We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

The purpose of the research is to evaluate the effect of albendazole as part of the supramolecular complex with disodium salt of glycyrrhizic acid obtained by solid-phase mechanical treatment.Materials and methods. The anthelmintic activity of the complex and its effect on the host organism was assessed on hamsters infected with Opisthorchis felineus by single and 7-fold administration at 45 days after infection. After 21 days, we counted the number of helminthes in the liver, and conducted a morphometric analysis of the liver and spleen, and detected biochemically the activity of alanine aminotransferase and aspartate aminotransferase in the animals’ blood serum.Results and discussion. The number of O. felineus significantly decreased after 7-fold, but not a single, administration of albendazole (ABZ) and ABZ-Na2GA complex (1 : 10). The administrated substances had no effect on the weight gain of the animals and the daily consumption of the pellets. At the same time, ABZ only as part of the complex normalized the weight of the liver and spleen in hamsters infected with O. felineus and reduced the alanine aminotransferase activity. Consequently, a longer administration of ABZ as part of the complex with disodium glycyrrhizinate has not only a pronounced anthelmintic effect, but also improves some of the physiological parameters of hamsters to a greater extent than a pure substance.

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.

Migrating cells need to coordinate distinct leading and trailing edge dynamics but the underlying mechanisms are unclear. Here, we combine experiments and mathematical modeling to elaborate the minimal autonomous biochemical machinery necessary and sufficient for this dynamic coordination and cell movement. RhoA activates Rac1 via DIA and inhibits Rac1 via ROCK, while Rac1 inhibits RhoA through PAK. Our data suggest that in motile, polarized cells, RhoA–ROCK interactions prevail at the rear, whereas RhoA-DIA interactions dominate at the front where Rac1/Rho oscillations drive protrusions and retractions. At the rear, high RhoA and low Rac1 activities are maintained until a wave of oscillatory GTPase activities from the cell front reaches the rear, inducing transient GTPase oscillations and RhoA activity spikes. After the rear retracts, the initial GTPase pattern resumes. Our findings show how periodic, propagating GTPase waves coordinate distinct GTPase patterns at the leading and trailing edge dynamics in moving cells.

Interleukin-1 beta (IL-1β) is an indicator of liver infl ammatory processes in opisthorchiasis. Administration of curcumin and new complex of praziquantel with disodium glycyrrhizin ate reduced the parasitic load in O. felineus-infected animals. Moreover, the complex prevented the development of pathomorphological changes and reduced the IL-1β expression in the liver.

Background Nowadays, it is still important to develop effective anti-opisthorchiasis agents. In this work, we tested a complex of praziquantel (PZQ) with a plant origin compound-disodium glycyrrhizinate-in the ratio 1:10 PZQ:Na(2)GA, containing 11-fold less of the active ingredient. Our aim was to study various ways to treat trematode Opisthorchis felineus with this complex in vitro. Additionally, an in vitro comparison of the anthelmintic action was made among racemic-PZQ, (R)-PZQ, and (S)-PZQ on juvenile and adult maritae of O. felineus. Methods Worms extracted from the hamsters were subjected to various regimens of administration of the complex: once a day for 3 days or three times within 1 day. Moreover, mature maritae and juvenile worms of O. felineus were subjected to the comparison the anthelmintic effectiveness of racemic-PZQ, (R)-PZQ, and (S)-PZQ. Results The O. felineus maritae that received PZQ:Na(2)GA (1:10) thrice within 1 day were most strongly affected by the drug. Their motility substantially decreased already on the second day after the last dose, and the percentage of live worms by the end of the experimental period was the lowest. These results indicate a cumulative anthelmintic effect of this substance under the regimen "three times within 1 day." For the first time, we report that among the three substances (racemic-PZQ and two enantiomers), (R)-PZQ has the highest anthelmintic activity, toward both juvenile and sexually mature maritae of O. felineus. Conclusion These findings suggest that the development of a supramolecular complex of (R)-PZQ with disodium glycyrrhizinate and administration of this complex three times within 1 day are promising approaches.

