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.     

Robust and efficient design of experiments for the Monod model

In this paper the problem of designing experiments for the Monod model, which is frequently used in microbiology, is studied. The model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics, environmental research, pharmacokinetics, and plant physiology. The designs presented so far in the literature are local optimal designs, which depend sensitively on a preliminary guess of the unknown parameters, and are for this reason in many cases not robust with respect to their misspecification. Uniform designs and maximin optimal designs are considered as a strategy to obtain robust and efficient designs for parameter estimation. In particular, standardized maximin D- and E-optimal designs are determined and compared with uniform designs, which are usually applied in these microbiological models. It is demonstrated that maximin optimal designs are substantially more efficient than uniform designs. Parameter variances can be decreased by a factor of two by simply sampling at optimal times during the experiment. Moreover, the maximin optimal designs usually provide the possibility for the experimenter to check the model assumptions, because they have more support points than parameters in the Monod model.     

Validity of Monod kinetics at different sludge ages – Peptone biodegradation under aerobic conditions

The study presented an evaluation of the effect of culture history (sludge age) on the growth kinetics of a mixed culture grown under aerobic conditions. It involved an experimental setup where a lab-scale sequencing batch reactor was operated at steady-state at two different sludge ages (θ_X) of 2 and 10 days. The system sustained a mixed culture fed with a synthetic substrate mainly consisting of peptone. The initial concentration of substrate COD was selected around 500 mg COD/L. Polyhydroxyalkanoate (PHA) storage occurred to a limited extent, around 30 mg COD/L for θ_X = 10 days and 15 mg COD/L for θ_X = 2 days. Evaluation of the experimental data based on calibration of two different models provided consistent and reliable evidence for a variable Monod kinetics where the maximum specific growth rate, was assessed as 6.1/day for θ_X = 2 days and 4.1/day for θ_X = 10 days. A similar variability was also applicable for the hydrolysis and storage kinetics. The rate of storage was significantly lower than the levels reported in the literature, exhibiting the ability of the microorganisms to regulate their metabolic mechanisms for adjusting the rate of microbial growth and storage competing for the same substrate. This adjustment evidently resulted in case-specific, variable kinetics both for microbial growth and substrate storage.     

基质穿膜传递过程和Monod方程的耦合模型

在分析基质进入细胞穿膜传质机理的基础上,提出了相应的简单传质模型。以此讨论了传递过程对Monod方程的影响,得出了传递过程不影响Monod方程的形式,但影响其动力学参数的结论。这和文献结果和实验数据一致。     

一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型的周期解

研究了一类带Monod增长率及脉冲状态反馈控制的微生物杀虫剂模型．证明了无脉冲系统的负向全局渐近稳定性及带有脉冲状态反馈控制系统具有阶一周期解,并且给出阶一周期解存在和稳定的充分条件．数值模拟验证了理论结果．     

Characterization of gas-liquid mass transfer phenomena in microtiter plates

Gas-liquid mass transfer properties of shaken 96-well microtiter plates were characterized using a recently described method. The maximum oxygen transfer capacity (OTR(max)), the specific mass transfer area (a), and the mass transfer coefficient (k(L)) in a single well were determined at different shaking intensities (different shaking frequencies and shaking diameters at constant filling volume) and different filling volumes by means of sulfite oxidation as a chemical model system. The shape (round and square cross-sections) and the size (up to 2 mL maximum filling volume) of a microtiter plate well were also considered as influencing parameters. To get an indication of the hydrodynamic behavior of the liquid phase in a well, images were taken during shaking and the liquid height derived as a characteristic parameter. The investigations revealed that the OTR(max) is predominantly dependent on the specific mass transfer area (a) for the considered conditions in round-shaped wells. The mass transfer coefficient (k(L)) in round-shaped wells remains at a nearly constant value of about 0.2 m/h for all shaking intensities, thus within the range reported in the literature for surface-aerated bioreactors. The OTR(max) in round-shaped wells is strongly influenced by the interfacial tension, determined by the surface tension of the medium used and the surface properties of the well material. Up to a specific shaking intensity the liquid surface in the wells remains horizontal and no liquid movement can be observed. This critical shaking intensity must be exceeded to overcome the surface tension and, thus, to increase the liquid height and enlarge the specific mass transfer area. This behavior is solely specific to microtiter plates and has not yet been observed for larger shaking bioreactors such as shaking flasks. In square-shaped microtiter plate wells the corners act as baffles and cause a significant increase of OTR(max), a, and k(L). An OTR(max) of up to 0.15 mol/L/h can be reached in square-shaped wells.     

