Derivation of Hill's Equation from Scale Invariance
Andrés Felipe Giraldo Ortíz, Vladik Kreinovich · 2012
In biochemistry, the amount of the ligand that gets attached to macromolecule is often well described by an empirical formula with an empirical parameter n; this formula is called Hill’s equation. In this paper, we provide a theoretical derivation of this formula from the generalized equations of chemical kinetics. In the traditional chemical kinetics equations, the reaction rate is proportional to the product of the concentrations [c] of all reactants c. For example, for the reaction A +B! C, the rate is proportional to the product [A] [B]; for the reaction 2A + B ! C, the rate is proportional to [A] [A] [B], etc. These formulas explain specic cases of Hill’s equation, corresponding to the case when n is an integer. To describe the general case, we assume a more general dependence f([A]; [B]) of the rate on the component concentrations. The numerical description of this dependence depends on the units that we use to describe the concentrations [A] and [B]. We make a natural assumption that this dependence is scale-invariant, in the sense that if we change units for measuring the concentrations [A] and [B], the formula remains the same if we appropriately change the unit for measuring the rate. We show that this assumption leads to an explicit formula for the generalized chemical kinetics which, in turn, leads to an expression for the amount of the bound macromolecule, an expression that includes the Hill’s equation as an important specic case. Thus, we get the desired theoretical derivation of the empirical Hill’s equation. c 2013 World Academic Press, UK. All rights reserved.