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# Geometric localization of thermal fluctuations in red blood cells

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved January 24, 2017 (received for review August 8, 2016)

## Significance

Cell shapes are related to their biological function. More generally, membrane morphology plays a role in the segregation and activity of transmembrane proteins. Here we show geometric implications regarding how cellular mechanics plays a role in localizing thermal fluctuations on the membrane. We show theoretically that certain geometric features of curved shells control the spatial distribution of membrane undulations. We experimentally verify this theory using discocyte red blood cells and find that geometry alone is sufficient to account for the observed spatial distribution of fluctuations. Our results, based on statistical physics and membrane elasticity, have general implications for the use of membrane shape to control thermal undulations in a variety of nanostructured materials ranging from cell membranes to graphene sheets.

## Abstract

The thermal fluctuations of membranes and nanoscale shells affect their mechanical characteristics. Whereas these fluctuations are well understood for flat membranes, curved shells show anomalous behavior due to the geometric coupling between in-plane elasticity and out-of-plane bending. Using conventional shallow shell theory in combination with equilibrium statistical physics we theoretically demonstrate that thermalized shells containing regions of negative Gaussian curvature naturally develop anomalously large fluctuations. Moreover, the existence of special curves, “singular lines,” leads to a breakdown of linear membrane theory. As a result, these geometric curves effectively partition the cell into regions whose fluctuations are only weakly coupled. We validate these predictions using high-resolution microscopy of human red blood cells (RBCs) as a case study. Our observations show geometry-dependent localization of thermal fluctuations consistent with our theoretical modeling, demonstrating the efficacy in combining shell theory with equilibrium statistical physics for describing the thermalized morphology of cellular membranes.

Geometric mechanics, and in particular the century-old theory of thin shells, has seen a resurgence in recent technological applications at length scales spanning several orders of magnitude. Thin elastic surfaces that are curved in their stress-free state display a host of intriguing and useful properties, such as geometry-induced rigidity, bistability, and anisotropic momentum transport (1⇓⇓⇓–5). The general applicability of the mechanics of curved surfaces has wide-ranging consequences for biological functionality as well; it has been used to describe the desiccation of pollen grains, the mechanics of viral capsids, and RBCs (6⇓⇓–9).

The effect of geometry on biological membranes is particularly interesting, because these structures are typically soft enough to support large undulations in thermal equilibrium. Moreover, biology provides a plethora of complex membrane shapes, including the endoplasmic reticulum (10, 11) and the membrane of RBCs. The role of geometry in determining the spatial distribution of their surface undulations is not currently understood, and there may be important implications for biomembrane morphology arising from the use of geometry to control the spatial distribution of thermal undulations.

The case of RBCs is particularly instructive. It provides a unique testing ground for understanding the effect of geometry on thermal undulations of elastic shells because RBCs are both soft enough to have significant thermal undulations and naturally have a complex geometry. The RBC membrane is made up of a lipid bilayer containing transmembrane proteins linked into a 2D triangular network on the cytosolic side of the membrane by spectrin proteins. However, on scales much larger than either the thickness of the membrane or the lattice constant of the spectrin network (12, 13), the composite membrane may be treated as an elastic shell. This shell controls the elasticity of RBCs, because they lack a space-filing internal cytoskeleton. Based on this simplified elastic description and the assumption of flat membranes, a basic theory for RBC undulatory dynamics was proposed by Brochard and Lennon (12). Subsequent exploration of RBC membrane elasticity has included micropipette aspiration (14), electric field-induced deformation (15), optical tweezers (16), and microrheology, which uses the observed thermal undulations of the membrane to infer elastic moduli (9, 17⇓⇓–20). These last studies, which did not fully account for RBC geometry, found an unexpectedly complex spatial distribution of membrane undulations. Very little was understood about the effects of curvature in altering the mechanical properties and equilibrium fluctuation spectrum of the membrane.

In this paper we develop an elasticity theory of curved surfaces subject to thermal fluctuations and describe how this framework can be applied to geometrically complex objects. Although our theory is developed quite generally for any elastic shell, we consider specifically its application to RBC fluctuations, using data collected from diffraction phase microscopy (DPM) measurements of RBCs. These data produce high-resolution images of RBC flicker maps, which show a spatial distribution of membrane undulations. This distribution is correlated with the curvature of the cells. We demonstrate that this distribution can be quantitatively explained by the theory, without an appeal to active forces or heterogeneous membrane composition.

