Single-Particle Orbit Tracking - Setup, Characterisation and Application

Single-Particle Orbit Tracking Setup, Characterisation and Application Der Universität Bayreuth zur Erlangung des Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung von Dominique Ernst geboren am 28. April 1982 in Tirschenreuth 1. Gutachter: Prof. Dr. J. Köhler 2. Gutachter: Prof. Dr. M. Weiss Tag der Einreichung: 27.09.2012 Tag des Kolloquiums: 14.12.2012 Abstract Tracking of single nanoobjects (e.g. beads, proteins or molecules) is of fundamental interest in many research fields, particularly in the fields of biophysics and material sciences. Information such as the local viscosity or the structure around the tracer particle can be gathered on the nanometer scale. Further, the photophysical properties or conformational dynamics of the tracer can be studied without the need of immobilising them. Moreover, fundamental research with respect to diffusion processes like the normal Brownian motion or anomalous diffusion can be examined with the acquisition of single-particle trajectories. In this thesis, the development and experimental realisation of an optical setup which records the 2-dimensional trajectories of single fluorescently labeled polystyrene beads, either 20 nm or 50 nm in diameter, with a high spatial and temporal resolution is introduced. Combining single molecule fluorescence techniques with a new method called single-particle orbit tracking the spatial position of the beads could be determined with an accuracy of less than 10 nm at a time resolution of 4 ms. The idea is to manipulate the excitation light spatially and temporally to locate a particle. In order to do so, special optics which deflect a laser beam and guide it on a circular path were used. Subsequently, this rotating beam is projected by a microscope into the sample with the diffusing particles. Due to the spatially and periodically modulated excitation light, the emission signal of the bead is modulated with the frequency of the rotation of the laser focus. The amplitude of the modulated emission signal depends on the position of the particle within the excitation orbit. An ingeniously developed algorithm calculates the position of the particle with respect to the centre of the orbit by demodulating the emission signal and restores the particle back to the orbit centre. Applying this method successively, the trajectory of the diffusing bead can be reconstructed. Besides the experimental realisation, the characterisation of the setup in terms of the spatial and temporal accuracy as well as the experimental shortcomings that influences the measured trajectories and hence, the interpretation of the data, were also the main topics of this work. For this purpose a reference sample of 20 nm sized beads in glycerol was used. The accuracies were studied mainly by computer simulations and the artifacts by experiments. The technical details of the setup and the characterisation results were published (publication P1). The recorded trajectories were analysed with various methods, among which the commonly used mean squared displacement (MSD) yields the results with highest information. The diffusion coefficient as well as the diffusion behaviour could be quantified. With this method the obtainable accuracy in measuring the diffusion coefficient by the acquisition of single-particle trajectories was studied as a function i of the length of the trajectories and as a function of the number of fitting points that were used for a linear fit to the experimentally determined MSD-curves. As expected, the relative error of the determined diffusion coefficient gets better for longer trajectories. Further, an optimal number of fitting points for the linear approximation to the MSD-curves was found, which yields the most exact values for the diffusion coefficients and which is independent of the trajectory length. For the first time, experimental results on that issue were compared with theoretical predictions, where a good agreement was found. These findings were published (publication P2). By the use of the Stokes-Einstein relation the diffusion coefficients could further be converted to particle radii. A closer examination of these radii emphasises the influence of the afore mentioned number of fitting points. For the optimal value, significantly precise radii could be determined. Finally, an application of the new setup is presented. In cooperation with the chair of experimental physics I (group of Prof. Dr. M. Weiss) of the University of Bayreuth, the diffusion behaviour of single nanoparticles in a complex fluid was studied. Background hereto is the investigation of biochemical reactions in a biological cell, whose kinetic is given by the diffusion of the corresponding reaction partners. Due to the high crowding of the cell compartments the diffusion is hindered. The diffusion behaviour in these systems is called anomalous and more exactly subdiffusive. Several theoretical models have been developed to explain this phenomenon, but yet without experimental verifications. Here, the diffusion of 50 nm sized polymer beads in the model system dextran (a highly branched biopolysaccaride) is investigated experimentally with high spatial and temporal resolution. The data were analysed in the group of the cooperation partner which yields a very good agreement with the model of “fractional Brownian motion”. These results were also published (publication P3). A final outlook concerns possible technical advancements of the experimental setup, in particular to measure 3-dimensional trajectories, and several applications at which the tracking of single particles might be helpful for a better understanding of the system of interest. ii Kurzdarstellung Die räumliche Verfolgung einzelner Nanopartikel (z.B. Polystyrolkolloide, Proteine oder Moleküle) ist für viele Forschungsgebiete, vor allem in der Biophysik und den Materialwissenschaften, von großem Interesse. So können unter anderem Informationen über die lokale Viskosität oder über Strukturen in der Umgebung des Teilchens auf einer Nanometerskala gewonnen werden. Weiterhin können die photophysikalischen Eigenschaften oder die Konformationsdynamik der verfolgten Teilchen selbst studiert werden, ohne sie zu immobilisieren. Auch die Grundlagen von Diffusionsprozessen, wie zum Beispiel die Brownsche Bewegung oder anomales Diffusionsverhalten, können durch die Messungen von Trajektorien einzelner Teilchen untersucht werden. In dieser Dissertation wird die Entwicklung und experimentelle Umsetzung eines optischen Aufbaus zur Messung von zweidimensionalen Trajektorien einzelner fluoreszenzmarkierter Polystyrolbeads mit einem Durchmesser von 20 nm bzw. 50 nm mit hoher räumlicher und zeitlicher Auflösung vorgestellt. Durch die Kombination von Einzelmolekülfluoreszenztechniken mit einer neuartigen Methode mit der englischen Bezeichnung „single-particle orbit tracking“ konnte eine räumliche Auflösung in der Positionsbestimmung der Beads von weniger als 10 nm bei einer Zeitauflösung von 4 ms erzielt werden. Die Idee dabei ist, das Anregungslicht räumlich und zeitlich zu manipulieren, um die Position eines Teilchens zu bestimmen. Dazu werden spezielle optische Bauelemente verwendet die einen Laserstrahl auf einen Kegelmantel ablenken. Nachfolgend wird dieser rotierende Laserstrahl mit Hilfe eines Mikroskops in die Probe mit den diffundierenden Teilchen projiziert. Aufgrund der räumlichen und periodischen Modulation des Anregungslichts ist auch das Emissionssignal des Teilchens mit der Frequenz des rotierenden Laserfokus moduliert. Die Amplitude des modulierten Emissionssignals ist von der Teilchenposition innerhalb des Orbits abhängig, welcher durch das fokussierte Anregungslicht erzeugt wird. Ein speziell entwickelter Algorithmus berechnet die Teilchenposition bezüglich des Mittelpunktes des Orbits indem das Emissionssignal demoduliert wird. Anschließend wird das Teilchen um den berechneten Wert zurück in die Mitte des Orbits verschoben. Sukzessive Anwendung dieser Berechnungsmethode liefert die rekonstruierte Trajektorie des Teilchens. Schwerpunkt dieser Arbeit war neben der technischen Realisierung, die Charakterisierung des Aufbaus in Bezug auf die räumliche und zeitliche Auflösung der Trajektorien, sowie auf experimentelle Unzulänglichkeiten, welche die gemessenen Trajektorien und damit auch die Interpretation der Messdaten, beeinflussen. Dazu wurde die Referenzprobe von 20 nm großen Teilchen in Glycerin verwendet. Die erreichbaren Auflösungsgrenzen wurden hauptsächlich durch den Einsatz computergestützter Simulationen verifiziert, wohingegen die Artefakte iii experimentell untersucht wurden. Die diesbezüglich erzielten Ergebnisse sowie die technischen Details des Aufbaus wurden veröffentlicht (Publikation P1). Die aufgenommenen Trajektorien wurden auf verschiedene Weise analysiert, wobei die weitverbreitete Methode des mittleren Verschiebungsquadrats (engl.: mean squared displacement, MSD), Ergebnisse mit dem höchsten Informationsgehalt lieferte. Sowohl der Diffusionskoeffizient als auch das Diffusionsverhalten konnte quantifiziert werden. Mit Hilfe dieser Analyse wurde die erreichbare Genauigkeit von Diffusionskoeffizienten durch Messungen von Einzelteilchentrajektorien in Abhängigkeit der Trajektorienlänge und der Anzahl an Fitpunkten, die für eine lineare Kurvenanpassung an die experimentell bestimmten MSD-Kurven verwendet wurde, untersucht. Die Analyse der Messdaten zeigte erwartungsgemäß, dass der relative Fehler des Diffusionskoeffizienten für längere Trajektorien kleiner ist. Weiterhin wurde eine optimale Anzahl an Fitpunkten für die MSD-Kurvenanpassung gefunden, die unabhängig von der Trajektorienlänge ist und die genauesten Werte für die Diffusionskoeffizienten liefert. Die experimentellen Ergebnisse dieser Untersuchung wurden erstmals mit theoretischen Vorhersagen verglichen, wobei eine gute Übereinstimmung gefunden wurde. Die Resultate wurden veröffentlicht (Publikation P2). Mit Hilfe der Stokes-Einstein Beziehung konnten weiterhin die Diffusionskoeffizienten in Teilchenradien umgerechnet werden. Eine genaue Betrachtung der Radien verdeutlicht den Einfluss der Anzahl an Fitpunkten. Für die optimale Anzahl an Fitpunkten wurden wesentlich präzisere Werte ermittelt. Als Anwendung des neuen Aufbaus wurde in Kooperation mit dem Lehrstuhl Experimentalphysik I (Arbeitsgruppe von Prof. Dr. M. Weiss) der Universität Bayreuth das Diffusionsverhalten von einzelnen Polystyrolbeads in einer komplexen Flüssigkeit studiert. Hintergrund hierbei ist die Untersuchung biochemischer Reaktionen innerhalb einer biologischen Zelle, deren Kinetik durch die Diffusion der entsprechenden Reaktionspartner gegeben ist. Diese Diffusion ist durch die hohe Dichte an Zellkompartimenten stark eingeschränkt. Man spricht deshalb von einem anomalem Diffusionverhalten, genauer gesagt von Subdiffusion. Verschiedene theoretische Modelle zur Beschreibung dieses Phänomens wurden entwickelt, wobei eine experimentelle Verifikation noch nicht möglich war. In dieser Arbeit wird die Diffusion von 50 nm großen Polystyrolbeads in dem Modellsystem Dextran (hochverzweigtes Biopolysaccharid) mit hoher räumlicher und zeitlicher Auflösung untersucht. Die Messdaten wurden in der Arbeitsgruppe des Kooperationspartners analysiert und zeigten eine sehr gute Übereinstimmung mit dem Modell „fractional Brownian motion“. Die Ergebnisse wurden ebenfalls veröffentlicht (Publikation P3). Ein abschließender Ausblick befasst sich mit der technischen Weiterentwicklung des experimentellen Aufbaus, speziell mit der Messung 3-dimensionaler Trajektorien, und mit verschiedenen Anwendungsmöglichkeiten, bei denen die Bewegung einzelner Teilchen aufschlussreiche Erkenntnisse liefern können. iv Contents I Introduction 1 1 Motivation 3 2 Theoretical & experimental background 5 2.1 Theory.................................. 5 2.1.1 Diffusion equation . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Brownianmotion........................ 6 2.1.3 Trajectory analysis . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Trackingtechniques........................... 10 2.2.1 CCD-Tracking.......................... 11 2.2.2 Orbit-Tracking . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.3 Alternative approaches . . . . . . . . . . . . . . . . . . . . . 13 3 Materials 15 3.1 Beadsandsamples ........................... 15 3.2 Samplepreparation........................... 17 4 Simulations 19 4.1 Generalprocedure ........................... 19 4.2 Spatio-temporal accuracy . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Loosingaparticle............................ 25 5 The tracking setup 29 5.1 Optics&Hardware........................... 29 5.2 Astaticparticle............................. 30 5.2.1 Simulation............................ 31 5.2.2 Experiment ........................... 33 5.3 Artifacts & Corrections . . . . . . . . . . . . . . . . . . . . . . . . . 34 6 Single-particle tracking: results & discussion 37 6.1 Analysing diffusion based on single-particle trajectories . . . . . . . 38 6.1.1 Mean squared displacement analysis . . . . . . . . . . . . . . 38 6.1.2 Alternative approaches . . . . . . . . . . . . . . . . . . . . . 42 6.2 Accuracy of diffusion coefficients . . . . . . . . . . . . . . . . . . . . 47 6.3 Particlesizes .............................. 48 v Contents 6.4 CrowdedFluids............................. 52 7 Outlook 57 Appendix 59 A Programcode.............................. 59 B Harmonic approximation . . . . . . . . . . . . . . . . . . . . . . . . 61 Bibliography 63 List of publications 69 Danksagung 71 Erklärung 73 II Publications 75 P1. Setup for single-particle orbit tracking: artifacts and corrections 77 P2. Measuring a diffusion coefficient by single-particle tracking: Statistical analysis of experimental mean-squared-displacement curves 90 P3. Fractional Brownian Motion in Crowded Fluids 102 vi List of abbreviations The following list of abbreviations summarizes all used parameters and gives an explanation of the symbols. Rradius of the orbit νfrequency of the rotating focus ωangular velocity of the rotating focus (ω= 2πν) wfull width at half maximum of the focussed laser spot I0maximum emission of a fluorescent particle placed directly in the focal spot Ibbackground emission S0maximum number of emitted photons of a fluorescent particle placed directly in the focal spot Sbnumber of background photons Sntheoretical number of emitted photons during the sampling intervall δt Sn,poiss simulated number of emitted photons by the use of a poisson distribution Imean emission intensity of the fluorescent tracer xp, ypreal x and y coordinates of the particle xc, yccalculated x and y coordinates of the particle xs, ysx and y coordinates of the piezo stage aradius of the particle Ddiffusion coefficient kBBoltzmann constant Ttemperature, at which the experiments were performed ηvisosity of the used fluids αanomaly parameter nnumber of fitting points used for a linear fit to the MSD curves ttime τlag time ∆ttime resolution of the experiments ∆rspatial resolution δt sampling time of the experiments and the simulations NSnumber of sampling data points Ntotal number of data points of a trajectory Nseg number of data points of a cut segment Nens number of trajectories an ensemble of segments consists of vii 2 Theoretical & experimental background current density j, which arises due to a concentration gradient ∇c. j=−D∇c(2.1) This equation is called the first Fickian law. Beside the concentration gradient, the particle current density depends further on the diffusion coefficient D. The latter was developed in the works of Stokes and Einstein [4] and is defined as: D=kBT 6πηa (2.2) Here, kBis the Boltzmann constant, Tthe temperature of the system, ηthe viscosity and athe hydrodynamic radius of the diffusing particles. With the use of the continuity equation (2.3) d dtc+∇j= 0,(2.3) which has its origin in the law of conservation of particles, the diffusion equation is obtained: ∂ ∂tc=D∇2c(2.4) In this form the diffusion coefficient is regarded as constant. The diffusion equation describes the dynamics of the concentration of particles or molecules. Strictly speaking, this equation holds true for a continuum of particles in an infinite space. To study single particle phenomena, the concentration has to be interpreted as a probability density to find a particle in space. However, in this work the diffusion of single particles is studied by measuring the trajectory of the particle. Hence, in the following the theoretical description of this stochastic motion is introduced. 2.1.2 Brownian motion At this point it is worth mentioning, that the following theoretical derivations can be found in great detail in the book An introduction to Dynamics of Colloids written by J. Dhont [33]. The diffusion of a particle in a static viscous fluid can be understood as a random walk, also called Brownian motion. The mathematical description of such a stochastic process is based on the Langevin equation (eq. (2.5)). m¨r(t) = −γ˙r(t) + Fs(t)(2.5) Here, the vector r(t)represents the position of a particle with mass mat the time t. The particle, that moves with respect to the liquid, experiences an accelerating force Fs(t)and a friction force −γ˙r(t)where γdenotes the friction coefficient. For spherical particles with radius ain a fluid with viscosity ηthe friction coefficient is given by: γ= 6πηa (2.6) 6 2.1 Theory The origin of the accelerating force are thermal fluctuations of the liquid molecules and the concomitant collisions with the particle. The force Fs(t)in equation (2.5) can be separated in a strength parameter K, and a stochastic variable (t), representing the random orientation. Fs(t) = K(t)(2.7) The stochastic variable (t), also known as white noise, fullfils two conditions. First, it is isotropic in space and second, two consecutive values (forces) in time are uncorrelated, i.e. h(t)i= 0 (2.8) h(t)(t0)i=δ(t−t0)(2.9) (t)2= 1 (2.10) where h·i denotes averaging over time or an ensemble. A distinct averaging method will be written as h·iTfor time averaging and h·iEfor ensemble averaging, respectively. The strength can be calculated from the fluctuation-dissipation-theorem: hFs(t)Fs(t0)i=K2h(t)(t0)i(2.11) = 2nγkBTδ(t−t0)(2.12) The fluctuation strength depends on the friction coefficient and the temperature. nis the number of dimensions. Here a 2-dimensional random walk is analysed, i.e. 2n= 4. Combining equations (2.7) and (2.12) the stochastic force results to: Fs(t) = p4γkBT(t)(2.13) The total force (m¨r(t)) is rapidly fluctuating on time scales of 10−14 s. Due to the normally relative large mass of the particle, the Brownian motion covers a typical time scale of 10−9s. The system is highly overdamped and we can neglect the left-hand side of equation (2.5). The Langevin equation reduces to: ˙r(t) = 1 γFs(t)(2.14) Because of the random nature of the force, every realisation of equation 2.14 leads to a new trajectory of the particle. The Langevin equation provides discrete steps of the particle movement with random orientation, that make up the trajectory. Such a trajectory can be simulated with an iterative Euler method. For a small time step τthe position of the particle at the time t+τcan be calculated from its position r(t)at time tby r(t+τ) = r(t) + ˙r(t)τ. (2.15) 7 2 Theoretical & experimental background Using equations (2.13) and (2.14), this yields r(t+τ) = r(t) + s4kBT γτ (t)τ(2.16) which incorporates an additional factor 1/√τfor a proper description of the movement. This equation describes the Brownian motion of a particle and was applied in this thesis to simulate trajectories using a home-written Matlab program. 2.1.3 Trajectory analysis In the following I will give an overview on possible methods to analyse a singleparticle trajectory. In particular, these are i) the mean squared displacement as a function of a lag time τ(MSD(τ)) [34], ii) the spatial extend and shape, also termed asphericity [35, 36] and iii) the cumulative distribution function (CDF) of squared displacements [37, 38]. All of these methods will be used in section 6 for the analysis of experimental single-particle trajectories. i) Mean squared displacement. The MSD can be calculated either timeaveraged or ensemble-averaged, i.e. h∆r(t)2iTor h∆r(t)2iE. The latter one requires a statistical relevant ensemble of trajectories, while the first one is commonly used for a few single trajectories with a high number of x,y-position pairs and is hence more suited for single-particle tracking experiments. The MSD of a particle after a time step τis determined according to: h∆r(t)2iT=(r(t+τ)−r(t))2T=4kBT γτ(2.17) Here the equations (2.16) and (2.10) were used to calculate the MSD. With the Stokes-Einstein equation (2.2) and equation (2.6) the diffusion coefficient can be determined. h∆r(t)2iT= 4Dτ (2.18) The linear dependence in time holds true for normal (Brownian) diffusion. But, for systems showing anomalous diffusion, the linear dependence breaks down and a power law with a scaling exponent αis introduced [39]. h∆r(t)2iT= 4 ˜ Dτα(2.19) Here, the diffusion coefficient has to be interpreted as a generalized diffusion coefficient ˜ D, that explains the diffusion in the system under investigation. The anomaly parameter αis regarded as a strength for the anomaly and can be used to group the diffusion behaviour. Processes with an exponent α > 1are called superdiffusive 8 2.1 Theory and those with α < 1subdiffusive. Only if α= 1 Brownian motion is obtained. This value can be determined easily with logarithmic calculus of equation (2.19). logh∆r(t)2iT=αlog τ+ log 4 ˜ D(2.20) The slope represents the scaling exponent, which is obtained by a linear fit to the MSD data points, plotted in a logarithmic scale. In an experiment only discrete positions are available, i.e. the time-averaged MSD of a single trajectory r(t)constituting Nposition determinations, has to be calculated for consecutive lag times τ=k∆t(k= 1..