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Fluorescence fluctuation spectroscopy in living cells R.A. Migueles-Ramirez 1 , A.G. Velasco-Felix 1 , R. Pinto-Cámara 1 , C.D. Wood 1 , A. Guerrero* ,1 1 Laboratorio Nacional de Microscopía Avanzada, Instituto de Biotecnología. Universidad Nacional Autónoma de México, 62210 Cuernavaca, Morelos, México. * Author for correspondence ([email protected]). Recent developments in fluorescence microscopy have opened the path to quantitatively describe cellular processes in living cells at the molecular level, through characterization of concentration, stoichiometry, diffusion, transport and binding amongst other parameters. Here we review the basic principles of fluorescence fluctuation spectroscopy techniques and for each section we present application examples of these techniques to living cells. Altogether, this mini review provides a comprehensive panorama of the principles behind microscopic spectroscopy techniques as well as some practical considerations for their application. Keywords: Fluorescence Fluctuation Spectroscopy, Fluorescence Correlation Spectroscopy (FCS), Raster Image Correlation Spectroscopy (RICS), Cross-Correlation Spectroscopy (FCCS, RICCS), image Mean Square Displacement (iMSD), Number and Brightness Analysis (N&B), Photon Counting Histogram (PCH). 1. Introduction Living systems are subject to tireless and incessant change. Because life’s processes occur at the molecular scale, the thermally-driven random motion of molecules is a major influence governing biochemical reactions. The rate and location at which these reactions occur are affected by molecular crowding [1]. The diffusive behavior of a molecule in the cell environment depends on the structure of the molecule itself, on the physical properties of its environment and on its molecular interactions. Diffusive motion is always present at molecular length scales, and biological systems must exploit, tolerate, or inhibit Brownian motion to perform directed and timed biochemical processes. A host of techniques have been developed to scrutinize diffusive dynamics within cells, for example fluorescence recovery after photobleaching (FRAP), single-particle tracking (SPT), and fluorescence correlation spectroscopy (FCS). FRAP depends on the capability to rapidly photobleach fluorophores within a defined region (usually in the range of a micron) in the imaging field using a high-energy laser pulse. After the laser pulse, photobleached fluorophores within the irradiated region are replaced by diffusive exchange with unbleached fluorophores from outside the irradiated region. By analyzing the dynamics of the recovery of fluorescence, it is possible to extract features of the diffusive properties of the fluorophore [2], [3]. Despite its potential, the phototoxicity of the laser pulse makes FRAP a rather invasive technique [4]. A more direct technique for monitoring diffusive dynamics is the explicit tracking of particles, through individual trajectory mapping [5]. The experimental design involves video microscopy, in which moving species of interest are visualized at fixed time intervals. Single particle tracking has been applied to the study of a wide variety of phenomena ranging from single cell locomotion to the wandering of individual proteins in organelles [5]. Fluorescence correlation spectroscopy is a non-invasive optical technique of great utility for probing diffusive dynamics within cells. In FCS, the fluorescence intensity in a small volume (~1 fl) is measured as a function of time [6]– [9]. The intensity of the recorded signal fluctuates as the fluorescent molecules enter and leave the region under observation. By analyzing the temporal fluctuations of the intensity using time- or space- dependent correlation functions, the diffusion coefficient and other characteristics of the molecular motion can be unveiled. In this mini-review, we present a selection of FCS-derived techniques which are valuable for studying how crowding alters both molecular equilibria and kinetics within the milieu of living cells. We will explore specific cases of how cell structure information can be recovered by examining deviations from the expected diffusive behavior of fluorophores in dilute solutions. Depending on the phenomenon of interest, we might be interested in the diffusion, the molecular interactions, or a combination of both. We will explain how this information can be obtained through statistical analysis of either the duration or the amplitude of the fluorescence fluctuations. 2. Unveiling molecular mobility The number of particles diffusing through an open volume fluctuates according to a Poisson distribution [10]. The average number of particles depends on the concentration and size of the volume. By considering independency between molecular displacements, the probability for several molecules to simultaneously leave is small compared to the probability of having a single molecule leave the observation volume. The former probability is lesser, as it results from the product of the probabilities of individual molecules leaving the observation volume. The information relating to molecular displacement is stored in the temporal structure of the signal itself. Hence, the analysis of a fluctuation Microscopy and imaging science: practical approaches to applied research and education (A. Méndez-Vilas, Ed.) 138 ___________________________________________________________________________________________

Fluorescence fluctuation spectroscopy in living cells · Fluorescence fluctuation spectroscopy in living cells R.A. Migueles-Ramirez1, A.G. Velasco-Felix1, R. Pinto-Cámara 1, C.D

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Page 1: Fluorescence fluctuation spectroscopy in living cells · Fluorescence fluctuation spectroscopy in living cells R.A. Migueles-Ramirez1, A.G. Velasco-Felix1, R. Pinto-Cámara 1, C.D

Fluorescence fluctuation spectroscopy in living cells

R.A. Migueles-Ramirez1, A.G. Velasco-Felix1, R. Pinto-Cámara1, C.D. Wood1, A. Guerrero*,1 1Laboratorio Nacional de Microscopía Avanzada, Instituto de Biotecnología. Universidad Nacional Autónoma de México,

62210 Cuernavaca, Morelos, México. *Author for correspondence ([email protected]).

Recent developments in fluorescence microscopy have opened the path to quantitatively describe cellular processes in living cells at the molecular level, through characterization of concentration, stoichiometry, diffusion, transport and binding amongst other parameters. Here we review the basic principles of fluorescence fluctuation spectroscopy techniques and for each section we present application examples of these techniques to living cells. Altogether, this mini review provides a comprehensive panorama of the principles behind microscopic spectroscopy techniques as well as some practical considerations for their application.

