The physics of flagellar motion of E. coli during chemotaxis

Flagellar motion has been an active area of study right from the discovery of bacterial chemotaxis in 1882. During chemotaxis, E. coli moves with the help of helical flagella in an aquatic environment. Helical flagella are rotated in clockwise or counterclockwise direction using reversible flagellar motors situated at the base of each flagellum. The swimming of E. coli is characterized by a low Reynolds number that is unique and time reversible. The random motion of E. coli is influenced by the viscosity of the fluid and the Brownian motion of molecules of fluid, chemoattractants, and chemorepellants. This paper reviews the literature about the physics involved in the propulsion mechanism of E. coli. Starting from the resistive-force theory, various theories on flagellar hydrodynamics are critically reviewed. Expressions for drag force, elastic force and velocity of flagellar elements are derived. By taking the elastic nature of flagella into account, linear and nonlinear equations of motions are derived and their solutions are presented.     

Experiences with the Coulter Counter in Bacteriology

Viable and killed suspensions of Staphylococcus aureus SM, Escherichia coli, and Serratia marcescens, as well as polystyrene spheres, 0.81 and 2.85 μ in diameter, were counted electronically with a model A Coulter Counter. Simultaneous counts by standard bacteriological methods and microscopy were done for purposes of control and comparison with the data from the Coulter Counter. Results indicated: (i) electrical characteristics of different bacterial populations are different; (ii) electronic counts were consistent for species used; (iii) live S. aureus exhibits a denser pattern of thick bright pulses on the cathode-ray tube than does live E. coli; (iv) killed bacteria resemble inert particles in pulse pattern; and (v) some viable bacteria do not react independently of current flow, as do inert particles and killed bacteria.     

Cyclic AMP waves during aggregation of Dictyostelium amoebae

During the aggregation phase of their life cycle, Dictyostelium discoideum amoebae communicate with each other by traveling waves of cyclic AMP. These waves are generated by an interplay between random diffusion of cyclic AMP in the extracellular milieu and the signal-reception/signal/relaying capabilities of individual amoebae. Kinetic properties of the enzymes, transport proteins and cell-surface receptor proteins involved in the cyclic AMP signaling system have been painstakingly worked out over the past fifteen years in many laboratories. Recently Martiel & Goldbeter (1987) incorporated this biochemical information into a unified mathematical model of communication among Dictyostelium amoebae. Numerical simulations of the mathematical model, carried out by Tyson et al. (1989), agree in quantitative detail with experimental observations of cyclic AMP traveling waves in Dictyostelium cultures. Such mathematical modeling and numerical experimentation provide a necessary link between detailed studies of the molecular control mechanism and experimental observations of the intact developmental system.     

The Genetic Basis of Escherichia coli Pathoadaptation to Macrophages

Antagonistic interactions are likely important driving forces of the evolutionary process underlying bacterial genome complexity and diversity. We hypothesized that the ability of evolved bacteria to escape specific components of host innate immunity, such as phagocytosis and killing by macrophages (MΦ), is a critical trait relevant in the acquisition of bacterial virulence. Here, we used a combination of experimental evolution, phenotypic characterization, genome sequencing and mathematical modeling to address how fast, and through how many adaptive steps, a commensal Escherichia coli (E. coli) acquire this virulence trait. We show that when maintained in vitro under the selective pressure of host MΦ commensal E. coli can evolve, in less than 500 generations, virulent clones that escape phagocytosis and MΦ killing in vitro, while increasing their pathogenicity in vivo, as assessed in mice. This pathoadaptive process is driven by a mechanism involving the insertion of a single transposable element into the promoter region of the E. coli yrfF gene. Moreover, transposition of the IS186 element into the promoter of Lon gene, encoding an ATP-dependent serine protease, is likely to accelerate this pathoadaptive process. Competition between clones carrying distinct beneficial mutations dominates the dynamics of the pathoadaptive process, as suggested from a mathematical model, which reproduces the observed experimental dynamics of E. coli evolution towards virulence. In conclusion, we reveal a molecular mechanism explaining how a specific component of host innate immunity can modulate microbial evolution towards pathogenicity.     

