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.     

Relationship between substrate concentration,growth rate,and respiration rate of Escherichia coli in continuous culture

Summary Using a continuous flow technique the relationship between growth rate and substrate concentration was investigated with glucose as the limiting factor of a culture of Escherichia coli. Graphical and numerical analysis of the experimental data demonstrated that the application of the Michaelis-Menten equation produced erroneous results, whereas, the constants obtained from the Teissier equation were in agreement with the experimental data. On this basis, new equations defining the steady state cell and substrate concentration in continuous flow cultures were developed and tested against experimental data.Comparison of the specific growth rates, substrate uptake rates and oxygen consumption rates demonstrated that all were directly proportional to each other and could be related to each other by mathematical equations. Specifically it was shown that as the growth rate increased from 0.06 to k _m =0.76 the substrate uptake rate increased from 134 to 1420 mg glucose per gram cell weight per hour and the oxygen consumption rate increased from 48.6 to 505 mg O₂ per gram cell weight per hour. Independent of the growth rate 37% of the carbohydrate consumed were oxidized. The yield factor varied from 0.44 at low growth rates to 0.54 at high growth rates. Analysis of the growth rate-substrate uptake rate relationship indicated that a minimum substrate uptake rate of 55 mg glucose per gram cell weight per hour existed below which cell reproduction would cease. This was supported by the fact that steady state conditions could not be maintained in the culture at D values below 0.02 when the substrate supply rate decreased below 45 mg glucose per gram cell weight per hour.Material contained in this paper was submitted as a thesis in partial fulfillment of the requirements for the Ph. D. degree of Dr. R. S. Lipe.     

Influence of body weight and temperature on growth rates of Arctic charr, Salvelinus alpinus (L.)

When reared for a period of 6 months at a temperature of 10°C Arctic charr, Salvelinus alpinus , increased in weight from 18 g to approximately 135 g. Specific growth rates decreased as the fish increased in size and the relationship between size and growth rate could be described by the equation:
where G _w is specific growth rate and W is fish weight in grams. Temperature effects upon growth were examined using previously published data. Below the optimum growth temperature, the growth rate of a fish of given size could be predicted using the equation:
where T is the rearing temperature.
Rates of growth of Arctic charr were as high as those reported for other salmonid species reared under similar conditions. Preliminary results suggested that growth rates of charr may be lower in salt water than in fresh water.     

A mechanism of microbial cell growth

Presently empirical expressions, especially the Monod equation, are used to quantitatively relate microbial growth rate to limiting substrate concentration in the solution. In this paper microbial growth is postulated to occur by a mechanism involving a mass transfer or assimilation process. The assimilation process is assumed to be substrate mass transfer limited and hence proportional to the limiting substrate concentration. The ingestion is assumed independent of limiting substrate concentration and only dependent upon internal reaction rates. The quantitative relationship between limiting substrate and microbial growth rate resulting from this mechanism is developed. Under certain limiting conditions this expression is shown to reduce to the Monod equation and under other conditions it reduces to the Lotka-Volterra relationship. This mechanism is applied to batch and continuous cultures and the results obtained are compared quantitatively with experiment.     

A theoretical derivation of the Contois equation for kinetic modeling of the microbial degradation of insoluble substrates

Although developed as an empirical model to describe microbial growth on soluble substrates, the Contois equation has been widely accepted for kinetic modeling of insoluble substrate degradation. Yet, the mechanistic basis underlining these successful applications remains unanswered. Unlike soluble substrates that mainly cultivate suspended cultures, microbes cultivated on insoluble substrates have the populations that attach to the substrate surface or remain suspended in the bulk solution, while those attached usually grow faster than those suspended due to their proximity to food resources. This imbalanced growth provides a plausible explanation to the inverse relationship between microbial concentration and their specific growth rate as conveyed in the Contois equation. Based on a theoretical derivation, this study revealed that the Contois equation holds true only when attached microbes substantially obstruct the access of food to their suspended counterparts. On the other hand, when plentiful insoluble substrate surfaces are exposed for cell attachment, the Contois equation will be reduced back to the classic Monod equation.     

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.     

Cancer chemotherapy: Optimal control using the Verhulst-Pearl equation

General (deterministic) ordinary differential equations for the representation of cancer growth are presented when the growth is perturbed due to the action of a chemotherapeutic agent. The Verhulst-Pearl equation is introduced as a particular example of a growth equation applicable to human tumors. An optimal control problem with general performance criterion and state equation is formulated and shown to possess a novel feedback control relationship. This relationship is used in two continuous drug delivery problems involving the Verhulst-Pearl equation.     

