Evaluation of ratiometric FRET experiments with rFRET

Written by Peter Nagy ([email protected])

If you find the program useful and write a publication using it, please cite the original paper decribing the program:

Aim and concept

Ratiometric or intensity-based FRET is one of the most widely used techniques to measure the clustering and conformation of fluorescently labeled proteins or other molecules of biological interest. In a ratiometric FRET experiment three intensities are measured for a sample double-labeled with donor and acceptor fluorophores:

  designation excitation emission detection
donor channel
I1
donor absorption wavelength
donor emission wavelength range
FRET channel
I2
donor absorption wavelength
acceptor emission wavelength range
acceptor channel
I3
acceptor absorption wavelength
acceptor emission wavelength range

This Matlab application evaluates the FRET efficiency and calculates all the necessary parameters (overspill factors and parameter α, described in detail below). In general, three different approaches are used for the calculations:

The program was written using Matlab R2013a and upgraded for R2015a.

The main panel of the program displays the parameters required for the FRET calculations (S1-4 and α) which are automatically filled in by the appropriate modules or can be adjusted manually. The main panel can be used to launch the four processes required for the evaluation of an intensity-based FRET measurement:

Many of the functionalities of the program are available when calling rFRET with command-line arguments as described at the end of the help file.

In general, the white fields are editable in the program, whereas the blue ones are not.

By selecting mean, trimmed mean or median in the "Statistic to calculate" popup menu in the main panel the user can specify which measure of central tendency will be calculated if a parameter (e.g. overspill parameter or FRET efficiency) is calculated on a pixel-by-pixel basis.

The bottom of the main panel contains a message box in which a concise report about the results is posted. Right-clicking on a message copies the selected line to the clipboard.

Accepted image types

The program accepts images in Matlab (2D array) and DipImage format. Therefore, DipImage is advised to be installed.

Updating

The program automatically checks for updates when started in GUI mode (i.e. it won't check for updates when used in command-line mode) and warns you to press the Update button to download the latest version. If you would like to prevent the program from checking for updates in the GUI mode, you have to start it by typing rfret('noupdate').

Saving and loading results

When pressing the "Save results" button the results of the calculations and the status of the program can be saved to a Matlab variable or to a file. The window shown below is displayed in which the following options are available:

When pressing the "Load results" button the structure variable saved to the Matlab workspace or to a file can be reloaded into the program in order to restore the settings and to reload the results of calcualtion.

Registration

The program requires a machine-specific unlock code which can be obtained free of charge from the developer. You will have to send the machine specific code the program generates to [email protected] and you will recieve the unlock code by email.

Introduction to the principles of FRET

You can find a couple of links to resources describing the principles of FRET and methods for measuring it:

Book chapters

Papers

Presentations

Enter parameters in the graphical user interface

Interaction with the graphs and the result field

Determination of the background

After entering the Matlab variables corresponding to the donor, FRET and acceptor images (and optionally the mask image), the background has to be determined, if the images have not been background-corrected before. The background is the mean fluorescence intensity of such an area in the image where there is no fluorescently-labeled structure. The background will be subtracted from each pixel during the rest of the calculation.

There are two ways of entering the background:

If the "Delete images" button is pressed, the images on which the background was drawn will be deleted.

Determination of overspill parameters

Although the excitation and detection wavelengths of the donor, FRET and acceptor channels are chosen so that the donor and acceptor fluorescence is optimally detected in the donor and acceptor channels, respectively, and acceptor emission arising as a result of FRET is mainly seen in the FRET channel, there is a practically inevitable overspill of these fluorescence intensities to the other fluorescence channels. As it can be seen from the equations in the FRET section it is assumed that every kind of fluorescence (donor, FRET, acceptor) is detected in each fluorescence channel. For the FRET equation set there are four overspill factors to be determined. Two of them, S1 and S3, characterize the overspill of the donor fluorescence to the FRET and acceptor channels, respectively. These parameters are to be meausured on a sample having only donor fluorescence:

Two other overspill parameters (S2 and S4), to be measured on a sample having only acceptor fluorescence, characterize how large fraction of the acceptor fluorescence spills over to FRET and donor channels, respectively:

The panel for calculations with the donor-only sample is shown below. The panel for calculating the acceptor overspill factors looks the same.

The principle of the calculations is decribed for the donor overspill factors, but calculation of the acceptor-related parameters is principally the same.

First, the images recorded in the donor, FRET and acceptor channels have to be specified by entering the name of the Matlab variable holding them. The mask image is a binary image in which 0s correspond to pixels which will be omitted from the calculation, i.e. only those pixels in the images recorded in the donor, FRET and acceptor channels will be used whose corresponding pixel in th mask image has a value of 1. Although the mask image is optional, it is strongly advised to be used to limit the calculation to the relevant pixels. A mask image can be generated by simple thresholding or by more complex methods, e.g. watershed segmentation for membrane proteins (watershed_segment program).

The size of the donor, FRET, acceptor and mask images has to be identical.

Determination of the background is described in a separate section.

Pixel-by-pixel method

You can start doing the calculations by pressing the "Do it" button. In the pixel-by-pixel method the ratio of two intensities (i.e. parameters S1 and S3 for the donor, and S2 and S4 for the acceptor) is calculated for every pixel in the mask (i.e. for every pixel for which the corresponding pixel is 1 in the mask). You can interact with the graphs as described in the "Interaction with the graphs" section. The mean of the pixelwise overspill parameters is calculated for the gated pixels, i.e. for the pixels labeled with orange in the graphs.

