The Role of Optical Projection in the Analysis of Membrane Fluctuations

We propose a methodology to measure the mechanical properties of membranes from their fluctuations and apply this to optical microscopy measurements of giant unilamellar vesicles of lipids. We analyze the effect of the projection of thermal shape undulations across the focal depth of the microscope. We derive an analytical expression for the mode spectrum that varies with the focal depth and accounts for the projection of fluctuations onto the equatorial plane. A comparison of our model with existing approaches, that use only the apparent equatorial fluctuations without averaging out of this plane, reveals a significant and systematic reduction in the inferred value of the bending rigidity. Our results are in full agreement with the values measured through X-ray scattering and other micromechanical manipulation techniques, resolving a long standing discrepancy with these other experimental methods.

The spectroscopy of shape fluctuations is a general technique to infer the mechanical properties of soft and biological matter. Specifically, fluid membranes are very soft and analysis of shape conformations allows one to characterise physical properties and aspects of the biochemical activity: the ''flickering'' of giant unilamellar vesicles (GUVs) and red blood cells (RBC) is a classic example. 1 In flicker spectroscopy, the equatorial fluctuations of GUVs are imaged through video microscopy and reconstructed by image analysis, yielding a power spectrum that is then compared to a theoretical model to retrieve the membrane tension s and the bending rigidity k. The latter is an intrinsic property of the membrane that affects its dynamics and structure, as well as morphology, motility, and endocytosis in living cells. 2 Flickering experiments are widely used to assess the differences in the membrane rigidity of various lipid compositions, since only a basic microscope is needed, the sample preparation procedures are well established and direct manipulation of the membrane can be avoided. However, the absolute value of k measured via flickering is systematically larger than those obtained by X-ray scattering, direct manipulation of GUVs and active driving, 3 as reported in Table 1 for 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC). Here we show that this discrepancy arises from the inadequacy of the theoretical models currently used to fit experimental data. Specifically, these ignore the finite focal depth of the setups used to image the membranes, which ultimately projects some out-of-focus fluctuations onto the imaging plane. We introduce a theoretical model that accurately accounts for these projections, and test it on experimental data collected with a fast scanning confocal microscope, in which the focal depth is precisely set. 4 When compared with standard flickering analysis on DOPC vesicles (that gives k = 28 AE 3k B T), we observe a 30% reduction in the measured values of k and find an excellent agreement with the alternative experimental methods. Our technique can be applied to any optical microscopy setup, once the focal depth is known; a proof-of-concept analysis of a phase contrast imaging experiment is reported in the ESI. † A fluctuating GUV can be described as a quasi-spherical shell S R[1 + u(y,j)]r, where (y,j) are the spherical angular coordinates, u(y,j) is a small deviation about a sphere of radius  10 21 AE 3 Pulling membrane tethers 12, 13 20 AE 4 X-ray scattering on bilayer stacks [14][15][16][17] 18 AE 2 Micropipette aspiration of GUVs [18][19][20] 20 AE 2 R, and r is the radial unit vector (see Fig. 1a). The free-energy of the vesicles is described by the Helfrich Hamiltonian, which depends on k and s. 5 By expanding this Hamiltonian to second order in u, written in the basis of the spherical harmonics Y m n as uðy; jÞ ¼ P where n and m are the integer mode numbers with n Z 2 and |m| r n, s = sR 2 /k À 2H 0 R + 2H 0 2 R 2 is the reduced tension, H 0 is the mean spontaneous curvature, k B is the Boltzmann constant, and T is the temperature. The decay time of each mode is given by with Z in and Z out as the viscosities of the fluid found inside and outside of the vesicle, respectively. Membrane fluctuations are recorded by microscopy imaging in 2D, normally around the equatorial plane. Thus, experimental data have been typically compared with mode amplitudes obtained by projecting the modes in eqn (1) onto the plane y = p/2; see ref. 7 and the ESI † for a full derivation. However, due to finite depth of focus, the experiments cannot isolate the signal from the equatorial plane alone. What is actually observed is a projection over the strip of membrane that lies within a focal region near the equator, as shown in Fig. 1(a); this strip can support a spectrum of surface modes that are partially averaged out in projection. This averaging effect is expected to be particularly strong for modes q \ 1/D, where D is a non-dimensional parameter given by the ratio of the focal depth of the microscope to the vesicle radius R.
