Monod rate expressions

This simple approach was adopted in order to circumvent the complications that are introduced by the fact that the volume of the liquid phase in the reactor varies with time. When the volume of the aqueous growth medium varies during the course of the reaction, an approach based on integration of a proposed rate law is problematic, although numerical integration would be possible. An additional reason for employing the differential approach below is that for rate laws that are other than those of the simple nth-order form (such as a Monod rate expression) a differential method of data analysis is often adequate for preliminary considerations involved in the design of a bioreactor that is intended to operate in a batch mode. 

The subscripts on the usual process parameters indicate the positions in the process to which these parameters refer. The kinetics of yeast growth can be described by a Monod rate expression with = 0.625 h and Kg = 2 g/L. Cell death and cell maintenance effects are negligible, as is formation of products other than yeast cells. The yield coefficient x/s is 0.44. The effluent from the bioreactor flows directly to a membrane filtration apparams. The membrane is completely permeable to the substrate, so the concentrations of the substrate in the CSTBR, the effluent from the bioreactor, and the permeate from the membrane and in the recycle stream are all identical. The membrane rejects a substantial proportion of the yeast cells so that the ratio of the concentration of yeast in the recycle stream is a factor of 4 larger than that in the effluent from the CSTBR. Volumetric expansion and contraction effects may be considered negligible. 

The most frequently used rate expression for balanced growth was proposed by Jacques Monod in 1942[28] and is known as the Monod rate expression. 

Baughman and colleagues145 derive a second-order kinetic rate expression as a special case of the Monod kinetic equation. It appears to describe biodegradation of organics in natural surface waters reasonably well ... 

The rate expressions can be simply given by the Monod Equation ... 

Inserting the Monod-type rate expressions gives For the cell balance... 

Frequently, the specific byproduct formation rate is presented as a function of specific substrate consumption rate and substrate-to-product yield (see Equation 12), but other structures can be assumed. The specific production rate can be limited by a precursor substrate and modeled by a Monod-type expression, as in Equation 73, or it may be inhibited by a substrate that is not, in principle, linked to its production, as in Equation... 

This type of rate expression is often used in models for water treatment, and many environmental factors can be included (the effect of, e.g., phosphate, ammonia, volatile fatty acids, etc.). The correlation between parameters in such complicated models is, however, severe, and very often a simple Monod model (7-92) with only one limiting substrate is sufficient. 

Table 3 lists the kinetic rate expressions for each of the hydrolysis and fermentation reaction rates shown in Fig. 5 and in the mass balance equations of Tables 1 and 2. Each of the reaction rates were found to fit the data through trial and error, starting with the simplest model. For the hydrolysis reaction rates (rs,arch and / maltose), the simplest form was the Michaelis-Menten model without inhibition. For all other reaction rates which described fermentation kinetics, the simplest form was the Monod model without inhibition. More descriptive models were found in the literature and tested one by one until the set of kinetic rate equations with the best fit to the experimental data were determined. This was completed with the hydrolysis datasets first, then the complete SSF datasets. 

If cell death and maintenance metabolic effects are negligible, and if the Monod rate law is applicable to this system, the steady-state material balance on the biomass can be expressed as... 

Elucidating the dynamics of this uptake system is an essential step to critically assess the reliability of approximations based on the unstructured Monod type of rate expression. To address this issue, we applied the tools of measurements of metabolite concentrations and in vivo diagnosis to E. coli (Noissomit-Rizzi et al., to be published). Both sampling techniques were used during continuous culture of Escherichia coli W3110. In what follows, only those metabolites will be discussed which are related to the uptake system. Also, a first attempt towards kinetic analysis of the PTS system will be presented. 

For the Nernst-Monod term expression, the f and Eka variables were originally defined in MATLAB and needed to be first sent to COMSOL using the fem.const command as shown previously. The variable E is a solution component (i.e., dependent variable) of a generic PDE application mode that is described in detail later. It is only defined in COMSOL, and thus, it was not necessary to declare it in MATLAB or send it via the fem.const command. Likewise, for the specific rate of substrate utilization expression, q 5, and were defined in MATLAB, whereas S and Mo are the solution components of the diffusion application mode. Note that in MATLAB and COMSOL, we do not use subscripts instead, we use regular fonts. Because the expression contains both COMSOL defined and MATLAB defined variables, the expression must be in single quotes. 

Specific detail on Michaelis-Menten kinetics, quasi steady-state approximations, competitive and non-competitive inhibitions, substrate inhibition, rate expressions for enzyme catalysis and deactivations, Monod growth kinetics, etc. are not presented in an extensive manner although additional information is available in the work of Vasudevan for the interested leader. " Also note that the notation adopted by Vasudevan is employed throughout this chapter. 

Several experimental studies have established that extracellular glucose concentration modulates both cell division times and migration speeds [132-134]. Cheng et al. [35] used a Monod-type expression to describe the dependence of cell-doubling rates on extracellular nutrient concentration ... 

Note that p. has units of reciprocal time—for example, h. Model parameter p max is referred to as the maximum growth rate, because p. has a maximum value of p max when 5 Ks. The second model parameter, Ks, is called the Monod constant. The Monod equation has the same form as the Michaelis-Menten equation, a standard rate expression for enzyme reactions (Bailey and Ollis, 1986 Fogler, 2006). 

The reaction rate is expressed by a Monod-type equation... 

Conversion rate data obtained under a wide range of operating conditions may be worked out to provide a kinetic expression, most typically expressed according to well established models for bioprocess kinetics first and second order, Monod, Haldane, product-inhibited, etc. 

The relationship between i. and S as depicted in Figure 2.7 is relevant because it quantifies the importance of a substrate in terms of its concentration on the growth rate. As seen from Equation (2.16), X= 1/2 imax for S=Ks. For this reason, Ks is also named the half saturation constant. Equation (2.16) and the corresponding curves shown in Figure 2.7 are called the Monod expression and Monod curve, respectively. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

The classic expression for measurement of growth rate is the Monod Equation. 

Many other kinetic forms have been proposed and have been used in the past however, they have all been forgotten since Monod came out with his expression. Its simplicity won the day. So we will use this type of expression throughout to relate the rate of cell growth to substrate concentration. 

The rate of cell growth is influenced by temperature, pH, composition of medium, rate of air supply, and other factors. In the case that all other conditions are kept constant, the specific growth rate may be affected by the concentration of a certain specific substrate (the limiting substrate). The simplest empirical expression for the effect ofthe substrate concentration on the specific growth rate is the following Monod equation, which is similar in form to the Michaelis-Menten equation for enzyme reactions ... 

Monod(40) proposed the use of a saturation-isotherm type of equation to relate the growth rate of a micro-organism culture to the prevailing feed concentration. This has become known as the Monod equation and is usually expressed as ... 

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