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Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells EL Elson, Washington University School of Medicine, St. Louis, MO, USA r 2016 Elsevier Inc. All rights reserved.
Glossary Observation volume or area The open volume (or area for two-dimensional systems) that is illuminated by the
Introduction To understand dynamic molecular processes of metabolism and regulation at the cellular level it is necessary to determine how specific molecules move and interact on the surfaces and within living cells. To accomplish this task requires methods that have the following characteristics: 1. Chemical selectivity – the ability to measure the concentrations of specified molecules present at low concentration within an environment, for example, cytoplasm, that contains high concentrations of many other kinds of molecules 2. A wide dynamic range from omicroseconds to 4minutes 3. Relatively short time required for the measurements (seconds) 4. High spatial resolution to enable measurements within dimensions o1 mm 5. Sensitivity, even down to the single molecule level 6. Lack of toxicity and interference with normal cell function 7. Ability to localize the measurement within identifiable cell structures and features 8. Applicability to a range of concentrations from nM to mM. These specifications suggest the use of fluorescence, which has the required rapidity of measurement, sensitivity, and selectivity, and, when based on a confocal fluorescence microscope, the spatial resolution and ability to localize the measurement relative to cell structures. Furthermore, when used judiciously, fluorescence of labeled molecules can be measured with little damage to living cells. Two closely related and complementary methods, fluorescence correlation spectroscopy (FCS) and fluorescence photobleaching recovery (FPR, also known as fluorescence recovery after photobleaching, FRAP) take advantage of these properties of fluorescence to measure rates of molecular transport, for example, diffusion and convection, and of chemical reaction over a wide dynamic range of concentrations and time in living cells. FCS was introduced in 1972 (Magde et al., 1972). The history of its origins and development have been related from the perspectives of the two groups that first demonstrated the feasibility of using fluorescence fluctuations to measure translational diffusion and chemical kinetics (Elson, 2013a,b) and rotational motion (Rigler, 2009). FCS remained a specialized subject of study for a number of years until technological advances enabled more rapid and robust measurements and this in turn led to the production of
Encyclopedia of Cell Biology, Volume 2
doi:10.1016/B978-0-12-394447-4.20094-1
excitation laser and from which fluorescence is detected. The observation volume is typically limited by a confocal aperture in the microscope optical system.
commercial instruments by a number of companies and widespread use of FCS in many laboratories. Important improvements in FCS measurements resulted from using confocal microscopy to limit the volume from which fluorescence is detected (Koppel et al., 1976; Rigler et al., 1993) and which allowed single molecule detection (Mets and Rigler, 1994; Rigler, 2009). Improvements in microscope technology (infinity-corrected optics) and stability of fluorophores and computation of correlation functions (Schatzel et al., 1988) also were important contributions. Improvements in microscopy minimized the detection volume thereby diminishing interfering background fluorescence while increasing the detected number of photons emitted per fluorphore.
Basic Concepts and Theory Correlation of Fluctuations of Concentration and Fluorescence Most conventional methods to measure chemical rate constants determine their values by perturbing the molecular system from its equilibrium state, for example, by a temperature jump that displaces the system from equilibrium, and then observing the rate of relaxation back to equilibrium (Eigen and De Maeyer, 1963). Similarly, conventional measurements of diffusion are typically carried out by observing the dissipation of a concentration gradient that has been created in the system by the experimenter (Tanford, 1961). In contrast, FCS measures these rates without perturbing systems that remain in equilibrium or nonequilibrium steady states during the measurement. In an open volume of a chemical system defined by a focused laser beam, the numbers of molecules of the system components continually fluctuate due either to their diffusing into and out of the volume or to their participating in chemical reactions. The fluctuation of the concentration of chemical component, j, at position r and time t is δCj(r,t)¼ Cj(r,t) 〈Cj〉 where 〈Cj〉 is the equilibrium (or nonequilibrium) steady state value of the concentration. The concentration fluctuations, δCj(r,t), causes corresponding fluorescence fluctuations, δFj(t), that are measured directly, typically using a confocal fluorescence microscope (Koppel et al., 1976; Qian and Elson, 1991). A fluorescence fluctuation is related to a concentration fluctuation by adding the contributions from different locations in the observation region weighted by the excitation intensity,
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Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
R IðrÞ : δFj ðtÞ ¼ Qj IðrÞδCj ðr,tÞdr, where Qj is a constant that accounts for the absorption coefficient and fluorescence quantum yield of the fluorophore (component j) and properties of the optical measurement system. The durations of the fluctuations depend on the size of the volume and the diffusion rate as well as on the chemical reaction rate(s) and for nonlinear chemical reactions, the concentrations of the reactants. It is not sufficient, however, simply to measure one or a few fluctuations to derive the diffusion coefficients and rate constants because the fluctuations are stochastic. Each fluctuation can differ substantially from the others. Therefore, it is necessary to carry out a statistical analysis of the time courses of many fluctuations to obtain accurate values of transport and reaction rate parameters. The statistical analysis of the fluorescence fluctuations is most often carried out by calculating a fluorescence fluctuation autocorrelation function: Gjj ðτÞ ¼
〈δFj ðt ÞδFj ðt þ τÞ〉 〈Fj ðtÞ2 〉
dimensional system, for example, 〈Nj〉¼ πw2〈Cj〉, the average number of fluorescent molecules of the j’th chemical species in the observation area πw2 (Elson, 2013a,b). This property of G(0) provides an absolute measurement of the concentration of the fluorescent component and also the ‘brightness’ 〈F 〉 〈Qj 〉 ¼ 〈Njj 〉 ¼ Gð0Þ〈F〉, the average number of fluorescence photons detected per unit time per fluorescent molecule. Many reviews and primary articles provide more details about the forms of correlation functions for different dynamic processes (e.g., Elson and Magde, 1974; Digman and Gratton, 2011; Elson, 2011, 2013a,b; Ries and Schwille, 2012). In systems with M fluorescent components, the contributions of all the fluorophores must be included in the correlation function. The total fluorescence fluctuation is δF ðt Þ ¼ ∑M j ¼ 1 δFj ðt Þ and the correlation function is GðτÞ ¼
,
where 〈…〉 denotes an average over many time points and the division by 〈Fj(t)〉2 yields an expression independent of the excitation laser power and fluorophore brightness for a single component system. The fluorescence fluctuation at some time t is multiplied by a fluctuation at a later time t þ τ. Averaging many of these products obtained for different times, t, yields an indication of the extent to which the fluorescence/concentration fluctuations have relaxed toward the steady state value for the system during the delay time τ. Then, carrying out this operation for a range of τ values gives a functional form for Gjj(τ) that is characteristic for specific transport processes and chemical reactions (Elson, 2011). Up to now the fluorescence fluctuations and the fluctuation autocorrelation function has been discussed from a conventional chemical perspective. It may also be of interest to frame the discussion from the point of view of contemporary single molecule studies. This discussion is presented in the Appendix. The final results of the derivations, whether from a chemical or a single molecule perspective, must, of course, yield the same equations for the fluorescence fluctuation autocorrelation function. The form of Gjj(τ) also depends on the ‘shape’ of the volume or area from which the fluorescence is detected. This is frequently approximated as a Gaussian function with a characteristic width, w, in the focal (x ,y ) plane and a width wz h i 2 2 2 Þ expðw2zz 2 Þ. As along the optical (z ) axis: I ¼ I0 exp 2ðxwþy 2 an example, for fluorescent compound j with diffusion coefficient Dj that is diffusing in a two-dimensional plane, for example, a biological membrane: Gjj ð0Þ w2 ; the characteristic time is τDj ¼ : Gjj ðτÞ ¼ 1 þ τDτ 4Dj j
For chemical reaction kinetics Gjj(τ) can have a number of exponential relaxation components. The amplitude of the correlation function depends only on the concentration and size of the observation area/volume defined by the focused laser. The relative size of the fluctuations is determined by a Poisson distribution that governs the number of molecules in an open volume. Thus, Gjj ð0Þ ¼ 〈N1j 〉, where for a two-
M ∑M j ¼ 1 ∑k ¼ 1 〈δFj ðtÞδFk ðt þ τÞ〉 : 2 ∑M j ¼ 1 〈Fj ðtÞ〉
For the simple example of a multicomponent system in which the only dynamic process is the independent diffusion of noninteracting molecules 〈δFj(0)δFk(τ)〉¼ 0 for jak. Then, Q2j 〈Cj 〉
∑M j¼1 GðτÞ ¼
1þτ τ
Dj
2 : πw2 ∑m J ¼ 1 Qj 〈Cj 〉
If system components interact, as in chemical reactions, the situation is more complicated, especially if the kinetics of the reactions must be taken into account (Elson and Magde, kf 1974). For example, for the simplest reaction, A ⟶ B, in a ⟵ kb
closed system the chemical relaxation can be described in simple terms δCB(t) ¼ δCB(0)exp( Rt) ¼ δCA(t), where R¼ kf þ kb. In contrast for an FCS measurement in the open observation region, the kinetics must be analyzed as a two component coupled reaction-diffusion system because the A and B molecules can independently diffuse into and out of the observation region (Elson and Magde, 1974). Although the reaction–diffusion systems are more complicated to analyze, there are standard methods for solving the equations describing them, and so the mathematic analysis does not present a fundamental problem. More important for measuring chemical kinetics by FCS is to find a substantial change in fluorescence that indicates the reaction progress.
