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.     

Re-interpretation of the logistic equation for batch microbial growth in relation to Monod kinetics

Aims:  To determine the underlying substrate utilization mechanism in the logistic equation for batch microbial growth by revealing the relationship between the logistic and Monod kinetics. Also, to determine the logistic rate constant in terms of Monod kinetic constants.
Methods and Results:  The logistic equation used to describe batch microbial growth was related to the Monod kinetics and found to be first-order in terms of the substrate and biomass concentrations. The logistic equation constant was also related to the Monod kinetic constants. Similarly, the substrate utilization kinetic equations were derived by using the logistic growth equation and related to the Monod kinetics.
Conclusion:  It is revaled that the logistic growth equation is a special form of the Monod growth kinetics when substrate limitation is first-order with respect to the substrate concentration. The logistic rate constant ( k ) is directly proportional to the maximum specific growth rate constant ( μ _m) and initial substrate concentration ( S ₀) and also inversely related to the saturation constant ( K _s).
Significance and Impact of the Study:  The semi-empirical logistic equation can be used instead of Monod kinetics at low substrate concentrations to describe batch microbial growth using the relationship between the logistic rate constant and the Monod kinetic constants.     

Updated Models for the Effects of Temperature,Initial pH,NaCl, and NaNO2 on the Aerobic and Anaerobic Growth of Listeria monocytogenes

Data from several studies were appended to aerobic and anaerobic data sets that had been previously used to develop response surface models describing the growth kinetics of Listeria monocytogenes (Buchanan and Phillips, 1990). The expanded data sets included 709 aerobic and 358 anaerobic growth curves fitted with the Gompertz equation, and representing 189 and 150 unique combinations of four variables (temperature, pH, NaCl, NaNO₂), respectively. Response surface models were developed for (1) the Gompertz B and M terms and (2) lag phase durations (LPD) and generation times (GT). In addition to modeling NaCl as a variable, a second set of response surface models was developed by substituting calculated water activity as a variable. A number of data transformations were evaluated in an attempt to better utilize no-growth data. Full quadratic models of the natural logarithm transformation of the data (no-growth data excluded) predicted values that fit the observed data well. The assignment of GT=50[emsp4 ]h and LPD=600[emsp4 ]h (the approximate maximum duration of experiments) for the variable combinations that did not support growth proved to be the most effective means of making use of the no-growth data. However, this approach did not offer any clear advantage over quadratic models where the no-growth data were excluded. Error matrices were developed for the LPD and GT models to provide 95 % confidence intervals. The agreement between observed and predicted growth kinetics was excellent considering the number and ranges of the variables encompassed in the models. The models provide reasonable predictions of the growth of L. monocytogenes in foods. The full quadratic models of LPD and GT without inclusion of the no-growth data were selected for inclusion in the USDA Pathogen Modeling Program, release 5.1.     

The use of automated tubidimetric data for the construction of kinetic models

Summary An automated tubidimetric instrument (Bioscreen) was used to observe the growth response ofListeria monocytogenes to combinations of temperature (15–30°C), hydrogen-ion (0.1–21.9 m) (equivalent pH 4.66–7.0) and NaCl concentration (0.5–9.5% w/v). Compared to traditional plate count techniques, the technique allowed many more data points to be captured and replicates to be used, with less expenditure of effort. Optical density curves were filtered (smoothed) to minimize the effect of signal noise and the mean signal from uninoculated wells was subtracted to minimize the effect of signal draft. A novel procedure for fitting growth curves to optical density data has been developed. The procedure involves the use of the logistic function and a calibration equation for fitting, in a single step, in the dimension of optical density. This approach allowed the four parameters of the logistic equation to be derived at each set of experimental conditions. A quadratic response surface was then fitted to the curve parameters using temperature, NaCl and hydrogen-ion concentration as three independent variables. Predicted time to 1000-fold increase in cell numbers compared well to predictions from predictive microbial growth equations generated in other laboratories using traditional plate counting. We propose that this technique should be further evaluated as a method for generating data for modeling the kinetics of microbial growth.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Kinetics of substrate inhibition of exponential yeast growth