Our analysis uncovers two generic geometric features that control the mechanics of membranes: the sign of the Gaussian curvature, which qualitatively affects cell deformation, and the existence of singular lines (SLs) where the Gaussian and normal curvatures simultaneously vanish. The former has been shown to determine localized and extended static deformations and to guide the propagation of undulatory waves on curved surfaces (1, 4). The importance of the latter, particularly in regard to the mechanics of cell membranes, has not been adequately appreciated, although it has been discussed in the context of isometric deformations of axisymmetric shells (21⇓–23) and the folding of creased shells (5). The RBC geometry includes an SL, which leads to the localization of undulations in its vicinity and dominates the structure of the RBC flicker maps. Neither flat membranes, where stretching and bending deformation modes decouple, nor shells of strictly positive Gaussian curvature admit these geometrically induced, anomalously soft regions.

## Thermalized Shallow Shell Theory

We begin by examining the response of an elastic shell subjected to normal forces. The resulting Green’s function provides insight into the spectrum of undulatory modes of the surface. We will later address the spatial distribution of thermal fluctuations on RBCs using this understanding of the equilibrium population of undulatory fluctuations. In general, the elastic theory of shells (materials that are curved in their unstressed state) is remarkably complex (24, 25). Solving the full problem is formidable, if not impossible; for our purposes we consider small-amplitude undulations on a surface that is only gently curved (i.e., the local radius of curvature is much greater than the wavelength of deformation). In this case, the full nonlinear problem of elastostatics is known to reduce to the linearized Donnell–Mushtari–Vlasov (DMV) equations for a “shallow shell” (25) (see *Supporting Information* for a detailed derivation):

Here

Eq. **1** ensures normal stress balance across the shell. In-plane stress is identically satisfied due to the function **2**, which ensures the surface is physically realizable. In the limit of a flat membrane where

The operator **3** that the coupling of normal to in-plane stress vanishes in the limit of a flat reference state

The mechanics of the shell are controlled by two elastic constants. For thin shells treated as elastic continua, these are related to the Young’s modulus

Consider first the case of uniform membrane uniform curvature. It is convenient to work in the Fourier domain, such that **1**

Using Eqs. **1** and **2** to eliminate the stress function in favor of the normal displacement and making use of Eq. **4** we write a Langevin equation for the membrane undulations in terms of the response function

We see from the response function that the spectrum of undulatory modes is gapped by

We now compute the equilibrium height fluctuations by integrating over all frequencies and wavenumbers. To do so we introduce a short distance cutoff set by the thickness of the membrane *Supporting Information* for details):

Asymptotically we may evaluate how the magnitude of fluctuations scale with both the curvature anisotropy *Supporting Information* for details). In the limit of large system size, **7** and find equilibrium height fluctuations as a function of

## The Elastic Erythrocyte Model

We have so far examined the effect of spatially uniform surface geometry on undulations. This simplification is reasonable in cases where the wavelength of the undulations is very small compared with both the smallest radius of curvature and the scale over which that curvature is changing. In the presence of boundary conditions or spatially heterogeneous curvature, as is certainly the case in RBCs, a more complete shell formulation is required.

We begin numerically by creating a finite element model for a linearly elastic shell of mixed type. For simplicity, we use an axisymmetric shell that is generated from a planar curve. This curve is composed of two circular arcs of the same radius *A*, *Inset* shows the axisymmetric shell generated by revolving the curve about the *z* axis for

Using this 3D model we perform a mode analysis in ABAQUS (Dassault Systèmes), calculating the eigenvalues and eigenmodes associated with free vibration. For a free-standing membrane, we find that there are two distinct classes of eigenvalues (Fig. 2*A*). This distinction becomes sharper at high *C*, *Inset*). In these lowest-frequency modes, deformation is localized near boundaries between negative and positive Gaussian curvature. Based on our above analysis, we expect to see such large deformation in regions where the Gaussian curvature approaches zero, as is necessarily found near the boundaries on the membrane where the Gaussian curvature changes sign.

However, there are two such boundaries in the elastic erythrocyte model, and the localization of large-amplitude bending occurs near only one of them. To understand this, one must consider the asymptotic curves of a surface and a special subset of such curves that we call SLs (21⇓–23). Asymptotic curves on the surface are those along which the osculating plane of the curve is locally tangent to the surface (Fig. 2*B*, *Left*), where the asymptotic curves are shown as black lines. These directions correspond to the special directions *B*, *Left*) it is an SL [although they have been referred to as “crowns” or “rigidifying curves” in other contexts (21⇓–23)]. Along such lines, the shell equations become geometrically singular in that the undulations there are enhanced just as they are on a flat

To further examine the peculiar behavior of undulations near SLs we consider the two curves on an elastic RBC where the Gaussian curvature of the shell changes sign (Fig. 2*B*). Defining a local coordinate system

