The effects of subdiffusion on the NTA sizemeasurements of extracellular vesicles in

biological samples

M. Majka,∗,† M. Durak-Kozica,† A. Kamińska,† A. Opalińska,‡ M. Szczęch,¶ andE. Stępień†

†Marian Smoluchowski Institute of Physics, Jagiellonian University, ul. prof. StanisławaŁojasiewicza 11, 30-348 Kraków, Poland

‡Institute of High Pressure Physics of the Polish Science Academy, ul. Sokołowska 29/37,01-142, Warsaw, Poland

¶Jerzy Haber Institute of Catalysis and Surface Chemistry Polish Academy of Sciences, ul.Niezapominajek 8, 30-239 Kraków, Poland

E-mail: maciej.majka@uj.edu.pl

AbstractThe interest in the extracellular vesicles (EVs)is rapidly growing as they became reliablebiomarkers for many diseases. For this reason,fast and accurate techniques of EVs size char-acterization are the matter of utmost impor-tance. One increasingly popular technique isthe Nanoparticle Tracking Analysis (NTA), inwhich the diameters of EVs are calculated fromtheir diffusion constants. The crucial assump-tion here is that the diffusion in NTA followsthe Stokes-Einstein relation, i.e. that the MeanSquare Displacement (MSD) of a particle growslinearly in time (MSD ∝ t). However, we showthat NTA violates this assumption in both arti-ficial and biological samples, i.e. a large popu-lation of particles show a strongly sub-diffusivebehaviour (MSD ∝ tα, 0 < α < 1). To sup-port this observation we present a range of ex-perimental results for both polystyrene beadsand EVs. This is also related to another prob-lem: for the same samples there exists a hugediscrepancy (by the factor of 2-4) between thesizes measured with NTA and with the directimaging methods, such as AFM. This can beremedied by e.g. the Finite Track Length Ad-

justment (FTLA) method in NTA, but its ap-plicability is limited in the biological and poly-disperse samples. On the other hand, the mod-els of sub-diffusion rarely provide the direct re-lation between the size of a particle and the gen-eralized diffusion constant. However, we solvethis last problem by introducing the logarith-mic model of sub-diffusion, aimed at retrievingthe size data. In result, we propose a novelprotocol of NTA data analysis. The accuracyof our method is on par with FTLA for small('200nm) particles. We apply our method tostudy the EVs samples and corroborate the re-sults with AFM. Incorporating the sub-diffusiveeffects reduces the average measured EV diam-eter by '50% in comparison to the normal dif-fusion models. The remaining discrepancy be-tween NTA and AFM can be explained withseveral known effects, which we also discuss inthis article.

IntroductionExtracellular microvesicles (EVs) are the frag-ments of cell membranes generated by bothprokaryotic and eukaryotic cells.1 EVs diame-ter varies between 50 and 1000nm.2 Their pres-

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ence was firstly reported using hemocytometrymethods in late 60’s and suggested a proco-agulant potential.3 Neglected for the next 20years, EVs drew back the scientists’ attentionin late 90s, as possible thrombotic activators.4Microscopically, they consist of cell componentsincluding a lipid bilayer, cytoplasm and ribo-somal fragments.5 It has been commonly ac-cepted that EVs can be produced either bybudding the cell membrane fragments (ecto-somes), or by the subsequent exocytosis.2,6,7In blood, the main source of EVs are plateletsand endothelial cells. Nevertheless, in stress-ing conditions (hypoxia, inflammation or hyper-glycemia), also macrophages and neutrophilscan release a number of EVs.8,9 The rapidlygrowing interest in EVs has a significant im-pact on the clinical and basic science, provid-ing the novel potential for nanomedicine withnew biomarkers, therapeutic targets and cell-to-cell communication vehicles.10 The majorityof clinical data, obtained from the flow cytom-etry studies, show the heterogeneousness of EVsurface antigen profile.1,4,8,9 However, the sig-nificant progress in nanotechnology has recentlyallowed a systemic approach to isolation, enu-meration and the characterization of EVs size,shape and molecular components.11,12Present research has demonstrated that, the

larger population of EVs - ectosomes - rep-resents a rather heterogeneous population ofvesicles whose size ranges from 100 to 1000nm in diameter.11,13 Their size distributioncan be determined with a range of meth-ods, e.g.: scanning and transmission electronmicroscopy (EM),5,8,14 tunable resistive pulsesensing (TRPS),15–17 atomic force microscopy(AFM)18 and nanoparticle tracking analysis(NTA).19,20 Fig. 1 illustrates their applicationto EVs studies.NTA is of particular interest in this article,

as it is involved in 58% of the current EVs-related research.21 In this technique a sampleof a biological fluid is observed through a mi-croscope. A laser beam illuminates the samplefrom the direction perpendicular to the opticalaxis. The particles diffusing in the plane of thelaser beam are visible thanks to the scatteredlight and their trajectories can be recorded (Fig.

Figure 1: Extracellular vesicles visualizationmethods. (A) Image showing typical tracks ofEVs moving under Brownian motion recordedby means of the NTA method. (B,C) Transmis-sion Electron Microscopy (TEM) images of EVsisolated from human plasma using the ultra-centifugation method. Samples were fixed with2.5% glutaraldehyde in 0.1M cacodylic bufferand then postfixed in 1% osmium tetroxide so-lution. (D) Topography of EVs placed on a poli-L-lysine coated slide. Samples were analysedby means of Atomic Force Microscopy (AFM).The yellow line shows the cross-section, the col-orimetric scale indicates the Z dimension pre-sented as a height profile across the section line(E).

