Traveling waves in yeast extract and in cultures of Dictyostelium discoideum

From pattern to process? Dual travelling waves, with contrasting propagation speeds, best describe a self-organised spatio-temporal pattern in population growth of a cyclic rodent

The dynamics of cyclic populations distributed in space result from the relative strength of synchronising influences and the limited dispersal of destabilising factors (activators and inhibitors), known to cause multi‐annual population cycles. However, while each of these have been well studied in isolation, there is limited empirical evidence of how the processes of synchronisation and activation–inhibition act together, largely owing to the scarcity of datasets with sufficient spatial and temporal scale and resolution. We assessed a variety of models that could be underlying the spatio‐temporal pattern, designed to capture both theoretical and empirical understandings of travelling waves using large‐scale (>35,000 km²), multi‐year (2011–2017) field monitoring data on abundances of common vole (Microtus arvalis), a cyclic agricultural rodent pest. We found most support for a pattern formed from the summation of two radial travelling waves with contrasting speeds that together describe population growth rates across the region.

"Ballistic" waves among chemically oscillating micromotors

Coordinating a group of chemically powered micromotors holds great importance in potential applications that involve a large population in a complex environment, yet information transmission at a population scale remains challenging. To this end, we demonstrate how propagating waves emerge among a population of spontaneously oscillating micromotors that dash toward a direction prescribed by their Janus orientations (termed a "ballistic" wave). Moreover, chemical communication among these micromotors enables the tuning of the speed and frequency of individual micromotors and their waves, by varying the population density or the viscosity of the medium.

Synthetic quorum sensing in model microcapsule colonies

Biological quorum sensing refers to the ability of cells to gauge their population density and collectively initiate a new behavior once a critical density is reached. Designing synthetic materials systems that exhibit quorum sensing-like behavior could enable the fabrication of devices with both self-recognition and self-regulating functionality. Herein, we develop models for a colony of synthetic microcapsules that communicate by producing and releasing signaling molecules. Production of the chemicals is regulated by a biomimetic negative feedback loop, the "repressilator" network. Through theory and simulation, we show that the chemical behavior of such capsules is sensitive to both the density and number of capsules in the colony. For example, decreasing the spacing between a fixed number of capsules can trigger a transition in chemical activity from the steady, repressed state to large-amplitude oscillations in chemical production. Alternatively, for a fixed density, an increase in the number of capsules in the colony can also promote a transition into the oscillatory state. This configuration-dependent behavior of the capsule colony exemplifies quorum-sensing behavior. Using our theoretical model, we predict the transitions from the steady state to oscillatory behavior as a function of the colony size and capsule density.

Extracellular and intracellular factors regulating the migration direction of a chemotactic cell in traveling-wave chemotaxis

This report presents a simple model that describes the motion of a single Dictyostelium discoideum cell exposed to a traveling wave of cyclic adenosine monophosphate (cAMP). The model incorporates two types of responses to stimulation by cAMP: the changes in the polarity and motility of the cell. The periodic change in motility is assumed to be induced by periodic cAMP stimulation on the basis of previous experimental studies. Consequently, the net migration of the cell occurs in a particular direction with respect to wave propagation, which explains the migration of D. discoideum cells in aggregation. The wave period and the difference between the two response times are important parameters that determine the direction of migration. The theoretical prediction compared with experiments presented in another study. The transition from the single-cell state of the population of D. discoideum cells to the aggregation state is understood to be a specific example of spontaneous breakage of symmetry in biology.

Spiral defect drift in the wave fields of multiple excitation patterns

Spiral waves in excitable systems decay to drifting defects if forced by high-frequency wave trains. Using the Barkley model we analyze the drift velocity in planar wave trains as a function of wave frequency. Within two antiparallel, planar wave trains of equal frequency a defect is pushed into the collision region where it stops. Within two circular wave fields, however, it continues its drift in a direction perpendicular to the axis connecting the pacemakers. Depending on the forcing frequency and the initial position, this motion occurs either away from or toward the pacemaker axis. Three circular wave fields can be used to position the defect at a unique point close to the center of the pacemaker triangle. The results are also observed in experiments with the Belousov-Zhabotinsky reaction.