Kinetic model of Proteus mirabilis swarm colony development

 Proteus mirabilis colonies display striking symmetry and periodicity. Based on experimental observations of cellular differentiation and group motility, a kinetic model has been developed to describe the swarmer cell differentiation-dedifferentiation cycle and the spatial evolution of swimmer and swarmer cells during Proteus mirabilis swarm colony development. A key element of the model is the age dependence of swarmer cell behaviour, in particular specifying a minimal age for motility and maximum age for septation and dedifferentiation to swimmer cells. Density thresholds for collective motility by mature swarmer cells serve to synchronize the movements of distinct swarmer cell groups and thus help provide temporal coherence to colony expansion cycles. Numerical computations show that the model fits experimental data by generating a complete swarming plus consolidation cycle period that is robust to changes in parameters which affect other aspects of swarmer cell migration and colony development. The kinetic equations underlying this model provide a different mathematical basis for a temporal oscillator from reaction-diffusion partial differential equations. The modelling shows that Proteus colony geometries arise as a consequence of macroscopic rules governing collective motility. Thus, in this case, pattern formation results from the operation of an adaptive bacterial system for spreading on solid substrates, not as an independent biological function. Kinetic models similar to this one may be applicable to periodic phenomena displayed by other biological systems with differentiated components of defined lifetimes. Received 3 July 1996; received in revised form 9 December 1996     

Transient pulse formation in jasmonate signaling pathway

The jasmonate (JA) signaling pathway in plants is activated as defense response to a number of stresses like attacks by pests or pathogens and wounding by animals. Some recent experiments provide significant new knowledge on the molecular detail and connectivity of the pathway. The pathway has two major components in the form of feedback loops, one negative and the other positive. We construct a minimal mathematical model, incorporating the feedback loops, to study the dynamics of the JA signaling pathway. The model exhibits transient gene expression activity in the form of JA pulses in agreement with experimental observations. The dependence of the pulse amplitude, duration and peak time on the key parameters of the model is determined computationally. The deterministic and stochastic aspects of the pathway dynamics are investigated using both the full mathematical model and a reduced version of it. We also compare the mechanism of pulse formation with the known mechanisms of pulse generation in some bacterial and viral systems.     

Kinetics and models of the Drosophila heat-shock system

The puffing of Drosophila heat-shock genes after (1) a step-wise temperature increase (“heat-shock”); (2) recovery from anaerobiosis; and (3) incubation with uncoupling reagents was expressed as percent of the maximal size and normalized to the time scale. Data were taken from the literature and new measurements. In addition, puffing was measured after a 30-min temperature pulse and after two 30-min pulses. The latter experiment revealed a second, smaller increase in puff-size. Data on RNA and protein synthesis in Drosophila cells were collected from the literature and also normalized. From the available data, a feed-back control system is derived that consists of a controlled variable x, possibly a metabolic function of the mitochondria, interacting with an activator molecule which exists in an active (A⁺) and an inactive (A^?) configuration. A⁺ activates the heat-shock genes which in turn produce their mRNA (y) and proteins (z) which then change the controlled variable x into a new steady state. A modified version of this model assumes a feed-back control of the heat-shock proteins on the activator molecule. A mathematical model of this system (Goodwin, 1965) was simulated by computer and compared with the experimental results.     

A Traveling Wave Model for Invasion by Precursor and Differentiated Cells

We develop and investigate a continuum model for invasion of a domain by cells that migrate, proliferate and differentiate. The model is applicable to neural crest cell invasion in the developing enteric nervous system, but is presented in general terms and is of broader applicability. Two cell populations are identified and modeled explicitly; a population of precursor cells that migrate and proliferate, and a population of differentiated cells derived from the precursors which have impaired migration and proliferation. The equation describing the precursor cells is based on Fisher’s equation with the addition of a carrying-capacity limited differentiation term. Two variations of the proliferation term are considered and compared. For most parameter values, the model admits a traveling wave solution for each population, both traveling at the same speed. The traveling wave solutions are investigated using perturbation analysis, phase plane methods, and numerical techniques. Analytical and numerical results suggest the existence of two wavespeed selection regimes. Regions of the parameter space are characterized according to existence, shape, and speed of traveling wave solutions. Our observations may be used in conjunction with experimental results to identify key parameters determining the invasion speed for a particular biological system. Furthermore, our results may assist experimentalists in identifying the resource that is limiting proliferation of precursor cells.     

Occurrence and Treatment of Bone Atrophic Non-Unions Investigated by an Integrative Approach

Recently developed atrophic non-union models are a good representation of the clinical situation in which many non-unions develop. Based on previous experimental studies with these atrophic non-union models, it was hypothesized that in order to obtain successful fracture healing, blood vessels, growth factors, and (proliferative) precursor cells all need to be present in the callus at the same time. This study uses a combined in vivo-in silico approach to investigate these different aspects (vasculature, growth factors, cell proliferation). The mathematical model, initially developed for the study of normal fracture healing, is able to capture essential aspects of the in vivo atrophic non-union model despite a number of deviations that are mainly due to simplifications in the in silico model. The mathematical model is subsequently used to test possible treatment strategies for atrophic non-unions (i.e. cell transplant at post-osteotomy, week 3). Preliminary in vivo experiments corroborate the numerical predictions. Finally, the mathematical model is applied to explain experimental observations and identify potentially crucial steps in the treatments and can thereby be used to optimize experimental and clinical studies in this area. This study demonstrates the potential of the combined in silico-in vivo approach and its clinical implications for the early treatment of patients with problematic fractures.     

