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.     

Kinetics of potato starch fermentation by Schwanniomyces castellii

The growth behaviour of Schwanniomyces castellii in slurry fermentation systems using untreated potato starch as substrate was studied in order to asses the eventual effect of the initial concentration of substrate (S_o) on cell growth rate. By applying the elementary balance method in combination with a Monod-type kinetic equation it was possible to formulate not only an unstructured model, but also the stoichiometry for such a yeast fermentation process. From a kinetic viewpoint, the Monod model was found to be redundant with respect to the pseudo-first order one, it being impossible to discriminate the contribution of v _M and K _S on the overall fermentation kinetics. Whereas the main yield coefficients appeared to be independent of S _O, the pseudo-first order rate constant was found to be inversely proportional to S _O. Therefore, cell growth appears to be controlled by the initial amount of amylolytic enzymes, that is to some extent proportional to the inoculum size, instead of the initial concentration of potato starch, at least within the experimental range of 3 to 30 g dm³.     

PREDICTING THE KINETICS OF DISSOLVED INORGANIC CARBON LIMITED GROWTH FROM THE SHORT-TERM KINETICS OF PHOTOSYNTHESIS IN SYNECHOCOCCUS LEOPOLIENSIS (CYANOPHYTA)1

The blue-green alga (Cyanobacterium) Synechococcus leopoliensis (Racib.) Komarek was grown in dissolved inorganic carbon [DIC]-limited chemostats over the entire range of growth rates. At each growth rate, the kinetics of photosynthesis with respect to [DIC] and the maximal rate of photosynthesis (P_max) were determined. The half-saturation constant for [DIC]-limited photosynthesis (K_1/2^DIC) for cells growing below 1.7 d^?1 was constant (4.7 μM) whereas for growth rates between 1.7 d^?1 and 2.1 d^?1 (μ_max) the kinetics of photosynthesis were multiphasic with an apparent K_1/2^DIC between 1.5–2.0 mM. P_max increased in a linear fashion with growth rate for growth rates below 1.7 d^?1. No trend in P_max was apparent for growth rates greater than 1.7 d^?1. These kinetic parameters were used to predict a growth rate versus [DIC] relationship. Results show that the Monod relationship is a physiologically valid expression of growth as a function of [DIC] provided (K_1/2^DIC) remains constant. The major change in (K_1/2^DIC) as μ approaches μ_max results in the conclusion that two separate and distinct Monod equations must be used to describe growth as a function of DIC over the entire growth range. These results point to a major discontinuity in the μ vs. [DIC] curve at 1.7 d^?1 which corresponds to the change from high to low affinity photosynthetic kinetics. We believe these results account for the previously described deficiencies of the Monod equation in describing [DIC]-limited algal growth.     

Growth of Saccharomycopsis fibuligera in a continuous-stirred-tank fermentor

Growth rates and amylase production rates were determined for the yeast Saccharomycopsis fibuligera grown on a simulated potato processing waste in a continuous-stirred-tank fermentor. S. fibuligera formed large multicellular flocs in the fermentor, and cell growth was reduced at low dilution rates because of mass-transfer resistance. The average Thiele modulus, which is the measure of extent of substrate diffusion, had a value ranging from ?_av = 2.2 for D = 0.10 to 0.3 for D = 0.40. Growth rates were described by the Monod equation modified to include mass-transfer effects. This modified Monod equation was used to predict growth rates from measured floc-size distribution. Experimentally determined growth rates were in close agreement with these predicted values.     

Kinetic behavior of heterogeneous populations in completely mixed reactors

The kinetic behavior of heterogeneous microbial populations was studied in a continuous flow completely mixed reactor operated at various dilution rates. Glucose was used as the growth-limiting nutrient. The physiological growth parameters for cells harvested from continuous flow reactors were determined using batch experiments. It, was found that the growth parameters, maximum growth rate (μ_m), saturation constant (k_s), and cell yield (Y) vary for each dilution rate, and cannot be considered as precise constants in depicting the kinetic behavior of heterogeneous populations. In addition, it was found that the yield coefficients obtained from batch experiments were always lower than those obtained from continuous flow experiments. Levels of substrate and biological solids calculated for different dilution rates using growth constants from batch experiments did not agree with the experimental values observed in steady-state experiments. However, when the yield values from, the continuous flow experiments were used in conjunction with batch values for μ_m and k_s the theoretical and experimental dilute-out curves agreed fairly closely (within the range needed for engineering prediction) until the culture began to wash out of the unit. In general, the data substantiated the use of the single phase relationship between growth rate and substrate concentration described by the Monod equation, μ = μ_mS/(k_s + s).     

