Hyperbola equations

The data have been fitted using the equilateral hyperbola equation. The calculated equation fits the... 

If we plot the fractional coverage 9 against the partial pressure P, we obtain a hyperbolae. Equation 5.107 can be written in the alternative form as... 

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 graph of n/n against plp° will thus be obtained as the difference between the two hyperbolae represented by the equations... 

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. ... 

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

In this form, Alhas the units of torr.) The relationship defined by Equation (A15.4) plots as a hyperbola. That is, the MbOg saturation curve resembles an enzyme substrate saturation curve. For myoglobin, a partial pressure of 1 torr for jbOg is sufficient for half-saturation (Figure A15.1). We can define as the partial pressure of Og at which 50% of the myoglobin molecules have a molecule of Og bound (that is, F= 0.5), then... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

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... 

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. 

For reverse titration, i.e., of arsenite with hexacyanoferrate(III), the situation will be the same as in Fig. 3.79 the part of the curve beyond the end-point is a hyperbola again, for which Kies gave the following equation ... 

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

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. 

If [S] = Km, the Michaelis-Menten equation says that the velocity will be one-half of Vmax. (Try substituting [S] for Km in the Michaelis-Menten equation, and you too can see this directly.) It s really the relationship between Km and [S] that determines where you are along the hyperbola. Like most of the rest of biochemistry, Km is backward. The larger the Km, the weaker the interaction between the enzyme and the substrate. Km is also a collection of rate constants. It may not be equal to the true dissociation constant of the ES complex (i.e., the equilibrium constant for ES E + S). 

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. 

As depicted in Figure 4, all the solutions for can be found graphically at each intersection of / ss (which is a straight line of slope — DM/ro) with the curve for /u = /u i + /u,2 (which is the sum of two hyperbolae with their corresponding vertical asymptotes at = Km,i and at = Km,2)- Due to the positive character of all the physical constants, one concludes that there is only one positive (physically meaningful) solution of equation (23). 

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. 

Figure 4. Precipitation became visible at the points marked by the dots. The shape of the curve (hyperbola) is described by the equation CaxP = rrf. Solutions with CaxP products above the curve show precipitation, while solutions with CaxP products below the curve remain free of precipitate. (Reproduced with permission from Ref. 4. Copyright 1983 American Society for Parenteral and Enteral Nutrition.)...

From this equation for the pressure, it is obvious that the Hugoniot curve is a hyperbola. Its asymptotes are the lines... 

Velocity data may be plotted in any one of a number of ways to illustrate the relation between v and [S], and plotting procedures are detailed later in this chapter. However, the Michaelis-Menten equation is an equation for a rectangular hyperbola, and plotting v versus [S] yields a hyperbolic curve, in the absence of cooperative or other unusual behaviors (O Figure 4-3). 

Fitting velocity data directly to a hyperbohc curve has several advantages over linear methods, transformed or otherwise. The major advantages are that no transformation of data is necessary, curves are fitted easily with currently available graphing software, and variations in behavior from a simple Michaelis-Menten one-substrate equation usually result in an equation which still describes a hyperbola, thus requiring no change in the analytical approach. 

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