Modeling mechanical interactions in growing populations of rod-shaped bacteria

Is it selfish to be filamentous in biofilms? Individual-based modeling links microbial growth strategies with morphology using the new and modular iDynoMiCS 2.0

Microbial communities are found in all habitable environments and often occur in assemblages with self-organized spatial structures developing over time. This complexity can only be understood, predicted, and managed by combining experiments with mathematical modeling. Individual-based models are particularly suited if individual heterogeneity, local interactions, and adaptive behavior are of interest. Here we present the completely overhauled software platform, the individual-based Dynamics of Microbial Communities Simulator, iDynoMiCS 2.0, which enables researchers to specify a range of different models without having to program. Key new features and improvements are: (1) Substantially enhanced ease of use (graphical user interface, editor for model specification, unit conversions, data analysis and visualization and more). (2) Increased performance and scalability enabling simulations of up to 10 million agents in 3D biofilms. (3) Kinetics can be specified with any arithmetic function. (4) Agent properties can be assembled from orthogonal modules for pick and mix flexibility. (5) Force-based mechanical interaction framework enabling attractive forces and non-spherical agent morphologies as an alternative to the shoving algorithm. The new iDynoMiCS 2.0 has undergone intensive testing, from unit tests to a suite of increasingly complex numerical tests and the standard Benchmark 3 based on nitrifying biofilms. A second test case was based on the "biofilms promote altruism" study previously implemented in BacSim because competition outcomes are highly sensitive to the developing spatial structures due to positive feedback between cooperative individuals. We extended this case study by adding morphology to find that (i) filamentous bacteria outcompete spherical bacteria regardless of growth strategy and (ii) non-cooperating filaments outcompete cooperating filaments because filaments can escape the stronger competition between themselves. In conclusion, the new substantially improved iDynoMiCS 2.0 joins a growing number of platforms for individual-based modeling of microbial communities with specific advantages and disadvantages that we discuss, giving users a wider choice.

Spatial exclusion leads to "tug-of-war" ecological dynamics between competing species within microchannels

Competition is ubiquitous in microbial communities, shaping both their spatial and temporal structure and composition. Classical minimal models of competition, such as the Moran model, have been employed in ecology and evolutionary biology to understand the role of fixation and invasion in the maintenance of population diversity. Informed by recent experimental studies of cellular competition in confined spaces, we extend the Moran model to incorporate mechanical interactions between cells that divide within the limited space of a one-dimensional open microchannel. The model characterizes the skewed collective growth of the cells dividing within the channel, causing cells to be expelled at the channel ends. The results of this spatial exclusion model differ significantly from those of its classical well-mixed counterpart. The mean time to fixation of a species is greatly accelerated, scaling logarithmically, rather than algebraically, with the system size, and fixation/extinction probability sharply depends on the species' initial fractional abundance. By contrast, successful takeovers by invasive species, whether through mutation or immigration, are substantially less likely than in the Moran model. We also find that the spatial exclusion tends to attenuate the effects of fitness differences on the fixation times and probabilities. We find that these effects arise from the combination of the quasi-neutral "tug-of-war" diffusion dynamics of the inter-species boundary around an unstable equipoise point and the quasi-deterministic avalanche dynamics away from the fixed point. These results, which can be tested in microfluidic monolayer devices, have implications for the maintenance of species diversity in dense bacterial and cellular ecosystems where spatial exclusion is central to the competition, such as in organized biofilms or intestinal crypts.

Simulation of various biofilm fractal morphologies by agent-based model

Biofilms are clusters of bacteria wrapped in extracellular matrix and polymers. The study of biofilm morphological transformation has been around for a long time and has attracted widespread attention. In this paper, we present a model for biofilm growth based on the interaction force, in which bacteria are treated as tiny particles and locations of particles are updated by calculating the repulsive forces among particles. We adapt a continuity equation to indicate nutrient concentration variation in the substrate. Based on the above, we study the morphological transformation of biofilms. We find that nutrient concentration and nutrient diffusion rate dominate different biofilm morphological transition processes, in which biofilms would grow into fractal morphology under the conditions of low nutrient concentration and nutrient diffusivity. At the same time, we expand our model by introducing a second particle to mimic extracellular polymeric substances (EPS) in biofilms. We find that the interaction between different particles can lead to phase separation patterns between cells and EPSs, and the adhesion effect of EPS can attenuate this phenomenon. In contrast to single particle system models, branches are inhibited due to EPS filling in dual particle system models, and this invalidation is boosted by the enhancement of the depletion effect.

