## Abstract

The multiplexing of fluorescence correlation spectroscopy (FCS), especially in imaging FCS using fast, sensitive array detectors, requires the handling of large amounts of data. One can easily collect in excess of 100,000 FCS curves a day, too many to be treated manually. Therefore, ImFCS, an open-source software which relies on standard image files was developed and provides a wide range of options for the calculation of spatial and temporal auto- and cross-correlations, as well as differences in Cross-Correlation Functions (ΔCCF). ImFCS permits fitting of standard models to correlation functions and provides optimized histograms of fitted parameters. Applications include the measurement of diffusion and flow with Imaging Total Internal Reflection FCS (ITIR-FCS) and Single Plane Illumination Microscopy FCS (SPIM-FCS) in biologically relevant samples. As a compromise between ITIR-FCS and SPIM-FCS, we extend the applications to Imaging Variable Angle-FCS (IVA-FCS) where sub-critical oblique illumination provides sample sectioning close to the cover slide.

©2010 Optical Society of America

**1.** Introduction

Fluorescence Correlation Spectroscopy (FCS) records fluorescence fluctuations from small observation volumes and extracts information about molecular processes which underlie these fluctuations. FCS is typically performed in confocal systems at a single spot at a time. Multiplexing started with the first two-foci measurements by Brinkmeier *et al* [1], later followed by 4 foci measurements on a CMOS device with 4 detection elements [2]. This was followed by the usage of electron multiplying charge coupled device (EMCCD) cameras for the first time as detection elements with sufficient high read-out rates to perform confocal FCS [3]. Although the time resolution could be improved by using selected regions of the EMCCD [4], the number and density of confocal spots was limited since neighboring confocal volume elements cross-talk to each other in dependence of their distance. Therefore a minimum distance between focal volume elements of at least 10-15 confocal diameters was required to perform FCS [3]. Therefore, despite a density of 250,000 pixels on a CCD, not more than 3-400 confocal spots could be used. This was improved upon by spinning disk FCS [5] at the expense of the observation time per pixel. With the introduction of total internal reflection (TIR) [6–8], single plane illumination microscopy (SPIM) [9] and critical angle illumination [10,11] in FCS, the creation of the observation volumes was facilitated by selectively illuminating only a thin layer of the sample which lies in the focal plane of the detection objective. Recently, a stationary Nipkow disc based multiplexed FCS has also been demonstrated [12]. Since these techniques only illuminate the parts of the sample which are observed [9,13], the background and cross-talk between the detection elements is greatly reduced making FCS in an imaging mode possible even on live cells and within living organisms. Multiplexing can also be achieved by scanning the beam in a pre-determined manner in a confocal laser scanning microscope. A wide variety of scanning FCS techniques have been reported [14]. Scanning FCS has the advantages of reduced cross-talk since the pixels are spatially well separated and higher temporal resolution whereas Imaging FCS has the advantage of obtaining more measurements per sample per time interval with less phototoxicity due to a low light exposure allowing measurements over for a longer time [9].

Imaging FCS is now routinely capable of recording FCS measurements on regions of interest (ROI) of 4000 points or more. The limit of the ROI is here set by two requirements: i) the frame rate of the camera, which decreases with increasing size of the ROI, but has to be kept at a minimum value so that the time resolution is sufficient to observe single molecule events of interest. This is typically between 250 and 3000 frames per second, for ROIs on the order of 400-4000 pixels. The highest frame rate here is determined by the maximum available camera speed at the moment. ii) The capacity of computers of handling the acquired data in an acceptable time. At the time of writing, 32 × 32 pixels can be handled on a standard PC (3 GHz Dual Core, 4 GB RAM, 32 bit Windows XP) in an acceptable time (~1 min for correlation and ~4 mins for curve fitting). Higher pixel numbers can considerably slow down calculations as well as data fitting.

