Probing short-range protein Brownian motion in the cytoplasm of living cells

The translational motion of molecules in cells deviates from what is observed in dilute solutions. Theoretical models provide explanations for this effect but with predictions that drastically depend on the nanoscale organization assumed for macromolecular crowding agents. A conclusive test of the nature of the translational motion in cells is missing owing to the lack of techniques capable of probing crowding with the required temporal and spatial resolution. Here we show that fluorescence-fluctuation analysis of raster scans at variable timescales can provide this information. By using green fluorescent proteins in cells, we measure protein motion at the unprecedented timescale of 1 μs, unveiling unobstructed Brownian motion from 25 to 100 nm, and partially suppressed diffusion above 100 nm. Furthermore, experiments on model systems attribute this effect to the presence of relatively immobile structures rather than to diffusing crowding agents. We discuss the implications of these results for intracellular processes.

comparison with cytoplasm for a time range below 2x10 -5 seconds. This shows that short-range diffusion of GFP in the nucleoplasm is never unobstructed in the range observed. Linear interpolation of this part of the iMSD yields an apparent diffusivity of D app =70±4 μm 2 s -1 , that is statistically different from the same quantity calculated for GFP diffusion in the cytoplasm Supporting Information for more details). b, iMSD derived by fitting simulated correlation functions to Eq. 15. c, Quantification of the relative error between theoretical and measured iMSD quantities. As expected, the maximum relative error occurs when the particle reaches a displacement comparable to the characteristic spatial scale of the obstacles. Figure 12. A dimensionless particle is left free to diffuse in each direction adding for each time step 't 0 ' a Gaussian-distributed random number of variance '2D 0 t 0 '. The space is considered with circular boundaries and it is filled with a variable number of randomly distributed overlapping disks (radius 'r' and height 'h') (see Supporting Information for additional details).

Supplementary
Here we show data fitting to Eq. 15 for correlation functions calculated from the trajectories of particles. Analogously to what observed in cells, correlation functions are well fitted by Eq. 15 both for short-and long-range particle displacements. Conversely, as expected, deviation from the The iMSD values at the different time scales are reported in a double-logarithmic representation as average of N=1 experiments, n=9 cells (black dots). D 0 (65±5 μm 2 s -1 ) and D inf (7.5±2 μm 2 s -1 ) are estimated by a linear fit of the iMSD below 2x10 -5 s and above 1x10 -3 s respectively. The green lines represent the expected iMSD for normal diffusion with diffusivity D 0 and D inf . Data are mean values ± s.d.

Supplementary Note 1
Comparison with other correlation techniques capable of extracting the MSD. The approach presented here shares many similarities with the multiple scan-speed image correlation spectroscopy (msICS) strategy presented by Groner et al. 1 . However it is worth noting that their aim is completely different: Groner et al. used msICS to build a diffusion map in the cell. Here, instead of using the different pixel dwell/line scan times to generate a temporal autocorrelation function for every pixel shift, we extract the characteristic molecular mean square displacement for each scan speed (iMSD). In other words, each scan speed is used as a filter to select the characteristic temporal scale of molecular displacements that significantly contributes to the measured correlation function. Such a conceptual difference leads to different methods of data analysis and data interpretation (see main text). Regarding the comparison with spot-variation FCS 2,3 , some connections with our strategy deserve to be highlighted. The two approaches share the concept that selectively observing different (spatial and/or temporal) parts of the whole dynamics can unveil hidden features of particle mobility. Particularly, in spot-variation FCS the size of the spatial filter (the observation volume) is changed and the local dynamics is then obtained by extrapolation. In fact, as elegantly demonstrated by Wawrezinieck et al. 2 , local dynamics such as trapping in isolated domains or hopping between adjacent transient confinement zones are very difficult to be distinguished from simple diffusion by means of classical FCS. On the contrary, the possibility to sample different wavenumbers of the scattering function by changing the waist size represents a very successfully strategy. By the present approach, as described above, we are applying temporal filters through which we reconstruct the complete dynamics. A different successful approach to exploit the huge amount of information contained in the FCS correlation function has been presented by Shusterman et al. 4 . The authors technically use the equivalent of Eq. 11 of the present work to directly extract the MSD of a particle diffusing in 3D. Despite the analogy, the obtained information can be substantially different in the two cases. In fact, the MSD measured by Shusterman et al. represents local information related to the spatial scale of the observation volume waist. Although in a homogenous environment (e.g. dilute solution) this local information (averaged in time during the acquisition) can be considered as representative of the whole sample, the same paradigm does not apply in cells where structures such as membranes, cytoskeleton, vesicles and organelles may regulate molecular diffusion in a different way depending on the particular position in space. Thus we decided to follow a spatio-temporal approach able to average together all the possible positions in space. In fact, with our approach the particle displacement is measured as averaged on a wide cell portion of several microns while each point of the iMSD plot represents an independent measurement (although performed on the same portion of the cell).

