Overview of mathematical approaches used to model bacterial chemotaxis I: the single cell

Mathematical modeling of bacterial chemotaxis systems has been influential and insightful in helping to understand experimental observations. We provide here a comprehensive overview of the range of mathematical approaches used for modeling, within a single bacterium, chemotactic processes caused by changes to external gradients in its environment. Specific areas of the bacterial system which have been studied and modeled are discussed in detail, including the modeling of adaptation in response to attractant gradients, the intracellular phosphorylation cascade, membrane receptor clustering, and spatial modeling of intracellular protein signal transduction. The importance of producing robust models that address adaptation, gain, and sensitivity are also discussed. This review highlights that while mathematical modeling has aided in understanding bacterial chemotaxis on the individual cell scale and guiding experimental design, no single model succeeds in robustly describing all of the basic elements of the cell. We conclude by discussing the importance of this and the future of modeling in this area.

Modeling the variability of single-cell lag times for Listeria innocua populations after sublethal and lethal heat treatments

Optical density measurements were used to estimate the effect of heat treatments on the single-cell lag times of Listeria innocua fitted to a shifted gamma distribution. The single-cell lag time was subdivided into repair time ( the shift of the distribution assumed to be uniform for all cells) and adjustment time (varying randomly from cell to cell). After heat treatments in which all of the cells recovered (sublethal), the repair time and the mean and the variance of the single-cell adjustment time increased with the severity of the treatment. When the heat treatments resulted in a loss of viability (lethal), the repair time of the survivors increased with the decimal reduction of the cell numbers independently of the temperature, while the mean and variance of the single-cell adjustment times remained the same irrespective of the heat treatment. Based on these observations and modeling of the effect of time and temperature of the heat treatment, we propose that the severity of a heat treatment can be characterized by the repair time of the cells whether the heat treatment is lethal or not, an extension of the F value concept for sublethal heat treatments. In addition, the repair time could be interpreted as the extent or degree of injury with a multiple-hit lethality model. Another implication of these results is that the distribution of the time for cells to reach unacceptable numbers in food is not affected by the time-temperature combination resulting in a given decimal reduction.

Single-cell proteomic analysis of glucosinolate-rich S-cells in Arabidopsis thaliana

Single-cell analysis is essential for understanding the processes of cell differentiation and metabolic specialisation in rare cell types. The amount of single proteins in single cells can be as low as one copy per cell and is for most proteins in the attomole range or below; usually considered as insufficient for proteomic analysis. The development of modern mass spectrometers possessing increased sensitivity and mass accuracy in combination with nano-LC-MS/MS now enables the analysis of single-cell contents. In Arabidopsis thaliana, we have successfully identified nine unique proteins in a single-cell sample and 56 proteins from a pool of 15 single-cell samples from glucosinolate-rich S-cells by nanoLC-MS/MS proteomic analysis, thus establishing the proof-of-concept for true single-cell proteomic analysis. Dehydrin (ERD14_ARATH), two myrosinases (BGL37_ARATH and BGL38_ARATH), annexin (ANXD1_ARATH), vegetative storage proteins (VSP1_ARATH and VSP2_ARATH) and four proteins belonging to the S-adenosyl-l-methionine cycle (METE_ARATH, SAHH1_ARATH, METK4_ARATH and METK1/3_ARATH) with associated adenosine kinase (ADK1_ARATH), were amongst the proteins identified in these single-S-cell samples. Comparison of the functional groups of proteins identified in S-cells with epidermal/cortical cells and whole tissue provided a unique insight into the metabolism of S-cells. We conclude that S-cells are metabolically active and contain the machinery for de novo biosynthesis of methionine, a precursor for the most abundant glucosinolate glucoraphanine in these cells. Moreover, since abundant TGG2 and TGG1 peptides were consistently found in single-S-cell samples, previously shown to have high amounts of glucosinolates, we suggest that both myrosinases and glucosinolates can be localised in the same cells, but in separate subcellular compartments. The complex membrane structure of S-cells was reflected by the presence of a number of proteins involved in membrane maintenance and cellular organisation.