Since we record typically 10,000 frames per measurement at a frame rate of 250-3000 frames per second, our recording time is between 3 and 40 s. This means that a typical experimenter records easily 100,000 correlation curves per day. This is an amount of data that cannot be treated manually anymore. Presently, to the best of our knowledge, no commercially available software can read in image stacks and calculate correlations in each pixel of the image stack. The goal of this work was therefore to provide a program that allows the user to read-in the intensity files from different CCDs, to automatically calculate the temporal autocorrelations and temporal and spatial cross-correlations, to fit all data with a set of predefined models and to display images and histograms of all parameters. The program, ImFCS, provided here is written in C++ for Windows XP/Vista, and is linked to the widely available commercial software Igor Pro (WaveMetrics Inc, Lake Oswego, OR, USA) to provide a graphical interface for the user. It should be noted though that the C++ routines can as well be easily implemented into any other graphic user interface or adapted for any other operating system. The article is divided into two parts, the first part deals with the description of the software and the second part deals with examples of application of the software in various camera-based FCS techniques (ITIR-FCS, IVA-FCS and SPIM-FCS).

**2.** Description of ImFCS

ImFCS is a data analysis tool for camera based FCS. Upon recording of a time series or stack of images by a camera and providing the stacks to ImFCS, the software presents intensity time traces for each pixel and calculates auto– and cross-correlations of and between pixels. The correlation curves are fitted with suitable models to extract parameters. The fitted parameters are displayed to the user as individual images. Figures S1-S3 in the supplement provide an overview of the program functionalities, structure and screen shot respectively.

#### 2.1 Input to the program

Correlations are performed on a stack of multiple images acquired at different time points. Each image is made up of a certain number of pixels and each pixel has an associated intensity value. The format which is required as input by ImFCS is the Tagged Image File Format (tiff). The present specification of Tiff files 6.0 [15] allows an entire stack of frames to be stored as a single “multi-plane” Tiff file and most commercial softwares allow saving data in this format. Conversion of other file formats for storing frame stacks into *.tif is available with ImageJ [16]. The intensity values from the multi-plane tiff file are written into an intensity array of dimensions *n, w, l* where *n* is the number of frames, *w* is the number of rows in the image and *l* is the number of columns in the image. A detailed description of the program is given in the supplement (Secs. 2.1-2.3). Each measurement has a background value (*bg*) associated which originates from camera, environment and sample related issues. The user has three options to remove the background. The background value can be determined by a background file which was acquired without excitation of the fluorophores or the background value can be entered directly into software or can be set to the minimum value of the stack being correlated. For a full frame data treatment, the correlation is performed at each pixel and upon completion the output consists of *w × l* number of correlation curves. Note that ImFCS allows choosing subregions or cross-correlations between pixels in which case the number of correlation curves will vary accordingly. The program provides as an option to bin the data which is a process in which the adjacent pixels are added up. The number of pixels to be binned (*bin*) is determined by the user. In case, binning is performed, the output consists of $\lfloor \frac{w}{bin}\rfloor \times \lfloor \frac{l}{bin}\rfloor $ number of correlations where $\lfloor x\rfloor $ is the largest integer less than or equal to *x*.

#### 2.2 Correlation: Types and architecture

Correlations are performed between pixels which had been acquired at different times and/or locations. Assuming stationary processes, the acquisition time of the first frame can be set to $t=0$. The pixels in the frame are correlated individually with pixels in another frame that was acquired at $t=\tau $. The difference between the acquisition times of these two frames being correlated, *τ*, is referred to as lag time. The cross-correlation *G _{AB}(τ)* between the fluorescent intensities in pixels A and B (

*F*and

_{A}*F*) is defined as

_{B}Autocorrelation is a special case of cross-correlation when the correlation is performed on the fluorescent intensity for a single pixel. The above formula is modified by replacing B with A. The program calculates various types of cross-correlations, for instance, those between the centre pixel and the pixels along the central row or central column or the leading and trailing diagonal and the cross-correlation of the central pixels with the surrounding rectangular region of pixels. It calculates the differences in forward and backward correlations referred to as ΔCCF. Presently two formats of the calculation are permitted in the software.