The calculated overspill parameters are displayed in blue boxes above the graphs.

You can simulate the distribution of the intensities and the parameters (overspill factor) arising from the Poissonian nature of photon detection by pressing the "Simulate parameter distributions" button. For more information on this feature read the corresponding paragraph of the help.

"Back-mapping"

You can save a binary image into a variable specified in the "Map variable" box in which 1s correspond to gated pixels, whereas ungated pixels are displayed as 0s.

"Save output image "

A 3D output image is saved to a user-defined variable in the Matlab workspace. The structure of the output image is the following:

The gray-scale images (first and second slice) contain NaN (not-a-number) values at pixels corresponding to 0s in the mask image.

Deming regression

Deming regression is a method where a line is fitted to the measured data set (Iy-Ix) assuming that

For the donor sample a straight line is fitted to the I2-I1 (IFRET-Idonor) and I3-I1 (Iacceptor-Idonor) plots whose slope is equal to S1 and S3, respectively. For the acceptor-labeled sample a straight line is fitted to the I2-I3 (IFRET-Iacceptor) and I1-I3 (Idonor-Iacceptor) plots whose slope is equal to S2 and S4, respectively. These lines are shown in purple in the graphs. If background correction has been carried out correctly, the intercept of the fitted lines should be zero. Therefore, in these cases the "constrain intercept to 0" option should be selected. If the "constrain intercept to 0" option is not selected, the intercept and the slope are both fitted parameters. However, if the intercept is markedly different from zero, there must have been some problem with the background subtraction.

The "var(y)/var(x)" parameter is the ratio of the variance of the measurement of the parameter plotted on the "y" axis divided by that of the parameter plotted on the "x" axis, i.e. how many times the measurement of single pixel intensities plotted on the y axis is more uncertain than the uncertainty in the measurement of parameter x if uncertainty is characterized by the variance. This parameter is required for Deming regression. The variance in this formula is not the variance of all the pixel intensities plotted on one of the axes, but the squared error of determination of individual pixel intensities. Therefore, there is no straightforward way to determine either the variance or the variance ratio. However, it turns out that the slope (S factors) is rather insensitive to changes in the variance ratio, especially in the practically relevant case when the intercept is constrained to zero. Therefore, it is advisable to use the default value of 1 for the variance ratio unless there is a good reason to assume otherwise.

In addition to the Deming regression trend lines are also shown in the graphs in color green. These lines display the average Y parameter calculated in a certain number of bins on the X axis. The number of bins is specified by the "Number of bins" box. This feature allows the user to determine intensity range-dependent overspill factors, i.e. to check if the overspill factor displays any intensity dependence as suggested previously.

You can simulate the distribution of the intensities and the parameters (overspill factor) arising from the Poissonian nature of photon detection by pressing the "Simulate parameter distributions" button. For more information on this feature read the corresponding paragraph of the help. Using this module reveals that the Poissonian nature of photon detection can in itself lead to some apparent intensity-dependence of overspill factors as described previously.

Maximum likelihood estimation (MLE)

The algorithm finds the overspill factor (or FRET efficiency for MLE in the FRET panel) at which the likelihood of the measured intensities is the highest assuming Poisson photon detection statistics. For this reason the algorithm can only be applied for data obtained with photon counting detectors. For the same reason pixels with negative intensities would lead to an error in the MLE algorithm since it is impossible to assign a probability to negative values in a Poisson distribution. Therefore, such pixels are ignored in the estimation process. Principles of the algorithm can be found in the original publication. The algorithm finds a single estimate for the overspill parameter (or FRET efficiency in the FRET panel), i.e. biological variation is disregarded. You can still get an impression on biological variability by

"initial values"

The fitting algorithm starts searcing for the optimal fitted parameters from initial values given by the user. A reasonable initial value is calculated by the program using calculation with summed intensities.

"Calc. off/on" button

If you would like to tweak around with the adjustment of different parameters, it is possible to turn calculations off, i.e. to disable refreshing the plots and the fitted parameters. Fitting will only be performed if you turn calculation back on.

"maximum number of steps", "tolerance"

The "maximum number of steps" determines how many times the iteration is allowed to be performed. If the relative change in the likelihood is smaller than "tolerance", the algorithm stops, because it assumes that a good estimate has been found..

"estimate confidence interval?"

If confidence interval estimation is turned on, the normalized likelihood will be calculated and plotted against the fitted parameter. This option can be turned off, since confidence interval estimation can be slow especially if a 3D confidence interval is to be calculated when two parameters (e.g. slope and intercept, FRET and α) are estimated.

"thresholded MLE"

One of the advantages of the MLE algorithm is that the likelihood of every pixel is calculated making the elimination of outlier pixels possible. A pixel is considered to be an outlier if its likelihood is smaller than the threshold given in the "threshold likelihood" fields. These outlier pixels are eliminated from the calculation; therefore, they will not distort the estimated parameters. You can manually enter threshold likelihoods or you can use the "Get threshold likelihood" functionality of the program.

"Back-mapping"

If non-thresholded MLE is carried out, only "Nothing" and "Log-likelihood image" can be selected in this drop-down menu. If log-likelihood image is selected, the log-likelihood of every pixel is saved into the variable given in the "Map variable" field. The two images correspond to the first and second estimation (S1 and S3 for the donor sample, and S2 and S4 for the acceptor sample). It is called mapping because the likelihoods are saved in an image, i.e. they are mapped back to the original image.

