Ultrahigh-Speed Imaging of Rotational Diffusion on a Lipid Bilayer

We studied the rotational and translational diffusion of a single gold nanorod linked to a supported lipid bilayer with ultrahigh temporal resolution of two microseconds. By using a home-built polarization-sensitive dark-field microscope, we recorded particle trajectories with lateral precision of 3 nm and rotational precision of 4°. The large number of trajectory points in our measurements allows us to characterize the statistics of rotational diffusion with unprecedented detail. Our data show apparent signatures of anomalous diffusion such as sublinear scaling of the mean-squared angular displacement and negative values of angular correlation function at small lag times. However, a careful analysis reveals that these effects stem from the residual noise contributions and confirms normal diffusion. Our experimental approach and observations can be extended to investigate diffusive processes of anisotropic nanoparticles in other fundamental systems such as cellular membranes or other two-dimensional fluids.

B rownian motion in a thermally excited fluid, wherein small particles experience continuous random displacements and independent molecular collisions, has a profound impact in various areas of nature, science, and technology. 1−5 The translational component of this diffusive motion for a nonspherical particle with sufficient symmetry, such as a cylinder or an ellipsoid, can be characterized by the generalized Stokes−Einstein relation D i = k B T/γ i . 6−8 It links diffusion constant D i along the direction constants i set by the principal axes of the particle with the corresponding hydrodynamic friction coefficient γ i where k B is the Boltzmann constant and T is the temperature of the solvent. This process has been interrogated by decades of research in a variety of contexts and regimes. For instance, motion of tracer particles has been used to examine the properties of complex systems, ranging from turbulent fluids 9,10 to living cells. 11−15 Furthermore, the role of anomalous diffusion in stochastic transport phenomena has awakened much interest. 16−19 Reports on rotational diffusion have been less frequent although several studies have recently appeared. 20−26 The free rotational diffusion of an uniaxial anisotropic particle rotating about the axis orthogonal to its symmetry axis can be characterized by a single diffusion coefficient, D θ , associated with the rotational diffusion time τ θ = 1/(2D θ ). 20,27−30 In one of the most recent efforts, rotational diffusion of gold nanorods (GNRs) with dimensions of 100 nm × 20 nm was investigated in the context of microrheology, 31 where imaging speeds limited to a few thousand frames per second were sufficient to follow the slow diffusion in a viscoelastic polyethylene glycol solution. However, studying rotational diffusion in aqueous solutions or at interfaces such as biomembranes requires more advanced experimental techniques and data analysis. Most importantly, higher temporal and spatial resolutions are desirable.
An effective approach to resolving the angular orientation of a particle is to exploit the polarization-dependent optical response. Dye molecules provide a convenient platform for this purpose because they usually possess linear dipole transitions. However, fluorescence measurements not only suffer from photobleaching but also from saturation, leading to a limited rate of photon emission and thus measurement speed. A powerful alternative to fluorescence is Rayleigh scattering from rod-shaped nanoparticles. 32−34 In this work, we apply darkfield microscopy to image two-dimensional rotational and translational diffusion of GNRs linked to supported lipid bilayers with an ultrahigh temporal resolution of 2.3 μs and precision of 4°. The rich statistics of the resulting angular trajectories allows us to reliably estimate rotational diffusion constants from each individual particle trajectory.
