Simultaneous interferometric determination of Gaussian, tilt and bending moduli of biomimetic membranes

Ayantika Khanra^a and Prerna Sharma*^ab
aDepartment of Physics, Indian Institute of Science, Bangalore 560012, India
^bDepartment of Bioengineering, Indian Institute of Science, Bangalore 560012, India. E-mail: prerna@iisc.ac.in

First published on 12th August 2025

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1. Introduction

Cell membranes undergo large changes in the curvature driven by insertion and spatial reorganization of proteins and lipids during vital processes such as cell division, organelle biogenesis, endo- and exocytosis, intra-cellular trafficking and locomotion.^1–5 Consequently, cell membrane remodeling has become a prominent field of research, and to systematically study it under reduced structural and compositional complexity, various simplified model systems, such as lipid bilayers, unilamellar vesicles and block copolymer based synthetic membranes, have been introduced.^6–8 The membrane curvature energy involves two elastic moduli: the bending modulus and the Gaussian modulus, which control the mean and Gaussian curvature generation, respectively. The bending modulus has been extensively researched and experimentally measured in both biological and model membranes.^9–14 In contrast, measurement of the Gaussian modulus has been very challenging due to the Gauss–Bonnet theorem, which implies that the total Gaussian curvature energy of a closed membrane remains constant unless its topology changes.¹⁵ A few approaches have been proposed for Gaussian modulus measurement based on membrane closure probabilities, buckling instabilities, and edge fluctuations.^15–19 However they have been limited to theoretical modeling or molecular dynamics simulations, while experimental studies have remained sparse. The tilt modulus is another important yet unexplored parameter that controls the energy cost of membrane deformation when the constituent axis is tilted relative to membrane surface normal. While the tilt modulus has been studied using molecular dynamics simulations^20–23 and X-ray scattering data,²⁴ experimental work has been limited as tilt deformation is primarily observed at constituent length scales (nanometer scale in biomembranes and model membranes).

To address the limitations of conventional model membranes, it is useful to study an alternative model membrane with open edges and optically resolvable constituent size, which enables simultaneous measurement of Gaussian, tilt and bending moduli using optical microscopy. Colloidal membranes, which are fluidic monolayers composed of aligned nano-rods with lengths ranging from 200 nm to 1200 nm, fulfill both these criteria and offer powerful alternative systems for studying membrane mechanics.^25–28 These one-rod-length-thick self-assembled membranes (Fig. 1a–c) with liquid-like in-plane order and dynamics²⁵ are governed by the same underlying physics as biomembranes, despite being different in microstructure. The customizable rod length, diameter, and surface properties allow the control of colloidal membrane elastic properties, leading to diverse membrane morphologies.^29–33 A striking example is the assembly of curved colloidal membranes, such as saddles, catenoids, sponge phases³⁴ and vesicles, the latter demonstrating liposome-like cargo encapsulation capabilities.³⁵ Colloidal membranes have been relevant not only in membrane research, but also in shaping thin sheets, geometric frustration in 2D materials, self-assembly of nanorods and nanotubes, and the study of biological liquid crystals.^36–41

In this work, we focus on a thermal fluctuation-based approach for measurement of membrane elastic moduli. Traditional thermal fluctuation spectroscopy measures fluctuation at the equator of vesicle-shaped membranes, but sampling of fluctuation only at a single cross-section and contribution of light from the out of focus planes are its significant drawbacks. Interference reflection microscopy (IRM) is a powerful technique to quantify the complete 3D surface profile of a microscopic object with nanometer height accuracy and millisecond temporal resolution.^42–49 Precise, instantaneous height maps of colloidal membranes acquired by the IRM technique can help separate the tilt modes from the bending modes and enable independent estimation of tilt and bending moduli. Recently a theoretical study of colloidal membrane edge fluctuations has predicted that in the achiral limit, the amplitude spectrum of the out-of-plane component of edge fluctuations can be related to the Guassian modulus.¹⁹ Therefore, analysis of instantaneous height maps of membrane edges using IRM is a method for easy experimental determination of the Gaussian modulus, which we also explore in this study.

