Theory of the influence of changes in salt concentration on the configuration of intrinsically curved, impenetrable, rod-like structures modeling DNA minicircles

DNA molecules in the familiar double helical B form are treated here as though they have rod-like structures obtained by stacking the nearly planar base pairs comprising them one on top of another with each rotated by approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. As each base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson-Crick structure, and each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair. Thus, each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to electrostatic forces, of sequentially remote origin, that, because they are only partially screened out by positively charged counter ions, can render the molecule's equilibrium configurations sensitive to changes in the concentration of salt in the medium. As there are cases in which, even though the intramolecular electrostatic forces of repulsion are strong, the distance of closest approach has value equal to that of the impenetrable diameter of the molecule, the theory presented here takes into account self-contact. Examples are given of cases in which the theory predicts that the radius of gyration of the minimum energy configuration of a small (549 base pair) circularized DNA molecule (called a “DNA minicircle”) has a remarkably strong dependence on the salt concentration.     

On Bifurcations of Equilibria of Intrinsically Curved, Electrically Charged, Rod-like Structures that Model DNA Molecules in Solution

DNA molecules in the familiar Watson–Crick double helical B form can be treated as though they have rod-like structures obtained by stacking dominoes one on top of another with each rotated by approximately one-tenth of a full turn with respect to its immediate predecessor in the stack. These “dominoes” are called base pairs. A recently developed theory of sequence-dependent DNA elasticity (Coleman, Olson, & Swigon, J. Chem. Phys. 118:7127–7140, 2003) takes into account the observation that the step from one base pair to the next can be one of several distinct types, each having its own mechanical properties that depend on the nucleotide composition of the step. In the present paper, which is based on that theory, emphasis is placed on the fact that, as each base in a base pair is attached to the sugar-phosphate backbone chain of one of the two DNA strands that have come together to form the Watson–Crick structure, and each phosphate group in a backbone chain bears one electronic charge, two such charges are associated with each base pair, which implies that each base pair is subject to not only the elastic forces and moments exerted on it by its neighboring base pairs but also to long range electrostatic forces that, because they are only partially screened out by positively charged counter ions, can render the molecule’s equilibrium configurations sensitive to changes in the concentration c of salt in the medium. When these electrostatic forces are taken into account, the equations of mechanical equilibrium for a DNA molecule with N + 1 base pairs are a system of μN non-linear equations, where μ, the number of kinematical variables describing the relative displacement and orientation of adjacent pairs is in general 6; it reduces to 3 when base-pair steps are assumed to be inextensible and non-shearable. As a consequence of the long-range electrostatic interactions of base pairs, the μN × μN Jacobian matrix of the equations of equilibrium is full. An efficient numerically stable computational scheme is here presented for solving those equations and determining the mechanical stability of the calculated equilibrium configurations. That scheme is employed to compute and analyze bifurcation diagrams in which c is the bifurcation parameter and to show that, for an intrinsically curved molecule, small changes in c can have a strong effect on stable equilibrium configurations. Cases are presented in which several stable configurations occur at a single value of c.      

Elliptical inhomogeneity with polynomial eigenstrains embedded in orthotropic materials

A closed-form solution for elastic field of an elliptical inhomogeneity with polynomial eigenstrains in orthotropic media having complex roots is presented. The distribution of eigenstrains is assumed to be in the form of quadratic functions in Cartesian coordinates of the points of the inhomogeneity. Elastic energy of inhomogeneity–matrix system is expressed in terms of 18 real unknown coefficients that are analytically evaluated by means of the principle of minimum potential energy and the corresponding elastic field in the inhomogeneity is obtained. Results indicate that quadratic terms in the eigenstrains induce zeroth-order elastic strain components, which reflect the coupling effect of the zeroth- and second-order terms in the polynomial expressions on the elastic field. In contrast, the first-order terms in the eigenstrains only produce corresponding elastic fields in the form of the first-order terms. Numerical examples are given to demonstrate the normal and shear stresses at the interface between the inhomogeneity and the matrix. Furthermore, the solution reduces to known results for the special cases.     

