Anomalous subdiffusion in living cells: bridging the gap between experiments and realistic models through collaborative challenges
Maxime Woringer, Ignacio Izeddin, Cyril Favard, Hugues Berry
AAnomalous subdiffusion in living cells:bridging the gap between experiments andrealistic models through collaborativechallenges.
Maxime Woringer , Ignacio Izeddin , Cyril Favard (cid:0) , and Hugues Berry (cid:0) Unité Imagerie et Modélisation, CNRS UMR 3691, and C3BI (Center of Bioinformatics, Biostatistics and Integrative Biology), CNRS USR 3756, Institut Pasteur, 75015Paris, France Sorbonne Universités, CNRS, 75005 Paris, France Department of Molecular and Cell Biology, Li Ka Shing Center for Biomedical and Health Sciences, and CIRM Center of Excellence in Stem Cell Genomics, University ofCalifornia, Berkeley, California 94720, USA Institut Langevin, ESPCI Paris, CNRS, PSL University, 1 rue Jussieu, Paris 75005, France Membrane Domains and Viral Assembly, Institut de Recherche en Infectiologie de Montpellier, CNRS UMR 9004, Montpellier, France Inria, Lyon, F-69603, Villeurbanne, France, and Universite de Lyon, LIRIS UMR5205, F-69621, Villeurbanne, France GDR Imabio, CNRS, Lille, France
The life of a cell is governed by highly dynamical microscopicprocesses. Two notable examples are the diffusion of membranereceptors and the kinetics of transcription factors governing therates of gene expression. Different fluorescence imaging tech-niques have emerged to study molecular dynamics. Amongthem, fluorescence correlation spectroscopy (FCS) and single-particle tracking (SPT) have proven to be instrumental to ourunderstanding of cell dynamics and function. The analysis ofSPT and FCS is an ongoing effort, and despite decades of work,much progress remains to be done. In this paper, we give a quickoverview of the existing techniques used to analyze anomalousdiffusion in cells and propose a collaborative challenge to fos-ter the development of state-of-the-art analysis algorithms. Wepropose to provide labelled (training) and unlabelled (evalua-tion) simulated data to competitors all over the world in an openand fair challenge. The goal is to offer unified data benchmarksbased on biologically-relevant metrics in order to compare thediffusion analysis software available for the community.
Diffusion in cells | continuous-time random walks | fractional Brownian Mo-tion | fluorescence correlation spectroscopy | single-particle tracking.Correspondence: [email protected], [email protected]
Introduction
The life of a cell is governed by highly dynamical micro-scopic processes occurring at different space and time scalesfrom single macromolecules up to organelles. Optical mi-croscopy provided four decades ago the first measurements ofbiomolecule motion in cells. First by fluorescence recovery after photobleaching (FRAP) (1) and fluorescence correlationspectroscopy (FCS) (2), and more recently with the help ofsingle particle tracking (SPT) (3, 4). Several factors havecolluded to popularize these techniques in many biophysicsand biology labs: i) the development of highly sensitive de-tectors, ii) the emergence of genetically encoded fluorescentprotein labelling in the late 90s (5–7), and iii) the advent in theyears 2000-2010 of far-field super-resolution microscopy(8–12). All these technological efforts have granted us accessto the monitoring of molecular motion in cells with unprece-dented spatial (down to single molecule) and temporal reso-lution (13, 14). The adoption of these techniques has beenparamount in the advancement of the understanding of cellorganisation and dynamics (15–17).While acquiring sufficient experimental data sets used tobe a limiting factor, these technological advances combinedwith data acquisition parallelization provide nowadays hugeamounts of data available for analysis of molecular motioninside the cell. In turn, the richness of this data has unravelledan unforeseen complexity and diversity of mechanisms forbiomolecule motion in cells. Therefore, many efforts are de-voted to analyze data provided by FCS or SPT with direct orinference approaches.However, choosing the appropriate algorithms to analyse thecomplexity of the observed phenomena is still an importantchallenge. Indeed, the richness of experimental data oftenmakes it difficult to determine which are the physical models
