Mathematical Model Of Microbial Growth Research Articles

Time-Resolved Studies of Bioluminescence From Photobacterium Leiognathi and Rapid Antimicrobial Susceptibility Testing on E. Coli Using Tunable Diode Laser Spectroscopy

We demonstrate that tunable diode laser spectroscopy (TDLS) can be used as a reliable, convenient and versatile investigative technique for non-invasive, high-sensitivity and time-resolved studies of the expression of various traits by bacterial strains. We use a 2004 nm vertical cavity surface emitting laser and a 4320 nm mid-infrared quantum cascade laser to show that the bioluminescence and simultaneous CO2 emission from Photobacterium leiognathi are correlated. The bioluminescence peaks when the growth rate is maximum. This supports the hypothesis that bioluminescence is a cell density-dependent phenomenon. We also show that TDLS can be used for ultra-rapid antimicrobial susceptibility testing without any need of sample preparation. The mole fraction of CO2 emitted during the growth of E. coli was recorded for different standardized doses of ampicillin to show the drastic suppression of growth as the antibiotic dose was increased to the recommended minimum inhibitory drug concentration. Growth suppression was detectable within 2 h, which is much shorter than the time required by conventional methods. We conclude that TDLS is a potent technique for validation and refinement of mathematical models of microbial growth that are necessary to better understand the dynamics of the expression of various traits at specific stages of growth. It is also important for rapid antimicrobial susceptibility testing.

Mathematical models of microbial growth and metabolism: a whole-organism perspective.

We review the principles underpinning the development of mathematical models of the metabolic activities of micro-organisms. Such models are important to understand and chart the substantial contributions made by micro-organisms to geochemical cycles, and also to optimise the performance of bioreactors that exploit the biochemical capabilities of these organisms. We advocate an approach based on the principle of dynamic allocation. We survey the biological background that motivates this approach, including nutrient assimilation, the regulation of gene expression, and the principles of microbial growth. In addition, we discuss the classic models of microbial growth as well as contemporary approaches. The dynamic allocation theory generalises these classic models in a natural manner and is readily amenable to the additional information provided by transcriptomics and proteomics approaches. Finally, we touch upon these organising principles in the context of the transition from the free-living unicellular mode of life to multicellularity.

Growth potential of Yersinia enterocolitica in pasteurised cow's and goat's milk stored at 8 °C and 24 °C

Milk and dairy products are among the most commonly consumed foods. The shelf life of pasteurised milk and many other dairy products is guaranteed, among others, when stored at 4–8 °C. These temperatures, however, will not prevent the growth of psychrotrophic bacteria, including some foodborne pathogens such as Yersinia enterocolitica. The study examines the growth potential of Y. enterocolitica in pasteurised cow's and goat's milk at different storage conditions (at a refrigerated temperature of 8 °C and an improper storage temperature of 24 °C). Y. enterocolitica growth dynamics was studied using a mathematical model for microbial growth. The behaviour of Y. enterocolitica depends on storage temperature and inoculum size and does not differ between cow's and goat's milk. The growth curve of Y. enterocolitica cultivated at 8 °C included the lag, exponential, and stationary phases. Both, the duration of the lag phase and growth rate did not differ between milk samples inoculated with the low (approximately 1 log cfu ml−1) or high (3 log cfu ml−1) inoculum size. The stationary phase was reached (with the peak number of the bacteria produced) within eight and six days with the low and high inoculum size, respectively. The growth of the bacteria at 24 °C was exponential from the first hour, peaking within two to three days or slightly later with the low inoculum size. The maximum cell concentration did not depend on inoculum size and cultivation temperature, reaching 9 log cfu ml−1 in all cases. Due to its composition and absence of competitive microflora, pasteurised milk provides a favourable environment for the growth of Y. enterocolitica. In case of contamination of pasteurised milk, even with a low amount of Y. enterocolitica, the bacterium multiplies to an infective dose within hours or days, depending on the storage temperature.

Antimicrobial Activity and Mechanism of Inhibition of Silver Nanoparticles against Extreme Halophilic Archaea.

