Spatial dynamics of invasion in sinusoidally varying environments

Range expansions of invading species in homogeneous environments have been extensively studied since the pioneer works by Fisher (Ann Eugen 7:255–369, 1937) and Skellam (Biometrika 38:196–218, 1951). However, environments for living organisms are often fragmented by natural or artificial habitat destruction. Here we address how such environmental heterogeneity affects the range expansion of invading species. We consider a single-species invasion in heterogeneous environments whose habitat parameters vary in a sinusoidal or quasi-sinusoidal manner. Accordingly, Fisher’s model is modified to make the intrinsic growth rate and diffusion coefficient spatially variable. By numerically solving the model, we examine the spatio-temporal pattern of propagating waves, and predict the speed as a function of the amplitude and the wave length of the diffusion coefficient and the intrinsic growth rate. Firstly, the results demonstrate that in the sinusoidally varying environment, if the intrinsic growth rate solely oscillates, the speed increases with increases in the amplitude of oscillation. Conversely, if the diffusion coefficient solely oscillates, the speed decreases with increases in the amplitude of oscillation. When both the intrinsic growth rate and diffusion coefficient oscillate, the speed is synergistically accelerated if the oscillations are in anti-phase, whereas it is decelerated if the oscillations are in same phase. Secondly, the increase in the wave length in either the intrinsic growth rate or the diffusion coefficient leads to decreases in the speed. Thirdly, in the irregularly varying environment, the irregularity in the amplitude of the intrinsic growth rate enhances the speed, while that of the diffusion coefficient attenuates the speed.     

On the roles of Ca2+ diffusion, Ca2+ buffers, and the endoplasmic reticulum in IP3-induced Ca2+ waves.

We have investigated the effects of Ca2+ diffusion, mobile and stationary Ca2+ buffers in the cytosol, and Ca2+ handling by the endoplasmic reticulum on inositol 1,4,5-trisphosphate-induced Ca2+ wave propagation. Rapid equilibration of free and bound Ca2+ is used to describe Ca2+ sequestration by buffers in both the cytosol and endoplasmic reticulum (ER) lumen. Cytosolic Ca2+ regulation is based on a kinetic model of the inositol 1,4,5-trisphosphate (IP3) receptor of De Young and Keizer that includes activation and inhibition of the IP3 receptor Ca2+ channel in the ER membrane and SERCA Ca2+ pumps in the ER. Diffusion of Ca2+ in the cytosol and the ER and the breakdown and diffusion of IP3 are also included in our calculations. Although Ca2+ diffusion is severely limited because of buffering, when conditions are chosen just below the threshold for Ca2+ oscillations, a pulse of IP3 or Ca2+ results in a solitary trigger wave that requires diffusion of Ca2+ for its propagation. In the oscillatory regime repetitive wave trains are observed, but for this type of wave neither the wave shape nor the speed is strongly dependent on the diffusion of Ca2+. Local phase differences lead to waves that are predominately kinematic in nature, so that the wave speed (c) is related to the wavelength (lambda) and the period of the oscillations (tau) approximately by the formula c = lambda/tau. The period is determined by features that control the oscillations, including [IP3] and pump activity, which are related to recent experiments. Both solitary waves and wave trains are accompanied by a Ca2+ depletion wave in the ER lumen, similar to that observed in cortical preparations from sea urchin eggs. We explore the effect of endogenous and exogenous Ca2+ buffers on wave speed and wave shape, which can be explained in terms of three distinct effects of buffering, and show that exogenous buffers or Ca2+ dyes can have considerable influence on the amplitude and width of the waves.     

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.     

Relationships between amplitudes and kinetics of rapid cytosolic free calcium fluctuations in GH4C1 rat pituitary cells: roles for diffusion and calcium-induced calcium release.

