Effects of cell motility and chemotaxis on microbial population growth.

A mathematical model is developed to elucidate the effects of biophysical transport processes (nutrient diffusion, cell motility, and chemotaxis) along with biochemical reaction processes (cell growth and death, nutrient uptake) upon steady-state bacterial population growth in a finite one-dimensional region. The particular situation considered is that of growth limitation by a nutrient diffusing from an adjacent phase not accessible to the bacteria. It is demonstrated that the cell motility and chemotaxis properties can have great influence on steady-state population size. In fact, motility effects can be as significant as growth kinetic effects, in a manner analogous to diffusion- and reaction-limited regimes in chemically reacting systems. In particular, the following conclusions can be drawn from our analysis for bacterial populations growing at steady-state in a confined, unmixed region: (a) Random motility may lead to decreased population density; (b) chemotaxis can allow increased population density if the chemotactic response is large enough; (c) a species with superior motility properties can outgrow a species with superior growth kinetic properties; (d) motility effects become greater as the size of the confined growth region increases; and (e) motility effects are diminished by significant mass-transfer limitation of the nutrient from the adjacent source phase. The relationships of these results for populations to previous conclusions for individual cells is discussed, and implications for microbial competition are suggested.     

Traveling bands of chemotactic bacteria in the context of population growth

A mathematical model for traveling bands of motile and chemotactic bacteria in the presence of cell growth and death is examined. It is found that asymptotic traveling wave solutions exist in the absence of chemotaxis, due to the balance of growth, death and random motility. Thus random motility confers the ecological advantage of population propagation through migration into nutrient-rich regions. The presence of chemotaxis amplifies this advantage by moving more cells into higher nutrient concentration regions, resulting in larger and faster bands. Therefore there seem to be two types of traveling bands that can be attained by chemotactic bacteria in the presence of growth and death: (1) these growth/death/motility bands; and (2) pure chemotactic ‘Keller-Segel'-type bands. Comparison to experimental observations by Chapman in 1973 indicate that the latter seem to be formed. The relationship between these two types of solution is at present uncertain. The growth/death/motility bands may have relevance on longer time or distance scales characteristic of microbial ecological systems.     

Effects of random motility on growth of bacterial populations

A spatially distributed mathematical model is developed to elucidate the effects of chemical diffusion and cell motility as well as cell growth, death, and substrate uptake on steady-state bacterial population growth in a finite, one-dimensional, nonmixed region. The situation considered is growth limited by a diffusing substrate from an adjacent phase not accessible to the bacteria. Chemotactic movement is not considered in this paper; we consider only randomwalk-type random motility behavior here. The following important general concepts are suggested by the results of our theoretical analysis: (a) The significance of random motility effects depends on the magnitude of the ratio/kL ², where is the bacterial random motility coefficient,k is the growth rate constant, andL is the linear dimension of the confined growth region. (b) In steady-state growth in a confined region, the bacterial population size decreases as increases. (c) The effect of on population size can be great; in fact, sometimes relative population sizes of two species can be governed primarily by the relative values of rather than by the relative values ofk.     

On the fisher and the cubic-polynomial equations for the propagation of species properties

For precise boundary conditions of biological relevance, it is proved that the steadily propagating plane-wave solution to the Fisher equation requires the unique (eigenvalue) velocity of advance 2(Df)^1/2, whereD is the diffusivity of the mutant species andf is the frequency of selection in favor of the mutant. This rigorous result shows that a so-called “wrong equation”, i.e. one which differs from Fisher's by a term that is seemingly inconsequential for certain initial conditions, cannot be employed readily to obtain approximate solutions to Fisher's, for the two equations will often have qualitatively different manifolds of exact solutions. It is noted that the Fisher equation itself may be inappropriate in certain biological contexts owing to the manifest instability of the lowerconcentration uniform equilibrium state (UES). Depicting the persistence of a mutantdeficient spatial pocket, an exact steady-state solution to the Fisher equation is presented. As an alternative and perhaps more faithful model equation for the propagation of certain species properties through a homogeneous population, we consider a reaction-diffusion equation that features a cubic-polynomial rate expression in the species concentration, with two stable UES and one intermediate unstable UES. This equation admits a remarkably simple exact analytical solution to the steadily propagating plane-wave eigenvalue problem. In the latter solution, the sign of the eigenvelocity is such that the wave propagates to yield the “preferred” stable UES (namely, the one further removed from the unstable intermediate UES) at all spatial points ast→∞. The cubic-polynomial equation also admits an exact steady-state solution for a mutant-deficient or mutant-isolated spatial pocket. Finally, the perpetuating growth of a mutant population from an arbitrary localized initial distribution, a mathematical problem analogous to that for ignition in laminar flame theory, is studied by applying differential inequality analysis, and rigorous sufficient conditions for extinction are derived here.     