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1A). However, the size of the particles is notconcluded from the direct observation (which isconstrained by the diffraction limit), but fromthe estimation of their diffusion constant. Inthe simplest approach, for each trajectory themean square displacement (MSD) can be cal-culated and the Stokes-Einstein theory of diffu-sion predicts that MSD grows linearly in time,i.e.:

MSD(t) = 2εD1t D1 =kBT

3πη0d(1)

where ε is dimensionality (ε = 2 for NTA), kB isthe Boltzmann constant, T is temperature, η0 isthe viscosity of a sample and d is the diameterof a particle. Equation (1) relates the diameterd of a particle to the diffusion constant D1 andmakes it possible to find the diameter distribu-tion from the set of trajectories. The diffusiondescribed by the relation (1) is called normal.NTA has entered the field of EVs research rel-

atively recently and the evaluation of its appli-cability to biological samples is still in progress.Most comparative studies involving NTA fo-cuses on its advantages over e.g. flow cytom-etry and dynamic light scattering (DLS).12,18,19NTA also proves to be competitive againstAFM or EM, as it is cheap, fast and doesnot require much additional preparation of thesample that could affect the biological state ofEVs.12,18 Thus, it is looked at as a ’method-of-choice’ in the EVs characterization with apossibly broad use in the future diagnostics.However, few studies examine whether the as-sumptions of NTA are satisfied in the biologicalcontext. In particular, the presence of normaldiffusion is usually taken for granted.In this article we show that the assumption

of normal diffusion is neither true in the bio-logical samples nor, more surprisingly, for ar-tificial mono-disperse beads. In all cases a re-markably huge population of particles has theMSD that grows sub-linearly in time. This is astrong signature of sub-diffusion. We obtainedthese results consistently in several different ex-periments involving EVs from human plasmaand artificial beads. Measurements were per-formed on two different NanoSight NTA ana-

lyzers (Malvern Instruments) in two separateinstitutes, which make us believe that the issueis inherent at least to this popular producer.Thus, the main goal of this paper is to proposethe new procedure of NTA data analysis thataccounts for the sub-diffusive effects.The claim that the fundamental assumption

of NTA, i.e. the relation (1), is not satisfied,might be surprising, as the NanoSight performswell in the industrial applications. The rea-son is that, by default, the NanoSight soft-ware applies the finite track length adjustment(FTLA)22,23 algorithm to the data. The actualimplementation of FTLA in this software is notspecified, but much can be inferred from theunderlying study by Savney et al.22 The pro-cedure assumes that the observed distributionof sizes is the convolution of the ideal size dis-tribution and the broadening effects from thefinite length of recorded tracks. The parame-ters of the ideal distribution are fitted via themaximum likelihood method and this ideal dis-tribution is provided as the final result. Twosets of assumptions are involved here. One isthe form of the ideal distribution itself. Fora mono-disperse sample it is expected to be amono-peaked function. However, when the un-known number of different-sized species is in-volved, one cannot predict how many peaks arenecessary. Indeed, FTLA performance exacer-bates in the poly-disperse model systems.31 Theother assumption is that the broadening effectsfollow from the normal diffusion model. As thisis not satisfied, the corrections are only a crudeapproximation for these effects. Nevertheless,FTLA significantly improves the results for themono-disperse model systems. However, its in-fluence on the much more complex EVs samplesis uncontrolled and, in fact, it was reported tocause artefacts.12 For these reasons a more di-rect method is desirable.There is also another problem related to the

biological applications of NTA. When the samesample is simultaneously measured with the di-rect imaging methods (e.g. AFM or EM) andNTA, NTA turns out to significantly overesti-mate the average size. This discrepancy is ob-served in the study of EVs,24,25 but also for thecoated gold nano-particles26 and, to a minor ex-

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tend, model polystyrene beads.20 Our own mea-surements of EVs from human plasma, whichwe discuss further in this article, indicate thatthe difference between AFM and NTA can beof a factor 3-4 for normal diffusion models andabout 2 for FTLA.A few effects contributing to this difference

are already recognized. One important reasonis that NTA has low sensitivity to the smallEVs (d < 50nm), which cuts out the low-endof the size distribution.18,24 In our AFM study,excluding all the particles smaller than 50nm in-creases the mean EV diameter by roughly 34%-40% (see Table 2). Another contribution comesfrom the fact that NTA measures the hydrody-namic radius, which might appear larger thanthe inelastic core measured in AFM and EM.This discrepancy is well-documented for the la-tex and polystyrene beads in monodisperse so-lutions, studied by both NTA and DLS.27–29However, to see this effect in AFM one mustusually reduce the widening due to the tip con-volution effect (algorithmically or by using aproperly sharp tip),30 as it might easily domi-nate the results. In general, the hydrodynamicwidening overestimates the size by 50% for verysmall particles (d ' 16nm),27,29 by 30% ford ' 50nm, by 5-20% for 100nm particles27–29and for d > 400nm the effect is negligible.27There are few systematic studies on this effectin the polydisperse systems, but from the workof Filipe et al. one can conclude that the dif-ference does not exceed 30%.27 Finally, NTA isinfluenced by the lengths of recorder trajecto-ries, e.g. the 100nm particles can be measuredas 15-25% larger when the extremely short (lessthan 5 points) trajectories are used.31 On theother hand, the binding to a surface and tip-induced deformation might increase the appar-ent size of EVs in AFM. For a sphere with radiusrs which is flatten into the disc with radius rd,but conserves its area (so 4πr2s = 2πr2d), rs is atworst 30% smaller than rd. The deformationscan be minimized by applying the ’tapping’ or’non-contact’ mode in AFM. Summarizing allof these effects, the size of particles measuredin NTA should be roughly 1.5-2.5 times largerthan the one resulting from AFM. Therefore,the sub-diffusive model for NTA should predict