Keywords: Fluorescence Fluctuation Spectroscopy, Fluorescence Correlation Spectroscopy (FCS), Raster Image Correlation Spectroscopy (RICS), Cross-Correlation Spectroscopy (FCCS, RICCS), image Mean Square Displacement (iMSD), Number and Brightness Analysis (N&B), Photon Counting Histogram (PCH).

1. Introduction

Living systems are subject to tireless and incessant change. Because life’s processes occur at the molecular scale, the thermally-driven random motion of molecules is a major influence governing biochemical reactions. The rate and location at which these reactions occur are affected by molecular crowding [1]. The diffusive behavior of a molecule in the cell environment depends on the structure of the molecule itself, on the physical properties of its environment and on its molecular interactions. Diffusive motion is always present at molecular length scales, and biological systems must exploit, tolerate, or inhibit Brownian motion to perform directed and timed biochemical processes. A host of techniques have been developed to scrutinize diffusive dynamics within cells, for example fluorescence recovery after photobleaching (FRAP), single-particle tracking (SPT), and fluorescence correlation spectroscopy (FCS). FRAP depends on the capability to rapidly photobleach fluorophores within a defined region (usually in the range of a micron) in the imaging field using a high-energy laser pulse. After the laser pulse, photobleached fluorophores within the irradiated region are replaced by diffusive exchange with unbleached fluorophores from outside the irradiated region. By analyzing the dynamics of the recovery of fluorescence, it is possible to extract features of the diffusive properties of the fluorophore [2], [3]. Despite its potential, the phototoxicity of the laser pulse makes FRAP a rather invasive technique [4]. A more direct technique for monitoring diffusive dynamics is the explicit tracking of particles, through individual trajectory mapping [5]. The experimental design involves video microscopy, in which moving species of interest are visualized at fixed time intervals. Single particle tracking has been applied to the study of a wide variety of phenomena ranging from single cell locomotion to the wandering of individual proteins in organelles [5]. Fluorescence correlation spectroscopy is a non-invasive optical technique of great utility for probing diffusive dynamics within cells. In FCS, the fluorescence intensity in a small volume (~1 fl) is measured as a function of time [6]– [9]. The intensity of the recorded signal fluctuates as the fluorescent molecules enter and leave the region under observation. By analyzing the temporal fluctuations of the intensity using time- or space- dependent correlation functions, the diffusion coefficient and other characteristics of the molecular motion can be unveiled. In this mini-review, we present a selection of FCS-derived techniques which are valuable for studying how crowding alters both molecular equilibria and kinetics within the milieu of living cells. We will explore specific cases of how cell structure information can be recovered by examining deviations from the expected diffusive behavior of fluorophores in dilute solutions. Depending on the phenomenon of interest, we might be interested in the diffusion, the molecular interactions, or a combination of both. We will explain how this information can be obtained through statistical analysis of either the duration or the amplitude of the fluorescence fluctuations.

2. Unveiling molecular mobility

The number of particles diffusing through an open volume fluctuates according to a Poisson distribution [10]. The average number of particles depends on the concentration and size of the volume. By considering independency between molecular displacements, the probability for several molecules to simultaneously leave is small compared to the probability of having a single molecule leave the observation volume. The former probability is lesser, as it results from the product of the probabilities of individual molecules leaving the observation volume. The information relating to molecular displacement is stored in the temporal structure of the signal itself. Hence, the analysis of a fluctuation

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experiment as a function of time provides single molecule information and can be unveiled by calculating its auto-correlation function. In this section we will discuss three different techniques that focus on the temporal behavior of the fluorescence fluctuations to determine the diffusion coefficient and the average number of particles. As mentioned before, FCS is based on the autocorrelation function to analyze the diffusive behavior of fluorescent particles within a fixed observation volume. Based on the FCS principle and taking advantage of the commercially available laser scanning microscopes, another technique called Raster Image Correlation Spectroscopy was developed [11]. In this technique, the laser beam is used to scan a defined area of the sample thus allowing us to obtain not only temporal, but spatial information [11]. In the cellular environment, molecular diffusion can sometimes present anomalous features which can be caused by crowding, binding or molecular confinement. The Image Mean Squared displacement is a powerful technique to help characterize the molecular diffusion by analyzing the distance a molecule travels in a given lapse of time.

