From intracellular signaling to population oscillations: bridging size- and time-scales in collective behavior

Collective oscillatory signaling in Dictyosteliumdiscoideum acts as a developmental timer initiated by weak coupling of a noisy pulsatile signal

Oscillatory phenomena play widespread roles in the control of biological systems. In D. discoideum, oscillatory cyclic adenosine monophosphate (cAMP) signaling drives collective behavior and induces a temporal developmental gene expression program. How collective cAMP oscillations emerge or how they encode temporal transcriptional information is still poorly understood. To address this, we identified a transcription factor required for the initiation of collective behavior. Hbx5 activity is cAMP dependent and provides a sensitive single-cell readout for cAMP signaling. Extensive stochastic pulsatile cAMP signaling is found to precede collective oscillations. Stochastic signaling induces Hbx5-dependent transcriptional feedback, which enhances signal sensitivity and cell-cell coupling. This results in the emergence of synchronized collective oscillations, which subsequently activates the GtaC transcription factor and triggers shifts in developmental gene expression. Our results suggest this temporal coordination is encoded by changes in the amplitude of cAMP oscillations and differential sensitivity of these transcription factors to the cAMP-regulated kinase ErkB.

Self-extinguishing relay waves enable homeostatic control of human neutrophil swarming

Neutrophils collectively migrate to sites of injury and infection. How these swarms are coordinated to ensure the proper level of recruitment is unknown. Using an ex vivo model of infection, we show that human neutrophil swarming is organized by multiple pulsatile chemoattractant waves. These waves propagate through active relay in which stimulated neutrophils trigger their neighbors to release additional swarming cues. Unlike canonical active relays, we find these waves to be self-terminating, limiting the spatial range of cell recruitment. We identify an NADPH-oxidase-based negative feedback loop that is needed for this self-terminating behavior. We observe near-constant levels of neutrophil recruitment over a wide range of starting conditions, revealing surprising robustness in the swarming process. This homeostatic control is achieved by larger and more numerous swarming waves at lower cell densities. We link defective wave termination to a broken recruitment homeostat in the context of human chronic granulomatous disease.

Cell-to-cell signaling in cell populations with large cell size variability

Sizes of multiple cells vary when they communicate with each other. Differences in cell size result in variations in the cell surface area and volume, as well as the number of enzymes and receptors involved in signal transduction. Although heterogeneity in cell size may inhibit uniformity in signaling, cell-to-cell signaling is still possible. The outcome when cell size changes to an extreme degree remains unclear. Hence, we inhibited cell division in Dictyostelium cells, a model organism for signal transduction, to gain insights into the consequences of extreme cell size variations. Measurements of cell signals in this population using fluorescence microscopy indicated that the giant cells can communicate with normal-sized cells by suppressing the signal level. Simulations of signal transduction based on the FitzHugh-Nagumo model also suggested similar results. Our findings suggest that signaling mechanism homogenizes cell-to-cell signaling in response to cell size.

The diversification of methods for studying cell-cell interactions and communication

No cell lives in a vacuum, and the molecular interactions between cells define most phenotypes. Transcriptomics provides rich information to infer cell-cell interactions and communication, thus accelerating the discovery of the roles of cells within their communities. Such research relies heavily on algorithms that infer which cells are interacting and the ligands and receptors involved. Specific pressures on different research niches are driving the evolution of next-generation computational tools, enabling new conceptual opportunities and technological advances. More sophisticated algorithms now account for the heterogeneity and spatial organization of cells, multiple ligand types and intracellular signalling events, and enable the use of larger and more complex datasets, including single-cell and spatial transcriptomics. Similarly, new high-throughput experimental methods are increasing the number and resolution of interactions that can be analysed simultaneously. Here, we explore recent progress in cell-cell interaction research and highlight the diversification of the next generation of tools, which have yielded a rich ecosystem of tools for different applications and are enabling invaluable discoveries.

Syncytial Assembly Lines: Consequences of Multinucleate Cellular Compartments for Fungal Protein Synthesis

Fast growth and prodigious cellular outputs make fungi powerful tools in biotechnology. Recent modeling work has exposed efficiency gains associated with dividing the labor of transcription over multiple nuclei, and experimental innovations are opening new windows on the capacities and adaptations that allow nuclei to behave autonomously or in coordination while sharing a single, common cytoplasm. Although the motivation of our review is to motivate and connect recent work toward a greater understanding of fungal factories, we use the analogy of the assembly line as an organizing idea for studying coordinated gene expression, generally.

Microbial models of development: Inspiration for engineering self-assembled synthetic multicellularity

While the field of synthetic developmental biology has traditionally focused on the study of the rich developmental processes seen in metazoan systems, an attractive alternate source of inspiration comes from microbial developmental models. Microbes face unique lifestyle challenges when forming emergent multicellular collectives. As a result, the solutions they employ can inspire the design of novel multicellular systems. In this review, we dissect the strategies employed in multicellular development by two model microbial systems: the cellular slime mold Dictyostelium discoideum and the biofilm-forming bacterium Bacillus subtilis. Both microbes face similar challenges but often have different solutions, both from metazoan systems and from each other, to create emergent multicellularity. These challenges include assembling and sustaining a critical mass of participating individuals to support development, regulating entry into development, and assigning cell fates. The mechanisms these microbial systems exploit to robustly coordinate development under a wide range of conditions offer inspiration for a new toolbox of solutions to the synthetic development community. Additionally, recreating these phenomena synthetically offers a pathway to understanding the key principles underlying how these behaviors are coordinated naturally.

