Biological soliton in multicellular movement

Structure formation induced by non-reciprocal cell-cell interactions in a multicellular system

Collective cellular behavior plays a crucial role in various biological processes, ranging from developmental morphogenesis to pathological processes such as cancer metastasis. Our previous research has revealed that a mutant cell of Dictyostelium discoideum exhibits collective cell migration, including chain migration and traveling band formation, driven by a unique tail-following behavior at contact sites, which we term "contact following locomotion" (CFL). Here, we uncover an imbalance of forces between the front and rear cells within cell chains, leading to an additional propulsion force in the rear cells. Drawing inspiration from this observation, we introduce a theoretical model that incorporates non-reciprocal cell-cell interactions. Our findings highlight that the non-reciprocal interaction, in conjunction with self-alignment interactions, significantly contributes to the emergence of the observed collective cell migrations. Furthermore, we present a comprehensive phase diagram, showing distinct phases at both low and intermediate cell densities. This phase diagram elucidates a specific regime that corresponds to the experimental system.

A Mutant of Dictyostelium discoideum, KI-Cell, as a Model of Collective Cell Migration Independent of Chemotaxis

Collective cell migration occurs when the orientation of cell polarity is aligned with each other in a group of cells. Such collective polarization depends on a reciprocal process between cell intrinsic mechanisms such as cell-cell adhesion and extracellular guidance mechanism such as wound healing and chemotaxis. As part of its development life cycle, individual single cells of Dictyostelium discoideum exhibit chemotaxis toward cAMP, which is secreted from a certain population of cells. During the formation of multicellular body by chemotaxis-dependent cell aggregation, D. discoideum is also known to relay on multiple cell-cell adhesion mechanisms. In particular, tail-following behavior at the contact site, called contact following of locomotion (CFL), plays a pivotal role on the formation of the multicellular body. However, whether and how CFL alone can lead to a formation of collective behavior was not well understood. KI cell is a mutant of D. discoideum that lacks all chemotactic activity. Yet, it can exhibit the CFL activity and show nontrivial collective cell migration. This mutant provides an excellent model system to analyze the mechanism of the CFL and the macroscopic phenomena brought by the CFL. This chapter describes protocols for using KI cell to understand the biophysics and cell biology behind the collective cell migration induced by CFL.

Jetting and Droplet Formation Driven by Interfacial Electrohydrodynamic Effects Mediated by Solitons in Liquid Crystals

Solitons are highly confined, propagating waves that arise from nonlinear feedback in natural (e.g., shallow and confined waters) and engineered systems (e.g., optical wave propagation in fibers). Solitons have recently been observed in thin films of liquid crystals (LCs) in the presence of ac electric fields, where localized LC director distortions arise and propagate due to flexoelectric polarization. Here we report that collisions between LC solitons and interfaces to isotropic fluids can generate a range of interfacial hydrodynamic phenomena. We find that single solitons can either generate single droplets or, alternatively, form jets of LC that subsequently break up into organized assemblies of droplets. We show that the influence of key parameters, such as electric field strength, LC film thickness, and LC-oil interfacial tension, map onto a universal state diagram that characterizes the transduction of soliton flexoelectric energy into droplet interfacial energy. Overall, we reveal that solitons in LCs can be used to focus the energy of nonlocalized electric fields to generate a new class of nonlinear electrohydrodynamic effects at fluid interfaces, including jetting and emulsification.

A cell membrane model that reproduces cortical flow-driven cell migration and collective movement

Many fundamental biological processes are dependent on cellular migration. Although the mechanical mechanisms of single-cell migration are relatively well understood, those underlying migration of multiple cells adhered to each other in a cluster, referred to as cluster migration, are poorly understood. A key reason for this knowledge gap is that many forces-including contraction forces from actomyosin networks, hydrostatic pressure from the cytosol, frictional forces from the substrate, and forces from adjacent cells-contribute to cell cluster movement, making it challenging to model, and ultimately elucidate, the final result of these forces. This paper describes a two-dimensional cell membrane model that represents cells on a substrate with polygons and expresses various mechanical forces on the cell surface, keeping these forces balanced at all times by neglecting cell inertia. The model is discrete but equivalent to a continuous model if appropriate replacement rules for cell surface segments are chosen. When cells are given a polarity, expressed by a direction-dependent surface tension reflecting the location dependence of contraction and adhesion on a cell boundary, the cell surface begins to flow from front to rear as a result of force balance. This flow produces unidirectional cell movement, not only for a single cell but also for multiple cells in a cluster, with migration speeds that coincide with analytical results from a continuous model. Further, if the direction of cell polarity is tilted with respect to the cluster center, surface flow induces cell cluster rotation. The reason why this model moves while keeping force balance on cell surface (i.e., under no net forces from outside) is because of the implicit inflow and outflow of cell surface components through the inside of the cell. An analytical formula connecting cell migration speed and turnover rate of cell surface components is presented.

