QSoas tips and tricks: generating smooth curves from a fit

Often, one would want to generate smooth data from a fit over a small number of data points. For an example, take the data in the following file. It contains (fake) experimental data points that obey to Michaelis-Menten kinetics: $$v = \frac{v_m}{1 + K_m/s}$$ in which \(v\) is the measured rate (the y values of the data), \(s\) the concentration of substrate (the x values of the data), \(v_m\) the maximal rate and \(K_m\) the Michaelis constant. To fit this equation to the data, just use the fit-arb fit:

QSoas> l michaelis.dat
QSoas> fit-arb vm/(1+km/x)

After running the fit, the window should look like this:

Now, with the fit, we have reasonable values for \(v_m\) (vm) and \(K_m\) (km). But, for publication, one would want to generate "smooth" curve going through the lines... Saving the curve from "Data.../Save all" doesn't help, since the data has as many points as the original data and looks very "jaggy" (like on the screenshot above)... So one needs a curve with more data points.

Maybe the most natural solution is simply to use generate-buffer together with apply-formula using the formula and the values of km and vm obtained from the fit, like:

QSoas> generate-buffer 0 20
QSoas> apply-formula y=3.51742/(1+3.69767/x)

By default, generate-buffer generate 1000 evenly spaced x values, but you can change their number using the /samples option. The two above commands can be combined to just one call to generate-buffer:

QSoas> generate-buffer 0 20 3.51742/(1+3.69767/x)

This works, but it is quite cumbersome and it is not going to work well for complex formulas or the results of differential equations or kinetic systems...

This is why to each fit- command corresponds a sim- command that computes the result of the fit using a "saved parameters" file (here, michaelis.params, but you can also save it yourself) and buffers as "models" for X values:

QSoas> generate-buffer 0 20
QSoas> sim-arb vm/(1+km/x) michaelis.params 0

This strategy works with every single fit ! As an added benefit, you even get the fit parameters as meta-data, which are displayed by the show command:

QSoas> show 0
Dataset generated_fit_arb.dat: 2 cols, 1000 rows, 1 segments, #0
Flags: 
Meta-data:	commands =	 sim-arb vm/(1+km/x) michaelis.params 0	fit =	 arb (formula: vm/(1+km/x))	km =	 3.69767
	vm =	 3.5174

They also get saved as comments if you save the data.

Important note: the sim-arb command will be available only in the 3.0 release, although you can already enjoy it if you use the github version.

About QSoas

QSoas is a powerful open source data analysis program that focuses on flexibility and powerful fitting capacities. It is released under the GNU General Public License. It is described in Fourmond, Anal. Chem., 2016, 88 (10), pp 5050–5052. Current version is 2.2. You can download its source code and compile it yourself or buy precompiled versions for MacOS and Windows there.

QSoas quiz #1 : averaging spectra

Here is the first QSoas quiz ! I recently measured several identical spectra in a row to evaluate the noise of the setup, and so I wanted to average all the spectra and also determine the standard deviation in the absorbances. Averaging the spectra can simply be done taking advantage of the average command:

QSoas> load Spectrum*.dat /flags=spectra
QSoas> average flagged:spectra

However, average does not provide means to make standard deviations, it just takes the average of all but the X column. I wanted to add this feature, but I realized there are already at least two distinct ways to do that...

Quiz

Your task is to determine the average and standard deviations of the three spectra located there, (Spectrum-1.dat, Spectrum-2.dat and Spectrum-3.dat). There are at least two ways:

• One that relies simply on average and on apply-formula, and which requires that you remember how to compute standard deviations.
• One that is a little more involved, that requires more data manipulation (take a look at contract for instance) and relies on the fact that you can use statistics in apply-formula (and in particular you can use y_stddev to refer to the standard deviation of \(y\)), but which does not require you to know exactly how to compute standard deviations.

To help you, I've added the result in Average.dat. The figure below shows a zoom on the data superimposed to the average (bonus points to find how to display this light red area that corresponds to the standard deviation !).

I will post the answer later. In the meantime, feel free to post your own solutions or attempts, hacks, and so on !

About QSoas

QSoas is a powerful open source data analysis program that focuses on flexibility and powerful fitting capacities. It is released under the GNU General Public License. It is described in Fourmond, Anal. Chem., 2016, 88 (10), pp 5050–5052. Current version is 2.2. You can download its source code or buy precompiled versions for MacOS and Windows there.