(N−1)) according to MSDT(τ) = ∆r(k∆t)2T=1 N−k N−k X n=0 [r(n∆t)−r((n+k)∆t)]2(2.21) Here h·iTsymbolizes time-averaging over the respective lag time. The alternative ensemble-average calculation is given as follows: MSDE(τ) = ∆r(k∆t)2E=1 Nens Nens X m=1 [rm(k∆t)−rm(0)]2(2.22) where Nens denotes the number of trajectories the ensemble consists of and rm(0) represents the starting position of each trajectory m. ii) Shape of a trajectory. The average spatial extend of a trajectory can be estimated by the radii of gyration Rxand Ry. Hence, the gyration tensor Tof a 2-dimensional trace has to be calculated [35, 36]. Tij =1 N N X n=1 (ri(n∆t)−hrii) (rj(n∆t)−hrji)(2.23) Here, the indices i, j denote the xand y-component of a position vector r(t)and the brackets hriirepresent the corresponding centre of masses for the x-trace and the y-trace. hri,ji=1 N N X n=1 ri,j(n∆t)(2.24) Diagonalisation of Tyields the eigenvalues, i.e. the squared radii of gyration. T=R2 x0 0R2 y(2.25) The eigenvectors defines the orientation of the respective gyration ellipse. The spatial extend of a trajectory changes with the number of data points and the mobility 9 2 Theoretical & experimental background of the tracer particle. An unambiguous criterion regarding the diffusion behaviour remains elusive. A more suitable value is the asphericity A, providing a single parameter that determines the shape of a random walk [36]. A=DR2 y−R2 x2E DR2 y+R2 x2E(2.26) The calculation of Arequires averaging over a sub-ensemble (h·i)that can be obtained from cutting the trajectory into consecutive segments with an equal number of positions Nseg. For each of the segments, the radii of gyration were calculated (cf. eqn. 2.23 - 2.25) and the asphericity was determined according to equation (2.26). Trivial values for the asphericity are given for a perfect rod-like shape, where one of the radii of gyration is 0, leading to A= 1, and for a perfect spheric shape with equal radii, i.e. A= 0. For a random walk one finds A= 4/7[35]. iii) Cumulative distribution function. An alternative method to investigate the diffusion process of a single particle is to calculate the (discrete) cumulative distribution function of the squared displacements ∆r2at a certain lag time τ, i.e. CDF(∆r2, τ)[37, 38]. In order to do so, the number of squared displacements smaller or equal to a given ∆r2is counted according to CDF ∆r2, τ=X ∆r2(τ)≤∆r2 P∆r2, τ(2.27) where P(∆r2, τ)denotes the empirical distribution of the squared displacements for a lag time τ. This is done consecutively throughout a trajectory until the highest ∆r2is reached. 2.2 Tracking techniques Numerous kinds of setups have been invented to follow the 2and even 3-dimensional motion of single particles [13, 17, 18, 20, 21, 23, 24, 29, 40]. Mostly, fluorescent techniques were used, which requires fluorescent particles. Therefore, either dye molecules, polymer beads that are loaded with dyes, or labeled proteins are possible tracers. For non-fluorescing particles, other microscopy techniques, e.g. dark field microscopy is a possible method. In the early years, simple video microscopy [17] was used to follow the 2-dimensional motion of single lipids and bigger molecules. Further developments leads to high standard CCD-tracking techniques that record the motion of a single particle with a high spatial and temporal resolution [18]. Besides the common CCD-tracking technique, other methods emerged, where the 10 2.2 Tracking techniques spatial and temporal resolution as well as the total observation time have been improved further. In this section I will give a brief introduction to the widely used CCD-tracking method, explain the technique of single-particle orbit tracking, which was exploited in this thesis and give finally an overview to some alternative experimental approaches. 2.2.1 CCD-Tracking Tracking techniques, that use a charged coupled device (CCD) as a detection unit are called CCD-tracking, which is the most wide-spread method for single-particle tracking [13, 17, 18, 20, 23, 29]. The principle is to record successive images of the sample with the moving particles. Each particle in a CCD-image is displayed as a diffraction limited spot, which is typically spread over some 10 pixels on the CCD-chip. A two-dimensional Gaussian fit to the spot is applied, where the centre of this fit gives the actual particle position with an accuracy of better than the classical diffraction limit of light. By doing so successively with all recorded frames, the trajectories of all the particles within the CCD-images are obtained. The advantage of this technique is the relatively simple experimental setup and the high position accuracy that can be obtained. Further, multiple single particles can be tracked simultaneously. Yet, this experimental approach often lacks a high temporal resolution, because the data storage of the images with a high information density is time consuming. At least sophisticated improvements have to be applied to avoid this shortcoming. Not only the time resolution is restricted, also the total observation time is, because the CCD-images need large computer memory. A video of about ten minutes can easily exceed 100 Gigabyte of hard drive space. To circumvent these limitations, other tracking techniques have been invented. 2.2.2 Orbit-Tracking The basic idea of this method is to focus a laser beam in the focal plane of the sample and let this focal point rotates around a fluorescent particle. By acquiring the emission signal, which is modulated by the frequency of the rotating laser beam, the position of the particle can be traced. By demodulation of this signal the actual particle position can be calculated. Theoretically this approach was studied by Enderlein [41, 42] and successfully implemented by the groups of Gratton [40], Mabuchi [21] and Lamb [43]. In figure 2.1 the principle is shown schematically for two different particle positions (left hand side) with the corresponding emission signals (right hand side). For particles, that move between the centre and the rim of the orbit, the emission signal is periodically modulated as it is depicted in figure 2.1 (right hand side). A high modulation occurs for particles that are apart from the centre of the orbit (see fig. 2.1a), while it gets weak for particles that are close to the centre (see fig. 2.1b). The amplitude of the modulation changes as a function 11 2 Theoretical & experimental background Figure 2.1: Schematic explanation of the orbit tracking technique. All necessary parameters, i.e. the radius of the orbit R, the angular velocity ω, the width of the focus wand the position of the particle xpand yp, drawn as a red sphere, are displayed. This scheme is not drawn to scale. In a) the particle is far off-centred, whereas it is close to the centre of the orbit in b). On the right hand side of the figure the corresponding emission intensities of the particle are sketched. Adapted from publication P1. of the position. Hence, from the demodulation of the emission, the direction as well as the absolute distance from the centre of the orbit can be calculated. The emission signal I(t)depends on the relative position of the particle (xpand yp) and the rotating laser focus and can be written as follows [41, 42]: I(t) = I0exp −2 w2(xp−Rcos(ωt))2exp −2 w2(yp−Rsin(ωt))2+Ib(2.28) Here, I0is the maximum emission intensity, i.e. the particle is at the position of the focal spot, Rthe radius of the orbit, wthe 1/e2-width of the laser focus, ω= 2πν the angular velocity of the rotating focus and Ibthe background intensity. By using lock-in techniques, the position of the particle can be calculated from this emission 12 2.2 Tracking techniques signal according to the equations xp(t) = w2 2RRT 0I(t) cos(ωt)dt RT 0I(t)dt , yp(t) = w2 2RRT 0I(t) sin(ωt)dt RT 0I(t)dt .(2.29) The integration boundaries are from 0 (begin of the position determination) to T, which is a time that corresponds to a multiple of the cycling time. Once the position of a particle with respect to the centre of the orbit is calculated, a feedback mechanism has to be implemented, that restores the particle back in the centre of the orbit. Generally, two possibilities exist to do so. On the one hand, this is beam scanning, where the whole light orbit is moved according to the new positions of the particle by applying the corresponding feedback signals to the optical elements that are responsible for the generation of the orbit. This are typically scanning mirrors or acousto optical deflectors. On the other hand, this is sample scanning, where the feedback signals are applied to hardware elements, that move the whole sample, which is mostly realised by a piezostage. The feedback loop is as follows: acquire emission signal, calculate position, restore particle in the centre of the orbit, and so on. Doing so successively, the whole trajectory of a particle can be reconstructed. This new setup for the recording of single-particle trajectories is quite powerful as it combines the spatial accuracy of a CCD camera with the temporal resolution of a single photon detector. 2.2.3 Alternative approaches In the following some alternative approaches and new developments in the research field of single-particle tracking are introduced. All of them have in common, that they want to push forward the temporal and spatial accuracy of the position determinations. In the research group of W. E. Moerner two ingenious methods have been invented. The first one is called the ABEL (Anti-Brownian ELectrokinetic) trap [15, 44, 45]. Here, a particle is trapped between four electrodes. As soon as the particle moves towards one of them a voltage to the corresponding electrode is applied that induces a flow field in the medium where the particle is diffusing and which pushes the particle back to the centre of the trap. The feedback mechanism is implemented by recording the motion of the particle with a CCD-camera. In a newer version of this setup, the afore explained orbit tracking is used to record the position of the particle [46], which is faster. The second setup, that was developed in the group of Moerner modulates the point spread function (PSF) [24, 47]. By a spatial light modulator a double-helix PSF is generated, where the particle of interest is located within that double helix. In other words, a spatial cavity of light surrounds the particle. A movement in any direction is detected by the emission signal of the fluorescent particle. Computer 13 2 Theoretical & experimental background software is then able to reconstruct the position in three dimensions. Another method was put forward by the group of H. Yang, in which the emitted light of a particle was split four times by prism mirrors [22]. Each part of the emission was acquired by a separate avalanche photo diode (APD). From the intensity ratios between the four detectors the position of the particle can be extracted. By a 3-dimensional piezostage this motion is compensated. The feedback signal of the piezo is used to determine the trajectory. In principle all techniques have in common, that they probe the space around a particle, either by trapping the particle, by modulation of the excitation light, or by splitting the emission signal. From the acquired signal of the tracer particles, feedback mechanisms restore the initial position. Successive calculations and the use of sophisticated algorithms provide the reconstructed trajectory. 14 The good thing about science is that it’s true whether or not you believe in it. Neil deGrasse Tyson 3 Materials For the characterisation and tracking experiments, two sizes of beads (20 nm and 50 nm) loaded with two different fluorescent dye molecules (nile red and rhodamine) and four kinds of sample substances (poly-vinyl-alcohol (PVA), glycerol, sucrose and dextran) were used. Among these, the combination of 20 nm sized beads in PVA is used for static experiments, where the beads are immobile. Further, another five combinations of beads/samples were used for the characterisation of the setup including the determination of the dynamic accuracies and for the investigation of anomalous diffusion. In the following paragraphs I will describe the characteristics of the beads as well as the preparation of the sample. The detailed preparation of the bead/sample-mixtures for the respective experiments can be found in the corresponding publications (publication P1: 20 nm beads in PVA and in glycerol, publication P2: 20 nm beads in glycerol, publication P3: 50 nm beads in sucrose and dextran). 3.1 Beads and samples In order to characterise the tracking performance of the experimental setup and to measure diffusion processes, two different sizes of dye labeled beads were used. On the one hand these are 20 nm (diameter) sized polystyrene beads (Molecular Probes) that are loaded with the dye nile red and are further stabilized with carboxylate groups attached to the surface to avoid aggregation. According to the manufacturer, the beads are suspended in water at a concentration of 20 mg/ml. On the other hand these are polystyrene-based latex microspheres (Polysciences) with a diameter of 50 nm, that are labeled with the dye rhodamine. Here, the concentration of the stock solution is 200 mg/ml with no additional stabilizers. The molecular structure as well as the normalized fluorescence excitation and emission spectra of the two highly fluorescent molecules are displayed in fig. 3.1a,b. The spectra have been recorded with a commercial fluorescence spectrometer (Cary Eclipse, Varian). Therefore the respective beads were dissolved in millipore water and this solution was then filled in cuvettes. The fluorescence excitation and emis15 4 Simulations was set to w= 270 nm, the radius of the orbit to R= 190 nm, which is the optimal value according to [50] where R=w/21/2was determined, and the frequency of the focus rotation was set to ν=ω/2π= 1 kHz. The sampling time was δt = 2 µs, which results to a number of sampling intervals of NS= ∆t/δt = 2000. This corresponds to 4 periods of rotation of the focus. (Later in this thesis we will see, that this parameters match the experimental conditions.) From the number of simulated photons, the mean emission intensity of the tracer results to I= 57kcps and the background intensity to Ib= 0.5 kcps (kcps: kilo counts per second). To calculate the coordinates xcand ycaccording to the equations (4.3), the simulated signal Sn,poiss as well as the values Sn,poiss cos(ωnδt)and Sn,poiss sin(ωnδt) are accumulated during each 4 ms period (Snin eqn. (4.3) was replaced by Sn,poiss). By successively repeating this procedure, the trajectory of the particle, i.e. r(t)=(x(t), y(t)) can be reconstructed. Figure 4.1c displays both, the trajectory generated with the Langevin equation (black) and the trajectory, that is reconstructed. Beside some slight deviations according to the position uncertainties (noise and movement during signal acquisition), the “simulated reconstructed” trajectory matches the “simulated real” one. To summarize, first a “fast” trajectory on a time scale of δt = 2 µsis generated, that corresponds to the movement of the particle during the integration time ∆t. Out of the positions of this trajectory, the signal Sn,poiss is calculated, accumulated and multiplied with a cosine and sin function, necessary for the calculation of the particle position. Finally, the positions xcand ycare calculated according to 4.3. All simulations were performed using home-written Matlab programs. The source code of the generation of the “simulated real” and “simulated reconstructed” trajectories is given in the appendix A. This simulations are an effective tool •to scan the parameter range (e.g. rotation frequency ν, time resolution ∆t, orbit radius R, etc.), which is appropriate for the experiment to achieve the best tracking performance, •to compare the results of simulations and experiments, •and to study the influences of several experimental artifacts (e.g. noise and position averaging) or the influences of diffusion parameters (e.g. viscosity ηor bead size a) on the analysis of the mean squared displacement and the concomitant interpretations. 4.2 Spatio-temporal accuracy With the use of the afore described simulations of the reconstructed trajectories the tracking performance by means of the spatial and temporal accuracy is studied 22 4.2 Spatio-temporal accuracy as a function of the emission intensity and as a function of the diffusion coefficient. Starting point is the simulation of the number of photons Sn,poiss that is used by the feedback algorithm to determine the position. The theoretical and Poisson distributed number of photons, i.e. Snand Sn,poiss, during a time interval that corresponds to the time resolution of ∆t= 4 ms for a moving bead is displayed in fig. 4.2a, represented in red and black, respectively. For a better visualization a low emission signal of I= 30 kcps was chosen, otherwise the Poisson distributed number of photons are too crowded. The diffusion coefficient was set to D= 43 ×10−3µm2/s. The signal for the theoretical number of photons Sn is periodic, but heavily fluctuating. The reason is the movement of the particle, during the integration time ∆t. As already mentioned, the emitted photons depend on the position of the particle within the orbit. As soon as the movement happens on a faster time scale than the integration time, the periodic emission fluctuates accordingly. Because the Poisson distributed number of photons is calculated from the theoretical number of photons, the described behaviour (periodic and fluctuating emission) is carried forward to Sn,poiss. The density of photons (Sn,poiss) is high when Snis high. However, despite the fluctuations of the emitted photons, the feedback algorithm is still able to calculate the position of the particle, with an accuracy, that is determined by photon statistics (noise) and position averaging during data acquisition. Figure 4.2b shows two 1-dimensional “simulated reconstructed” trajectories with N= 5000 data points for the time resolutions ∆t= 0.5 ms (blue) and ∆t= 9.0 ms (red), both with a mean emission intensity of I= 157 kcps and a diffusion coefficient of D= 400 ×10−3µm2/s. It is trivial, that the particle simulated with the higher time resolution of ∆t= 9.0 ms can move a larger distance, because the total observation time for the movement is t=N∆tand hence, longer than for a particle simulated with ∆t= 0.5 ms. However, rather than the covered distance, the position error in the trajectory is considered. Despite a larger number of photons, which favours the position determination, the “simulated reconstructed” trajectory determined with the time resolution of ∆t= 9.0 ms appears more noisy than the trajectory determined with ∆t= 0.5 ms. Obviously, the motion during the acquisition time is dominant. Dependent on the emission intensity and the mobility of the particle (diffusion coefficient), both effects contribute to the spatial accuracy. To determine this accuracy, the positional error of a trajectory has to be calculated. In order to do so, the root-mean-square (rms) error between the positions of the “simulated real” trajectory, xc, and the positions of the “simulated reconstructed” trajectory, xp, has to be determined. σx=σrms =v u u t1 N N X i=1 (xci −xpi)2(4.4) 23 4 Simulations Figure 4.2: Simulation of the dynamic position accuracy. a) theoretical (red) and Poisson distributed (black) number of emitted photons within one time bin of ∆tb) calculated position as a function of the number of position determinations Nfor a time resolution of ∆t= 0.5 ms (blue) and ∆t= 9.0 ms c),d) dynamic position accuracy for a high mean emission intensity (c) and a low mean emission intensity (d) as a function of the time resolution for various diffusion coefficients between the static case of D= 0 µm2/sand a fast particle motion of D= 6.4µm2/s. The two trajectories, simulated in b) are indicated in d) by the numbers 1 and 2. More details see text. In fig. 4.2c,d the 1-dimensional spatial accuracy σxas a function of the time resolution ∆tfor diffusion coefficients between D= 0 µm2/s(static) and D= 6.4µm2/s are shown for a low (fig. 4.2c) and a high (fig. 4.2d) mean emission intensity of I= 30 kcps and I= 157 kcps, respectively. The low emission signal corresponds to a typical emission of a single molecule, while the high signal corresponds to the typical emission of a fluorescent bead. Qualitatively, both graphs show the same results. With an increasing value of the time resolution (in everyday language: the time resolution gets worse) the spatial accuracy rapidly gets better until a minimum value is reached, from which it grows slowly. To explain this behaviour let us consider a given diffusion coefficient. For very low time resolutions the particle can be regarded as static and the position accuracy is getting better according to σx∝1/√∆t. This is evident for the spatial accuracy of the static particle (black data points in fig. 4.2c,d). For higher values of ∆tthe position of the particle gets blurred due to its movement during one time resolution step, i.e. the positional er24 4.3 Loosing a particle ror is growing. Hence, it exists a minimum, where the best spatial accuracy can be achieved. For fast particles, i.e. high diffusion coefficients, the position averaging effect is more pronounced than for slow particles. A common comparison is the diffusion of a 20 nm bead in water, which has a diffusion coefficient of D= 21.5µm2/s. According to the presented simulations, the particle can not be tracked or at least a very high time resolution is required. For a successful tracking experiment, the particle has to be larger and/or the viscosity of the surrounding fluid has to be higher. However, this two graphs (fig. 4.2c,d) serve as an orientation for setting suitable experimental parameters and to know the theoretical limits of the tracking performance. In advance, the experimentalist can check if the particles in the system he wants to investigate can be tracked and if so, which spatial accuracy can be expected. A problem occurs, if the simulated accuracies are compared with experimental ones. In an experiment the “real” position, which was necessary for this calculations, is not accessible and other methods for the determination of the spatial accuracy has to be used. One possibility is discussed in detail in publication P1, where the offset of the MSD-curve was used as an indicator for the spatial accuracy. 