Stochastic models for phototaxis

Amplitude responses of swarmalators

Swarmalators are entities that swarm through space and sync in time and are potentially considered to replicate the complex dynamics of many real-world systems. So far, the internal dynamics of swarmalators have been taken as a phase oscillator inspired by the Kuramoto model. Here we examine the internal dynamics utilizing an amplitude oscillator capable of exhibiting periodic and chaotic behaviors. To incorporate the dual interplay between spatial and internal dynamics, we propose a general model that keeps the properties of swarmalators intact. This adaptation calls for a detailed study, which we present in this paper. We establish our study with the Rössler oscillator by taking parameters from both chaotic and periodic regions. While the periodic oscillator mimics most of the patterns in the previous phase oscillator model, the chaotic oscillator brings some fascinating states.

Antiphase synchronization in a population of swarmalators

Swarmalators are oscillatory systems endowed with a spatial component, whose spatial and phase dynamics affect each other. Such systems can demonstrate fascinating collective dynamics resembling many real-world processes. Through this work, we study a population of swarmalators where they are divided into different communities. The strengths of spatial attraction, repulsion, as well as phase interaction differ from one group to another. Also, they vary from intercommunity to intracommunity. We encounter, as a result of variation in the phase coupling strength, different routes to achieve the static synchronization state by choosing several parameter combinations. We observe that when the intercommunity phase coupling strength is sufficiently large, swarmalators settle in the static synchronization state. However, with a significant small phase coupling strength the state of antiphase synchronization as well as chimeralike coexistence of sync and async are realized. Apart from rigorous numerical results, we have been successful to provide semianalytical treatment for the existence and stability of global static sync and the antiphase sync states.

Phototaxis in Cyanobacteria: From Mutants to Models of Collective Behavior

Cyanobacteria rely on photosynthesis, and thus have evolved complex responses to light. These include phototaxis, the ability of cells to sense light direction and move towards or away from it. Analysis of mutants has demonstrated that phototaxis requires the coordination of multiple photoreceptors and signal transduction networks. The output of these networks is relayed to type IV pili (T4P) that attach to and exert forces on surfaces or other neighboring cells to drive “twitching” or “gliding” motility. This, along with the extrusion of polysaccharides or “slime” by cells, facilitates the emergence of group behavior. We evaluate recent models that describe the emergence of collective colony-scale behavior from the responses of individual, interacting cells. We highlight the advantages of “active matter” approaches in the study of bacterial communities, discussing key differences between emergent behavior in cyanobacterial phototaxis and similar behavior in chemotaxis or quorum sensing.

Scattered migrating colony formation in the filamentous cyanobacterium, Pseudanabaena sp. NIES-4403

Background

Bacteria have been reported to exhibit complicated morphological colony patterns on solid media, depending on intracellular, and extracellular factors such as motility, cell propagation, and cell-cell interaction. We isolated the filamentous cyanobacterium, Pseudanabaena sp. NIES-4403 (Pseudanabaena, hereafter), that forms scattered (discrete) migrating colonies on solid media. While the scattered colony pattern has been observed in some bacterial species, the mechanism underlying such a pattern still remains obscure.

Results

We studied the morphology of Pseudanabaena migrating collectively and found that this species forms randomly scattered clusters varying in size and further consists of a mixture of comet-like wandering clusters and disk-like rotating clusters. Quantitative analysis of the formation of these wandering and rotating clusters showed that bacterial filaments tend to follow trajectories of previously migrating filaments at velocities that are dependent on filament length. Collisions between filaments occurred without crossing paths, which enhanced their nematic alignments, giving rise to bundle-like colonies. As cells increased and bundles aggregated, comet-like wandering clusters developed. The direction and velocity of the movement of cells in comet-like wandering clusters were highly coordinated. When the wandering clusters entered into a circular orbit, they turned into rotating clusters, maintaining a more stable location. Disk-like rotating clusters may rotate for days, and the speed of cells within a rotating cluster increases from the center to the outmost part of the cluster. Using a mathematical modeling with simplified assumption we reproduced some features of the scattered pattern including migrating clusters.

Conclusion

Based on these observations, we propose that Pseudanabaena forms scattered migrating colonies that undergo a series of transitions involving several morphological patterns. A simplified model is able to reproduce some features of the observed migrating clusters.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12866-021-02183-5.

