bioRxiv preprint doi: https://doi.org/10.1101/359240; this version posted June 29, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Biophysical Journal Volume: 00 Month Year 1–12 1 Anomalous diffusion in inverted variable-lengthscale fluorescence correlation spectroscopy M. Stolle and C. Fradin Abstract Using fluorescence correlation spectroscopy (FCS) to distinguish between different types of diffusion processes is often a per- ilous undertaking, as the analysis of the resulting autocorrelation data is model-dependant. Two recently introduced strategies, however, can help move towards a model-independent interpretation of FCS experiments: 1) the obtention of correlation data at different length-scales and 2) its inversion to retrieve the mean-squared displacement associated with the process under study. We use computer simulations to examine the signature of several biologically relevant diffusion processes (simple diffusion, continuous-time random walk, caged diffusion, obstructed diffusion, two-state diffusion and diffusing diffusivity) in variable-lengthscale FCS. We show that, when used in concert, lengthscale variation and data inversion permit to identify non-Gaussian processes and, regardless of Gaussianity, to retrieve their mean-squared displacement over several orders of magnitude in time. This makes unbiased discrimination between different classes of diffusion models possible. Received for publication xxx and in final form xxx. 1 Introduction characteristic decay time of the FCS autocorrelation func- tion (ACF), τ (5). The diffusion law provides a proxy Quantifying the motions of macromolecules in cells, while 1=2 for the particles MSD. Initially, VLS-FCS was achieved by an important task, is complicated. The cellular environment enlarging the usually diffraction-limited point-spread func- is crowded and heterogenous, and many biomolecules tran- tion (PSF) that sets the value of w (5–7). Since then, several siently interact with others. Because of this, macromolecular methods have been proposed to extend the range of avail- diffusion in cells can take many forms and is seldom sim- able w below the diffraction limit (8–10). Another interest- ply Brownian. It may exhibit a mean-squared displacement ing development has been the generation of VLS-FCS data (MSD) that is not linear in time, a distribution of displace- from imaging modalities (total internal reflection, single- ments that is not Gaussian, or both (1). Langowski’s study of plane illumination), in which case the observation volume the diffusion of the green fluorescent protein in cells, using size can be varied by binning pixels (11, 12). fluorescence correlation spectroscopy (FCS), was one of the The ultimate limit on FCS spatial resolution, however, first to put this anomalous behaviour in the spotlight (2). is not set by the size of the observation volume, but by the FCS, a technique based on the statistical analysis of the temporal resolution of the signal from which the ACF is fluorescence signal recorded from a well-defined observa- computed (13–15). The displacements of fluorophores over tion volume (radius w) is perfectly suited to the range of distance smaller than w still result in changes in fluorescence concentrations (1 to 100 nM) and diffusion coefficients (1 intensity - with low contrast -, and they are therefore cap- to 100 µm2/s) typically encountered for soluble proteins in tured in the short-time regime of the ACF. Thus even with cells (3, 4). Further, the systematic variation of w has made a fixed-size observation volume, FCS can resolve motions this technique especially useful for the study of anomalous over a range of lengthscales on either sides of the diffrac- diffusion processes, which are often lengthscale-dependent. tion limit. This capacity was demonstrated in a cluster of This scheme, known as spot variation FCS or more generally studies exploring DNA segments dynamics, where the ACF variable-lengthscale FCS (VLS-FCS), allows the construc- was inverted to directly recover the MSD of the segments tion of the so-called ”diffusion law”, that is the relationship (16–18). The same ACF inversion procedure was later used between projected area of the observation volume, w2, and to study the effect of crowding in membranes and polymer © 2013 The Authors 0006-3495/08/09/2624/12 $2.00 doi: 10.1529/biophysj.106.090944 bioRxiv preprint doi: https://doi.org/10.1101/359240; this version posted June 29, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. 