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.     

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.     

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.     

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.     

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.     

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.     

Characterization of Cell Boundary and Confocal Effects Improves Quantitative FRAP Analysis

Fluorescence recovery after photobleaching (FRAP) is an important tool used by cell biologists to study the diffusion and binding kinetics of vesicles, proteins, and other molecules in the cytoplasm, nucleus, or cell membrane. Although many FRAP models have been developed over the past decades, the influence of the complex boundaries of 3D cellular geometries on the recovery curves, in conjunction with regions of interest and optical effects (imaging, photobleaching, photoswitching, and scanning), has not been well studied. Here, we developed a 3D computational model of the FRAP process that incorporates particle diffusion, cell boundary effects, and the optical properties of the scanning confocal microscope, and validated this model using the tip-growing cells of Physcomitrella patens. We then show how these cell boundary and optical effects confound the interpretation of FRAP recovery curves, including the number of dynamic states of a given fluorophore, in a wide range of cellular geometries—both in two and three dimensions—namely nuclei, filopodia, and lamellipodia of mammalian cells, and in cell types such as the budding yeast, Saccharomyces pombe, and tip-growing plant cells. We explored the performance of existing analytical and algorithmic FRAP models in these various cellular geometries, and determined that the VCell VirtualFRAP tool provides the best accuracy to measure diffusion coefficients. Our computational model is not limited only to these cells types, but can easily be extended to other cellular geometries via the graphical Java-based application we also provide. This particle-based simulation—called the Digital Confocal Microscopy Suite or DCMS—can also perform fluorescence dynamics assays, such as number and brightness, fluorescence correlation spectroscopy, and raster image correlation spectroscopy, and could help shape the way these techniques are interpreted.     

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.     

Investigating complexity of protein-protein interactions in focal adhesions

The formation of focal adhesions governs cell shape and function; however, there are few measurements of the binding kinetics of focal adhesion proteins in living cells. Here, we used the fluorescence recovery after photobleaching (FRAP) technique, combined with mathematical modeling and scaling analysis to quantify dissociation kinetics of focal adhesion proteins in capillary endothelial cells. Novel experimental protocols based on mathematical analysis were developed to discern the rate-limiting step during FRAP. Values for the dissociation rate constant k_OFF ranged over an order of magnitude from 0.009 ± 0.001/s for talin to 0.102 ± 0.010/s for FAK, indicating that talin is bound more strongly than other proteins in focal adhesions. Comparisons with in vitro measurements reveal that multiple focal adhesion proteins form a network of bonds, rather than binding in a pair-wise manner in these anchoring structures in living cells.     

A closed-form analytic expression for FRAP formula for the binding diffusion model

One of the most dominant methods cells use for a large class of cellular processes is reaction (or binding) diffusion kinetics, which are controlled by kinetic constants such as diffusion coefficients and on/off binding rate constants. Fluorescence recovery after photobleaching (FRAP) can be used to determine these kinetic constants in living cells. While an analytic expression for FRAP formulae for pure diffusion has been available for some time, an analytic FRAP formula for the binding diffusion model has not been reported yet. Here, we present an analytic FRAP formula for the binding diffusion model in an explicit form allowing for diffusion of the bound complex for either a uniform circle laser profile or a Gaussian laser profile.     

Reversibility and binding kinetics of Thermobifida fusca cellulases studied through fluorescence recovery after photobleaching microscopy

Cellulases are enzymes capable of depolymerizing cellulose. Understanding their interactions with cellulose can improve biomass saccharification and enzyme recycling in biofuel production. This paper presents a study on binding and binding reversibility of Thermobifida fusca cellulases Cel5A, Cel6B, and Cel9A bound onto Bacterial Microcrystalline Cellulose. Cellulase binding was assessed through fluorescence recovery after photobleaching (FRAP) at 23, 34, and 45 °C. It was found that cellulase binding is only partially reversible. For processive cellulases Cel6B and Cel9A, an increase in temperature resulted in a decrease of the fraction of cellulases reversibly bound, while for endocellulase Cel5A this fraction remained constant. Kinetic parameters were obtained by fitting the FRAP curves to a binding-dominated model. The unbinding rate constants obtained for all temperatures were highest for Cel5A and lowest for Cel9A. The results presented demonstrate the usefulness of FRAP to access the fast binding kinetics characteristic of cellulases operating at their optimal temperature.     