the EV size that fits into this range.The NTA measurements are based on the ob-

servation of the diffusive motion, which mightbe affected by a plethora of effects. For bothmono- and poly-disperse samples these factorsare e.g. confinement, molecular crowding, in-homogeneous distribution of particles and in-teractions between particles (electrostatic, hy-drodynamic etc.). Poly-dispersity additionallyintroduces the unspecific excluded volume in-teractions32,33 and EVs might also bind bio-chemically with other EVs and proteins.9,34 Allof these aspects violate the assumptions of theStokes-Einstein theory35–37 (1), which describesthe freely diffusing particle in an uncorrelated,molecularly homogeneous environment. An-other issue might be the design of the NTAinstrument itself, e.g. it might induce tempera-ture gradients, counter-sedimentation flows andthe field of view might be distorted. In whatfollows, the experimentally measured MSDs candeviate significantly from the linear dependenceMSD ∝ t. This can be seen in Fig. 2A, wherewe show MSD(t) for three exemplary trajecto-ries from our measurements. For the short-to-intermediate time MSD(t) resembles the freediffusion behavior, but the slope of the dataon the log-log plot is clearly sub-linear, i.e.MSD(t) ∝ tα, where 0 < α < 1. This is charac-teristic for the sub-diffusion. From Fig. 2A onecan see that fitting the normal diffusion model,i.e. MSD(t) = 2εD1t + C2 (where C2 accountsfor the localization noise, see Berglund38), tothe sub-diffusive data leads to an underesti-mated D1 and, in result, an overestimated d.The free-diffusion regime is usually culminatedwith a local maximum, after which MSD(t) de-creases and fluctuates strongly. This behaviourusually indicates that the boundary effects be-come dominant.The sub-diffusion (or, more generally, anoma-

lous diffusion) is a phenomenon ubiquitouslyobserved in the biological systems,39–43 whichgains more and more recognition in the bio-and life- sciences.44 This is particularly impor-tant since the sub-diffusion affects the efficiencyof transport in the molecularly crowded envi-ronment, especially cytoplasm.39–41 Microscop-ically, sub-diffusion is usually caused by the

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Figure 2: A: MSD(t) plots for three exemplaryEVs trajectories, representing normal diffusion(blue circles), intermediate sub-diffusion (greensquares) and strong sub-diffusion (red trian-gles). The vertical lines show the cut-off timetaken into analysis. The data are fitted with thelinear model (8) (short-dashed lines), power-law model (9) (dashed lines) and logarithmicmodel (10) (solid lines). For a well pronouncedsub-diffusion the linear model clearly deviatesfrom the data (red and green plots), leading tothe underestimation of diffusion constant. B:The mean position increment < ∆x(t) > and< ∆y(t) > for each trajectory from panel B,showing the influence of drift. The incrementsare fitted with the power-law model ∝ tγ to testfor linearity.

geometrical confinement, trapping events thatintervene with the normal diffusion or by thehighly inhomogeneous viscosity of a system.39The mathematical models of sub-diffusion arenumerous.35–37,45–48 In particular, three classesstand out, i.e. Fractional Brownian Motion,48Continuous Time Random Walks47 and ScaledBrownian Motion.49–51 The former two are ca-pable of ergodicity breaking,51–54 which can beexamined in the single particle tracking experi-ments. However, most of these models are pre-occupied with obtaining the anomalous expo-nent α, i.e. reproducing the asymptotic depen-dence:

MSD(t) = 2εDαtα (2)

where the generalized diffusion constant Dα isusually assumed as a fitable parameter. Al-though (2) and (1) seem deceptively similar,they cannot be applied in the same manner.While D1 is straightforwardly related to η0 andd, there is no such simple dependence for Dα.This can be seen from the dimensional analy-sis, i.e. the units of D1 and Dα are m2/s andm2/sα, respectively. Thus, one might interpretthat there is an additional time scale τ involvedin the sub-diffusion, such that Dα = D1/τ

α−1.However, τ and D1 cannot be simultaneouslydetermined from the experimental data via sim-ple fitting, i.e. if one fits MSD with D1t

α/τα−1

model, it results in any pair of D1 and τ thatcan be recombined intoDα corresponding to thedirect fitting of model (2). In what follows, it isalso not possible to measure the distribution ofdiameters d with the aid of the model (2). Nev-ertheless, since α is a very convenient measureof anomalousness, we will frequently resort toit. However, for the purpose of application inNTA we must consider an alternative approach.It should be emphasized that while the power-

law model (2) is a popular description of sub-diffusion, it is mainly phenomenological. Infact, sub-diffusion is always a composition ofmultiple microscopic processes, as indicated bythe Mori-Zwanzig type models48 or by the inge-nious visco-elastic approach, proposed by Goy-chuk.46 Thus, different models of similar behav-ior can be justified on this basis. To remedythe problem of physical interpretation for Dα,

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we first attempted to use the modified power-law model: MSD(t) = 2εD1t/(1 + t