Three-dimensional spiral waves in an excitable reaction system: initiation and dynamics of scroll rings and scroll ring pairs

We report experimental results on spiral and scroll waves in the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction. The propagating concentration waves are detected by two-dimensional photometry and optical tomography. Wave pulses can disappear in front-to-front and front-to-back collisions. This anomaly causes the nucleation of vortices from collisions of three nonrotating waves. In three-dimensional systems, these vortices are scroll rings that rotate around initially circular filaments. Depending on reactant concentrations, the filaments shrink or expand indicating positive and negative filament tensions, respectively. Shrinkage results in vortex annihilation. Expansion is accompanied by filament buckling and bending, which is interpreted as developing Winfree turbulence. We also describe the initiation of scroll ring pairs in four-wave collisions. The two filaments are stacked on top of each other and their motion suggests filament repulsion.

Wave-pinned filaments of scroll waves

Scroll waves are three-dimensional excitation patterns that rotate around one-dimensional space curves. Typically these filaments are closed loops or end at the system boundary. However, in excitable media with anomalous dispersion, filaments can be pinned to the wake of traveling wave pulses. This pinning is studied in experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction and a three-variable reaction-diffusion model. We show that wave-pinned filaments are related to the coexistence of rotating and translating wave defects in two dimensions. Filament pinning causes a continuous expansion of the total filament length. It can be ended by annihilating the pinning pulse in a frontal wave collision. Following such an annihilation, the filament connects itself to the system boundary. Its postannihilation shape that is initially the exposed rim of the scroll wave unwinds continuously over numerous rotation periods.

Scroll Wave Filaments Terminate in the Back of Traveling Fronts

Experiments with the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction demonstrate that three-dimensional scroll waves can rotate around filaments that end in the wake of a traveling excitation pulse. The vortex structures nucleate during the collision of three nonrotating excitation pulses. The nucleation process and the wave-termination of filaments are direct consequences of the system's anomalous dispersion relation. Vortex filaments are found to expand with about twice the speed of their anchoring wave fronts. Filament expansion is accompanied by the build-up of phase differences in spiral rotation creating strongly twisted wave structures. Experiments employ optical tomography for the reconstruction of the three-dimensional wave patterns.

Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry

A mathematical framework for modeling biological cells from a physicochemical perspective is described. Cells modeled within this framework consist of at least two regions, including a cytosolic volume encapsulated by a membrane surface. The cytosol is viewed as a well-stirred chemical reactor capable of changing volume while the membrane is assumed to be an oriented 2-D surface capable of changing surface area. Two physical properties of the cell, namely volume and surface area, are determined by (and determine) the reaction dynamics generated from a set of chemical reactions designed to be occurring in the cell. This framework allows the modeling of complex cellular behaviors, including self-replication. This capability is illustrated by constructing two self-replicating prototypical whole-cell models. One protocell was designed to be of minimal complexity; the other to incorporate a previously reported well-known mechanism of the eukaryotic cell cycle. In both cases, self-replicative behavior was achieved by seeking stable physically possible oscillations in concentrations and surface-to-volume ratio, and by synchronizing the period of such oscillations to the doubling of cytosolic volume and membrane surface area. Rather than being enforced externally or artificially, growth and division occur naturally as a consequence of the assumed chemical mechanism operating within the framework.

Synchronization in coupled cells with activator-inhibitor pathways

The functional dynamics exhibited by cell collectives are fascinating examples of robust, synchronized, collective behavior in spatially extended biological systems. To investigate the roles of local cellular dynamics and interaction strength in the spatiotemporal dynamics of cell collectives of different sizes, we study a model system consisting of a ring of coupled cells incorporating a three-step biochemical pathway of regulated activator-inhibitor reactions. The isolated individual cells display very complex dynamics as a result of the nonlinear interactions common in cellular processes. On coupling the cells to nearest neighbors, through diffusion of the pathway end product, the ring of cells yields a host of interesting and unusual dynamical features such as, suppression of chaos, phase synchronization, traveling waves, and intermittency, for varying interaction strengths and system sizes. But robust complete synchronization can be induced in these coupled cells with a small degree of random coupling among them even where regular coupling yielded only intermittent synchronization. Our studies indicate that robustness in synchronized functional dynamics in tissues and cell populations in nature can be ensured by a few transient random connections among the cells. Such connections are being discovered only recently in real cellular systems.

