A multiscale model for red blood cell mechanics · A multiscale model for red blood cell mechanics D. Hartmann Center for Modelling and Simulation in the Biosciences (BIOMS) ... periments

A multiscale model for red blood cell mechanics

D. Hartmann

Center for Modelling and Simulation in the Biosciences (BIOMS)

University of Heidelberg

BQ 00 21 BIOQUANT

Im Neuenheimer Feld 267

69120 Heidelberg, Germany

[email protected]

submitted to Biomechanics and Modeling in MechanobiologySeptember 2008

Abstract

The object of this article is the derivation of a continuum model for mechanics of redblood cells via multiscale analysis. On the microscopic level, we consider realistic discretemodels in terms of energy functionals defined on networks / lattices. Using concepts ofΓ-convergence, convergence results as well as explicit homogenisation formulae are derived.Based on a characterisation via energy functionals, appropriate macroscopic stress-strainrelationships (constitutive equations) can be determined. Further, mechanical moduli of thederived macroscopic continuum model are directly related to microscopic moduli.

As a test case, we consider optical tweezers experiments, one of the most common ex-periments to study mechanical properties of cells. Our simulations of the derived continuummodel are based on finite element methods and account explicitly for membrane mechanicsand its coupling with bulk mechanics. Since the discretisation of the continuum model can bechosen freely, rather than it is given by the topology of the microscopic cytoskeletal network,the approach allows a significant reduction of computational efforts.

Our approach is highly flexible and can be generalised to many other cell models, alsoincluding biochemical control.

Keywords: biomechanics; red blood cells; cytoskeleton; multiscale modelling; effectiveproperties

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1 Introduction

Biological systems belong to the most complicated ones studied in the natural sciences. Theinvestigation of mechanical issues in biological systems has a long tradition dating back to theseminal book On Growth and Form by D’Arcy Thompson [1].

From a mechanist’s point of view, cells are well characterised by a continuum theory on amacroscopic level, e.g. cells are often described by hyperelastic (or Green elastic) materials. Ona microscopic level, e.g. on the scale of the cytoskeleton of a cell, continuous descriptions are oftennot appropriate and discrete descriptions typically in terms of energy functionals are preferable.At the same time, different physical concepts like entropic forces have to be considered. On onehand microscopic models allow an approach considering as many details as possible. On theother hand often biologically interesting length scales are not accessible by microscopic modelsdue to limited computational capacities.

The object of this article is the extension of multiscale techniques to biomechanical applica-tions. These techniques allow the rigorous upscaling of appropriate basic microscopic descrip-tions to macroscopic continuum models which inherit the microscopic properties. Using conceptsfrom Γ-convergence [2, 3] allows to close the gap between those two descriptions systematically:based on a microscopic description an explicit continuous macroscopic description can be de-rived in terms of energy functionals. From the latter, stress-strain relationships used in theframework of hyperelasticity can be calculated, which allows a description in terms of mass andlinear momentum conservation.

In this article we generalise the approach of Alicandro and Cicalese [4] to more general energies,e.g. describing the mechanical properties of the membrane skeleton of red blood cells. To ourknowledge this is the first approach, which systematically derives a continuum mechanical modelfor red blood cells based on discrete molecular models. A simple heuristic approach linkingmechanical properties of a thin shell with finite thickness with a discrete molecular model canbe found in [5].

The continuum model derived in this article considers the membrane as a hypersurface withmechanical properties resisting bending and stretching. Membrane mechanics can be coupledwith the cytosol (a simple fluid) allowing the calculation of energy minimising shapes (observedshapes), as well as an extension to more complex setups where dynamics play a crucial role. Ourmodel is challenged by the quantitative comparison of simulations of the derived macroscopicmodel with microscopic simulations as well as optical tweezers experiments, one of the mostcommon experiments in biophysics. To do so, an appropriate finite element framework is set up,which is implemented in Gascoigne [6]. Simulations of the macroscopic continuum model showa good qualitative and quantitative agreement with the microscopic approach and experiments..

