A model for traveling bands of chemotactic bacteria.

A theoretical model is used to study band formation by chemotactic populations of Escherichia coli. The model includes the bacterial response to attractant gradients, the chemotactic sensitivity of the bacteria to the concentration of the attractant, and population growth. For certain values of the parameters in the model, traveling bands of bacteria form and propagate with or without growth. Under specific growth conditions the band profile is maintained and the band propagates at constant speed. These predictions are in general agreement with the experiment results of J. Adler and earlier theoretical work by L. Segel and his collaborators. However, our theory differs in several important respects from the latter efforts. Suggestions are made for further experiments to test the proposed model and to clarify the nature of the processes which lead to band formation.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

Instability in a generalized Keller–Segel model

We present a generalized Keller–Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous stationary states of this generalized model, we consider the eigenvalues of a linearized system. We are able to reduce this infinite dimensional eigenproblem to a parametrized finite dimensional eigenproblem. By matrix theoretic tools, we then provide easily verifiable sufficient conditions for destabilizing the homogeneous stationary states. In particular, one of the sufficient conditions is that the chemotactic feedback is sufficiently strong. Although this mechanism was already known to exist in the original Keller–Segel model, here we show that it is more generally applicable by significantly enlarging the class of models exhibiting this instability phenomenon which may lead to pattern formation.     

Travelling Waves in Hyperbolic Chemotaxis Equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235–248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.     

Analytical solution to the initial-value problem for traveling bands of chemotactic bacteria

The governing parabolic partial differential equations for the diffusion and chemotactic transport of a distribution of bacteria and for the diffusion and bacterial degradation of a distribution of chemotactic agent are supplemented with boundary and initial conditions that model the recent capillary tube experiments on the formation and propagation of traveling bands of chemotactic bacteria. An iteration procedure that takes the exact solution to the “diffusionless” problem as a first approximation is applied to solve the equations of the complete theoretical model. It is shown that satisfactory agreement with experiment obtains for the analytical results of the first approximation which relate the velocity of propagation and total number of bacteria cells per unit cross-sectional area in a traveling band to the constant parameters in the governing equations and supplementary conditions. The second approximation is shown to yield approximate analytical expressions for the solution functions which are in close correspondence with previously derived traveling band solutions for values of time after the initial period of formation.     

Traveling wave solutions from microscopic to macroscopic chemotaxis models

In this paper, we study the existence and nonexistence of traveling wave solutions for the one-dimensional microscopic and macroscopic chemotaxis models. The microscopic model is based on the velocity jump process of Othmer et al. (SIAM J Appl Math 57:1044–1081, 1997). The macroscopic model, which can be shown to be the parabolic limit of the microscopic model, is the classical Keller–Segel model, (Keller and Segel in J Theor Biol 30:225–234; 377–380, 1971). In both models, the chemosensitivity function is given by the derivative of a potential function, Φ(v), which must be unbounded below at some point for the existence of traveling wave solutions. Thus, we consider two examples: F(v) = lnv{\Phi(v) = \ln v} and F(v) = ln[v/(1-v)]{\Phi(v) = \ln[v/(1-v)]}. The mathematical problem reduces to proving the existence or nonexistence of solutions to a nonlinear boundary value problem with variable coefficient on \mathbb R{\mathbb R}. The main purpose of this paper is to identify the relationships between the two models through their traveling waves, from which we can observe how information are lost, retained, or created during the transition from the microscopic model to the macroscopic model. Moreover, the underlying biological implications of our results are discussed.     

Spatio-temporal structure of migrating chemotactic band of Escherichia coli. I. Traveling band profile.

We developed a rapid-scanning, light-scattering densitometer by which extensive measurements of band migration speeds and band profiles of chemotactic bands of Escherichia coli in motility buffer both with and without serine have been made. The purpose is to test the applicability of the phenomenological model proposed by Keller and Segel (J. Theor. Biol. 1971. 30:235) and to determine the motility (mu) and chemotactic (delta) coefficients of the bacteria. We extend the previous analytical solution of the simplified Keller-Segel model by taking into account the substrate diffusion which turns out to be significant in the case of oxygen. We demonstrate that unique sets of values of mu and delta can be obtained for various samples at different stages of migration by comparing the numerical solution of the model equation and the experimental data. The rapid-scanning technique also reveals a hitherto unobserved time-dependent fine structure in the bacterial band. We give a qualitative argument to show that the fine structure is an example of the dissipative structure that arises from a nonlinear coupling between the bacterial density and the oxygen concentration gradient. Implications for a further study of the dissipative structure in testing the Keller-Segel model of chemotaxis are briefly discussed.     