These correlations are schematically displayed in Figs. S1 B and C in the supplement. The software allows the user to draw region(s) on an average intensity map of the stack, which can be auto- or cross-correlated. The details of how the above options are programmed are given in Sec. 2.4 in the supplement.

There are a number of important time scales for the calculation of the correlations. First, the frame rate of the camera limits the time resolution, and this time per frame is referred to as *Δτ*. Note that this time includes the illumination of the camera as well as the readout time (in our case, illumination times is between 0.2 and 1 ms and the readout time between 0.3 and 4.6 ms, resulting in overall frame rates between 171 and 2000 frames per second). All other time scales are multiples of this basic unit time *Δτ*. Second, the measurement has to be taken over a certain acquisition time *t _{acq}*. Third, the correlations are calculated for different lagtimes

*τ*(0<

*τ*<

*t*). Fourth, at different lagtimes

_{acq}*τ*, the width over which the intensity signal is integrated before the correlation is calculated can vary and is referred to as the bin width [17,18]. The program supports two correlator architectures

#### 2.2.1 Linear correlation

In linear correlation mode, the correlations are calculated at linearly increasing lagtimes $\tau =m\mathrm{\Delta}\tau $ where m ranges from *0* to *M-1*, if the correlations are calculated for *M* lagtimes. The bin width for each lagtime is kept constant at *Δτ*. Theoretically, the last point of the correlation is the acquisition time (*t _{acq}*). It is not advisable to calculate the correlation till

*t*since the number of data points to average are very few as the lagtime approaches

_{acq}*t*. To display correlations from $t=0$to$t={t}_{end}\left[{t}_{end}<{t}_{acq}\right],{t}_{end}/\mathrm{\Delta}\tau $+ 1 number of calculations need to be done. Substituting typical values,

_{acq}*Δτ*= 0.5 ms,

*t*= 1.0235 s, 2048 correlations at individual lag times need to be performed. For linear correlation, the lagtime is

_{end}*ℕ*is the set of natural numbers.

#### 2.2.2 Semi-logarithmic correlation

The semi-logarithmic correlator architecture is used more frequently since this architecture covers a larger range of lagtimes than the linear correlator using less number of computations. This correlator architecture is based on the multi-tau algorithm [17]. In the most common configuration, the first 16 correlations are at linearly increasing lagtimes $\tau =m\mathrm{\Delta}\tau $ where m ranges from *0* to *15* with a binwidth of *Δτ*. The next set of 8 correlations possess linearly spaced lagtimes at intervals of *2Δτ* beginning with (*15 + 2)Δτ* and a bin width of *2Δτ* . The next set of 8 correlations possess lagtimes at intervals of *4Δτ* beginning with *(31 + 4)Δτ* and a bin width of *4Δτ*. This is repeated for bin widths of *8Δτ, 16Δτ, 32Δτ, 64Δτ* and *128Δτ*. The last calculated lag time is at *(2048-1)Δτ*. Substituting *Δτ* = 0.5 ms, a lag time of 1.0235 s can be achieved by just *72 (16 + (8-1) × 8)* correlations. The same lagtime needs 2048 correlations in the linear configuration. The above example was for the configuration of a (16, 8) multi tau correlator but can be directly extended to any *(p, q)* correlator structure. In a *(p, q)* correlator, the first *p* correlations are at linearly increasing lagtimes $\tau =m\mathrm{\Delta}\tau $where m ranges from *0* to *p-1* with a binwidth of *Δτ*. The next *q* groups possess *p*/2 lagtimes with bin width and lagtime intervals which double from group to group. In this way a particular lagtime is always the sum of all the bin widths of the previous lagtimes. A *(p, q)* correlator calculates a correlation function at *h = [p + (q-1) × q/2]* number of lagtimes. The minimum number of frames (*Fr _{min}*), needed for a