Controlling periodic long-range signalling to drive a morphogenetic transition

Cells use signal relay to transmit information across tissue scales. However, the production of information carried by signal relay remains poorly characterised. To determine how the coding features of signal relay are generated, we used the classic system for long-range signalling: the periodic cAMP waves that drive Dictyostelium collective migration. Combining imaging and optogenetic perturbation of cell signalling states, we find that migration is triggered by an increase in wave frequency generated at the signalling centre. Wave frequency is regulated by cAMP wave circulation, which organises the long-range signal. To determine the mechanisms modulating wave circulation, we combined mathematical modelling, the general theory of excitable media, and mechanical perturbations to test competing models. Models in which cell density and spatial patterning modulate the wave frequency cannot explain the temporal evolution of signalling waves. Instead, our evidence leads to a model where wave circulation increases the ability for cells to relay the signal, causing further increase in the circulation rate. This positive feedback between cell state and signalling pattern regulates the long-range signal coding that drives morphogenesis.

Rate thresholds in cell signaling have functional and phenotypic consequences in non-linear time-dependent environments

All cells employ signal transduction pathways to respond to physiologically relevant extracellular cytokines, stressors, nutrient levels, hormones, morphogens, and other stimuli that vary in concentration and rate in healthy and diseased states. A central unsolved fundamental question in cell signaling is whether and how cells sense and integrate information conveyed by changes in the rate of extracellular stimuli concentrations, in addition to the absolute difference in concentration. We propose that different environmental changes over time influence cell behavior in addition to different signaling molecules or different genetic backgrounds. However, most current biomedical research focuses on acute environmental changes and does not consider how cells respond to environments that change slowly over time. As an example of such environmental change, we review cell sensitivity to environmental rate changes, including the novel mechanism of rate threshold. A rate threshold is defined as a threshold in the rate of change in the environment in which a rate value below the threshold does not activate signaling and a rate value above the threshold leads to signal activation. We reviewed p38/Hog1 osmotic stress signaling in yeast, chemotaxis and stress response in bacteria, cyclic adenosine monophosphate signaling in Amoebae, growth factors signaling in mammalian cells, morphogen dynamics during development, temporal dynamics of glucose and insulin signaling, and spatio-temproral stressors in the kidney. These reviewed examples from the literature indicate that rate thresholds are widespread and an underappreciated fundamental property of cell signaling. Finally, by studying cells in non-linear environments, we outline future directions to understand cell physiology better in normal and pathophysiological conditions.

Reproducible and sensitive micro-tissue RNA sequencing from formalin-fixed paraffin-embedded tissues for spatial gene expression analysis

Spatial transcriptome analysis of formalin-fixed paraffin-embedded (FFPE) tissues using RNA-sequencing (RNA-seq) provides interactive information on morphology and gene expression, which is useful for clinical applications. However, despite the advantages of long-term storage at room temperature, FFPE tissues may be severely damaged by methylene crosslinking and provide less gene information than fresh-frozen tissues. In this study, we proposed a sensitive FFPE micro-tissue RNA-seq method that combines the punching of tissue sections (diameter: 100 μm) and the direct construction of RNA-seq libraries. We evaluated a method using mouse liver tissues at two years after fixation and embedding and detected approximately 7000 genes in micro-punched tissue-spots (thickness: 10 μm), similar to that detected with purified total RNA (2.5 ng) equivalent to the several dozen cells in the spot. We applied this method to clinical FFPE specimens of lung cancer that had been fixed and embedded 6 years prior, and found that it was possible to determine characteristic gene expression in the microenvironment containing tumor and non-tumor cells of different morphologies. This result indicates that spatial gene expression analysis of the tumor microenvironment is feasible using FFPE tissue sections stored for extensive periods in medical facilities.

Calcium responses to external mechanical stimuli in the multicellular stage of Dictyostelium discoideum

Calcium acts as a second messenger to regulate many cellular functions, including cell motility. In Dictyostelium discoideum, the cytosolic calcium level oscillates synchronously, and calcium waves propagate through the cell population during the early stages of development, including aggregation. In the unicellular phase, the calcium response through Piezo channels also functions in mechanosensing. However, calcium dynamics during multicellular morphogenesis are still unclear. Here, live imaging of cytosolic calcium revealed that calcium wave propagation, depending on cAMP relay, disappeared at the onset of multicellular body (slug) formation. Later, other forms of occasional calcium bursts and their propagation were observed in both anterior and posterior regions of migrating slugs. This calcium signaling also occurred in response to mechanical stimuli. Two pathways—calcium release from the endoplasmic reticulum via IP3 receptor and calcium influx from outside the cell—were involved in calcium signals induced by mechanical stimuli. These data suggest that calcium signaling is involved in mechanosensing in both the unicellular and multicellular phases of Dictyostelium development using different molecular mechanisms.

The macroscopic limit to synchronization of cellular clocks in single cells of Neurospora crassa

We determined the macroscopic limit for phase synchronization of cellular clocks in an artificial tissue created by a “big chamber” microfluidic device to be about 150,000 cells or less. The dimensions of the microfluidic chamber allowed us to calculate an upper limit on the radius of a hypothesized quorum sensing signal molecule of 13.05 nm using a diffusion approximation for signal travel within the device. The use of a second microwell microfluidic device allowed the refinement of the macroscopic limit to a cell density of 2166 cells per fixed area of the device for phase synchronization. The measurement of averages over single cell trajectories in the microwell device supported a deterministic quorum sensing model identified by ensemble methods for clock phase synchronization. A strong inference framework was used to test the communication mechanism in phase synchronization of quorum sensing versus cell-to-cell contact, suggesting support for quorum sensing. Further evidence came from showing phase synchronization was density-dependent.

