Improvement of new logistic model for bacterial growth

Quantifying Geobacter sulfurreducens growth: A mathematical model based on acetate concentration as an oxidizing substrate

We developed a mathematical model to simulate dynamics associated with the proliferation of Geobacter and ultimately optimize cellular operation by analyzing the interaction of its components. The model comprises two segments: an initial part comprising a logistic form and a subsequent segment that incorporates acetate oxidation as a saturation term for the microbial nutrient medium. Given that four parameters can be obtained by minimizing the square root of the mean square error between experimental Geobacter growth and the mathematical model, the model underscores the importance of incorporating nonlinear terms. The determined parameter values closely align with experimental data, providing insights into the mechanisms that govern Geobacter proliferation. Furthermore, the model has been transformed into a scaleless equation with only two parameters to simplify the exploration of qualitative properties. This allowed us to conduct stability analysis of the fixed point and construct a co-dimension two bifurcation diagram.

Electromagnetic fields effects on microbial growth in cocoa fermentation: A controlled experimental approach using established growth models

Understanding the effects of electromagnetic fields is crucial in the fermentation of cocoa beans, since through precise control of fermentation conditions the sensory and nutritional properties of cocoa beans could be improved. This study aimed to evaluate the effect of oscillating magnetic fields (OMF) on the kinetic growth of the core microbial communities of the Collections Castro Naranjal (CCN 51) cocoa bean. The data was obtained by three different models: Gompertz, Baranyi, and Logistic. The cocoa beans were subjected to different OMF strengths ranging from 0 mT to 80 mT for 1 h using the Helmholtz coil electromagnetic device. The viable microbial populations of lactic acid bacteria (LAB), acetic acid bacteria (AAB), and yeast (Y) were quantified using the colony-forming unit (CFU) counting method. The logistic model appropriately described the growth of LAB and Y under magnetic field exposure. Whereas the Baranyi model was suitable for describing AAB growth. The microbial populations in cocoa beans exposed to magnetic fields showed lower (maximum specific growth rate (μmax), values than untreated controls, with AAB exhibiting the highest average growth rate value at 5 mT and Y having the lowest average maximum growth rate value at 80 mT. The lower maximum specific growth rates and longer lag phases when exposed to magnetic fields compared to controls demonstrate the influence of magnetic fields on microbial growth kinetics.

A compressed logistic equation bacteria growth: Inferring time-dependent growth rate

We propose a compressed logistic model for bacterial growth by invoking a time-dependent rate instead of the intrinsic growth rate (constant), which was adopted in traditional logistic models. The new model may have a better physiological basis than the traditional ones, and it replicates experimental observations, such as the case example for E. coli, Salmonella, and Staphylococcus aureus. Stochastic colonial growth at a different rate may have a fractal-like nature, which should be an origin of the time-dependent reaction rate. The present model, from a stochastic viewpoint, is approximated as a Gaussian time evolution of bacteria (error function).

A parametric logistic equation with light flux and medium concentration for cultivation planning of microalgae

Microalgae are considered to be promising producers of bioactive chemicals, feeds and fuels from carbon dioxide by photosynthesis. Thus, the prediction of microalgal growth profiles is important for the planning of cost-effective and sustainable cultivation–harvest cycles. This paper proposes a mathematical model capable of predicting the effect of light flux into culture and medium concentration on the growth profiles of microalgae by incorporating these growth-limiting factors into a logistic equation. The specific form of the equation is derived based on the experimentally measured growth profiles of Monoraphidium sp., a microalgal strain isolated by the authors, under 16 conditions consisting of combinations of incident light fluxes into culture and initial medium concentrations. Using a cross-validation method, it is shown that the proposed model has the ability to predict necessary incident light flux into culture and initial medium concentration for harvesting target biomass at a target time. Finally, model-guided cultivation planning is performed and is evaluated by comparing the result with experimental data.

Optimization of immunoglobulin substitution therapy by a stochastic immune response model

Background

The immune system is a complex adaptive system of cells and molecules that are interwoven in a highly organized communication network. Primary immune deficiencies are disorders in which essential parts of the immune system are absent or do not function according to plan. X-linked agammaglobulinemia is a B-lymphocyte maturation disorder in which the production of immunoglobulin is prohibited by a genetic defect. Patients have to be put on life-long immunoglobulin substitution therapy in order to prevent recurrent and persistent opportunistic infections.

