Phasor Graph

The Phasor Graph is the central tool for phasor analysis as implemented in AlliGator. This page discusses how to use the many functions associated with the Phasor Graph, including those related to:

• Phasor Calibration

  • Phasor Ratio Analysis

  • Phase Lifetime

The Phasor Graph tab represents data selected in the image ROI in a complementary way to the fluorescence decay plot. Discussion of the phasor representation of fluorescence decay data can be found in many places in the literature and will not be repeated here.

A phasor point is added to the plot each time the ROI Analysis (Crtl + A) menu item is used, but also when phasor-specific actions are performed, as described below.

The value of the Current Phasor (g, s) are displayed to the right of the plot, together with the corresponding modulus and phase (m, ϕ) as well as the apparent Phase Lifetime, defined as:

                                 (1)

Changing g or s automatically updates m, ϕ and the Lifetime, while changing m or ϕ automatically updates g, s and the Lifetime.

1. Phasor Calibration

Uncalibrated phasors are not particularly useful in general, and it is therefore recommended to provide a sample with known lifetime and acquired in the same experimental conditions as the sample of interest. To use a ROI within this sample as calibration:

• Use the Analysis >> ROI Analysis (Ctrl+A) menu item

• Specify the Calibration Lifetime value

• Right-click on the Phasor Graph and select Phasor Calibration >> Use Current Phasor as Calibration:

As illustrated in the figure below (Before Applying Calibration red arrow), an uncalibrated phasor will in general be far from the universal circle (UC).

Once the calibration is stored, the corresponding status indicator (first green LED) will light up (bottom right of the panel).

To use the newly defined calibration, check the Use Calibration Phasor checkbox if it not already checked. The red LED next to the status indicator will light up, indicating that this calibration is now used (and will be for future calculations).

Note that it is possible to enter an arbitrary phasor value in the Current Phasor control and right-click the label and choose Use as Calibration Phasor in order to use this specific value as the calibration phasor.

A new calibration will override the old one and be applied to all data points within the current phasor plot. Previous plots will not be updated automatically, but can be updated manually using either the Update Single Plot Calibration or Update All Plots Calibrations context menu items.

As visible on the shortcut menu shown above, it is also possible to clear a previously defined calibration.

Note that it is possible to load or save calibration using the following (main menu bar) Calibration menu items:

• Save Phasor Calibration: saves the current Phasor Calibration to a file. The file extension is automatically set to .cal.

• Load Phasor Calibration: loads a Phasor Calibration from a file.

Note: 

A Phasor Calibration includes the phasor value, the lifetime and phasor frequency. Trying to load a Phasor Calibration saved with a different phasor frequency than the currently defined one will fail and result in a warning dialog box.

Note also that the right-click menu gives access to other sets of functions:

• for time series calibration: Use as Phasor Calibration Curve and Clear Phasor Calibration Curve. These items are discussed in the Time Series Analysis manual page.

• for location-dependent calibration: Use as Phasor Calibration Map and Clear Phasor Calibration Map. These functions are used with a Phasor Plot that consist of a several phasor points pertaining to the same dataset, i.e. phasors corresponding to different ROIs in a dataset (by contrast with the previous case, where a phasor time series involves a single ROI in each dataset, but several datasets. This is typically used with ROI grids covering the field of view of interest, and allows accounting for spatial variation in the sample IRF or detector response.

2. Phasor Ratio References

Phasor Ratio Analysis (discussed in more detail in the corresponding manual page) requires two reference points, defined on the Phasor Graph.

To define Reference 1 or 2, right-click on (or close to) the selected point and use the corresponding menu item (Use as Reference 1 or Use as Reference 2):

The Reference 1 point will appear as a green dot (see figure below), while Reference 2 will appear as a red dot. The corresponding task indicator LEDs will light up (bottom right of the panel).

Note that the phasors defined as reference in this manner need only be displayed in the Phasor Graph.

In particular, they can be points on the Universal Circle (UC plot), generally represented by a dashed line (see figure above). In other words, they do not need to be experimental points. The next section will describe another way to define reference points, which does not use experimental data.

Another way to define reference phasors is by using the Current Phasor control's shortcut menu (right-click on the Current Phasor label):

This method allows typing in any phasor value or use the current position of one of the cursors and set it as Reference 1 or Reference 2.