The model as well as the simulation approach can be extend easily to include much more de-tails, e.g. chemical processes controlling the membrane elasticity of the red blood cell. Using thehomogenisation approach allows a significant reduction of computational complexity. Consider-ing a realistic number of filaments in a discrete microscopic model, the system is hardly solvable.However using a continuum approach finite element methods allow the choice of relatively coarsediscretisations speeding up simulations. The efficiency of computations for continuum modelscan be further increased by appropriate mesh refinement strategies. Another significant advan-tage of continuum models is that they can be easily coupled with reaction-diffusion equations,which are typically used to model biochemical reaction networks in cells. This allows a straightforward approach investigating interactions between mechanics and biochemistry. And last butnot least continuum models are usually more accessible to mathematical analysis than discretemodels.

The structure of the article is as follows: First we review in Section 1.1 biomechanics of red

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A B

Figure 1: (A) Spread membrane skeleton examined by negative-staining electron microscopy. Itclearly shows the RBC’s hexagonal lattice of junctional complexes. (reprinted from [9], c©1987Rockefeller University Press) (B) Schematic presentation of a red blood cell in an optical tweezers(according to [11], c©1997 Biophysical Society)

blood cells as well as optical tweezers experiments. In Section 2 we introduce a realistic staticmicroscopic model for optical tweezers experiments based on existing microscopic approaches.A corresponding static continuum model is then systematically derived in Section 3. Usinga variational approach, allows us to determine energy minimisers via relaxation. To do so acorresponding continuum mechanical model in terms of mass and liner momentum conservationis set up for optical tweezers experiments (Section 4). Finally, we introduce in Section 5 anappropriate numerical approach for solving the derived model and close with a discussion of theresults in Section 6.

1.1 Biology and mechanics of red blood cells

Red blood cells (RBC) have a simple structure and therefore often serve as model systems forthe development of theoretical and experimental methods in biophysics. Under physiologicalconditions, a normal human RBC in an unstressed state assumes a biconcave discoid shapeapproximately 8µm in diameter [7]:

z = R√

1− (x2 + y2)/R2[c0 + c1(x2 + y2)/R2 + c2(x2 + y2)2/R4] (1)

with R = 3.91µm, c0 = 0.1035805, c1 = 1.001270, and c2 = −0.561381. The nucleus andother organelles that are present in RBCs during their development are expelled before andshortly after the cells are released into the circulatory system, leaving the mature cells withno internal structural components other than the membrane-associated spectrin cytoskeleton[8], a quasi two-dimensional network (Fig. 1). Basic building blocks of the spectrin networkare 200nm long spectrin tetramers (edges) which crosslink the junctional complexes of actin(vertexes). The average length between two vertexes is 80 nm [9]. Hence, the end-to-enddistance of spectrin tetramers is significantly smaller than their contour length, which stronglyunderlines that mechanics are due to entropic effects. According to Liu and co-workers [9] thereare over 80% degree-6 vertexes in spectrin networks extracted from healthy human RBCs, whichsuggest a relative regular hexagonal structure of the spectrin network (c.f. Fig. 1). However,recent experiments indicate that the network might be more disordered with a significant loweraverage vertex degree [10].

Functional actin complexes are affected by ATP, which is usually present in RBCs, by inducingspectrin-actin dissociations [12]. Creation and motion of such defects allows the network torearrange constantly. Therefore, it is postulated that the spectrin vertices of RBCs behave likea liquid on large time scales. Thus dynamics would allow a relaxation of the in-plane shearelastic energy [10]. However, for periods of around 30 min and temperatures up to 37C large

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cytoskeletal deformations have shown to be stable [13], i.e. during this timescale RBCs canmanifest large shear.

One of the most common experiments to study mechanics of RBCs as well as other cell typesare optical tweezers experiments (Fig. 1) [14, 15]. Using focused laser beams, optical tweezersallow the application of forces in the range of pico Newtons to dielectric microbeads. Sinceforces applied to the optical tweezers can be determined quite well, the experiments allow theestimation of involved mechanical moduli. Usually, the deformation rate is faster than therelaxation rate of stresses, such that relaxation can be neglected [10].