Are thermal constants constant? A test using two species of ladybird

There is a controversy about whether the thermal constants, lower developmental threshold, rate of development and corresponding degree days required for development, change when a species is reared under different developmental conditions. We present a more precise way of measuring these constants using the linear relationship between the rate of development and temperature. First we use the equation proposed by Ikemoto and Takai (2000) to determine the linear phase of development and then a generalised linear model having a different variance at low and high temperatures, specific for each condition, to estimate the parameters of the linear relationship. Using this method, we show that providing the difference in food quality is sufficiently great, an aphidophagous ladybird develops significantly faster and starts developing at a significantly lower temperature on a good than on a poor quality diet. Adaptive significance of the thermal constants not remaining constant is discussed in terms of a trade-off between growth and rate of development, when temperature and food quality varies.     

Levels and rates of synthesis of ribosomal ribonucleic acid, transfer ribonucleic acid, and protein in Neurospora crassa in different steady states of growth.

The levels of ribosomes, tRNA molecules, and total protein per genome in Neurospora mycelia have been determined in eight different conditions of exponential growth. By increasing the rate of growth the number of ribosomes per genome increases dramatically while the level of total protein remains almost unchanged and the level of tRNA increases only slightly. The rates of synthesis of each of the macromolecules have been estimated. Increasing the rate of growth (mu) up to 0.5, the ratio between the rates of synthesis of tRNA and rRNA decreases reaching a constant value. The equations that best describe the dependence of the rate of synthesis of the macromolecules on the rate of growth (mu) have been determined. The rate of rRNA synthesis (rr), expressed as nucleotides polymerized, min- minus 1 per genome, is given by the equation: rr equals 6.51 times 10-7 mu-2-19. The rate of protein synthesis (rp), expressed as amino acids polymerized, min- minus 1 per genome is given by the following relationship: rp equals -1.43 times 10-7 + 3.43 times 10-8 mu. The equation describing the tRNA synthesis (rt) expressed as nucleotides, min- minus 1 per genome is rt equals 6.45 times 10-5 times exp 2.30 mu; however, more accurate determinations appear to be required for a firmer assignment of this latter equation. The significance of these equations for the studies on the regulation of rRNA and protein synthesis is discussed. For instance the rate of rRNA synthesis may set the limit for the maximal growth rate attainable by a cell, as the maximal rate of rRNA synthesis that may take place in a given cell is limited by the degree of redundancy of the rRNA genes.     

Growth analysis of soil-grown spinach plants at different N-regimes

Spinach plants were grown in pots under controlled conditions in three different soils (a loamy sand, a silt loam at low mineral-N level and a silt loam at the double mineral-N level). The nitrogen uptake pattern varied considerably between the three soil types and was used to validate an equation between the relative growth rate and nitrogen content. This equation is based on the growth response of spinach plants grown hydroponically at equal environmental conditions either at optimum nitrogen supply (complete nutrient solution) or with a relative nitrate addition rate of 0.30 day^–1, 0.225 day^–1 or 0.15 day^–1 effecting an exponential increase in nitrogen uptake. Growth in potted soil was slightly overestimated. Part of this bias was explained by the lower shoot weight ratio observed for the soil grown plants. This was demonstrated by the improvement in growth predictions when using net assimilation rate rather than relative growth rate as the driving variable in the model.     

On the distribution of state values of reproducing cells

Characterizing a cell state by measuring the degree of gene expression as well as its noise has gathered much attention. The distribution of such state values (e.g., abundances of some proteins) over cells has been measured, and is not only a result of intracellular process, but is also influenced by the growth in cell number that depends on the state. By incorporating the growth-death process into the standard Fokker-Planck equation, a nonlinear temporal evolution equation of distribution is derived and then solved by means of eigenfunction expansions. This general formalism is applied to the linear relaxation case. First, when the growth rate of a cell increases linearly with the state value x, the shift of the average x due to the growth effect is shown to be proportional to the variance of x and the relaxation time, similar to the biological fluctuation-response relationship. Second, when there is a threshold value of x for growth, the existence of a critical growth rate, represented again by the variance and the relaxation time, is demonstrated. The relevance of the results to the analysis of biological data on the distribution of cell states, as obtained for example by flow cytometry, is discussed.     

Effect of Nutrient Concentration on the Growth of Escherichia coli

The relationship between specific growth rate of Escherichia coli and the concentration of limiting nutrient (glucose or phosphate or tryptophan) has been determined for populations in a steady state. At high concentrations the specific growth rate is independent of the concentration of nutrient, but at low concentrations the specific growth rate is a strong function of the nutrient concentration. Such a relationship was predicted by Monod; however, Monod's equation does not predict the relationship over the entire range of nutrient concentration. If parameters of the equation are estimated from the results obtained at low concentrations, then at high concentrations of nutrient, the specific growth rate is significantly higher than that predicted by Monod's equation. These results were interpreted on the basis that the rate of growth is controlled by at least two parallel reactions and that the affinities of the enzymes catalyzing these reactions are different. The relationship between specific growth rate and mean cell volume was also measured, and the results indicate that mean cell volume depends not only on the specific growth rate but also on the nature of the limiting nutrient. There are different mean cell volumes at the same specific growth rate established by different limiting nutrients. Therefore, the mean cell volume is not uniquely determined by the specific growth rate.     

