Modelling the growth response of Staphylococcus xylosus to changes in temperature and glycerol concentration/water activity

The growth response of Staphylococcus xylosus strain CM21/3 to changes in temperature and water activity (glycerol concentration) was similar to that observed when water activity was adjusted by added NaCl. At each water activity level the effect of temperature on bacterial growth rate was described well by the square root model. T _MIN (the notional minimum temperature for growth) was found to be constant and was similar to the value obtained for the same organism grown in media containing NaCl. Growth rate was proportional to glycerol concentration/water activity allowing the combined effect of this factor and temperature to be modelled by substitution of the constant b in the basic square root model by a term for water activity. The observed minimum water activity for growth at the optimum temperature was close to that predicted by the model.     

Modelling the growth response of Staphylococcus xylosus to changes in temperature and glycerol concentration/water activity

The growth response of Staphylococcus xylosus strain CM21/3 to changes in temperature and water activity (glycerol concentration) was similar to that observed when water activity was adjusted by added NaCl. At each water activity level the effect of temperature on bacterial growth rate was described well by the square root model. TMIN (the notional minimum temperature for growth) was found to be constant and was similar to the value obtained for the same organism grown in media containing NaCl. Growth rate was proportional to glycerol concentration/water activity allowing the combined effect of this factor and temperature to be modelled by substitution of the constant b in the basic square root model by a term for water activity. The observed minimum water activity for growth at the optimum temperature was close to that predicted by the model.     

The effect of pH, acidulant and temperature on the survival of Yersinia enterocolitica

The survival of Yersinia enterocolitica at sub-optimal temperatures (0–23°C) and growth inhibitory pH values, achieved using a range of acidulants, was investigated. At a given pH, survival was greater the lower the temperature. Sulphuric and citric acids had lower bactericidal activity than acetic and lactic acids and in nearly all cases where the four acids could be compared at the same pH the order of bactericidal activity was acetic > lactic > citric > sulphuric. Attempts to model this behaviour by a negative square root relationship gave good correlation coefficients for plots of the square root of death rate against temperature at different combinations of pH and acidulant but so too did several other functions of death rate. The high coefficient of variation for T ₀ determined from square root plots prevented construction of a combined temperature/pH model similar to that already described for growth.     

Comparison of a quadratic response surface model and a square root model for predicting the growth rate of Yersinia enterocolitic

Polynomial equations, relating the growth rate of Yersinia enterocolitica to temperature (0–25°C) and pH (4.5–6-5) in a liquid medium were constructed for four different acidulants. The logarithm of the time for a 100-fold increase in bacterial numbers could be represented by a quadratic response surface function of pH and temperature. The interactions between pH and temperature on growth rate were found to be additive. Values for a 2 log cycle increase in growth derived from the model were in good agreement with experimental values. Predictions from the quadratic model and from a square root model were compared with experimental values in laboratory media and UHT milk. The mean square error (MSE) for the quadratic response surface model was smaller than that for the square root model in 81% of cases. In UHT milk the square root model increasingly underestimated growth rate, as the temperature decreased and would 'fail dangerous' if used for predictive purposes. This indicated that the response surface model is more reliable for predicting the growth of Y. enterocolitica under conditions of sub-optimal temperature and pH.     

Model for combined effect of temperature and salt concentration/water activity on the growth rate of Staphylococcus xylosus

The combined effect of temperature and NaCl concentration/water activity on the growth rate of a strain of halotolerant Staphylococcus is described by the square-root models which had been used previously to model temperature dependence only. The model square root r = b(T-T min) is shown to be a special case of the B?lehrádek temperature function which is given by r = a(T-alpha)d. The constant alpha is the socalled 'biological zero' and equivalent to T min in the square-root models. This and the exponent d = 2 were unaffected by changing NaCl concentration/water activity. The B?lehrádek-type equations are preferable to the Arrhenius equation in that their parameters do not change with temperature. The constancy of T min allows derivation of a simple expression relating growth rate of strain CM21/3 to temperature and salt concentration/water activity within the range of linear response to temperature predicted by the square-root model.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Utility of phenomenological models for describing temperature dependence of bacterial growth.