We consider a piecewise linear approximation of the diffusive Morris-Lecar model of neuronal activity, the Tonnelier-Gerstner model. Exact analytical solutions for one-dimensional excitation waves are derived. The dynamics of traveling waves is related to two basic regimes of wave propagation: excitable and oscillatory cases. In the first case we describe mathematically the structure of a solitary pulse and in the second case—the form of a periodic sequence of pulses (a periodic wave train).

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes. In the first case, we find fronts that display a monotonic profile of the kink type, whereas in the second case the oscillations give rise to fronts with wavy tails. Further, the oscillations may be damped with exponential decay or undamped so that a saw-shaped pattern forms. Finally, we observe an unexpected feature in the behavior of both types of the oscillatory waves: the coexistence of several fronts with different profile shapes and propagation speeds for the same parameter values of the model, i.e., a multifront regime of wave propagation.

The mechanochemical preparation of solid compositions of praziquantel with plant saponin (glycyrrhizic acid disodium salt) is described. The study of a number of physicochemical parameters showed that dissolving solid compositions in water is accompanied by the inclusion of praziquantel molecules into micelles, which are formed in the solution of the glycyrrhizic acid disodium salt. Using the opisthorchiasis model caused by Opisthorchis felineus, we found a 4- to 11-fold increase in the anthelmintic activity of praziquantel in the composition as compared to the official praziquantel. According to the pharmacokinetic data, the use of the composition increased the bioavailability of praziquantel 3 times.

We explore traveling waves with oscillatory tails in a bistable piecewise linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross diffusion. These waves differ fundamentally from the standard simple fronts of the kink type. In contrast to kinks, the waves studied here have a complex shape profile with a front-back-front (a pulse-front) pattern. The characteristic feature of such pulse-front waves is a hybrid type of the speed diagram, which on the one hand reflects the typical dynamical behavior of the fronts in the FitzHugh-Nagumo model, related to the nonequilibrium Ising-Bloch bifurcation, and on the other hand exhibits also the solitary pulse scenario where several waves appear simultaneously with different speeds of propagation. We describe analytically the wave profiles and heteroclinic trajectories in the phase plane and discuss their morphology and transformation. The phenomena of wave formation and propagation are also studied by numerical simulations of the model partial differential equations. These simulations support the view that the pulse-front waves are constructed of fronts and pulses.

Opisthorchiasis is a dangerous parasitic disease caused by trematodes in the family Opisthorchiidae. One of the causes of this infection is the species Opisthorchis felineus, which is common in the Russian Federation and Western Europe. The disease has a large number of complications and relatively few effective treatments, so nowadays it is relevant to look for new drugs for the treatment of opisthorchiasis, with the maximum antiparasitic and minimal side effect. In this work, a potentially anthelmintic effect of the methanol extract of the golden chanterelle mushroom (Cantharellus cibarius) was investigated. In in vitro experiments, the significantly reduced mobility and survival rates of juvenile O. felineus specimens with increasing concentrations (10-1000 pg/ml) of the C. cibarius extract were shown. In in vivo studies, administration of the C. cibarius extract on the first day after parasitic infection of inbred C57BL/6 mice resulted in a decrease of the number of helminths in the bile ducts of the liver, evaluated 6 weeks after infection. In another series of experiments, administration of the C. cibarius extract for 7 days to mice infected with O. felineus for five weeks had no anthelmintic effect. In both cases, the state of the infected hosts, evaluated by a number of physiological and biochemical parameters (relative weight of organs, blood indices), did not deteriorate, indicating that there was no adverse effect of the C. cibarius extract. The results obtained suggest that the C. cibarius extract might have anthelmintic properties if applied as parasite larvae excyst.

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, "excitable" systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems.