Co-ordinate single-cell expression of LEE4- and LEE5-encoded proteins of Escherichia coli O157:H7

Escherichia coli O157:H7 is a zoonotic pathogen that can express a type III secretion system (TTSS) considered important for colonization and persistence in ruminants. E. coli O157:H7 strains have been shown to vary markedly in levels of protein secreted using the TTSS and this study has confirmed that a high secretion phenotype is more prevalent among isolates associated with human disease than isolates shed by healthy cattle. The variation in secretion levels is a consequence of heterogeneous expression, being dependent on the proportion of bacteria in a population that are actively engaged in protein secretion. This was demonstrated by indirect immunofluorescence and eGFP fusions that examined the expression of locus of enterocyte effacement (LEE)-encoded factors in individual bacteria. In liquid media, the expression of EspA, tir::egfp, intimin, but not map::egfp were co-ordinated in a subpopulation of bacteria. In contrast to E. coli O157:H7, expression of tir::egfp in EPEC E2348/69 was equivalent in all bacteria although the same fusion exhibited variable expression when transformed into an E. coli O157:H7 background. An E. coli O157:H7 strain deleted for the LEE demonstrated weak but variable expression of tir::egfp indicating that the elements controlling the heterogeneous expression lie outside the LEE. The research also demonstrated the rapid induction of tir::egfp and map::egfp on contact with bovine epithelial cells. This control in E. coli O157:H7 may be required to limit exposure of key surface antigens, EspA, Tir and intimin during colonization of cattle but allow their rapid production on contact with bovine gastrointestinal epithelium at the terminal rectum.

Adaptation-induced collective dynamics of a single-cell protozoan

We investigate the behavior of a single-cell protozoan in a narrow tubular ring. This environment forces them to swim under a one-dimensional periodic boundary condition. Above a critical density, single-cell protozoa aggregate spontaneously without external stimulation. The high-density zone of swimming cells exhibits a characteristic collective dynamics including translation and boundary fluctuation. We analyzed the velocity distribution and turn rate of swimming cells and found that the regulation of the turing rate leads to a stable aggregation and that acceleration of velocity triggers instability of aggregation. These two opposing effects may help to explain the spontaneous dynamics of collective behavior. We also propose a stochastic model for the mechanism underlying the collective behavior of swimming cells.

Patterning and detailed study of human hNT astrocytes on parylene-C/silicon dioxide substrates to the single cell level

It is estimated that the adult human brain contains 100 billion neurons with 5–10 times as many astrocytes. Although it has been generally considered that the astrocyte is a simple supportive cell to the neuron, recent research has revealed new functionality of the astrocyte in the form of information transfer to neurons of the brain. In our previous work we developed a protocol to pattern the hNT neuron (derived from the human teratocarcinoma cell line (hNT)) on parylene-C/SiO2 substrates. In this work, we report how we have managed to pattern hNT astrocytes, on parylene-C/SiO2 substrates to single cell resolution. This article disseminates the nanofabrication and cell culturing steps necessary for the patterning of such cells. In addition, it reports the necessary strip lengths and strip width dimensions of parylene-C that encourage high degrees of cellular coverage and single cell isolation for this cell type. The significance in patterning the hNT astrocyte on silicon chip is that it will help enable single cell and network studies into the undiscovered functionality of this interesting cell, thus, contributing to closer pathological studies of the human brain.

First human hNT astrocytes patterned to single cell resolution on parylene-C/Silicon dioxide substrates

In our previous work we developed a successful protocol to pattern the human hNT neuron (derived from the human teratocarcinoma cell line (hNT)) on parylene-C/SiO2 substrates. This communication, reports how we have successfully managed to pattern the supportive cell to the neuron, the hNT astrocyte, on such substrates. Here we disseminate the nanofabrication, cell differentiation and cell culturing protocols necessary to successfully pattern the first human hNT astrocytes to single cell resolution on parylene-C/SiO2 substrates. This is performed for varying parylene strip widths providing excellent contrast to the SiO2 substrate and elegant single cell isolation at 10μm strip widths. The breakthrough in patterning human cells on a silicon chip has widespread implications and is valuable as a platform technology as it enables a detailed study of the human brain at the cellular and network level.

