Rectangular hyperbolic

FIGURE 2.16 Effects of successive rectangular hyperbolae on receptor stimulus, (a) Stimulus to three agonists, (b) Three rectangular hyperbolic stimulus-response functions in series. Function 1 ((3 = 0.1) feeds function 2 ((3 = 0.03), which in turn feeds function 3 ((3 = 0.1). (c) Output from function 1. (d) Output from function 2 (functions 1 and 2 in series), (e) Final response output from function 3 (all three functions in series). Note how all three are full agonists when observed as final response. 

Successive rectangular hyperbolic equations necessarily lead to amplification (2.11.2)... 

FIGURE 3.2 General curve for an input/output function of the rectangular hyperbolic form (y = 50x/( 1 Ox + 100)). The maximal asymptote is given by A/B and the location parameter (along the x axis) is given by C/B (see text). 

In general, a model will express a relationship between an independent variable (input by the operator) and one or more dependent variables (output, produced by the model). A ubiquitous form of equation for such input/output functions are curves of the rectangular hyperbolic form. It is worth illustrating some general points about models with such an example. Assume that a model takes on the general form... 

As described above, mechanism-based inactivation conforms to a two-step reaction and should therefore display saturation behavior. The value of should be a rectangular hyperbolic function of [/]. This was described in detail above in Section 8.1. 

Figure 5. Dependence of rate of dissolution of 5pM Y-FeOOH in pH 4.0, 0.01M NaCl on concentration of a) tartaric acid, and b) salicylic acid. Fitted parameters obtained for rectangular hyperbolic model are given. Light source mercury arc lamp with 365nm band-pass filtering.

Most commonly, the rate of formation of the inactivated enzyme, under steady-state conditions, can be described by the rectangular hyperbolic function often associated with the traditional Henri-Michaelis-Menten function (25,26) ... 

Figure 3.27 illustrates a rectangular hyperbolic plot [Equation (3.169)], doublereciprocal plot [Equation (3.170)], Scatchard plot [Equation (3.171a)], and Hames plot [Equation (3.171b)] for the binding of NADH to rabbit muscle lactate dehydrogenase. 

When [S]o>the enzyme concentration, is usually directly proportional to the enzyme concentration in the reaction mixture, and for most enzymes v0 is a rectangular hyperbolic function... 

Under typical experimental conditions, the enzyme system is saturated with O2 and H+. Thus this enzyme system includes four substrates and four products. However, the initial steady state kinetics of this enzyme system obeys a simple Michaelis-Menten equation (a rectangular hyperbolic relation) for each kinetic phase of the two phases at low and high ferrocytochrome c concentrations as described above. This result indicates that the four ferrocytochromes c react with the enzyme in a ping-pong fashion in each substrate concentration range. That is, each ferroferrocytochrome c reacts with the enzyme after the previous cytochrome c in the oxidized state is released from the enzyme. Cytochrome c... 

Successive Rectangular Hyperbolic Equations Necessarily Lead to Amplification... 

A plot of V versus [S] fits a rectangular hyperbolic function (Figure 6-4). 

The complex rate constant in eq 15 is a rectangular hyperbolic function. It is exactly equal to k) at infinite dilution (the low-[M ] limit) and asymptotically approaches kMi as [M1 increases to large values. ... 

It is apparent from inspection of eq 27 that the observed rate constant, kobs, possesses a rectangular hyperbolic functional dependence on [M ] ... 

By analogy to the derivation of Equation (7.23), we can then substitute for the sum of [RL], terms with a sum of rectangular hyperbolic expressions as follows ... 

As we shall see later in this chapter, this rectangular hyperbolic expression provides a useful model for microbial growth kinetics. At this point we should examine the two constants that determine the shape of the V versus [S] curve. The value of Vmax is the maximum rate of reaction, that is, the rate attained at high values of [S], When is reached, further increases in [S] have no effect on reaction rate. 

When [SJo the enzyme concentration, Vq is usually directly proportional to the enzyme concentration in the reaction mixture, and for most enzymes Vq is a rectangular hyperbolic function of [S]q (see Fig. 5-15). If there are other (co-) substrates, then these are usually held constant during the series of experiments in which [SJo is varied. 

The first term In equation (14), representing the gross rate of biomass production, Is Identical with the function Monod (25) originally adopted "to express conveniently the relation between exponential growth rate and concentration of an essential nutrient." Such a rectangular hyperbolic function has been derived many times from various reaction mechanisms (26-30). but none has addressed the present case of continuous culture systems where y j and K have been observed to vary with temperature and dilution rate. 

The rectangular hyperbolic nature of the relationship, having asymptotes at v= and [S]=K, ... 

At a concentration equal to the Michaelis-Menten constant, half of the enzyme population will have substrate associated with it. Therefore, the Michaelis-Menten constant itself is an inflection point. As substrate concentrations exceed the Michaelis-Menten constant, the fraction of the enzyme population interacting with substrate is pushed towards 100%. This term produces the characteristic rectangular hyperbolic profile associated with the Michaelis-Menten equation shown above. 

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