Monod’s equation

Unstructured distributed models such as Monod s equation satisfactorily predict the growth behavior in many situations. However, they cannot account for lag phases, sequential uptake of substrates, or changes in mean cell size during the growth cycle of a batch culture. Structured models recognize the multiplicity of cell components and their interactions. Many different models have been proposed based on the assumptions made for cell components and their interactions. 

It is of interest to note that Eqs. (170-172) reduce to Monod s model [Eqs. (41-42)] under certain conditions. These can be established by multiplying Eq. (170) by m and integrating the resulting equation over all w. One finds that Monod s equations are obtained if ... 

Cj = concentration of limiting substrate), then growth of a culture of rodshaped cells should follow Monod s equations. On the other hand, if Condition (ii) above is valid but Condition (i) is not, then one can show that substitution of Eq. (178) into Eq. (170) leads to Herbert s generalization [Eq. (Ill)] of Monod s equations. 

From this, we conclude that if Monod s equations (or Herbert s generalization thereof) and von Bertalanffy s equation [Eq. (176)] are valid, then the rate of transport of limiting substrate from environment to cell interior is proportional to CJ K + C ). However, if this transport were Fickian, the rate should be a linear function of C. Hence the simultaneous validity of Eq. (176) and Monod s equations for rod-shaped bacteria would appear to rule out ordinary transport and point instead to some more complicated mechanism. 

Thus, one sees from Eq. (180) that cultures of spherical bacteria should not follow Monod s equations. 

In this type of dispersed system some of the cells will be adsorbed at the drop surface and the others will be suspended in the continuous phase also. However if the dispersed phase is pure substrate, those cells attached to the drop surface will have a specific growth rate equal to the maximum specific growth rate, and those cells suspended in the continuous phase will have a specific growth rate that can be represented by Monod s equation. When mass transfer is important, the rate of growth in the continuous phase is limited by the quantity of substrate that diffuses into that phase. Since the growth at the drop... 

This equation simplifies to Monod s equation if one of the forward reaction rates is much slower than the others. A more complex situation obtains if two slow reactions are assumed. In the case that the second is slower than the first, a three-parameter equation results... 

This equation is also sometimes written as Monod s equation, except the reaction velocity is replaced with the specific growth rate ( a ) for the microorganism... 

A kinetic model originally derived by Nyholm is distinguished from Monod s model by the fate of a hmiting substrate. Instead of immediate metabolism, the substrate in Nyholm s model is sequestered. The governing equations are ... 

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 ... 

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 ... 

An understanding of the influence of environmental conditions on the kinetics of enzyme reactions is essential for the design of processes based upon the use of these materials as catalysts. Their growth kinetics are also governed by similar kinetic equations (c.f. Monod growth equations, Section S.9). 

Certainly, Monod s formula has been used extensively in phenomenological (unstructured) models, although the literature presents other equations for one limiting substrate systems (Equations 17 and 18). In Moser s formulation it was necessary to introduce a third parameter ( n in Equation 17) to represent experimental data. 

In the dynamics of any population, some members are bom, some members die, or some members simply grow en masse. In addition, in the absence of food, the population may cannibalize each other. The dynamics or kinetics of death and cannibalization may simply be mathematically represented as a mass decay of the population A [X, where is the rate of decay. Incorporating Monod s concept and the kinetics of death into Equation (87), the model for the net rate of increase of [X now becomes... 

It was considered a stirred tank reactor where the Desulfovibrio Alaskensis (DA) bacteria growth is carried out. The DA bacterium is a sulfate reducing bacteria used, in this case, to degrade some undesirable sulfate compounds to sulfides. It was previously determined that the Monod s kinetic equation adequately describes our reactor [2], where its kinetics parameters were determined via standard methodology in a batch culture [3]. 

While the Monod equation is an oversimplification of the complicated mechanism of cell growth, it often adequately describes fermentation kinetics. The Monod kinetic parameters can be determined by making a series of ideal continuous stirred-tank bioreactors, which will be discussed later. Table 19.6 shows the typical values of the Monod s kinetic parameters when glucose is a limiting substrate. 

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... 

As pointed out above, Tetrahymena pyrlformls consistently falls to clear batches of water of bacteria, and the explanation for this may be that some of the bacteria are too small or too large to be captured by the protozoans. If this Is the case, then there is no threshold density of the bacteria, of course, so there would be little justification for using models like Equations (1) or (2). In this circumstance, what one needs to do is to divide the bacteria into sub-populations, one which is eaten by the protozoans and one or more which is (are) not eaten by them. Monod s model might be assumed for feeding of the protozoans on the subpopulation of bacteria that they can eat, and models for transfer of bacteria from one sub-population to another would be needed, also. 

The present-day theory of microbial growth kinetics stems from, and is still dominated by, Monod s formulation (1942, 1949) of the function fi = /x(s), given in Equ. 2.54. Also, this relation is a homologue of the Michaelis-Menten equation Monod derived it empirically, and thus this is a formal kinetic equation. The consequence is a different interpretation of the parameters and K. The microbial growth rate is... 

In most cases of inhibition, the formal kinetic model equations are, like Monod s relationship, derived from theories of the inhibition of single enzymes. The equations are, however, only hypotheses they may be replaced by any other adequate model. 

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 ... 

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. 

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

Substrate-limited growth in terms of reduced availability of both the electron donor and the electron acceptor is common in wastewater of sewer systems. Based on the concept of Michaelis-Menten s kinetics for enzymatic processes, Monod (1949) formulated, in operational terms, the relationship between the actual and the maximal specific growth rate constants and the concentration of a limiting substrate [cf. Equation (2.14)] ... 

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. 

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