Tuesday, 17th January 2012

Power Saturation Curves

Feature implemented in EPRtoolbox or download now

Power saturation overview

Determination of the effect of increasing microwave powers on the saturation behaviour of a sample can give information regarding the relaxation properties of the sample.

At low powers the amount of signal is linearly proportional to the microwave power. However, at high powers the signal dies away as a function of the inverse square root. Essentially you are saturating the sample, by giving it too much power the spin states cannot relax in the measureable time frame.

In a typical power saturation experiment, the microwave power is increased and the behavior is monitored by examining the peak-to-peak intensity of the signal as a function of the square root of the power.  The results can give an indication as to the identity of an unknown signal or determine whether two signals are originating from the same species.

More usefully, observing the power saturation behavior of a sample in the presence of relaxation enhancing substances like oxygen or NiEDDA can give useful information about the location of a spin label.

Since oxygen is nonpolar, it more readily enters a membrane bilayer than the aqueous solvent.  This means that radicals within the bilayer will see a more pronounced relaxation enhancement than those outside of the bilayer.  Conversely, NiEDDA is polar and will not enter the membrane bilayer and so will only interact with the spin label when the label is exposed to the solvent.

PowerSat

The main PowerSat window in action

PowerSat is a MATLAB program that will load raw Bruker EPR spectra (.DTA/.DSC or .spc/.par) from folders of experiments (nitrogen, oxygen, NiEDDA and a standard (usually DPPH)) or a 3 dimensional Bruker BES3T file (.DTA/.DSC/.YGF).

The user loads a dataset by clicking “Load Datasets” and then the file or folder option for the type of experiment. Upon loading, the program will automatically show all the spectra loaded, plotted on top of each other, as in the figure above.

The largest peak is automatic found after using a simple averaging method of all the spectra and centred. The display at the top of the screen is updated to show the peak and trough position (in Gauss) of these points. If the user chooses then they can manually change the position of the peaks.

Spectra fitting

Single point

In the original releases of PowerSat (v11.11 and before), the nearest data point to the peak and trough were taken forward for further calculation.

Single point

However, as the peak and trough position was calculated from the “average” spectrum of the dataset, the fitting is not good for the extreme datasets (shown above). Additionally for noisy spectra, the highest data point may not be the actual point of interest it could well be noise. For this reason some fitting options were developed.

Gaussian

This was the first fitting method introduced into PowerSat. It was included as a tickbox in v12.7 but is now the default option in v12.8.

By auto-zeroing the spectra we can then create two datasets from the spectrum. By using a simple logic operator on the first dataset we turn all negative datapoints to equal zero on the second we turn all positive datapoints to equal zero. This gives us 2 datasets, one containing only the peak and one the trough. We can then fit each to a Gaussian distribution.

Gaussian fitting by splitting the peak and trough

We can then take the point of each Gaussian forward into the calculations. As you can see above this gives a much better fit to the data and accommodates noisy data much better. However, the disadvantage is that the fitting algorithms require some time to process many spectra.

​Lorentzian

The Lorentzian function functions in an almost identical way to the Gaussian function described above. However, strictly speaking EPR spectra are(/should be) closer to Lorentzian distributions than Gaussian.

Lorentzian fitting by splitting the peak and trough

As you can see, the Lorentzian function does indeed fit the data better than a Gaussian function.

However, this function needs to be used with caution; Lorentzian distributions by their very nature take the tails of any peak much more into consideration than a Gaussian distribution. As a result of the logic operator to split the peak and trough the peaks are asymmetric and this can occasionally cause the fitting function to fail and instead fit a straight line across zero. This will result in some very odd power saturation curves.

This however, need not be an issue. Gaussian distributions are more concerned with the peak than the tails of the peak and it is the peak position/size that we are concerned with in the later calculations. For this reason Gaussian should be more than sufficient, and the Lorentzian function is only included here for completeness. As always, if you do not understand the processes behind a function then dont be suprised when it gives you the occasional odd result.

​Gaussian derivative

For the most accurate fitting we should consider each spectrum in its entirety rather than using crude splitting methods and independently fitting the results.

EPR spectra are themselves presented as first derivative absorbance spectra so they should fit well to a 1st derivative distribution.

By fitting the 1st derivative Gaussian distribution to the data we can pull out the coefficients (peak centre, height, width), by then taking the 2nd derivative and solving for y = 0 we can find the exact magnetic field positions of the peak and trough.