Relationship between FCS and Fluorescence Photobleaching Measurements Both the interpretive theory (Elson, 1985) and the implementation of measurements on a confocal fluorescence microscope (Koppel et al., 1976) are very similar for FCS and fluorescence photobleaching. The main difference in principle between the two kinds of measurements is that FCS relies on spontaneous microscopic concentration (fluorescence) fluctuations in mesoscopic systems, i.e., systems in which there are so few molecules that the concentration fluctuations are readily measured compared to the mean concentrations. In contrast, photobleaching recovery uses a pulse of intense light to photolyze fluorophores irreversibly within a small
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
observation volume. This produces a localized macroscopic concentration gradient, i.e., a relatively large difference between the concentration in the observation volume and the mean concentration in the surrounding regions of the system. The measured rate of relaxation of the concentration in the observation volume back to its equilibrium value due to diffusion or chemical reaction then provides the desired kinetic parameters. For FCS many stochastic fluorescence fluctuations, no matter how accurately each is measured, must be analyzed statistically via the correlation function to obtain accurate values of phenomenological diffusion coefficients and rate constants. In contrast, in a photobleaching measurement, a single macroscopic relaxation is sufficient to provide the phenomenological coefficients to an accuracy determined by the accuracy of the single measurement. The theoretical basis for supposing that the coefficients determined by each approach are the same is Onsager’s regression hypothesis (Onsager, 1931). During its early days FCS was difficult to use to measure processes in cells because the signals were relatively small and could be easily corrupted by cellular motions and other changes of cell properties during the time required to acquire a sufficient number of fluorescence fluctuations. Photobleaching was then more useful for measurements on live cells (e.g., Peters et al., 1974; Edidin et al., 1976; Schlessinger et al., 1976a,b). For the last two decades, however, due to technological advances, especially the use of confocal microscopy to minimize the observation volume (Koppel et al., 1976; Rigler, 1995), FCS has also proved to be useful for studies in living cells. Initial FRAP measurements were carried out using a nonscanning confocal microscope. Fluorescence emission was measured from a single diffraction-limited spot in the sample defined by the incident laser beam and the confocal aperture. The duration of the bleach pulse, controlled by a fast mechanical or acousto-optical shutter could be very brief (Bo1 ms) and data could be essentially continually recorded from the single spot (Koppel et al., 1976). More recently confocal laser scanning microscopes have been used for photobleaching measurements (Braga et al., 2004; Seiffert and Oppermann, 2005; Kang et al., 2012). Although this approach has the advantages of availability on a commercially available platform and of the possibility of bleaching and acquiring recovery data from arbitrarily shaped observation regions, very fast processes are less accessible by this approach due to its dependence on the scan speed of the microscope. Moreover, the interpretation of the recovery can be complicated by the need to account for the scanning rate of the microscope (e.g., Braeckmans et al., 2003).
measurements to the level of detecting a single fluorophore by minimizing the observation volume (and therefore the interfering background fluorescence) was a major step in facilitating FCS measurements (Mets and Rigler, 1994; Rigler, 2009).
Experimental Considerations Instrumentation and Observation Volume The primary task in an FCS measurement is to record the number of photon counts measured during each of a sequence of fixed time intervals or ‘bins.’ Subtraction of the mean photon count from the counts for each bin yields the time sequence of count fluctuations, the digital representation of the fluctuations of the theoretical continuous fluorescence intensity. The experimental correlation function is calculated from these photon count fluctuations. Fitting the theoretical correlation function, calculated as described above, to the experimental correlation function tests the adequacy of the model and yields values for the model’s parameters. At a minimum an instrument for measuring FCS requires an excitation light source (typically a laser), an optical system to convey the excitation light to the sample and the emitted fluorescence to a detector such as a photomultiplier or avalanche photodiode, and the means to compute the correlation function from record of the fluctuating fluorescence intensity. The optical system is typically a confocal fluorescence microscope (Koppel et al., 1976; Krichevsky and Bonnet, 2002). Very capable and versatile systems for FCS measurements are available commercially from suppliers such as Zeiss, Leica, Olympus, ISS, and others. A critical function of the optical system is to determine the shape of the observation volume or area, i.e., the region of the sample from which emitted fluorescence is detected. This volume is determined by the set of lenses and apertures that shape the excitation beam and by the confocal aperture that determines what emitted photons reach the detector (Koppel et al., 1976; Qian and Elson, 1991). It is common to approximate the observation volume as a threedimensional Gaussian function defined by beam width, w, in the focal plane and a different beam width, wz, along the optical(z)-axis, as defined above. This yields a simple correlation function for diffusion of a single fluorophore component (Aragon and Pecora, 1976): GðτÞ ¼
FCS provides a bridge between population and single-molecule approaches. Simple diffusion measurements are instructive. Because FCS measurements are carried out in very dilute solutions, interactions among diffusing fluorophores are minimal, and so the solution is essentially ideal. As a result a molecule moving in this kind of system correlates only with itself. (In this respect FCS is a single molecule measurement.) It is also worth noting that increasing the sensitivity of
1 1 〈N〉 1 þ τ τD
2
FCS as a Single Molecule Approach
101
wz 2 4D .
1 1 þ τDτ
12 :
z
w As before τD ¼ 4D and τDz ¼ This approximation can lead to inaccurate measurements for observation volumes defined by diffraction limited laser illumination (Hess and Webb, 2002). One way to minimize this effect is to use two-photon excitation (2PE) to define the observation volume (Berland et al., 1995; Schwille et al., 1999). In addition to providing an observation volume that conforms more closely to a threedimensional ellipsoidal Gaussian (Hess and Webb, 2002), 2PE also can penetrate deeper into cells while exposing them to less damaging low wavelength excitation light outside the observation volume.
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Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
Nanophotonic and Super-Resolution Approaches The size of the observation region, defined by the values of w and wz, determine the spatial resolution of an FCS measurement. Because of diffraction in the range of visible light w4B300 250 nm and wz4B500 700 nm. It would be desirable to reduce the values of w and wz to make possible measurements in small compartments such as endocytic vesicles, to reduce background fluorescence, to enable measurements at higher fluorophore concentration, and to diminish the time needed to measure very slow diffusion. For more than 30 years total internal reflection microscopy (TIRFM) has been used to confine the observation region to within very small distances from a glass substratum for both FCS and photobleaching measurements (Thompson et al., 1981; Lieto et al., 2003; Ohsugi et al., 2006; Vobornik et al., 2008). TIRFM does not, however, confine the observation region parallel to the optical axis. Among several methods developed to provide ‘super resolution’ (Toomre and Bewersdorf, 2010), stimulated emission depletion (STED) microscopy holds great promise for FCS measurements (Hell and Wichmann, 1994). The particular advantage of STED microscopy is that provides detection volumes with linear dimensions o50 nm while retaining the ability to make rapid measurements (o0.1 ms). For example, STED FCS measurements have revealed that cholesterol-rich nanodomains (o20 nm) transiently trap proteins anchored to a cell membrane by glycophosphatidylinositol tethers (Eggeling et al., 2009). A general discussion of the application of STED FCS to membrane structure and an extension of the approach to raster scanning have recently appeared (Hedde et al., 2013; Mueller et al., 2013). Further extensions of FCS technology are also possible using nanophotonics (Wenger and Rigneault, 2010).
Errors Systematic errors in FCS measurements are consistent from one measurement to another and can result from faults in the alignment of the optics, anomalies due to photobleaching and photophysical effects such as blinking, optical saturation, and triplet state decay, mischaracterization of the shape and size of the observation volume, detector artifacts, and other sources (Ries and Schwille, 2012). Random errors due to shot noise can be minimized by increasing the rate of fluorescence photon counts acquired. Eventually, however, there will be a point of diminishing returns when increasing excitation intensity causes unacceptable photobleaching of the flurophores. Even if shot noise were negligible and each fluorescence fluctuation could be measured with high accuracy, the accurate determination of the phenomenological rate coefficients would nevertheless require measuring many fluctuations due to the stochastic behavior of the mesoscopic molecular system being observed. Evaluation of the random noise for FCS measurements can be based on theoretical calculation of the variance of the correlation function (Koppel, 1974; Qian, 1990; Kask et al., 1997; Saffarian and Elson, 2003). Another approach is to use multiple measurements to estimate variance empirically (Wohland et al., 2001). An apparently good fit of a measured to a theoretical correlation function guarantees neither that the model correctly represents the experimental system nor that
the derived parameters are accurate. It is important to test the model in as many ways as possible (e.g., Saffarian et al., 2004). It is also important to evaluate systematic and random errors of FCS measurements both to optimize experimental accuracy and to have a quantitative sense of the accuracy of the evaluated phenomenological parameters.
Measurement of Molecular Association and Aggregation FCS and its extensions provide a variety of ways to detect and measure the association of different molecular species with one another and the aggregation of molecules in and on cells. An elementary approach is through the diffusion rate, which depends on the size of the diffusing particle. In a simple solution the diffusion coefficient, D is given by the Einstein relation: D ¼ kBf T , where kB, T, and f are Boltzmann’s constant, the absolute temperature and the frictional coefficient, respectively. For a spherical particle of radius a in a solution of viscosity η, f¼ 6πηa, and so the diffusion coefficient varies only as the cube root of the molecular weight. Then, if two spherical molecules of the same volume combine to form a particle, also regarded as spherical, of twice the volume, the diffusion coefficient will decrease by only B26% (21/3 1). Unfortunately, mainly because diffusion correlation functions even for single components are spread over a wide time range (due to the dependence on
GðτÞ Gð0Þ
¼ 1þ1 τ ), it is difficult to resolve τD
diffusion coefficients that are not very different from one another. It has been shown that even with good data (high photon count rates) it is possible to distinguish two components of comparable brightness by FCS only if their diffusion times differ by at least a factor of 1.6 (Meseth et al., 1999). Thus a fourfold volume/weight change would be required to detect an aggregation or association process. Nevertheless, this approach can be useful for detecting the binding of a small fast diffusing molecule to a larger slow diffusing one. Several other approaches provide more sensitive indications of binding or aggregation.