High CO2 levels reduce ethylene production in kiwifruit

We examined the roles of turgor potential and osmotic adjustment in plant growth by comparing the growth of spring wheat ( Triticum aestivum cv. Siete cerrors) and sudangrass ( Sorghum vulgare var. Piper) seedlings in response to soil water and temperature stresses. The rates of leaf area expansion, leaf water potential and osmotic potential were measured at combinations of 5 soil water potentials ranging from −0.03 to −0.25 MPa and 6 soil temperatures ranging from 14 to 36°C. Spring wheat exhibited little osmotic adjustment while sudangrass exhibited a high degree of osmotic adjustment. However, the rate of leaf area growth for sudangrass was more sensitive to water stress than that of spring wheat. These results were used to evaluate the relationship between growth and turgor potential. The modified Arrhenius equation based on thermodynamic considerations of the growth process was evaluated. This equation obtains growth rate as a function of activation energy, enthalpy difference between active and inactive states of enzymes, base growth rate and optimum temperature. Analyses indicate that the modified Arrhenius equation is consistent with the Lockhart equation with a metabolically controlled cell wall extensibility.     

Control of Growth Rate by Initial Substrate Concentration at Values Below Maximum Rate

The hyperbolic relationship between specific growth rate, mu, and substrate concentration, proposed by Monod and used since as the basis for the theory of steady-state growth in continuous-flow systems, was tested experimentally in batch cultures. Use of a Flavobacterium sp. exhibiting a high saturation constant for growth in glucose minimal medium allowed direct measurement of growth rate and substrate concentration throughout the growth cycle in medium containing a rate-limiting initial concentration of glucose. Specific growth rates were also measured for a wide range of initial glucose concentrations. A plot of specific growth rate versus initial substrate concentration was found to fit the hyperbolic equation. However, the instantaneous relationship between specific growth rate and substrate concentration during growth, which is stated by the equation, was not observed. Well defined exponential growth phases were developed at initial substrate concentrations below that required for support of the maximum exponential growth rate and a constant doubling time was maintained until 50% of the substrate had been used. It is suggested that the external substrate concentration initially present "sets" the specific growth rate by establishing a steady-state internal concentration of substrate, possibly through control of the number of permeation sites.     

STEADY-STATE LUXURY CONSUMPTION AND THE CONCEPT OF OPTIMUM NUTRIENT RATIOS: A STUDY WITH PHOSPHATE AND NITRATE LIMITED SELENASTRUM MINUTUM (CHLOROPHYTA)1

Selenastrum minutum (Naeg.) Collins was grown over a wide range of growth rates under phosphate or nitrate limitation with non-limiting nutrients added to great excess. This resulted in saturated luxury consumption. The relationships between growth rate and cell quota for the limiting nutrients were well described by the Droop relationship. The observed variability in N cell quota under N limitation as reflected in k_Q·Q_max^?1*, was similar in magnitude to previously reported values but k_Q·Q_max^?1* for P under P limitation was greater than previously reported for other species. These results were evaluated in light of the optimum ratio hypothesis. Our findings support previous work suggesting that the use of a single optimum ratio (k_Qi·K_Qj^?1) is inappropriate for dealing with a species growing under steady-state nutrient limitation. Under these conditions the optimum ratio should be viewed as a growth rate dependent variable. Two approaches for testing the growth rate dependency of optimum ratios are proposed. The capacity for luxury consumption differed between nutrients and was growth rate dependent. At low growth rates, the coefficient of luxury consumption (R_sat) for P was ca. four times that for N. The set of all possible relationships between N and P cell quota under these conditions was reported and these values were then used to establish the cellular N:P niche boundaries for S. minutum. Cell quotas of non-limiting nutrients were not described by the Droop equation. Analysis showed that as the cellular N:P ratio deviates from the optimum ratio, the ability of the Droop equation to describe the relationship between growth rate and non-limiting cell quotas decreases. When non-limiting nutrient cell quotas are saturated, the Droop equation appears to be invalid. Previously reported patterns of non-limiting nutrient utilization are summarized in support of this conclusion. The physiological and ecological consequences of luxury consumption and growth rate dependent optimum ratios are considered.     

