Improving Parameter Inference from FRAP Data: an Analysis Motivated by Pattern Formation in the Drosophila Wing Disc

What's past is prologue: FRAP keeps delivering 50 years later

Fluorescence recovery after photobleaching (FRAP) has emerged as one of the most widely utilized techniques to quantify binding and diffusion kinetics of biomolecules in biophysics. Since its inception in the mid-1970s, FRAP has been used to address an enormous array of questions including the characteristic features of lipid rafts, how cells regulate the viscosity of their cytoplasm, and the dynamics of biomolecules inside condensates formed by liquid-liquid phase separation. In this perspective, I briefly summarize the history of the field and discuss why FRAP has proven to be so incredibly versatile and popular. Next, I provide an overview of the extensive body of knowledge that has emerged on best practices for quantitative FRAP data analysis, followed by some recent examples of biological lessons learned using this powerful approach. Finally, I touch on new directions and opportunities for biophysicists to contribute to the continued development of this still-relevant research tool.

A Random Walk Approach to Transport in Tissues and Complex Media: From Microscale Descriptions to Macroscale Models

The biological processes necessary for the development and continued survival of any organism are often strongly influenced by the transport properties of various biologically active species. The transport phenomena involved vary over multiple temporal and spatial scales, from organism-level behaviors such as the search for food, to systemic processes such as the transport of oxygen from the lungs to distant organs, down to microscopic phenomena such as the stochastic movement of proteins in a cell. Each of these processes is influenced by many interrelated factors. Identifying which factors are the most important, and how they interact to determine the overall result is a problem of great importance and interest. Experimental observations are often fit to relatively simple models, but in reality the observations are the output of complicated functions of the physicochemical, topological, and geometrical properties of a given system. Herein we use multistate continuous-time random walks and generalized master equations to model transport processes involving spatial jumps, immobilization at defined sites, and stochastic internal state changes. The underlying spatial models, which are framed as graphs, may have different classes of nodes, and walkers may have internal states that are governed by a Markov process. A general form of the solutions, using Fourier-Laplace transforms and asymptotic analysis, is developed for several spatially infinite regular lattices in one and two spatial dimensions, and the theory is developed for the analysis of transport and internal state changes on general graphs. The goal in each case is to shed light on how experimentally observable macroscale transport coefficients can be explained in terms of microscale properties of the underlying processes. This work is motivated by problems arising in transport in biological tissues, but the results are applicable to a broad class of problems that arise in other applications.

Parameter estimation in fluorescence recovery after photobleaching: quantitative analysis of protein binding reactions and diffusion

Fluorescence recovery after photobleaching (FRAP) is a common experimental method for investigating rates of molecular redistribution in biological systems. Many mathematical models of FRAP have been developed, the purpose of which is usually the estimation of certain biological parameters such as the diffusivity and chemical reaction rates of a protein, this being accomplished by fitting the model to experimental data. In this article, we consider a two species reaction–diffusion FRAP model. Using asymptotic analysis, we derive new FRAP recovery curve approximation formulae, and formally re-derive existing ones. On the basis of these formulae, invoking the concept of Fisher information, we predict, in terms of biological and experimental parameters, sufficient conditions to ensure that the values all model parameters can be estimated from data. We verify our predictions with extensive computational simulations. We also use computational methods to investigate cases in which some or all biological parameters are theoretically inestimable. In these cases, we propose methods which can be used to extract the maximum possible amount of information from the FRAP data.