Simple and complex spatiotemporal structures in a glycolytic allosteric enzyme model

Pattern formation in glycolysis is studied with a classical reaction-diffusion allosteric enzyme model. It is found that, similar to recent experimental reports in the yeast extracts, a small magnitude local perturbation can induce transient target waves in a two dimensional oscillatory medium. An above threshold stimulation generates target waves which eventually evolve into spatiotemporal chaos upon collisions with the boundary or other wave activities. Detailed simulation studies show that the studied simple glycolytic reaction-diffusion model can support three types of spatiotemporal behaviors which are independent of the boundary conditions: (1) a spatially uniform stable steady state, (2) periodic global oscillations and (3) spatiotemporal chaos.

Singular perturbation analysis of cAMP signalling in Dictyostelium discoideum aggregates

In this paper, we use singular perturbation methods to study the structure of travelling waves for some reaction-diffusion models obtained from the Martiel-Goldbeter and Goldbeter-Segel's models of cAMP signalling in Dictyostelium discoideum. As a consequence, we derive analytic formulae for quantities like wave speed, maximum concentration and other magnitudes in terms of the different biochemical constants that appear in the model.

An elegant method to study an isolated spiral wave in a thin layer of a batch Belousov–Zhabotinsky reaction under oxygen-free conditions

A method to prepare a uniform thin layer of a batch Belousov-Zhabotinsky (BZ) reaction under oxygen-free conditions for the study of an isolated spiral wave is presented. After a first layer of gel soaked with the BZ solution has been delivered into the reactor, a single spiral wave was initiated, and finally the remaining reactor volume was filled with gel and BZ medium. The completely filled reactor is sealed gas-tightly, yielding oxygen-free, and thus more controlled, reaction conditions. A systematic study of the behaviour of an isolated spiral wave in a ferroin-catalyzed BZ reaction under batch conditions has been performed. Recipes for BZ media that support a slowly rotating meandering spiral were developed. In cases of extremely low excitability (i.e., relative large stimuli are required to induce a propagating wave), the number of petals in the trajectory of a spiral tip decreased due to aging of the reaction system. Since oxygen-free conditions are necessary for the study of the dynamics in three-dimensional excitable media, and the wave velocities of a spiral are sufficiently low, the developed chemical recipes are suitable for studies of the behaviour of scroll waves in three-dimensional systems by optical tomography.

Micropatterning Chemical Oscillations: Waves, Autofocusing, and Symmetry Breaking

Arrays of chemical oscillators are micropatterned by Wet Stamping. The technique is used to demonstrate that chemical waves can be initiated and controlled in oscillatory systems and that such waves can give rise to phenomena not observed in excitable media. Interoscillator coupling and synchronization, kinetic autofocusing, and twist-symmetry breaking are a consequence of the dependence of the oscillation phase on the local concentrations of reagents and on systems' geometry. Conditions under which a generic oscillatory system would exhibit such behaviors are determined.

Front aggregation in multiarmed excitation vortices

Using the Belousov-Zhabotinsky reaction, we study the pinning of multiarmed spiral waves to nonexcitable obstacles. With increasing obstacle size, the individual arms switch from a repulsive to an attractive state. This transition yields densely aggregated spiral arms and is caused by anomalous dispersion. A kinematic model reproduces the measurements quantitatively and identifies the transition as a supercritical pitchfork bifurcation.