4.3 Loosing a particle As long as the particle can be tracked, it stays inside the light orbit, generated by the rotating focus, and is restored to the centre of the orbit after every time period ∆t. Hence, this means, that the emission intensity of the particle is kept constant during tracking. As soon as the particle gets lost, the emission intensity decreases immediately to the background intensity. Possible reasons to loose a particle are a weak emission signal or a high mobility. The situations for a successful and a failed tracking were simulated for a bead with radius a= 10 nm at a temperature of T= 294 K. The top graph of fig. 4.3a displays the 1-dimensional “simulated real” (black) and “simulated reconstructed” (red) trajectory of a slow diffusing bead with a high emission intensity. The parameters were set to I= 157 kcps and η= 1.2 Pa s, which corresponds to a diffusion coefficient of D= 17.95 ×10−3µm2/s. The reconstructed positions follow nicely the real trace. A detailed view is shown in the inset of fig. 4.3. The corresponding intensity trace (bottom graph of fig. 4.3a) yields the expected constant emission intensity, i.e. the particle is tracked continuously. To simulate a tracking experiment, where the particle gets lost, a fast bead with a low emission intensity is used. The trajectories (real and reconstructed) are shown in the upper part of fig. 4.3b. Here, I= 21 kcps and η= 0.1 Pa s, i.e. D= 215.3×10−3µm2/s, were chosen. The diffusion is too fast and the emission is too low, to follow the movement of the particle. At a time of about 2.5 s the posi25 4 Simulations Figure 4.3: Results for the simulations of a fast and high emitting particle (a) and a slow and low emitting particle (b). The top graphs display the time traces of the true (black) and the calculated positions (red) and the bottom graphs of the intensity time trace. The inset in (a) is an enlarged view of the two trajectories. The blue dashed line in (b) indicates the time, where the particle is lost. tion can not be calculated anymore and as mentioned above, the emission intensity drops instantly to the background level (indicated by the blue dashed line in fig. 4.3b). This happens for example, if the bead moves that fast, that it exceeds the rim of the orbit or if the emission is not high enough to calculate the position. This example explains nicely the criterion for a termination of a tracking experiment. As soon as the emission signal drops to the background level the measurement is stopped. With a rough estimation, a limiting case for the 2-dimensional tracking performance can be determined for a high emission signal. The particle can definitely not be tracked, if the mean displacement between two consecutive positions is larger than the value for the radius of the orbit R, i.e. if ph∆r(k∆t)2i ≥ R for k = 1 (4.5) ⇔4D∆t≥R2 ⇒D≥R2 4∆t By chance it could happen, that the particle can be tracked partially, but with a mean displacement larger than the radius of the orbit, a permanent tracking is not possible. For the parameters ∆t= 4 ms and an orbit radius of R= 190 nm, the limiting diffusion coefficient is determined to D= 2.26 µm2/s. This can also be compared with the values given in fig. 4.2d. Systems, that exhibits a larger diffusion coefficient than the determined one, can not be tracked with this settings. Coming back to the example of a 20 nm bead in water, successful tracking would 26 4.3 Loosing a particle require a time resolution of about ∆t= 0.4 ms, according to equation 4.6. In the following section the parameter limitations for the experimental setup are introduced. At this point I want to mention, that at the present stage of the setup it is not possible to track a 20 nm bead in water. The temporal accuracy has to be improved by a factor of 10. 27 A learning experience is one of those things that say, “You know that thing you just did? Don’t do that.” Douglas Adams 5 The tracking setup The main work of this thesis was the development of a new experimental setup that is capable of measuring single-particle trajectories with a high spatio-temporal resolution. It started from an empty optical table and ended with the successful implementation of single-particle tracking experiments. Among the afore mentioned various techniques, the orbit tracking method with sample scanning was chosen. This enabled the recording of successive positions of a particle for more than 10 minutes with a time resolution of 4 ms, resulting in trajectories with more than 1.5×105positions. A spatial resolution of better than 10 nm was achieved. However, prior to the acquisition of evaluable trajectories, numerous characterisation experiments have to be performed, to identify possible shortcomings that disturb the measurements. This work was substantial, so that it was suited for a publication in the Journal of the optical society of America A (JOSA A) with the titel “Setup for single-particle orbit tracking: artifacts and corrections”, which can be found in part II (publication P1) and which is the major publication of this thesis. It includes the description of the experimental setup, its full characterisation and the first successful tracking experiments with this setup. In the following sections the content of this publication is summarized and additional results on the spatial tracking accuracy of a static particle are given. 5.1 Optics & Hardware In principle, the experimental setup consists of a home-build confocal fluorescence microscope, a detection unit, and a laser beam deflection unit, which is the core part of the setup. Two acousto-optical deflectors (AOD) guide the laser beam (514 nm) on a circular path. This light orbit is projected into the microscope and is further reflected towards a water-immersion objective, that focuses the orbit into the plane of the sample, where the dye loaded particles are excited. The emission signal of the particles is then collected by the same objective and directed to either a CCD camera for widefield imaging or to an avalanche photo diode (APD) for tracking with high time resolution. The optical pathway was calculated with the aid of ray 29 5 The tracking setup transfer matrix analysis, because the mutual distances between all optical elements have to be as exact as possible. To operate the setup, an ingenious home-written program was used to implement the tracking algorithm for the calculation of the particle positions. This was done with a programmable measuring instrument, that is also responsible for the generation of all output signals, e.g. the piezo and AOD signals, and the acquisition of the emission signal, recorded by the APD. The use of only one single communication instrument between the PC and the setup made it possible to avoid hardware, like frequency generators, lock-in amplifiers or counter cards for the PC, and to have a perfect time synchronisation between all output and input signals, which is a prerequisite for this tracking method. Only if the output signals for the generation of the light orbit are synchronized with the data acquisition of the emission signal, a meaningful position of the particle can be calculated. The presented orbit tracking technique is able to reconstruct the 2-dimensional motion of the particle as long as this motion takes place in the plane of the orbit, which is also the xy-plane. The problem is now to keep the particle in this plane, e.g. by compensating for the particle motion in the 3rd dimension (zaxis). This was done by implementing a z-tracking algorithm, which was developed together with the diploma student Stefan Hain. The piezo with the mounted sample is wobbled up and down along the z-axis, i.e. also the particle is moved up and down and penetrates the orbit once in each direction. For every up (down) movement the z-position at which the particle features the highest emission is stored and serves as the new origin for the subsequent down (up) movement. This enabled the tracking over an extended period of time. Typically, tracking times of 10 minutes were used to analyse the diffusion processes, but it is possible to follow the motion of the tracer for more than 30 minutes. At the present stage of the setup the total recording time depends only on bleaching of the tracer beads and on the maximum scan range of the piezo (100 µmper axis). The important parameters of the setup are the orbit radius R, the frequency of the focus rotation νand the number of rotation periods P, that were used for setting the time resolution (∆t=P/ν) and the parameters for the z-tracking algorithm. The latter ones are a number of steps with a defined step size for the up and down movements of the piezo. These values have to be optimized separately for every new sample. Combining simulations, literature research and characterisation measurements allowed to define optimal settings for the remaining parameters. The values were found to be R= 190 nm, ν= 1 kHz and P= 4. Having a suitable set of parameters, a characterisation is necessary to learn the capability of the setup. 5.2 A static particle The most basic characterisation is the investigation of a static particle. It is useful for studying the position accuracy without any disturbing effects due to the motion 30 5.2 A static particle of the particle. First this situation is again simulated to know the theoretical predictions, which are then compared with experimental data. Therefore, in the following two subsections I will explain the simulated as well as the experimental determined position accuracy of a static particle, followed by a comparison. 5.2.1 Simulation The procedure of the simulation is similar to the already described one in chapter 4, with the difference, that the “simulated real” trajectory is now a fixed position. Here, it was set to (xp, yp) = (0,0), which corresponds to the centre of the orbit. With the feedback mechanism switched on, it does not matter at which starting position within the orbit the particle is placed. Because after the first position calculation, the feedback loop restores the particle back in the centre of the orbit with an accuracy, that is determined by acquisition limits (noise and position averaging during data acquisition). The simulation of the theoretical and the Poisson distributed number of photons, i.e. Snand Sn,poiss are shown in fig. 5.1a during the acquisition time of ∆t= 4 ms. A particle at the (0,0) position should result in a constant value for the theoretical number of emitted photons, because in an ideal case, also xsis zero and the expression xp−xsin equation (4.2) vanishes. Sn=δtS0exp   −2 w2(−Rcos(ωnδt))2+ (−Rsin(ωnδt))2 | {z } =R2   +Sbδt (5.1) =δtS0exp −2R2 w2+Sbδt =const. The reason for the periodic signal are the photon statistics and the feedback mechanism. Due to the Poisson distributed emission, the calculated position is not exactly zero. Via xsthis non-zero position is fed back to the generation of the number of photons. Hence, the expression xp−xsin equation 4.2 is non-zero, leading to a weak periodic signal Sn. The emission of photons, i.e. Sn,poiss stays more or less constant, because the absolute values of Snitself as well as the amplitude are quite small. From the simulated number of photons, the position of the particle is reconstructed. Figure 5.1b displays the x-position time-trace for N= 5000 position determinations, for a time resolution of ∆t= 0.5 ms (blue) and ∆t= 9.0 ms (red). Both simulations were performed for a mean emission intensity of I= 157 kcps. The high fluctuations in the position at a time resolution of ∆t= 0.5 ms compared to ∆t= 9.0 ms are ascribed to the reduced number of photons, that can be acquired during the time resolution of ∆t= 0.5 ms. A higher photon count rate leads to a more accurate position determination. To quantify this positional noise, the error in calculating a position in one dimension, which also defines the spatial accuracy, 31 6 Single-particle tracking: results & discussion Figure 6.1: a) Example of a trajectory of a 20 nm sized bead in pure glycerol, recorded for t= 608 s (N= 1.52 ×105data points). b) Extended view of a sequence with Nseg = 2000 positions (t= 8 s). 6.1 Analysing diffusion based on single-particle trajectories From seven different beads each with a nominal diameter of 20 nm diffusing in pure glycerol, trajectories of about 1.5×105positions were recorded. For every run a particle was selected from the widefield image that was sufficiently separated from other particles and from the surface of the coverslips. An example for a typical trajectory is displayed in fig. 6.1a. It represents 1.52×105data points recorded with a time resolution of ∆t= 4 ms and corresponds to an elapsed time of 608 s which is indicated by the colour code where blue corresponds to the begin of the trajectory and red to its end. In the following the analysis of the mean squared displacements and three alternative approaches were discussed on one representative trajectory. 6.1.1 Mean squared displacement analysis From the recorded trajectory r(t) = (xp(t), yp(t)), consisting of Npositions, the time-averaged MSD after a time lag τ=k∆tis calculated according to equation 2.21 [34]. For the trajectory displayed in fig. 6.1a the inset in fig. 6.2a shows the full MSD as a function of the lag time. For lag times τ < 50 s the MSD features a linear increase followed by strong fluctuations at longer times that reflect the progressively decreasing averaging. Therefore the analysis of the MSD curves is usually limited to the first few data points. Having more than 105data points at 38 6.1 Analysing diffusion based on single-particle trajectories Figure 6.2: a) Time-averaged MSD of the full trajectory shown in fig. 6.1a. The inset displays the full MSD, while the main figure shows the first 50 data points. The red line corresponds to a linear fit to the data points. b) Double-logarithmic plot of the data shown in a). The red line corresponds to a linear fit with a slope of 1.05 c) MSDs of 76 subtrajectories that result from cutting the full trajectory into pieces of 2000 data points each. Similarly to part a) of the figure, the inset shows the full MSDs and the main figure displays the first 50 data points of each MSD. The dotted green line corresponds to the MSD curve shown in a). d) Distributions of the diffusion coefficient and the anomaly parameter as obtained from the MSDs shown in c). hand, the MSD shown in fig. 6.2a was restricted to the first 50 data points corresponding to τ= 0.2 s. It is worth to note that even the 50th data point represents an average over more than 150000 entries. The full line in fig. 6.2a corresponds to a linear fit, which is in very good agreement with the data and from which a diffusion coefficient of hDiT= 17.19 ×10−3µm2/sis extracted. For this analysis, the first data point was neglected due to remaining experimental artifacts (oscillations of the piezo, details see publication P1). The notation hDiTwas used to indicate that this value stems from a time-averaged MSD according to equation 2.21. This number can be compared with the prediction from the Stokes-Einstein relation DSE =kBT/6πηa [5]. For T= 294 K,η= 1.2 Pa ·s[39], and a mean bead radius of a= 10 nm as given by the manufacturer, DSE = 17.94 ×10−3µm2/swas found. Although the agreement between hDiTand DSE is better than 5%, one can 39 6 Single-particle tracking: results & discussion argue that the discrepancy might reflect deviations of the radius of the bead from the value of 10 nm. Changing the point of view and taking the theoretical (StokesEinstein) diffusion coefficient as “real”, a= 10.4 nm for the actual size of the bead is found. Figure 6.2b shows the same data on a double logarithmic plot that can be fitted with a linear slope of hαiT= 1.05, which is in line with the expected α= 1 for Brownian motion and justifies the linear fit to the MSD data as explained in the theory part (section 2.1.3). Often it is not possible to measure that long trajectories due to bleaching processes of the tracer particle, due to weak emission signals or simply, because the particle moves out of the detectable area. In order to mimic to have only shorter trajectories with less data points the long trajectory was chopped into 76 subtrajectories, also called segments, of 2000 data points each (corresponding to a time range of 8 s), see figure 6.1b. From these subtrajectories the time-averaged MSDs were calculated. The result for all 76 data sets is shown in the inset of figure 6.2c and reveals very large variations of the MSDs with respect to each other. For the same reasons as detailed above, figure 6.2c displays only the first 50 data points of the MSDs from all 76 subtrajectories. A bundle of linear MSDs that vary in slope was found. Since these data have been recorded from the same individual polymer bead it can be excluded that this spread reflects variations in the bead size. The different slopes rather result from the reduction of the statistical weight for the time-averaged MSDs of the subtrajectories. This is in line with the fact that the time-averaged MSD from the total trajectory of 1.52 ×105data points runs nicely through the centre of the bundle of partial MSDs, as shown in figure 6.2c by the dotted green line for comparison. Analysing the slopes of the individual MSDs in terms of a diffusion coefficient and an anomaly parameter yields the distributions shown in figure 6.2d. The mean (empirical standard deviation) for the diffusion coefficient is hhDiTiE= 17.15 ×10−3µm2/s(sD= 2.39 ×10−3µm2/s) and for the anomaly parameter hhαiTiE= 1.01 (sα= 0.05). The notation hh·iTiEsymbolizes that the data have been evaluated from (short) time-averaged MSDs that have been averaged over an ensemble (here 76 subtraces). The histograms presented in fig. 6.2d can be interpreted as the empirical probability density functions (PDFs) to measure a distinct range of values for these parameters. Mathematical statistics tells us, that the best estimator for the “real” value of the diffusion coefficient is provided from the first moment of this probability density which corresponds to hhDiTiE. The good agreement between the numerical values found for hDiT, i.e. the result from the full trajectory of 1.5×105data points, and hhDiTiEconfirms that the statistical shortcomings of the trajectories with only 2000 data points are averaged out in the long trajectory. However, it shows as well that an accurate determination of the diffusion coefficient requires either a very long trajectory