2. Results and discussion

2.1. Fourier analysis of membrane surface profiles

Rod shaped bacteriophages with lengths ranging from 200 to 1200 nm were grown and purified to obtain a highly mono-disperse suspension (see the Experimental section). The rods were denoted as nano_D(nm), with the subscript representing rod lengths (D). Each rod has a diameter of 6.6 nm, a persistence length of 2.2 μm, left-handed chirality, and a surface charge density of 7 e⁻¹/nm at pH 8.0.⁵⁰ The rod suspensions prepared in 20 mM Tris-HCl buffer (pH 8.0) have repulsive electrostatic interactions, and salt (100 mM NaCl) was added to screen them. Adding the non-adsorbing polymer dextran induces inter-rod attractive depletion interactions that lead to the lateral association of rods into flat, disc shaped colloidal membranes, with thickness equal to the length of the constituent nanorods.

The colloidal membranes appeared as transparent, uniform disks in DIC (differential interference contrast) imaging (Fig. 1c). However interference reflection microscopy (IRM) movies showed fluctuating bright and dark patches on the membrane, caused by constructive and destructive interference due to variation of membrane height from the coverslip (Fig. 1d and Video S1, SI). From dual wavelength IRM⁴⁶ analysis (see the ‘Experimental methods’ section, and Sections S1.2–S1.4 of SI) each frame of the movie was converted into a relative height map, h^R (Fig. 2a). This map represents the height measured not relative to the coverslip surface, but rather relative to another horizontal reference plane. The height maps were decomposed into a 2D Fourier series of plane waves.⁵¹ The time-averaged, squared Fourier amplitude spectrum of the measured height maps, , was calculated, with being the wave-vector, and imaging noise contribution was subtracted (Section S1.5 of SI). The observed intensity pattern in a microscope is a convolution of the object's true intensity pattern with the microscope's point spread function (PSF), which turns into a simple multiplication operation in Fourier space. Consequently, the measured amplitude fluctuation spectrum was corrected to obtain the true fluctuation spectrum as,⁵²

	(1)

where the modulation transfer function (MTF) is the Fourier transform of PSF normalized to unity at q = 0, which was theoretically calculated and experimentally validated (refer to Section S1.6 of SI).^53,54

The corrected fluctuation spectrum for the thickest membrane studied, nano₁₂₀₀, has a dependence (Fig. 2c, red line), and spectra of nano₁₀₃₈ and nano₈₈₀ membranes were nearly coincident with the same dependence. However, thinner membranes like nano₃₈₅ and nano₂₀₇ deviated from this 1/||² dependence, with steeper slopes (Fig. 2c, green and blue lines, respectively). The fluctuation spectrum of nano₂₀₇ had a dependence at the lowest q region probed. This suggests the existence of distinct regimes of fluctuations with specific q dependencies, and which one dominates depends on the membrane thickness.

2.2. Theory of membrane thermal fluctuation

We now build upon the theoretical works on lipid bilayers to interpret these observations. Surface tension is zero in a free, unattached membrane.^55,56 Helfrich energy of such a membrane is given by,

	(2)

where the surface integral represents the energy cost of mean curvature (H) and Gaussian curvature (K) of the surface, with bending rigidity (κ_c) and the Gaussian modulus (κ_g) as the respective moduli. Analysis of this energy for a thermally fluctuating membrane reveals the simple expression of thermal fluctuation amplitudes:⁵¹ , for edgeless membranes or regions of flat membranes away from the edge. As the second term of eqn (2) integrates to zero in these cases, the Gaussian modulus doesn't affect the thermal fluctuation. During pure bending fluctuations, the constituent rods maintain alignment perpendicular to the membrane surface, resulting in stretching/compression in various regions of the membrane (Fig. 2b, bottom panel). We derive an expression for the bending rigidity from this strain energy cost⁵⁷ (see Section S2.2 of SI),

