Fluorescence correlation spectroscopy for particle sizing: A notorious challenge

Abstract

In many quantitative investigations of biological systems, including, e.g., the study of biomolecular interactions, assembly and disassembly, aggregation, micelle and vesicle formation, or drug encapsulation, accurate determination of particle sizes is of key interest. Fluorescence correlation spectroscopy (FCS), with its exceptional sensitivity for molecular diffusion properties, has long been proposed as a valuable method to size small, freely diffusible particles with superior precision. It is conceptually related to the more widespread particle sizing technique dynamic light scattering (DLS) but offers greater selectivity and sensitivity due to the use of fluorescence rather than scattered light. However, in spite of these apparent advantages, FCS has never become established as a biophysical routine for particle sizing. This is due to the fact that sensitivity can, under certain conditions, indeed be disadvantageous, as it renders the technique error prone and overly susceptible to signal disturbances. Here, we discuss the systematic challenges, as well as the advances made over the past decades, to employing FCS in polydisperse samples. The problematic role of large particles, a common issue in DLS and FCS, and the effect of fluorescent labeling are discussed in detail, along with strategies for respective error mitigation in experiments and data analysis. We expect this overview to guide future users in successfully applying FCS to their particle sizing problems in the hope of fostering a more widespread and routine use of FCS-based technology.

Correlation spectroscopy using scattered light and fluorescence

Particle sizing is an extremely common challenge in (bio)physical chemistry. In polymer science, the size of a particle is a key parameter for understanding both the properties of the polymer and its interaction with other particles (1,2,3,4). Changes in particle size are a convenient readout for complex formation of two or more molecules in biochemistry, e.g., ligand binding or protein-protein association/aggregation (5,6,7,8,9). For nanoparticle-based drug carriers, monitoring the carrier size is considered important for targeting drug distribution in the body and for their uptake by target cells (10,11).

As for any other problem, there is no universal method of choice for all particle sizing applications. Many methods are available, with their specific strengths and weaknesses (12,13). One approach is to physically fractionate particles by size, followed by a secondary readout such as UV absorbance. Analytical ultracentrifugation is a powerful approach to fractionate particles based on sedimentation in solution, which can be interpreted based on well-established theories of hydrodynamics (11). (Micro)fluidic approaches like asymmetric flow field fractionation separate particles by exploiting predictable relations between flow velocity profiles, particle diffusion coefficients, and resulting distributions of particles within the channel (14,15). Among the more direct approaches, nanoscale imaging, e.g., by electron microscopy, is an obvious choice for image-based characterization of particle size and shape but generally suffers from low throughput (14,16). Other single-particle approaches include tunable resistive pulse sensing, in which particles passing through a microscopic pore transiently reduce electrolyte currents, which are converted into particle volume estimates (17), or nanoparticle tracking analysis, another image-based technique in which diffusing particles are imaged with high time resolution to extract particle sizes from particle mobility (13).

Perhaps the most widespread technique, particularly popular for combining simple and fast measurements with typically high signal/noise ratios, is dynamic light scattering (DLS) (18,19,20). In DLS, laser light elastically scattered at moving particles is statistically analyzed in time through the intensity autocorrelation function (ACF):

$$G\left(\tau \right)=\frac{{⟨I\left(t\right)I\left(t+\tau \right)⟩}_t}{{{⟨I\left(t\right)⟩}_t}^2}\text{.}$$ (1)

The temporal profile of the ACF represents the characteristic profiles of all processes that change the detected intensity. In essence, it reflects on the probability for a signal fluctuation that had been observed at an arbitrary time point $t$ to persist until a later time point $t+\tau$. For homodisperse systems and at low noise levels, the ACF reveals the diffusion coefficient $D$ of the particles via the time constant of an exponential decay. Similar to the other diffusion-based techniques mentioned above, DLS exploits the fact that $D$ depends on the particle size, where the Stokes-Einstein equation describes the relation for (approximately) spherical particles with hydrodynamic radius $r_H$:

$$D=\frac{k_{B}T}{6\pi \eta r_H}\text{,}$$ (2)

with solvent viscosity $\eta$. Obviously, not every particle is spherical, and nonspherical particles of the same volume have different diffusion coefficients than spherical ones (21). Alternative models exist to replace Eq. 2 for other particle shapes, for example, that of elongated cylinders (22,23). By extending the models used in the analysis of DLS data to describe a multi-species mixture with a range of diffusion coefficients $D_j$, one can also characterize a distribution of particle sizes. This allows using DLS to quantify not only particle suspensions of a single particle size, but also polydisperse ones that display a range of particle sizes, at least in principle.