or knowledge about the (empirical) PDF for the outcome of an experiment on a shorter trajectory. A single experiment on a short trajectory is not sufficient and the result for the diffusion coefficient can differ by up to a factor 40 6.1 Analysing diffusion based on single-particle trajectories Figure 6.3: Ensemble-averaged mean squared displacement of the cut sequences of the single trajectory shown in fig. 6.1a. The inset displays the double-logarithmic plotted data, whereas the main figure shows the linear ones. The red solid lines are linear fits to the data points. of 2. Assuming that ergodicity is preserved, the cut sequences of the sub-ensemble can also be used to calculate the ensemble-averaged MSD according to the equation 2.22. This method is an alternative way to investigate a diffusion process, but is less suited for single trajectory experiments, because the number of traces for a statistical relevant ensemble, recorded under identical experimental conditions is mostly too small to yield proper results. However, for a comparison with the presented results also Dand αwere determined with this method. From line fits (red solid lines in fig. 6.3) to the linear as well as the double-logarithmic data hDiE= 17.06×10−3µm2/sand hαiE= 0.97 was achieved (cf. fig. 6.3). Again both values fit quite well with the values determined by the above mentioned analysing methods. Slight differences stem from the rather weak ensemble. No violation of the ergodic principle is observed. The presented MSD analysis was finally performed for all the seven measured trajectories. The results from the time-averaged MSDs (hDiT,hαiT), the mean values of the ensemble of subtrajectories (hhDiTiE,hhαiTiE) as well as the results from the ensemble-averaged MSDs (hDiE,hαiE) are summarized in table 6.1. Beside the values obtained from the pure ensemble-average analysis, which are normally not used because of its weak statistics, all the values from the time-averaged analysis agree quite well with respect to each other. For all the following discussions regarding this sample (20 nm bead in glycerol), the measured normal Brownian motion with α= 1 is used. 41 6 Single-particle tracking: results & discussion Exp. nr. hDiT/hhDiTiE/hDiE/hαiThhαiTiEhαiE 10−3µm2/s 1 17.19 17.15 17.06 1.04 1.01 0.97 2 21.58 21.64 21.03 1.02 1.02 0.99 3 16.70 16.60 17.71 1.03 1.01 0.96 4 17.49 17.47 18.18 1.04 1.01 1.15 5 15.01 14.98 14.95 1.05 0.94 0.91 6 24.14 24.10 24.57 1.04 1.01 1.02 7 12.32 12.34 13.30 1.01 1.04 9.94 Table 6.1: Summary of the diffusion coefficients and the anomaly parameters obtained from time-averaged as well as ensemble-averaged analysis. The entries are the timeaveraged diffusion coefficient (anomaly parameter) hDiT(hαiT), the mean value determined from the ensemble of subtrajectories hhDiTiE(hhαiTiE) and the ensemble-averaged diffusion coefficient (anomaly parameter) hDiE(hαiE). The analysis of single-particle trajectories with the use of the mean squared displacements (MSD) is the most common method. Among the here presented techniques of trajectory analysis, this is the only one, where the diffusion coefficient as well as the anomaly parameter can be determined. The most informations regarding a diffusion process can be extracted. 6.1.2 Alternative approaches Cumulative distribution function This method supplies an alternative approach to analyse a trajectory [37, 38]. According to the explanation in section 2.1.3 the empirical cumulative distribution function (CDF) for the square displacements at the lag times τ= 4 ms −0.2 s (τ= (1−50)∆t) was determined consecutively for a single trajectory. An example, calculated at a lag time of τ= 40 ms is shown in fig. 6.4a (black squares). The data points can be modeled with a double-exponential function as given by CDF ∆r2, τ= 1 −βexp −∆r2 ∆r2 1+ (1 −β)exp −∆r2 ∆r2 2 (6.1) Here, the ∆r2 iare related to a fast (1) and slow (2) component of the diffusion coefficient, i.e. ∆r2 i= 4Diτ, weighted by a fraction β. For systems, where pure Brownian motion is present, βequals 1 and the CDF reduces to a single-exponential with a single diffusion coefficient D1=Dcdf . Yet, for systems showing subdiffusive behaviour βclearly differs from 1 and a double-exponential with a so called slow component of the diffusion coefficient (∆r2 2)is better suited to describe the data. Hence, this method is powerful in distinguishing normal from anomalous diffusion, just by determining β. The afore mentioned fit function (eq. 6.1) was applied to 42 6.1 Analysing diffusion based on single-particle trajectories Figure 6.4: a) Example of a cumulative density function at a lag time of τ= 40 ms. b) diffusion coefficient and fraction βtaken from double-exponential fits to the empirical CDF. The arrow marks the values that correspond to the CDF shown in a). Due to remaining experimental shortcomings the first data point was neglected for the interpretation. the data points (cf. fig. 6.4a, red solid line). The shown example (τ= 40 ms) yields β= 1.002 which is in nearly perfect agreement with Brownian motion. The value of the fraction as a function of the lag time can be seen in fig. 6.4b, upper part. For this single trajectory on average a fraction of hβi= 0.99 was found, which is close to unity and resembles Brownian motion with a single-exponential for the CDF. Further, the mean diffusion coefficient was calculated with this method to hDcdf i= 17.44 ×10−3µm2/s, which is shown graphically in fig. 6.4b, lower part. The calculated value for the example CDF, shown in fig. 6.4a are marked with an arrow. Again, the first data points for τ= 4 ms was neglected due to experimental shortcomings. Compared to the value obtained from the time-averaged MSD (cf. table 6.1, exp. nr. 1), a good agreement is achieved. This procedure was done for all the 7 recorded trajectories. The results can be found in table 6.2. From the overall average of the seven trajectories of the fraction hβi= 0.96 one can conclude that the here used system reflects Brownian motion. Additionally, the diffusion coefficient fits quite well with the already determined ones from the mean squared displacement analysis. The slight discrepancy to theoretical/expected values can be addressed to the usage of partly uncorrected trajectory data (only the setup induced artifacts were corrected, i.e. the systematic error in the tracking algorithm and the (mis)alignment of the piezo stage) which was necessary for this evaluation. The advantage of this technique is the capability to study multi-component diffusion phenomena, e.g. a two-component diffusion can be interpreted as a slow and a fast mobility of the tracked particle. The fraction between these two mobilities can then be used to distinguish normal from anomalous diffusion. One has to be careful, that this fraction is not related to the anomaly parameter α. Another big advantage is, that this method can also be applied to relative short trajectories 43 6 Single-particle tracking: results & discussion Exp. nr. hβi hDicdf / 10−3µm2/s 1 0.99 17.44 2 0.90 22.15 3 0.98 16.81 4 0.92 18.19 5 0.97 16.46 6 1.01 24.05 7 0.97 11.98 Table 6.2: Summary of the results from the analysis according to the cumulative distribution function from 7 individual polymer beads. The entries denote the experiment nr., the fraction hβiT, and the diffusion coefficients hDicdf . with a few position determinations [37]. Spatial extend & shape Another method to extract information from a tracking experiment is to determine the spatial extend by means of a gyration ellipse around a trajectory and its shape, i.e. the asphericity. The method, how to calculate this value is discussed in detail in section 2.1.3. In fig. 6.5a the gyration ellipse with its radii of gyration is overlayed to a representative trajectory with N= 1.52 ×105data points. The overall spatial extend of the trajectory is rather elongated than circular. The latter one would be expected for a perfect random walk and holds true for an ensemble of traces. But, for a single trajectory the elongated version is found [35]. This behaviour is better described by the shape of the trace instead of the spatial extend, which can vary (mainly) due to the mobility of the particle and the length of the trajectory. The shape is determined by the asphericity, which is a measure for the deviation from a spheric (3D) or a circular (2D) shape. According to eqn. 2.26 the asphericity A of an individual trajectory is calculated for a certain length (number of positions Nseg) of cut segments. (As a reminder, the calculation of Arequires averaging over an ensemble of traces.) It is quite hard to decide, at which segment length the value of Ais reliable. Therefore the asphericity of a trace was calculated as a function of the length Nseg, i.e. A(Nseg). The result of the 7 individual measured traces is shown in fig. 6.5b. For a segment length of Nseg = 2 the shape of the trajectory is perfect rod-like and leads a value of A(2) = 1. Within the first 15 data points the asphericity converges towards a value of A= 4/7(grey line in fig. 6.5b), that is the analytical value for a 2-dimensional random walk [35, 36]. To verify this behaviour for the here presented measurements in a more quantitative way, a statistical relevant window (grey shaded area in fig. 6.5b) for the calculation of a mean asphericity hAiwas chosen. This region was restricted to 20 ≤Nseg ≤250 where at the one end (Nseg >20)A(Nseg)is regarded as fully converged and on the other 44 6.1 Analysing diffusion based on single-particle trajectories Figure 6.5: Asphericity of all trajectories plotted against the sequence length Nseg. The grey shaded region is the window, where the asphericity is averaged. The grey solid line represents the analytical value for a 2-dimensional random walk, i.e. A=4/7. hand (Nseg <250) the averaging in calculating the asphericity is high enough. The values for the 7 experiments are summarized in table 6.3. On average, the mean asphericity was determined to hAi7= 0.566 which is in very good agreement with the analytical value of A= 4/7≈0.571 for a Brownian motion. The index 7 indicates the average over all seven experiments. The nearly perfect coincidence reflects furthermore, that no significant signs of drift or flow is overlayed to the trajectories that would lead to a distortion and concomitantly influence the asphericity. For the investigation of systems showing anomalous diffusion, this method can be used to clarify the type of sub-diffusion, which is described in publication P3 (see part II). With this method, neither the diffusion coefficient nor the anomaly parameter can be determined. Hence, it yields minimal informations about a diffusion process. But, in addition to the time-dependent parameters (D,α) a structural parameter Exp. nr. hAi 1 0.555 2 0.575 3 0.564 4 0.563 5 0.540 6 0.569 7 0.595 Table 6.3: Summary of the determined mean asphericities hAiof the seven individual trajectories. 45 6 Single-particle tracking: results & discussion is obtained, that describes the deviations from a circular shape (in case of a 2D trajectory). In combination with the anomaly this technique is powerful in investigating subiffusive behaviour, e.g. of crowded fluids like it is discussed in section 6.4 of this thesis. Ergodicity breaking parameter Recently a technical parameter for determining the degree of ergodicity breaking has been published [51, 52]. E(τ) = Dh∆r(τ)2i2 TEE−hh∆r(τ)2iTi2 E hh∆r(τ)2iTi2 E (6.2) The calculation of this ergodicity breaking parameter E(τ)requires again an ensemble of short trajectories, that can be obtained by splitting a long trace into short segments with equal length. From these short segments the time-averaged MSDs are calculated. According to equation 6.2 the time-averaged MSDs, i.e. h∆r(τ)2iT, are on the one hand squared and then ensemble-averaged, symbolized by h·iE, and on the other hand first ensemble-averaged and then squared. For values of Eclose to 0 the system is regarded as non ergodicity breaking [51, 52]. In an analog way to the determination of the asphericity the ergodicity breaking parameter was calculated. For averaging E(τ)we used a time window that corresponds to the analysis of the diffusion coefficient and the anomaly parameter (2-50 data points). The results for the averaged ergodicity breaking parameter hEi of every single trajectory is listed in table 6.4. All values are smaller than 0.020 which is interpreted as non-ergodicity breaking, compared to values found in the literature for weak ergodicity breaking system. There, E= 0.57 [51] and hence a factor of more than 20 times larger than the here obtained values. This additional parameter reflects the expected Brownian motion, where ergodicity is preserved. Exp. nr. hEi 1 0.017 2 0.019 3 0.020 4 0.020 5 0.018 6 0.017 7 0.020 Table 6.4: Summary of the determined mean ergodicity breaking parameter hEiof the seven individual trajectories. 46 6.2 Accuracy of diffusion coefficients 6.2 Accuracy of diffusion coefficients How well a diffusion coefficient can be determined by a mean squared displacement analysis of experimental single-particle trajectories is an important issue. In the previous chapter it was explained, that long trajectories results in accurate diffusion coefficients. However, one can imagine, that not only the length of the trajectory is important, also uncertainties in the position determination or the number of fitting points used in the MSD analysis have an effect on the accuracy of the measured diffusion coefficient. To what extend these parameters influence the accuracy is discussed in great detail in the literature on the basis of simulations and numerical calculations [53–55]. First it was studied by Qian and Saxton and was further developed and subtilized by Michalet who took also experimental localisation errors into account. Up to now a comparison with experimental data remained elusive. In this section the problem is treated from the experimental point of view and the results were compared with the predictions made in the afore mentioned literature. The findings were published in Physical Chemistry Chemical Physics (see publication P2). In the following I will give a summary of this work. Many studies that use single-particle tracking to investigate diffusion processes, especially in the field of biophysics, often lack of sufficiently long trajectories, either because the particle diffuses out of the detectable area or due to photobleaching of the fluorescent tracer particles. Hence, the MSD points of a trajectory are less averaged which results in a less accurate diffusion coefficient. In order to investigate this accuracy, many trajectories are required to perform a statistical analysis. Usually, it is not possible to acquire a statistically relevant number of trajectories with identical experimental conditions. While it is rather simple to keep the viscosity and temperature between several independent tracking experiments constant, the bead size determined from each recorded trajectory can vary drastically. Therefore, in the literature simulations with a high number of fictitious trajectories with identical parameters (trajectory length N, temperature T, viscosity η, bead radius a) were used to clarify this issue [53–55]. In this work an experimental approach to measure the accuracy of a diffusion coefficient is presented. Exploiting single-particle orbit tracking, very long trajectories comprising more than N= 1.5×105data points were recorded from individual 20 nm sized particles in glycerol that undergo Brownian motion. Each of these large data sets can be decomposed into an ensemble of segments with a various number of data points Nseg, providing the same ensemble of mean squared displacement curves from the same individual particle. By a linear fit to each MSD curve of an ensemble with a given value of Nseg, the slopes D∗as a function of the number of fitting points nwere determined (nstarts from the MSD point 2 and ends at n+ 1, details see publication P2). Subsequently for each value of n, the relative error sD∗/D∗was calculated from the histograms of the slopes and plotted against each other. Here, the (empirical) mean D∗and standard deviation sD∗were determined. Figure 6.6 47 6 Single-particle tracking: results & discussion Figure 6.9: Results of the anomaly parameter αas a function of the determined bead radius a. The full blue (red) squares are the nominal 20 nm sized beads in the sample sucrose (dextran), while the open blue (red) squares are the nominal 50 nm beads in sucrose (dextran). The grey line serves as a guide for the eye. i.e. the viscosity stays constant (see equation 6.4). 1 D=6πη kbTa=γa (6.4) This presentation is suitable enough, because only the dependence in the particle sizes and not the absolute values for the radii are important. The experiments were performed in two fluids with different viscosities, i.e. for a proper visualization two x-axis were introduced, where the bottom (blue) axis in figure 6.9 corresponds to the measurements in sucrose and the top (red) axis to the measurements in dextran. To compare the two data sets, the axis were further scaled according to the smallest measured values of 1/D for the nominal 20 nm sized beads. Despite the nominal values of 20 nm and 50 nm for the bead sizes a distribution was determined, which was discussed in detail in section 6.3. For this measurement, a broad distribution was helpful, because the size effect on the diffusion behaviour was studied. The measurements in sucrose showed independent of the size of the beads an anomaly value close to 1, while the measurements in dextran showed a clear dependence on the size of the beads. However, all measurements for the 50 nm sized beads in dextran showed an unambiguous subdiffusive behaviour. The mean anomaly values for the 50 nm sized beads were determined to hαsuci= 0.98 for sucrose and 54 6.4 Crowded Fluids to hαdexi= 0.82 for the crowded dextran solution. The clear separation between these two values is evident, i.e. the subdiffusion is not just an artifact. A plausible explanation is the size ratio of the beads and the mesh size of the dextran network. Small beads can diffuse nearly freely through the network structure, while larger particles are hindered. Hence, in the following only the data obtained for the measurements with bead sizes of 50 nm are discussed. The experimentally determined trajectories of the 50 nm sized beads in dextran were compared with simulated trajectories based on the three theoretical models mentioned afore. These models have in common, that they can describe subdiffusion. But, while the CTRW model shows slight ergodicity breaking according to the recently developed ergodicity breaking parameter [51, 52] (see eq. (6.2), the other two models do not. Also the analysis of the experimentally measured trajectories do not show any signs of ergodicity breaking. This rules out the CTRW model for a proper description of the crowding induced subdiffusion in dextran. The remaining two models (OD and FBM) show ergodic behaviour and therefore another criterium has to be found to decipher between those two theories. The idea is to calculate the asphericity Aand the anomaly αof experimentally determined trajectories and compare those results with the results obtained from simulated trajectories that repose on the theories of OD and FBM. For these simulations, trajectories with αvalues between 0.5and 1.0were generated and the asphericity was calculated subsequently, yielding the asphericity as a function of the anomaly, i.e. A(α). This analysis was performed in the group of the cooperation partner. The details are described in the publication P3. The mean values of the experimentally determined anomaly and asphericity are hαi= 0.82 (vide supra) and hAi= 0.46, respectively. For the comparison, the αvalue from the experiment (0.82) was inserted in both functions A(α), obtained by the simulations with OD and FBM. This yields AOD = 0.56 and AFBM = 0.47. A nearly perfect agreement between the experimental data and the model of fractional Brownian motion was obtained. With this work a clear statement regarding the theoretical description of a subdiffusive process in a complex fluid was made, which helps to better understand the reaction kinetics in living cells. It is worth to note, that it is of crucial importance for the analysis of the experimental data to measure very long trajectories with a high spatial and temporal accuracy. Having those data a comparison with simulations was possible. 