Hazard analysis applied to microbial growth in foods: Development of mathematical models describing the effect of water activity

Mathematical models have been developed which describe the effect of lowering the water activity on the growth kinetics of Staphylococcus aureus and Salmonella typhimurium. By treating the lag phase and exponential phase kinetics separately predictions can be made on the extent of microbial growth over successive time-temperature cycles. Staph. aureus was far more tolerant than Salm. typhimurium to lowered water activity and under near growth limiting conditions of water activity and temperature was showing lag periods as long as ca 40 d. The maximum lag period observed for Salm. typhimurium was ca 5 d. Under these conditions the predicted generation times for Staph. aureus were 2–3 d and for Salm. typhimurium 1–4 d.     

Comparison of several mathematical models for describing the joint effect of temperature and ph on glucanex activity

The aim of the present work was to evaluate with different statistical criteria the suitability of nine equations for describing and optimizing the simultaneous effect of temperature and pH on glucanex activity using two characteristic polysaccharides (curdlan and laminarin) as substrates. The most satisfactory solutions were found with an empirical equation constituted with parameters of practical interest (Rosso model), and a hybrid model between the Arrhenius equation and the mathematical expression generated by the protonation-hydroxylation mechanism (Tijskens model). The joint optimal values of pH and temperature calculated with the Rosso model were obtained at 4.64 and 50°C with curdlan and 4.64 and 48°C using laminarin as substrate.     

Two mathematical models for the development of a single microbial pellet

A mathematical model for pellet development of filamentous microorganisms is presented, which simulates in detail location and growth of single hyphal elements. The basic model for growth, septation and branching of discrete hyphae is adopted from Yang et al. [2, 23]. Exact solutions to the intracellular mass-balance equations of a growth-limiting key component is given for two types of either branched or unbranched cellular compartments. Furthermore, the growth model was extended in regard to the external mass-balance equations of limiting substrates (oxygen, glucose) under the assumption that the substrates can enter the denser regions of the pellet only diffusively. Penetration of the substrates into the more porous outer regions of the pellet occurs more easily due to microeddies in the surrounding fluid. Chipping of hyphae from the pellet surface by shear forces was included in the model as well. The application of shear forces leads to a marked smoothing of the simulated pellet surface. The development of pellets from spore germination up to late stages with cell-lysis due to shortage of substrates in the pellet centre can be described. The effects of various model parameters are discussed.List of Symbols A _i algebraic coefficient (i = 1, 2,..., 6) - B _i algebraic coefficient (i = 1, 2,..., 6) - C _i mass-concentration of component i (i = O₂, S) (gl^–1) - C _i,crit concentration of substance i critical for lysis (i=O₂, S) (gl^–1) - C _i,stop concentration of substance i below which cells are inactivated (gl^–1) - C(l _i,t) intracellular concentration of the key component at site l _i and time t (gl^–1) - C _m maximal intracellular concentration of the key component (gl^–1) - C _X Concentration of dry biomass (gl^–1) - D intracellular diffusion coefficient of the key component (m² h^–1) - D _max,i maximal molecular diffusion coefficient of substrate i (i = O₂, S) (m² h^–1) - D _eff,i effective diffusion coefficient of component i (i = O₂) (m² h^–1) - d _h cross-sectional diameter of hyphae (m) - k production coefficient for the key component (h^–1) - K _s Monod coefficient for glucose (gl^–1) - k ₀ Monod coefficient for oxygen (gl^–1) - L _c total length of a compartment (m) - L _i total length of branch i (i=1, 2, 3) (m) - l _i position on branch i (i=1, 2, 3) - L _m maximal length of a segment (m) - m _i maintenance coefficient of substrate i (h^–1) - N _m maximal number of segments in a compartment - n _iR number of tips of type i in layer R, i=1, 2 - p auxiliary variable (see Eq. (7)) - P _Br probability that a hypha is chipped off (%h^–1) - pO ₂ partial pressure of oxygen in the liquid phase (%) - Q auxiliary variable (see Eq. (8)) - Q _i uptake rate of substrate i (i = O₂, S) (gl^–1 h^–1) - q auxiliary variable (see Eq. (7)) - R index of radial layer (R=1, 2, 3,..., R _max) - r radius (m) - r _crit critical radius, Eq. (15) (m) - r _max pellet radius (m) - r _tip distance from the pellet centre to the tip position (m) - r _thr threshold radius (m) - s auxiliary variable (see Eq. (7)) - S index for glucose - t time (h) - v _R volume of layer R (1) - Y _Mi observable yield coefficient of biomass on substrate i (gg^–1) - Y _Xi yield coefficient of biomass on substrate i (gg^–1) Greek Letters _i actual tip expansion rate (m h^–1) - _i,m actual maximal extension rate of tip i (i=1, 2) (m h^–1) - _1y lysis rate (h^–1) - _m maximal tip extension rate (m h^–1) - auxiliary variable in Eq. (2) - auxiliary variable in Eq. (3) - auxiliary variable defined in Eq. (4) (m^–1) - _shear shear force parameter - _R overall specific growth rate in layer R (h^–1) - _m maximal specific hyphal growth rate (h^–1) - cell volume density (l cell volume per 1) - _crit critical cell volume density in Eq. (15) - _S shear force parameter - _X cell mass density (g dry weight per 1 wet cells) - (C _i) growth kinetics on substrate i - proportional factor in Eq. (34) (l g^–1) We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.We thank the Deutsche Forschungsgemeinschaft (DFG) for financially supporting parts of this work.     

