Phosphate‐limited growth of the marine diatom Thalassiosira weissflogii (Bacillariophyceae): evidence of non‐monod growth kinetics1

The marine diatom Thalassiosira weissflogii (Grunow) G. A. Fryxell & Hasle was grown in a chemostat over a series of phosphate‐limited growth rates. Ambient substrate concentrations were determined from bioassays involving picomolar spikes of ³³P‐labeled phosphate, and maximum uptake rates were determined from analogous bioassays that included the addition of micromolar concentrations of unlabeled phosphate and tracer concentrations of ³³P. The relationship between cell phosphorus quotas and growth rates was well described by the Droop equation. Maximum uptake rates of phosphate spikes were several orders of magnitude higher than steady state uptake rates. Despite the large size of the T. weissflogii cells, diffusion of phosphate through the boundary layer around the cells had little effect on growth kinetics, in part because the cellular N:P ratios exceeded the Redfield ratio at all growth rates. Fitting the Monod equation to the experimental data produced an estimate of the nutrient‐saturated growth rate that was ~50% greater than the maximum growth rate observed in batch culture. A modified hyperbolic equation with a curvature that is a maximum in magnitude at positive growth rates gave a better fit to the data and an estimate of the maximum growth rate that was consistent with observations. The failure of the Monod equation to describe the data may reflect a transition from substrate to co‐substrate limitation and/or the presence of an inducible uptake system.     

NON-STEADY STATE DYNAMICS OF ALGAL POPULATION GROWTH: EXPERIMENTS WITH TWO CHLOROPHYTES1

The relations among dissolved phosphorus, cell quota of phosphorus, and population growth rate were determined for two Chlorophytes, Chlorella sp. and Scenedesmus quadricauda var. longispina (Chod.) G. M. Smith, in two types of non-steady state continuous culture. One of these types had relatively smooth transitions between growth under different degrees of phosphorus limitation. Under these conditions, two equations often applied to growth kinetics in steady state cultures were found to apply to non-steady state growth. Monad's equation described the relation between dissolved phosphorus concentration and population growth rate, and Droop's equation described the relation between cell quota and population growth rate. The second type of culture received phosphorus only as periodic pulses, leading to sharp changes in dissolved phosphorus, cell quota, and growth rate. A simulation model based on Droop's equation described much of the observed dynamics of cell numbers and quotas in these cultures. Droop's equation could not be convincingly fitted directly to the data, however, due to its incorrect prediction of an immediate growth response to phosphorus pulses. A third relation, predicting that saturated rates of phosphorus uptake would depend on the recent nutrient history of the cells as reflected by the cell quota, was not supported.     

A dynamic mathematical model of the chemostat

A number of experimental studies on the dynamic, behavior of the chemostat have shown that the specific growth rate does not, instantaneously adjust to changes in the concentration of limiting substrate in the chemostat following disturbances in the steady state input limiting substrate concentration or in the steady state dilution rate. Instead of an instantaneous response, as would be predicted by the Monod equation, experimental studies have shown that the specific growth rate experiences a dynamic lag in responding to the changes in the concentration of limiting substrate in the culture vessel. The observed dynamic lag has been recognized by researchers in such terms as an inertial phenomenon and as a hysteresis effect, but as yet a systems engineering approach has not been applied to the observed data. The present paper criticizes the use of the Monod equation as a dynamic relationship and offers as an alternative a dynamic equation relating specific growth rate to the limiting substrate concentration in the chemostat. Following the development of equations, experimental methods of evaluating parameters are discussed. Dynamic responses of analog simulations (incorporating the newly derived equations) are compared with the dynamic responses predicted by the Monod equation and with the dynamic responses of experimental chemostats.     

PHOSPHATE AND SILICATE GROWTH AND UPTAKE KINETICS OF THE DIATOMS ASTERIONELLA FORMOSA AND CYCLOTELLA MENEGHINIANA IN BATCH AND SEMICONTINUOUS CULTURE1

Information on the nutrient kinetics of Asterionella formosa Hass. and Cyclotella meneghiniana Kutz. under either phosphate or silicate limitation was obtained for use in a Monod model and in a variable internal stores model of growth. Short-term batch culture growth experiments were fit to the Monod model and long-term semicontinuous culture experiments and short-term uptake experiments were fit to the variable internal stores model. Mathematical analysis indicates that the parameters of the 2 models may be expressed in terms of each other at steady state. The qualitative results of both batch and steady state culture methods agree. For limiting phosphate experiments. A. formosa is better able to grow at low PO₄-P concentrations than C. meneghiniana, as shown by its lower K for PO₄-P limited growth. The k_Q of A. formosa compared to C. meneghiniana found in long-term semicontinuous culture indicates that A. formosa is almost an order of magnitude more efficient at using internal phosphate for growth. The qualitative results under silicate-limited growth of C. meneghiniana is less than that of A. formosa. The k_Q from semicontinuous culture experiments indicates that C. meneghiniana is the more efficient at using internal silicate for growth. Nutrient uptake experiments showed more variability from a Michaelis-Menten relationship than short-term growth experiments. There were no significant differences between the 2 species in half saturation constants for either phosphate or silicate uptake. We observed a marked dependence of the coefficient of luxury consumption (R) of phosphate on the steady state growth rate. A. formosa has a higher R than C. meneghiniana.     