Liquid-phase mass transfer coefficients in bioreactors

Liquid-phase mass transfer coefficient in bioreactors have been examined. A theoretical model based on the surface renewal concept has been devloped. The predicted liquid-phase mass transfer coefficients are compared with the experimental data for a mycelial fermentation broth (Chaetomium cellulolyticum) and model media (carboxymethyl cellulose) in a bench-scale bubble column reactor. The liquid-phase mass transfer coefficient is evaluated by dividing the volumetric mass transfer coefficient obtained experimentally by the specific surface area estimated using the available correlations. The available literature data in bubble column and stirred tank bioreactors is also used to test the validity of the proposed model. A reasonable agreement between the model and the experimental data is found.     

The oxygen transfer rate influences the molecular mass of the alginate produced by Azotobacter vinelandii

The influence of oxygen transfer rate (OTR) on the molecular mass of alginate was studied. In batch cultures without dissolved oxygen tension (DOT) control and at different agitation rates, the DOT was nearly zero and the OTR was constant during biomass growth, hence the cultures were oxygen-limited. The OTR reached different maximum levels (OTR_max) and enabled to establish various relative respiration rates. Overall, the findings showed that OTR influences alginate molecular mass. The mean molecular mass (MMM) of the alginate increased as OTR_max decreased. The molecular mass obtained at 3.0 mmol l⁻¹ h⁻¹ was 7.0 times higher (1,560 kDa) than at 9.0 mmol l⁻¹ h⁻¹ (220 kDa). An increase in molecular mass can be a bacterial response to adverse nutritional conditions such as oxygen limitation.     

从莫诺方程谈起——生物类本科专业教学中3个相似方程的探讨

介绍并比较探讨了生物类相关本科专业课程教学中涉及的3个相似方程：微生物细胞生长动力学中的莫诺方程、酶促反应动力学中的米氏方程、吸附分离过程中的等温吸附方程,以供相关专业教师的教学和学生的学习参考。     

Kinetics and mass transfer for lactic acid recovered with anion exchange method in fermentation solution

An anion exchange method for lactic acid recovered from lactic acid-glucose solution in an ion-exchange membrane-based extractive fermentation system was examined. The exchange isotherms of anion exchange resins for lactic acid recovered were measured batchwise, and the exchange-desorption kinetics of lactic acid passing through the exchange column was investigated. The determined typical breakthrough and elution curves were measured and simulated by conventional mode. The mass transfer coefficients were identified by numberical method. The effects of the velocity of the fluid on the dynamics were studied. Aqueous NaOH solution was found to be the best solvent for elution. An experiment on anioun exchange from clarified lactic acid fermentation broth was carried out to obtain knowledge of the performance of the ion exchange system from a borth. The ion-exchange mass-transfer coefficient and efficiency from the fermentation broth is found to be lower when compared with aqueous solutions of pure lactic acid. The results show that the separation method with anion exchange resins may be used in the production of lactic acid by fermentation.(c) 1995 John Wiley & Sons, Inc.     