*(p,q)*correlator is

_{min}= 2047. Although 2047 frames are sufficient to carry out the correlations, it is advised to perform the correlation with higher number of frames in order to increase the precision of the calculated correlation. Here, all the correlations have been calculated using 10000 frames. A detailed description of the semi-logarithmic correlator is available in [18,19]. Thus the lagtime in this architecture can be represented by the formula below

#### 2.3 Implementation in the program

There are two ways, by which the correlation can be calculated, using the sums of products method or by using Fourier transforms [20]. In this software, the correlation is calculated using the former method. The continuous expression for correlation in Eq [1]. is converted to discrete form and implemented in the program as in Eq [6]. for the linear and the first cycle of the semi-logarithmic architecture. Symmetric normalization is performed where each correlation is normalized by only those intensity values used in the calculation of the autocorrelation [17].

*Δτ*are created by summation and the correlations are calculated in these stacks as shown in Eq [7]. The correlations are calculated for

*p/2*points at arithmetically progressing time intervals at twice (

*2*) the time resolution,

^{1}*Δτ*, of the camera. This is followed for

*p/2*points with a time difference of four times (

*2*) the time resolution. This is repeated till the time difference has reached

^{2}*2*times the time resolution. The procedure by which the formula below is computed in the program is described in the supplement (Sec. 2.5).

^{q-1}#### 2.4 Fitting model

Two fitting models are included in the software to fit the data obtained above and to extract the parameters. The generalized fitting model for cross-correlation of diffusion and flow processes for square regions separated by *r _{x}* and

*r*in the

_{y}*x*and

*y*axes respectively, for TIRF based camera FCS [8] is given by Eq. [S1] in the supplement where

*N = <C>a*where

^{2}*a*is the pixel size in object space,

*C*is the surface concentration,

*D*is diffusion coefficient,

*v*and

_{x}*v*are the velocities in the

_{y}*x*and

*y*axes respectively.

*σ*is the width of the Gaussian Point Spread Function (PSF) [21,22] of the microscope in the

*x-y*plane defined by Eq. [S2] in the supplement. By setting

*r*and

_{x}*r*to zero, the model can be simplified to describe the autocorrelation as provided in [7,23]. In the case of SPIM-FCCS [9] with a light sheet of considerable thickness (

_{y}*σ*in

_{z})*z*direction, the same model was modified to Eq. [S6] where

*N = 2<C>a*Upon fitting with each of the models, the fitting parameters are displayed as image plots. Using the histogram button in the software, a histogram of the fitted parameters can be plotted. The number of histogram bins is calculated using the Freedman-Diaconis rule [24] where

^{2}σ_{z}.*max*and

*min*are the sample maximum and minimum respectively,

*s*is the sample size and

*Q1*and

*Q3*are the first and the third quartile respectively.

## 3. Data collection modes in Imaging FCS analyzed by ImFCS

One of the critical needs of FCS is to create a small observation volume (order of 10^{−15} l) to observe the fluctuations of fluorescence. This is achieved in imaging FCS by restricting the volume in which the sample is excited. In ITIR-FCS, a ~100 nm thin section close to the cover slip is illuminated by an evanescent wave which is created when a laser beam impinges on an interface coming from an optically more dense (*µ _{1}*) to an optical less dense (

*µ*) medium at angles greater than the critical angle (critical angle = sin

_{2}^{−1}(

*µ*)). In SPIM-FCS, the volume isolation is provided by a diffraction limited light sheet created in the focal-plane of the detection objective illuminating regions away from the cover slide [9].

_{2}/µ_{1}Recently, other related illumination schemes with sectioning capability have been introduced for imaging and FCS. Variable Angle Epi-fluorescent Microscopy (VAEM) [25] and Highly Inclined and Laminated Optical Sheet (HILO) microscopy were utilized for imaging of plant cells and single molecule imaging, respectively [26,27]. Critical angle illumination based FCS was demonstrated on fluorescent beads [10]. These techniques make use of sub-critical illumination. At sub-critical, oblique angles of illumination, the refracted light is just above the surface of separation sufficient to illuminate fluorophores away from the surface in the bulk sample. The use of sub-critical angles reduces the background considerably and provides volume isolation in the bulk suitable to perform FCS. Performing FCS in such illumination conditions is referred to as IVA-FCS (Imaging Variable Angle-FCS). IVA-FCS does not need any separate add-on apparatus to a TIRF microscope. Using IVA-FCS, diffusion of fluorescent beads in solution was studied. In comparison with ITIR-FCS, IVA-FCS has the advantage of increased penetration depth into bulk of the sample away from the surface of separation. A schematic of the illumination schemes is presented in Fig. 1 .