The time-correlation concept underlying DLS is generic and has been successfully used using other contrast methods besides scattered light, including, but not limited to, fluorescence emission (24,25). The latter approach, known as fluorescence correlation spectroscopy (FCS), is the focus of this article. Here, laser light is used to excite fluorescent dyes attached to a particle of interest, and the fluorescence emission is correlated and analyzed in time analogous to the scattered laser light intensity in DLS. ACFs recorded in FCS from suitable samples also primarily report on diffusion dynamics, like those in DLS. Also, similarly to DLS, the FCS ACF for a multi-species mixture containing a range of diffusion coefficients for the respective species $j$ in FCS is generally described by three key parameters: the species-wise diffusion time ${\tau }_{D,j}$, the particle number $⟨N_j⟩$, and the molecular brightness ${\epsilon }_j$:

$$G\left(\tau \right)=\gamma \sum _{j=1}^{j_{\mathit{max}}}{G}_{0,j}{g}_{D,j}\left(\tau \right)\text{,}$$ (3a)

$$G_{0,j}=\frac{⟨N_j⟩{{\epsilon }_j}^2}{{\left(\sum _{j=1}^{j_{\mathit{max}}}⟨N_j⟩{\epsilon }_j\right)}^2},\text{and}$$ (3b)

$$g_{D,j}\left(\tau \right)=\frac{1}{1+\frac{\tau }{{\tau }_{D,j}}}\frac{1}{\sqrt{1+\frac{\tau }{S^2{\tau }_{D,j}}}}\text{,}$$ (3c)

where the usually confocal observation volume is described by its width in the xy plane $w_{xy}$, by the aspect ratio $S=w_{\mathrm{z}}/w_{xy}$, and by the shape factor $\gamma =const.\approx 0.354$. $w_{xy}$ enters the model implicitly through ${\tau }_{D,j}$ via the mean-squared displacement $⟨{{\Delta }_j}^2⟩$ in the Einstein-Smoluchowski equation: ${w_{xy}}^2=⟨{{\Delta }_j}^2⟩=4D_j{\tau }_{D,j}$.

The overall correlation amplitude is proportional to the inverse of the total particle number, meaning that FCS also probes absolute particle concentrations, as long as the size of the observation volume is known. If background levels are significant even in confocal detection, then the evaluation of absolute concentrations requires correction of the background, which distorts the correlation amplitude (26). Depending on the hardware and software available, one can either employ techniques like fluorescence lifetime correlation spectroscopy (27) or explicitly correct for the amplitude if the background level if known (25,26,28). Of note, for especially low signal/background ratios, the correlation amplitude increases with increasing total particle number rather than being inversely related. Thus, without appropriate correction, this relation is not always trivial.

In spite of the obvious conceptual similarities and a superior signal to the background level, FCS remains less routinely used for particle sizing than DLS. The purpose of this article is first to discuss the key limitations, i.e., the particular challenges FCS faces for particle sizing applications. We also highlight what advances have been made and offer an overview of various considerations to make in such experiments. We will focus on the analysis of polydisperse samples, i.e., samples that are not adequately described by a single particle size. As both FCS and DLS generally determine particle sizes indirectly via the diffusion coefficients using Eq. 2 or similar relations, we will not strictly distinguish between particle size measurements and diffusion coefficient measurements. We start by discussing some practical reasons why one would be interested in using FCS as an alternative to DLS in the first place. We then emphasize challenges that are specific to FCS. In the remaining sections, we address how fluorescent labeling affects the interpretation of FCS data acquired for particle sizing, followed by a short treatise on advances in experiments and data processing. While the technical considerations are generic in nature, we will focus on (self-assembling) proteins and other biomolecules.