Lattice-Boltzmann model for bacterial chemotaxis

We present a new numerical approach for modeling bacterial chemotaxis and the fate and transport of a chemoattractant in bulk liquids. This Lattice-Boltzmann method represents the microorganisms and the chemoattractant by quasi-particles that move, collide, and react with each other on a two-dimensional numerical lattice. We use the model to simulate traveling bands of bacteria along self-generated gradients in substrate concentration in bulk liquids. Particularly, we simulate Pseudomonas putida that respond chemotactically to naphthalene dissolved in water. We find that only a fraction of a bacterial slug injected into a domain containing the chemoattractant at constant concentration forms a traveling band as the slug length exceeds a critical value. An expanding bacterial ring forms as one injects a droplet of bacteria into a two-dimensional domain.     

Inactivation of Pseudomonas putida by Pulsed Electric Field Treatment: A Study on the Correlation of Treatment Parameters and Inactivation Efficiency in the Short-Pulse Range

An important issue for an economic application of the pulsed electric field treatment for bacterial decontamination of wastewater is the specific treatment energy needed for effective reduction of bacterial populations. The present experimental study performed in a field amplitude range of 40 > E > 200 kV/cm and for a suspension conductivity of 0.01 = κ _e > 0.2 S/m focusses on the application of short pulses, 25 ns > T > 10 μs, of rectangular, bipolar and exponential shape and was made on Pseudomonas putida, which is a typical and widespread wastewater microorganism. The comparison of inactivation results with calculations of the temporal and azimuthal membrane charging dynamics using the model of Pauly and Schwan revealed that for efficient inactivation, membrane segments at the cell equator have to be charged quickly and to a sufficiently high value, on the order of 0.5 V. After fulfilling this basic condition by an appropriate choice of pulse field strength and duration, the log rate of inactivation for a given suspension conductivity of 0.2 S/m was found to be independent of the duration of individual pulses for constant treatment energy expenditure. Moreover, experimental results suggest that even pulse shape plays a minor role in inactivation efficiency. The variation of the suspension conductivity resulted in comparable inactivation performance of identical pulse parameters if the product of pulse duration and number of pulses was the same, i.e., required treatment energy can be linearly downscaled for lower conductivities, provided that pulse amplitude and duration are selected for entire membrane surface permeabilization.     

Growth-dependent bacterial susceptibility to ribosome-targeting antibiotics

Bacterial growth environment strongly influences the efficacy of antibiotic treatment, with slow growth often being associated with decreased susceptibility. Yet in many cases, the connection between antibiotic susceptibility and pathogen physiology remains unclear. We show that for ribosome-targeting antibiotics acting on Escherichia coli, a complex interplay exists between physiology and antibiotic action; for some antibiotics within this class, faster growth indeed increases susceptibility, but for other antibiotics, the opposite is true. Remarkably, these observations can be explained by a simple mathematical model that combines drug transport and binding with physiological constraints. Our model reveals that growth-dependent susceptibility is controlled by a single parameter characterizing the ‘reversibility’ of ribosome-targeting antibiotic transport and binding. This parameter provides a spectrum classification of antibiotic growth-dependent efficacy that appears to correspond at its extremes to existing binary classification schemes. In these limits, the model predicts universal, parameter-free limiting forms for growth inhibition curves. The model also leads to non-trivial predictions for the drug susceptibility of a translation mutant strain of E. coli, which we verify experimentally. Drug action and bacterial metabolism are mechanistically complex; nevertheless, this study illustrates how coarse-grained models can be used to integrate pathogen physiology into drug design and treatment strategies.     

Soliton-Like Regimes and Excitation Pulse Reflection (Echo) in Homogeneous Cardiac Purkinje Fibers: Results of Numerical Simulations

On the basis of numerical simulations of the partial McAllister-Noble-Tsien equations quantitatively describing the dynamics of electrical processes in conductive cardiac Purkinje fibers we reveal unusual – soliton-like – regimes of interaction of nonlinear excitation pulses governing the heart contraction rhythm: reflection of colliding pulses instead of their annihilation. The phenomenological mechanism of the reflection effects is that in a narrow (but finite) range of the system parameters the traveling pulse presents a doublet consisting of a high-amplitude leader followed by a low-amplitude subthreshold wave. Upon collisions of pulses the leaders are annihilated, but subthreshold waves summarize becoming superthreshold and initiating two novel echo-pulses traveling in opposite directions. The phenomenon revealed presents an analogy to the effect of reflection of colliding nerve pulses, predicted recently, and can be of use in getting insight into the mechanisms of heart rhythm disturbances.     

