Improving Parameter Inference from FRAP Data: an Analysis Motivated by Pattern Formation in the <Emphasis Type="BoldItalic">Drosophila</Emphasis> Wing Disc

Fluorescence recovery after photobleaching (FRAP) is used to obtain quantitative information about molecular diffusion and binding kinetics at both cell and tissue levels of organization. FRAP models have been proposed to estimate the diffusion coefficients and binding kinetic parameters of species for a variety of biological systems and experimental settings. However, it is not clear what the connection among the diverse parameter estimates from different models of the same system is, whether the assumptions made in the model are appropriate, and what the qualities of the estimates are. Here we propose a new approach to investigate the discrepancies between parameters estimated from different models. We use a theoretical model to simulate the dynamics of a FRAP experiment and generate the data that are used in various recovery models to estimate the corresponding parameters. By postulating a recovery model identical to the theoretical model, we first establish that the appropriate choice of observation time can significantly improve the quality of estimates, especially when the diffusion and binding kinetics are not well balanced, in a sense made precise later. Secondly, we find that changing the balance between diffusion and binding kinetics by changing the size of the bleaching region, which gives rise to different FRAP curves, provides a priori knowledge of diffusion and binding kinetics, which is important for model formulation. We also show that the use of the spatial information in FRAP provides better parameter estimation. By varying the recovery model from a fixed theoretical model, we show that a simplified recovery model can adequately describe the FRAP process in some circumstances and establish the relationship between parameters in the theoretical model and those in the recovery model. We then analyze an example in which the data are generated with a model of intermediate complexity and the parameters are estimated using models of greater or less complexity, and show how sensitivity analysis can be used to improve FRAP model formulation. Lastly, we show how sophisticated global sensitivity analysis can be used to detect over-fitting when using a model that is too complex.     

A mathematical model of actin filament turnover for fitting FRAP data

A novel mathematical model of the actin dynamics in living cells under steady-state conditions has been developed for fluorescence recovery after photobleaching (FRAP) experiments. As opposed to other FRAP fitting models, which use the average lifetime of actins in filaments and the actin turnover rate as fitting parameters, our model operates with unbiased actin association/dissociation rate constants and accounts for the filament length. The mathematical formalism is based on a system of stochastic differential equations. The derived equations were validated on synthetic theoretical data generated by a stochastic simulation algorithm adapted for the simulation of FRAP experiments. Consistent with experimental findings, the results of this work showed that (1) fluorescence recovery is a function of the average filament length, (2) the F-actin turnover and the FRAP are accelerated in the presence of actin nucleating proteins, (3) the FRAP curves may exhibit both a linear and non-linear behaviour depending on the parameters of actin polymerisation, and (4) our model resulted in more accurate parameter estimations of actin dynamics as compared with other FRAP fitting models. Additionally, we provide a computational tool that integrates the model and that can be used for interpretation of FRAP data on actin cytoskeleton.     

Methods for measuring rates of protein binding to insoluble scaffolds in living cells: histone H1-chromatin interactions

Understanding of cell regulation is limited by our inability to measure molecular binding rates for proteins within the structural context of living cells, and many systems biology models are hindered because they use values obtained with molecules binding in solution. Here, we present a kinetic analysis of GFP-histone H1 binding to chromatin within nuclei of living cells that allows both the binding rate constant k(ON) and dissociation rate constant k(OFF) to be determined based on data obtained from fluorescence recovery after photobleaching (FRAP) analysis. This is accomplished by measuring the ratio of bound to free concentration of protein at steady state, and identifying the rate-determining step during FRAP recovery experimentally, combined with mathematical modeling. We report k(OFF) = 0.0131/s and k(ON) = 0.14/s for histone H1.1 binding to chromatin. This work brings clarity to the interpretation of FRAP experiments and provides a way to determine binding kinetics for nuclear proteins and other cellular molecules that interact with insoluble scaffolds within living cells.     