On a mathematical model describing the aggregation of amoebae

In this paper an extension of a mathematical model of Keller and Segel (1970) describing the aggregation of amoebae is presented. In their paper (Keller and Segel, 1970) they showed that the onset of the aggregation could be viewed as a spatial instability. Their instability condition involved diffusion constants of the cyclic AMP and of the amoebae as well as a constant describing the chemotactic behavior of the amoebae. In our case we consider a temporal instability that depends only on the kinetics of cyclic AMP production, degradation and transport through the cell wall. Our model then explains the oscillatory behavior of the cyclic AMP in well-stirred suspensions of amoebae. In addition we discuss existence and non-existence of nonuniform steady states of the nonlinear parabolic system involved.     

Bacterial chemotaxis in a two-dimensional attractant gradient

The motion of a population of chemotactic bacteria in a radial exponential gradient of attractant in a cylindrical container has been calculated using a mathematical model suggested by Keller and Segel. Numerical solutions for the equations of bacterial migration have been found which give for all times the cell density at distances from the center of the cylinder. The ultimate distribution of bacteria is a simple stationary exponential function of the distance. Experiments to verify the theoretical predictions are suggested.     

G-CSF Control of Neutrophils Dynamics in the Blood

White blood cell neutrophil is a key component in the fast initial immune response against bacterial and fungal infections. Granulocyte colony stimulating factor (G-CSF) which is naturally produced in the body, is known to control the neutrophils production in the bone marrow and the neutrophils delivery into the blood. In oncological practice, G-CSF injections are widely used to treat neutropenia (dangerously low levels of neutrophils in the blood) and to prevent the infectious complications that often follow chemotherapy. However, the accurate dynamics of G-CSF neutrophil interaction has not been fully determined and no general scheme exists for an optimal G-CSF application in neutropenia. Here we develop a two-dimensional ordinary differential equation model for the G-CSF—neutrophil dynamics in the blood. The model is built axiomatically by first formally defining from the biology the expected properties of the model, and then deducing the dynamic behavior of the resulting system. The resulting model is structurally stable, and its dynamical features are independent of the precise form of the various rate functions. Choosing a specific form for these functions, three complementary parameter estimation procedures for one clinical (training) data set are utilized. The fully parameterized model (6 parameters) provides adequate predictions for several additional clinical data sets on time scales of several days. We briefly discuss the utility of this relatively simple and robust model in several clinical conditions. Dedicated to Lee Segel who guided us to apply mathematics for the benefit of mankind—a teacher, a colleague, a friend. L.A. Segel passed away on 31 January 2005.     

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

Spatial stability of traveling wave solutions of a nerve conduction equation.

A simplified FitzHugh-Nagumo nerve conduction equation with known traveling wave solutions is considered. The spatial stability of these solutions is analyzed to determine which solutions should occur in signal transmission along such a nerve model. It is found that the slower of the two pulse solutions is unstable while the faster one is stable, so the faster one should occur. This agrees with conjectures which have been made about the solutions of other nerve conduction equations. Furthermore for certain parameter values the equation has two periodic wave solutions, each representing a train of impulses, at each frequency less than a maximum frequency wmax. The slower one is found to be unstable and the faster one to be stable, while that at wmax is found to be neutrally stable. These spatial stability results complement the previous results of Rinzel and Keller (1973. Biophys. J. 13: 1313) on temporal stability, which are applicable to the solutions of initial value problems.     

Computerized study of interactions among factors and their optimization through response surface methodology for the production of tannin acyl hydrolase by Aspergillus niger PKL 104 under solid state fermentation

