Measuring fast stochastic displacements of bio-membranes with dynamic optical displacement spectroscopy

Stochastic displacements or fluctuations of biological membranes are increasingly recognized as an important aspect of many physiological processes, but hitherto their precise quantification in living cells was limited due to a lack of tools to accurately record them. Here we introduce a novel technique—dynamic optical displacement spectroscopy (DODS), to measure stochastic displacements of membranes with unprecedented combined spatiotemporal resolution of 20 nm and 10 μs. The technique was validated by measuring bending fluctuations of model membranes. DODS was then used to explore the fluctuations in human red blood cells, which showed an ATP-induced enhancement of non-Gaussian behaviour. Plasma membrane fluctuations of human macrophages were quantified to this accuracy for the first time. Stimulation with a cytokine enhanced non-Gaussian contributions to these fluctuations. Simplicity of implementation, and high accuracy make DODS a promising tool for comprehensive understanding of stochastic membrane processes.


Supplementary Figure 2: Axial intensity scans and ACFs for GUVs
(a)-(d) Recorded axial scans of GUVs with different fluorophore concentrations. Average intensities (grey dots), Gaussian fit (grey line) and correlation amplitude ξ for fluorophore concentration 0.001% (yellow), 0.01% (orange), 0.1% (purple) and 1% (light blue) are shown. ξ changes from a single-peaked to a double-peaked shape when the GUV is deflated and the fluorophore concentration increased. , is calculated from these traces and exhibits a flattening of the amplitude.

Supplementary Figure 3: Axial intensity scans and ACFs for SLB
Recorded axial scan of a supported lipid bilayer (SLB) as control measurement. Average intensities (grey dots), Gaussian fit (grey line) and correlation amplitude ξ (green dots). ξ remains flat over the whole scan. Since a SLB cannot have spatial fluctuations, the maximal value, 0024 . 0 det   , marks the detection limit of our DODS setup.

Supplementary Figure 7: Error in height conversion
Top: displacement traces of a GUV membrane at the GUV distal site. Measurements were recorded at the inflection point of the confocal detection volume and converted via the linear intensity dependence (blue), for the 2 nd (green) and 3rd (black) order expansion (see Supplementary Eq. 2). Bottom: absolute deviation between higher and first order conversion, 1 2 h h    (green) and

Supplementary Note 1: Converting intensity flickers to displacements
Here we derive the theoretical expressions linking first the measured intensity flickers ) (t I  to the real membrane displacements ) (t h  , and second, the intensity autocorrelation function (ACF) to the displacement autocorrelation function (dACF). We assume that the signal from diffusion of fluorophores is negligible, and that the confocal volume has a Gaussian intensity distribution given by Here, z is the average vertical position of the membrane within the confocal volume (see also Fig. 2 for an explanation of coordinates and symbols), c 0 is the fluorophore concentration, Bg I is the background intensity, and ω 0 and z 0 are the radial and axial dimensions of the confocal volume (ω 0 = 280 ± 5 nm (N = 10) and z 0 = 1285 ± 10 nm (N = 10) 1 , as measured using fluorescent beads of sub-resolution size: data not shown). 2 is the maximal intensity detected in a vertical scan through the confocal volume. For a fluctuating membrane, the vertical position is the mean membrane position and ) (t h  is the instantaneous membrane displacement (see Fig. 2).
Intensity-displacement relation: For membrane displacements that are small compared to the size of the confocal volume ( is the instantaneous measured intensity. Using Supplementary Eq. 1 and expanding If the membrane is placed at the inflection point of the intensity profile, as is the case with typical DODS measurements, the second order term vanishes and a first order approximation for ) (t I  is sufficient (see also Supplementary Note 2 for validation of this approximation). Supplementary Eq. 2 then reads In a DODS experiment, the slope m is measured from axial scans in the vertical ( z ) direction as described in the main text. In the form Supplementary Eq. 3 can be used as long as the intensity distribution in the axial scan exhibits a well-defined inflection point. no net displacement over time), the system is amenable to correlation analysis. In general, the correlation amplitude and the autocorrelation function (ACF) of the intensity fluctuations are defined as Consequently, in the limit of small membrane fluctuations, the correlation amplitude and the ACF of the intensity fluctuations become (up to second order in defining the amplitude of fluctuations 2 h    and the displacement autocorrelation . Note again that for DODS the membrane is placed at the inflection point of the Gaussian intensity profile. Here, the intensity profile is the steepest (the slope m is maximum) and thus, the correlation amplitude is the highest. Note, that the above conversion of intensity into heights has only a unique solution as long as fluctuations remain on one side of the Gaussian illumination profile. In all the cases considered here, this condition is easily fulfilled since the fluctuation amplitudes are much smaller than the axial confocal radius The raw data is recorded at rate of 200 ns. This is then averaged in the acquisition software (see Methods section) to 1 µs to build the intensity ACF. The absolute time resolution of the system is therefore 1 µs. However, for the systems studied here we found that the signal to noise ratio at such short times were unacceptably low. To determine the appropriate time cut-off, we examined dACF curves from optimal GUV data and chose the time point at which the value of the noise seen in the dACF data (defined as the standard deviation) exceeds 10% of dACF value. This corresponds to a lag time of 10 µs. Thus, 10 µs is the appropriate time resolution in the present set-up.