2.1. FCS principles and theory

Fluorescence correlation spectroscopy (FCS) is a technique with high spatial and temporal resolution used to analyze the kinetics of particles diffusing at low concentrations [6]– [9]. Fluorescently labeled molecules in solution move randomly, with each particle following a unique trajectory. These molecules are excited as they transit through a small focal volume (~1 fl) which is defined by the Point Spread Function (PSF) (Fig 1a). This function is the result of the microscope focused optical system convolution with the point source emission of the fluorophore and is therefore characteristic of the experimental setup and fluorophore. The detected fluorescence intensity as a function of time is defined as: ( ) = ( ) ( , ), (1) where k is the detector's sensitivity, Q is the fluorescence quantum yield, W(r) describes the illumination profile and C(r, t) is a function of the fluorophore concentration over time. In the case of single-photon confocal microscopy, the illumination profile W(r) follows a Gaussian distribution whereas for two-photon excitation, W(r) is polynomial, i.e. Gaussian-Lorentzian. In order to perform FCS tens of millions of measurements must be acquired at high frequency, generally in the kHz to MHz range. Fluctuations around the mean value ⟨ ( )⟩ contain information regarding the particles' diffusive nature (Fig. 1b) [12] ( ) = ( ) − ⟨ ( )⟩ (2) The correlation between ( ) and ( + ) is calculated for a range of delay times. The resulting autocorrelation function G(τ) represents the self-similarity of the signal (Fig. 1c). The autocorrelation function of the fluorescence fluctuations is defined as [12] : ( ) = ⟨ ( ) ( )⟩⟨ ( )⟩ , (3)

where time t refers to the time point of intensity acquisition, τ to the time delay between acquisitions and ⟨… ⟩ indicates average. The autocorrelation function contains information on the molecular diffusion coefficient and the number of molecules occupying the observation volume. To interpret such data in molecular terms we need a model to describe the fluctuations. An example is the model of free diffusion in three dimensions, which is defined as [3]: ( ) = (0) 1 + 1 + /

, with = , (4)

where is the diffusion time, s is the radius and u is the half-length of the observation volume. The parameter is usually expressed as = , with as the eccentricity of the observation volume, which typically has a radius of 0.25 and a half-length of 1.0 . In the simplest case, two parameters define the autocorrelation function: the amplitude of the fluctuation when τ tends to 0 ( ( )), and the characteristic relaxation time of the fluctuations. The amplitude of the autocorrelation function ( ) is proportional to the number of molecules , by / (0) where is a geometric factor that depends on the

illumination profile. For a Gaussian volume =1 √8 ≈ 0.35⁄ , whereas for a Gaussian-Lorentzian volume ≈ 0.076. The decay of the autocorrelation curve is directly proportional to the diffusion coefficient (length2 time-1) (Fig. 1e).

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Figure 1. Simulations to illustrate the principle behind FCS. a) Fluctuations in the fluorescence intensity are caused by occupancy changes of the fluorophores as particles diffuse in and out of the observation volume over time. b) The fluorescence intensity function ( ), is expressed as photon counts per integration time. The average number of particles is reflected in the fluorescence intensity mean value ⟨ ( )⟩ (red dotted line), whereas the diffusion is reflected in the fluctuations around the mean value ( ) =( ) − ⟨ ( )⟩.c) The autocorrelation function of a single diffusing species is usually of sigmoidal form and contains information about the number of molecules occupying the illumination volume and the diffusion coefficient. Correlation curve simulated using a three-dimensional model of free diffusion. d) Correlation curve of multiple diffusing species simulated using a three-dimensional model of free diffusion. e) The autocorrelation value when → 0 is inversely proportional to the occupation number so the amplitude of the function decreases as the molecular concentration increases (blue arrow). The autocorrelation function decay time is directly proportional to the diffusion coefficient (green arrow). The simulations were performed using the Sim tools of Stowers plugins of ImageJ developed by Jay Unruh. Data analysis was performed in R statistical software v3.2.4. There may be several mechanisms that cause the fluorescence signal to fluctuate: diffusion dynamics (i.e. variations in the number of fluorescent molecules in the volume of observation); conformational dynamics; rotational motion – if polarizers are used either in the source of excitation light or prior to the detector; protein folding; transitions between excited states, among others. In this context it is important to state that Eq. (4) ( ) = ( ) represents a scenario of simple three-dimensional diffusion.

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In the case where two or more independent processes contribute to fluctuations in fluorescence, the product of the correlation should be used. For example, if we assume that discontinuous signal generation of the particles is independent of the movement (blinking, i.e. due to intersystem crossing), the correlation function splits into the product of the two independent processes [3]. ( ) = ( ) ( ) , (5) = ⁄ , (6) where B is the fraction of blinking molecules (in the triplet state) and is their characteristic blinking rate. FCS has been applied to study physical processes such as translational and rotational diffusion, chemical reactions and fluorescence lifetime, to name a few [13]– [15]. It can be used to study molecular binding dynamics by fitting an appropriate model (Fig. 1d). The autocorrelation function of binding dynamics can be written as follows [3]:

( ) = ( ) + ( ) (7)

For which an unbound fluorescently-tagged molecule has a diffusion coefficient and the same molecule bound to a protein has a diffusion coefficient , represents the unbound fraction of molecules and represents the bound fraction of molecules. Although FCS is a powerful technique, one of its limitations is that it can only distinguish differences in diffusion coefficients, therefore in solutions containing multiple fluorescent species if two or more species have similar diffusion coefficients, FCS in unable to differentiate between them.

2.2. Raster Image Correlation Spectroscopy

Based on the FCS principle, a series of techniques have been developed in which the observation volume, instead of being stationary, is in movement, allowing us to measure correlation not only in time but also in space [11]. These techniques, collectively named scanning FCS, can be distinguished mainly by the form of the laser trajectory: line scanning FCS, orbital scanning FCS and raster scanning FCS. In the latter, also known as Raster Image Correlation Spectroscopy (RICS), the laser beam scans the sample from left to right then travels back to the left of the image without collecting any data and begins a second row just underneath the first [11], [16], [17]. These lines are usually, although not exclusively, collected in a similar way to occidental reading: from left to right and from top to bottom. Once the whole sample is scanned, the laser beam returns to its initial position to begin a new scan. This acquisition method results in a series of frames in which the time delay between pixels (µs), lines (ms) and frames (sec) provides temporal correlation whereas the pixel coordinates (nm) provide spatial correlation.