Molecular Association Measured by Two-Color CrossCorrelation Spectroscopy Fluorescence cross-correlation spectroscopy (FCCS) measures interactions among fluorophores that can be detected at different wavelengths. Suppose that the fluorescence of components A and B can be distinguishably measured at well-separated emission wavelengths λA and λB. The cross correlation function is GAB ðτÞ ¼
〈δFA ðt ÞδFB ðt þ τÞ〉 : 〈FA ðt Þ〉〈FB ðt Þ〉
If A and B molecules move independently of each other, there is no correlation, and so GAB(τ)¼ 0. If, however, A and B are bound together, their fluorescence fluctuations are correlated. Then the amplitude GAB(0) indicates the faction of A and B in complex and can also provide information about the stoichiometry of the complex (Kim et al., 2005). To achieve accurate results, however, there are many practical issues that must be
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
addressed, including optimizing the overlap of the observation volumes for the two fluorescence wavelengths, saturation of the fluorophores at high excitation intensities, cross talk between the detected emission of the two fluorophores that can produce false cross-correlation, and the effects of photobleaching (Bacia et al., 2006; Bacia and Schwille, 2007).
Aggregation Measured by the Photon Count Histogram Formation of molecular aggregates, for example, of triskelions in the formation of coated pits (Pearse and Crowther, 1987), and of polymers such as actin microfilaments and tubulin microtubules (Frieden, 1985; Cooper, 1991), and also intermediate filaments that have tissue-specific monomers (Herrmann and Aebi, 2004), are essential processes in establishing cell structure and function. As a simple formal example, consider the aggregation reaction, nA1 ⇄An . In contrast to the weak dependence of diffusion coefficient on molecular size, the brightness scales linearly with the number of fluorescent particles in an aggregate. The brightness of An, containing n fluorescent molecules (‘monomers’) would be n-fold greater than that of the individual monomers, A1, as long as there were no electronic interactions among the monomers in the aggregate that quenched or enhanced their fluorescence. Therefore, A1 and An can much more readily be distinguished by their brightness than by their diffusion coefficients. As we have seen the brightness of an individual component is readily determined as 〈Q〉¼ 〈F(t)〉G(0). For typical aggregation or polymerization systems, however, assuming that the system contains only A1 or An is a gross oversimplification. Rather there will typically be not only both A1 and An but also intermediate species Aj, j¼ 2…n 1. Therefore one ought to determine the set {〈Nj〉, 〈Qj〉}, the average molecule numbers and brightnesses of all species j in the system. This can be done in principle using the photon count histogram (PCH). The PCH is the probability distribution, P(k), of finding k photons in any time bin. As with the correlation function the PCH can be calculated theoretically for different models of {〈Nj〉, 〈Qj〉} and then fitted to the experimental PCH to determine the optimal values of 〈Nj〉 and 〈Qj〉. There are several practical approaches to comparing experimental data to the model calculations including via the moments of the distribution (Qian and Elson, 1990) and via direct comparison with the PCH (Chen et al., 1999) or its generating function (Kask et al., 1999). PCH analysis has proved useful in a number of applications including among many others, studies of the formation of oligomeric complexes in cells (Chen et al., 2003), the clustering of EGF receptors in the absence of EGF (Saffarian et al., 2007), the formation of virus-like capsids in infected cells (Chen et al., 2009). Although the methods derived from the PCH can in principle yield values of 〈Nj〉 and 〈Qj〉 for systems with many components with different brightnesses and concentrations, in practice there are limits on the numbers of components for which these parameters can be accurately evaluated (Pryse et al., 2012). It appears that one of the underlying factors is a kind of ambiguity that results from the Gaussian laser excitation profile almost universally used in these kinds of measurements. A dim particle passing through the center of the
103
laser beam will contribute a fluorescence pulse that is comparable to a bright particle passing through the dimmer parts of the beam. Computational modeling shows that this apparent ambiguity can be resolved by acquiring a sufficient number of fluctuation measurements but the large number required may be impractical to obtain, especially on labile systems such as cells.
Aggregation by Image Correlation Spectroscopy Imaging methods can measure fluctuation amplitudes that provide information about aggregation of fluorescent cellular constituents. Similar in principle to a time record of fluorescence fluctuations, spatial fluorescence fluctuations can be measured and auto-correlated from a two-dimensional image of a cell. This approach, called Image Correlation Spectroscopy, supplies a two-dimensional spatial fluorescence fluctuation autocorrelation function GICS(x,y) (Petersen et al., 1993). Then, (as for the temporal autocorrelation discussed 1 where 〈N〉 above) for a one component system, GICS ð0,0Þ ¼ 〈N〉 is the average number of particles in the observation area. For a one component aggregation system in which the aggregates contain m ‘monomers,’ the brightness of the aggregate is Qm ¼ GICS(0,0)〈F〉 and so the number of monomers in the aggregate is m ¼ QQm1 where Q1 is the brightness of the monomer. One should note that while FCS measures fluctuations over time and so requires that the fluorescent particles of interest to move through the observation region, ICS is independent of dynamic properties and can measure aggregation of immobile structures. Both static and dynamic properties of aggregates are observable using more complex image correlation methods that include spatiotemporal ICS measurements carried out on a temporal sequence of images (Kolin and Wiseman, 2007). For example, this approach can show the direction and velocity of convective transport of fluorescent particles (Hebert et al., 2005). A related method been cast in a form that readily takes into account fluctuations due to detector noise and autofluorescence to provide the number and brightness of fluorescent particles taken to comprise a onecomponent system (Digman et al., 2008a,b). As an example, this approach has been applied to the determination of the clustering of ErbB1 (the EGF receptor) and ErbB2 in the presence and absence of EGF (Nagy et al., 2010).
Summary 1. FCS can be used to measure both dynamic processes such as diffusion, convection, and chemical reaction kinetics and equilibrium properties such as concentrations and extents of aggregation of fluorescent molecules in very small systems. The laser-illuminated observation region can be of diffraction limited size (femtoliters) providing very high spatial resolution that allows comparison of properties of different regions and compartments of cells. The dynamic range is very broad, from microseconds to seconds. 2. FCS is based on measuring processes in mesoscopic systems, i.e., systems that contain small numbers of the fluorescent molecules of interest. Therefore it is necessary to
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Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
acquire a large number of fluctuations that must be averaged, usually by calculating a fluorescence fluctuation autocorrelation function. The time needed for the acquisition of these fluctuations determines how rapidly the measurements can be accomplished. 3. FCS is optimal for measurements of systems with fluorophores at very low concentration (nanomolar range) and with characteristic times, for example, τD, ranging from microseconds to seconds. Photobleaching recovery is suited to systems with higher concentrations and with times ranging from B100 ms to many seconds. In contrast to the requirement by FCS for many fluctuations to achieve accuracy, a single photobleaching recovery measurement is sufficient to determine rate coefficients to the precision of the measurement of the recovery transient. Therefore, photobleaching is advantageous for measuring on moving cells. Also photobleaching is useful for measuring the time course of exchange reactions (McNally et al., 2000). Nevertheless, FCS can provide information such as absolute fluorophore concentrations and extents of aggregation that are inaccessible to photobleaching recovery. 4. FCS has given rise to a large number of extensions that are useful for specific purposes, for example, two-color FCCS to measure association of fluorescent ‘monomers’ and image correlation spectroscopy to measure aggregation. 5. FCS has been applied to a wide range of cell biological questions, for example, among many others, detection and measurement of receptor clustering (Saffarian et al., 2007; Nagy et al., 2010), probing the endyocytic pathway using two-color cross-correlation spectroscopy (Bacia et al., 2002), measurement of the dynamics of paxillin molecules during cell adhesion (Digman et al., 2008a,b).
