Quantitative analysis of dose-effect relationships: the combined effects of multiple drugs or enzyme inhibitors

The following simple and generalized equation not involving kinetic constants for substrates (Km) or inhibitors (Km) describes the summation of the effects of two mutually exclusive and reversible inhibitors on enzyme systems obeying Michaelis-Menten kinetics: 



where fv and fi(= 1 −fv) are the fractional velocity and the fractional inhibition of the reaction, respectively, and I50 is the concentration of the inhibitor (I) that is required to produce 50% inhibition (i.e. the median effect). More specifically, (fv)1 and (fv)2 are the fractional velocities in the presence of inhibitors I1, and I2, respectively, and (fv)1,2 the fractional velocity in the simultaneous presence of both inhibitors.

For mutually nonexclusive reversible inhibitors that follow Michaelis-Menten kinetics, the relationship becomes: 



Similar relationships apply to situations involving more than two inhibitors, for which generalized equations are given in the text. The above relationships express summation of inhibitory effects, irrespective of the number of substrates, the number of inhibitors, the type of reversibie inhibition (competitive, noncompetitive, or uncompetitive), or the kinetic mechanisms (sequential or ping-pong) of the enzyme reaction under consideration.

These concepts have been extended to higher-order (Hill-type) systems in which each reversible inhibitor has more than one binding site. If such inhibitors are mutually exclusive: 



where m is a Hill-type coefficient, assumed to be the same for each inhibitor.

If the inhibitors, I1, and I2, are mutually nonexclusive, this relationship becomes: 



It may be seen that (for higher-order mutually exclusive and nonexclusive inhibitors), when (fi)1,2= 0.5, the above relationships are equal to unity, and hence require no knowledge of the magnitude of the values of m for each inhibitor. The above relations at (fi)1,2= 0.5 describe I50-isobolograms (curves for isoeffective combinations of inhibitors).

The above-described relationships lend themselves to simple graphical representations, directly applicable to experimental results. Thus, plots of log [(= 0.5 describe I50-isobolograms (curves for isoeffective combinations of inhibitors).

The above-described relationships lend themselves to simple graphical representations, directly applicable to experimental results. Thus, plots of log [(fi)−1– 1]−1 versus log [I] for each inhibitor individually lead to their corresponding m values (i.e. slope) and log I50, values (i.e. intercept at y= 0). Linearity of such plots is diagnostic of Michaelis-Menten (where m= 1) or Hill (where m > 1) kinetics. Furthermore, if the two inhibitors are mutually exclusive and have the same m values, the plot of log [(fi)−1– 1]−1 with respect to log [I1+I2] will be linear and parallel to the plots for each component inhibitor. For mutually nonexclusive inhibitors, if the plot of log [(fi)−1– 1]−1 with respect to log [I] are linear for each individual inhibitor, then the plot of log [(fi)−1– 1]−1 with respect to log [I1+I2] will be concave upward, and will intersect ihe plot for the morc potent of the two inhibitors.

https://febs.onlinelibrary.wiley.com/doi/pdf/10.1111/j.1432-1033.1981.tb06218.x

Generalized Equations for the Analysis of Inhibitions of Michaelis‐Menten and Higher‐Order Kinetic Systems with Two or More Mutually Exclusive and Nonexclusive Inhibitors

The mechanisms of verapamil and tetraethylammonium (TEA) inhibition of voltage-gated K+ channels in LNCaP human prostate cancer cells were studied in whole-cell and outside/inside-out patch-clamp configurations. Rapidly activating outward K+ currents (I(K)) exhibited neither C-type, nor rapid (human ether a go-go-related gene-type) inactivation. With 2 mM [Mg(2+)](o), I(K) activation kinetics was independent of holding potential, suggesting the absence of ether a go-go-type K+ channels. Extracellular applications of TEA and verapamil (IC(50) = 11 microM) rapidly (12 s) inhibited I(K) in LNCaP cells. Blocking was also rapidly reversible. Intracellular TEA (1 mM), verapamil (1 mM), and membrane-impermeable N-methyl-verapamil (25 microM) did not influence whole-cell I(K), although both phenylalkylamines inhibited single-channel currents in inside-out patches. Extracellular application of N-methyl-verapamil (25 microM) had no influence on I(K). Our results are compatible with the hypothesis that, in LNCaP cells expressing C-type inactivation-deficient voltage-activated K+ channels, phenylalkylamines interact with an intracellular binding site, and probably an additional hydrophobic binding site that does not bind charged phenylalkylamines. The inhibiting effects of verapamil and TEA on I(K) were additive, suggesting independent K+-channel blocking mechanisms. Indeed, TEA (1 mM) reduced a single-channel conductance (from 7.3 +/- 0.5 to 3.2 +/- 0.4 pA at a membrane potential of +50 mV, n = 6), whereas verapamil (10 microM) reduced an open-channel probability (from 0.45 +/- 0.1 in control to 0.1 +/- 0.09 in verapamil-treated cells, n = 9). The inhibiting effects of verapamil and TEA on LNCaP cell proliferation were not multiplicative, suggesting that both share a common antiproliferative mechanism initiated through a K+ channel block.

Verapamil inhibits proliferation of LNCaP human prostate cancer cells influencing K+ channel gating.

Geometric analogue of holographic reduced representation

FDA-approved drugs and other compounds tested as inhibitors of human glutathione transferase P1-1.

A complete theoretical system of double inhibitions has been evolved where the inhibitors may be purely or partially competitive, noncompetitive, or purely uncompetitive with the substrate. A method is presented for the determination of interaction constants (a) for inhibitors on (i) the free enzyme, or (ii) the enzyme–substrate complex; and (b) in some cases for the substrate and one of the inhibitors on the enzyme complex formed with the other inhibitor. This method permits determination of the type of a pure inhibition in the presence of an unseparable inhibitor in the enzyme preparation. The different possible mechanisms of action of two inhibitors on the same enzyme are discussed and the method of their determination is described.

The system of double inhibitions

A theory is presented concerning the possible arrangements of protomers in tetrameric molecules. Isoenzymes may exist even in the case of homotetramers if the asymmetry of the identical protomers is detectable. The number of tetrahedral isoenzymes that can be isolated depends on the nature of the intersubunit bonds and on the level of the asymmetry of the protomers. Five isoenzymes can be distinguished only if two different types of protomers form tetrahedral tetramers and the asymmetry of protomers is not detectable with the method used. If the two types of protomers can bind each other by any pairs of binding sets and the asymmetry of both protomer types reaches the level of detection with the method used, we obtain 117 isoenzymes: 15 individual ones, and 51 stereoisomeric pairs.

T. Keleti

Papers

The system of double inhibitions

Isoenzymes of tetrameric proteins.