Figure 1.
In 1, 2 and 3 the geometric logic underlying the analysis of the CA hypothesis through the equation (4) is shown. Type 4 isoboles arise in many real responses corresponding to the IA hypothesis with null interaction, and illustrate the limitations of the relation between factual and formal aspects of isobole analysis beyond a particular case of the CA mode of action (see results, section 5).
Figure 2.
Ambiguity of the IA-CA dualism.
Rates of the enzymatic reactions yielding products P1 to P3 from substrates S1 and S2 are affected by the inhibiting effectors E1 and E2 with the specified inhibition constants kij. Under these conditions, responses measured as drops of the levels of any of the products are dependent of the nature (competitive, non-competitive, acompetitive) of the inhibition and the values of the inhibition constants. But any result can be unambiguously attributed to IA or CA modes of action.
Figure 3.
Possible response modifications involving alterations of the effective dose or the number of active receptors are illustrated (see Table 1). Notice that the alterations of the effector-receptor affinity (for simplicity reasons only perturbations are illustrated) do not modify the response to a given dose, but the response to a given time.
Table 1.
A possible systematics on the modifications of the response to an effector.
Figure 4.
Effect of a perturbator on the response (R) to a same dose series (D) of an effector and the parameters of the model (22).
The three cases in which the perturbation depresses the response are illustrated: reduction of the effective dose corresponding to the nominal one (left), reduction of the number of active receptors (canter), and increase of the threshold (right). Dots are simulated results in the absence (○) and presence (•) of increasing concentrations of the perturbator, and lines the respective fittings to model (22). See also tables 2 and 3.
Table 2.
Variations (+: increase; −: decrease; 0: no change) in the parameters of the response to an effector, as described by the equation (7), due to the presence of an agent which produces the specified perturbations.
Figure 5.
Possible variations of a parameter (θ) of the response to an effector, as a function of the concentration of a perturbator (P).
Any of the functions (19), (20) and (21) can produce all the profiles. Parametric value can be increased (+) or decreased (−), with constant (L), decreasing (A) or increasing (C) slope.
Table 3.
Simulation conditions of the responses to an effector as perturbed according to the three modalities that cause response drop, and respective fittings to the model (22).
Figure 6.
Joint response to two effectors in the four suppositions resulting from combining the two implicit key conditions of the IA hypothesis (see text).
In the first column, dots are the result of simulations and surfaces the respective fittings to the model (26). Isobolograms, correlations between observations and predictions and parametric variations (Ki: ○, mi: •) of the response to an effector as a function of the dose of the another are added. D1 and D2: doses; R: response. Numerical data in Table 4.
Table 4.
Simulation conditions and respective fittings (α = 0.05) to the generalized IA model in the specified examples.
Figure 7.
Some examples of joint responses to two effectors under IA mode of action.
Concrete types of interactions are specified and adjusted to the generalized model (26). Keys and graphic criteria as in Figure 6. See also Table 4.
Figure 8.
Joint response to two effectors under CA mode of action.
Concrete types of interactions are specified and adjusted to the generalized model (32). Keys and graphic criteria as in Figure 6. See also Table 5.
Table 5.
Simulation conditions and respective fittings (α = 0.05) to the generalized CA model in the specified examples.
Figure 9.
More examples of joint responses to two effectors under CA mode of action.
Concrete types of interactions are specified and adjusted to the generalized model (32). Keys and graphic criteria as in Figure 6. See also Table 5.
Figure 10.
An example of ambiguous interpretation of a response through IA and CA hypothesis.
A: IA response surface (defined by K1 = K2 = 1.00; m1 = m2 = 0.30; a1 = a2 = 2.00). B: result obtained when this surface is interpreted, through CA hypothesis, as a case of symmetrical antagonism (K = 1.00; m = 0.316; a = 2.348; cD2 = cD1 = 1.311. All coefficients statistically significant, with α = 0.05). Notice the lack of residual randomness. C: differences between CA minus IA (C1) and CA null interaction minus CA antagonism (C2) responses.
Figure 11.
Joint effect of ascorbic acid (A1) and trolox (A2) on crocin oxidation under different hypotheses.
Keys and graphic criteria as in Figure 6, with parametrtic variations replaced by residuals, which are more informative in this case. See details in text, and numerical results in.
Table 6.
Antioxidant joint action of ascorbic acid (A1) and trolox (A2) on crocin oxidation.