Reactive/Less-cooperative individuals advance population's synchronization: Modeling of Dictyostelium discoideum concerted signaling during aggregation phase

Orchestrated chemical signaling of single cells sounds to be a linchpin of emerging organization and multicellular life form. The social amoeba Dictyostelium discoideum is a well-studied model organism to explore overall pictures of grouped behavior in developmental biology. The chemical waves secreted by aggregating Dictyostelium is a superb example of pattern formation. The waves are either circular or spiral in shape, according to the incremental population density of a self-aggregating community of individuals. Here, we revisit the spatiotemporal patterns that appear in an excitable medium due to synchronization of randomly firing individuals, but with a more parsimonious attitude. According to our model, a fraction of these individuals are less involved in amplifying external stimulants. Our simulations indicate that the cells enhance the system’s asymmetry and as a result, nucleate early sustainable spiral territory zones, provided that their relative population does not exceed a tolerable threshold.

Towards a physical understanding of developmental patterning

The temporal coordination of events at cellular and tissue scales is essential for the proper development of organisms, and involves cell-intrinsic processes that can be coupled by local cellular signalling and instructed by global signalling, thereby creating spatial patterns of cellular states that change over time. The timing and structure of these patterns determine how an organism develops. Traditional developmental genetic methods have revealed the complex molecular circuits regulating these processes but are limited in their ability to predict and understand the emergent spatio-temporal dynamics. Increasingly, approaches from physics are now being used to help capture the dynamics of the system by providing simplified, generic descriptions. Combined with advances in imaging and computational power, such approaches aim to provide insight into timing and patterning in developing systems.

Arnold tongues in oscillator systems with nonuniform spatial driving

Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing qualitative implications of our results for controlling coupled oscillator systems such as the social amoebae Dictyostelium and populations of Belousov Zhabotinsky (BZ) catalytic particles using spatially structured external drives.

Nontrivial amplification below the threshold for excitable cell signaling

In many asymptotically stable fluid systems, arbitrarily small fluctuations can grow by orders of magnitude before eventually decaying, dramatically enhancing the fluctuation variance beyond the minimum predicted by linear stability theory. Here using influential quantitative models drawn from the mathematical biology literature, we establish that dramatic amplification of arbitrarily small fluctuations is found in excitable cell signaling systems as well. Our analysis highlights how positive and negative feedback, proximity to bifurcations, and strong separation of timescales can generate nontrivial fluctuations without nudging these systems across their excitation thresholds. These insights, in turn, are relevant for a broader range of related oscillatory, bistable, and pattern-forming systems that share these features. The common thread connecting all of these systems with fluid dynamical examples of noise amplification is non-normality.

Spatial gradient sensing and chemotaxis via excitability in Dictyostelium discoideum

The social amoeba Dictyostelium discoideum performs chemotaxis under starvation conditions, aggregating towards clusters of cells following waves of the signaling molecule cAMP. Cells sense extracellular cAMP and produce internal caches of cAMP to be released, relaying the signal. These events lead to traveling waves of cAMP washing over the population of cells. While much research has been performed to understand the functioning of the chemotaxis network in D. discoideum, limited work has been done to link the operation of the signal relay network with the chemotaxis network to provide a holistic view of the system. We take inspiration from D. discoideum and propose a model that directly links the relaying of a chemical message to the directional sensing of that signal. Utilizing an excitable dynamical systems model that has been previously validated experimentally, we show that it is possible to have both signal amplification and perfect adaptation in a single module. We show that noise plays a vital role in chemotaxing to static gradients, where stochastic tunneling of transient bursts biases the system towards accurate gradient sensing. Moreover, this model also automatically matches its internal time scale of adaptation to the naturally occurring periodicity of the traveling chemical waves generated in the population. Numerical simulations were performed to study the qualitative phenomenology of the system and explore how the system responds to diverse dynamic spatiotemporal stimuli. Finally, we address dynamical instabilities that impede chemotactic ability in a continuum version of the model.

Waiting on the Fringe: cell autonomy and signaling delays in segmentation clocks

The rhythmic and sequential segmentation of the vertebrate body axis into somites during embryogenesis is governed by a multicellular, oscillatory patterning system called the segmentation clock. Despite many overt similarities between vertebrates, differences in genetic and dynamic regulation have been reported, raising intriguing questions about the evolution and conservation of this fundamental patterning process. Recent studies have brought insights into two important and related issues: (1) whether individual cells of segmentation clocks are autonomous oscillators or require cell-cell communication for their rhythm; and (2) the role of delays in the cell-cell communication that synchronizes the population of genetic oscillators. Although molecular details differ between species, conservation may exist at the level of the dynamics, hinting at rules for evolutionary trajectories in the system.

Intracellular signaling dynamics and their role in coordinating tissue repair

Tissue repair is a complex process that requires effective communication and coordination between cells across multiple tissues and organ systems. Two of the initial intracellular signals that encode injury signals and initiate tissue repair responses are calcium and extracellular signal-regulated kinase (ERK). However, calcium and ERK signaling control a variety of cellular behaviors important for injury repair including cellular motility, contractility, and proliferation, as well as the activity of several different transcription factors, making it challenging to relate specific injury signals to their respective repair programs. This knowledge gap ultimately hinders the development of new wound healing therapies that could take advantage of native cellular signaling programs to more effectively repair tissue damage. The objective of this review is to highlight the roles of calcium and ERK signaling dynamics as mechanisms that link specific injury signals to specific cellular repair programs during epithelial and stromal injury repair. We detail how the signaling networks controlling calcium and ERK can now also be dissected using classical signal processing techniques with the advent of new biosensors and optogenetic signal controllers. Finally, we advocate the importance of recognizing calcium and ERK dynamics as key links between injury detection and injury repair programs that both organize and execute a coordinated tissue repair response between cells across different tissues and organs. This article is categorized under: Models of Systems Properties and Processes > Mechanistic Models Biological Mechanisms > Cell Signaling Laboratory Methods and Technologies > Imaging Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models.