Finally, the phasor references are not corrected in any way when computing a phasor ratio. In other words, if they are obtained from experimental data, this data needs to be calibrated before the reference points are selected from it.

Clearing the phasor data (see next section) does not clear the reference phasors, which are cleared using their respective menu items (Clear Reference 1 and Clear Reference 2, or to clear both, Clear All References).

Each (or both) reference can be saved in a XML file using the Save/Load References context menu items. The file extension is automatically set to .ref.

3. Phasor Ratio Line

When the Show Line checkbox is unchecked, the cursors can be used to explore the Phasor Graph (their coordinates are displayed in the lower right box below the graph).

When it is checked, a dashed gray line connecting both cursors is displayed:

In addition, the lifetimes of the intersections of this line with the universal circle are displayed in the bottom right corner of the panel (not shown above). This can be useful to get an idea of limiting values of phasors in a series of data points aligned along an approximately straight direction. While this does not guarantee that the experimental phasors are a linear combination of these two limiting values, it might be a valid hypothesis.

In this case, it is possible to use the Phasor Ratio References >> Use UC/Cursor-Defined Line Intersections context menu item to define these two points as the reference phasors to be use in Phasor Ratio Analysis. Alternatively, the Phasor Ratio References >> Use UC/Fitted Line Intersections context menu item can be used to fit a line through the set of phasor points displayed in the Phasor Graph, and use the intersections of this line with the universal circle as reference phasors.

4. Managing plots in the Phasor Graph

The Phasor Graph is comprised of one or more plots. The Universal Circle (UC) is always present and cannot be removed (however, its style and appearance can be modified, and in particular, it can be hidden by checking off its visibility checkbox in the legend).

Other plots are either created automatically or upon user request, as explained next.

When starting a Time-Series Analysis (see the corresponding manual page for details), a new plot is created and added to the list, and is used to store all phasors computed during the analysis. Aborting the analysis and restarting it will preserve the first, incomplete plot and start a new one for the new analyzed phasors.

Alternatively, it is possible to group different computed phasors into different plots. For instance, this can be useful when manually analyzing a dataset comprised of different samples which you may want to group into different series. To do so, perform the analysis of the first series of sample in the dataset (which will automatically create a new plot) and when done, use the New Plot context menu item:

This will add a new (empty) plot to the Phasor Graph, to which future phasors will be added. An arbitrary number of plots can be added in this manner.

Occasionally, it can be necessary to cleanup the Phasor Graph by removing all or a particular plot from it, or to remove a particular phasor point from a specific plot.

This is done using the right-click context menu:

• Remove Data Point removes the point closest to the right-click location from whichever plot it belongs to.

• Delete Plot removes the plot pointed at in the Phasor Graph legend.

Note that in the first case, it might be necessary to zoom in to be able to isolate the point to remove from neighboring ones.

5. Loading and Saving Phasor Plots

In addition to the standard ASCII file saving/loading functions available in other graphs (Save Plot, Save Selected Plots, Save All Plots), the Phasor Graph offers special functions to save the extra information necessary to make sense of them for phasor analysis (mostly phasor harmonic frequency). Since most phasor plots are used in conjunction with a phasor calibration, and this can be changed at will (see Section 1 above), phasor plots are saved as uncalibrated phasors. In other words, when saving a calibrated phasor plot, it is a good idea to also save the phasor plot used to calibrated it, so that it can be reloaded later on an reused as calibration. The Phasor Graph specific file functions are:

• Save Phasor Plot saves a file of the uncalibrated phasor plot. The file extension is automatically set to .phplot. The first time a phasor plot is saved, this function asks the user to select the folder in which to save the plot (as well as offers the ability to name the file). Any subsequent phasor plot saved will be saved in that folder and will be named Plot Name_Phasor Plot.phplot, where "Plot Name" is the name of the plot in the Phasor Graph. Any already existing file will be overwritten.

• Save Phasor Plot as... will offer the user a chance to define the name and location of the plot to be saved (the file extension is still enforced to be .phplot).

• Save All Phasor Plots will accomplish for all plots what Save Phasor Plot does for an individual plot. The UC plot is not included in this action.

• Load Phasor Plot(s) allows selecting one or more files (.phplot file extension) and load them as uncalibrated phasor plots in the Phasor Graph.

