Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection.

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.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains.

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.

Periodic propagating waves coordinate RhoGTPase network dynamics at the leading and trailing edges during cell migration.

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.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type.

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.

Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion.

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.

Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion.

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.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion.

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.

Bistability in the Rac1, PAK, and RhoA Signaling Network Drives Actin Cytoskeleton Dynamics and Cell Motility Switches.

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.

Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion.

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion 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.

Control of the G-protein cascade dynamics by GDP dissociation inhibitors.

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.

The topology design principles that determine the spatiotemporal dynamics of G-protein cascades.

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.

Long-range signaling by phosphoprotein waves arising from bistability in protein kinase cascades.

A hallmark of protein kinase/phosphatase cascades, including mitogen-activated protein kinase (MAPK) pathways, is the spatial separation of their components within cells. The top-level kinase, MAP3K, is phosphorylated at the cell membrane, and cytoplasmic kinases at sequential downstream levels (MAP2K and MAPK) spread the signal to distant targets. Given measured protein diffusivity and phosphatase activities, signal propagation by diffusion would result in a steep decline of MAP2K activity and low bisphosphorylated MAPK (ppMAPK) levels near the nucleus, especially in large cells, such as oocytes. Here, we show that bistability in a two-site MAPK (de)phosphorylation cycle generates a novel type of phosphoprotein wave that propagates from the surface deep into the cell interior. Positive feedback from ppMAPK to cytoplasmic MAP2K enhances the propagation span of the ppMAPK wave, making it possible to convey phosphorylation signals over exceedingly long distances. The finding of phosphorylation waves traveling with constant amplitude and high velocity may solve a long-standing enigma of survival signaling in developing neurons.