Overview of Mathematical Approaches Used to Model Bacterial Chemotaxis I: The Single Cell

Homeostasis in input-output networks: Structure, Classification and Applications

Homeostasis is concerned with regulatory mechanisms, present in biological systems, where some specific variable is kept close to a set value as some external disturbance affects the system. Many biological systems, from gene networks to signaling pathways to whole tissue/organism physiology, exhibit homeostatic mechanisms. In all these cases there are homeostatic regions where the variable is relatively to insensitive external stimulus, flanked by regions where it is sensitive. Mathematically, the notion of homeostasis can be formalized in terms of an input-output function that maps the parameter representing the external disturbance to the output variable that must be kept within a fairly narrow range. This observation inspired the introduction of the notion of infinitesimal homeostasis, namely, the derivative of the input-output function is zero at an isolated point. This point of view allows for the application of methods from singularity theory to characterize infinitesimal homeostasis points (i.e. critical points of the input-output function). In this paper we review the infinitesimal approach to the study of homeostasis in input-output networks. An input-output network is a network with two distinguished nodes 'input' and 'output', and the dynamics of the network determines the corresponding input-output function of the system. This class of dynamical systems provides an appropriate framework to study homeostasis and several important biological systems can be formulated in this context. Moreover, this approach, coupled to graph-theoretic ideas from combinatorial matrix theory, provides a systematic way for classifying different types of homeostasis (homeostatic mechanisms) in input-output networks, in terms of the network topology. In turn, this leads to new mathematical concepts, such as, homeostasis subnetworks, homeostasis patterns, homeostasis mode interaction. We illustrate the usefulness of this theory with several biological examples: biochemical networks, chemical reaction networks (CRN), gene regulatory networks (GRN), Intracellular metal ion regulation and so on.

Dimensionless Parameters Define Criteria for Optimal Flow Velocity in Enhancing Chemotactic Response toward Residual Contaminants in Porous Media

Chemotactic bacteria may overcome challenges posed by nonaqueous-phase liquid (NAPL) contaminants of low solubility in groundwater and limited bioavailability in tight pores by preferentially migrating to NAPL sources. We explored the transport of chemotactic bacteria to NAPL ganglia at varying pore water velocities in a dual-permeability microfluidic device and using computer-simulated solutions of transport equations. In our experiments, bacteria exhibited a chemotactic response toward NAPL ganglia at the junctures of low- and high-permeability regions (i.e., micropockets), and the extent of retention initially increased with velocity and then decreased at the highest velocity. A dimensional analysis revealed that maximum accumulations occurred at moderate values of the Péclet number P e c = v m d p χ o ∼10 in our system. We also found that accumulation dynamics in micropockets can be represented by a logistic equation incorporating convection and chemotaxis time scales τ f = L i v f and τ che = l a 2 χ o , respectively. By analyzing seven literature studies on chemotaxis, we identified an exposure time scale to chemicals τ exp = d p v f that was useful for evaluating the chemotaxis efficiency. Our study provided unique insights into the effect of fluid flow on chemotaxis in porous media by demonstrating that increasing the fluid velocity to some extent can promote chemotaxis. The dimensionless parameters inform the design of efficient bioremediation strategies for contaminated porous media.

Quantifying Plant Signaling Pathways by Integrating Luminescence-Based Biosensors and Mathematical Modeling

Plants have evolved intricate signaling pathways, which operate as networks governed by feedback to deal with stressors. Nevertheless, the sophisticated molecular mechanisms underlying these routes still need to be comprehended, and experimental validation poses significant challenges and expenses. Consequently, computational hypothesis evaluation gains prominence in understanding plant signaling dynamics. Biosensors are genetically modified to emit light when exposed to a particular hormone, such as abscisic acid (ABA), enabling quantification. We developed computational models to simulate the relationship between ABA concentrations and bioluminescent sensors utilizing the Hill equation and ordinary differential equations (ODEs), aiding better hypothesis development regarding plant signaling. Based on simulation results, the luminescence intensity was recorded for a concentration of 47.646 RLUs for 1.5 μmol, given the specified parameters and model assumptions. This method enhances our understanding of plant signaling pathways at the cellular level, offering significant benefits to the scientific community in a cost-effective manner. The alignment of these computational predictions with experimental results emphasizes the robustness of our approach, providing a cost-effective means to validate mathematical models empirically. The research intended to correlate the bioluminescence of biosensors with plant signaling and its mathematical models for quantified detection of specific plant hormone ABA.

Multiscale Porosity Microfluidics to Study Bacterial Transport in Heterogeneous Chemical Landscapes

Microfluidic models are proving to be powerful systems to study fundamental processes in porous media, due to their ability to replicate topologically complex environments while allowing detailed, quantitative observations at the pore scale. Yet, while porous media such as living tissues, geological substrates, or industrial systems typically display a porosity that spans multiple scales, most microfluidic models to date are limited to a single porosity or a small range of pore sizes. Here, a novel microfluidic system with multiscale porosity is presented. By embedding polyacrylamide (PAAm) hydrogel structures through in-situ photopolymerization in a landscape of microfabricated polydimethylsiloxane (PDMS) pillars with varying spacing, micromodels with porosity spanning several orders of magnitude, from nanometers to millimeters are created. Experiments conducted at different porosity patterns demonstrate the potential of this approach to characterize fundamental and ubiquitous biological and geochemical transport processes in porous media. Accounting for multiscale porosity allows studies of the resulting heterogeneous fluid flow and concentration fields of transported chemicals, as well as the biological behaviors associated with this heterogeneity, such as bacterial chemotaxis. This approach brings laboratory studies of transport in porous media a step closer to their natural counterparts in the environment, industry, and medicine.