Spatial resolution:
The spatial resolution is related to the shape of the confocal volume and the background noise. A rough estimate can be obtained by considering the traces from SLB presented in Supplementary Fig. 2 kcps and the usually measured background intensity of  1 kcps, allow for an accuracy of  20 nm. Note that utilization of phenol red containing medium raised Bg I  to 2 kcps, its use was therefore avoided.

Stationarity and recording time:
Stationarity of the signal and sufficient recording time are essential prerequisites for correlation analysis. To test their validity in DODS measurements, deflated GUVs of high fluorophor concentration were used and intensity traces of  180 s duration at the IP acquired. Each measurement was subdivided into  36 short intervals of 5 s duration, so-called repeats. The ACF of each repeat was calculated and, subsequently, average ACFs for an increasing number of repeats were generated. Supplementary Fig. 5 illustrates examples of ACF, for a single repeat, an average of 5 repeats, as well as the total average ACF of all repeats after 3 min recording time. Stationarity of the signal is tested for by plotting the deviation between the total average ACF and an ACF of shorter recording time. While a symmetric variation around 0 indicates stationarity, the square of this deviation averaged over all  specifies the shape stability of the ACF. In our setup, stationarity and a shape precision comparable to the experimental error is achieved in case of 5 repeats. Thus, the minimal recording time of DODS measurements is 25 s, which roughly amounts to 1000 correlation times (cf. Supplementary Fig. 5a).

Error introduced by linear approximation:
The theoretical correlation amplitude, which is defined as the extrapolated ACF at zero lag time 0   , is calculated to first order in 2 ) (t h  , see Supplementary Eq. 5. The theoretical correlation amplitude calculated up to different orders of approximation is shown in Supplementary Fig. 6. Supplementary Fig. 7 depicts an example of intensity-displacement conversion via a Taylor expansion up to 1st, 2nd or 3rd order in ) (t h  . The absolute deviation between higher and first order trace is calculated and averaged. Despite the obvious deviation in 1 3 h h    the temporal average yields an error of