Localized bacterial infection in a distributed model for tissue inflammation

Phagocyte motility and chemotaxis are included in a distributed mathematical model for the inflammatory response to bacterial invasion of tissue. Both uniform and non-uniform steady state solutions may occur for the model equations governing bacteria and phagocyte densities in a macroscopic tissue region. The non-uniform states appear to be more dangerous because they allow large bacteria densities concentrated in local foci, and in some cases greater total bacteria and phagocyte populations. Using a linear stability analysis, it is shown that a phagocyte chemotactic response smaller than a critical value can lead to a non-uniform state, while a chemotactic response greater than this critical value stabilizes the uniform state. This result is the opposite of that found for the role of chemotaxis in aggregation of slimemold amoebae because, in the inflammatory response, the chemotactic population serves as an inhibitor rather than an activator. We speculate that these non-uniform steady states could be related to the localized cell aggregation seen in chronic granulomatous inflammation. The formation of non-uniform states is not necessarily a consequence of defective phagocyte chemotaxis, however. Rather, certain values of the kinetic parameters can yield values for the critical chemotactic response which are greater than the normal response.Numerical computations of the transient inflammatory response to bacterial challenge are presented, using parameter values estimated from the experimental literature wherever possible.     

A gradually slowing travelling band of chemotactic bacteria

A model for describing the motion of chemotactic bacteria in a capillary tube containing substrate is treated. Chemotactic substrate threshold effects are included in the chemotactic response coefficient. The ratio of the substrate threshold, s _T, to the substrate level far ahead of the travelling band, s , is used as a small parameter in developing an asymptotic solution of near travelling wave form.Also at: Membrane Filtration Technology, Kiryat Weizmann, Rehovot, Israel.Also at: Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12181, USA.     

Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation

We consider a simple cell-chemotaxis model for spatial pattern formation on two-dimensional domains proposed by Oster and Murray (1989,J. exp. Zool. 251, 186–202). We determine finite-amplitude, steady-state, spatially heterogeneous solutions and study the effect of domain growth on the resulting patterns. We also investigate in-depth bifurcating solutions as the chemotactic parameter varies. This numerical study shows that this deceptively simple-chemotaxis model can produce a surprisingly rich spectrum of complex spatial patterns.     

Production of elemental sulfur by green and purple sulfur bacteria

The utilization of sulfide by phototrophic sulfur bacteria temporarily results in the accumulation of elemental sulfur. In the green sulfur bacteria (Chlorobiaceae), the sulfur is deposited outside the cells, whereas in the purple sulfur bacteria (Chromatiaceae) sulfur is found intracellularly. Consequently, in the latter case, sulfur is unattainable for other individuals. Attempts were made to analyze the impact of the formation of extracellular elemental sulfur compared to the deposition of intracellular sulfur.According to the theory of the continuous cultivation of microorganisms, the steady-state concentration of the limiting substrate is unaffected by the reservoir concentration (S _R).It was observed in sulfide-limited continuous cultures ofChlorobium limicola f.thiosulfatophilum that higherS _R values not only resulted in higher steady-state population densities, but also in increased steady-state concentrations of elemental sulfur. Similar phenomena were observed in sulfide-limited cultures ofChromatium vinosum.It was concluded that the elemental sulfur produced byChlorobium, althouth being deposited extracellularly, is not easily available for other individuals, and apparently remains (in part) attached to the cells. The ecological significance of the data is discussed.Non-standard abbreviations RP reducing power - BChl bacteriochlorophyll - N_cell cell material - specific growth rate - {ie52-1} maximal specific growth rate - D dilution rate - K _s saturation constant - s concentration of limiting substrate - S _R same ass but in reservoir bottle - Y yield factor - iS^o intracellular elemental sulfur - eS^o extracellular elemental sulfur - PHB poly-beta-hydroxybutyric acid     

Effects of leukocyte random motility and chemotaxis in tissue inflammatory response.

A rapidly growing body of experimental evidence indicates that defects in leukocyte motility and chemotactic response correlate with increased susceptibility to and severity of bacterial infection in tissue. While this is understandable in qualitative terms, the sensitivity of the correlation is remarkable.In the present study, a theoretical analysis has been developed to relate the dynamics of bacterial growth to the growth and transport parameters of bacteria and leukocytes in tissue. The model considers a local tissue region in the vicinity of a venule and applies continuum unsteady state species conservation equations to the bacterial population, the phagocytic leukocytes, and a chemotactically active chemical mediator assumed to be produced by the bacteria. The analysis quantifies the effects of key parameters, such as leukocyte random motility and chemotactic coefficients, phagocytic and growth rate constants, and leukocyte vessel wall permeability, upon host ability to eliminate the bacteria.As an example, the model's predictions are compared to experimental results correlating inhibition of leukocyte chemotaxis by hemoglobin with its adjuvant action in experimental peritoneal infection by E. coli.     