If thresholded MLE is carried out, the "Binary map" option is also available in the drop-down list. If it is selected, a binary image will be saved into the variable given in the "Map variable" field in which 0s and 1s correspond to pixels whose likelihood is smaller and higher, respectively, than the threshold likelihood (these pixels are excluded from and included, respectively, in the estimation).

"Use optimization toolbox"

This option is only available if the Optimization toolbox is installed in Matlab. If this is the case, the Optimization toolbox, more specifically the fmincon function performing constrrained optimization, is used in the MLE algorithm by default. Negative values are not allowed for the overspill parameters, the FRET efficiency and α in the MLE algorithm. Although these are non-sense values anyway, there is another reason for constraining the optimization to non-negative values since negative parameter values could result in negative intensities whose probability cannot be interpreted according to the Poisson distribution used in MLE. Although in most cases this constained optimization produces optimal results, sometimes it does not converge to a minimum. If the MLE algorithm performed with the Optimization toolbox fails, it is worth trying unticking the "Use optimization toolbox" check box and performing the MLE algorithm without the Optimization toolbox, more specifically with the fminsearch function of Matlab.

"Get threshold likelihood"

The distribution of the estimated parameters (overspill factors) is simulated assuming

The likelihood distribution of all the pixels will be determined. A certain percentile value, given by the user, of the likelihood distribution is calculated and is returned as a threshold likelihood to the MLE panel. In addition, the likelihood distribution of the simulated pixels is also returned to the MLE panel and it will be plotted in the lower right graph. The legend of these lines starts with "Simulated".

For more information about this part of the program read the "Simulation of parameter distributions" section of the help.

"Delete simulated plot"

By pressing this button the simulated log-likelihood distribution plots will be deleted.

"constrain intercepts to 0"

In the "constrain intercepts to 0" check-box is ticked, a fitted line crossing the origin is found. In this case only the slope, i.e. the overspill parameter, is estimated. If the intercept is not constrained to 0, both the intercept and the slope are fitted parameters. In this case the confidence interval plot is 3D, i.e. it shows the normalized likelihood as a function of the slope and the intercept. You can rotate the graph by

"Function undefined" error

The error which is displayed in the panel in red means that the optimization function could not be evaluated. Although this phenomenon can be the consequence of many factors, in most cases it is caused by too high threshold likelihood: if the threshold likelihood is adjusted to a too high value in thresholded MLE, all pixels can have likelihoods below the threshold. In such a case no pixel remains to be evaluated. If this is the case, decrease the threshold likelihood.

Ratio of sums

In the "ratio of sums" approach the ratio of the summed (or mean) intensities is calculated (e.g. for parameter S1):

as opposed to the pixelwise calculation where the mean of the pixelwise ratios is calculated (e.g. for parameter S1):

At high intensities (i.e. intensities well above the background) the two esimators are equally optimal. However, at low intensities the pixelwise approach has problems due to error propagating into the overspill parameters from the detected intensities. See the paper of Van Kempen and the one about maximum likelihood estimation.

Although the ratio of the summed (or mean) intensities is calculated, gating is still possible. The summed intensities will be calculated for those pixels only which are inside the gates.

"Back-mapping"

You can save a binary image into a variable specified in the "Map variable" box in which 1s correspond to gated pixels, whereas ungated pixels are displayed as 0s.

Determination of the α parameter

The α parameter describes how efficiently an excited acceptor molecule is detected in the FRET channel compared to an excited donor molecule in the donor channel. α is required for FRET calculations because an excited donor molecule disappears and an excited acceptor molecule is generated during FRET, and the relative efficiency of their detection has to be known. α can be expressed according to the following equation:

where QA and QDstand for the fluorescence quantum efficiency of the acceptor and donor, respectively, and ηA,2 and ηD,1 designate the detection efficiency of the acceptor in the second (FRET) channel and that of the donor in the first (donor) channel, respectively.

The program can determine the α parameter by six different methods. In the top part of the panel you can choose one of the methods and the bottom panel will display the required input fields. The middle panel designated "Source data" determines what kind of input is used for the intensities (donor, FRET and acceptor channel, mask image).

For all other parameters (overspill factors, labeling ratio, ε1 and ε2 ratio, expression ratio, max step, tolerance) numeric input is expected. For the "Spectral" method the required input types will be defined in the corresponding part of the help.

Ma/Md (a method for samples labeled with the same number of donor- or acceptor-tagged antibodies)

Since the α parameter describes how well an excited acceptor molecule can be detected in the FRET channel compared to an excited donor molecule in the donor channel, it seems straightforward that we intend to create samples with an equal number of donor and acceptor molecules. To this end two samples are labeled:

Therefore, the data of two separate images must be entered into the panel. Since two images recorded of two different samples are used, the masks and the background intensities must be determined separately for the two images.

Even though this labeling strategy ensures that the two samples have the same number of bound antibodies/cell, we have to compare the intensities of an equal number of excited donor and acceptor fluorophores. Therefore, two additional corrections have to be introduced:

Calculation of α is affected by fluorophore saturation, briefly described in another section. If you would like to determine α with regard to saturation phenomena, you have to choose one of the saturation correction options in the "Saturation panel".