Note: Prior to version 0.17, a XML file format was used, which turned out to be a bit verbose and slow to load. From 0.17 onward, the file format was changed to JSON to mitigate these issues. Older XML files can still be loaded, but it is recommended to save them (e.g. overwriting the older version) in the new format.

6. Computing the Average Phasor of a Phasor Plot

In some circumstances, it is desirable to compute the average phasor of a phasor plot. The corresponding reight-click menu item provides this functionality:

The result is displayed in the Current Phasor control and can be used, for instance, as calibration or reference.

7. Computing a Phasor Ratio and its associated Lifetime

Once phasor references have been defined, it is possible to compute the Phasor Ratio of any point in the graph, as discussed in more details in the Phasor Ratio Analysis manual page. An associated lifetime is then automatically computed, and added to the Lifetime Graph, as described in the Lifetime Analysis manual page. Here, we briefly summarize what this involves.

The Phasor Ratio Calculation right-click menu:

gives access to two functions. The first one is interactive and processes the last computed phasor data point, while the second deals with the last computed phasor series and does not require any user input.

In the interactive mode (Compute Single Phasor Ratio), the calculated phasor ratio is added to the last plot created in the Phasor Ratio Graph.If you want to create a new plot for a select number of phasor points, you should first create a new (empty) plot in the Phasor Ratio Graph. A new (empty) plot will then automatically be added to the Lifetime Graph.

Alternatively, you can create a new plot in the Lifetime Graph, which will automatically create a new plot in the Phasor Ratio Graph.

The Time-Series mode requires that a phasor time-series has been computed (or more generally, a plot containing at least one phasor). A new plot is automatically created in both the Phasor Ratio Graph and the Lifetime Graph.

8. Computing a Phase Lifetime Plot

The Phase Lifetime sub-menu gives access to different functions described below:

8.1 Phase Lifetime Trajectory

In case there is no way to simply define phasor references, and therefore no way to define a phasor ratio and associated lifetime, AND if the phasors of interest happen to be located on or close to the universal circle, it can be useful to compute the phase lifetime (Eq. (1) above) of phasors in a Phasor Graph plot.

The Phase Lifetime Trajectory menu item allows doing just that. Right-click on the target plot's name in the Phasor Graph's legend and select this menu item: this will automatically create a plot in the Lifetime Graph of the Lifetime Analysis panel. The term "Trajectory" refers to the fact that this is applicable to phasor plots corresponding to time series, but this is equally applicable to phasor plots resulting from multiple ROIs analyzed in one go (using the Analysis>>All ROIs Analysis menu).

8.2. Phase Lifetime vs Total Intensity/Background-Subtracted Intensity/Background/Fitted Lifetime/Amplitude

As discussed in the Fluorescence Decay page of the manual (section 4), background subtraction from decays results in a number of derived parameters (depending on the method used):

• Background per Gate (B)

• Amplitude (A) also equal to the theoretical Background-Subtracted Intensity

  • Lifetime (tau_fit)

Additionally, each decay is associated with the total intensity (I) in the corresponding ROI.

The Phase Lifetime vs Total Intensity menu item allows plotting the (tau, I) pairs for all phasor points in a phasor plot. The resulting scatter plot is represented in the Lifetime Graph of the Lifetime Analysis panel.

Similarly, the Phase Lifetime vs Background-Subtracted Intensity menu item allows plotting the (tau, I-GB) pairs for all phasor points in a phasor plot (G: number of gates), while the Phase Lifetime vs Background menu item allows plotting the (tau, B) pairs.

In the specific case, where the background subtraction method is "Square-Gated Single-Exponential", two additional parameters can be studied: Amplitude (A) and Lifetime (tau_fit).

The Phase Lifetime vs Fitted Lifetime menu item allows plotting the (tau, tau_fit) pairs, while the Phase Lifetime vs Amplitude menu item allows plotting the (tau, A) pairs for all phasor points in a phasor plot. Note that the amplitude A is also equal to the integrated intensity of the decay (and should therefore be comparable to the I - GB output, although the latter is the result of subtracting GB from the measured total intensity.

All these scatter plots, once transposed and histogrammed, gives access to the distribution of the conjugated parameters for all phasor points in the phasor plot.