Collision-induced Hopf-type bifurcation reversible transitions in a dual-wavelength femtosecond fiber laser

Collisions refer to a striking nonlinear interaction process in dissipative systems, revealing the particle-like properties of solitons. In dual-wavelength mode-locked fiber lasers, collisions are inherent and periodic. However, how collisions influence the dynamical transitions in the dual-wavelength mode-locked state has not yet been explored. In our work, dispersion management triggers the complex interactions between solitons in the cavity. We reveal the smooth or Hopf-type bifurcation reversible transitions of dual-color soliton molecules (SMs) during the collision by the real-time spectral measurement technique of time-stretch Fourier transform. The reversible transitions between stationary SMs and vibrating SMs, reveal that the cavity parameters pass through a bifurcation point in the collision process without active external intervention. The numerical results confirm the universality of collision-induced bifurcation behavior. These findings provide new insights into collision dynamics in dual-wavelength ultrafast fiber lasers. Furthermore, the study of inter-molecular collisions is of great significance for other branches of nonlinear science.

Dissipative solitons stabilized by nonlinear gradient terms: Time-dependent behavior and generic properties

We study the time-dependent behavior of dissipative solitons (DSs) stabilized by nonlinear gradient terms. Two cases are investigated: first, the case of the presence of a Raman term, and second, the simultaneous presence of two nonlinear gradient terms, the Raman term and the dispersion of nonlinear gain. As possible types of time-dependence, we find a number of different possibilities including periodic behavior, quasi-periodic behavior, and also chaos. These different types of time-dependence are found to form quite frequently from a window structure of alternating behavior, for example, of periodic and quasi-periodic behaviors. To analyze the time dependence, we exploit extensively time series and Fourier transforms. We discuss in detail quantitatively the question whether all the DSs found for the cubic complex Ginzburg-Landau equation with nonlinear gradient terms are generic, meaning whether they are stable against structural perturbations, for example, to the additions of a small quintic perturbation as it arises naturally in an envelope equation framework. Finally, we examine to what extent it is possible to have different types of DSs for fixed parameter values in the equation by just varying the initial conditions, for example, by using narrow and high vs broad and low amplitudes. These results indicate an overlapping multi-basin structure in parameter space.

Oscillatory dissipative solitons stabilized by nonlinear gradient terms: The transition to localized spatiotemporal disorder

We investigate properties of oscillatory dissipative solitons (DSs) in a cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. As a main result we find a transition to dissipative solitons with spatiotemporal disorder as a function of the diffusion coefficient. This transition proceeds via quasiperiodicity and shows incommensurate satellites next to the fundamental frequency and its harmonics indicating a possible route to localized spatiotemporal chaos. The transition back to oscillatory DSs follows a similar scenario.

Dynamic self-organization of migrating cells under constraints by spatial confinement and epithelial integrity

Understanding how migrating cells can establish both dynamic structures and coherent dynamics may provide mechanistic insights to study how living systems acquire complex structures and functions. Recent studies revealed that intercellular contact communication plays a crucial role for establishing cellular dynamic self-organization (DSO) and provided a theoretical model of DSO for migrating solitary cells in a free space. However, to apply those understanding to situations in living organisms, we need to know the role of cell-cell communication for tissue dynamics under spatial confinements and epithelial integrity. Here, we expand the previous numerical studies on DSO to migrating cells subjected spatial confinement and/or epithelial integrity. An epithelial monolayer is simulated by combining the model of cellular DSO and the cellular vertex model in two dimensions for apical integrity. Under confinement to a small space, theoretical models of both solitary and epithelial cells exhibit characteristic coherent dynamics, including apparent swirling. We also find that such coherent dynamics can allow the cells to overcome the strong constraint due to spatial confinement and epithelial integrity. Furthermore, we demonstrate how epithelial cell clusters behave without spatial confinement and find various cluster dynamics, including spinning, migration and elongation.