Non-slipping adhesive contact between mismatched elastic spheres: A model of adhesion mediated deformation sensor

A generalized JKR model is established for non-slipping adhesive contact between two dissimilar elastic spheres subjected to a pair of pulling forces and a mismatch strain. We discuss the full elastic solution to the problem as well as the so-called non-oscillatory solution in which tension and shear tractions along the contact interface is decoupled from each other. The model indicates that the mismatch strain has significant effect on the contact area and the pull-off process. Under a finite pulling force, a pair of adhering spheres is predicted to break apart spontaneously at a critical mismatch strain. This study suggests an adhesion mediated deformation sensing mechanism by which cells and molecules can detect mechanical signals in the environment via adhesive interactions.     

Remarks on cavity formation in fiber-reinforced incompressible non-linearly elastic solids

In this paper we focus on cavity formation in fiber-reinforced incompressible non-linearly elastic solids. In particular, the material under consideration is a base neo-Hookean augmented by a function that accounts for the existence of a unidirectional reinforcing. This function characterizes the anisotropy of the material and is referred to as reinforcing model. Previous works has dealt with the analysis of a specific reinforcing model, the so called standard reinforcing model, that is a quadratic function that depends only on the fiber reinforcement stretch. Here, two different reinforcing models are examined: a power law that depends only on the fiber stretch and a quadratic function that depends simultaneously on the fiber stretch and fiber shearing. Closed form analytic solutions are found for the classical problem of cavity formation in a sphere under uniform radial tensile dead-load with the fiber in the radial direction. It is shown that for some of the new reinforcing models under study the cavitation instabilities obtained in previous works for the standard reinforcing model are not possible.     

Application of moment interaction to the construction of a stable model of graphite crystal lattice

The aim of the present paper is to construct and study a model of pair moment interaction between carbon atoms in the two-dimensional graphite lattice. The carbon atom is modeled by a structure consisting of three rigidly connected mass points located at the vertices of an equilateral triangle. The interaction between mass points is described by a pair force potential, but the total interatomic interaction contains moment components owing to the finite size of the structure modeling the atom. We compute rank 4 tensors characterizing the elastic properties of the graphite crystal lattice constructed on the basis of our model. We determine lattice stability criteria depending on the number of coordination spheres taken into account. We show that this model permits one to ensure stability of the graphite lattice but significantly underestimates the transverse-to-longitudinal interatomic coupling rigidity ratio. We construct a generalized moment potential that permits one to obtain a rigidity ratio consistent with experimental data.     

A REDUCED QUADRATIC PROGRAMMING METHOD FOR ELASTIC CONTACT PROBLEMS

基于势能原理以节点位移为设计变量、以接触条件为约束方程构建了无摩擦弹性接触问题的二次规划数学模型,在此基础之上利用力平衡线性约束方程的特解和由基础解向量构成的奇异模态矩阵,提出一种新的基于奇异坐标变换的自由度缩减方法,大大降低了二次规划的规模,并使得二次规划模型不再含显性等式约束;根据弹性接触力学体系的特点,通过人为假定接触自由度位移模式,提出了一种简单高效的奇异模态矩阵的计算方法。通过两圆柱接触、轴孔间隙配合接触两个数值算例的对比分析,验证了对于弹性接触问题的求解,缩减二次规划方法有效克服了传统方法计算量大、对求解参数设置敏感、收敛困难的问题。     

刚-柔耦合动力学系统的建模理论研究

刚－柔耦合动力学系统的传统的混合坐标方法是零次近似方法,在建模过程中,直接套用的结构动力学的小变形假设,忽略了变形位移的高次耦合变形量．本文对柔性梁建立较零次近似更精确的高次耦合动力学模型,从连续介质力学理论出发,在变形位移中,计及横向位移引起的轴向缩短,导出变形位移的二次耦合量．用一致质量有限元方法对梁进行离散,基于Jourdain速度变分原理导出大范围运动为自由的柔性梁的刚－柔耦合动力学方程．计算了柔性重力摆的角速度和摆端点的横向变形,揭示零次近似模型和耦合模型的刚－柔耦合动力学性质的根本差异．     

