Hyperbola rectangular

It is interesting to note that the BET equation is equivalent to the difference between the upper branches of two rectangular hyperbolae, as may be seen by breaking up the right-hand side of Equation (2.12) into partial fractions ... 

The Michaelis-Menten equation is, like Eq. (3-146), a rectangular hyperbola, and it can be cast into three linear plotting forms. The double-reciprocal form, Eq. (3-152), is called the Lineweaver-Burk plot in enzyme kinetics. ... 

FIGURE 14.7 Substrate saturation curve for au euzyme-catalyzed reaction. The amount of enzyme is constant, and the velocity of the reaction is determined at various substrate concentrations. The reaction rate, v, as a function of [S] is described by a rectangular hyperbola. At very high [S], v= Fnax- That is, the velocity is limited only by conditions (temperature, pH, ionic strength) and by the amount of enzyme present becomes independent of [S]. Such a condition is termed zero-order kinetics. Under zero-order conditions, velocity is directly dependent on [enzyme]. The H9O molecule provides a rough guide to scale. The substrate is bound at the active site of the enzyme. 

The Michaelis-Menten equation (14.23) describes a curve known from analytical geometry as a rectangular hyperbola. In such curves, as [S] is increased,... 

FIGURE 2.9 Amplification of stimulus through successive rectangular hyperbolae. The output from the first function (P = 0.3) becomes the input of a second function with the same coupling efficiency (P = 0.3), to yield a more efficiently coupled overall function (P = 0.069). Arrows indicate the potency for input to yield 50% maximal output for the first function and the series functions. 

It can be seen from Equation 2.2 that for positive non-zero values of p2, ptotai < pi- Therefore, the location parameter of the rectangular hyperbola of the composite set of reactions in series is shifted to the left (increased... 

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. 

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 ... 

The rates of many catalyzed reactions depend upon substrate concentrations, as shown in Fig. 4-7. The rate at high substrate concentrations is zeroth-order with respect to [S], falling until it shows a first-order dependence in the limit of low [S], This pattern is that of a rectangular hyperbola, defined by an empirical relation known as the Michaelis-Menten equation. 

Hill found a rectangular hyperbola fitted this mechanical relationship quite well ... 

Figure 4. When a muscle contracts isotonically or a constant resisting force is imposed on it during a contraction, the velocity at which it shortens quickly comes to a constant. The force-velocity curve shows the relationship between the force applied to a muscle and the steady-state velocity of shortening. As in all other muscles, the force-velocity curve of smooth muscle is a rectangular hyperbola for all positive shortening velocities. In order to compare the behavior of muscles of different lengths and diameters, it is common to normalize force and velocity by dividing each by its maximum value and expressing the result as a percentage, nd...

For uncompetitive inhibition, the value of kobs will increase as a rectangular hyperbola with increasing substrate concentrations according to Equation (6.17) ... 

With the logarithmic scale, the slope of the line initially increases. The curve has the form of an elongated S and is said to be sigmoidal. In contrast, with a linear (arithmetic) scale for [A], sigmoidicity is not observed the slope declines as [A] increases, and the curve forms part of a rectangular hyperbola. 

Typical units for B are pmol.mg protein-1, pmol.mg dry tissue, etc. A curve of B vs. [L] has the form of a rectangular hyperbola, exactly equivalent to the curve describing receptor occupancy presented in Chapter 1, Figure 1.1. 

Mathematically, the Michaelis-Menten equation is the equation of a rectangular hyperbola. Sometimes you ll here reference to hyperbolic kinetics, this means it follows the Michaelis-Menten equation. A number of other names also imply that a particular enzyme obeys the Michaelis-Menten equation Michaelis-Menten behavior, saturation kinetics, and hyperbolic kinetics. 

The Km is a landmark to help you find your way around a rectangular hyperbola and your way around enzyme behavior. When [S] < Km (this means [S] + Km = Km), the Michaelis-Menten equation says that the velocity will be given by v = CVnvdepends linearly on [S], Doubling [S] doubles the rate. 

The same solution presented in Fig. 14 can easily be recast (see Ref. 22) into another form, as in Fig. 15. Note here that the maximum scaled response curve is now essentially a rectangular hyperbola with one asymptote which depends only on the level of applied peak force and another asymptote which depends only on the level of applied total impulse. In the intermediate loading regime (the "knee" of the hyperbola), response determination requires knowledge of both peak force and total impulse. 

Plotting the parabola (broken curve) on the left of (1.40) with the rectangular hyperbola (solid curve) on the right (Fig. 1.4) shows that this cubic equation in eJlk has three roots given by the intersections pi, P2 and p3. However, since eJlk 1 at pi, this root is rejected, so (1.40) has two real solutions, P2 and p3, whose corresponding /rfc-values give rise to two P-states. 

When the rate of reaction is measured at fixed [E], but varying [S] and the results plotted, the Michaelis—Menten graph is obtained (below). This rectangular hyperbola indicates saturation of the enzyme with substrate. 

This curve describes any inverse relationship. The commonest value for the constant, k, in anaesthetics is 1, which gives rise to a curve known as a rectangular hyperbola. The line never crosses the x or the y axis and is described as asymptotic to them (see definition below). Boyle s law is a good example (volume = 1/pressure). This curve looks very similar to an exponential decline but they are entirely different in mathematical terms so be sure about which one you are describing. 

C The critical temperature line. This climbs from right to left as a rectangular hyperbola with a small flattened section at its midpoint. This is where a small amount of gas is liquidized. It climbs rapidly after this section as before. 

C A true rectangular hyperbola representing Boyle s law. The pressure doubles as the volume halves. As it is above the critical temperature, it is a gas and obeys the gas laws. 

The shape of the curve is an inverted rectangular hyperbola approaching Vmax. Ensure you mark on the curve at the correct point. The portion of the curve below on the x axis is where the reaction follows first-order kinetics, as shown by a fairly linear rise in the curve with increasing [S]. The portion of the curve to the far right is where the reaction will follow zero-order kinetics, as shown by the almost horizontal gradient. The portion in between these two extremes demonstrates a mixture of properties. 

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