Quantifying dynamic pro-inflammatory gene expression and heterogeneity in single macrophage cells

Macrophages must respond appropriately to pathogens and other pro-inflammatory stimuli in order to perform their roles in fighting infection. One way in which inflammatory stimuli can vary is in their dynamics-that is, the amplitude and duration of stimulus experienced by the cell. In this study, we performed long-term live cell imaging in a microfluidic device to investigate how the pro-inflammatory genes IRF1, CXCL10, and CXCL9 respond to dynamic interferon-gamma (IFNγ) stimulation. We found that IRF1 responds to low concentration or short duration IFNγ stimulation, whereas CXCL10 and CXCL9 require longer or higherconcentration stimulation to be expressed. We also investigated the heterogeneity in the expression of each gene and found that CXCL10 and CXCL9 have substantial cell-to-cell variability. In particular, the expression of CXCL10 appears to be largely stochastic with a subpopulation of nonresponding cells across all the stimulation conditions tested. We developed both deterministic and stochastic models for the expression of each gene. Our modeling analysis revealed that the heterogeneity in CXCL10 can be attributed to a slow chromatin-opening step that is on a similar timescale to that of adaptation of the upstream signal. In this way, CXCL10 expression in individual cells can remain stochastic in response to each pulse of repeated stimulation, which we also validated by experiments. Together, we conclude that pro-inflammatory genes in the same signaling pathway can respond to dynamic IFNγ stimulus with very different response features and that upstream signal adaptation can contribute to shaping heterogeneous gene expression.

Enhanced cellular longevity arising from environmental fluctuations

Cellular longevity is regulated by both genetic and environmental factors. However, the interactions of these factors in the context of aging remain largely unclear. Here, we formulate a mathematical model for dynamic glucose modulation of a core gene circuit in yeast aging, which not only guided the design of pro-longevity interventions, but also revealed the theoretical principles underlying these interventions. We introduce the dynamical systems theory to capture two general means for promoting longevity - the creation of a stable fixed point in the "healthy" state of the cell and the dynamic stabilization of the system around this healthy state through environmental oscillations. Guided by the model, we investigate how both of these can be experimentally realized by dynamically modulating environmental glucose levels. The results establish a paradigm for theoretically analyzing the trajectories and perturbations of aging that can be generalized to aging processes in diverse cell types and organisms.

Engineering longevity-design of a synthetic gene oscillator to slow cellular aging

Synthetic biology enables the design of gene networks to confer specific biological functions, yet it remains a challenge to rationally engineer a biological trait as complex as longevity. A naturally occurring toggle switch underlies fate decisions toward either nucleolar or mitochondrial decline during the aging of yeast cells. We rewired this endogenous toggle to engineer an autonomous genetic clock that generates sustained oscillations between the nucleolar and mitochondrial aging processes in individual cells. These oscillations increased cellular life span through the delay of the commitment to aging that resulted from either the loss of chromatin silencing or the depletion of heme. Our results establish a connection between gene network architecture and cellular longevity that could lead to rationally designed gene circuits that slow aging.

Statistical theory of asymmetric damage segregation in clonal cell populations

Asymmetric damage segregation (ADS) is ubiquitous among unicellular organisms: After a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. Previous studies demonstrated that ADS provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. In this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ADS models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. We show that for a large class of deterministic ADS rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. By exploiting the analogy of linear ADS models with the Iterated Function Systems known in chaos theory, we derive the Frobenius-Perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. We also investigate nonlinear and stochastic variants of ADS models and show the robustness of the salient features of the damage distributions.

Statistical Theory of Asymmetric Damage Segregation in Clonal Cell Populations

Asymmetric damage segregation (ADS) is ubiquitous among unicellular organisms: After a mother cell divides, its two daughter cells receive sometimes slightly, sometimes strongly different fractions of damaged proteins accumulated in the mother cell. Previous studies demonstrated that ADS provides a selective advantage over symmetrically dividing cells by rejuvenating and perpetuating the population as a whole. In this work we focus on the statistical properties of damage in individual lineages and the overall damage distributions in growing populations for a variety of ADS models with different rules governing damage accumulation, segregation, and the lifetime dependence on damage. We show that for a large class of deterministic ADS rules the trajectories of damage along the lineages are chaotic, and the distributions of damage in cells born at a given time asymptotically becomes fractal. By exploiting the analogy of linear ADS models with the Iterated Function Systems known in chaos theory, we derive the Frobenius-Perron equation for the stationary damage density distribution and analytically compute the damage distribution moments and fractal dimensions. We also investigate nonlinear and stochastic variants of ADS models and show the robustness of the salient features of the damage distributions.

Optimal transcriptional regulation of dynamic bacterial responses to sudden drug exposures

Cellular responses to the presence of toxic compounds in their environment require prompt expression of the correct levels of the appropriate enzymes, which are typically regulated by transcription factors that control gene expression for the duration of the response. The characteristics of each response dictate the choice of regulatory parameters such as the affinity of the transcription factor to its binding sites and the strength of the promoters it regulates. Although much is known about the dynamics of cellular responses, we still lack a framework to understand how different regulatory strategies evolved in natural systems relate to the selective pressures acting in each particular case. Here, we analyze a dynamical model of a typical antibiotic response in bacteria, where a transcriptionally repressed enzyme is induced by a sudden exposure to the drug that it processes. We identify strategies of gene regulation that optimize this response for different types of selective pressures, which we define as a set of costs associated with the drug, enzyme, and repressor concentrations during the response. We find that regulation happens in a limited region of the regulatory parameter space. While responses to more costly (toxic) drugs favor the usage of strongly self-regulated repressors, responses where expression of enzyme is more costly favor the usage of constitutively expressed repressors. Only a very narrow range of selective pressures favor weakly self-regulated repressors. We use this framework to determine which costs and benefits are most critical for the evolution of a variety of natural cellular responses that satisfy the approximations in our model and to analyze how regulation is optimized in new environments with different demands.

