Abstract: A simple model of a circularly closed double-stranded DNA in a poor solvent is considered as an
example of a semi-ï¬exible polymer with self-attraction. To ï¬nd the ground states, the conformational
energy is computed as a sum of the bending and torsional elastic components and the effective self-attraction energy. The model includes a relative orientation or sequence dependence of the effective attraction forces between different pieces of the polymer chain. Two series of conformations are analysed: a multicovered circle (a toroid) and a multifold two-headed racquet. The results are presented as a diagram of state. It is suggested that the stability of particular conformations may be controlled by proper adjustment of the primary structure. Application of the model to other semi-ï¬exible polymers is considered.
Abstract: When twisting a strip of paper or acetate under high longitudinal tension, one observes, at some critical load, a buckling of the strip into a regular triangular pattern. Very similar triangular facets have recently been found in solutions to a new set of geometrically exact equations describing the equilibrium shape of thin inextensible elastic strips. Here, we formulate a modified boundary-value problem for these equations and construct post-buckling solutions in good agreement with the observed pattern in twisted strips. We also study the forceâextension and momentâtwist behaviour of these strips by varying the mode number n of triangular facets and find critical loads with jumps to higher modes.
Abstract: We persist in considering that, for a wide range of (experimentally available) forces and torques, evaluating the writhe of a DNA molecule in magnetic tweezers experiments should not be done with Fuller's formula. We propose a tentative plot of the limit of applicability of Fuller's formula in the (force, torque) plane.
Notes: See also: J.Samuel, S.Sinha, A. Ghosh.
Comment on âWrithe formulas and antipodal points in plectonemic DNA configurationsâ.
Phys. Rev. E 80, 063901 (2009)
and S.Neukirch, E.L.Starostin.
Writhe formulas and antipodal points in plectonemic DNA configurations.
Phys. Rev. E 78, 041912 (2008).
Abstract: We study the force vs. extension behaviour of a helical spring made of a thin torsionally stiff anisotropic elastic rod. Our focus is on springs of very low helical pitch. For certain parameters of the problem such a spring is found not to unwind when pulled but rather to form hockles that pop out one by one and lead to a highly non-monotonic forceâextension curve. Between abrupt loop pop-outs this curve is well described by the planar elastica whose relevant solutions are classified. Our results may be relevant for tightly coiled nanosprings in future micro- and nano(electro)mechanical devices.
Abstract: We consider geometric variational problems for a functional defined on a curve in a three-dimensional space. The functional is assumed to be written in a form invariant under the group of Euclidean motions. We present the Euler-Lagrange equations as equilibrium equations for the internal force and moment. Examples are discussed to illustrate our approach. This form of the equations particularly serves to promote the study of biofilaments and nanofilaments.
Abstract: An exact formula for the minimal coordination numbers of the parallel packed bundle of rods is presented based on an optimal thickening scenario. Hexagonal and square lattices are considered.
Abstract: We study the nonmonotonic force-extension behavior of helical ribbons using a new model for inextensible elastic strips. Unlike previous rod models, our model predicts hysteresis behavior for low-pitch ribbons of arbitrary material properties. Associated with it is a first-order transition between two different helical states as observed in experiments with cholesterol ribbons. Numerical solutions show nonuniform uncoiling with hysteresis also occurring under controlled tension. They furthermore reveal a new uncoiling scenario in which a ribbon of very low pitch shears under tension and successively releases a sequence of almost planar loops. Our results may be relevant for nanoscale devices such as force probes.
Abstract: The linking and writhing numbers are key quantities when characterizing the structure of a piece of supercoiled DNA. Defined as double integrals over the shape of the double helix, these numbers are not always straightforward to compute, though a simplified formula was established in a theorem by Fuller [Proc. Natl. Acad. Sci. U.S.A. 75, 3557 (1978)]. We examine the range of applicability of this widely used simplified formula, and show that it cannot be employed for plectonemic DNA. We show that inapplicability is due to a hypothesis of Fuller theorem that is not met. The hypothesis seems to have been overlooked in many works.
Abstract: A variational geometrical approach is applied to find the characteristic shape of the Möbius strip made of an inextensible rectangular sheet.
Abstract: In most cases the hexagonal packing of fibrous structures or rods extremizes the energy of interaction between strands. If the strands are not straight, then it is still possible to form a perfect hexatic bundle. Conditions under which the perfect hexagonal packing of curved tubular structures may exist are formulated. Particular attention is given to closed or cycled arrangements of the rods like in the DNA toroids and spools. The closure or return constraints of the bundle result in an allowable group of automorphisms of the cross-sectional hexagonal lattice. The structure of this group is explored. Examples of open helical-like and closed toroidal-like bundles are presented. An expression for the elastic energy of a perfectly packed bundle of thin elastic rods is derived. The energy accounts for both the bending and torsional stiffnesses of the rods. It is shown that equilibria of the bundle correspond to solutions of a variational problem formulated for the curve representing the axis of the bundle. The functional involves a function of the squared curvature under the constraints on the total torsion and the length. The Euler–Lagrange equations are obtained in terms of curvature and torsion and due to the existence of the first integrals the problem is reduced to the quadrature. The three-dimensional shape of the bundle may be readily reconstructed by integration of the Ilyukhin-type equations in special cylindrical coordinates. The results are of universal nature and are applicable to various fibrous structures, in particular, to intramolecular liquid crystals formed by DNA condensed in toroids or packed inside the viral capsids.