Woringer et al. | aR χ iv | April 3, 2020 | 1–10 a r X i v : . [ q - b i o . Q M ] A p r o be considered and which are the relevant biophysical pa-rameters to be estimated from them. We review and addressthis issue in this perspective.We will first briefly review key anomalous diffusion mod-els relevant to cell biology and summarily describe some ofthe existing techniques to either infer model parameters orto perform model selection. We will discuss the relevanceof numerical simulations and the importance of designingrealistic data sets closely mimicking the results obtained inexperiments on biological samples. We will also highlight theoften overlooked limitations in current acquisition methodsand emphasize the role of experimental noise and biases ofthe aforementioned techniques. Finally, we will present andadvocate in favour of the development of comprehensive setsof simulated data and metrics, allowing the community to ob-jectively evaluate existing and new analysis tools. Our hope isthat this work will instigate an open discussion about the limi-tations and challenges of analysing and modelling diffusionof molecules in the complex environment of the cell. Brownian vs anomalous diffusion
Maybe one of the best-known result of the theory of Browniandiffusion is that the mean squared displacement (MSD) of arandom walker scales linearly with time, and is proportionalto the diffusion coefficient of the fluid in which diffusion takesplace. With x ( t ) being the position of the random walker attime t (in one dimension), this means that the MSD D x ( t ) E =2 Dt , where h·i denotes ensemble averaging and x (0) = 0 .However, Brownian diffusion does not explain the physics ofdisordered systems. Interestingly, an ubiquitous observationin cell biology is that the diffusive motion of macromoleculesand organelles is anomalous, i.e. the MSD change with timeis typically characterized by a sublinear increase. In mostinstances, this sublinear increase of the MSD with time can befitted to a power-law relation D x ( t ) E ∝ t α with exponent α < , which justifies the vocable of “subdiffusion”. Subdiffusionis usually attributed to cellular crowding, spatial heterogeneityor molecular interactions. Another possibility of anomalousdiffusion is superdiffusion, with < α < . Indeed a lot ofprocess in biology exhibit active transport or combinations ofactive and random motions.Anomalous diffusion in cells is therefore a very active area ofresearch involving biophysics, cell biology, statistical physicsand mathematical modelling.When confronted to a set of data retrieved from FCS or SPTexperiments, the first question that one needs to answer is whether the measured subdiffusion is indeed a manifestationof an anomalous process. Often, a combination of severalnormal diffusion mechanisms or experimental artefacts givesrise to an apparent subdiffusion. If an anomalous diffusion–characterized by a power law scaling of the MSD with time–can be identified, establishing the physical model behind thediffusion process can shed light on the molecular mechanismsdriving the motion of the molecule of interest.Below, we will first focus on three classical models for anoma-lous subdiffusion and their common biological interpretation,namely the continuous-time random walk (CTRW) model, thefractional Brownian motion (fBm) model, and random walkson fractal and disordered systems (for a review, see, e.g. (18)),then we will briefly describe different models covering super-diffusion processes that can be encountered in cells such asrun and tumble model, Levy flights and super-diffusive fBm.The continuous-time random walk model is a generaliza-tion of a random walk in which the diffusing particle waitsfor a random time between jumps. More generally, when thedistribution φ ( τ ) of waiting times τ is long-tailed and cannotbe averaged (with e.g. φ ( τ ) ∝ τ − (1+ α ) and < α < ), theensemble-averaged MSD shows anomalous scaling with apower law. A straightforward interpretation of a CTRW in thecontext of molecular biology is assimilating the waiting timesto interactions of the molecule with an immobile substrate (atthe relevant temporal and spatial scales). It is important tonote that an interaction with a characteristic residence timedoes not fulfill the conditions of the model. Interestingly,however, the waiting-time distribution of non-specific interac-tions, abundant in the cell, might be non-averageable and thusCTRW a good microscopic model for one type of anomaloussubdiffusion in the cell. It has been proposed to govern thecytosolic diffusion of nanosized objects in mammalian cells(19) and it has also been used to explain the lateral motion ofpotassium channels in the plasma membrane of cells (20).The fractional Brownian motion model is a different gener-alization of Brownian diffusion in which the jumps betweenlag times follow a normal distribution but respect a correlationfunction given by h x ( t ) x ( s ) i = 1 / t H + s H − ( t − s ) H ) for t > s > . A fBm process is thus characterized by theHurst index H , ranging between and . The value of H determines the type of jump dependence in the fBm process,such that H > / indicates a positive correlation between theincrements, Brownian motion is achieved for H = 1 / , andthe increments are negatively correlated when H < / . TheMSD of a fBm is given by D x ( t ) E ∝ t H , which, again, en- χ iv Woringer et al.et al.
