Quantitative relation of twitch and tonic tensions to intracellular Na ion was studied by describing model equations and measuring simultaneously the electrical, mechanical, and intracellular Na ion activities in electrically driven cardiac Purkinje fibers exposed to strophanthidin, tetrodotoxin (TTX), and varied [K+]0. In each experiment a plot of tension (T) vs. intracellular Na ion activity (aiNa) on logarithmic coordinates showed a good fit of the data to a single line described by the equation of T = beta (aiNa) gamma, in which beta and gamma represent the intercept and slope of the log T-log aiNa relation. Implication and average values of beta were presented. The average value of gamma obtained was 6.1 +/- 0.9 (SD, n = 8) in the experiments with strophanthidin (5 X 10(-7) - 10(-6) M) that somewhat depolarized transmembrane potential (Vm). The gamma value was 6.6 +/- 1.4 (n = 6) in the experiments with TTX (10(-6) - 5 X 10(-6) M) that shortened action potential duration. The gamma of 4.3 +/- 0.7 (n = 5) and 4.0 +/- 0.5 (n = 6) were obtained with the low [K+]0 of 1.0 and 2.0 mM, respectively, that hyperpolarized diastolic membrane potential by 12.7 +/- 6.3 mV (n = 5) and 13.7 +/- 2.4 mV (n = 6). The gamma value was 7.0 +/- 1.7 (n = 10) in the experiments with 8.1 mM [K+]0 that depolarized diastolic potential by 10.8 +/- 1.4 mV (n = 10). K+-free solution resulted in the gamma values of 6.0 +/- 0.9 (n = 6) for the twitch-aiNa relation and 5.3 +/- 1.4 (n = 9) for the tonic tension-aiNa relation. Except the experiments with the low [K+]0 the gamma values obtained are reasonably close to the value of 6.0 that is given on the basis of the 3Na-1Ca exchange and Ca2+-tension relation. In the experiments with the low [K+]0, the gamma values lower than 6.0 may be explained by the large hyperpolarization of diastolic membrane potential that could reduce intracellular calcium by means of the Na-Ca exchange. In conclusion, aiNa is a powerful determinant of twitch and tonic tensions which, in most instances, are linear functions of (aiNa)approximately 6.