When can sigmoidal data be fit to a Hill curve?

Abstract

The Hill equation is a fundamental expression in chemical ikinetics relating velocity of response to concentration. It is known that the Hill equation is parameter identifiable in the sense that perfect data yield a unique set of defining parameters. However not all sigmoidal curves can be well fit by Hill curves. In particular the lower part of the curve can't be too shallow and the upper part can't be too steep. In this paper an exact mathematical criterion is derived to describe the degree of shallowness allowed.

A Simple Soft Limiter Describing Function for Biomedical Applications

“…• Some criteria developed by Heidel and Moloney [20] suggest that the Hill function, developed primarily for applications in pharmacology, is not a good candidate for the baroreflex curve data, following application to a range of test points on the curve. Our proposal is to use the arctangent function, of the form (8) This function provides a simple, symmetrical description of a sigmoid-like function and can, without approximation, be manipulated in a DF calculation to yield an attractive simple result.…”

“…The DF, representing the gain between the fundamental of the output and the input, can now be defined as (12) For the sigmoid of (8), with (13) Equation (13) can be recast as (14) and integrated, by parts, to give (15) (16) with the second term of the "integration by parts" disappearing at both upper and lower limits. Equation (16) is easily recast as (17) The remaining integral in (17) may be evaluated [21] as (18) Though the integral appears to be zero at both limits, observation of the symmetry of the integrand in (17) allows the integral to be recast as (19) and, with the final observation that (20) the DF in (17) may be evaluated as …”

Tensile fatigue performance of pultruded glass fiber reinforced polymer profiles

“…Despite the large dispersion of the measurements, frequency dependant fits with a Hill function were possible to describe the observed behavior. The basic equation of the function used is [10] DT…”

M ichaelis‐ M enten equation

“…Note that (1.1) has two parameters V m and K m , (1.2) has three parameters V m , K m and n while (1.3) has four parameters V m , K f , g s and g e . Recently, the authors [1] have made a data-fitting analysis of (1.1) and (1.2). For the Michaelis-Menten equation (1.1) let .V p1 ; S 1 / and .V p2 ; S 2 / be two points on the hyperbolic (concave down) response curve described by (1.1).…”

“…It depends on a precisely determined but complicated relationship between the three data points [1].…”