Monod growth model

In biochemical engineering we are often faced with the problem of estimating average apparent growth or uptake/secretion rates. Such estimates are particularly useful when we compare the productivity of a culture under different operating conditions or modes of operation. Such computations are routinely done by analysts well before any attempt is made to estimate true kinetics parameters like those appearing in the Monod growth model for example. 

Figure 2. (A) Monod Growth Model (B) model for growth inhibition and (C) model for substrate activation and inhibition of growth.

Attempts to model the phytoplankton changes include those of Dippner (1998) who was able to simulate observed diatom and flagellate changes using a Monod growth model with a fixed ratio of C Si P of 106 16 1, forced with a combination of phosphate increase and silicate decrease. The ecosystem model ERSEM II (with nutrient kinetics that uncoupled uptake of N, P and Si) was used (Patsch Radach, 1997) to... 

A Monod rate model is used to demonstrate the rate of biomass generation. We neglect the cell death rate. Let us denote the ratio of biomass rate of generation to biomass concentration, rJX, that is the specific growth rate /a also denotes the dilution rate D is defined as number of tank volumes passed through per unit time, FIV. After substitution of D and /a into (6.8.1), the following equation is obtained ... 

The Monod rate model is valid for a CSTR bioreactor with maximum specific growth rate of 0.5 li 1 and K, 2 g-1. What would be a suitable dilution rate at steady-state condition, where there is no cell death if initial substrate concentration is 50g-l-1 and yield of biomass on substrate is 100%. 

The parameters of the Monod cell growth model are needed i.e. the maximum specific growth rate and the Michaelis-Menten constant are required for a suitable rate equation. Based on the data presented in Tables 10.1 and 10.2, obtain kinetic parameters for... 

Tlie biochemical reaction rate followed the Monod rate model with a Monod rate constant of ks = 6.2 X 10 6g-cm 3 and a specific growth rate of vmlx 6.67 X 10 7g-cm 3-s. Design the bioreactor with a suitable heat transfer area. 

Odour will return in treated slurry as a result of post treatment fermentation. The concentration of readily fermentable substrates, measured as BOD5, provide an indicator of this problem. In continuous culture without oxygen limitation the BOD5 can be described by a model derived from the Monod (13) model of microbial growth (14). The supernatant BOD5 (g/1) from treatment at 15 to 45°C, was described by equation 3 and the whole BOD5 by equations 4 and 5(15). 

If the growth follows the Monod kinetic model, then equation 5.62 may be substituted into equation 5.115 to give ... 

Two continuous stirred-tank fermenters are connected in series, the first having an operational volume of 100 1 and that of the second being 50 1. The feed to the first fermenter is sterile and contains 5000 mg/1 of substrate, being delivered to the fermenter at 18 1/h. If the microbial growth can be described by the Monod kinetic model with /x, = 0.25 h l and Ks = 120 mg/l, calculate the steady-state substrate concentration in the second vessel. What would happen if the flow were from the 50 I fermenter to the 100 1 fermenter ... 

The equations describing increase in cell density [Eqs. (8.3)-(8.8)] so far do not contain any information about the nature and concentration of any substrate such as the C-source. As the specific growth rate /i tends to depend on quality and amount of substrate, however, we require a growth model which provides the function /i = jU([S]). The most widely used growth model is the Monod model (Monod, 1950) which assumes that only one substrate limits cell growth and proliferation. The corresponding equation [Eq. (8.9), in which /imax is the maximum specific growth rate [h-1]] reads very similarly to the Michaelis-Menten equation. 

Vallino, J. J., C. S. Hopkinson, and J. E. Hobbie. 1996. Modeling bacterial utilization of dissolved organic matter Optimization replaces Monod growth kinetics. Limnology and Oceanography 41 1591-1609. 

Moser (Ml4) applied extensions of Monod s model to growth of mixed... 

