Correlation Analysis

The Correlation page provides tools to compute the autocorrelation function (ACF) and cross-correlation function (CCF) of photon streams over the whole data set or limited to single-molecule bursts (either meeting constraints defined in the Burst Analysis page, or selected in the ALEX Histogram of the ALEX Analysis page). Several types of corrections can be applied (afterpulsing, background, alternation) as described in section 5. Finally, a number of operations  can be applied to the computed curves, available via the right-click contextual menu appearing when selecting the name of a plot in one of the graphs' legend (see section 6).

Note: The labVIEW core routines used to compute these correlation functions were kindly provided by Dr. Ted Laurence (Ref. 6).

1. Correlation Curves Binning

4 types of time lag binning can be chosen using the CF Bin Type pull-down menu (blue box in the figure above):

The corresponding set of parameters changes accordingly, as discussed next.

In the following, b_0 is the lowest bound of the first bin, b_1 is the largest bound (excluded) of the first bin, and generally, [b_i, b_(i+1)[ is the i+1 th bin.

1.1 Multitau

The parameters are Min Time, Max Time, Bins/Group and Use Bin Center:

# Bins indicates how many correlation bins will be computed. Large # Bins result in long computation time.

Using the following definitions:

b_0 = 0

b_1 = Min Time

G = Bins/Group

The first 2G bins are defined as follows:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + Min Time

where i = 0, ...,2G 

The next G bins are defined as follows:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + 2 x Min Time

where i = 2G+1, ...,3G 

And generally:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + 2^(k-1) x Min Time

where i = kG+1, ...,(k + 1)G 

and k = 2, ..., M

and M is such that the last bin encompasses Max Time.

The Use Bin Center checkbox specifies whether the function value F(i) is plotted at b_i (unchecked) or (b_i + b_(i + 1))/2 (checked).

The smallest acceptable value of Min Time is the timing resolution of the readout electronics. 

The traditional definition uses G = 8

1.2. Multitau-10

The parameters are Min Time, Max Time and Use Bin Center:

# Bins indicates how many correlation bins will be computed. Large # Bins result in long computation time.

Using the following definition:

b_0 = 0

b_1 = Min Time

G = 10

The first G bins are defined as follows:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + Min Time

where i = 0, ...,G

b_G = G x Min time

The next G bins are defined as follows:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + G x Min Time

where i = G+1, ...,2G 

And generally:

[b_i, b_(i + 1) [, where b_(i + 1) = b_i + G^(k-1) x Min Time

where i = kG+1, ...,(k + 1)G 

and k = 2, ..., M

and M is such that the last bin encompasses Max Time.

The Use Bin Center checkbox specifies whether the function value F(i) is plotted at b_i (unchecked) or (b_i + b_(i + 1))/2 (checked).

The smallest acceptable value of Min Time is the timing resolution of the readout electronics. 

1.3. Logarithmic

The parameters are Min Time, Max Time, Bins/Decade and Use Bin Center:

# Bins indicates how many correlation bins will be computed. Large # Bins result in long computation time.

Using the following definition:

b_0 = 0

b_1 = Min Time

G = Bins/Decade

The bins boundaries are defined as follows:

b_(i + 1) = b_i x 10^(1/G)

where i = 1, ...,M

and M is such that the last bin encompasses Max Time.

The Use Bin Center checkbox specifies whether the function value F(i) is plotted at b_i (unchecked) or (b_i + b_(i + 1))/2 (checked).

The smallest acceptable value of Min Time is the timing resolution of the readout electronics. 

1.4. Linear

The parameters are Min Time, Max Time, Bin Step and Use Bin Center:

# Bins indicates how many correlation bins will be computed. Large # Bins result in long computation time.

In this binning mode, the first bin boundary b_0 is equal to Min Time and does not need to be equal to zero. IT is critical to pay attention to the # Bins indicator, as large bin number result in very long computation (and memory usage). In general, linear binning should be used locally, to "zoom" in on some feature of the correlation curve, and not to construct a full correlation curve from 0 to the max time lag.

A comparison of different types of binning is shown on the figure below, where the ACF of a us-ALEX photon stream was computed using different bin types:

The parameters for the 3 binning types where set to the values shown in the illustrations above.