Learning black- and gray-box chemotactic PDEs/closures from agent based Monte Carlo simulation data

We propose a machine learning framework for the data-driven discovery of macroscopic chemotactic Partial Differential Equations (PDEs)-and the closures that lead to them- from high-fidelity, individual-based stochastic simulations of Escherichia coli bacterial motility. The fine scale, chemomechanical, hybrid (continuum-Monte Carlo) simulation model embodies the underlying biophysics, and its parameters are informed from experimental observations of individual cells. Using a parsimonious set of collective observables, we learn effective, coarse-grained "Keller-Segel class" chemotactic PDEs using machine learning regressors: (a) (shallow) feedforward neural networks and (b) Gaussian Processes. The learned laws can be black-box (when no prior knowledge about the PDE law structure is assumed) or gray-box when parts of the equation (e.g. the pure diffusion part) is known and "hardwired" in the regression process. More importantly, we discuss data-driven corrections (both additive and functional), to analytically known, approximate closures.

Tuning chemotactic and diffusiophoretic spreading via hydrodynamic flows

The transport of microorganisms by chemotaxis is described by the same "log-sensing" response as colloids undergoing diffusiophoresis, despite their different mechanistic origins. We employ a recently-developed macrotransport theory to analyze the advective-diffusive transport of a chemotactic or diffusiophoretic colloidal species (both referred to as "colloids") in a circular tube under a steady pressure-driven flow (referred to as hydrodynamic flow) and transient solute gradient. First, we derive an exact solution to the log-sensing chemotactic/diffusiophoretic macrotransport equation. We demonstrate that a strong hydrodynamic flow can reduce spreading of solute-repelled colloids, by eliminating super-diffusion which occurs in an otherwise quiescent system. In contrast, hydrodynamic flows always enhance spreading of solute-attracted colloids. Second, we generalize the exact solution to show that the above tunable spreading phenomena by hydrodynamic flows persist quantitatively for decaying colloids, as may occur with cell death, for example. Third, we examine the spreading of chemotactic colloids by employing a more general model that captures a hallmark of chemotaxis, that log-sensing occurs only over a finite range of solute concentration. Apart from demonstrating for the first time the generality of the macrotransport theory to incorporate an arbitrary chemotactic flow model, we reveal via numerical solutions new regimes of anomalous spreading, which match qualitatively with experiments and are tunable by hydrodynamic flows. The results presented here could be employed to tailor chemotactic/diffusiophoretic colloid transport using hydrodynamic flows, which are central to applications such as oil recovery and bioremediation of aquifers.

Leave or Stay: Simulating Motility and Fitness of Microorganisms in Dynamic Aquatic Ecosystems

Motility is a key adaptation factor in scarce marine environments inhabited by bacteria. The question of how a capacity for adaptive migrations influences the success of a microbial population in various conditions is a challenge addressed in this study. We employed the agent-based model of competition of motile and sedentary microbial populations in a confined aquatic environment supplied with a periodic batch nutrient source to assess the fitness of both. Such factors as nutrient concentration in a batch, batch period, mortality type and energetic costs of migration were considered to determine the conditions favouring different strategies: Nomad of a motile population and Settler of a sedentary one. The modelling results demonstrate that dynamic and nutrient-scarce environments favour motile populations, whereas nutrient-rich and stagnant environments promote sedentary microorganisms. Energetic costs of migration determine whether or not the Nomad strategy of the motile population is successful, though it also depends on such conditions as nutrient availability. Even without penalties for migration, under certain conditions, the sedentary Settler population dominates in the ecosystem. It is achieved by decreasing the local nutrient availability near the nutrient source, as motile populations relying on a local optimizing strategy tend to follow benign conditions and fail, enduring stress associated with crossing the valleys of suboptimal nutrient availability.

Receptor Models of Phagocytosis: The Effect of Target Shape

Phagocytosis is a remarkably complex process, requiring simultaneous organisation of the cell membrane, the cytoskeleton, receptors and various signalling molecules. As can often be the case, mathematical modelling is able to penetrate some of this complexity, identifying the key biophysical components and generating understanding that would take far longer with a purely experimental approach. This chapter will review a particularly important class of phagocytosis model, championed in recent years, that primarily focuses on the role of receptors during the engulfment process. These models are pertinent to a host of unsolved questions in the subject, including the rate of cup growth during uptake, the role of both intra- and extracellular noise, and the precise differences between phagocytosis and other forms of endocytosis. In particular, this chapter will focus on the effect of target shape and orientation, including how these influence the rate and final outcome of phagocytic engulfment.