Age-dependent aggregation of ribosomal RNA-binding proteins links deterioration in chromatin stability with challenges to proteostasis

Chromatin instability and protein homeostasis (proteostasis) stress are two well-established hallmarks of aging, which have been considered largely independent of each other. Using microfluidics and single-cell imaging approaches, we observed that, during the replicative aging of Saccharomyces cerevisiae, a challenge to proteostasis occurs specifically in the fraction of cells with decreased stability within the ribosomal DNA (rDNA). A screen of 170 yeast RNA-binding proteins identified ribosomal RNA (rRNA)-binding proteins as the most enriched group that aggregate upon a decrease in rDNA stability induced by inhibition of a conserved lysine deacetylase Sir2. Further, loss of rDNA stability induces age-dependent aggregation of rRNA-binding proteins through aberrant overproduction of rRNAs. These aggregates contribute to age-induced proteostasis decline and limit cellular lifespan. Our findings reveal a mechanism underlying the interconnection between chromatin instability and proteostasis stress and highlight the importance of cell-to-cell variability in aging processes.

Publisher's Note: Suppression of Beneficial Mutations in Dynamic Microbial Populations [Phys. Rev. Lett. 118, 028102 (2017)]

This corrects the article DOI: 10.1103/PhysRevLett.118.028102.

A programmable fate decision landscape underlies single-cell aging in yeast

Chromatin instability and mitochondrial decline are conserved processes that contribute to cellular aging. Although both processes have been explored individually in the context of their distinct signaling pathways, the mechanism that determines which process dominates during aging of individual cells is unknown. We show that interactions between the chromatin silencing and mitochondrial pathways lead to an epigenetic landscape of yeast replicative aging with multiple equilibrium states that represent different types of terminal states of aging. The structure of the landscape drives single-cell differentiation toward one of these states during aging, whereby the fate is determined quite early and is insensitive to intracellular noise. Guided by a quantitative model of the aging landscape, we genetically engineered a long-lived equilibrium state characterized by an extended life span.

Genetically engineered control of phenotypic structure in microbial colonies

Rapid advances in cellular engineering^1,2 have positioned synthetic biology to address therapeutic^3,4 and industrial⁵ problems, but a substantial obstacle is the myriad of unanticipated cellular responses in heterogeneous real-world environments such as the gut^6,7, solid tumours^8,9, bioreactors¹⁰ or soil¹¹. Complex interactions between the environment and cells often arise through non-uniform nutrient availability, which generates bidirectional coupling as cells both adjust to and modify their local environment through phenotypic differentiation^12,13. Although synthetic spatial gene expression patterns^14-17 have been explored under homogeneous conditions, the mutual interaction of gene circuits, growth phenotype and the environment remains a challenge. Here, we design gene circuits that sense and control phenotypic structure in microcolonies containing both growing and dormant bacteria. We implement structure modulation by coupling different downstream modules to a tunable sensor that leverages Escherichia coli's stress response and is activated on growth arrest. One is an actuator module that slows growth and thereby alters nutrient gradients. Environmental feedback in this circuit generates robust cycling between growth and dormancy in the interior of the colony, as predicted by a spatiotemporal computational model. We also use the sensor to drive an inducible gating module for selective gene expression in non-dividing cells, which allows us to radically alter population structure by eliminating the dormant phenotype with a 'stress-gated lysis circuit'. Our results establish a strategy to leverage and control microbial colony structure for synthetic biology applications in complex environments.

Advances in quantitative biology methods for studying replicative aging in Saccharomyces cerevisiae

Aging is a complex, yet pervasive phenomenon in biology. As human cells steadily succumb to the deteriorating effects of aging, so too comes a host of age-related ailments such as neurodegenerative disorders, cardiovascular disease and cancer. Therefore, elucidation of the molecular networks that drive aging is of paramount importance to human health. Progress toward this goal has been aided by studies from simple model organisms such as Saccharomyces cerevisiae. While work in budding yeast has already revealed much about the basic biology of aging as well as a number of evolutionarily conserved pathways involved in this process, recent technological advances are poised to greatly expand our knowledge of aging in this simple eukaryote. Here, we review the latest developments in microfluidics, single-cell analysis and high-throughput technologies for studying single-cell replicative aging in S. cerevisiae. We detail the challenges each of these methods addresses as well as the unique insights into aging that each has provided. We conclude with a discussion of potential future applications of these techniques as well as the importance of single-cell dynamics and quantitative biology approaches for understanding cell aging.

Coexistence and Pattern Formation in Bacterial Mixtures with Contact-Dependent Killing

Multistrain microbial communities often exhibit complex spatial organization that emerges because of the interplay of various cooperative and competitive interaction mechanisms. One strong competitive mechanism is contact-dependent neighbor killing enabled by the type VI secretion system. It has been previously shown that contact-dependent killing can result in bistability of bacterial mixtures so that only one strain survives and displaces the other. However, it remains unclear whether stable coexistence is possible in such mixtures. Using a population dynamics model for two interacting bacterial strains, we found that coexistence can be made possible by the interplay of contact-dependent killing and long-range growth inhibition, leading to the formation of various cellular patterns. These patterns emerge in a much broader parameter range than that required for the linear Turing-like instability, suggesting this may be a robust mechanism for pattern formation.