We compared three unstructured mathematical models, the master reaction, the square root, and the damage/repair models, for describing the relationship between temperature and the specific growth rates of bacteria. The models were evaluated on the basis of several criteria: applicability, ease of use, simple interpretation of model parameters, problem-free determination of model parameters, statistical evaluation of goodness of fit (chi 2 test), and biological relevance. Best-fit parameters for the master reaction model could be obtained by using two consecutive nonlinear least-square fits. The damage/repair model proved to be unsuited for the data sets considered and was judged markedly overparameterized. The square root model allowed nonproblematical parameter estimation by a nonlinear least-square procedure and, together with the master reaction model, was able to describe the temperature dependence of the specific growth rates of Klebsiella pneumoniae NCIB 418, Escherichia coli NC3, Bacillus sp. strain NCIB 12522, and the thermotolerant coccobacillus strain NA17. The square root and master reaction models were judged to be equally valid and superior to the damage/repair model, even though the square root model is devoid of a conceptual basis.     

Effect of altering the root-zone temperature on growth, translocation, carbon exchange rate, and leaf starch accumulation in the tomato

Tomato seedlings (Lycopersicon esculentum Mill. cv Vendor) were grown hydroponically with their root systems maintained at a constant temperature for a 2-week period commencing with the appearance of the first true leaf. Based on fresh and dry weight and leaf area, the optimal root-zone temperature for seedling growth was 30°C. The carbon exchange rate of the leaves was also found to increase with rising root-zone temperature up to 30°C. However, a more complex relationship seems to exist between root-zone temperature and the accumulation of ¹⁴C-labeled assimilates in the roots; inasmuch as there is no enhancement in this accumulation at the most growth promoting root-zone temperatures (22-30°C).     

Modeling Surface Growth of Escherichia coli on Agar Plates

Surface growth of Escherichia coli cells on a membrane filter placed on a nutrient agar plate under various conditions was studied with a mathematical model. The surface growth of bacterial cells showed a sigmoidal curve with time on a semilogarithmic plot. To describe it, a new logistic model that we presented earlier (H.Fujikawa et al., Food Microbiol. 21:501-509, 2004) was modified. Growth curves at various constant temperatures (10 to 34°C) were successfully described with the modified model (model III). Model III gave better predictions of the rate constant of growth and the lag period than a modified Gompertz model and the Baranyi model. Using the parameter values of model III at the constant temperatures, surface growth at various temperatures was successfully predicted. Surface growth curves at various initial cell numbers were also sigmoidal and converged to the same maximum cell numbers at the stationary phase. Surface growth curves at various nutrient levels were also sigmoidal. The maximum cell number and the rate of growth were lower as the nutrient level decreased. The surface growth curve was the same as that in a liquid, except for the large curvature at the deceleration period. These curves were also well described with model III. The pattern of increase in the ATP content of cells grown on a surface was sigmoidal, similar to that for cell growth. We discovered several characteristics of the surface growth of bacterial cells under various growth conditions and examined the applicability of our model to describe these growth curves.     

Comparison of temperature effects on soil respiration and bacterial and fungal growth rates

Temperature is an important factor regulating microbial activity and shaping the soil microbial community. Little is known, however, on how temperature affects the most important groups of the soil microorganisms, the bacteria and the fungi, in situ. We have therefore measured the instantaneous total activity (respiration rate), bacterial activity (growth rate as thymidine incorporation rate) and fungal activity (growth rate as acetate-in-ergosterol incorporation rate) in soil at different temperatures (0-45 degrees C). Two soils were compared: one was an agricultural soil low in organic matter and with high pH, and the other was a forest humus soil with high organic matter content and low pH. Fungal and bacterial growth rates had optimum temperatures around 25-30 degrees C, while at higher temperatures lower values were found. This decrease was more drastic for fungi than for bacteria, resulting in an increase in the ratio of bacterial to fungal growth rate at higher temperatures. A tendency towards the opposite effect was observed at low temperatures, indicating that fungi were more adapted to low-temperature conditions than bacteria. The temperature dependence of all three activities was well modelled by the square root (Ratkowsky) model below the optimum temperature for fungal and bacterial growth. The respiration rate increased over almost the whole temperature range, showing the highest value at around 45 degrees C. Thus, at temperatures above 30 degrees C there was an uncoupling between the instantaneous respiration rate and bacterial and fungal activity. At these high temperatures, the respiration rate closely followed the Arrhenius temperature relationship.     

Rate constants of primary reversible reactions of electron transfer do not depend on temperature. Numerical experiment

The process of irreversible photochemical charge separation in photosynthetic bacterial reaction centers is proposed to be characterized by the effective rate constant. A formula to compute this effective rate constant is derived. Similar rate constant was previously considered by R.A. Marcus (Marcus R.A. 1956. J. Chem. Phys. 24, 966–978) in order to describe nonphotochemical intermolecular electron transfer. The effective rate constant of the irreversible charge separation in photosynthetic bacterial reaction centers is shown to depend on the temperature. In contrast, rate constants of the forward electron transfer from the excited singlet primary donor to the bacteriochlorophyll and from its ion-radical to the bacteriopheophytin acceptor do not depend on temperature.     