α is calculated according to this equation without taking saturation effects into consideration (conventional mode):

where MA designates the mean intensity of the acceptor-labeled sample measured in the FRET channel,MD is the mean intensity of the donor-labeled sample measured in the donor channel, LD and LA are the labeling ratios of the donor- and acceptor-conjugated antibodies, respectively, and εD and εA are the molar absorption coefficients of the donor and the acceptor at the excitation wavelength of the donor channel, respectively. The εDA ratio is designated by ε1 in the program. You can read about the determination of this absorption ratio in the corresponding part of the help.

α is calculated according to this equation with taking saturation effects into consideration:

where τD and τA are the fluorescence lifetimes of the donor and the acceptor, respectively, and Dsat,D and Asat,D are the fractional saturation of the donor and the acceptor, respectively, at the excitation wavelength of the donor.

Remember that a relatively large number of cells has to be measured so that the MA / MD ratio is a reliable estimate of the intensity ratio of the two samples. Therefore, if you choose the "Image to sum" option for "Source data", you have to make sure that a large number of cells are present in the images. Since the determination of mean intensities of a couple of dozen cells, present in a single image, is not reliable, it is advisable to repeat the above procedure several times and calculate α from averaged MA and MD intensities according to the following formula:

where areaA(i) and areaD(i) are the number of pixels in the mask for the ith acceptor and donor mask, respectively. The left and righ formulas are for calculation without and with taking saturation effects into consideration, respectively.

This procedure can be carried out in two different ways by the program.

In addition, you have to enter the labeling ratios and the absorption ratio (e1). The latter can be estimated using the program (see the corresponding part of the help).

"Back-mapping"

When performing back-mapping in this panel two images are saved in a cell array. The first one shows the gated pixels in the donor channel, while the second one displays the gated pixels in the FRET or acceptor channel (depending on the calculation method).

Iterative for FPs (an iterative method for a sample expressing a fusion construct of a donor and an acceptor fluorescent protein)

A method has been published in Biophys. J. for the simultaneous determination of parameter α and the FRET efficiency. The approach is only applicable for cells expressing a fusion construct of a donor and an acceptor fluorescent protein, i.e. when the number of donor and acceptor fluorophores is equal.

The method is summarized briefly below:

  1. The FRET equation set is solved for the unquenched donor (ID) and the directly excited acceptor (IA) intensities by eliminating α and assuming a FRET value of 0 followed by calculating α according to the following equation:

    where εD and εA are the molar absorption coefficients of the donor and the acceptor, respectiely, at the excitation wavelength of the donor channel. The ratio εDA is designated by ε1 in the program.
  2. Using this α value the FRET efficiency is calculated by solving the FRET equation set and this FRET efficiency is used for re-estimating α (step 1). The iteration is continued until the relative change in the FRET efficiency between successive iterations is smaller than the "Tolerance" specified in the input panel.

The maximum number of iteration steps as well as the source images, the overspill parameters and the absorption ratios have to be provided.

Closed-form for FPs (a non-iterative method for a sample expressing a fusion construct of donor and acceptor fluorescent proteins)

This approach requires a sample expressing a fusion construct of a donor and an acceptor fluorescent protein for the simultaneous determination of parameter α and the FRET efficiency. It has been published in Cytometry.

The algorithm is implemented in the program in several different ways. You can decide to take saturation effects into consideration or disregard them. The program allows the user to define the expression ratio, i.e. the ratio of the number of donor and acceptor molecules ("Expression ratio, D/A"). In full saturation correction mode this option is not available. In that case the expression ratio is assumed to be one.

In this mode of α calculation the FRET equation set is supplemented with a fourth equation defining parameter α knowing the expression ratio of the donor and the acceptor. The equation shown below disregards saturation phenomena:

where IA and ID stand for the directly excited intensity of the acceptor in the acceptor channel and the unquenched intensity of the donor in the donor channel, respectively,, εD and εA are the molar absorption coefficients of the donor and the acceptor, respectiely, at the excitation wavelength of the donor channel. The ratio εDA is designated by ε1 in the program. Rexp is the average number of donor molecules (ND) devided by the average number of acceptor molecules (NA) in a cell. The equation set consisting of four equations can be solved for the FRET efficiency and α.

Simultaneous maximum likelihood estimation of FRET efficiency and α is also available when selecting this method. The parameters required and the controls available in the panel for maximum likelihood estimation are described in the section for MLE-based estimation of overspill parameters.

Fit for many FPs (a method for a series of donor-acceptor fusion constructs differing from each other in FRET efficiency)

A robust method for the simultaneous determination of parameter α and the ε1 ratio has been published for a series of fusion constructs of donor and acceptor fluorescent proteins.

The only available data source in this method is "variable", i.e. the "Donor ch.", "FRET ch." and "Acceptor ch." fields must contain Matlab variable names. These Matlab variables have to be vectors containing the intensities of the first, second, third, etc. donor-acceptor fusion construct. In the example below there were three different donor-acceptor fusion constructs, and the values 143, 179 and 109 in variable "i1" correspond to the intensities of the first, second and third donor-acceptor construct measured in the donor channel.

The algorithm can find parameter α, the ε1 abosrption ratio or both. If you would like the algorithm to find either α or ε1, you have to provide the other one.

If both α and ε1 are found during the fitting, a graph shown below is generated. By clicking on the individual data points on the graph on the left, you can include or exclude individual points from the fitting. Excluded points will be marked with emtpy symbols. The graphs in the middle and on the right show the confidence plots of the two fitted parameters.