Dynamic interactions between RhoA and Rac1, members of the Rho small GTPase family, play a vital role in the control of cell migration. Using predictive mathematical modeling, mass spectrometry-based quantitation of network components, and experimental validation in MDA-MB-231 mesenchymal breast cancer cells, we show that a network containing Rac1, RhoA, and PAK family kinases can produce bistable, switch-like responses to a graded PAK inhibition. Using a small chemical inhibitor of PAK, we demonstrate that cellular RhoA and Rac1 activation levels respond in a history-dependent, bistable manner to PAK inhibition. Consequently, we show that downstream signaling, actin dynamics, and cell migration also behave in a bistable fashion, displaying switches and hysteresis in response to PAK inhibition. Our results demonstrate that PAK is a critical component in the Rac1-RhoA inhibitory crosstalk that governs bistable GTPase activity, cell morphology, and cell migration switches.

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reactiondiffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system. We completely describe the dynamics of wavefronts and discuss the properties of the speed equation. The nonequilibrium Ising-Bloch bifurcation of traveling fronts is found to occur in the hyperbolic case as well as in the parabolic system. Waves in the hyperbolic case typically propagate with lower speeds, in absolute value, than waves in the parabolic one. We find the interesting feature that the hyperbolic and parabolic front trajectories coincide in the phase plane for the FitzHugh-Nagumo model with a diagonal diffusion matrix, which is the case of self-diffusion, and differ for the system with cross diffusion.

We consider principal properties of various wave regimes in two selected excitable systems with linear cross diffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multienvelope waves, and intermediate regimes appearing as waves propagating at a fixed shape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the ``quasisoliton'' property of reflection of boundaries and penetration through each other. We also present some examples of the behavior of envelope quasisolitons in two spatial dimensions.

Formation and interaction of the one-dimensional excitation waves in a reaction-diffusion system with the piecewise linear reaction functions of the Tonnelier-Gerstner type are studied. We show that there exists a parameter region where the established regime of wave propagation depends on initial conditions. Wave phenomena with a complex behavior are found: (i) the reflection of waves at a growing distance (the remote reflection) upon their collision with each other or with no-flux boundaries and (ii) the periodic transformation of waves with the jumping from one regime of wave propagation to another (the periodic trigger wave).

A network of the Rho family GTPases, which cycle between inactive GDP-bound and active GTP-bound states, controls key cellular processes, including proliferation and migration. Activating and deactivating GTPase transitions are controlled by guanine nucleotide exchange factors (GEFs), GTPase activating proteins (GAPs) and GDP dissociation inhibitors (GDIs) that sequester GTPases from the membrane to the cytoplasm. Here we show that a cascade of two Rho family GTPases, RhoA and Rac1, regulated by RhoGDI1, exhibits distinct modes of the dynamic behavior, including abrupt, bistable switches, excitable overshoot transitions and oscillations. The RhoGDI1 abundance and signal-induced changes in the RhoGDI1 affinity for GTPases control these different dynamics, enabling transitions from a single stable steady state to bistability, to excitable pulses and to sustained oscillations of GTPase activities. These RhoGDI1-controlled dynamic modes of RhoA and Rac1 activities form the basis of cell migration behaviors, including protrusion-retraction cycles at the leading edge of migrating cells.

Small monomeric G-proteins control cellular behavior, cycling between inactive GDP-bound and active GTP-bound states. Activating and deactivating transitions are regulated by guanine nucleotide exchange factors (GEFs) and GTPase activating proteins (GAPs), respectively. G-proteins can control different GEF and GAP activities, thereby creating GTPase signaling cascades. Here, we characterize all 128 different wiring topologies of two-layer cascades, which include feedforward/feedback interactions and an auto-regulatory loop. Exclusion of “mirror” designs leaves 64 topologies, which are classified into eight groups. We demonstrate that eight different cascades in each group generate the same number of steady states and similar spatiotemporal dynamics. Two groups (featuring 16 topologies) can generate three distinct dynamics: (i) bistable switches, (ii) excitable behavior, and (iii) sustained oscillations, giving rise to propagating waves of G-protein activation switches and pulses. Four other groups can produce switch-like, bistable behaviors and trigger waves. The remaining two groups have a single steady state. This first, complete classification of all possible interaction circuitries systematically links topological design to the spatiotemporal dynamics of G-protein cascades, predicting and explaining experimentally observed behavior.