Methodology

We formulate an immune response model in terms of stochastic differential equations and perform a systematic analysis of empirical therapy protocols that differ in the treatment frequency. The model accounts for the immunoglobulin reduction by natural degradation and by antigenic consumption, as well as for the periodic immunoglobulin replenishment that gives rise to an inhomogeneous distribution of immunoglobulin specificities in the shape space. Results are obtained from computer simulations and from analytical calculations within the framework of the Fokker-Planck formalism, which enables us to derive closed expressions for undetermined model parameters such as the infection clearance rate.

Conclusions

We find that the critical value of the clearance rate, below which a chronic infection develops, is strongly dependent on the strength of fluctuations in the administered immunoglobulin dose per treatment and is an increasing function of the treatment frequency. The comparative analysis of therapy protocols with regard to the treatment frequency yields quantitative predictions of therapeutic relevance, where the choice of the optimal treatment frequency reveals a conflict of competing interests: In order to diminish immunomodulatory effects and to make good economic sense, therapeutic immunoglobulin levels should be kept close to physiological levels, implying high treatment frequencies. However, clearing infections without additional medication is more reliably achieved by substitution therapies with low treatment frequencies. Our immune response model predicts that the compromise solution of immunoglobulin substitution therapy has a treatment frequency in the range from one infusion per week to one infusion per two weeks.

Model comparison for Escherichia coli growth in pouched food

We recently studied the growth characteristics of Escherichia coli cells in pouched mashed potatoes (Fujikawa et al., J. Food Hyg. Soc. Japan, 47, 95-98 (2006)). Using those experimental data, in the present study, we compared a logistic model newly developed by us with the modified Gompertz and the Baranyi models, which are used as growth models worldwide. Bacterial growth curves at constant temperatures in the range of 12 to 34 degrees C were successfully described with the new logistic model, as well as with the other models. The Baranyi gave the least error in cell number and our model gave the least error in the rate constant and the lag period. For dynamic temperature, our model successfully predicted the bacterial growth, whereas the Baranyi model considerably overestimated it. Also, there was a discrepancy between the growth curves described with the differential equations of the Baranyi model and those obtained with DMfit, a software program for Baranyi model fitting. These results indicate that the new logistic model can be used to predict bacterial growth in pouched food.

Characteristics and modeling of Escherichia coli growth in pouched food

Characteristics of the growth kinetics of Escherichia coli cells in pouched mashed potatoes under various conditions were studied with a mathematical model. Bacterial cells were inoculated in sterile mashed potatoes and then sealed in vinyl pouches, in which a very small amount of air was included. The growth curves of cells in the pouched mashed potatoes at constant temperature (12-34 degrees C) were sigmoidal with time on a semi-logarithmic plot and were successfully described with a new logistic model recently developed by us. The rate constant of growth showed a highly linear relationship to the temperature with the square-root model, and the lag period was longer at lower temperatures. The growth curve in glass tubes containing a large volume of air was similar to that in pouches, showing that the rate of growth was not affected by the volume of the surrounding air. The growth curves in pouched mashed potatoes were very similar to those in nutrient broth or on the surface of nutrient agar, which we previously reported. These results suggested that the growth kinetics of the bacterial cells under various conditions of rich nutrition might be almost identical, and can be described with a simple growth model like ours.

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 degrees 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.

Development of a Predictive Program for Microbial Growth under Various Temperature Conditions

A predictive program for microbial growth under various temperature conditions was developed with a mathematical model. The model was a new logistic model recently developed by us. The program predicts Escherichia coli growth in broth, Staphylococcus aureus growth and its enterotoxin production in milk, and Vibrio parahaemolyticus growth in broth at various temperature patterns. The program, which was built with Microsoft Excel (Visual Basic Application), is user-friendly; users can easily input the temperature history of a test food and obtain the prediction instantly on the computer screen. The predicted growth and toxin production can be important indices to determine whether a food is microbiologically safe or not. This program should be a useful tool to confirm the microbial safety of commercial foods.