A kinetic model incorporating energy spilling for substrate removal in substrate-sufficient batch culture of activated sludge

Batch assays are currently used to study the kinetic behavior of microbial growth. However, it has been shown that the outcome of batch experiments is greatly influenced by the initial ratio of substrate concentration (S _o) to biomass concentration (X _o). Substrate-sufficient batch culture is known to have mechanisms of spilling energy that lead to significant nongrowth-associated substrate consumption, and the Monod equation is no longer appropriate. By incorporating substrate consumption associated with energy spilling into the balance of the substrate oxidation reaction, a kinetic model for the observed specific substrate consumption rate was developed for substrate-sufficient batch culture of activated sludge, and was further verified by experimental data. It was demonstrated that the specific substrate consumption rate increased with the increase of the S _o/X _o ratio, and the majority of substrate was consumed through energy spilling at high S _o/X _o ratios. It appears that the S _o/X _o ratio is a key parameter in regulating metabolic pathways of microorganisms. Received: 18 January 1999 / Received revision: 7 May 1999 / Accepted: 28 May 1999     

Analytical solution for a hybrid Logistic-Monod cell growth model in batch and continuous stirred tank reactor culture

Monod and Logistic growth models have been widely used as basic equations to describe cell growth in bioprocess engineering. In the case of the Monod equation, the specific growth rate is governed by a limiting nutrient, with the mathematical form similar to the Michaelis–Menten equation. In the case of the Logistic equation, the specific growth rate is determined by the carrying capacity of the system, which could be growth-inhibiting factors (i.e., toxic chemical accumulation) other than the nutrient level. Both equations have been found valuable to guide us build unstructured kinetic models to analyze the fermentation process and understand cell physiology. In this work, we present a hybrid Logistic-Monod growth model, which accounts for multiple growth-dependent factors including both the limiting nutrient and the carrying capacity of the system. Coupled with substrate consumption and yield coefficient, we present the analytical solutions for this hybrid Logistic-Monod model in both batch and continuous stirred tank reactor (CSTR) culture. Under high biomass yield (Y_x/s) conditions, the analytical solution for this hybrid model is approaching to the Logistic equation; under low biomass yield condition, the analytical solution for this hybrid model converges to the Monod equation. This hybrid Logistic-Monod equation represents the cell growth transition from substrate-limiting condition to growth-inhibiting condition, which could be adopted to accurately describe the multi-phases of cell growth and may facilitate kinetic model construction, bioprocess optimization, and scale-up in industrial biotechnology.     

Use of continuous-culture techniques for determining the growth kinetics of a celluloytic Thermoactinomyces sp.

In order to better understand the kinetics of cellulose degradation by Thermoactinomyces sp., continuous-culture experiments were performed utilizing the various intermediates of cellulose degradation as the feed substrates. Steady-state data from the glucose runs suggest that this organism has a growth yield of 0.42 g cell/g glucose, and a specific maintenance of 0.24 g glucose/g cell/hr. The Monod equation did not seen to model the growth well, since a plot of 1/D vs. 1/S gave a maximum specific growth rate that was even lower than one of the steady-state dilution rates. A dynamic washout experiment suggested a maximum specific specific growth rate of 0.36 hr^?1 and indicated that glucose is only slightly growth inhibitory as the inhibition constant, K_i, is 19 g glucose/liter. An equation for substrate concentration for washout conditions was derived. This equation predicted the transient glucose concentration relatively well. A fill-and-draw technique was investigated for determination of the growth parameters. It was not successful because of difficulties in contamination and accurately monitoring the dissolved oxygen in the small highly agitated vessel. However, the technique could be useful in studying the growth characteristics of sludge in a waste treatment system where contamination is not a worry. One could cover the medium surface and use a nonsterilizable dissolved oxygen probe of high sensitivity membrane to overcome these difficulties.     

