Analysis of Active Transport by Fluorescence Recovery after Photobleaching.

Article Analysis of Active Transport by Fluorescence Recovery after Photobleaching Maria-Veronica Ciocanel,1 Jill A. Kreiling,2 James A. Gagnon,2,3 Kimberly L. Mowry,2 and Björn Sandstede1,* 1 Division of Applied Mathematics and 2Department of Molecular Biology, Cell Biology, and Biochemistry, Brown University, Providence, Rhode Island; and 3Department of Molecular and Cellular Biology, Harvard University, Cambridge, Massachusetts ABSTRACT Fluorescence recovery after photobleaching (FRAP) is a well-established experimental technique to study binding and diffusion of molecules in cells. Although a large number of analytical and numerical models have been developed to extract binding and diffusion rates from FRAP recovery curves, active transport of molecules is typically not included in the existing models that are used to estimate these rates. Here we present a validated numerical method for estimating diffusion, binding/unbinding rates, and active transport velocities using FRAP data that captures intracellular dynamics through partial differential equation models. We apply these methods to transport and localization of mRNA molecules in Xenopus laevis oocytes, where active transport processes are essential to generate developmental polarity. By providing estimates of the effective velocities and diffusion, as well as expected run times and lengths, this approach can help quantify dynamical properties of localizing and nonlocalizing RNA. Our results confirm the distinct transport dynamics in different regions of the cytoplasm, and suggest that RNA movement in both the animal and vegetal directions may influence the timescale of RNA localization in Xenopus oocytes. We also show that model initial conditions extracted from FRAP postbleach intensities prevent underestimation of diffusion, which can arise from the instantaneous bleaching assumption. The numerical and modeling approach presented here to estimate parameters using FRAP recovery data is a broadly applicable tool for systems where intracellular transport is a key molecular mechanism. INTRODUCTION Fluorescence recovery after photobleaching (FRAP) is a widely used tool for investigating protein mobility and local molecular transport in living cells (1,2). Because fluorescence is visualized as diffuse staining, FRAP data cannot be used to distinguish or track individual particles; to make predictions about mobility and local transport, further analysis is needed. A large number of diffusion and reaction-diffusion models have been proposed for the quantitative analysis of FRAP recovery data (reviewed in (1,2)). Most previous work makes use of linear reaction-diffusion partial differential equation (PDE) models to predict diffusion and binding in cells. Depending on the relative timescales of diffusion and binding, these methods involve estimating diffusion coefficients and binding rates by fitting the fluorescence recovery data to analytical solutions of the equations (3–12) or by using optimization and numerical solutions of the PDEs for more complex geometries and Submitted September 6, 2016, and accepted for publication February 27, 2017. *Correspondence: Editor: Anne Kenworthy. http://dx.doi.org/10.1016/j.bpj.2017.02.042 Ó 2017 Biophysical Society. 1714 Biophysical Journal 112, 1714–1725, April 25, 2017 models (13–15). FRAP data analysis typically involves modeling two particle states, and making assumptions about diffusion, the number and type of binding interactions, and their respective timescales in cells (6). In addition to movement by diffusion, macromolecules are actively transported on cytoskeletal networks by molecular motors such as myosin, kinesin, or dynein in many cell types. However, in systems with active movement, FRAP data analysis may overestimate diffusion rates if active transport is not taken into account (1). The goal of this work is to develop an approach for extracting movement information from FRAP data in cells where transport is a key mechanism for the dynamics. To capture active transport, we propose advection-reaction-diffusion PDEs that account for binding, diffusion, and active transport of particles. Parameter estimation is carried out through optimization of numerical solutions of the PDE models considered. We demonstrate that our approach allows efficient extraction of consistent estimates for movement, diffusion, and transition rate parameters from FRAP data using models of twoor four-particle states (see Fig. 1). We note that advection has been included in active transport models to describe spatial localization of RNA in Drosophila oocytes and Analysis of Active Transport by FRAP FRAP data PDE Models (advection-reaction-diffusion) Parameter estimation state parameters PDE Model Analysis effective speed, diffusion Comparison with experimental quantities Predictions for particle dynamics FIGURE 1 Cartoon of the proposed approach to drawing predictions for particle dynamics from FRAP data. To see this figure in color, go online. embryos (16,17), neurofilament transport along axons in neurons (18,19), and motor-driven transport along filaments (20,21). However, to our knowledge, such models were not applied to FRAP experimental data. Here we design general techniques for FRAP parameter estimation using transport PDE modeling frameworks and demonstrate their efficacy. In addition, given estimates of the coefficients in the transport models considered, we show how these parameters can be used to predict effective velocities and diffusion rates for long-term dynamics (see Fig. 1). Consider, for instance, a system where particles switch between movement and diffusion. In one state, particles move with speed c, and in the other, they diffuse with rate d (see Eq. 1 for details). Because the particles can switch between states with transition rates b1 and b2, the effective velocity and diffusion of the particles in the long run are different from the individual state parameters. Dynamical systems analysis of general advection-reaction-diffusion models allows us to provide general formulas for these large time quantities that go beyond specific examples (18,19) and extend the treatment of reaction-hyperbolic systems (20,22–24) and of diffusion in one population (21). Mathematical derivations of these large time solutions, as well as calculation of expected run lengths of motor-cargo complexes on microtubules, allow us to compare parameter estimation predictions with experimental observations (see Fig. 1). Our approach is applicable to many systems, and we validate it here through the study of active transport mechanisms including bidirectional transport on microtubules (MTs) for mRNA dynamics in Xenopus oocytes (25,26). In oocytes of the frog Xenopus laevis, mRNA transport and localization drive the developmental polarity (27). In particular, restricted expression of Vg1 protein in the vegetal hemisphere of the egg is critical for correct patterning of the embryo (2 (...truncated)