1.2 Models of red blood cells

Quite a variety of models for mechanics of RBCs are found in the literature. These can be dividedinto two classes: microscopic molecular based models (i.e. discrete models), e.g. [11, 16, 10, 17],and macroscopic continuum models, e.g. [7, 18, 15, 19, 5, 20]. Among them, we can distinguishsolely static models [16], i.e. models based on energy minimisation, or dynamic models [17],mainly concerned with RBCs embedded in fluid flow. Further we can distinguish betweenmodels which treat the membrane as a two-dimensional hypersurface [16] and those which treatit as a shell [19] with a small but finite thickness. The list of references given here is by nomeans complete, for a more detailed list we refer to [21].

The objective of this paper is to show how the different models can be rigorously linked. Here,we restrict ourselves to a slightly simplified variant of the molecular based static model proposedby [22] and [16]. A corresponding continuum model is derived in Section 3 using Γ-convergence.The static continuum model can be linked to a dynamic continuum model using a variationalapproach, as shown in Section 4.

2 Static molecular based model

The major advantage of discrete models is their simplicity, which allows straight forward numer-ical schemes using Monte Carlo methods. Most microscopic models go back to the work of Boey,Boal and Discher [22, 16] considering a micropipette aspiration experiment. Their approach hasbeen also extended to optical tweezers experiments [10]. Let us review the original model in thecase of an optical tweezers experiment using a slightly different notation adopted to our setup:

Model 1 The stationary shape of the RBC in an optical tweezers minimises the discrete freeenergy

Fε(χn) = Fε,tweezers + Fε,in-plane + Fε,bending + Fε,surface + Fε,volume.

Here ε is the typical length scale of the spectrin network. The degrees of freedom of the modelare the actin vertex coordinates χn = χ(Xn)n∈1...N (see Fig. 1), where Xn is the position ofa network vertex in the reference configuration. Its motion / deformation is thus given by thefunction

x = χ : R3 3X 7→ χ(X) ∈ R3.

Since we are interested in mechanics, we will restrict us to deformations preserving the orienta-tion.

Based on the observations of [9], Discher and co-workers propose to model the mechanics ofRBCs using a quasi-two-dimensional network with hexagonal symmetry (including 12 topologicaldefects, which are needed to cover a sphere with a hexagonal net). This approach has beengeneralised by [10] to more general networks. In this section we will not rely on a special

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geometry, however in Section 3 we will restrict ourselves to a purely hexagonal geometry inorder to derive a corresponding continuum model, which can be explicitly specified.

All energies considered in Model 1 are non-negative, apart from Fε,tweezers modelling theeffect of optical tweezers. Let us assume that Ntweezers actin vertex coordinates are boundto the microbeads of the optical tweezers on both sides. Therefore they experience a forcecorresponding to the energy:

Fε,tweezers(χn) = −∑

Xi bound

1Ntweezers

(χ(Xi)−Xi) · ftweezers,

where ftweezers is the force applied by the optical tweezers.The dominant term of the membrane energy in the context of large deformations, e.g. in

optical tweezers experiments, is the in-plane free energy Fε,in-plane of the membrane-bound cy-toskeleton:

Fε,in-plane(χn) =∑edges

L2(m,n)

√3

2VWLC

(L2

(m,n)

L2(m,n)

)(2)

+∑faces

2A(l,m,n)C

(A(l,m,n)/A(l,m,n))q + η,

where L(l,m) = |χm − χl| is the length of the spectrin link / edge connecting vertexes l, m andA(l,m,n) = |(χm−χl)×(χn−χl)|/2 is the area of the triangular face enclosed by the spectrin links(l,m), (m,n), (n, l) in the deformed state.L(l,m) = O(ε) and A(l,m,n) = O(ε2) are the edge lengths and face areas in the reference co-ordinate system. The different terms in (2) are explained in detail below.

