A Highly Accurate Pixel-Based FRAP Model Based on Spectral-Domain Numerical Methods.

Article A Highly Accurate Pixel-Based FRAP Model Based on Spectral-Domain Numerical Methods €ck,2 and Niklas Lore n1,3 Magnus Röding,1,* Leander Lacroix,1 Annika Krona,1 Tobias Geba 1 RISE Research Institutes of Sweden, Bioscience and Materials, Göteborg, Sweden; 2Mathematical Sciences and 3Department of Physics, Chalmers University of Technology, Göteborg, Sweden ABSTRACT We introduce a new, to our knowledge, numerical model based on spectral methods for analysis of fluorescence recovery after photobleaching data. The model covers pure diffusion and diffusion and binding (reaction-diffusion) with immobile binding sites, as well as arbitrary bleach region shapes. Fitting of the model is supported using both conventional recoverycurve-based estimation and pixel-based estimation, in which all individual pixels in the data are utilized. The model explicitly accounts for multiple bleach frames, diffusion (and binding) during bleaching, and bleaching during imaging. To our knowledge, no other fluorescence recovery after photobleaching framework incorporates all these model features and estimation methods. We thoroughly validate the model by comparison to stochastic simulations of particle dynamics and find it to be highly accurate. We perform simulation studies to compare recovery-curve-based estimation and pixel-based estimation in realistic settings and show that pixel-based estimation is the better method for parameter estimation as well as for distinguishing pure diffusion from diffusion and binding. We show that accounting for multiple bleach frames is important and that the effect of neglecting this is qualitatively different for the two estimation methods. We perform a simple experimental validation showing that pixel-based estimation provides better agreement with literature values than recovery-curve-based estimation and that accounting for multiple bleach frames improves the result. Further, the software developed in this work is freely available online. INTRODUCTION Diffusive transport properties in complex, soft matter fluctuate spatially and temporally and depend strongly on the degree of heterogeneity, obstruction effects, structural dynamics, and interactions with a matrix, e.g., binding effects (1). Understanding complex diffusion phenomena is a recurring problem in several fields, and fluorescence recovery after photobleaching (FRAP) has emerged as a powerful technique to this end (2). Having been used for estimation of diffusion coefficients since the 1970s (3), FRAP has later been put to use on reaction-diffusion systems, joint estimation of diffusion coefficients, and (on and off) binding rate constants, i.e., association and disassociation rate constants. Different approaches to FRAP for quantifying diffusion and binding interactions have shed light on how proteins interact with binding sites within the cell and nucleus (4–6), the transcription factor mobility in the nucleus (7) and its interaction with chromatin (8,9), interactions of membrane-associated proteins (10–12), and Submitted October 29, 2018, and accepted for publication February 25, 2019. *Correspondence: Editor: Nathan Baker. https://doi.org/10.1016/j.bpj.2019.02.023
2019 Biophysical Society. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/). 1348 Biophysical Journal 116, 1348–1361, April 2, 2019 probe diffusion in b-lactoglobulin gels and solutions (13), just to mention a few. In a typical FRAP experiment, a fluorescent species is irreversibly photobleached in either a circular or a rectangular bleach region. Unbleached particles will move into the bleach region at a rate governed by the mobility and interaction parameters. This leads to a recovery of fluorescence in the bleach region. Assuming that the bleaching does not significantly change the total amount of fluorescence in the sample and that no particles are immobile, the recovery will eventually be complete. A confocal laser scanning microscope (CLSM) is typically used to image the time evolution of the recovery, using the same laser for imaging and bleaching but with different intensity. Quantitative information is obtained by fitting a model for the fluorescence recovery to the experimental data. The physical/mathematical assumptions of the FRAP models as well as how the fitting is performed vary greatly between different approaches but boil down to representing the solution to a (reaction-)diffusion equation for the fluorescent species, analytically or numerically. We give a brief account of the literature for the factors that matter for the work below but refer the reader to the review in (2) for a more detailed account. Pixel-Based Numerical FRAP Model First, the prototypical FRAP approaches assuming pure diffusion are heavily used, but the generalizations incorporating interactions with binding sites facilitate the use of FRAP in more complex systems like cells and hydrogels. The governing model becomes a reaction-diffusion equation system with a ‘‘free’’ diffusion coefficient, binding and unbinding rate parameters, and possibly a ‘‘bound’’ diffusion coefficient, the latter depending on whether the binding sites are modeled as mobile (11,12,14) or immobile (6–9). Second, the bleach region theoretically has uniform intensity and is typically either a circle or a rectangle. For a uniform circular bleach region, the average intensity in the bleach region as a function of time after bleaching (i.e., the recovery curve) can be expressed in closed form using Bessel functions (15,16). However, to obtain a closedform expression for the full diffusion equation, i.e., the spatiotemporal evolution of the fluorescence intensity, a circular bleach region has to be approximated, e.g., by a Gaussian (17) or a nonparametric profile (18). For the rectangular case, the full solution to the diffusion equation is available in closed form (19,20). Arbitrary bleach region shapes are in principle not a problem if numerical or Monte Carlo methods are used (21). Some approaches account for the effective, finite bleach resolution (because of a nonuniform laser beam) by convolving the bleach region by a Gaussian (16,19). Third, the duration of bleaching is non-negligible, and therefore diffusion (and binding) during bleaching affects the observed fluorescence recovery (22). The fact that very often multiple bleach frames are used (to increase the amount of bleaching and hence the contrast and signal/ noise) and the fact that the laser moves in a raster scan pattern during bleaching (and imaging) both contribute to this effect. Diffusion during a single bleach frame can be accounted for implicitly to some extent by incorporating a bleach resolution parameter as a free fitting parameter (because both diffusion and the ‘‘smearing’’ of the bleach region by convolution with a Gaussian are mathematically equivalent) (19). It can also be accounted for explicitly by modifying the diffusion equ (...truncated)