## Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting

Optics Express, Vol. 11, Issue 26, pp. 3583-3591 (2003)

http://dx.doi.org/10.1364/OE.11.003583

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### Abstract

Fluorescence correlation spectroscopy (FCS) is a powerful spectroscopic technique for studying samples at dilute fluorophore concentrations down to single molecules. The standard way of data acquisition, at such low concentrations, is an asynchronous photon counting mode that generates data only when a photon is detected. A significant problem is how to efficiently convert such asynchronously recorded photon count data into a FCS curve. This problem becomes even more challenging for more complex correlation analysis such as the recently introduced combination of FCS and time-correlated single-photon counting (TCSPC). Here, we present, analyze, and apply an algorithm that is highly efficient and can easily be adapted to arbitrarily complex correlation analysis.

© 2003 Optical Society of America

## 1. Introduction

*et al*. [9

9. J.S. Eid, J.D. Müller, and E. Gratton “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. **71**, 361–368 (2000). [CrossRef]

10. D. Magatti and F. Ferri “25 ns software correlator for photon and fluorescence correlation spectroscopy,” Rev. Sci. Instrum. **74**, 1135–1144 (2003). [CrossRef]

## 2. Theory

### 2.1 Asynchronous single-photon counting data

*t*,

_{1}*t*, …,

_{2}*t*} of the detected photons, where

_{N}*N*is the total number of detected photons during a given measurement. As special feature of these detection times is that they are integer multiples of some minimal time δ

*t*, determined by the temporal resolution of the detection electronics. Without restriction of generality, it will be assumed that all times are measured in units of δ

*t*, so that all the numbers

*t*take integer values. The value

_{j}*g*(

*τ*) of the autocorrelation function for a given lag time

*τ*is defined as

*I*(

*t*) is the photon detection intensity at a given time

*t*, and the brackets denote averaging over all possible values of time

*t*. For a photon detection measurement with temporal resolution δ

*t*, the intensity values

*I*(

*t*) within consecutive time intervals can only take the values 1/δ

*t*or 0, depending on whether there was a photon detection event during a time interval of width δ

*t*or not. The average in Eq. (1) is then calculated as the sum over all consecutive time intervals of width δ

*t*, divided by the total number of summed intervals.

### 2.2 Autocorrelation with logarithmic time-scale

^{8}possible lag time value. Calculation of

*g*(τ) for all of these values would be an enormous time-consuming numerical effort. Instead, the autocorrelation is calculated for only few, approximately logarithmically spaced values of τ, which is sufficient for most FCS applications. This calculation at selected, approximately logarithmically spaced lag times was first used for hardware correlators, due to restrictions in practical complexity [11

11. K. Schätzel “Correlation techniques in dynamic light scattering,” Appl. Phys. B. **42**, 193–213 (1987). [CrossRef]

*j*taking integer values starting with one and running up to some maximum number

*j*=

_{max}*n*; where

_{casc}B*B*is some integer base number; the bracket ⌊ ⌋ gives the integer part of the enclosed expression. The resulting lag times are grouped into

*n*cascades with equal spacing of 2

_{casc}^{⌊j/B⌋}. The advantage of such a choice of lag times is that all τ

*have integer values so that fast integer arithmetic can be used in subsequent computations. For example, when using a base number value of*

_{j}*B*=10 and

*n*=3, one obtains the lag time sequence {

_{casc}*τ*}={1,2,…,9,10,12,…,28,30,34,…,70}.

_{j}### 2.3 Synchronous photon intensity data:bin-and-correlate algorithm

*t*-

_{N}*t*, into intervals of unit length δ

_{1}*t*, and to sort the detected photons into these intervals corresponding to their arrival times

*t*. The result is a synchronous photon detection intensity file

_{j}*I*with

_{j}*j*running from 1 through

*t*-

_{N}*t*, where the

_{1}*I*can only adopt the values one or zero. The fluorescence autocorrelation can then be calculated as given by Eq. (1). In practice, such an approach is prohibitively memory demanding and computationally expensive. As an example, consider an experiment with an average count rate of 10

_{j}^{5}counts per second and a typical time resolution of δ

*t*=100 ns. A measurement lasting one minute would result in an average number of 6·10

^{6}detected photons. Converting the photon arrival data into a synchronous binary intensity file with 100 ns temporal resolution results in the huge number of 6·10

^{8}time intervals, or little less than 100 MByte of data, whereas only 1 % of these will contain a nonzero value. In contrast, the asynchronous file of photon arrival times takes only an amount of 12 MByte, if one assumes that every photon is tagged with a 2 Byte long label containing its arrival time and some overflow information.