Inconstancy of Taylor'sb: Simulated sampling with different quadrat sizes and spatial distributions

Taylor 's power law, s²=am^b, provides a precise summary of the relationship between sample variance (s²) and sample mean (m) for many organisms. The coefficient b has been interpreted as an index of aggregation, with a characteristic value for a given species in a particular environment, and has been thought to be independent of the sample unit. Simulation studies were conducted that demonstrate that the value of b may vary with the size of the sample unit in quadrat sampling, and this relationship, in turn, depends on the underlying spatial distribution of the population. For example, simulated populations with hierarchical aggregation on a large scale produced values of b that increased with the size of the sample unit. In contrast, for a simulated population with randomly distributed clusters of individuals, the value of b eventually decreased with increasing quadrat size, as sample counts became more uniform. A single value ofTaylor 's b, determined with a particular sample unit, provides neither a fixed index of aggregation nor a complete picture of a species' spatial distribution. Rather, it describes a consistent relationship between sample variance and sample mean over a range of densities, on a spatial scale related to the size of the sample unit. This relationship may reflect, but not uniquely define, density-dependent population and behavioral processes governing the spatial distribution of the organism. Interpretation ofTaylor 'sb for a particular organism should be qualified by reference to the sample unit, and comparisons should not be made between cases in which different sample units were used. Whenever possible, a range of sample units should be used to provide information about the pattern of distribution of a population on various spatial scales.     

Model of steady-state-biofilm kinetics

A steady-state biofilm is defined as one that has neither net growth nor decay over time. The model, developed for steady-state-biofilm kinetics with a single substrate, couples the flux of substrate into a biofilm to the mass (or thickness) of biofilm that would exist at steady-state for a given bulk substrate concentration. Based on kinetic and energetic constraints, this model predicts for a single substrate that a steady-state bulk concentration, S_min, exists below which a steady-state biofilm cannot exist. Thus, in the absence of adsorption of bacteria from the bulk water and for substrate concentration below S_min, substrate flux and biofilm thickness are zero. Equations are provided for calculating the steady-state substrate flux and biofilm thickness for S greater than S_min. An example is provided to demonstrate the use of the steadystate model.     

Kinetics of biopolymerization on nucleic acid templates

The kinetics of biopolymerization on nucleic acid templates is discussed. The model introduced allows for the simultaneous synthesis of several chains, of a given type, on a common template, e.g., the polyribosome situation. Each growth center [growing chain end plus enzyme(s)] moves one template site at a time, but blocks L adjacent sites. Solutions are found for the probability n_j(t) that a template has a growing center that occupies the sites j — L + 1,…, j at time t. Two special sets of solutions are considered, the uniform-density solutions, for which n_j(t) = n, and the more general steady-state solutions, for which dn_j(t)/dt = 0. In the uniform-density case, there is an upper bound to the range of rates of polymerization that can occur. Corresponding to this maximum rate, there is one uniform solution. For a polymerization rate less than this maximum, there are two uniform solutions that give the same rate. In the steady-state case, only L = 1 is discussed. For a steady-state polymerization rate less than the maximum uniform-density rate, the steady-state solutions consist of either one or two regions of nearly uniform density, with the density value(s) assumed in the uniform region(s) being either or both of the uniform-density solutions corresponding to that polymerization rate. For a steady-state polymerization rate equal to or slightly larger than the maximum uniform-density rate, the steady-state solutions are nearly uniform to the single uniform-density solution for the maximum rate. The boundary conditions (rate of initiation and rate, of release of completed chains from the template) govern the choice among the possible solutions, i.e., determine the region(s) of uniformity and the value(s) assumed in the uniform region(s).     

