Multi‐bit Boolean model for chemotactic drift of Escherichia coli

Dynamic biological systems can be modelled to an equivalent modular structure using Boolean networks (BNs) due to their simple construction and relative ease of integration. The chemotaxis network of the bacterium Escherichia coli (E. coli ) is one of the most investigated biological systems. In this study, the authors developed a multi‐bit Boolean approach to model the drifting behaviour of the E. coli chemotaxis system. Their approach, which is slightly different than the conventional BNs, is designed to provide finer resolution to mimic high‐level functional behaviour. Using this approach, they simulated the transient and steady‐state responses of the chemoreceptor sensory module. Furthermore, they estimated the drift velocity under conditions of the exponential nutrient gradient. Their predictions on chemotactic drifting are in good agreement with the experimental measurements under similar input conditions. Taken together, by simulating chemotactic drifting, they propose that multi‐bit Boolean methodology can be used for modelling complex biological networks. Application of the method towards designing bio‐inspired systems such as nano‐bots is discussed.Inspec keywords: cell motility, microorganisms, Boolean functionsOther keywords: multibit Boolean approach, conventional BNs, high‐level functional behaviour, steady‐state responses, chemoreceptor sensory module, drift velocity, chemotactic drifting, multibit Boolean methodology, complex biological networks, bio‐inspired systems, multibit Boolean model, chemotactic drift, dynamic biological systems, equivalent modular structure, Boolean networks, simple construction, chemotaxis network, bacterium Escherichia coli, biological systems     

Digital clocks: simple Boolean models can quantitatively describe circadian systems

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock''s oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.     

A farewell to R: time-series models for tracking and forecasting epidemics

The time-dependent reproduction number, R_t, is a key metric used by epidemiologists to assess the current state of an outbreak of an infectious disease. This quantity is usually estimated using time-series observations on new infections combined with assumptions about the distribution of the serial interval of transmissions. Bayesian methods are often used with the new cases data smoothed using a simple, but to some extent arbitrary, moving average. This paper describes a new class of time-series models, estimated by classical statistical methods, for tracking and forecasting the growth rate of new cases and deaths. Very few assumptions are needed and those that are made can be tested. Estimates of R_t, together with their standard deviations, are obtained as a by-product.     

Searching for motifs in the behaviour of larval Drosophila melanogaster and Caenorhabditis elegans reveals continuity between behavioural states

We present a novel method for the unsupervised discovery of behavioural motifs in larval Drosophila melanogaster and Caenorhabditis elegans. A motif is defined as a particular sequence of postures that recurs frequently. The animal''s changing posture is represented by an eigenshape time series, and we look for motifs in this time series. To find motifs, the eigenshape time series is segmented, and the segments clustered using spline regression. Unlike previous approaches, our method can classify sequences of unequal duration as the same motif. The behavioural motifs are used as the basis of a probabilistic behavioural annotator, the eigenshape annotator (ESA). Probabilistic annotation avoids rigid threshold values and allows classification uncertainty to be quantified. We apply eigenshape annotation to both larval Drosophila and C. elegans and produce a good match to hand annotation of behavioural states. However, we find many behavioural events cannot be unambiguously classified. By comparing the results with ESA of an artificial agent''s behaviour, we argue that the ambiguity is due to greater continuity between behavioural states than is generally assumed for these organisms.     

Fractal analysis of behaviour in a wild primate: behavioural complexity in health and disease