	(3)

where K_A ≈ YD^58,59 is the area compressibility modulus and Y is Young's modulus. We estimate Y = 2.3 × 10⁶k_BT μm⁻³ using a previously reported bending rigidity of 11000k_BT for nano₃₈₅ membranes from analysis of sagging vesicles.^60,61

The continuum Helfrich model breaks down at wavelengths comparable to membrane thickness.⁶² In a previous study, Yang et al.⁶³ simulated the fluctuation spectrum of a 0.7 μm thick colloidal membrane, allowing constituent rods to displace vertically, creating a rod tilt field with respect to the instantaneous membrane surface normals. This resulted in a fluctuation spectrum. Allowing the possibility of additional orientational freedom for the rods did not change the spectrum. In the Hamm–Kozlov model,^64–67 this tilt fluctuation adds a second term in the fluctuation spectrum,⁶⁵

	(4)

where κ_θ is the tilt modulus. The centroids of depletants with diameter d can approach a colloidal membrane's surface up to d/2 distance as the membrane is impenetrable to the depletant molecules. This volume inaccessible to depletants is called excluded volume, V_ex (Fig. 2b, middle panel). Compared to flat membranes, excluded volume increases with tilt fluctuation, resulting in a proportional energy cost,^68,69 leading to tilt modulus estimation (see Section S2.2 of SI):

κθ = nkBTd,	(5)

where n is the number density of depletant molecules. With depletant dextran's diameter of 30 nm and concentration of 50 mg mL⁻¹, κ_θ ≈ 1800k_BT.

Eqn (4) implies that crossover from the constituent tilt mode to the continuum bending mode occurs at . Substituting eqn (5) and (3), we find that this crossover happens below the maximum probed q in our study, when membrane thickness, D, is lower than 450 nm. Thus, for nano₁₂₀₀, nano₁₀₃₈, and nano₈₈₀ membranes, only tilt fluctuations are expected, whereas for nano₃₈₅ and nano₂₀₇ membranes effects of bending fluctuations are expected, which aligns with our experimental observations.

2.3. Dependence of membrane bending rigidity on its thickness

All experimental data presented in Fig. 2c were fitted (up to q = 6 μm⁻¹) to eqn (4) to extract bending rigidity (κ_c) and tilt modulus (κ_θ) values, which are summarized in Table 1. Thinner membranes have lower bending rigidities as predicted by eqn (3).

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A secondary estimation of membrane bending rigidity was obtained from vesicle size distribution.³⁵ Large nano₃₈₅ and nano₂₀₇ membranes initially form bulges (Fig. S7a, SI) and eventually fold into vesicles. Boal et al.'s criterion⁷⁰ relates the smallest size of vesicles obtained with the membrane bending rigidity. From experimental vesicle data, we obtained bending rigidity values of 7140k_BT for nano₃₈₅ and 2680k_BT for nano₂₀₇ membranes, respectively (see Section S2.3 of SI), which are in reasonable agreement with measurements from fluctuation analysis. However, a limitation of vesicle size distribution measurements is that they can be biased by experimental artifacts such as membrane adhesion to the coverslip and gravitational sagging of large vesicles, leading to larger vesicle diameters being observed than expected. The estimates from vesicle size distribution are best treated as approximate values instead of precise measurements.

The scaling relationship κ_c ∝ D³ was predicted for colloidal membranes (eqn (3)). Using the bending rigidity data in Table 1, we find a 95% confidence interval for the exponent b in the equation κ_c ∝ D^b as [2.8783, 3.5294], which agrees with the expected exponent of 3. Note that this scaling relationship does not hold for all types of membranes. The general relationship for membranes is given by, κ_c ∝ K_AD², where K_A is the area expansion modulus. When colloidal membranes stretch their thickness must remain constant. As the total energy cost of stretching is the sum of energy contributions from each identical slice along the membrane thickness, K_A is proportional to D, which leads to κ_c ∝ D³. In contrast to colloidal membranes, when a lipid bilayer is stretched the lipid molecules change their conformation so that its thickness decreases, due to volume incompressibility. The primary contribution to the energy cost of stretching comes from the water–lipid interfacial energy, resulting in a thickness-independent area compressibility modulus K_A, and thus κ_c ∝ D².^62,71–74