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Figure 2.The RICS approach. a) In classical FCS analysis, fluctuations occurring at a fixed illumination volume are considered. In the raster scanning mode of RICS, correlations between adjacent (or distant) observation volumes are considered. The probability density of finding a particle at a different location and at a different time is then introduced into the correlation analysis of the fluorescence fluctuations. b) In the RICS method, the correlation is dependent on how fast molecules are moving as well as how fast the volume of observation shifts its position with time. For slow diffusing species the spatial correlation function decays slowly compared with the scanning velocity. In the same way, the analysis of fast diffusing species provides a correlation function characterized by a faster decay constant. c) To gain statistical significance, the RICS methodology requires acquisition of several images in laser-scanning mode. Depending on the experimental scenario, the dwell time (the time to collect the fluorescence signal at a given observation volume) must be adjusted. For fast diffusing species, such as molecules in solution (D > 100 µms-1) it is recommended to use dwell times between 2 – 5µs. For proteins moving inside of living cells (D = 1-100 µm s-1) it is recommended to use a dwell time in the range of 10 – 32µs, and for slower species (0.1- 10 µm s-1) such as molecular complexes, or molecules bound to matrixes, a dwell time in the range of 32 – 100 µs is recommended. d) An important element of RICS analysis is the subtraction of the immotile fraction, which is calculated as the time series mean intensity projection. e) The resulting image contains all the information of the dynamics f) The autocorrelation function of each frame must be calculated (i-ii), this can be performed to the entire data set (r0) or at particular regions of interest (r1 and r2), also shown in panel e. The RICS autocorrelation function results from the averaging of several auto-correlation functions of the same spatial domain (iii), this function contains information from two temporal domains of the diffusing species, which usually are in the scales of µs and ms, ξ and ψ, respectively (iii-iv). A quantitative description of the dynamics is obtained from the fitting of a mathematical model which encompasses the physical properties of the phenomena in study (v), which for illustrative purposes was chosen to be the tridimensional diffusion of two species with distinct diffusion coefficients (1 and 30 µm s-1), which were segregated in space (compare r1 with r2). Simulations were performed using the Sim tools of Stowers plugins of ImageJ 1.51h developed by Jay Unruh.

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A critical step for RICS analysis is the subtraction of the immotile fraction of molecules present, which can be performed by subtracting the average fluorescence intensity from each frame (Fig. 2). This process can be performed using the whole image series or a smaller subset of frames (moving average). This is particularly useful when dealing with slow moving objects or photobleaching. For each interval, the moving average is subtracted from the image in the middle of the considered interval. The interval size effect on the RICS outcome can be used to understand molecular diffusion at different time scales ranging from microseconds to seconds or in particular cases minutes Once the background has been subtracted, the image autocorrelation function is calculated as follows [11], [16], [18]. ( , ) = ⟨ ( , ) ( , )⟩⟨ ( , )⟩ , (8)

where ( , ) is the intensity of each pixel of the image without the immotile fraction, ξ and ψ are shifts in x and y respectively, and the angled bracket ⟨… ⟩ indicates the average over all the spatial locations in both x and y directions (Fig. 2 c-f). Once the image autocorrelation functions have been calculated for each frame, they are averaged (Fig. 2f), and the result is fitted with a RICS model for, i.e., diffusion dynamics. As correlation is computed more efficiently in the frequency domain rather than by the formulas of Equations (3,8), the vast majority of the implementations of FCS and RICS approaches use the FFT algorithm [19]. The global RICS function can be expressed in the following form [16]: ( , ) = ( , ) ( , ) + , (9) where ( , ) accounts for diffusion dynamics, and ( , ) for the raster scanning mode, b is a constant commonly added to scale up the RICS function to avoid negative values. ( , ) = 1 + ( ) 1 + ( ) /

, (10)

( , ) = − ( ) , (11)

These are the basic forms of these equations, which can be extended to account for factors such as two-photon excitation and membrane-bound diffusion [11], [18]. In Equations (10) and (11), D is the diffusion coefficient (in units of length2 time-1), x and y are the horizontal and vertical coordinates of the image, ξ and ψ are the x and y shifts, respectively; and and are the pixel sampling time and time between lines, respectively. The spatial part of the correlation function ( , ) is expressed directly in terms of the distance between adjacent points in the line (the pixel size, typically in the 0.05–0.2 µm range); pixel residence time, (typically in the 2–100 ms range); line repetition time, , (typically in the range of 2–100 ms). Note the similarity of Eqs. (9-10) with Eqs. (3,4). Equations 10 and 11 combined with Eq. 9 are used to fit the autocorrelation functions from RICS data (Eq. 8) to extract the diffusion coefficient, and the number of particles using the geometrical factors that describe the 1/e2 Gaussian illumination profile (w0 and wz). As the motion of molecules must be measured over a certain area, the spatial resolution of the RICS method is limited by how fast the molecules are moving. For molecules wandering in the crowded environment of the intracellular milieu, this area has a size in the order of a micron. RICS methodology can be successively applied locally to subregions within the acquired image to ultimately obtain a spatial map of diffusion dynamics and the local concentration of molecules. RICS can be performed using commercially available laser scanning microscopes, becoming one of the most used fluorescence fluctuation spectroscopy techniques [11]. RICS has been applied to the study of a wide variety of biological phenomena, such as paxillin dynamics in focal adhesions, ATP diffusion in rat cardiomyocytes, lipid diffusion in membranes, among others [11], [20], [21]. In terms of the implementation, an excellent protocol was provided by Rossow et al [16].