This work concerns mathematical modeling of the rate of microbial growth on inhibitory levels of nutrients as affected by pH, concentration of the nutrients, temperature, cultivation method, and method of data analysis. Candida utilis (ATCC 9226) was grown with sodium acetate as growth-limiting carbon and energy source in mineral salts medium in shake flask and continuous cultures to study inhibition by excess acetate. Differential shake flask cultures were grown at low yeast concentrations at temperatures (T) of 25 and 30°C, pH's between 5.5 and 7.0, and acetate concentrations (S) between 0.25 and 3.0% (w/v). Growth data were exponential with correlation coefficients greater than 0.995 in 49 of 56 experiments; the lowest correlation coefficient was 0.986. Specific growth rates (μ) determined by graphical methods showed only fair correlation with those determined by regression analysis. Both sets of specific growth rate data were grouped at constant T and pH and fitted to the three parameter equation, The improvement in use of the fitted equation instead of the mean value was significant with greater than 70% confidence in all (14 groups) and 90% confidence in only half of the data groups; the correlation does not improve with the increasing acetate inhibition at lower pH. Both defects in the model and insufficient data at each pH are responsible. A modified six parameters with hydrogen ion concentration(H⁺) as follows: Specific growth rates calculated with the six parameter equation matched observed values in all groups of isothermal data better than the means with greater than 99% confidence. The six parameter model adequately represents effects of acetate and hydrogen ion concentrations under constant or slowly changing environmental conditions and balanced growth; although better models probably exist. Thus steady-stste and transient continuous culture experiments agreed with many published growth yields, but specific growth rates could only be predicted qualitatively from the model fit to the shake flask data. The data and present models could be incorporated into published models for transient growth at low nutrient concentrations to correlate and perhaps predict microbial growth kinetics over a much wider range of growth conditions than now possible.     

Batch kinetics of microbial polysaccharide biosynthesis

A modified form of logistic equation has been proposed to quantity the batch kinetics of microbial growth during the biosynthesis of extra- and intracellular polymers. Based on the experimental data developed in this study, the proposed model appeared to provide adequate growth and fermentation kinetics of Aureobasidium pullulans. The model was also applicable for representing the reported data on pullulan, xanthan, and poly-beta-hydroxybutyricacid. In comparison to the logistic and Monod kinetics, this model fitted the data better and more accurately described the overall fermentation, both concentrations and fermentation time.     

Derivation of a growth rate equation describing microbial surface colonization

A surface growth rate equation is derived which describes simultaneous growth and attachment during microbial surface colonization. The equation simplifies determination of attachment and growth rate, and does not require a computer program for solution. This rate equation gives the specific growth rate (Μ) as a function of the number of cells on the surface (N), the incubation period (t), and the number of colonies (C_i) containing either one cell, two cells, four cells, etc, as shown below. $$\mu = \frac{{\ln (\frac{N}{{C_i }} + 1)}}{t}$$ The attachment rate (A) is given by the following relationship: $$A = \mu C_i $$ The proposed colonization kinetics are compared with exponential growth kinetics using 3-dimensional computer plots. Colonization kinetics diverged most from exponential kinetics when the growth rate was low or the attachment rate was high. Using these kinetics, it is possible to isolate the effects of growth and attachment on microbial surface colonization.     

Biodegradation kinetics of peptone and 2,6-dihydroxybenzoic acid by acclimated dual microbial culture

This study evaluated the kinetics of simultaneous biodegradation of peptone mixture and 2,6-dihydroxybenzoic acid (2,6-DHBA) by an acclimated dual microbial culture under aerobic conditions. A laboratory-scale sequencing batch reactor was sustained at steady-state with peptone mixture feeding. During the study period, peptone mixture feeding was continuously supplemented with 2,6-DHBA. Related experimental data were derived from three sets of parallel batch reactors, the first fed with the peptone mixture, the second with 2,6-DHBA and the third one with the two substrates, after acclimation of microbial culture and simultaneous biodegradation of both organics. A mechanistic model was developed for this purpose including the necessary model components and process kinetics for the model calibration of relevant experimental data. Model evaluation provided all biodegradation characteristics and kinetics for both peptone mixture and 2,6-DHBA. It also supported the development of a dual microbial community through acclimation, with the selective growth of a second group of microorganisms specifically capable of metabolizing 2,6-DHBA as an organic carbon source.     

Growth kinetics for shelf-life prediction: Theory and practice

Summary Predictive microbiology can be used to determine and predict the shelf-life of perishable foods under commercial distribution conditions based on microbial growth kinetics. This paper presents general microbial growth kinetics with the Monod model and the Gompertz function. Additional models are given to describe effects of food composition (e. g.a _w) and environmental conditions (e.g. temperature, gas atmosphere) as well as their interaction on the growth kinetic parameters (lag time and specific growth rate). These models can be used to predict the time to reach a critical level under any constant conditions within the range tested. A combination of microbial kinetics with an engineering accumulation approach can be used to predict the final microbial level in a food, or the loss of shelf-life, for any known time-temperature sequence, if there is no history effect or the history effect is negligible. A time-temperature indicator, could be used for predicting the remaining shelf-life of perishable foods under any distribution condition based on microbial growth kinetics.Mention of brand or firm names does not constitute an endorsement by the US Department of Agriculture over others of a similar nature not mentioned.     