We have examined statistical relationships between the amplitudes and the kinetics (rise times, fall times, and decay constants) of cytosolic free calcium fluctuations (spikes) in a population of 353 individual GH4C1 rat pituitary cells. The fast falling phase was approximated by a single exponential decay, and the decay time constant, tau, increased linearly with spike amplitude in 80% of the cells studied. The slope of the tau versus amplitude plot for each cell was inversely related to the cell's mean spike amplitude. Thus, some process responsible for prolonging the decay phase of spikes appeared to operate strongly in cells with spikes of low amplitude, but to become less prominent in cells with high amplitude spikes. Mean tau correlated more strongly with mean rise and fall times than with mean spike amplitude, indicating that the kinetic properties of spikes were not tightly coupled to spike amplitude. These findings are consistent with a model wherein the rise phase corresponds to entry of extracellular calcium via L-type calcium channels into localized sub-plasmalemmal domains, followed by diffusion of subplasmalemmal calcium into the cell interior; and the falling phase corresponds to further calcium diffusion combined with activation of cytoplasmic calcium-induced calcium release, which prolongs the falling phase.     

Generation of fast charged particles in plasma by a wave with a stochastically jumping phase

The interaction between charged plasma particles and an electromagnetic wave with a stochastic jumping phase is analyzed by numerical simulations. It is demonstrated that, in the course of interaction, the particle energy can increase by more than one order of magnitude. Optimal conditions for efficient interaction of charged plasma particles with a wave having a stochastically jumping phase are determined. According to the simulation results, substantial acceleration of charged plasma particles by a wave with a stochastically jumping phase takes place both at fixed time intervals between phase jumps and when these intervals are random. The influence of the wave parameters, such as the wave amplitude, the characteristic time interval between phase jumps, and the characteristic magnitude of these jumps, on the acceleration dynamics is analyzed.     

Ponderomotive Self-focusing of Surface Plasma Wave

A large amplitude surface plasma wave (SPW) propagating over a conductor–vaccum interface with Gaussian intensity profile transverse to the direction of propagation ( $ \widehat{z} $ ) and surface normal ( $ \widehat{x} $ ) is shown to undergo periodic self-focusing due to ponderomotive nonlinearity. The ponderomotive force on electrons arises due to the rapid decline in surface wave amplitude with the depth inside the conductor. In case of plasma, this leads to ambipolar diffusion of plasma, whereas in metals, only electron displacement occurs until the space charge balances the ponderomotive force on electrons. For a surface plasma wave, having Gaussian amplitude profile in y, the maximum electron density depression occurs on the axis (y?=?0) and the effect weakens as |y| increases. The axial portion of SPW thus travels with slower phase velocity than the nonaxial portion leading to self-focusing.     

Evolution of the optimum bidirectional (+/- biphasic) wave for defibrillation

Introduction of the asymmetric bidirectional (+/- biphasic) current waveform has made it possible to achieve ventricular defibrillation with less energy and current than are needed with a unidirectional (monophasic) waveform. The symmetrical bidirectional (sinusoidal) waveform was used for the first human-heart defibrillation. Subsequent studies employed the underdamped and overdamped sine waves, then the trapezoidal (monophasic) wave. Studies were then undertaken to investigate the benefit of adding a second identical and inverted wave; little success rewarded these efforts until it was discovered that the second inverted wave needed to be much less in amplitude to lower the threshold for defibrillation. However, there is no physiologic theory that explains the mechanism of action of the bidirectional wave, nor does any theory predict the optimum amplitude and time dimensions for the second inverted wave. The authors analyze the research that shows that the threshold defibrillation energy is lowest when the charge in the second, inverted phase is slightly more than a third of that in the first phase. An ion-flux, spatial-K+ summation hypothesis is presented that shows the effect on myocardial cells of adding the second inverted current pulse.     

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.     

