Z theo~ BioL (1989) 136, 151-170 Neurotransmitter Release: Development of a Theory for Total Release Based on Kinetics CORNEL LUSTIGt, HANNA PARNAS~ AND LEE A. SEGELt§ Department of Applied Mathematics, Weizmann Institute of Science, 76100 Rehovot, Israei, t and Department of Neurobiology, Institute of Life Sciences, Hebrew University, 91904 Jerusalem, lsrael$ (Received 8 July 1986, and accepted in revised form 26 April 1988) According to the calcium-voltage hypothesis for the control of neurotransmitter release, a molecule (or molecular complex) must be activated by membrane depolarization, after which the activated molecule can bind calcium and initiate release. In this study, we have examined properties of the kinetics of phasic release resulting from a set of differential equations that characterize the calcium-voltage hypothesis. It was found that, in accord with experiments, an important feature is the approximate constancy of the shape of the graph for the kinetics of phasic release at various depolarizations and extracellular calcium concentrations. The shape constancy allowed us to obtain an explicit and relatively simple analytical formula for the total transmitter release (quantal content) by approximating the differential equations of the model. This formula shows a saturating sigmoidal dependence on both intracellular and extracellular calcium concentrations. The formula thus agrees with various experiments. Moreover, it agrees with, and provides meaning to, earlier phenomenological expressions for the dependence of release on calcium concentration. In particular, the formula provides an expression for the maximal release in terms of kinetic parameters from the calcium-voltage model, and thereby allows one to supplement earlier kinetic tests of the calcium-voltage hypothesis with further tests focused upon the dependence of total release on depolarization. 1. Introduction It is well-established that release of neurotransmitter from nerve terminals requires calcium. More precisely, release requires the accumulation of intracellular calcium, which enters the terminal by virtue of a depolarization-controlled opening of calcium channels. The intracellular calcium concentration is difficult to manipulate in a controlled manner, so that experiments have concentrated on observing the effects of changing the extracellular calcium concentration Ce and the depolarization D. The experimental dependence on C~ of the average amount of transmitter released per pulse, i.e. of the total release LT, has been empirically summarized. The empirical formula for LT exhibits a sigmoid dependence on Ce (Dodge & Rahamimoff, 1967; Parnas et al., 1982a, b) and was suggested to show the same dependence on intracel- lular Ca 2+ (Parnas & Segel, 1980, 1981). § To whom all correspondence should be addressed. 15! 0022-5193/89/020151 + 20 $03.00/0 © 1989 Academic Press Limited