Multiplicative noise can induce a velocity change of propagating dissipative solitons

We investigate the influence of spatially homogeneous multiplicative noise on propagating dissipative solitons (DSs) of the cubic complex Ginzburg-Landau equation stabilized by nonlinear gradient terms. Here we focus on the nonlinear gradient terms, in particular on the influence of the Raman term and the delayed nonlinear gain. We show that a fairly small amount of multiplicative noise can lead to a change in the mean velocity for such systems. This effect is exclusively due to the presence of the stabilizing nonlinear gradient terms. For a range of parameters we find a velocity change proportional to the noise intensity for the Raman term and for delayed nonlinear gain. We note that the dissipative soliton decreases the modulus of its velocity when only one type of nonlinear gradient is present. We present a straightforward mean field analysis to capture this simple scaling law. At sufficiently high noise strength the nonlinear gradient stabilized DSs collapse.

Interaction of dissipative solitons stabilized by nonlinear gradient terms

We study the interaction of stable dissipative solitons of the cubic complex Ginzburg-Landau equation which are stabilized only by nonlinear gradient terms. In this paper we focus for the interactions in particular on the influence of the nonlinear gradient term associated with the Raman effect. Depending on its magnitude, we find up to seven possible outcomes of theses collisions: Stationary bound states, oscillatory bound states, meandering oscillatory bound states, bound states with large-amplitude oscillations, partial annihilation, complete annihilation, and interpenetration. Detailed results and their analysis are presented for one value of the corresponding nonlinear gradient term, while the results for two other values are just mentioned briefly. We compare our results with those obtained for coupled cubic-quintic complex Ginzburg-Landau equations and with the cubic-quintic complex Swift-Hohenberg equation. It turns out that both meandering oscillatory bound states as well as bound states with large-amplitude oscillations appear to be specific for coupled cubic complex Ginzburg-Landau equations with a stabilizing cubic nonlinear gradient term. Remarkably, we find for the large-amplitude oscillations a linear relationship between oscillation amplitude and period.

Dynamic Self-Organization of Idealized Migrating Cells by Contact Communication

This Letter investigates what forms of cellular dynamic self-organization are caused through intercellular contact communication based on a theoretical model in which migrating cells perform contact following and contact inhibition and attraction of locomotion. Tuning those strengths causes varieties of dynamic patterns. This further includes a novel form of collective migration, snakelike dynamic assembly. Scrutinizing this pattern reveals that cells in this state can accurately respond to an external directional cue but have no spontaneous global polar order.

Kinetics of motile solitons in nematic liquid crystals

The generation of spatially localized, soliton-like hydrodynamic disturbances in microscale fluidic systems is an intriguing challenge. Herein, we introduce nonequilibrium solitons in nematic liquid crystals stimulated by an electric field. These dynamic solitons are robust as long as the electric field is maintained. Interestingly, their kinetic behaviours depend on the field condition-Tuning of the amplitude and frequency of the applied electric field alters the solitons to self-assemble into lattice ordering like physical particles or to command them to various dynamic states. Our key property to the realisation is the electrohydrodynamic instability due to the coupling between the fluid elasticity and the background convection. This paper describes a new mechanism for realising dynamic solitons in fluid systems on the basis of the electrohydrodynamic phenomena.

Polar pattern formation induced by contact following locomotion in a multicellular system

Biophysical mechanisms underlying collective cell migration of eukaryotic cells have been studied extensively in recent years. One mechanism that induces cells to correlate their motions is contact inhibition of locomotion, by which cells migrating away from the contact site. Here, we report that tail-following behavior at the contact site, termed contact following locomotion (CFL), can induce a non-trivial collective behavior in migrating cells. We show the emergence of a traveling band showing polar order in a mutant Dictyostelium cell that lacks chemotactic activity. We find that CFL is the cell-cell interaction underlying this phenomenon, enabling a theoretical description of how this traveling band forms. We further show that the polar order phase consists of subpopulations that exhibit characteristic transversal motions with respect to the direction of band propagation. These findings describe a novel mechanism of collective cell migration involving cell-cell interactions capable of inducing traveling band with polar order.