Representative examples of data collected by any of the 3 aforementioned illumination schemes and analyzed using ImFCS are presented here. The “materials and methods” are available in the supplement in Sec. 5. Three different processes are probed using Imaging FCS. The first part is an analysis of diffusive behavior of lipids and membrane proteins on artificial and cell membranes respectively using ITIR-FCS and beads in solution using IVA-FCS. The second part is an analysis of flow process using ΔCCF imaging in ITIR-FCCS. In the last part, coupled diffusion and flow processes in a model system and in a living zebrafish embryo are studied using ITIR-FCCS and SPIM-FCCS respectively.

**4.** Results and Discussion

#### 4.1 Analysis of diffusion by ITIR-FCS and IVA-FCS using ImFCS

A POPC bilayer doped with fluorescently labeled lipids was prepared and stacks were acquired using ITIR-FCS. 21x21 pixels of 10000 images were acquired and analyzed by the software. The entire set of 441 ACFs are shown along with two representative autocorrelations in Fig. 2A
. The fitting model fits properly to the experimental data. All the correlation curves can be fitted and the parameters can be retrieved. Two of those parameters namely, *D* and *N* are shown in Figs. 2B and 2C. All the values hereinafter are shown as mean ± SD. The average value of *D* and *N* are 1.37 ± 0.44 µm^{2}/s and 8 ± 2 (441 values) respectively. The values of the fitted parameters can be viewed as a histogram.

To demonstrate the applicability in live cells, a membrane protein called Epidermal Growth Factor Receptor (EGFR) fused with EGFP at the C terminus was expressed in CHO cells using transfection. The average value of *D* and *N* are 0.07 ± 0.04 µm^{2}/s and 50 ± 12 (451 values) respectively (Figs. 2D-2F). In contrast to the previous case, not all the curves were fitted properly. 7% of the curves exhibited bleaching and were not included in computing the average *D* and *N*. This was done by discarding values with diffusion coefficients less than 0.01 µm^{2}/s. Upon inspection, it can be seen that most of these values fall into four distinct pockets in the *D* and *N* map. The regions centered at rows 5, 11 and 21 are characterized by a sudden rise in intensity lasting for around 2 seconds during the 40 second acquisition period. The intensity rise is twice the average intensity during all the other times. The sudden increase in intensity could be attributed to aggregates of fluorescent proteins diffusing on the membrane. Such problems can be overcome by implementing automatic FCS analysis algorithms in ImFCS for removal of unwanted peaks corrupting the curves [28]. Half of the pixels in the region centered at row 2 are characterized by intensities traces which could be aggregates of fluorophores. On the contrary, the remaining half shows traces which exhibit a loss in fluorescence with time. This may be the case where the molecule is immobile on the cell membrane and hence the pixels exhibit bleaching. An option to correct for loss incurred due to bleaching is available in the software. The details of this option can be found in the supplement in Sec. 6.