The linear binning parameters were:

Min Time: 0.1 ms

Max Time: 1 ms

Bin Step: 1 us

The latter binning provides much more detail on the 50 us period oscillations, but uses 900 bins for this limited window and therefore takes significantly more time to compute. The CPU time used by the 4 calculations are:

Multitau: 3,008 ms

Multitau-10: 1,116 ms

Logarithmic: 3,699 ms

Linear: 13,887 ms

2. ACF Analysis

ACF analysis is performed for the set of photon streams entered in the ACF Photon Streams array located to the left of the ACF Graph (red box in the figure at the top) by clicking on the ACF button. New plots are added to the graph.

The Modulation Function ACF checkbox below this list indicates whether the computed ACF is that of the photon stream (unchecked) or that of the alternation period histogram limited to that stream (checked). The latter is used to correct the ACF for us-ALEX-related oscillations (shown in the figure above). Computing this modulation function ACF is not needed to apply modulation corrections, and is provided as a tool for the interested user.

3. CCF Analysis

CCF analysis works similarly to ACF analysis, except that two streams need to be defined for each curve.

The streams are defined in the Photon Stream Pairs array to the left of the CCF Graph (red box in the figure below).

New plots are added to the graph.

Note that the CCF curves use the same bin definitions (CF Bin Type) as the ACF curves (blue box in the top figure).

The Modulation Function ACF checkbox below this list indicates whether the computed ACF is that of the photon stream (unchecked) or that of the alternation period histogram limited to that stream (checked). The latter is used to correct the ACF for us-ALEX-related oscillations (shown in the figure above). Computing this modulation function ACF is not needed to apply modulation corrections, and is provided as a tool for the interested user.

4. Burst Correlation Analysis

While correlation analysis can be performed as soon as photon streams have been defined, the calculation can be limited to bursts verifying contraints defined in the Burst Statistics Definitions tab of the Burst Analysis page, by checking the Constrained Bursts checkbox (there is one checkbox for each graph, see green boxes in the figures above). This only works if the Burst Analysis has been performed.

Moreover, if ALEX analysis has been performed, checking the ALEX Selected Bursts checkbox will limit the analysis to the selected bursts (there is one checkbox for each graph, see green boxes in the figures above).

Correlation analysis of burst photons simply selects photons belonging to bursts before proceeding with the correlation function calculation. There are a couple of caveats with this definition:

• artificial time lags are introduced, corresponding to the separation between the last photon of a burst and the first photon of the following burst. In general, the effect of these few hundreds to few thousands time lags on the correlation function is negligible.

• the characteristic diffusion time is sensitive to the burst search and burst selection parameters.

The latter problem is easy to understand, as stringent search criteria will tend to reduce the duration of bursts, therefore reducing the apparent diffusion time. Burst selection criteria can in principle have effects in either direction.

Moreover, the correlation amplitude (which, loosely speaking, reflects the concentration of molecules in solution) depends strongly on those parameters as well. For instance, using an increasingly large Min Burst Separation burst search parameter will eventually bring down the correlation amplitude to the value corresponding to the correlation function of the whole time trace.

5. Corrections

Three types of corrections can be applied (simultaneously or separately) while calculating the autocorrelation functions, using one or more of the corresponding checkboxes at the bottom right of the graph (green box in the top figure):

• afterpulsing (AP) correction (AP Correction)

• background correction (Normalize by Signal/Background)

• laser alternation correction (Normalize by Modulation Function ACF)

Unless two identical photon streams are chosen, no afterpulsing correction is necessary when computing their cross-correlation function, therefore only two options are available for CCFs:

• background correction (Normalize by Signal/Background)

• laser alternation correction (Normalize by Modulation Functions CCF)

5.1. Afterpulsing Correction

The effect of detector afterpulsing on the ACF can be easily corrected (Ref. 7) provided the time scale of the phenomenon of interest resulting in a measurable ACF amplitude is well separated from that of afterpulses (which is typically limited to short time scales, and in most cases, below 1 us).

The necessary inputs for this correction to be applied are:

1. the uncorrected ACF of the signal

2. the mean count rate of the signal

3. the ACF of an uncorrelated signal

4. the corresponding mean count rate of that signal

The first term is obtained as part of the calculation. The second term is obtained from the Background Analysis page (Average Rate (Hz) array).