Divergent Aging of Isogenic Yeast Cells Revealed through Single-Cell Phenotypic Dynamics

Although genetic mutations that alter organisms' average lifespans have been identified in aging research, our understanding of the dynamic changes during aging remains limited. Here, we integrate single-cell imaging, microfluidics, and computational modeling to investigate phenotypic divergence and cellular heterogeneity during replicative aging of single S. cerevisiae cells. Specifically, we find that isogenic cells diverge early in life toward one of two aging paths, which are characterized by distinct age-associated phenotypes. We captured the dynamics of single cells along the paths with a stochastic discrete-state model, which accurately predicts both the measured heterogeneity and the lifespan of cells on each path within a cell population. Our analysis suggests that genetic and environmental factors influence both a cell's choice of paths and the kinetics of paths themselves. Given that these factors are highly conserved throughout eukaryotes, divergent aging might represent a general scheme in cellular aging of other organisms.

Flavin-based metabolic cycles are integral features of growth and division in single yeast cells

The yeast metabolic cycle (YMC) is a fascinating example of biological organization, in which cells constrain the function of specific genetic, protein and metabolic networks to precise temporal windows as they grow and divide. However, understanding the intracellular origins of the YMC remains a challenging goal, as measuring the oxygen oscillations traditionally associated with it requires the use of synchronized cultures growing in nutrient-limited chemostat environments. To address these limitations, we used custom-built microfluidic devices and time-lapse fluorescence microscopy to search for metabolic cycling in the form of endogenous flavin fluorescence in unsynchronized single yeast cells. We uncovered robust and pervasive metabolic cycles that were synchronized with the cell division cycle (CDC) and oscillated across four different nutrient conditions. We then studied the response of these metabolic cycles to chemical and genetic perturbations, showing that their phase synchronization with the CDC can be altered through treatment with rapamycin, and that metabolic cycles continue even in respiratory deficient strains. These results provide a foundation for future studies of the physiological importance of metabolic cycles in processes such as CDC control, metabolic regulation and cell aging.

Rational engineering of synthetic microbial systems: from single cells to consortia

One promise of synthetic biology is to provide solutions for biomedical and industrial problems by rational design of added functionality in living systems. Microbes are at the forefront of this biological engineering endeavor due to their general ease of handling and their relevance in many potential applications from fermentation to therapeutics. In recent years, the field has witnessed an explosion of novel regulatory tools, from synthetic orthogonal transcription factors to posttranslational mechanisms for increased control over the behavior of synthetic circuits. Tool development has been paralleled by the discovery of principles that enable increased modularity and the management of host-circuit interactions. Engineered cell-to-cell communication bridges the scales from intracellular to population-level coordination. These developments facilitate the translation of more than a decade of circuit design into applications.

Multigenerational silencing dynamics control cell aging

Cellular aging plays an important role in many diseases, such as cancers, metabolic syndromes, and neurodegenerative disorders. There has been steady progress in identifying aging-related factors such as reactive oxygen species and genomic instability, yet an emerging challenge is to reconcile the contributions of these factors with the fact that genetically identical cells can age at significantly different rates. Such complexity requires single-cell analyses designed to unravel the interplay of aging dynamics and cell-to-cell variability. Here we use microfluidic technologies to track the replicative aging of single yeast cells and reveal that the temporal patterns of heterochromatin silencing loss regulate cellular life span. We found that cells show sporadic waves of silencing loss in the heterochromatic ribosomal DNA during the early phases of aging, followed by sustained loss of silencing preceding cell death. Isogenic cells have different lengths of the early intermittent silencing phase that largely determine their final life spans. Combining computational modeling and experimental approaches, we found that the intermittent silencing dynamics is important for longevity and is dependent on the conserved Sir2 deacetylase, whereas either sustained silencing or sustained loss of silencing shortens life span. These findings reveal that the temporal patterns of a key molecular process can directly influence cellular aging, and thus could provide guidance for the design of temporally controlled strategies to extend life span.

Criticality and Adaptivity in Enzymatic Networks

The contrast between the stochasticity of biochemical networks and the regularity of cellular behavior suggests that biological networks generate robust behavior from noisy constituents. Identifying the mechanisms that confer this ability on biological networks is essential to understanding cells. Here we show that queueing for a limited shared resource in broad classes of enzymatic networks in certain conditions leads to a critical state characterized by strong and long-ranged correlations between molecular species. An enzymatic network reaches this critical state when the input flux of its substrate is balanced by the maximum processing capacity of the network. We then consider enzymatic networks with adaptation, when the limiting resource (enzyme or cofactor) is produced in proportion to the demand for it. We show that the critical state becomes an attractor for these networks, which points toward the onset of self-organized criticality. We suggest that the adaptive queueing motif that leads to significant correlations between multiple species may be widespread in biological systems.

Coupled feedback loops control the stimulus-dependent dynamics of the yeast transcription factor Msn2

Information about environmental stimuli often can be encoded by the dynamics of signaling molecules or transcription factors. In the yeast Saccharomyces cerevisiae, different types of stresses induce distinct nuclear translocation dynamics of the general stress-responsive transcription factor Msn2, but the underlying mechanisms remain unclear. Using deterministic and stochastic modeling, we reproduced in silico the different dynamic responses of Msn2 to glucose limitation and osmotic stress observed in vivo and found that a positive feedback loop on protein kinase A mediated by the AMP-activated protein kinase Snf1 is coupled with a negative feedback loop to generate the characteristic pulsatile dynamics of Msn2. The model predicted that the stimulus-specific positive feedback loop could be responsible for the difference between Msn2 dynamics induced by glucose limitation and osmotic stress. This prediction was further verified experimentally by time-lapse microscopic examinations of the snf1Δ strain. In this mutant lacking the Snf1-mediated positive feedback loop, Msn2 responds similarly to glucose limitation and osmotic stress, and its pulsatile translocation is largely abrogated. Our combined computational and experimental analysis reveals a regulatory mechanism by which cells can encode information about environmental cues into distinct signaling dynamics through stimulus-specific network architectures.