We rescale variables *C*. We may also calculate the energy content of this boundary layer. Because bending energy regularizes the divergence in the stress, the bending energy dominates the contribution in the layer. Bending energy in a curved shell with vertical displacement *C*, *Inset*) is given by

## DPM

Finally, we compare our geometric model to measurements performed using DPM, which allows for high temporal and spatial resolution of the RBC thermal fluctuations (35, 36) (see *Supporting Information* for details and Fig. S1 for experimental schematic). DPM measures the phase change accumulated through fluctuating surfaces. Because the index of refraction of the RBC interior is spatially homogeneous, changes in the optical path length *A*) correspond directly to changes in the cell’s thickness projected along the path of the light. Thus, DPM can observe membrane fluctuations on the order of *C*, *Inset*) is thus invisible to DPM because of the pseudorigid body deformation. Attaching the cell to a substrate introduces pinned boundary conditions and the points of contact changing the fundamental deformation mode [compare the sketches at the bottom of Fig. 3*A* (pinned at the substrate) to that in Fig. 2*A*, *Inset* (no substrate)].

Typical datasets for both the average height profile *B*. The height data reproduce the characteristic biconcave shape of most healthy RBCs, whereas the fluctuations are localized in a band at finite radius from the axis of symmetry. We collected data from a small ensemble of five cells and obtained averages (over all cells and over azimuthual directions about the cells’ symmetry axis) and SDs for *C*, *Top*, where the dashed black lines represent the average height and the blue bands represent sample SDs). In Fig. 3*C*, *Bottom* we show the SD of the fluctuations about these mean heights

To quantitatively compare with our model predictions we construct a more specific finite element model using the precise geometric data of the RBC. Taking the ensemble-averaged height field from the experimental data in Fig. 3*C* we generate a smoothed, axisymmetric shell model. We first extract the height field corresponding to the dashed black line and truncate the model at *C*, *Bottom* corresponds to these predicted fluctuations. We find that they are consistent with those observed by DPM with no free fitting parameters. We conclude that one can understand the spatial distribution of thermal undulations on the RBC with a minimal model that assumes all spatial variation results from geometry alone; the elastic properties of the cell membrane may be assumed to be spatially homogeneous.

## Summary

The geometry of the undeformed reference state of an elastic shell strongly affects the spectrum of its undulatory modes. In particular, the existence of SLs on surfaces with spatially inhomogeneous curvature, ones that include boundaries between regions of positive and negative Gaussian curvature, introduces a set of low-frequency modes of the surface. One can understand the appearance of these low-frequency states from an analysis of linearized shallow shell theory, as expressed in the DMV equations of the surface.

Because the spectrum of soft modes dominates the fluctuations and linear response of many-body systems, the existence and distribution of sets of SLs on membranes of complex geometry is the principal feature through which geometry controls the statistical physics of such structures.

There is an ongoing discussion regarding whether RBC fluctuations are strongly affected by nonequilibrium processes, specifically ATP-consuming pumps (12, 13, 17, 20, 32, 33, 37⇓⇓–40). This question is difficult to address experimentally because it is clear that ATP depletion has a number of effects on the RBC membrane, including large-scale geometric transitions such as the formation of spherocytes. The agreement of our experiments with an analysis of the predicted fluctuation spectrum in thermal equilibrium suggests that the large-scale deformation modes, which form the dominant contribution to the observed local height fluctuations, can be accounted for without recourse to nonequilibrium noise. This does not imply that nonequilibrium processes are irrelevant but suggests that they might couple strongly only to the short-wavelength modes, to which our experiments are less sensitive. Studying these nonequilibrium dynamics will be challenging, because one is then required to account for hydrodynamic dissipation in the cytosol and the fluid surrounding the cell. Such hydrodynamic interactions have been studied only for the simpler cases of flat (12) and spherical membranes (8, 33). In addition, dissipation within the membrane (membrane viscoelasticity) would have to be addressed.

The most direct implication of this work is that membrane microrheology experiments must take into account the global geometry of the membrane. Because local geometry controls fluctuations, one may also imagine that there is selective pressure on cell membrane morphology to control the spatial distribution of its thermal (and nonthermal) motion. For example, intercellular junctions (e.g., synapses) may be engineered to suppress fluctuations and thereby minimize the disjoining pressure at these junctions.

Our results have direct implications on engineering membrane geometry to localize or guide thermal undulations in both biological and synthetic systems. One synthetic system, graphene sheets, is of particular interest. Here one finds a direct coupling between geometry and their electronic properties (41, 42). We anticipate that one may be able to modify the coupled fluctuations of the surface and local electrochemical potential through curvature in graphene. Finally, we observe that the coupling of curvature to mechanics in the presence of thermal fluctuations suggests that renormalization of area and bending moduli due to nonlinear terms in equations of motion may be affected by nontrivial curvature of the elastic reference state of the membrane, and thus provide a way in which complex membrane geometry at long length scales serves to create spatial variations in the effective elastic moduli of the thermalized membrane.