Prediction of methane yield at optimum pH for anaerobic digestion of organic fraction of municipal solid waste

A concept of methane yield at optimum pH was advanced and subsequently a mathematical model that simulates the optimal pH of a batch process for anaerobic digestion of organic fraction of municipal solid waste (MSW) was developed and validated. The model was developed on the basis of the microbial growth kinetics and was divided into three processes: hydrolysis of substrates by hydrolytic bacteria, consumption of soluble substrate by acidogenic bacteria, and finally consumption of acetate and methane generated by methanogenic bacteria. Material balance and liquid phase equilibrium chemistry were used in this study. A series of experiments were conducted to validate the model. The model simulation results agreed reasonably with experimental data in different temperatures and total solid (TS) concentrations under uncontrolled pH. A computer circulation program was used to predict the optimal pH in different conditions. Experiments in different temperatures and TS were run under optimal pH which predicted by the model. The model was succeeded in increasing the methane production and the cumulative methane production had an average increment about 35% in optimal pH of different temperatures and TS.     

Modeling Microbially Induced Carbon Degradation in Redox-Stratified Subsurface Environments: Concepts and Open Questions

The degradation of organic matter, including organic contaminants, in subsurface environments is controlled by the abundances and functional capabilities of the resident microorganisms. As a consequence, modeling approaches simulating the fate of organics and related changes in redox conditions have to account for the effects of microbial activity on the degradation kinetics, as well as for the spatial and temporal distributions of the chemical species (e.g., terminal electron acceptors, nutrients or toxic substances) that control microbial activity. The present paper reviews the principal modeling approaches that are used to simulate the degradation of organic matter in water-saturated porous media. Special attention is devoted to modeling the bioavailability of chemical substrates of microbial reactions, and the sequential occurrence of terminal electron accepting pathways. While the various model approaches found in the literature are capable of reproducing field data sets from various environmental settings, they are rarely compared in terms of performance and predictive ability. Most approaches incorporate simplifications or empirical rate laws, which limit their range of application. Thus, there remains a need for further development of more general, process-based modeling concepts to represent microbially mediated reactive processes.     

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.     

Kinetics and mathematical modeling of homoacetic fermentation of lactate by Clostridium formicoaceticum

Fermentation kinetics of Clostridium formicoaceticum grown on lactate at pH 7.0 and 35 degrees C was studied. Acetate was the only fermentation product and its production was growth associated. The growth of this bacterium was insensitive to the lactate concentrations studied, but was inhibited by acetic acid. A Monod-type expression with product inhibition similar to the noncompetitive inhibition of enzyme kinetics was used to model the batch fermentation. An integrated equation was developed and used to help estimating the kinetic parameters in the model. This mathematical model can be used to simulate the homoacetic fermentation of lactate by C. formicoaceticum at pH 7.0 and 35 degrees C.     

Establishing equivalence for microbial-growth-inhibitory effects ("iso-hurdle rules") by analyzing disparate listeria monocytogenes data with a gamma-type predictive model

Preservative factors act as hurdles against microorganisms by inhibiting their growth; these are essential control measures for particular food-borne pathogens. Different combinations of hurdles can be quantified and compared to each other in terms of their inhibitory effect ("iso-hurdle"). We present here a methodology for establishing microbial iso-hurdle rules in three steps: (i) developing a predictive model based on existing but disparate data sets, (ii) building an experimental design focused on the iso-hurdles using the model output, and (iii) validating the model and the iso-hurdle rules with new data. The methodology is illustrated with Listeria monocytogenes. Existing data from industry, a public database, and the literature were collected and analyzed, after which a total of 650 growth rates were retained. A gamma-type model was developed for the factors temperature, pH, a(w), and acetic, lactic, and sorbic acids. Three iso-hurdle rules were assessed (40 logcount curves generated): salt replacement by addition of organic acids, sorbic acid replacement by addition of acetic and lactic acid, and sorbic acid replacement by addition of lactic/acetic acid and salt. For the three rules, the growth rates were equivalent in the whole experimental domain (γ from 0.1 to 0.5). The lag times were also equivalent in the case of mild inhibitory conditions (γ ≥ 0.2), while they were longer in the presence of salt than acids under stress conditions (γ < 0.2). This methodology allows an assessment of the equivalence of inhibitory effects without intensive data generation; it could be applied to develop milder formulations which guarantee microbial safety and stability.     