Optimization of five parameters (initial moisture, initial pH, incubation temperature, inoculum ratio and fermentation period), as per central composite rotable design falling under the response surface methodology, was attempted in a total of 32 experimental sets, after fitting the experimental data to the polynomial model of a suitable degree, for tannin acyl hydrolase production by Aspergillus niger PKL 104 in solid state fermentation system. The quantitative relation between the enzyme production and different levels of these factors was exploited to work out optimized levels of these parameters by flexible polyhedron search method and confirmed by further experimentations. The best set required 5% inoculum, 6.5 initial pH, 28 °C fermentation temperature, 62% initial moisture and 3 days fermentation time. The optima were worked out under the additional constraints for temperature ( 30 °C) and fermentation time (not more than 3 days) which are essential from industrial conditions and to pre-empt contamination, respectively. The best set resulted in 1.34 times more enzyme production than that was obtained before this optimization. Three dimensional plots, relating the enzyme production to paired factors (when other three factors were kept at their optimal levels) best described the behaviour of solid state fermentation system and the interactions between factors under optimized conditions. The model showed that the enzyme production was affected by all the five factors studied. The initial pH exhibited a positive interaction with moisture but no interaction with other factors. Initial moisture level and inoculum ratio showed negative interaction in contrast to positive interaction between inoculum ratio and fermentation period. It is thus apparent that the response surface methodology not only gives valuable information on interactions between the factors but also leads to identification of feasible optimum values of the studied factors, in addition to 99% (or more) savings on resources as compared to a full factorial traditional optimization method. Response surface methods have not been used earlier for optimizing parameters in solid state fermentation system.The authors thank Dr. S. R. Bhowmik, Director, CFTRI for the interest shown in the work. P. K. Lekha is thankful to the Council of Scientific and Industrial Research, New Delhi, India, for the award of a research fellowship.     

The role of initial inoculum on epidemic dynamics

Transient dynamics are important in many epidemics in agricultural and ecological systems that are prone to regular disturbance, cyclical and random perturbations. Here, using a simple host-pathogen model for a sessile host and a pathogen that can move by diffusion and advection, we use a range of mathematical techniques to examine the effect of initial spatial distribution of inoculum of the pathogen on the transient dynamics of the epidemic. We consider an isolated patch and a group of patches with different boundary conditions. We first determine bounds on the host population for the full model, then non-dimensionalizing the model allows us to obtain approximate solutions for the system. We identify two biologically intuitive groups of parameters to analyse transient behaviour using perturbation techniques. The first parameter group is a measure of the relative strength of initial primary to secondary infection. The second group is derived from the ratio of host removal rate (via infection) to pathogen removal rate (by decay and natural mortality) and measures the infectivity of initial inoculum on the system. By restricting the model to mimic primary infection only (in which all infections arise from initial inoculum), we obtain exact solutions and demonstrate how these depend on initial conditions, boundary conditions and model parameters. Finally, we suggest that the analyses on the balance of primary and secondary infection provide the epidemiologist with some simple rules to predict the transient behaviours.     

Oscillations and waves of cyclic AMP in Dictyostelium: A prototype for spatio-temporal organization and pulsatile intercellular communication

The amoebae Dictyostelium discoideum aggregate after starvation in a wavelike manner in response to periodic pulses of cyclic AMP (cAMP) secreted by cells which behave as aggregation centers. In addition to autonomous oscillations, the cAMP signaling system that controls aggregation is also capable of excitable behavior, which consists in the transient amplification of suprathreshold pulses of extracellular cAMP. Since the first theoretical model for slime mold aggregation proposed by Keller and Segel in 1970, many theoretical studies have addressed various aspects of the mechanism and function of cAMP signaling in Dictyostelium. This paper presents a brief overview of these developments as well as some reminiscences of the author's collaboration with Lee Segel in modeling the dynamics of cAMP relay and oscillations. Considered in turn are models for cAMP signaling in Dictyostelium, the developmental path followed by the cAMP signaling system after starvation, the frequency encoding of cAMP signals, and the origin of concentric or spiral waves of cAMP.     

Spiky and transition layer steady states of chemotaxis systems via global bifurcation and Helly’s compactness theorem

The most important phenomenon in chemotaxis is cell aggregation. To model this phenomenon we use spiky or transition layer (step-function-like) steady states. In the case of one spatial dimension, we carry out global bifurcation analysis on the Keller–Segel model and several variants of it, showing that positive steady states exist if the chemotactic coefficient ${\chi}$ is larger than a bifurcation value ${\bar{\chi}_1}$ which can be explicitly expressed in terms of the parameters in the models; then we use Helly’s compactness theorem to obtain the profiles of these steady states when the ratio of the chemotactic coefficient and the cell diffusion rate is large, showing that they are either spiky or have the transition layer structure. Our results provide insights on how the biological parameters affect pattern formation, and reveal the similarities and differences of some popular chemotaxis models.     