2.4. Dependence of membrane bending rigidity and tilt modulus on the self-assembly environment

Now, we focus on 1D height profiles to understand the effect of self-assembly conditions on membrane elastic properties. The Fourier series coefficients of the observed 1D height fluctuation, , are related to the observed 2D Fourier series coefficients by, ,⁵¹ where is related to elastic moduli by eqn (1) and (4). The observed 1D fluctuation spectra, , were calculated at various depletant polymer (dextran) concentrations for nano₂₀₇ membranes (Fig. 3a), imaging noise contribution was deducted, and fit parameters, tilt modulus and bending rigidity were extracted.

Higher depletant concentrations result in higher depletion attraction between rods, which increases the tilt modulus of nano₂₀₇ membranes (Fig. 3b), a trend also observed in thicker membranes like nano₁₂₀₀ (Fig. S4b, SI). We observed disc-shaped fluidic nano₂₀₇ membranes, characterized by diffusive lateral rod movement, up to a depletant concentration of 80 mg mL⁻¹ (Fig. 3d). However, at 81 mg mL⁻¹ depletant concentration the rod movement and edge fluctuations stopped, indicating membrane solidification; and we observed membranes with irregular edges (Fig. 3d and Fig. S7b, SI). Surprisingly, no difference in the surface fluctuation spectrum was observed between 80 and 81 mg mL⁻¹ depletant concentration (Fig. 3a). A further 5 mg mL⁻¹ increase in depletant concentration drastically reduced surface fluctuation (Fig. 3a, black open circles). We observed a rapid rise in the tilt modulus near the membrane crystallization point (Fig. 3b), which is not captured by the simplified expression in eqn (5) as it applies only to purely fluid membranes far from any phase transition. Fig. S4b, SI shows a comprehensive dataset of tilt moduli measured in all the colloidal membranes at various depletant concentrations, suggesting that membrane thickness has minimal effect on tilt moduli. The bending rigidity had a linear dependence on the depletant concentration (Fig. 3c). Changing salt concentration at a constant depletant concentration did not affect the elastic moduli under discussion, given the membrane fluctuation spectrum did not change (Fig. S4a, SI).

Two potential sources of error in the fluctuation spectrum were considered. Finite camera exposure time can lead to the averaging of thermal fluctuations; however, the chosen exposure time of 1 ms caused only a 7% error in the determined elastic parameters (Fig. S6b and Section S1.7 of SI); thus the exposure time effect was ignored in this study. For a membrane at an absolute height of ≈450 nm, typical for IRM measurements, amplitudes in the q range explored in this study were not impeded by the presence of the coverslip (Section S1.8 of SI).

2.5. Saddle-like fluctuations at the membrane edge yield the Gaussian modulus

So far, all the measurements of the fluctuation spectrum were done away from the membrane edge. However, increased height fluctuations (root mean squared value of h^R(t)) were observed as we approach the edge (Fig. 4a). This edge-specific fluctuation decayed exponentially with the distance from the edge with a 2.7 μm decay length for nano₃₈₅ membranes. Measurement of surface profiles of entire membranes reveals saddle-shaped ripples at the edges (Fig. 4b). We captured a time-series of instantaneous shapes in a 4 × 4 μm area at the nano₃₈₅ membrane interior and edges using the dual wavelength IRM technique (Fig. 4c inset) to better understand this aspect. Subtracting the time-averaged surface detected only the fluctuations, and second-order polynomial fits provided averaged mean and Gaussian curvatures of each detected instantaneous surface. While histograms of the instantaneous mean curvature were identical at the membrane interior and edge (Fig. 4d), histograms of the Gaussian curvature had an increased probability of a negative Gaussian curvature at the edge (Fig. 4c). The direct observation of saddle-like ripples and the subsequent curvature analysis suggest that the edge-specific fluctuations create a negative Gaussian curvature, and thus the Gaussian modulus should control the amplitude of this fluctuation.