Improved FRAP Measurements on Biofilms

We expand the standard fluorescence recovery after photobleaching (FRAP) model introduced by Axelrod et al. in 1976. Our goal is to capture some of the following common artifacts observed in the fluorescence measurements obtained with a confocal laser scanning microscope in biofilms: 1) linear drift, 2) exponential decrease (due to bleaching during the measurements), 3) stochastic Gaussian noise, and 4) uncertainty in the exact time point of the onset of fluorescence recovery. To fit the resulting stochastic model to data from FRAP measurements and to estimate all unknown model parameters, we apply a suitably adapted Metropolis-Hastings algorithm. In this way, a more accurate estimation of the diffusion coefficient of the fluorophore is achieved. The method was tested on data obtained from FRAP measurements on a cultivated biofilm.     

Stochastic and deterministic simulations of heterogeneous cell population dynamics

A Monte Carlo algorithm, which can accurately simulate the dynamics of entire heterogeneous cell populations, was developed. The algorithm takes into account the random nature of cell division as well as unequal partitioning of cellular material at cell division. Moreover, it is general in the sense that it can accommodate a variety of single-cell, deterministic reaction kinetics as well as various stochastic division and partitioning mechanisms. The validity of the algorithm was assessed through comparison of its results with those of the corresponding deterministic cell population balance model in cases where stochastic behavior is expected to be quantitatively negligible. Both algorithms were applied to study: (a) linear intracellular kinetics and (b) the expression dynamics of a genetic network with positive feedback architecture, such as the lac operon. The effects of stochastic division as well as those of different division and partitioning mechanisms were assessed in these systems, while the comparison of the stochastic model with a continuum model elucidated the significance of cell population heterogeneity even in cases where only the prediction of average properties is of primary interest.     

Fluorescence-based methods to image palmitoylated proteins

A well known function of palmitoylation is to promote protein binding to cell membranes. Until recently, it was unclear what additional roles, if any, palmitoylation has in controlling protein localization in cells. Recent studies of palmitoylated forms of the small GTPase Ras have now revealed that palmitoylation plays multiple roles in the regulation of protein trafficking, including targeting proteins into the secretory pathway and recycling proteins between the plasma membrane and Golgi complex. We here describe how quantitative fluorescence microscopy and photobleaching approaches can be used to study the intracellular targeting and trafficking of GFP-tagged palmitoylated proteins in living cells. We discuss (1) general considerations for fluorescence recovery after photobleaching (FRAP) measurements of GFP-tagged proteins; (2) FRAP-based assays to test the strength of binding of palmitoylated proteins to cell membranes; (3) methods to establish the kinetics and mechanisms of recycling of palmitoylated proteins between the Golgi complex and the plasma membrane; (4) the use of the palmitoylation inhibitor 2-bromo-palmitate as a tool to study the dynamic regulation of protein targeting and trafficking by palmitate turnover.     

Accurate quantification of diffusion and binding kinetics of non-integral membrane proteins by FRAP

Non-integral membrane proteins frequently act as transduction hubs in vital signaling pathways initiated at the plasma membrane (PM). Their biological activity depends on dynamic interactions with the PM, which are governed by their lateral and cytoplasmic diffusion and membrane binding/unbinding kinetics. Accurate quantification of the multiple kinetic parameters characterizing their membrane interaction dynamics has been challenging. Despite a fair number of approximate fitting functions for analyzing fluorescence recovery after photobleaching (FRAP) data, no approach was able to cope with the full diffusion-exchange problem. Here, we present an exact solution and matlab fitting programs for FRAP with a stationary Gaussian laser beam, allowing simultaneous determination of the membrane (un)binding rates and the diffusion coefficients. To reduce the number of fitting parameters, the cytoplasmic diffusion coefficient is determined separately. Notably, our equations include the dependence of the exchange kinetics on the distribution of the measured protein between the PM and the cytoplasm, enabling the derivation of both k(on) and k(off) without prior assumptions. After validating the fitting function by computer simulations, we confirm the applicability of our approach to live-cell data by monitoring the dynamics of GFP-N-Ras mutants under conditions with different contributions of lateral diffusion and exchange to the FRAP kinetics.     

Sexual reproduction and parent-offspring conflict in the RNA "Tumour" virus/host relationship: implications for vertebrate oncogene evolution.

Longmuir and co-workers have reported that respiration of certain tissue slices is approximated by Michaelis-Menten kinetics. From this and other experimental findings, Longmuir proposed that a carrier is involved in tissue oxygen transport. Gold developed a deterministic model to examine this hypothesis. This report presents a stochastic model for a fixed site carrier in a more general framework that includes the stochastic counter-part to Gold's deterministic model as a special case. The kinetics of tissue oxygen consumption predicted by the model are examined for various cases.