Maybe one of the best-known result of the theory of Browniandiffusion is that the mean squared displacement (MSD) of arandom walker scales linearly with time, and is proportionalto the diffusion coefficient of the fluid in which diffusion takesplace. With x ( t ) being the position of the random walker attime t (in one dimension), this means that the MSD D x ( t ) E =2 Dt , where h·i denotes ensemble averaging and x (0) = 0 .However, Brownian diffusion does not explain the physics ofdisordered systems. Interestingly, an ubiquitous observationin cell biology is that the diffusive motion of macromoleculesand organelles is anomalous, i.e. the MSD change with timeis typically characterized by a sublinear increase. In mostinstances, this sublinear increase of the MSD with time can befitted to a power-law relation D x ( t ) E ∝ t α with exponent α < , which justifies the vocable of “subdiffusion”. Subdiffusionis usually attributed to cellular crowding, spatial heterogeneityor molecular interactions. Another possibility of anomalousdiffusion is superdiffusion, with < α < . Indeed a lot ofprocess in biology exhibit active transport or combinations ofactive and random motions.Anomalous diffusion in cells is therefore a very active area ofresearch involving biophysics, cell biology, statistical physicsand mathematical modelling.When confronted to a set of data retrieved from FCS or SPTexperiments, the first question that one needs to answer is whether the measured subdiffusion is indeed a manifestationof an anomalous process. Often, a combination of severalnormal diffusion mechanisms or experimental artefacts givesrise to an apparent subdiffusion. If an anomalous diffusion–characterized by a power law scaling of the MSD with time–can be identified, establishing the physical model behind thediffusion process can shed light on the molecular mechanismsdriving the motion of the molecule of interest.Below, we will first focus on three classical models for anoma-lous subdiffusion and their common biological interpretation,namely the continuous-time random walk (CTRW) model, thefractional Brownian motion (fBm) model, and random walkson fractal and disordered systems (for a review, see, e.g. (18)),then we will briefly describe different models covering super-diffusion processes that can be encountered in cells such asrun and tumble model, Levy flights and super-diffusive fBm.The continuous-time random walk model is a generaliza-tion of a random walk in which the diffusing particle waitsfor a random time between jumps. More generally, when thedistribution φ ( τ ) of waiting times τ is long-tailed and cannotbe averaged (with e.g. φ ( τ ) ∝ τ − (1+ α ) and < α < ), theensemble-averaged MSD shows anomalous scaling with apower law. A straightforward interpretation of a CTRW in thecontext of molecular biology is assimilating the waiting timesto interactions of the molecule with an immobile substrate (atthe relevant temporal and spatial scales). It is important tonote that an interaction with a characteristic residence timedoes not fulfill the conditions of the model. Interestingly,however, the waiting-time distribution of non-specific interac-tions, abundant in the cell, might be non-averageable and thusCTRW a good microscopic model for one type of anomaloussubdiffusion in the cell. It has been proposed to govern thecytosolic diffusion of nanosized objects in mammalian cells(19) and it has also been used to explain the lateral motion ofpotassium channels in the plasma membrane of cells (20).The fractional Brownian motion model is a different gener-alization of Brownian diffusion in which the jumps betweenlag times follow a normal distribution but respect a correlationfunction given by h x ( t ) x ( s ) i = 1 / t H + s H − ( t − s ) H ) for t > s > . A fBm process is thus characterized by theHurst index H , ranging between and . The value of H determines the type of jump dependence in the fBm process,such that H > / indicates a positive correlation between theincrements, Brownian motion is achieved for H = 1 / , andthe increments are negatively correlated when H < / . TheMSD of a fBm is given by D x ( t ) E ∝ t H , which, again, en- χ iv Woringer et al.et al. | A challenge for anomalous subdiffusion in living cells ompasses Brownian diffusion for H = 1 / and yields subdif-fusion for H < / or superdiffusion for H > / (see below).The fBm model describes faithfully the diffusion of particlesin a viscoelastic fluid (21), and it has been often argued thatmolecular crowding in the cell gives rise to microviscosityand therefore to anomalous diffusion. It was proposed as themodel of telomere diffusion in nucleus (22, 23).Another possible model for anomalous diffusion in the cellis that of random walks on fractal media and disorderedsystems . Fractals are self-similar mathematical objects builtupon the repetition of simple rules and characterized by anon-integer number: the fractal dimension . Although stillunder debate, some authors have proposed that chromatinorganization follows, as a first order approximation, a fractalstructure, and estimates of its fractal dimension have beenproposed (24). Random walks on fractals are subdiffusivedue to the spatial correlation of displacements, and the powerlaw scaling factor of the MSD with time is given by /d w ,where d w is the dimension of the walk that is specific to thefractal. Although the pertinence of a fractal network model todescribe molecular diffusion is still up to debate, it is justifiedto attempt to integrate the multiscale characteristics of the cellorganization to such fractal model.Amongst the existing superdiffusive motion in cells is the run-and-tumble process , which consists of alternating phases offast active and slow passive motion leading to transient anoma-lous diffusion (25). Initially observed for bacteria motion ithas recently been used to describe molecular motions in cellssuch as the motion of motors along cytoskeletal filaments.Motor proteins perform a number of steps (run) until they ran-domly unbind from the filaments and diffuse in the crowdedcytoplasm (tumble) before rebinding (26). The same couldalso stand for transcription factors in the nucleus searchingfor their initiation codon, alternating successively diffusionand 1D sliding along the DNA. Superdiffusive fBm which ischaracterized by an Hurst index
H > / has been describedas the intracellular motion of particles in the super-crowdedcytoplasm of a amibae (27). Finally, Levy flights , has pre-viously been proposed for intracellular actin-based transportmediated by molecular motors (28) and recently in the caseof a membrane targeting C2 protein (29).Note that by no means the above described models exhaus-tively cover the range of models that are known to exhibitanomalous diffusion (see e.g. (30–32)). However the CTRW,fBM, and random walks in a fractal models have been ex-tensively studied; more importantly, they have the potential to map parameters of the model to relevant biological andbiophysical features. Therefore, we will limit our discussionto the aforementioned cases, and how they can be used toanalyse and interpret experimental data obtained by FCS andSPT.