Growth control in the Drosophila wing disk

The regulation of size and shape is a fundamental requirement of biological development and has been a subject of scientific study for centuries, but we still lack an understanding of how organisms know when to stop growing. Imaginal wing disks of the fruit fly Drosophila melanogaster, which are precursors of the adult wings, are an archetypal tissue for studying growth control. The growth of the disks is dependent on many inter- and intra-organ factors such as morphogens, mechanical forces, nutrient levels, and hormones that influence gene expression and cell growth. Extracellular signals are transduced into gene-control signals via complex signal transduction networks, and since cells typically receive many different signals, a mechanism for integrating the signals is needed. Our understanding of the effect of morphogens on tissue-level growth regulation via individual pathways has increased significantly in the last half century, but our understanding of how multiple biochemical and mechanical signals are integrated to determine whether or not a cell decides to divide is still rudimentary. Numerous fundamental questions are involved in understanding the decision-making process, and here we review the major biochemical and mechanical pathways involved in disk development with a view toward providing a basis for beginning to understand how multiple signals can be integrated at the cell level, and how this translates into growth control at the level of the imaginal disk. This article is categorized under: Analytical and Computational Methods > Computational Methods Biological Mechanisms > Cell Signaling Models of Systems Properties and Processes > Cellular Models.

A Model for the Hippo Pathway in the Drosophila Wing Disc

Although significant progress has been made toward understanding morphogen-mediated patterning in development, control of the size and shape of tissues via local and global signaling is poorly understood. In particular, little is known about how cell-cell interactions are involved in the control of tissue size. The Hippo pathway in the Drosophila wing disc involves cell-cell interactions via cadherins, which lead to modulation of Yorkie, a cotranscriptional factor that affects control of the cell cycle and growth, and studies involving over- and underexpression of components of this pathway reveal conditions that lead to tissue over- or undergrowth. Here, we develop a mathematical model of the Hippo pathway that can qualitatively explain these observations, made in both whole-disc mutants and mutant-clone experiments. We find that a number of nonintuitive experimental results can be explained by subtle changes in the balances between inputs to the Hippo pathway and suggest some predictions that can be tested experimentally. We also show that certain components of the pathway are polarized at the single-cell level, which replicates observations of planar cell polarity. Because the signal transduction and growth control pathways are highly conserved between Drosophila and mammalian systems, the model we formulate can be used as a framework to guide future experimental work on the Hippo pathway in both Drosophila and mammalian systems.

Fluorescence techniques in developmental biology

Advanced fluorescence techniques, commonly known as the F-techniques, measure the kinetics and the interactions of biomolecules with high sensitivity and spatiotemporal resolution. Applications of the F-techniques, which were initially limited to cells, were further extended to study in vivo protein organization and dynamics in whole organisms. The integration of F-techniques with multi-photon microscopy and light-sheet microscopy widened their applications in the field of developmental biology. It became possible to penetrate the thick tissues of living organisms and obtain good signal-to-noise ratio with reduced photo-induced toxicity. In this review, we discuss the principle and the applications of the three most commonly used F-techniques in developmental biology: Fluorescence Recovery After Photo-bleaching (FRAP), Fo¨ rster Resonance Energy Transfer (FRET), and Fluorescence Correlation and Cross-Correlation Spectroscopy (FCS and FCCS).

Quantitative diffusion measurements using the open-source software PyFRAP

Fluorescence Recovery After Photobleaching (FRAP) and inverse FRAP (iFRAP) assays can be used to assess the mobility of fluorescent molecules. These assays measure diffusion by monitoring the return of fluorescence in bleached regions (FRAP), or the dissipation of fluorescence from photoconverted regions (iFRAP). However, current FRAP/iFRAP analysis methods suffer from simplified assumptions about sample geometry, bleaching/photoconversion inhomogeneities, and the underlying reaction-diffusion kinetics. To address these shortcomings, we developed the software PyFRAP, which fits numerical simulations of three-dimensional models to FRAP/iFRAP data and accounts for bleaching/photoconversion inhomogeneities. Using PyFRAP we determined the diffusivities of fluorescent molecules spanning two orders of magnitude in molecular weight. We measured the tortuous effects that cell-like obstacles exert on effective diffusivity and show that reaction kinetics can be accounted for by model selection. These applications demonstrate the utility of PyFRAP, which can be widely adapted as a new extensible standard for FRAP analysis.

FRAP analysis often relies on simplified assumptions that can affect measurement accuracy. Here the authors present a Python-based FRAP analysis software using simulations instead of simplified theoretical models to fit the data, which accounts for complex sample geometries and bleach conditions.