Derivation and properties of Michaelis-Menten type and Hill type equations for reference ligands

doi:10.1016/0022-5193(76)90169-7

Journal of Theoretical Biology

Volume 59, Issue 2, 7 July 1976, Pages 253-254, IN1, 255-276

https://doi.org/10.1016/0022-5193(76)90169-7 Get rights and content

Abstract

The dose-effect relationships of the reference ligands (e.g. inhibitors) in the presence of the primary ligands (e.g. substrates) in enzyme kinetic models have been analyzed systematically. Enzyme reactions with different number of substrates, different reaction mechanisms, and different types and mechanisms of inhibition yield, equation (A), $f_{a} f_{u} = D D_{m}$ , for the first-order interactions; and equation (B), $f_{a} f_{u} = (D D_{m})^{m}$ , for the m-order interactions; where D is dose, D_m is the D required for the median-effect, and f_a and f_u are the fractions of the system that are affected and unaffected by D, respectively. Conversion of equation (A) gives $f_{a} = 1 [1 + (D_{m} D)]$ which has the same form as the Michaelis-Menten equation, $v V_{max} = 1 [1 + (K_{m} S)]$ . The logarithmic form of equation (B) gives the Hill type equation. In equations (A) and (B), the fractional velocity is expressed with respect to the control velocity rather than the maximal velocity; and D can be the reference ligands or environmental factors rather than the restriction to the primary ligands.

It is concluded that a plot of log (D) v. log [(f_u)⁻¹ − 1] or log [(f_a)⁻¹ − 1]⁻¹ will be a useful procedure for transforming dose-effect relationships and for analysing mass action characteristics in various systems. The plot will give the slope m, and the intercepts of the dose-effect lines with the median-effect axis (i.e. log [(f_u)⁻¹ − 1] = 0) will give D_m values. Any cause-consequence relationships that give straight lines in the plot will give m and D_m values, and thus give empirical equation for the system. Both equations (A) and (B) have been compared with relevant existing equations in terms of limitations and applicabilities. It is concluded that mechanism-dependent equations involving kinetic or equilibrium constants can be transformed to a mechanism-independent general equation involving the median-effect value.

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^☆: This work was supported in part by National Cancer Institute Grants CA-08748 and CA-16534.

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Derivation and properties of Michaelis-Menten type and Hill type equations for reference ligands☆

Abstract

J. biol. Chem

Biochem. J

Molec. Pharmacol

Arch. Biochem

Eur. J. Biochem

Br. J. Pharmacol

J. gen. Physiol

Eur. J. Biochem

J. biol. Chem

Biochemistry, N. Y

J. theor. Biol

J. Physiol

Biochim. Biophys. Acta

Biochim. Biophys. Acta

Biochim. Biophys. Acta

J. Cell. Comp. Physiol

J. Physiol

Acta Biol. Med. Germ