Transport-limited growth in the chemostat and its competitive inhibition; A theoretical treatment

Summary An equation expressing the specific growth rate of heterotrophic cell populations in terms of yield factor and transport rate is proposed. From this equation expressions are derived for the specific growth rate when the transport of the energy source is growth0limiting. These expressions are applied to cell population growth in the chemostat limited by the transport of the energy source or of other substrates and simple mathematical tools are provided for obtaining estimates of the transport parameters. An equation is derived which predicts that at constant dilution rate in the chemostat the concentration of any substrate (whether or not the source of energy) the transport of which is growth limiting, is a linear function of the concentration of a competitive inhibitor of its transport. With this equation estimates of the Michaelis constants of competitive transport inhibitors can be obtained. The growth rate equation of Monod (1942) is discussed.     

Population growth rate and its determinants: an overview

We argue that population growth rate is the key unifying variable linking the various facets of population ecology. The importance of population growth rate lies partly in its central role in forecasting future population trends; indeed if the form of density dependence were constant and known, then the future population dynamics could to some degree be predicted. We argue that population growth rate is also central to our understanding of environmental stress: environmental stressors should be defined as factors which when first applied to a population reduce population growth rate. The joint action of such stressors determines an organism's ecological niche, which should be defined as the set of environmental conditions where population growth rate is greater than zero (where population growth rate = r = log(e)(N(t+1)/N(t))). While environmental stressors have negative effects on population growth rate, the same is true of population density, the case of negative linear effects corresponding to the well-known logistic equation. Following Sinclair, we recognize population regulation as occurring when population growth rate is negatively density dependent. Surprisingly, given its fundamental importance in population ecology, only 25 studies were discovered in the literature in which population growth rate has been plotted against population density. In 12 of these the effects of density were linear; in all but two of the remainder the relationship was concave viewed from above. Alternative approaches to establishing the determinants of population growth rate are reviewed, paying special attention to the demographic and mechanistic approaches. The effects of population density on population growth rate may act through their effects on food availability and associated effects on somatic growth, fecundity and survival, according to a 'numerical response', the evidence for which is briefly reviewed. Alternatively, there may be effects on population growth rate of population density in addition to those that arise through the partitioning of food between competitors; this is 'interference competition'. The distinction is illustrated using a replicated laboratory experiment on a marine copepod, Tisbe battagliae. Application of these approaches in conservation biology, ecotoxicology and human demography is briefly considered. We conclude that population regulation, density dependence, resource and interference competition, the effects of environmental stress and the form of the ecological niche, are all best defined and analysed in terms of population growth rate.     

Kinetics of microbial growth on pentachlorophenol

Batch and fed-batch experiments were conducted to examine the kinetics of pentachlorophenol utilization by an enrichment culture of pentachlorophenol-degrading bacteria. The Haldane modification of the Monod equation was found to describe the relationship between the specific growth rate and substrate concentration. Analysis of the kinetic parameters indicated that the maximum specific growth rate and yield coefficients are low, with values of 0.074 h-1 and 0.136 g/g, respectively. The Monod constant (Ks) was estimated to be 60 micrograms/liter, indicating a high affinity of the microorganisms for the substrate. However, high concentrations (KI = 1,375 micrograms/liter) were shown to be inhibitory for metabolism and growth. These kinetic parameters can be used to define the optimal conditions for the removal of pentachlorophenol in biological treatment systems.     

Comparative analysis of growth of edible mussel Mytilus edulis from different White Sea regions

Growth properties were studied in edible mussel Mytilus edulis from different biotopes of the Kandalaksha Bay in the White Sea.Growth curves of the mussel shell were approximated by the von Bertalanffy equation. The highest growth rate, primarily dependent on hydrological conditions, was observed in mussels from the Chupa Bay (Levin Navolok fishery), while the lowest rate was observed in mussels from the Umba region (Turii Cape, Murmansk Region). Allometric relationships of the mussel shell were determined. Analysis of the relationship between the maximum width/convexity and their length demonstrated that this relationship is described by a single allometric equation for mussels from the Umba region, which indicates the homogeneity of this population. For mussels from the Chupa Bay, this relationship cannot be described by a single equation, which points to the population heterogeneity. For 90% mussels from this region, the shell width/convexity ratio ranged from 0.5 to 0.9; while for the remaining 10%, it ranged from 0.9 to 1.15.     

Kinetics of microbial growth on pentachlorophenol.

Batch and fed-batch experiments were conducted to examine the kinetics of pentachlorophenol utilization by an enrichment culture of pentachlorophenol-degrading bacteria. The Haldane modification of the Monod equation was found to describe the relationship between the specific growth rate and substrate concentration. Analysis of the kinetic parameters indicated that the maximum specific growth rate and yield coefficients are low, with values of 0.074 h-1 and 0.136 g/g, respectively. The Monod constant (Ks) was estimated to be 60 micrograms/liter, indicating a high affinity of the microorganisms for the substrate. However, high concentrations (KI = 1,375 micrograms/liter) were shown to be inhibitory for metabolism and growth. These kinetic parameters can be used to define the optimal conditions for the removal of pentachlorophenol in biological treatment systems.