Abrupt transitions in kinetic plots: an artifact of plotting procedures.

Kinetic data for enzymes that do not obey MichaelisMenten kinetics have often been displayed in ways that suggest abrupt transitions between straight portions of double-reciprocal plots or other plots that are straight in the Michaelis-Menten case, a notable example being provided by glutamate dehydrogenase (EC 1.4.1.3) from cattle liver (Engel & Dalziel, 1969). Taken literally, such plots indicate that the first derivative of the rate with respect to the substrate concentration is a discontinuous function of the substrate concentration, an interpretation that raises serious difficulties for the underlying physical causes of the behaviour. It seems preferable, therefore, unless the existence of a saltus in the derivative is overwhelmingly demonstrated, to interpret the appearance of one as an optical illusion that is strongly reinforced by drawing a pair of straight lines (which are, of course, not data but an interpretation placed on data) on a kinetic plot [see Fig. 3 of Cornish-Bowden & Koshland (1975)]. Presenting new data for glutamate dehydrogenase of Clostridium symbiosum, Syed & Engel (1987) have recently reasserted the reality of saltuses in kinetic data, dismissing earlier doubts about them (Cornish-Bowden & Koshland, 1975; Bardsley, 1977) as being 'essentially on semantic grounds'. Semantic considerations are not, however, the primary source of difficulty in accepting the reality of saltuses; moreover, the data presented by Syed & Engel (1987) fit a single saltus-free model almost as well as they fit a pair of straight lines. The relationships between different plots of kinetic data have been thoroughly explored by Bardsley and coworkers (e.g. Bardsley & Childs, 1975). Here it will be sufficient to note that the slope of an Eadie-Hofstee plot is defined if the slope of a plot of rate against substrate concentration is defined, and vice versa. A saltus in the slope of an Eadie-Hofstee plot therefore requires a saltus in the first derivative of rate with respect to substrate concentration, as assumed implicitly above. This is, however, impossible for any system in which the rate is a rational function of the substrate concentration with all denominator terms positive, i.e. for any rate equation that can be derived by the method of King & Altman (1956). It follows, therefore, that if a saltus exists the system cannot be in a true steady state or some other requirement for a valid steady-state analysis is not met. This point is not semantic and needs to be discussed by anyone proposing the existence of a saltus in kinetic data. The only semantic aspect lies in the meaning attached to a word like 'simpler' as applied to kinetic behaviour: is behaviour 'simpler' if it can be explained by reference to a well-defined and plausible model but requires a curve for its expression on an Eadie-Hofstee plot, or is it 'simpler' if it is easy to represent by an exercise in ruler-and-pencil geometry but has no basis in the known laws of chemistry? Data for glutamate dehydrogenase from Clostridium symbiosum were plotted in Fig. 1 of Syed & Engel (1987) assuming Michaelis-Menten kinetics with limiting rate V = 19 and Michaelis constant Km = 0.25 at NAD+ concentrations (s) less than 0.06, but with an abrupt change to V = 5.5, Km = 0.0289 at s greater than 0.06. (Rates are in ,uM/min per /sg enzyme, and concentrations are in mm, but as units complicate the discussion to no purpose they will be omitted.) Reading the co-ordinates of the 27 data points from their graph one can estimate a root-mean-square relative deviation of 4.6 %. As plotted originally, there is an obvious saltus at v/s = 60, v = 36, i.e. at s = 0.06 (referred to twice in their paper as s = 0.04, but this appears to be a typographical error). However, there are very few observations in the vicinity of the saltus, which can thus readily be interpreted as an illusion created by one or two erratic points on the plot, and the data are consistent with a model generating a smooth curve. In Fig. 1, the curve was calculated as the sum of two Michaelis-Menten expressions with (V, Km) = (18.2, 0.333) and (1.60, 0.000941). It does not