In the case of fluorescence microscopy, where the membrane is assumed to be uniformly labelled with fluorophores that emit isotropically (see Fig. 1b), we idealise the acquired optical signal as a convolution of the membrane shape with a Gaussian of width equal to the focal depth. Namely, light arriving from height z above (or below) the focal plane has intensity scaled by Thus, the projected intensity field on the equatorial plane is where Ð dO Ð 1 0 dr 0 r 02 Ð p 0 dy 0 sin y 0 Ð 2p 0 dj 0 , r and j are the polar coordinates, and d is the Dirac delta function. Experiments detect I(r,j) and locate the apparent membrane contour as the first radial moment r D ðjÞ ¼ Ð 1 0 rIðr; jÞdr Ð 1 0 Iðr; jÞdr. Deviations of r D (j) are analyzed in Fourier space, where they are written as m q (t) and are non-dimensionalised by R. By defining m q ðtÞ t À1 Ð t 0 dt 0 m q t þ t 0 ð Þ to account for microscope exposure time t, the mean square amplitude can be exactly computed where L n;q ¼ Ð 1 0 doP q n ðoÞf D ðoÞ 1 þ ðÀ1Þ nþq ½ , with P q n (cos y) Y q n (y,j = 0), and f D (o) is defined by where erf is the error function, and I 0 is the modified Bessel function of the first kind of order zero. 8 A complete derivation of eqn (4) and (5), and the comparison with the standard formula (which is recovered in the limit D -0) is provided in the ESI. † The experiments are performed on GUVs prepared by means of electroformation as in ref. 9, with DOPC and the fluorescent lipid Texas Red 1,2-dihexadecanoyl-sn-glycero-3-phosphoethanolamine (DHPE) in proportions of 99.2% and 0.8%, respectively. Videos of the equatorial fluctuations of GUVs are collected on a Leica TCS SP5 II scanning confocal microscope, using a HCX-PL-APO-CS 40.0Â oil immersion objective with numerical aperture NA = 1.30. The focal depth is defined by the pinhole size of the microscope, which allows us to experimentally obtain the value of D by fitting the intensity profile along the z-direction to a Gaussian form of standard deviation RD. The position of the equatorial contours in every frame is determined with sub-pixel precision by correlating the radial intensity profile with a template. By Fourier transforming the contours and averaging over B2000 frames, we extract the mean-squared amplitude of the equatorial modes F q (D), which can be directly compared with the model (eqn (4)); see ESI † for full experimental details. Note that a cutoff condition in the mode spectrum, q t Q, arises due the finite temporal resolution of the microscope, particularly the rate at which the illumination spot is raster-scanned across the field of view. With our setup we typically obtain Q E 20 (see ESI †). The accessible wave number is also limited by the spatial resolution of the setup to q t q w = R/W, where W is the lateral width of the diffraction-limited illumination spot.
Using eqn (4), the best-fit values of k and s to the experimental spectrum F q (D) are found by means of a maximum posterior estimate, 21 assuming a uniform prior and that the measurement errors are independent and Gaussian; namely, we seek to minimise where S q (D) is the standard error in the mean associated with F q (D).
Here, q min and q max define the lower and upper bounds of the fitting range, respectively, with the former chosen to be q min = 3. Due to the rapid convergence to zero of L n,q 2 , the sum in eqn (4) is truncated at the mode n = q + 30. On the other hand, the upper bound of the fitting range is selected as one that maximises the posterior probability P(q max |D D ) based on data D D = {F q (D)} q . This can be numerically computed, but further analytical progress can be made by expanding w D 2 (k, s) to second order around the best-fit values of k and s, which yields the following expression: where w min 2 = min[w D 2 (k, s)], and H min is the Hessian matrix of eqn (6) evaluated at the best-fit values. We also impose that q max must be greater than the crossover q c R ffiffiffiffiffiffiffiffi s=k p , ‡ and less than the cutoffs q w and Q. In other words, the optimal fit is achieved when w D 2 (k, s) is minimal and simultaneously its upper bound q max A (q c , min(q w ,Q)] maximises the probability in eqn (7). If q max lies outside the interval discussed above, then the dataset D D is rejected. See ESI † for details, code and example files. The maximum posterior estimate can be applied to datasets D containing spectra from N different vesicles imaged with different D. In this case, since s is not a material property of the bilayer, the posterior probability becomes an (N + 1)-dimensional function P(k, s 1 ,. . . s N |D), with a different s c for each c-th vesicle. As in eqn (6), we assume a uniform prior and that the measurement errors are independent and Gaussian; thus, maximising this posterior probability function is equivalent to minimizing w D 2 S D S c w D 2 (k, s c ), where each individual q max is given by the maximum of eqn (7).
By imaging three GUVs with radii between 6-17 mm at various pinhole sizes, the fluctuation spectrum associated with each D yields an individual estimate for k and s. A systematic decrease in the inferred value of k is found when the data is fitted with the model in eqn (4) in comparison with the ''standard'' model that considers only equatorial fluctuations, i.e. D = 0 (see Fig. 2a). Using a maximum posterior estimate based on the data of all the vesicles imaged at different D, we find k = 19 AE 1k B T (Fig. 2b), in perfect agreement with literature values obtained with X-rays and micromanipulation techniques (see Table 1). In contrast, by fitting all spectra with the conventional model (D = 0), we find that k = 27 AE 1k B T, a value compatible with the literature estimates of k obtained by conventional flickering analysis (see Table 1).
Large discrepancies in the inferred surface tension values are also to be expected. To illustrate the dependence of the inferred values of k with the focal depth, the conventional fitting procedure is repeated at arbitrary non-zero values of D for all of the spectra in D, yielding an interpolated curve depicted by the green line in Fig. 2b. This shows how increasing the focal depth can result in an apparent increase in k if its effects are not properly analysed. s c is the corresponding reduced surface tension of c-th vesicle. This yields k = 19 AE 1k B T (the blue band is the 95% confidence interval). The green curve shows k inferred by fitting the data D at fixed D. Large errors arise if D is not accounted for correctly; the value k E 27k B T, found at D = 0, corresponds to the use of the ''standard'' model to fit the experiments. The correction is also important for GUVs of varying radius, even when imaged with same optics.
In conclusion, a new methodology is developed for the analysis of flickering data that can accurately account for the finite focal depth of the optical imaging setups. Thus, this resolves a systematic inconsistency in the estimated values of bending rigidity, bringing flickering into full quantitative agreement with other methods such as X-ray scattering and micromanipulation techniques. Neglecting this correction leads to a systematic and size-dependent error which is unfortunately present in all (hundreds) published papers. Using our approach, flickering analysis is now competitive in terms of accuracy with the aforementioned methods, allowing the user to exploit its several advantages in terms of easiness of use and simplicity, as it relies on general purpose and easily accessible equipment, it is noninvasive, and can be integrated into microfluidic devices.