A three-dimensional random network model of the cytoskeleton and its role in mechanotransduction and nucleus deformation

We have developed a three-dimensional random network model of the intracellular actin cytoskeleton and have used it to study the role of the cytoskeleton in mechanotransduction and nucleus deformation. We use the model to predict the deformation of the nucleus when mechanical stresses applied on the plasma membrane are propagated through the random cytoskeletal network to the nucleus membrane. We found that our results agree with previous experiments utilizing micropipette pulling. Therefore, we propose that stress propagation through the random cytoskeletal network can be a mechanism to effect nucleus deformation, without invoking any biochemical signaling activity. Using our model, we also predict how nucleus strain and its relative displacement within the cytosol vary with varying concentrations of actin filaments and actin-binding proteins. We find that nucleus strain varies in a sigmoidal manner with actin filament concentration, while there exists an optimal concentration of actin-binding proteins that maximize nucleus displacement. We provide a theoretical analysis for these nonlinearities in terms of the connectivity of the random cytoskeletal network. Finally, we discuss laser ablation experiments that can be performed to validate these results in order to advance our understanding of the role of the cytoskeleton in mechanotransduction.     

A calcium‐mediated actin redistribution at egg activation in Drosophila

Egg activation is the essential process in which mature oocytes gain the competency to proceed into embryonic development. Many events of egg activation are conserved, including an initial rise of intracellular calcium. In some species, such as echinoderms and mammals, changes in the actin cytoskeleton occur around the time of fertilization and egg activation. However, the interplay between calcium and actin during egg activation remains unclear. Here, we use imaging, genetics, pharmacological treatment, and physical manipulation to elucidate the relationship between calcium and actin in living Drosophila eggs. We show that, before egg activation, actin is smoothly distributed between ridges in the cortex of the dehydrated mature oocytes. At the onset of egg activation, we observe actin spreading out as the egg swells though the intake of fluid. We show that a relaxed actin cytoskeleton is required for the intracellular rise of calcium to initiate and propagate. Once the swelling is complete and the calcium wave is traversing the egg, it leads to a reorganization of actin in a wavelike manner. After the calcium wave, the actin cytoskeleton has an even distribution of foci at the cortex. Together, our data show that calcium resets the actin cytoskeleton at egg activation, a model that we propose to be likely conserved in other species.     

Resonance interaction of electrons with a slow transverse wave in a weakly inhomogeneous plasma

Excitation of a circularly polarized slow wave by external sources and its subsequent propagation in a weakly inhomogeneous plasma with a positive density gradient are described in terms of the adiabatic approach. It is shown that the wave dispersion is mainly determined by the ratio between the contributions of trapped and nonresonant untrapped electrons to the total wave current. The relationship between the wave amplitude and its phase velocity and the limiting phase velocity above which the wave is strongly damped are found using the energy balance equation and the dispersion relation.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

Traveling waves and pulses in a one-dimensional network of excitable integrate-and-fire neurons

 We study the existence and stability of traveling waves and pulses in a one-dimensional network of integrate-and-fire neurons with synaptic coupling. This provides a simple model of excitable neural tissue. We first derive a self-consistency condition for the existence of traveling waves, which generates a dispersion relation between velocity and wavelength. We use this to investigate how wave-propagation depends on various parameters that characterize neuronal interactions such as synaptic and axonal delays, and the passive membrane properties of dendritic cables. We also establish that excitable networks support the propagation of solitary pulses in the long-wavelength limit. We then derive a general condition for the (local) asymptotic stability of traveling waves in terms of the characteristic equation of the linearized firing time map, which takes the form of an integro-difference equation of infinite order. We use this to analyze the stability of solitary pulses in the long-wavelength limit. Solitary wave solutions are shown to come in pairs with the faster (slower) solution stable (unstable) in the case of zero axonal delays; for non-zero delays and fast synapses the stable wave can itself destabilize via a Hopf bifurcation. Received: 27 October 1998     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Traveling wave solutions of a nonlinear reaction–advection equation

 We establish the existence of traveling wave solutions for a nonlinear partial differential equation that models a logistically growing population whose movement is governed by an advective process. Conditions are presented for which traveling wave solutions exist and for which they are stable to small semi-finite domain perturbations. The wave is of mathematical interest because its behavior is determined by a singular differential equation and those with small speed of propagation steepen into a shock-like solutions. Finally, we indicate that the smoothing presence of diffusion allows wave persistence when an advective slow moving wave may collapse. Received: 24 November 1997 / Revised version: 13 July 1998     

Existence of Traveling Waves for the Generalized F–KPP Equation

Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher–Kolmogorov–Petrovskii–Piskunov (F–KPP) equation. This generalized F–KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F–KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth–decay rate in the generalized F–KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.     