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.

Isolating single primary rat hippocampal neurons & astrocytes on ultra-thin patterned parylene-C/silicon dioxide substrates

We report here the patterning of primary rat neurons and astrocytes from the postnatal hippocampus on ultra-thin parylene-C deposited on a silicon dioxide substrate, following observations of neuronal, astrocytic and nuclear coverage on strips of different lengths, widths and thicknesses. Neuronal and glial growth was characterized ‘on’, ‘adjacent to’ and ‘away from’ the parylene strips. In addition, the article reports how the same material combination can be used to isolate single cells along thin tracks of parylene-C. This is demonstrated with a series of high magnification images of the experimental observations for varying parylene strip widths and thicknesses. Thus, the findings demonstrate the possibility to culture cells on ultra-thin layers of parylene-C and localize single cells on thin strips. Such work is of interest and significance to the Neuroengineering and Multi-Electrode Array (MEA) communities, as it provides an alternative insulating material in the fabrication of embedded micro-electrodes, which can be used to facilitate single cell stimulation and recording in capacitive coupling mode.

First human hNT neurons patterned on parylene-C/silicon dioxide substrates: Combining an accessible cell line and robust patterning technology for the study of the pathological adult human brain

In this communication, we describe a new method which has enabled the first patterning of human neurons (derived from the human teratocarcinoma cell line (hNT)) on parylene-C/silicon dioxide substrates. We reveal the details of the nanofabrication processes, cell differentiation and culturing protocols necessary to successfully pattern hNT neurons which are each key aspects of this new method. The benefits in patterning human neurons on silicon chip using an accessible cell line and robust patterning technology are of widespread value. Thus, using a combined technology such as this will facilitate the detailed study of the pathological human brain at both the single cell and network level.

Quantitative single-molecule localization microscopy combined with rule-based modeling reveals ligand-induced TNF-R1 reorganization toward higher-order oligomers

We report on the assembly of tumor necrosis factor receptor 1 (TNF-R1) prior to ligand activation and its ligand-induced reorganization at the cell membrane. We apply single-molecule localization microscopy to obtain quantitative information on receptor cluster sizes and copy numbers. Our data suggest a dimeric pre-assembly of TNF-R1, as well as receptor reorganization toward higher oligomeric states with stable populations comprising three to six TNF-R1. Our experimental results directly serve as input parameters for computational modeling of the ligand-receptor interaction. Simulations corroborate the experimental finding of higher-order oligomeric states. This work is a first demonstration how quantitative, super-resolution and advanced microscopy can be used for systems biology approaches at the single-molecule and single-cell level.

The phytocannabinoid Delta(9)-tetrahydrocannabivarin modulates inhibitory neurotransmission in the cerebellum

Background and purposeThe phytocannabinoid Delta(9)-tetrahydrocannabivarin (Delta(9)-THCV) has been reported to exhibit a diverse pharmacology; here, we investigate functional effects of Delta(9)-THCV, extracted from Cannabis sativa, using electrophysiological techniques to define its mechanism of action in the CNS.Experimental approachEffects of Delta(9)-THCV and synthetic cannabinoid agents on inhibitory neurotransmission at interneurone-Purkinje cell (IN-PC) synapses were correlated with effects on spontaneous PC output using single-cell and multi-electrode array (MEA) electrophysiological recordings respectively, in mouse cerebellar brain slices in vitro.Key resultsThe cannabinoid receptor agonist WIN 55,212-2 (WIN55) decreased miniature inhibitory postsynaptic current (mIPSC) frequency at IN-PC synapses. WIN55-induced inhibition was reversed by Delta(9)-THCV, and also by the CB(1) receptor antagonist AM251; Delta(9)-THCV or AM251 acted to increase mIPSC frequency beyond basal values. When applied alone, Delta(9)-THCV, AM251 or rimonabant increased mIPSC frequency. Pre-incubation with Delta(9)-THCV blocked WIN55-induced inhibition. In MEA recordings, WIN55 increased PC spike firing rate; Delta(9)-THCV and AM251 acted in the opposite direction to decrease spike firing. The effects of Delta(9)-THCV and WIN55 were attenuated by the GABA(A) receptor antagonist bicuculline methiodide.Conclusions and implicationsWe show for the first time that Delta(9)-THCV acts as a functional CB(1) receptor antagonist in the CNS to modulate inhibitory neurotransmission at IN-PC synapses and spontaneous PC output. Delta(9)-THCV- and AM251-induced increases in mIPSC frequency beyond basal levels were consistent with basal CB(1) receptor activity. WIN55-induced increases in PC spike firing rate were consistent with synaptic disinhibition; whilst Delta(9)-THCV- and AM251-induced decreases in spike firing suggest a mechanism of PC inhibition.British Journal of Pharmacology advance online publication, 3 March 2008; doi:10.1038/bjp.2008.57.