We used streptavidin-conjugated GNRs of nominal dimension 71 nm × 25 nm (Creative Diagnostic) with plasmon resonance along the long rod axis centered at a wavelength of about 650 nm. The GNRs were linked to an artificial lipid bilayer at the interface between a physiological buffer and a glass cover slide (see inset in Figure 1a). The lipid bilayer was composed of 99% 1,2-dioleoyl-sn-glycero-3phosphocholine (DOPC) and 1% biotinylated 1,2 dioleoylsn-glycero-3-phosphoethanolamine (cb-DOPE). We used the vesicle fusion method to produce supported lipid bilayers on a cleaned cover glass. 35 After formation of the membrane, which is in the fluid state at room temperature, streptavidinconjugated GNRs were added to the aqueous solution and bound to the membrane. Figure 1a depicts a home-built polarization-sensitive darkfield microscope based on total internal reflection illumination. Light from a tapered-amplified diode laser at a wavelength of λ = 670 nm was coupled to a single-mode polarizationmaintaining fiber to ensure a high-quality wavefront. The resulting laser beam was p-polarized and focused in the back focal plane of an oil-immersion objective (Zeiss, alpha Plan-Apochromat 100×/1.46 with 90% transmittance at 670 nm) to generate an evanescent electric field in the substrate plane. The polarization of the incident light was carefully chosen so that the evanescent electric field was parallel to the substrate. We note in passing that longitudinal field components acting on a circularly polarized dipole can affect its emission pattern to introduce a systematic error in the localization of its pointspread function. 36 We verified that our imaging system does not suffer from unexpected systematic errors (see Supplementary). The scattered light from the sample was collected by the same objective, and after passing a birefringent element (calcite beam displacer Thorlabs-BD27) split into two orthogonal polarization components denoted x and y. The displaced beams were imaged on two adjacent regions of a high-speed camera (Phantom v1610). The camera had a pixel size of 28 μm, corresponding to an effective pixel size of around 60 nm in our setup. We typically used regions of 256 × 64 pixels to acquire data at 442 105 frames per second (fps) and an exposure time of 1.7 μs.
To minimize the background, we blocked the reflected light from the detection path at the closest distance accessible to the back focal plane of the objective with a small circular mask coated on a coverslip. The remaining residual background features which stem from the sample surface or wavefront imperfections were eliminated by subtracting reference images of cases where there was no signal from the GNR. By adjusting the x and y detection axes at π/4 rad from the incident polarization and comparing the scattering signals in the two images, we could capture the rotational angle θ of a single GNR (see inset in Figure 2a and SI. Images in Figure 1b illustrate examples recorded at different rod angles. Figure 2a shows the theoretically modeled contrasts C x and C y for detection along the x and y directions at ±π/4 rad with respect to the s-polarized illumination as a function of the rod angle θ. The model accounts for the complex polarizability of the GNR and the collection efficiency of the setup for different polarizations (see SI). Figure 2b displays the experimentally recorded time traces of C x and C y , whereas Figure 2c depicts a heat map of their variations with the color of each bin representing the frequency of events. The solid blue line shows the fit to the measured data based on a maximum likelihood algorithm and variation of the polarizability parameters in the model. We can thus assign an angle between [0, π] to each data point. Nano Letters pubs.acs.org/NanoLett Letter To reconstruct the full angular trajectory of a nanorod, one has to distinguish between orientation angles θ ± nπ for different integers n, which yield the same contrasts C x and C y . Here, we first calculated the angular difference between two consecutive frames and plotted the step size distribution. Since the intensities along the two polarization axes are periodic, we could also map θ in the interval of [−π/2, π/2]. Then by comparing the step size distributions obtained for the intervals [0, π] and [−π/2, π/2], we chose the smaller step size for each consecutive frame (see SI). In doing so, we greatly benefited from our high-speed imaging which lets us assume that it is highly unlikely for half revolutions to occur between consecutive frames.
The black curve in Figure 3a shows a typical trace of GNR angles as a function of time measured at the maximum speed. On the other hand, when emulating the trajectories from the same GNR recorded at 1/10th and 1/100th of the original frame rate by binning several consecutive frames, we see clear deviations from the original trajectory (Figure 3a), which again demonstrates the necessity for high-speed imaging. It is evident that even a relatively high frame rate of several kilohertz is not sufficient to capture the fine rotational motion of the nanoparticle. Figure 3b displays 30 further examples of such trajectories recorded from different GNRs. Figure 3c shows an intuitive polar diagram of the same data as in Figure 3a to visualize the random angular motion of a GNR in a strong temporal zoom represented along the radial direction. By evaluating the histogram of the GNR orientations, we can verify an overall isotropic motion (Figure 3d). We now explore several different analysis methods to examine the nature of the observed rotational diffusion.
First, we plot a probability distribution function (PDF) of angular steps. A typical angular PDF of a GNR on a supported lipid bilayer is depicted in Figure 3e. Here, we have mapped θ from 66 530 consecutive frames as explained above (see Figure  S2, SI). The step size distribution has a recognizable bell shape. A fit with a single Gaussian yields D r = 3980 rad 2 /s although it clearly does not provide an ideal description of the experimental data. Indeed, a double Gaussian fit with D r 1 = 5130 rad 2 /s and D r 2 = 464 rad 2 /s characterizes the distribution much better. An overview of the measured values of D r 1 and D r 2 from 30 particles is provided in the SI.
