Michaelis-Menten kinetics, the operator-repressor system, and least squares approaches

The Michaelis-Menten (MM) function is a fractional linear function depending on two positive parameters.These can be estimated by nonlinear or linear least squares methods.The non-linear methods, based directly on the defect of the MM function, can fail and not produce any minimizer.The linear methods always produce a unique minimizer which, however, may not be positive. Here we give sufficient conditions on thedata such that the nonlinear problem has at least one positive minimizer and also conditions for the minimizer of the linearproblem to be positive.
    We discuss in detail the models and equilibrium relations of a classical operator-repressor system,and we extend our approach to the MM problem with leakage and to reversible MM kinetics.Thearrangement of the sufficient conditions exhibits the important role of data that have aconcavity property (chemically feasible data).