What advantages do we expect from FCS over DLS?

While built on the same conceptual basis, FCS and DLS have different strengths and weaknesses in practice. The first obvious differences is the use of fluorescent labeling in FCS. Fluorescent labeling has the advantage of a very high signal/background ratio. With contemporary high-performance fluorescent dyes and modern detector technology, discriminating a single fluorescent molecule’s signal over the (usually solute) background from ca. ${10}^{10}$ molecules within a 1 fL observation volume has become possible on systems of all major commercial manufacturers of confocal microscopes. This specificity allows performing FCS routinely in complex environments that would hardly allow meaningful DLS measurements, such as intact cells (29,30) or even organisms (31).

Further, the single-molecule sensitivity also allows FCS to be performed with very low sample concentrations and volumes (29). Another obvious advantage of using fluorescent labeling lies in multi-color experiments, namely fluorescence cross correlation spectroscopy (FCCS) (32). FCCS distinguishes signal fluctuations that are (cross) correlated between two signal channels from single-channel fluctuations. The former indicate molecular complex formation, the latter unbound particles. While not typically used as a particle sizing technique, FCCS is a valuable extension for studying exactly what components interact to form a particle.

Besides these obvious strengths of fluorescence as a readout, there are more subtle advantages to FCS. FCS is applicable to practically any particle that is large enough to attach a fluorescent dye without significantly altering the particle dynamics. What constitutes a “significant alteration” has to be decided and checked in suitable control experiments on a case-by-case basis. Typically, attaching a synthetic dye to a protein of interest adds ca. 1 kDa molecular mass, a fluorescent protein tag, or a self-labeling protein tag ca. 10–30 kDa (33). If labeling limitations are a serious concern, then DLS may be preferred. However, DLS may struggle to detect small particles over the background, especially if the particles are not dissolved in simple buffers: the contrast from scattered light scales sharply with the particle volume. However, the limit of detection in DLS depends on multiple factors (18). Thus, although the required attachment of a fluorophore seems to be a conceptual limitation, FCS is a particularly attractive technique for studying small particles or complex environments but also to add specificity in samples of unknown composition.

What keeps us from routinely using FCS for dispersity characterization?

In spite of these obvious methodological advantages, FCS is not used very frequently for particle sizing. A first, trivial reason is perhaps found in history: DLS is significantly older than FCS. While the concept of FCS was proposed in the 1970s (24), the technique reached maturity only in the 1990s following technical breakthroughs in confocal microscopy (5,34) and is still marketed mostly as an optional add-on to confocal imaging platforms. In contrast, DLS has been commercialized since the late 70s and is nowadays performed in affordable benchtop systems (an outline of the early history of DLS can be found, for example, in (20) and for FCS in (35,36)). Thus, FCS is lagging behind in endorsement by the community and by instrument manufacturers by about two decades.

Furthermore, the required use of fluorophores creates complications beyond the added effort of attaching fluorophores to the particles of interest. While the specificity of fluorescence is an enormous advantage, one should not confuse the high signal/background ratio with a high signal/noise ratio. In contrast to elastic scattering, the excitation/emission photocycle of fluorescence is a saturable process under realistic measurement conditions. Therefore, one cannot arbitrarily increase the signal/noise ratio in any fluorescence technique by increasing the laser power. In addition, saturation tends to introduce artifacts: in FCS in particular, saturating the photocycle can lead to an overestimation of particle sizes (37,38).

Photochemical side processes of the photocycle can lead to off switching of the fluorophore. Reversible off switching (“blinking”) mostly makes data interpretation more complicated (39). Irreversible off switching (“photobleaching”) has rather complex consequences: photobleaching leads to a compromised signal/noise ratio, introduces artificial additional slow-diffusing kinetics into the ACF, and/or leads to an overestimation of diffusion coefficients, i.e., underestimation of particle sizes (28,40). While the first two effects of photobleaching can be controlled or compensated rather easily, the underestimation of particle sizes due to photobleaching often remains cryptic unless controlled via an experimental laser power series and is especially severe for large, slow-moving particles.

