Measuring the Size Distribution of Nanomaterials

A Fluorescence Recovery after Photobleaching Study

  • Fig.1 (a) Confocal images (field of view 154 μm by 154 μm) are shown of a cFRAP experiment in a FITC-dextran solution. Images are shown before (t < 0), during (t = 0) and after photobleaching (t > 0). cFRAP analysis is performed on the indicated Region of Interest (ROI). The REF region indicates the reference area that is used in the analysis to correct for potential laser fluctuations and bleaching during imaging. (b) For analysis according to the ‘ring-based method’, the ROI is divided into n equally spaced rectangular ring areas. NC is the number of ring. (c) Recovery images were simulated according to Eq (2) for a single component system of D = 10 µm2/s. D-distributions are shown for the ring-based method for different numbers of equally spaced rings (NC=1,10 and 24) and the pixel-based method.
  • Fig. 2 (a) The ability of cFRAP analysis to recover continuous D distributions was tested for two simulated polydisperse systems. The top three graphs are for a sample with single lognormal distribution of diffusion coefficients with increasing polydispersity. The bottom graph is for a sample with double lognormal distribution. Black bars indicated the actual simulated values and the orange curve is the measured result by cFRAP. (b) cFRAP was tested experimentally on mixtures of two (top) and five (bottom) FITC-dextrans of increasing Mw. The gray dotted lines are the size distributions of the individual components and the black dotted line is the sum of those, representing the expected size distribution of the mixture. The orange line is the cFRAP result, which matches well with the expected size distribution.
  • Fig. 3 Sizing protein aggregates in serum by cFRAP. Aggregates of fluorescently labeled BSA were prepared through heat-stress. Unstressed (black lines) and heat-stressed (orange lines) samples were prepared in buffer solution (a) and in 90% serum (b). Dashed lines indicate the standard deviation on independent repeats (n=10). The size distributions as determined by cFRAP in serum nicely correspond to those in buffer solution. This demonstrates that cFRAP is very well capable of analyzing protein aggregation in (undiluted) serum.

FRAP is widely applicable in the biophysical, pharmaceutical and material sciences to study diffusion of molecules and nanoparticles on a micrometer scale. Here we extend the capability of FRAP to measuring polydisperse samples with a continuous distribution of diffusion coefficients (cFRAP). We demonstrate that cFRAP is a new analytical technique that allows to measure the size distribution of polydisperse nanomaterials, not only in buffer solutions but also in biological fluids.


Fluorescence recovery after photobleaching (FRAP) has been used extensively to study the local mobility of molecules and nanoparticles in terms of an average local diffusion coefficient [1]. In a FRAP experiment, fluorescently labeled molecules in a micron sized area of the sample are photobleached by a powerful laser excitation pulse. Through this photochemical process, fluorescent molecules lose their fluorescence properties. After photobleaching, the photobleached fluorescent molecules will diffuse out of the beach area and are replaced by unbleached fluorescent molecules from the surrounding region. A gradual recovery of the fluorescence inside the area will occur due to this diffusional exchange, as can be observed from confocal time-lapse images. The rate of fluorescence recovery is proportional to the rate of diffusion of the fluorescently labeled molecules. Fitting a suitable FRAP model to the observed fluorescence recovery profile can yield the physical quantities describing the local diffusion in the sample, such as the average diffusion coefficient [2].

To date, quantitative interpretation of FRAP data is limited to the analysis of the average intensity in the bleach area over time, as it reduces the mathematical complexity of the diffusion model. As such the spatial information of the diffusion profiles is essentially lost, which results in limited precision of FRAP analyses. Recently, we developed a fast and straightforward FRAP model that makes use of the full temporal and spatial diffusion process after photobleaching of an arbitrary rectangular area (rFRAP) [3]. While it offers improved precision, rFRAP with least squares fitting still limits data interpretation to a single-component diffusion process.

Generally, however, diffusional systems may have a continuous distribution of diffusion coefficients. Therefore, we extended the rFRAP model to analyze continuous distributions of diffusion coefficients (termed as cFRAP). We evaluated the new cFRAP method both by simulations and experiments.

cFRAP Method

We start from the rFRAP (rectangle FRAP, fig. 1a) model developed before for measuring a single average diffusion coefficient, which makes use of both time and spatial information in the recovery images [3]:

where t is the time after photobleaching, K0 the photobleaching parameter (which determines the extent of bleaching), D is the isotropic diffusion coefficient of the diffusing species, lx and ly are the width and height of the rectangular photobleaching area, and r2 is the mean square resolution of the bleaching and imaging point-spread function. In case of N independent diffusing components, we can simply make a superposition of the individual fluorescence recovery profiles:

where αi is the relative fraction of the ith component and εi is the corresponding relative fluorescence brightness. Evidently,

the multicomponent rFRAP model becomes:

Eq. (4) can be used for direct fitting to the pixel values in the recovery images with the D-space discretized in n components (e.g. with equal interval in log(D) space) in the range of Dmin  to Dmax. Instead of performing a standard least squares fitting of Eq. (7) to the experimental data, the Maximum Entropy Method (MEM) is introduced to find the ‘best-fit’ solution with maximum entropy. MEM ensures that the fitting result (i.e. the distribution of diffusion coefficients) contains the least possible information in order to avoid over-interpretation of noise due to limited sampling statistics. In other words, it looks for the smoothest best fit solution in the maximum entropy sense. The ‘historic MEM’ approach was implemented in this work, which means maximizing the Shannon-Jaynes entropy: 

Rather than applying the fitting procedure to the pixel values in the images, the recovery data can be analyzed based on the average intensities in ring areas instead (fig. 1b). This considerably reduces computation time while retaining the essential spatial information. The average intensity in ring Ri is calculated as:

where F(x,y,t) is defined in Eq. (4) and Mi is the number of pixels inside ring Ri. As the results show in Fig. 1c for a simulated system with a single diffusion coefficient of 10 µm2/s, the ring-averaged method has the same precision as full pixel based fitting when the number of rings is ~20. Increasing the number of rings further did not result in more precision. Most importantly, the calculation time is reduced by 3 orders of magnitude when using the ring method versus pixel-based fitting.

cFRAP for Analyzing Samples with a Broad Range of Diffusion Coefficients

The cFRAP method was developed with the aim to analyze continuous distributions of diffusion coefficients. Therefore, we went on to simulate recovery images for a polydisperse sample with a continuous distribution of diffusion coefficients according to an increasingly broad lognormal distribution (black bars in fig. 2a). The orange curves are the result from the cFRAP analysis and nicely correspond to the expected distributions. Going one step further we also simulated a double lognormal distribution as shown at the bottom of figure 2a. Again cFRAP is very well capable of retrieving the expected distribution. We next prepared mixtures of FD (FITC-dextran) with a gradually increasing range of MW to see if cFRAP can measure the full size distribution correctly. As the results in figure 2b show, cFRAP can accurately retrieve the expected size distributions from ~2 to 80 nm (after converting the distribution of diffusion coefficients to a size distribution with the Stokes-Einstein equation). This demonstrates that cFRAP is very well suited for the intended task of analyzing complex polydisperse systems.

cFRAP Applied to the Characterization of Protein Aggregates in Biological Fluids

As an application, we used cFRAP-sizing to analyze protein aggregates in the sub 0.1 µm range in full serum. This is of current interest since protein aggregation has emerged as a key issue underlying multiple deleterious effects in the use of protein therapeutics, including loss of efficacy, altered pharmacokinetics, reduced stability and shelf life, and induction of unwanted immunogenicity. Aggregates were prepared from fluorescein labeled bovine serum albumin (BSA) by applying heat stress. cFRAP was used to analyze samples prepared of BSA monomers and BSA aggregates in 90% serum. Comparison with the size distributions in buffer solution shows that both monomers and aggregates could be correctly sized in serum by cFRAP (fig. 3b).


The presented cFRAP method can precisely measure the distribution of diffusion coefficients of polydisperse systems, and can for instance be used to measure the size of polydisperse fluorescent nanomaterials, such as proteins, in biological fluids.

[1] N. Lorén , J. Hagman, J.K. Jonasson , H. Deschout , D. Bernin , F. Cella-Zanacchi , A. Diaspro,  J.G. McNally,  M. Ameloot , N. Smisdom, M. Nydén, A.M. Hermansson, M. Rudemo, K. Braeckmans, Fluorescence recovery after photobleaching in material and life sciences: putting theory into practice. Quarterly Reviews of Biophysics, 48(3):323 (2015) DOI: 10.1017/S0033583515000013
[2] T.K.L Meyvis, S.C. De Smedt, P. Van Oostveldt,. & J. Demeester, Fluorescence recovery after photobleaching: A versatile tool for mobility and interaction measurements in pharmaceutical research. Pharm Res-Dordr 16, 1153-1162 (1999).
[3] Hendrik Deschout, Joel Hagman, Sophia Fransson, Jenny Jonasson, Mats Rudemo, Niklas Lorén, and Kevin Braeckmans, Straightforward FRAP for quantitative diffusion measurements with a laser scanning microscope. Opt Express 18, 22886-22905 (2010) DOI 10.1364/OE.18.022886

Ranhua Xiong1, Stefaan C. De Smedt2, Kevin Braeckmans1

1Biophotonic Imaging Group, Laboratory for General Biochemistry and Physical Pharmacy, Ghent University, Ghent, Belgium
2Ghent Research Group on Nanomedicines, Laboratory of General Biochemistry and Physical Pharmacy, Ghent University, Ghent, Belgium

Prof. Dr. Kevin Braeckmans
Bio-Photonic Imaging Group
Laboratory for General Biochemistry and Physical Pharmacy
Ghent University
Ghent, Belgium

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