It is shown that the extent of sigmoidicity in kinetic or binding curves can always be characterized by two parameters, omega and delta, even when substrate inhibition or other causes of deviations from hyperbolic character are present. 2. The parameter omega is defined in such a way as to measure the fraction of the vertical span of the curve that is sigmoid. 3. The parameter delta is defined in such a way as to measure the S-shaped character in the sigmoid region. 4. It is shown that limits exist to the maximum values of omega and delta for degree n : n and so any individual v(S) or y(x) curve can be described as being barely sigmoid or very sigmoid by a comparison of measured omega and delta values with the limiting values. 5. Monte Carlo simulations of 12 kinetic mechanisms and 6 binding models were performed and the probability density functions and cumulative distribution functions for omega and delta were calculated. 6. An empirical study was performed on the delta values required before experimentalists can recognize a set of data points with error as being sigmoid. 7. The probability with which representative kinetic mechanisms or binding schemes give rise to complex curve shape features has been estimated before. Here using the parameters omega and delta, we calculate the conditional probabilities that sigmoid curves can occur in physiological ranges of substrate or ligand concentration and be sufficiently exaggerated to be recognized as S-shaped. 8. It is shown that some mechanisms, e.g. the random bi bi one, are very unlikely to give strongly sigmoid curves. Such exaggerated curves were found to be more typical of simple sequential kinetic schemes and binding models. 9. It is shown that omega and delta values measured experimentally can sometimes be used in model discrimination and fixing the minimum degree of rate equations. 10. A very powerful result is that in saturation functions of order n the maximum height of the curve that can be sigmoid is (n - 1)/n, and a possible evolutionary significance for this is suggested.