55 The future belongs to those who believe in the beauty of their dreams Eleanor Roosevelt 7 Outlook The described experimental setup for single-particle orbit tracking, the data analysis and the applications are on a level were fascinating experiments and investigations can be performed. The setup is well characterised and first results showed its capability. But some improvements in all of the mentioned fields are helpful to push this research forward and to get a highly developed setup. In the following a short outlook for possible applications and setup improvements are given. Instead of concentrating on tracking experiments in the research sector of biophysics, where most of the groups are working, the field of material sciences can be investigated, where tracking of single nanoobjects can help to better understand processes on a nanometer length scale. For example the switching behaviour of a liquid crystal is challenging when working on a submicrometer scale. If traceable molecules or particles can be attached to the molecules of a liquid crystal, the timedependent motion of these molecules can be recorded with a high precision. This would help to better understand the phase transition of liquid crystals. Further, if molecules with a defined transition dipole moment are used, also the rotational diffusion can be investigated. Another possibility to use particle tracking in material sciences is the examination of diffusion processes through nanoporous membranes. Recently successful experiments were published, where the diffusion through switchable nanopores that vary in size were measured [64]. With the single-particle tracking experiments, the structure of the pores can be determined, with a resolution far beyond the diffraction limit of light. This helps to construct filter systems on a microand even nanometer scale. With the here presented setup, at the moment only 2-dimensional tracking experiments can be performed with a high spatio-temporal resolution. But with a lower time resolution also 3-dimensional trajectories can be recorded. The 3rd dimension becomes accessible through the z-tracking algorithm. Here it was only used to keep the particle of interest in the focal plane, but the algorithm stores the z-position of the particle with a lower time resolution. In figure 7.1 the 3-dimensional trace with a number of N= 15000 positions of a 20 nm sized bead, diffusing in glycerol is shown for a time resolution of ∆t= 40 ms, which is a factor of 10 slower than 57 7 Outlook Figure 7.1: 3-dimensional trajectory of a single particle with a size of 20 nm, that diffuses in the fluid glycerol. The trace consists of N= 15000 data points with a time resolution of ∆t= 40 ms. The green, blue and red trajectory corresponds to the projections to the xz, yz, and xy plane, respectively. the rest of the results. Regarding the position accuracies of the z-position, this has to be investigated in a future work, the 2-dimensional accuracy (xy plane) is the same like in the presented work (≈10 nm). To improve also the temporal resolution for the z dimension, a fast scanning in this direction has to implemented. One option is to install a third AOD, that is responsible for the generation of a second light orbit. The optics has to be aligned in a manner, that one of the light orbits is slightly above the particle and the other is slightly below. By switching the excitation intensities of these two orbits periodically on and off with a different frequency than the one used for the focus rotation, in an analog way the z-position of the particle can be calculated be demodulating the emission signal. The temporal resolution of all of the axis can be improved by implementing a beam scanning technique, rather than sample scanning with the piezo. By moving the laser beam with the AODs a much higher temporal resolution should be obtained, because no mechanical elements restrict the scanning mechanism. These were just some improvements that can be done to bring the setup to a new level. Especially with the higher time resolution, diffusion processes on a faster time scale becomes accessible. 58 Appendix A Program code In the following the Matlab source code for the generation of the “simulated real” and “simulated reconstructed” trajectories is displayed. 1 2% Simulation of the " real " and the " reconstructed " trajectory 3% *********************************************************** 4 5clc; 6clear ; 7format long 8nr_trace =1; 9 10 %**** tracking parameter ************************************ 11 dt =2 E -6; % sampling time [s] 12 f=1000; % orbit frequency [ Hz] 13 ro_x =190E -9; % orbit radius x [ nm ] 14 ro_y =190E -9; % orbit radius y [ nm ] 15 w =270E -9; %1/e^2 of focus [nm] 16 gb =500; % background emission [ cps] 17 g0 =450000; % emission in the focus 18 omega =2* pi*f*dt; % orbit frequency 19 s =100; % samples ( length of trajectory ) 20 21 22 for i2 =0: nr_trace -1 23 24 % ***** diffusion parameter ** ********** ********** ******* 25 eta =1.2; % viscosity [ Pas] 26 r =10 E -9; %radius of sphere [m] 27 kb =1.3806504 E -23; % Boltzmann constant [J/K] 28 T =294; % Temperature [K] 29 g =6* pi* eta*r; % friction coefficient ( Stokes ) 30 fc= sqrt(2* kb *T/g/dt); % prefactor stochastic force 31 D=kb*T/g; % diffusion coefficient 32 P=4; % orbit periods 33 a=P /(f* dt); % sampling intervals 34 tr=P/f; % time resolution [s] 35 36 37 % ***** initial values ******* ************** ************* 38 x=0E -9; %x of "real" trace 39 y=0E -9; %y of "real" trace 40 dx =0; %x step of "real " trace 41 dy =0; %y step of "real " trace 42 xc =0; %x of " reconstructed " trace 43 yc =0; %y of " reconstructed " trace 44 dxc =0; %x step of " reconstructed " trace 45 dyc =0; %y step of " reconstructed " trace 46 xs =0; %x of piezostage 47 ys =0; %y of piezostage 48 x_a = zeros (1,s); % array for " real " x values 49 y_a = zeros (1,s); % array for " real " y values 59 7 Outlook 50 xc_a=zeros (1,s); % array for " reconstructed " x values 51 yc_a=zeros (1,s); % array for " reconstructed " y values 52 time=zeros (1,s); % array for time 53 int = zeros (1,s); % array for intensity 54 randn (’state ’,sum (1000*clock )); 55 56 57 %** generation of the "real " and " reconstructed " trace * 58 for i3 =1:s 59 60 S= zeros (1 ,a); % theoretical nr. of photons 61 Sp= zeros(1,a); % poisson dist. nr. of photons 62 Scos=zeros (1,a); 63 Ssin=zeros (1,a); 64 cos_v = zeros (1,a); 65 sin_v = zeros (1,a); 66 67 68 for n=1: a 69 % ********* " real " trajectory ********** 70 fs_x=randn ; 71 fs_y=randn ; 72 73 dx= fs_x *fc*dt; 74 dy= fs_y *fc*dt; 75 76 x=x+dx ; 77 y=y+dy ; 78 79 % ***** " reconstructed " trajectory ***** 80 81 S(n)=dt* gb+dt*g0*exp ( -2/w ^2*(( x-xs - ro_x * cos( omega *n)) ^2+ 82 (y-ys -ro_y* sin( omega *n)) ^2) ); 83 cos_v (n) =cos( omega * n); 84 sin_v (n) =sin( omega * n); 85 end 86 87 x_a (i3)=x *1 E9; 88 y_a (i3)=y *1 E9; 89 90 Sp= poissrnd (S); 91 Scos = Sp .* cos_v ; 92 Ssin = Sp .* sin_v ; 93 94 % ***** calculation of the position ******** 95 96 if (sum(Sp)==0) 97 dxc =0; 98 dyc =0; 99 else 100 dxc =w ^2/(2* ro_x )* sum( Scos ) /sum( Sp); 101 dyc =w ^2/(2* ro_x )* sum( Ssin ) /sum( Sp); 102 103 xc=xc+ dxc; 104 yc=yc+ dyc; 105 106 xc_a ( i3)= xc *1 E9 ; 107 yc_a ( i3)= yc *1 E9 ; 108 109 % ******** feedback mechanism ************** 110 xs=xc; 111 ys=yc; 112 60 B Harmonic approximation 113 % ******** time and intensity ************** 114 115 time (i3)=i3*tr; 116 int( i3)= sum(Sp)/tr; 117 118 end 119 end B Harmonic approximation In this section the derivation of the harmonic modulation of the emission signal for particle positions close to the centre of the light orbit is shown. Therefore equation (2.28) from the section 2.2.2 is used. I(t) = I0exp −2 w2(xp−Rcos(ωt))2exp −2 w2(yp−Rsin(ωt))2+Ib(B.1) A 2-dimensional Taylor series of this equation in the neighborhood of the position (x0, y0) = (0,0) gives: I(t)≈I(t)|x0,y0+∂ ∂xp I(t)x0,y0 xp+∂ ∂yp I(t)x0,y0 yp+O(x2, y2) =Ib+I0e2 w2R2cos2(ωt)e2 w2R2sin2(ωt) +I0e−2 w2(xp−Rcos(ωt))2e−2 w2(yp−Rsin(ωt))2·−4 w2(xp−Rcos(ωt))x0,y0 xp +I0e−2 w2(xp−Rcos(ωt))2e−2 w2(yp−Rsin(ωt))2·−4 w2(yp−Rsin(ωt))x0,y0 yp +O(x2, y2) =Ib+I0e2R2 w2+I0e2R2 w2·4R w2cos(ωt)xp+I0e2R2 w2·4R w2sin(ωt)yp +O(x2, y2) =Ib+I0e2R2 w21 + 4R w2(xpcos(ωt) + ypsin(ωt))+O(x2, y2) The emission signal is modulated with a cosand a sinfunction. 61 Bibliography [1] R. Brown, The Miscellaneous Botanical Works of Robert Brown (Hardwicke, Robert, 1866). [2] University of Strathclyde, Diffusion processes: proceedings of the Thomas Graham Memorial Symposium (Gordon and Breach, London, 1971). [3] A. Fick, “Über Diffusion,” Annalen der Physik (Berlin) 170, 59 (1855). [4] A. Einstein, “Zur Theorie der Brownschen Bewegung,” Annalen der Physik (Berlin) 324, 371–381 (1906). [5] A. Einstein, “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen,” Annalen der Physik (Berlin) 322, 549–560 (1905). [6] M. von Smoluchowski, “Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen,” Annalen der Physik 326, 756–780 (1906). [7] R. Bird, W. Stewart, and E. Lightfoot, Transport Phenomena (Wiley, 2002). [8] H. Schmalzried, Chemical kinetics in solids (VCH, Weinheim, 1995). [9] B. Alberts, A. Johnson, P. Walter, J. Lewis, M. Raff, and K. Roberts, Molecular Biology of the Cell (Taylor & Francis, 2007). [10] E. L. Elson, “Fluorescence correlation spectroscopy and photobleaching recovery,” Annu. Rev. Phys. Chem. 36, 379–406 (1985). [11] E. L. Elson and D. Madge, “Fluorescence Correlation Spectroscopy. I. Conceptual Basis and Theory,” Biopolymers 13, 1–27 (1974). [12] E. L. Elson, “Fluorescence correlation spectroscopy: Past, present, future,” Biophys. J. 101, 2855–2870 (2011). [13] J. Gelles, B. J. Schnapp, and M. P. Sheetz, “Tracking kinesin-driven movements with nanometre-scale precision,” Nature 331, 450–453 (1988). [14] Q. Wang and W. Moerner, “Optimal strategy for trapping single fluorescent molecules in solution using the abel trap,” Appl. Phys. B 99, 23–30 (2010). [15] A. E. Cohen and W. E. Moerner, “Suppressing Brownian motion of individual biomolecules on solution,” Proc. Natl. Acad. Sci. USA 103, 4362–4365 (2006). 63 Danksagung Zum Schluss möchte ich mich noch bei einigen Leuten bedanken, die auf vielfältige Weise zum Gelingen dieser Arbeit beigetragen haben. Als Erstes möchte meinem Doktorvater Prof. Dr. Jürgen Köhler einen Dank aussprechen. Durch sein Vertrauen in meine Arbeit hat er es mir ermöglicht, einen neuen Aufbau zu entwickeln, der nach einigen Hochs und Tiefs letztendlich ausgereift war und neue Perspektiven für die weitere Forschung eröffnet. Er hat mir stets die nötige Freiheit gegeben meine Ideen umzusetzen und hat mir immer durch zahlreiche Diskussionen über meine Messergebnisse weitergeholfen, wenn ich seine Unterstützung brauchte. Ein weiterer großer Dank geht an unseren Kooperationspartner Prof. Dr. Matthias Weiss. Von Matthias konnte ich viel über anomale Diffusionsprozesse und neue Auswertemethoden lernen. Er war immer diskussionsbereit und hat sich für meine Probleme viel Zeit genommen. Weiterhin möchte ich Dr. Marcel Hellmann danken, der mich in programmtechnischen Fragen zur Auswertung der Messdaten unterstützt hat. Der Serviceabteilung der Firma Jäger Messtechnik GmbH möchte ich für den hervorragenden Support danken. Insbesondere ist hier Stefan Plappert zu erwähnen, der mir des öfteren die nötigen Tipps zur Softwareansteuerung des Aufbaus gab und auch fortwährend ein reges Interesse am Fortschritt meiner Arbeit zeigte. Danke auch an Prof. Dr. Werner Köhler, der mir bei grundlegenden Fragen zu Diffusionsprozessen weiterhalf. Unsere technischen Angestellten, ohne die ein Lehrstuhl meiner Meinung nach nie richtig funktionieren kann, waren immer hilfsbereit und sorgten an vielen Stellen für eine Erleichterung der Arbeit. Stefan Schlicht war bei elektrotechnischen Fragen stets ein guter Ansprechpartner. Waltraud Joy ist mir bei chemischen Fragen bezüglich der Probenpräparation mit Rat und Tat zur Seite gestanden. Werner „Wörni“ Reichstein ist immer für einen zur Stelle und hat mir vor Allem in der Anfangsphase durch einige, teils größere, Umbauten im Labor geholfen. Weiterhin möchte ich der Mechanikwerkstatt und insbesondere Frank Neumann und Olli ein Lob aussprechen. Ohne sie wären einige Konstruktionen wohl nicht zustande gekommen. Ein großer Dank geht an die „Verwaltungszentrale“ des Lehrstuhls, das Sektretariat. Besonders erwähnen möchte ich hier Evelyn Hülsmann, die sich um sämtliche Verwaltungsangelegenheiten gekümmert hat und immer ein offenes Ohr für die „Problemchen“ der Mitarbeiter hat. Vielen Dank auch an unsere PostDocs Richard Hildner und Abey Issac für viele interessante Diskussionen und das Korrekturlesen meiner Arbeit. Ich möchte an dieser Stelle auch „meinem“ ersten Bachelor Daniel Zalami für neue Ideen zur Weiterentwicklung des Aufbaus und für die gute Zusammenarbeit danken. Ein Dankeschön 71 geht auch an die Mitstreiter der Doktoranden während meiner Zeit am Lehrstuhl, Tobias Pflock und Florian Spreitler, die den Sprung in die „echte“ Welt bereits hinter sich gebracht haben. Vor allem Florian möchte ich für die große Unterstützung durch viele Diskussionen über PC-Programme, angewandte Optik und Interpretation von Messungen danken. Nicht zu vergessen sind natürlich Ralf Kunz und Paul Böhm, die mit mir die Zeit am Lehrstuhl begonnen haben und langsam auch dem Ende der Promotion entgegensehen. Auf geht’s Jungs! Marc Jendrny danke ich für eine coole Banz-Tagung. Besonders hervorheben möchte ich Steffen Hartmann, Matthias Gebhart, Axel Herrmann, Vanessa Weith, Laura Winterling, Lisa Krapf und Florian Schwaiger. Ein Teil davon ist zwar schon lange nicht mehr am Lehrstuhl, sie haben aber entscheidend dazu beigetragen, dass ich heute der bin, der ich bin. Ich danke euch für eine Menge schöner Erlebnisse in Bayreuth, diverse Wanderungen, exzessive Parties, lustige Kochabende, den Kaffeeeckentratsch und vieles mehr. Was wären die Miamiam-Besuche ohne euch gewesen. Vor allem Florian möchte ich großen Dank aussprechen. Die unzähligen Diskussionen mit ihm waren für mich nicht nur physikalisch sehr wertvoll, auch das Humorniveau wurde auf ungeahnte Ebenen gelenkt. Mann, mann, mann. Highlights waren London (Baby!) und das Pokalfinale in Dortmund (Heja BVB!). Es war eine geniale Zeit. Alle weiteren Kollegen, die hier nicht namentlich aufgeführt sind (bitte nicht böse sein), möchte ich natürlich auch für das angenehme Arbeitsklima danken. Der Lehrstuhl ist ein bunter „Haufen“ aus chaotischen und organisierten, ruhigen und aufbrausenden, engagierten und weniger engagierten, aber immer gut aufgelegten Kollegen. Ich möchte mich bei Allen für die interessanten fachlichen und (zum Teil sehr abstrusen) nicht fachlichen Gespräche in der Kaffeeecke, bei diversen Grillfeiern und abendlichen Kneipenund Biergartenbesuchen herzlichst bedanken. Das Arbeitsumfeld war durchwegs positiv und es hat sehr viel Spaß gemacht in dieser Gruppe zu arbeiten. Auch meinen Freunden außerhalb der Universität gebührt ein Dank für die Ablenkung vom teilweise stressigen Alltag. Außerdem möchte ich auch der Familie meiner Freundin für viele schöne Stunden und Ausflüge danken. Schließlich möchte ich mich bei meiner Familie und insbesondere bei meinen Eltern für die jahrelange und uneingeschränkte Unterstützung bedanken. Ihr habt mir immer den nötigen Rückhalt gegeben, was mir ein sorgenfreies Studium und eine anschließende Promotion ermöglichte. Ihr wart immer für da und dafür möchte einfach nur DANKE sagen. Ohne euch wäre vieles nicht möglich gewesen. Zu guter Letzt möchte ich mich bei meiner Freundin Nadine bedanken. Die letzten Jahre mit Dir waren wunderschön und hoffentlich folgen noch viele weitere. Du warst immer für mich da, hast mich bei meiner Arbeit immer unterstützt und mein Leben auf vielfältige Weise bereichert. Danke, für all die schönen Momente und dass du immer für mich da bist. ILD 72 Erklärung Hiermit versichere ich an Eides statt, dass ich die vorliegende Arbeit selbstständig verfasst und keine anderen als die von mir angegebenen Quellen und Hilfsmittel benutzt habe. Ich erkläre, dass ich keine früheren Promotionsversuche unternommen habe. Die vorgelegte Abhandlung wurde weder in gleicher noch in ähnlicher Form einer anderen Prüfungsbehörde zur Erlangung eines akademischen Grades vorgelegt. Desweiteren erkläre ich, dass ich Hilfe von gewerblichen Promotionsberatern bzw. -vermittlern oder ähnlichen Dienstleistern weder bisher in Anspruch genommen habe noch künftig in Anspruch nehmen werde. Bayreuth, den 27.09.2012 Dominique Ernst 73 Part II Publications Publication P1 Setup for single-particle orbit tracking: artifacts and corrections Dominique Ernst, Stefan Hain, and Jürgen Köhler published in: J. Opt. Soc. Am. A 29, 1277-1287, (2012) c 2012 Optical Society of America http://dx.doi.org/10.1364/JOSAA.29.001277 Setup for single-particle orbit tracking: artifacts and corrections Dominique Ernst, Stefan Hain, and Jürgen Köhler* Experimental Physics IV and Bayreuth Institute of Macromolecular Research (BIMF), University of Bayreuth, 95440 Bayreuth, Germany *Corresponding author: juergen.koehler@uni‐bayreuth.de Received March 16, 2012; accepted March 30, 2012; posted April 3, 2012 (Doc. ID 164958); published June 7, 2012 We report on an experimental setup for single-particle orbit tracking, which allows following fluorescent nanoparticles for more than 10 min with a temporal resolution of 4 ms and a dynamic position accuracy of better than 10 nm. On a model sample—20 nm sized fluorescent polymer beads in glycerol—we will illustrate how artifacts caused by unavoidable experimental shortcomings (might) obscure the experimental result and how misinterpretations can be prevented. © 2012 Optical Society of America OCIS codes: 180.2520, 180.5810, 300.6280. 1. INTRODUCTION The study of transport processes on molecular length scales is of great importance in many fields of research [1–9]. In the life sciences, reaction kinetics may depend crucially on the diffusion of the reactants [10–12], or in material engineering the flow of material through an interface is of great importance for filtering and catalysis [6,13]. Many methods for studying such processes rely on fluorescence microscopy. Initially, this was largely restricted to fluorescence recovery after photobleaching [14], which is an ensemble technique that prohibits probing of transport processes beyond the classical diffraction limit of light microscopy. Moreover, due to ensemble averaging, complex diffusion behavior might get masked in photobleaching experiments. The situation changed drastically with the advent of singlemolecule techniques. Confocal microscopy lies at the heart of fluorescence-correlation spectroscopy (FCS) and its variants [9,15]. The disadvantage of this method is that the average transition time of a particle through the detection volume of about 1μm3is very short. Hence, one is either limited to short observation times or one averages sequentially over many particles, making it difficult to extract information about temporal or spatial inhomogeneities. Alternatively, wide-field fluorescence microscopy has been employed to follow the diffusion of individual nano-objects. The general idea of single-particle tracking (SPT) is to determine the position of an individual particle by fitting its diffraction-limited image on a CCD camera to the known point-spread function of the microscope or, more pragmatically, by fitting it to a twodimensional Gaussian. Associating the center of the fit with the position of the particle allows determination of the spatial position with an accuracy far beyond the classical diffraction limit of light microscopy and to follow the diffusion of this particle with high precision [1,3,16–21]. Fascinating experiments have been reported using SPT in combination with fluorescently labeled particles. Examples are the diffusion of individual lipids in membranes [4,22], the movement of proteins or quantum dots in cells [5,21,23], or the study of biomolecular motors [24]. It even has become possible to trace the infection pathways of individual viruses [25]. While SPT allows for determining the spatial position of the tracer particle with exceptional accuracy, it provides only limited temporal resolution, mainly given by the readout time of the CCD camera. Instead of following the spatial position of a particle by recording a sequence of images in epifluorescence microscopy, Enderlein proposed a method nowadays termed single-particle orbit tracking in [26,27]. There, the excitation light is focused into the plane of the sample and rotates on a circle that encloses the particle of interest; see Fig. 1. As long as the particle is located precisely at the center of the orbit, it experiences a constant excitation intensity, despite the variation of the position of the focus. Upon any movement of the particle away from the center, this situation changes drastically and the excitation intensity, and concomitantly the intensity of the emitted fluorescence Itfrom the particle, becomes modulated with the rotation frequency of the light orbit, according to ItI0exp −2 w2xp−Rcosωt2 · exp −2 w2yp−Rsinωt2IB:(1) Here I0corresponds to the maximum emission intensity, Rto the radius of the light orbit, wto the waist of the focused excitation beam, xPand yPto the coordinates of the particle in the plane of the light orbit, ωto the cycle frequency of the light orbit, and IBto the background intensity. Demodulation of the emission signal yields the coordinates of the particle xpw2 2RRT 0Itcosωtdt RT 0Itdt;y pw2 2RRT 0Itsinωtdt RT 0Itdt;(2) and a feedback loop can be implemented that follows the trajectory of the particle rtxpt;y pt as a function of time. Ernst et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1277 1084-7529/12/071277-11$15.00/0 © 2012 Optical Society of America that the corrections (3) and (4) are significant only at short time scales or small movements, respectively. As soon as τ≫ Δtor MSD ≫σ2is fulfilled, the influence of these corrections on the results is negligible. 