Parasitism and other stressors are ubiquitous in nature but their effects on animal behaviour can be difficult to identify. We investigated the effects of nematode parasitism and other indicators of physiological impairment on the sequential complexity of foraging and locomotion behaviour among wild Japanese macaques (Macaca fuscata yakui). We observed all sexually mature individuals (n = 28) in one macaque study group between October 2007 and August 2008, and collected two faecal samples/month/individual (n = 362) for parasitological examination. We used detrended fluctuation analysis (DFA) to investigate long-range autocorrelation in separate, binary sequences of foraging (n = 459) and locomotion (n = 446) behaviour collected via focal sampling. All behavioural sequences exhibited long-range autocorrelation, and linear mixed-effects models suggest that increasing infection with the nodular worm Oesophagostomum aculeatum, clinically impaired health, reproductive activity, ageing and low dominance status were associated with reductions in the complexity of locomotion, and to a lesser extent foraging, behaviour. Furthermore, the sequential complexity of behaviour increased with environmental complexity. We argue that a reduction in complexity in animal behaviour characterizes individuals in impaired or ‘stressed’ states, and may have consequences if animals cannot cope with heterogeneity in their natural habitats.     

Directional reversals enable Myxococcus xanthus cells to produce collective one-dimensional streams during fruiting-body formation

The formation of a collectively moving group benefits individuals within a population in a variety of ways. The surface-dwelling bacterium Myxococcus xanthus forms dynamic collective groups both to feed on prey and to aggregate during times of starvation. The latter behaviour, termed fruiting-body formation, involves a complex, coordinated series of density changes that ultimately lead to three-dimensional aggregates comprising hundreds of thousands of cells and spores. How a loose, two-dimensional sheet of motile cells produces a fixed aggregate has remained a mystery as current models of aggregation are either inconsistent with experimental data or ultimately predict unstable structures that do not remain fixed in space. Here, we use high-resolution microscopy and computer vision software to spatio-temporally track the motion of thousands of individuals during the initial stages of fruiting-body formation. We find that cells undergo a phase transition from exploratory flocking, in which unstable cell groups move rapidly and coherently over long distances, to a reversal-mediated localization into one-dimensional growing streams that are inherently stable in space. These observations identify a new phase of active collective behaviour and answer a long-standing open question in Myxococcus development by describing how motile cell groups can remain statistically fixed in a spatial location.     

Lag,lock, sync,slip: the many ‘phases’ of coupled flagella

In a multitude of life''s processes, cilia and flagella are found indispensable. Recently, the biflagellated chlorophyte alga Chlamydomonas has become a model organism for the study of ciliary motility and synchronization. Here, we use high-speed, high-resolution imaging of single pipette-held cells to quantify the rich dynamics exhibited by their flagella. Underlying this variability in behaviour are biological dissimilarities between the two flagella—termed cis and trans, with respect to a unique eyespot. With emphasis on the wild-type, we derive limit cycles and phase parametrizations for self-sustained flagellar oscillations from digitally tracked flagellar waveforms. Characterizing interflagellar phase synchrony via a simple model of coupled oscillators with noise, we find that during the canonical swimming breaststroke the cis flagellum is consistently phase-lagged relative to, while remaining robustly phase-locked with, the trans flagellum. Transient loss of synchrony, or phase slippage, may be triggered stochastically, in which the trans flagellum transitions to a second mode of beating with attenuated beat envelope and increased frequency. Further, exploiting this alga''s ability for flagellar regeneration, we mechanically induced removal of one or the other flagellum of the same cell to reveal a striking disparity between the beatings of the cis and trans flagella, in isolation. These results are evaluated in the context of the dynamic coordination of Chlamydomonas flagella.     

Three-dimensional tracking and behaviour monitoring of multiple fruit flies

The increasing interest in the investigation of social behaviours of a group of animals has heightened the need for developing tools that provide robust quantitative data. Drosophila melanogaster has emerged as an attractive model for behavioural analysis; however, there are still limited ways to monitor fly behaviour in a quantitative manner. To study social behaviour of a group of flies, acquiring the position of each individual over time is crucial. There are several studies that have tried to solve this problem and make this data acquisition automated. However, none of these studies has addressed the problem of keeping track of flies for a long period of time in three-dimensional space. Recently, we have developed an approach that enables us to detect and keep track of multiple flies in a three-dimensional arena for a long period of time, using multiple synchronized and calibrated cameras. After detecting flies in each view, correspondence between views is established using a novel approach we call the ‘sequential Hungarian algorithm’. Subsequently, the three-dimensional positions of flies in space are reconstructed. We use the Hungarian algorithm and Kalman filter together for data association and tracking. We evaluated rigorously the system''s performance for tracking and behaviour detection in multiple experiments, using from one to seven flies. Overall, this system presents a powerful new method for studying complex social interactions in a three-dimensional environment.     