The phytocannabinoid Δ9-tetrahydrocannabivarin modulates inhibitory neurotransmission in the cerebellum

Background and purpose: The phytocannabinoid Delta(9)-tetrahydrocannabivarin (Delta(9)-THCV) has been reported to exhibit a diverse pharmacology; here, we investigate functional effects of Delta(9)-THCV, extracted from Cannabis sativa, using electrophysiological techniques to define its mechanism of action in the CNS. Experimental approach: Effects of Delta(9)-THCV and synthetic cannabinoid agents on inhibitory neurotransmission at interneurone-Purkinje cell (IN-PC) synapses were correlated with effects on spontaneous PC output using single-cell and multi-electrode array (MEA) electrophysiological recordings respectively, in mouse cerebellar brain slices in vitro. Key results: The cannabinoid receptor agonist WIN 55,212-2 (WIN55) decreased miniature inhibitory postsynaptic current (mIPSC) frequency at IN-PC synapses. WIN55-induced inhibition was reversed by Delta(9)-THCV, and also by the CB1 receptor antagonist AM251; Delta(9)-THCV or AM251 acted to increase mIPSC frequency beyond basal values. When applied alone, Delta(9)-THCV, AM251 or rimonabant increased mIPSC frequency. Pre-incubation with Delta(9)-THCV blocked WIN55-induced inhibition. In MEA recordings, WIN55 increased PC spike firing rate; Delta(9)-THCV and AM251 acted in the opposite direction to decrease spike firing. The effects of Delta(9)-THCV and WIN55 were attenuated by the GABA(A) receptor antagonist bicuculline methiodide. Conclusions and implications: We show for the first time that Delta(9)-THCV acts as a functional CB1 receptor antagonist in the CNS to modulate inhibitory neurotransmission at IN-PC synapses and spontaneous PC output. Delta(9)-THCV- and AM251-induced increases in mIPSC frequency beyond basal levels were consistent with basal CB1 receptor activity. WIN55-induced increases in PC spike firing rate were consistent with synaptic disinhibition; whilst Delta(9)-THCV-and AM251-induced decreases in spike firing suggest a mechanism of PC inhibition.

Relation between neuronal morphology and electrophysiology in the Kainate lesion model of Alzheimer's Disease

This paper describes a computational and statistical study of the influence of morphological changes on the electrophysiological response of neurons from an animal model of Alzheimer's Disease (AD). We combined experimental morphological data from rat hippocampal CA1 pyramidal cells with a well-established model of active membrane properties. Dendritic morphology and the somatic response to simulated current clamp conditions were then compared for cells from the control and the AD group. The computational approach allowed us to single out the influences of neuromorphology on neuronal response by eliminating the effects of active channel variability. The results did not reveal a simple relationship between morphological changes associated with AD and changes in neural response. However, they did suggest the existence of more complex than anticipated relationships between dendritic morphology and single-cell electrophysiology.

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. WIREs Syst Biol Med 2012 doi: 10.1002/wsbm.1168 For further resources related to this article, please visit the WIREs website.