Cortical activity in vertebrate eggs. I: The activation waves

We present a physical model for the propagation of chemical and mechanical waves on the surface of vertebrate eggs. As a first step we analyzed the propagation of the calcium wave observed to sweep over the surface of the Medaka egg (Gilkey et al., 1978). It has been assumed that this wave is driven by a mechanism of calcium-stimulated-calcium-release. By formulating this hypothesis mathematically we can use the observed wavefront data to obtain a map of cortical reactivity. This map indicates a gradient of reactivity along the egg: highest in the animal hemisphere and tapering off towards the vegetal hemisphere. The cortex of Xenopus eggs is also capable of propagating a calcium wave (Busa & Nuccitelli, 1985). At about the same time a wave of expansion followed by a wave of contraction sweeps across the egg surface (Takeichi et al., 1984). We have proposed a mechanism for this wave pair based on the physical chemistry of actomyosin gels. The calcium wave activates solation factors which sever some of the actin chains which leads to an osmotic swelling of the gel. Calcium also activates the contractile machinery of the actomyosin system which causes the gel to contract. The contraction lags the swelling because of the nature of the kinetics: solation and swelling is a more rapid process than contraction. By writing the equations for gel expansion and contraction we can mimic the mechanical and chemical wave propagation by a computer simulation. If the model is correct this provides a method for using the waves as a diagnostic of the mechanochemical properties of the egg cortex.     

Traveling waves in the discrete fast buffered bistable system

We study the existence and uniqueness of traveling wave solutions of the discrete buffered bistable equation. Buffered excitable systems are used to model, among other things, the propagation of waves of increased calcium concentration, and discrete models are often used to describe the propagation of such waves across multiple cells. We derive necessary conditions for the existence of waves, and, under some restrictive technical assumptions, we derive sufficient conditions. When the wave exists it is unique and stable.      

Finite element modeling of impulsive excitation and shear wave propagation in an incompressible,transversely isotropic medium

Elastic properties of materials can be measured by observing shear wave propagation following localized, impulsive excitations and relating the propagation velocity to a model of the material. However, characterization of anisotropic materials is difficult because of the number of elasticity constants in the material model and the complex dependence of propagation velocity relative to the excitation axis, material symmetries, and propagation directions. In this study, we develop a model of wave propagation following impulsive excitation in an incompressible, transversely isotropic (TI) material such as muscle. Wave motion is described in terms of three propagation modes identified by their polarization relative to the material symmetry axis and propagation direction. Phase velocities for these propagation modes are expressed in terms of five elasticity constants needed to describe a general TI material, and also in terms of three constants after the application of two constraints that hold in the limit of an incompressible material. Group propagation velocities are derived from the phase velocities to describe the propagation of wave packets away from the excitation region following localized excitation. The theoretical model is compared to the results of finite element (FE) simulations performed using a nearly incompressible material model with the five elasticity constants chosen to preserve the essential properties of the material in the incompressible limit. Propagation velocities calculated from the FE displacement data show complex structure that agrees quantitatively with the theoretical model and demonstrates the possibility of measuring all three elasticity constants needed to characterize an incompressible, TI material.     

Ion acoustic cnoidal waves in a dusty plasma with a critical dust density

The propagation of nonlinear periodic ion acoustic waves in a dusty plasma is considered for conditions in which the coefficient in the nonlinear equation that describes the quadratic nonlinearity of the medium is zero. An equation that accounts for the cubic nonlinearity of the system is derived, and its solution is found. The dependence of the phase velocity of a cnoidal wave on its amplitude and modulus is determined. In describing the effect of higher order nonlinearities on the properties of a dust ion acoustic wave, two coupled equations for the first- and second-order potentials are obtained. It is shown that the nonlinear ion flux generated by a cnoidal wave propagating in a medium with a cubic nonlinearity is proportional to the fourth power of the wave amplitude.     

Arabidopsis VILLIN5, an Actin Filament Bundling and Severing Protein, Is Necessary for Normal Pollen Tube Growth

A dynamic actin cytoskeleton is essential for pollen germination and tube growth. However, the molecular mechanisms underlying the organization and turnover of the actin cytoskeleton in pollen remain poorly understood. Villin plays a key role in the formation of higher-order structures from actin filaments and in the regulation of actin dynamics in eukaryotic cells. It belongs to the villin/gelsolin/fragmin superfamily of actin binding proteins and is composed of six gelsolin-homology domains at its core and a villin headpiece domain at its C terminus. Recently, several villin family members from plants have been shown to sever, cap, and bundle actin filaments in vitro. Here, we characterized a villin isovariant, Arabidopsis thaliana VILLIN5 (VLN5), that is highly and preferentially expressed in pollen. VLN5 loss-of-function retarded pollen tube growth and sensitized actin filaments in pollen grains and tubes to latrunculin B. In vitro biochemical analyses revealed that VLN5 is a typical member of the villin family and retains a full suite of activities, including barbed-end capping, filament bundling, and calcium-dependent severing. The severing activity was confirmed with time-lapse evanescent wave microscopy of individual actin filaments in vitro. We propose that VLN5 is a major regulator of actin filament stability and turnover that functions in concert with oscillatory calcium gradients in pollen and therefore plays an integral role in pollen germination and tube growth.     