Let us rewrite the relative lengths L(l,m)/L(l,m) and areas A(l,m,n)/A(l,m,n) used in formula (2)in terms of discrete finite difference quotientsDξLχ(X) ≡ (χ(Xl + Lξ)− χ(X))/L (in direction of the vector ξ), which will facilitate ourmathematical analysis in Section 3:

L(l,m)

L(l,m)=|χ(Xl + L(l,m)ξ(l,m))− χ(Xl)|

L(l,m)= |Dξ(l,m)

L(l,m)χ(Xl)|, (3)

A(l,m,n)

A(l,m,n)= det

(Dξ(l,m)

L(l,m)χ(Xl)⊗D

ξ(l,n)

L(l,n)χ(Xl)

)= J(l,m,n)(χ(Xl)), (4)

where ξ(l,m) = (Xm −Xl)/|Xm −Xl| is the unit vector pointing from vertex Xl to the vertexXm in the undeformed coordinate system. J(l,m,n)(χ(Xl)) is the discrete Jacobian correspondingto the triangle spanned by the vertexes l, m, and n. Since we consider only orientation preservingdeformations J(l,m,n)(χ(Xl)) and equivalently A(l,m,n)/A(l,m,n) are always positive.

The first term in (2) is the entropic energy stored in the spectrin links. Discher and co-workersassume that the energy is given by the worm-like chain model introduced by [23], which is basedon experiments with DNA:

L2

√3

2VWLC

(L2

L2

)= kBT

4pLmax

L2(2L−3Lmax)(L−Lmax) . (5)

Since limL→Lmax VWLC(L2/L2) = ∞, we consider here a pth-order Taylor-expansion VWLC ofVWLC around the rest state L = L due to mathematical restrictions (as we will see below).Hence, VWLC is a positive super-linearly growing function with p-growth, i.e. it satisfies thefollowing growth condition

c1(|z|p − 1) ≤ VWLC(z2) ≤ c2(|z|p + 1) (6)

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for constant c1, c2 ∈ R+ 1 ≤ p < ∞. Dimensional analysis reveals that VWLC(L2/L2) is anenergy density, respectively VWLC(L/L).

Considering only VWLC in (2), the minimum of Fε,in-plane corresponds to a collapsed network.However, due to repulsive forces of steric interactions (i.e. entropic forces) the network doesnot collapse: The end-to-end distance of spectrin filaments is much smaller than their contourlength. Hence, spectrin fibres are polymer coils with a non-negligible width leading to repulsion.Therefore, the second sum is introduced into the model. It is of a phenomenological origin andaccounts for steric interactions [16]. C > 0 and q > 0 are constants, typically the case q = 1 isadopted [16, 10]. Due to mathematical restrictions (see below) the constant η > 0 is introducedto ensure that the contribution of the repulsive forces is bounded. Dimensional analysis showsthat C/[(A(l,m,n)/A(l,m,n))q + η] is an energy density.

The introduction of the regularisation of repulsive steric interactions in (2), i.e. the introduc-tion of η > 0, as well as the restriction to Taylor expansions of VWLC is necessary to obtainrigorous convergence results following the work of Alicandro and Cicalese [4]. Without thisregularisation our approach yields only formal results which would still needed to be verified bymathematical analysis.

Due to the fluid character of the lipid bilayer it cannot sustain shear stress, nevertheless itpossesses a bending stiffness and a large compressional stiffness. Often, it is assumed to beincompressible. Since the building blocks of the lipid bilayer are much smaller than the spectrinlinks, the resistance to bending of the lipid bilayer is well described by continuum models, e.g.the the Canham and Helfrich functional [24, 25]:

FCanham-Helfrich =κ

2

∫Γ(H −H0)2dµ+ κg

∫ΓKdµ, (7)

with cell membrane Γ, bending elasticity moduli κ, κg, mean curvature H = C1 + C2, Gausscurvature K = C1C2, principal curvatures C1, C2, and the constant H0, which representsthe spontaneous curvature. The last term in (7) can be neglected, since due to the Gauss-Bonnet theorem the integral is constant for a given topology (We allow only variations overa fixed topology). The Canham-Helfrich functional is well established for intermediate surfacecurvatures based on molecular dynamic simulations [26] on the one hand as well as experiments[27] on the other hand. It is a sufficient approximation considering optical tweezer experiments,which exhibit relative mild curvatures.