### 2.4 Asynchronous photon intensity data time-tag-to-correlation algorithm

*t*,

_{1}*t*, …,

_{2}*t*}, without converting them into time-binned data. In its simplest form, our algorithm is rather straightforward. For a given lag time τ, a second vector of arrival times {

_{N}*t*′

_{1},

*t*′

_{2},…,

*t*′

_{N}} is generated, containing the time values

*t*′

*=*

_{j}*t*+τ. In the beginning, the value of the autocorrelation at lag time τ is set to zero. The algorithm starts with the time

_{j}*t*in the original vector and moves to consecutive time entries in that vector until it encounters a value

_{1}*t*that is equal to or larger than

_{j}*t*′

_{1}. If

*t*=

_{j}*t*′

_{1}, the value of the autocorrelation at lag time τ is increased by one. Next, the algorithm switches to the entries of the second vector and, starting with

*t*′

_{1}, moves to consecutive time entries in that vector until it encounters a value

*t*+τ if there was a photon detection event at time

*t*.

### 2.5 Correlation time coarsening

*t*when coming to the calculation of the autocorrelation function at increasingly larger lag time. This is equivalent to the multiple-tau and multiple-sampling time correlation method employed in hardware correlators [12

_{j}12. K. Schätzel, M. Drewel, and S. Stimac “Photon correlation measurements at large lag times: Improving statistical accuracy,” J. Mod. Opt. **35**, 711–718 (1988). [CrossRef]

*t*,

_{1}*t*,…,

_{2}*t*} and {

_{N}*t*′

_{1},

*t*′

_{2},…,

*t*′

*}, all time entries*

_{N}*t*and

_{j}*t*′

*are associated with weight values*

_{j}*w*and

_{j}*w*′

_{j}that are all set to one at the start of the algorithm. In case of an equality

*t*=

_{j}*t*′

_{k}, the autocorrelation is increased by the weight product

*w*

_{j}*w*′

_{k}and not by one. A time coarsening step is inserted each time when finishing the calculations for one cascade of

*B*lag times

*τ*with equal spacing and before starting with the next cascade of

_{j}*B*lag times with doubled spacing: All values {

*t*,

_{1}*t*,…,

_{2}*t*} used in the previous cascade are divided by two and rounded to the nearest lower integer value, which will occasionally leads to the occurrence of consecutive entries with the same time value. Before continuing the autocorrelation computation, such double entries are reduced to one entry, and the corresponding weight of that remaining entry is increased by the weight of the eliminated one. Thus, with increasing lag time τ

_{N}_{j}, the time scale underlying the autocorrelation calculation becomes increasingly coarser, and the total number of time entries to be processed increasingly smaller. To correct for the varying time scale of the autocorrelation calculation, one has finally to divide, at each lag time τ

_{j}, the calculated autocorrelation value by the corresponding time scale factor 2

^{⌊j/B⌋}. As pointed out in Ref. [12

12. K. Schätzel, M. Drewel, and S. Stimac “Photon correlation measurements at large lag times: Improving statistical accuracy,” J. Mod. Opt. **35**, 711–718 (1988). [CrossRef]

### 2.6 Generation of cross-correlation data

13. P. Schwille, F.J. Meyer-Almes, and R. Rigler “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. **72**, 1878–1886 (1997). [CrossRef] [PubMed]

14. M. Höbel and J. Ricka “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors,” Rev. Sci. Instrum. **65**, 2326–2336 (1994). [CrossRef]

*t*,

_{1}*t*,…,

_{2}*t*} and the time values of the second channel for calculating the lag-time shifted vector {

_{N}*t*′

_{1},

*t*′

_{2},…,

*t*′

_{N}}. In that case, one obtains the cross-correlation of the second channel with positive lag times against the first channel. By reversing the order, i.e., assigning the time values of the second channel to {

*t*,

_{1}*t*,…,

_{2}*t*} and using the time values of the first channel for {

_{N}*t*′

_{1},

*t*′

_{2},…,

*t*′

_{N}}, one obtains the cross-correlation of the first channel with positive lag times against the second channel.