Measurement of local mass transfer coefficient in biofilms

Local mass transfer rates for an electrochemically formed microsink in an aerobic biofilm was measured by a mobile microelectrode using limiting current technique. Mass transfer coefficients varied both horizontally and vertically in the biofilm. The results implied the existence of an irregular biofilm structure consisting of microbial cell clusters surrounded by tortuous water channels. An unexpected increase of the local mass transfer coefficient just above the biofilm surface suggested the existence, of local flow instability in this region. As expected, the influence of bulk flow velocity on the local mass transfer rate decreased with increasing depth into the biofilm. Mass transfer coefficients fluctuated significantly inside microbial cell clusters, suggesting the existence of internal channels through which liquid could flow. A new conceptual model of biofilm microbial cluster structure is proposed to account for such biofilm microstructure irregularities. (c) 1995 John Wiley & Sons, Inc.     

Revisiting Verhulst and Monod models: analysis of batch and fed-batch cultures

The paper re-evaluates Verhulst and Monod models. It has been claimed that standard logistic equation cannot describe the decline phase of mammalian cells in batch and fed-batch cultures and in some cases it fails to fit somatic growth data. In the present work Verhulst, population-based mechanistic growth model was revisited to describe successfully viable cell density (VCD) in exponential and decline phases of batch and fed-batch cultures of three different CHO cell lines. Verhulst model constants, K, carrying capacity (VCD/ml or μg/ml) and r, intrinsic growth factor (h⁻¹) have physical meaning and they are of biological significance. These two parameters together define the course of growth and productivity and therefore, they are valuable in optimisation of culture media, developing feeding strategies and selection of cell lines for productivity. The Verhulst growth model approach was extended to develop productivity models for batch and fed-batch cultures. All Verhulst models were validated against blind data (R² > 0.95). Critical examination of theoretical approaches concluded that Monod parameters have no physical meaning. Monod-hybrid (pseudo-mechanistic) batch models were validated against specific growth rates of respective bolus and continuous fed-batch cultures (R² ≈ 0.90). The reduced form of Monod-hybrid model C_L/(K_L + C_L) describes specific growth rate during metabolic shift (R² ≈ 0.95). Verhulst substrate-based growth models compared favourably with Monod-hybrid models. Thus, experimental evidence implies that the constants in the Monod-hybrid model may not have physical meaning but they behave similarly to the biological constants in Michaelis–Menten enzyme kinetics, the basis of the Monod growth model.     

酶反应速率方程的普适形式

酶反应速率方程的普适形式是应用于相互关联的大规模代谢途径动力学建模的重要方法.把酶反应速率方程写成Michaelis-Menten-King-Altman方程形式可以使得动力学参数(或函数)容易与数据库中的实验数据相接轨,并可以处理任意数量的底物和产物,有利于大规模的计算.普适形式可以同时描述正、负反应方向,并能精确地用于准稳态条件.展示了在三类生物体系中广泛存在的酶反应机制中普适方程的严格推导过程,并讨论了普适方程的特点,针对不可逆反应酶反应产生的产物抑制效应可以自然消除,总结了在普适速率方程中体现调节剂的作用和协同作用.     

Oxygen requirements and mass transfer in hairy-root culture

Oxygen mass transfer in clumps of Atropa belladonna hairy roots was investigated as a function of root density and external flow conditions. Convection was the dominant mechanism for mass transfer into root clumps 3.5 to 5.0 cm in diameter; Peclet numbers inside the clumps ranged from 1.4 x 10(3) to 7.1 x 10(4) for external superficial flow velocities between 0.4 and 1.4 cm s(-1). Local dissolved-oxygen levels and rates of oxygen uptake were measured in aflow chamber and in bubble column and stirred bioreactors. When air was used as oxygen source, intraclump dissolved-oxygen tensions ranged from90% to 100% air saturation at high external flow velocity andlow root density, to less than 20% air saturation in dense root clumps. Specific oxygen-uptake rate declined with increasing root density. When external boundary layers around individual roots were eliminated byforcing liquid through the clumps at superficial velocities between 0.2 and1.0 cm s(-1), internal dissolved-oxygen tension was maintained at 95% to 100% air saturation and rate of oxygen uptake at 1.6 x 10(-6) g g(-1) s(-1) dry weight. Liquid culture of single A. belladonna hairy roots was used to investigate the effect of dissolved-oxygen tensionon root growth and morphology. Total root length and number of root tips increased exponentially at oxygen tensions between 70% and 100%air saturation. Specific growth rate increased with oxygen tension up to 100% air saturation; this result demonstrates that hairy roots aeratedwithout oxygen supplementation are likely to be oxygenlimited. No growth occurred at 50% air saturation. Growth of hairy roots proceeded with an average length per tip of about 1 cm; this value was essentially independent of dissolved-oxygen tension between 70% and 100% air saturation. (c) 1994 John Wiley & Sons, Inc.     