Therefore the microscopic models proposed by [16] and [10] rely on the Canham-Helfrich en-ergy, although more complex approches can be found in the litterature, e.g. considering non-localbending energies [28, 29, 30, 31]. To work with a fully discrete approach, Discher and co-workersdiscretised the Canham-Helfrich model (7) using network vertex-coordinates as degrees of free-dom yielding a discrete free energyFε,bending(χn). For the discrete formulation we refer to the original paper [16]. In the followingwe will work directly with the continuum description (7), which is possible since Γ-convergenceis stable to continuous pertubations [32]. Of course, also more complex models for membranemechanics, e.g. non-local bending energies, can be included in our approach. Here, we howeverrely on the somewhat simpler Canham-Helfrich energy to allow an exact quantitative compari-sion between simulations of the microscopic model and the upscaled macroscopic model. Precisemeasurements of bilayer bending elasticity can be found in the literature [33, 34, 35, 36, 37].

The non-physical energies Fε,volume and Fε,surface account phenomenologically for the incom-pressibility of the cytosol and of the lipid bilayer:

Fε,volume(χn) = kvolume(|Vcell| − Vdesired)2,

Fε,surface(χn) = ksurface(|Acell| − Adesired)2,

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ε1

2

g

Ω

2

g

1ξ

2ξ3ξ

4ξ

5ξ 6ξ

∆1

∆6∆5

∆4

∆3

∆

Figure 2: Abstraction of the two dimensional membrane skeleton of a red blood cell spread ona flat 2 dimensional domain Ω shown in Fig 1.

where the total area of the cell is given by Acell =∑

facesA(l,m,n) and the total volume byVcell. The latter can be directly related to the areas A(l,m,n), since the divergence theorem|Ω| = 1

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∫∂Ω(x · n)dµ must hold. kvolume and ksurface are constants.

According to the adopted hypothesis, the stationary shape of the RBC, e.g. stretched inan optical tweezers, minimises the free energy. Boey, Boal and Discher [22, 16] use MonteCarlo schemes for energy minimisation and Li and co-workers [10] use coarsegrained moleculardynamics to determine the energy minimum. Both start with a given reference shape and let theenergy relax until a minimum is reached. Using these approaches the experimentally observedshapes are recovered.

The choice of an appropriate reference shape is however subtle, since the computed shapesdepend usually on the chosen reference shape [10] (initial or rest shapes). Li and co-workers [10]have invoked the physical hypothesis that the spectrin network undergoes constant remodellingto always relax the in-plane shear elastic energy to zero at some slow characteristic time scale.Therefore they suggest to use an initial shape which minimises the energy Fε,bending +Fε,surface +Fε,volume (in-plane energy is neglected). Using this approach the biconcave shape (1) is recovered[10] (see also Section 5.3.1).

Starting from this reference shape the energy minimum of the full Model 1 is determined (seeFig. 7). For computations as well as a more details we refer to the original paper [10].

3 From molecular based to continuum models

Different static continuum models in terms of energy functionals for mechanics of RBCs can befound in the literature [31, 38, 18]. These have the same structure as the microscopic modelconsidered above, but they are however gernerally of a heuristic type.

In the following, we derive systematicly a continuous energy functional F for the mechanics ofRBCs based on the microscopic model Fε given in Model 1. Because we are generally interestedin deformations with minimal energies, Γ-convergence is an appropriate framework.

In the following we will concentrate on the energies Fε,bending + Fε,in-plane, since Fε,surface,Fε,volume are of a non-physical type and Fε,tweezers has an obvious continuous counterpart. Fur-ther, we replace Fε,bending with FCanham-Helfrich, i.e. expression (7). Since Γ-convergence is stablewith respect to continuous perturbations [3], it is sufficient to restrict ourselves to Fε,in-plane,which represents the energy contribution of the discrete spectrin cytoskeleton / membrane skele-ton.