An ultrasound-aided study of temporal relationships between the patterns of LH/FSH secretion, development of ovulatory-sized antral follicles and formation of corpora lutea in ewes

To characterize the pulsatile secretion of LH and FSH and their relationships with various stages of follicular wave development (follicles growing from 3 to > or =5 mm) and formation of corpora lutea (CL), 6 Western white-faced ewes underwent ovarian ultrasonography and intensive blood sampling (every 12 min for 6 h) each day, for 10 and 8 consecutive days, commencing 1 and 2 d after estrus, respectively. Basal serum concentrations of LH and LH pulse frequency declined, whereas LH pulse duration and FSH pulse frequency increased by Day 7 after ovulation (P<0.05). LH pulse amplitude increased (P<0.05) at the end of the growth phase of the largest ovarian follicles in the first follicular wave of the cycle. The amplitude and duration of LH pulses rose (P<0.05) 1 d after CL detection. Mean and basal serum FSH concentrations increased (P<0.05) on the day of emergence of the second follicular wave, and also at the beginning of the static phase of the largest ovarian follicles in the first follicular wave of the cycle. FSH pulse frequency increased (P<0.05) during the growth phase of emergent follicles in the second follicle wave. The detection of CL was associated with a transient decrease in mean and basal serum concentrations of FSH (P<0.05), and it was followed by a transient decline in FSH pulse frequency (P<0.05). These results indicate that LH secretion during the luteal phase of the sheep estrous cycle reflects primarily the stage of development of the CL, and only a rise in LH pulse amplitude may be linked to the end of the growth phase of the largest follicles of waves. Increases in mean and basal serum concentrations of FSH are tightly coupled with the days of follicular wave emergence, and they also coincide with the end of the growth phase of the largest follicles in a previous wave, but FSH pulse frequency increases during the follicle growth phase, especially at mid-cycle.     

Cytogenetic damage in human lymphocytes following GMSK phase modulated microwave exposure.

The present study investigated, using in vitro experiments on human lymphocytes, whether exposure to a microwave frequency used for mobile communication, either unmodulated or in presence of phase only modulation, can cause modification of cell proliferation kinetics and/or genotoxic effects, by evaluating the cytokinesis block proliferation index and the micronucleus frequency. In the GSM 1800 mobile communication systems the field is both phase (Gaussian minimum shift keying, GMSK) and amplitude (time domain multiple access, TDMA) modulated. The present study investigated only the effects of phase modulation, and no amplitude modulation was applied. Human peripheral blood cultures were exposed to 1.748 GHz, either continuous wave (CW) or phase only modulated wave (GMSK), for 15 min. The maximum specific absorption rate (approximately 5 W/kg) was higher than that occurring in the head of mobile phone users; however, no changes were found in cell proliferation kinetics after exposure to either CW or GMSK fields. As far as genotoxicity is concerned, the micronucleus frequency result was not affected by CW exposure; however, a statistically significant micronucleus effect was found following exposure to phase modulated field. These results would suggest a genotoxic power of the phase modulation per se.     

Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

The effect of spatial scale of resistive inhomogeneity on defibrillation of cardiac tissue

Defibrillation of cardiac tissue can be viewed in the context of dynamical systems theory as the attempt to move a dynamical system from the basin of attraction of one attractor (fibrillation) to another (the uniform rest state) by applying a stimulus whose form is physically constrained. Here we give an introduction to the physical mechanism of cardiac defibrillation from this dynamical perspective and examine the role of resistive inhomogeneity on defibrillation efficacy. Using numerical simulations with rotating waves on a one-dimensional periodic ring, we study the role of the spatial scale of resistive inhomogeneity on defibrillation. For a rotating wave on a periodic ring there are three stable attractors, namely the uniform rest state, a wave traveling clockwise and a wave traveling counterclockwise. As a result, the application of a stimulus has the potential for three different outcomes, namely elimination of the wave, phase resetting of the wave, and reversal of the wave. The results presented here show that with resistive inhomogeneities of large spatial scale, all three of these transitions are possible with large amplitude shocks, so that the probability of defibrillation is bounded well below one, independent of stimulus amplitude. On the other hand, resistive inhomogeneities of small spatial scale produce a defibrillation threshold that is qualitatively consistent with that found experimentally, namely the probability of defibrillation success is an increasing function that approaches one for large enough stimulus amplitude. Extending these results to higher dimensions, we describe conditions for successful defibrillation of functional reentry with large scale spatial inhomogeneity, but find that elimination of anatomical reentry is quite difficult. With small spatial scale inhomogeneity, there are no similar restrictions.     

