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.     

In vitro Approach to Bacterial Chemotaxis

An in vitro approach to study bacterial motility and chemotaxis is described. The approach is based on a preparation of flagellated cell envelopes. The envelopes are prepared from bacteria by a penicillin treatment and subsequent osmotic lysis. When the envelopes are energized, their flagella rotate. The direction of rotation in wild type envelopes is counterclockwise. Inclusion of the CheY protein within the envelopes may restore clockwise rotation. The advantages and disadvantages of this approach are pointed out.     

The Study of Bacterial Chemotaxis to Naphthalene

The chemotaxis of two pseudomonads,Pseudomonas putida AZ (Naph⁺) and P. putida AZ (Naph^–), differing in their ability to metabolize naphthalene was studied by the known capillary method of Adler and the densitometric method devised in our laboratory. The migration of P. putida AZ (Naph⁺) cells toward increasing levels of naphthalene was accompanied by the formation of a migrating front of converted naphthalene. P. putida AZ (Naph^–) cells also exhibited positive chemotaxis to naphthalene, but they did not form the front of converted naphthalene. The analysis of experimental data in terms of a kinetic model of bacterial chemotaxis showed that the densitometric method is a potential tool for studying bacterial chemotaxis to hydrophobic organic substances.     

Mathematical Modeling of Chemotactic Bacterial Transport through a Two-Dimensional Heterogeneous Porous Medium

The impact of bacterial chemotaxis on in situ ground-water bioremediation remains an unanswered question. Although bacteria respond to chemical gradients in aqueous environments and under no-flow conditions, it is unclear whether they can also respond in porous media with advective flow to improve overall contaminant degradation. The effect of chemotaxis is most profound in regions with sharp chemical gradients, most notably around residual nonaqueous phase liquid (NAPL) ganglia and surrounding clay lenses or aquitards with trapped contamination. The purpose of this study is to simulate bacterial transport through a two-dimensional subsurface environment, containing one region of low permeability with trapped contaminant surrounded above and below by two regions of higher permeability. Using mathematical predictions of the effect of pore size on measured bacterial transport parameters, the authors observe a 50% decrease in both motility and chemotaxis in the finer-grained, low-permeability porous medium. The authors simulate how chemotaxis affects bacterial migration to the contaminated region under various flow and initial conditions. Results indicate that bacteria traveling through a high-permeability region with advective flow can successfully migrate toward and accumulate around a contaminant diffusing from a lower permeability region.     

Mathematical Modeling of Chemotactic Bacterial Transport through a Two-Dimensional Heterogeneous Porous Medium

The impact of bacterial chemotaxis on in situ ground-water bioremediation remains an unanswered question. Although bacteria respond to chemical gradients in aqueous environments and under no-flow conditions, it is unclear whether they can also respond in porous media with advective flow to improve overall contaminant degradation. The effect of chemotaxis is most profound in regions with sharp chemical gradients, most notably around residual nonaqueous phase liquid (NAPL) ganglia and surrounding clay lenses or aquitards with trapped contamination. The purpose of this study is to simulate bacterial transport through a two-dimensional subsurface environment, containing one region of low permeability with trapped contaminant surrounded above and below by two regions of higher permeability. Using mathematical predictions of the effect of pore size on measured bacterial transport parameters, the authors observe a 50% decrease in both motility and chemotaxis in the finer-grained, low-permeability porous medium. The authors simulate how chemotaxis affects bacterial migration to the contaminated region under various flow and initial conditions. Results indicate that bacteria traveling through a high-permeability region with advective flow can successfully migrate toward and accumulate around a contaminant diffusing from a lower permeability region.     