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.     

Overview of some theoretical approaches for derivation of the Monod equation

The Monod equation has been widely applied to describe microbial growth, but it has no mechanistic basis and is purely empirical. Extensive efforts have been dedicated to develop theoretical approaches for derivation of the Monod equation, which can be classified into three major groups, i.e., kinetic, thermodynamic, and substance transport approaches. In this review, four representative approaches are thus discussed. Due to the fact that different assumptions are made in each approach, no universal physical meaning of the Monod constant (K (s)) can be revealed. However, it seems that the Monod constant would be free energy-dependent and have nonequilibrium thermodynamic characteristics.     

Methanogenic degradation kinetics of phenolic compounds in aquifer-derived microcosms

In this segment of a larger multidisciplinary study of the movement and fate of creosote derived compounds in a sand-and-gravel aquifer, we present evidence that the methanogenic degradation of the major biodegradable phenolic compounds and concomitant microbial growth in batch microcosms derived from contaminated aquifer material can be described using Monod kinetics. Substrate depletion and bacterial growth curves were fitted to the Monod equations using nonlinear regression analysis. The method of Marquardt was used for the determination of parameter values that best fit the experimental data by minimizing the residual sum of squares. The Monod kinetic constants ( _max , K _s, Y, and k _d) that describe phenol, 2-, 3-, and 4-methylphenol degradation and concomitant microbial growth were determined under conditions that were substantially different from those previously reported for microcosms cultured from sewage sludge. The K _s values obtained in this study are approximately two orders of magnitude lower than values obtained for the anaerobic degradation of phenol in digesting sewage sludge, indicating that the aquifer microorganisms have developed enzyme systems that are adapted to low nutrient conditions. The values for k _d are much less than _max, and can be neglected in the microcosms. The extremely low Y values, approximately 3 orders of magnitude lower than for the sewage sludge derived cultures, and the very low numbers of microorganisms in the aquifer derived microcosms suggest that these organisms use some unique strategies to survive in the subsurface environment.Abbreviations GC gas chromatography - HPLC high performance liquid chromatography - LBSSB likelihood based sum of squares boundaries - MPN most probable number - NLR nonlinear regression analysis - OFAG oxygen free Argon gas - PCP pentachlorophenol - RSS residual sum of squares - SRB sulfate reducing bacteria     

Effect of temperature on ethanol tolerance of thermotolerant Isshatchenkia orientalis IPE100

Tolerance to high temperature and ethanol is a major factor in high‐temperature bio‐ethanol fermentation. The inhibitory effect of exogenously added ethanol (0–100 g L^?1) on the growth of the newly isolated thermotolerant Issatchenkia orientalis IPE100 was evaluated at a range of temperatures (30–45°C). A generalized Monod equation with product inhibition was used to quantify ethanol tolerance, and it correlated well with the experimental data on microbial growth inhibition of ethanol at the temperatures of 30–45°C. The maximum inhibitory concentration of ethanol for growth (P_m) and toxic power (n) at the optimal growth temperature of 42°C were estimated to be 96.7 g L^?1 and 1.23, respectively. The recently isolated thermotolerant I. orientalis IPE100 shows therefore a strong potential for the development of future high‐temperature bio‐ethanol fermentation technologies. This study provides useful insights into our understanding of the temperature‐dependent inhibitory effects of ethanol on yeast growth.     