Numerical simulation and graphical analysis of in vitro benign tumor growth: application of single-particle state bosonic matter equation with length scaling

We describe the application of a non-linear single-particle state bosonic condensate equation to simulate multicellular tumor growth by treating it as a coupling of two classical wave equations with real components. With one component representing the amplitude of the cells in their volume growth phase and the other representing the amplitude of the cells in their proliferation or mitosis phase, the two components of the coupled equation feed each other during the time evolution and are coupled together through diffusion and other linear and non-linear terms. The features of quiescent and necrotic cells, which result from poor nutrient diffusion into a tumor, have been found to correspond quite well to experimental data when they are modeled as depending on higher cell density. Classical hallmarks of benign tumor growth, such as the initial rapid growth, followed by a dramatic collapse in the proliferating cell count and a strong re-growth thereafter appear quite encouragingly in the theoretical results. A tool for graphical analysis of the tumor simulation results has been developed to provide morphological information about tumors at various growth stages. The model and the graphical analysis can be extended further to create an effective tool to predict/monitor tumor growth. 1 Screen shot from the graphical analysis tool showing simulation results after ten days: clustering of cells of the tumor (up); cell density profile (down) Dedicated to Professor Dr. Paul von Ragué Schleyer on the occasion of his 75th birthday     

Time-dependent measurement of differential digital plethysmogram in bicycle ergometer exercise

Time-dependent measurements of differential digital plethysmogram during exercise were made on five male subjects. The results obtained were as follows; Differential digital plethysmogram (delta DPG) was obtained by using biophysical amplifier with the time constant of 0.03 seconds which minimized the fluctuation of the baseline in digital plethysmogram (DPG). A linear relationship was shown in P wave amplitude of both delta DPG and DPG. The decrease in delta DPG-P waves was observed in visual concentrations, mental learning and arithmetic, initial inspiratory phase with tachycardia, maximal inspiratory and/or expiratory breath holding, and head-up tilt at 60 degrees or over. The increase in delta DPG-P waves was obtained at the expiratory phase with bradycardia and in the effect of alcohol intake. During 15 minutes of bicycle ergometer exercise at 750 kpm/min, the P wave amplitude in delta DPG decreased to 77% of the control in the first one minute of exercise, and then gradually increased to 218% at the final stage of exercise (p less than 0.01). Heart rate measured simultaneously increased, as compared from the beginning of exercise. P wave amplitude and heart rate after exercise decreased progressively to the control level. It is suggested that the initial decrease in P wave amplitude of delta DPG couples with the dominant activity of the sympathetic vasoconstrictor, and the final increase in P waves is due to the compound factors of the increased cardiac output and arteriolar vasodilation.     

Trapping of high-energy electrons into regime of surfatron acceleration by electromagnetic waves in space plasma

The phenomenon of trapping of weakly relativistic charged particles (with kinetic energies on the order of mc ²) into a regime of surfatron acceleration by an electromagnetic wave that propagates in plasma across a weak external magnetic field has been studied using nonlinear numerical calculations based on a solution of the relativistic equations of motion. Analysis showed that, for the wave amplitude above a certain threshold value and the initial wave phase outside the interval favorable for the surfing regime, the trajectory of a charged particle initially corresponds to its cyclotron rotation in the external magnetic field. For the initial particle energies studied, the period of this rotation is relatively short. After a certain number (from several dozen to several thousand and above) of periods of rotation, the wave phase takes a value that is favorable for trapping of the charged particle on its trajectory by the electromagnetic wave, provided the Cherenkov resonance conditions are satisfied. As a result, the wave traps the charged particle and imparts it an ultrarelativistic acceleration. In momentum space, the region of trapping into the regime of surfing on an electromagnetic wave turns out to be rather large.     