Ring states in swarmalator systems

Synchronization is a universal phenomenon, occurring in systems as disparate as Japanese tree frogs and Josephson junctions. Typically, the elements of synchronizing systems adjust the phases of their oscillations, but not their positions in space. The reverse scenario is found in swarming systems, such as schools of fish or flocks of birds; now the elements adjust their positions in space, but without (noticeably) changing their internal states. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed, and analyzed in the continuum limit. Here, we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density we calculate explicitly. These states may be observable in groups of Japanese tree frogs, ferromagnetic colloids, and other systems with an interplay between swarming and synchronization.

Phototaxis as a Collective Phenomenon in Cyanobacterial Colonies

Cyanobacteria are a diverse group of photosynthetic bacteria that exhibit phototaxis, or motion in response to light. Cyanobacteria such as Synechocystis sp. secrete a mixture of complex polysaccharides that facilitate cell motion, while their type 4 pili allow them to physically attach to each other. Even though cells can respond individually to light, colonies are observed to move collectively towards the light source in dense finger-like projections. We present an agent-based model for cyanobacterial phototaxis that accounts for slime deposition as well as for direct physical links between bacteria, mediated through their type 4 pili. We reproduce the experimentally observed aggregation of cells at the colony boundary as a precursor to finger formation. Our model also describes the changes in colony morphology that occur when the location of the light source is abruptly changed. We find that the overall motion of cells toward light remains relatively unimpaired even if a fraction of them do not sense light, allowing heterogeneous populations to continue to mount a robust collective response to stimuli. Our work suggests that in addition to bio-chemical signalling via diffusible molecules in the context of bacterial quorum-sensing, short-ranged physical interactions may also contribute to collective effects in bacterial motility.

Maintenance of motility bias during cyanobacterial phototaxis

Signal transduction in bacteria is complex, ranging across scales from molecular signal detectors and effectors to cellular and community responses to stimuli. The unicellular, photosynthetic cyanobacterium Synechocystis sp. PCC6803 transduces a light stimulus into directional movement known as phototaxis. This response occurs via a biased random walk toward or away from a directional light source, which is sensed by intracellular photoreceptors and mediated by Type IV pili. It is unknown how quickly cells can respond to changes in the presence or directionality of light, or how photoreceptors affect single-cell motility behavior. In this study, we use time-lapse microscopy coupled with quantitative single-cell tracking to investigate the timescale of the cellular response to various light conditions and to characterize the contribution of the photoreceptor TaxD1 (PixJ1) to phototaxis. We first demonstrate that a community of cells exhibits both spatial and population heterogeneity in its phototactic response. We then show that individual cells respond within minutes to changes in light conditions, and that movement directionality is conferred only by the current light directionality, rather than by a long-term memory of previous conditions. Our measurements indicate that motility bias likely results from the polarization of pilus activity, yielding variable levels of movement in different directions. Experiments with a photoreceptor (taxD1) mutant suggest a supplementary role of TaxD1 in enhancing movement directionality, in addition to its previously identified role in promoting positive phototaxis. Motivated by the behavior of the taxD1 mutant, we demonstrate using a reaction-diffusion model that diffusion anisotropy is sufficient to produce the observed changes in the pattern of collective motility. Taken together, our results establish that single-cell tracking can be used to determine the factors that affect motility bias, which can then be coupled with biophysical simulations to connect changes in motility behaviors at the cellular scale with group dynamics.

Probabilistic cellular automata

Cellular automata are binary lattices used for modeling complex dynamical systems. The automaton evolves iteratively from one configuration to another, using some local transition rule based on the number of ones in the neighborhood of each cell. With respect to the number of cells allowed to change per iteration, we speak of either synchronous or asynchronous automata. If randomness is involved to some degree in the transition rule, we speak of probabilistic automata, otherwise they are called deterministic. With either type of cellular automaton we are dealing with, the main theoretical challenge stays the same: starting from an arbitrary initial configuration, predict (with highest accuracy) the end configuration. If the automaton is deterministic, the outcome simplifies to one of two configurations, all zeros or all ones. If the automaton is probabilistic, the whole process is modeled by a finite homogeneous Markov chain, and the outcome is the corresponding stationary distribution. Based on our previous results for the asynchronous case-connecting the probability of a configuration in the stationary distribution to its number of zero-one borders-the article offers both numerical and theoretical insight into the long-term behavior of synchronous cellular automata.