Computation of FRAP recovery times for linker histone – chromatin binding on the basis of Brownian dynamics simulations

BackgroundFluorescence recovery after photobleaching (FRAP) studies can provide kinetic information about proteins in cells. Single point mutations can significantly affect the binding kinetics of proteins and result in variations in the recovery half time (t₅₀) measured in FRAP experiments. FRAP measurements of linker histone (LH) proteins in the cell nucleus have previously been reported by Brown et al. (2006) and Lele et al. (2006).MethodsWe performed Brownian dynamics (BD) simulations of the diffusional association of the wild-type and 38 single or double point mutants of the globular domain of mouse linker histone H1.0 (gH1.0) to a nucleosome. From these simulations, we calculated the bimolecular association rate constant (k_on), the Gibbs binding free energy (ΔG) and the dissociation rate constant (k_off) related to formation of a diffusional encounter complex between the nucleosome and the gH1.0.ResultsWe used these parameters, after application of a correction factor to account for the effects of the crowded environment of the nucleus, to compute FRAP recovery times and curves that are in good agreement with previously published, experimentally measured FRAP recovery time courses.ConclusionsOur computational analysis suggests that BD simulations can be used to predict the relative effects of single point mutations on FRAP recovery times related to protein binding.General SignificanceBD simulations assist in providing a detailed molecular level interpretation of FRAP data.     

Characterizing fluorescence recovery curves for nuclear proteins undergoing binding events

Fluorescence recovery after photobleaching (FRAP) is an experimental technique used to measure the mobility of proteins within the cell nucleus. After proteins of interest are fluorescently tagged for their visualization and monitoring, a small region of the nucleus is photobleached. The experimental FRAP data are obtained by recording the recovery of the fluorescence in this region over time. In this paper, we characterize the fluorescence recovery curves for diffusing nuclear proteins undergoing binding events with an approximate spatially homogeneous structure. We analyze two mathematical models for interpreting the experimental FRAP data, namely a reaction-diffusionmodel and a compartmental model. Perturbation analysis leads to a clear explanation of two important limiting dynamical types of behavior exhibited by experimental recovery curves, namely, (1) a reduced diffusive recovery, and (2) a biphasic recovery characterized by a fast phase and a slow phase. We show how the two models, describing the same type of dynamics using different approaches, relate and share common ground. The results can be used to interpret experimental FRAP data in terms of protein dynamics and to simplify the task of parameter estimation. Application of the results is demonstrated for nuclear actin and type H1 histone.     

Challenges and artifacts in quantitative photobleaching experiments

Confocal fluorescence recovery after photobleaching (FRAP) is today the prevalent tool when studying the diffusional and kinetic properties of proteins in living cells. Obtaining quantitative data for diffusion coefficients via FRAP, however, is challenged by the fact that both bleaching and scanning take a finite time. Starting from an experimental case, it is shown by means of computer simulations that this intrinsic temporal limitation can lead to a gross underestimation of diffusion coefficients. Determining the binding kinetics of proteins to membranes with FRAP is further shown to be severely hampered by additional diffusional contributions, e.g. diffusion-limited binding. In some cases, the binding kinetics may even be masked entirely by diffusion. As current efforts to approach biological problems with biophysical models have to rely on experimentally determined model parameters, e.g. binding rates and diffusion constants, it is proposed that the accuracy in evaluating FRAP measurements can be improved by means of accompanying computer simulations.     

Improving parameter estimation for cell surface FRAP data

Fluorescence Recovery After Photobleaching (FRAP) using the confocal laser scanning microscope has become a standard method used to determine the diffusion coefficient and mobile fraction of cell surface proteins. A common experimental approach is to bleach a stripe on the cell surface and fit the ensuing FRAP curve to a 1D diffusion model. This model is derived from the time course of recovery to an infinitely long stripe bleached on an infinite flat plane. This choice of model dictates the use of a long bleach stripe. We demonstrate that, in the case of a long bleach stripe, the finite extent of the cell leads to significant errors in parameter estimation. We further show that these errors are reduced when a relatively small stripe is bleached. Unfortunately, diffusion to such a region is fundamentally two dimensional and therefore applying the 1D model of diffusion leads to significant errors. We derive an equation suitable for fitting to FRAP data acquired from small bleach regions and analyze its accuracy using simulated data. We propose that the use of a small bleach region along with a two dimensional diffusion model is the ideal protocol for cell surface FRAP.     