τ)1−α, which

asymptotically (t� 0) behaves like (2) and sep-arates D1 and τ . However, fitting this model toMSD data did not result in the stable values ofD1 and τ , similarly as in the case of (2). Mostprobably this is because NTA do not provideenough information on the t → 0 behavior, forwhich these two models mainly differ.Our solution is provided by the logarithmic

model, based on the following, qualitative rea-soning. In the dispersed colloidal solution thereis a certain average distance between particles.At the short time scale, a particle diffuses freely,but over a longer period, as

√MSD(t) becomes

comparable to this average length-scale, a par-ticle is more likely to encounter obstacles (i.e.other particles). Thus, the viscosity of a system’perceived’ by this particle effectively grows intime. This can also happen if the particles aresystematically carried by a drift into the denserregion (i.e. near a barrier). In order to accountfor these effects, we introduce the time depen-dent viscosity η(t). In the lowest order, it canbe approximated as η(t) ' (1 + vt)η0, where vis some effective speed of viscosity change. As-suming that the short-time diffusion is still nor-mal, the increment of MSD over some dt reads:

dMSD(t) =2εkT

3πη(t)ddt ' 2εD1

1 + vtdt (3)

One can integrate this expression over time, toobtain:

MSD(t) =

∫ t

0

ds2εD1

1 + vs=

2εD1

vln(1 + vt) (4)

The logarithmic model can mimic the power-law behavior over several orders of magnitude,when tuned properly. In fact, it can be seen asan example of the ultra-slow scaled Brownianmotion49,50 and a similar result is encounteredin e.g. the granular gas.51 This model preservesthe standard interpretation of D1 and it keepsD1 and v separated enough to allow reliable andrepetitive MSD fits. One might also check thatin this model limt→0 MSD(t)/t = 2εD1, so itpredicts the non-zero diffusivity for every t, asdesired. It also leads to the normal diffusion for

v = 0, i.e. limv→0 MSD(t) = 2εD1t. The qual-ity of fits provided by this model is comparableto the power-law model.

Samples and measurementsFor the purpose of this work we gathered theMSD data from NTA for the polystyrene (PS)beads with diameters 203 and 453 nm (ThermoScientific) and EVs extracted from blood sam-ples. Two series of experiments were carriedout: one, involving EVs and PS beads, atthe Institute of High Pressure Physics (PolishAcademy of Science, Warsaw) and the other,involving only PS beads, but at several concen-trations, at the Jerzy Haber Institute of Catal-ysis and Surface Chemistry (Polish Academy ofScience, Krakow). The measurements were per-formed by two different teams and involved twodifferent NTA instruments.

PSK: NTA measurements of PS beadsin Krakow. The 1% aqua solution of the203 nm PS beads (Thermo Scientific) was di-luted 103 and 104 times. The 1% solution of453 nm PS beads was diluted 103, 2 × 103

and 5 × 103 times. For each concentrationand size the NTA measurements were repeatedthree times. For 103 dilution runs lasted 25sand 60s for the others. The experiments wereperformed with NanoSight NS500 instrument(Malvern Instruments Ltd. United Kingdom)equipped with 405 nm laser, at a room temper-ature of 20 − 24◦C (varying between measure-ments). The shutter and gain were adjustedmanually. The NTA 3.2 (dev build 3.2.16) soft-ware was used for capturing the data as well asthe preliminary and FTLA analysis.

PSW : NTA measurements of PS beadsin Warsaw. The 1% solutions of 203 nm and453 nm polystyrene beads (Thermo Scientific)were diluted 7× 103 and 5× 103 times, respec-tively. The NTA measurement lasted 90 s forbigger particles and 60 s for the smaller, eachrun was repeated three times. The temperatureread 29◦C. The NanoSight NS500 instrument(Malvern Instruments Ltd. United Kingdom)equipped with 405 nm laser was used. Theshutter and gain were adjusted manually and

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the NTA 2.3 (build 0025) software was appliedto capture the data as well as for the initial andFTLA analysis.

EVs: blood donors. Samples from 3healthy donors (coded further as P1, P2 andP3) were used for this study. The samples hasbeen collected in the conjunction with our pre-vious study on diabetics.55

Ethics. Bioethical Committee at Jagiel-lonian University Medical College (JUMC) ac-cepted all project’s protocols and forms, in-cluding an information for patients form anda consent form for participation in a researchstudy. The permission No. KBET/206/B/2013is valid until 31st of December 2017.

Blood collection and platelet poorplasma (PPP) preparation. All blood sam-ples were drawn at the same time of the day(between 08:00 and 10:00 am) with venipunc-ture with > 21-gauge needle in the antecubitalvein following the application of a light tourni-quet. Citrate blood was centrifuged twice at2500 g for 15 min to obtain PPP. The plasmasamples were aliquoted and frozen at −80◦Cuntil further analysis. Before measurementssamples were thawed in a 37◦C water bath andvortexed for 30 s.

Nanoparticle Tracking Analysis (NTA)of plasma EVs. To avoid EVs aggregation,all samples were diluted 100x in HEPES buffer(10 mM Hepes/NaOH, 140 mM NaCl, 2.5 mMCaCl2, pH 7.4). The NTA measurement wasperformed with a NanoSight NS500 instrument(Malvern Instruments Ltd, United Kingdom),equipped with a sample chamber with a 405-nm laser. The assay was performed at roomtemperature 23.3± 0.1◦C. For each donor (P1-P3) the measurements has been carried out 3times. The samples were measured for 30s withthe manual shutter and gain adjustments in ad-vanced settings. The NTA 2.3 Build 0025 soft-ware was used for capturing of the data as wellas their preliminary and FTLA analysis.