Mutant-Dependent Local Orientational Correlation in Biofilms of Vibrio campbellii Revealed through Digital Processing of Light Microscopy Images

Biofilms are key bacterial communities in genetic and adaptive resistance to antibiotics as well as disease control strategies. The mature high-coverage biofilm formations of the Vibrio campbellii strains (wild type BB120 and isogenic derivatives JAF633, KM387, and JMH603) are studied here through the unstraightforward digital processing of morphologically complex images without segmentation or the unrealistic simplifications used to artificially simulate low-density formations. The main results concern the specific mutant- and coverage-dependent short-range orientational correlation as well as the coherent development of biofilm growth pathways over the subdomains of the image. These findings are demonstrated to be unthinkable based only on a visual inspection of the samples or on methods such as Voronoi tessellation or correlation analyses. The presented approach is general, relies on measured rather than simulated low-density formations, and could be employed in the development of a highly efficient screening method for drugs or innovative materials.

A mechanistic understanding of microcolony morphogenesis: coexistence of mobile and sessile aggregates

Most bacteria in the natural environment self-organize into collective phases such as cell clusters, swarms, patterned colonies, or biofilms. Several intrinsic and extrinsic factors, such as growth, motion, and physicochemical interactions, govern the occurrence of different phases and their coexistence. Hence, predicting the conditions under which a collective phase emerges due to individual-level interactions is crucial. Here we develop a particle-based biophysical model of bacterial cells and self-secreted extracellular polymeric substances (EPS) to decipher the interplay of growth, motility-mediated dispersal, and mechanical interactions during microcolony morphogenesis. We show that the microcolony dynamics and architecture significantly vary depending upon the heterogeneous EPS production. In particular, microcolony shows the coexistence of both motile and sessile aggregates rendering a transition towards biofilm formation. We identified that the interplay of differential dispersion and the mechanical interactions among the components of the colony determines the fate of the colony morphology. Our results provide a significant understanding of the mechano-self-regulation during biofilm morphogenesis and open up possibilities of designing experiments to test the predictions.

Emergent spatiotemporal population dynamics with cell-length control of synthetic microbial consortia

The increased complexity of synthetic microbial biocircuits highlights the need for distributed cell functionality due to concomitant increases in metabolic and regulatory burdens imposed on single-strain topologies. Distributed systems, however, introduce additional challenges since consortium composition and spatiotemporal dynamics of constituent strains must be robustly controlled to achieve desired circuit behaviors. Here, we address these challenges with a modeling-based investigation of emergent spatiotemporal population dynamics using cell-length control in monolayer, two-strain bacterial consortia. We demonstrate that with dynamic control of a strain's division length, nematic cell alignment in close-packed monolayers can be destabilized. We find that this destabilization confers an emergent, competitive advantage to smaller-length strains-but by mechanisms that differ depending on the spatial patterns of the population. We used complementary modeling approaches to elucidate underlying mechanisms: an agent-based model to simulate detailed mechanical and signaling interactions between the competing strains, and a reductive, stochastic lattice model to represent cell-cell interactions with a single rotational parameter. Our modeling suggests that spatial strain-fraction oscillations can be generated when cell-length control is coupled to quorum-sensing signaling in negative feedback topologies. Our research employs novel methods of population control and points the way to programming strain fraction dynamics in consortial synthetic biology.

Agent Based Models of Polymicrobial Biofilms and the Microbiome-A Review

The human microbiome has been a focus of intense study in recent years. Most of the living organisms comprising the microbiome exist in the form of biofilms on mucosal surfaces lining our digestive, respiratory, and genito-urinary tracts. While health-associated microbiota contribute to digestion, provide essential nutrients, and protect us from pathogens, disturbances due to illness or medical interventions contribute to infections, some that can be fatal. Myriad biological processes influence the make-up of the microbiota, for example: growth, division, death, and production of extracellular polymers (EPS), and metabolites. Inter-species interactions include competition, inhibition, and symbiosis. Computational models are becoming widely used to better understand these interactions. Agent-based modeling is a particularly useful computational approach to implement the various complex interactions in microbial communities when appropriately combined with an experimental approach. In these models, each cell is represented as an autonomous agent with its own set of rules, with different rules for each species. In this review, we will discuss innovations in agent-based modeling of biofilms and the microbiota in the past five years from the biological and mathematical perspectives and discuss how agent-based models can be further utilized to enhance our comprehension of the complex world of polymicrobial biofilms and the microbiome.