Investigation of binding mechanisms of nuclear proteins using confocal scanning laser microscopy and FRAP

Fluorescence recovery after photobleaching (FRAP) measurements offer an important tool towards analysing diffusion processes within living biological cells. A model is presented that aims to provide a rigorous theoretical framework from which binding information of proteins from FRAP data can be extracted. A single binding reaction is considered and a set of mathematical equations is introduced that incorporates the concentration of free proteins, vacant binding sites and bound complexes in addition to the on- and off-rates of the proteins. To allow a realistic FRAP model, characteristics of the instruments used to perform FRAP measurements are included in the equation. The proposed model has been designed to be applied to biological samples with a confocal scanning laser microscope (CSLM) equipped with the feature to bleach regions characterised by a radially Gaussian distributed profile. Binding information emerges from FRAP simulations considering the diffusion coefficient, radial extent of the bleached volume and bleach constant as parameters derived from experimental data. The proposed model leads to FRAP curves that depend on the on- and off-rates. Analytical expressions are used to define the boundaries of on- and off-rate parameter space in simplified cases when molecules can move on an infinite domain. A similar approach is ensued when movement is restricted in a compartment with a finite size. The theoretical model can be used in conjunction to experimental data acquired by CSLM to investigate the biophysical properties of proteins in living cells.     

Are Assumptions about the Model Type Necessary in Reaction-Diffusion Modeling? A FRAP Application

At present, fluorescence recovery after photobleaching (FRAP) data are interpreted using various types of reaction-diffusion (RD) models: the model type is usually fixed first, and corresponding model parameters are inferred subsequently. In this article, we describe what we believe to be a novel approach for RD modeling without using any assumptions of model type or parameters. To the best of our knowledge, this is the first attempt to address both model-type and parameter uncertainties in inverting FRAP data. We start from the most general RD model, which accounts for a flexible number of molecular fractions, all mobile, with different diffusion coefficients. The maximal number of possible binding partners is identified and optimal parameter sets for these models are determined in a global search of the parameter-space using the Simulated Annealing strategy. The numerical performance of the described techniques was assessed using artificial and experimental FRAP data. Our general RD model outperformed the standard RD models used previously in modeling FRAP measurements and showed that intracellular molecular mobility can only be described adequately by allowing for multiple RD processes. Therefore, it is important to search not only for the optimal parameter set but also for the optimal model type.     

Using FRAP and mathematical modeling to determine the in vivo kinetics of nuclear proteins

Fluorescence recovery after photobleaching (FRAP) has become a popular technique to investigate the behavior of proteins in living cells. Although the technique is relatively old, its application to studying endogenous intracellular proteins in living cells is relatively recent and is a consequence of the newly developed fluorescent protein-based living cell protein tags. This is particularly true for nuclear proteins, in which endogenous protein mobility has only recently been studied. Here we examine the experimental design and analysis of FRAP experiments. Mathematical modeling of FRAP data enables the experimentalist to extract information such as the association and dissociation constants, distribution of a protein between mobile and immobilized pools, and the effective diffusion coefficient of the molecule under study. As experimentalists begin to dissect the relative influence of protein domains within individual proteins, this approach will allow a quantitative assessment of the relative influences of different molecular interactions on the steady-state distribution and protein function in vivo.     

A reaction-diffusion model to study RNA motion by quantitative fluorescence recovery after photobleaching

Fluorescence recovery after photobleaching (FRAP) is a powerful technique to study molecular dynamics inside living cells. During the past years, several laboratories have used FRAP to image the motion of RNA-protein and other macromolecular complexes in the nucleus and cytoplasm. In the case of mRNAs, there is growing evidence indicating that these molecules assemble into large ribonucleoprotein complexes that diffuse throughout the nucleus by Brownian motion. However, estimates of the corresponding diffusion rate yielded values that differ by up to one order of magnitude. In vivo labeling of RNA relies on indirect tagging with a fluorescent probe, and here we show how the binding affinity of the probe to the target RNA influences the effective diffusion estimates of the resulting complex. We extend current reaction-diffusion models for FRAP by allowing for diffusion of the bound complex. This more general model can be used to fit any fluorescence recovery curve involving two interacting mobile species in the cell (a fluorescent probe and its target substrate). The results show that interpreting FRAP data in light of the new model reconciles the discrepant mRNA diffusion-rate values previously reported.     

