Signature of a Nonharmonic Potential as Revealed from a Consistent Shape and Fluctuation Analysis of an Adherent Membrane

The interaction of fluid membranes with a scaffold, which can be a planar surface or a more complex structure, is intrinsic to a number of systems from artificial supported bilayers and vesicles to cellular membranes. In principle, these interactions can be either discrete and protein mediated, or continuous. In the latter case, they emerge from ubiquitous intrinsic surface interaction potentials as well as nature-designed steric contributions of the fluctuating membrane or from the polymers of the glycocalyx. Despite the fact that these nonspecific potentials are omnipresent, their description has been a major challenge from experimental and theoretical points of view. Here, we show that a full understanding of the implications of the continuous interactions can be achieved only by expanding the standard superposition models commonly used to treat these types of systems, beyond the usual harmonic level of description. Supported by this expanded theoretical framework, we present three independent, yet mutually consistent, experimental approaches to measure the interaction potential strength and the membrane tension. Upon explicitly taking into account the nature of shot noise as well as the nature of finite experimental resolution, excellent agreement with the augmented theory is obtained, which finally provides a coherent view of the behavior of the membrane in the vicinity of a scaffold.

Popular Summary

In embryogenesis, vertebrate cells assemble into organized tissues. In cancer, metastasis tumor cells spreading in the circulatory system use mechanisms of adhesion to establish new tumors. At the root of these life-forming or life-threatening biological phenomena is cell adhesion, the binding of a biological cell to other cells or to a material substrate or scaffold. The most obvious fundamental question to ask is then as follows: What factors control or govern cell adhesion? For a long time, the paradigmatic answer to this question was that specific protein molecules embedded in the cell wall (or membrane) were responsible for cell adhesion, in either a key-lock fashion (in cell-cell adhesion) or a suction-cup fashion (in cell-substrate adhesion). But, a new realization has emerged during the past two decades that the cell membrane itself, being a “floppy” sheet, adds another unavoidable, yet not fully understood, interaction with the opposing surface it binds to. Although this interaction does not at all depend on any specific proteins, it can have a major impact on the protein-mediated adhesion and can be viewed as a mechanism that controls the binding affinity to the cell-adhesion molecules.

A great deal of scientific effort has then been made to measure and understand this nonspecific interaction. However, consistent results to fully explain its nature are still lacking, as are the tools for obtaining them. In this paper, we present a set of state-of-the-art tools and demonstrate their power by establishing a concrete, consistent understanding of the interaction between a model cell and a sculpted substrate.

One of the “tools” is the concept and construction of a well-defined membrane-adhesion system: A substrate is patterned to have a protruding latticework-like structure that binds a cell membrane. A model cell, a “bag” of about 20 microns in diameter, made of a cell membrane that is free of protein molecules, becomes pinned on the substrate by the pinning latticework structure, leaving identical unpinned square parts of the floppy membrane to experience nonspecific interaction with the base of the substrate. Consequently, the average shape of the unpinned membranes and their minute flopping (“fluctuations”) contain all the information about the membrane-substrate interaction. Another tool, a state-of-the-art dual wavelength reflection interference contrast microscopy setup, enables us to directly and easily measure the average membrane shape and fluctuations. Complementing the experiment with the tool of theoretical modeling, we have obtained systematic and quantitative results on the strength of the nonspecific membrane adhesion.

Our work represents a key step towards the ultimate understanding of the cell recognition process by providing both a self-consistent set of tools to tackle the problem and substantive knowledge of an interaction that should be highly relevant in the final, yet-to-be-completed picture of cell adhesion.

Figure 1

RICM image of a vesicle pinned to a patterned substrate, with squares within which the membrane fluctuates in the nonspecific potential (upper-left diagram). The lower-left diagram is the schematic view. The reconstruction of the average membrane shape within one square is shown in the right panel. Only segments of a nearly planar membrane were processed to maintain accuracy in the height reconstruction [25]. The color code indicates the height above the substrate positioned at $h=0$, while $l_0$ denotes the thickness of the adhesive pattern on the glass substrate. The vertical axis in the right figure is in units of nanometer.

Figure 2

Experimental height probability distribution (black line) and the respective effective potential (black symbols) are shown in the left and right panels, respectively. Fitting the data with a potential of the Mie or the harmonic form (right), and their Boltzmann factors (left), yields the red dashed and the blue dotted curves, respectively.

Figure 3

Left panel: Fluctuation amplitude as a function of the membrane tension (red line). Approximations characteristic of the bending- and tension-dominated regime are shown in green and blue, respectively. Right panel: Fluctuation amplitude as a function of the tension and potential strength. Solid lines are contour lines of constant fluctuation amplitude.

Figure 4

Membrane in a nonspecific harmonic potential, with the minimum at the height $h_0$, pinned to a square of an edge length $d$ (in units of ${\xi }_{\parallel }$). The universal mean membrane shape and the associated profile of the mean fluctuation amplitude (normalized by ${\xi }_{\perp }^2$) are shown in the top and bottom rows, respectively. All profiles are calculated for $\kappa =20{\mathrm{k}}_{\mathrm{B}}\mathrm{T}$, $\sigma =0$, and $\gamma =2\times 10^7\mathrm{J}/{\mathrm{m}}^4$.