Effect of Bacterial Chemotaxis on Biodegradation in a Porous Medium

This research investigates the effect of bacterial chemotaxis on biodegradation rate in an experimental model for in situ bioremediation. The novel experimental methodology of this work has provided for the systematic evaluation of the effect of the chemotaxis phenomenon in a saturated porous medium. The methodology has been developed to measure enhancement of degradation rate of serine, a simulated contaminant and chemoattractant. Escherichia coli RP437 was used as a representative, chemotactic, in situ bacterium, whereas E. coli RP5700, a tsr? mutant strain of RP437 that lacks the serine chemoreceptor, was used as the specifically nonchemotactic control strain. These two strains were highly characterized for this work. Swimming speeds, run lengths, and turn angles were compared using a tracking microscope and were statistically similar, as were serine uptake rates, making this pair of strains an excellent choice for chemotaxis studies. For these experiments, a model aquifer introduces bacteria to serine in saturated sand via a sharp gradient. The aquifer successfully achieved biodegradation at an 18% level; however, the degradation rate of serine was similar for both strains over 21 h, indicating that chemotaxis enhancement did not occur. This result is in agreement with certain prior works which did not detect enhanced chemotactic migration.     

The Mechanism of Fractal-Like Structure Formation by Bacterial Populations

Three types of population growth and development of chemotaxic motile bacteria Escherichia coli on semi-solid nutrient media are investigated: a) stable development – circular symmetrical waves; b) bursts; c) fractal-like self-organization. Experimental investigation of the burst formation is presented. The microscopic analysis of growing, fractal-like structures is carried out, and a mechanism for such structure formation is suggested. It is supposed that fractal-like bacterial structures growth is based on the principle of successively forming multiple micro-bursts. A mathematical model has been suggested to reproduce the experimental results. The structures obtained by numerical modeling of population growth in the parameter space substrate concentration - bacterial movement rate reproduce the corresponding experimental structures in the space nutrient concentration in the media – the density of the media.     

Chemotaxis of Fungal Zoospores,with Special Reference to Aphanomyces cochlioides

Zoospores of phytopathogenic fungi accumulate at the potential infection sites of host roots by chemotaxis. The aggregated spores then adhere, encyst, germinate, and finally penetrate into the root tissues to initiate infection. Some of the host-specific attractants have already been identified. The host-specific attractants also induce cell differentiation of certain zoospores under laboratory conditions. This indicates that a signal released from the roots of the host plant guides the pest propagules for orientation and prepares them for establishing a host-pathogen relationship by necessary physiological changes. Some non-host plant secondary metabolites were found to markedly regulate behavior and viability of zoospores, suggesting that non-host compounds may also play a role in protecting the non-host plants from the attack of zoosporic fungi. We hypothesized that zoospores perceive the host signal(s) by specific G-protein-coupled receptors and translate it into responses by way of the phosphoinositide-Ca²⁺ signaling cascade. The details of the signal transduction mechanism in fungal zoospores are yet to be discovered. In this report, we review the signaling and communications between phytopathogenic fungal zoospores and host and non-host plants with special reference to Aphanomyces cochlioides.     

Mathematical modeling of tumor-induced angiogenesis

Journal of Mathematical Biology -     

A Neural Network Model of Chemotaxis Predicts Functions of Synaptic Connections in the Nematode Caenorhabditis elegans

The anatomical connectivity of the nervous system of the nematode Caenorhabditis elegans has been almost completely described, but determination of the neurophysiological basis of behavior in this system is just beginning. Here we used an optimization algorithm to search for patterns of connectivity sufficient to compute the sensorimotor transformation underlying C. elegans chemotaxis, a simple form of spatial orientation behavior in which turning probability is modulated by the rate of change of chemical concentration. Optimization produced differentiator networks capable of simulating chemotaxis. A surprising feature of these networks was inhibitory feedback connections on all neurons. Further analysis showed that feedback regulates the latency between sensory input and behavior. Common patterns of connectivity between the model and biological networks suggest new functions for previously identified connections in the C. elegans nervous system.     

A Multiple-Relaxation-Time Lattice-Boltzmann Model for Bacterial Chemotaxis: Effects of Initial Concentration,Diffusion, and Hydrodynamic Dispersion on Traveling Bacterial Bands