Breaking of symmetry of interacting dissipative solitons can lead to partial annihilation

We show that for a large range of approach velocities and over a large interval of stabilizing cubic cross-coupling between counterpropagating waves, a collision of stationary pulses leads to a partial annihilation of pulses via a spontaneous breaking of symmetry. This result arises for coupled cubic-quintic complex Ginzburg-Landau equations for traveling waves for sufficiently large values of the stabilizing cubic cross-coupling and for large enough approach velocities of the pulses. Briefly, we show in addition that the collision of counterpropagating pulses in a system of two coupled cubic Ginzburg-Landau equations with nonlinear gradients (Raman effect) might also lead to partial annihilation, indicating that this breaking of symmetry is generic. Systems of experimental interest include surface reactions, convective onset, biosolitons, and nonlinear optics.

Collective cell migration of Dictyostelium without cAMP oscillations at multicellular stages

In Dictyostelium discoideum, a model organism for the study of collective cell migration, extracellular cyclic adenosine 3',5'-monophosphate (cAMP) acts as a diffusible chemical guidance cue for cell aggregation, which has been thought to be important in multicellular morphogenesis. Here we revealed that the dynamics of cAMP-mediated signaling showed a transition from propagating waves to steady state during cell development. Live-cell imaging of cytosolic cAMP levels revealed that their oscillation and propagation in cell populations were obvious for cell aggregation and mound formation stages, but they gradually disappeared when multicellular slugs started to migrate. A similar transition of signaling dynamics occurred with phosphatidylinositol 3,4,5-trisphosphate signaling, which is upstream of the cAMP signal pathway. This transition was programmed with concomitant developmental progression. We propose a new model in which cAMP oscillation and propagation between cells, which are important at the unicellular stage, are unessential for collective cell migration at the multicellular stage.

Traveling pulse emerges from coupled intermittent walks: A case study in sheep

Monitoring small groups of sheep in spontaneous evolution in the field, we decipher behavioural rules that sheep follow at the individual scale in order to sustain collective motion. Individuals alternate grazing mode at null speed and moving mode at walking speed, so cohesive motion stems from synchronising when they decide to switch between the two modes. We propose a model for the individual decision making process, based on switching rates between stopped / walking states that depend on behind / ahead locations and states of the others. We parametrize this model from data. Next, we translate this (microscopic) individual-based model into its density-flow (macroscopic) equations counterpart. Numerical solving these equations display a traveling pulse propagating at constant speed even though each individual is at any moment either stopped or walking. Considering the minimal model embedded in these equations, we derive analytically the steady shape of the pulse (sech square). The parameters of the pulse (shape and speed) are expressed as functions of individual parameters. This pulse emerges from the non linear coupling of start/stop individual decisions which compensate exactly for diffusion and promotes a steady ratio of walking / stopped individuals, which in turn determines the traveling speed of the pulse. The system seems to converge to this pulse from any initial condition, and to recover the pulse after perturbation. This gives a high robustness to this coordination mechanism.

Bio-soliton model that predicts non-thermal electromagnetic frequency bands, that either stabilize or destabilize living cells

Solitons, as self-reinforcing solitary waves, interact with complex biological phenomena such as cellular self-organization. A soliton model is able to describe a spectrum of electromagnetism modalities that can be applied to understand the physical principles of biological effects in living cells, as caused by endogenous and exogenous electromagnetic fields and is compatible with quantum coherence. A bio-soliton model is proposed, that enables to predict which eigen-frequencies of non-thermal electromagnetic waves are life-sustaining and which are, in contrast, detrimental for living cells. The particular effects are exerted by a range of electromagnetic wave eigen-frequencies of one-tenth of a Hertz till Peta Hertz that show a pattern of 12 bands, and can be positioned on an acoustic reference frequency scale. The model was substantiated by a meta-analysis of 240 published articles of biological electromagnetic experiments, in which a spectrum of non-thermal electromagnetic waves were exposed to living cells and intact organisms. These data support the concept of coherent quantized electromagnetic states in living organisms and the theories of Fröhlich, Davydov and Pang. It is envisioned that a rational control of shape by soliton-waves and related to a morphogenetic field and parametric resonance provides positional information and cues to regulate organism-wide systems properties like anatomy, control of reproduction and repair.