Atomic Force Microscopy (AFM) anal-ysis of plasma EVs. Poly-L-Lysine Slidesfrom Thermo Scientific (cat no J2800AMNZ)were previously cleaved into 1 × 1cm plates,rinsed both sides with distilled water andcleansed by compressed air. Samples with EVs

were spread on glass slide and incubated for1 hour in a humid chamber at room temper-ature. Then slides were rinsed three times bygently dipping in PBS (pH=7.4) and fixed in2.5% glutaraldehyde in PBS for 30 minutesat room temperature. After that, slides wererinsed and analyzed by means of the AFMtechnique. To determine the size distributionof EVs, 20 × 20µm topographical images ofthe samples were performed by Atomic ForceMicroscopy - Nanoscope IIIa Multimode-SPM(Veeco Instruments, Santa Barbara, CA, USA)in contact mode. AFM images were recordedin liquid (PBS) using fluid chamber and imag-ing conditions included a 0.4 Hz scan rate, 256points collected per line (pixel resolution). As aprobe, non-conductive pyramidal silicon nitridetip (MLCT, Bruker) with a resonance frequency10-20 kHz, nominal spring constant 0.01 N/mand with a radius 100 nm was used. The topo-graphical analysis of AFM images was carriedout using SPIP software version 6.5.2 (ImageMetrology A/S, Horsholm, Denmark). In thissoftware the Particle and Pore Analysis modewas used to asses the EVs diameters. An exem-plary image is shown in Fig. 1D and E. Imagefeatures that were brighter (higher) than thebackground were recognized as particles, whilethe darker (lower) were identified as pores. Thedetection threshold was set to 0.2 nm. The al-gorithm identifies the area occupied by a par-ticle and calculates the diameter of a disc withthe same area. This results is provided as anEV’s diameter. Finally, the "Post processing"was applied to classify EVs and the size distri-bution histograms were prepared. Because theresolution of AFM images performed in contactmode depends on the radius and shape of thetip (the tip convolution effect) and the tip usedin this study was relatively wide, we also de-cided to recalculate the size of observed parti-cles basing on correction method from Engel etal.56

NTA data analysisIn the preliminary stage of analysis the NTAsoftware identifies the trajectories of observed

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particles and this raw data are available to theuser (alltracks files). Each trajectory is given asa sequence of positions ~ri = (xi, yi) (in pixels)on the consecutive frames indexed by i, cap-tured by the camera. Thanks to the frame-rateand calibration parameter provided by the soft-ware it is possible to recalculate these data intothe actual physical units, i.e. nanometers andseconds. NTA software is also able to distin-guish the valid trajectories from the artefacts onthe basis of combined length and scattered lightintensity criteria. These trajectories are labeledas included in the distribution in the output .csvfiles. We restrict our analysis solely to these tra-jectories and obtain the size distributions withFTLA algorithm for further comparison. How-ever, the NanoSight software gives us no con-trol over the truncation of the trajectories anddrift treatment. Since our goal is to explicatethe difference between the Stokes-Einstein andsub-diffusive models, we decided to perform theMSD analysis on our own, maintaining the fullcontrol over the data processing.Our first goal is to determine the MSD func-

tion individually, for each trajectory. In thefirst step of our analysis we introduce the in-crements :

∆~ri(n) = ~ri+n − ~ri (5)

where ∆~ri(n) = (∆xi(n),∆yi(n)) and calculatetheir average value:

< ∆~rn >=1

N − n

N−n∑i=1

(~ri+n − ~ri) (6)

N is the maximal number of frames on which aparticular molecule is recognized. Having foundthe increments, we can calculate the MSD(n),which reads:

MSD(n) =1

N − n

N−n∑i=1

(∆~ri(n)− < ∆~rn >)2

(7)One should note that < ∆~rn > is also the mea-sure of the average drift, so our definition re-moves the influence of a flow along a trajectory.In the Fig. 2A, we present the log-log plots ofMSD for three exemplary trajectories rescaled

to the actual physical units, i.e. nanometersand seconds. These plots show several featureswhich are representative for our data. In gen-eral, MSD consists of an initial period of thelinear growth on the log-log plot up to the lo-cal maximum, followed by the interval of de-crease and strong fluctuations. Most of the freediffusion models, normal or sub-diffusion, can-not describe the decrease in MSD, which usu-ally indicates the presence of boundaries in thesystem.45 One possible exception is the scaledBrownian motion,49,50 which has a tendency toform a local maximum in MSD, but even inthis case the effect is most pronounced in theconfinement. Thus, we conclude that the post-maximum behavior is induced by the bound-aries. For this reason, we must propose thecriteria to exclude it from the further analysisbased on the free diffusion models.The features we just described are most rep-

resentative for the long (N > 9) trajecto-ries. Unfortunately, it is well known that whileNTA measures a massive number of trajecto-ries, there is no control over their length and,usually, only a mere 10− 15% of them has thelength of N ≥ 5. This is particularly severin the EVs samples, in which the number oftrajectories is generally low. Due to the poorstatistics, the shortest trajectories have verynoisy MSD plots,57 which might significantlydeviate from what we described in the previ-ous section. Additionally, the minimal trajec-tory length can also significantly broaden thediameter distribution.31,57 In our case we facethe same problem, but, in addition, we wantto analyze only those trajectories for which theinitial interval of MSD at least vaguely resem-ble the free-diffusion behavior. This decreasesthe number of trajectories even further. Thus,for the purpose of our study, a trajectory isaccepted if its first three MSD points satisfyMSD(1) < MSD(2) < MSD(3). This ensuresthe existence of a minimal, strictly growing se-quence. This also means that we use trajecto-ries of N = 4 at least, which is still close tothe N = 5, recommended by Gardiner. Fordonors P1, P2 and P3 the NanoSight softwarehas admitted respectively 919, 679 and 366 tra-jectories of which 73%, 62% and 58% has been