Kinetic behavior of mixed populations of activated sludge

The kinetic behavior of heterogeneous microbial populations of sewage origin was studied in a single-stage isothermal continuous flow completely mixed aeration tank. A series of experiments were carried out at various dilution rates using glucose as the growth limiting substrate. The steady-state behavior of the system was observed at each dilution rate and the results were found to fit fairly well with the steady-state equation bayed on the Monod model with an endogenous respiration term included, i.e., μ = μ_mS/(K_s + S) ? K_d. The growth kinetics of cells harvested at steady state for each dilution rate were studied using batch experiments. The multiple response data of the system as functions of time were used to estimate the parameter values in the above kinetic model. It was found that values of the growth parameters changed significantly and systematically with cell population. For example, values of μ_m were high at high dilution rates and low at low dilution rates. It was also found that only those batch growth parameters from cells obtained at fairly high dilution rates are comparable with those estimated by the results of steady-state operations. The results of this investigation suggest that (1) different cell populations pre dominated at different steady-state dilution rates, with high dilution rates resulting in predominantly fast-growing organisms and low dilution rates resulting in predominantly slow-growing cells, and (2) risk exists in any randomly picked batch experiment to predict the steady-state behavior of the system when heterogeneous microbial populations must be used.     

Kinetics of Growth and Amylase Production of Saccharomycopsis fibuligera on Potato Processing Wastewater

The kinetics of growth and amylase production of Saccharomycopsis fibuligera were studied in a chemostat on a synthetic potato processing blancher water. Dilution rates (D) from 0.101 to 0.480 h⁻¹ were examined. A mathematical model based on the Monod equation was developed. The yield of cell mass from carbohydrates was constant and equal to 0.84. The maximum specific growth rate and the Monod constant were determined to be 0.596 h⁻¹ and 0.226 mg/ml, respectively. An equation for the steady-state starch concentrations was empirically derived. The steady-state noncarbohydrate carbon levels rose linearly with D. Reducing sugars were the growth-limiting substrate, and their steady-state levels conformed to Monod kinetics. The yield of amylase from the cell mass (Y_z) declined as D rose and was described by the equation Y_z = (−8.005D + 4.076). The model predicted that the maximum production of cell mass should occur at D = 0.35 h⁻¹ and the maximum production of amylase should occur at D = 0.22 h⁻¹. The mathematical model presented agreed with the experimental results in its prediction of the steady-state level of reducing sugar, starch, cell mass, and amylase concentrations as well as the productivity of amylase.     

Evaluation of Various Unstructured Kinetic Models for the Production of Protease by Bacillus sphaericus MTTC511

Bacillus sphaericus MTCC511 was used for the production of protease in submerged batch fermentation. Maximum protease activity of 1010 U/L was obtained during a fermentation period of 24 h under optimized conditions of 30 °C in a medium with an initial pH of 7 and at a shaking rate of 120 rpm. The maximum biomass obtained in the batch fermentation was 2.55 g/L after 16 h. Various unstructured models were analyzed to simulate the experimental values of microbial growth, protease activity and substrate concentration. The unstructured models, i.e. the Monod model for microbial growth, the Monod incorporated Luedeking‐Piret model for the production of protease and the Monod‐incorporated modified Luedeking‐Piret model for the utilization of substrate were capable of predicting the fermentation profile with high coefficient of determination (R²) values of 0.9967, 0.9402 and 0.9729, respectively. The results indicated that the unstructured models were able to describe the fermentation kinetics more effectively.     

Kinetic analysis of the inhibitory effect of trichloroethylene (TCE) on nitrification in cometabolic degradation

In this study, the inhibitory effect of TCE on nitrification process was investigated with an enriched nitrifier culture. TCE was found to be a competitive inhibitor of ammonia oxidation and the inhibition constant (K _I ) was determined as 666–802 μg/l. The TCE affinity for the AMO enzyme was significantly higher than ammonium. The effect of TCE on ammonium utilization was evaluated with linearized plots of Monod equation (e.g., Lineweaver–Burk, Hanes–Woolf and Eadie–Hofstee plots) and non-linear least square regression (NLSR). No significant differences were found among these data evaluation methods in terms of kinetic parameters obtained.     