Which methods to correctly analyse diffusiveprocess?
Fluorescence Correlation Spectroscopy.
The principleof FCS consists in measuring the temporal variations of molec-ular concentration at a given position within the volume of abiological sample. This is achieved by monitoring the tempo-ral fluctuations of fluorescence signal emitted by the moleculespresent in the observation volume, which is excited with afocused laser. The underlying assumption of FCS is that thesystem is in a dynamic equilibrium and therefore the signalfluctuation can be correlated to the diffusion of moleculeswithin the observation volume. While the amplitude of thefluctuations relates to the number of molecules in the observa-tion volume, the decay of their autocorrelation in time dependson their mobility.A typical FCS set-up consists of an illumination laser and aconfocal microscope with a fast single-channel single-photondetector. The laser beam illuminates the detection volumewith, usually, a Gaussian intensity profile and excites thefluorophores in the focal volume. The emitted fluorescent lightis collected by the detector and it depends on the fluctuationsof the local concentration of the labelled molecules.Parameters such as the average number of molecules (N) andtheir mean residence time ( τ d ) in the confocal volume (sur-face) can be obtained either directly from this fluorescenceintensity fluctuation measurement or indirectly by a temporalauto-correlation of this fluctuation. The second method is themost popular approach for FCS data analysis (see Fig. 1). Themain drawback of standard FCS is the lack in directly monitor-ing possible spatial and/or temporal heterogeneities that willgive rise to deviation from pure Brownian motion. Several ap-proaches have been proposed to overcome this issue includingspot variation FCS (sv-FCS) (14, 33), line scanning FCS andSTED-FCS (34, 35) , as well as imaging approaches such as(spatio)-temporal imaging correlation spectroscopy ((S)TICS),raster imaging correlation spectroscopy (RICS) (36) or morerecently whole plane Imaging FCS (Im-FCS)(37). With thedevelopment of commercial microscopes coupled to FCS ca-pabilities, this technique and its derivatives are now becomingmore and more popular in biology labs. Woringer et al. | A challenge for anomalous subdiffusion in living cells aR χ iv | 3 (ms) t d t d : residence timeN: number of molecules F l uo r e s ce n ce F ( t ) timeDichroic mirrorObjective A P D C o rr e l a t o r Laser G ( t ) CCD
Dichroic mirrorObjectiveConfocalPinhole time P o s iti on ( x , y , z ) M S D ( µ m ² ) Time Lag (ms)
Laser Tube lensTube lens
A B
Fig. 1. Figure 1. Schematic view of the typical setup used in fluorescence correlation spectroscopy (A) and single/multiple particle tracking (B) experiments. A :A laser is focused on the fluorescently labelled sample by the objective of a microscope. The fluorescence is then collected by the objective and focused in a confocal way(using a pinhole) on a single photon counting detector (avalanche photodiode, APD). This detector records the fluctuation of fluorescence emission within the confocal volumeof the sample. A direct link to an electronic correlator authorize on line generation of the autocorrelogram. B : A laser is focused at the back focal plane of a microscopeobjective in order to obtain a full field illumination of the sample. The fluorescence emitted by each single particle present in the illumination field is then directly imaged on asensitive camera (Charge Coupled Device, CCD). A movie is obtained and the post processing of this movie allow tracking of the individual emitter and latter on, generationof Mean Square Displacement (MSD) as a function of lag time curves. A large range of dynamic processes leading to concentrationfluctuations (i.e, diffusion, flow, chemical reactions and differ-ent combinations of these) has been investigated to generatecorresponding analytical expressions of the temporal autocor-relation curve G ( t ) in the case of Gaussian (laser confocal)illumination/detection geometry (for a review, see (38) andreferences therein). For instance, in the case of a Brownian mo-tion in 2D, G ( t ) = 1 / { ¯ N (1 + 4 Dt/w ) } where w is the sizeof the beam waist and ¯ N is the average number of moleculesin the observation volume. The main approach to diffusiveprocess identification and quantification in FCS consists innon linear least square fitting of experimental autocorrelationcurves using above described analytical expressions and dis-criminate amongst these models which suits the best usingvarious statistical test. Although it can deliver quantitativevalues of the parameters of the statistically chosen model of motion, it could be strongly biased, in particular for complexmotions. A Bayesian approach to single spot FCS correlogramanalysis has been proposed to discriminate between differentmodels without bias (39, 40)Another way to discriminate between different types of mo-tion is to explore space and time with FCS using svFCS forexample. svFCS offers the opportunity to generate so-called"diffusion-laws" by plotting changes in the residence time( τ d ) as a function of the surface (i.e. laser waist) explored w . This has enabled to directly identify deviations from pureBrownian motion in the plasma membrane of cells (41) oranomalous diffusion occurring, either during first order lipidphase transition (42) or in non-homogeneous fluids, gels andcrowded solutions (43, 44). It has been recently extended tothe line-scanning STED-FCS (45) and to Im-FCS (46). χ iv Woringer et al.et al.