From indication to decision: A hierarchical approach to model the chemotactic behavior of Escherichia coli

Reducing the complex behavior of living entities to its underlying physical and chemical processes is a formidable task in biology. Complex behaviors can be characterized as decision making: the ability to process the incoming information via an intracellular network and act upon this information to choose appropriate strategies. Motility is one such behavior that has been the focus many modeling efforts in the past. Our aim is to reduce the chemotactic behavior in Escherichia coli to its molecular constituents in order to paint a comprehensive and end-to-end picture of this intricate behavior. We utilize a hierarchical approach, consisting of three layers, to achieve this goal: at the first level, chemical reactions involved in chemotaxis are simulated. In the second level, the chemical reactions give rise to the mechanical movement of six independent flagella. At the last layer, the two lower layers are combined to allow a digital bacterium to receive information from its environment and swim through it with verve. Our results are in concert with the experimental studies concerning the motility of E.coli cells. In addition, we show that our detailed model of chemotaxis is reducible to a non-homogeneous Markov process.

Modelling bacterial chemotaxis for indirectly binding attractants

In bacterial chemotaxis, chemoattractant molecules may bind either directly or indirectly with receptors within the cell periplasmic space. The indirect binding mechanism, which involves an intermediate periplasmic binding protein, has been reported to increase sensitivity to dilute attractant concentrations as well as range of response. Current mathematical models for bacterial chemotaxis at the population scale do not appear to take the periplasmic binding protein (BP) concentration or the indirect binding mechanics into account. We formulate an indirect binding extension to the existing Rivero equation for chemotactic velocity based on fundamental reversible enzyme kinetics. The formulated indirect binding expression accounts for the periplasmic BP concentration and the dissociation constants for binding between attractant and periplasmic BP, as well as between BP and chemoreceptor. We validate the indirect-binding model using capillary assay simulations of the chemotactic responses of E. coli to the indirectly-binding attractants maltose and AI-2. The predicted response agrees well with experimental data from a number of maltose capillary assay studies conducted in previous literature. The model is also able to achieve good agreement with AI-2 capillary assay data of one study out of two tested. The chemotactic response of E. coli towards AI-2 appears to be of higher complexity due to reports of variable periplasmic BP concentration as well as the low concentration of periplasmic BP relative to the total receptor concentration. Our current model is thus suitable for indirect binding chemotactic response systems with constant periplasmic BP concentration that is significantly larger than the total receptor concentration, such as the response of E. coli towards maltose. Further considerations may be taken into account to model the chemotactic response towards AI-2 with greater accuracy.

COMPUTATIONAL MODELING OF SELF-ORGANIZATION OF BACTERIAL POPULATION CONSISTING OF SUBPOPULATIONS OF ACTIVE AND PASSIVE CELLS

This paper deals with the computational modeling of the bioluminescence pattern formation in suspensions of Escherichia coli bacteria. The aim was to develop a computational model for simulating the bacterial populations consisting of two subpopulations of active and passive cells. A suitable model based on Keller–Segel and Fisher equations was proposed and the spatiotemporal patterns were simulated using the finite difference technique. The influence of cell activation, deactivation, chemotactic sensitivity, growth rate and saturating signal production parameter values on the pattern formation was investigated. The proposed model can be used to effectively simulate quasi-one-dimensional spatiotemporal patterns. We provide a simple qualitative explanation of the experimental results and estimated model parameters. In particular, it is argued that the effective model simulates patterns of evaporation-driven convection in open-to-air suspensions of cells that can be either active or passive.

Adjustment in tumbling rates improves bacterial chemotaxis on obstacle-laden terrains

The mechanisms of bacterial chemotaxis have been extensively studied for several decades, but how the physical environment influences the collective migration of bacterial cells remains less understood. Previous models of bacterial chemotaxis have suggested that the movement of migrating bacteria across obstacle-laden terrains may be slower compared with terrains without them. Here, we show experimentally that the size or density of evenly spaced obstacles do not alter the average exit rate of Escherichia coli cells from microchambers in response to external attractants, a function that is dependent on intact cell-cell communication. We also show, both by analyzing a revised theoretical model and by experimentally following single cells, that the reduced exit time in the presence of obstacles is a consequence of reduced tumbling frequency that is adjusted by the E. coli cells in response to the topology of their environment. These findings imply operational short-term memory of bacteria while moving through complex environments in response to chemotactic stimuli and motivate improved algorithms for self-autonomous robotic swarms.

Bacterial chemotaxis in a microfluidic T-maze reveals strong phenotypic heterogeneity in chemotactic sensitivity

Many microorganisms have evolved chemotactic strategies to exploit the microscale heterogeneity that frequently characterizes microbial habitats. Chemotaxis has been primarily studied as an average characteristic of a population, with little regard for variability among individuals. Here, we adopt a classic tool from animal ecology - the T-maze - and implement it at the microscale by using microfluidics to expose bacteria to a sequence of decisions, each consisting of migration up or down a chemical gradient. Single-cell observations of clonal Escherichia coli in the maze, coupled with a mathematical model, reveal that strong heterogeneity in the chemotactic sensitivity coefficient exists even within clonal populations of bacteria. A comparison of different potential sources of heterogeneity reveals that heterogeneity in the T-maze originates primarily from the chemotactic sensitivity coefficient, arising from a distribution of pathway gains. This heterogeneity may have a functional role, for example in the context of migratory bet-hedging strategies.