2.3. Image Mean Squared Displacement

Mean Squared Displacement (MSD), as the name suggests, is a measure of the distance traveled by a freely diffusing particle during a time interval. It arises from the argument that displacement of a Brownian particle is proportional to the square root of the elapsed time. The MSD can be expressed in terms of the diffusion coefficient as follows [22]. = 2 , (10) where d is the number of spatial dimensions under consideration, t is the time and D is the diffusion coefficient.

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By measuring the MSD of a particle it is possible to calculate its diffusion coefficient. Furthermore, by analyzing how the MSD changes over time, information of the particle’s environment can be obtained. It is with this approach and the use of Single Particle Tracking (SPT) that concepts such as the Fences and pickets model of plasma membrane structure were proposed [23]. The current concept of cellular diffusion is constructed from diverse diffusion modes. Among them the simplest scenario is isotropic diffusion also known as Brownian diffusion, in which a particle moves unobstructed by the environment and its mean squared displacement increases linearly with time. Confined diffusion is often observed in cellular compartments, in which particles move randomly but cannot go farther than a limit and the mean squared displacement cannot increase beyond the dimensions of this distance [24]. Transiently confined particles move freely within a confinement space, when they reach the border of their confinement space they may or may not go through the barrier. The smaller diffusion coefficient at longer times can be reported as a macroscopic diffusion coefficient and the point where the behavior of the particle changes is an approximation of the dimensions of the transient confinement space [25]. The MSD of a super diffusing particle increases nonlinearly with time, when this behavior is found inside the cell, it is usually due to its interaction with cellular active transport mechanisms [26] (Fig. 3a).

Figure 3.The iMSD approach. a) Image Mean Squared Displacement versus Time plot of four diffusing modes (left) and their random walk representation (right). From top to bottom: Intracellular super diffusion can be caused by active transport (blue). Isotropic diffusion in the living cells is observed when particles diffuse freely or data is acquired at high frame rates (red). Transiently confined diffusion, also known as hop diffusion of plasma membrane particles; the initial slope corresponds to the diffusion coefficient at distances shorter than the barrier indicated by the dotted line (TC), the second corresponds to the diffusion coefficient at distances longer than the barrier (yellow). In confined diffusion, particles diffusing inside organelles cannot displace distances longer than the dimensions of their confinement, dotted line (C) (green). b) For iMSD analysis, image series is acquired (left). To measure the particles’ displacement between each frame the correlation between the first frame of the image series and each subsequent frame is calculated (STICS) yielding a series of correlation functions (right). A Gaussian is fitted to each correlation function and its variance or iMSD is extracted from it. The iMSD can be plotted versus time resulting in a plot similar to that shown in a) (left); a model composed of the diffusion modes mentioned in a) is fitted to the iMSD series to obtain a micro- and macro-diffusion coefficients and the dimensions of any existing barriers. The versatility of iMSD derives partly from not requiring special tagging methods or microscopy equipment, i.e. MSD can be measured through image analysis of fluorescent particles in live cells, with conventional fluorescent proteins, probes, conventional microscopes and using CCD or sCMOS cameras. However certain microscopy techniques offer specific advantages when implementing iMSD, for instance, Total Internal Reflection Fluorescence Microscopy (TIRF) excites fluorophores located at a few hundred nanometers away from the coverslip, reducing out-of-focus blur from molecules at higher planes, and thus greatly improving the signal to noise ratio and spatial selectivity when observing fluorophores in the plasma membrane [24]. In another example Selective Plane Illumination Microscopy (SPIM) employs a lateral thin excitation beam that similarly avoids exciting particles in planes above and

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below, which is useful when studying diffusion in or between cell compartments such as the nucleus or mitochondria [27]. If using a laser scanning microscope it is possible to vary the frame rate over a wide range to obtain information of the fluorophores at different displacement magnitudes, facilitating identification of diffusion barriers [25]. If we consider a population of fluorescent proteins in the cytosol (i.e. fluorophores in solution inside an observation field) that diffuse over time, acquiring an image series of the observation field yields a recording of the particles’ displacement between each frame or time shift. The displacement of the population of fluorophores between the first frame and each subsequent frame corresponds to the image Mean Square Displacement. To calculate the iMSD the fluorescence intensity fluctuations in the image series are analyzed using Spatiotemporal Image Correlation Spectroscopy (STICS) (Fig. 3b) [28]. ( , , ) = ⟨ ( , , )⋅( ( , , )⟩⟨ ( , , )⟩ − 1, (12)

where ξ, χ represents the distance in the x and y directions respectively, I(x, y, t) represents the fluorescence intensity in the position (x, y) at time t, and ⟨… ⟩ represents the average. In the correlation function series, the decrease of spatial correlation after each time shift is due to the constant diffusion or displacement of the fluorescent particles, which is in turn governed by Fick’s second law. = , (13)

where φ is the concentration in a given point, D is the diffusion coefficient and is the Laplacian of the concentration. In cases without flux or angular anisotropy, Fick’s second law is solved and rewritten as a probability distribution function that describes the probability to find a particle at a distance ξ, χ from its origin (x, y, t) after a time shift τ. ( , , ) = − (14)

