numerical accuracy of, a simple factorization that separates the part of the apparent tangent-tangent correlation decay [Formula: see text] attributable to intrinsic shape, from a part [Formula: see text] attributable purely to stiffness, i.e., a sequence-dependent version of what has been called sequence-averaged dynamic persistence length l̅ (=58.8 nm within the cgDNA model). For ensembles of both random and λ-phage fragments, the apparent persistence length [Formula: see text] has a standard deviation of 4 nm over sequence, whereas our dynamic persistence length [Formula: see text] has a standard deviation of only 1 nm. However, there are notable dynamic persistence length outliers, including poly(A) (exceptionally straight and stiff), poly(TA) (tightly coiled and exceptionally soft), and phased A-tract sequence motifs (exceptionally bent and stiff). The results of our numerical simulations agree reasonably well with both molecular dynamics simulation and diverse experimental data including minicircle cyclization rates and stereo cryo-electron microscopy images.

Abstract. 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.

The different types of naturally occurring, normal human hemoglobins vary in their tetramer-dimer subunit interface strengths (stabilities) by three orders of magnitude in the liganded (CO or oxy) state. The presence of embryonic z-subunits leads to an average 20-fold weakening of tetramer-dimer interfaces compared to corresponding hemoglobins containing adult a-subunits. The dimer-monomer interfaces of these hemoglobins differ by at least 500-fold in their strengths; such interfaces are weak if they contain z-subunits and exchange with added b-subunits in the form of b 4 (HbH) significantly faster than do those with a-subunits. Subunit exchange occurs at the level of the dimer, although tetramer formation reciprocally influences the amount of dimer available for exchange. Competition between subunit types occurs so that pairs of weak embryonic hemoglobins can exchange subunits to form the stronger fetal and adult hemoglobins. The dimer strengths increase in the order Hb Portland-2 (z 2 b 2 ) < Hb Portland-1 (z 2 g 2 ) ffi Hb Gower-1 (z 2 e 2 ) < Hb Gower-2 (a 2 e 2 ) < HbF 1 < HbF (a 2 g 2 ) < HbA 2 (a 2 d 2 ), i.e., from embryonic to fetal to adult types, representing maturation from weaker to stronger monomermonomer subunit contacts. This increasing order recapitulates the developmental order in which globins are expressed (embryonic ! fetal ! adult), suggesting that the intrinsic binding properties of the subunits themselves regarding the strengths of interfaces they form with competing subunits play an important role in the dynamics of protein assemblies and networks.

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.