4. EXAMPLE As an example, we present an experiment where we studied the diffusion of a 20 nm sized tracer particle in pure glycerol. This sample was chosen, because glycerol does not form network structures and normal, Brownian diffusion can be expected [42], i.e., a linear dependence of the MSD of the traces as a function of the lag time. Or more formally MSDτ∼τα, with α1. We recorded trajectories with 150,000 positions, corresponding to an experimental time of 10 min. In order to compensate for the decrease of the emission signal in the course of time, for example due to bleaching effects, the excitation power was varied between 130 nW–1.5 μW, which ensured a rather constant emission of about 1.5×105cps (counts per second) during the experiment. In most of the experiments, the mean excitation power was about Pex 560 nW, which corresponds to an excitation intensity of Iex 250 W∕cm2 in the focus of the microscope objective. The rotation frequency of the light orbit was set to ν1kHz with P4 periods of rotation, resulting in a bin time of Δt4ms. The parameters for the ztracking were Nz10 and s5 (≈15 nm). We recorded trajectories from seven different particles in order to compensate for the (slight) dispersion in the diameters of the beads and for possible variations of their shapes. In Fig. 8the first 50 data points of the “ensemble”average of time-averaged MSDs are shown for different stages of the corrections. The MSD that has been calculated after carrying out corrections (1) and (2) of the coordinates is shown by the black color code (squares), the one that results after step (3) is given by the blue color code (circles), and finally the fully corrected MSD [after step (4)], is indicated by the red color code (triangles). In order to show the differences between the three MSDs more clearly, some data points are shown on an expanded scale in the top right inset of Fig. 8(a). The solid lines in Fig. 8(a) refer to linear fits, which differ only by their offset; see inset, top left. From the slope we obtain the diffusion coefficient Dexp 0.0178 μm2∕s that can be compared with the prediction according to the Stokes– Einstein relation Dtheo kBT∕6πηa. Here arefers to the radius of the particle, ηto the viscosity of the medium, Tto the temperature, and kBto the Boltzmann constant. Using T294 K, η1.2Pa · s [42], and a10 nm, we obtain Dtheo 0.0179 μm2∕s in nearly perfect agreement with the experimental value. The differences in the offsets of the three MSD curves become clear in the top left inset of Fig. 8(a), which shows the extrapolation of the three fitted curves start end x y -47 nm -38 nm Fig. 7. (Color online) Measuring the flow in the sample chamber. (a) Sequence of 1000 data points (4 s) taken from a long trajectory of 150,000 data points. The gray arrow corresponds to the displacement vector of the particle during the 4 s. (b) Scatterplot of 1050 displacement vectors from seven independent trajectories. The vectors have been shifted with their starting point to a common origin. (c) Histogram of the xcomponent of the displacement vectors. The full line corresponds to a Gaussian fit centered at a mean of −47 nm. (d) Histogram of the ycomponent of the displacement vectors. The full line corresponds to a Gaussian fit centered at a mean of −38 nm. 1284 J. Opt. Soc. Am. A / Vol. 29, No. 7 / July 2012 Ernst et al. toward τ0. Once the diffusion coefficient is obtained, the other two corrections, i.e., adding 4∕3DΔtand subtracting the remaining offset are trivial. From the last correction, we obtain 2σ2, which provides the dynamic position accuracy of σ7.5nm for the example shown here. In order to enhance the visibility of any deviation from α1, Fig. 8(b) displays the same data on a double logarithmic scale. The lines are linear fits with slopes α12 1.00 (black squares), α30.98 (blue circles), and α41.03 (red triangles), which all agree very well with both the data and the expectation. Here the indices refer to the level of corrections. However, it should be noted that only the fully corrected MSD obeys log MSDταlog τlog 4D, which allows extracting the scaling exponent αby linear fitting. The other two exponents (α12 and α3) should be regarded only as apparent scaling parameters αfollowing the notation in [40]. Nevertheless, for any level of correction, the deviations of these parameters from α1are within the experimental accuracy, testifying that the expected Brownian motion is revealed, and that the experiments were performed in a regime where corrections (3) and (4) were of minor importance. An alternative method to analyze single-particle trajectories with respect to the diffusion behavior, relies on the cumulative distribution function [CDF(r2τ)] of the squared displacements r2for a certain lag time τ[43,44]. The analysis of our data according to this protocol is given in detail in Appendix A.4 and reproduces Brownian motion. 5. CONCLUSION We have described the details of an experimental setup that exploits orbit tracking to follow the movement of an individual fluorescent particle with a position accuracy far beyond the classical diffraction limit. The figures of merit depend on the dynamics of the system and on the number of detected photons per unit time. Typical values for diffusion in viscous media that we achieved are a position accuracy better than 10 nm, a temporal resolution of 4 ms, and a total observation time of more than 600 s. This allowed recording of trajectories of the particle movement consisting of more than 105data points providing excellent statistics for data evaluation. However, the experimental realization of orbit tracking was hindered by several shortcomings leading to artifacts in the data that might misleadingly be interpreted as an underlying anomalous diffusion process. We have presented a detailed characterization of these problems, elucidated the origin of several artifacts, and showed how to correct the data accordingly. The operation of the setup and the influence of the artifacts on the data were illustrated for a model system from which it is known that it features normal diffusion. We have demonstrated that the normal diffusion can be revealed on all experimentally accessible time scales, if the corrections of the artifacts are properly taken into account. The agreement between the measured and calculated diffusion coefficient is better than 1%. APPENDIX A 1. Beam Path Calculation The optical system for the excitation light was designed using ray transfer matrix analysis [45], which has been proven to be a powerful tool to trace a Gaussian beam through a complex optical setup. The beam is represented by a two-component vector bb; φwhere bcorresponds to the distance of the beam from the optical axis and φto the angle between the propagation direction of the beam and the optical axis. Each optical element can be represented by a transfer matrix that relates a given input vector bin to a distinct output vector bout. The optical path of a series of optical elements is then simply calculated by multiplying the respective transfer matrices. For the setup described in the text, we find for the transfer matrix of the full system bout φout −f7f5 f6f−1 2 f7f5 f6 2f6 f7f50·bin φin ;(A1) where the microscope objective has been approximated as a single lens with f73mm. All other focal distances are given in the text. This simple set of linear equations allows to calculate the radius of the light orbit (Rout bout) and its orientation with respect to the optical axis (φout) as a function of the incoming beam radius bin and deflection angle φin. Here we used bin 0;0.76 mradstarting at the center of the first AOD and obtain bin 190 nm;0, i.e., a focused laser beam that rotates on a cylinder having the optical axis as symmetry axis; see Fig. 9. Fig. 8. (Color online) Ensemble average of time-averaged MSDs from seven trajectories taken from different 20 nm sized beads in glycerol at different stages of the correction. (a) Linear plot: raw data (black squares), data corrected for position averaging (blue circles), and fully corrected data (red triangles). All lines correspond to linear fits. The top right inset shows the same data at a scale that has been expanded by a factor of 100. The top left inset shows the extrapolation of the fits toward τ0and the respective intercepts with the MSD axis. For this example, we obtain Dand σas given in the figure. (b) Same data on double logarithmic axis. The lines are linear fits to the data with slopes of 1.00 (black squares), 0.98 (blue circles), and 1.03 (red triangles). For more details, see the text. Ernst et al. Vol. 29, No. 7 / July 2012 / J. Opt. Soc. Am. A 1285 2. Radiation Pressure The force exerted on a fluorescent particle due to the incident photons is given by FPPex∕c, where Pex denotes the excitation power and cthe speed of light. For Pex ≈1μW, which is typical for our experiments, and the extreme assumption that all photons would be absorbed, this yields a force of about 10−15 N. This figure has to be compared with the force exerted on the particle (radius a) due to friction in a medium (viscosity η), which is given by FStokes 6πηa. Even for a low-viscosity medium like water (η10−3Pa · s), this yields for a particle with a radius of 10 nm a force of FStokes ≈10−10 N, exceeding the force induced by the radiation pressure by several orders of magnitude. 3. Optical Trapping The potential of an optical trap is given by Utrap 4Pexn4 ma3 w2cm2−1 m22;(A2) where Pex refers to the excitation power, nmto the index of refraction of the medium, ato the radius of the particle, wto the beam waist, and mnp∕nmto the ratio of the indices of refraction of the particle and the medium, respectively [46,47]. Using the numerical values from our experiments, i.e., Pex 1μW, nm1.33,np1.65,a10 nm, and w 270 nm, we obtain a trapping potential of about Utrap ≈ 10−26 J, which is orders of magnitude smaller than the thermal energy at room temperature, which amounts to Utherm  kBT≈10−21 J. 4. Cumulative Distribution Function The empirical CDF (r2,τ) for a certain lag time τ(data not shown) is determined by counting the number of squared displacements smaller or equal to r2. For normal diffusion a single-exponential curve is expected, while in systems with anomalous diffusion deviations from this function are taken into account by a double-exponential curve given as CDFr2;τ1−βexp −r2 r2 11−βexp −r2 r2 2:(A3) Here, the r2 iare fit parameters that are related to the diffusion coefficients, commonly interpreted as a fast and a slow component, weighted with a parameter β. For Brownian motion, this reduces to the single-exponential; i.e., β1[43,44]. 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Chem.Phys., 2013, 15, 845 Measuring a diffusion coefficient by single-particle tracking: statistical analysis of experimental mean squared displacement curves† Dominique Ernst and Ju ¨rgen Ko ¨hler* We provide experimental results on the accuracy of diffusion coefficients obtained by a mean squared displacement (MSD) analysis of single-particle trajectories. We have recorded very long trajectories comprising more than 1.5 10 5 data points and decomposed these long trajectories into shorter segments providing us with ensembles of trajectories of variable lengths. This enabled a statistical analysis of the resulting MSD curves as a function of the lengths of the segments. We find that the relative error of the diffusion coefficient can be minimized by taking an optimum number of points into account for fitting the MSD curves, and that this optimum does not depend on the segment length. Yet, the magnitude of the relative error for the diffusion coefficient does, and achieving an accuracy in the order of 10% requires the recording of trajectories with about 1000 data points. Finally, we compare our results with theoretical predictions and find very good qualitative and quantitative agreement between experiment and theory. 1 Introduction Diffusion is often exploited to examine interactions and movements of individual nanoscopic objects in soft matter and/or biological environments on a molecular length scale. 1–10 Early experimental work was carried out using fluorescence recovery after photobleaching (FRAP) 11 which yields the ensemble average of the diffusing particles and which is, dictated by the diffraction of light, restricted in spatial resolution to length scales of about 200–300 nm. Alternatively, researchers employed fluorescence correlation spectroscopy (FCS), 12,13 which provides the average over a number of individual objects that are registered sequentially and from which it is assumed that they behave uniformly. Since about two decades single-particle tracking (SPT) became a valuable tool to map out the movement of an individual particle with high spatial and temporal resolution. 1–4,14–21 The developed methodology covers techniques where the movement of an individual particle can be followed by recording its diffractionlimited image on a sequence of CCD frames, 1 sophisticated approaches that compensate the Brownian motion using electrokinetic forces, 14 techniques that use structured illumination by actively designing the point-spread function of the microscope, 21,22 as well as methods that rely on a spatial modulation of the light that travels to or comes from the particle. 19 Fascinating results have been obtained, for example in biophysics the movement of molecules, viruses, or motor proteins could be made visible, 4–6 and in the materials science transport processes through nanoporous structures 7,8 or the manifestation of diffusion anomalies in liquid crystals and mesoporous structures could be followed. 9,23,24 Typically the fluorescence of a particle is monitored as a function of time and the position of the particle is extracted from the data with sub-diffraction limited accuracy. This provides the trajectory r(t) of the particle that is commonly analysed in terms of the mean squared displacement (MSD). For a 2-dimensional diffusion process the MSD generally scales with a power law according to MSD(t)=4D ˜t a ,whereD ˜is the generalized diffusion coefficient, and athe anomaly parameter. For a= 1 the underlying process corresponds to normal diffusion (Brownian motion) 25 and D ˜reduces to the diffusion coefficient Dknown from Einstein. 26 Otherwise the process is called subdiffusive (ao1) or superdiffusive (a>1). For obvious reasons an experimental trajectory can only be recorded with a distinct temporal resolution, it suffers from localisation errors due to the movement of the particle during data acquisition, 27 it is affected by unavoidable signal-to-noise Experimental Physics IV and Bayreuth Institute of Macromolecular Research (BIMF), University of Bayreuth, 95440 Bayreuth, Germany. E-mail: [email protected]; Fax: +49 921 55 4002; Tel: +49 921 55 4000 † Electronic supplementary information (ESI) available: Analysis of all measured trajectories. See DOI: 10.1039/c2cp43433d Received 28th September 2012, Accepted 13th November 2012 DOI: 10.1039/c2cp43433d www.rsc.org/pccp PCCP PAPER 846 Phys. Chem. Chem. Phys., 2013, 15, 845--849 This journal is cthe Owner Societies 2013 limitations 28 and last but not least it is inherently of finite length. Hence, it is of crucial importance to understand how accurate the diffusion coefficient can be extracted from a real experimental MSD curve. 25,29,30 Although the mathematical framework for the MSD analysis is known for many years, the implications of the experimental limitations on the accuracy of the measured parameters have been considered in detail only recently. 29,31 These studies address the achievable precision for the diffusion coefficient that can be obtained from a given MSD curve as a function of the experimental parameters mentioned above. In order to test their theoretical results the authors had to rely on fictitious tracking experiments based on simulations rather than on experimental data. The reason is that a sophisticated statistical analysis of the tracking data requires a very large data set which is difficult to obtain, because the particle might get lost for tracking due to diffusion out of the focal volume or due to photobleaching. Often it is already a great challenge to register trajectories consisting of some hundred data points. Naively speaking, a trajectory of arbitrary length could be acquired by repeating a tracking experiment under exactly the same experimental conditions on several nanoparticles. However, since even nominally identical nanoparticles are slightly different in shape and size, the statistics of the (unknown) size distribution of the nanoparticles will be superimposed on the statistics of the diffusion coefficient. Moreover, the precision with which the diffusion coefficient can be determined from a MSD curve depends on the accuracy of the MSD data points and on the number of fitting points that are taken into account. 29,30 Therefore, the numerous theoretical and numerical developments that allow assessment of the experimental shortcomings still await experimental verification. In this work we use single-particle orbit tracking, which allows us to obtain single-particle trajectories that consist of more than 1.5 10 5 data points with a temporal resolution of 4 ms and a spatial accuracy of better than 10 nm. 32 Such a long trajectory can be divided into a sequence of segments, where each segment can be considered as an individual trajectory that, by definition, has been recorded on exactly the same particle under identical experimental conditions. This enables us to evaluate the statistics of the diffusion coefficient extracted from the segments as a function of the length of the segments and to compare the results with the theoretical predictions made in the literature. 25,29 2 Experimental section 2.1 Sample preparation For the single-particle tracking experiments we use fluorescent beads with a diameter of 20 nm that are loaded with nile red (Molecular Probes, 20 mg ml 1 dissolved in water). This solution is further diluted in water to a concentration of 0.1 nM and subsequently mixed with pure glycerol (Sigma) resulting in a concentration of 2 pM for the tracers. From that solution a drop of about 25 ml is sandwiched between two microscope coverslips that are cleaned with acetone. In order to prevent evaporation of the solvent (and the resulting flow field in the sample) the edges of the coverslips are sealed with grease (High-Vacuum Grease, Wacker). This construct is mounted on top of a 3-axis piezo stage (Tritor 102, piezosystem Jena) providing a scan range of 100 mmfor each axis. All experiments are performed at room temperature, i.e. (21 0.5) 1C. 2.2 Experimental setup The home-built setup for single-particle orbit tracking has been described in great detail in a separate paper. 32 Briefly, the output from an Ar/Kr-ion laser (Innova 70C Spectrum, Coherent) operated at 514 nm is guided through a deflection unit consisting of two mutually perpendicular arranged acousto optical deflectors (AOD, DTSX-400-532, Pegasus) that generate a rotating light orbit. This orbit is projected via a dichroic beam splitter (z532RDC, AHF) towards an infinity-corrected water-immersion objective (UPLSAPO, 60, NA = 1.2, Olympus). This results in a focussed laser beam with a waist of w= 270 nm that rotates on an orbit with aradiusofR= 190 nm in the focal plane of the objective. The frequency of the rotation can be adjusted by the AODs and is set to 1 kHz. The emission of the fluorescent nanoparticles is collected with the same objective, passes the dichroic and is focussed either onto the chip of a CCD (sensicam qe, PCO) or an avalanche photo diode (SPCM-AQR-14, Perkin Elmer). Residual laser light that passes the dichroic is suppressed by a dielectric optical filter (HQ545LP, OD = 6 at 514 nm, AHF). To spot the location of the tracers we operate the setup in widefield mode. Therefore the deflection unit is switched off and an additional lens in the excitation path defocusses the laser light to an area of 80 80 mm 2 . With the aid of the piezo stage an appropriate particle is moved close to the position where the light orbit will appear (centre of the field of view). Subsequently, the optics are switched to confocal mode, the light orbit is generated and the algorithm for automated tracking is started. We record the emission intensity of the fluorescent particle which is modulated by the frequency of the rotating laser focus. By demodulating this emission signal we are able to calculate the x-, y-position of the particle with respect to the centre of the orbit. The position provides a feedback signal for the piezo and the particle is moved (together with the sample) back to the centre of the orbit. These steps (collect emission – calculate position – move piezo) are repeated continuously, which allows us to reconstruct the movement of a fluorescent tracer particle for more than 10 minutes with a spatial resolution of better than 10 nm. The temporal resolution of the experiments is Dt= 4 ms which results in trajectories of N= 1.5 10 5 x-, y-position pairs. 