Phenotypic delay in the evolution of bacterial antibiotic resistance: Mechanistic models and their implications

Phenotypic delay-the time delay between genetic mutation and expression of the corresponding phenotype-is generally neglected in evolutionary models, yet recent work suggests that it may be more common than previously assumed. Here, we use computer simulations and theory to investigate the significance of phenotypic delay for the evolution of bacterial resistance to antibiotics. We consider three mechanisms which could potentially cause phenotypic delay: effective polyploidy, dilution of antibiotic-sensitive molecules and accumulation of resistance-enhancing molecules. We find that the accumulation of resistant molecules is relevant only within a narrow parameter range, but both the dilution of sensitive molecules and effective polyploidy can cause phenotypic delay over a wide range of parameters. We further investigate whether these mechanisms could affect population survival under drug treatment and thereby explain observed discrepancies in mutation rates estimated by Luria-Delbrück fluctuation tests. While the effective polyploidy mechanism does not affect population survival, the dilution of sensitive molecules leads both to decreased probability of survival under drug treatment and underestimation of mutation rates in fluctuation tests. The dilution mechanism also changes the shape of the Luria-Delbrück distribution of mutant numbers, and we show that this modified distribution provides an improved fit to previously published experimental data.

Cellular Dialogues: Cell-Cell Communication through Diffusible Molecules Yields Dynamic Spatial Patterns

Cells form spatial patterns by coordinating their gene expressions. How a group of mesoscopic numbers (hundreds to thousands) of cells, without pre-existing morphogen gradients and spatial organization, self-organizes spatial patterns remains poorly understood. Of particular importance are dynamic spatial patterns such as spiral waves that perpetually move and transmit information. We developed an open-source software for simulating a field of cells that communicate by secreting any number of molecules. With this software and a theory, we identified all possible "cellular dialogues"-ways of communicating with two diffusing molecules-that yield diverse dynamic spatial patterns. These patterns emerge despite widely varying responses of cells to the molecules, gene-expression noise, spatial arrangements, and cell movements. A three-stage, "order-fluctuate-settle" process forms dynamic spatial patterns: cells form long-lived whirlpools of wavelets that, following erratic dynamics, settle into a dynamic spatial pattern. Our work helps in identifying gene-regulatory networks that underlie dynamic pattern formations.

Emergence of collective oscillations in adaptive cells

Collective oscillations of cells in a population appear under diverse biological contexts. Here, we establish a set of common principles by categorising the response of individual cells against a time-varying signal. A positive intracellular signal relay of sufficient gain from participating cells is required to sustain the oscillations, together with phase matching. The two conditions yield quantitative predictions for the onset cell density and frequency in terms of measured single-cell and signal response functions. Through mathematical constructions, we show that cells that adapt to a constant stimulus fulfil the phase requirement by developing a leading phase in an active frequency window that enables cell-to-signal energy flow. Analysis of dynamical quorum sensing in several cellular systems with increasing biological complexity reaffirms the pivotal role of adaptation in powering oscillations in an otherwise dissipative cell-to-cell communication channel. The physical conditions identified also apply to synthetic oscillatory systems.

There are many examples of cell populations exhibiting density-dependent collective oscillatory behaviour. Here, the authors show that sustained collective oscillations emerge when cells anticipate variation in signal and attempt to amplify it, a property that can be linked to adaptation.

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.

Bi-functional biochemical networks

Understanding the relationship between the topology of a network and its function remains an important question in biological physics. However, this is not a one-to-one mapping. Often the behavior of a signaling system varies with the input signal it receives. For example, some biological systems show adaptation when they receive a low input signal while they show oscillation with a high input signal. We therefore set out to find all possible two-node and three-node networks that can perform both adaptation and oscillation with transcriptional regulation and enzymatic reactions. For two-node networks, we identified all bi-functional topologies by analyzing the Jacobean matrix. For three-node networks, they were identified by enumeration. We further investigated how the system can be transformed between these two functions. We found that the switching of functions can be achieved through changing anyone of the several key parameters, including the input signal level.

Statistical Dynamics of Spatial-Order Formation by Communicating Cells

Communicating cells can coordinate their gene expressions to form spatial patterns, generating order from disorder. Ubiquitous "secrete-and-sense cells" secrete and sense the same molecule to do so. Here we present a modeling framework-based on cellular automata and mimicking approaches of statistical mechanics-for understanding how secrete-and-sense cells with bistable gene expression, from disordered beginnings, can become spatially ordered by communicating through rapidly diffusing molecules. Classifying lattices of cells by two "macrostate" variables-"spatial index," measuring degree of order, and average gene-expression level-reveals a conceptual picture: a group of cells behaves as a single particle, in an abstract space, that rolls down on an adhesive "pseudo-energy landscape" whose shape is determined by cell-cell communication and an intracellular gene-regulatory circuit. Particles rolling down the landscape represent cells becoming more spatially ordered. We show how to extend this framework to more complex forms of cellular communication.