Model and analysis of chemotactic bacterial patterns in a liquid medium

 A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals. Received: 10 March 1998 / Revised version: 7 June 1998     

Cell balance equation for chemotactic bacteria with a biphasic tumbling frequency

Alts three-dimensional cell balance equation characterizing the chemotactic bacteria was analyzed under the presence of one-dimensional spatial chemoattractant gradients. Our work differs from that of others who have developed rather general models for chemotaxis in the use of a non-smooth anisotropic tumbling frequency function that responds biphasically to the combined temporal and spatial chemoattractant gradients. General three-dimensional expressions for the bacterial transport parameters were derived for chemotactic bacteria, followed by a perturbation analysis under the planar geometry. The bacterial random motility and chemotaxis were summarized by a motility tensor and a chemotactic velocity vector, respectively. The consequence of invoking the diffusion-approximation assumption and using intrinsic one-dimensional models with modified cellular swimming speeds was investigated by numerical simulations. Characterizing the bacterial random orientation after tumbles by a turn angle probability distribution function, we found that only the first-order angular moment of this turn angle probability distribution is important in influencing the bacterial long-term transport. Mathematics Subject Classification (2000):60G05, 60J60, 82A70     

Finite time blow-up in some models of chemotaxis

We consider a class of models of chemotactic bacterial populations, introduced by Keller-Segel. For those models, we investigate the possibility of chemotactic collapse, in other words, the possibility that in finite time the population of predators aggregates to form a delta-function. To study this phenomenon, we construct self-similar solutions, which may or may not blow-up (in finite time), depending on the relative strength of three mechanisms in competition: (i) the chemotactic attraction of bacteria towards regions of high concentration in substrate (ii) the rate of consumption of the substrate by the bacteria and (iii) (possibly) the diffusion of bacteria. The solutions we construct are radially symmetric, and therefore have no relation with the classical traveling wave solutions. Our scaling can be justified by a dimensional analysis. We give some evidence of numerical stability.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Chemosensory Responses of Acanthamoeba castellanii: Visual Analysis of Random Movement and Responses to Chemical Signals

A visual assay slide chamber was used in conjunction with time-lapse videomicroscopy to analyze chemotactic behavior of axenically grown Acanthamoeba castellanii. Data were collected and analyzed as vector scatter diagrams and cell tracks. Amebas responded to a variety of bacterial products or potential bacterial products by moving actively toward the attractant. Responses to the chemotactic peptide formyl-methionyl-leucyl-phenylalanine (fMLP), lipopolysaccharide, and lipid A were statistically significant (P≤ 0.03), as was the response to fMLP benzylamide (P≤ 0.05). Significant responses to cyclic AMP, lipoteichoic acid, and N-acetyl glucosamine were also found. Chemotactic peptide antagonists, mannose, mannosylated bovine serum albumin, and N-acetyl muramic acid all yielded nonsignificant responses (P > 0.05). There was no single optimal concentration for response to any of the attractants tested, and amebas responded equally over the range of concentrations tested. Pretreatment of amebas with chemotactic peptides, bacterial products, and bacteria reduced the directional response to attractants. Amebas that had been grown in the presence of bacteria appeared more responsive to chemotactic peptides. Treatment of amebas with trypsin reduced the response of cells to chemotactic peptides, though sensitivity was restored within a couple of hours. This suggests the ameba membrane may have receptors, sensitive to these bacterial substances, which are different from the mannose receptors involved in binding bacteria to the membrane during phagocytosis. The rate of movement was relatively constant (ca. 0.40 μm/s), indicating that the locomotor response to these signals is a taxis, or possibly a klinokinesis, but not an orthokinesis. Studies of the population diffusion rate in the absence of signals indicate that the basic population motility follows the pattern of a Levy walk, rather than the more familiar Gaussian diffusion. This suggests that the usual mathematical models of ameboid dispersion may need to be modified.     

Reproduction rate,feeding process,and liebig limitations in cell populations—II

A biological system consisting of a population of cells suspended in a liquid substrate is considered. The general problem addressed in the paper is the derivation of the kinetic pattern of population growth as a statistical effect of a very large number of elementary interactions between a single cell and the molecules of nutrient in substrate. Solution of the problem is obtained in the form of equation expressing the population growth ratec as a function of substrate concentration,C _s. The analytical expression derived is applied to a real bacterial population (Escherichi coli) and kinetic patterns are theoretically computed. The major findings, expressed roughly, without nuances, are: (i) the concentration of nutrient at the cell membrane,C _c, can only be equal to either 0 (for theC _s below some threshold valueC ^*) orC _s (forC _s>C ^*); (ii) the Michaelis-Menten-Monod kinetics observed in experiments is an artifact: the pure (not contaminated by foreign factors) dependence ofc onC _s is actually such that the functionc=c(C _s) has practically linear increase whenC _s<C ^*, and is constant,c=c(C ^*)=const, whenC _s>C ^*; (iii) the Liebig principle is strictly fulfilled: up to a feasible accuracy of observation, under no circumstances can population growth be limited (controlled) by more than one substrate component—replacement of a limiting component for another one is an instant event rather than a gradual process.