Combined influence of hydrodynamics and chemotaxis in the distribution of microorganisms around spherical nutrient sources

We study how the interaction between hydrodynamics and chemotaxis affects the colonization of nutrient sources by microorganisms. We use an individual-based model and perform probabilistic simulations to ascertain the impact of important environmental and motility characteristics on the spatial distribution of microorganisms around a spherical nutrient source. In general, we unveil four distinct regimes based on the distribution of the microorganisms: (i) strong surface colonization, (ii) rotary-diffusion-induced "off-surface" accumulation, (iii) a depletion zone in the spatial distribution, and (iv) no appreciable aggregation, with their occurrence being contingent on the relative strengths of hydrodynamic and chemotactic effects. More specifically, we show that the extent of surface colonization first increases, then reaches a plateau, and finally decreases as the nutrient availability is increased. We also show that surface colonization reduces monotonically as the mean run length of the chemotactic microorganisms increases. Our study provides insight into the interplay of two important mechanisms governing microorganism behavior near nutrient sources, isolates each of their effects, and thus offers greater predictability of this nontrivial phenomenon.

Modeling Transport of Chemotactic Bacteria in Granular Media with Distributed Contaminant Sources

Chemotaxis has the potential to improve bioremediation strategies by enhancing the transport of pollutant-degrading bacteria to the source of contamination, leading to increased pollutant accessibility and biodegradation. This computational study extends work reported previously in the literature to include predictions of chemotactic bacterial migration in response to multiple localized contaminant sources within porous media. An advection-dispersion model, in which chemotaxis was represented explicitly as an additional advection-like term, was employed to simulate the transport of bacteria within a sand-packed column containing a distribution of chemoattractant sources. Simulation results provided insight into attractant and bacterial distributions within the column. In particular, it was found that chemotactic bacteria exhibited a distinct biased migration toward contaminant sources that resulted in a 30% decrease in cell recovery, and concomitantly an enhanced retention within the sand column, compared to the nonchemotactic control. Model results were consistent with experimental observations. Parametric studies were conducted to provide insight into the influence of chemotaxis parameters on bacterial migration and cell percent recovery. The model results provide a better understanding of the effect of chemotaxis on bacterial transport in response to distributed contaminant sources.

How cells engulf: a review of theoretical approaches to phagocytosis

Phagocytosis is a fascinating process whereby a cell surrounds and engulfs particles such as bacteria and dead cells. This is crucial both for single-cell organisms (as a way of acquiring nutrients) and as part of the immune system (to destroy foreign invaders). This whole process is hugely complex and involves multiple coordinated events such as membrane remodelling, receptor motion, cytoskeleton reorganisation and intracellular signalling. Because of this, phagocytosis is an excellent system for theoretical study, benefiting from biophysical approaches combined with mathematical modelling. Here, we review these theoretical approaches and discuss the recent mathematical and computational models, including models based on receptors, models focusing on the forces involved, and models employing energetic considerations. Along the way, we highlight a beautiful connection to the physics of phase transitions, consider the role of stochasticity, and examine links between phagocytosis and other types of endocytosis. We cover the recently discovered multistage nature of phagocytosis, showing that the size of the phagocytic cup grows in distinct stages, with an initial slow stage followed by a much quicker second stage starting around half engulfment. We also address the issue of target shape dependence, which is relevant to both pathogen infection and drug delivery, covering both one-dimensional and two-dimensional results. Throughout, we pay particular attention to recent experimental techniques that continue to inform the theoretical studies and provide a means to test model predictions. Finally, we discuss population models, connections to other biological processes, and how physics and modelling will continue to play a key role in future work in this area.

Modeling of active swimmer suspensions and their interactions with the environment

In this article, we review mathematical models used to study the behaviour of suspensions of micro-swimmers and the accompanying biophysical phenomena, with specific focus on stimulus response. The methods discussed encompass a range of interactions exhibited by the micro-swimmers; including passive hydrodynamic (gyrotaxis) and gravitational (gravitaxis) effects, and active responses to chemical cues (chemotaxis) and light intensities (phototaxis). We introduce the simplest models first, and then build towards more sophisticated recent developments, in the process, identifying the limitations of the former and the new results obtained by the latter. We comment on the accuracy/validity of the models adopted, based on the agreement between theoretical results and experimental observations. We conclude by identifying some of the open problems and associated challenges faced by researchers in the realm of active suspensions.

Stable swarming using adaptive long-range interactions

Sensory mechanisms in biology, from cells to humans, have the property of adaptivity, whereby the response produced by the sensor is adapted to the overall amplitude of the signal, reducing the sensitivity in the presence of strong stimulus, while increasing it when it is weak. This property is inherently energy consuming and a manifestation of the nonequilibrium nature of living organisms. We explore here how adaptivity affects the effective forces that organisms feel due to others in the context of a uniform swarm, in both two and three dimensions. The interactions between the individuals are taken to be attractive and long-range and of power-law form. We find that the effects of adaptivity inside the swarm are dramatic, where the effective forces decrease (or remain constant) with increasing swarm density. Linear stability analysis demonstrates how this property prevents collapse (Jeans instability), when the forces are adaptive. Adaptivity therefore endows swarms with a natural mechanism for self-stabilization.

