Minimal models of growth and decline of microbial populations

Dynamics of growth and decline of microbial populations were analysed and respective models were developed in this investigation. Analysis of the dynamics was based on general considerations concerning the main properties of microorganisms and their interactions with the environment which was supposed to be affected by the activity of the population. Those considerations were expressed mathematically by differential equations or systems of the equations containing minimal sets of parameters characterizing those properties. It has been found that: (1) the factors leading to the decline of the population have to be considered separately, namely, accumulation of metabolites (toxins) in the medium and the exhaustion of resources; the latter have to be separated again into renewable (‘building materials’) and non-renewable (sources of energy); (2) decline of the population is caused by the exhaustion of sources of energy but no decline is predicted by the model because of the exhaustion of renewable resources; (3) the model determined by the accumulation of metabolites (toxins) in the medium does not suggest the existence of a separate ‘stationary phase’; (4) in the model determined by the exhaustion of energy resources the ‘stationary’ and ‘decline’ phases are quite discernible; and (5) there is no symmetry in microbial population dynamics, the decline being slower than the rise. Mathematical models are expected to be useful in getting insight into the process of control of the dynamics of microbial populations. The models are in agreement with the experimental data.     

Hopf bifurcation for a family of two-state microbial growth models

We describe a family of two-state microbial growth models, in which growth and maintenance are assigned to two different cell states. The way of splitting periodic solutions for low dilution rates of a continuous fermentation is shown. The existence of these periodie solutions is mainly influenced by the properties of substrate consumption, which the maintenance rate in the second cell state amounts to.     

Comparative analysis of some models of gene regulation in mixed-substrate microbial growth

Mixed-substrate microbial growth is of fundamental interest in microbiology and bioengineering. Several mathematical models have been developed to account for the genetic regulation of such systems, especially those resulting in diauxic growth. In this work, we compare the dynamics of three such models (Narang, 1998a. The dynamical analogy between microbial growth on mixtures of substrates and population growth of competing species. Biotechnol. Bioeng. 59, 116-121; Thattai and Shraiman, 2003. Metabolic switching in the sugar phosphotransferase system of Escherichia coli. Biophys. J. 85(2), 744-754; Brandt et al., 2004. Modelling microbial adaptation to changing availability of substrates. Water Res. 38, 1004-1013). We show that these models are dynamically similar--the initial motion of the inducible enzymes in all the models is described by the Lotka-Volterra equations for competing species. In particular, the prediction of diauxic growth corresponds to "extinction" of one of the enzymes during the first few hours of growth. The dynamic similarity occurs because in all the models, the inducible enzymes possess properties characteristic of competing species: they are required for their own synthesis, and they inhibit each other. Despite this dynamic similarity, the models vary with respect to the range of dynamics captured. The Brandt et al. model always predicts the diauxic growth pattern, whereas the remaining two models exhibit both diauxic and non-diauxic growth patterns. The models also differ with respect to the mechanisms that generate the mutual inhibition between the enzymes. In the Narang model, mutual inhibition occurs because the enzymes for each substrate enhance the dilution of the enzymes for the other substrate. The Brandt et al. model superimposes upon this dilution effect an additional mechanism of mutual inhibition. In the Thattai and Shraiman model, the mutual inhibition is entirely due to competition for the phosphoryl groups. For quantitative agreement with the data, all models must be modified to account for specific mechanisms of mutual inhibition, such as inducer exclusion.     

On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities

The reason for difficulties in obtaining unique estimates of the parameters μ_m and K_s of the Michaelis-Menten equation are analysed for a microbial batch growth process. With the aid of simulation studies in which the influences of different types of noise on the parameter estimates are compared, it is shown that, although theoretically identifiable in the deterministic case with ideal measurements, the parameters cannot in general be correctly determined from noisy measurements. The difficulties are further illuminated by estimation examples using real data. It certain situations, in which the value of the ratio K_s/s_o is high or in which only few and noisy measurements are available, the linear approximation of the Michaelis-Menten equation gives a better fit. The practical difficulties in obtaining correct values of the model parameters do not limit the applicability of the Michaelis-Menten model, which in most cases explains the bacterial growth behavior excellently. Rather, they underline the fact that care must be taken when utilizing parameter estimates for biological interpretations.     

Use of stochastic models to assess the effect of environmental factors on microbial growth

We present a novel application of a stochastic ecological model to the study and analysis of microbial growth dynamics as influenced by environmental conditions in an extensive experimental data set. The model proved to be useful in bridging the gap between theoretical ideas in ecology and an applied problem in microbiology. The data consisted of recorded growth curves of Escherichia coli grown in triplicate in a base medium with all 32 possible combinations of five supplements: glucose, NH(4)Cl, HCl, EDTA, and NaCl. The potential complexity of 2(5) experimental treatments and their effects was reduced to 2(2) as just the metal chelator EDTA, the presumed osmotic pressure imposed by NaCl, and the interaction between these two factors were enough to explain the variability seen in the data. The statistical analysis showed that the positive and negative effects of the five chemical supplements and their combinations were directly translated into an increase or decrease in time required to attain stationary phase and the population size at which the stationary phase started. The stochastic ecological model proved to be useful, as it effectively explained and summarized the uncertainty seen in the recorded growth curves. Our findings have broad implications for both basic and applied research and illustrate how stochastic mathematical modeling coupled with rigorous statistical methods can be of great assistance in understanding basic processes in microbial ecology.     