Appendix We have provided the conventional definition of the correlation function in terms of chemical concentrations, but it is instructive also to consider an equivalent definition from the perspective of contemporary single molecule studies. We will be concerned with mesoscopic systems in which there are a small number of fluorescent molecules of interest the positions of which fluctuate continuously over time. We therefore adopt a probabilistic point of view: p(r,t) is the probability that a fluorescent particle is at position r at time t. The fluorescence intensity measured from a single molecule can be R represented as FðtÞ ¼ Q IðrÞpðr,tÞd3 r where I(r) is the fluorescence intensity detected from position r, and Q is a brightness parameter, i.e., the emitted intensity measured from a single fluorescent molecule, as defined above. Then, for a system that contains a single fluorescent molecule the normalized and non-normalized correlation functions, G1(τ) and g1(τ), respectively, are G1 ðτÞ ¼
g 1 ðτ Þ 〈F ðt ÞFðt þ τÞ〉 ¼ 〈F ðt Þ〉2 〈F ðt Þ〉2
# jr r0 j2 exp W ðrjr0 ,τÞ ¼ : 4Dτ ð4πDτÞ3=2 1
"
For simplicity we now restrict ourselves to two-dimensional systems; the results are readily extended to three dimensions by taking into account the detected intensity along the z-axis of light propagation. FCS measurements commonly use a Gaussian laser intensity profile, that yields adetection function 2 with r¼ (x,y) represented in the focal plane as Iðr Þ ¼ exp 2r w2 and w, the radius of the Gaussian detection area. This yields, g1 ðτÞ ¼
Q2 V
¼
Q2 V
Z exp Z
2r02 2r 2 exp Wðr r0 ,τÞd2 r0 d2 r 2 2 w w
2r02 2r 2 1 exp exp w2 w2 ð4πDτÞ " # jr r0 j2 2 2 exp d r0 d r: 4Dτ
Here we have used W(r|r0,τ) for a two-dimensional system. Extension of the correlation function for a one-particle system to a system containing Nt fluorescent particles is complicated by the fact that the probability distribution function for N particles is the N-fold convolution of the distributions for single particles. It is useful to avoid convolutions by noting that g1(τ) can be regarded as a moment of a bivariate distribution the two dimensions of which correspond to the fluctuating fluorescence intensity record and the same record delayed by lag time τ (Melnykov and Hall, 2009). Then, one may use the additive properties of cumulants, a related type of moment, so that g1(τ)¼ κ1(τ) where κ1(τ)is a cumulant (Qian and Elson, 1990; Wu et al., 2008; Melnykov and Hall, 2009). Since cumulants are additive, for a system containing N molecules, κN(τ) ¼ Nκ1(τ). This leads to the following expression for the correlation function: Z Nd Q2 2r02 2r 2 1 exp exp gN ðτÞ ¼ Ad w2 w2 ð4πDτÞ " exp
# jr r0 j2 2 2 d r0 d r: 4Dτ
Here, Ad ¼ πw2 is the area from which the fluorescence is detected, and Nd is the number of molecules in that area. (Thus the concentration of the fluorescent particles is C ¼ Nd/Ad.) It can further be shown that carrying out the integrations in this equation yields the expected form of the correlation function for simple diffusion (Elson and Magde, 1974).
Acknowledgments
Z ¼ Q2
In this equation, and W(r|r0,τ) is the probability that the molecule initially at r0 will be at position r at a later time τ. A freely diffusing molecule is equally likely to be anywhere in the sample volume, and so pðr0 ,0Þ ¼ V1 , where V is the total volume of the system. The transition probability for simple diffusion with diffusion coefficient D is
Iðr0 ÞIðr Þpðr0 ,0ÞW ðrjr0 ,τÞd3 r0 d3 r 〈F ðt Þ〉2
This work was financially supported by NIH via grant R01 HL 109505.
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
See also: Imaging the Cell: Light Microscopy: Fluorescence Lifetime Imaging − Applications and Instrumental Principles
References Aragon, S.R., Pecora, R., 1976. Fluorescence correlation spectroscopy as a probe of molecular-dynamics. Journal of Chemical Physics 64 (4), 1791–1803. Bacia, K., Kim, S.A., Schwille, P., 2006. Fluorescence cross-correlation spectroscopy in living cells. Nature Methods 3 (2), 83–89. Bacia, K., Majoul, I.V., Schwille, P., 2002. Probing the endocytic pathway in live cells using dual-color fluorescence cross-correlation analysis. Biophysical Journal 83 (2), 1184–1193. Bacia, K., Schwille, P., 2007. Practical guidelines for dual-color fluorescence crosscorrelation spectroscopy. Nature Protocols 2 (11), 2842–2856. Berland, K.M., So, P.T., Gratton, E., 1995. Two-photon fluorescence correlation spectroscopy: Method and application to the intracellular environment. Biophysical Journal 68 (2), 694–701. Braeckmans, K., Peeters, L., Sanders, N.N., De Smedt, S.C., Demeester, J., 2003. Three-dimensional fluorescence recovery after photobleaching with the confocal scanning laser microscope. Biophysical Journal 85 (4), 2240–2252. Braga, J., Desterro, J.M., Carmo-Fonseca, M., 2004. Intracellular macromolecular mobility measured by fluorescence recovery after photobleaching with confocal laser scanning microscopes. Molecular Biology of the Cell 15 (10), 4749–4760. Chen, Y., Muller, J.D., So, P.T., Gratton, E., 1999. The photon counting histogram in fluorescence fluctuation spectroscopy. Biophysical Journal 77 (1), 553–567. Chen, Y., Wei, L.N., Muller, J.D., 2003. Probing protein oligomerization in living cells with fluorescence fluctuation spectroscopy. Proceedings of the National Academy of Sciences of the United States of America 100 (26), 15492–15497. Chen, Y., Wu, B., Musier-Forsyth, K., Mansky, L.M., Mueller, J.D., 2009. Fluorescence fluctuation spectroscopy on viral-like particles reveals variable gag stoichiometry. Biophysical Journal 96 (5), 1961–1969. Cooper, J.A., 1991. The role of actin polymerization in cell motility. Annual Review of Physiology 53, 585–605. Digman, M.A., Brown, C.M., Horwitz, A.R., Mantulin, W.W., Gratton, E., 2008a. Paxillin dynamics measured during adhesion assembly and disassembly by correlation spectroscopy. Biophysical Journal 94 (7), 2819–2831. Digman, M.A., Dalal, R., Horwitz, A.F., Gratton, E., 2008b. Mapping the number of molecules and brightness in the laser scanning microscope. Biophysical Journal 94 (6), 2320–2332. Digman, M.A., Gratton, E., 2011. Lessons in fluctuation correlation spectroscopy. Annual Review of Physical Chemistry 62, 645–668. Edidin, M., Zagyansky, Y., Lardner, T.J., 1976. Measurement of membrane protein lateral diffusion in single cells. Science 191 (4226), 466–468. Eggeling, C., Ringemann, C., Medda, R., et al., 2009. Direct observation of the nanoscale dynamics of membrane lipids in a living cell. Nature 457 (7233), 1159–1162. Eigen, M., De Maeyer, L.C., 1963. In: Freiss, S.L., Lewis, E.S., Weissberger, A. (Eds.), Techniques of Organic Chemistry, vol. 8, Part II. New York, NY: Interscience, pp. 895. Elson, E., Magde, D., 1974. Fluorescence correlation spectroscopy. I. Conceptual basis and theory. Biopolymers 13, 1–27. Elson, E.L., 1985. Fluorescence correlation spectroscopy and photobleaching recovery. Annual Review of Physical Chemistry 36, 379–406. Elson, E.L., 2011. Fluorescence correlation spectroscopy: Past, present, future. Biophysical Journal 101 (12), 2855–2870. Elson, E.L., 2013a. 40 Years of FCS: How it all began. In: Tetin, S.Y. (Ed.), Methods in Enzymology, vol. 518. Burlington: Elsevier (Academic Press), pp. 1–10. Elson, E.L., 2013b. Brief introduction to fluorescence correlation spectroscopy. In: Tetin, S.Y. (Ed.), Methods in Enzymology, vol. 518. Burlington: Elsevier (Academic Press), pp. 11–41. Frieden, C., 1985. Actin and tubulin polymerization: The use of kinetic methods to determine mechanism. Annual Review of Biophysics and Biomolecular Structure 14, 189–210. Hebert, B., Costantino, S., Wiseman, P.W., 2005. Spatiotemporal image correlation spectroscopy (STICS) theory, verification, and application to protein velocity mapping in living CHO cells. Biophysical Journal 88 (5), 3601–3614.