Synchronization of electrically induced calcium firings in self-assembled cardiac cells

We study the adaptive changes of a population of cells responding to external stimulus. Two-dimensionally distributed cardiac cells were homogeneously subjected to periodic electrical stimulus and intracellular calcium concentration ([Ca(2+)](i)) changes were simultaneously observed. In the absence of stimulation, coupled cells in monolayer formed groups of several cells oscillating in similar phase, while isolated cells showed irregular periodicity. In both systems, [Ca(2+)](i) oscillations were modulated by periodic stimulation, and ascending degrees of synchronization among [Ca(2+)](i) oscillations were shown as stimulation intensity increased. In a population of coupled cells, the cells act like a single robust oscillator. These results are evaluated using statistical calculations, comparing the response manner of isolated cells.

Voronoi diagrams generated by regressing edges of precipitation fronts

Reaction-diffusion systems where one of the reagents (outer electrolyte) penetrates into a gel by diffusion and forms a precipitate with the other reagent (inner electrolyte) homogenized in the gel, are able to produce various complex precipitation patterns. The previously studied NaOH + AgNO3 and recently discovered CuCl2 + K3[Fe(CN)6] processes, (where the first reagent is the outer electrolyte and the other is the inner electrolyte homogenized in the gel), when reacted using the above mentioned method, are able to generate tessellations of a plane by a mechanism dependant on the dynamics of so-called regressing edges of the reaction fronts. The spontaneous partitioning of the reacted phases results in the construction of a pattern analogous to a Voronoi diagram or one of their generalizations.

Long-range signal transmission in autocrine relays

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.

Eukaryotic cell locomotion depends on the propagation of self-organized reaction-diffusion waves and oscillations of actin filament assembly

Actin filament (F-actin) assembly kinetics determines the locomotion and shape of crawling eukaryotic cells, but the nature of these kinetics and their determining reactions are unclear. Live BHK21 fibroblasts, mouse melanoma cells, and Dictyostelium amoebae, locomoting on glass and expressing Green Fluorescent Protein-actin fusion proteins, were examined by confocal microscopy. The cells demonstrated three-dimensional bands of F-actin, which propagated throughout the cytoplasm at rates usually ranging between 2 and 5 microm/min in each cell type and produced lamellipodia or pseudopodia at the cell boundary. F-actin's dynamic behavior and supramolecular spatial patterns resembled in detail self-organized chemical waves in dissipative, physico-chemical systems. On this basis, the present observations provide the first evidence of self-organized, and probably autocatalytic, chemical reaction-diffusion waves of reversible actin filament assembly in vertebrate cells and a comprehensive record of wave and locomotory dynamics in vegetative-stage Dictyostelium cells. The intensity and frequency of F-actin wavefronts determine locomotory cell projections and the rotating oscillatory waves, which structure the cell surface. F-actin assembly waves thus provide a fundamental, deterministic, and nonlinear mechanism of cell locomotion and shape, which complements mechanisms based exclusively on stochastic molecular reaction kinetics.

Shock structures and bunching fronts in excitable reaction-diffusion systems

We report experimental results on the dynamics of excitation waves in a modified Belousov-Zhabotinsky reaction. The waves in this system obey nonmonotonic dispersion relations. This anomaly induces the stacking of excitation fronts into patterns with stable interpulse distances. The stacking process creates either a traveling shock structure or a cascade of bunching events in which metastable wave packets are formed. The direction and the speed of the shock are explained in terms of a simple geometrical analysis. We also present experimental evidence for the corresponding instabilities in two-dimensional systems. Here, wave stacking generates atypical structures in the collision of target patterns and wave bunching is accompanied by complex front deformations.

Influence of an external electric field on cAMP wave patterns in aggregating Dictyostelium discoideum

Effects of an external electric field on the propagation of excitable cyclic adenosin monophosphate (cAMP) pulses in aggregating cells of Dictyostelium discoideum have been studied by means of numerical simulations. The mathematical model includes a three-variable version of the Martiel-Goldbeter kinetic model of cAMP production by cells, and diffusion and electromigration of cAMP through intercellular space. Electric field effects on the planar pulses propagating in a spatially one-dimensional system and on circular and spiral pulses propagating in a spatially two-dimensional system include changes in the propagation velocities of pulses initiated in the field-free system, the annihilation of pulses propagating towards the cathode by electric fields of high intensities, and the formation of region(s) of spontaneous excitation permanently generating trains of cAMP pulses. The spiral patterns were found to drift in the electric fields of low intensities, fields of high intensities were found to decompose the spiral arms on the side facing the negative electrode.

