Diffusive-stochastic-viscoelastic model on specific adhesion of viscoelastic solids via molecular bonds

A diffusive-stochastic-viscoelastic model is proposed for the specific adhesion of viscoelastic solids via stochastically formed molecular bonds. In this model, we assumed that molecular level behaviours, including the diffusion of mobile adhesion molecules and stochastic reaction between adhesion molecules and binding sites, obey the Markovian stochastic processes, while mesoscopic deformations of the viscoelastic media are governed by continuum mechanics. Through Monte Carlo simulations of this model, we systematically investigated how the competition between time scales of molecular diffusion, reaction, and deformation creep of the solids may influence the lifetime and dynamic strength of the adhesion. We revealed that there exists an optimal characteristic time of molecule diffusion corresponding to the longest lifetime and largest adhesion strength, which is in good agreement with experimental observed characteristic time scales of molecular diffusion in cell membranes. In addition, we identified that the media viscosity can significantly increase the lifetime and dynamic strength, since the deformation creep and stress relaxation can effectively reduce the concentration of interfacial stress and increases the rebinding probability of molecular bonds.

Coiflet solution of strongly nonlinear p-Laplacian equations

A new boundary extension technique based on the Lagrange interpolat- ing polynomial is proposed and used to solve the function approximation defined on an interval by a series of scaling Coiflet functions, where the coefficients are used as the single-point samplings. The obtained approximation formula can exactly represent any polynomials defined on the interval with the order up to one third of the length of the compact support of the adopted Coiflet function. Based on the Galerkin method, a Coifiet-based solution procedure is established for general two-dimensional p^Laplacian equations, following which the equations can be discretized into a concise matrix form. As examples of applications, the proposed modified wavelet Galerkin method is applied to three typical p-Laplacian equations with strong nonlinearity. The numerical results justify the efficiency and accuracy of the method.

Wavelet method applied to specific adhesion of elastic solids via molecular bonds

We propose a wavelet method to analyze the stochastic-elastic problem of specific adhesion between two elastic solids via ligand-receptor bond clusters, which is governed by a nonlinear integro-differential equation with a sin- gular Cauchy kernel to describe the mean-field coupling between deformation of elastic materials and stochastic behavior of the molecular bonds. To solve this problem, Galerkin method based on a wavelet approximation scheme is adopted, and special treatment which transforms the singular Cauchy kernel into a smooth one has been proposed to avoid the cumbersome calculation of singular integrals. Numerical results demonstrate that the method is fully capable of solving the specific adhesion problems with complex nonlinear and singular equations. Based on the proposed method, investigations are performed to reveal the relation between steady-state pulling force and mean surface separation under different stress concentration indexes, which is crucial for assembling the overall constitutive relations for multicellular tumor spheroids and polymer-matrix microcomposites.

A simultaneous space-time wavelet method for nonlinear initial boundary value problems

A high-precision and space-time fully decoupled numerical method is developed for a class of nonlinear initial boundary value problems. It is established based on a proposed Coiflet-based approximation scheme with an adjustable high order for the functions over a bounded interval, which allows the expansion coefficients to be explicitly expressed by the function values at a series of single points. When the solution method is used, the nonlinear initial boundary value problems are first spatially discretized into a series of nonlinear initial value problems by combining the proposed wavelet approximation and the conventional Galerkin method, and a novel high-order step-by-step time integrating approach is then developed for the resulting nonlinear initial value problems with the same function approximation scheme based on the wavelet theory. The solution method is shown to have the N th-order accuracy, as long as the Coiflet with [0, 3 N-1]compact support is adopted, where N can be any positive even number. Typical examples in mechanics are considered to justify the accuracy and efficiency of the method.

Multiresolution method for bending of plates with complex shapes

A high-accuracy multiresolution method is proposed to solve mechanics problems subject to complex shapes or irregular domains.To realize this method,we design a new wavelet basis function,by which we construct a fifth-order numerical scheme for the approximation of multi-dimensional functions and their multiple integrals defined in complex domains.In the solution of differential equations,various derivatives of the unknown function are denoted as new functions.Then,the integral relations between these functions are applied in terms of wavelet approximation of multiple integrals.Therefore,the original equation with derivatives of various orders can be converted to a system of algebraic equations with discrete nodal values of the highest-order derivative.During the application of the proposed method,boundary conditions can be automatically included in the integration operations,and relevant matrices can be assured to exhibit perfect sparse patterns.As examples,we consider several second-order mathematics problems defined on regular and irregular domains and the fourth-order bending problems of plates with various shapes.By comparing the solutions obtained by the proposed method with the exact solutions,the new multiresolution method is found to have a convergence rate of fifth order.The solution accuracy of this method with only a few hundreds of nodes can be much higher than that of the finite element method(FEM)with tens of thousands of elements.In addition,because the accuracy order for direct approximation of a function using the proposed basis function is also fifth order,we may conclude that the accuracy of the proposed method is almost independent of the equation order and domain complexity.

Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix.The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities,including transcendental ones,in which the discretization process is as simple as that in solving linear problems,and only common two-term connection coefficients are needed.All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method,which does not require numerical integration in the resulting nonlinear discrete system.The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers.The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids,and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids.In addition,Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method,including the initial guess far from real solutions.

