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

— Owing to the complexity of biological membranes, many model systems have been studied in order to gain insight into the molecular mechanism of specific functions. One such model membrane extensively investigated in the past decade is the so-called bilayer lipid membrane (BLM). With suitable modifications, a BLM less than 100 A thick separating two aqueous solutions has been used as a model for a variety of biological membranes. This paper is devoted to a review of the properties and electronic processes of modified BLM.



Recent experiments using these membranes which contain photosynthetic pigments or dyes have demonstrated that, upon illumination, an EMF and a current can be generated. The connection between the photoelectric BLM and light-sensitive biological membranes and the rationale for this work are described.



Additionally, the effects of physical chemical parameters such as electric field, temperature, light intensity, duration of illumination and chemical agents (electron acceptors, donors, uncouplers, etc.) on the photoresponses of BLM are discussed. Other results indicate that BLM containing photoactive compounds behave similar to that of a silicon solar cell with one side of the membrane reducing and the other side oxidizing. The transverse pathway for the electron across the BLM could be provided by carotenoids such as β-carotene. Photoelectric BLM of this type represents a unique kind of energy transducing system and may well be useful in the conversion of solar energy into electricity and/or other forms of energy.

Electronic processes and photoelectric aspects of bilayer lipid membranes

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

Earlier many investigations were carried out to characterize the so-called crystalline chlorophyll because this highly ordered form of the main photosynthetic pigments (chlorophyll a and h) was believed to have some structural features in common with the in v im pigment states (Ke, 1966). Recently an aggregated chlorophyll form absorbing at about 740 nm, where the crystalline chlorophyll has a characteristic absorption band, was observed in intact plants (Thomas et al., 1970; lnoue et ul., 1972). Using artificial thin lipid membranes containing chlorophyll Tien (1968) observed a band at about 745 nm in the photoelectric action spectrum of the membrane. Although the membrane forming conditions seem to meet the requirements needed for the formation of the socalled crystalline chlorophyll, investigations revealing the presence of crystalline chlorophyll in model membranes are lacking. The present work was designed to study the formation of the so-called crystalline chlorophyll in membranes of various compositions.

Black lipid membranes (BLM) containing crystalline chlorophyll.

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.

Cs. Fajszi

Papers

The system of double inhibitions

Black lipid membranes (BLM) containing crystalline chlorophyll.

Isoenzymes of tetrameric proteins.