Bacterial chemotaxis can enhance the bioremediation of contaminants in aqueous and subsurface environments if the contaminant is a chemoattractant that the bacteria degrade. The process can be promoted by traveling bands of chemotactic bacteria that form due to metabolism-generated gradients in chemoattractant concentration. We developed a multiple-relaxation-time (MRT) lattice-Boltzmann method (LBM) to model chemotaxis, because LBMs are well suited to model reactive transport in the complex geometries that are typical for subsurface porous media. This MRT-LBM can attain a better numerical stability than its corresponding single-relaxation-time LBM. We performed simulations to investigate the effects of substrate diffusion, initial bacterial concentration, and hydrodynamic dispersion on the formation, shape, and propagation of bacterial bands. Band formation requires a sufficiently high initial number of bacteria and a small substrate diffusion coefficient. Uniform flow does not affect the bands while shear flow does. Bacterial bands can move both upstream and downstream when the flow velocity is small. However, the bands disappear once the velocity becomes too large due to hydrodynamic dispersion. Generally bands can only be observed if the dimensionless ratio between the chemotactic sensitivity coefficient and the effective diffusion coefficient of the bacteria exceeds a critical value, that is, when the biased movement due to chemotaxis overcomes the diffusion-like movement due to the random motility and hydrodynamic dispersion.     

Autocatalytic Loop,Amplification and Diffusion: A Mathematical and Computational Model of Cell Polarization in Neural Chemotaxis

The chemotactic response of cells to graded fields of chemical cues is a complex process that requires the coordination of several intracellular activities. Fundamental steps to obtain a front vs. back differentiation in the cell are the localized distribution of internal molecules and the amplification of the external signal. The goal of this work is to develop a mathematical and computational model for the quantitative study of such phenomena in the context of axon chemotactic pathfinding in neural development. In order to perform turning decisions, axons develop front-back polarization in their distal structure, the growth cone. Starting from the recent experimental findings of the biased redistribution of receptors on the growth cone membrane, driven by the interaction with the cytoskeleton, we propose a model to investigate the significance of this process. Our main contribution is to quantitatively demonstrate that the autocatalytic loop involving receptors, cytoplasmic species and cytoskeleton is adequate to give rise to the chemotactic behavior of neural cells. We assess the fact that spatial bias in receptors is a precursory key event for chemotactic response, establishing the necessity of a tight link between upstream gradient sensing and downstream cytoskeleton dynamics. We analyze further crosslinked effects and, among others, the contribution to polarization of internal enzymatic reactions, which entail the production of molecules with a one-to-more factor. The model shows that the enzymatic efficiency of such reactions must overcome a threshold in order to give rise to a sufficient amplification, another fundamental precursory step for obtaining polarization. Eventually, we address the characteristic behavior of the attraction/repulsion of axons subjected to the same cue, providing a quantitative indicator of the parameters which more critically determine this nontrivial chemotactic response.     

Humanized Mouse Model to Study Bacterial Infections Targeting the Microvasculature

Neisseria meningitidis causes a severe, frequently fatal sepsis when it enters the human blood stream. Infection leads to extensive damage of the blood vessels resulting in vascular leak, the development of purpuric rashes and eventual tissue necrosis. Studying the pathogenesis of this infection was previously limited by the human specificity of the bacteria, which makes in vivo models difficult. In this protocol, we describe a humanized model for this infection in which human skin, containing dermal microvessels, is grafted onto immunocompromised mice. These vessels anastomose with the mouse circulation while maintaining their human characteristics. Once introduced into this model, N. meningitidis adhere exclusively to the human vessels, resulting in extensive vascular damage, inflammation and in some cases the development of purpuric rash. This protocol describes the grafting, infection and evaluation steps of this model in the context of N. meningitidis infection. The technique may be applied to numerous human specific pathogens that infect the blood stream.     

Bacterial motility: links to the environment and a driving force for microbial physics

Bacterial motility was recognized 300 years ago. Throughout this history, research into motility has led to advances in microbiology and physics. Thirty years ago, this union helped to make run and tumble chemotaxis the paradigm for bacterial movement. This review highlights how this paradigm has expanded and changed, and emphasizes the following points. The absolute magnitude of swimming speed is ecologically important because it helps determine vulnerability to Brownian motion, sensitivity to gradients, the type of receptors used and the cost of moving, with some bacteria moving at 1 mm s(-1). High costs for high speeds are offset by the benefit of resource translocation across submillimetre redox and other environmental gradients. Much of environmental chemotaxis appears adapted to respond to gradients of micrometres, rather than migrations of centimetres. In such gradients, control of ion pumps is particularly important. Motility, at least in the ocean, is highly intermittent and the speed is variable within a run. Subtleties in flagellar physics provide a variety of reorientation mechanisms. Finally, while careful physical analysis has contributed to our current understanding of bacterial movement, tactic bacteria are increasingly widely used as experimental and theoretical model systems in physics.     