Mechanics of DNA packaging and ejection from elastic phage capsid

This study intends to investigate how the elasticity of a bacterial phage can affect the process of DNA packaging and ejection. For this purpose, we propose a unified continuum and statistical mechanics model by taking into account the effects of DNA bending deformation, electrostatic repulsion between DNA-DNA strands and elastic deformation of the phage capsid. Based on such a model, we derive the quantitative relations between packaging force, elasticity of capsid, DNA length remaining in the capsid, osmotic pressure and ejection time. The theoretically predicted results are found to agree very well with in vitro experimental observations in the lit-erature.

Specific adhesion of a soft elastic body on a wavy surface

This paper aims at developing a stochastic-elastic model of a soft elastic body adhering on a wavy surface via a patch of molecular bonds. The elastic deformation of the system is modeled by using continuum contact mechanics, while the stochastic behavior of adhesive bonds is modeled by using Bell＇s type of exponential bond association/dissociation rates. It is found that for sufficiently small adhesion patch size or stress concentration index, the adhesion strength is insensitive to the wavelength but decreases with the amplitude of surface undulation, and that for large adhesion patch size or stress concentration index, there exist optimal values of the surface wavelength and amplitude for maximum adhesion strength.

Creep effect on cellular uptake of viral particles

Receptor diffusion on cell membrane is usually believed as a major factor that controls how fast a virus can enter into host cell via endocytosis.However,when receptors are densely distributed around the binding site so that receptor recruiting through diffusion is no longer energetically favorable,we thus hypothesize that another effect,the creep deformation of cytoskeleton,might turn to play the dominant role in relaxing the engulfing process.In order to deeply understand this mechanism,we propose a viscoelastic model to investigate the dynamic process of virus engulfment retarded by the creep deformation of cytoskeleton and driven by the binding of ligand-receptor bonds after overcoming resistance from elastic deformation of lipid membrane and cytoskeleton.Based on this new model,we predict the lower bound of the ligand density and the range of virus size that allows the complete engulfment,and an optimal virus size corresponding to the smallest wrapping time.Surprisingly,these predictions can be reduced to the previous predictions based on simplified membrane models by taking into account statistical thermodynamic effects.The results presented in this study may be of interest to toxicologists,nanotechnologists,and virologists.

A WAVELET METHOD FOR BENDING OF CIRCULAR PLATE WITH LARGE DEFLECTION

A wavelet method for solving strongly nonlinear boundary value problems is described, which has been demonstrated early to have a convergence rate of order 4, almost independent of the nonlinear intensity of the equations. By using such a method, we study the bending problem of a circular plate with arbitrary large deflection. As the deflection increases, the bending behavior usually exhibits a so-called plate-to-membrane transition. Capturing such a transition has ever frustrated researchers for decades. However, without introducing any addi- tional treatment, we show in this study that the proposed wavelet solutions can naturally cover the plate-membrane transition region as the plate deflection increases. In addition, the high accuracy and efficiency of the wavelet method in solving strongly nonlinear problems is numerically confirmed, and applicable scopes for the linear, the membrane and the yon Karman plate theories are identified with respect to the large deformation bending of circular plates.

Measuring Viscoelastic Properties of Living Cells

Precise measurement of mechanical properties of living cells is important in understanding their mechanics-biology relations.In this study,we adopted the atomic force microscope to measure the creep deformation and stress relaxation of six different human cell lines.We examined whether the measured creep and relaxation trajectories satisfy a verification relation derived based on the linear viscoelastic theory.We compared the traditional spring-dashpot and the newly developed power-law-type constitutive relations in fitting the experimental measurements.We found that the human normal liver(L02),hepatic cancer(HepG2),hepatic stellate(LX2)and gastric cancer(NCI-N87)cell lines are linear viscoelastic materials,and human normal gastric(GES-1)and gastric cancer(SGC7901)cell lines are nonlinear due to failing in satisfying the verification relation for linear viscoelastic theory.The three-parameter power-law-type const计utive relation can fit the experimentai measurements better than that of the five-parameter classical spring-dashpot.

A sixth-order wavelet integral collocation method for solving nonlinear boundary value problems in three dimensions

A sixth-order accurate wavelet integral collocation method is proposed for solving high-order nonlinear boundary value problems in three dimensions.In order to realize the establishment of this method,an approximate expression of multiple integrals of a continuous function defined in a three-dimensional bounded domain is proposed by combining wavelet expansion and Lagrange boundary extension.Through applying such an integral technique,during the solution of nonlinear partial differential equations,the unknown function and its lower-order partial derivatives can be approximately expressed by its highest-order partial derivative values at nodes.A set of nonlinear algebraic equations with respect to these nodal values of the highest-order partial derivative is obtained using a collocation method.The validation and convergence of the proposed method are examined through several benchmark problems,including the eighth-order two-dimensional and fourth-order three-dimensional boundary value problems and the large deflection bending of von Karman plates.Results demonstrate that the present method has higher accuracy and convergence rate than most existing numerical methods.Most importantly,the convergence rate of the proposed method seems to be independent of the order of the differential equations,because it is always sixth order for second-,fourth-,sixth-,and even eighth-order problems.

Biomechanics in"Sino-Italian Joint"

As originally proposed by professor Alberto Corigliano of Politecnico di Milano and professor Jizeng Wang of Lanzhou University,the Chinese Society of Theoretical and Applied Mechanics(CSTAM)and the Italian Association for Theoretical and Applied Mechanics(AIMETA)signed a memorandum of understanding on strengthening exchange and cooperation between mechanics scholars of the two countries in January,2018,officially opening the activities of bilateral academic exchange.