As Gaussian curvature energy integrates into a constant for closed vesicles, the Gaussian modulus does not affect their shape. However, previous studies^19,68 on flat colloidal membranes have demonstrated that the Gaussian modulus plays a role in shaping their open edges. Specifically, theoretical work conducted by Jia et al.¹⁹ has suggested that enhanced height fluctuation of the membrane edge is governed by elastic parameters of the edge and the membrane's Gaussian modulus. This theoretical work incorporated the membrane's Helfrich surface energy (eqn (2)) and the full edge energy, including contributions from edge lengthening, bending, and torsion.¹⁹ At small chirality limit and high bending rigidity (κ_c ≫ κ_g) approximation, Jia et al. derived the following expressions of the averaged squared amplitude spectrum of in-plane (〈u_q²〉) and out-of-plane (〈v_q²〉) components of edge fluctuation,

	(6)

	(7)

where γ is the line tension and B is the edge bending rigidity.

The 1D height fluctuation spectrum was measured on concentric arcs (with angles below 100°, Fig. 4a, inset) at various distances (ΔR) from the nano₃₈₅ membrane edge (Fig. 4e). The spectrum at ΔR = 8 μm was representative of the bulk membrane, free from edge effects. Higher amplitudes of low-frequency (q < 3 μm⁻¹) fluctuations were observed near the edge. However, the membrane thickness reduces at the edge due to the twist of rods (Fig. 1a, b and Fig. S9 in SI), resulting in a 500 nm width region with altered IRM intensity (Fig. 4a, inset) at the edge, which was excluded from analysis. Thus to estimate fluctuation amplitudes at the exact membrane edge (ΔR = 0), an exponential fit, 〈(a^obs_q)²〉 = a + bexp(−ΔR/L_e), was performed instead. The decay length of the fluctuation amplitude with the distance from the edge, L_e, was smaller for high q fluctuations (Fig. S8b, SI), indicating that high q fluctuations are localized very close to the edge.

Subtracting the membrane interior fluctuations (ΔR = 8 μm) from the edge fluctuations (ΔR = 0), we found the edge-specific out-of-plane fluctuations, 〈v_q²〉 (see Fig. 4f), exhibiting a peak at q = 0.6 μm⁻¹. The theoretical prediction for the vertical component of edge fluctuations shows a peak at,

qc = κg/2B	(8)

From eqn (8), we estimated the Gaussian modulus, κ_g = 50k_BT. A previous theoretical calculation⁶⁸ has predicted a comparable κ_g = D²dnk_BT/6 = 56k_BT for nano₃₈₅ membranes. Note that local variations in moduli values within the very narrow ≈0.5 μm edge region (Fig. S9 in SI), caused by reduced membrane thickness, were not considered here, as this region is a negligible fraction of the 30–60 μm diameter membrane.

The Gaussian modulus can also be estimated from the peak height of vertical edge fluctuations, using the relationship κ_g²/2Bk_BT = [1/q²〈v_q²〉]_q→0 − [1/q²〈v_q²〉]_q=qc derived from eqn (7). But we avoided this due to three issues in determination of the vertical component of edge fluctuation: (i) height fluctuation at the exact edge was inferred from height fluctuation data 0.8 μm away using a simplified single exponential decay model, (ii) edge specific fluctuations that decay within 0.8 μm were not observed by us, and (iii) lateral fluctuation of the edge was ignored. These factors likely led to an underestimation of the vertical edge fluctuations, which is evident from our observation that [q²〈v_q²〉]_q→0 = [q²〈u_q²〉]_q→0/3.7, which contradicts the expected result [q²〈v_q²〉]_q→0 = [q²〈u_q²〉]_q→0 from eqn (6) and (7).