Rf and Ri have been defined in the publication describing the method.

where FAD and FA are the fluorescence intensities of the acceptor in the presence and absence of the donor, respectively, excited at the excitation wavelength of the donor channel. εD and εA are the molar absorption coefficients of the donor and the acceptor, respectively, at the excitation wavelength of the donor channel. The ratio εDA is designated by ε1 in the program. cD and cA are the molar concentrations of the donor and the acceptor, respectively.

Ri can be defined according to the solution of the FRET equation set for the FRET efficiency:

i.e. Ri is a function of the intensities and overspill factors.

It was shown that

and the linearized form of this equation is

Consequently, a plot of 1/(Rf-1) against 1/Ri will produce a line with an intercept of εAD=1/ε1 (if cD=cA) and a slope of intercept × α.

The fitted parameters are not obtained by linear regression, so the graph of 1/(Rf-1) against 1/Ri is only used for displaying purposes.

Briefly, the fitting approach is based on the fact that there are two equations, the one containing Rf and the other containing Ri, from which the FRET efficiency can be derived. The α or ε1 parameters, or both of them, are obtained so that the difference between the FRET efficiencies calculated using the two approaches is minimal.

If only one of the parameters is found by fitting (α or ε1), the squared error of the minimized function is shown on the right.

Bleaching (a method utilizig partial photobleaching of the acceptor)

Parameter α can also be determined by partial photobleahing of the acceptor and observing the consequent increase in the donor fluoresnence due to dequenching. According to the equation in the original publication

where subscript "post" refers to intensities after partial photobleaching of the acceptor, while intensities without the "post" subscript designate fluorescence measured before acceptor photobleaching. Since the method can only be applied if overspill factors S3 and S4 are negligible, the equation describes how much the FRET-sensitized emission of the acceptor decreases compared to the accompanying increase in the donor fluorescence upon partial acceptor photobleaching.

Please note that there is a slight mistake in this equation (Eq. 19) in the original publication.

On the panel defining the required parameters you have to provide the pre- and post-bleach intensities (as values, Matlab variables or images). Since the same area is observed before and after partial acceptor bleaching, the same mask image is used for the pre-bleach and post-bleach images (if image is chosen as the input type).

Since S3 and S4 have to be zero, only S1 and S2 have to be provided.

Spectral (a method calculating α according to the equation using quantum yields and detection efficiencies)

The most direct way of determining parameter α is based on the equation defining the parameter shown at the beginning of the chapter about the determination of α. In order to make it experimentally applicable the equation has to be extended:

In this equation QA and QD stand for the fluorescence quantum efficiency of the acceptor and the donor, respectively, and ηA,2 and ηD,1 designate the detection efficiency of the acceptor in the second (FRET) channel and that of the donor in the first (donor) channel, respectively. This equation can only be used if photon-counting detectors are used.

The detection efficiencies can be obtained by integrating the product of the normalized fluorescence emission spectrum (f) of the fluorophore, the transmission of the optics (T) and the quantum efficiency of the detector (DQ) in the fluorescence channel detecting the fluorescence. Subscripts A and D designate the acceptor and the donor, respectively, whereas subscripts 1 and 2 stand for the first (donor) and second (FRET) detection channels, respectively.

Format of the required input:

Determination of the absorption ratio (ε1)

You can reach this section of the program by pressing the "Absorp. ratio" button in the top right corner of the panel for the determination of α.

The absorption ratio (ε1) is the ratio of the molar absorption coefficients of the donor and the acceptor at the excitation wavelength of the donor channel:

This parameter is required in some methods for estimating the α parameter. ε1 can be determined

Determination of the FRET efficiency

Basic equations

Calculation of FRET is affected by fluorophore saturation. From version 1.37 rFRET is able to consider these saturation phenomena. The equations in this section do NOT take saturation phenomena into consideration. The effect of fluorophore saturation on FRET calculations is briefly summarized in a separate section.

In practice, there are two kinds of equation sets describing a ratiometric FRET experiment. If the detected wavelength ranges in the FRET and acceptor channels are identical (if they are detected with the same detector using the same detector settings), the following equation set is to be used:

The solution of this set of equations for FRET, Id and Ia:

If, on the other hand, the detected wavelength ranges in the FRET and acceptor channels differ, another equation set holds:

The solution of this second equation set for FRET, Id and Ia:

In these equations Id and Ia are the unquenched donor and directly excited acceptor intensities, respectively, and ε2 is a ratio of the molar absorption coefficients of the donor and the acceptor:

Strictly speaking the second equation set is valid even if the detected wavelength ranges in the FRET and acceptor channels are identical. However, the second equation set requires the determination of ε2, typically from absorption spectra. Therefore, if the conditions for applying the first equation set are met, it is advisable to use it.

You can decide which equation set to use in the program by selecting one of them in the "FRET equation set" panel. The box for entering ε2 is enabled and disabled according to your selection since it is only required for the second kind of equation set.

The blue field under a popup menu displayed the calculated FRET, Id or Ia values.

Since the principle of how to adjust the settings on the FRET panel is similar to what has been described for the determination of overspill parameters, only a brief summary is given below.

Pixel-by-pixel method

The FRET efficiency is calculated for every pixel in the mask according to the equations above and the distribution of the FRET efficiency and its dependence on other parameters are displayed along with intensity dot plots. Gating on the histograms and dot plots is possible as described for the determination of overspill parameters.

Back-mapping and saving an output image are also available. They are described at the determination of overspill parameters. The structure of the output image saved in the Matlab workspace is as follows:

The gray-scale images (first and second slice) contain NaN (not-a-number) values at pixels corresponding to 0s in the mask image.

