Monod function

Some formulations of jointly NO3/NH4 limited growth have not relied on inhibition. For example, Jamart et al. (1977) defined preference functions that divided the N sink between NH4 and NO3, while the two jointly regulate the phytoplankton growth rate by a single Monod function of their aggregate concentration. Similarly, Frost (1993) defined a Monod function for NH4 uptake, but assumed that N is not limiting to phytoplankton growth, and that any additional N requirements would be met by NO3. 

These assumptions are really quite mild the / need only be increasing and sufficiently smooth. It is not even required that f, be bounded on IR. From a biological perspective, of course, the model loses relevance for really large values of S. In addition to the Monod functions, other functions that have been suggested include the exponential kinetics AM(1—exp(—S log 2/a)), hyperbolic kinetics m tanh(S log 3/2a), and piece-wise linear kinetics given by mS/2a for S<2a and by w for S > 2a. The piecewise linear kinetics fails to satisfy the strict monotonocity of (iii) and fails to satisfy (iv) at one point, but these assumptions could be weakened so as to include this case. 

In Other words, p has the Monod form in S but the saturation value of the Monod function, p ,ax. decreases with cell quota g. Cunningham and Nisbet [CNl CN2] take p ,ax to be constant. Therefore, we assume that p is continuously differentiable in (S, g) for S > 0 and g > P and satisfies... 

Inhibition Function. Incorporation of an inhibition function into the model in lieu of the Monod function is essential for process failure to be caused by high concentrations of volatile acids at residence times exceeding the wash-out residence time as is observed in the field. Koga and Humphry (IJ) have shown that continuous cultures obeying the Monod function are stable except at the wash-out residence time. The function proposed by Haldane (12) for the inhibition of enzymes by high substrate concentration will be used as an inhibition function and may be expressed as ... 

Andrews (13) has discussed the basis for the use of this function, and a more detailed treatment of its properties is given by Dixon and Webb (14). Yano and Koga (15) and Edwards (16) have discussed the properties of other functions which may be used to reflect inhibition by substrate. Figure 3 illustrates the properties of the inhibition function with the Monod function shown for comparison. For low values of substrate concentration or high values of Ki (less inhibitory substrates) the inhibition function reduces to the Monod function. There is a considerable reduction in the maximum growth rate attainable as compared with the case without inhibition. The maximum rate attainable may be obtained by setting the first derivative of Equation 9 equal to zero and can be expressed as ... 

Both the Lotka-Volterra and the Monod functions for predator specific growth rate are dependent upon a single variable, the concentration of prey organisms. A third function, proposed by Contois O) for bacterial growth as an alternative to that of Monod, when applied to predator growth takes the form ... 

Figure 2.82 Rate of growth vs. biomass concentration substata [COD] according to the Monod function.

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

More than 30 years ago Jacob and Monod introduced the Escherichia coli lac operon as a model for gene regulation. The lac repressor molecule functions as a switch, regulated by inducer molecules, which controls the synthesis of enzymes necessary for E. coli to use lactose as an energy source. In the absence of lactose the repressor binds tightly to the operator DNA preventing the synthesis of these enzymes. Conversely when lactose is present, the repressor dissociates from the operator, allowing transcription of the operon. 

Monod kinetics Kinetics of microbial cell growth as a function of substrate concentration proposed by Jacques Monod and widely used to understand growth-substrate relationships. 

FIGURE 15.9 Monod-Wyman-Changeux (MWC) model for allosteric transitions. Consider a dimeric protein that can exist in either of two conformational states, R or T. Each subunit in the dimer has a binding site for substrate S and an allosteric effector site, F. The promoters are symmetrically related to one another in the protein, and symmetry is conserved regardless of the conformational state of the protein. The different states of the protein, with or without bound ligand, are linked to one another through the various equilibria. Thus, the relative population of protein molecules in the R or T state is a function of these equilibria and the concentration of the various ligands, substrate (S), and effectors (which bind at f- or Fj ). As [S] is increased, the T/R equilibrium shifts in favor of an increased proportion of R-conformers in the total population (that is, more protein molecules in the R conformational state). 

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 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... 

Inhibitory substrates at high concentrations reduce the specific growth rate below that predicted by the simple Monod equation. The inhibition function may... 

Linked-function mechanisms for cooperative binding interaction of metabolites and/or drugs, based on the presence of two or more different conformational states of the protein or receptor. See Adair Equation Cooperative Ligand Binding Hemoglobin Hill Equation Plot Koshland-Nemethy-Filmer Model Monod-Wyman-Changeux Model Negative Cooperativity Positive Cooperativity... 

Reduced concentrations are also useful in the Monod-Wyman-Changeux cooperativity model, where a = [F]/ Kj and ca = [F]/X t. This makes polynomial functions simpler to handle. For example, if ligand F binds exclusively to the R-state, then the ligand F saturation function, Tf, for an n-site protein equals (1 + +... 

COOPERATIVITY ALLOSTERISM INDUCED FIT MODEL MONOD-WYMAN-CHANGEUX MODEL INDEPENDENT BINDING LINKED FUNCTIONS ADAIR EQUATION POSITIVE COOPERATIVITY NEGATIVE COOPERATIVITY KOSMOTROPES CHAOTROPIC AGENTS HOFMEISTER SERIES Kp, Kq, K .. . ... 

Substituting Monod s mathematical description of microbial population growth, p, as a function of substrate concentration, [j], (Eq. 17-61) into Eq. 17-64, we have ... 

Noncompetitive inhibition cannot be completely reversed by very high substrate concentrations. Monod et al. defined for an allosteric enzyme a function of state R (Eq. 9-71) which is the fraction of total enzyme in the R (B) conformation ... 

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