Inferences from FRAP data are model dependent: A subdiffusive analysis

Fluorescence recovery after photobleaching (FRAP) is a widely used biological experiment to study the kinetics of molecules that react and move randomly. Since the development of FRAP in the 1970s, many reaction-diffusion models have been used to interpret FRAP data. However, intracellular molecules are widely observed to move by anomalous subdiffusion instead of normal diffusion. In this article, we extend a popular reaction-diffusion model of FRAP to the case of subdiffusion modeled by a fractional diffusion equation. By analyzing this reaction-subdiffusion model, we show that FRAP data are consistent with both diffusive and subdiffusive motion in many scenarios. We illustrate this general result by fitting our model to FRAP data from glucocorticoid receptors in a cell nucleus. We further show that the assumed model of molecular motion (normal diffusion or subdiffusion) strongly impacts the biological parameter values inferred from a given experimentally observed FRAP curve. We additionally analyze our model in three simplified parameter regimes and discuss parameter identifiability for varying subdiffusion exponents.     

Real time kinetic studies of the interaction between folded antisense and target RNAs using surface plasmon resonance

Antisense RNAs interact with their complementary target RNAs as folded structures. The formation of early binding intermediates is the most important step in determining the overall rates of stable complex formation in vitro and the efficiency of control in vivo. In the case of CopA and CopT (antisense/target RNA pair of plasmid R1), recent studies have identified a four-way junction structure as the major binding intermediate. Previously, the kinetics of antisense/target RNA interaction was studied by indirect methods. Here we have used surface plasmon resonance to follow the binding of CopI (a truncated variant of CopA) to CopT in real time. A protocol was developed that permitted the determination of association and dissociation rate constants for wild-type and mutant CopI-CopT pairs. The K(D)-values calculated from these rate constants were in good agreement with the results obtained by indirect methods. In comparison to earlier model studies of interactions between simple complementary nucleic acids, we observe a different temperature dependence for dissociation rate constants. This may be indicative of the complexity of the steps required for interacting folded RNAs; intramolecular structure competes with intermolecular helix progression during complex formation. The association rate constants were not significantly dependent on temperature. The analysis presented shows that the stability of a kissing complex is not the primary determinant of the rate of stable CopA/CopT complex formation.     

Promiscuous and Selective: How Intrinsically Disordered BH3 Proteins Interact with Their Pro-survival Partner MCL-1

The BCL-2 family of proteins plays a central role in regulating cell survival and apoptosis. Disordered BH3-only proteins bind promiscuously to a number of different BCL-2 proteins, with binding affinities that vary by orders of magnitude. Here we investigate the basis for these differences in affinity. We show that eight different disordered BH3 proteins all bind to their BCL-2 partner (MCL-1) very rapidly, and that the differences in sequences result in different dissociation rates. Similarly, mutation of the binding surface of MCL-1 generally affects association kinetics in the same way for all BH3 peptides but has significantly different effects on the dissociation rates. Importantly, we infer that the evolution of homologous, competing interacting partners has resulted in complexes with significantly different lifetimes.     

FRAP Analysis: Accounting for Bleaching during Image Capture

The analysis of Fluorescence Recovery After Photobleaching (FRAP) experiments involves mathematical modeling of the fluorescence recovery process. An important feature of FRAP experiments that tends to be ignored in the modeling is that there can be a significant loss of fluorescence due to bleaching during image capture. In this paper, we explicitly include the effects of bleaching during image capture in the model for the recovery process, instead of correcting for the effects of bleaching using reference measurements. Using experimental examples, we demonstrate the usefulness of such an approach in FRAP analysis.     

Model of For3p-Mediated Actin Cable Assembly in Fission Yeast

Formin For3p nucleates actin cables at the tips of fission yeast cells for polarized cell growth. The results of prior experiments have suggested a possible mechanism for actin cable assembly that involves association of For3p near cell tips, For3p-mediated actin polymerization, retrograde flow of actin cables toward the cell center, For3p dissociation from cell tips, and cable disassembly. We used analytical and computational modeling to test the validity and implications of the proposed coupled For3p/actin mechanism. We compared the model to prior experiments quantitatively and generated predictions for the expected behavior of the actin cable system upon changes of parameter values. We found that the model generates stable steady states with realistic values of rate constants and actin and For3p concentrations. Comparison of our results to previous experiments monitoring the FRAP of For3p-3GFP and the response of actin cables to treatments with actin depolymerizing drugs provided further support for the model. We identified the set of parameter values that produces results in agreement with experimental observations. We discuss future experiments that will help test the model''s predictions and eliminate other possible mechanisms. The results of the model suggest that flow of actin cables may establish actin and For3p concentration gradients in the cytoplasm that could be important in global cell patterning.