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accepted according to our criterion. The afore-mentioned problems are incomparably less se-vere for the PS beads, for which NTA providedusually between 1500 and 6000 relevant trajec-tories of which 60-80% passed our criteria.We also had to decide on how many points

of an individual MSD sequence are accountedfor the free-diffusion behavior, i.e. we had tochoose the truncation time T . Since we ex-pect to encounter a local maximum in MSD,we propose that T is equal to the first n vi-olating MSD(n + 2) > MSD(n). This con-dition can detect the systematic decrease inMSD, which usually occurs after the local max-imum. However, it also allows some acciden-tal detours from the strictly growing characterof MSD (e.g. MSD(n + 1) < MSD(n) whileMSD(n + 2) > MSD(n)). The random vari-ability of MSD must be expected, especially forshort trajectories, since the sample size fromwhich an individual value of MSD(n) is calcu-lated reads N−n.57 This number is usually low(especially for EVs) and decreases with subse-quent n.Having proposed the criteria of data selection

and truncation, we address the central issue ofthis letter, which is the sub-diffusive dynamicsof EVs. All the selected trajectories were fittedwith the three following models:

MSD(n) = 4D1t+ C2 (8)MSD(n) = 4Dαt

α + C2 (9)

MSD(n) = 4D1

vln(1 + vt) + C2 (10)

where the additional constant C2 accountsfor the localization noise as advocated byBerglund.38 In Fig. 2A the exemplary fits ob-tained with all the three models are shownfor weak and strongly sub-diffusive trajectories.While for normal diffusion they perform com-parably well, it is evident that for strong sub-diffusion the linear model visibly deviates fromthe data. While the localization noise is able topartially remedy the quality of fits, it is still in-sufficient, leading to the underestimated valueof D1, and, in turn, to the overestimated d. Itshould be denoted that the sub-diffusive modelspredict extremely low level of the localization

Figure 3: Anomalous exponent α in EVs sam-ples: A: The histogram of α resulting fromthe fitting of all accepted MSD data (see text)with the power-law model. For each donor thehistogram has a maximum near α ' 0.7, in-dicating the sub-diffusive behaviour. B: Thehistogram of α obtained only for long (N >50) trajectories. The maximum shifts towardsα ' 0.8. C: The joint histogram of exponentγ resulting from fitting the average increments< ∆xn > and < ∆yn > with the power lawmodel ∝ tγ. For all donors the distributionshave distinct peaks at γ ' 1 and γ ' 0.

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Table 1: The summary for anomalous exponent α. Abbreviations: EV - biological samples (extra-cellular vesicles), PSW - polystyrene beads measured in Warsaw, PSK - polystyrene beads measuredin Krakow, C0 = 1% - initial concentration of sample solution, d - the certified diameter of a bead,Nall - number of all accepted trajectories with length N ≥ 4, < α > - mean anomalous exponentfor all accepted trajectories, SD - standard deviation of < α >, NN>50 - number of accepted tra-jectories longer than N = 50, < α >N>50 anomalous exponent calculated only for long trajectories,SDN>50 - standard deviation of < α >N>50.

Id. C0 × 103 d[nm] Nall < α > SD NN>50 < α >N>50 SDN>50

EV - - 673 0.71 0.22 80 0.77 0.13EV - - 420 0.71 0.24 50 0.81 0.16EV - - 212 0.73 0.26 22 0.75 0.14PSW 8 200 1872 0.74 0.20 437 0.80 0.16PSW 5 453 1861 0.73 0.20 611 0.80 0.16PSK 1 203 6807 0.69 0.22 566 0.80 0.16PSK 10 203 4141 0.73 0.21 1054 0.81 0.17PSK 1 453 1201 0.70 0.24 376 0.81 0.16PSK 2 453 1670 0.71 0.21 610 0.80 0.16PSK 5 453 537 0.74 0.19 268 0.80 0.16

noise.The power-low model provides a conventional

test for anomalous diffusion. Fig. 3 and Fig.4 show the histograms of the anomalous ex-ponents α for EVs and PS beads, respectively.The numerical results are also summarized inthe Table 1. In general, the distributions ofα calculated for all the accepted trajectories(Fig. 3A and Fig. 4A), have a mono-peakedshape with a maximum at < α >' 0.69− 0.74and the standard deviation of approximately0.19−0.26. This is common for all samples, i.e.both PS beads and EVs, which shows the ubiq-uity of the sub-diffusive behavior in NTA. How-ever, this data include a huge number of shorttrajectories with low predictive power,57 so tominimized their influence we also plot the samedistributions, but only for the long (N > 50)trajectories (Fig. 3B and Fig. 4B). While thisslightly shifts < α > to 0.75-0.81, the pres-ence of sub-diffusion clearly persists. Furtherincrease in the minimal length of admitted tra-jectories do not lead to a significant growth in< α >. The experiments on the bigger beadsalso suggest that there is a dependence between< α > and concentration, i.e. as the dilutiongrows by 5 times, < α > increases from 0.7 to0.74 (see Table 1). This is in agreement with