Fig. 1. Figure 1. Schematic view of the typical setup used in fluorescence correlation spectroscopy (A) and single/multiple particle tracking (B) experiments. A :A laser is focused on the fluorescently labelled sample by the objective of a microscope. The fluorescence is then collected by the objective and focused in a confocal way(using a pinhole) on a single photon counting detector (avalanche photodiode, APD). This detector records the fluctuation of fluorescence emission within the confocal volumeof the sample. A direct link to an electronic correlator authorize on line generation of the autocorrelogram. B : A laser is focused at the back focal plane of a microscopeobjective in order to obtain a full field illumination of the sample. The fluorescence emitted by each single particle present in the illumination field is then directly imaged on asensitive camera (Charge Coupled Device, CCD). A movie is obtained and the post processing of this movie allow tracking of the individual emitter and latter on, generationof Mean Square Displacement (MSD) as a function of lag time curves. A large range of dynamic processes leading to concentrationfluctuations (i.e, diffusion, flow, chemical reactions and differ-ent combinations of these) has been investigated to generatecorresponding analytical expressions of the temporal autocor-relation curve G ( t ) in the case of Gaussian (laser confocal)illumination/detection geometry (for a review, see (38) andreferences therein). For instance, in the case of a Brownian mo-tion in 2D, G ( t ) = 1 / { ¯ N (1 + 4 Dt/w ) } where w is the sizeof the beam waist and ¯ N is the average number of moleculesin the observation volume. The main approach to diffusiveprocess identification and quantification in FCS consists innon linear least square fitting of experimental autocorrelationcurves using above described analytical expressions and dis-criminate amongst these models which suits the best usingvarious statistical test. Although it can deliver quantitativevalues of the parameters of the statistically chosen model of motion, it could be strongly biased, in particular for complexmotions. A Bayesian approach to single spot FCS correlogramanalysis has been proposed to discriminate between differentmodels without bias (39, 40)Another way to discriminate between different types of mo-tion is to explore space and time with FCS using svFCS forexample. svFCS offers the opportunity to generate so-called"diffusion-laws" by plotting changes in the residence time( τ d ) as a function of the surface (i.e. laser waist) explored w . This has enabled to directly identify deviations from pureBrownian motion in the plasma membrane of cells (41) oranomalous diffusion occurring, either during first order lipidphase transition (42) or in non-homogeneous fluids, gels andcrowded solutions (43, 44). It has been recently extended tothe line-scanning STED-FCS (45) and to Im-FCS (46). χ iv Woringer et al.et al. | A challenge for anomalous subdiffusion in living cells ingle/Multiple Particle(s) Tracking. While the concentra-tion of the subset of fluorescent molecules within a confocalvolume in FCS experiments is close to the single-moleculeregime, the measurement gauges the average motion of theensemble of molecules diffusing in and out the observationspot. Conversely, SPT is by construction a single-moleculeapproach, monitoring thus the motion of individual molecules.One of the strengths of SPT is the potential to capture rareevents or behaviours that would otherwise be buried within anaverage.The principle of SPT experiments is simple, it consists inretrieving the changes in position of individual moleculeswithin the sample of interest, i.e. the time series of two-dimensional or three-dimensional coordinates of the moleculelocation. This is achieved in two stages: firstly by estimatingthe centroid of the measured point spread function (PSF) ofeach detected individual emitter, and secondly by linking thetrajectory of the same molecule between consecutive images.Importantly, the accuracy at which one is able to pinpointthe molecule position depends only on the signal-to-noiseratio of the measured PSF, obtaining sub-wavelength accuracytypically in the order of ∼
10 nm.The basic SPT experimental setup consists of an excitationlaser, a high NA objective, a set of dichroic and filters to sep-arate the excitation and emission wavelengths, a tube lens,and a highly sensitive camera capable of detecting single flu-orophores (see Fig. 1). The laser is focused on the backfocal plane of the objective to obtain a wide-field illuminationconfiguration, which can be adjusted to total internal reflec-tion (TIRF) or highly inclined illumination (HILO) (47) toincrease the SNR when studying molecular dynamics in cel-lular membranes or at the interior of cells, respectively. Thefluorescence light is collected by the same objective, and animage of the single emitters is formed on the camera plane viathe tube lens (13, 48).The amount of retrieved information about the biological sys-tem from an SPT assay depends on the nature of the experi-ment. The study of a slowly diffusing transmembrane proteinwill yield much longer traces than a fast diffusing transcrip-tion factor in the nucleus. In the latter case, the traces will belimited to the number of images in which the tracked particleremains within the depth of focus around the image plane,unlike the former case where photobleaching is the limitingfactor.The classical analysis of a set of trajectories consists in com-puting the dependence of the MSD (time-average or ensemble- average) over time from the distribution of jumps at increasinglag times defined by the camera acquisition, typically in theorder of tens of ms. However, as we will see in the follow-ing section, different approaches and estimators have beenproposed in order to analyze and interpret SPT data to itsfull extent. In comparison to FCS, the analysis of SPT hasbeen intensively investigated, and one can distinguish severalfamilies of techniques (see also for reviews: (24, 49, 50)).In the field of stochastic processes, the inference of a diffu-sion coefficient from a sampled process is a common problem(see for instance (51, 52)). However, this theory cannot beapplied when moving to experimental trajectories, and otherapproaches have been proposed.