The measured Gaussian and bending moduli lead to a κ_g/κ_c ratio of ≈10⁻² for the nano₃₈₅ membrane. This positive ratio contrasts with κ_g/κ_c ≈ −1 typically observed in lipid bilayers.^15–17,75 However, simulations have shown that positive values of the Gaussian modulus, and thus a positive κ_g/κ_c ratio, can also occur in certain lipid bilayers.⁷⁶ A high and positive Gaussian modulus is known to promote the formation of sponge-like structures with saddle-like curvatures in colloidal membranes.³⁴ Similar morphologies in lipid bilayers have also been hypothesized to partly result from anomalously positive values of the Gaussian modulus.⁷⁶

3. Conclusion

To summarize, we have circumvented the challenges associated with measuring tilt and Gaussian moduli in biological and other biomimetic membranes by using colloidal membranes as an unconventional model system with open edges and optically resolvable constituents. We adapted interference reflection microscopy based imaging for measurement of dynamic height maps of the fluctuating colloidal membrane surface. We identified two regimes of fluctuations: continuum bending modes prominent at long wavelengths, and molecular tilt fluctuations observed at wavelengths comparable to the membrane thickness. The tilt and bending moduli were determined by fitting the fluctuation spectra. The measured 3D surface profiles also allowed local measurements of mean and Gaussian curvatures generated by thermal fluctuations. Compared to fluctuation at the membrane bulk, we observed increased surface fluctuation at the membrane edge with negative Gaussian curvatures. This led to the measurement of the Gaussian modulus, which has been a challenging parameter to measure in the past.

To our knowledge, this is the first experimental study to simultaneously measure Gaussian, tilt, and bending moduli using a well-designed model system. The interferometric imaging and data analysis technique presented here for nanometer precision height map measurement can be generalized to any membrane of uniform thickness and refractive index for thermal fluctuation based determination of curvature moduli.

Furthermore, the underlying self-assembly principles of colloidal membranes can be extended to other soft matter and materials science problems. For example, self-assembled ordered arrays of phage rods can serve as a template for fabrication of highly ordered nanostructures of metallic compounds, which have been used in construction of solar cells and batteries.^77,78 Formation of self-assembled and ordered structures may increase detection efficiency in biosensor applications of functionalized phages.^33,79 Studying the curvature modulus of self-assembled arrays of phages can lead to fabrication of optimised curved surfaces in these applications.

4. Experimental section

4.1. Purification of bacteriophages and preparation of colloidal membrane samples

Bacteriophages M13KO7, M137560, and M13-wt, abbreviated in the main text as nano₁₂₀₀, nano₁₀₃₈, nano₈₈₀, respectively, were grown in E. coli strain ER2738 using standard protocols.⁸⁰ Gel electrophoresis revealed the presence of multimer rods in M13-wt and M137560 suspensions. Dextran solution was added in the virus suspension to drive it to isotropic-nematic phase coexistence,⁸¹ where multimers remain in the nematic phase. After centrifugation, the nematic phase precipitated at the bottom, and the monomer rich isotropic phase was collected and ultracentrifuged to pellet monomers.

Litmus 38i phagemid DNA was transformed into E. coli strain ER2738, and nano₃₈₅ rods were produced from it using M13K07 as the helper phage. The pScaf-1512.1 phagemid (Addgene plasmid # 111402 by Shawn Douglas lab³²) and HP17-KO7 (Addgene plasmid # 120346 by Hendrik Dietz lab⁸²) helper plasmid were transformed into the E. coli strain XL1-blue and then grown using a standard protocol³² to prepare nano₂₀₇ rods. Differential PEG precipitation was performed to isolate the target phage from longer byproduct rods as the longer rods precipitate at lower PEG concentrations.⁸³ Supernatants were thoroughly removed from phage pellets during PEG precipitation processes to avoid any salt carryover from growth media. All purified phages were suspended in 100 mM NaCl and 20 mM Tris HCl buffer (pH 8.0). The buffer pH was measured using a pH meter and was chosen to ensure the long-term stability of phages.

The sample chamber was made with a clean slide, a poly-acrylamide coated coverslip (to prevent adhesion), and parafilm as a spacer. Dextran of 500 kDa molecular weight (Sigma-Aldrich) was added to the purified buffered phage suspensions and the resulting mixture was injected into the sample chamber. The chamber was sealed with optical glue (Norland) and stored coverslip-side down for 8–24 hours to allow membrane formation. The samples were sealed this way to prevent evaporation and maintain consistent dextran, salt and pH over several days.