In principle, particle size overestimation due to fluorescence saturation and particle size underestimation due to photobleaching always counteract each other. In practice, their balance is system specific and hard to predict, and the safe option is to perform experiments at low laser power where both effects remain negligible, albeit at the cost of reduced signal/noise ratio.

Fluorescence emission can also change in response to the local environment of the particle, and one should be careful about photophysical processes possibly affecting the measured diffusion dynamics. The increasingly widespread combination of FCS with time-correlated single-photon counting helps to detect and correct such effects through the measurement of accompanying changes in fluorescence lifetime (41).

As none of these limitations apply to the scattered light used in DLS, interpretation of DLS data seems more robust. However, caveats about similar effects resulting from local heating effects and internal dynamics within particles have been raised for DLS (19).

Besides these rather practical issues, there are further challenges that are specific to the use of FCS for the characterization of polydisperse samples and are of a more conceptual nature. Here, progress can be made by improving the strategy of FCS data acquisition and analysis as described in further sections.

Why are large particles such a problem and how can they be dealt with?

A challenge that is common to both FCS and DLS and deserves explicit mention is an overrepresentation of large particles in the data. In DLS, the amplitude of a particle species in the intensity correlation function scales with the square of the particle mass. Similarly, in FCS, the species amplitude usually scales with the square of the particle brightness (Eqs. 3a, 3b, and 3c). While this scaling is predictable, it is unfavorable for many applications for two reasons: firstly, the bias of the correlation function toward large particles makes it hard to capture the particle number distribution of polydisperse samples. It also strongly biases analysis strategies based on single-species approximations toward larger species. Secondly, these large particles are often rare and, therefore, weakly sampled in the experimental data. This obscures the dynamics of interest below high noise levels.

Different, to some degree complementary approaches can be taken to obtain data that are more representative of the large number of smaller particles in the sample. A first, obvious, and widespread approach is to discard data that show signal “bursts” from especially bright particles. This approach is very established and powerful, and different partially or fully automated strategies are available (42,43,44). Such “burst removal” performs well for the characterization of low-disperse systems with the occasional formation of unwanted aggregates. However, for intrinsically polydisperse systems, the decision of what is a desired signal and what is undesired remains somewhat arbitrary, posing a severe risk of distorting the results and rendering them useless.

A strategy frequently applied by FCS experimenters when working with disperse self-assembling systems is to dilute fluorescently labeled monomers in an excess of unlabeled ones (8,45,46). This incomplete labeling reduces the average brightness of large particles and the noise they introduce into the data. While this strategy is successful, it has an often-overlooked side effect (47). Consider a system in which the number of fluorophore moieties within a particle scales linearly with particle mass, which is realistic for many self-assembling systems. If we assume that the fluorescence quantum yield of fluorophores is independent of particle size and that every monomer has a stoichiometry-independent probability $p$ to carry a fluorophore moiety, the normalized (to ${\epsilon }_{monomer}$) brightness of a $j$-mer oligomer species is characterized by a binomial distribution with average relative brightness $jp$ (Fig. 1 A) but also, and easily overlooked, variance $jp\left(1-p\right)$. Inserting this into Eq. 3b and following the calculations in (47) yields

$$G_{0,j,p}=\frac{⟨N_j⟩j^2\left(1+\frac{1-p}{jp}\right)}{{\left(\sum _{j=1}^{j_{\mathit{max}}}⟨N_j⟩j\right)}^2}\text{.}$$ (4)

Figure 1.