Maximum likelihood estimation (MLE)

A single estimate is calculated for the FRET efficiency with which the likelihood of the observed intensities in the donor, FRET and acceptor channels is the highest assuming Poisson photon detection statistics.

FRET calculated from summed intensities

The same expression which is applied during the pixel-by-pixel calculations is used for determining FRET, but in this case summed intensities are used. Summation is carried out for pixels in the mask and in the gates. Consequently, a single FRET value is calculated.

The effect of fluorophore saturation on FRET calculations

The number of photons emitted by a dye only increase linearly with excitation intensity at low excitation intensities, then fluorescence begins to saturate.

Such saturation already takes place at laser intensities commonly applied in confocal microscopy. Saturation of the donor and the acceptor influence FRET calculations for several reasons.

The principles of FRET calculations under conditions of fluorophore saturation was published in Anal. Chem.

The effect of fluorophore saturation on FRET calculations can be taken into consideration by selecting one of the modes in the "Saturation panel". The saturation panel is only displayed in the FRET panel, and two alpha panels ("Ma/Md" and "Closed-form for FPs"). Since calculation of overspill parameters (S1 - S4) is not affected by saturation effects, the saturation panel is not shown in the panels for calculating donor and acceptor parameters.

Saturation effects to consider:

The equation sets taking saturation phenomena into consideration are too long to be presented here.

Limitations as far as calculation modes are concerned when taking saturation phenomena into consideration:

In order to consider saturation effects several parameters must be specified in a panel shown after clicking on the "Parameters" button in the saturation panel.

You can choose if you specify the molar absorption coefficient or the absorption cross-section of the dyes at both excitation wavelengths. The fluorescence lifetime and the photon flux of the donor-exciting and acceptor-exciting lasers must also be given. Pay attention to the units of parameters shown in square brackets. By clicking on the calculation button the fractional saturation of the donor and the acceptor at the two excitation wavelengths is calcualted.

Simulation of parameter distributions

Depending on from which panel the simulation has been called, the simulation panel looks differently. The one simulating parameter distributions for the donor overspill parameters is shown in the figure below. The simulation algorithm generates normally distributed donor intensities (Id) from which the Ifret and Ia intensities are generated according to the specified overspill factors:

Then, Poisson noise is added to the simulated intensities. Due to the underlying normal distribution of expression levels negative intensities may be generated. Since the Poisson distribution is not interpreted for negative values, these intensities will be neglected. The nunber of remaining simulated pixels with non-negative intensities is shown in the "Remaining pixels" box.

You have to give the number of pixels to simulate. Usually, 100000 simulated pixels will yield relatively smooth curves.

Some parameters have to be given for the simulation. It is assumed that the expression level of the fluorescent label follows a normal distribution whose mean and SD have to be given. Poissonian noise will be added to these expression levels resulting in the simulated photon numbers. These photon numbers have Gaussian and Poissonian noise. If the SD of the normal distribution is adjsted to 0, the simulated intensities will be characterized by pure Poisson distributions.

The percental value to be determined from the likelihood distributions is also to be given. This will be displayed in the threshold likelihood (Thr. L) boxes and also in the likelihood distributions as vertical red dashed lines.

You can start the simulation by pressing the "Simulate" button. The likelihood of every pixel is calculated using the simulated parameters and the histogram of the likelihoods is displayed. The distribution of the calculated parameter (overspill factor, FRET efficiency) and the dot plots of the intensity distributions are also displayed. A purple line is fitted using Deming regression to the simulated intensities providing the "fitted S1 and "fitted S3" parameters. During Deming regression the intercept is constrained to zero, and it is assumed that the variance ratio is 1. The Deming regression lines are shown in purple. Green trend lines display the average Y parameter for bins on the X axis. The number of bins is specified by the user in the "Number of bins" text box.

You can rescale the graphs by double-clicking on their axes.

If the simulation has been called from an MLE window, the "Return" button will return the threshold likelihood values to the MLE panel as well as the likelihood distribution plots. The caption of the likelihood distribution of the simulated pixels and that of the original data set will be "simulated" and "measured", respectively.

Command-line usage

rFRET can also be called from the command-line with arguments. In this case the graphical user interface won't be displayed, and the analysis is performed automatically. The syntax of the command is the following:

[different number of outputs depending on 'mode']=rfret(Name1,Value1,Name2,Value2,...)

If rfret is used with command-line arguments, no background correction is carried out. In this case background correction must be performed by yourself.

Name Values
'mode' It determines what is calculated.
Possible values:
'donor' - calculation of donor parameters
'acceptor' - calculation of acceptor parameters
'alpha1' - calculation of parameter α using the Ma/Md method
'alpha2' - calculation of parameter α using the iterative method for fluorescent proteins
'alpha3' - calculation of parameter α using the closed form solution for fluorescent proteins
'alpha4' - calculation of parameter α using the fitting approach for many pairs of fluorescent proteins
'alpha5' - calculation of parameter α using the bleaching approach
'alpha6' - calculation of parameter α using the spectral approach
'fret' - calculation of FRET
'method'

It determines how the calculation is performed.
Possible values:
'sums' - calculations are performed from summed intensities
'pixelwise' - calculations are performed on a pixel-by-pixel basis and the mean of pixelwise values is returned
'pixelwiseall' - the same as 'pixelwise', but all pixelwise values are returned in an array
'pixelwisemap' - the same as 'pixelwise', but the values are returned as an image
'deming' - Deming regression
'mle' - maximum likelihood estimation

'stat'

It determines what kind of statistic is returned. It is only interpreted for the 'pixelwise' method. The default value is 'mean', i.e. if 'stat' is not specified, the mean of all pixelwise values is returned.
Possible values:
'mean' - the mean of the pixelwise values is returned
'median' - the median of the pixelwise values is returned
'trimmedmean' - the trimmed mean of the pixelwise values is returned. It this case the syntax is the following: 'stat', 'trimmedmean', 5, ... The number after 'trimmedmean' determines how many percent of the highest and lowest values is excluded.