Convex Fung-type potentials for biological tissues

The anisotropic, non-linear elastic behavior of biological soft tissue is typically accounted for by the hypothesis of hyperelasticity, i.e., the existence of an elastic potential. Fung-type potentials, based on the exponential of a quadratic form in the components of the Green-Lagrange strain, have been widely used in soft tissue modeling, and have inspired potentials in which the exponential was replaced by other monotonically increasing functions. It has been shown that simple fitting of the parameters of a Fung-type potential to experimental stress-strain curves may lead to non-convexity, with undesirable effects on the reliability of the algorithms used in Finite Element simulations. In this paper, we prove that the necessary and sufficient condition for the strict convexity of a Fung-type potential is that the quadratic form in the exponential is positive definite. This result provides a clear physical meaning for the parameters featuring in the quadratic form, and their relationship with the small-strain elastic moduli. This consistency relationship must be respected in order to guarantee that the Fung-type potential correctly reduces to the quadratic potential of classic linear elasticity in the small-strain approximation. Furthermore, we show that, when the conditions of convexity and consistency with the linear theory are respected, Fung-type potentials become a one-parameter family, and we discuss the consequences of this result for when fitting experimental data. An erratum to this article can be found at     

Non-slipping adhesive contact between mismatched elastic cylinders

We have recently proposed a generalized JKR model for non-slipping adhesive contact between two elastic spheres subjected to a pair of pulling forces and a mismatch strain (Chen, S., Gao, H., 2006c. Non-slipping adhesive contact between mismatched elastic spheres: a model of adhesion mediated deformation sensor. J. Mech. Phys. Solids 54, 1548–1567). Here we extend this model to adhesion between two mismatched elastic cylinders. The attention is focused on how the mismatch strain affects the contact area and the pull-off force. It is found that there exists a critical mismatch strain at which the contact spontaneously dissociates. The analysis suggests possible mechanisms by which mechanical deformation can affect binding between cells and molecules in biology.     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

Surface Green function for a soft elastic half-space: Influence of surface stress

Surface Green function for incompressible, elastically isotropic half-space coupled with surface stress is derived by using double Fourier transform technique. The result indicates that the surface displacement induced by a force tangential to the surface is the same as the usual solution for elastic half-spaces where the effect of surface stress is ignored. However, the displacement caused by a force normal to the surface involves an additional parameter, i.e. the ratio of specific surface stress to shear modulus. The parameter has the dimension of length, and may provide a means to introduce an intrinsic length scale for some related problems regarding the surface of an elastic half-space. This is extremely true for soft elastic media with very low shear modulus, because in that situation the magnitude of the parameter is relatively large. As an illustrative example, the proposed Green function is adopted to analyze the interaction between two molecules with circular section adsorbed on the surface of a soft elastic half-space. It is shown that surface stress remarkably affects the pair interaction potential when the distance between the molecules is not larger than several times of the intrinsic length scale.     

Analysis of deformation localization based on proposed theory of ultrasonic wave velocity propagating in plastically deformed solids

The localization of plastic deformation is discussed as “stationary discontinuity” characterized by a vanishing velocity of an acceleration wave derived using the author’s proposed theory of ultrasonic wave velocities propagating in plastically deformed solids. To formulate the proposed theory, the elasto-plastic coupling effect was introduced to consider the elastic stiffness degradation due to the plastic deformation. The driving force of the deformation localization is caused by the yield vertex effect, which introduces a pronounced softening of the shear modulus, and geometrical softening due to double slip caused by lattice rotations. In the present paper, it is examined theoretically and experimentally that the diagonal terms of the introduced elasto-plastic coupling tensor represent a slight hardening followed by a pronounced softening of the elastic modulus induced by the point defect development caused by cross slides among dislocations at multiple slip stages similar to the yield vertex effects. The off-diagonal terms represent geometrical softening induced by lattice rotations such as texture evolution. Then, based on the coincidence of the onset strains between localization and acceleration waves of vanishing velocity, the diagrams of diffuse necking, localized necking and forming limit are analyzed by applying the proposed acoustic tensor, which is based on the generalized Christoffel tensor derived by the author, and solving cut off conditions of the quasi-longitudinal wave to determine the onset strains of deformation localization and localization modes. As a result, diagrams of diffuse necking, localized necking and forming limit were obtained. Moreover, the localization modes were determined and distinguished as the SH-mode, SV-mode, tearing mode and splitting mode.     