### 2.7 Combination with time-correlated single-photon counting

4. C. Eggeling, S. Berger, L. Brand, J.R. Fries, J. Schaffer, A. Volkmer, and C.A.M. Seidel “Data registration and selective single-molecule anaylsis using mulit-parameter fluorescence detection,” J. Biotechnol. **86**, 163–180 (2001). [CrossRef] [PubMed]

7. D.C. Lamb, A. Schenk, C. Röcker, C. Scalfi-Happ, and G.U. Nienhaus “Sensitivity Enhancement in Fluorescence Correlation Spectroscopy of Multiple Species Using Time-Gated Detection,” Biophys. J. **79**, 1129–1138 (2000). [CrossRef] [PubMed]

8. M. Böhmer, M. Wahl, H.J. Rahn, R. Erdmann, and J. Enderlein “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. **353**, 439–445 (2002). [CrossRef]

*t*,

_{1}*t*,…,

_{2}*t*} are eliminated where the lifetime tag of the corresponding photons lie outside the set time-gate. In case of applying a filter function to the recorded photons that acts on the lifetime tag, the procedure is similarly simple: the initial weight values

_{N}*w*are set to the filter output instead to one, subsequently applying the FCS algorithm without any alteration. Thus, the presented calculation method is most general and allows arbitrarily complex correlation analysis of the TTTR data.

_{j}## 3. Computation time

### 3.1 Bin-and-correlate algorithm

*T*, and neglecting the computational load of transforming the time-tagged data into binned data, one has to perform roughly

*T*/τ

_{k}multiplications for calculating the autocorrelation at τ=τ

_{k}(measurement time over minimum possible bin width). Thus, the total number of multiplications is estimated as

*k*runs over all lag times used.

### 3.2 Time-tag-to-correlation algorithm

*N*, the average rate of photon detection events is

*N*/

*T*. On a time scale of 2

^{k}, i.e., after

*k*time-coarsening steps leading to a temporal resolution of 2

^{k}(see above), the length of the vector {

*t*,

_{1}*t*,…,

_{2}*t*} will be approximately equal to the total number of time intervals at that temporal resolution times the probability to find at least one photon detection event during time interval of length 2

_{N}^{k}. The number of time intervals of width 2

^{k}equals the total measurement time

*T*divided by 2

^{k}. Assuming furthermore a Poissonian statistics of photon detection, the probability to find at least one photon within a time interval of length 2

^{k}is given by 1-exp(-2

^{k}

*N*/

*T*). Finally, assuming that the number of necessary multiplications for calculating the autocorrelation at a given lag time value is roughly proportional to the length of the vector {

*t*,

_{1}*t*,…,

_{2}*t*}, one arrives at the following estimation of the total number of multiplications for the calculation of the complete autocorrelation curve

_{N}*k*runs from zero up to

*n*

_{casc}- 1.

### 3.3 Comparison of bin-and-correlate and time-tag-to-correlation algorithms

*B*=10 and

*n*=17, thus covering a time range of nearly six orders of magnitude (from τ

_{casc}_{1}=1 up to τ

_{170}=1310710). Let the smallest lag time of τ

_{1}=1 correspond to 100 ns in experimental time, a temporal resolution typical for TTTR photon counting measurements. Typical average photon count rates range between 1 kHz and 1 MHz. The result for the ratio of Lbin to Ltttr is shown in Fig. 1. The ratio

*L*/

_{bin}*L*does not depend on total measurement time

_{tttr}*T*but only on the count rate

*N*/

*T*which can be seen when inserting Eq. (3) and Eq. (4) into

*L*/

_{bin}*L*and noticing that the final result contains only the ratio

_{tttr}*N*/

*T*but not

*T*alone. As can be seen, the time-tag-to-correlation algorithm is superior to the bin-and-correlate approach even at high count rates. Moreover, one has to bear in mind that the made estimates are very conservative and biased in favour of the bin-and-correlate method: Any computational load of converting the tagged into binned data was neglected, and the rather unphysical assumption of evenly spaced photons was considered for the time-tag-to-correlation approach. In a real experiment, such as the detection of sparse numbers of molecules within a confocal detection volume, photon arrivals will be rather bunched into bursts every time a molecule diffuses through the detection region.

