T hree separate approaches were applied to examine the general relationship between R, the rate of induction of specific enzymes (mitochondrial alpha-glycero-phosphate dehydrogenase and cytosolic malic enzyme) and q, the fractional nuclear occupancy by triiodothyronine in male Sprague-Dawley rats. Daily 200-microgram injections of triiodothyronine per 10u g body wt for 7 days resulted in saturation of the hepatic nuclear sites and the achievement of an apparent new steady state of enzyme levels. The increase achieved over base-line hypothyroid levels was then compared with the increment over hypothyroid base line characteristic of intact euthyroid animals with 47% of nuclear sites occupied. The maximal theoretical reate of steady-state enzyme induction could be protected on the basis of the observed maximal increase in enzyme activity observed 1 day after the injection of graded doses of hormone and lambda, the known fractional rate of enzyme dissipation. The 24-h dose-response studies were used to generate R as a continuous function of q, both in hypothyroid as well as in euthyroid animals. This approach involved the numerical solution of an ordinary differential equation describing the rate of change of enzyme as a function of R, which was assumed to be uniquely related to q. Results of these analyses indicated that the ratio of the maximal rate of induction of enzyme at full occupancy to the rate of induction under euthyroid conditions assumes a value between 9.0 and 19.5, depending on the precise analytic and experimental approach applied. This value is far in excess of the theoretical ratio 2.13 which on would anticipate if R were linearly related to q and 47% of the nuclear sites occupied under physiological conditions. Thus, the signal for enzyme induction appears to undergo progressjive amplification with increasing nuclear occupancy. Moreover, the curve describing the relationship between R and q appears highly nonlinear throughout (concave upwards). Although the molecular mechanism responsible for amplification is unknown, recognition of this phenomenon may be helpful in understanding tissue effects of thyroid hormone excess. Moreover, the analytic technique for determining R as a function of q may be of general applicability in studying hormonal response systems under nonsteady-state conditions.