Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

Simulation of deformable models with the Poisson equation

In this paper, we present a new methodology for the deformation of soft objects by drawing an analogy between the Poisson equation and elastic deformation from the viewpoint of energy propagation. The potential energy stored due to a deformation caused by an external force is calculated and treated as the source injected into the Poisson system, as described by the law of conservation of energy. An improved Poisson model is developed for propagating the energy generated by the external force in a natural manner. An autonomous cellular neural network (CNN) model is established by using the analogy between the Poisson equation and CNN to solve the Poisson model for the real-time requirement of soft object deformation. A method is presented to derive the internal forces from the potential energy distribution. The proposed methodology models non-linear materials with the non-linear Poisson equation and thus non-linear CNN, rather than geometric non-linearity. It not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying constitutive coefficients. A haptic virtual reality system has been developed for deformation simulation with force feedback. Examples are presented to demonstrate the efficiency of the proposed methodology.     

An Improved Mathematical Approach for Determination of Molecular Kinetics in Living Cells with FRAP

The estimation of binding constants and diffusion coefficients of molecules that associate with insoluble molecular scaffolds inside living cells and nuclei has been facilitated by the use of Fluorescence Recovery after Photobleaching (FRAP) in conjunction with mathematical modeling. A critical feature unique to FRAP experiments that has been overlooked by past mathematical treatments is the existence of an `equilibrium constraint': local dynamic equilibrium is not disturbed because photobleaching does not functionally destroy molecules, and hence binding-unbinding proceeds at equilibrium rates. Here we describe an improved mathematical formulation under the equilibrium constraint which provides a more accurate estimate of molecular reaction kinetics within FRAP studies carried out in living cells. Due to incorporation of the equilibrium constraint, the original non-linear kinetic terms become linear allowing for analytical solution of the transport equations and greatly simplifying the estimation process. Based on mathematical modeling and scaling analysis, two experimental measures are identified that can be used to delineate the rate-limiting step. A comprehensive analysis of the interplay between binding-unbinding and diffusion, and its effect on the recovery curve, are presented. This work may help to bring clarity to the study of molecular dynamics within the structural complexity of living cells.     

Polymer-mediated electrostatic interactions between charged lipid assemblies and electrolyte solutions: a tentative model of the polyethylene glycol-induced cell fusion

We developed a theoretical model to investigate the interaction between charged lipid aggregates and a water solution containing ions and uncharged polymers. The local concentration of ions and polymer chains around the lipid aggregate have been treated as variational parameters which can be found by minimizing the total energy of the system. We divided the energy into the following main contributions: (a) Solvation energy of the ions. This depends on the local polymer concentration through the variation of the solvent dielectric properties. (b) Ions-lipid aggregate interactions. These depend on the local concentrations both of the ion cloud and polymer chains. (c) Conformational energy of the polymer. This term is related to the inhomogeneous spatial density of the polymer segments. Any direct interaction between the charged lipid surface and the polymer coils has been intentionally neglected. The minimization procedure leads to a non-linear Poisson-Boltzmann equation coupled with a non-linear algebraic equation describing the polymer distribution. The solution of the above system allows one to calculate the ions and polymer spatial distribution around the lipid aggregate. The knowledge of such parameters is useful to predict the effect of non-ionic polymers on the structure and properties of lipid assemblies such as the mean area per lipid molecule, the aggregation number, the critical micellar concentration and the formation of immiscibility gaps in mixed lipid systems. A possible involvement of these parameters into the fusion process between lipid vesicles is discussed.     

Theory for the formation of intercellular junctions based on intramembranous particle patterns observed in the freeze fracture technique

A heterogeneous continuum theory of biological membrane interactions is presented which takes account of the spatial inhomogeneity of intramembranous particle patterns observed in ultrastructural studies of junctional complexes using the freeze-cleaved technique. The theory attempts to explain (i) how electrostatic and electrodynamic forces between particles of different biochemical composition in the same and opposing membranes might give rise to the specialized particle configurations characteristic of tight and gap junctions, (ii) how the spatial non-uniformity of the membrane proteins quantitatively modifies the local long range molecular level force field between adjacent membrane bilayers and (iii) how membrane elastic stresses and the modified molecular level forces combine to determine the equilibrium configurations of the various junctional complexes. The mathematical problem is highly non-linear since the molecular forces are a rapidly varying function of the local membrane spacing which is a priori unknown. The simplified dimensionless boundary value problem to determine the junction geometry has been reduced to a fourth order quasi one-dimensional equation with split end point conditions. The equation is singular in the vicinity of the junction complex and special solution techniques had to be employed. The numerical results are in reasonable quantitative agreement with electron microscopic observations of tight, gap and venous junctions.     

From micro to nano: recent advances in high-resolution microscopy

Improving the spatial resolution of optical microscopes is important for a vast number of applications in the life sciences. Optical microscopy allows intact samples and living cells to be studied in their natural environment, tasks that are not possible with other microscopy methods (e.g. electron microscopy). Major advances in the past two decades have significantly improved microscope resolution. By using interference and structured light methods microscope resolution has been improved to approximately 100 nm, and with non-linear methods a ten times improvement has been demonstrated to a current resolution limit of approximately 30 nm. These methods bring together old theoretical concepts such as interference with novel non-linear methods that improve spatial resolution beyond the limits that were previously assumed to be unreachable.     