Modeling Selective Local Interactions with Memory: Motion on a 2D Lattice

We consider a system of particles that simultaneously move on a two-dimensional periodic lattice at discrete times steps. Particles remember their last direction of movement and may either choose to continue moving in this direction, remain stationary, or move toward one of their neighbors. The form of motion is chosen based on predetermined stationary probabilities. Simulations of this model reveal a connection between these probabilities and the emerging patterns and size of aggregates. In addition, we develop a reaction diffusion master equation from which we derive a system of ODEs describing the dynamics of the particles on the lattice. Simulations demonstrate that solutions of the ODEs may replicate the aggregation patterns produced by the stochastic particle model. We investigate conditions on the parameters that influence the locations at which particles prefer to aggregate. This work is a two-dimensional generalization of [Galante & Levy, Physica D, http://dx.doi.org/10.1016/j.physd.2012.10.010], in which the corresponding one-dimensional problem was studied.

Modeling selective local interactions with memory

Recently we developed a stochastic particle system describing local interactions between cyanobacteria. We focused on the common freshwater cyanobacteria Synechocystis sp., which are coccoidal bacteria that utilize group dynamics to move toward a light source, a motion referred to as phototaxis. We were particularly interested in the local interactions between cells that were located in low to medium density areas away from the front. The simulations of our stochastic particle system in 2D replicated many experimentally observed phenomena, such as the formation of aggregations and the quasi-random motion of cells. In this paper, we seek to develop a better understanding of group dynamics produced by this model. To facilitate this study, we replace the stochastic model with a system of ordinary differential equations describing the evolution of particles in 1D. Unlike many other models, our emphasis is on particles that selectively choose one of their neighbors as the preferred direction of motion. Furthermore, we incorporate memory by allowing persistence in the motion. We conduct numerical simulations which allow us to efficiently explore the space of parameters, in order to study the stability, size, and merging of aggregations.

Motility enhancement through surface modification is sufficient for cyanobacterial community organization during phototaxis

The emergent behaviors of communities of genotypically identical cells cannot be easily predicted from the behaviors of individual cells. In many cases, it is thought that direct cell-cell communication plays a critical role in the transition from individual to community behaviors. In the unicellular photosynthetic cyanobacterium Synechocystis sp. PCC 6803, individual cells exhibit light-directed motility ("phototaxis") over surfaces, resulting in the emergence of dynamic spatial organization of multicellular communities. To probe this striking community behavior, we carried out time-lapse video microscopy coupled with quantitative analysis of single-cell dynamics under varying light conditions. These analyses suggest that cells secrete an extracellular substance that modifies the physical properties of the substrate, leading to enhanced motility and the ability for groups of cells to passively guide one another. We developed a biophysical model that demonstrates that this form of indirect, surface-based communication is sufficient to create distinct motile groups whose shape, velocity, and dynamics qualitatively match our experimental observations, even in the absence of direct cellular interactions or changes in single-cell behavior. Our computational analysis of the predicted community behavior, across a matrix of cellular concentrations and light biases, demonstrates that spatial patterning follows robust scaling laws and provides a useful resource for the generation of testable hypotheses regarding phototactic behavior. In addition, we predict that degradation of the surface modification may account for the secondary patterns occasionally observed after the initial formation of a community structure. Taken together, our modeling and experiments provide a framework to show that the emergent spatial organization of phototactic communities requires modification of the substrate, and this form of surface-based communication could provide insight into the behavior of a wide array of biological communities.

Modeling local interactions during the motion of cyanobacteria

Synechocystis sp., a common unicellular freshwater cyanobacterium, has been used as a model organism to study phototaxis, an ability to move in the direction of a light source. This microorganism displays a number of additional characteristics such as delayed motion, surface dependence, and a quasi-random motion, where cells move in a seemingly disordered fashion instead of in the direction of the light source, a global force on the system. These unexplained motions are thought to be modulated by local interactions between cells such as intercellular communication. In this paper, we consider only local interactions of these phototactic cells in order to mathematically model this quasi-random motion. We analyze an experimental data set to illustrate the presence of quasi-random motion and then derive a stochastic dynamic particle system modeling interacting phototactic cells. The simulations of our model are consistent with experimentally observed phototactic motion.