DODS was validated by comparing thermal fluctuations measured at the vesicle bottom with measurements obtained by Dual-wavelength Reflection Interference Contrast Microscopy (DW-RICM) [SI4, SI5
]. This microinterferometric technique is an advanced setup of RICM where two interferograms for different wavelengths are recorded simultaneously. RICM is highly sensitive to membrane-substrate distances and allows reconstruction of membrane topographic shapes with 5 nm axial resolution and some ~ 10 ms temporal resolution (for further details see Methods and [SI4, SI5]). Here, this well established technique is used to compare DW-RICM with DODS using equally prepared GUV thus substantiating the DODS approach.
In order to perform experiments on a precisely studied system we monitored GUVs interacting with special substrates [SI6]. To prepare these, cleaned glass substrates were stamped with micro-scale grids of BSA-biotin which was further functionalized with neutravidin (NAV). The GUV membrane which exhibits biotin moieties, adheres to these grids but in-between the grids, the membrane continues to fluctuate (as shown in Supplementary Fig. 8a and b). This geometry ensures well defined measurement conditions without any lateral drift of the vesicles. Following reference [SI2], the DW-RICM data was theoretically corrected to account for the limited time resolution. Since both DW-RICM and DODS require dedicated microscopes, it was not possible to make one-to-one correspondence of the height traces. We therefore report the average and standard deviation of the fluctuation amplitude measured for 14 vesicles in each case. The two measurements were done successively on the same sample. The value extracted from DW-RICM data matches very well with DODS data.

Supplementary Note 4: Calculation of tension and dissipation from dACF
Accounting for system point spread function: The cross-section between the confocal volume and the membrane has a finite size, instead of being a single point on the membrane. In order to make quantitative comparison with theory, we account for blurring of the fluctuations by the point spread function ) (r P of the system Here,  is the radial radius of the laser beam at the axial position 0 h which increases with distance from the center of the point spread function (for definition of the coordinates see also is used to deconvolve the dACF which is fitted with a theoretical expression to calculate the membrane bending rigidity  , membrane tension  (see following subsection). Note that if the dACF is not deconvoluted, both  and  are underestimated by about 20% (Authors, in preparation).

Tension and dissipation from fluctuations:
In a model membrane, the fluctuations are determined by the membrane bending elasticity,  , membrane tension,  , and a dissipation depending on the bulk liquid viscosity,  . The theoretical description of the dynamics of model membranes is well known in the literature (for a review see Seifert [SI3] and references therein), which combined with the deconvolution above yields The integral runs over all modes that can be excited on the membrane, where the natural lower limit of the wave vector is given by the vesicle radius R and R q / 1 min  . Note, that the Monge Gauge here assumes that on average the GUV membrane is flat (in fact, it is curved at large scale). Calculations in the quasi-spherical approximations yield values which can differ by up to 20% compared to an analysis performed in Monge Gauge. The systematic error can be reduced below 10% by choosing a slightly different lower cut off (Authors, in preparation).
Here, for simplicity, we retain the flat membrane approximation and the conventionally used lower cut off. By using vesicles of typically 30 µm diameter and with a beam diameter of about 0.6 µm we ensured the validity of this approximation. For SOPC membrane, 20   k B T [6] is well known and was kept constant throughout the analyses. This strategy provides a way to sensitively determine  and  . A typical fit is shown in Fig. 3. The corresponding tension is 23 . 0 µJ/m² and the viscosity is 1.15 mPas. As mentioned in the main text, this is in very good agreement with expectations. Note, however, that a refinement in the theoretical description (see [SI2] and Authors, in preparation), accounting for the viscosity of the inner buffer, which is 4 . 1  i  mPas in the presented system and different from the outer buffer PBS with 0 . 1 0   mPas, can further improve the agreement between the estimated value and the DODS measurements.

Supplementary Note 5: Axial scans for intensity gradient control
DODS can be used to measure membrane fluctuations at variable distances away from the substrate spanning 0-100 µm. This is demonstrated here for measurements at the rim and centre of erythrocytes, close to the cell body and in the lamellipodium of macrophages, at the distal surface of GUV and in control measurements with SLBs. This spatial flexibility is due to the fact that DODS requires only axial intensity gradient information around the membrane average position. This gradient intensity is measured by axial scans across the membrane, which -for the objects in this study-returned Gaussian intensity distributions. Measurements are performed at the inflection point of the Gaussian, where the highest intensity gradient is reached. Moreover, any systematic distortion of the confocal volume, asymmetric fluorophore distribution, etc. will directly be visible through changes in the Gaussian intensity distribution. Thus, performing axial scans prior to each measurement yields high control over the actual intensity variation for height fluctuation conversion.