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.     

Development of mathematical models (Logistic, Gompertz and Richards models) describing the growth pattern of Pseudomonas putida (NICM 2174)

Bacterial growth curve, which is asymptotic after a certain period, is described using three different mathematical models, namely, Logistic model, Gompertz model and Richards model. The equations for these three models are fitted by evaluating the mathematical parameters involved in these models. This is done by applying the method of partial sums to the data in Table 1 containing the optical density values for different cell mass at different time intervals. The sum of square of residuals between the expected optical density values and the experimental values is calculated for each of these models. In the cases tested, the Logistic model was found to be the best fit for the growth curve of Pseudomonas putida (NICM 2174) and was found to be easy to use. These results fit the data very well at 5% level for more than 70% of the readings.     

Role of microbial enzymes in flavor development in foods

There are four main sources of enzymes in foods—these being the inherent enzymes, enzymes from microbial contaminants, enzymes elaborated by microorganisms added to foods, and specific enzymes added to foods. This study primarily deals with the latter two sources of enzymes in food. Although both plants and animals serve as sources of enzymes, they are not as economical or versatile sources as are enzymes obtained from microorganisms. In the meat industry, proteases are used to tenderize muscle and to obtain flavor precursors. In the preparation of cured meat products such as sausages, lipases, and proteases from bacterial cultures are utilized. Similarly, proteases and lipases are used in the dairy industry to develop flavor compounds. Proteases and amylases also have applications in the baking and milling industries where they are used to produce precursors for the nonenzymatic browning reactions. Carbohydrases such as amylase, amyloglucosidase, and glucose isomerase have found usage in the starch and syrup industry for the production of high dextrose and high fructose syrups. Other enzymes such as glucose oxidase, pectinase, and naringinase are of value to the wine and fruit juice industries. A better understanding of the mode of action of enzymes as well as the mechanisms of development of flavor compounds will further enhance the use of microbial enzymes to develop specific and desired flavors in foods.     

A mathematical model applied to the fungal colony growth of Sclerotium rolfsii

Colonies of Sclerotium rolfsii growing on solid medium showed a linear increase in radius simultaneously with exponential changes in colonial weight. The mathematical model developed by Koch (Koch, A.L. 1975. J. Gen. Microbiol. 89, 209216) to describe similar kinetics of mycelial growth was applied. Most of the parameters defined in the mentioned model were directly measured or calculated, except the values of K (maximum mycelium carrying capacity) and S (initial mycelial density) which were fitted to the experimental data of growth at 30°C and 23°C. The ratios K/S at 30°C and at 23°C were 18.8 and 1200 respectively, being consistent with two-dimensional growth. The model was predictive only during the exponential increase of the colony weight.     