Confinement-induced self-organization in growing bacterial colonies

We investigate the emergence of global alignment in colonies of dividing rod-shaped cells under confinement. Using molecular dynamics simulations and continuous modeling, we demonstrate that geometrical anisotropies in the confining environment give rise to an imbalance in the normal stresses, which, in turn, drives a collective rearrangement of the cells. This behavior crucially relies on the colony's solid-like mechanical response at short time scales and can be recovered within the framework of active hydrodynamics upon modeling bacterial colonies as growing viscoelastic gels characterized by Maxwell-like stress relaxation.

Novel Tunable Spatio-Temporal Patterns From a Simple Genetic Oscillator Circuit

Multicellularity, the coordinated collective behavior of cell populations, gives rise to the emergence of self-organized phenomena at many different spatio-temporal scales. At the genetic scale, oscillators are ubiquitous in regulation of multicellular systems, including during their development and regeneration. Synthetic biologists have successfully created simple synthetic genetic circuits that produce oscillations in single cells. Studying and engineering synthetic oscillators in a multicellular chassis can therefore give us valuable insights into how simple genetic circuits can encode complex multicellular behaviors at different scales. Here we develop a study of the coupling between the repressilator synthetic genetic ring oscillator and constraints on cell growth in colonies. We show in silico how mechanical constraints generate characteristic patterns of growth rate inhomogeneity in growing cell colonies. Next, we develop a simple one-dimensional model which predicts that coupling the repressilator to this pattern of growth rate via protein dilution generates traveling waves of gene expression. We show that the dynamics of these spatio-temporal patterns are determined by two parameters; the protein degradation and maximum expression rates of the repressors. We derive simple relations between these parameters and the key characteristics of the traveling wave patterns: firstly, wave speed is determined by protein degradation and secondly, wavelength is determined by maximum gene expression rate. Our analytical predictions and numerical results were in close quantitative agreement with detailed individual based simulations of growing cell colonies. Confirming published experimental results we also found that static ring patterns occur when protein stability is high. Our results show that this pattern can be induced simply by growth rate dilution and does not require transition to stationary phase as previously suggested. Our method generalizes easily to other genetic circuit architectures thus providing a framework for multi-scale rational design of spatio-temporal patterns from genetic circuits. We use this method to generate testable predictions for the synthetic biology design-build-test-learn cycle.

Long-range temporal coordination of gene expression in synthetic microbial consortia

Synthetic microbial consortia have an advantage over isogenic synthetic microbes because they can apportion biochemical and regulatory tasks among the strains. However, it is difficult to coordinate gene expression in spatially extended consortia because the range of signaling molecules is limited by diffusion. Here, we show that spatio-temporal coordination of gene expression can be achieved even when the spatial extent of the consortium is much greater than the diffusion distance of the signaling molecules. To do this, we examined the dynamics of a two-strain synthetic microbial consortium that generates coherent oscillations in small colonies. In large colonies, we find that temporally coordinated oscillations across the population depend on the presence of an intrinsic positive feedback loop that amplifies and propagates intercellular signals. These results demonstrate that synthetic multicellular systems can be engineered to exhibit coordinated gene expression using only transient, short-range coupling among constituent cells.

Spatiotemporal Dynamics of Synthetic Microbial Consortia in Microfluidic Devices

Synthetic microbial consortia consist of two or more engineered strains that grow together and share the same resources. When intercellular signaling pathways are included in the engineered strains, close proximity of the microbes can generate complex dynamic behaviors that are difficult to obtain using a single strain. However, when a consortium is not cultured in a well-mixed environment the constituent strains passively compete for space as they grow and divide, complicating cell-cell signaling. Here, we explore the temporal dynamics of the spatial distribution of consortia cocultured in microfluidic devices. To do this, we grew two different strains of Escherichia coli in microfluidic devices with cell-trapping regions (traps) of several different designs. We found that the size of the traps is a critical determinant of spatiotemporal dynamics. In small traps, cells can easily signal one another, but the relative proportion of each strain within the trap can fluctuate wildly. In large traps, the relative ratio of strains is stabilized, but intercellular signaling can be hindered by distances between cells. This presents a trade-off between the trap size and the effectiveness of intercellular signaling, which can be mitigated by increasing the initial seeding of cells in larger traps. We also built a mathematical model, which suggests that increasing the number of seed cells can also increase the strain ratio variability due to an increased number of strain interfaces in the trap. These results help elucidate the complex behaviors of synthetic microbial consortia in microfluidic traps and provide a means of analysis to help remedy the spatial heterogeneity inherent to different trap types.