Fluorescent beads were diffusing in solution and were studied using IVA-FCS. The average value of *D* and *N* are 1.1 ± 0.7 µm^{2}/s and 0.01 ± 0.006 (944 values) respectively. 2% of the curves were not fitted. The theoretical value calculated from the Stokes Einstein’s equation is 2.4 µm^{2}/s. The deviations from the theoretical model are due to the facts that FCS is sensitive to bright and big particles and beads are prone to aggregation. Particles comparable to the size of PSF will increase the apparent PSF [9]. The histograms of D and N are provided in the supplement in Sec.8. The diffusion coefficients obtained in this example are similar to the D’s obtained using SPIM-FCS [9]. Unlike the case of EGFR on cell membranes, there are less number of curves which are not fitted. (Figs. 2G-2I). Illumination at angles lower than the critical angle helps one to perform measurements in 3D environments. This can be further extended for biological applications where measurements need to be performed in native state. Different pixels are characterized by different correlation curves. This variation is due to the inherent variability associated with the system under investigation. Generally, it is observed that measurements on cells and biological samples have an increased variability when compared to the measurements on solutions and model membranes. A few guidelines to perform Imaging FCS are provided in the Sec. 9 of the supplement. A systematic comparison of diffusion coefficients obtained from Imaging FCS and other fluorescent techniques is available in Sec. 10 of the supplement.

*4.2 Analysis of flow by* ΔCCF imaging *using ImFCS*

ΔCCF imaging is demonstrated on two different systems, those exhibiting isotropic phenomena like diffusion and anisotropic phenomena like flow. The cross-correlations were calculated between adjacent pixels. For studying diffusion, a Rho-PE labeled POPC/POPG (2:1 500 µM) lipid bilayer prepared using the protocol in [8] was used. The forward and backward correlations for an isotropic process are similar and hence when the correlations are subtracted, on an average, the area under such curves is zero.

Quantum dots were immobilized to coverslides and moved using a mechanical stage simulating a flow process. For non-isotropic processes, the correlation in the direction of the flow, exhibits a maximum at the time it takes to travel from the first region to the second region being correlated. The intensity observed in any pixel is a sum total of the intensity of the pixel and the contributions of cross-talk from pixels which are separated from each other at distances on the order of the PSF. This cross-talk leads to a pseudo-autocorrelation term. Hence, the correlation in the direction against the flow is a decaying curve which is only due to the pseudo-autocorrelation between these two regions. When such curves are subtracted, the area under the resulting curve is a non-zero number.

The distributions of these two processes are shown in Fig. 3 . Flow being an anisotropic process, shows two distinct populations of curves, the populations being the forward and the backward correlation. On the contrary, the forward and backward correlations overlap each other in the case of diffusion. Both the distributions are Gaussian which differs in the mean value. The average values of ΔCCF are 0 ± 0.02 (380 values) and 0.03 ± 0.01 (420 values) for diffusion and flow respectively. The Gaussian distribution of ΔCCF values of flow has a non-zero mean characteristic of anisotropic processes. As expected, the SD of ΔCCF values of a directed flow process is less than that of a random diffusion process. Thus ΔCCF distribution serves as a way to distinguish processes exhibiting directed transport alone or in combination with other processes. The reader is referred to Sec. 3.2 in the supplement for a theoretical treatment of the above phenomena.

#### 4.3 Analysis of diffusion and flow by ITIR-FCCS and SPIM-FCCS using ImFCS

In most of the cases, it is found that diffusion and flow processes are always seen in conjunction. This was the primary motive behind defining the fitting model as a generalized function to describe both processes. Imaging FCS was used to probe diffusion and flow processes together. As a model system, a Rho-PE labeled POPC bilayer was moved using a microscope stage and ITIR-FCCS measurements were performed. To extend the applicability of Imaging FCS to real, complex 3D environments; SPIM-FCS was used to study the diffusion of fluorescent beads injected into a zebrafish embryo.

All the curves were fitted with Eq. [S3] in the case of the model system described above. The average *D* and *N* obtained are 1.65 ± 0.48 µm^{2}/s and 152 ± 20 (400 values) respectively. As expected, the coefficients of variation (mean/SD) of the above two values are small since a model system is being studied here. Apart from the *D* value, the system is characterized by the velocity which is found to be 9.92 ± 0.11 µm/s (400 values). The microscope stage was moved at a speed of 10 µm/s and the expected and obtained values are close to each other. These values were obtained by fitting the autocorrelation data.