Issues relating to numerical modelling of creep crack growth

In this paper creep crack growth behaviour of P92 welds at 923 K are presented. Creep crack growth behaviour for P92 welds are discussed with C^* parameter. Creep crack growth behaviour of P92 welds has been compared with that of P91 welds with C^* parameter. NSW and NSW-MOD model were compared with the experimental creep crack growth data. Plane strain NSW model significantly overestimates the crack growth rate, and plane stress NSW model underestimates it. Whilst, NSW-MOD model for plane stress and plane strain conditions gives lower and upper bound of the experimental data, respectively.FE analysis of creep crack growth has been conducted. Constrain effect for welded joints has been examined with C^* line integrals of C(T) specimens. As a result, constant C^* value using the material data of welded joint gives 10 times lower than that of only HAZ property. Whilst, the predicted CCG rates for welded joint are 10 times higher than those for only HAZ properties. Compared with predicted CCG rate from FE analysis and the experimental CCG rate, it can be suggested that creep crack growth tests for lower load level or for large specimen should be conducted, otherwise the experimental data should give unconservative estimation for components operated in long years.     

Mapping the stereotyped behaviour of freely moving fruit flies

A frequent assumption in behavioural science is that most of an animal''s activities can be described in terms of a small set of stereotyped motifs. Here, we introduce a method for mapping an animal''s actions, relying only upon the underlying structure of postural movement data to organize and classify behaviours. Applying this method to the ground-based behaviour of the fruit fly, Drosophila melanogaster, we find that flies perform stereotyped actions roughly 50% of the time, discovering over 100 distinguishable, stereotyped behavioural states. These include multiple modes of locomotion and grooming. We use the resulting measurements as the basis for identifying subtle sex-specific behavioural differences and revealing the low-dimensional nature of animal motions.     

Effect of magnetically simulated zero-gravity and enhanced gravity on the walk of the common fruitfly

Understanding the effects of gravity on biological organisms is vital to the success of future space missions. Previous studies in Earth orbit have shown that the common fruitfly (Drosophila melanogaster) walks more quickly and more frequently in microgravity, compared with its motion on Earth. However, flight preparation procedures and forces endured on launch made it difficult to implement on the Earth''s surface a control that exposed flies to the same sequence of major physical and environmental changes. To address the uncertainties concerning these behavioural anomalies, we have studied the walking paths of D. melanogaster in a pseudo-weightless environment (0g*) in our Earth-based laboratory. We used a strong magnetic field, produced by a superconducting solenoid, to induce a diamagnetic force on the flies that balanced the force of gravity. Simultaneously, two other groups of flies were exposed to a pseudo-hypergravity environment (2g*) and a normal gravity environment (1g*) within the spatially varying field. The flies had a larger mean speed in 0g* than in 1g*, and smaller in 2g*. The mean square distance travelled by the flies grew more rapidly with time in 0g* than in 1g*, and slower in 2g*. We observed no other clear effects of the magnetic field, up to 16.5 T, on the walks of the flies. We compare the effect of diamagnetically simulated weightlessness with that of weightlessness in an orbiting spacecraft, and identify the cause of the anomalous behaviour as the altered effective gravity.     