The third term is defined in the ACF Correction Files array of the ALiX Settings >> Correlation Analysis tab. In other words, such an ACF needs to have been computed (with the same bin definitions used for the current calculation), saved and selected in the Settings (see the corresponding manual page for details on how to load such a file).

The fourth parameter is in fact part of this ACF Correction File and both are discussed next.

Selecting an ACF Correction File

Note that the ACF Correction File in the Settings window is associated with a Channel name. It is important to make sure that the name used in the current data set to be analyzed (visible in the Channel Names array in the Data File page) matches one of the names in the ACF Correction Files array. For instance, if the ACF of stream F_D^D is computed, and channel D (named for instance in the current file "Ch 1") was recorded with detector X, an ACF of an uncorrelated signal recorded with detector X needs to be loaded in the ACF Correction Files array and its Channel name needs to be changed to "Ch 1".

Creating an ACF Correction File

To create an ACF Correction File for detector X, it is recommended to acquire and load a file comprised of detector X's dark counts only, compute its average count rate (by performing a Background Analysis), compute its ACF and right-click on its legend and select Save as Afterpulsing ACF:

It is critical that the ACF is computed with as many photons as possible to minimize its variance (several millions are recommended). Unfortunately, since single-photon avalanche diodes have very low dark count rates, this means long recording time. However, such a file is usually valid for the whole life of the detector and justifies the effort. While it is possible to acquire data faster by exposing the detector to a constant light source, it is rather difficult to avoid residual intensity fluctuations of various origins.

With these steps accomplished, computing an afterpulsing-corrected ACF is simply a matter of checking the AP Correction checkbox and pressing the ACF button. An indication that the AP Correction is invalid (for instance because the ACF Correction File is inadequate, the average count rate has not been updated or the hypotheses of Ref. 7 are not met), is a dip in the ACF at low time lags or unexpected oscillations in the ACF.

5.2. Background Correction

Uncorrelated background contribution to a recorded signal results in a reduction of the ACF amplitude which can be easily compensated for by a simple renormalization.

Note that the background rate values computed in the Background Analysis page are in general not appropriate because some, if not most, of their amplitude is due to out-of-focus molecules, which contribute correlated background to the signal (and therefore count as signal).

The best strategy to obtain the appropriate uncorrelated background rate values is to record a buffer only sample in the exact same conditions as those of the data file being analyzed. Perform the analysis up to the Background Analysis step and obtain background rate values, then save them in memory as uncorrelated background rates as described in the corresponding page of the manual.

5.3. Alternation Correction

In the case of us-ALEX data, periods of donor laser excitation alternate with periods of acceptor laser excitation. This results in oscillations in the calculated ACFs and CCFs (see for instance the figure illustrating Section 1 above), which are difficult to take into account when fitting models to the correlation functions.

Fortunately, when basic assumptions are satisfied, a simple renormalization of the raw correlation function by that of the alternation period histogram (for the relevant photon stream) recovers the correlation function of all other phenomena (see  Ref. 10 for details). This correction work only for "pure" photon streams of the form F_X^Y and not for the Donor, Acceptor and All_Photons streams, as they comprise photons from both excitation periods..

An example of the effect of the first and last correction on a us-ALEX ACF is shown below:

Green: original us-ALEX ACF, red: alternation correction, blue: alternation and afterpulsing corrections.

6. Context menu operations

The ACF/CCF graphs context menus are similar and offer standard graph/plot functionalities plus some specific operations described next. A snapshot of the ACF Graph context menu showing up when right-clicking on the name of a plot (or in the graph region) is depicted below:

6.1. ACF/CCF Fit

The Fit menu gives access to a few 2D and 3D models described in the ALiX Reference>>Curve Fitting>>Fit Models>>Correlation Function Fit part of the manual.

6.2. Analysis

This menu gives access to the (ACF-1) Ratios Calculation script, which assumes that the ACF Graphs comprises N ACFs from molecules in a flow (ACF_f,i, i = 1, ..., N), followed by the corresponding N ACFs of the same molecules without flow (i.e. only diffusing, ACF_d,i, i = 1, ..., N).

The script then computes N curves equal to (ACF_f,i - 1)/(ACF_d,i - 1), which should correspond to the flow part of the correlation function.