Ideal nutrient productivities and nutrient proportions in plant growth

Abstract I propose that one single formulation can be applied to relate growth and content of several nutrients, including the most important macronutrients, of most plant species. The plant growth rate is proportional to the nutrient content minus a given minimal concentration of the nutrient in minimum. The proportionality factor, the nutrient productivity, and the minimum concentration are species specific properties. The nutrient productivity formulation is shown to apply for very different plant species and for different nutrients.     

Non-invasive mass transfer measurements in complex biofilm-coated structures

We demonstrate a novel application of (13)C pulsed field gradient (PFG) NMR to monitor mass transfer, due to both flow and diffusion, in a 3D complex porous support structure modified by biofilm growth. This enables timescales an order of magnitude larger than previously possible to be accessed with respect to displacement probability distribution (propagator) measurements. The evolution in the propagator shape with observation time to the Gaussian asymptote (constant dispersion coefficient) is consequently well resolved. We also simulated the measured displacement propagators with good agreement between experiment and prediction. The methodology has significant potential for the selective characterization of the transport of nutrients, metabolic products, pollutants and biocides in such complex biofilm-containing structures.     

The role of hydrogen mass transfer for the growth kinetics of Methanobacterium thermoautotrophicum in batch and chemostat cultures

Hydrogen concentration was determined in batch and chemostat cultures of Methanobacterium thermoautotrophicum, both in the headspace and in the medium using mass spectrometry. The calculated dissolved hydrogen concentration in the medium as derived from the headspace hydrogen concentration when equilibrium conditions between gas and liquid phase were assumed, was ten times higher than the experimentally determined hydrogen concentration. Variation of the partial pressure of hydrogen resulted in different values for substrate affinity for hydrogen (K_s) and yield (Y) of the cells. Upon hydrogen limitation, K_s decreased while the yield coefficient for hydrogen increased, indicating a change in the affinity of the cells towards hydrogen. Received 15 November 1996/ Accepted in revised form 21 July 1997     

Influence of mass transfer and surface ligand heterogeneity on quantitative BIAcoreTM binding data. Analysis of the interaction of NC10 Fab with an anti-idiotype Fab′

The interaction of monovalent Fab fragments of NC10, an antiviral neuraminidase antibody, and the anti-idiotype antibody 3-2G12 has been used as a model system to demonstrate experimentally the influence of non-ideal binding effects on BIAcore^TM binding data. Because the association rate constant for these two molecules was found to be relatively high (about 5×10⁵ M ⁻¹ s⁻¹), mass transfer was recognised as a potential source of error in the analysis of the interaction kinetics. By manipulation of the flow rate and the surface density of the immobilised ligand, however, the magnitude to this error was minimised. In addition, the application of site-specific immobilisation procedures was found to improve considerably the correlation of experimental binding data to the ideal 1:1 kinetic model such that the discrepancy between experimental and fitted curves was within the noise range of the instrument. Experiments performed to measure the equilibrium constant (K_D) in solution resulted in a value of similar magnitude to those obtained from the ratio of the kinetic rate constants, even those measured with a heterogeneous ligand or with a significant mass transfer component. For this system, the experimental complexities introduced by covalent immobilisation did not lead to large errors in the K_D values obtained using the BIAcore © 1997 John Wiley & Sons, Ltd.