Social Aggregation in Pea Aphids: Experiment and Random Walk Modeling

From bird flocks to fish schools and ungulate herds to insect swarms, social biological aggregations are found across the natural world. An ongoing challenge in the mathematical modeling of aggregations is to strengthen the connection between models and biological data by quantifying the rules that individuals follow. We model aggregation of the pea aphid, Acyrthosiphon pisum. Specifically, we conduct experiments to track the motion of aphids walking in a featureless circular arena in order to deduce individual-level rules. We observe that each aphid transitions stochastically between a moving and a stationary state. Moving aphids follow a correlated random walk. The probabilities of motion state transitions, as well as the random walk parameters, depend strongly on distance to an aphid''s nearest neighbor. For large nearest neighbor distances, when an aphid is essentially isolated, its motion is ballistic with aphids moving faster, turning less, and being less likely to stop. In contrast, for short nearest neighbor distances, aphids move more slowly, turn more, and are more likely to become stationary; this behavior constitutes an aggregation mechanism. From the experimental data, we estimate the state transition probabilities and correlated random walk parameters as a function of nearest neighbor distance. With the individual-level model established, we assess whether it reproduces the macroscopic patterns of movement at the group level. To do so, we consider three distributions, namely distance to nearest neighbor, angle to nearest neighbor, and percentage of population moving at any given time. For each of these three distributions, we compare our experimental data to the output of numerical simulations of our nearest neighbor model, and of a control model in which aphids do not interact socially. Our stochastic, social nearest neighbor model reproduces salient features of the experimental data that are not captured by the control.     

Collective Motion of Spherical Bacteria

A large variety of motile bacterial species exhibit collective motions while inhabiting liquids or colonizing surfaces. These collective motions are often characterized by coherent dynamic clusters, where hundreds of cells move in correlated whirls and jets. Previously, all species that were known to form such motion had a rod-shaped structure, which enhances the order through steric and hydrodynamic interactions. Here we show that the spherical motile bacteria Serratia marcescens exhibit robust collective dynamics and correlated coherent motion while grown in suspensions. As cells migrate to the upper surface of a drop, they form a monolayer, and move collectively in whirls and jets. At all concentrations, the distribution of the bacterial speed was approximately Rayleigh with an average that depends on concentration in a non-monotonic way. Other dynamical parameters such as vorticity and correlation functions are also analyzed and compared to rod-shaped bacteria from the same strain. Our results demonstrate that self-propelled spherical objects do form complex ordered collective motion. This opens a door for a new perspective on the role of cell aspect ratio and alignment of cells with regards to collective motion in nature.     

Collective motion of bacteria in two dimensions

Collective motion can be observed in biological systems over a wide range of length scales, from large animals to bacteria. Collective motion is thought to confer an advantage for defense and adaptation. A central question in the study of biological collective motion is how the traits of individuals give rise to the emergent behavior at population level. This question is relevant to the dynamics of general self-propelled particle systems, biological self-organization, and active fluids. Bacteria provide a tractable system to address this question, because bacteria are simple and their behavior is relatively easy to control. In this mini review we will focus on a special form of bacterial collective motion, i.e., bacterial swarming in two dimensions. We will introduce some organization principles known in bacterial swarming and discuss potential means of controlling its dynamics. The simplicity and controllability of 2D bacterial behavior during swarming would allow experimental examination of theory predictions on general collective motion.     

Modeling the Infection Dynamics of Bacteriophages in Enteric Escherichia coli: Estimating the Contribution of Transduction to Antimicrobial Gene Spread

Animal-associated bacterial communities are infected by bacteriophages, although the dynamics of these infections are poorly understood. Transduction by bacteriophages may contribute to transfer of antimicrobial resistance genes, but the relative importance of transduction among other gene transfer mechanisms is unknown. We therefore developed a candidate deterministic mathematical model of the infection dynamics of enteric coliphages in commensal Escherichia coli in the large intestine of cattle. We assumed the phages were associated with the intestine and were predominantly temperate. Model simulations demonstrated how, given the bacterial ecology and infection dynamics, most (>90%) commensal enteric E. coli bacteria may become lysogens of enteric coliphages during intestinal transit. Using the model and the most liberal assumptions about transduction efficiency and resistance gene frequency, we approximated the upper numerical limits (“worst-case scenario”) of gene transfer through specialized and generalized transduction in E. coli by enteric coliphages when the transduced genetic segment is picked at random. The estimates were consistent with a relatively small contribution of transduction to lateral gene spread; for example, generalized transduction delivered the chromosomal resistance gene to up to 8 E. coli bacteria/hour within the population of 1.47 × 10⁸ E. coli bacteria/liter luminal contents. In comparison, the plasmidic bla_CMY-2 gene carried by ∼2% of enteric E. coli was transferred by conjugation at a rate at least 1.4 × 10³ times greater than our generalized transduction estimate. The estimated numbers of transductants varied nonlinearly depending on the ecology of bacteria available for phages to infect, that is, on the assumed rates of turnover and replication of enteric E. coli.