Collisions of non-explosive dissipative solitons can induce explosions

We investigate the interaction of stationary and oscillatory dissipative solitons in the framework of two coupled cubic-quintic complex Ginzburg-Landau equation for counter-propagating waves. We analyze the case of a stabilizing as well as a destabilizing cubic cross-coupling between the counter-propagating dissipative solitons. The three types of interacting localized solutions investigated are stationary, oscillatory with one frequency, and oscillatory with two frequencies. We show that there is a large number of different outcomes as a result of these collisions including stationary as well as oscillatory bound states and compound states with one and two frequencies. The two most remarkable results are (a) the occurrence of bound states and compound states of exploding dissipative solitons as outcome of the collisions of stationary and oscillatory pulses; and (b) spatiotemporal disorder due to the creation, interaction, and annihilation of dissipative solitons for colliding oscillatory dissipative solitons as initial conditions.

Cell-alignment patterns in the collective migration of cells with polarized adhesion

Dictyostelium discoideum (Dd) utilizes inhomogeneities in the distribution of cell-cell adhesion molecules on cell membranes for collective cell migration. A simple example of an inhomogeneity is a front-side (leading-edge) polarization in the distribution at the early streaming stage. Experiments have shown that the polarized cell-cell adhesion induces side-by-side contact between cells [Beug et al., Nature (London) 274, 445 (1978)NATUAS0028-083610.1038/274445a0]. This result is counterintuitive, as one would expect cells to align front to front in contact with each other on the basis of front-side polarization. In this work, we theoretically examine whether front-side polarization induces side-by-side contact in collective cell migration. We construct a model for expressing cells with this polarization based on the two-dimensional cellular Potts model. By a numerical simulation with this model, we find cell-cell alignment wherein cells form lateral arrays with side-by-side contacts as observed in the experiments.

Multiplicative noise can lead to the collapse of dissipative solitons

We investigate the influence of spatially homogeneous multiplicative noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. We find that for sufficiently large multiplicative noise the formation of stationary and temporally periodic dissipative solitons is suppressed. This result is characterized by a linear relation between the bifurcation parameter and the noise amplitude required for suppression. For the regime associated with exploding dissipative solitons we find a reduction in the number of explosions for larger noise strength as well as a conversion to other types of dissipative solitons or to filling-in and eventually a collapse to the zero solution.

Non-unique results of collisions of quasi-one-dimensional dissipative solitons

We investigate collisions of quasi-one-dimensional dissipative solitons (DSs) for a large class of initial conditions, which are not temporally asymptotic quasi-one-dimensional DSs. For the case of sufficiently small approach velocity and sufficiently large values of the dissipative cross-coupling between the counter-propagating DSs, we find non-unique results for the outcome of collisions. We demonstrate that these non-unique results are intrinsically related to a modulation instability along the crest of the quasi-one-dimensional objects. As a model, we use coupled cubic-quintic complex Ginzburg-Landau equations. Among the final results found are stationary and oscillatory compound states as well as more complex assemblies consisting of quasi-one-dimensional and localized states. We analyse to what extent the final results can be described by the solutions of one cubic-quintic complex Ginzburg-Landau equation with effective parameters.

Noisy localized structures induced by large noise

We investigate the influence of large noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. The interaction of localization and noise can lead to filling in or noisy localized structures for fixed noise strength. To focus on the interaction between noise and localization we cover a region in parameter space, in particular, subcriticality, for which stationary stable deterministic pulses do not exist. Possible experimental tests of the work presented for autocatalytic chemical reactions and bioinspired systems are outlined.

Collision dynamics of traveling bands in systems of deformable self-propelled particles

We study the soliton-like character of traveling bands in systems of interacting deformable self-propelled particles in two dimensions. The collision dynamics of the model in which migration velocity increases with increasing local density is investigated numerically by changing the relaxation rate of deformations. The bands are unstable upon head-on collisions for larger relaxation rates. This clearly indicates that deformability plays a crucial role of the soliton-like behavior.

Class of compound dissipative solitons as a result of collisions in one and two spatial dimensions

We study the interaction of quasi-one-dimensional (quasi-1D) dissipative solitons (DSs). Starting with quasi-1D solutions of the cubic-quintic complex Ginzburg-Landau (CGL) equation in their temporally asymptotic state as the initial condition, we find, as a function of the approach velocity and the real part of the cubic interaction of the two counterpropagating envelopes: interpenetration, one compound state made of both envelopes or two compound states. For the latter class both envelopes show DSs superposed at two different locations. The stability of this class of compound states is traced back to the quasilinear growth rate associated with the coupled system. We show that this mechanism also works for 1D coupled cubic-quintic CGL equations. For quasi-1D states that are not in their asymptotic state before the collision, a breakup along the crest can be observed, leading to nonunique results after the collision of quasi-1D states.