Effect of fluorescent labeling efficiency on FCS signal. (A–C) Theoretical curves illustrating effects of changing labeling efficiency. (A) Simulated lognormal particle concentration and relative fluorescence signal over particle size. (B) Resulting species-wise correlation function amplitudes. (C) Obtained correlation functions with fits with Eqs. 3a, 3b, and 3c. Vertical lines in (A)–(C) indicate fitted diffusion times. (D and E) Experimental results from broad distributions of single-stranded RNA (ssRNA) fragment lengths with similar size distributions but different stochastic labeling efficiencies (compare also Fig. S2). (D) Experimental ACFs. (E) Reconstructed concentration profiles over particle size. Analysis was repeated with and without correction of labeling effect. Irregular distribution shape features at low fragment lengths are discretization artifacts. The smaller overall size in the 3% labeled sample compared to the others is consistent with gel electrophoresis (compare Fig. S2).

For $p\ll \frac{1}{j+1}$, the numerator is proportional to $j$ rather than $j^2$ as implied by Eq. 3b. In other words, the limited labeling efficiency not only reduces the noise caused by rare large particles but also shifts the weights with which different complex sizes are measured in FCS altogether (Fig. 1, B and C). While Eq. 4 is based on strong assumptions, it will often serve as a reasonable approximation (Fig. 1 E). Related effects of concentrations and brightness have been investigated in some detail in experiments on micelles (46). In that study, the concentrations of both the micelle constituent and the fluorescent label were systematically varied. This allowed to understand both the self-assembly kinetics of the micelles and the impact of labeling in the system. The experimental data from sparsely labeled detergent micelles supported analysis using assumptions analogous to those underlying Eq. 4. However, this work also presents considerations about the case of changing quantum yields with label density.

Another issue that arises with increasing particle sizes is that Eqs. 3a, 3b, 3c, and 4 implicitly assume that particles are small compared to the observation volume. Simply speaking, Eqs. 3a, 3b, and 3c only consider center-of-mass movement of particles, but for bigger particles, the center of mass leaving the observation volume does not mean that the entire particle left the observation volume. Particle sizes significantly exceeding ca. $w_{xy}/5$ lead to significant distortions of the effective diffusion time and particle number (48,49). A simple correction for particles of moderate size is to scale the effective beam waist parameter as $w_{xy,eff}^2=1+r^2$ with particle radius $r$. ${\tau }_{D,j}$ and $⟨N_j⟩$ then both increase by the same factor (48,49). Note that $r$, strictly speaking, refers to the spatial distribution of fluorophores in the particle, for which the hydrodynamic radius $r_H$ may or may not be a good approximation. (49) contains more accurate expressions for the large-particle correction, considering different particle shapes.

Experimental avenues beyond single-spot confocal FCS

Nowadays, FCS is mostly performed on confocal microscopes due to the convenience of its implementation in commercial instruments, with small and relatively well-defined detection volumes. As mentioned above, confocal microscopes from all major manufacturers offer adequate platforms for traditional single-spot confocal FCS measurements. However, besides the single-spot FCS acquisition that most of our discussion focuses on, other FCS concepts exist that promise to be especially attractive for particle sizing applications.

In single-spot FCS, deviations from the assumed Gaussian detection volume will lead to stretched decay patterns in the ACF that are easily confused with polydispersity (50). To increase the robustness of analysis, it is desirable to observe diffusion over fixed length scales that are robust against detection volume shape deviations. This has been part of the motivation behind the development of multifocus FCS approaches like dual-focus FCS (51,52). Dual-focus FCS was the method of choice in a study pushing the limits of sensitivity to mass changes upon protein-ligand binding in FCS (6).

A further extension of this concept is scanning FCS, which also increases particle throughput by sampling a larger volume without a large increase in the confocal detection volume (47,53). The concept of scanning FCS allows many different scanning modes regarding geometry and timescales. Raster image correlation spectroscopy (RICS) is very attractive for the analysis of polydisperse samples, as it samples a range of spatial and temporal scales in a single dataset (54,55,56). Given that the scan parameters in RICS are generally adaptable to the diffusion coefficients of the sample, an experimental approach globally analyzing RICS data from multiple scan speeds is likely to be of particular value for the analysis of disperse mixtures (57).

