Fitting concentration profile

Fitting the results of concentration titration is the ultimate and most quantitative way to analyze results of concentration titration. In general, fitting is a method of empirical search for parameters that best describe experimental data. You can read more on fitting here.

From Beer-Lambert law we know that absorption changes proportionally with concentration. This simplifies fitting by limiting dependence to a linear function:
O.D. = A + B × C.

Here A and B are coefficients of a linear function and C is concentration of substance that you have previously calculated. Strictly speaking, Beer-Lambert law does not contain coefficient A and in ideal situation it should equal zero. It can be left in the linear function as a control for possible offset of background, although this value should be small.

Coefficient B contains two parameters according to Beer-Lambert law: pathlength and coefficient of molar extinction. Because you used a standard 1 cm cuvette and molar extinction is measured per cm pathlength,   B is effectively reduced to extinction coefficient alone.

Before you can carry out fitting analysis you need to decide what data to use. Most data books characterize absorption properties by molar extinction at absorption peaks, therefore it will be best to do fitting at the position of maximum absorption.  Return to spectral overlay plot and, using cursor, find row index (point) of the absorption maximum in the 240-300 nm region. Remember distortions that occur at high concentration and use one of moderate concentration spectra.

Once you know the row index of the maximum, copy that row of a matrix into a new wave. This will make fitting easier because Igor does not allow to use fitting dialog with matrices. First, duplicate CP_ wave with a new name using Duplicate Waves in Data menu or by executing the following from the command line:
Duplicate CP_MyMatix MyMatixProfXXX

Here CP_MyMatrix is the name of concentration calibration wave, MyMatrixProfXXX is the name of new wave where XXX is row index of absorption maximum.

Once wave is created, you can copy data from matrix into the new wave. This has to be done from command line:

MyMatrixProfXXX = MyMatrix[XXX][p].

Again, here XXX is the row index. You can read more on addressing waves in Igor.

Display this new wave against CP_ wave in a new graph. For convenience, open trace properties by double-clicking on the trace and change style to markers as well as change color. You can choose any marker style and color that you like, but it is more convenient to change color from standard red so you can see fitting results.

On the graph open Info window (Ctrl +I) and put cursor on the trace. Move cursor left and right to find the last point with highest concentration that is still on a straight line.  This will allow you to limit data range that will be used in fitting and eliminate error due to saturation.

From Analysis menu open Curve Fitting dialog. Set function field to Line, Y Data field to MyMatrixProfXXX and X Data field to CP_ wave. On Data Option tab in the End field of Range group specify pcsr(A) or pcsr(B) depending on whether you selected the end point using round (A) or square (B) cursor. Start range can be left blank.

On the Output Options tab make sure Destination field is set to _auto_. Enable X Range Full Width … option if it is available.

You are now ready to do the fit – press Do It button. Igor will calculate the line that fits you data the best using least squares method and repot results in the history area. A new trace should appear on the graph that represents calculated line.

Examine results of fitting. First, check if calculated line overlays experimental points well. If the higher concentration points do not fall on a line (experimental points curve across calculated line) you may have placed upper data limit too far. Move cursor to lower concentration and try again. If absorption at the limit is below O.D. =1.5 and data points still do not fall on the line - there may be a problem with concentration calculations. Typical point of deviation from linear dependence is around OD=2 with saturation around OD=3, but may be affected by selected wavelength and light absorption by other components.

Once you select the linear range and carry out fitting, examine the second coefficient of your calculated line. As described above, at 1 cm pathlength second coefficient equals molar extinction coefficient in inverse units of calculated concentration.  Find tabulated value for extinction coefficient of Trp and absorption maximum wavelength. Compare tabulated values to yours.  Because absorption bands are relatively wide, difference in the wavelength of absorption maximum of 2-3 nm will not affect molar extinction significantly. How close is your calculated molar extinction to tabulated value? Their difference is a good measure of accuracy of sample preparation and pipetting.