Reconstructing the genotype-to-fitness map for the bacterial chemotaxis network and its emergent behavioural phenotypes

The signal-transduction network responsible for chemotaxis in Escherichia coli has been characterised in extraordinary detail. Yet, relatively little is known about eco-evolutionary aspects of chemotaxis, such as how the network has been shaped by selection and to what extent natural populations may fine-tune their chemotactic behaviour to the ecological conditions. To address these questions, we here develop an evolutionary-systems-biology model of the chemotaxis network of E. coli, which we apply to estimate the resource accumulation rate (here used as a proxy for fitness) of wildtype and a large number of potential mutant genotypes. Mutant genotypes differ from the wildtype in the concentrations of one or more constituent proteins of the chemotaxis signalling network or in one or more of its kinetic parameters. To guarantee model consistency across the genotype space, we explicitly incorporated biochemical constraints that underly observed phenotypic trade-offs. The model was validated by reconstructing the phenotypic properties of several known mutant genotypes. We also characterised differences in the fitness distribution between genotypes, and reconstructed adaptive walks in genotype space for populations exposed to different environmental conditions. We found that the local fitness landscape is rugged, due to non-additive interactions between mutations. When selection has a consistent direction, just a few adaptive mutations are required to reach a local peak, and different local peaks can be reached by adaptive walks starting from the same initial genotype. However, when the direction of selection is fluctuating, evolutionary paths are much longer and genotype space is explored further. Longer adaptive walks were also observed when evolution was started from a low-fitness genotype such as a CheZ knockout mutant. In line with empirical observations, the initial ΔcheZ mutant did not respond to a step-down stimulus, but a dynamic response similar to the wildtype was recovered following the fixation of compensatory mutations.

Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance σa2 in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that σa2 peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.

A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, [Formula: see text] and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant [Formula: see text] with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.

The Impact of Social Behavior on the Attenuation and Delay of Bacterial Nanonetworks

Molecular communication (MC) is a new paradigm for developing communication systems that exchanges information through the transmission and reception of molecules. One proposed model for MC is using bacteria to carry information encoded into DNA plasmids, and this is termed bacterial nanonetworks. However, a limiting factor in the models that have been studied so far is the environment considered only in ideal conditions with a single population. This is far from realistic in natural environments, where bacteria coexist in multiple populations of same and different species, resulting in a very complex social community. This complex community has social interactions that include cooperation, cheating, as well as competition. In this paper, the effects of these social interactions on the information delivery in bacterial nanonetworks are studied in terms of delay, attenuation and data rate. The numerical results show that the cooperative behavior of bacteria improves the performance of delay and attenuation leading to a higher data rate, and this performance can be degraded once their behavior switches towards cheating. The competitive social behavior shows that the performance can degrade delay as well as attenuation leading to slower data rates, as the population with the encoded DNA plasmids are prevented from reaching the receiver. The analysis of social interactions between the bacteria will pave the way for efficient design of bacterial nanonetworks enabling applications such as intrabody sensing, drug delivery, and environmental control against pollution and biological hazards.

Engineering Bacteria to Search for Specific Concentrations of Molecules by a Systematic Synthetic Biology Design Method

Bacteria navigate environments full of various chemicals to seek favorable places for survival by controlling the flagella’s rotation using a complicated signal transduction pathway. By influencing the pathway, bacteria can be engineered to search for specific molecules, which has great potential for application to biomedicine and bioremediation. In this study, genetic circuits were constructed to make bacteria search for a specific molecule at particular concentrations in their environment through a synthetic biology method. In addition, by replacing the “brake component” in the synthetic circuit with some specific sensitivities, the bacteria can be engineered to locate areas containing specific concentrations of the molecule. Measured by the swarm assay qualitatively and microfluidic techniques quantitatively, the characteristics of each “brake component” were identified and represented by a mathematical model. Furthermore, we established another mathematical model to anticipate the characteristics of the “brake component”. Based on this model, an abundant component library can be established to provide adequate component selection for different searching conditions without identifying all components individually. Finally, a systematic design procedure was proposed. Following this systematic procedure, one can design a genetic circuit for bacteria to rapidly search for and locate different concentrations of particular molecules by selecting the most adequate “brake component” in the library. Moreover, following simple procedures, one can also establish an exclusive component library suitable for other cultivated environments, promoter systems, or bacterial strains.

Understanding the link between single cell and population scale responses of Escherichia coli in differing ligand gradients

We formulate an agent-based population model of Escherichia coli cells which incorporates a description of the chemotaxis signalling cascade at the single cell scale. The model is used to gain insight into the link between the signalling cascade dynamics and the overall population response to differing chemoattractant gradients. Firstly, we consider how the observed variation in total (phosphorylated and unphosphorylated) signalling protein concentration affects the ability of cells to accumulate in differing chemoattractant gradients. Results reveal that a variation in total cell protein concentration between cells may be a mechanism for the survival of cell colonies across a wide range of differing environments. We then study the response of cells in the presence of two different chemoattractants. In doing so we demonstrate that the population scale response depends not on the absolute concentration of each chemoattractant but on the sensitivity of the chemoreceptors to their respective concentrations. Our results show the clear link between single cell features and the overall environment in which cells reside.

Inferring Models of Bacterial Dynamics toward Point Sources

Experiments have shown that bacteria can be sensitive to small variations in chemoattractant (CA) concentrations. Motivated by these findings, our focus here is on a regime rarely studied in experiments: bacteria tracking point CA sources (such as food patches or even prey). In tracking point sources, the CA detected by bacteria may show very large spatiotemporal fluctuations which vary with distance from the source. We present a general statistical model to describe how bacteria locate point sources of food on the basis of stochastic event detection, rather than CA gradient information. We show how all model parameters can be directly inferred from single cell tracking data even in the limit of high detection noise. Once parameterized, our model recapitulates bacterial behavior around point sources such as the “volcano effect”. In addition, while the search by bacteria for point sources such as prey may appear random, our model identifies key statistical signatures of a targeted search for a point source given any arbitrary source configuration.