Chemical warfare and survival strategies in bacterial range expansions

Dispersal of species is a fundamental ecological process in the evolution and maintenance of biodiversity. Limited control over ecological parameters has hindered progress in understanding of what enables species to colonize new areas, as well as the importance of interspecies interactions. Such control is necessary to construct reliable mathematical models of ecosystems. In our work, we studied dispersal in the context of bacterial range expansions and identified the major determinants of species coexistence for a bacterial model system of three Escherichia coli strains (toxin-producing, sensitive and resistant). Genetic engineering allowed us to tune strain growth rates and to design different ecological scenarios (cyclic and hierarchical). We found that coexistence of all strains depended on three strongly interdependent factors: composition of inoculum, relative strain growth rates and effective toxin range. Robust agreement between our experiments and a thoroughly calibrated computational model enabled us to extrapolate these intricate interdependencies in terms of phenomenological biodiversity laws. Our mathematical analysis also suggested that cyclic dominance between strains is not a prerequisite for coexistence in competitive range expansions. Instead, robust three-strain coexistence required a balance between growth rates and either a reduced initial ratio of the toxin-producing strain, or a sufficiently short toxin range.     

Anatomy of a negative feedback loop: the case of IκBα

The magnitude, duration and oscillation of cellular signalling pathway responses are often limited by negative feedback loops, defined as an ‘activator-induced inhibitor’ regulatory motif. Within the NFκB signalling pathway, a key negative feedback regulator is IκBα. We show here that, contrary to current understanding, NFκB-inducible expression is not sufficient for providing effective negative feedback. We then employ computational simulations of NFκB signalling to identify IκBα molecular properties that are critical for proper negative feedback control and test the resulting predictions in biochemical and single-cell live-imaging studies. We identified nuclear import and nuclear export of IκBα and the IκBα–NFκB complex, as well as the free IκBα half-life, as key determinants of post-induction repression of NFκB and the potential for subsequent reactivation. Our work emphasizes that negative feedback is an emergent systems property determined by multiple molecular and biophysical properties in addition to the required ‘activator-induced inhibitor’ relationship.     

Transmission and dose�Cresponse experiments for social animals: a reappraisal of the colonization biology of Campylobacter jejuni in chickens

Dose–response experiments characterize the relationship between infectious agents and their hosts. These experiments are routinely used to estimate the minimum effective infectious dose for an infectious agent, which is most commonly characterized by the dose at which 50 per cent of challenged hosts become infected—the ID₅₀. In turn, the ID₅₀ is often used to compare between different agents and quantify the effect of treatment regimes. The statistical analysis of dose–response data typically makes the assumption that hosts within a given dose group are independent. For social animals, in particular avian species, hosts are routinely housed together in groups during experimental studies. For experiments with non-infectious agents, this poses no practical or theoretical problems. However, transmission of infectious agents between co-housed animals will modify the observed dose–response relationship with implications for the estimation of the ID₅₀ and the comparison between different agents and treatments. We derive a simple correction to the likelihood for standard dose–response models that allows us to estimate dose–response and transmission parameters simultaneously. We use this model to show that: transmission between co-housed animals reduces the apparent value of the ID₅₀ and increases the variability between replicates leading to a distinctive all-or-nothing response; in terms of the total number of animals used, individual housing is always the most efficient experimental design for ascertaining dose–response relationships; estimates of transmission from previously published experimental data for Campylobacter spp. in chickens suggest that considerable transmission occurred, greatly increasing the uncertainty in the estimates of dose–response parameters reported in the literature. Furthermore, we demonstrate that accounting for transmission in the analysis of dose–response data for Campylobacter spp. challenges our current understanding of the differing response of chickens with respect to host-age and in vivo passage of bacteria. Our findings suggest that the age-dependence of transmissibility between hosts—rather than their susceptibility to colonization—is the mechanism behind the ‘lag-phase’ reported in commercial flocks, which are typically found to be Campylobacter free for the first 14–21 days of life.     

