A continuum model of contraction waves and protoplasm streaming in strands of Physarum plasmodium

We present a mathematical model for continuously distributed mechanochemical autooscillations (autowaves) in a protoplasmic strand of Physarum polycephalum. The model is based on a hypothesis of local positive feedback between deformation and contraction of the contractile apparatus. This feedback is mediated through a cell regulatory system whose kinetics involves coupling to mechanical strain. Mathematical analysis and computer simulations have demonstrated that the solutions of the model agree quantitatively with the available experimental data. In particular, hydrodynamic interaction alone, between different sections of the strand via the streaming endoplasm, is capable of inducing the characteristic contractile behavior.     

A continuum model for skeletal muscle contraction at homogeneous finite deformations

The contractile force in skeletal muscle models is commonly postulated to be the isometric force multiplied by a set of experimentally motivated functions which account for the muscle’s active properties. Although both flexible and simple, this approach does not automatically guarantee a thermodynamically consistent behavior. In contrast, the continuum mechanical model proposed herein is derived from fundamental principles in mechanics and guarantees a dissipative behavior. Further, the contractile force is associated with a friction clutch which provides a simple and well-defined macroscopic model for cycling cross-bridges. To show the performance of the model, it is specialized to standard experiments for rabbit tibialis anterior muscle. The results show that the model is able to capture important characteristics of skeletal muscle.     

A continuum model for root growth

This paper presents a study of a simple one-dimensional continuum model for growth of the plant root. A fundamental constitutive equation is derived. The model is studied by means of various special cases of increasing complexity. Asymptotic expansions are used to derive approximate solutions to the equation of the model under the fundamental assumption that cell wall thickness is small in comparison with the diameter of the cell. The basic results of the study may be summarized as follows. The observed growth pattern of the root cannot be modelled by a mechanical system whose properties are independent of position on the root. The observed pattern can be modelled by a simple mechanical system in which, for example, cell wall yield stress first decreases and then increases. Two fundamental observations are made based on the modelling study. The first is that any mechanical model must take into account the convective displacement from the tip of points along the root. The second is that in describing growth, data on cell wall mechanical properties are meaningless without corresponding data on cell water potential, and vice versa.     

A continuum model for coupled cells

A continuum model of diffusion-coupled cells that more accurately reflects the presence of low-permeability gap junctions between cells is analyzed. It is shown by a multi-scale analysis that to lowest order the slow evolution of the mean concentration is described by the usual ordinary differential equations for a discrete model. Furthermore, stable non-uniform steady solutions are shown to exist in the continuum model of a one component system, whereas this is impossible for the standard reaction-diffusion model of this system. It is also shown how to average the equations in this continuum model to obtain a system of reaction-diffusion equations with constant coefficients.     

A continuum model of elephant trunks

A continuum model is presented that relates the trunk parameters of loading, geometry, and muscle structure to the necessary conditions of static equilibrium. Linear theory for stress-strain behavior is used to describe an elephant trunk for an incremental displacement as the animal slowly lifts a weight at the trunk tip. With this analysis and experimental values for the trunk parameters, the apparent trunk stiffness Ea is estimated for the living animal. For an Asian elephant with a maximum compression strain of 33 percent, Ea is of the order of 10(6) N/m2. The continuum model is quite general and may be applied to similar nonskeletal appendages and bodies of other animals.     

A continuum model for pressure-flow relationship in human pulmonary circulation

A continuum model was introduced to analyze the pressure-flow relationship for steady flow in human pulmonary circulation. The continuum approach was based on the principles of continuum mechanics in conjunction with detailed measurement of vascular geometry, vascular elasticity and blood rheology. The pulmonary arteries and veins were considered as elastic tubes and the "fifth-power law" was used to describe the pressure-flow relationship. For pulmonary capillaries, the "sheet-flow" theory was employed and the pressure-flow relationship was represented by the "fourth-power law". In this paper, the pressure-flow relationship for the whole pulmonary circulation and the longitudinal pressure distribution along the streamlines were studied. Our computed data showed general agreement with the experimental data for the normal subjects and the patients with mitral stenosis and chronic bronchitis in the literature. In conclusion, our continuum model can be used to predict the changes of steady flow in human pulmonary circulation.     

A one-dimensional continuum elastic model for membrane-embedded gramicidin dimer dissociation

Membrane elastic properties, which are subject to alteration by compounds such as cholesterol, lipid metabolites and other amphiphiles, as well as pharmaceuticals, can have important effects on membrane proteins. A useful tool for measuring some of these effects is the gramicidin A channels, which are formed by transmembrane dimerization of non-conducting subunits that reside in each bilayer leaflet. The length of the conducting channels is less than the bilayer thickness, meaning that channel formation is associated with a local bilayer deformation. Electrophysiological studies have shown that the dimer becomes increasingly destabilized as the hydrophobic mismatch between the channel and the host bilayer increases. That is, the bilayer imposes a disjoining force on the channel, which grows larger with increasing hydrophobic mismatch. The energetic analysis of the channel-bilayer coupling is usually pursued assuming that each subunit, as well as the subunit-subunit interface, is rigid. Here we relax the latter assumption and explore how the bilayer junction responds to changes in this disjoining force using a simple one-dimensional energetic model, which reproduces key features of the bilayer regulation of gramicidin channel lifetimes.     