Current advances in systems and integrative biology

Systems biology has gained a tremendous amount of interest in the last few years. This is partly due to the realization that traditional approaches focusing only on a few molecules at a time cannot describe the impact of aberrant or modulated molecular environments across a whole system. Furthermore, a hypothesis-driven study aims to prove or disprove its postulations, whereas a hypothesis-free systems approach can yield an unbiased and novel testable hypothesis as an end-result. This latter approach foregoes assumptions which predict how a biological system should react to an altered microenvironment within a cellular context, across a tissue or impacting on distant organs. Additionally, re-use of existing data by systematic data mining and re-stratification, one of the cornerstones of integrative systems biology, is also gaining attention. While tremendous efforts using a systems methodology have already yielded excellent results, it is apparent that a lack of suitable analytic tools and purpose-built databases poses a major bottleneck in applying a systematic workflow. This review addresses the current approaches used in systems analysis and obstacles often encountered in large-scale data analysis and integration which tend to go unnoticed, but have a direct impact on the final outcome of a systems approach. Its wide applicability, ranging from basic research, disease descriptors, pharmacological studies, to personalized medicine, makes this emerging approach well suited to address biological and medical questions where conventional methods are not ideal.

Bacterial chemotaxis without gradient-sensing

Chemotaxis models are based on spatial or temporal gradient measurements by individual organisms. The key contribution of Keller and Segel (J Theor Biol 30:225-234, 1971a; J Theor Biol 30:235-248, 1971b) is showing that erratic measurements of individuals may result in an accurate chemotaxis phenomenon as a group. In this paper we provide another option to understand chemotactic behavior when individuals do not sense the gradient of chemical concentration by any means. We show that, if individuals increase their dispersal rate to find food when there is not enough food, an accurate chemotactic behavior may be obtained without sensing the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol 75:845-870, 2013) and was called starvation driven diffusion. This model is surprisingly similar to the original Keller-Segel model. A comprehensive picture of traveling bands and fronts is provided.

Fold-change detection in a whole-pathway model of Escherichia coli chemotaxis

There has been recent interest in sensory systems that are able to display a response which is proportional to a fold change in stimulus concentration, a feature referred to as fold-change detection (FCD). Here, we demonstrate FCD in a recent whole-pathway mathematical model of Escherichia coli chemotaxis. FCD is shown to hold for each protein in the signalling cascade and to be robust to kinetic rate and protein concentration variation. Using a sensitivity analysis, we find that only variations in the number of receptors within a signalling team lead to the model not exhibiting FCD. We also discuss the ability of a cell with multiple receptor types to display FCD and explain how a particular receptor configuration may be used to elucidate the two experimentally determined regimes of FCD behaviour. All findings are discussed in respect of the experimental literature.

Chemotactic clusters in confined run-and-tumble bacteria: a numerical investigation

We present a simulation study of pattern formation in an ensemble of chemotactic run-and-tumble bacteria, focussing on the effect of spatial confinement, either within traps or inside a maze. These geometries are inspired by previous experiments probing pattern formation in chemotactic strains of E. coli under these conditions. Our main result is that a microscopic model of chemotactic run-and-tumble particles which themselves secrete a chemoattractant is able to reproduce the main experimental observations, namely the formation of bacterial aggregates within traps and in dead ends of a maze. Our simulations also demonstrate that stochasticity plays a key role and leads to a hysteretic response when the chemotactic sensitivity is varied. We compare the results of run-and-tumble particles with simulations performed with a simplified version of the model where the active particles are smooth swimmers which respond to chemotactic gradients by rotating towards the source of chemoattractant. This class of models leads again to aggregation, but with quantitative and qualitative differences in, for instance, the size and shape of clusters.

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak-Keller-Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.

Evolution of dispersal toward fitness

It is widely believed that the slowest dispersal strategy is selected in the evolutional if the environment is temporally invariant but spatially heterogeneous. Authors claim in this paper that this belief is true only if random dispersals with constant motility are considered. However, if a dispersal strategy with fitness property is included, the size of the dispersal is not such a crucial factor anymore. Recently, a starvation driven diffusion has been introduced by Cho and Kim (Bull. Math. Biol., 2013), which is a random dispersal strategy with a motility increase on starvation. The authors show that such a dispersal strategy has fitness property and that the evolutional selection favors fitness but not simply slowness. Such a conclusion is obtained from a stability analysis of a competition system between two phenotypes with different dispersal strategies of linear and starvation driven diffusions.

Ultrasensitivity in independent multisite systems

Multisite modifications are widely recognized as an essential feature of many switch-like responses in signal transduction. It is usually assumed that the modification of one site directly or indirectly increases the rate of modification of neighboring sites. In this paper we provide a new set of assumptions for a multisite system to become highly ultrasensitive even in the absence of cooperativity or allostery. We assume that the individual sites are modified independently of each other, and that protein activity is an ultrasensitive function of the fraction of modified sites. These assumptions are particularly useful in the context of multisite systems with a large (8+) number of sites. We estimate the apparent Hill coefficient of the dose responses in the sequential and nonsequential cases, highlight their different qualitative properties, and discuss a formula to approximate dose responses in the nonsequential case. As an example we describe a model of bacterial chemotaxis that features robust ultrasensitivity and perfect adaptation over a wide range of ligand concentrations, based on non-allosteric multisite behavior at the level of receptors and flagella. We also include a model of the inactivation of the yeast pheromone protein Ste5 by cell cycle proteins.