Consider Monod s model of growth. It is often possible to arrange experimental conditions so that a single substrate does in fact limit growth. One can then proceed to test the more quantitative aspects of the model. Monod s model has been applied to two cases batch growth and continuous propagation. To fulfill the first requirement for a valid model, it should predict the results of (say) continuous propagation from batch data, if the model is to be accepted. 

Fig. 16. Comparison of Monod s model with data of Herbert et al. (HIO) on growth of Aerobacter cloacae. Solid line calculated from Monod s model with /a = 0.85 hr , K = 0.0123 g/liter, C,f = 2.5 g/liter, and a = 1.89 g/g. Replotted from J. Gen. Microbiol. 14, 601-622 (1956), by permission of Cambridge University Press.

Experimental results of Herbert et al. are compared with predictions of Monod s model in Fig. 16. At long holding times, agreement of the model with experiment is good this is not the case at holding times near the critical, where there is a definite trend not predicted by the model. Thus, it appears that the maximum specific growth rate (//) is faster than that determined from batch experiments also the stoichiometric coefficient a changes as 6 approaches the critical. 

The growth rate and the stoichiometric coefficients Oi used in Monod s model represent averages over the distribution of cell ages. Thus, from Eq. (131) and the definition of / , we find... 

In continuous, steady-state culture, and a,- will in general depend on the holding time, since U z) varies with holding time. Hence the model is able to account for dependence of stoichiometric coefficients and growth rates on the steady-state holding time in Monod s model, there is no dependence of these quantities on holding time. It is possible, then, for the newer model to rationalize data such as those shown in Fig. 16. 

Waltman of the University of Iowa recently pointed out to me in a personal communication that even when Lotka-Volterra concepts are discarded entirely and Monod s model is used for all growth rates, the resulting competition equations for two predators and one prey seem to have limit cycle solutions for certain conditions of operation. Mr. Basil Baltzls has found that use of a so-called multiple saturation model for the predators, which seems to be more appropriate than Monod s model for protozoans at any rate... 

If there is Indeed a threshold density of bacteria below which protozoans like Tetrahymena do not feed, then it follows that growth and feeding of such organisms cannot be described by Monod s model of growth (69) A model that suggests itself for situations like this is... 

For the net cell growth rate, we use Monod s model for growth and a first order rate of loss of viability ... 

Another deviation often observed is the so-called wall growth, which results in plots similar to Fig. 3.12. Topiwala and Hamer (1971) analyzed this effect of the adherence of microorganisms to glass or metal surfaces. When the part of adhered biomass that cannot wash out is denoted by x, the following modification of Monod s model results ... 

Biocatalyst growth is represented by the standard Monod kinetic model using model coefficients for E. coli found in the literature. The specific growth rate, saturation constant, and cell maintenance coefficients are 0.50 h 4.0 mg/L, and 0.045 h respectively (25). 

Product formation kinetics in mammalian cells has been studied extensively for hybridomas. Most monoclonal antibodies are produced at an enhanced rate during the Gq phase of the cell cycle (8—10). A model for antibody production based on this cell cycle dependence and traditional Monod kinetics for cell growth has been proposed (11). However, it is not clear if this cell cycle dependence carries over to recombinant CHO cells. In fact it has been reported that dihydrofolate reductase, the gene for which is co-amplified with the gene for the recombinant protein in CHO cells, synthesis is associated with the S phase of the cell cycle (12). Hence it is possible that the product formation kinetics in recombinant CHO cells is different from that of hybridomas. 

Table 3.1 shows the kinetic parameters for cell growth, rate models with or without inhibition and mass transfer coefficient calculation at various acetate concentrations in the culture media. The Monod constant value, KM, in the liquid phase depends on some parameters such as temperature, initial concentration of the carbon source, presence of trace metals, vitamin B solution, light intensity and agitation speeds. The initial acetate concentrations in the liquid phase reflected the value of the Monod constants, Kp and Kp. The average value for maximum specific growth rate (/xm) was 0.01 h. The value... 

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