Fig. 4: High-order Michaelis-Menten equations.

a Second order Michaelis-Menten equation. The moments combination, \({\Omega }_{2}={\left\langle {T}_{{turn}}\right\rangle }^{2}-\frac{\left\langle {T}_{{turn}}^{2}\right\rangle }{2}\), as a function of the reciprocal of the substrate concentration, \(\left[S\right]\) (measured in units of the Michaelis constant, \({K}_{M}\)). Solid lines come from Eq. (11) which is corroborated by numerical simulations (symbols). In all cases the linear relation predicted by Eq. (11) is shown to hold. b Third order Michaelis-Menten equation. The moments combination, \({\Omega }_{3}=\frac{\left\langle {T}_{{turn}}^{3}\right\rangle }{6}-\left\langle {T}_{{turn}}\right\rangle \left\langle {T}_{{turn}}^{2}\right\rangle+{\left\langle {T}_{{turn}}\right\rangle }^{3}\), as a function of the reciprocal of the substrate concentration (measured in units of the Michaelis constant). Solid lines come from Eq. (12) which is corroborated by numerical simulations (symbols). In all cases the linear relation predicted by Eq. (12) is shown to hold. For both (a) and (b), we assume a Markovian substrate binding process and consider three different cases for catalysis and unbinding: (i) exponential catalysis and unbinding times (\(\exp :\) blue), (ii) Gamma distributed catalysis time and exponential unbinding time (\({\gamma }_{{cat}}:\) orange), and (iii) Gamma distributed catalysis and unbinding times (\({\gamma }_{{cat}},\,{\gamma }_{{off}}:\) green) [see “Methods” section for details]. Source data are provided as a Source Data file. Codes employed for generating these data sets have been supplied as Supplementary Software.