Rectangular hyperbolic function

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. 

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) ... 

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... 

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 ] ... 

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. 

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. 

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... 

The function f is usually hyperbolic, which introduces the nonlinearity between receptor occupancy and response. A common experimentally observed relationship between receptor stimulus and response is a rectangular hyperbola (see Chapter 2). Thus, response can be thought of as a hyperbolic function of stimulus ... 

Hyperbolic functions are combinations of positive and negative exponentials. They resemble gonlometrlc functions and derive their names from the fact that they describe the coordinates of points on rectangular hyperbolas. They are often encountered In diffuse double layer theory. ... 

Unlike the circular functions, the ratios x/a, yja, when referred to the hyperbola, do not represent angles. An hyperbolic function expresses a certain relation between the coordinates of a given portion on the arc of a rectangular hyperbola. 

As an illustration, let us do the double integration over area involved in the geometric representation of hyperbolic functions (see Fig. 4.14). Referring to Fig. 10.3, it is clearly easier to first do the x integration over horizontal strips between the straight line and the rectangular hyperbola. The area is then... 

The spatial domain for problems normally encountered in chemical engineering are usually composed of rectangular, cylindrical, or spherical coordinates. Linear problems having these types of domain usually result in ODEs (after the application of separation of variables) that are solvable. Solutions of these ODEs normally take the form of trigonometric, hyperbolic, Bessel, and so forth. Among special functions, these three are familiar to engineers because they arise so frequently. They are widely tabulated in handbooks, for example, the handbook by Abramowitz and Stegun (1964) provides an excellent resource on the properties of special functions. 

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