温度与昆虫生长发育关系模型的发展与应用

昆虫作为变温动物,对温度变化更为敏感。研究温度变化对昆虫生长、发育的影响有重要理论和实践意义。目前已构建了多个描述温度与昆虫增长速率的关系模型,用于解释温度对昆虫发育速率的影响。这些模型大体可分为两类:没有热动力学基础的纯描述性模型和有热动力学基础的应用性模型。本文在对现有的有关温度变化与昆虫生长发育关系的11个模型进行评述的基础上,结合作者近年来的研究,重点介绍了迄今为止国际上最为合理的、用以反映温度对昆虫发育速率影响的Sharpe-Schoolfield-Ikemoto模型,并利用这些模型拟合了一组温发育速率数据用以展示这些模型的应用。     

Segmented linear regression models applied to the analysis of data from a cross-sectional growth experiment

Several different methods of analysis are applied to data consisting of weight measurements, taken at specified post-treatment times, of harvested thyroids from rats given one of four treatments. Previous studies of this type of data indicated that the growth is initially rapid, and that a second phase of less rapid growth is followed by a final phase in which little additional growth occurs. The data are further characterized by increasing variance through time. The primary purpose of the analysis is to study the effect of the treatments at the end of the study period. One-way analysis of variance tests among groups are performed on each day, but the results are not particularly helpful. However, results from two-way analyses of variance (over subsets of days and groups) are consistent with the three phase model and accordingly indicate significant group differences during each. Finally, maximum likelihood methods are used to fit a three part segmented linear regression model.     

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.     

Evaluation of mathematical models to describe testicular growth in Blackbelly ram lambs

The primary objective was to compare various mathematical models to describe scrotal circumference (SC) and paired testis volume development in Blackbelly ram lambs. The study was conducted in the state of Querétaro, México (20° 43' N, 100° 15' W). Spring-born Blackbelly ram lambs (n = 41) were housed outdoors and fed alfalfa hay and concentrate. Body weight, SC, and testis length, diameter, and volume were recorded every 2 wk from 24 to 172 d of age (June 18 to November 3). The following mathematical functions were used to model SC-age and testis volume-age relationship: Von Bertalanffy, Brody, Gompertz, Logistic, and Richards. The suitability of the models was evaluated based on parameter values and standard errors, residual mean square, the coefficient of determination (R²), and the average prediction error (APE). All models, except for Brody's, had good fit to SC (R² > 0.98) and testis volume (R² > 0.95), and produced similar growth curves in the range of ages studied. The logistic model predicted SC at maturity quite well, 33.6 ± 0.6 cm as compared with 33.9 ± 0.5 cm observed in adult animals; all models had APE's smaller than ±7% between 56 and 168 d of age. The Bertalanffy model predicted testis volume at maturity quite well, 513 ± 22 cm³ as compared with 488 ± 20 cm³ calculated for adult animals. The logistic model had a good fit to testis volume during the period of study, but underestimated the volume at maturity by 28%. All models, except for Brody's, had APE's smaller than ±14% between 98 and 168 d of age. The logistic and Bertalanffy models predicted the inflection point for SC at 83 and 59 d of age, and testis volume at 116 and 109 d of age, respectively. In conclusion, all models, except for Brody's, had good fit to actual SC and testis volume data in the range of age evaluated, whereas the logistic and Bertalanffy's models made the best predictions for adult SC and testis volume, respectively.     

Three-dimensional models of chick embryo somites obtained by mathematical methods applied to ultrastructural data

Summary A model of a thoracolumbar somite of a chick embryo at the 53rd incubation hour was obtained by mathematical methods, after identification of somite cell types by means of electron microscopy.Each specific district occupied by the cell types was precisely determined.On the basis of these observations, the somite was three-dimensionally reconstructed and the spatial positions of the primitive myotome, dermatome, sclerotome, undifferentiated mesoderm and myocele were precisely identified.     

A predictive model for combined temperature and water activity on microbial growth during the growth phase

An empirical and generalized model is presented, based on a modified Arrhenius equation, for predicting the combined effect of temperature and water activity on the growth rate of bacteria. When it was applied to seven separate sets of wide ranging published results, spanning some 50 years and including a spore-former and a silage micro-organism, predictions explained between 92.9 and 99.0% of the variation in the results with an overall mean of 96.6%. Advantages over existing models are that it is relatively easy to fit to data using least squares regression and requires only five coefficients. These, together with its simplicity and demonstrated wide application, will facilitate its practical use.     

A predictive model for combined temperature and water activity on microbial growth during the growth phase

An empirical and generalized model is presented, based on a modified Arrhenius equation, for predicting the combined effect of temperature and water activity on the growth rate of bacteria. When it was applied to seven separate sets of wide ranging published results, spanning some 50 years and including a spore-former and a silage micro-organism, predictions explained between 92.9 and 99.0% of the variation in the results with an overall mean of 96.6%. Advantages over existing models are that it is relatively easy to fit to data using least squares regression and requires only five coefficients. These, together with its simplicity and demonstrated wide application, will facilitate its practical use.