Mathematical model of a three-stage innate immune response to a pneumococcal lung infection

Pneumococcal pneumonia is a leading cause of death and a major source of human morbidity. The initial immune response plays a central role in determining the course and outcome of pneumococcal disease. We combine bacterial titer measurements from mice infected with Streptococcus pneumoniae with mathematical modeling to investigate the coordination of immune responses and the effects of initial inoculum on outcome. To evaluate the contributions of individual components, we systematically build a mathematical model from three subsystems that describe the succession of defensive cells in the lung: resident alveolar macrophages, neutrophils and monocyte-derived macrophages. The alveolar macrophage response, which can be modeled by a single differential equation, can by itself rapidly clear small initial numbers of pneumococci. Extending the model to include the neutrophil response required additional equations for recruitment cytokines and host cell status and damage. With these dynamics, two outcomes can be predicted: bacterial clearance or sustained bacterial growth. Finally, a model including monocyte-derived macrophage recruitment by neutrophils suggests that sustained bacterial growth is possible even in their presence. Our model quantifies the contributions of cytotoxicity and immune-mediated damage in pneumococcal pathogenesis.     

大鼠肝移植术后腹腔细菌感染模型的建立与评价

目的建立大鼠肝移植术后腹腔感染的模型。方法构建DA大鼠到LEW大鼠的肝移植模型,采用腹腔内细菌注射的方法建立感染模型,通过对大鼠肝功、血气、血细胞计数等各项指标的检测对模型进行综合评价。结果肝移植术后5 d注射细菌,大鼠死亡率高,不利后续研究;术后3 d注射细菌,并选定5×105cfu/mL为最终注射浓度,感染后大鼠的7d存活率累计可达到37.5%左右,随之感染的加重,大鼠状态逐渐变差,直肠温度不断升高,WBC计数也随之增加,pH下降,大鼠出现代谢性酸中毒,肝功能损害进行性加重,肝实质的损害重于胆道的损伤,大约在感染5 d左右相继死亡,多器官病理分析表明,大鼠死亡原因为肝损害,不并发肺脏、肾脏损害。结论采用的腹腔内大肠埃希菌注射建立肝移植术后腹腔细菌感染的模型是比较成功的,可用于相关领域的研究。     

Measurement of Tactile Allodynia in a Murine Model of Bacterial Prostatitis

Uropathogenic Escherichia coli (UPEC) are pathogens that play an important role in urinary tract infections and bacterial prostatitis¹. We have recently shown that UPEC have an important role in the initiation of chronic pelvic pain², a feature of Chronic prostatitis/Chronic pelvic pain syndrome (CP/CPPS)^3,4. Infection of the prostate by clinically relevant UPEC can initiate and establish chronic pain through mechanisms that may involve tissue damage and the initiation of mechanisms of autoimmunity⁵.A challenge to understanding the pathogenesis of UPEC in the prostate is the relative inaccessibility of the prostate gland to manipulation. We utilized a previously described intraurethral infection method⁶ to deliver a clinical strain of UPEC into male mice thereby establishing an ascending infection of the prostate. Here, we describe our protocols for standardizing the bacterial inoculum⁷ as well as the procedure for catheterizing anesthetized male mice for instillation of bacteria.CP/CPPS is primarily characterized by the presence of tactile allodynia⁴. Behavior testing was based on the concept of cutaneous hyperalgesia resulting from referred visceral pain^8-10. An irritable focus in visceral tissues reduces cutaneous pain thresholds allowing for an exaggerated response to normally non-painful stimuli (allodynia). Application of normal force to the skin result in abnormal responses that tend to increase with the intensity of the underlying visceral pain. We describe methodology in NOD/ShiLtJ mice that utilize von Frey fibers to quantify tactile allodynia over time in response to a single infection with UPEC bacteria.