Haloarchaea are salt-loving halophilic microorganisms that inhabit marine environments, sea water, salterns, and lakes. The resistance of haloarchaea to physical extremities that challenge organismic survival is ubiquitous. Metal and antibiotic resistance of haloarchaea has been on an upsurge due to the exposure of these organisms to metal sinks and drug resistance genes augmented in their natural habitats due to anthropogenic activities and environmental pollution. The efficacy of silver nanoparticles (SNPs) as a potent and broad spectrum inhibitory agent is known, however, there are no reports on the inhibitory activity of SNPs against haloarchaea. In the present study, we have investigated the antimicrobial potentials of SNPs synthesized using aqueous leaf extract of Cinnamomum tamala against antibiotic resistant haloarchaeal isolates Haloferax prahovense RR8, Haloferax lucentense RR15, Haloarcula argentinensis RR10 and Haloarcula tradensis RR13. The synthesized SNPs were characterized by UV-Vis spectroscopy, scanning electron microscopy, energy dispersive X-ray spectroscopy, dynamic light scattering, X-ray diffraction and Fourier transform infrared spectroscopy. The SNPs demonstrated potent antimicrobial activity against the haloarchaea with a minimum inhibitory concentration of 300–400 μg/ml. Growth kinetics of haloarchaea in the presence of SNPs was studied by employing the Baranyi mathematical model for microbial growth using the DMFit curve fitting program. The C. tamala SNPs also demonstrated cytotoxic activity against human lung adenocarcinoma epithelial cell line (A540) and human breast adenocarcinoma cell line (MCF-7). The mechanism of inhibition of haloarchaea by the SNPs was investigated. The plausible mechanism proposed is the alterations and disruption of haloarchaeal membrane permeability by turbulence, inhibition of respiratory dehydrogenases and lipid peroxidation causing cellular and DNA damage resulting in cell death.

Effect of C/N ratio and support material on heterotrophic denitrification of potable water in bio-filters using sugar as carbon source

This work studied heterotrophic denitrification in packed-bed reactors using sugar as electron donor and carbon source. Two pilot-scale bio-filters, loaded with different types and sizes of support media, were operated under sequencing batch (SBR) mode with recirculation. The materials used were silicic gravel placed in two layers (diameter 2.41 and 4.03 mm) within the first filter (R1 – multilayer), and hollow plastic tubes (diameter 1.6 cm) within the second filter (R2 – monolayer). Feed NO3−-N concentrations ranged between 20 and 400 mg/L, while the C/N ratio was kept constant at 13.5. Higher denitrification rates (4.06–8.76 kg N/(m3d)) and total sugar biodegradation were observed in bio-filter R1 than in R2. However, carbon residues were observed in the treated water of bio-filter R1 for feed concentrations above 70 mgNO3−-N/L. Bio-filter R1 was then used to study the effect of C/N ratio on its performance. Experiments showed that a decrease of the C/N ratio from 13.5 to 6.25 resulted in lower carbon residue in the treated water and high denitrification rates (3.21–6.64 kgN/(m3d)), however, significant NO2−-N accumulation was observed. Further reduction of the C/N ratio (i.e., 5.48 and 4.71) resulted in filter malfunction as insufficient removal of NO3−-N was achieved. Results suggest that under the more favourable conditions of C/N:13.5 and silicic gravel support material, satisfactory NO3−-N and COD removal could be achieved for feed concentrations up to 60 mgNO3−-N/L at a cost of 0.019 €/KgN or 1.13 × 10−3 €/m3, which is much lower than ever reported in the literature and confirms the economic viability of the process. To describe the heterotrophic denitrification process, a mathematical model of microbial growth on nitrate and nitrite treated as substitutable substrates and sugar as the carbon source is proposed. Model simulations could be used as a guide for the optimum design and operation of water treatment systems for nitrate removal.

Existence of traveling waves of auto-catalytic systems with decay

This article establishes the existence of traveling waves of a class of reaction–diffusion systems which model the pre-mixed isothermal autocatalytic chemical reaction of order m (m>1) between two chemical species, a reactant and an auto-catalyst, and a linear decay. Moreover, our result shows that the set of speed is contained in a bounded interval for any fixed initial value at x=−∞. This is in strong contrast to either the reaction–diffusion systems of autocatalytic chemical reaction of the order m without decay, or to the systems which have the same order of decay, which were shown by various authors (e.g. [8,13,17,26]) that the set of traveling wave speeds contains [c⁎,∞) for some c⁎>0.The same systems also appear in a mathematical model of microbial growth and competition in a flow reactor; see [2,24].

Effect of gamma irradiation and storage time on microbial growth and physicochemical characteristics of pumpkin (Cucurbita Moschata Duchesne ex Poiret) puree

The effect of gamma irradiation (0-2 kGy) and storage time (0-28 days) on microbial growth and physicochemical characteristics of a packed pumpkin puree was studied. For that purpose, a factorial design was applied. The puree contained potassium sorbate, glucose and vanillin was stored at 25°C . Gamma irradiation diminished and storage time increased microbial growth. A synergistic effect between both variables on microbial growth was observed. Storage time decreased pH and color of purees. Sorbate content decreased with storage time and gamma irradiation. Mathematical models of microbial growth generated by the factorial design allowed estimating that a puree absorbing 1.63 kGy would have a shelf-life of 4 days. In order to improve this time, some changes in the applied hurdles were assayed. These included a thermal treatment before irradiation, a reduction of irradiation dose to 0.75 kGy and a decrease in storage temperature at 20°C . As a result, the shelf-life of purees increased to 28 days.