The displacement can be separated in two components: the distance component follows a normal distribution and the direction component an equivalent distribution, therefore the displacement probability distribution function is similar to a Gaussian function, and its variance can be identified as the mean squared displacement of the particle population. When the particles are smaller than the microscope’s point spread function, the correlation function can be rewritten as: ( , , ) = ( , , ) ⊗ ( , ), (15)

where γ is a constant dependent on the observation volume’s shape, N represents the average particle number in the observation volume, ⊗ represents convolution and W(ξ, χ) represents the point spread function of the microscope. This function is fitted with a Gaussian function, if the particles move over time the peak of the Gaussian will decrease and its variance will increase as the particles continue move away from their origins. The variance of this final function can be identified as the image Mean Squared Displacement [24]. Lastly the iMSD is extracted from each frame, yielding a series of iMSDs. A model composed of the diffusion modes mentioned before is fitted to the iMSD series to calculate the micro and macro diffusion coefficients and the dimensions of barriers if any exist [24], [25], [28].

3. Unveiling molecular interaction

As previously discussed, the autocorrelation function can be used to determine the diffusion coefficient and the average number of particles in a small volume, allowing us to assess the molecular concentration. The Photon Counting Histogram (PCH) analysis and the Number and Brightness (N&B) method constitute two time-independent techniques that provide an alternative for determining molecular concentration, based on the statistical analysis of the fluorescence intensity fluctuations [29]– [31]. Additionally, these techniques allow the determination of the molecular brightness, i.e., the measured photon counts per molecule per second, which can be used to extract information on the oligomerization state of molecular complexes [32]. Furthermore, cross-correlation techniques, such as cross-FCS, cross-RICS and cross-N&B can provide information on the stoichiometry of the molecular complexes as well as their diffusion coefficient and their affinity constants, among many other applications [33], [34]. By providing information on molecular interactions using non-invasive optical approaches, these techniques constitute excellent tools to study systems in living cells [33].

3.1. The Photon Counting Histogram (PCH) and the Number and Brightness (N&B) methods

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If the diffusion coefficients of two species differ significantly, the autocorrelation function can be used to distinguish between them. When this is not the case, changes in the brightness can sometimes be more informative than changes in diffusion coefficients. In fact, an x-fold change in the molecular weight will only affect the diffusion coefficient by a factor of x-⅓. The most common example of molecular weight increase is oligomerization, which has little impact on the diffusion coefficient and the autocorrelation function [29]. Techniques such as the N&B or the PCH take advantage of the statistical analysis of the changes in the apparent brightness to provide the average number of molecules and their oligomerization state, although not providing any information on the temporal diffusing behavior of the particles. The observed fluorescence intensity is mainly due to the presence of fluorophores in the observation volume. More precisely, it is due to the position of the emitting fluorophore within the PSF, which is heterogeneous in illumination intensity and without sharp boundaries. For a given pixel in a time series, the mean intensity contains information on the number of fluorophores present in the observation volume. By determining the effective volume of the illumination profile, one can extract the fluorophore concentration [35]. The fluctuations in the fluorescence intensity are caused by both the shot noise of the detector and by the changes in the fluorophore position - entrance and exit of fluorophores in and out the observation volume [36]. The amplitude of these fluctuations - and therefore the variance of the time series - increases as the fluorophore oligomerization state increases [17].This is due to the increase in the apparent brightness of dimeric species compared to the brightness of monomeric species which results in an increased occurrence of high intensity detections. The molecular brightness and the number of particles in the observation volume both contribute to the fluorescence intensity. The mean value of the fluorescence intensity of two samples with different oligomerization states can therefore be the same if one has many dim monomers and the other few bright aggregates (Fig. 4). This concept is the basis of the PCH and the N&B methods. The Photon Counting Histogram. The PCH produced by the movement of a particle in a heterogeneous illumination profile is the sum of the Poissonian distributions of all possible positions of the particle within the PSF. This yields a super-Poissonian distribution, in which the variance is greater than the mean value [29]. For two or more independent moving particles, the broadening of the distribution is caused by the sum of the Poissonian distributions of every possible position of each particle. By fitting the data with an appropriate model, it is possible to extract information on the number of molecules in the observation volume and of their molecular brightness:

= ∑ ( ) ( )∙ ( )⋅ ( ), (16)

where M is the number of observations, kmax is the maximum number of counts and d is the number of fitting parameters [29]. Since the distribution of a single species can be uniquely characterized by the average number of molecules in the observation volume (N) and the molecular brightness (ε), it is possible to resolve mixtures of two or more different species in the observation volume [29]. If properly acquired, the same data can be used to perform FCS and PCH. Additionally, FCS data can also be used to perform a new analysis technique based on the PCH called Time-Integrated Fluorescence Cumulant Analysis (TIFCA) [37], [38]. This technique makes use of both the molecular brightness and diffusion time of freely diffusing fluorescent species and has been applied to determine the stoichiometry of antibody complexes and the oligomerization state of endophilins in live cells [38], [39]. The PCH analysis has been extended for the use of two separate channels, allowing discrimination of mixtures of different species based not only on their molecular brightness but on their emission wavelength. This analysis is particularly useful in the case of fluorophores with overlapping emission spectra [40], [41]. The Number and Brightness Method Unlike the PCH, which requires a large number of observations at each observation volume, and is therefore computationally too slow for the analysis of all pixels in an image, the N&B method provides molecular concentration and brightness for each pixel of a series of images in a reasonable time delay (in the order of milliseconds), thus preserving structural information [30]. In that sense, N&B can be considered the imaging equivalent of the PCH and is particularly useful to spatially resolve different oligomerization states formed in different parts of the same sample, revealing structural and functional insights (Fig. 4) [32], [42], [43]. If the acquisition time is fast enough, it is possible to observe aggregation dynamics [30]. A limitation of N&B is that it is unable to resolve a mixture of multiple species of different brightness in the same pixel, which can be attained using the PCH [30]. Compared to PCH, N&B does not require a non-linear fit of the data [32]. Instead, the average particle number and brightness are calculated directly from the mean value < > and variance of the fluorescence intensity data for a given pixel as follows: = ⟨ ⟩ , = ⟨ ⟩ (17, 18)