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.Research supported by NSF grant DMS-90-96279Research supported by NSERC grant A-9358     

New experimental models in microbial ecology

Peculiar features of dysbiosis development in persons under extreme conditions were studied. It was shown that a number of extreme factors participated in formation of dysbiotic disorders in intestinal microflora. Of paramount importance was the neuro-emotional stress. Lability of bifido- and lactoflora was considered as the starting mechanism in dysbacteriosis under the extreme conditions. In the experimental models with rats SPF and Primates during flights of biosatellites of the Kosmos series the role of indigenous++ microflora in maintaining the microecological homeostasis, as well as the need for development of artificial and controlled intestinal microflora promising in prophylaxis of dysbacteriosis under extreme conditions was shown. The theoretical and experimentally grounded necessity of maintaining constant intestine microbiocenosis was confirmed by the practice of using the system of measures for recovery, stabilization and optimization of microflora in persons under extreme conditions.     

Kinetics of microbial cell growth

As a rate equation of microbial cell growth, the Monod equation is widely used. However, this equation cannot fully correspond to real courses of microbial cell growth in many batch cultivations. Especially, predicted values based on this equation do not agree with observed values in many continuous cultivations. In this paper, which introduces new concepts of critical concentration and coefficient of consumption activity, the growth rate equation which corresponds to the whole period including lag period is newly derived and characteristics of microbial cell growth in batch cultivation are clarified. Further, applying the new rate equation to continuous cultivation, a general equation with which to calculate cell concentration is derived and characteristics of microbial cell growth in continuous cultivation are clarified. The calculated values of cell concentration based on the new theory showed quite good agreement with the observed values in both batch and continuous cultivation.     

New microbial growth factor.

A screening procedure was used to isolate from soil a Penicillium sp., two bacterial isolates, and a Streptomyces sp. that produced a new microbial growth factor. This factor was an absolute growth requirement for three soil bacteria. The Penicillium sp. and one of the bacteria requiring the factor, an Arthrobacter sp., were selected for more extensive study concerning the production and characteristics of the growth factor. It did not seem to be related to the siderochromes. It was not present in soil extract, rumen fluid, or any other medium component tested. It appears to be a glycoprotein of high molecular weight, and it has high specific activity. When added to the diets for a meadow vole mammalian test system, it caused an increased consumption of diet without a concurrent increase in rate of weight gain.     

New microbial growth factor.

A screening procedure was used to isolate from soil a Penicillium sp., two bacterial isolates, and a Streptomyces sp. that produced a new microbial growth factor. This factor was an absolute growth requirement for three soil bacteria. The Penicillium sp. and one of the bacteria requiring the factor, an Arthrobacter sp., were selected for more extensive study concerning the production and characteristics of the growth factor. It did not seem to be related to the siderochromes. It was not present in soil extract, rumen fluid, or any other medium component tested. It appears to be a glycoprotein of high molecular weight, and it has high specific activity. When added to the diets for a meadow vole mammalian test system, it caused an increased consumption of diet without a concurrent increase in rate of weight gain.     

Quantitation of microbial growth on surfaces

An equation describing the initial phases of microbial surface colonization is presented. Simultaneous microbial attachment and growth are considered as the primary components of colonization. A table is given that permits determination of growth rate from the density and distribution of cells present on surfaces after incubation in situ. Other methods used to calculate microbial growth rate on surfaces are evaluated. The new procedure is more accurate and less time consuming than those used previously. Published data on microbial surface colonization more closely follow the proposed colonization equation than the exponential growth equation, which overestimates the growth rate.     

Modeling substrate inhibition of microbial growth

This article presents a general equation for substrate inhibition of microbial growth using a statistical thermodynamic approach. Existing empirical models adapted from enzyme kinetics, for example, the Haldane-Andrews equation, often criticized for not being physically based for microbial growth, are shown to derive from the general equation in this article, and their empirical parameters are shown to be well defined physically. Three sets of experimental data from the literature are used to test the modeling abilities of the general equation to represent experimental data. The results are compared with those obtained by fitting the same data set to a widely used empirical model existing in the literature. The general equation is found to represent all three experimental data sets better than the alternative model tested. In addition, a graphical method existing in enzyme kinetics is successfully adapted and further developed to determine the number of inhibition sites of a basic functional unit of a bacterial cell. (c) 1996 John Wiley & Sons, Inc.