Another common quantifier of diffusion processes is the lag time dependence of the mean-squared displacements. Since the angular trajectory is unbounded, we can evaluate it in terms of the time-averaged mean-squared angular displacement (MSAD) defined as where τ denotes the lag of a time window swept along the time series, and T is the overall measurement time. In the case of normal diffusion, MSAD grows as δ τ τ ∝ T ( , ) 2 whereas for anomalous diffusion, MSAD can be modeled as a power law following δ τ τ ∝ α T ( , ) 2 with α < 1 and α > 1 indicating subdiffusive and superdiffusive behaviors, respectively. 17,37 The symbols in Figure 3f depict the measured MSAD as a function of lag time τ in double logarithmic scale. A global fit to these data yields α = 0.97, strongly suggesting normal diffusion at large times. However, there is a noticeable deviation from the fit at short lag times which we will address below. Extraction of the rotational diffusion coefficients from the linear fit to the MSAD curve is depicted in Figure 3g. One reason for the observed variations in the measured diffusion coefficient could be fluctuations in the number of bindings between the GNR and the lipid bilayer (see SI). Nano Letters pubs.acs.org/NanoLett Letter Next, we consider the angular velocity (ω) autocorrelation function (AVACF) defined as g ω (t i ) = ⟨ω(t i )ω(0)⟩, where t i represents time at step i. In the case of normal diffusion, the autocorrelation function starts from 1 for t i = 0 and decays exponentially to zero with a certain characteristic relaxation time. 38 Figure 3h depicts the normalized GNR angular velocity autocorrelation function on a semilogarithmic scale. We find that for very short times, the GNR angular velocity is negatively correlated, which indicates a possible deviation from normal diffusion. 39−41 Our findings of the three diffusion quantifiers discussed above seem to indicate deviations from normal diffusion at short times and have previously been used to provide evidence for a subdiffusive regime. 39,42,43 To scrutinize this conclusion, we checked the error propagation and noise contribution in our measurements by generating synthetic data. Here, we produced trajectories assuming a normal diffusion process with a diffusion coefficient and a temporal resolution corresponding to those of our experimental observations. The blue trace in Figure 4a shows the simulated data. Next, we included the experimental uncertainty and noise sources such as shot noise and camera readout noise. The latter is especially important for images with short exposure times and low signal-to-background ratio. To determine the degree of agreement between the noise-free modeled data and those including sources of error, we assign to each individual measurement point i at angle θ i a data point Θ i = θ i + ξ i,θ where ξ i,θ describes the measurement noise. We assume that ξ i,θ values are uncorrelated and independent and have a variance of σ ξ . The red curve in Figure 4a displays the resulting trajectory constructed by following the same procedure as in our experimental data processing. Figure 4b displays the scatter plot of the synthetic data equivalent to Figure 2b. Interestingly, Figure 4c shows that the MSAD calculated from the reconstructed trajectory deviates from normal diffusion in the same fashion as observed in the measurements discussed in Figure 3f. Similarly, Figure 4d shows the negative autocorrelation at a lag time of one frame, which was also observed in the measured data. To understand the origin of this feature, we consider the expression for AVACF used to calculate the data in Figure 4d: Given that measurement errors ξ i are uncorrelated, AVACF yields 2σ θ 2 + 2σ ξ 2 for n = 0 and −σ ξ 2 for n = 1 for lag times of zero and one frame, respectively (see SI). Indeed, Figure 4d shows that statistically independent and uncorrelated noise in θ can give rise to negative Δθ autocorrelation values. Hence, we find that the signatures in the acquired data that are typically attributed to subdiffusion (negative AVACF, deviation in MSAD) are also reproduced in the synthetic data. We thus conclude that our experimental results remain consistent with a normal rotational diffusion. Furthermore, we can now apply our knowledge of the role of noise to extract the uncertainty in our experimental measurements, arriving at an angular resolution of about 4°at a frame rate of almost half a million frames per second (see SI).