Collective behaviour and swarm intelligence in slime moulds

The study of collective behaviour aims to understand how individual-level behaviours can lead to complex group-level patterns. Collective behaviour has primarily been studied in animal groups such as colonies of insects, flocks of birds and schools of fish. Although less studied, collective behaviour also occurs in microorganisms. Here, we argue that slime moulds are powerful model systems for solving several outstanding questions in collective behaviour. In particular, slime mould may hold the key to linking individual-level mechanisms to colony-level behaviours. Using well-established principles of collective animal behaviour as a framework, we discuss the extent to which slime mould collectives are comparable to animal groups, and we highlight some potentially fruitful areas for future research.

Excitable Dynamics and Yap-Dependent Mechanical Cues Drive the Segmentation Clock

The periodic segmentation of the vertebrate body axis into somites, and later vertebrae, relies on a genetic oscillator (the segmentation clock) driving the rhythmic activity of signaling pathways in the presomitic mesoderm (PSM). To understand whether oscillations are an intrinsic property of individual cells or represent a population-level phenomenon, we established culture conditions for stable oscillations at the cellular level. This system was used to demonstrate that oscillations are a collective property of PSM cells that can be actively triggered in vitro by a dynamical quorum sensing signal involving Yap and Notch signaling. Manipulation of Yap-dependent mechanical cues is sufficient to predictably switch isolated PSM cells from a quiescent to an oscillatory state in vitro, a behavior reminiscent of excitability in other systems. Together, our work argues that the segmentation clock behaves as an excitable system, introducing a broader paradigm to study such dynamics in vertebrate morphogenesis.

Comprehensive benchmarking of Markov chain Monte Carlo methods for dynamical systems

BACKGROUND

In quantitative biology, mathematical models are used to describe and analyze biological processes. The parameters of these models are usually unknown and need to be estimated from experimental data using statistical methods. In particular, Markov chain Monte Carlo (MCMC) methods have become increasingly popular as they allow for a rigorous analysis of parameter and prediction uncertainties without the need for assuming parameter identifiability or removing non-identifiable parameters. A broad spectrum of MCMC algorithms have been proposed, including single- and multi-chain approaches. However, selecting and tuning sampling algorithms suited for a given problem remains challenging and a comprehensive comparison of different methods is so far not available.

RESULTS

We present the results of a thorough benchmarking of state-of-the-art single- and multi-chain sampling methods, including Adaptive Metropolis, Delayed Rejection Adaptive Metropolis, Metropolis adjusted Langevin algorithm, Parallel Tempering and Parallel Hierarchical Sampling. Different initialization and adaptation schemes are considered. To ensure a comprehensive and fair comparison, we consider problems with a range of features such as bifurcations, periodical orbits, multistability of steady-state solutions and chaotic regimes. These problem properties give rise to various posterior distributions including uni- and multi-modal distributions and non-normally distributed mode tails. For an objective comparison, we developed a pipeline for the semi-automatic comparison of sampling results.

CONCLUSION

The comparison of MCMC algorithms, initialization and adaptation schemes revealed that overall multi-chain algorithms perform better than single-chain algorithms. In some cases this performance can be further increased by using a preceding multi-start local optimization scheme. These results can inform the selection of sampling methods and the benchmark collection can serve for the evaluation of new algorithms. Furthermore, our results confirm the need to address exploration quality of MCMC chains before applying the commonly used quality measure of effective sample size to prevent false analysis conclusions.

Fold-change detection and scale invariance of cell-cell signaling in social amoeba

Cell-cell signaling is subject to variability in the extracellular volume, cell number, and dilution that potentially increase uncertainty in the absolute concentrations of the extracellular signaling molecules. To direct cell aggregation, the social amoebae Dictyostelium discoideum collectively give rise to oscillations and waves of cyclic adenosine 3',5'-monophosphate (cAMP) under a wide range of cell density. To date, the systems-level mechanism underlying the robustness is unclear. By using quantitative live-cell imaging, here we show that the magnitude of the cAMP relay response of individual cells is determined by fold change in the extracellular cAMP concentrations. The range of cell density and exogenous cAMP concentrations that support oscillations at the population level agrees well with conditions that support a large fold-change-dependent response at the single-cell level. Mathematical analysis suggests that invariance of the oscillations to density transformation is a natural outcome of combining secrete-and-sense systems with a fold-change detection mechanism.

A critical-like collective state leads to long-range cell communication in Dictyostelium discoideum aggregation

The transition from single-cell to multicellular behavior is important in early development but rarely studied. The starvation-induced aggregation of the social amoeba Dictyostelium discoideum into a multicellular slug is known to result from single-cell chemotaxis towards emitted pulses of cyclic adenosine monophosphate (cAMP). However, how exactly do transient, short-range chemical gradients lead to coherent collective movement at a macroscopic scale? Here, we developed a multiscale model verified by quantitative microscopy to describe behaviors ranging widely from chemotaxis and excitability of individual cells to aggregation of thousands of cells. To better understand the mechanism of long-range cell—cell communication and hence aggregation, we analyzed cell—cell correlations, showing evidence of self-organization at the onset of aggregation (as opposed to following a leader cell). Surprisingly, cell collectives, despite their finite size, show features of criticality known from phase transitions in physical systems. By comparing wild-type and mutant cells with impaired aggregation, we found the longest cell—cell communication distance in wild-type cells, suggesting that criticality provides an adaptive advantage and optimally sized aggregates for the dispersal of spores.

A multiscale model and imaging data show that cells of the slime mold Dictyostelium discoideum maximize their cell—cell communication range during aggregation by a critical-like state known from phase transitions in physical systems.