The free energy Fε,in-plane is independent of deformations perpendicular to the membrane.Therefore we consider only the two-dimensional tangent space of the membrane for the sake

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of simplicity. Further, we restrict our analysis to two-dimensional networks with hexagonalsymmetry (Fig. 2) in order to find an explicit expression for the continuum functional and notonly abstract convergence results. Let us introduce the considered networks

εG ∩ Ω

with

G = X ∈ R2 : X = µ1g1 + µ2g2 with µi ∈ Z, (8)

g1 = (1, 0), g2 = (1/2,√

3/4), and Ω ⊂ R2 open and bounded. Here, ε is the typical length scaleof the network. The basic building block of the network is a unit cell as illustrated in Fig. 2with links Gξ = ξi and triangles G4 = 4i. Since we restrict ourselves to the tangent space,the deformation of network vertexes X ∈ εG ∩ Ω is given by

x = χ : εG ∩ Ω 3X 7→ χ(X) ∈ R2,

which should be orientation preserving, as discussed above.Let us rewrite (2) as follows:

Fε,in-plane(χ,Ω) =∑

X∈vertexesin Ω

[12

∑ξ∈edges inΩ

connected to X

ε2

√3

2VWLC

((Dξεχ(X)

)2)

(9)

+13

∑4∈faces inΩ

connected to X

ε2

√3

2C

(Jε4(χ(X)))q + η

]

where Dξεχ(X) is the discrete finite difference quotient (3), i.e. the relative length change of theedge in the deformed confiuration, and Jε4(χ(X)) is the discrete Jacobian (4), i.e. the relativeface area change in the deformed configuration. The factor 1/2 in front of the pair interactionsaccounts for the fact that each interaction is counted twice. Similarly a factor 1/3 accountsfor the multiple counting of steric interactions. The scaling ε2

√3/2 is the natural geometrical

scaling.Neglecting steric interaction, i.e. choosing C = 0 and thus considering only pair-interactions,

Fε,in-plane is a special case of free energies describing atomistic interactions in crystal lattices.Such energies have been studied in several works [4, 32] using Γ-convergence as well as othertechniques [39, 40].

A constructive proof based on the concept of Γ-convergence, calculating the lim inf and lim supinequalities directly, is relatively straight forward in one dimension [32]. In higher dimensions,the direct calculation of the lim inf inequality is, however, highly non-trivial, unless the problemcan be reduced to several one-dimensional problems, e.g. in the direction of the coordinate axes.Due to the steric interaction energies, involving multiple dimensions, such a reduction is notpossible in our setup.

Considering chrystal lattices Alicandro and Cicalese [4] use a more abstract approach based onΓ-convergence results from the theory of homogenisation of integrals [41]. This abstract approachallows to prove the existence of appropriate continnum limit energy functionals. Consideringperiodic microscopic geometries, the continuum functionals can be characterised indirectly as alimit of discrete functionals on simple rectangular domains.

Since the steric interaction energies in (2) are positive and bounded the proof of Alicandro andCicalese [4] needs basically no modification. Therefore, we review here only the correspondingresults. Using convexity of the functional, we then derive an appropriate cell problem, i.e. ahomogenisation formula.

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3.1 Existence

To consider possible minimisers χ of (9) for arbitrary ε within one function space, we identifythe discrete maps χ : εG ∩ Ω → R2 with maps χ : Ω → R2 constant on each cell of the lattice.Therefore, let us introduce the following function spaces:

Fε(Ω) ≡u : Ω→ R2 : for any X ∈ εG,u is constant on

Y ∈ R2 : Y = X + µ1g1 + µ2g2, 0 ≤ µi < ε,

Fε,φ(Ω) ≡u ∈ Fε(Ω) : u(X) = φ(X) if d(X, ∂Ω) < 1

.