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The use of sensitive detectors capable of photon counting detection is desirable but not a requirement for both PCH and N&B analysis. If the acquisition is performed in analog mode or in a pseudo-photon counting mode, the Offset, shot noise and the photon-to-electron proportionality factor (S factor) of the detector must be measured and taken into consideration, yielding the following corrected equations for the average number of particles in the observation volume (n) and the molecular brightness (ε, in counts per second per molecule) [36]. = (⟨ ⟩ )(⟨ ⟩ ) , = (⟨ ⟩ )(⟨ ⟩ ) , (19, 20)

Additionally, the immobile fraction, sample bleaching and slow-moving features must be removed or corrected, for which different algorithms have been developed [30]. The laser intensity and the pixel dwell time are of paramount importance. Laser power must remain constant throughout calibrations and experiments and must avoid saturation of the detector. The pixel dwell time (usually in the range of microseconds), has to be long enough to observe particle fluctuations but short enough to avoid averaging out the fluctuations [30]. N&B can be performed using data collected with a confocal or TIRF microscope, as long as the pixel dwell time is less than the characteristic diffusion time of the particle [32], [42], [44]. If acquired with a laser scanning confocal microscope, the same data can be used for RICS and N&B analysis. The N&B method has been applied to the study of several biological processes in live cells such as cell adhesions, Parkinson’s disease, and Ebola virus, among many other application [34], [42], [45]. The N&B method has been extended by the use of the cross-correlation of two channels, allowing us not only to determine oligomerization state of heterocomplexes, but also their stoichiometry [34], [46].

Figure 4. Simulations to illustrate the N&B method. The N&B method is based on the statistical properties of the fluorescence fluctuation distribution. (a) Consider two different scenarios: monomers (green) and dimers (purple). These can be two different pixels of a single sample or two pixels from different samples. Note that the observed intensity of each fluorophore depends on its position within the PSF. (b) The fluorescence intensity at a given moment is caused by the particle's brightness, its position within the PSF and its concentration. The analysis of the fluorescence fluctuations in time allows us to extract information on the fluorophore concentration and oligomerization state. The particle concentration affects mainly the mean value of the distribution, whereas the increase of oligomerization (in this case dimerization) affects mainly the variance of the distribution (light green compared to dark green and purple in a and b). Based on these observations, the apparent number (N value) and apparent brightness (B value) can be calculated using equations 17 and 18. The N value is indicative of the number of particles (75 monomers and 75 dimers yield an N value of 14.4 and 15.8, respectively, while 150 monomers yield an N value of 34.7). The B value is representative of the oligomerization state (75 and 150 monomers yield a B value of 541 and 445, respectively, while 75 dimers yield a B value of 997). Simulations were performed using SimFCS 2.0 with the following parameters: monomer brightness = 100 000 cpsm, monomer diffusion coefficient = 40 µm2/s, dimer brightness = 200 000 cpsm, dimer diffusion coefficient= 32 µm2/s, 500 observations (only 250 are showed in b), PSF effective volume = 0.27 µm3. Data analysis was done using R for statistics v.3.2.3.

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3.2.Cross-correlation fluctuation correlation spectroscopy

We previously mentioned that auto-correlation analysis allows the comparison of a signal with itself to identify recurring patterns that can be interpreted as diffusion dynamics, binding, conformational changes, among others [3]. In fluorescence cross-correlation spectroscopy (FCCS or ccFCS) rapid simultaneous intensity fluctuations in two independently acquired signals are assumed to be due to the co-movement of two fluorescent particles (Fig. 5) [33], [34]. By comparing the signals of two fluorescently labeled molecules using the cross-correlation function, we can detect their molecular interaction, determine their diffusion coefficients, oligomerization state, concentration, affinity and the stoichiometry of the complex they form [10], [32], [47]. For signals that depend on each other, such as enzyme–substrate, or protein–protein interactions, correlating them allows the quantification of their ‘cross-talk’ or co-movement. Consider two independently measured fluorescence signals F1 and F2 we can generalize the normalized cross-correlation function in the following form:

( ) = ⟨ ( )⋅ ( )⟩⟨ ( )⋅ ( )⟩ , (21)

where ( ) and ( ) are the fluorescence intensity of each channel, < ( ) > and < ( ) > are the fluctuations around their mean values, which can be defined as ( ) = ( ) − ⟨ ( )⟩, and ( ) = ( ) − ⟨ ( )⟩, respectively. This equation measures the fraction of the fluctuations caused by simultaneous movement of two different fluorophores. This technique can be applied to different locations in a cell, allowing us to assess dynamic colocalization in subcellular compartments. In principle, ccFCS is only sensitive to the fluctuations due to molecular complexes carrying both molecular species, as the detector's noise is uncorrelated. However, deriving quantitative information is more challenging; in practice, the ccFCS method must be calibrated for the bleedthrough or cross-excitation between acquisition channels and fluorophore pairs. In addition, ccFCS, is also strongly affected by the presence of non-radiative transfer of energy, such as FRET. Nonetheless, ccFCS has been successfully used to uncover dynamic colocalization in living cells. For example, binding of different pairs of MAP kinases in yeast, which is coupled with the recruitment of the Ste5 scaffold protein to the cell cortex [48]. More examples regarding the application of the ccFCS approach are described in [33]. The ccRICS method is sensitive to the formation of molecular complexes, thus equation (7) becomes: ( , ) = ⟨ ( , ) ( , )⟩⟨ ( , )⟩⟨ ( , )⟩ , (22)

where I1 and I2 are the fluorescence intensity signals of channels 1 and 2 respectively, ξ and ψ are the shifts in x and y axis respectively. ccRICS is non-zero only when the intensity fluctuations of the studied signals are correlated, therefore if the equations (9 - 11) are applied to the cross-correlation function it is possible to unveil the diffusion dynamics of molecular complexes. An important aspect to consider when performing ccRICS is that channel bleedthrough (generally green into red) can contribute to the cross-correlation, resulting in a false positive cross-correlation [47]. The ccRICS method has been used to detect the formation of protein complexes in the focal adhesions in living cells [47], to study the process of exogenous DNA degradation in the cytoplasm, also in living cells [49], and to characterize the spatial organization of the Src-kinase family member Lck, which is controlled by downstream signaling events triggered by the activation of T cell receptors [50].

4. Perspectives

With advances in the sensitivity of detectors and increases in the stability and affordability of diode lasers, fluorescence fluctuation measurements have never been more accessible to scientific researchers than at the present time, especially as the refinement and availability of both commercial and open-source analysis software inexorably progress. Our ability to spatially and temporally map the dynamic characteristics of cellular components has opened the door to a much deeper understanding of the organization and interaction of the molecular processes that underlie biology, and are an important component underpinning research in systems biology and synthetic biology that require true quantitative data to validate their experimental and modeling approaches. As further improvements in access to instrumentation accumulate, and the number of reported applications continues to grow, we will surely see fluorescence fluctuation analysis emerge further into the mainstream and consolidate its position in the toolbox of optical microscopy techniques.

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Figure 5. The principle of ccFCS and ccRICS measurements. a) For ccFCS and ccRICS, two separate laser lines are used to excite fluorophores with distinct spectral properties. The emission fluorescence from the sample is collected through the same objective and is separated from the excitation by a dichroic mirror which sends the signal to two separate detectors. b) The fluctuating fluorescence signals of each detector are auto-correlated (green and red curves) to study the diffusion dynamics of each population of fluorophores (FCS). In addition, the cross-correlation function between the green and red channels is computed (black curve). The amplitude of the cross-correlation curve is indicative of the degree of binding or colocalization. Moreover, the cross-correlation signal also provides information of the mobility of the hetero-complex. For illustrative purposes two populations of fluorophores, green and red, diffusing at 40 µm s-1 are presented. Note that the formation of a dimer, with a diffusion coefficient of 32 µm s-1, is uncovered by calculating the cross-correlation signal. In the absence of molecular interaction or spectral bleedthrough, the cross-correlation signal tends to zero (not shown). In the ccRICS approach, independent image sequences in laser scanning mode are collected simultaneously. RICS analysis can be performed with either fluorescence channel (the details of RICS methodology are explained in Fig. 2, note that the present explanation starts from the dynamic fraction of both red and green channels). The per-frame cross-correlation function between the red and green channels can also be calculated. The ccRICS signal results from the time projection of all per-frame cross-correlations. A quantitative description of the dynamics is obtained from the fitting of a model, to either the auto- or cross- correlation functions, which encompasses the physical properties of the phenomenon studied, which for illustrative purposes was chosen to be the tridimensional diffusion of two species with the same diffusion coefficient (40 µm s-1), and with a sub-population of red and green molecules allowed to form heterodimers (D = 32 µm s-1). Data analysis and curve fitting was performed using ImageJ 1.51h and R statistical software v3.2.4.

Acknowledgements: We thank Michelle Digman, David Jameson, Francisco Romero and Andrés Saralegui for critically reading and commenting section 3.1.The authors apologize to colleagues whose publications were not cited owing to space limitations. This work is financially supported by CONACyT Ciencia Básica 252213 to AG, DGAPA-PAPIIT IA202417 to AG. DGAPA-PAPIIT IN211216 to CDW. RMR is recipient of CONACyT-PNPC master’s degree scholarship 414221, RP and AV thank DGAPA-PAPIIT IA202417 and CONACyT Fronteras de la Ciencia 71 for fellowships, respectively. AG thank to the Company of Biologist and to the Journal of Cell Science for a Travelling Fellowship to visit the laboratory of Satyajit Major at the National Centre for Biological Sciences, Bangalore, India (2012), to study FCS. The Nacional Laboratory of Advanced Microscopy was founded and is maintained through the generous support of CONACyT and the Coordinación de la Investigación Científica of UNAM.

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