Cells are often coupled to each other in cell collectives, such as aggregates during early development, tissues in the developed organism, and tumors in disease. How do cells communicate over macroscopic distances much larger than the typical cell—cell distance to decide how they should behave? Here, we developed a multiscale model of social amoeba, spanning behavior from individuals to thousands of cells. We show that local cell—cell coupling via secreted chemicals may be tuned to a critical value, resulting in emergent long-range communication and heightened sensitivity. Hence, these aggregates are remarkably similar to bacterial biofilms and neuronal networks, all communicating in a pulselike fashion. Similar organizing principles may also aid our understanding of the remarkable robustness in cancer development.

New insights into mammalian signaling pathways using microfluidic pulsatile inputs and mathematical modeling

Temporally modulated input mimics physiology. This chemical communication strategy filters the biochemical noise through entrainment and phase-locking. Under laboratory conditions, it also expands the observability space for downstream responses. A combined approach involving microfluidic pulsatile stimulation and mathematical modeling has led to deciphering of hidden/unknown temporal motifs in several mammalian signaling pathways and has provided mechanistic insights, including how these motifs combine to form distinct band-pass filters and govern fate regulation under dynamic microenvironment. This approach can be utilized to understand signaling circuit architectures and to gain mechanistic insights for several other signaling systems. Potential applications include synthetic biology and biotechnology, in developing pharmaceutical interventions, and in developing lab-on-chip models.

The Influence of Cellular Characteristics on the Evolution of Shape Homeostasis

The importance of individual cells in a developing multicellular organism is well known, but precisely how the individual cellular characteristics of those cells collectively drive the emergence of robust, homeostatic structures is less well understood. For example, cell communication via a diffusible factor allows for information to travel across large distances within the population, and cell polarization makes it possible to form structures with a particular orientation, but how do these processes interact to produce a more robust and regulated structure? In this study we investigate the ability of cells with different cellular characteristics to grow and maintain homeostatic structures. We do this in the context of an individual-based model where cell behavior is driven by an intracellular network that determines the cell phenotype. More precisely, we investigated evolution with 96 different permutations of our model, where cell motility, cell death, long-range growth factor (LGF), short-range growth factor (SGF), and cell polarization were either present or absent. The results show that LGF has the largest positive influence on the fitness of the evolved solutions. SGF and polarization also contribute, but all other capabilities essentially increase the search space, effectively making it more difficult to achieve a solution. By perturbing the evolved solutions, we found that they are highly robust to both mutations and wounding. In addition, we observed that by evolving solutions in more unstable environments they produce structures that were more robust and adaptive. In conclusion, our results suggest that robust collective behavior is most likely to evolve when cells are endowed with long-range communication, cell polarisation, and selection pressure from an unstable environment.

Progress and perspectives in signal transduction, actin dynamics, and movement at the cell and tissue level: lessons from Dictyostelium

Movement of cells and tissues is a basic biological process that is used in development, wound repair, the immune response to bacterial invasion, tumour formation and metastasis, and the search for food and mates. While some cell movement is random, directed movement stimulated by extracellular signals is our focus here. This involves a sequence of steps in which cells first detect extracellular chemical and/or mechanical signals via membrane receptors that activate signal transduction cascades and produce intracellular signals. These intracellular signals control the motile machinery of the cell and thereby determine the spatial localization of the sites of force generation needed to produce directed motion. Understanding how force generation within cells and mechanical interactions with their surroundings, including other cells, are controlled in space and time to produce cell-level movement is a major challenge, and involves many issues that are amenable to mathematical modelling.

Collective Signal Processing in Cluster Chemotaxis: Roles of Adaptation, Amplification, and Co-attraction in Collective Guidance

Single eukaryotic cells commonly sense and follow chemical gradients, performing chemotaxis. Recent experiments and theories, however, show that even when single cells do not chemotax, clusters of cells may, if their interactions are regulated by the chemoattractant. We study this general mechanism of "collective guidance" computationally with models that integrate stochastic dynamics for individual cells with biochemical reactions within the cells, and diffusion of chemical signals between the cells. We show that if clusters of cells use the well-known local excitation, global inhibition (LEGI) mechanism to sense chemoattractant gradients, the speed of the cell cluster becomes non-monotonic in the cluster's size-clusters either larger or smaller than an optimal size will have lower speed. We argue that the cell cluster speed is a crucial readout of how the cluster processes chemotactic signals; both amplification and adaptation will alter the behavior of cluster speed as a function of size. We also show that, contrary to the assumptions of earlier theories, collective guidance does not require persistent cell-cell contacts and strong short range adhesion. If cell-cell adhesion is absent, and the cluster cohesion is instead provided by a co-attraction mechanism, e.g. chemotaxis toward a secreted molecule, collective guidance may still function. However, new behaviors, such as cluster rotation, may also appear in this case. Co-attraction and adaptation allow for collective guidance that is robust to varying chemoattractant concentrations while not requiring strong cell-cell adhesion.

Self-organization of chemoattractant waves in Dictyostelium depends on F-actin and cell-substrate adhesion

In the social amoeba Dictyostelium discoideum, travelling waves of extracellular cyclic adenosine monophosphate (cAMP) self-organize in cell populations and direct aggregation of individual cells to form multicellular fruiting bodies. In contrast to the large body of studies that addressed how movement of cells is determined by spatial and temporal cues encoded in the dynamic cAMP gradients, how cell mechanics affect the formation of a self-generated chemoattractant field has received less attention. Here, we show, by live cell imaging analysis, that the periodicity of the synchronized cAMP waves increases in cells treated with the actin inhibitor latrunculin. Detail analysis of the extracellular cAMP-induced transients of cytosolic cAMP (cAMP relay response) in well-isolated cells demonstrated that their amplitude and duration were markedly reduced in latrunculin-treated cells. Similarly, in cells strongly adhered to a poly-l-lysine-coated surface, the response was suppressed, and the periodicity of the population-level oscillations was markedly lengthened. Our results suggest that cortical F-actin is dispensable for the basic low amplitude relay response but essential for its full amplification and that this enhanced response is necessary to establish high-frequency signalling centres. The observed F-actin dependence may prevent aggregation centres from establishing in microenvironments that are incompatible with cell migration.

