Logarithmic

Figure A4-9. Logarithmic function. The curve follows equation A4-7 with a - 2, b = 1.

Data with the behavior shown in Figure A4-9 can be fitted by the logarithmic equation A4-7.

The Trendline type is Logarithmic.

'Plateau" Curve. A relationship of the form ax y =

b + x exhibits the behavior shown in Figure A4-10. 1

Plateau Equations

Figure A4-10. Plateau curve. The curve follows equation A4-8 with a=\,b=\.

In biochemistry, this type of curve is encountered in a plot of reaction rate of an enzyme-catalyzed reaction of a substrate as a function of the concentration of the substrate, as in Figure A4-10. The behavior is described by the MichaelisMenten equation,

where V is the reaction velocity (typical units mmol/s), Km is the MichaelisMenten constant (typical units mM), Vmm is the maximum reaction velocity and [S] is the substrate concentration. Some typical results are shown in Figure A4-11.

[S], mM

Figure A4-11. Michaelis-Menten enzyme kinetics. The curve follows equation A4-9 with Fmax = 50, K,„ = 0.5.

Double Reciprocal Plot. The Michaelis-Menten equation can be converted to a straight line equation by taking the reciprocals of each side. This treatment is called a Lineweaver-Burk plot, a plot of the reciprocal of the enzymatic reaction velocity (1/F) versus the reciprocal of the substrate concentration (1/[S]).

A double-reciprocal plot of the data of Figure A4-11 is shown in Figure A4-12. The parameters Fmax and Km can be obtained from the slope and intercept of the straight line (Fmax = 1/intercept, Km = intercept/slope). However, relationships dealing with the propagation of error must be used to calculate the standard deviations of Fmax and Km from the standard deviations of slope and intercept. By contrast, when the Solver is used the expression does not need to be rearranged, yca)c is calculated directly from equation A4-19, the Solver returns the coefficients Fmax and K,„, and SolvStat.xls returns the standard deviations of Pmax and Km.

Hooke Law Graph

Figure A4-12. Double-reciprocal plot of enzyme kinetics. The curve follows equation A4-10 with Vmax = 50, Km = 0.5.

Logistic Function. The logistic equation or dose-response curve

1 + e-ax produces an S-shaped curve like the one shown in Figure A4-13.

X

Figure A4-13. Simple logistic curve. The curve follows equation A4-11 with a = 1.

In the dose-response form of the equation, the jy-axis (the response) is normalized to 100% and the x-axis (usually logarithmic) is normalized so that the midpoint (the half-maximum response or EC5o) occurs at jc = 0.

Logistic Curve with Variable Slope. In equation A4-11, the coefficient a determines the slope of the rising part of the curve; in biochemistry a is referred to as the Hill slope. Figure A4-14 illustrates the effect of varying Hill slope. At

Curve Hill Slope
Figure A4-14. Variable slopes of logistic curve. The three curves have a = 0.5, 1 and 2, respectively.

Logistic Curve with Additional Parameters. Equation A4-12 is the logistic equation with addition parameters that determine the height of the "plateau" and the offset of the mid-point from x = 0.

The height of the plateau is equal to b/c.

Sigmoid Activation Function

Figure A4-15. Logistic curve with additional variables. The curve follows equation A4-12 with a = 1, b = 0.5 and c = 5.

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