2 2 THEORY solutions (19, 20). A similar idea was later implemented for 2.2 ACF for an isotropic Gaussian diffusive process image correlation spectroscopy, using a different mathemat- If the propagator is both Gaussian and isotropic, it can be ical scheme to extract the MSD from spatiotemporal ACFs, expressed as a function of the MSD, hr2i(τ) : and used to characterize the anomalous behaviour of GFP −3=2 2 diffusion in cells (15, 21). Of note, both these MSD recov- 2π − 3r p(r; τ) = hr2i(τ) e 2hr2i(τ) ; (2) ery schemes work under the assumption of a process with a 3 Gaussian propagator. Following analytical work by Hofling¨ & Franosch (1, which leads to: 22), we recently showed that variation of the FCS obser- −1 − 1 G(τ; w) hr2i(τ) hr2i(τ) 2 vation volume size can be combined with ACF inversion = 1 + 2 1 + 2 : to obtain the MSD of tracer particles for over 5 orders of G(0; w) 3w =2 3(Sw) =2 (3) magnitude in time (20). The two model systems we studied The above equation is valid for any lag time τ at which (crowded dextran solutions and agarose gels) behaved very the propagator is Gaussian. It is the basis for the inver- differently in regard to the superimposition of the apparent sion procedure used in this work, and illustrated in Fig. 1, MSDs extracted from ACFs obtained at different length- ~2 scales (20). This dissimilarity reflects a qualitative differ- where hr i(τ) is obtained from G(τ; w)=G(0; w) by invert- ence in the nature of the propagator (a.k.a. distribution of ing Eq. 3 (the tilde is used as a reminder that the apparent displacements) of the underlying process. Since only pro- MSD extracted from the ACF might differ from the actual cesses with Gaussian propagators are expected to lead to a MSD when the propagator is not Gaussian). perfect superimposition of the apparent MSDs, the combi- nation of VLS-FCS with ACF inversion provides a test of 2.3 Relationship between ACF and MSD at short lag times the Gaussianity of the diffusive process (20, 23). Here we study the signature of different biologically rel- We show in this section that for short τ there is a simple lin- evant diffusive processes (continuous time random walk, ear relationship between G(τ; w)=G(0; w) − 1 and hr2i(τ), two-component diffusion, diffusing diffusivity, obstructed whether the propagator is Gaussian or not. The only neces- diffusion, caged diffusion). Since analytical solutions are sary assumption is that the propagator is isotropic, in which available only in a small number of limiting cases, we per- case it can be written p(r; τ). At lag times much smaller formed 3D single agent simulations for all these processes, than the ACF characteristic decay time (τ τ1=2), particles and obtained apparent MSDs from the inversion of ACFs have not yet diffused over distances comparable to the obser- simulated for a range of w. By highlighting the different vation volume radius, in other words hr2i(τ) w2. This signatures of each diffusion process, our work provides a means that p(r; τ) ' 0 for r > w, in which case O(~r + ~ρ) benchmark for model-independent interpretation of inverted can be replaced in Eq. 1 by the first two even terms of its VLS-FCS experiments. It also refines our understanding of Taylor series expansion in ρ/w (the odd terms are left out the relationship between apparent MSD and real MSD. We as they disappear when integrating over ~ρ). Performing the show in particular that the inversion procedure yields the cor- integration over ~r in Eq. 1 then yields: rect MSD at short lag times regardless of the nature of the G(τ; w) ZZZ ρ2 ρ2 ρ2 propagator, suggesting possible refinements for VLS-FCS x y z ' d~ρp(ρ, τ)(1 − 2 )(1 − 2 )(1 − 2 2 ): experiments. G(0; w) w w S w (4) Again ignoring higher order terms in (ρ/w)2, and using the 2 2 2 1 2 2 Theory equality hrxi = hryi = hrz i = 3 hr i valid for an isotropic 2 2 propagator, we get (for τ τ1=2, or hr i w ): 2.1 General form of the ACF G(τ; w) (2 + 1=S2) hr2i(τ) The fluorescence signal I(t) obtained in an FCS experiment ' 1 − 2 : (5) is correlated in time to give the ACF, G(τ) = hδI(t)δI(t + G(0; w) 3 w τ)i=hIi2, where δI(t) = I(t) − hIi. For a transport pro- For a Gaussian process, this linear relationship between ACF cess with propagator p(~r; τ), and a 3D Gaussian observation and MSD can be recovered directly from Eq. 3, by per- volume (radius w, aspect ratio S) with normalized profile forming a first-order Taylor expansion in hr2i=w2.