Approximation of continuous growth of Cephalotaxus harringtonia plant cell cultures using fed-batch operation

In Cephalotaxus harringtonia plant cell cultures, periods of batch growth that are limited by hexose uptake are too short to make an accurate estimate of the Monod saturation constant. Continuous cultures are infeasible on a laboratory scale, and semicontinuous cultures require too frequent sampling. Fed-batch operation, consisting of intermittent removal from a culture that is fed continuously, was investigated as a possible solution to these problems. For a constant feed rate, computer simulations showed that a steady state can be achieved which is useful for studying growth at different specific growth rates. In terms of the dilution rate it was confirmed that the operation is essentially equivalent to continuous culture when the samples represent a small fraction of the total culture volume. Experiments with glucose or fructose as the carbon source were carried out in shake flasks fed by a multichannel syringe pump. Results indicate that Monod kinetics based on medium glucose levels cannot adequately describe growth under these conditions. Monod's expression for specific growth rate using internal glucose concentration gives an improved correlation.     

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.     

PHOSPHATE‐LIMITED GROWTH OF PAVLOVA LUTHERI (PRYMNESIOPHYCEAE) IN CONTINUOUS CULTURE: DETERMINATION OF GROWTH‐RATE‐LIMITING SUBSTRATE CONCENTRATIONS WITH A SENSITIVE BIOASSAY PROCEDURE1

The relationship between steady‐state growth rate and phosphate concentration was studied for the marine prymnesiophyte Pavlova lutheri (Droop) J. C. Green grown in a chemostat at 22°C under continuous irradiance. A bioassay procedure involving short‐term uptake of 10 picomolar spikes of ³³P‐labeled phosphate was used to estimate the concentration of phosphate in the growth chamber. The relationship between growth rate and phosphate was well described by a simple rectangular hyperbola with a half‐saturation constant of 2.6 nM. The cells were able to take up micromolar spikes of phosphate at rates two to three orders of magnitude higher than steady‐state uptake rates. The kinetics of short‐term uptake displayed Holling type III behavior, suggesting that P. lutheri may have multiple uptake systems with different half‐saturation constants. Chl a:C ratios were linearly related to growth rate and similar to values previously reported for P. lutheri under nitrate‐limited conditions. C:N ratios, also linearly related to growth rate, were consistently lower than values reported for P. lutheri under nitrate‐limited conditions, a result presumably reflecting luxury assimilation of nitrogen under phosphate‐limited conditions. C:P ratios were linearly related to growth rate in a manner consistent with the Droop equation for growth rate versus cellular P:C ratio.     

The relationship between kinetics of substrate-limited transitions and steady-state growth in continuous cultures ofAquaspirillum autotrophicum limited by pyruvate

Heterotrophic growth at steady state and during transient states caused by the sudden change of the concentration of the limiting factor in the feed medium was investigated experimentally for continuous cultures ofAquaspirillum autotrophicum limited by pyruvate. A model for describing the growth at steady state was selected from three unstructured models after statistical tests of the data. This model postulates that the growth yield increases linearly with the growth rate. Growth during transitions where the substrate remained limiting at all times was fitted with first-order kinetics. Theoretical predictions of these kinetics were derived from the unstructured models used to describe steady state. The predicted rate coefficients of the transients were compared to the experimental coefficients. It appeared that the model which best described steady-state growth also provided the best predictions for growth during the transient state. It is a widespread opinion that unstructured models are adequate to describe growth under steady-state conditions but not to predict transitions in continuous culture. However, for the particular case studied here, no higher degree of complexity was required to describe transitions, provided the growth of the culture was always limited by the substrate.     

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.     

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.     

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.     

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.     

Continuous culture of Thiobacillus ferrooxidans on a zinc sulfide concentrate

A zinc sulfide concentrate was leached microbiologically by Thiobacillus ferrooxidans in a continuous stirred tank reactor. A model was developed to predict, the leaching kinetics when the bacterial growth rate was not limited by any substrate other than the zinc concentrate, and it was modified to explain the observed results. Stable steady sates were obtained over a range of dilution rates from 0.0171 to 0.1038 hr^?1. Because a solid substrate was used, the specific growth rate of the bacaeria was not a unique function of the subastrate concentration, and conventional contnuous culture theory based on the Monod equation did not apply to this system. The leaching rates and bacterial growth rates were first order in mineral surface area cocentration.     