Design of a cascade observer for a model of bacterial batch culture with nutrient recycling

A mathematical model of microbial growth on a single limited substrate in batch culture is proposed as an extension of the Monod's one. This model takes into account cell mortality, non-viable cell accumulation and cell recycling. We consider that only substrate concentration and total biomass are measured on-line. The parameters of the model are not identifiable at steady state, but we propose a design of an observer that reconstructs both parameters and state variable with a practical convergence, from any initial condition away from the equilibrium. The observer is built as a coupling of two non-linear observers in cascade with different time scales.

Development of a Program to Fit Data to a New Logistic Model for Microbial Growth

Recently we developed a mathematical model for microbial growth in food. The model successfully predicted microbial growth at various patterns of temperature. In this study, we developed a program to fit data to the model with a spread sheet program, Microsoft Excel. Users can instantly get curves fitted to the model by inputting growth data and choosing the slope portion of a curve. The program also could estimate growth parameters including the rate constant of growth and the lag period. This program would be a useful tool for analyzing growth data and further predicting microbial growth.

Bacterial Wall Attachment in a Flow Reactor

A mathematical model of microbial growth for limiting nutrients in a fully three dimensional flow reactor which accounts for the colonization of the reactor wall surface by the microbes is studied analytically. It can be viewed as a model of the large intestine or of the fouling of a commercial bioreactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each. The effects of adding an antimicrobial agent to the reactor are examined with and without wall growth.

A model of microbial growth in a plug flow reactor with wall attachment

A mathematical model of microbial growth for limiting nutrient in a plug flow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each, and for the long term persistence of the bacteria in the reactor.

Mechanistic mathematical models of microbial growth in bioreactors and in natural soils: Explanation of complex phenomena

Abstract The paper summarizes the development of so-called synthetic chemostat model (SCM) simulating complex dynamic behavior of microorganisms. The basic SCM represents microbial growth as a conversion of exosubstrate into cell macromolecules via pool of intermediates. The “quality” of biomass is characterized by scalar variable r. This variable denotes a relative amount of inducible cell constituents which provide intensive growth and display positive correlation with growth rate (RNA, ribosomal proteins, respiratory enzymes, etc.). Basic SCM (four differential equations) was able to simulate both steady-state growth and transient dynamics of such well studied microorganisms as Pseudomonas and Enterobacteria. To describe the behavior of other organisms having unconventional life strategy (sporeforming bacteria, K-selected and oligotrophic microorganisms), the basic SCM was augmented to different degree by incorporating additional state variables (reserve compounds, prospore compartment, signal metabolic products, etc.). The competition of 2–5 species for a common limiting substrate and growth of one population on several substrates were simulated by enlarged model containing up to 22 differential equations. Application of SCM in physiological and ecological studies provided mechanistic interpretation of many unusual aspects of microbial behavior both in vitro and in situ.

Predictive food microbiology: A probabilistic approach

Abstract An existing mathematical model for microbial growth and inactivation in dynamically changing temperature conditions has been modified to explain the experimentally observed variability between replicate growth curves. The model incorporates a random variable parameter representing the random physiological state of the inoculum. The model is validated using experimental growth data which were obtained in time-varying temperature conditions. It is shown by Monte Carlo analyses that broad confidence bands are associated with the model predictions.

Evaluation of Fourier-transform infrared spectroscopy for data capture in predictive microbiology

Chemical changes in the medium, induced by the fermentative species Lactobacillus plantarum and Lactobacillus brevis and by the enzymatic action of a proteolytic, spoilage species, Yarrowia lipolytica, were analysed using Fourier-transform i.r. spectroscopy (FTIR). Changes in the absorbance data over time could be modelled using one of the more current predictive, mathematical models of microbial growth, such as the Gompertz equation. Moreover, a linear correlation between FTIR data (expressed as absorbance of some selected peaks) and viability data (expressed as log10 c.f.u./g or ml) was observed during the fermentation process, both for L. plantarum and L. brevis.