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Hedde, P.N., Dörlich, R.M., Blomley, R., et al., 2013. Stimulated emission depletionbased raster image correlation spectroscopy reveals biomolecular dynamics in live cells. Nature Communications 4, 2093. Hell, S.W., Wichmann, J., 1994. Breaking the diffraction resolution limit by stimulated emission: Stimulated-emission-depletion fluorescence microscopy. Optics Letters 19 (11), 780–782. Herrmann, H., Aebi, U., 2004. Intermediate filaments: Molecular structure, assembly mechanism, and integration into functionally distinct intracellular Scaffolds. Annual Review of Biochemistry 73, 749–789. Hess, S.T., Webb, W.W., 2002. Focal volume optics and experimental artifacts in confocal fluorescence correlation spectroscopy. Biophysical Journal 83 (4), 2300–2317. Kang, M., Day, C.A., Kenworthy, A.K., DiBenedetto, E., 2012. Simplified equation to extract diffusion coefficients from confocal FRAP data. Traffic 13 (12), 1589–1600. Kask, P., Gunther, R., Axhausen, P., 1997. Statistical accuracy in fluorescence fluctuations experiments. European Biophysics Journal with Biophysics Letters 25, 163–169. Kask, P., Palo, K., Ullmann, D., Gall, K., 1999. Fluorescence-intensity distribution analysis and its application in biomolecular detection technology. Proceedings of the National Academy of Sciences of the United States of America 96 (24), 13756–13761. Kim, S.A., Heinze, K.G., Bacia, K., Waxham, M.N., Schwille, P., 2005. Two-photon cross-correlation analysis of intracellular reactions with variable stoichiometry. Biophysical Journal 88 (6), 4319–4336. Kolin, D.L., Wiseman, P.W., 2007. Advances in image correlation spectroscopy: Measuring number densities, aggregation states, and dynamics of fluorescently labeled macromolecules in cells. Cell Biochemistry and Biophysics 49 (3), 141–164. Koppel, D., 1974. Statistical accuracy in fluorescence correlation spectroscopy. Physical Review A 10, 1938–1945. Koppel, D.E., Axelrod, D., Schlessinger, J., Elson, E.L., Webb, W.W., 1976. Dynamics of fluorescence marker concentration as a probe of mobility. Biophysical Journal 16 (11), 1315–1329. Krichevsky, O., Bonnet, G., 2002. Fluorescence correlation spectroscopy: The technique and its applications. Reports on Progress in Physics 65 (2), 251–297. Lieto, A.M., Cush, R.C., Thompson, N.L., 2003. Ligand-receptor kinetics measured by total internal reflection with fluorescence correlation spectroscopy. Biophysical Journal 85 (5), 3294–3302. Magde, D., Elson, E.L., Webb, W.W., 1972. Thermodynamic fluctuations in a reacting system − Measurement by fluorescence correlation spectroscopy. Physical Review Letters 29, 705–708. McNally, J.G., Muller, W.G., Walker, D., Wolford, R., Hager, G.L., 2000. The glucocorticoid receptor: Rapid exchange with regulatory sites in living cells. Science 287 (5456), 1262–1265. Melnykov, A.V., Hall, K.B., 2009. Revival of high-order fluorescence correlation analysis: Generalized theory and biochemical applications. Journal of Physical Chemistry B 113 (47), 15629–15638. Meseth, U., Wohland, T., Rigler, R., Vogel, H., 1999. Resolution of fluorescence correlation measurements. Biophysical Journal 76 (3), 1619–1631. Mets, U., Rigler, R., 1994. Submillisecond detection of single rhodamine molecules in water. Journal of Fluorescence 4 (3), 259–264. Mueller, V., Honigmann, A., Ringemann, C., et al., 2013. FCS in STED microscopy: Studying the nanoscale of lipid membrane dynamics. Methods in Enzymology 519, 1–38. Nagy, P., Claus, J., Jovin, T.M., Arndt-Jovin, D.J., 2010. Distribution of resting and ligand-bound ErbB1 and ErbB2 receptor tyrosine kinases in living cells using number and brightness analysis. Proceedings of the National Academy of Sciences of the United States of America 107 (38), 16524–16529. Ohsugi, Y., Saito, K., Tamura, M., Kinjo, M., 2006. Lateral mobility of membranebinding proteins in living cells measured by total internal reflection fluorescence correlation spectroscopy. Biophysical Journal 91 (9), 3456–3464. Onsager, L., 1931. Reciprocal relations in irreversible processes. I..Physical Review 37, 405–426. Pearse, B.M., Crowther, R.A., 1987. Structure and assembly of coated vesicles. Annual Review of Biophysics and Biomolecular Structure 16, 49–68. Peters, R., Peters, J., Tews, K.H., Bahr, W., 1974. A microfluorimetric study of translational diffusion in erythrocyte membranes. Biochimica et Biophysica Acta 367 (3), 282–294. Petersen, N.O., Hoddelius, P.L., Wiseman, P.W., Seger, O., Magnusson, K.E., 1993. Quantitation of membrane receptor distributions by image correlation spectroscopy: Concept and application. Biophysical Journal 65 (3), 1135–1146.
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Pryse, K.M., Rong, X., Whisler, J.A., et al., 2012. Confidence intervals for concentration and brightness from fluorescence fluctuation measurements. Biophysical Journal 103 (5), 898–906. Qian, H., 1990. On the statistics of fluorescence correlation spectroscopy. Biophysical Chemistry 38, 49–57. Qian, H., Elson, E.L., 1990. On the analysis of high order moments of fluorescence fluctuations. Biophysical Journal 57 (2), 375–380. Qian, H., Elson, E.L., 1991. Analysis of confocal laser-microscope optics for 3 − Fluorescence correlation spectroscopy. Applied Optics 30, 1185–1195. Ries, J., Schwille, P., 2012. Fluorescence correlation spectroscopy. BioEssays 34 (5), 361–368. Rigler, R., 1995. Fluorescence correlations, single molecule detection and large number screening. Applications in biotechnology. Journal of Biotechnology 41 (2−3), 177–186. Rigler, R., 2009. FCS and single molecule spectroscopy. In: Graslund, A., Rigler, R., Widengren, J. (Eds.), Single Molecule Spectroscopy in Chemistry, Physics, and Biology. Heidleberg: Springer-Verlag, pp. 77–103. Rigler, R., Mets, U., Widengren, J., Kask, P., 1993. Fluorescence correlation spectroscopy with high count rate and low background: Analysis of translational diffusion. European Biophysics Journal 22, 169–175. Saffarian, S., Collier, I.E., Marmer, B.L., Elson, E.L., Goldberg, G., 2004. Interstitial collagenase is a Brownian ratchet driven by proteolysis of collagen. Science 306 (5693), 108–111. Saffarian, S., Elson, E.L., 2003. Statistical analysis of fluorescence correlation spectroscopy: The standard deviation and bias. Biophysical Journal 84 (3), 2030–2042. Saffarian, S., Li, Y., Elson, E.L., Pike, L.J., 2007. Oligomerization of the EGF receptor investigated by live cell fluorescence intensity distribution analysis. Biophysical Journal 93 (3), 1021–1031. Schatzel, K., Drewel, M., Stimac, S., 1988. Photon-correlation measurements at large lag times − Improving statistical accuracy. Journal of Modern Optics 35 (4), 711–718.
Schlessinger, J., Koppel, D.E., Axelrod, D., et al., 1976a. Lateral transport on cell membranes: Mobility of concanavalin A receptors on myoblasts. Proceedings of the National Academy of Sciences of the United States of America 73 (7), 2409–2413. Schlessinger, J., Webb, W.W., Elson, E.L., Metzger, H., 1976b. Lateral motion and valence of Fc receptors on rat peritoneal mast cells. Nature 264 (5586), 550–552. Schwille, P., Haupts, U., Maiti, S., Webb, W.W., 1999. Molecular dynamics in living cells observed by fluorescence correlation spectroscopy with one- and twophoton excitation. Biophysical Journal 77 (4), 2251–2265. Seiffert, S., Oppermann, W., 2005. Systematic evaluation of FRAP experiments performed in a confocal laser scanning microscope. Journal of Microscopy 220 (Pt 1), 20–30. Tanford, C., 1961. Physical Chemistry of Macromolecules. New York, NY: John Wiley & Sons. Thompson, N.L., Burghardt, T.P., Axelrod, D., 1981. Measuring surface dynamics of biomolecules by total internal reflection fluorescence with photobleaching recovery or correlation spectroscopy. Biophysical Journal 33 (3), 435–454. Toomre, D., Bewersdorf, J., 2010. A new wave of cellular imaging. Annual Review of Cell and Developmental Biology 26, 285–314. Vobornik, D., Banks, D.S., Lu, Z.F., et al., 2008. Fluorescence correlation spectroscopy with sub-diffraction-limited resolution using near-field optical probes. Applied Physics Letters 93 (16), 163904. Wenger, J., Rigneault, H., 2010. Photonic methods to enhance fluorescence correlation spectroscopy and single molecule fluorescence detection. International Journal of Molecular Sciences 11 (1), 206–221. Wohland, T., Rigler, R., Vogel, H., 2001. The standard deviation in fluorescence correlation spectroscopy. Biophysical Journal 80 (6), 2987–2999. Wu, B., Chen, Y., Muller, J.D., 2008. Fluorescence correlation spectroscopy of finite-sized particles. Biophysical Journal 94 (7), 2800–2808.
Glossary Observation volume or area The open volume (or area for two-dimensional systems) that is illuminated by the
Introduction To understand dynamic molecular processes of metabolism and regulation at the cellular level it is necessary to determine how specific molecules move and interact on the surfaces and within living cells. To accomplish this task requires methods that have the following characteristics: 1. Chemical selectivity – the ability to measure the concentrations of specified molecules present at low concentration within an environment, for example, cytoplasm, that contains high concentrations of many other kinds of molecules 2. A wide dynamic range from omicroseconds to 4minutes 3. Relatively short time required for the measurements (seconds) 4. High spatial resolution to enable measurements within dimensions o1 mm 5. Sensitivity, even down to the single molecule level 6. Lack of toxicity and interference with normal cell function 7. Ability to localize the measurement within identifiable cell structures and features 8. Applicability to a range of concentrations from nM to mM. These specifications suggest the use of fluorescence, which has the required rapidity of measurement, sensitivity, and selectivity, and, when based on a confocal fluorescence microscope, the spatial resolution and ability to localize the measurement relative to cell structures. Furthermore, when used judiciously, fluorescence of labeled molecules can be measured with little damage to living cells. Two closely related and complementary methods, fluorescence correlation spectroscopy (FCS) and fluorescence photobleaching recovery (FPR, also known as fluorescence recovery after photobleaching, FRAP) take advantage of these properties of fluorescence to measure rates of molecular transport, for example, diffusion and convection, and of chemical reaction over a wide dynamic range of concentrations and time in living cells. FCS was introduced in 1972 (Magde et al., 1972). The history of its origins and development have been related from the perspectives of the two groups that first demonstrated the feasibility of using fluorescence fluctuations to measure translational diffusion and chemical kinetics (Elson, 2013a,b) and rotational motion (Rigler, 2009). FCS remained a specialized subject of study for a number of years until technological advances enabled more rapid and robust measurements and this in turn led to the production of
Encyclopedia of Cell Biology, Volume 2
doi:10.1016/B978-0-12-394447-4.20094-1
excitation laser and from which fluorescence is detected. The observation volume is typically limited by a confocal aperture in the microscope optical system.
commercial instruments by a number of companies and widespread use of FCS in many laboratories. Important improvements in FCS measurements resulted from using confocal microscopy to limit the volume from which fluorescence is detected (Koppel et al., 1976; Rigler et al., 1993) and which allowed single molecule detection (Mets and Rigler, 1994; Rigler, 2009). Improvements in microscope technology (infinity-corrected optics) and stability of fluorophores and computation of correlation functions (Schatzel et al., 1988) also were important contributions. Improvements in microscopy minimized the detection volume thereby diminishing interfering background fluorescence while increasing the detected number of photons emitted per fluorphore.