A dynamic model of smooth muscle contraction

A dynamic model of smooth muscle contraction is presented and is compared with the mechanical properties of vascular smooth muscle in the rat portal vein. The model is based on the sliding filament theory and the assumption that force is produced by cross-bridges extending from the myosin to the actin filaments. Thus, the fundamental aspects of the model are also potentially applicable to skeletal muscle. The main concept of the model is that the transfer of energy via the cross-bridges can be described as a 'friction clutch' mechanism. It is shown that a mathematical formulation of this concept gives rise to a model that agrees well with experimental observations on smooth muscle mechanics under isotonic as well as isometric conditions. It is noted that the model, without any ad hoc assumptions, displays a nonhyperbolic force-velocity relationship in its high-force portion and that it is able to maintain isometric force in conditions of reduced maximum contraction velocity. Both these findings are consistent with new experimental observations on smooth muscle mechanics cannot be accounted for by the classical Hill model.     

Steady-state analysis of a continuum model for super-infection

A large system of N strains of parasite and a single host is analyzed as a function of the degree of virulence in the strains when there is super-infection between hosts (more virulent strains can infect hosts that are already infected) and within-host transition between strains that is neutral. When this small amount of local switching is allowed, steady-state solutions converge to a continuous distribution as the number of strains increases. The resulting nonlinear-nonautonomous integro-differential equation is reduced to a fourth order boundary value problem (BVP) and the existence of positive solutions is proven. The methods here and associated BVP allow for a thorough exploration of parameter space for this class of models.      

Analysis of a continuum model for root growth

We give an analysis by asymptotic expansions of a model for root growth. The model assumes growth is due entirely to mechanical forces created by the flux of water from the surrounding soil. We attempt to suggest ways in which the model could account for experimentally observed patterns of root growth. The model predicts that the rate of growth of the root is independent of the extensibility of the root wall.     

A phenomenological model for estimating metabolic energy consumption in muscle contraction

A phenomenological model for muscle energy consumption was developed and used in conjunction with a simple Hill-type model for muscle contraction. The model was used to address two questions. First, can an empirical model of muscle energetics accurately represent the total energetic behavior of frog muscle in isometric, isotonic, and isokinetic contractions? And second, how does such a model perform in a large-scale, multiple-muscle model of human walking? Four simulations were conducted with frog sartorius muscle under full excitation: an isometric contraction, a set of isotonic contractions with the muscle shortening a constant distance under various applied loads, a set of isotonic contractions with the muscle shortening over various distances under a constant load, and an isokinetic contraction in lengthening. The model calculations were evaluated against results of similar thermal in vitro experiments performed on frog sartorius muscle. The energetics model was then incorporated into a large-scale, multiple-muscle model of the human body for the purpose of predicting energy consumption during normal walking. The total energy estimated by the model accurately reflected the observed experimental behavior of frog muscle for an isometric contraction. The model also accurately reproduced the experimental behavior of frog muscle heat production under isotonic shortening and isokinetic lengthening conditions. The estimated rate of metabolic energy consumption for walking was 29% higher than the value typically obtained from gait measurements.     

Self-consistent model of a pulsed air discharge excited by surface waves

A complete self-consistent electrodynamic model of a pulsed gas discharge excited by surface waves is developed. The model allows one to calculate both the initial phase of the discharge front propagation and the parameters of the produced plasma. The spatiotemporal evolution of the electromagnetic field and plasma parameters at the discharge front is investigated for the first time. It is shown that discharge propagation is mainly governed by a breakdown wave in an inhomogeneous electric field at the leading edge of the ionization front. It is found that the effect of the electric field enhancement in the plasma resonance region significantly affects the velocity of the breakdown wave. The results of calculations agree well with experimental data.     

On the Lacker-Peskin model for muscular contraction

We study Lacker and Peskin's version of A.F. Huxley's model of muscular contraction. The model, which presumes a continuous (and independent) distribution of sites of attachment for a myosin cross-bridge, is presumed to apply to uniformly excited skeletal muscles under conditions of maximal overlap of fibres. We show that given velocity as a function of time there exists a unique force as a function of time, and that the velocity-force relation is continuous and one-to-one. We also suggest minor modifications of the model.     

A viscoelastic model for use in predicting arterial pulse waves

In nonlinear mathematical models of the arterial circulation, the viscoelasticity of the vessel walls has generally been neglected or only taken into account in a highly approximate manner. A new method is proposed to simulate the nonlinear viscoelastic properties of the wall material with the aid of a convolution integral of the creep function and the pressure history. With this simulation it is possible to properly describe the measured characteristics of arterial viscoelasticity. Moreover, it is utilized in a mathematical model of arterial pulse propagation to study the influence of the internal wall friction on the shape, amplitude and mean value of pressure and flow pulses. The corresponding predictions are in much better agreement with in-vivo measurements, especially for the distal part of the circulation, than those obtained without viscoelasticity.     

A dynamic continuum model for molecular regulation of the humoral immune response.

Based on current concepts of the nature of specific immunocompetent cell surface receptors, hematopoietic stem cell differentiation and membrane dynamics, a simple model is proposed by which the event of immunogen binding by cell surface receptors specifically stimulates a constitutive process. The concept that enhancement of process flows in two directions with time allows for correlation of recent experimental findings into a molecular theory for antibody induction. The Model helps explain antibody specificity, heterogeneity and affinity changes and rationalizes immune memory.