Gaussian fitting by derivative methods

This gives very accurate data, but the additional fitting and solving of equations takes longer computationally, and may not give much difference to the power saturation curves (or at least it should for good data)

​Lorentzian derivative

As previously mentioned in the Lorentzian section, EPR absorbances should appear more Lorentzian than Gaussian, so it logically follows that 1st derivative Lorentzian fit should be better than a 1st derivative Gaussian.

As with the Gaussian derivative method the spectrum’s peak is fitted to a 1st derivative. These fitting coefficients can then be used to construct the 2nd derivative which is solved for 0 to give the turning points (peaks) in the 1st derivative fit.

Lorentzian fitting by derivative fitting

Again the fit here is good, even for noisy data. The resulting power saturation curve from the spectrum above still gives a very good and close P1/2 to the other methods despite this fit not being the best.

Which fitting option do I use?

As with all fitting, the simple answer is whichever one works for you.

I have extensively tested each of the fitting methods and each has it’s own merits.

The single point method, is computationally the easiest to perform and works well as a “rough-and-ready” approach, with results displayed almost instantaniously. The problems with noise, however, limit this method to strong signals and experiments with no extremes in microwave power.

The Gaussian method is the default method for PowerSat. Whilst taking some computational time to fit each spectrum, it is still quite quick and it deals with noisy data well.

The Lorentzian method, whilst technically more correct than Gaussian, it is mostly included for completeness and not recommended as the fitting can often go get caught in local minimum thanks to the method’s focus on peak tails and return flat lines instead of peaks.

The derivative methods give theoretically the most accurate and true fitting, but this is at the expense of computational time. Taking approximately double the time of the non-derivate fitting methods.

For low motility samples (ie buried spin label sites) with large line broadening often the peak is sharpened but the trough is widened. For these types of sample it is recommended to avoid the derivative methods, as they assume a perfect distribution, but they may still work. In which case trial-and-error combined with professional judgement is required.

Fitting options on a DPPH standard

To emphasis the differences between the different fitting methods the following figures show the same DPPH data fitted using the different methods

Gaussian (P1/2 = 57.6) Lorentzian (P1/2 = 100.0)
Gaussian Derivative (P1/2 = 58.9) Lorentzian Derivative (P1/2 = 59.2)
Single point (P1/2 = 57.9)

​Viewing options

Fundamentally PowerSat has 2 methods for displaying raw data. The first shows all of one dataset at a time, the other shows a single spectrum at a time. Different datasets can be displayed by clicking the radio buttons in the Datasets box. Whilst viewing models can be toggled using the dropdown menu in the Viewing options box.

Showing all the spectra at once is good for visualising the entire experiment and will immediately show any outliers. Whilst the single spectrum view allows individual inspection of spectra and more importantly each spectrum’s fit.

Depending upon which fitting method has been selected the Spectra axes will display slightly different information. The information will always been shown in the colour appropiate to the dataset (oxygen = red, nitrogen = blue, NiEDDA = green, DPPH = orange)

View all spectra

With standard Gaussian or Lorentzian fitting methods the window shows 6 lines.

The coloured line corresponds to the peak, the black line the trough. Dashed lines show the peak position, whilst the dotted lines show the window around the dashed line.

With derivative Gaussian or Lorentzian fitting methods the window shows 2 lines (in dataset colours), corresponding to the fitting window limits about the centre point.

With single point fitting the window shows 2 lines (in dataset colours), corresponding to the peak and trough location

View single spectrum

With standard Gaussian or Lorentzian fitting methods the window shows the raw data plus 2 fitting curves in the experiment colours.

With derivative Gaussian or Lorentzian fitting methods the window shows 2 lines (in dataset colours), the dashed line shows the 1st derivative fit whilst the dotted line shows the calculated 2nd derivative spectrum.

With single point fitting the window shows 2 lines (in dataset colours), corresponding to the peak and trough location

Power Saturation Curves

Once the fitting routine has been completed the power saturation curves are automatically calculated. The peak-to-trough height for each spectrum is calculated and the power saturation curves drawn as peak-to-peak intensity against the square root of power.

This plot is fitted and the P1/2 calculated from it. When DPPH and Nitrogen experiments have already been loaded then the Oxygen and/or NiEDDA accessibility values (Π) are also calculated and displayed on screen.

Additionally the user can select the Error bars option in PowerSat Options this will show the figure with the fitting 95% confidence bounds.

By clicking the PowerSat Curves menu the user can export the fitting parameters to the MATLAB workspace. The user can also choose to export or save the figure.


This program requires the Mathworks Curve Fitting toolbox.

All figures shown on this webpage are direct screenshots from PowerSat or saved images from the PowerSat menu bar (and have not been editted in any way)

About Morgan Bye

Scientist, programmer and all around techie

Category

EPR toolbox