Modeling of the ComRS Signaling Pathway Reveals the Limiting Factors Controlling Competence in Streptococcus thermophilus

In streptococci, entry in competence is dictated by ComX abundance. In Streptococcus thermophilus, production of ComX is transient and tightly regulated during growth: it is positively regulated by the cell-cell communication system ComRS during the activation phase and negatively regulated during the shut-off phase by unidentified late competence gene(s). Interestingly, most S. thermophilus strains are not or weakly transformable in permissive growth conditions (i.e., chemically defined medium, CDM), suggesting that some players of the ComRS regulatory pathway are limiting. Here, we combined mathematical modeling and experimental approaches to identify the components of the ComRS system which are critical for both dynamics and amplitude of ComX production in S. thermophilus. We built a deterministic, population-scaled model of the time-course regulation of specific ComX production in CDM growth conditions. Strains LMD-9 and LMG18311 were respectively selected as representative of highly and weakly transformable strains. Results from in silico simulations and in vivo luciferase activities show that ComR concentration is the main limiting factor for the level of comX expression and controls the kinetics of spontaneous competence induction in strain LMD-9. In addition, the model predicts that the poor transformability of strain LMG18311 results from a 10-fold lower comR expression level compared to strain LMD-9. In agreement, comR overexpression in both strains was shown to induce higher competence levels with deregulated kinetics patterns during growth. In conclusion, we propose that the level of ComR production is one important factor that could explain competence heterogeneity among S. thermophilus strains.

Molecular-Level Tuning of Cellular Autonomy Controls the Collective Behaviors of Cell Populations

A rigorous understanding of how multicellular behaviors arise from the actions of single cells requires quantitative frameworks that bridge the gap between genetic circuits, the arrangement of cells in space, and population-level behaviors. Here, we provide such a framework for a ubiquitous class of multicellular systems-namely, "secrete-and-sense cells" that communicate by secreting and sensing a signaling molecule. By using formal, mathematical arguments and introducing the concept of a phenotype diagram, we show how these cells tune their degrees of autonomous and collective behavior to realize distinct single-cell and population-level phenotypes; these phenomena have biological analogs, such as quorum sensing or paracrine signaling. We also define the "entropy of population," a measurement of the number of arrangements that a population of cells can assume, and demonstrate how a decrease in the entropy of population accompanies the formation of ordered spatial patterns. Our conceptual framework ties together diverse systems, including tissues and microbes, with common principles.

Regulation of Spatiotemporal Patterns by Biological Variability: General Principles and Applications to Dictyostelium discoideum

Spatiotemporal patterns often emerge from local interactions in a self-organizing fashion. In biology, the resulting patterns are also subject to the influence of the systematic differences between the system’s constituents (biological variability). This regulation of spatiotemporal patterns by biological variability is the topic of our review. We discuss several examples of correlations between cell properties and the self-organized spatiotemporal patterns, together with their relevance for biology. Our guiding, illustrative example will be spiral waves of cAMP in a colony of Dictyostelium discoideum cells. Analogous processes take place in diverse situations (such as cardiac tissue, where spiral waves occur in potentially fatal ventricular fibrillation) so a deeper understanding of this additional layer of self-organized pattern formation would be beneficial to a wide range of applications. One of the most striking differences between pattern-forming systems in physics or chemistry and those in biology is the potential importance of variability. In the former, system components are essentially identical with random fluctuations determining the details of the self-organization process and the resulting patterns. In biology, due to variability, the properties of potentially very few cells can have a driving influence on the resulting asymptotic collective state of the colony. Variability is one means of implementing a few-element control on the collective mode. Regulatory architectures, parameters of signaling cascades, and properties of structure formation processes can be "reverse-engineered" from observed spatiotemporal patterns, as different types of regulation and forms of interactions between the constituents can lead to markedly different correlations. The power of this biology-inspired view of pattern formation lies in building a bridge between two scales: the patterns as a collective state of a very large number of cells on the one hand, and the internal parameters of the single cells on the other.