In addition, one may wonder whether the regular pixel grid usually employed in RICS and similar approaches is the most efficient grid for spatiotemporal diffusion analysis (55). Following the principles of pair correlation function analysis (35,58,59), a set of observation volumes could be engineered for optimal information content given the question at hand. Besides established implementations in scanning confocal systems and more recent advances exploiting array detectors (58,59), it will be interesting to explore what developments toward ultrafast laser scanning microscopes can contribute here (60,61,62).

Besides these ideas toward increased precision of FCS-based particle sizing, another interesting aspect is that of FCS automation. Another reason why FCS applications lag behind DLS is that nowadays, DLS is routinely and quickly performed with compact cuvette spectrometers or even plate reader systems. In contrast, the performance of FCS in confocal microscopes requires instrumentation that is expensive, physically large, and relatively complex in operation. This need not be the case. Using concepts that are widespread in automated high-throughput/high-content image acquisition, workflows and dedicated devices suitable for automated FCS acquisition have been developed, which effectively operate as plate readers and even automatically recognize cells as regions of interest for measurements (63,64,65,66). Alternatively, a cuvette-based FCS spectrometer was reported that exploits advances in objective lens design to go beyond measurements through standard-thickness coverslips (67). FCS hardware has reached technical maturity to the point of allowing highly automated apparatuses that strip away most of the complexity of standard confocal microscopy. Further efforts, also by commercial manufacturers, to implement and spread turn-key apparatuses for a nonexpert user community would be of great value.

How to make the most of available data?

For the analysis of a polydisperse system, a naive approach would be to evaluate Eqs. 3a, 3b, and 3c or 4, with many j-mer species up to a cutoff stoichiometry $j_{\mathit{max}}$. This would, however, require estimating a parameter number reaching or even surpassing the number of data points in an experimentally determined ACF. That is impossible in practice, especially given that successive data points in the ACF are strongly correlated and show few distinctive features. In this section, we will briefly discuss strategies for dealing with FCS data from disperse samples.

The first obvious solution is to determine only a single average value. However, the interpretation of this average is nontrivial due to the brightness dependence of amplitudes. The vertical lines in Fig. 1, A–C, illustrate the average diffusion time estimated by the single-component fits in comparison to the ground-truth distributions. The estimated diffusion time changes with labeling efficiency but under no circumstances is representative of the particle number distribution (Fig. 1 A) or particle mass distribution (equals fluorescence intensity distribution in Fig. 1 A). Fig. 1 D shows single-component FCS model fits (Eqs. 3a, 3b, and 3c with $j_{\mathit{max}}=1$), which display visible mismatch for some curves. An alternative is to modify Eq. 3c with an anomalous diffusion model ($⟨{\Delta }^2⟩=4\Gamma {\tau }^{\alpha }$) such that

$${\mathrm{G}}_{anom}\left(\tau \right)=G_0\frac{1}{1+{\left(\frac{\tau }{{\tau }_D}\right)}^{\alpha }}\frac{1}{\sqrt{1+\frac{1}{S^2}{\left(\frac{\tau }{{\tau }_D}\right)}^{\alpha }}}\text{and}$$ (5a)

$${\tau }_D={\left(\frac{{w_0}^2}{4\Gamma }\right)}^{\frac{1}{\alpha }}\text{,}$$ (5b)

with transport coefficient $\Gamma$, anomalous diffusion coefficient $\alpha$, and amplitude $G_0$. Originally intended to describe FCS data for non-Brownian motion inside living cells, the model allows the fit to adapt to deviations from the assumption of “random walk characterized by a single diffusion coefficient in an open Gaussian-shaped detection volume” (29,50,68). Obviously, a polydisperse sample violates this assumption. Analyzing the example in Fig. 1 C with Eqs. 5a and 5b with $\alpha <1$ improves the fits, but the change in ${\tau }_D$ compared to fitting with Eqs. 3a, 3b, and 3c was negligible (below 1% difference; Fig. 2 C). Note that for real data including noise, Eqs. 3a, 3b, 3c, 5a, and 5b can return significantly different diffusion times. Considering other factors besides dispersity affect $\alpha$ (50,68), anomalous motion models should generally only be used as an empirical approximation if no better model is available.