'channels'

The images to be evaluated. Depending on 'mode' different number of images are to be specified. See the examples below the table.
If 'mode' is 'donor', 'acceptor', 'fret', 'alpha2', 'alpha3' or 'alpha4', three channels are to be defined:
'channels',a,b,c where a,b,c are the images recorded in the donor, FRET and acceptor channels, respectively.
If 'mode' is 'alpha1', two channels are to be defined:
'channels',a,b where a is the image recorded of a donor-only sample in the donor channel, and b is the image recorded of an acceptor-only sample in the FRET channel (if 'alpha1method' is 1) or in the acceptor channel (if 'alpha1method' is 2).
If 'mode' is 'alpha5', six channels are to be defined:
'channels',aBefore,bBefore,cBefore,aAfter,bAfter,cAfter where the first three variables are the images recorded in the donor, FRET and acceptor channels before acceptor belaching, and the last three variables are the images recorded in the donor, FRET and acceptor channels after acceptor bleaching.

'mask' A binary image specifying which pixels are to be evaluated. Pixels with values 0 won't be analyzed. If a mask is not specified, the whole image will be evaluated.
'alpha1method'

The method of determination of Ma if 'mode' is 'alpha1' (Ma/Md method)
Possible values:
1 - Ma is determined from the intensity measured in the FRET channel of the acceptor-only sample ('Calculation method' I2 in the GUI)
2- Ma is determined by multiplying the intensity measured in the acceptor channel of the acceptor-only sample by S2 ('Calculation method' I3 S2 in the GUI). In this case S2 must also be specified.

'mask1' and 'mask2' Binary images defining the masks to be used if 'mode' is 'alpha1' (Ma/Md method). In this mode 'mask' is not used since two different masks, defined by 'mask1' and 'mask2', must be specified for the donor-only ('mask1') and the acceptor-only ('mask2') sample.
's1' It specifies parameter S1.
's2' It specifies parameter S2.
's3' It specifies parameter S3.
's4' It specifies parameter S4.
'alpha' It specifies parameter α.
'freteqtype' It determines which equation system is used for FRET calculations.
Possible values:
1 - the emission wavelength ranges detected by the FRET and acceptor detectors are the same
2 - the emission wavelength ranges detected by the FRET and acceptor detectors are different
'epsrat2' ε2 defined in the section "Basic FRET equations". It is required if 'freteqtype' is 2.
'tolerance' Tolerance used in recursive fitting approaches (e.g. MLE and the fitting approach for many pairs of fluorescent proteins for determining α.
'maxstep' The maximum number of iterations allowed in recursive fitting approaches (e.g. MLE and the fitting approach for many pairs of fluorescent proteins for determining α.
'ld' The degree of labeling of the donor-conjugated antibody required for the determination of α using the Ma/Md method.
'la' The degree of labeling of the acceptor-conjugated antibody required for the determination of α using the Ma/Md method.
'epsrat1' ε1 defined in the section "Determination of the absorption ratio (ε1)". It is required for the determination of α using the Ma/Md method.
'expratio' The expression ratio of the donor- and acceptor-tagged fluorescent proteins. It is required for the determination of α using the closed form solution for fluorescent proteins.
'alpha4method' It specifies how the determination of parameter α using the fitting approach for many pairs of fluorescent proteins is performed.
Possible values:
1 - only parameter α is determined. It this case ε1 ratio must be specified.
2 - both parameter α and ε1 ratio are determined. In this case neither α, nor ε1 ratio can be specified.
3 - only ε1 ratio is determined. It this case α must be specified.
'Qa' or 'qa' The quantum yield of the acceptor. Required for determination of parameter α using the spectral approach.
'Qd' or 'qd' The quantum yield of the donor. Required for determination of parameter α using the spectral approach.
'spectrumAcceptor' or 'spectrumacceptor' The emission spectrum of the acceptor. Required for determination of parameter α using the spectral approach.
'spectrumDonor' or 'spectrumdonor' The emission spectrum of the donor. Required for determination of parameter α using the spectral approach.
'unitSpectrumAcceptor' or 'unitspectrumacceptor'

Unit of the acceptor emission spectrum. Required for determination of parameter α using the spectral approach.
Possible values:
1 - power
2 - photon flux