Pattern formation is abundant in nature—from the rich ornaments of sea shells and the diversity of animal coat patterns to the myriad of fractal structures in biology and pattern-forming colonies of bacteria. Particularly fascinating are patterns changing with time, spatiotemporal patterns, like propagating waves and aggregation streams. Bacteria form large branched and nested aggregation-like patterns to immobilize themselves against water flow. The individual amoeba in Dictyostelium discoideum colonies initiates a transition to a collective multicellular state via a quorum-sensing form of communication—a cAMP signal propagating through the community in the form of spiral waves—and the subsequent chemotactic response of the cells leads to branch-like aggregation streams. The theoretical principle underlying most of these spatial and spatiotemporal patterns is self-organization, in which local interactions lead to patterns as large-scale collective”modes” of the system. Over more than half a century, these patterns have been classified and analyzed according to a”physics paradigm,” investigating such questions as how parameters regulate the transitions among patterns, which (types of) interactions lead to such large-scale patterns, and whether there are "critical parameter values" marking the sharp, spontaneous onset of patterns. A fundamental discovery has been that simple local interaction rules can lead to complex large-scale patterns. The specific pattern "layouts" (i.e., their spatial arrangement and their geometric constraints) have received less attention. However, there is a major difference between patterns in physics and chemistry on the one hand and patterns in biology on the other: in biology, patterns often have an important functional role for the biological system and can be considered to be under evolutionary selection. From this perspective, we can expect that individual biological elements exert some control on the emerging patterns. Here we explore spiral wave patterns as a prominent example to illustrate the regulation of spatiotemporal patterns by biological variability. We propose a new approach to studying spatiotemporal data in biology: analyzing the correlation between the spatial distribution of the constituents’ properties and the features of the spatiotemporal pattern. This general concept is illustrated by simulated patterns and experimental data of a model organism of biological pattern formation, the slime mold Dictyostelium discoideum. We introduce patterns starting from Turing (stripe and spot) patterns, together with target waves and spiral waves. The biological relevance of these patterns is illustrated by snapshots from real and theoretical biological systems. The principles of spiral wave formation are first explored in a stylized cellular automaton model and then reproduced in a model of Dictyostelium signaling. The shaping of spatiotemporal patterns by biological variability (i.e., by a spatial distribution of cell-to-cell differences) is demonstrated, focusing on two Dictyostelium models. Building up on this foundation, we then discuss in more detail how the nonlinearities in biological models translate the distribution of cell properties into pattern events, leaving characteristic geometric signatures.

Intracellular Information Processing through Encoding and Decoding of Dynamic Signaling Features

Cell signaling dynamics and transcriptional regulatory activities are variable within specific cell types responding to an identical stimulus. In addition to studying the network interactions, there is much interest in utilizing single cell scale data to elucidate the non-random aspects of the variability involved in cellular decision making. Previous studies have considered the information transfer between the signaling and transcriptional domains based on an instantaneous relationship between the molecular activities. These studies predict a limited binary on/off encoding mechanism which underestimates the complexity of biological information processing, and hence the utility of single cell resolution data. Here we pursue a novel strategy that reformulates the information transfer problem as involving dynamic features of signaling rather than molecular abundances. We pursue a computational approach to test if and how the transcriptional regulatory activity patterns can be informative of the temporal history of signaling. Our analysis reveals (1) the dynamic features of signaling that significantly alter transcriptional regulatory patterns (encoding), and (2) the temporal history of signaling that can be inferred from single cell scale snapshots of transcriptional activity (decoding). Immediate early gene expression patterns were informative of signaling peak retention kinetics, whereas transcription factor activity patterns were informative of activation and deactivation kinetics of signaling. Moreover, the information processing aspects varied across the network, with each component encoding a selective subset of the dynamic signaling features. We developed novel sensitivity and information transfer maps to unravel the dynamic multiplexing of signaling features at each of these network components. Unsupervised clustering of the maps revealed two groups that aligned with network motifs distinguished by transcriptional feedforward vs feedback interactions. Our new computational methodology impacts the single cell scale experiments by identifying downstream snapshot measures required for inferring specific dynamical features of upstream signals involved in the regulation of cellular responses.

Single cell studies have shown that differential patterns in the dynamics of signaling proteins, transcription factor activity, gene expression, etc. produce distinct downstream outcomes. The opposite also holds true where particular cellular outcomes have been found to be associated with the dynamical pattern of one or more signaling molecules. Signaling pathways, therefore, serve as signal processing units to inform specific downstream regulation. However, the functional capabilities of the dynamic aspects of signaling are not well understood. To address this issue, we developed a new approach that evaluates information processing between dynamic features in signaling patterns and transcriptional regulatory activity. Our work demonstrates that the information transfer occur through decoding of temporal history of signals rather than only through instantaneous correlations. Moreover, our results identify regulatory network motifs as the critical components in the information processing and filtering of variability in signaling dynamics to produce distinct patterns of downstream transcriptional responses. Our methodology can be broadly applied to single cell scale data on experimentally accessible downstream measures to infer dynamic aspects of upstream signaling.

A collective path toward regeneration

How do cells collectively control an organ's behavior? By plucking various numbers of hairs from the mouse skin, Chen et al. show that hairs regenerate only when a sufficiently high density of them are plucked. Remarkably, a hair follicle can only regenerate in concert with other follicles, but not autonomously.

Modeling oscillations and spiral waves in Dictyostelium populations

Unicellular organisms exhibit elaborate collective behaviors in response to environmental cues. These behaviors are controlled by complex biochemical networks within individual cells and coordinated through cell-to-cell communication. Describing these behaviors requires new mathematical models that can bridge scales-from biochemical networks within individual cells to spatially structured cellular populations. Here we present a family of "multiscale" models for the emergence of spiral waves in the social amoeba Dictyostelium discoideum. Our models exploit new experimental advances that allow for the direct measurement and manipulation of the small signaling molecule cyclic adenosine monophosphate (cAMP) used by Dictyostelium cells to coordinate behavior in cellular populations. Inspired by recent experiments, we model the Dictyostelium signaling network as an excitable system coupled to various preprocessing modules. We use this family of models to study spatially unstructured populations of "fixed" cells by constructing phase diagrams that relate the properties of population-level oscillations to parameters in the underlying biochemical network. We then briefly discuss an extension of our model that includes spatial structure and show how this naturally gives rise to spiral waves. Our models exhibit a wide range of novel phenomena. including a density-dependent frequency change, bistability, and dynamic death due to slow cAMP dynamics. Our modeling approach provides a powerful tool for bridging scales in modeling of Dictyostelium populations.