Figure 2.

Analysis strategies for FCS of polydisperse samples. (A) Illustration of conventional two-component FCS fitting that attempts to find amplitude and diffusion time for each component. (B) Illustration of method of histograms fitting, with fixed array of diffusion times and optimization of associated amplitudes. (C) Same theoretical ACFs as in Fig. 1C but fitted with Eqs. 5a and 5b. Vertical lines are fitted diffusion times. (D) Same as Fig. 1E but comparing resulting particle size distributions between fits using the maximum entropy method (MEM) and a stretched-exponential model (based on Eq. 4) of concentration over particle size. (E) Fit residuals for the fits that (B) is based on, as well as fits with Eqs. 3a, 3b, and 3c with a single component and with Eqs. 5a and 5b.

Two-component approximations are another popular analysis strategy. Here, the diffusion coefficient of one species, for example, the monomer, should be known and fixed in the fitting. Two-component models (Eqs. 3a, 3b, and 3c with $j_{\mathit{max}}=2$) have the same, or even greater, difficulties in interpretation as single-component approximations. Thus, while single- or two-component diffusion or anomalous diffusion models can be used to quantify trends in data from polydisperse samples, one should be cautious about what the average represents.

When going beyond single-component approximations, one encounters the issue that the ACF alone is not sufficient to distinguish differences in $⟨N_j⟩$ from differences in ${\epsilon }_j$ (Eqs. 3a, 3b, and 3c). Some assumption must generally be made about these two parameters. One possible assumption is that all particles are of equal brightness, a trivial case in which ${\epsilon }_j$ disappears from the calculation. While for some systems, this is justified, for many, this assumption is unphysical. A better approach is often that described in the previous section, considering labeled monomer units within the oligomer particle (Eq. 4). When combining this with assumptions about the link of particle size (${\tau }_{d,j}$ as a function of $j$) and brightness ${\epsilon }_j$ as mentioned before, this drastically reduces the number of model parameters. For example, one can combine the before-discussed impact of labeling (Eq. 4) with models of particle mobility, such as Eq. 2, to arrive at

$$\frac{{\tau }_{d,j}}{{\tau }_{d,monomer}}={\left(\frac{{\epsilon }_j}{{\epsilon }_{monomer}}\right)}^{\frac{1}{3}}=j^{\frac{1}{3}}\text{.}$$ (6)

This scaling is different for other models of particle shape, of course (22,23). In that case, by determining ${\tau }_{d,monomer}$ from control measurements, one can reduce the model to $⟨N_j⟩$, being the only remaining parameter to determine for each species. One can then fit $⟨N_j⟩$ using the “method of histograms” (69): this approach uses a regular-spaced array of ${\tau }_{d,j}$ (considering associated bin widths) and optimizes an array of associated amplitudes as opposed to diffusion times and amplitudes (Fig. 2, A and B). That way, one only needs to estimate $⟨N_j⟩$ for a relatively modest-sized subset of particle sizes. For robust results, this is combined with suitable fit constraints, e.g., non-negativity of amplitudes, and with methods to exploit the fact that for most polydisperse systems, no sharp jumps in the profile of concentration over particle size are expected. There are at least two different techniques to ensure a smooth profile of $G_{0,j}$. Firstly, one can regularize the fit without specific assumptions about the underlying distribution via a modified cost function that penalizes unphysically spiky profiles of $⟨N_j⟩$ using, for example, the maximum entropy method (70) or minimization of the squared second derivative (71). These and similar approaches have been successfully introduced into FCS (68,69,72,73,74,75). A downside is that these methods require very high signal/noise ratios (Figs. 2 D and 1 D, note the skewed distributions at low labeling efficiency). Also, secondary analysis of the obtained particle size distribution is required for quantitative interpretation. As a simpler alternative, we can force the distribution of $⟨N_j⟩$ to follow a predefined profile over stoichiometry with tuneable parameters, such as Gaussian, lognormal, or stretched-exponential functions. This allows us to describe the entire distribution with a small number of parameters (three for the mentioned functions). In some cases, such fits can offer a direct link to physical models of self-assembly (2). In our experience, the use of regularization techniques offered no notable advantage over the use of parameterized models (Fig. 2 D), in agreement with previous reports (72,76). Suitable parameterized models and the maximum entropy method or minimization of the squared second derivative fits typically yield similar quality of fit and overall similar distributions, and parameterized models handle data of limited signal/noise ratio more robustly (Fig. 2, D and E).