'unitSpectrumDonor' or 'unitspectrumdonor' Unit of the donor emission spectrum. Required for determination of parameter α using the spectral approach.
Possible values:
1 - power
2 - photon flux
'transmissionFretChannel' or 'transmissionfretchannel' Transmission spectrum of the detection optics in the FRET channel. Required for determination of parameter α using the spectral approach.
'transmissionDonorChannel' or 'transmissiondonorchannel' Transmission spectrum of the detection optics in the donor channel. Required for determination of parameter α using the spectral approach.
'unitTransmissionFretChannel' or 'unittransmissionfretchannel' Unit of the transmission spectrum of the optics in the FRET channel. Required for determination of parameter α using the spectral approach.
Possible values:
1 - %
2 - optical density (OD)
'unitTransmissionDonorChannel' or 'unittransmissiondonorchannel' Unit of the transmission spectrum of the optics in the donor channel. Required for determination of parameter α using the spectral approach.
Possible values:
1 - %
2 - optical density (OD)
'fretDetectorQ' or 'fretdetectorq' Quantum yield spectrum of the detector in the FRET channel. Required for determination of parameter α using the spectral approach.
'donorDetectorQ' or 'donordetectorq' Quantum yield spectrum of the detector in the donor channel. Required for determination of parameter α using the spectral approach.
'sat'

It determines if fluorophore saturation and FRET frustration are taken into consideration in the calculations.
Possible values:
'nosat' - saturation effects are not considered at all (conventional calculation)
'flsat' - saturation of the donor and the acceptor are considered without taking FRET frustration into consideration
'fullsat' - complete correction for saturation effects taking FRET frustration into consideration

'fullsat' is only allowed if 'mode' is 'fret' or 'alpha3'.
'flsat' is allowed if 'mode' is ''fret', 'alpha1' or 'alpha3.

'sigmaD' Parameter required for considering fluorophore saturation phenomena.
The absorption cross-section of the donor at the donor excitation wavelength. Unit: cm2.
'sigmaDa' Parameter required for considering fluorophore saturation phenomena.
The absorption cross-section of the donor at the acceptor excitation wavelength. Unit: cm2.
'sigmaA' Parameter required for considering fluorophore saturation phenomena.
The absorption cross-section of the acceptor at the acceptor excitation wavelength. Unit: cm2.
'sigmaAd' Parameter required for considering fluorophore saturation phenomena.
The absorption cross-section of the acceptor at the donor excitation wavelength. Unit: cm2.
'tauD' Parameter required for considering fluorophore saturation phenomena.
Fluorescence lifetime of the donor (in the absence of FRET). Unit: s.
'tauA' Parameter required for considering fluorophore saturation phenomena.
Fluorescence lifetime of the acceptor. Unit: s.
'phiD' Parameter required for considering fluorophore saturation phenomena.
Photon flux of the donor-exciting laser. Unit: 1/(cm2 s).
'phiA' Parameter required for considering fluorophore saturation phenomena.
Photon flux of the acceptor-exciting laser. Unit: 1/(cm2 s).

Examples for command-line usage:

[s1,s3]=rfret('mode','donor','method','sums','channels',a,b,c,'mask',m)

[s1,s3]=rfret('mode','donor','method','sums','channels',a,b,c,'mask',m,'stat','median')
[s1,s3]=rfret('mode','donor','method','sums','channels',a,b,c,'mask',m,'stat',trimmedmean',5)

[s1image,s3image]=rfret('mode','donor','method','pixelwisemap','channels',a,b,c,'mask',m)

[fretefficiency,id,ia]=rfret('mode','fret','method','sums','channels',a,b,c,'mask',m,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'alpha',0.94,'freteqtype',1)

[fretefficiency,id,ia]=rfret('mode','fret','method','sums','channels',a,b,c,'mask',m,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'alpha',0.94,'freteqtype',2,'epsrat2',0.004)

alpha=rfret('mode','alpha1','alpha1method',1,'channels',a,b,'mask1',ma,'mask2',mb,'ld',1.5,'la',1.9,'epsrat1',3.3)

If channels are 3D stacks, each slice is processed separately as described in the section Ma/Md.

alpha=rfret('mode','alpha1','alpha1method',2,'channels',a,b,'mask1',ma,'mask2',mb,'ld',1.5,'la',1.9,'epsrat1',3.3,'s2',0.36)

[alpha,fretefficiency]=rfret('mode','alpha2','channels',a,b,c,'mask',m,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'freteqtype',2,'epsrat2',0.004,'epsrat1',3.3)

[alpha,fretefficiency]=rfret('mode','alpha3','method','mle','channels',a,b,c,'mask',m,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'freteqtype',2,'epsrat2',0.004,'epsrat1',3.3,'expratio',1)

'method' can be 'mle' or 'sums' in this calculation mode.

alpha=rfret('mode','alpha4','channels',i1,i2,i3,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'freteqtype',2,'epsrat2',0.004,'alpha4method',1,'epsrat1',3.3)

[alpha,epsrat1]=rfret('mode','alpha4','channels',i1,i2,i3,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'freteqtype',2,'epsrat2',0.004,'alpha4method',2)

epsrat1=rfret('mode','alpha4','channels',i1,i2,i3,'s1',0.12,'s2',0.1,'s3',0.04,'s4',0,'freteqtype',2,'epsrat2',0.004,'alpha4method',3,'alpha',2.1)

alpha=rfret('mode','alpha5','channels',a,b,c,apost,bpost,cpost,'mask',m,'s1',0.12,'s2',0.1)

alpha=rfret('mode','alpha6','Qa',0.8,'Qd',0.9,'spectrumAcceptor',acceptorEmSp,'spectrumDonor',donorEmSp,...
'unitSpectrumAcceptor',1,'unitSpectrumDonor',1,'transmissionFretChannel',fretChSp,'transmissionDonorChannel',donorChSp,...
'unitTransmissionFretChannel',1,'unitTransmissionDonorChannel',1,'fretDetectorQ',fretDetSp,'donorDetectorQ',donorDetSp)