Basic Concepts and Theory Correlation of Fluctuations of Concentration and Fluorescence Most conventional methods to measure chemical rate constants determine their values by perturbing the molecular system from its equilibrium state, for example, by a temperature jump that displaces the system from equilibrium, and then observing the rate of relaxation back to equilibrium (Eigen and De Maeyer, 1963). Similarly, conventional measurements of diffusion are typically carried out by observing the dissipation of a concentration gradient that has been created in the system by the experimenter (Tanford, 1961). In contrast, FCS measures these rates without perturbing systems that remain in equilibrium or nonequilibrium steady states during the measurement. In an open volume of a chemical system defined by a focused laser beam, the numbers of molecules of the system components continually fluctuate due either to their diffusing into and out of the volume or to their participating in chemical reactions. The fluctuation of the concentration of chemical component, j, at position r and time t is δCj(r,t)¼ Cj(r,t) 〈Cj〉 where 〈Cj〉 is the equilibrium (or nonequilibrium) steady state value of the concentration. The concentration fluctuations, δCj(r,t), causes corresponding fluorescence fluctuations, δFj(t), that are measured directly, typically using a confocal fluorescence microscope (Koppel et al., 1976; Qian and Elson, 1991). A fluorescence fluctuation is related to a concentration fluctuation by adding the contributions from different locations in the observation region weighted by the excitation intensity,
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R IðrÞ : δFj ðtÞ ¼ Qj IðrÞδCj ðr,tÞdr, where Qj is a constant that accounts for the absorption coefficient and fluorescence quantum yield of the fluorophore (component j) and properties of the optical measurement system. The durations of the fluctuations depend on the size of the volume and the diffusion rate as well as on the chemical reaction rate(s) and for nonlinear chemical reactions, the concentrations of the reactants. It is not sufficient, however, simply to measure one or a few fluctuations to derive the diffusion coefficients and rate constants because the fluctuations are stochastic. Each fluctuation can differ substantially from the others. Therefore, it is necessary to carry out a statistical analysis of the time courses of many fluctuations to obtain accurate values of transport and reaction rate parameters. The statistical analysis of the fluorescence fluctuations is most often carried out by calculating a fluorescence fluctuation autocorrelation function: Gjj ðτÞ ¼
〈δFj ðt ÞδFj ðt þ τÞ〉 〈Fj ðtÞ2 〉
dimensional system, for example, 〈Nj〉¼ πw2〈Cj〉, the average number of fluorescent molecules of the j’th chemical species in the observation area πw2 (Elson, 2013a,b). This property of G(0) provides an absolute measurement of the concentration of the fluorescent component and also the ‘brightness’ 〈F 〉 〈Qj 〉 ¼ 〈Njj 〉 ¼ Gð0Þ〈F〉, the average number of fluorescence photons detected per unit time per fluorescent molecule. Many reviews and primary articles provide more details about the forms of correlation functions for different dynamic processes (e.g., Elson and Magde, 1974; Digman and Gratton, 2011; Elson, 2011, 2013a,b; Ries and Schwille, 2012). In systems with M fluorescent components, the contributions of all the fluorophores must be included in the correlation function. The total fluorescence fluctuation is δF ðt Þ ¼ ∑M j ¼ 1 δFj ðt Þ and the correlation function is GðτÞ ¼
,
where 〈…〉 denotes an average over many time points and the division by 〈Fj(t)〉2 yields an expression independent of the excitation laser power and fluorophore brightness for a single component system. The fluorescence fluctuation at some time t is multiplied by a fluctuation at a later time t þ τ. Averaging many of these products obtained for different times, t, yields an indication of the extent to which the fluorescence/concentration fluctuations have relaxed toward the steady state value for the system during the delay time τ. Then, carrying out this operation for a range of τ values gives a functional form for Gjj(τ) that is characteristic for specific transport processes and chemical reactions (Elson, 2011). Up to now the fluorescence fluctuations and the fluctuation autocorrelation function has been discussed from a conventional chemical perspective. It may also be of interest to frame the discussion from the point of view of contemporary single molecule studies. This discussion is presented in the Appendix. The final results of the derivations, whether from a chemical or a single molecule perspective, must, of course, yield the same equations for the fluorescence fluctuation autocorrelation function. The form of Gjj(τ) also depends on the ‘shape’ of the volume or area from which the fluorescence is detected. This is frequently approximated as a Gaussian function with a characteristic width, w, in the focal (x ,y ) plane and a width wz h i 2 2 2 Þ expðw2zz 2 Þ. As along the optical (z ) axis: I ¼ I0 exp 2ðxwþy 2 an example, for fluorescent compound j with diffusion coefficient Dj that is diffusing in a two-dimensional plane, for example, a biological membrane: Gjj ð0Þ w2 ; the characteristic time is τDj ¼ : Gjj ðτÞ ¼ 1 þ τDτ 4Dj j
For chemical reaction kinetics Gjj(τ) can have a number of exponential relaxation components. The amplitude of the correlation function depends only on the concentration and size of the observation area/volume defined by the focused laser. The relative size of the fluctuations is determined by a Poisson distribution that governs the number of molecules in an open volume. Thus, Gjj ð0Þ ¼ 〈N1j 〉, where for a two-
M ∑M j ¼ 1 ∑k ¼ 1 〈δFj ðtÞδFk ðt þ τÞ〉 : 2 ∑M j ¼ 1 〈Fj ðtÞ〉
For the simple example of a multicomponent system in which the only dynamic process is the independent diffusion of noninteracting molecules 〈δFj(0)δFk(τ)〉¼ 0 for jak. Then, Q2j 〈Cj 〉
∑M j¼1 GðτÞ ¼
1þτ τ
Dj
2 : πw2 ∑m J ¼ 1 Qj 〈Cj 〉
If system components interact, as in chemical reactions, the situation is more complicated, especially if the kinetics of the reactions must be taken into account (Elson and Magde, kf 1974). For example, for the simplest reaction, A ⟶ B, in a ⟵ kb
closed system the chemical relaxation can be described in simple terms δCB(t) ¼ δCB(0)exp( Rt) ¼ δCA(t), where R¼ kf þ kb. In contrast for an FCS measurement in the open observation region, the kinetics must be analyzed as a two component coupled reaction-diffusion system because the A and B molecules can independently diffuse into and out of the observation region (Elson and Magde, 1974). Although the reaction–diffusion systems are more complicated to analyze, there are standard methods for solving the equations describing them, and so the mathematic analysis does not present a fundamental problem. More important for measuring chemical kinetics by FCS is to find a substantial change in fluorescence that indicates the reaction progress.
Relationship between FCS and Fluorescence Photobleaching Measurements Both the interpretive theory (Elson, 1985) and the implementation of measurements on a confocal fluorescence microscope (Koppel et al., 1976) are very similar for FCS and fluorescence photobleaching. The main difference in principle between the two kinds of measurements is that FCS relies on spontaneous microscopic concentration (fluorescence) fluctuations in mesoscopic systems, i.e., systems in which there are so few molecules that the concentration fluctuations are readily measured compared to the mean concentrations. In contrast, photobleaching recovery uses a pulse of intense light to photolyze fluorophores irreversibly within a small
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
observation volume. This produces a localized macroscopic concentration gradient, i.e., a relatively large difference between the concentration in the observation volume and the mean concentration in the surrounding regions of the system. The measured rate of relaxation of the concentration in the observation volume back to its equilibrium value due to diffusion or chemical reaction then provides the desired kinetic parameters. For FCS many stochastic fluorescence fluctuations, no matter how accurately each is measured, must be analyzed statistically via the correlation function to obtain accurate values of phenomenological diffusion coefficients and rate constants. In contrast, in a photobleaching measurement, a single macroscopic relaxation is sufficient to provide the phenomenological coefficients to an accuracy determined by the accuracy of the single measurement. The theoretical basis for supposing that the coefficients determined by each approach are the same is Onsager’s regression hypothesis (Onsager, 1931). During its early days FCS was difficult to use to measure processes in cells because the signals were relatively small and could be easily corrupted by cellular motions and other changes of cell properties during the time required to acquire a sufficient number of fluorescence fluctuations. Photobleaching was then more useful for measurements on live cells (e.g., Peters et al., 1974; Edidin et al., 1976; Schlessinger et al., 1976a,b). For the last two decades, however, due to technological advances, especially the use of confocal microscopy to minimize the observation volume (Koppel et al., 1976; Rigler, 1995), FCS has also proved to be useful for studies in living cells. Initial FRAP measurements were carried out using a nonscanning confocal microscope. Fluorescence emission was measured from a single diffraction-limited spot in the sample defined by the incident laser beam and the confocal aperture. The duration of the bleach pulse, controlled by a fast mechanical or acousto-optical shutter could be very brief (Bo1 ms) and data could be essentially continually recorded from the single spot (Koppel et al., 1976). More recently confocal laser scanning microscopes have been used for photobleaching measurements (Braga et al., 2004; Seiffert and Oppermann, 2005; Kang et al., 2012). Although this approach has the advantages of availability on a commercially available platform and of the possibility of bleaching and acquiring recovery data from arbitrarily shaped observation regions, very fast processes are less accessible by this approach due to its dependence on the scan speed of the microscope. Moreover, the interpretation of the recovery can be complicated by the need to account for the scanning rate of the microscope (e.g., Braeckmans et al., 2003).
measurements to the level of detecting a single fluorophore by minimizing the observation volume (and therefore the interfering background fluorescence) was a major step in facilitating FCS measurements (Mets and Rigler, 1994; Rigler, 2009).