our conjecture that the sub-diffusion is at leastpartially caused by the crowding in a sample,though the effect is rather weak.The drift of particles was also analyzed. The

mean increments < ∆xn > and < ∆yn > (seeFig. 2B) were fitted with the power law model∝ tγ to test their linearity. In Fig. 3C and Fig.4C the histograms of γ are shown. A distinctpeak at γ ' 1 indicate that, indeed, the driftwas approximately constant during the obser-vation. The additional peak at γ = 0 was asso-ciated with the particles experiencing no flow,i.e. < ∆~rn >= 0. Since the majority of parti-cles were carried by an approximately constantdrift, it is possible that they were systematicallyconveyed into the denser regions, in agreementwith the assumptions of our logarithmic model.Another step is to compare the efficiency of

size prediction between the normal diffusionmodel, logarithmic sub-diffusion and FTLA al-gorithm. As explained before, the power-lawmodel cannot be used for this purpose. Themono-disperse solutions of PS beads with 203nm and 453 nm diameters and at several differ-ent concentrations were used as the test sam-ples. In this paragraph we will refer to Warsawmeasurements as PSW and to experiments inKrakow as PSK . The size histograms obtained

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Table 2: Size determination in the NTA and AFM measurements of EVs and PS beads of differentnominal size d and at different dilutions (dil.). The NTA data were analysed with the normaldiffusion model, sub-diffusion model and the FTLA algorithm. For AFM the mean size < d > wascalculated for both: entire dataset (column all) and for the particles with d ≥ 50nm (for directcomparison with optical methods). The table summarizes < d >, standard deviation SD of d andthe position of the distribution maximum, dmax. All parameters were determined by fitting theH(d) histograms (see Fig. 5 and 6) with the log-normal function (except for FTLA, which wasprovided by the NanoSight software).

dil. < d > ± SD[nm] dmax[nm]Id. × d NTA AFM NTA AFM

103 [nm] lin dif. sub-dif. FTLA d ≥ 50 all lin. diff sub-diff. FTLAP1 - - 400±361 223±293 180±78 101±73 74±65 253 123 - 10P2 - - 285±341 155±263 159±66 91±61 66±54 117 54 - 13P3 - - 278±263 143±189 130±57 90±56 67±51 129 65 - 16PSW 7 203 375±188 200±133 202±47 - - 254 140 199 -PSW 5 453 598±200 364±162 450±86 - - 527 299 459 -PSK 1 203 367±272 200±158 180±25 - - 250 127 185 -PSK 10 203 354±479 192±137 185±19 - - 242 138 185 -PSK 1 453 541±209 323±190 307±112 - - 473 237 385 -PSK 2 453 539±207 338±191 348±82 - - 455 244 395 -PSK 5 453 589±200 348±174 367±72 - - 504 293 395 -

for each sample are shown in the Fig. 5 and thenumerical results are summarized in Tab. 2. Ingeneral, the distributions of diameters obtainedwith the normal and logarithmic diffusion mod-els are significantly asymmetric and can be ef-ficiently fitted with the log-normal function.Thus, one can consider the maximum and theaverage of this distribution as the two possi-ble measures of the particle size. On the otherhand, FTLA usually predicts the distributiondominated by a single, mostly symmetric peak.Let us consider the 203 nm beads first (Fig. 5A,C and D). FTLA recovers this size almost per-fectly (202 nm) for PSW and with 91% accuracy(180 and 185 nm, depending on the concentra-tion) for PSK . Conversely, the normal diffu-sion model performs very poorly in both series,predicting the mean size < d > which is twiceover-sized. The maximum of these distributionsis also shifted by '+25% in comparison to theactual size. Similarly, the maximum of the log-arithmic sub-diffusive model indicates the sizewhich is underestimated by 30-40%. However,there is a remarkable agreement between theexpected size and < d > in the sub-diffusivemodel. For PSW sub-diffusive model indicates

200 nm (98% accuracy) and for PSK it predicts200 and 192 nm. These last two results are evenmore accurate than the FTLA predictions. Thesituation is much different for the bigger par-ticles, with 453 nm diameter. FTLA works al-most perfectly (Fig. 5B) in PSW measurements,predicting 450 nm. However, in PSK series theartefacts appear, decreasing < d > to 307-367nm. If the position of the main peak is used,the prediction improves to 385-395 nm. Thisdifference in FTLA performance between PSWand PSK is astonishing, as the measurementconditions and sample concentrations in bothcases are comparable. However, two differentversions of NanoSight software are involved inthe analysis, which might be the reason. Thesub-diffusive model proved less accurate thanFTLA, resulting in < d >= 323−348 nm. Sur-prisingly, the best estimate is provided by thenormal diffusion model, when the maximum ofthe distribution is considered. In this case theerror is lower than 11%. In general, these re-sults strongly suggest that the diffusion modelapplied to the NTA data should be chosen ac-cording to the expected particle diameter, as,apparently, the accuracy of methods changes

11

Figure 4: Anomalous exponent α in the samplesof polystyrene beads. PSW - beads measuredin Warsaw, PSK - beads measured in Krakow,dil. - sample dilution. A: The histogram of αcalculated for all accepted trajectories B: Thehistogram of α for long (N > 50) trajectories.C: See caption of Fig. 3C.