Individual Based Model Links Thermodynamics, Chemical Speciation and Environmental Conditions to Microbial Growth

Individual based Models (IbM) must transition from research tools to engineering tools. To make the transition we must aspire to develop large, three dimensional and physically and biologically credible models. Biological credibility can be promoted by grounding, as far as possible, the biology in thermodynamics. Thermodynamic principles are known to have predictive power in microbial ecology. However, this in turn requires a model that incorporates pH and chemical speciation. Physical credibility implies plausible mechanics and a connection with the wider environment. Here, we propose a step toward that ideal by presenting an individual based model connecting thermodynamics, pH and chemical speciation and environmental conditions to microbial growth for 5·105 individuals. We have showcased the model in two scenarios: a two functional group nitrification model and a three functional group anaerobic community. In the former, pH and connection to the environment had an important effect on the outcomes simulated. Whilst in the latter pH was less important but the spatial arrangements and community productivity (that is, methane production) were highly dependent on thermodynamic and reactor coupling. We conclude that if IbM are to attain their potential as tools to evaluate the emergent properties of engineered biological systems it will be necessary to combine the chemical, physical, mechanical and biological along the lines we have proposed. We have still fallen short of our ideals because we cannot (yet) calculate specific uptake rates and must develop the capacity for longer runs in larger models. However, we believe such advances are attainable. Ideally in a common, fast and modular platform. For future innovations in IbM will only be of use if they can be coupled with all the previous advances.

Moran Model of Spatial Alignment in Microbial Colonies

We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise as a result of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap's long side. This modeling approach and analysis can be extended to directionally-growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior.

Statistics of noisy growth with mechanical feedback in elastic tissues

Tissue growth is a fundamental aspect of development and is intrinsically noisy. Stochasticity has important implications for morphogenesis, precise control of organ size, and regulation of tissue composition and heterogeneity. However, the basic statistical properties of growing tissues, particularly when growth induces mechanical stresses that can in turn affect growth rates, have received little attention. Here, we study the noisy growth of elastic sheets subject to mechanical feedback. Considering both isotropic and anisotropic growth, we find that the density-density correlation function shows power law scaling. We also consider the dynamics of marked, neutral clones of cells. We find that the areas (but not the shapes) of two clones are always statistically independent, even when they are adjacent. For anisotropic growth, we show that clone size variance scales like the average area squared and that the mode amplitudes characterizing clone shape show a slow [Formula: see text] decay, where n is the mode index. This is in stark contrast to the isotropic case, where relative variations in clone size and shape vanish at long times. The high variability in clone statistics observed in anisotropic growth is due to the presence of two soft modes-growth modes that generate no stress. Our results lay the groundwork for more in-depth explorations of the properties of noisy tissue growth in specific biological contexts.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Modeling cell population dynamics

 Quantitative modeling is quickly becoming an integral part of biology, due to the ability of mathematical models and computer simulations to generate insights and predict the behavior of living systems. Single-cell models can be incapable or misleading for inferring population dynamics, as they do not consider the interactions between cells via metabolites or physical contact, nor do they consider competition for limited resources such as nutrients or space. Here we examine methods that are commonly used to model and simulate cell populations. First, we cover simple models where analytic solutions are available, and then move on to more complex scenarios where computational methods are required. Overall, we present a summary of mathematical models used to describe cell population dynamics, which may aid future model development and highlights the importance of population modeling in biology.

Computer simulation study of early bacterial biofilm development

Most bacteria form organized sessile communities, known as biofilms. Their ubiquity and relevance have stimulated the development of efficient mathematical models able to predict biofilm evolution and characteristics at different conditions. Here we present a study of the early stages of bacterial biofilm formation modeled by means of individual cell-based computer simulation. Simulation showed that clusters with different degrees of internal and orientational order were formed as a function of the aspect ratio of the individual particles and the relation between the diffusion and growth rates. Analysis of microscope images of early biofilm formation by the Gram-negative bacterium Pseudomonas putida at varying diffusion rates revealed a good qualitative agreement with the simulation results. Our model is a good predictor of microcolony morphology during early biofilm development, showing that the competition between diffusion and growth rates is a key aspect in the formation of stable biofilm microcolonies.