Carbohydrate Level and Growth of Tomato Plants: II. The Effect of Irradiance and Temperature

The growth response of tomato (Lycopersicon esculentum L.) to temperature and irradiance may be related to carbohydrate concentration. Plants in the exponential phase of vegetative growth were grown under temperatures ranging from 9 to 36°C and under low or high irradiances of approximately 110 or 370 microeinsteins per square meter per second photosynthetically active radiation for a 12 hour photoperiod. The relative growth rate, leaf area ratio, net assimilation rate and whole plant carbohydrate levels were measured. At high irradiance, relative growth rate was 43% faster and total nonstructural carbohydrate concentration was 41% greater than at low irradiance. The change in carbohydrate with irradiance could explain the growth response. Plant growth was fastest at 25°C and decreased parabolically at lower and higher temperatures with a half-maximal rate at 13 and 36°C. Total nonstructural carbohydrate decreased between 13 and 23°C and remained constant at higher temperatures. Soluble sugar concentrations varied little with temperature above 13°C except for sucrose, whose level rose above 30°C. The change in carbohydrate with temperature could not explain the growth response. Above 23°C tomato plants appeared to regulate growth rate to maintain a relatively constant nonstructural carbohydrate concentration.     

Studies on the origin of bacterial viruses. V. The effect of temperature on the terramycin-resistant and phage-producing cells of Bacillus megatherium cultures

The growth rates, the mutation frequency rate constants of the terramycin-resistant cells, the burst size of the phage-producing cells, and the ratio of phage to cells all have a temperature coefficient of about 2 from 20 to 35° (µ = 9 x 10³ calories), with a maximum at 40°. The mutation frequency rate constant (or time rate constant) of the phage-producing cells increases from 20 to 45° with a temperature coefficient of about 3 (µ = 2 to 3 x 10⁴ cal.). The change in the values for the growth rate, mutation rate, and cell volume occurs in less than 1 hour, after the temperature is changed. The value for the burst size of phage-producing cells changes for 3 to 4 hours. Prolonged growth of megatherium 899 at 48 to 50° results in the production of C + S phage, in place of T. Returning the culture to 25° results in the production of small T phage.     

Denitrification and a numerical modelling approach for shallow waters

A numerical model (`DiffDeni') has been developed to describe the disappearance of nitrate from the water column of 10–200 cm deep waters. The disappearance is caused by bacterial denitrification in the sediments. The model employs the molecular diffusion constant, an acceleration factor describing eddy diffusion, and three bacterial growth constants, viz. the inoculum size, the maximum growth rate and the half saturation constant for the hyperbolic process. The values of these system-constants were varied over a wide range. The curves obtained were compared with the curves for well-defined situations, viz. in which diffusion takes place without any or with a complete, immediate reaction. These cases have analytical solutions, and were simulated closely by the model `DiffDeni', though this model is based on different assumptions. It is shown that, when the bacterial growth rate is above a critical value, a negative exponential curve describes the nitrate disappearance well. On the other hand, a more complicated negative exponential equation can be used to describe the first phase of this denitrification in which bacterial activity is low and nitrate behaves as a conservative compound. The change-over period from phase 1 (no reaction) to phase 2 (complete, immediate reaction) which may vary between <1 and 50 days cannot be described analytically (mathematically correctly). The influence of temperature on denitrification is assessed and it is shown that both bacterial activity and diffusion may influence the denitrification rate.     

Effect of temperature on nitrogenase functioning in cowpea nodules

Nitrogenase (EC 1.7.99.2) activity of a cowpea (Vigna unguiculata (L.) Walp cv Caloona) symbiosis formed with a Rhizobium strain (176A27) lacking uptake hydrogenase and maintained under conditions of a 12-hour day at an air temperature of 30°C (800-1000 microeinsteins per square meter per second) and a 12-hour night at an air temperature of 20°C showed a marked diurnal variation in ratio of nitrogen fixed to hydrogen evolved. As little as 0.3 micromole nitrogen was fixed per micromole hydrogen evolved in the photoperiod versus up to 0.6 in the dark period. In plants maintained under the same diurnal illumination regime but at constant (day and night) air temperature (30°C), this difference was abolished and a relatively constant ratio of nitrogen fixed to hydrogen evolved (around 0.3 micromole per micromole) was observed day and night. Exposure of nodulated roots to a range of temperatures maintained for 2 hours in a single photoperiod indicated that, whereas hydrogen evolution increased with increasing temperature from 15°C to a maximum around 35°C, nitrogen fixation was largely unaffected over this temperature range. Both functions of the enzyme declined sharply at temperatures above 38°C. A similar general response of nitrogen fixation to root temperature was observed in glasshouse-grown, sand-cultured plants maintained under a range of temperatures (from 15 to 35°C) for a 14-day period in mid vegetative growth. The effect of temperature on the proportion of electrons allocated to proton reduction compared with nitrogen reduction showed a linearly increasing relationship (correlation coefficient = 0.96) between 15°C and 47°C.     