An interesting feature of a GNR attached to a lipid bilayer is that it not only undergoes two-dimensional rotational diffusion, but it also experiences translational diffusion. Indeed, already in the first part of the last century Perrin pointed out that particle anisotropy can lead to dissipative coupling of translational and rotational motions. 7 This effect was recently observed for the diffusion of an isolated micrometer-sized ellipsoid, and the existence of non-Gaussian probability density functions in the laboratory frame was demonstrated. 20 To investigate this phenomenon in our system, we extracted the lateral trajectory of each GNR by localizing its point-spread function in each frame with precision better than 3 nm. Figure  5 shows two examples of trajectory snapshots over a very short duration of 340 μs, resolving both nanometer displacements  Nano Letters pubs.acs.org/NanoLett Letter and full rotations of single nanorods. Next, we decomposed the displacements δx i and δy i into δx′(t i ) and δy′(t i ) denoting the long and short axes of the GNR in its rest frame, respectively. The step size distributions along the short and long axes of the GNR are shown in Figure 5c,d. The similarity of the two distributions suggests that the shape of the GNR in our system has little effect on its lateral diffusion and that lateral diffusion is isotropic with respect to its geometry.
Considering the finite size of the nanorod and its contact with an aqueous surrounding, we now examine the hydrodynamic effects of the environment on the rotational diffusion of a rigid cylinder with flat ends. At short lag times, the diffusion coefficient can be expressed as a power series in terms of the aspect ratio ϵ = b/a: . 44 Thus, the rotational and translational diffusion coefficients for a cylinder of the same dimensions as our GNRs in water and at 25°C would yield D r ∼ 7 × 10 4 rad 2 /s, D ⊥ ∼ 20 μm 2 /s, and D ∥ ∼ 24 μm 2 /s. These quantities are all higher than our measured values of D r ∼ 2 × 10 3 rad 2 /s, D ⊥ ∼ 1 μm 2 /s, and D ∥ ∼ 1 μm 2 /s. Therefore, we conclude that neither the rotational nor the translational dynamics of the GNRs in our measurements are restricted by the hydrodynamic properties of water but are rather governed by the viscosity of the lipid bilayer to which the GNRs are attached via multiple binding sites. The Saffman−Delbruck model states that the rotational diffusion constant of an inclusion of radius a, thickness h, and viscosity μ on a lipid bilayer is given by k B T/4πμa 2 h 45 (see SI and refs 46−48) for discussion on the validity conditions). It follows that one expects D r ∼ 10 7 rad 2 /s and D t ∼ 10 μm 2 /s for the rotational and translational diffusion coefficients of a single DOPE lipid molecule in a bilayer interfaced with water. The fact that our measured quantities are smaller than these values indicates that our GNRs are linked to the bilayer via more than one lipid. Indeed, an inclusion of the order of 25−30 nm in radius would lead to rotational and translational diffusion coefficients comparable to our measured values. To check the hypothesis that our GNRs interact with the lipid membrane over a considerable portion of their geometry, we repeated our measurements on another sample of GNRs where we changed the average number of conjugated streptavidin per GNR from 36 to 15. As seen in Figure S4 of the SI, we find that the lower number of binding sites results in an increase of the mean rotational diffusion coefficients beyond 10 000 rad 2 /s. We also observe a higher standard deviation from the mean, which is consistent with the fact that there is now a larger variation in the number of possible arrangements of streptavidin molecules on the GNR surface or, equivalently, in the effective binding surface area. Our findings are also in line with the outcome of recent experiments on gold nanoparticles attached to lipid bilayers, which have reported lateral diffusion coefficients in the order of 1 μm 2 /s. 35,49−51 We point out that the illumination power of about 40 mW spread over an area of 10 × 10 μm 2 /s in our experiment is not expected to raise the temperature of the gold nanorods by more than a few degrees centigrade (see SI). We, thus, neglect heat-related systematic effects in our analysis.
While the basic concepts in diffusion of micrometer-sized particles have been studied extensively in the laboratory, 52 quantitative experiments with nanoscopic particles require more advanced methods. In particular, both data acquisition and analysis have to be sophisticated enough to search for deviations from conventional Brownian motion. In this work, we have pushed the experimental limits to smaller particles, faster time scales, and higher spatial precisions and have verified normal diffusion of gold nanorods in a true twodimensional system. Further advances could involve measurements on even smaller nanorods and in different media. In particular, extension to biological membranes could shed light on rotational processes, for example, during the uptake of proteins or viruses, as recently explored using spherical particles. 53 ■ ASSOCIATED CONTENT  Nano Letters pubs.acs.org/NanoLett Letter