Suppression of Beneficial Mutations in Dynamic Microbial Populations

Quantitative predictions for the spread of mutations in bacterial populations are essential to interpret evolution experiments and to improve the stability of synthetic gene circuits. We derive analytical expressions for the suppression factor for beneficial mutations in populations that undergo periodic dilutions, covering arbitrary population sizes, dilution factors, and growth advantages in a single stochastic model. We find that the suppression factor grows with the dilution factor and depends nontrivially on the growth advantage, resulting in the preferential elimination of mutations with certain growth advantages. We confirm our results by extensive numerical simulations.

Synchronized cycles of bacterial lysis for in vivo delivery

The pervasive view of bacteria as strictly pathogenic has given way to an appreciation of the widespread prevalence of beneficial microbes within the human body^1–3. Given this milieu, it is perhaps inevitable that some bacteria would evolve to preferentially grow in environments that harbor disease and thus provide a natural platform for the development of engineered therapies^4–6. Such therapies could benefit from bacteria that are programmed to limit bacterial growth while continually producing and releasing cytotoxic agents in situ^7–10. Here, we engineer a clinically relevant bacterium to lyse synchronously at a threshold population density and to release genetically encoded cargo. Following quorum lysis, a small number of surviving bacteria reseed the growing population, thus leading to pulsatile delivery cycles. We use microfluidic devices to characterize the engineered lysis strain and we demonstrate its potential as a drug delivery platform via co-culture with human cancer cells in vitro. As a proof of principle, we track the bacterial population dynamics in ectopic syngeneic colorectal tumors in mice. The lysis strain exhibits pulsatile population dynamics in vivo, with mean bacterial luminescence that remained two orders of magnitude lower than an unmodified strain. Finally, guided by previous findings that certain bacteria can enhance the efficacy of standard therapies¹¹, we orally administer the lysis strain, alone or in combination with a clinical chemotherapeutic, to a syngeneic transplantation model of hepatic colorectal metastases. We find that the combination of both circuit-engineered bacteria and chemotherapy leads to a notable reduction of tumor activity along with a marked survival benefit over either therapy alone. Our approach establishes a methodology for leveraging the tools of synthetic biology to exploit the natural propensity for certain bacteria to colonize disease sites.

Considering the kinetics of mRNA synthesis in the analysis of the genome and epigenome reveals determinants of co-transcriptional splicing

When messenger RNA splicing occurs co-transcriptionally, the potential for kinetic control based on transcription dynamics is widely recognized. Indeed, perturbation studies have reported that when transcription kinetics are perturbed genetically or pharmacologically splice patterns may change. However, whether kinetic control is contributing to the control of splicing within the normal range of physiological conditions remains unknown. We examined if the kinetic determinants for co-transcriptional splicing (CTS) might be reflected in the structure and expression patterns of the genome and epigenome. To identify and then quantitatively relate multiple, simultaneous CTS determinants, we constructed a scalable mathematical model of the kinetic interplay of RNA synthesis and CTS and parameterized it with diverse next generation sequencing (NGS) data. We thus found a variety of CTS determinants encoded in vertebrate genomes and epigenomes, and that these combine variously for different groups of genes such as housekeeping versus regulated genes. Together, our findings indicate that the kinetic basis of splicing is functionally and physiologically relevant, and may meaningfully inform the analysis of genomic and epigenomic data to provide insights that are missed when relying on statistical approaches alone.

Synchronization of degrade-and-fire oscillations via a common activator

The development of synthetic gene oscillators has not only demonstrated our ability to forward engineer reliable circuits in living cells, but it has also proven to be an excellent testing ground for the statistical behavior of coupled noisy oscillators. Previous experimental studies demonstrated that a shared positive feedback can reliably synchronize such oscillators, though the theoretical mechanism was not studied in detail. In the present work, we examine an experimentally motivated stochastic model for coupled degrade-and-fire gene oscillators, where a core delayed negative feedback establishes oscillations within each cell, and a shared delayed positive feedback couples all cells. We use analytic and numerical techniques to investigate conditions for one cluster and multicluster synchrony. A nonzero delay in the shared positive feedback, as expected for the experimental systems, is found to be important for synchrony to occur.

Noise in biology

Noise permeates biology on all levels, from the most basic molecular, sub-cellular processes to the dynamics of tissues, organs, organisms and populations. The functional roles of noise in biological processes can vary greatly. Along with standard, entropy-increasing effects of producing random mutations, diversifying phenotypes in isogenic populations, limiting information capacity of signaling relays, it occasionally plays more surprising constructive roles by accelerating the pace of evolution, providing selective advantage in dynamic environments, enhancing intracellular transport of biomolecules and increasing information capacity of signaling pathways. This short review covers the recent progress in understanding mechanisms and effects of fluctuations in biological systems of different scales and the basic approaches to their mathematical modeling.