We performed the experiment in a temperature-controlled laboratory room set to 24 °C. Temperature was monitored during experiments and it remained within ±2 °C of the setpoint. The incident light was passed through an infrared filter during microscopy to prevent sample heating.

4.2. Dual wavelength IRM setup and imaging

An inverted microscope was adapted for interference reflection microscopy (IRM) (see Fig. 1e).⁴⁶ White light from a mercury arc lamp enters the microscope's epi-illumination path and a 50R/50T plate beamsplitter positioned at a 45° angle directs it vertically towards an oil immersion objective. The objective focuses the light on the sample and light reflected from all interfaces of the sample, which creates the interference pattern, was collected by the objective. This reflected white light entered a two-way image splitter (Cairn OptoSplit II), where it was divided into two beams by a 605 nm dichroic mirror, passed through bandpass filters of 560 nm (green) and 660 nm (red) wavelengths (with 10 nm bandwidths), and projected side by side on a high speed camera sensor, allowing simultaneous imaging of interference patterns at two wavelengths. During imaging, the microscope field diaphragm was closed to its narrowest setting, and the objective's numerical aperture (NA) was adjusted to its lowest setting, NA = 0.55. Image series with 3000-frames were captured with 1 ms exposure time and 4.75 ms intervals. The exposure time and frame interval were increased to 4 ms and 27 ms, respectively, for measurement of long wavelength edge fluctuations at a higher signal-to-noise ratio. The image processing involves carefully aligning them spatially and correcting for uneven illumination, stray light contributions and back-reflections from the sample ceiling. Further details are provided in Section S1.1 of SI.

4.3. Conversion of interference images into surface profiles

By modeling a locally flat membrane of uniform thickness and refractive index in a dextran environment hovering at height h above a glass coverslip using finite aperture theory⁴⁴ at a numerical aperture of 0.55, we find that in general the calculated interference intensity had a damped sinusoidal type relationship with height h (see Fig. 5a, b, and Section S2.1 of SI). However, a segment of the interference intensity plot in green light illumination (subscript ‘g’) between intensity maxima (I_max) and its neighboring intensity minima (I_min) can be approximated by a simple sinusoidal function with negligible error (see Fig. 5c for numerical results; see Fig. S11, SI for experimental validation),

	(9)

where I_avg = (I_max + I_min)/2, I_mod = (I_max − I_min)/2, and Δh_g = 110.6 nm was calculated from finite aperture theory calculations. To convert a known green light intensity profile (I_g) of a target region into a relative height (h^R) profile, I_max,g and I_min,g are determined from the nearest bright and dark fringes, respectively (Fig. 5d), and eqn (9) is solved using the principle branch of the arc sin function. However, when I_g approaches I_max,g or I_min,g, this approach becomes error-prone, and a dual-wavelength approach of IRM⁴⁶ is more suitable. Here, the interference intensity in red light illumination (subscript ‘r’) is also expressed in a sinusoidal form,

	(10)

where Δh_r = 128.3 nm and the phase ϕ in eqn (10) is the unknown phase difference between I_g and I_r. The process of determining ϕ and simultaneously solving eqn (9) and (10) to find the relative height is detailed in Section S1.3 of SI.

Author contributions

A. K. conducted experiments, analyzed data and wrote the manuscript. P. S. conceived study design, analyzed data and wrote the manuscript.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data supporting the findings of this study are provided within this article and its SI. Supplementary information contains text and figures providing details of interference microscopy calibration and validation methods, optical modelling of membrane systems, intensity-to-height conversion process and correction of height fluctuation spectrum. The supplementary movie shows interference microscopy video of a fluctuating membrane alongside the calculated dynamic surface map. See DOI: https://doi.org/10.1039/d5sm00511f

Acknowledgements

All authors acknowledge funding support from grants CRG/2019/000855 and WEA/2023/000006. All authors thank Zvonimir Dogic, Leroy L. Jia, Bidisha Sinha and Arikta Biswas for useful discussions.

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