Hyperbolic functions

Inverse Hyperbolic Functions If x = sinh y, then y is the inverse hyperbolic sine of x written y = sinh" x or arcsinh x. sinh" x = log x + + 1)... 

The cluster properties of the reactants in the MM model at criticality have been studied by Ziff and Fichthorn [89]. Evidence is given that the cluster size distribution is a hyperbolic function which decays with exponent r = 2.05 0.02 and that the fractal dimension (Z)p) of the clusters is Dp = 1.90 0.03. This figure is similar to that of random percolation clusters in two dimensions [37], However, clusters of the reactants appear to be more solid and with fewer holes (at least on the small-scale length of the simulations, L = 1024 sites). 

Directed Angles 27. Basic Trigonometric Functions 28. Radian Measure 28. Trigonometric Properties 29. Hyperbolic Functions 33. Polar Coordinate System 34. 

The inverse hyperbolic functions, sinh" x, etc., are related to the logarithmic functions and are particularly useful in integral calculus. These relationships may be defined for real numbers x and y as... 

While individual stimulus-response pathways are extremely complicated, they all can be mathematically described with hyperbolic functions. 

Series hyperbolae can be modeled by a single hyperbolic function (2.11.1)... 

Series Hyperbolae Can Be Modeled by a Single Hyperbolic Function... 

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 ... 

FIGURE 3.6 Classical model of agonism. Ordinates response as a fraction of the system maximal response. Abscissae logarithms of molar concentrations of agonist, (a) Effect of changing efficacy as defined by Stephenson [24], Stimulus-response coupling defined by hyperbolic function Response = stimulus/(stimulus-F 0.1). (b) Dose-response curves for agonist of e = 1 and various values for Ka. 

The operational model, as presented, shows dose-response curves with slopes of unity. This pertains specifically only to stimulus-response cascades where there is no cooperativity and the relationship between stimulus ([AR] complex) and overall response is controlled by a hyperbolic function with slope = 1. In practice, it is known that there are experimental dose-response curves with slopes that are not equal to unity and there is no a priori reason for there not to be cooperativity in the stimulus-response process. To accommodate the fitting of real data (with slopes not equal to unity) and the occurrence of stimulus-response cooperativity, a form of the operational model equation can be used with a variable slope (see Section 3.13.4) ... 

The transducer function defines the efficiency of the system to translate receptor stimulus into response and defines the efficacy of the agonist. Specifically, it is the fitting parameter of the hyperbolic function linking receptor... 

In terms of classical receptor theory—where response is a hyperbolic function of stimulus (Response = Stimulus/ (Stimulus 4- [3), [3 is a transducer function reflecting the efficiency of the stimulus-response mechanism of the system), and stimulus is given by Stimulus = [A] e/([A] + KA) (e is the efficacy of the agonist)— Response is given by... 

The retention index value is tenperature dependent and when eui index value is required at another tenperature it can be obtained by Interpolation using an Antoine-type hyperbolic ] function... 

A plot of HETP as a function of nobile phase velocity is a hyperbolic function (Figure 1.3) nost generally described by the van Deenter e< atlon (1.31). 

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. 

The first of these functions is effectively the sum of two sin expatta a shown in Fig. 14a, while the hyperbolic sine (sink) is the Btehce . (W) and Rg. 14b]. It should be noted that the hyperbolic functions have do period. They are periodic in the imaginary argument 2nr/. 

Because of this duality, every relation involving circular functions btf formal counterpart in foe corresponding hyperbolic functions, and vice ... 

On the other hand if x/m < h2/4m2, the eqffetion for z(t) is of the form of Eq. (25) and the solutions are in terms of expbhential functions of retil arguments or hyperbolic functions. In this case jr(r) is not oscillatory and will simply decrease exponentially with time. 

Thus, the various relations between the hyperbolic functions can be derived as carried out above for the circular functions. For example,... 

Equation 12.3.59 can thus be rewritten in terms of hyperbolic functions as... 

Hyperbolic Functions sinh z = (e2 — e )/2 cosh z = e + 2)/2 tanh z = sinh z/cosh z coth z = cosh z/sinh z csch z = 1/sinh z sech z = 1/coshz. Identities are cosh2z - sinh2z = 1 sinh (z, + z2) = sinh z, cosh z2 + cosh Zi sinh z2 cosh (z, + z2) = cosh Zi cosh z2 + sinh Zi sinh z2 cosh z + sinh z =et cosh z - sinh z = e. The hyperbolic sine and hyperbolic cosine are periodic functions with the imaginary period 27ti. That is, sinh (z + 2iti) = sinh z. 

One can also state [02] in terms of pressure described by the hyperbolic function of equation 4.9 giving the noncooperative, hyperbolic Mb curve in Figure 4.9 ... 

Different from conventional chemical kinetics, the rates in biochemical reactions networks are usually saturable hyperbolic functions. For an increasing substrate concentration, the rate increases only up to a maximal rate Vm, determined by the turnover number fccat = k2 and the total amount of enzyme Ej. The turnover number ca( measures the number of catalytic events per seconds per enzyme, which can be more than 1000 substrate molecules per second for a large number of enzymes. The constant Km is a measure of the affinity of the enzyme for the substrate, and corresponds to the concentration of S at which the reaction rate equals half the maximal rate. For S most active sites are not occupied. For S >> Km, there is an excess of substrate, that is, the active sites of the enzymes are saturated with substrate. The ratio kc.AJ Km is a measure for the efficiency of an enzyme. In the extreme case, almost every collision between substrate and enzyme leads to product formation (low Km, high fccat). In this case the enzyme is limited by diffusion only, with an upper limit of cat /Km 108 — 109M. v 1. The ratio kc.MJKm can be used to test the rapid... 

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