Equations of substrate-limited growth: the case for blackman kinetics

A simplified model of cell metabolism, consisting of a series of linked reversible enzymatic reactions dependent on the concentration of a single external substrate has been developed. The general mathematical solution for this system of reactions is presented. This general solution confirms the concept of a rate-limiting step, or “master reaction”, in biological systems as first proposed by Blackman. The maximum rate of such a process is determined by, and equal to, the maximum rate of the slowest forward reaction in the series. Of practical interest in modeling the growth rate of cells are three cases developed from the general model. The simplest special case results in the Monod equation when the maximum forward rate of one enzymatic reaction in the cell is much less than the maximum forward rate of any other enzymatic reactions. More realistic is the case where the maximum forward rates of more than one enzymatic reaction are slow. When two slow enzymatic reactions are separated from each other by any number of fast reactions that overall can be described by a large equilibrium constant, the Blackman form results: and A third case is that in which two slow enzymatic steps are separated by an equilibrium constant that is not large. Unlike the Monod and Blackman forms, which contain only two arbitrary constants, this model contains three arbitrary constants: The Monod and Blackman forms are special cases of this three constant form. In comparing equations with two arbitrary constants the Monod equation gave poorer fit of the data in most cases than the Blackman form. It is concluded that workers modeling the growth of microorganisms should give a t least as much consideration to the Blackman form as is given to the Monod equation.     

Fundamental aspects of waste sewage sludge treatment: Microbial solids biodegradation in an aerobic thermophilic semi-continuous system

The solubilization and biodegradation of whole microbial cells by an aerobic thermophilic microbial population was investigated over a 72 h period. Various parameters were followed including total suspended solids reduction, changes in the dissolved organic carbon, protein and carbohydrate concentrations, and carboxylic acid production and utilisation. From the rates of removal of the various fractions a simple model for the biodegradation processes is proposed and verified with respect to acetic acid production and utilization, and total suspended solids removal. The process is initiated by enzymic degradation of the substrate microbe cell walls followed by growth on the released soluble substrates at low dissolved oxygen concentration with concommitant carboxylic acid production. Subsequent utilization of the unbranched, lower molecular weight carboxylic acids allows additional energy supply following exhaustion of the easily utilisable soluble substrate from microbial cell hydrolysis.List of Symbols Y _Xp/Xs kg/kg yield process microbes on substrate yeast cells - Y _Xp/Ac kg/kg yield process microbes on acetate - Y _Ac/Ss kg/kg yield acetate produced by process microbes growing on substrate yeast cells - Y _Ss/Xs kg/kg yield soluble substrate from lysis of yeast cells - Y _Ss/Xp kg/kg yield soluble substrate from lysis of process microbes - Y _P/Xs kg/kg yield particulates from lysis of yeast cells - Y _P/Xp kg/kg yield particulates from lysis of process microbes - _{max (Ss)} h^–1 maximum specific growth rate constant for growth of process microbes on soluble substrate - _{max (Ac)} h^–1 maximum specific growth rate constant for growth of process microbes on acetate - Ks _Ss kg/m³ saturation coefficient for growth of process microbes on soluble substrate - Ks _Ac kg/m³ saturation coefficient for growth of process microbes on acetate - K _d h^–1 death/lysis rate constant for process microbes - K _i kg/m³ inhibition constant for growth of process microbes on acetate - K _L h^–1 lysis rate constant for whole yeast cells - K _h h^–1 hydrolysis rate constant for particulate biomass     

Studies on the relationship between specific growth rate and concentration of nitrogen source for heterogeneous microbial populations of sewage origin

Batch experiments were run using heterogeneous populations to determine whether a hyperbolic equation of the type suggested by Monod could be used to depict the relation between specific growth rate, μ, and NH₃-N concentration when ammonia N was the growth-limiting nutrient. The heterogeneous populations employed were developed from sewage seed grown on glucose at various levels of nitrogen and various dilution rates in completely mixed continuous flow reactors. It was found that the hyperbolic function could be used. Values of μ_m in the range of 0.4–0.7 hr^?1 were observed, and values of K_s, in general, ranged from 1.5 to 4.0 mg/l. Variation in the values of these growth “constants” did not follow any discernible pattern related to past growth history (i.e., COD:N ratio or dilution rate at which the cells were previously grown).