Equation Monod

The Monod equation is the relation most commonly applied to describe the rate at which a microbe metabolizes its substrate (e.g., Panikov, 1995). Taking ace-totrophic sulfate reduction as an example, the redox reaction, 

If sulfate rather than acetate is limiting, the equation is written, 

As substrate concentration varies, the Monod equation behaves like the Michaelis-Menten equation discussed in the previous chapter (Eqn. 17.18). If mAc 

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Various functional relationships between [L and S have been proposed, but the Monod equation is used almost exclusively ... 

This is an old, familiar analysis that applies to any continuous culture with a single growth-limiting nutrient that meets the assumptions of perfect mixing and constant volume. The fundamental mass balance equations are used with the Monod equation, which has no time dependency and should be apphed with caution to transient states where there may be a time lag as [L responds to changing S. At steady state, the rates of change become zero, and [L = D. Substituting ... 

Since there are various specific growth rates and different values of rate constants while substrate concentration varies, therefore mix inhibition exists. Andrew26 incorporated a substrate inhibition model27 in the Monod equation the modified Monod equations with second-order substrate inhibition are presented in (3.14.5.1) and (3.14.5.2).16,17... 

The effect of substrate concentration on specific growth rate (/i) in a batch culture is related to the time and p,max the relation is known as the Monod rate equation. The cell density (pcell) increases linearly in the exponential phase. When substrate (S) is depleted, the specific growth rate (/a) decreases. The Monod equation is described in the following equation ... 

The reaction rate for simple fermentation systems is normally given by the Monod equation. This model indicates that the specific conversion rate is constant when applied to an immobilised cell system (Table 8.7). If a first-order rate equation for sugar consumption is used, (8.7.4.2) yields ... 

It has been suggested that fungi grow in filamentous form at an exponential rate with a constant specific growth rate (ji) until some substrate becomes growth limiting, according to the Monod equation 4 6... 

When microorganisms use an organic compound as a sole carbon source, their specific growth rate is a function of chemical concentration and can be described by the Monod kinetic equation. This equation includes a number of empirical constants that depend on the characteristics of the microbes, pH, temperature, and nutrients.54 Depending on the relationship between substrate concentration and rate of bacterial growth, the Monod equation can be reduced to forms in which the rate of degradation is zero order with substrate concentration and first order with cell concentration, or second order with concentration and cell concentration.144... 

The Monod equation assumes a single carbon source. The difficulty in handling multiple carbon sources, which are typical in nature, has led to the use of an empirical biodegradation rate constant k. ... 

The basic biofilm model149,150 idealizes a biofilm as a homogeneous matrix of bacteria and the extracellular polymers that bind the bacteria together and to the surface. A Monod equation describes substrate use molecular diffusion within the biofilm is described by Fick s second law and mass transfer from the solution to the biofilm surface is modeled with a solute-diffusion layer. Six kinetic parameters (several of which can be estimated from theoretical considerations and others of which must be derived empirically) and the biofilm thickness must be known to calculate the movement of substrate into the biofilm. 

Microbial Biotransformation. Microbial population growth and substrate utilization can be described via Monod s (35) analogy with Michaelis-Menten enzyme kinetics (36). The growth of a microbial population in an unlimiting environment is described by dN/dt = u N, where u is called the "specific growth rate and N is microbial biomass or population size. The Monod equation modifies this by recognizing that consumption of resources in a finite environment must at some point curtail the rate of increase (dN/dt) of the population ... 

The Monod equation differs from the Michaelis-Menten equation in that it includes as a factor biomass concentration [X], which can change with time. A microbe as it catalyzes a redox reaction harvests some of the energy liberated, which it uses to grow and reproduce, increasing [X], At the same time, some microbes in the population decay or are lost to predation. The time rate of change in biomass... 

To account for reverse as well as forward reaction, the Monod (and dual Monod) equation can be modified by appending to it a thermodynamic potential factor, as shown by Jin and Bethke (2005), in which case the equation predicts the net rate of reaction. The thermodynamic factor Ft, which can vary from zero to one, is given... 

Appending Ft to the Monod equation, we write a thermodynamically consistent rate law,... 

If the concentrations of only the electron donor and acceptor are considered to vary, each mD+ is invariant and the term ] [ n/ 1 in Equation 18.23 reverts to a half-saturation constant K[y Similarly, the corresponding term in Equation 18.24 may be represented by K A. Now, we see the dual Monod equation (Eqn. 18.16) is a specific simplification of the general rate law (Eqn. 18.22). 

The bacteria in our example promote Reaction 18.7 at a rate given by Equation 18.15, the thermodynamically consistent form of the Monod equation,... 

Fig. 18.2. Results of modeling at 25 °C bacterial sulfate reduction using acetate as the electron donor, according to a thermodynamically consistent form of the Monod equation. Labels identify values and line slopes after seven days of reaction.

The acetotrophic sulfate reducers proceed at a rate r (mol s 1) given by Equation 18.16, the thermodynamically consistent dual Monod equation,... 

A continuous fermenter is operated at a series of dilution rates though at constant, sterile, feed concentration, pH, aeration rate and temperature. The following data were obtained when the limiting substrate concentration was 1200 mg/1 and the working volume of the fermenter was 9.8 1. Estimate the kinetic constants Km, //, and kd as used in the modified Monod equation ... 

When the yield coefficient yxs can be found by inspection of the data, the Monod equation can be put in the linearized form,... 

The exponential and limiting regions of cell growth can be described by a single relation, in which /x is a function of substrate concentration, i.e., the Monod equation... 

Although very simple, the Monod equation frequently describes experimental growth rate data very well. The form of this relation is shown in Fig. 1.23. 

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