An example of the stages in the development of a predictive mathematical model for microbial growth: the effects of NaCl, pH and temperature on the growth of Aeromonas hydrophila

The stages involved in developing a predictive model are illustrated using data describing the effects of temperature (3–20°C), NaCl concentration (0.5–4.5% w/v) and pH (4.6–7.0) on the aerobic growth of Aeromonas hydrophila (cocktail of 6 strains). Optical density measurements using micro-titre plates were used as an initial screen, to determine the appropriate sampling times for viable counts to be made and to determine the approximate boundaries for growth. Growth curves were generated from viable counts and fitted using a modified Gompertz equation. Quadratic response surface equations were fitted to the log of lag and generation times, in response to the variables of temperature, NaCl and pH (in terms of hydrogen ion concentration). The effects of various combinations of these controlling factors are described. Comparisons between predicted growth rates and lag times from our response surface equations and other models for growth of a A. hydrophila, developed with viable count data and optical density measurements, are made, together with comparisons with data from the literature on the growth of this bacterium in foods.

Kinetic investigations of microbial metabolism by means of flow calorimeters

A mathematical model for microbial growth is presented which combines a Michaelis—Menten kinetics for oxygen and a substrate inhibition kinetics of the Haldane-type of phenol. It is applied to flow calorimetric experiments of the growth of Pseudomonas putida on phenol and other aromatic compounds. The model describes features of the growth well and makes unexpected predictions which have been experimentally verified and compared with as yet unexplained observations from the literature. Possible applications to batch and flow calorimetric investigations of microbial growth are discussed with respect to the critical evaluation of the chosen instrumental set-up.

An error estimation of Michaelis-Menten (Monod)-type kinetics

Due to research on biochemistry and genetic engineering, mathematical models of microbial growth have become more complicated but Michaelis-Menten or Monod type expressions have still been used for conversion rates, uptake rates, etc. It is worth examining the error that can be caused by these quasi-steady-state-hypotheses. This paper presents a simple but very effective rationale function that describes the error of the quasi-steady-state hypothesis in enzyme kinetics. A simplified fermentation kinetic model was used for comparison of microbial growth but no analytical error function has been found for batch cultivation. In the case of continuous fermentation the error can be given in an analytical form. Many simulations, based on real SCP experiments, show a significant effect of the quasi-steady-state hypothesis. Since the rate constants of intracellular events are not really known, we have to be very careful when taking into account Michaelis-Menten type expressions in the building of complicated models.

Mathematical models of microbial growth and competition in the chemostat regulated by cell-bound extracellular enzymes

A mathematical model of growth and competitive interaction of microorganisms in the chemostat is analyzed. The growth-limiting nutrient is not in a form that can be directly assimilated by the microorganisms, and must first be transformed into an intermediate product by cell-bound extracellular enzymes. General monotone functions, including Michaelis-Menten and sigmoidal response functions, are used to describe nutrient conversion and growth due to consumption of the intermediate product. It is shown that the initial concentration of the species is an important determining factor for survival or washout. When there are two species whose growth is limited by the same nutrient, three different modes of competition are described. Competitive coexistence steady states are shown to be possible in two of them, but they are always unstable. In all of our numerical simulations, the system approaches a steady state corresponding to the washout of one or both of the species from the chemostat.

Segregated, structured, distributed models and their role in microbial ecology: A case study based on work done on the filter-feeding ciliateTetrahymena pyriformis

Microbial populations are composed of individual organisms each of which, if environmental circumstances are favorable, is undergoing change of its internal state through the operation of the set of processes that we call the cell cycle. The rate of progression through the cycle is subject to internal controls as well as external influences, and exhibits random as well as deterministic features. Microorganisms of the same species in different stages of the cell cycle have different internal states, and thus, the operation of the cell cycle is by itself sufficient to produce a distribution of states among the individual organisms of a population. In turn, the distribution of states produces distributions of the rates at which the cells of a population carry on their activities. Mathematical models of microbial growth that take the operation of the cell cycle and its consequences into account are more complicated than the kinds of models that are often used in microbial ecology. This paper gives some account of the nature, formulation, and uses of complex growth models. The account is illustrated by work done by the author and his collaborators H. M. Tsuchiya and more recently F. Srienc, as well as by others, on the filter-feeding ciliateTetrahymena pyriformis.

Antimicrobial Activity of Undissociated Sorbic Acid in Vacuum Packaged Beef

ABSTRACTInhibitory effectiveness of sorbic acid was analyzed in vacuum packaged refrigerated beef by studying the modifications of bacterial growth parameters. Total counts, Lactobacillus sp., Pseudomonas sp., Enrerobacreriaceae, Brochorhrix thermosphacta and yeasts in samples of different pH values (5.70–6.20) which had been treated with potassium sorbate (27–3125 ppm) showed variations in the lag phase, growth rate, stationary phase and inhibition index. For refrigerated beef the contribution of sorbic acid dissociated fraction to the inhibitory effect was negligible, Considering growth parameters’dependence on undissociated sorbic acid concentration a mathematical model of microbial growth in treated beef was derived, which could be used to predict product shelf‐life.