Bacterial chemotaxis: introverted or extroverted? A comparison of the advantages and disadvantages of basic forms of metabolism-based and metabolism-independent behavior using a computational model

Using a minimal model of metabolism, we examine the limitations of behavior that is (a) solely in response to environmental phenomena or (b) solely in response to metabolic dynamics, showing that basic forms of each of these kinds of behavior are incapable of driving survival-prolonging behavior in certain situations. Inspired by experimental evidence of concurrent metabolism-based and metabolism-independent chemotactic mechanisms in Escherichia coli and Rhodobacter sphaeroides, we then investigate how metabolism-independent and metabolism-based sensitivities can be integrated into a single behavioral response, demonstrating that a simple switching mechanism can be sufficient to effectively integrate metabolism-based and metabolism-independent behaviors. Finally, we use a spatial simulation of bacteria to show that the investigated forms of behavior produce different spatio-temporal patterns that are influenced by the metabolic-history of the bacteria. We suggest that these patterns could be a way to experimentally derive insight into the relationship between metabolism and chemotaxis in real bacteria.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

Starvation driven diffusion as a survival strategy of biological organisms

The purpose of this article is to introduce a diffusion model for biological organisms that increase their motility when food or other resource is insufficient. It is shown in this paper that Fick's diffusion law does not explain such a starvation driven diffusion correctly. The diffusion model for nonuniform Brownian motion in Kim (Einstein's random walk and thermal diffusion, preprint http://amath.kaist.ac.kr/papers/Kim/31.pdf , 2013) is employed in this paper and a Fokker-Planck type diffusion law is obtained. Lotka-Volterra type competition systems with spatial heterogeneity are tested, where one species follows the starvation driven diffusion and the other follows the linear diffusion. In heterogeneous environments, the starvation driven diffusion turns out to be a better survival strategy than the linear one. Various issues such as the global asymptotic stability, convergence to an ideal free distribution, the extinction and coexistence of competing species are discussed.

Theoretical insights into bacterial chemotaxis

Research into understanding bacterial chemotactic systems has become a paradigm for Systems Biology. Experimental and theoretical researchers have worked hand-in-hand for over 40 years to understand the intricate behavior driving bacterial species, in particular how such small creatures, usually not more than 5 µm in length, detect and respond to small changes in their extracellular environment. In this review we highlight the importance that theoretical modeling has played in providing new insight and understanding into bacterial chemotaxis. We begin with an overview of the bacterial chemotaxis sensory response, before reviewing the role of theoretical modeling in understanding elements of the system on the single cell scale and features underpinning multiscale extensions to population models.

Spatial and stochastic cellular modeling with the Smoldyn simulator

This chapter describes how to use Smoldyn, which is a computer program for modeling cellular systems with spatial and stochastic detail. Smoldyn represents each molecule of interest as an individual point-like particle. These simulated molecules diffuse, interact with surfaces (e.g., biological membranes), and undergo chemical reactions much as they would in real biochemical systems. Smoldyn has been used to model signal transduction within bacterial cells, pheromone signaling between yeast cells, bacterial carboxysome function, diffusion in crowded spaces, and many other systems. A new "rule-based modeling" feature automatically generates chemical species and reactions as they arise in simulations due to protein modifications and complexation. Smoldyn is easy to use, quantitatively accurate, and computationally efficient. It is generally best for systems with length scales between nanometers and several microns, time scales from tens of nanoseconds to tens of minutes, and up to about 10(5) individual molecules. Smoldyn runs on Macintosh, Linux, or Windows systems, is open source, and can be downloaded from http://www.smoldyn.org.

Modeling a self-propelled autochemotactic walker

We develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis. This means that microorganisms, such as bacteria, amoebae, or cells, follow the gradient of a chemical that they produce themselves to attract or repel each other. A microorganism is represented as a self-propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle. We study the autochemotactic response of a single self-propelled walker whose dynamics is non-Markovian. We show that its long-time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak- and strong-coupling case. We confirm our findings by numerical simulations.

Adaptive response and enlargement of dynamic range

Many membrane channels and receptors exhibit adaptive, or desensitized, response to a strong sustained input stimulus, often supported by protein activity-dependent inactivation. Adaptive response is thought to be related to various cellular functions such as homeostasis and enlargement of dynamic range by background compensation. Here we study the quantitative relation between adaptive response and background compensation within a modeling framework. We show that any particular type of adaptive response is neither sufficient nor necessary for adaptive enlargement of dynamic range. In particular a precise adaptive response, where system activity is maintained at a constant level at steady state, does not ensure a large dynamic range neither in input signal nor in system output. A general mechanism for input dynamic range enlargement can come about from the activity-dependent modulation of protein responsiveness by multiple biochemical modification, regardless of the type of adaptive response it induces. Therefore hierarchical biochemical processes such as methylation and phosphorylation are natural candidates to induce this property in signaling systems.

