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Appendix

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APPENDIX A

Curve Fitting by Using Hill's Characteristic Equation

With reference to Eq. 1, the SSR is given by

(6)

where n is the number of data points in a set of F-V data. Curve fitting can be considered as searching for the combination of F_max, a, and b that will give a minimal SSR.

To start a search, the input for F_max, a, and b were 100-105% of F_max at intervals of 1%, 50-200% at intervals of 4%, and 100-400% at intervals of 4%, respectively. These initial inputs for a and b were based on the results of the searches performed on several sets of data. In the subsequent rounds of search, the inputs for the F_max remained the same. The inputs for a and b were plus and minus 50 of the respective values obtained in the previous round at intervals of 1%. A search would be terminated when the combinations obtained in two consecutive rounds of search were identical. In general, three to five rounds of search were required before the optimal combination was found.

Let be the sample mean of the forces

(7)

and SSY be the sum of the squares of deviation associated with

(8)

the coefficient of determination (r²) was obtained by

(9)

APPENDIX B

Determination of the Relationship Between a Hill Constant and the Activation Level

Mathematically, a parabola (2nd-order polynomial) in a two-dimensional rectangular coordinate system (x-y plane) can be expressed as

(10)

where (p, q) is the coordinate of the parabola's vertex and f is a constant related to the focus of the parabola. With reference to Fig. 4, a parabola passing through the data point for 100% AL (100, 59) is

(11)

Rearranging the terms, we get

(12)

Substituting Eq. 12 into Eq. 10, we have

(13)

Therefore, a second-order polynomial passing through (100, 59) will have the form of Eq. 13.

J Appl Physiol 86: 1025-1031, 1999

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