Mathematical modelling and quantification of the autoinhibitory feedback control of noradrenaline release in brain slices

Summary

Concentration-response curves, reflecting α₂-autoreceptor-mediated inhibition of [³H]-noradrenaline release by exogenous noradrenaline in rat cerebral cortex and rabbit hippocampus slices, were analysed in order to test the usefulness of a mathematical model describing the relation between the independent variable, exogenous noradrenaline, and the dependent variable, inhibition of release. This model was based on the assumption of direct proportionality between receptor occupation and response, implying that there is correspondence between the shape of a concentration-binding curve and a concentration-response curve. The experimental concentration-response curves were obtained by different approaches: noradrenaline release from brain slices prelabelled with [³H]-noradrenaline was elicited electrically either by pseudo-one-pulse (POP) stimulation or by stimulation with 36 pulses applied with a frequency of 3 Hz. POP stimulation avoids autoinhibition by released noradrenaline and, therefore, was a suitable touchstone for the applied mathematical model which evaluates by nonlinear regression analysis two primary parameters: the dissociation constant between noradrenaline and the α₂-adrenoceptor and the biophase concentration of noradrenaline which reflects the extent of autoinhibition and should be zero under POP conditions. In rat cerebral cortex tissue, the corresponding biophase concentration of endogenous noradrenaline was indeed estimated to be zero and the dissociation constant was K _d = 10^{−7.62±0.14} mol/l. With 3 Hz stimulation, the biophase concentration was 10^{−7.80±0.05} mol/l, which has to be interpreted with respect to a simultaneously estimated K _d of 10^{−7.63±0.12} mol/l. Since the K _d-values under POP or 3 Hz conditions were similar, the biophase concentration obviously had no influence on the estimate of the other primary parameter, K _d. With rabbit hippocampus, however, the main prerequisite of the mathematical model; i.e. direct proportionality between receptor occupation and response, was not established since the slope of the POP concentration-response curve (estimated as slope parameter c) did not correspond to that of a concentration-binding curve.

In conclusion, mathematical modelling by nonlinear regression analysis of the autoinhibitory circuit of noradrenaline release allows the estimation of a parameter c of this feedback regulation which supports or rejects the assumption of direct proportionality between receptor occupation and functional response. When the given requirement of direct proportionality is shown to hold, this analysis allows the feedback circuit to be described quantitatively in terms of the affinity of noradrenaline for, and of the biophase concentration of noradrenaline at, the presynaptic α₂-adrenoceptors.