How mathematical modelling elucidates signalling in Bacillus subtilis

Appropriate stimulus perception, signal processing and transduction ensure optimal adaptation of bacteria to environmental challenges. In the Gram-positive model bacterium Bacillus subtilis signalling networks and molecular interactions therein are well-studied, making this species a suitable candidate for the application of mathematical modelling. Here, we review systems biology approaches, focusing on chemotaxis, sporulation, σ(B) -dependent general stress response and competence. Processes like chemotaxis and Z-ring assembly depend critically on the subcellular localization of proteins. Environmental response strategies, including sporulation and competence, are characterized by phenotypic heterogeneity in isogenic cultures. The examples of mathematical modelling also include investigations that have demonstrated how operon structure and signalling dynamics are intricately interwoven to establish optimal responses. Our review illustrates that these interdisciplinary approaches offer new insights into the response of B. subtilis to environmental challenges. These case studies reveal modelling as a tool to increase the understanding of complex systems, to help formulating hypotheses and to guide the design of more directed experiments that test predictions.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Attractant binding induces distinct structural changes to the polar and lateral signaling clusters in Bacillus subtilis chemotaxis

Bacteria employ a modified two-component system for chemotaxis, where the receptors form ternary complexes with CheA histidine kinases and CheW adaptor proteins. These complexes are arranged in semi-ordered arrays clustered predominantly at the cell poles. The prevailing models assume that these arrays are static and reorganize only locally in response to attractant binding. Recent studies have shown, however, that these structures may in fact be much more fluid. We investigated the localization of the chemotaxis signaling arrays in Bacillus subtilis using immunofluorescence and live cell fluorescence microscopy. We found that the receptors were localized in clusters at the poles in most cells. However, when the cells were exposed to attractant, the number exhibiting polar clusters was reduced roughly 2-fold, whereas the number exhibiting lateral clusters distinct from the poles increased significantly. These changes in receptor clustering were reversible as polar localization was reestablished in adapted cells. We also investigated the dynamic localization of CheV, a hybrid protein consisting of an N-terminal CheW-like adaptor domain and a C-terminal response regulator domain that is known to be phosphorylated by CheA, using immunofluorescence. Interestingly, we found that CheV was localized predominantly at lateral clusters in unstimulated cells. However, upon exposure to attractant, CheV was found to be predominantly localized to the cell poles. Moreover, changes in CheV localization are phosphorylation-dependent. Collectively, these results suggest that the chemotaxis signaling arrays in B. subtilis are dynamic structures and that feedback loops involving phosphorylation may regulate the positioning of individual proteins.

Adaptation and control circuits in bacterial chemotaxis

Bacteria are capable of sensing and responding to changes in their environment. One of the ways they do this is via chemotaxis, regulating swimming behaviour. The chemotaxis pathway senses chemoattractant gradients and uses a feedback loop to change the bacterial swimming pattern; this feedback loop differs in detail between species. In the present article, we summarize the current understanding of the regulatory mechanisms in three species and how these pathways can be viewed and analysed through the ideas of feedback control systems engineering.

Modeling chemotaxis reveals the role of reversed phosphotransfer and a bi-functional kinase-phosphatase

Understanding how multiple signals are integrated in living cells to produce a balanced response is a major challenge in biology. Two-component signal transduction pathways, such as bacterial chemotaxis, comprise histidine protein kinases (HPKs) and response regulators (RRs). These are used to sense and respond to changes in the environment. Rhodobacter sphaeroides has a complex chemosensory network with two signaling clusters, each containing a HPK, CheA. Here we demonstrate, using a mathematical model, how the outputs of the two signaling clusters may be integrated. We use our mathematical model supported by experimental data to predict that: (1) the main RR controlling flagellar rotation, CheY₆, aided by its specific phosphatase, the bifunctional kinase CheA₃, acts as a phosphate sink for the other RRs; and (2) a phosphorelay pathway involving CheB₂ connects the cytoplasmic cluster kinase CheA₃ with the polar localised kinase CheA₂, and allows CheA₃-P to phosphorylate non-cognate chemotaxis RRs. These two mechanisms enable the bifunctional kinase/phosphatase activity of CheA₃ to integrate and tune the sensory output of each signaling cluster to produce a balanced response. The signal integration mechanisms identified here may be widely used by other bacteria, since like R. sphaeroides, over 50% of chemotactic bacteria have multiple cheA homologues and need to integrate signals from different sources.

Chemotactic bacteria sense nutrient gradients and swim towards better environments for growth. A cluster of receptors in the cell membrane detects nutrient levels and signals via a cytoplasmic signaling pathway to the flagellum. The complexity of this signaling pathway varies across different bacterial species. The relatively simple pathway used by Escherichia coli is well understood; however, many bacteria, for example Rhodobacter sphaeroides, have more sophisticated pathways that, as well as being able to detect nutrients, are also able to assess the metabolic state of the cell. The receptors that detect metabolic state are located within an additional cluster that is physically distinct from the one that senses nutrients. In this work, we use a combination of experimentation and mathematical modeling to gain insight into the complex decision-making mechanisms that enable bacteria to weigh-up different stimuli and decide upon an appropriate response. We find novel communication mechanisms between the two signaling clusters that allow the outputs of the signaling pathways to be balanced and tuned according to the prevailing environmental conditions. The signaling principles identified here are likely to be used in other complex sensory networks.