with the particle size. Nevertheless, < d > ob-tained from the logarithmic sub-diffusion modelproves to be a very accurate measure for thesmaller particles.As the final test, we applied the normal

diffusion model, logarithmic sub-diffusion andFTLA to predict the < d > of EVs. To corrob-orate this analysis, the samples were also exam-ined with AFM, which provided the size distri-bution measured by a direct imaging method.The results for patients P1, P2 and P3 are plot-ted in the Fig. 6 and the numerical results aresummarized in the Table 2. The size distribu-tion obtained with AFM is an asymmetric andmono-peaked function with < d > equal 74, 66and 67 nm, depending on the sample. Rejectingthe particles with d < 50nm shifts this averagesto 101, 91 and 90 nm, respectively. Let us recallthat, according to our discussion in the intro-ductory section, the factor of discrepancy be-tween NTA and AFM reads approximately 1.5-2.5. Using this value and the AFM data onecan estimate to expected EVs size in NTA tobe roughly equal 150-250nm. This is preciselythe size range in which the sub-diffusive modelis reliable. Indeed, the sub-diffusive model re-sults in a d histogram whose general shape issimilar to that of the AFM distribution, with< d >=223, 155 and 143 nm. For P2 and P3these results neatly fall into the expected rangeof discrepancy, while for P1 the difference is bythe factor of ' 3.0, which is still close to ourexpectations. On the other hand, the normaldiffusion model leads to a distribution with anappropriate shape, but predicts < d > which is4.0-5.5 times greater than < d > from AFM.Finally, FTLA provides < d > which is compa-rable to the sub-diffusive results, but the shapeof the distribution is much different. In partic-ular, FTLA predicts a few maxima of a similarheight that weakly coincide with the structureof the normal diffusion distribution. Suppos-edly, this is the manifestation of the problemswith FTLA in the highly poly-disperse samples,i.e. it tries to map the actual size distributionon a limited number of peaks. However, thesepeaks do not seem to be present in the AFMdata.Summarizing the experimental part, we have

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Figure 5: Histograms of diameter d for PS beads at different dilutions (dil.), measured with NTA.Each panel represents the size distribution obtained with the normal diffusion model (red solidline), logarithmic sub-diffusion (black solid line) and with FTLA algorithm (green solid). Dashedlines show the log-normal fits applied to the data. Distributions are normalized to have the sameheight at the maximum. Vertical lines indicate the position of < d >. PSW , PSK - measurementscarried out, respectively, in Warsaw and Krakow.

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Figure 6: The histograms of EVs diameter dfor three donors (A - P1,B - P2, C - P3) ob-tained for 3 different models. Shaded blacksolid line - the logarithmic sub-diffusive model,red solid line - the normal (linear) diffusionmodel, green line - FTLA algorithm. For com-parison, the size distribution measured withAFM (blue dashed line), for the same samples,are shown. The vertical lines indicate the meanvalue for each distribution. Histograms are nor-malized to have the same height at the maxi-mum.

shown that the sub-diffusive behavior occurs inboth artificial and biological samples and at arange of concentrations. The diffusion modelapplied to the MSD data proves to have a crit-ical impact on the particles size measured withNTA. The experiments were performed on twodifferent instruments, thus we can suspect thatthe issue is not inherent to a specific experimen-tal setup. The method of data selection andthe logarithmic sub-diffusive model which wepropose seem to be acceptable tools to improvethe analysis of NTA data in the biological con-text. We have shown that for the small particles(d ' 200nm) the sub-diffusive predictions areon par with the state-of-the-art FTLA meth-ods. However, our model does not introduce thespurious manipulation of data that can causeartefacts in the poly-disperse samples. Never-theless, the idea behind FTLA is sound and thisapproach might be improved in the future. Es-pecially, combining the FTLA-type correctionswith the theory of sub-diffusion might lead toa significant progress in NTA data processingfor the biological applications. We have also il-lustrated the problem of discrepancy betweenAFM and NTA size prediction for EVs. Ourresults confirm that this difference exists and itcan be minimized only by applying the analysismethods beyond the normal diffusion models.

SummaryNTA is a technique of huge potential, but itsbroad application to the biological research isstill in development and requires the clarifica-tion of many ambiguities. In this article weanalysed the problem of anomalous diffusionaffecting NTA measurements in both artificialand biological poly-disperse systems. We pro-posed the protocol of data processing and selec-tion that allows us to stay in agreement with therequirements of the free-diffusion model. Wealso introduced a simple model of sub-diffusionthat can be ready-applied to retrieve the EVssize distributions. The application of the sub-diffusion models leads to the results compara-ble with FTLA approach, but is more reliable inthe biological samples. It should be emphasized

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that our protocol and model can be applied inthe NTA study of any poly-disperse system pro-vided that the sub-diffusion is observed in theMSD data. The notorious discrepancy betweenNTA and AFM results was also attributed tothe several well-recognized effects. Finally, ourapproach opens a way for further developmentof the NTA method.

Acknowledgement Experiments have beenfinanced by the Polish National Science Centre(NCN) grant No. 2012/07/B/NZ5/02510 (toE.S.). TEM images were performed by dr inż.Olga Woźnicka from the Department of Cell Bi-ology and Imaging Institute of Zoology, Jagiel-lonian University. M. Majka acknowledges theNational Science Center, Poland for the grantsupport (2014/13/B/ST2/02014).

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