The influence of the reference electrode location on the M−wave characteristics in the quadriceps

IntroductionIn the compound muscle action potential (M wave) recorded using the belly-tendon configuration, the contribution of the tendon electrode is assumed to be negligible compared to the belly electrode. We tested this assumption by placing the reference electrode at a distant (contralateral) site, which allowed separate recording of the belly and tendon contributions.MethodsM waves were recorded at multiple selected sites over the right quadriceps heads and lower leg using two different locations for the reference electrode: the ipsilateral (right) and contralateral (left) patellar tendon. The general parameters of the M wave (amplitude, area, duration, latency, and frequency) were measured.Results(1) The tendon potential had a small amplitude (<30%) compared to the belly potential; (2) Changing the reference electrode from the ipsilateral to the contralateral patella produced moderate changes in the M wave recorded over the innervation zone, these changes affecting significantly the amplitude of the M−wave second phase (p = 0.006); (3) Using the contralateral reference system allowed recording of short-latency components occurring immediately after the stimulus artefact, which had the same latency and amplitude (p = 0.18 and 0.25, respectively) at all recording sites over the leg.ConclusionsThe potential recorded at the “tendon” site after femoral nerve stimulation is small (compared to the belly potential), but not negligible, and makes a significant contribution to the second phase of belly-tendon M wave. Adopting a distant (contralateral) reference allowed recording of far-field components that may aid in the understanding of the electrical formation of the M wave.     

Agonist-induced calcium waves in oscillatory cells: A biological example of Burgers' equation

Oscillations in cytosolic Ca²⁺ concentrations in living cells are often a manifestation of propagating waves of Ca²⁺. Numerical simulations with a realistic model of inositol 1, 4, 5-trisphosphate (IP₃)-induced Ca²⁺ wave trains lead to wave speeds that increase linearly at long times when (a) IP₃ levels are in the range for Ca²⁺ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP₃, and (c) IP₃ concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP₃ receptor that includes the influence of Ca²⁺ buffers. For concentrations of IP₃ in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca²⁺ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca²⁺. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca²⁺. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.     

Multi-species interactions with hereditary effects and spatial diffusion

Summary We study the effect of spatial diffusion on oscillatory states in arbitrary multi-species growth models having hereditary terms. We show that it is a general principle that the addition of spatial diffusion to a stable oscillatory ecological community induces a periodic diffusion wave in which the original wavenumber (or phase) evolves according to a nonlinear evolution equation of generalized Burgers' type.Supported in part by the U.S. Army Research Office (Durham) under Contract DAHC-04-68-C-0006 and by the National Science Foundation under Grant GP-32157X2     

Effect of the ponderomotive force on the development of beam-plasma instability

The amplitude of the wave generated in a plasma during the development of beam-plasma instability is nonuniform in the longitudinal direction. The ponderomotive force associated with this nonuniformity leads to a redistribution of the plasma density; as a result, the wave amplitude and its spatial distribution change. As the beam current grows, the ponderomotive force plays an increasingly important role and radically changes the mechanism by which the beam-plasma instability saturates. Ion acoustic waves generated by the ponderomotive force propagate in the direction opposite to the propagation direction of the beam, thereby ensuring distributed feedback and giving rise to a strong low-frequency self-modulation of the wave amplitude and phase. Results are presented from experimental investigations of the self-modulation regime of the beam-plasma instability in a magnetized plasma waveguide. Theoretical estimates of the parameters of the low-frequency self-modulation agree well with the experimental data.     

Quasi-elastic laser light scattering from solutions and gels of hemoglobin S.

Quasi-elastic light scattering has been used to examine solutions and gels of deoxyhemoglobin S. The autocorrelation function is found to decay with a characteristic exponential relaxation which can be ascribed to the diffusion of monomer (64,000 molecular weight) hemoglobin S molecules. In the absence of polymers, the relaxation time is in good agreement with previous measurements of the diffusion coefficient for solutions of normal human hemoglobin. In the presence of the polymer phase, a large (greater than 200-fold) increase in the scattered intensity is observed but no contribution to the decay of the autocorrelation function from the motion of the aligned polymer phase can be detected. Heterodyning between the time-independent scattering amplitude from the polymers and the time-dependent scattering of the diffusing monomers results in a twofold increase in the relaxation time arising from monomer diffusion.