Supplementary Methods
From RICS in tunable time scales to iMSD: theory. The theoretical foundation of RICS for the case of normal Brownian diffusion has been published previously 5 . Here we shall consider a general case of transport in order to highlight the potential of our approach to address molecular dynamics with very high spatio-temporal resolution. In particular, we will follow the approach already presented by Shusterman et al. 4 and further generalized by Hofling et al. 6 for FCS. We will tackle the case of mono-dimensional scanning in the x-direction. In particular, we will derive the equations in a reference system with the origin is set at the center of the observation volume. We define as a random variable that represents the displacement of the molecule during the time from its position at time 0; thus, we can define as the displacement in the chosen reference system, where v represents the scan speed. Now we define the measured fluorescence as: the concentration of the fluorescent molecule in time and space (this term contains the physics of the transport process) and  the total efficiency of the acquisition. Furthermore, we define the correlation function as: where  represents the average and Now we can introduce the commonly used Gaussian profile approximation of the observation volume as: represents the anisotropy ratio for the illumination volume ( 2 0 z is the waist size in the axial dimension). Thus, the correlation function can be expressed in its general form as: that represents, as expected, the well-known spatial correlation function 7 . Moreover, if we consider the case of 0  v , the de-correlating contribution due to the molecular movement ) will dominate and Eq. 7 reduces to the FCS case.
Particularly, if the central limit theorem holds, we can consider a Gaussian transport by defining the van Hove self-correlation function as: represents the Mean Square Displacement. Following the approach we presented in a previous report 8 , we define the iMSD as the width of the van Hove self-correlation function, and consequently: If we consider the case of simple Brownian diffusion, where: then Eq.12 can be reduced to a mono-dimensional analogous of the well-known RICS equation 5 .

From RICS in tunable time scales to iMSD: experimental realization.
In classical RICS, in order to describe the shape of the correlation function in space, we need to set the distance between adjacent pixels lower than the instrumental waist. However, the maximum scan speed achievable is limited by the pixel size. In order to maximize the speed, we fix the pixel  Figure 4, for pixel dwell times ≥100 μs). In this context, in order to obtain a more robust strategy that can be easily and extensively applicable in a live cell, we take advantage of the spatiotemporal sampling of the molecular displacement using different scanning speeds.
Particularly, at each scanning speed selected, only the molecular displacements at the timescale comparable to the pixel dwell-time ( p  ) significantly contribute to determine the shape of the correlation function (with no contribution arising from displacements that take place on a time scale significantly greater or smaller than p  ). In other words, the scanning speed acts as a temporal filter selecting the scale of the intermediate scattering function that significantly contributes to the measured correlation function. Following this reasoning, we define for each scanning speed: which allows in turn to recast Eq. 12 into: From RICS in tunable time scales to iMSD: data interpretation. In this regime a significant deviation from Gaussian transport is observed. Instead, for average displacements significantly smaller or greater than this characteristic threshold, particle movement is not distinguishable from a Brownian motion. To quantitatively assess the consistency of our approximations in this case, we set a series of simulations following a Monte Carlo approach.
Particularly, a dimensionless particle is left free to diffuse in each direction adding for each time step t 0 =50 ns a Gaussian-distributed random number of variance 2D 0 t 0 (D 0 =130 μm 2 s -1 ). The space was considered with circular boundaries and it was filled of a variable number of randomly distributed overlapping disks (radius r=300 nm and height h=100 nm) following the approach previously introduced by Novak et al. 14 to reproduce the obstructed environment imposed by cellular structures (e.g. membrane layers) in the cytoplasm. The obstructed motion was reproduced as follows: when the particle attempts to enter into an obstacle, it is stopped and a new step is run.
In order to characterize the simulated dynamics, the particle trajectories were used to calculate the is the mean quartic displacement. This is a simple dimensionless indicator for transport beyond the Gaussian approximation. Such phenomena are expected to play a role, for instance, in the case of particle diffusion within obstructed environments 12 . The correlation functions obtained for all the selected scanning speeds (using Eq. 7) are then fitted to Eq. 15 in order to measure the iMSD by our approach. Different obstacles concentration were characterized by the ratio between the imposed diffusivity (D 0 ) and the apparent diffusivity ( t t iMSD D 4 ) ( app  ) measured at t=10 milliseconds. Supplementary Figure 12 shows the agreement between the simulated correlation function and Eq. 15. In particular, for all simulated conditions Eq. 15 well describes the correlation function for a wide range of pixel dwell times. At slower scan speeds, significant deviation from Gaussian transport can be observed only for the highest obstacle density simulated (that reduces diffusivity of a factor of 10, bottom-right corner in Supplementary Figure   12). Also, please note that such deviation is transient as it disappears for long-range particle displacements. Supplementary Figure 11 compares the measured iMSD with the expected one.
Notably, the measured iMSD correctly unveils the deviation from Brownian dynamics with only a slight underestimation of the expected values. Moreover, particularly for the reduction in diffusivity of a factor of 3, as estimated in live cell by us and others (see main text), this deviation is at maximum 20%, lower than the experimental error. Based on these simulations we believe that our strategy can probe the correct dynamics of GFP in live cells.