An analysis of a fractal Michaelis-Menten curve

Simple chemical reactions can be described by the Michaelis-Menten response curve relating the velocity V of the reaction and the concentration [S] of the substrate S. To handle more complicated reactions without introducing general polynomial response curves, the rate constants can be considered to be scale dependent. This leads to a new response curve with characteristic sigmoidal shape. But not all sigmoidal curves can be accurately fit with three parameters. In order to get an accurate fit, the lower part of the integral shaped curve cannot be too shallow and the upper part can't be too steep. This paper determines an exact mathematical expression for the steepness and shallowness allowed.