Translational cross talk in gene networks

It has been shown experimentally that competition for limited translational resources by upstream mRNAs can lead to an anticorrelation between protein counts. Here, we investigate a stochastic model for this phenomenon, in which gene transcripts of different types compete for a finite pool of ribosomes. Throughout, we utilize concepts from the theory of multiclass queues to describe a qualitative shift in protein count statistics as the system transitions from being underloaded (ribosomes exceed transcripts in number) to being overloaded (transcripts exceed ribosomes in number). The exact analytical solution of a simplified stochastic model, in which the numbers of competing mRNAs and ribosomes are fixed, exhibits weak positive correlations between steady-state protein counts when total transcript count slightly exceeds ribosome count, whereas the solution can exhibit strong negative correlations when total transcript count significantly exceeds ribosome count. Extending this analysis, we find approximate but reasonably accurate solutions for a more realistic model, in which abundances of mRNAs and ribosomes are allowed to fluctuate randomly. Here, ribosomal fluctuations contribute positively and mRNA fluctuations contribute negatively to correlations, and when mRNA fluctuations dominate ribosomal fluctuations, a strong anticorrelation extremum reliably occurs near the transition from the underloaded to the overloaded regime.

Measuring competitive fitness in dynamic environments

Most yeast genes are dispensable for optimal growth in laboratory cultures. However, this apparent lack of fitness contribution is difficult to reconcile with the theory of natural selection. Here we use stochastic modeling to show that environmental fluctuations can select for a genetic mechanism that does not affect growth in static laboratory environments. We then present a novel experimental platform for measuring the fitness levels of specific genotypes in fluctuating environments. We test this platform by monitoring a mixed culture of two yeast strains that differ in their ability to respond to changes in carbon source yet exhibit the same fitness level in static conditions. When the sugar in the growth medium was switched between galactose and glucose, the wild-type strain gained a growth advantage over the mutant strain. Interestingly, both our computational and experimental results show that the strength of the adaptive advantage conveyed by the wild-type genotype depends on the total number of carbon source switches, not on the frequency of these fluctuations. Our results illustrate the selective power of environmental fluctuations on seemingly slight phenotypic differences in cellular response dynamics and underscore the importance of dynamic processes in the evolution of species.

Dual delayed feedback provides sensitivity and robustness to the NF-κB signaling module

Many cellular stress-responsive signaling systems exhibit highly dynamic behavior with oscillatory features mediated by delayed negative feedback loops. What remains unclear is whether oscillatory behavior is the basis for a signaling code based on frequency modulation (FM) or whether the negative feedback control modules have evolved to fulfill other functional requirements. Here, we use experimentally calibrated computational models to interrogate the negative feedback loops that regulate the dynamic activity of the transcription factor NF-κB. Linear stability analysis of the model shows that oscillatory frequency is a hard-wired feature of the primary negative feedback loop and not a function of the stimulus, thus arguing against an FM signaling code. Instead, our modeling studies suggest that the two feedback loops may be tuned to provide for rapid activation and inactivation capabilities for transient input signals of a wide range of durations; by minimizing late phase oscillations response durations may be fine-tuned in a graded rather than quantized manner. Further, in the presence of molecular noise the dual delayed negative feedback system minimizes stochastic excursions of the output to produce a robust NF-κB response.

Typical trajectories of coupled degrade-and-fire oscillators: from dispersed populations to massive clustering

We consider the dynamics of a piecewise affine system of degrade-and-fire oscillators with global repressive interaction, inspired by experiments on synchronization in colonies of bacteria-embedded genetic circuits. Due to global coupling, if any two oscillators happen to be in the same state at some time, they remain in sync at all subsequent times; thus clusters of synchronized oscillators cannot shrink as a result of the dynamics. Assuming that the system is initiated from random initial configurations of fully dispersed populations (no clusters), we estimate asymptotic cluster sizes as a function of the coupling strength. A sharp transition is proved to exist that separates a weak coupling regime of unclustered populations from a strong coupling phase where clusters of extensive size are formed. Each phenomena occurs with full probability in the thermodynamics limit. Moreover, the maximum number of asymptotic clusters is known to diverge linearly in this limit. In contrast, we show that with positive probability, the number of asymptotic clusters remains bounded, provided that the coupling strength is sufficiently large.

Continuum modeling of clustering of myxobacteria

In this paper we develop a continuum theory of clustering in ensembles of self-propelled inelastically colliding rods with applications to collective dynamics of common gliding bacteria Myxococcus Xanthus. A multiphase hydrodynamic model that couples densities of oriented and isotropic phases is described. This model is used for the analysis of an instability that leads to spontaneous formation of directionally moving dense clusters within initially dilute isotropic "gas" of myxobacteria. Numerical simulations of this model confirm the existence of stationary dense moving clusters and also elucidate the properties of their collisions. The results are shown to be in a qualitative agreement with experiments.

Vacuum-assisted cell loading enables shear-free mammalian microfluidic culture

Microfluidic perfusion cultures for mammalian cells provide a novel means for probing single-cell behavior but require the management of culture parameters such as flow-induced shear stress. Methods to eliminate shear stress generally focus on capturing cells in regions with high resistance to fluid flow. Here, we present a novel trapping design to easily and reliably load a high density of cells into culture chambers that are extremely isolated from potentially damaging flow effects. We utilize a transient on-chip vacuum to remove air from the culture chambers and rapidly replace the volume with a liquid cell suspension. We demonstrate the ability of this simple and robust method to load and culture three commonly used cell lines. We show how the incorporation of an on-chip function generator can be used for dynamic stimulation of cells during long-term continuous perfusion culture.

Fast stochastic algorithm for simulating evolutionary population dynamics

MOTIVATION

Many important aspects of evolutionary dynamics can only be addressed through simulations. However, accurate simulations of realistically large populations over long periods of time needed for evolution to proceed are computationally expensive. Mutants can be present in very small numbers and yet (if they are more fit than others) be the key part of the evolutionary process. This leads to significant stochasticity that needs to be accounted for. Different evolutionary events occur at very different time scales: mutations are typically much rarer than reproduction and deaths.