Wave speed in excitable random networks with spatially constrained connections

Very fast oscillations (VFO) in neocortex are widely observed before epileptic seizures, and there is growing evidence that they are caused by networks of pyramidal neurons connected by gap junctions between their axons. We are motivated by the spatio-temporal waves of activity recorded using electrocorticography (ECoG), and study the speed of activity propagation through a network of neurons axonally coupled by gap junctions. We simulate wave propagation by excitable cellular automata (CA) on random (Erdös-Rényi) networks of special type, with spatially constrained connections. From the cellular automaton model, we derive a mean field theory to predict wave propagation. The governing equation resolved by the Fisher-Kolmogorov PDE fails to describe wave speed. A new (hyperbolic) PDE is suggested, which provides adequate wave speed that saturates with network degree , in agreement with intuitive expectations and CA simulations. We further show that the maximum length of connection is a much better predictor of the wave speed than the mean length. When tested in networks with various degree distributions, wave speeds are found to strongly depend on the ratio of network moments rather than on mean degree , which is explained by general network theory. The wave speeds are strikingly similar in a diverse set of networks, including regular, Poisson, exponential and power law distributions, supporting our theory for various network topologies. Our results suggest practical predictions for networks of electrically coupled neurons, and our mean field method can be readily applied for a wide class of similar problems, such as spread of epidemics through spatial networks.     

Modeling Compositionality by Dynamic Binding of Synfire Chains

This paper examines the feasibility of manifesting compositionality by a system of synfire chains. Compositionality is the ability to construct mental representations, hierarchically, in terms of parts and their relations. We show that synfire chains may synchronize their waves when a few orderly cross links are available. We propose that synchronization among synfire chains can be used for binding component into a whole. Such synchronization is shown both for detailed simulations, and by numerical analysis of the propagation of a wave along a synfire chain. We show that global inhibition may prevent spurious synchronization among synfire chains. We further show that selecting which synfire chains may synchronize to which others may be improved by including inhibitory neurons in the synfire pools. Finally we show that in a hierarchical system of synfire chains, a part-binding problem may be resolved, and that such a system readily demonstrates the property of priming. We compare the properties of our system with the general requirements for neural networks that demonstrate compositionality.     

Self-organizing actin waves as planar phagocytic cup structures

Actin waves that travel on the planar membrane of a substrate-attached cell underscore the capability of the actin system to assemble into dynamic structures by the recruitment of proteins from the cytoplasm. The waves have no fixed shape, can reverse their direction of propagation, and can fuse or divide. Actin waves separate two phases of the plasma membrane that are distinguished by their lipid composition. The area circumscribed by a wave resembles in its phosphoinositide content the interior of a phagocytic cup, leading us to explore the possibility that actin waves are in-plane phagocytic structures generated without the localized stimulus of an attached particle. Consistent with this view, wave-forming cells were found to exhibit a high propensity for taking up particles. Cells fed rod-shaped particles produced elongated phagocytic cups that displayed a zonal pattern that reflected in detail the actin and lipid pattern of free-running actin waves. Neutrophils and macrophages are known to spread on surfaces decorated with immune complexes, a process that has been interpreted as “frustrated” phagocytosis. We suggest that actin waves enable a phagocyte to scan a surface for particles that might be engulfed.     

Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory

Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes' cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the pressure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.     

Supersonic and near-sonic solitary ion-acoustic waves in magnetized plasma

The propagation of large-amplitude solitary ion-acoustic waves in magnetized plasma is analyzed. The problem is solved without assuming plasma quasineutrality within the pulse, and the wave potential is described by Poisson’s equation. Solutions in the form of supersonic and near-sonic solitary waves propagating obliquely to the magnetic field are found. The pulses have several peaks and exist for a discrete set of the wave parameters. The amplitude and oscillation frequency of a solitary wave are determined as functions of the Mach number and the propagation angle with respect to the magnetic field.     

Actin polymerizability is influenced by profilin, a low molecular weight protein in non-muscle cells

We have previously isolated and crystallized a complex from calf spleen, containing actin and a smaller protein which we call profilin. In this paper we describe some properties of this complex, and show that association with profilin is sufficient to explain the persistent monomeric state of some of the actin in spleen extracts; moreover, spleen profilin will recombine with skeletal muscle actin to form a non-polymerizable complex resembling that isolated from spleen. Profilin is not restricted to spleen, but is found in a variety of other tissues and tissue-cultured cell lines. We propose that reversible association of actin with profilin in the cell may provide a mechanism for storage of monomeric actin and controlled turnover of microfilaments.