甲烷利用菌培养条件的优化及其初步应用

利用统计学实验设计（RSM）对能够利用甲烷的假单胞菌菌株ME16主要培养条件进行了优化。以液体无机盐和甲烷气体作为培养基分别进行了温度、接种量、甲烷含量和培养pH对细菌生长影响的研究,并在此基础上,利用响应面法分析优化了ME16菌株的主要培养条件,得到最佳培养条件为：温度29.4℃,接种量1.8%,甲烷含量25%。采用优化培养条件进行培养,细菌生物量增大0.8倍,达到稳定期的培养时间缩短了50h。该菌株初步应用于甲烷气体的脱除,脱除率达65.7 %,表明该菌株能良好的脱除空气中甲烷。     

Regulation of mammalian cell growth by autocrine growth factors: analysis of consequences for inoculum cell density effects

Effects of inoculum cell density on mammalian cell growth in culture have been observed in a variety of experimental systems. Although these effects have been attributed generally to medium conditioning by the cells, there has previously been no quantitative theory proposed for this phenomenon based on developments in molecular and cell biology. In this article, we offer such a theory founded on the regulatory action of autocrine growth factors. A particularly relevant example of these is platelet- derived growth factor (PDGF), which is produced by fibroblastic cells in response to stimulation by transforming growth factor beta (TGFbeta), a common serum constituent, and provides a mitogenic signal for the same cells. A simple mathematical model for the production, diffusive transport, and binding of autocrine growth factors to cell surface receptors, coupled to a model for the dependence of cell proliferation on growth factor receptor binding allows prediction of initial cell population growth rate as a function of inoculum cell density. We focus on situations involving anchorage-dependent cell growth, in which the cells are attached to a surface. A number of clear results are obtained, most notably the following: 1) for cells cultured on spherical microcarrier bead surfaces, the inoculum cell density needed to produce a given growth rate is linearly proportional to the bead radius; and 2) all other factors being equal, the inoculum cell density on a unit surface area basis needed to produce a given growth rate is greater for spherical microcarrier surfaces than for flat culture dish surfaces. These two results are consistent with the experimental observations of Hu and coworkers(1,2) for fibroblast growth in minimal medium plus serum. The model also allows elucidation of the influence of other system parameters, both biological and physical, on initial cell proliferation rate and the inoculum cell density dependence.     

Estimation of vaginal probiotic lactobacilli growth parameters with the application of the Gompertz model

Lactobacilli are widely described as probiotic microorganisms used to restore the ecological balance of different animal or human tracts. For their use as probiotics, bacteria must show certain characteristics or properties related to the ability of adherence to mucosae or epithelia or show inhibition against pathogenic microorganisms. It is of primary interest to obtain the highest biomass and viability of the selected microorganisms. In this report, the growth of seven vaginal lactobacilli strains in four different growth media and at several inoculum percentages was compared, and the values of growth parameters (lag phase time, maximum growth rate, maximum optical density) were obtained by applying the Gompertz model to the experimental data. The application and estimation of this model is discussed, and the evaluation of the growth parameters is analyzed to compare the growth conditions of lactobacilli. Thus, these results in lab experiments provide a basis for testing different culture conditions to determine the best conditions in which to grow the probiotic lactobacilli for technological applications.     

Efficient spatial coverage by a robot swarm based on an ant foraging model and the Lévy distribution

This work proposes a control law for efficient area coverage and pop-up threat detection by a robot swarm inspired by the dynamical behavior of ant colonies foraging for food. In the first part, performance metrics that evaluate area coverage in terms of characteristics such as rate, completeness and frequency of coverage are developed. Next, the Keller–Segel model for chemotaxis is adapted to develop a virtual-pheromone-based method of area coverage. Sensitivity analyses with respect to the model parameters such as rate of pheromone diffusion, rate of pheromone evaporation, and white noise intensity then identify and establish noise intensity as the most influential parameter in the context of efficient area coverage and establish trends between these different parameters which can be generalized to other pheromone-based systems. In addition, the analyses yield optimal values for the model parameters with respect to the proposed performance metrics. A finite resolution of model parameter values were tested to determine the optimal one. In the second part of the work, the control framework is expanded to investigate the efficacy of non-Brownian search strategies characterized by Lévy flight, a non-Brownian stochastic process which takes variable path lengths from a power-law distribution. It is shown that a control law that incorporates a combination of gradient following and Lévy flight provides superior area coverage and pop-up threat detection by the swarm. The results highlight both the potential benefits of robot swarm design inspired by social insect behavior as well as the interesting possibilities suggested by considerations of non-Brownian noise.