Of course other embeddings, e.g. assuming piecewise linear functions, can be considered equiv-alently.

Theorem 1 For every sequence (εj) of positive real numbers converging to 0, there exists a sub-sequence (εjk) and a continuous quasi-convex functionΨ : R2×2 → [0,∞) satisfying

c(|M|p − 1) ≤ Ψ(M) ≤ C(|M|p + 1)

with 0 < c < C, such that (Fεjk,in-plane(·, ·)) given in (9) Γ-converges with respect to the

Lp(Ω; R2)-topology to Fin-plane : Lp(Ω; R2)× A ⊂ Ω : A open → [0,∞] defined as

Fin-plane(χ, A) =

∫A Ψ(∇Xχ)dµ if χ ∈W 1,p(A; R2)

∞ otherwise.

Here, W 1,p(A; R2) is given by the standard definition of Sobolev spaces [42].As already mentioned Alicandro and Cicalese [4] consider energies of the type (9) without

steric interactions, i.e. C ≡ 0. They show that the upper and lower Γ-limits of these functionalsdefined on pairs function-set are inner-regular increasing set functions. This allows the use of acorresponding compactness and integral representation result [41]. Considering the case C 6= 0the arguments can be repeated. The pair interactions in (9) fulfil the required growth conditions,c.f. growth condition (6), and the repulsive energy of steric interactions in (9) is bounded, suchthat the proof of Theorem 1 goes along the lines of the original proof.

The embedding of W 1,p(Ω; R2) to Lp(Ω; R2) is compact. We use the standard definition ofLp spaces [42]. Hence, Theorem 1 implies also the convergence of minimisers to a minimiserof the limit functional. Theorem 1 considers an unconstrained Γ-limit. However, often one isinterested in problems with prescribed boundary conditions. Corresponding convergence resultswith restrictions to χ ∈ Fε,φ, and accordingly χ − φ ∈ W 1,p

0 (Ω; R2), can be proven. Here, weuse the standard definition for W 1,p

0 (Ω; R2), i.e. Sobolev functions of the space W 1,p(Ω; R2) witha vanishing trace on the boundary of Ω [42]. Similar results hold also for periodic boundaryconditions. For more details, we refer to the work of Alicandro and Cicalese [4].

3.2 Homogenisation

Let us first define a rhombus which is spaned by N times the vector g1 and N times the vecotorg2, c.f. definition (8):

QN = X : X = µ1g1 + µ2g2 with 0 ≤ µi < N, i = 1, 2.

Minor modifications of the approach of Alicandro and Cicalese [4], yield the following homogeni-sation result:

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Theorem 2 For every sequence (εj) of positive real numbers converging to 0, the sequence(Fεj ,in-plane) given in (9) Γ-converges with respect to the Lp(Ω; R2)-topology to Fin-plane : Lp(Ω; R2)×A ⊂ Ω : A open → [0,∞] defined as

Fin-plane(χ, A) ≡

∫A Ψhom(∇Xχ)dµ if χ ∈W 1,p(A; R2)

∞ otherwise,

where the integrand Ψ : R2×2 → [0,∞) satisfies the rescaled homogenisation formula

Ψhom(M) ≡ limN→∞

1N2

minF1,in-plane(χ, QN ) : χ = M ·X + u

(10)with (N − 2)-periodic u ∈ F1(QN )

with F1,in-plane defined by (9).

An analogous result holds also in the case of Dirichlet or periodic boundary conditions with thesame characterisation of Ψhom. Above, (N − 2)-periodic functions u, i.e. u(X + (N − 2)g1) =u(X) as well as u(X+(N−2)g2) = u(X), are considered to ensure that the discrete derivativeof the perturbation on the boundary is zero.