RESULTS

We introduce a new exact algorithm for fast fully stochastic simulations of evolutionary dynamics that include birth, death and mutation events. It produces a significant speedup compared to direct stochastic simulations in a typical case when the population size is large and the mutation rates are much smaller than birth and death rates. The algorithm performance is illustrated by several examples that include evolution on a smooth and rugged fitness landscape. We also show how this algorithm can be adapted for approximate simulations of more complex evolutionary problems and illustrate it by simulations of a stochastic competitive growth model.

Recent advances in single-cell studies of gene regulation

A mechanistic understanding of gene regulatory network dynamics requires quantitative single-cell data of multiple network components in response to well-defined perturbations. Recent advances in the development of fluorescent biomarkers for proteins, detection of RNA and interactions, microfluidic technology, and high-resolution imaging have set the stage for a host of new studies that elucidate the important roles of stochasticity and cell-cell variability in response to external perturbations. In this review, we briefly describe methods for high-resolution visualization and the control of gene expression, along with application of these novel methods to recent studies involving gene networks.

Corepressive interaction and clustering of degrade-and-fire oscillators

Strongly nonlinear degrade-and-fire (DF) oscillations may emerge in genetic circuits having a delayed negative feedback loop as their core element. Here we study the synchronization of DF oscillators coupled through a common repressor field. For weak coupling, initially distinct oscillators remain desynchronized. For stronger coupling, oscillators can be forced to wait in the repressed state until the global repressor field is sufficiently degraded, and then they fire simultaneously forming a synchronized cluster. Our analytical theory provides necessary and sufficient conditions for clustering and specifies the maximum number of clusters that can be formed in the asymptotic regime. We find that in the thermodynamic limit a phase transition occurs at a certain coupling strength from the weakly clustered regime with only microscopic clusters to a strongly clustered regime where at least one giant cluster has to be present.

Buckling instability in ordered bacterial colonies

Bacterial colonies often exhibit complex spatio-temporal organization. This collective behavior is affected by a multitude of factors ranging from the properties of individual cells (shape, motility, membrane structure) to chemotaxis and other means of cell-cell communication. One of the important but often overlooked mechanisms of spatio-temporal organization is direct mechanical contact among cells in dense colonies such as biofilms. While in natural habitats all these different mechanisms and factors act in concert, one can use laboratory cell cultures to study certain mechanisms in isolation. Recent work demonstrated that growth and ensuing expansion flow of rod-like bacteria Escherichia coli in confined environments leads to orientation of cells along the flow direction and thus to ordering of cells. However, the cell orientational ordering remained imperfect. In this paper we study one mechanism responsible for the persistence of disorder in growing cell populations. We demonstrate experimentally that a growing colony of nematically ordered cells is prone to the buckling instability. Our theoretical analysis and discrete-element simulations suggest that the nature of this instability is related to the anisotropy of the stress tensor in the ordered cell colony.

Correlation resonance generated by coupled enzymatic processing

A major challenge for systems biology is to deduce the molecular interactions that underlie correlations observed between concentrations of different intracellular molecules. Although direct explanations such as coupled transcription or direct protein-protein interactions are often considered, potential indirect sources of coupling have received much less attention. Here we show how correlations can arise generically from a posttranslational coupling mechanism involving the processing of multiple protein species by a common enzyme. By observing a connection between a stochastic model and a multiclass queue, we obtain a closed form expression for the steady-state distribution of the numbers of molecules of each protein species. Upon deriving explicit analytic expressions for moments and correlations associated with this distribution, we discover a striking phenomenon that we call correlation resonance: for small dilution rate, correlations peak near the balance-point where the total rate of influx of proteins into the system is equal to the maximum processing capacity of the enzyme. Given the limited number of many important catalytic molecules, our results may lead to new insights into the origin of correlated behavior on a global scale.

Streaming instability in growing cell populations

Flows of cells growing as a quasimonolayer in a confined space can exhibit streaming, with narrow streams of fast-moving cells flowing around clusters of slowly moving cells. We observed and analyzed this phenomenon experimentally for E. coli bacteria proliferating in a microfluidic cell trap using time-lapse microscopy. We also performed continuum modeling and discrete-element simulations to elucidate the mechanism behind the streaming instability. Our analysis demonstrates that streaming can be explained by the interplay between the slow adaptation of a cell to its local microenvironment and its mobility due to changes of cell-substrate contact forces.

Cell cycle-dependent variations in protein concentration

Computational modeling of biological systems has become an effective tool for analyzing cellular behavior and for elucidating key properties of the intricate networks that underlie experimental observations. While most modeling techniques rely heavily on the concentrations of intracellular molecules, little attention has been paid to tracking and simulating the significant volume fluctuations that occur over each cell division cycle. Here, we use fluorescence microscopy to acquire single cell volume trajectories for a large population of Saccharomyces cerevisiae cells. Using this data, we generate a comprehensive set of statistics that govern the growth and division of these cells over many generations, and we discover several interesting trends in their size, growth and protein production characteristics. We use these statistics to develop an accurate model of cell cycle volume dynamics, starting at cell birth. Finally, we demonstrate the importance of tracking volume fluctuations by combining cell division dynamics with a minimal gene expression model for a constitutively expressed fluorescent protein. The significant oscillations in the cellular concentration of a stable, highly expressed protein mimic the observed experimental trajectories and demonstrate the fundamental impact that the cell cycle has on cellular functions.

