Biological rate equation

This equation illustrates that when there is incomplete S penetration in thick films, the observed rate depends only on the magnitude of the depth of penetration and not of the total thickness (La Motta, 1976b). This so-called active depth was specified by Kornegay and Andrews (1969), and it is often used for design purposes. The significance of internal mass transfer resistance to the interpretation of the kinetic data of suspended growth systems will be explained quantitatively. The biological rate equation, written as... 

Finally, the practicability of the rjj. concept will be illustrated for STRs. Here it is known that due to the gradients in shear rates, floe size distribution will be of consequence. Therefore, estimation of the biological rate equation coefficient is not possible in STRs—it is possible only in biofilm reactors. Nevertheless, the rj. concept remains a useful approach because it has been shown that a single floe size, dp (i.e., the mean value), is sufficient to characterize a given distribution function (Atkinson and Ur-Rahman, 1979). The mean floe size closely corresponds to the surface mean floe size ( Sauter diameter )... 

Figure 5.71. Equations derived from the biological rate equation (Atkinson, 1974) that can be used for the quantification of different cases of biofloc processing. ...

Most biological reactions fall into the categories of first-order or second-order reactions, and we will discuss these in more detail below. In certain situations the rate of reaction is independent of reaction concentration hence the rate equation is simply v = k. Such reactions are said to be zero order. Systems for which the reaction rate can reach a maximum value under saturating reactant conditions become zero ordered at high reactant concentrations. Examples of such systems include enzyme-catalyzed reactions, receptor-ligand induced signal transduction, and cellular activated transport systems. Recall from Chapter 2, for example, that when [S] Ku for an enzyme-catalyzed reaction, the velocity is essentially constant and close to the value of Vmax. Under these substrate concentration conditions the enzyme reaction will appear to be zero order in the substrate. 

Of course, the fractional calculus does not in itself constitute a physical/ biological theory however, one requires such a theory in order to interpret the fractional derivatives and integrals in terms of physical/biological phenomena. We therefore follow a pedagogical approach and examine the simple relaxation process described by the rate equation... 

However, as we will see in the next chapter, more complicated rate equations are often found for biological reactions. 

Figure 7.13 Ligand A collides with the biological macromolecular receptor B by traversing through a spherical surface of area 4jrr/g that is concentric about B. The total flux of A through this spherical surface, taking into account that B is also moving leads to a derivation of the encounter rate (equation 7.15).

Cohen and Monod (C2) have summarized experimental evidence which shows indeed that special mechanisms of transport of organic nutrients occur in bacteria. They call such transport systems permeases. This term ending in -use implies that the system involves enzymes—an implication not yet proved by available data. At any rate, it is found that permease systems can lead to transport against an apparent rise in concentration, as well as other effects not possible with Fickian diffusion. Various hypothetical mechanisms for operation of permease systems yield rates of permeation which exhibit the Michaelis-Menten type of dependence on substrate (including water) concentration. Perhaps it is in the occurrence of one of these mechanisms that the rate equation [Eq. (38)] assumed by Monod and almost all subsequent workers finds its justification. [For further information on biological transport, see, e.g., Christensen (Cl).]... 

Looking carefully at eqn [2], one sees that the solution depends on the ratio KJw but not or w separately. Similarly the equation for oxygen gives us information on the relative rates of upwelling and remineralization. It is only by the inclusion of radiocarbon, with its independent clock due to radioactive decay, that we can solve for the absolute physical and biological rates. The solutions to eqns [2]-[4] can be derived analytically, and as shown in Figure 2 parameter values of w = 2.3x10 cms K =... 

Most biological processes can be characterized by three types of rate equations, two of which are limiting cases of the third more general rate equation. The first of these is the first order rate equation represented by Equation 1. In this rate process, k represents the first order rate constant and C... 

Starred species are held constant by buffering, or reservoirs, or flows. A biological example will be given shortly. The macroscopic rate equations are given by... 

In situ formation. The chemical and biological processes generating a chemical species (T) In natural surface waters may be symbolically expressed by the rate equation... 

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