任意轴对称弹性体吸附接触的广义Maugis模型

通过线性叠加Sneddon方法和Lowengrub-Sneddon方法分别给出的解, 得到了一个弹性半空间 轴对称混合边值问题的一般解，进而研究了两个一般轴对称弹性体的正向无摩擦吸附接触问 题. 考虑任意有效的表面形状(要求中心部分首先进入接触)和任意的表面吸附作用，推广 得到了广义Maugis模型. 该模型是一个半解析的模型，它可以分解成表面形状和表面吸附 作用的分别独立影响的两部分，以及一个关联变形和吸附作用的式子. 利用Dugdale模型近 似表面吸附作用，得到了具有任意有效的表面形状的广义M-D模型. 它在强吸附或软材料条 件下的极限形式是广义JKR模型，而在弱吸附或硬材料下的另一个极限形式是广义DMT模型.     

Primary resonance of coupled cantilevers subjected to magnetic interaction

Based on a distributed-parameter model, the forced vibration of a cantilever pair excited by a sinusoidal base movement is analyzed. Two cantilevers are coupled at their free ends by a linear spring. A nonlinear concentrated magnetic force acts on the tip of one cantilever, serving at the nonlinear boundary condition of the continuous model. The magnetic force is modeled as a fractional function, strongly dependent on the distance between two magnets. Via the method of multiple scales, the primary resonance is analyzed for all modes. A second-order approximate solution and its stability condition are analytically captured. It is revealed that the frequency–response curves are sensitive to the distance between the two magnets. The curve may exhibit the hardening-type, softening-type or linear behavior due to the existence of the quadratic nonlinearity. The outcomes are supported by the numerical simulations very well.     

From Discrete Visco-Elasticity to Continuum Rate-Independent Plasticity: Rigorous Results

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a visco-elastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.     

Elastica of an axially loaded column/ footing system resting on elastic base

The exact non-linear differential equation of large deformations of an axially loaded column fixed into a rigid eccentric footing, resting on elastic base, is formulated, along with the accompanying exact non-linear boundary conditions. The above differential equation is solved using the theory of elliptic integrals and elliptic functions. Employing a quadratic polynomial approximation in the expression for the curvature of the central line of the deformed column, an extended parametric study is carried out, in which as basic geometric parameters the height as well as the eccentricity of the footing have been adopted. The content of the parametric study consists in determining the relations among the basic variables: axial load, rotation angle of the footing and footing rotational stiffness. These relations are presented in the form of families of curves.     

A mathematical model and numerical method for thermoelectric DNA sequencing

Single nucleotide polymorphisms (SNPs) are single base pair variations within the genome that are important indicators of genetic predisposition towards specific diseases. This study explores the feasibility of SNP detection using a thermoelectric sequencing method that measures the heat released when DNA polymerase inserts a deoxyribonucleoside triphosphate into a DNA strand. We propose a three-dimensional mathematical model that governs the DNA sequencing device with a reaction zone that contains DNA template/primer complex immobilized to the surface of the lower channel wall. The model is then solved numerically. Concentrations of reactants and the temperature distribution are obtained. Results indicate that when the nucleoside is complementary to the next base in the DNA template, polymerization occurs lengthening the complementary polymer and releasing thermal energy with a measurable temperature change, implying that the thermoelectric conceptual device for sequencing DNA may be feasible for identifying specific genes in individuals.