## 4. Experimental application

3. M. Böhmer, F. Pampaloni, M. Wahl, H.J. Rahn, R. Erdmann, and J. Enderlein “Advanced Time-Resolved Confocal Scanning Device For Ultrasensitive Fluorescence Detection,” Rev. Sci. Instrum. **72**, 4145–4152 (2001). [CrossRef]

^{-9}M). The fluorescence light is collected through the same objective and, after passing a dichroic mirror (DRLP 650, Omega Optical), imaged onto a confocal aperture (50 µm diameter). After splitting the transmitted light into two detection channels by a 50/50-beam splitter, it is refocused onto the active area of two single-photon avalanche diodes (SPAD, SPCM-AQR 13, Perkin Elmer). The photon count signals of the SPADs are processed by a fast TCSPC electronics (TimeHarp 200, PicoQuant) and converted into a TTTR data stream of photon arrival and TCSPC times. The macroscopic temporal resolution of the TTTR data is 100 ns. The FCS algorithm was applied to data obtained from buffered aqueous solutions of fluorescently labelled latex beads (Crimson Red FluoSpheres® by Molecular Probes of ~36 nm diameter). These beads deliver a bright fluorescence signal and a well-defined slowly decaying autocorrelation curve due to their slow diffusion. Figure 2 shows the result of different autocorrelation algorithms applied to 10 seconds of a measurement at 200 µW excitation power.

*without*(red dots) and

*with*time-scale coarsening (light-green line). The figure demonstrates that the bin-and-correlate and the time-tag-to-correlation algorithms yield identical results, as it should be. It also shows the effect of time-scale coarsening: The resulting autocorrelation curve is smoothed at larger time scales (at the lowest lag-time values, where no time-scale coarsening applies, all algorithms give of-course the same result).

## 5. Conclusion

## Acknowledgments

## References

1. | C. Zander, J. Enderlein, and R.A. Keller (Eds.) |

2. | W. Becker, H. Hickl, C. Zander, K.H. Drexhage, M. Sauer, S. Siebert, and J. Wolfrum “Time-resolved detection and identification of single analyte molecules in microcapillaries by time-correlated single-photon counting (TCSPC),” Rev. Sci. Instrum. |

3. | M. Böhmer, F. Pampaloni, M. Wahl, H.J. Rahn, R. Erdmann, and J. Enderlein “Advanced Time-Resolved Confocal Scanning Device For Ultrasensitive Fluorescence Detection,” Rev. Sci. Instrum. |

4. | C. Eggeling, S. Berger, L. Brand, J.R. Fries, J. Schaffer, A. Volkmer, and C.A.M. Seidel “Data registration and selective single-molecule anaylsis using mulit-parameter fluorescence detection,” J. Biotechnol. |

5. | M. Böhmer and J. Enderlein “Single molecule detection on surfaces with the confocal laser scanning microscope,” in ref.[1], pp.145–183. |

6. | R. Rigler and E. Elson (Eds.) |

7. | D.C. Lamb, A. Schenk, C. Röcker, C. Scalfi-Happ, and G.U. Nienhaus “Sensitivity Enhancement in Fluorescence Correlation Spectroscopy of Multiple Species Using Time-Gated Detection,” Biophys. J. |

8. | M. Böhmer, M. Wahl, H.J. Rahn, R. Erdmann, and J. Enderlein “Time-resolved fluorescence correlation spectroscopy,” Chem. Phys. Lett. |

9. | J.S. Eid, J.D. Müller, and E. Gratton “Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,” Rev. Sci. Instrum. |

10. | D. Magatti and F. Ferri “25 ns software correlator for photon and fluorescence correlation spectroscopy,” Rev. Sci. Instrum. |

11. | K. Schätzel “Correlation techniques in dynamic light scattering,” Appl. Phys. B. |

12. | K. Schätzel, M. Drewel, and S. Stimac “Photon correlation measurements at large lag times: Improving statistical accuracy,” J. Mod. Opt. |