Short prokaryotic DNA fragment binning using a hierarchical classifier based on linear discriminant analysis and principal component analysis

Metagenomics is an emerging field in which the power of genomic analysis is applied to an entire microbial community, bypassing the need to isolate and culture individual microbial species. Assembling of metagenomic DNA fragments is very much like the overlap-layout-consensus procedure for assembling isolated genomes, but is augmented by an additional binning step to differentiate scaffolds, contigs and unassembled reads into various taxonomic groups. In this paper, we employed n-mer oligonucleotide frequencies as the features and developed a hierarchical classifier (PCAHIER) for binning short (≤ 1,000 bps) metagenomic fragments. The principal component analysis was used to reduce the high dimensionality of the feature space. The hierarchical classifier consists of four layers of local classifiers that are implemented based on the linear discriminant analysis. These local classifiers are responsible for binning prokaryotic DNA fragments into superkingdoms, of the same superkingdom into phyla, of the same phylum into genera, and of the same genus into species, respectively. We evaluated the performance of the PCAHIER by using our own simulated data sets as well as the widely used simHC synthetic metagenome data set from the IMG/M system. The effectiveness of the PCAHIER was demonstrated through comparisons against a non-hierarchical classifier, and two existing binning algorithms (TETRA and Phylopythia).     

Protein crystal growth. Growth kinetics for tetragonal lysozyme crystals

A method for immobilizing protein crystals has been devised for determining face growth rates, and used to investigate the growth kinetics of hen egg white lysozyme crystals. Growth rates were determined at 22 degrees C in 0.1 M sodium acetate, 5% NaCl, pH 4.0, on the visually identified (110) face of tetragonal lysozyme crystals. Protein concentrations ranged from 13 to 57 mg/ml (saturation concentration = 1.7 mg/ml). Growth rate data were fit to the equation R = kappa sigma ri, where R = rate in cm/s; kappa = constant; sigma i = solute growth interface supersaturation; and r = rate dependence upon super-saturation, with the result that kappa = 0.146 X 10(-8) cm/s and r = 2.0. A model of the growth process was developed and the experimental data were used to determine the relative roles of transport and interfacial kinetics in the growth of this crystal. Values for the width of the boundary layer delta, the interfacial concentration Ci, and growth rate R were determined. The model may be used to extrapolate to other growth conditions. The relative role of transport and interfacial kinetics can be expressed by the coefficient gamma = (CB - Ci)/(CB - Cs), when CB is the bulk concentration and Cs the saturation. Values for gamma were found to range from much less than 0.1 for submicron-size crystals to approximately 0.15 for cm sizes. The results indicate that attachment or surface effects are rate-limiting in lysozyme crystal growth in Earth's gravity because solutal convection always provides more transport of solute than can be accommodated by the interface. In order to grow such crystals under transport limiting conditions, it would be necessary to suppress this solutal convection.     

The estimation of growth yield and maintenance parameters for photoautotrophic growth

Methods are presented for examining the consistency of experimental data for microbial growth where light energy is converted to chemical energy through photosynthesis. True growth yield and maintenance parameters are estimated for several sets of available experimental data. Methods of parameter estimation are presented which allow all of the measured variables to be used simultaneously for parameter estimation. The results show that a wide range of values have been found for the true growth yield and maintenance parameters. Values of the true growth yield range from 0.04 to values above those predicted by the Z-scheme model for photosynthesis.     

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.     

乳酸链球菌肽发酵动力学的研究

提出了在恒定不同pH的发酵条件下,乳酸链球菌SM526的菌体生长、底物消耗、乳酸及Nisin产生的动力学模型。菌体生长、乳酸及Nisin产生用逻辑方程描述,而底物消耗是菌体生长和乳酸产生速率的函数。模型表明,乳酸链球菌SM526菌体生长和乳酸产生的最佳pH为7．0,而Nisin产生的最佳pH却为6．5。     

A decision support system for prediction of microbial spoilage in foods

Summary A method is developed to combine qualitative and quantitative information for the prediction of growth of microorganisms in foods. pH, water activity, temperature and oxygen availability of foods are coupled to growth characteristics of microorganisms. For that purpose, a database with characteristics of foods and a database of kinetic parameters of microorganisms are built. The first database has a tree structure, based on physical similarity of food products. This structure makes it possible to estimate information about a food product which is not listed by comparison with similar products at the same level of the tree or the level above. A method is developed to make an estimation of the microbial growth kinetics on the basis of models. This is done by introducing a growth factor, which can be calculated on the basis of readily available data from literature. Finally, qualitative knowledge is added. Since any bit of information can be changed, the system will give better predictions when more and more accurate information is added.