MSD-based techniques.
A first family of SPT analysis algo-rithms tries to perform robust MSD inference. The use ofMSD to study diffusion was introduced by Einstein in 1906,and was revived in biology by (53). MSD analysis can eitherbe performed by inferring a diffusion coefficient from a singletrajectory (a setting studied in (54)) or by pooling varioustrajectories (55), and many refinements and estimators basedon the MSD have been proposed (56, 57).When inferring kinetic parameters from a series of singletrajectories, one faces the issue that for common trajectorylengths obtained in nuclear SPT (length of << points pertrack) and common localization error, inaccuracy might reach100% (54, 58). As such, any approach that uses MSD on shorttrajectories should be evaluated with great care. For longertrajectories (such as diffusion in a membrane), approacheshave been proposed that can segment trajectories based on thetype of motion (59). Hidden Markov Models (HMMs).
A second family of SPT anal-ysis algorithms derives from Markov models and HiddenMarkov Models. Most of them were derived to perform trajec-tory segment classification, the hidden variable inferred beingthe state of diffusion, or the current diffusion coefficient. Forinstance, (60) introduces the HMM-Bayes technique to inferwhether a trajectory segment is in one (or several) diffusiveor active transport states. Moreover, (61) implemented theinference of localization noise to infer switches in diffusioncoefficient within one trajectory. A similar approach was usedto detect confinement (62).These methods often rely on a fixed number of states, whichcomes from significant mathematical limitations. Some ofthese limitations were overcome using so-called variationalBayesian inference (63). The prototypical algorithm perform-
Woringer et al. | A challenge for anomalous subdiffusion in living cells aR χ iv | 5 ng variational Bayesian inference on a HMM is vbSPT (64).This algorithm can estimate the number of diffusive states andprogressively consolidate increasing information about thesestates as trajectories are analyzed. The algorithm was furtherrefined to incorporate the estimate of localization error (65). Inferring maps of diffusion coefficients.
A third family ofSPT analysis algorithms not only infers the diffusion coef-ficient over the population of diffusing molecules, but also aspatial map of diffusivity (66, 67). This approach has beenpioneered in membranes, where a high density of tracks caneasily be obtained. An extension of this approach using anoverdamped Langevin equation of the single molecule motionhas shed new lights on HIV-1 assembly within living cells(68). These promising techniques have not been tested beyondmembrane molecules, but the high diffusion coefficients offreely diffusing cellular proteins might render such a mapdifficult to establish. Moreover, unlike in membranes, pro-teins can reside at the same location with different diffusioncoefficients, depending on whether they are interacting with agiven structure or not.
Inferring anomalous diffusion.