Please note that the autocorrelation expression as given in Eq. [S3] is an even function in both *v _{x}* and

*v*. Hence the autocorrelation cannot reveal the direction of the flow and the cross correlation needs to be calculated which is not an even function in

_{y}*v*and

_{x}*v*. The cross-correlation of the pixel in the center was performed with 4 other pixels, one each along the co-ordinate axes separated from the center pixel by 4 pixel units. The cross-correlation curves with pixels separated by 4 pixel units along the positive and negative

_{y}*y*axis are identical and close to the background noise. This indicates that there is no flow along these directions. On the contrary, the cross-correlation curves of the center pixel with pixels separated by 4 pixel units along the positive and negative

*x*axis are different from each other. The cross-correlation along the positive

*x*axis shows a peak characteristic of processes exhibiting flow, indicating the fact that the flow is along this axis. This is supported by the fact that, the cross-correlation along the negative

*x*axis is insignificant. This flow velocity in magnitude and direction is displayed as an arrow plot.

To demonstrate the applicability of analyzing coupled diffusion and flow data in biological systems, SPIM-FCCS was performed in the veins in a zebrafish embryo by injecting fluorescent beads. The average *D* and *N* of the beads were found to be 1.18 ± 0.7 µm^{2}/s and 0.38 ± 0.19 (356 values). 11% of the curves were not fitted. The curves which were not fitted fall into 2 main regions, the regions in the top left and the region in the bottom middle (Figs. 4H
-4J). The curves in the top left region do not fit because there is no blood flow in the top region as seen in the time-series raw data. The absence of flow in that region, leads to correlation curves which are characteristic of diffusion only. As a result, the fitting model fails to fit the data properly. The curves at the bottom middle failed to converge.

To obtain vectorial information about blood flow, the direction of flow is determined by computing the cross-correlations. The four cross-correlations described in the previous case were determined. The cross-correlations along *x* axis show that the flow is along the negative *x* axis since a peak is seen only along this curve and not in the other. Unlike the case above, where the flow was only along one of the directions, the flow here had a component along the *y* axis as well. By similar reasoning, it was concluded that the flow is along the negative y axis. This is confirmed by visual inspection of the time series movie from which the correlations were calculated. The flow is at an angle of −45° from the horizontal. This agrees with the finding the velocity has both *x* and *y* components. The exact angle was computed by *tan ^{−1}(v_{y}/v_{x}).* The magnitude and direction of the flow velocity was found to be (13 ± 5 µm/s) and (−44 ± 15°) respectively.

## 5. Conclusion

Camera based FCS technologies provide the user with a multiplexing advantage and can be used to probe dynamics on 2D model/cell membranes and 3D living cells/embryos. Here, ImFCS, an open source software is described which is a data analysis software to evaluate image stacks acquired in imaging FCS. The software calculates a variety of spatiotemporal auto- and cross-correlations and differences between spatial forward and backward cross-correlations. Quantitative vectorial parameters can be retrieved by curve fitting using the software. Selected applications of the software were demonstrated on data acquired using 3 different modes of imaging FCS (ITIR-FCS, IVA-FCS and SPIM-FCS). This user-interactive and fast open-source software to evaluate imaging FCS data makes the analysis easier and accessible for a larger community interested in the dynamic behavior of molecules in model and living systems.

## 6. Supplementary material

The computational aspects of the program are described in detail in the supplement. The supplement, the manual to the program ImFCS, the source code in VC++ .net2003 and the Igor Pro Procedure files are available at http://staff.science.nus.edu.sg/~chmwt/ImFCS.html.

## Acknowledgements

The authors thank Lin Guo for his help in sample preparation and imaging. The authors thank Dr. Hwee Kuan Lee (Bioinformatics Institute, Singapore) and Dr. Joachim Wuttke (Forschungszentrum Juelich GmbH) for helpful discussions. TW acknowledges funding from the Alexander von Humboldt Foundation. This work was supported by a grant from the Ministry of Education of Singapore (R-143-000-358-112). JS is supported by a scholarship of the Singapore-MIT Alliance.

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