Abstract: A problem is formulated about how many unit-radius tubes can touch a ball of given radius from the outside and from the inside. Upper bounds for the maximum numbers of contacts are obtained for both interior and exterior contacts. It is also shown that the maximum number of unit-radius tubes touching the same orthogonal cross-section of a particular tube of radius P is [Ï (arcsin(P+1)â1)â1] and if the number of contacts takes on its maximum, then all tubes are locally aligned.
Abstract: A variational approach is used to find the shortest curves confined to lie in two orthogonal planes and separated by a constant distance. The method is applicable to constructing tight shapes of linked structures each component of which is known to be planar. The shapes of the Borromean rings and two clasped pieces of rope are two examples. A concept of tight periodic structures is introduced and discussed.
Abstract: Spatial equilibria of a closed thin isotropic elastic rod are considered. The thin elastic rod is a classical model for the large-scale structure of relatively long DNA molecules. Particular attention is paid to the shapes with self-contacts which are assembled from the elementary loops.
Abstract: We consider free manifolds of dynamic billiards that allow constructing mathematical billiards equivalent to original dynamic billiards. It is shown that free manifolds of dynamic billiards in constant and Newtonian force field are surfaces of rotation in 3D Euclidean space. It is demonstrated that parabolic billiards in Newtonian attracting force field are equivalent to plane mathematical billiards.
Abstract: The equilibrium shapes of a closed DNA are investigated by employing a model of a thin, homogeneous, isotropic, linearly elastic rod of circular cross section. An equilibrium configuration of such an initially straight and twisted rod, submitted to external forces and moments at its ends only, obeys equations identical to those governing the rotation of a symmetric gyrostat spinning about a fixed point in a gravitational field (the Kirchhoff analogy). To represent the equilibrium of the looped DNA, the model rod must be smoothly closed into a ring. The corresponding BVP results in a system of four nonlinear equations with respect to four parameters. The perturbation analysis and the parameter continuation approach are used to find nonplanar solutions. The conformation change is discussed for various values of parameters.
Abstract: Three-dimensional motion of the Small Space Lab (SSL) spacecraft about its centre of mass under the solar radiation torques is considered. The centre of mass of the spacecraft moves in a circular heliocentric orbit. The spacecraft has an axially symmetrical solar stabilizer and eight solar paddles arranged like a windmill. The paddle's slope angles are made controllable. In the design condition, the spacecraft fore-and-aft axis should be directed to the Sun and its spin rate should have a value required.
The equations in evolutionary variables are studied. The dynamical model takes into account an effect of re-radiating solar electromagnetic energy by the thin films of the stabilizer. A phenomenology model of re-radiation torque is suggested. It is shown, that the re-radiation torque leads to nutation damping.
An algorithm to control the solar paddles is developed. It provides bringing the spacecraft to the nominal Sun-pointing orientation with a given spin rate. The algorithm requires no synchronization with the spacecraft's spin rate and only the mean deviation of the fore-and-aft axis from the direction to the Sun has to be input. Thus, the algorithm may readily be implemented on board.
The results of numerical simulation are presented.
Abstract: Satellite oscillations about its centre of mass in the circular orbit plane are dealt with. The satellite is assumed symmetrical about a plane permanently coinciding with the orbit plane. A gravity-gradient torque and a torque of solar radiation pressure on an unshadowed flat plateâa part of the satelliteâare taken into account. The centre of pressure is supposed to belong to the principal axis of inertia. Effects of entering the Earth's shadow are neglected. A simplification that the orbit lies in the ecliptic plane is adopted. Under the assumptions made, the satellite motion is described by a non-autonomous differential second-order equation. A problem is to find symmetrical and nonsymmetrical periodic motions of orbital period and to determine their stability. For the case of small radiation disturbance, the Krylov-Bogolyubov asymptotic approach is used in the analysis. The libration in the vicinity of the main resonance has been elaborated. For the satellite dynamically resembling a sphere the investigation is treated with the Volosov-Morgunov averaging method. A resonant value of the radiation torque parameter has been found. A question of periodic motions bifurcation is cleared up. For the satellite with an arbitrary tensor of inertia under non-small radiation disturbance the problem has been solved numerically. The main results are represented as a chart graphically demonstrating regions of existence and stability of possible periodic librations of the satellite on parametric plane.
Abstract: We present a new theory to describe a wide class of equilibrium conï¬gurations of a ply made of two Cosserat rods. The
rods, of circular cross-section, are assumed to be in continuous contact. The axis of the ply is free to adopt any spatial conï¬guration
under the action of end loads. Local interaction of the rods (e.g., of electrostatic nature) is incorporated into the formulation. The theory
is illustrated on two examples that allow for analytical treatment. Applications to plectonemic DNA are discussed.
Abstract: The problem of determining spatial position and orientation of several cameras, knowing corresponding coordinates obtained by perspective projections onto the camera planes, is considered. Input data for calibration also include distances between some points in space. The calibration is carried out in two stages. In the first stage, position and orientation for pairs of images (stereo pairs) are determined. Every image is calibrated being included in one calibrated stereo pair. A special tree-like structure is built up as a result of the first stage. This structure contains input data and some information about links between the images. The calibration parameters obtained after each stereo pair calibration are considered as an initial approximation for the second stage. In this stage, the simultaneous calibration of all the images is performed to provide consistency and compatibility of final results. The proposed approach permits the user to avoid possible conflicts between calibration parameters and alleviates the problem of obtaining a good initial approximation for simultaneous calibration of multiple cameras. The experiments with real images produced promising results.
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