A diffusion model for the evolution of metalloporphyrin

We give a mathematical model of the evolution of enzymes, the molecular structure of which is like metalloporphyrins or chlorophylls. We show, for this model, that even a small amount of these enzymes at the first stage is sufficient to increase and dominate the majority in a cell (like phenomena of gene fixation). For this purpose we use Kimura's equation, which has been explored for the study of evolution of genetics and has been known as a neutral theory of molecular evolution. Our model is a non-linear, non-equilibrium and non-closed (open to the external world) model.     

Issues in electromagnetic field-biointeractions

Low level electromagnetic fields have been found to produce a variety of biological effects, though the mechanism of such interaction is still not completely understood. Cell membrane of the brain is a critical structure perceiving the action of microwaves, which has received greater attention in the recent past. The interactions of EMF with the living cells are considered as stochastic resonance, cooperative effects, non-equilibrium thermodynamic process and non-linear interactions. The living cells derive the energy from noise and pumps it into the modes of excitation at the driving frequency of an electromagnetic wave which give sufficient amplification of the signal and increase the signal to noise ratio. The non-linear mechanism plays their main role in the process of transmembrane coupling of the signal to the cytoplasm. The criteria for safe exposure limits of electromagnetic field to humans is also discussed.     

A comparison of methods for estimating the kinetic parameters of two simple types of transport process

Sets of experimental data, with known characteristics and error structures, have been simulated for the Michaelis-Menten equation plus a second term, either for linear transport or for competitive inhibition. The Michaelis-Menten equation plus linear term was fitted by several methods and the accuracy and the precision of the parameter estimates from the several methods were compared. The model-fitting methods were: three for least-squares non-linear regression, computer versions of two graphical methods and of two non-parametric methods. The most precise and accurate method was that of D.W. Marquardt (J. Soc. Ind. Appl. Math. 11 (1963) 431–441). The Michaelis-Menten equation with competitive inhibition was also fitted by several methods, viz., two for least-squared non-linear regression, a non-parametric method and four variants of the Preston-Schaeffer-Curran plot (Preston, R.L. et al. (1974) J. Gen. Physiol. 64, 443–467). The most precise and accurate of these was the non-linear regression method of W.W. Cleland (Adv. Enzymol. 29 (1967) 1–32). For both these models, the various graphical methods and non-parametric methods gave poor results and are not recommended.     

Calculation of the Volterra kernels of non-linear dynamic systems using an artificial neural network

The Volterra series is a well-known method of describing non-linear dynamic systems. A major limitation of this technique is the difficulty involved in the calculation of the kernels. More recently, artificial neural networks have been used to produce black box models of non-linear dynamic systems. In this paper we show how a certain class of artificial neural networks are equivalent to Volterra series and give the equation for the nth order Volterra kernel in terms of the internal parameters of the network. The technique is then illustrated using a specific non-linear system. The kernels obtained by the method described in the paper are compared with those obtained by a Toeplitz matrix inversion technique. Received: 4 June 1993/Accepted in revised form: 2 March 1994     

Impulses and Physiological States in Theoretical Models of Nerve Membrane

Van der Pol''s equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting “BVP model” has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer''s theoretical model for the iron wire model of nerve. This BVP model serves as a simple representative of a class of excitable-oscillatory systems including the Hodgkin-Huxley (HH) model of the squid giant axon. The BVP phase plane can be divided into regions corresponding to the physiological states of nerve fiber (resting, active, refractory, enhanced, depressed, etc.) to form a “physiological state diagram,” with the help of which many physiological phenomena can be summarized. A properly chosen projection from the 4-dimensional HH phase space onto a plane produces a similar diagram which shows the underlying relationship between the two models. Impulse trains occur in the BVP and HH models for a range of constant applied currents which make the singular point representing the resting state unstable.     

A non-ideal replacement for the Boyle van't Hoff equation

A non-ideal osmotic equilibrium equation is proposed as a replacement for the Boyle van’t Hoff equation to describe the equilibrium volume of a living cell as a function of external osmolality. Contrary to common understanding, the Boyle van’t Hoff equation is only thermodynamically correct for ideal, dilute solutions. However, the Boyle van’t Hoff equation is commonly used to determine the osmotically inactive fraction of the cell. This involves extrapolating to infinite osmolality, which violates the ideal, dilute solution constraint. It has been noted that the osmotically inactive fractions obtained from the Boyle van’t Hoff equation for human erythrocytes are markedly larger than measured values of the dry volume fraction of the cell. Using the new osmotic equilibrium equation to analyze experimental osmotic equilibrium data reduces the inferred osmotically inactive fraction of human erythrocytes by approximately 20%.     

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.     

Lateral organization of membranes and cell shapes.