13. | P. Schwille, F.J. Meyer-Almes, and R. Rigler “Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,” Biophys. J. |

14. | M. Höbel and J. Ricka “Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors,” Rev. Sci. Instrum. |

**OCIS Codes**

(170.6280) Medical optics and biotechnology : Spectroscopy, fluorescence and luminescence

(180.1790) Microscopy : Confocal microscopy

(300.2530) Spectroscopy : Fluorescence, laser-induced

**ToC Category:**

Research Papers

**History**

Original Manuscript: November 20, 2003

Revised Manuscript: December 12, 2003

Published: December 29, 2003

**Citation**

Michael Wahl, Ingo Gregor, Mattias Patting, and Jörg Enderlein, "Fast calculation of fluorescence correlation data with asynchronous time-correlated single-photon counting," Opt. Express **11**, 3583-3591 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3583

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### References

- C. Zander, J. Enderlein, R.A. Keller (Eds.) Single-Molecule Detection in Solution - Methods and Applications (VCH-Wiley, Berlin/New York, 2002). [CrossRef]
- W. Becker, H. Hickl, C. Zander, K.H. Drexhage, M. Sauer, S. Siebert, J. Wolfrum �??Time-resolved detection and identification of single analyte molecules in microcapillaries by time-correlated single-photon counting (TCSPC),�?? Rev. Sci. Instrum. 70, 1835-41 (1999). [CrossRef]
- M. Böhmer, F. Pampaloni, M. Wahl, H.J. Rahn, R. Erdmann, J. Enderlein �??Advanced Time-Resolved Confocal Scanning Device For Ultrasensitive Fluorescence Detection,�?? Rev. Sci. Instrum. 72, 4145-52 (2001). [CrossRef]
- C. Eggeling, S. Berger, L. Brand, J.R. Fries, J. Schaffer, A. Volkmer, C.A.M. Seidel �??Data registration and selective single-molecule anaylsis using mulit-parameter fluorescence detection,�?? J. Biotechnol. 86, 163-80 (2001). [CrossRef] [PubMed]
- M. Böhmer, J. Enderlein �??Single molecule detection on surfaces with the confocal laser scanning microscope,�?? in ref.[1], pp.145-83.
- R. Rigler, E. Elson (Eds.) Fluorescence Correlation Spectroscopy (Springer, New York/Berlin, 2001). [CrossRef]
- D.C. Lamb, A. Schenk, C. Röcker, C. Scalfi-Happ, G.U. Nienhaus �??Sensitivity Enhancement in Fluorescence Correlation Spectroscopy of Multiple Species Using Time-Gated Detection,�?? Biophys. J. 79, 1129-38 (2000). [CrossRef] [PubMed]
- M. Böhmer, M. Wahl, H.J. Rahn, R. Erdmann, J. Enderlein �??Time-resolved fluorescence correlation spectroscopy,�?? Chem. Phys. Lett. 353, 439-45 (2002). [CrossRef]
- J.S. Eid, J.D. Müller, E. Gratton �??Data acquisition card for fluctuation correlation spectroscopy allowing full access to the detected photon sequence,�?? Rev. Sci. Instrum. 71, 361-8 (2000). [CrossRef]
- D. Magatti, F. Ferri �??25 ns software correlator for photon and fluorescence correlation spectroscopy,�?? Rev. Sci. Instrum. 74, 1135-44 (2003). [CrossRef]
- K. Schätzel �??Correlation techniques in dynamic light scattering,�?? Appl. Phys. B. 42, 193-213 (1987). [CrossRef]
- K. Schätzel, M. Drewel, S. Stimac �??Photon correlation measurements at large lag times: Improving statistical accuracy,�?? J. Mod. Opt. 35, 711-8 (1988). [CrossRef]
- P. Schwille, F.J. Meyer-Almes, R. Rigler �??Dual-color fluorescence cross-correlation spectroscopy for multicomponent diffusional analysis in solution,�?? Biophys. J. 72, 1878-86 (1997). [CrossRef] [PubMed]
- M. Höbel, J. Ricka �??Dead-time and afterpulsing correction in multiphoton timing with nonideal detectors,�?? Rev. Sci. Instrum. 65, 2326-36 (1994). [CrossRef]

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