CHARACTERISTICS OF PHOSPHORUS LIMITATION IN CHLAMYDOMONAS REINHARDTH (CHLOROPHYCEAE) AND ITS PALMELLOIDS1

Chlamydomonas reinhardtii Dang, was grown in a chemostat culture under phosphate limitation. The steady state concentration of phosphate was below the detection limit (< 1 μg P/L) in all runs. The cellular content of phosphorus (Q_p), polyphosphate (Q_pp) and chlorophyll a increased with increasing dilution rate, and the growth rate of the alga was described by Q_p as well as Q_pp in the Droop model. The ratio Q_pp/Q_p and the activity of alkaline phosphatase were maximal at high and low growth rates, respectively. Palmelloids of Chlamydomonas were found at high dilution rates (D > 0.12 h^?1) and became attached to the wall of the culture vessel. They differed from the vegetative stage in both chemical composition and growth rate. Their contents of phosphorus and chlorophyll a were low, as in the vegetative cells, which grew at a low growth rate, whereas the ration Q_pp/Q_p and the activity of alkaline phosphatase were comparable with those of fast growing vegetative cells. The growth rate of the palmelloids was 0.03 h^?1 whereas maximum growth rate (μ_m) for the vegetative cells was 0.21 h^?1.     

The attractiveness of the Droop equations.

The Droop equations are a system of three coupled, nonlinear ordinary differential equations describing the growth of a microorganism in a chemostat. The growth rate of the organism is limited by the availability of a single nutrient. In contrast to the better known Monod equations, the nutrient is divided into external and internal cellular pools. Only the internal pool can catalyze growth. This paper proves that the Droop equations are globally stable. Based on a single combination of parameters, either the chemostat organism goes extinct or it tends to a fixed, positive concentration.     

PHOSPHATE-LIMITED GROWTH KINETICS OF SELENASTRUM CAPRICORNUTUM (CHLOROPHYCEAE)1

Most theoretical studies of phytoplankton growth in aquatic environments assume that relative nutrient utilization abilities regulate species composition. The steady-state phosphate-limited growth kinetics of Selenastrum capricornutum Printz were examined using continuous cultures to characterize the green alga's ability to compete for orthophosphate (Pi) when Pi limits growth. The maximal specific growth rate for Selenastrum at 20 C was 1.20 day^?1, and the concentration where half maximal growth rate occurs was 40 nM Pi. There was an apparent threshold of 10 nM Pi. Cell yields varied inversely with growth rate; thus ability to utilize Pi could not be characterized in terms of the Monod half-saturation constant and maximal growth rate. Instead, we computed the Pi affinity from steady-state flux vs. external Pi concentrations. This affinity was 2.8 l·mg dry wt^?1· day^?1 for Selenastrum. Kinetic evidence from this study suggests that Selenastrum will not be growth competitive with some other common aquatic heterotrophs and autotrophs when Pi limits microbial growth in lakes.     

Steady state and transient growth of autotrophic nitrifying bacteria

Growth of the autotrophic nitrifying bacteria Nitrosomonas europaea and Nitrobacter sp. was studied in continuous culture. Steady state growth kinetics of both organisms conformed with that predicted by chemostat theory, modified to account for maintenance energy requirement. Steady state data were used to calculate the maximum specific growth rate, the saturation constant for growth, the true growth yield and the maintenance coefficient. Transient growth was studied by imposing step changes in dilution rate. Step increases resulted in overshoots and oscillations in substrate concentration before establishment of a new steady state while step decreases in dilution rate were followed by monotonic changes in substrate concentration. The size of overshoots in substrate concentration following step increases in dilution rate was dependent on both the magnitude of the increase and of the dilution rate prior to the change.     

A comparison of two methods for measuring phosphate uptake by Monochrysis lutheri droop grown in continuous culture

Two methods for measuring phosphate uptake by phosphate-limited continuous cultures of Monochrysis lutheri Droop are found to yield comparable results. Data from both isotopic tracer (³³P) and disappearance experiments strongly support a Michaelis- Menten-type hyperbolic relationship between instantaneous uptake rate and ambient phosphate concentration. The data show no systematic linear trend in either the maximum uptake rate per cell or its associated half saturation constant with pre-conditioning steady state growth rate; a weak non-linear trend with pre-conditioning may, however, be present in the maximum uptake rate with an apparent maximum at intermediate growth rates.     

The substrate constant for the ammonium ion of growing

The relation between ammonium concentration and growth rate was studied in steady state continuous cultures of Saccharomyces cerevisiae in nitrogenlimited glucose ammonium medium. This relation could be described by the Monod equation. A maximum specific growth rate of 0.41 h^-1 and a substrate constant for ammonium of 5–11 M were calculated. Ammonium was determined by a modification of the phenol hypochlorite method. A discussion of the results in view of literature data on the substrate constants for other nutrients is given.