Experimental Considerations Instrumentation and Observation Volume The primary task in an FCS measurement is to record the number of photon counts measured during each of a sequence of fixed time intervals or ‘bins.’ Subtraction of the mean photon count from the counts for each bin yields the time sequence of count fluctuations, the digital representation of the fluctuations of the theoretical continuous fluorescence intensity. The experimental correlation function is calculated from these photon count fluctuations. Fitting the theoretical correlation function, calculated as described above, to the experimental correlation function tests the adequacy of the model and yields values for the model’s parameters. At a minimum an instrument for measuring FCS requires an excitation light source (typically a laser), an optical system to convey the excitation light to the sample and the emitted fluorescence to a detector such as a photomultiplier or avalanche photodiode, and the means to compute the correlation function from record of the fluctuating fluorescence intensity. The optical system is typically a confocal fluorescence microscope (Koppel et al., 1976; Krichevsky and Bonnet, 2002). Very capable and versatile systems for FCS measurements are available commercially from suppliers such as Zeiss, Leica, Olympus, ISS, and others. A critical function of the optical system is to determine the shape of the observation volume or area, i.e., the region of the sample from which emitted fluorescence is detected. This volume is determined by the set of lenses and apertures that shape the excitation beam and by the confocal aperture that determines what emitted photons reach the detector (Koppel et al., 1976; Qian and Elson, 1991). It is common to approximate the observation volume as a threedimensional Gaussian function defined by beam width, w, in the focal plane and a different beam width, wz, along the optical(z)-axis, as defined above. This yields a simple correlation function for diffusion of a single fluorophore component (Aragon and Pecora, 1976): GðτÞ ¼
FCS provides a bridge between population and single-molecule approaches. Simple diffusion measurements are instructive. Because FCS measurements are carried out in very dilute solutions, interactions among diffusing fluorophores are minimal, and so the solution is essentially ideal. As a result a molecule moving in this kind of system correlates only with itself. (In this respect FCS is a single molecule measurement.) It is also worth noting that increasing the sensitivity of
1 1 〈N〉 1 þ τ τD
2
FCS as a Single Molecule Approach
101
wz 2 4D .
1 1 þ τDτ
12 :
z
w As before τD ¼ 4D and τDz ¼ This approximation can lead to inaccurate measurements for observation volumes defined by diffraction limited laser illumination (Hess and Webb, 2002). One way to minimize this effect is to use two-photon excitation (2PE) to define the observation volume (Berland et al., 1995; Schwille et al., 1999). In addition to providing an observation volume that conforms more closely to a threedimensional ellipsoidal Gaussian (Hess and Webb, 2002), 2PE also can penetrate deeper into cells while exposing them to less damaging low wavelength excitation light outside the observation volume.
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Nanophotonic and Super-Resolution Approaches The size of the observation region, defined by the values of w and wz, determine the spatial resolution of an FCS measurement. Because of diffraction in the range of visible light w4B300 250 nm and wz4B500 700 nm. It would be desirable to reduce the values of w and wz to make possible measurements in small compartments such as endocytic vesicles, to reduce background fluorescence, to enable measurements at higher fluorophore concentration, and to diminish the time needed to measure very slow diffusion. For more than 30 years total internal reflection microscopy (TIRFM) has been used to confine the observation region to within very small distances from a glass substratum for both FCS and photobleaching measurements (Thompson et al., 1981; Lieto et al., 2003; Ohsugi et al., 2006; Vobornik et al., 2008). TIRFM does not, however, confine the observation region parallel to the optical axis. Among several methods developed to provide ‘super resolution’ (Toomre and Bewersdorf, 2010), stimulated emission depletion (STED) microscopy holds great promise for FCS measurements (Hell and Wichmann, 1994). The particular advantage of STED microscopy is that provides detection volumes with linear dimensions o50 nm while retaining the ability to make rapid measurements (o0.1 ms). For example, STED FCS measurements have revealed that cholesterol-rich nanodomains (o20 nm) transiently trap proteins anchored to a cell membrane by glycophosphatidylinositol tethers (Eggeling et al., 2009). A general discussion of the application of STED FCS to membrane structure and an extension of the approach to raster scanning have recently appeared (Hedde et al., 2013; Mueller et al., 2013). Further extensions of FCS technology are also possible using nanophotonics (Wenger and Rigneault, 2010).
Errors Systematic errors in FCS measurements are consistent from one measurement to another and can result from faults in the alignment of the optics, anomalies due to photobleaching and photophysical effects such as blinking, optical saturation, and triplet state decay, mischaracterization of the shape and size of the observation volume, detector artifacts, and other sources (Ries and Schwille, 2012). Random errors due to shot noise can be minimized by increasing the rate of fluorescence photon counts acquired. Eventually, however, there will be a point of diminishing returns when increasing excitation intensity causes unacceptable photobleaching of the flurophores. Even if shot noise were negligible and each fluorescence fluctuation could be measured with high accuracy, the accurate determination of the phenomenological rate coefficients would nevertheless require measuring many fluctuations due to the stochastic behavior of the mesoscopic molecular system being observed. Evaluation of the random noise for FCS measurements can be based on theoretical calculation of the variance of the correlation function (Koppel, 1974; Qian, 1990; Kask et al., 1997; Saffarian and Elson, 2003). Another approach is to use multiple measurements to estimate variance empirically (Wohland et al., 2001). An apparently good fit of a measured to a theoretical correlation function guarantees neither that the model correctly represents the experimental system nor that
the derived parameters are accurate. It is important to test the model in as many ways as possible (e.g., Saffarian et al., 2004). It is also important to evaluate systematic and random errors of FCS measurements both to optimize experimental accuracy and to have a quantitative sense of the accuracy of the evaluated phenomenological parameters.
Measurement of Molecular Association and Aggregation FCS and its extensions provide a variety of ways to detect and measure the association of different molecular species with one another and the aggregation of molecules in and on cells. An elementary approach is through the diffusion rate, which depends on the size of the diffusing particle. In a simple solution the diffusion coefficient, D is given by the Einstein relation: D ¼ kBf T , where kB, T, and f are Boltzmann’s constant, the absolute temperature and the frictional coefficient, respectively. For a spherical particle of radius a in a solution of viscosity η, f¼ 6πηa, and so the diffusion coefficient varies only as the cube root of the molecular weight. Then, if two spherical molecules of the same volume combine to form a particle, also regarded as spherical, of twice the volume, the diffusion coefficient will decrease by only B26% (21/3 1). Unfortunately, mainly because diffusion correlation functions even for single components are spread over a wide time range (due to the dependence on
GðτÞ Gð0Þ
¼ 1þ1 τ ), it is difficult to resolve τD
diffusion coefficients that are not very different from one another. It has been shown that even with good data (high photon count rates) it is possible to distinguish two components of comparable brightness by FCS only if their diffusion times differ by at least a factor of 1.6 (Meseth et al., 1999). Thus a fourfold volume/weight change would be required to detect an aggregation or association process. Nevertheless, this approach can be useful for detecting the binding of a small fast diffusing molecule to a larger slow diffusing one. Several other approaches provide more sensitive indications of binding or aggregation.
Molecular Association Measured by Two-Color CrossCorrelation Spectroscopy Fluorescence cross-correlation spectroscopy (FCCS) measures interactions among fluorophores that can be detected at different wavelengths. Suppose that the fluorescence of components A and B can be distinguishably measured at well-separated emission wavelengths λA and λB. The cross correlation function is GAB ðτÞ ¼
〈δFA ðt ÞδFB ðt þ τÞ〉 : 〈FA ðt Þ〉〈FB ðt Þ〉
If A and B molecules move independently of each other, there is no correlation, and so GAB(τ)¼ 0. If, however, A and B are bound together, their fluorescence fluctuations are correlated. Then the amplitude GAB(0) indicates the faction of A and B in complex and can also provide information about the stoichiometry of the complex (Kim et al., 2005). To achieve accurate results, however, there are many practical issues that must be
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
addressed, including optimizing the overlap of the observation volumes for the two fluorescence wavelengths, saturation of the fluorophores at high excitation intensities, cross talk between the detected emission of the two fluorophores that can produce false cross-correlation, and the effects of photobleaching (Bacia et al., 2006; Bacia and Schwille, 2007).