3 Results and discussion An example for a typical trajectory measured with our setup is displayed in Fig. 1. It represents 1.52 10 5 data points and corresponds to an elapsed time of 608 s which is indicated by the colour code, where blue corresponds to the start of the trajectory and red to its end. In order to mimic to have only shorter trajectories with less data points we cut the long trajectory into segments that were treated as independent Paper PCCP This journal is cthe Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 845--849 847 trajectories of shorter length. In the following we denote the length of the full trajectory as N T (here N T = 1.52 10 5 ) and the length of a segment as N seg . For our study we choose N seg = 10, 20, 40, 60, 80, 100, 200, 400, 600, 800, and 1000, which yields ensembles of N T /N seg short trajectories of equal length. The idea is now to determine the diffusion coefficient Dfrom the slope of the time-averaged MSDs of each segment and to examine the statistical variation of Dwithin each ensemble of trajectories. Yet, according to ref. 29 there exists an optimum number of data points of the MSD that should be considered to obtain the best result for the diffusion coefficient. This can be understood as follows. For increasing lag times the accuracy of the data points in the MSD decreases due to the progressively decreasing averaging of the available data. For example, the first data point of the MSD represents an average over (N seg 1) positions of the particle whereas the last data point has not been averaged at all. Hence, fitting the slope of the MSD curve by taking too many data points into account leads to a deterioration rather than an improvement of the result. On the other hand, the very first points of the MSD are stronger subjected to localisation errors, either due to noise (static error) or due to blurring of the position of the particle during data acquisition (dynamic error). Both effects average out for MSD points at longer lag times. As a consequence of this, we first have to find out the optimum number of data points that should be considered for fitting the slope of the MSD. In the following, the protocol for doing so will be explained on the example of N seg = 1000 which yields an ensemble of 152 trajectories of equal length and the same number of MSD curves. For this ensemble we fitted the slope, D*, of each MSD curve by an unweighted linear fit to the first ndata points. More precisely, we have skipped the very first data point of the MSDs, because it turned out that it is strongly affected by residual oscillations of the piezo. These oscillations affect the position determination and reduce the accuracy of the first point of the MSD curve, whereas the influence of these oscillations on the accuracy of the succeeding MSD points level off (for details see Experimental section and ref. 32). Therefore the fit was applied to the data points from 2 to (n+ 1) and the slope D* of the MSD curves was determined as a function of n. In order to be compatible with the existing literature we prefer the slope D* of the MSD curves rather than the diffusion coefficient D=D*/4. 29 An example for the distribution of D* is shown in the top right inset of Fig. 2 for n=4,i.e. taking only the data points 2–5 for fitting the MSDs into account as indicated schematically in the top left inset of Fig. 2. Subsequently, we determined from each histogram the first and the second moment providing the empirical mean Dand the empirical standard deviation s D * for this parameter, and plotted the ratio sD=Das a function of the number of fitting points n. The result of this procedure is shown in Fig. 2 for the examples of N seg = 100 and N seg = 1000. For both samples, the relative error sD=Dfirst decreases for growing nand then rapidly increases if more fitting points are taken into account. Here we find an optimum for the accuracy of the slope of the MSDs for n=4. While the relative accuracy that can be achieved for D*(about8% for N seg = 1000, and about 25% for N seg = 100) clearly depends on the lengths of the segments, it is interesting to note that the number of fitting points nthat yield the optimum result does not. In order to facilitate a quantitative comparison of the data shown in Fig. 2 with the theoretical predictions in the literature 29 we have to resort to the reduced localisation error x=s 2 /DDt, where sis the localisation error, Dthe diffusion Fig. 1 Example of a trajectory of a 20 nm sized bead in pure glycerol. The colour code refers to the elapsed time of 608 s (N= 1.52 10 5 data points; blue corresponds to the start and red corresponds to the end). Fig. 2 Relative error of the slope D* obtained from unweighted linear fits to the MSD curve as a function of the number of fitting points nfor the segment length N seg = 100 (open symbols) and N seg = 1000 (full symbols). The inset top left displays schematically a MSD curve as a function of the lag time tand the data points that are considered for the linear fit (red) to obtain D*. For all fits the first data point of the MSD (brackets) is ignored (for details see text). The inset top right shows as an example for the distribution of the slopes within the ensemble of N T /N seg trajectories for n= 4 and N seg = 100, from which D(first moment; empirical mean value) and s D* (second moment; empirical standard deviation) can be calculated. PCCP Paper Publication P3 Fractional Brownian Motion in Crowded Fluids Dominique Ernst, Marcell Hellmann, Jürgen Köhler and Matthias Weiss published in: Soft Matter 8, 4886-4889, (2012) Electronic supplementary information (ESI) available c Royal Society of Chemistry 2012 http://dx.doi.org/10.1039/C2SM25220A Fractional Brownian motion in crowded fluids† Dominique Ernst,‡ a Marcel Hellmann,‡ b J€ urgen K€ ohler* a and Matthias Weiss* b Received 30th January 2012, Accepted 22nd March 2012 DOI: 10.1039/c2sm25220a Diffusion in crowded fluids, e.g. in the cytoplasm of living cells, has frequently been reported to show anomalous characteristics (socalled ‘subdiffusion’). Several random walk models have been proposed to explain these observations, yet so far an experimentally supported decision in favor of one of these models has been lacking. Here, we show that experimentally obtained trajectories in a prototypical crowded fluid show an asphericity that is most consistent with the predictions of fractional Brownian motion, i.e. an anticorrelated, anti-persistent generalization of normal Brownian motion that is related to the fluid’s viscoelasticity. Macromolecular crowding, i.e. a total concentration of a variety of macromolecules up to 400 mg ml 1 , is a common phenomenon in intracellular fluids. 1 Crowding can have a considerable impact on (bio)chemical reactions, 2 hence challenging insights derived from biochemical assays in dilute aqueous solutions. The phosphorylation pattern of the mitogen-activated protein kinase (MAPK), for example, has been shown to vary greatly with the degree of cytoplasmic crowding: 3 In dilute solutions, MAPK was twice phosphorylated by its kinase in a distributive manner, whereas adding artificial crowding agents resulted in a processive phosphorylation and hence a more efficient activation of MAPK. Recently, a theoretical explanation of these results has been given in terms of crowding-induced anomalous diffusion. 4 Indeed, crowding is known to strongly alter the diffusional mobility of macromolecules. 5 Apart from a mere reduction of the diffusion coefficient, i.e. an increased viscosity of the fluid, anomalous diffusion has also been frequently observed in crowded fluids in vivo 6–10 and in vitro. 11–15 Here, the mean square displacement (MSD) of a diffusing particle was shown to scale over several decades as hr(t) 2 it a with a< 1 (‘subdiffusion’). In spite of the frequent observation of subdiffusion, even in fairly unstructured fluids in vitro, an experimentally supported and unambiguous explanation of the effect in terms of a random walk model has remained elusive. So far, three types of random walks have been considered as an explanation of crowding-induced subdiffusion: (1) Obstructed diffusion (OD), i.e. the motion of a tracer particle in a maze of immobile obstacles, 16 (2) fractional Brownian motion (FBM) due to the viscoelasticity of the crowded fluid, 15 and (3) a continuous time random walk (CTRW) in which the diffusing tracer takes power-law distributed rests between periods of free diffusion. The CTRW model is special since it shows weak ergodicity breaking 17,18 whereas OD and FBM are ergodic random processes with stationary increments. Recent experimental data have indicated that CTRW may be less well suited to explain crowding-induced subdiffusion 15,19 at least on short and intermediate time scales. 20 The main problem in relating experimental data to the above models is a lack of detailed information on the diffusion process: several techniques, e.g. fluorescence correlation spectroscopy, only report the MSD and leave all higher moments of the diffusion propagator undetermined. Single-particle tracking (SPT) techniques allow one to record individual trajectories and hence can overcome this limitation. 21,22 However, precise position determination in SPT requires the collection of many photons of the moving tracer which sets limitations to the temporal resolution and the overall length of the recorded trajectory (due to bleaching of the dye). Yet, an unambiguous deciphering of the random walk model from fairly short SPT trajectories, often accompanied by an unfavorable spatial and temporal resolution, is challenging. Here, we have utilized a fast and precise single-particle tracking technique to record particle trajectories with a length of 10 5 positions and a spatio-temporal resolution of 10nm and 4ms. From trajectories in prototypical crowded and purely viscous fluids, we have determined the timeand ensemble-averaged MSD of the diffusing particle as well as the random walk’s asphericity. As a result, we have found that a transient, yet long-lasting subdiffusion emerged in a crowded but not in a purely viscous fluid. The anomaly was associated with an ergodic mode of motion as evidenced by a recently introduced ergodicty breaking parameter. Comparing the random walks’ asphericity with those predicted by computer simulations of normal Brownian motion, FBM, CTRW, and OD, we have found that our experimental data in crowded fluids are best described by the FBM model. Since FBM is closely related to viscoelasticity, we put forward the hypothesis that macromolecular crowding equips fluids with viscoelastic properties that enforce a fractional Brownian motion of diffusing tracer particles. Single-particle tracking (SPT) is frequently limited by a poor temporal and/or spatial resolution as well as fairly short trajectories. These limitations can be overcomeusingatrackingtechniquethat has been developed within the last few years: 23–26 AGaussianfocus circles at high speed around a fluorescent particle with the particle a Experimental Physics IV, University of Bayreuth, 95440 Bayreuth, Germany. E-mail: [email protected] b Experimental Physics I, University of Bayreuth, 95440 Bayreuth, Germany. E-mail: [email protected] † Electronic supplementary information (ESI) available: Experimental and Numerical Methods. See DOI: 10.1039/c2sm25220a ‡ These authors contributed equally to this work. 4886 | Soft Matter, 2012, 8, 4886–4889 This journal is ªThe Royal Society of Chemistry 2012 Dynamic Article LinksC < Soft Matter Cite this: Soft Matter, 2012, 8, 4886 www.rsc.org/softmatter COMMUNICATION Downloaded by UNIVERSITAT BAYREUTH on 12 April 2012 Published on 03 April 2012 on http://pubs.rsc.org | doi:10.1039/C2SM25220A View Online / Journal Homepage / Table of Contents for this issue located at the position of the steepest gradient of the excitation intensity. Taking a diffusion step to escape this position is compensated by moving the sample stage via a negative feedback loop. Hence, the two-dimensional center-of-mass motion can be tracked with a high spatial and temporal resolution. Using this approach (see the ESI† for the schematic setup and technical details), we were able to track fluorescent beads (diameter 50nm) for up to ten minutes with a temporal resolution of Dt¼4ms and a spatial accuracy of Dr¼10 nm. We have tracked particles in two prototypical fluids: (i) in a purely viscous solution obtained by mixing 60% sucrose (per weight) into water, and (ii) in a crowded fluid, where 30% dextran (500 kDa) was dissolved in water. For the latter, anomalous diffusion has already been reported previously. 6,12 From a slightly higher number of acquired trajectories, we have retained for each fluid only those 21 SPT trajectories for further analysis that contained 4.5 10 4 –1.5 10 5 positions without blanks, i.e. we discarded those few trajectories in which a weak emission signal lead to a transient loss of the tracked bead. The chosen trajectories did not show any signs of drift. Representative trajectories for both fluids are shown in the ESI.† As a first step in the analysis, we calculated for each of the selected time traces r i hr(t¼iDt) the time-averaged MSD, DrðtÞ2ET¼1 NkX Nk i¼1 ðririþkÞ2 :(1) Representative time-averaged MSDs for sucrose and dextran solutions are shown in Fig. 1a. To highlight the emergence of a diffusion anomaly, we have divided out the leading order of the MSD, i.e. we have plotted D(t)¼hr(t) 2 i T /tas a function of t. While the purely viscous sucrose solution yielded a horizontal line, D(t)¼ const., a transient power-law decay emerged for the crowded fluid. From the transient scaling D(t)1/t 0.2 (obtained within the grey shaded region), we inferred hr(t) 2 i T t 0.8 for small and intermediate time scales. This observation is in quantitative agreement with previous reports 15 on similar probes. Beyond tz1 s a crossover towards normal diffusion emerges, i.e. D(t) tends towards a horizontal line. Indeed, this behavior is expected for all of the above mentioned random walk models for subdiffusion since adapting them to a physical sample requires specification of a minimum and maximum length/time scale. To determine the anomaly for each trajectory, we restricted the fitting process to the temporal range 50 ms #t#500 ms which is not affected by some remaining inertia traces of the setup (t<50ms;see discussion in the ESI†) but also does not suffer from the emerging crossover to normal diffusion at large time scales. The resulting anomaly values, a, for all trajectories are summarized in Fig. 1b. A clear separation of the data for the purely viscous sucrose solution (average: hai¼0.98) and the results for a crowded dextran solution (average: hai¼0.82) can be seen. From the observation hr(t) 2 i T t a we can already infer that the CTRW model with its distinct weak ergodicity breaking cannot describe the experimental data since it predicts 17,18 hr(t) 2 i T t. Indeed, even for a truncated CTRW model with only a transient scaling p(s) s (1 + a) of the distribution of waiting times one observes hr(t) 2 i T t (cf. ESI†). Hence, even a more realistic adaptation of the CTRW model appears incompatible with our experimental data. Next, we calculated for all trajectories an ergodicity parameter 17 that vanishes if ergodicity is preserved: EðtÞ¼ DrðtÞ2E2 T  E DrðtÞ2ET2 E DrðtÞ2ET2 E (2) To this end, we have cut each trajectory into segments of N¼3000 time steps and used these segments for the ensemble averaging h.i E . As a result, we observed that for all trajectories hEi#0.03 (Fig. 1c). Here, the average of E(t) was taken in the same temporal window in which awas also determined. This result strongly supports the notion that all trajectories were ergodic. In particular, our data separates well from the predictions of a non-truncated CTRW model that yields a lower bound E(a#0.9) $0.1. 17 However, for the truncated CTRW model (cf. ESI†) we also obtained Ez0.03 on the experimentally relevant time scale. We attribute this effect to the truncation of p(s) which narrows the distribution of apparent diffusion constants in hr(t) 2 i T . 17,18 Hence, based only on E, a clear-cut decision that our experimental data is inconsistent with a truncated CTRW model is not possible. We next inspected the trajectories’ shape to gain deeper insights into the underlying type of random walk. The asphericity provides a simple yet powerful parameter to quantify the shape of fractal objects like random walks. 27 Diagonalizing the random walk’s gyration tensor T ij (cf. ESI†) yields the principal axes of gyration and Fig. 1 (a) Representative time-averaged MSD, shown as D(t)¼hr(t) 2 i T /t to highlight the asymptotic scaling. Data for sucrose solutions (blue circles) follows the anticipated scaling for normal diffusion (D(t)¼ const.). In contrast, data for dextran solutions (red squares) shows a transient subdiffusion (dashed line, D(t)1/t 0.2 ). For t> 1 s a crossover to the asymptotic scaling (a¼1, D(t)¼const.) is visible. The grey shaded region indicates the temporal window in which the curves were fitted to extract the anomaly a. (b) Anomaly values afor each trajectory as obtained from fitting the time-averaged MSD in the indicated time window. A clear separation between a sucrose solution (blue circles, hai¼0.98) and a crowded dextran solution (red squares, hai¼0.82) is evident. (c) The ergodicity breaking parameter hEi[eqn (2)] of all trajectories was very small, indicating ergodicity. This journal is ªThe Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 4886–4889 | 4887 Downloaded by UNIVERSITAT BAYREUTH on 12 April 2012 Published on 03 April 2012 on http://pubs.rsc.org | doi:10.1039/C2SM25220A View Online the corresponding eigenvalues, i.e. the squared principal radii of gyration, R 2 i . Restricting ourselves to two dimensions (the experimental trajectories are two-dimensional objects), the asphericity is defined as A¼h(R 2 1 R 2 2 ) 2 i/h(R 2 1 +R 2 2 ) 2 i. (3) We note that Ainvolves an averaging over the ensemble of walks (indicated by h.i). The limiting cases A¼0andA¼1 resemble a perfect sphere and a simple rod, respectively. For Brownian motion in two dimensions an exact value is available: 27 A¼4/7. Hence, even an individual trajectory of a two-dimensional Brownian random walk differs drastically from a circular shape at each instant of time. The time-averaged orientation of the longest principal axis of gyration, however, is isotropic. Moreover, the isotropy of diffusion is also recovered in an ensemble of particles due to the uncorrelated random orientations of the gyration axes. We have determined via simulations the values of Afor FBM, OD, and a truncated CTRW at varying anomaly values a(see ESI† for details). Our simulation results revealed that for 0.5 #a#1, which is the experimentally relevant regime, the asphericity changes almost linearly with a,i.e. A ¼m 1 a+b 1 . For OD we found m 1 ¼0.120  0.006, b 1 ¼0.458 0.004 whereas for FBM we obtained m 1 ¼ 0.638 0.009,b 1 ¼0.057 0.006. For the truncated CTRW model we found Az4/7 irrespective of a(cf. ESI†). This result can be rationalized by bearing in mind that a CTRW trajectory at any instance of time looks similar to the path of normal Brownian motion. To compare our experimental trajectories to these predictions, we assigned the previously determined anomaly ato each trajectory (cf. Fig. 1b). Then, we calculated the accompanying asphericity: Since the anomaly reflects a scaling for short and intermediate times, a consistent estimate of the random walk’s asphericity must relate to the same time scale. Therefore, each trajectory was cut into sequences of N¼ 3000 time steps of length Dt, and the average over these sub-trajectories yielded the (mean) asphericity [eqn (3)] of the entire trajectory on the length and time scales during which anomalous diffusion was observed. As can be seen from Fig. 2a, the cloud of data points for sucrose solutions overlaps well with the anticipated result for normal Brownian motion, i.e. the mean of all 21 data points (hai¼0.98 and hAi¼0.58) agrees quantitatively with the expectation a¼1andA¼ 4/7 z0.57. Hence, sucrose solutions indeed feature normal Brownian trajectories also from the geometric perspective. In dextran solutions, however, we obtained hai¼0.82 and hAi¼0.46 which is most consistent with the simulation results for the FBM model that predicts locally a more spherical shape of the trajectory due to the anti-persistence of the random walk. Given that FBM is closely related to the viscoelasticity of nonNewtonian and crowded fluids, 13,28 the emergence of subdiffusion may be traced back to transient restoring forces on short length and time scales. It is hence meaningful to translate the SPT trajectories into the fluid’s complex shear modulus, 28 G(u)¼G0(u)+iG00(u). Here, the real (imaginary) part of G(u) represents the elastic (viscous) modulus of the fluid. Employing a semi-analytical approach, we have fitted the time-averaged MSD of each trajectory by an empirical expression w(t)¼a 0 t a +a 1 tto capture the transient anomaly and the asymptotic normal diffusion. The resulting fit parameters were then used to determine the complex shear modulus as described earlier. 