Dissipative metabolic patterns respond during neutrophil transmembrane signaling

Self-organization is a common theme in biology. One mechanism of self-organization is the creation of chemical patterns by the diffusion of chemical reactants and their nonlinear interactions. We have recently observed sustained unidirectional traveling chemical redox [NAD(P)H - NAD(P)(+)] waves within living polarized neutrophils. The present study shows that an intracellular metabolic wave responds to formyl peptide receptor agonists, but not antagonists, by splitting into two waves traveling in opposite directions along a cell's long axis. Similar effects were noted with other neutrophil-activating substances. Moreover, when cells were exposed to an N-formyl-methionyl-leucyl-phenylalanine (FMLP) gradient whose source was perpendicular to the cell's long axis, cell metabolism was locally perturbed with reorientation of the pattern in a direction perpendicular to the initial cellular axis. Thus, extracellular activating signals and the signals' spatial cues are translated into distinct intracellular dissipative structures.

Reaction-diffusion waves of actin filament polymerization/depolymerization in Dictyostelium pseudopodium extension and cell locomotion

Cell surface movements and the intracellular spatial patterns and dynamics of actin filament (F-actin) were investigated in living and formalin-fixed cells of Dictyostelium discoideum by confocal microscopy. Excitation waves of F-actin assembly developed and propagated several micrometers at up to 26 microm/min in cells which had been intracellularly loaded with fluorescently labeled actin monomer. Wave propagation and extinction corresponded with the initiation and attenuation of pseudopodium extension and cell advance, respectively. The identification of chemical waves was supported by the ring, sphere, spiral and scroll wave patterns, which were observed in the extensions of fixed cells stained with phalloidin-rhodamine, and by the similar, asymmetrical [F-actin] distribution in wavefronts in living and fixed cells. These F-actin patterns and dynamics in Dictyostelium provide evidence for a new supramolecular state of actin, which propagates as a self-organized, reaction-diffusion wave of reversible F-actin assembly and affects pseudopodium extension. Actin's properties of oscillation and self-organization might also fundamentally determine the nature of the eukaryotic cell's reactions of adaptation, timing and signal response.

Imaging sustained dissipative patterns in the metabolism of individual living cells

Theoretical studies have predicted spatiotemporal organization of cell metabolism. Using a rapidly gated CCD camera, we demonstrate for the first time sustained traveling waves of NAD(P)H autofluorescence and protons in individual morphologically polarized living cells. Chemical concentration fronts moved in the direction of cell orientation, thus correlating dissipative structures with cell shape.

Protein-assisted pericyclic reactions: an alternate hypothesis for the action of quantal receptors

The rules for allowable pericyclic reactions indicate that the photoisomerizations of retinals in rhodopsins can be formally analogous to thermally promoted Diels-Alder condensations of monoenes with retinols. With little change in the seven-transmembrane helical environment these latter reactions could mimic the retinal isomerization while providing highly sensitive chemical reception. In this way archaic progenitors of G-protein-coupled chemical quantal receptors such as those for pheromones might have been evolutionarily plagiarized from the photon quantal receptor, rhodopsin, or vice versa. We investigated whether the known structure of bacteriorhodopsin exhibited any similarity in its active site with those of the two known antibody catalysts of Diels-Alder reactions and that of the photoactive yellow protein. A remarkable three-dimensional motif of aromatic side chains emerged in all four proteins despite the drastic differences in backbone structure. Molecular orbital calculations supported the possibility of transient pericyclic reactions as part of the isomerization-signal transduction mechanisms in both bacteriorhodopsin and the photoactive yellow protein. It appears that reactions in all four of the proteins investigated may be biological analogs of the organic chemists' chiral auxiliary-aided Diels-Alder reactions. Thus the light receptor and the chemical receptor subfamilies of the heptahelical receptor family may have been unified at one time by underlying pericyclic chemistry.