Remote automated multi-generational growth and observation of an animal in low Earth orbit

The ultimate survival of humanity is dependent upon colonization of other planetary bodies. Key challenges to such habitation are (patho)physiologic changes induced by known, and unknown, factors associated with long-duration and distance space exploration. However, we currently lack biological models for detecting and studying these changes. Here, we use a remote automated culture system to successfully grow an animal in low Earth orbit for six months. Our observations, over 12 generations, demonstrate that the multi-cellular soil worm Caenorhabditis elegans develops from egg to adulthood and produces progeny with identical timings in space as on the Earth. Additionally, these animals display normal rates of movement when fully fed, comparable declines in movement when starved, and appropriate growth arrest upon starvation and recovery upon re-feeding. These observations establish C. elegans as a biological model that can be used to detect changes in animal growth, development, reproduction and behaviour in response to environmental conditions during long-duration spaceflight. This experimental system is ready to be incorporated on future, unmanned interplanetary missions and could be used to study cost-effectively the effects of such missions on these biological processes and the efficacy of new life support systems and radiation shielding technologies.     

Nearby boundaries create eddies near microscopic filter feeders

We show through calculations, simulations and experiments that the eddies often observed near sessile filter feeders are frequently due to the presence of nearby boundaries. We model the common filter feeder Vorticella, which is approximately 50 µm across and which feeds by removing bacteria from ocean or pond water that it draws towards itself. We use both an analytical stokeslet model and a Brinkman flow approximation that exploits the narrow-gap geometry to predict the size of the eddy caused by two parallel no-slip boundaries that represent the slides between which experimental observations are often made. We also use three-dimensional finite-element simulations to fully solve for the flow around a model Vorticella and analyse the influence of multiple nearby boundaries. Additionally, we track particles around live feeding Vorticella in order to determine the experimental flow field. Our models are in good agreement both with each other and with experiments. We also provide approximate equations to predict the experimental eddy sizes owing to boundaries both for the case of a filter feeder between two slides and for the case of a filter feeder attached to a perpendicular surface between two slides.     

Modelling and analysis of bacterial tracks suggest an active reorientation mechanism in Rhodobacter sphaeroides

Most free-swimming bacteria move in approximately straight lines, interspersed with random reorientation phases. A key open question concerns varying mechanisms by which reorientation occurs. We combine mathematical modelling with analysis of a large tracking dataset to study the poorly understood reorientation mechanism in the monoflagellate species Rhodobacter sphaeroides. The flagellum on this species rotates counterclockwise to propel the bacterium, periodically ceasing rotation to enable reorientation. When rotation restarts the cell body usually points in a new direction. It has been assumed that the new direction is simply the result of Brownian rotation. We consider three variants of a self-propelled particle model of bacterial motility. The first considers rotational diffusion only, corresponding to a non-chemotactic mutant strain. Two further models incorporate stochastic reorientations, describing ‘run-and-tumble’ motility. We derive expressions for key summary statistics and simulate each model using a stochastic computational algorithm. We also discuss the effect of cell geometry on rotational diffusion. Working with a previously published tracking dataset, we compare predictions of the models with data on individual stopping events in R. sphaeroides. This provides strong evidence that this species undergoes some form of active reorientation rather than simple reorientation by Brownian rotation.     

The importance of accurate muscle modelling for biomechanical analyses: a case study with a lizard skull

Computer-based simulation techniques such as multi-body dynamics analysis are becoming increasingly popular in the field of skull mechanics. Multi-body models can be used for studying the relationships between skull architecture, muscle morphology and feeding performance. However, to be confident in the modelling results, models need to be validated against experimental data, and the effects of uncertainties or inaccuracies in the chosen model attributes need to be assessed with sensitivity analyses. Here, we compare the bite forces predicted by a multi-body model of a lizard (Tupinambis merianae) with in vivo measurements, using anatomical data collected from the same specimen. This subject-specific model predicts bite forces that are very close to the in vivo measurements and also shows a consistent increase in bite force as the bite position is moved posteriorly on the jaw. However, the model is very sensitive to changes in muscle attributes such as fibre length, intrinsic muscle strength and force orientation, with bite force predictions varying considerably when these three variables are altered. We conclude that accurate muscle measurements are crucial to building realistic multi-body models and that subject-specific data should be used whenever possible.