Effect of noise on solid-to-liquid transition in small granular systems under shear

The effect of noise on the solid-to-liquid transition of a dense granular assembly under planar shear is studied numerically using soft-particle molecular dynamics simulations in two dimensions. We focus on small systems in a thin planar Couette cell, examining the bistable region while increasing shear, with varying amounts of random noise, and determine statistics of the shear required for fluidization. In the absence of noise, the threshold value of the shear stress depends on the preparation of the system and has a broad distribution. However, adding force fluctuations both lowers the mean threshold value of the shear stress and decreases its variability. This behavior is interpreted as thermoactivated escape through a fluctuating barrier.

Delay-induced degrade-and-fire oscillations in small genetic circuits

Robust oscillations have recently been observed in a synthetic gene network composed of coupled positive and negative feedback loops. Here we use deterministic and stochastic modeling to investigate how a small time delay in such regulatory networks can lead to strongly nonlinear oscillations that can be characterized by "degrade-and-fire" dynamics. We show that the period of the oscillations can be significantly greater than the delay time, provided the circuit components possess strong activation and tight repression. The variability of the period is strongly influenced by fluctuations near the oscillatory minima, when the number of regulatory molecules is small.

A fast, robust and tunable synthetic gene oscillator

One defining goal of synthetic biology is the development of engineering-based approaches that enable the construction of gene-regulatory networks according to “design specs” generated from computational modeling^1–6. This approach provides a systematic framework for exploring how a given regulatory network generates a particular phenotypic behavior. Several fundamental gene circuits have been developed using this approach, including toggle switches⁷ and oscillators^8–10, and these have been applied in novel contexts such as triggered biofilm development¹¹ and cellular population control¹². Here we describe an engineered genetic oscillator in Escherichia coli that is fast, robust, and persistent, with tunable oscillatory periods as fast as 13 minutes. The oscillator was designed using a previously modeled network architecture comprising linked positive and negative feedback loops^1,13. Using a microfluidic platform tailored for single-cell microscopy, we precisely control environmental conditions and monitor oscillations in individual cells through multiple cycles. Experiments reveal remarkable robustness and persistence of oscillations in the designed circuit; almost every cell exhibited large-amplitude fluorescence oscillations throughout observation runs. The oscillatory period can be tuned by altering inducer levels, temperature, and media source. Computational modeling demonstrates that the key design principle for constructing a robust oscillator is a time delay in the negative feedback loop, which can mechanistically arise from the cascade of cellular processes involved in forming a functional transcription factor. The positive feedback loop increases the robustness of the oscillations and allows for greater tunability. Examination of our refined model suggested the existence of a simplified oscillator design without positive feedback, and we construct an oscillator strain confirming this computational prediction.

Nonlocal rheological properties of granular flows near a jamming limit

We study the rheology of sheared granular flows close to a jamming transition. We use the approach of partially fluidized theory (PFT) with a full set of equations extending the thin layer approximation derived previously for the description of the granular avalanches phenomenology. This theory provides a picture compatible with a local rheology at large shear rates [G. D. R. Midi, Eur. Phys. J. E 14, 341 (2004)] and it works in the vicinity of the jamming transition, where a description in terms of a simple local rheology comes short. We investigate two situations displaying important deviations from local rheology. The first one is based on a set of numerical simulations of sheared soft two-dimensional circular grains. The next case describes previous experimental results obtained on avalanches of sandy material flowing down an incline. Both cases display, close to jamming, significant deviations from the now standard Pouliquen's flow rule [O. Pouliquen, Phys. Fluids 11, 542 (1999); 11, 1956 (1999)]. This discrepancy is the hallmark of a strongly nonlocal rheology and in both cases, we relate the empirical results and the outcomes of PFT. The numerical simulations show a characteristic constitutive structure for the fluid part of the stress involving the confining pressure and the material stiffness that appear in the form of an additional dimensionless parameter. This constitutive relation is then used to describe the case of sandy flows. We show a quantitative agreement as far as the effective flow rules are concerned. A fundamental feature is identified in PFT as the existence of a jammed layer developing in the vicinity of the flow arrest that corroborates the experimental findings. Finally, we study the case of solitary erosive granular avalanches and relate the outcome with the PFT analysis.

Athermal dynamics of strongly coupled stochastic three-state oscillators

We study the behavior of globally coupled ensembles of cyclic stochastic three-state units with transition rates from i-1 to i proportional to the number of units in state i. Contrary to mean-field theory predictions, numerical simulations show significant stochastic oscillations for sufficiently large coupling strength. The order parameter characterizing units synchrony increases monotonically with coupling while the coherence of oscillations has a maximum at a certain coupling strength. We find the exact formulas for the stationary probability distribution and the order parameter.

Swarming and swirling in self-propelled polar granular rods

Using experiments with anisotropic vibrated rods and quasi-2D numerical simulations, we show that shape plays an important role in the collective dynamics of self-propelled (SP) particles. We demonstrate that SP rods exhibit local ordering, aggregation at the side walls, and clustering absent in round SP particles. Furthermore, we find that at sufficiently strong excitation SP rods engage in a persistent swirling motion in which the velocity is strongly correlated with particle orientation.

Interactions of semiflexible filaments and molecular motors

This paper summarizes the results of numerical simulations of the interaction of a pair of biofilaments mediated by a molecular motor. The filaments are modeled as flexible rods, and the results are applicable to microtubules, which are relatively stiff, as well as to much softer filaments, such as actin. The results provide insight into the effects of flexibility on cytoskeleton formation and the rheology of semiflexible filament networks. The simulations are based on a nonlinear elasticity equation. The results show that flexibility enhances the tendency of filaments to align. The enhancement in turn favors the formation of large-scale structures in multifilament systems. Simulations for soft filaments show that the action of the motor can result in the formation of multiple loops of the filaments as a result of buckling, which can affect the structure of a cross-linked network and thereby its rheology. The estimate for the minimal buckling length as a function of the motor speed, the viscosity of the solvent, and the bending stiffness of the filament is derived analytically.