Another option is the use of alternative data analysis frameworks besides ACF fitting, such as photon counting histograms and related approaches (36,77,78). These methods use other representations of the fluctuation signal besides the ACF. Their key advantage over ACF fitting is that they can discriminate changes in $⟨N_j⟩$ or ${\epsilon }_j$, thus removing ambiguity in the fitting. While these methods have proven their value, they generally come with increased computational cost compared to ACF fitting, especially when many species must be evaluated. Thus, they tend to encounter practical limitations in use on data from polydisperse systems.

Outlook: From ACF fitting to neural networks?

Particle sizing of polydisperse samples is a challenging problem and probably will remain one for the foreseeable future. Nonetheless, since the inception of FCS in the 1970s, the FCS community has introduced many concepts into experiments and analysis that are tremendously useful for the task. An exciting avenue for such a challenge is obviously the recent progress in machine learning, especially convolutional neural networks (CNNs). A few recent papers explored the capabilities that machine learning brings into FCS (43,79,80). Besides applications for artifact removal, specifically for the above-mentioned burst removal problem (43), of key interest are papers that report the use of machine learning for parameter estimation in FCS (79,80). The good understanding of the physical basis of FCS was exploited to generate large amounts of realistic training data in silico based on random walk simulations. Different models were trained to analyze ACFs previously calculated using traditional approaches (CNN in (79) or gradient boosting models in (80)) or to analyze the raw data directly (CNN in (79)). Good results could be obtained from significantly shortened acquisition times. Remarkably, the CNN utilizing raw data compensated experimental artifacts that it was not explicitly trained to deal with (79). ACF-based analysis and related frameworks are essentially data compressions that emphasize patterns of interest but also discard a lot of information. Machine learning techniques utilizing the raw data can retrieve and exploit information that is lost in the ACF calculation. Similar ideas of exploiting uncompressed raw data have been developed in Bayesian statistics frameworks (81,82). Analysis of uncompressed data promises to overcome many of the difficulties of ACF-based analysis, as discussed above. Fortunately, simulations incorporating various aspects of the physics of FCS allow us to generate a large body of training data for such models (83,84).

To conclude, while FCS is, for many practical reasons, still not the primary method of choice for routine particle sizing, it performs quite well when some of the major error sources can be routinely dealt with, as demonstrated by a number of referenced studies. That these improvements have not yet been widely implemented and standardized in FCS routines is regrettable but understandable due to the persistent lack of affordable turnkey instrumentation. We argue that, particularly with respect to the immense potential of implementing machine learning in FCS data analysis, the time has come for instrument developers to revisit FCS as an easily accessible technology that opens fully new perspectives on a fast and widely applicable large-scale analysis of molecular sizes. With respect to the omics era, particle sizes and diffusion coefficients are highly valuable complementary parameters to map the dynamic proteome and interactome on a cell-wide scale—comprehensive dynamic information being the next big challenge in our holistic understanding of living systems.

Acknowledgments

This work was supported by funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 – 390783311. We would like to thank Béla Frohn, Martin Spitaler, and the Imaging Facility team (all MPI of Biochemistry, Martinsried) and Don C. Lamb (Ludwig Maximilians University Munich) for helpful discussions. We dedicate this work to the memory of Watt W. Webb, whose pioneering contributions to a large selection of biophysical methods, including FCS, have left a lasting impact on our scientific community.

Declaration of interests

The authors declare no competing interests.

Editor: Elizabeth Rhoades.

Supporting references

References (85,86,87,88) appear in the supporting material.