## DPM

The experimental setup is described in Fig. S1. The optical layout of DPM is essentially a 4f telecentric system. A frequency-doubled Nd-YAG laser, with wavelength 532 nm, was used as the light source for the microscope (Z2 Axio Imager; Zeiss). Output of the laser was coupled to a single-mode (SM) fiber and subsequently expanded and collimated using a collimation lens. A polarizer was placed in the optical path to control the polarization and intensity of the illuminating beam. A blazed diffraction grating (300 lines per mm) was placed at the image plane (IP1) of the microscope, which generates multiple diffraction orders, with the positive first order having the highest intensity (blazed order). The diffracted waves were passed through a 4f system formed by lenses L1 (f1 = 75 mm) and L2 (f2 = 400 mm). The magnification of the 4f system was M = − f2/f1. The lens L1 performs the Fourier transform of the diffracted waves in the Fourier plane (FP). In plane FP, a mask is placed, which consists of a pinhole (10 ^{2}) CCD at the plane IP2. Interferograms of 512 × 512 pixels^{2} size were imaged at the speed of 16.35 Hz with 10-ms exposure time. A 40× (0.95 N.A.) objective was used for the microscope imaging. Further details on the DPM system are given in ref. 36.

## Blood Sample Preparation

A blood sample from the hospital was first diluted in a phosphate-buffered saline (Life Technologies) solution containing 0.1% BSA (Sigma-Aldrich) to achieve a concentration of 0.2% RBC by volume. A sample chamber was created by punching a hole in double-sided tape and sticking one side of the tape onto a poly-l-lysine–coated coverslip (Neuvitro). The sample was then pipetted into the chamber created by the hole and it was sealed on the top using another coverslip. RBCs were allowed to settle for 45 min on the poly- l-lysine–coated coverslip to avoid any cell movement before fast RBC imaging. More details can be found in ref. 43.

## Image Processing

The intensity recorded by the detector has the form

## Shallow Shell Theory

Starting with a surface described by a 3D vector

For a shell, an elastic surface with a reference state defined by

We will consider deformations away from the reference state ** r** in terms of a displacement given by

We will approximate

The energy for a thin shell using linear elasticity is given by

On the condition that the deformation of the surface yields an extremum of the energy,

We may write this equation in terms of the displacement by introducing Hooke’s law for a shell:

Although this simplifies matters, by turning the tensorial stress into a scalar function, we need to introduce an additional equation so that our system remains well-posed. Fortunately, there are conditions on the metric and curvature tensors that ensure that the surface we have defined is actually derivable from a real, physical surface. Although the equations of equilibrium are completely described by specifying three strain components and three bending components, in terms of coordinate deformations the metric and curvature tensors are interdependent. To ensure that the metric and curvature tensors arise from a real surface, we have an additional set of relationships, the equations of Gauss and Codazzi, that must be satisfied so that the surface is “compatible” (25, 46):** W** is an arbitrary vector. This is, in general, a very complicated system of equations. In the limit of small displacements (i.e., linear deformations), a laborious but straightforward calculation yields the following equivalent equation in terms of strain and curvature:

## Correlations and Fluctuations of a Shallow Shell

With these shallow-shell equations in hand, we write the DMV equations for a patch of shell that has uniform radii of curvature

To determine the influence that thermal forces have on the normal displacement **6** in the main text.

We find the dynamic spatial correlation function by using the fluctuation-dissipation theorem:

For a general surface, the hydrodynamic function

The equilibrium variance of the normal displacement is found by taking the inverse Fourier transform of the dynamic spatial correlation function:

After performing the integral over

## Scaling Arguments

To estimate how the equilibrium variance of normal displacement scales with physical parameters we asymptotically estimate the integral in Eq. **8** using Laplace’s method (49). Consider an integral of the form

The case for general

Changing to complex variables such that

*q* may be evaluated:

## Acknowledgments

This work was supported by National Science Foundation (NSF) Grants CBET-0939511 STC, DBI 14-50962 EAGER, IIP-1353368 (to G.P.), DMR-1309188 (to A.J.L.), and DMR-1121288 (MRSEC) (to A.A.E.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: artio.evans{at}gmail.com.

Author contributions: A.A.E., G.P., and A.J.L. designed research; A.A.E., B.B., G.P., and A.J.L. performed research; A.A.E. analyzed data; and A.A.E. and A.J.L. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1613204114/-/DCSupplemental.

Freely available online through the PNAS open access option.

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