Many approaches have beenproposed to infer anomalous diffusion in cells; some of themare reviewed in (69). A direct technique can be used by fittingthe MSD with a power law to estimate the anomalous diffusioncoefficient α . However, alternative techniques have beenproposed, many of them focused on the inference of model-specific parameters, or on techniques to distinguish betweentypes of anomalous diffusion.Several methods have been proposed to infer diffusion param-eters for several anomalous diffusion models. For the caseof diffusion in disordered (fractal) media, (70) proposes es-timators that can be applied to SPT, FCS and FRAP. For thecase of fractional Brownian motion, techniques to infer boththe anomalous diffusion coefficient ( α ) and the generalizeddiffusion coefficient ( D α ) have been proposed. The formerapproach (71) takes into account noise (localization error) anddrift, and uses Bayesian inference. The latter (72) relies onsquared displacements and uses least squares to estimate D α .Conversely, instead of trying to estimate the parameters of aknown model, a key question is to distinguish between var-ious anomalous diffusion models. A prototypical approach(73) used Bayesian inference to distinguish between Brown-ian, anomalous, confined and directed diffusion, and uses thepropagators associated with each different diffusion model.However, (74) found using simulations that it is very hard to distinguish between fBm and diffusion on a fractal when lo-calization noise is present, both in SPT and FCS. The authorsused a combination of techniques for the inference, includingMSD and p -variation techniques. In (23), the authors pro-pose a series of tests to "unambiguously" identify fBm, byprogressively proving that several other models are wrong.Other tests were proposed to distinguish fBm from a CTRWusing a test based on p -variations (75). The p -variations arethe finite sum of the p -th powers of the increments of thetrajectory. Finally, approaches inferring the mean first passagetime of a particle were used to distinguish between CTRWand diffusion in fractals (76, 77).Many other families of techniques to identify types of diffu-sion have been proposed. Some relied on maximum likelihoodestimates (78), auto-correlation functions (79) or on more ex-otic estimators (80). Another line of progress was made in thetype of models being simulated. For instance, (81) introduceda model in which TFs can bind and rebind in a dense chro-matin mesh. This model was successively fitted to explainanomalous diffusion of CTCF dynamics (82).Finally, we note that many models were developed to infertrapping potential in membranes ((83, 84) for instance). Wedo not review them here since their application seems limitedto membranes. Strengths & limitations of the two techniques.
A stronglimitation is that the experimental context, either in FCS or inSPT, may lead to spurious determination of anomalous diffu-sion. In other words, specific experimental parameters (lowstatistics, location noise, spatial confinement, etc.) and/or in-appropriate anaysis of the data can lead to incorrectly concludethat the diffusion exponent α = 1 . Those artifacts concernboth SPT (85) and FCS (86). This is for instance the case if α is determined by a fit of the MSD or the autocorrelationwith time and the statistical power is low (low sampling ofthe time points or short trajectories in SPT, low signal/noiseat small or large times in FCS). To avoid such caveats, modelselection must use more elaborate approaches to unambigu-ously demonstrate and characterize an underlying complexdiffusion process.So far, most of the inference tools available in the literatureonly partially account for the biases detailed above, and areusually limited in terms of the anomalous diffusion modelsthey consider. For instance, in (58), the authors showed that analgorithm not taking into account localization error was likelyto improperly estimate diffusion coefficients. Similarly, the χ iv Woringer et al.et al.
A stronglimitation is that the experimental context, either in FCS or inSPT, may lead to spurious determination of anomalous diffu-sion. In other words, specific experimental parameters (lowstatistics, location noise, spatial confinement, etc.) and/or in-appropriate anaysis of the data can lead to incorrectly concludethat the diffusion exponent α = 1 . Those artifacts concernboth SPT (85) and FCS (86). This is for instance the case if α is determined by a fit of the MSD or the autocorrelationwith time and the statistical power is low (low sampling ofthe time points or short trajectories in SPT, low signal/noiseat small or large times in FCS). To avoid such caveats, modelselection must use more elaborate approaches to unambigu-ously demonstrate and characterize an underlying complexdiffusion process.So far, most of the inference tools available in the literatureonly partially account for the biases detailed above, and areusually limited in terms of the anomalous diffusion modelsthey consider. For instance, in (58), the authors showed that analgorithm not taking into account localization error was likelyto improperly estimate diffusion coefficients. Similarly, the χ iv Woringer et al.et al. | A challenge for anomalous subdiffusion in living cells act that the observed proteins diffuse in a confined volumeleads to a sublinear MSD, a phenomenon that has been widelydocumented and that needs to be taken into account to properlydistinguish between genuine anomalous diffusion and mereconfinement effect. Similarly, tracking errors (misconnectionsbetween tracks) can also look like anomalous diffusion.Some of these biases can be minimized at the acquisition step(for instance by using fast frame rates and low labeling den-sity (58)), other need to be explicitly taken into account in themodel. As of today, most inference algorithms available havenot been benchmarked against realistic imaging conditions.Furthermore, a general realistic inference algorithm is stillmissing. Conclusion: the need for controlled bench-marks