Swirling motion in a system of vibrated elongated particles

Large-scale collective motion emerging in a monolayer of vertically vibrated elongated particles is studied. The motion is characterized by recurring swirls, with the characteristic scale exceeding several times the size of an individual particle. Our experiments identified a small horizontal component of the oscillatory acceleration of the vibrating plate in combination with orientation-dependent bottom friction (with respect to horizontal acceleration) as a source for the swirl formation. We developed a continuum model operating with the velocity field and local alignment tensor, which is in qualitative agreement with the experiment.

Theory of self-assembly of microtubules and motors

We derive a model describing spatiotemporal organization of an array of microtubules interacting via molecular motors. Starting from a stochastic model of inelastic polar rods with a generic anisotropic interaction kernel, we obtain a set of equations for the local rods concentration and orientation. At large enough mean density of rods and concentration of motors, the model describes an orientational instability. We demonstrate that the orientational instability leads to the formation of vortices and (for large density and/or kernel anisotropy) asters seen in recent experiments. We derive the specific form of the interaction kernel from the detailed analysis of microscopic interaction of two filaments mediated by a moving molecular motor and extend our results to include variable motor density and motor attachment to the substrate.

Thermal collapse of a granular gas under gravity

Free cooling of a gas of inelastically colliding hard spheres represents a central paradigm of kinetic theory of granular gases. At zero gravity the temperature of a freely cooling homogeneous granular gas follows a power law in time. How does gravity, which brings inhomogeneity, affect the cooling? We combine molecular dynamics simulations, a numerical solution of hydrodynamic equations and an analytic theory to show that a granular gas cooling under gravity undergoes thermal collapse: it cools down to zero temperature and condenses on the bottom of the container in a finite time.

Stochastically driven genetic circuits

Transcriptional regulation in small genetic circuits exhibits large stochastic fluctuations. Recent experiments have shown that a significant fraction of these fluctuations is caused by extrinsic factors. In this paper we review several theoretical and computational approaches to modeling of small genetic circuits driven by extrinsic stochastic processes. We propose a simplified approach to this problem, which can be used in the case when extrinsic fluctuations dominate the stochastic dynamics of the circuit (as appears to be the case in eukaryots). This approach is applied to a model of a single nonregulated gene that is driven by a certain gating process that affects the rate of transcription, and to a simplified version of the galactose utilization circuit in yeast.

Origins of extrinsic variability in eukaryotic gene expression

Variable gene expression within a clonal population of cells has been implicated in a number of important processes including mutation and evolution, determination of cell fates and the development of genetic disease. Recent studies have demonstrated that a significant component of expression variability arises from extrinsic factors thought to influence multiple genes simultaneously, yet the biological origins of this extrinsic variability have received little attention. Here we combine computational modelling with fluorescence data generated from multiple promoter-gene inserts in Saccharomyces cerevisiae to identify two major sources of extrinsic variability. One unavoidable source arising from the coupling of gene expression with population dynamics leads to a ubiquitous lower limit for expression variability. A second source, which is modelled as originating from a common upstream transcription factor, exemplifies how regulatory networks can convert noise in upstream regulator expression into extrinsic noise at the output of a target gene. Our results highlight the importance of the interplay of gene regulatory networks with population heterogeneity for understanding the origins of cellular diversity.

Delay-induced stochastic oscillations in gene regulation

The small number of reactant molecules involved in gene regulation can lead to significant fluctuations in intracellular mRNA and protein concentrations, and there have been numerous recent studies devoted to the consequences of such noise at the regulatory level. Theoretical and computational work on stochastic gene expression has tended to focus on instantaneous transcriptional and translational events, whereas the role of realistic delay times in these stochastic processes has received little attention. Here, we explore the combined effects of time delay and intrinsic noise on gene regulation. Beginning with a set of biochemical reactions, some of which are delayed, we deduce a truncated master equation for the reactive system and derive an analytical expression for the correlation function and power spectrum. We develop a generalized Gillespie algorithm that accounts for the non-Markovian properties of random biochemical events with delay and compare our analytical findings with simulations. We show how time delay in gene expression can cause a system to be oscillatory even when its deterministic counterpart exhibits no oscillations. We demonstrate how such delay-induced instabilities can compromise the ability of a negative feedback loop to reduce the deleterious effects of noise. Given the prevalence of negative feedback in gene regulation, our findings may lead to new insights related to expression variability at the whole-genome scale.

Regularization of dynamics in an ensemble of nondiffusively coupled chaotic elements

We investigate the dynamics in an ensemble of chaotic elements with nondiffusive coupling. First, we analyze the case of global coupling. The type of coupling we consider leads to the suppression of oscillations in the whole ensemble at a high coupling strength. A distinct feature of this transition from high-dimensional chaos at a low coupling strength to the stationary state is that there is no partially ordered phase characterized by a large number of coexisting synchronized clusters. A two-cluster mode emerges abruptly, replacing the asynchronous mode. We focus on the influence of connectivity on the dynamics in the two-cluster modes and their domains of existence. We introduce a parameter that characterizes the connectivity: the range of coupling. Our computational and analytical results indicate that the most significant changes in the dynamics occur in the case of local coupling, when extra connections are added. By contrast, if the range of coupling is high, even substantial changes in this range have a small influence on the dynamics.