Aggregation Measured by the Photon Count Histogram Formation of molecular aggregates, for example, of triskelions in the formation of coated pits (Pearse and Crowther, 1987), and of polymers such as actin microfilaments and tubulin microtubules (Frieden, 1985; Cooper, 1991), and also intermediate filaments that have tissue-specific monomers (Herrmann and Aebi, 2004), are essential processes in establishing cell structure and function. As a simple formal example, consider the aggregation reaction, nA1 ⇄An . In contrast to the weak dependence of diffusion coefficient on molecular size, the brightness scales linearly with the number of fluorescent particles in an aggregate. The brightness of An, containing n fluorescent molecules (‘monomers’) would be n-fold greater than that of the individual monomers, A1, as long as there were no electronic interactions among the monomers in the aggregate that quenched or enhanced their fluorescence. Therefore, A1 and An can much more readily be distinguished by their brightness than by their diffusion coefficients. As we have seen the brightness of an individual component is readily determined as 〈Q〉¼ 〈F(t)〉G(0). For typical aggregation or polymerization systems, however, assuming that the system contains only A1 or An is a gross oversimplification. Rather there will typically be not only both A1 and An but also intermediate species Aj, j¼ 2…n 1. Therefore one ought to determine the set {〈Nj〉, 〈Qj〉}, the average molecule numbers and brightnesses of all species j in the system. This can be done in principle using the photon count histogram (PCH). The PCH is the probability distribution, P(k), of finding k photons in any time bin. As with the correlation function the PCH can be calculated theoretically for different models of {〈Nj〉, 〈Qj〉} and then fitted to the experimental PCH to determine the optimal values of 〈Nj〉 and 〈Qj〉. There are several practical approaches to comparing experimental data to the model calculations including via the moments of the distribution (Qian and Elson, 1990) and via direct comparison with the PCH (Chen et al., 1999) or its generating function (Kask et al., 1999). PCH analysis has proved useful in a number of applications including among many others, studies of the formation of oligomeric complexes in cells (Chen et al., 2003), the clustering of EGF receptors in the absence of EGF (Saffarian et al., 2007), the formation of virus-like capsids in infected cells (Chen et al., 2009). Although the methods derived from the PCH can in principle yield values of 〈Nj〉 and 〈Qj〉 for systems with many components with different brightnesses and concentrations, in practice there are limits on the numbers of components for which these parameters can be accurately evaluated (Pryse et al., 2012). It appears that one of the underlying factors is a kind of ambiguity that results from the Gaussian laser excitation profile almost universally used in these kinds of measurements. A dim particle passing through the center of the
103
laser beam will contribute a fluorescence pulse that is comparable to a bright particle passing through the dimmer parts of the beam. Computational modeling shows that this apparent ambiguity can be resolved by acquiring a sufficient number of fluctuation measurements but the large number required may be impractical to obtain, especially on labile systems such as cells.
Aggregation by Image Correlation Spectroscopy Imaging methods can measure fluctuation amplitudes that provide information about aggregation of fluorescent cellular constituents. Similar in principle to a time record of fluorescence fluctuations, spatial fluorescence fluctuations can be measured and auto-correlated from a two-dimensional image of a cell. This approach, called Image Correlation Spectroscopy, supplies a two-dimensional spatial fluorescence fluctuation autocorrelation function GICS(x,y) (Petersen et al., 1993). Then, (as for the temporal autocorrelation discussed 1 where 〈N〉 above) for a one component system, GICS ð0,0Þ ¼ 〈N〉 is the average number of particles in the observation area. For a one component aggregation system in which the aggregates contain m ‘monomers,’ the brightness of the aggregate is Qm ¼ GICS(0,0)〈F〉 and so the number of monomers in the aggregate is m ¼ QQm1 where Q1 is the brightness of the monomer. One should note that while FCS measures fluctuations over time and so requires that the fluorescent particles of interest to move through the observation region, ICS is independent of dynamic properties and can measure aggregation of immobile structures. Both static and dynamic properties of aggregates are observable using more complex image correlation methods that include spatiotemporal ICS measurements carried out on a temporal sequence of images (Kolin and Wiseman, 2007). For example, this approach can show the direction and velocity of convective transport of fluorescent particles (Hebert et al., 2005). A related method been cast in a form that readily takes into account fluctuations due to detector noise and autofluorescence to provide the number and brightness of fluorescent particles taken to comprise a onecomponent system (Digman et al., 2008a,b). As an example, this approach has been applied to the determination of the clustering of ErbB1 (the EGF receptor) and ErbB2 in the presence and absence of EGF (Nagy et al., 2010).
Summary 1. FCS can be used to measure both dynamic processes such as diffusion, convection, and chemical reaction kinetics and equilibrium properties such as concentrations and extents of aggregation of fluorescent molecules in very small systems. The laser-illuminated observation region can be of diffraction limited size (femtoliters) providing very high spatial resolution that allows comparison of properties of different regions and compartments of cells. The dynamic range is very broad, from microseconds to seconds. 2. FCS is based on measuring processes in mesoscopic systems, i.e., systems that contain small numbers of the fluorescent molecules of interest. Therefore it is necessary to
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Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
acquire a large number of fluctuations that must be averaged, usually by calculating a fluorescence fluctuation autocorrelation function. The time needed for the acquisition of these fluctuations determines how rapidly the measurements can be accomplished. 3. FCS is optimal for measurements of systems with fluorophores at very low concentration (nanomolar range) and with characteristic times, for example, τD, ranging from microseconds to seconds. Photobleaching recovery is suited to systems with higher concentrations and with times ranging from B100 ms to many seconds. In contrast to the requirement by FCS for many fluctuations to achieve accuracy, a single photobleaching recovery measurement is sufficient to determine rate coefficients to the precision of the measurement of the recovery transient. Therefore, photobleaching is advantageous for measuring on moving cells. Also photobleaching is useful for measuring the time course of exchange reactions (McNally et al., 2000). Nevertheless, FCS can provide information such as absolute fluorophore concentrations and extents of aggregation that are inaccessible to photobleaching recovery. 4. FCS has given rise to a large number of extensions that are useful for specific purposes, for example, two-color FCCS to measure association of fluorescent ‘monomers’ and image correlation spectroscopy to measure aggregation. 5. FCS has been applied to a wide range of cell biological questions, for example, among many others, detection and measurement of receptor clustering (Saffarian et al., 2007; Nagy et al., 2010), probing the endyocytic pathway using two-color cross-correlation spectroscopy (Bacia et al., 2002), measurement of the dynamics of paxillin molecules during cell adhesion (Digman et al., 2008a,b).
Appendix We have provided the conventional definition of the correlation function in terms of chemical concentrations, but it is instructive also to consider an equivalent definition from the perspective of contemporary single molecule studies. We will be concerned with mesoscopic systems in which there are a small number of fluorescent molecules of interest the positions of which fluctuate continuously over time. We therefore adopt a probabilistic point of view: p(r,t) is the probability that a fluorescent particle is at position r at time t. The fluorescence intensity measured from a single molecule can be R represented as FðtÞ ¼ Q IðrÞpðr,tÞd3 r where I(r) is the fluorescence intensity detected from position r, and Q is a brightness parameter, i.e., the emitted intensity measured from a single fluorescent molecule, as defined above. Then, for a system that contains a single fluorescent molecule the normalized and non-normalized correlation functions, G1(τ) and g1(τ), respectively, are G1 ðτÞ ¼
g 1 ðτ Þ 〈F ðt ÞFðt þ τÞ〉 ¼ 〈F ðt Þ〉2 〈F ðt Þ〉2
# jr r0 j2 exp W ðrjr0 ,τÞ ¼ : 4Dτ ð4πDτÞ3=2 1
"
For simplicity we now restrict ourselves to two-dimensional systems; the results are readily extended to three dimensions by taking into account the detected intensity along the z-axis of light propagation. FCS measurements commonly use a Gaussian laser intensity profile, that yields adetection function 2 with r¼ (x,y) represented in the focal plane as Iðr Þ ¼ exp 2r w2 and w, the radius of the Gaussian detection area. This yields, g1 ðτÞ ¼
Q2 V
¼
Q2 V
Z exp Z
2r02 2r 2 exp Wðr r0 ,τÞd2 r0 d2 r 2 2 w w
2r02 2r 2 1 exp exp w2 w2 ð4πDτÞ " # jr r0 j2 2 2 exp d r0 d r: 4Dτ
Here we have used W(r|r0,τ) for a two-dimensional system. Extension of the correlation function for a one-particle system to a system containing Nt fluorescent particles is complicated by the fact that the probability distribution function for N particles is the N-fold convolution of the distributions for single particles. It is useful to avoid convolutions by noting that g1(τ) can be regarded as a moment of a bivariate distribution the two dimensions of which correspond to the fluctuating fluorescence intensity record and the same record delayed by lag time τ (Melnykov and Hall, 2009). Then, one may use the additive properties of cumulants, a related type of moment, so that g1(τ)¼ κ1(τ) where κ1(τ)is a cumulant (Qian and Elson, 1990; Wu et al., 2008; Melnykov and Hall, 2009). Since cumulants are additive, for a system containing N molecules, κN(τ) ¼ Nκ1(τ). This leads to the following expression for the correlation function: Z Nd Q2 2r02 2r 2 1 exp exp gN ðτÞ ¼ Ad w2 w2 ð4πDτÞ " exp
# jr r0 j2 2 2 d r0 d r: 4Dτ
Here, Ad ¼ πw2 is the area from which the fluorescence is detected, and Nd is the number of molecules in that area. (Thus the concentration of the fluorescent particles is C ¼ Nd/Ad.) It can further be shown that carrying out the integrations in this equation yields the expected form of the correlation function for simple diffusion (Elson and Magde, 1974).
Acknowledgments
Z ¼ Q2
In this equation, and W(r|r0,τ) is the probability that the molecule initially at r0 will be at position r at a later time τ. A freely diffusing molecule is equally likely to be anywhere in the sample volume, and so pðr0 ,0Þ ¼ V1 , where V is the total volume of the system. The transition probability for simple diffusion with diffusion coefficient D is
Iðr0 ÞIðr Þpðr0 ,0ÞW ðrjr0 ,τÞd3 r0 d3 r 〈F ðt Þ〉2
This work was financially supported by NIH via grant R01 HL 109505.
Fluorescence Correlation Spectroscopy: A Tool for Measuring Dynamic and Equilibrium Properties of Molecules in Cells
See also: Imaging the Cell: Light Microscopy: Fluorescence Lifetime Imaging − Applications and Instrumental Principles
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