Global Analysis of Concentration-Response Curves

 

Using SigmaPlot to perform global curve fitting is described in the enzyme kinetics article. A previous approach used an index variable and index values placed in a column in the worksheet. But now the global fit method is part of the nonlinear regression engine and this can be used directly without the use of index values.

Wiener and Thalody describe the contraction of rabbit aorta to methaxomine and subsequent relaxation to dibenamine. Families of concentration-response curves were measured for varying concentrations of both methaxomine and dibenamine. The family of data sets was then globally fit to data using the following operational model of agonism:

where E is the measured response, A is the agonist concentration. The curve fit parameters are Em, KA, n and τ. Em, KA and n are global parameters that are fit using all data. τ is a local parameter that will be different for each data set; i.e., for two data sets there will be two parameters τ1 and τ2.

The SigmaPlot worksheet shows two data sets. The control data set (no dibenamine) is in columns 1 and 2. The data set with dibenamine is in columns 3 and 4. This data is not concatenated.

The curve fit equations for global analysis of this data are

[Variables]
A = col(1)   ‘methoxamine concentration
Y = col(2   ‘contractile response

[Parameters]
K = 5e-5   ‘initial estimate for Ka
E = 8.0   ‘initial estimate for Em
T1 = 12   ‘initial estimate for tau before and after dibenamine alkylation
n = 2   ‘initial estimate for slope factor

[Equation]
F = E*(Ti^n)*(A^n)/((K+A)^n + (Ti^n)*(A^n))   ‘concentration-response equation
fit F to Y

[Constraints]
Ti>0

[Options]
iterations=500

Right click on a graph data point and select Global Curve Fit

Then select E, n and K to share these parameters across the two data sets

Click Next and select the two data sets to be fit

Double click the curve fit icon to select it and click Next

Work the rest of the way through the Wizard to show the numeric results, select numeric output and define the graph characteristics.

The three shared parameter values and the two independently fit τ values for the first and second data sets, τ1 and τ2, are:

The graph showing the fits for the two data sets is:

The relaxation effect of dibenamine is shown and the global curve fit (solid lines) is excellent.

Download the file global_fit_to_concentration-response and double click on it to obtain the SigmaPlot notebook with this example containing the global curve fit equations.

1. Weiner, H.L. and Thalody, G.P. Differential attenuation of the responses to Adenosie and Methoxamine in isolated rabbit aorta. J. Pharm. Exp. Therap. 267 (1993) 828-837.

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