Physiology of Root-Associated Nitrogenase Activity in Oryza sativa

An intact method for measuring immediately linear rates of acetylene reduction was used to investigate the relationship between temperature, pH, O₂ concentration, and light intensity with the rate of root-associated nitrogenase activity in rice (Oryza sativa L.). Nitrogenase activity varied over a temperature range of 10 to 50°C and optimal rates of acetylene reduction were recorded at 35°C. Nitrogenase activity was also influenced by the pH of the liquid surrounding the roots prior to assay. Maximal rates of acetylene reduction were recorded over a pH range from 5.8 to 7.5. Nitrogenase activity was significantly reduced by concentrations of O₂ 0.5% (v/v) or more when the intact plant assay method was used, and no optimum was detected. However, when the plant tops were removed and the cut ends sealed from the atmosphere for 4 hours, acetylene reduction rates were maximal at 0.25% O₂ (v/v). When plants were moved from sunlight (1,400 microeinsteins per square meter per second) to shade (9.6) root-associated nitrogenase activity at 35° C significantly decreased 15 min later to one-fourth the rate and recovered upon return to sunlight. When the light intensity reaching the leaf canopy was progressively reduced from 1,050 to 54 microeinsteins per square meter per second the rate of root-associated nitrogenase activity decreased from 550 ± 135 to 192 ± 55 nanomoles ethylene per gram dry root per hour. The study suggests that the rate of root-associated nitrogenase activity in rice at constant temperature may well be mediated by variations in the concentration of O₂ resulting from changes in the rate of photosynthesis as well as variations in the rate of transport of photosynthate.     

Combined effect of temperature and salt concentration/water activity on the growth rate of Halobacterium spp.

Growth responses of the halophilic bacteria, Halobacterium sp. strain HB9 and Halobacterium salinarium strain CM42/12, to temperature and water activity/sodium chloride concentration were described by the square root model and T _min (the theoretical minimum temperature for growth) was fixed. Little change in growth rate was observed in response to added NaCl at water activities below which cell lysis was avoided. Hence, growth of halobacteria on products such as salted, dried fish at water activities below 0.85 may be based on the square root temperature response without the need to incorporate a water activity term.     

A New Secondary Model Developed for the Growth Rate of <Emphasis Type="Italic">Escherichia coli</Emphasis> O157:H7 in Broth

This study was attempted to develop a new exponential sum model to describe the effect of temperature on growth rate (GR) of Escherichia coli O157:H7 in broth. The growth rates of E. coli O157:H7 at different storage temperatures (4, 10, 15, 20, 25, 30, and 35°C) estimated by fitting with the modified Gompertz model were used to develop secondary models such as square root model, Ratkowsky model and exponential sum model. Measures of coefficient of determination (R ²), root mean square error (RMSE) and the sum of squares due to error (SSE) were employed to compare the performances of these three secondary models. Based on these criteria, the developed exponential sum model showed the better goodness-of-fit and performance.     

THE RELATIONSHIP BETWEEN BACTERIAL GROWTH AND PHAGE PRODUCTION

1. The effects of temperature and H-ion concentration on the reaction between antistaphylococcus phage and a susceptible staphylococcus have been studied. 2. The temperature optimum for phage production is in the neighborhood of 35°C. and that for bacterial growth is approximately 40°C. 3. With increasing H-ion concentrations there occur: (a) an increase in the lag phase of bacterial growth without any corresponding increase in the lag phase of phage production; (b) a diminution in the total bacterial population accumulating in the medium without any corresponding drop in the total amount of phage formed. 4. With increasing alkalinity there is no pronounced change in the curves of bacterial growth and phage formation. At pH 8.5 the lytic threshold is increased to about 1000 phage units per bacterium instead of 100–140 as is usually the case and the time of lysis is delayed. 5. By adjusting the medium to pH 6 and 28°C. bacterial growth can be completely inhibited while phage production continues at a rapid rate. 6. Apparently, the previously stressed importance of bacterial growth as the prime conditioning factor for phage formation does not hold, for under certain experimental conditions the two mechanisms can be dissociated.