Figure 5

Comparison of the mean membrane profile of a membrane residing in a Mie potential (left panel) and a harmonic potential of identical curvature (middle panel). The cross section through the center of the shapes is also shown, enveloped by the mean fluctuations of the shape (right panel). The profile associated with the Mie potential is shown in red, while the profile in the harmonic potential is shown in green. The inset shows a detailed view of the membrane shape near the pinning point. The shapes are determined for $\sigma =6.6\times 10^{-6}\mathrm{J}/{\mathrm{m}}^2$ and $\gamma =3.3\times 10^7\mathrm{J}/{\mathrm{m}}^4$, and $h_0-l_0=65\mathrm{nm}$, as for vesicle segments shown later in Fig. 10.

Figure 6

Normalized apparent fluctuation amplitude as a function of temporal (upper left panel) and spatial resolution (upper right panel). The tension increases (in the direction of the arrow) from $\sigma =0$ to $5\sqrt{\kappa \gamma }$. The true and the apparent fluctuation amplitudes are shown in the bottom left and bottom right panels, respectively. The latter has been calculated for a fixed integration time of $\tau =51\mathrm{ms}$ and is averaged over an area of $A=0.5\mu \mathrm{m}\times 0.5\mu \mathrm{m}=0.25\mu {\mathrm{m}}^2$. Contour lines of constant mean-square fluctuation amplitudes are indicated. Specifically, the contour lines of the true and apparent fluctuation amplitude of $50{\mathrm{nm}}^2$ are presented by the thick lines.

Figure 7

The mean-square amplitude of the camera noise averaged over an area $A$ containing $N$ pixels of area $A_{\mathrm{px}}$. The noise decreases exceptionally well with $1/A\sim 1/N$, proving the camera noise is independent for different pixels.

Figure 8

Determining the tension and the potential strength by systematic spatial averaging over a square of area $A$. The best fit results in $\sigma =5.0\times 10^{-6}\mathrm{J}/{\mathrm{m}}^2$ and $\gamma =3.7\times 10^7\mathrm{J}/{\mathrm{m}}^4$.

Figure 9

Measured temporal correlation function (every second data point presented) is fitted with the expression given in Eq. (19). The fitting procedure provides $\sigma =13.0\times 10^{-6}\mathrm{J}/{\mathrm{m}}^2$ and $\gamma =1.0\times 10^7\mathrm{J}/{\mathrm{m}}^4$.

Figure 10

Fitting the theoretically obtained mean shape (colored) to the experimentally determined mean profile of the membrane (gray). The best fit is found for $\sigma =6.6\times 10^{-6}\mathrm{J}/{\mathrm{m}}^2$ and $\gamma =3.0\times 10^7\mathrm{J}/{\mathrm{m}}^4$.

Figure 11

A comparison of fitting procedures for a single vesicle is shown in the table and for four vesicles in the graph. The vesicle discussed in the manuscript and fittings shown in Figs. 7, 8, 9, 10 is indicated by black lines and symbols. Circles denote results of determining the potential strength and the tension by the method of systematic averaging, while crosses and squares are obtained by fitting the shape and the time-dependent correlation function, respectively. Results of all fitting procedures for each vesicle lie very close to the appropriate contour line.

Figure 12

Analysis of the reliability of the three approaches presented for extracting the potential strength and the tension in the membrane. The apparent fluctuation amplitude as a function of the potential strength and tension is shown as a background of each graph. The color code is given on the right. Contours associated with the apparent mean fluctuation amplitudes that are particular to each data set are displayed with solid lines. In all panels, the spatial and temporal averaging is performed with $A=0.25\mu {\mathrm{m}}^2$ and $\tau =0.51\mathrm{ms}$, respectively. (a) Mean values and standard deviations (error bars) associated with performing the analysis on several data recordings from the same patch. Results are presented for one patch with the low (red) and one patch with the high (black) mean apparent fluctuation amplitude (indicated by numbers on the relevant contour line). (b) Mean values and standard deviations associated with averaging the fit results over the eight patches on one vesicle. The patches and the vesicle are shown in the inset by a white square. The numbers within the squares indicate the apparent mean membrane fluctuation amplitude $⟨\mathrm{\Delta }{\overline{h}}^2⟩$ for the particular membrane segment in units of ${\mathrm{nm}}^2$. The analysis was performed on two short data sequences (two subsamples shown in red and yellow) and on a long sequence (black symbols). The results of averaging over three squares with the mean apparent fluctuation amplitude between 14 and $18{\mathrm{nm}}^2$ are shown in blue. (c) Mean values and standard deviations associated with averaging over fitting results from membrane patches with similar mean apparent fluctuation amplitudes on different vesicles. The results for patches with small fluctuation amplitudes, ranging from 6 to $11{\mathrm{nm}}^2$, are shown in red, while the results for large fluctuations, ranging from 18 to $20{\mathrm{nm}}^2$, are shown in black.