The relations among membrane structure, mechanical properties, and cell shape have been investigated. The fluid mosaic membrane models used contains several components that move freely in the membrane plane. These components interact with each other and determine properties of the membrane such as curvature and elasticity. A free energy equation is postulated for such a multicomponent membrane and the condition of free energy minimum is used to obtain differential equations relating the distribution of membrane components and the local membrane curvature. The force that moves membrane components along the membrane in a variable curvature field is calculated. A change in the intramembrane interactions can bring about phase separation or particle clustering. This, in turn, may strongly affect the local curvature. The numerical solution of the set of equations for the two dimensional case allows determination of the cell shape and the component distribution along the membrane. The model has been applied to describe certain erythrocytes shape transformations.     

运用改进单纯形法拟合Logistic曲线的研究

Ｌｏｇｉｓｔｉｃ方程是研究有限空间内种群增长规律的重要工具之一本文运用改进单纯形法最优拟合Ｌｏｇｉｓｔｉｃ曲线,结果表明改进单纯形法具有较强的拟合非线性方程的能力,对生物实验及生态、生理学中诸多非线性曲线的参数估计具有普遍意义．     

Equation of state and life span of poikilothermic animals

An equation of state of a living organism that corresponds to the Kleiber's empirical dependence has been found using the conditions of entropy and energy balance. Life span of animals has been calculated using the equation obtained and it was found to depend inversely on metabolism intensity.     

Two-dimensional standing gradient osmotic flow: a generalization of the “isotonic convection approximation”

A linear two-dimensional model of a flow system for solute and fluid transport in intercellular spaces has been obtained by using the so called isotonic convection approximation, already employed in the one-dimensional case (Segel (1970)). This is equivalent to ignoring the convective components in the relevant differential equation. The solutions found by means of the eigenfunctions of a Sturm-Liouville system have been compared with the numerical solutions of the general non-linear model, in which also the convective terms are present.The results of the linear model agree fairly well with those of the non-linear one, in the range of interest of parameters. This fact shows that the ignorance of the profile of fluid velocity does not introduce significant errors in the evaluation of solute and fluid fluxes.The differences of results between the one-dimensional and the twodimensional, the linear and non-linear models, give a way to evaluate the relative effects of the diffusive and convective terms in the differential equation and to estimate the errors introduced by the one-dimensional approximation.     

A study of the effect of non-linearities in the equation of bone remodeling

In this paper, we introduced a high-order non-linear equation of bone remodeling to combine with FEM by introducing two non-linearities, i.e. the remodeling coefficient B(t) and the order of non-linear remodeling equation. The influence of each non-linearity was tested based on its mechanical and physiological implications discussed. We use two finite element models to investigate the influences of non-linearities in this equation: a plate subjected to a ramp load, and a 2D model of the cross-section of a vertebra. By importing the idea of topology optimization in engineering, their external shapes and internal density distributions were simulated from unfixed configurations. To a certain extent, the high-order non-linear equation of bone remodeling we suggested here can control the remodeling processes of bones in different stages of growth or at different anatomic sites more effectively, and make it more consistent with physiological reality, i.e. express the remodeling characteristic that bone's best morphology is adapted to its mechanical environment. Furthermore, it is likely to describe the process of bone growth and evolution.     

Non-Linear Neuronal Responses as an Emergent Property of Afferent Networks: A Case Study of the Locust Lobula Giant Movement Detector

In principle it appears advantageous for single neurons to perform non-linear operations. Indeed it has been reported that some neurons show signatures of such operations in their electrophysiological response. A particular case in point is the Lobula Giant Movement Detector (LGMD) neuron of the locust, which is reported to locally perform a functional multiplication. Given the wide ramifications of this suggestion with respect to our understanding of neuronal computations, it is essential that this interpretation of the LGMD as a local multiplication unit is thoroughly tested. Here we evaluate an alternative model that tests the hypothesis that the non-linear responses of the LGMD neuron emerge from the interactions of many neurons in the opto-motor processing structure of the locust. We show, by exposing our model to standard LGMD stimulation protocols, that the properties of the LGMD that were seen as a hallmark of local non-linear operations can be explained as emerging from the dynamics of the pre-synaptic network. Moreover, we demonstrate that these properties strongly depend on the details of the synaptic projections from the medulla to the LGMD. From these observations we deduce a number of testable predictions. To assess the real-time properties of our model we applied it to a high-speed robot. These robot results show that our model of the locust opto-motor system is able to reliably stabilize the movement trajectory of the robot and can robustly support collision avoidance. In addition, these behavioural experiments suggest that the emergent non-linear responses of the LGMD neuron enhance the system''s collision detection acuity. We show how all reported properties of this neuron are consistently reproduced by this alternative model, and how they emerge from the overall opto-motor processing structure of the locust. Hence, our results propose an alternative view on neuronal computation that emphasizes the network properties as opposed to the local transformations that can be performed by single neurons.     

On the numerical integration of non-local terms for age-structured population models

We formulate explicit second-order finite difference schemes for the numerical integration of non-linear age-dependent population models. These methods have been designed by means of a representation formula for the theoretical solution of the integro-differential equation joint with open quadrature formulae for the numerical approximation of non-local terms. The schemes are analyzed and some numerical experiments are also reported in order to show numerically their accuracy.