Confronted with the variety of approaches described above,one would like to know the performance of each approachon typical representative datasets. For the comparison to befair, this demands two main ingredients: (i) the existence ofa reference dataset, or benchmark – possibly one referencedataset for each main classes of experimental methods and(ii) a fair, objective, transparent and open comparison pro-cess, with datasets, comparison procedures and performanceresults that are clearly stated and publicly available. Severalfields in computer science have been using open commu-nity competitions to organize the process and produce openbenchmarks for the community. Computer vision, appliedmachine learning or time series forecasting, among manyothers, have a long tradition of leveraging these competi-tions. The strategy has been widely successful because itparallelizes research along a vast community of high-skilledresearchers. Internet platforms or services are even avail-able to that purpose, including, among many others, Kaggle( ) or DrivenData ( ). This increases further the size of the competing com-munity, and the richness of the proposals. In fact, in addi-tion to providing reference datasets and benchmarks, opencompetitive challenges can also foster the emergence of radi-cally new approaches to the open problem at hand. Many ofthese competitive challenges are concerned with biomedicalapplications (for instance, http://dreamchallenges.org or https://grand-challenge.org ), includingseveral revolving around microscopy (see e.g. https://cremi.org ). Recently, a series of consecutive commu-nity competitions for single-molecule imaging have involved dozens of labs and focused on tracking algorithms (87), and2D and 3D localization for super-resolution (88). Finally,another challenge has also been set up recently to inferthe anomalous diffusion exponent from particle trajectories ( https://competitions.codalab.org/competitions/23601).In practice, an important feature of competitive challenges isto provide labelled data examples that the participants willbe able to use as a training set. Indeed according to standardmachine learning practice, this training dataset must be dis-tinct from the test set, that includes the data used to estimatethe performance of the algorithm. The organizers thereforeusually publish two datasets (training dataset and test), ofwhich only the training dataset comes with the label of eachexamples – only the organizers know the true label of the testdataset. After training, the results of the challenge is basedon some quantification of the performance of the participanttools on the test set, although performance on the learningset can also be communicated as a way to judge overtrain-ing/generalization capacities. In many cases however, it isnot possible to provide the “true” label of experimental data,because such a gold standard does not exist. In this case, com-puter simulations can be used to generate synthetic data, aslong as these simulations are realistic enough that the perfor-mance of the algorithms is not different than their performanceon real experimental measurements. In the recent challengeson super-resolution, training and test data were a combinationof computer-generated data and experimental data. Computer-generated data gives a clear access to ground truth whereasexperimental data incorporate uncharacterized biases that canaffect the inference process.Here we propose to organize an international open collabora-tive challenge for the quantification and analysis of moleculemovements in living cells via SPT and FCS. To date, thegeneration of realistic computer-simulated data has been ham-pered by the number of experimental biases to be taken intoaccount, and by the diversity of the diffusion models, in par-ticular for anomalous diffusion. For the challenge, we willgenerate both SPT and FCS data from the same set of sim-ulated trajectories and in different modalities (2D in mem-branes and 3D in the nucleus) using a dedicated open sourcesimulation software, simSPT ( https://gitlab.com/tjian-darzacq-lab/simSPT ), that is freely availableto the participants to generate their own additional trainingsets if needed.The challenge will be organized around various sub-challengesthat represent the main classes of experimental situations Woringer et al. | A challenge for anomalous subdiffusion in living cells aR χ iv | 7 high-density short trajectories in membranes, less denselong trajectories in membranes, very short trajectories inthe nucleus) and the main types of Brownian and anoma-lous diffusion (Brownian motion, fractional Brownian mo-tion, continuous-time random walks and diffusion on fractals),and mixtures thereof. In the long run, we will also proposesub-challenges where the molecule dynamics depends on thelocation, to emulate localized spatial heterogeneity in thedynamics (local potentials, position-dependent diffusion co-efficients). Moreover, we will progressively propose twochallenge categories. In parameter inference challenges, themodels used to generate the trajectories (Brownian motion,anomalous diffusion, ...) will be given and the task will beto infer as precisely as possible the value of the parametersused for the generation. In model selection challenges, thegoal will be to infer what model was used to generate the datagiven a known limited list of models.Finally, we are aware that it may well be that no generictool is able to solve all the sub-challenges evoked above.We are also aware that the difficulty of each sub-challengescan be quite variable. We therefore propose to start withthe simple challenges and work in collaboration with thecommunity involved in the analysis of molecular dynam-ics in living cells, to progressively climb the steps towardthe more difficult sub-challenges. In this strategy, maintain-ing an open communication channel between the organiz-ers and the participants is paramount. To this aim, we pro-pose to start with a mailing list that will be used to supportthis communication. Every interested individual is there-fore welcome to subscribe to the mailing list of the chal-lenge by visiting https://listes.services.cnrs.fr/wws/info/diffusion.challenge . Once regis-tered in the mailing list through this website, participants willbe able to exchange with the organizers and they will receivethe instructions to access the datasets of the challenge. Conflict of Interest Statement
The authors declare that the research was conducted in the ab-sence of any commercial or financial relationships that couldbe construed as a potential conflict of interest.
Author Contributions
MW, II, CF and HB developed these perspectives and wrotethe manuscript.
Funding
This work was partly funded by the CNRS-supported GDRImaBio, http://imabio-cnrs.fr . Bibliography
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