13 From the ensemble of complex shear moduli for each fluid, we have determined the minimum and maximum values of G0and G00.As expected, sucrose showed a vanishing elastic contribution whereas the crowded dextran solution showed a significant viscoelasticity for large frequencies (Fig. 2b). Since high frequencies are related to small times, this viscoelastic behavior is intimately linked to the transient subdiffusion observed for small and intermediate times. A similar viscoelastic behavior (related to subdiffusion) has been observed for the cytoplasm and nucleoplasm of living cells. 13,29 In conclusion, we have shown with an advanced SPT approach that a purely viscous sucrose solution features normal Brownian motion of tracer particles with an asphericity of the random walk that agrees very well with analytical predictions. In contrast, diffusion in a crowded dextran solution was anomalous (‘subdiffusion’). Trajectories showed no signs of ergodicity breaking and their asphericity was in quantitative agreement with predictions of the FBM model. In contrast, obstructed diffusion (i.e., a standard random site percolation model) and CTRW were incompatible with the experimental data. This result is corroborated by the associated complex shear modulus: A strong viscoelastic behavior of the crowded dextran solution was seen at high frequencies as expected due to the relation of FBM with viscoelastic media. It is tempting to speculate about the reasons and consequences of our finding in the context of living matter. Since the degree of Fig. 2 (a) Asphericity Aas a function of the anomaly a(data for sucrose and dextran shown as blue circles and red squares, respectively). Mean values (standard deviation) are indicated by cross hairs. Dashed lines indicate simulation results for OD and FBM. Data for sucrose solutions are in very good agreement with the asymptotic value A¼4/7 for a¼1 (dash-dotted mark), whereas data for dextran compare favorably to the predictions of FBM. (b) Elastic (red) and viscous (grey) moduli, G0and G00, as obtained from the ensemble of trajectories in a crowded dextran solution. Shown are the minimum and maximum values for G0and G00 at each frequency u,i.e. all trajectories lie within the indicated bands. For low frequencies the fluid is almost completely viscous whereas for u> 100 s 1 , a clear viscoelastic behavior emerges. 4888 | Soft Matter, 2012, 8, 4886–4889 This journal is ªThe Royal Society of Chemistry 2012 Downloaded by UNIVERSITAT BAYREUTH on 12 April 2012 Published on 03 April 2012 on http://pubs.rsc.org | doi:10.1039/C2SM25220A View Online cytoplasmic crowding appears to be conserved, 29 cells might have adapted to highly crowded conditions and aim at maintaining this state (cf. also discussion in ref. 2 and 29). Indeed, a potential benefit of FBM-like subdiffusion in cells is the increased return probability to a position in three-dimensional space. In particular, FBM with a<2/ 3 yields a bulk-filling random walk that can massively increase the capture probability to a target as compared to normal diffusion. 30 Moreover, an enhanced rebinding due to FBM most likely is the explanation for the recently observed phosphorylation enhancement of MAPK under crowded conditions. 3,4 As an enhanced recurrence is a generic feature of FBM-like subdiffusion, we expect that the behavior of a multitude of biochemical pathways in cells will have to be revisited and interpreted in light of our findings. Acknowledgements DE and JK gratefully acknowledge financial support by Research Unit FOR608. MH was partly financed by the German-Israeli Project Cooperation GA309/10. We would like to thank Stefan Hain for technical support. References 1 J. Ellis and A. Minton, Nature, 2003, 425, 27–28. 2 H. Zhou, G. Rivas and A. Minton, Annu. Rev. Biophys., 2008, 37, 375–397. 3 K. Aoki, M. Yamada, K. Kunida, S. Yasuda and M. Matsuda, Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 12675–12680. 4 M. Hellmann, D. Heermann and M. Weiss, Europhys. Lett., 2012, 97, 58004. 5 J. Dix and A. Verkman, Annu. Rev. Biophys., 2008, 37, 247–263. 6 M. Weiss, M. Elsner, F. Kartberg and T. Nilsson, Biophys. J., 2004, 87, 3518–24. 7 I. Tolic-Norrelykke, E. Munteanu, G. Thon, L. Oddershede and K. Berg-Sorensen, Phys. Rev. Lett., 2004, 93, 078102. 8 I. Golding and E. Cox, Phys. Rev. Lett., 2006, 96, 098102. 9 S. Weber, A. Spakowitz and J. Theriot, Phys. Rev. Lett., 2010, 104, 238102. 10 V. Tejedor, O. B enichou, R. Voituriez, R. Jungmann, F. Simmel, C. Selhuber-Unkel, L. Oddershede and R. Metzler, Biohys. J., 2010, 98, 1364–1372. 11 I. Wong, M. Gardel, D. Reichman, E. Weeks, M. Valentine, A. Bausch and D. Weitz, Phys. Rev. Lett., 2004, 92, 178101. 12 D. Banks and C. Fradin, Biophys. J., 2005, 89, 2960–2971. 13 G. Guigas, C. Kalla and M. Weiss, Biophys. J., 2007, 93, 316–23. 14 W. Pan, L. Filobelo, N. Pham, O. Galkin, V. Uzunova and P. Vekilov, Phys. Rev. Lett., 2009, 102, 058101. 15 J. Szymanski and M. Weiss, Phys. Rev. Lett., 2009, 103, 038102. 16 M. Saxton, Biophys. J., 1994, 66, 394–401. 17 Y. He, S. Burov, R. Metzler and E. Barkai, Phys. Rev. Lett., 2008, 101, 058101. 18 A. Lubelski, I. Sokolov and J. Klafter, Phys. Rev. Lett., 2008, 100, 250602. 19 M. Magdziarz, A. Weron, K. Burnecki and J. Klafter, Phys. Rev. Lett., 2009, 103, 180602. 20 J.-H. Jeon, V. Tejedor, S. Burov, E. Barkai, C. Selhuber-Unkel, K. Berg-Sorensen, L. Oddershede and R. Metzler, Phys. Rev. Lett., 2011, 106, 048103. 21 B. Hicks and K. Angelides, J. Membr. Biol., 1995, 144, 231–244. 22 T. Schmidt, G. Schutz, W. Baumgartner, H. Gruber and H. Schindler, Proc. Natl. Acad. Sci. U. S. A., 1996, 93, 2926–2929. 23 J. Enderlein, Appl. Phys. B: Lasers Opt., 2000, 71, 773–777. 24 K. Kis-Petikova and E. Gratton, Microsc. Res. Tech., 2004, 63, 34. 25 A. Berglund and H. Mabuchi, Appl. Phys. B: Lasers Opt., 2004, 78, 653. 26 Y. Katayama, O. Burkacky, M. Meyer, C. Brauchle, E. Gratton and D. Lamb, ChemPhysChem, 2009, 10, 2458–2464. 27 J. Rudnick and G. Gaspari, Science, 1987, 237, 384–9. 28 T. Mason and D. Weitz, Phys. Rev. Lett., 1995, 74, 1250–1253. 29 G. Guigas, C. Kalla and M. Weiss, FEBS Lett., 2007, 581, 5094–98. 30 G. Guigas and M. Weiss, Biophys. J., 2008, 94, 90–94. This journal is ªThe Royal Society of Chemistry 2012 Soft Matter, 2012, 8, 4886–4889 | 4889 Downloaded by UNIVERSITAT BAYREUTH on 12 April 2012 Published on 03 April 2012 on http://pubs.rsc.org | doi:10.1039/C2SM25220A View Online Fractional Brownian Motion in Crowded Fluids – Supplement Dominique Ernst 1, Marcel Hellmann 2, J¨urgen K¨ohler 1, and Matthias Weiss 2 1Experimental Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany and 2Experimental Physics I, University of Bayreuth, D-95440 Bayreuth, Germany I. EXPERIMENTAL APPROACH Fluids for SPT experiments were obtained by dissolving dextran (500 kDa, Sigma) and sucrose (342 Da, Roth) in millipore water at concentrations of about 430 mg/ml (30% w/w) and 1500 mg/ml (60% w/w), respectively. Rhodamine-tagged tracer beads (50 nm, Polysciences) were added from a predissolved solution, resulting in a typical concentration of about 2 pM. Hence, 3-5 beads were observed in the focal plane of the microscope’s widefield image (80 ×80 µm2). About 35 µl of each sample was placed between acetone-cleaned coverslips and sealed on the edges with highly viscous grease to prevent evaporation or adhesion forces that would induce a flow field. Tracking experiments were performed with a homebuilt single-particle tracking setup (Fig. 1) using a novel tracking technique [1–5]. Here we only describe the basic concept, technical details will be presented elsewhere. The output of an Ar/Kr-Ion laser (Innova 70C Spectrum, Coherent) at a wavelength of 514nm with a circular polarisation (due to a λ/4-waveplate) was used as an excitation light source. The laser beam was directed through a series of two perpendicularly arranged acousto optical deflectors (AOD, DTSX-400-532, Pegasus), responsible for the generation of an orbiting laser beam with rotation frequency f. The rotating laser beam was then passed through a telecentric lens system and directed into a home-built confocal microscope. The laser light was reflected by a dichroic beamsplitter (z532RDC, AHF) towards an infinity corrected water-immersion objective (UPLSAPO, 60x, NA=1.2, Olympus). The sample with the diffusing tracer particles was mounted on top of a three-dimensional piezo stage. This setup allowed us to create an orbit radius Rin the range of 0 to 5 µm in the focal plane of the objective. Suitable dye-labeled particles in the vicinity of the rotating focal spot were excited. The emitted light was collected by the same objective, passed the dicroic beamsplitter and a further dielectric filter (HQ525LP, OD=6 @ 514nm, AHF) to suppress remaining laser light. Finally it was focussed either onto the chip of a CCD (sensicam qe, PCO) or an avalanche photodiode (APD, SPCM-AQR-14, Perkin Elmer) with a sensitive area of 180 µm in diameter. The setup was capable of working in a widefield and a confocal operation mode. For the widefield mode an optional lens in front of the microscope was flipped into the optical path to defocus the excitation to an area of about 80 ×80 µm2in the plane of the sample. In this mode the deflection unit is set to a neutral state (no deflection). The diffusing particles are located within the 0s 400s 2 m sucrose 2 m dextran μ μ Ar/Kr-Ion laser AOD AOD Piezo APD CCD Microscope & sample stage L4L3L2L1LpLp L5L6 L8 L7 F l/4 Calculation unit A cos(wt) A sin(wt) emission x y LWF Piezo stage (a) (b) FIG. 1: (a) Sketch of the experimental setup with lenses for beam profile optimization (Lp), a widefield lense (LWF), a dielectric filter (F), and an acousto optical deflector (AOD). Signals were collected either with a CCD camera or an avalanche photodiode (APD). The calculation unit provided the drive signals for the AODs, gathered the emission intensity, calculated the positions xand y, and fed the negated position to a piezo stage. (b) Representative trajectories in a purely viscous sucrose solution (left) and a crowded dextran fluid (right); color-coding blue to red highlights the temporal direction of the trajectory. The gyration ellipsoids which reflect the random walks’ asphericity are superimposed in grey. Due to a higher mobility in the sucrose solution, the ellipsoids differ in size. CCD image and are moved by the piezo stage to a proper position near the centre of the laser orbit. The confocal mode is subsequently used to perform the measurements. By flipping the optional lens back, the emission is now detected with the APD and the deflection unit is switched on. The emission intensity of the moving particle is modulated with the known frequency of the laser orbit. Using a tracking software based on a lock-in technique [1, 3] we were able to reconstruct the two-dimensional motion of particles from the frequency-modulated fluorescence signal: From the detected photons the position with respect to the orbit center was calculated, and the piezo stage was fed with a signal corresponding to the Electronic Supplementary Material (ESI) for Soft Matter This journal is © The Royal Society of Chemistry 2012 2 negated position. The whole trajectory of the tracer can be reconstructed by monitoring the feedback signal of the piezo. Experiments were done with an orbit frequency of f= 1kHz. Every four periods of rotation the position was calculated resulting in a time resolution of ∆t= 4 ms. The radius for best tracking perfomance depends on the beam waist wof the focal spot and was found to be R= w/√2 [6]. A typical width of w= 270 nm lead to a radius of R= 190 nm. II. EVALUATION WINDOW OF EXPERIMENTAL DATA We have chosen to evaluate the scaling properties of the experimental data in the temporal range 50-500ms, e.g. when inspecting the MSD. Enlarging the window in which the anomaly αand the corresponding asphericty A were determined will alter the stated numbers. Extending the fit range to larger times will include more of the crossover towards the asymptotic normal diffusion and hence α→1 and A→4/7. This underlines the transient nature of the anomaly. In better words, significant elastic restoring forces are only present in crowded fluids for large frequencies, i.e. for rapid motion on short length and time scales. Extending the fit range to smaller times will include particular features of the measurement process. The accuracy of the position measurement in SPT depends on the number of photons acquired. If too few photons are detected, small diffusion steps are masked by noise and the MSD appears to converge to a constant for t→0 [7, 8]. This behavior may mimic a subdiffusive characteristics at small times (’false positives’). In our experiments photon statistics was sufficiently high to make this effect negligible, i.e. the scaling exponent of a purely viscous fluid deviated from unity only by less than 2%. But even when having enough photon statistics and normal diffusion, an apparently anomalous characteristics may emerge: Since diffusion does not stop during the acquisition process (given by the acquisition time ∆t), the MSD will take on a form ⟨r(t)2⟩= 4D(t−∆t/3) [9]. Due to the subtraction of a constant, the MSD hence may mimic a superdiffusive scaling for short times. To avoid all these contributions, we have restricted ourselves to the indicated fit window which is least affected by the above mentioned processes. From the two-dimensional trajectory with Nposition and a time resolution of ∆t, we obtained the gyration tensor via Tij =1 N N ∑ n=1 (ri(n∆t)−⟨ri⟩) (rj(n∆t)−⟨rj⟩).(1) Here, ⟨ri⟩denotes the i-th component of the center of mass. Diagonalizing Tij yields the principal axes of gyration and the corresponding eigenvalues, i.e. the squared principal radii of gyration, R2 i. III. SIMULATIONS We have considered two different models for anomalous diffusion, namely diffusion in a percolation system (obstructed diffusion, OD) and fractional Brownian motion (FBM). Computer simulations of the respective process provide numerical values for the shape parameters that can be compared to experimental data. Obstructed diffusion was simulated on a square lattice (350 ×350 sites) with periodic boundary conditions. A fixed fraction fof randomly chosen sites were occupied by static obstacles and tracer particles were allowed to move on the remaining free sites according to the blind ant algorithm (see, e.g. [10]). Depending on the occupied volume fraction f, the support becomes a fractal [11], and diffusion can become (transiently) anomalous. For a critical concentration of obstacles fp= 0.40726 [12], the percolation threshold in two dimensions, subdiffusion with α≈0.69 is observed on all time scales whereas for f < fp a transient, yet long-lasting subdiffusion with a finite-size corrected anomaly αemerges. Indeed, for f < fpnormal diffusion is asymptotically restored. For f > fp, tracers are confined to finite domains, i.e. an initial subdiffusion is observed but asymptotically the particle is bound to a certain region in space. In our simulations, we varied the occupied volume fraction in the range 0.33 ≤f≤0.42 which resulted in straight power laws of the particles’ MSD within the simulation period. For every value of f, we simulated 1.5×106random walks, where for every 1000th run a new environment was created. Each random walk was started at a randomly chosen vacant site. Occasionally, particles were trapped in a small subvolume of the lattice due to the random placement of obstacles. We identified such situations and removed trapped trajectories from the analysis. For the simulation of FBM we used the circulant method [13] which is in principle exact, i.e. the deviations between ’true’and simulated FBM are due to computational limitations like finite numerical accuracy. The method relies on the embedding of the covariance matrix of FBM into a circulant matrix that is diagonalized by a discrete Fourier transform. Using a fast Fourier transform (FFT), the simulation time for a trajectory of length Nscales as Nlog N. We generated 106independent trajectories, each having N= 213 positions. The anomaly was varied in the range 0.5≤α≤0.9. For normal diffusion (α= 1), we relied on Brownian Dynamics simulations [14] that are based on the overdamped Langevin equation, r(t+∆t) = r(t)+ξ(∆t) with ξbeing a random variable with white noise characteristics. IV. RESULTS ON A TRUNCATED CTRW MODEL To overcome the somewhat artificial features of the CTRW model due to its asymptotic scaling of the distribution of waiting times, p(τ)∼1/τ1+α, we have conElectronic Supplementary Material (ESI) for Soft Matter This journal is © The Royal Society of Chemistry 2012 3 (a) (b) (c) 10-2 10-1 100101 t [s] 100 101 D(t) ~1/t0.2 10-3 10-2 10-1 100 10-5 10-3 10-1 101 τ p(τ) ~1/τ1.8 0.7 0.8 0.9 1.0 1.1 α 0.3 0.4 0.5 0.6 0.7 A OD FBM FIG. 2: (a) Representative time-averaged MSD of truncated CTRW model (α= 0.8), shown as D(t) = ⟨r(t)2⟩/t to highlight the asymptotic scaling. Data for the timeand ensemble-averaged MSD are shown in blue and red, respectively. For the ensemble-averaged quantity a transient scaling D(t)∼1/t0.2in the experimentally relevant interval is observed before asymptotically reaching the normal diffusion limit (D→const.). Hence, the truncated CTRW model only shows transiently a weak ergodicity breaking. (b) The probability distribution function of waiting times, p(τ) used for the truncated CTRW model. After a transient power-law scaling, the distribution displays an exponential tail that enforces an asymtotic convergence of the MSD to normal diffusion. (c) The asphericity of the truncated CTRW model for α= 0.8 (red dot) deviates considerably from the predictions of the OD and FBM models (dashed lines). It is most consistent with the limiting value A= 4/7 for normal diffusion and hence incompatible with the experimental data found for a crowded dextran solution. structed a truncated CTRW model. In particular, we followed previous reports that had implied exponentially truncated power-law distributions [15]. A truncated p(τ) is expected to yield a long-lasting transient subdiffusion which asymptotically converges to normal diffusion. We therefore have simulated a two-dimensional CTRW with α= 0.8 and a truncated distribution of waiting times with parameters that yielded a close match with the experimental MSD data. The chosen waiting time distribution and the resulting behavior of the MSD (again shown as D(t) = ⟨r(t)2⟩/t) are reported in Fig. 2. As can be seen in Fig. 2a, the time-averaged D(t) is approximately constant whereas the ensemble-averaged quantity shows a transient scaling ∼1/t0.2(i.e. ⟨r(t)2⟩E∼t0.8) before converging to the asymptotic limit of normal diffusion. Hence, the CTRW’s feature of a linear scaling of the time-averaged MSD persists even for the truncated model. The associated wating time distribution (Fig. 2b) shows a power-law decay over several orders of magnitude before being exponentially truncated. We next determined the asphericity for the truncated CTRW model. Since the trajectory of a CTRW at any instance of time geometrically looks like the path of normal Brownian motion, we expected a value A≈4/7 for the truncated CTRW model. Indeed, our expectation turned out to be correct (Fig. 2c). Furthermore, A did not change significantly with the imposed anomaly α(data not shown). Shifting the truncation to larger and larger times resulted in a slight increase of Arather than reducing the value. Therefore, based on the scaling of ⟨r(t)2⟩Tand the asphericity, we can not only rule out OD and the full CTRW model but also a (truncated) CTRW model as an explanation of the experimentally observed anomalous diffusion. [1] J. Enderlein, Appl. Phys. B 71, 773 (2000). [2] J. Enderlein, Single Mol. 1, 225 (2000). [3] K. Kis-Petikova and E. Gratton, Microsc. Res. Tech. 63, 34 (2004). [4] K. McHale, A. J. Berglund, and H. Mabuchi, Nano Lett. 7, 3535 (2007). [5] A. Berglund and H. Mabuchi, Appl. Phys. B 78, 653 (2004). [6] A. Berglund and H. Mabuchi, Appl. 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