3.3 Cell problem

The homogenisation formula (10) in Theorem 2 is given as a minimisation problem over agrowing rectangle QN (N →∞). Since the contribution from pair as well as steric interactionsare convex the homogenisation formula can be reduced to a cell problem, i.e. a homogenisationformula:

Theorem 3 For every sequence (εj) of positive real numbers converging to 0, the sequence(Fεj ,in-plane) given in (9) Γ-converges with respect to theLp(Ω; R2)-topology to Fin-plane : Lp(Ω; R2)× A ⊂ Ω : A open → [0,∞] defined as

Fin-plane(χ, A) ≡

∫A Ψ(∇Xχ)dµ if χ ∈W 1,p(A; R2)

∞ otherwise,

where the integrand Ψ : R2×2 → [0,∞) is given by the following problem on a unit-cell

Ψ ≡ 12

∑i=1...6

VWLC(ξT · (∇Xχ)T · (∇Xχ) · ξ) + 2C

(det(∇Xχ)q + η)(11)

with vectors ξi ∈ Gξ (|ξi| = 1) (c.f. Fig 2) corresponding to the spectrin edges in the undeformedunit cell.

Proof : For simplicity, we split the proof into two parts: (a) the case C ≡ 0 and (b) VWLC ≡ 0.For all N , we show that in both cases χ = M ·X is a minimiser of (10). Since the energyFε,in-plane is non-negative, the minimum in the general case, C 6= 0 and VWLC 6= 0, is alsorealised by χ = M ·X, i.e. formula (11) holds.

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Case (a)

Set C ≡ 0 and let us show that χmin = M ·X + umin, with umin constant, is a solution of theminimisation problem (10) independent of N . That is, among all (N − 2)-periodic functionsu ∈ F1(QN ) the energy

E1(M ·X + u, QN ) ≡∑

X∈vertexesin QN

12

∑ξ∈edges inΩ

connected to X

√3

2VWLC

((Dξ1 (M ·X + u(X))

)2)

(12)

is minimised by umin constant. Let us choose an arbitrary minimiser umin ∈ F1(QN ) of E1

given by (12). By definition, the minimiser is at least N − 2 periodic. Hence, the functionu(X) ∈ F1(QN ) defined by

u(X) =1

(N − 2)2

∑i,j∈[0,N−2]

umin(X + ig1 + jg2)

with g1 and g2 defined in (8) is one periodic and thus constant by construction of F1(QN ). Bystrict convexity of VWLC and hence strict convexity of E1, given by (12), the function u satisfiesthe inequality

E1(M ·X + u, QN ) < 1(N−2)2

∑i,j∈[0,N−2]E1 (M ·X + umin (X + ig1 + jg2) , QN ) .

Since E1 depends only on the gradient of χ, it is invariant under a shift, e.g. under the shiftX + ig1 + jg2. Hence, we can conclude

E1(M ·X + u, QN ) < 1(N−2)2

∑i,j∈[0,N−2]E1 (M ·X + umin (X) , QN ) .

Therefore also u(X) is a minimiser of E1, and accordingly χmin(X) = M ·X + u(X). Sinceu(X) is an arbitrary constant and E1(χ, QN ) depends only on the discrete gradient of χ, wecan choose χmin(X) = M ·X. The inequalities are strict, thus that uniqueness of the minimiser(up to a constant) is guaranteed.

Case (b)

Choose VWLC ≡ 0 and let us prove that χmin = M ·X is also a solution of the minimisationproblem (10) for all N . That is, it minimises

E1(χ, QN ) ≡∑

X∈vertexesin QN

13

∑ξ∈edges inQN

connected to X

√3

2C

(J4 (χ (X)))q + η(13)

among all χ ∈ F1,M·X(QN ).Let us fix N . For all deformations χ ∈ F1,M·X(QN ) the total area of QN after the deformation

equals N2√

3/2 det M, since |QN | = N2√

3/2 and the total area depends only on the value of χon ∂QN . Hence, it follows

N2

√3

2det M =

∑X∈vertexes

in QN

∑4∈faces inQNconnected to X

√3

4J4 (χ (X)) (14)

for all χ ∈ F1,M·X(QN ). Thus energy E1(χ, QN ) given by (13) can be minimised only by avariation of the triangular face areas

√3

4 J4 (χ (X)) under the constraint that the sum (total

11

area) is constant. Using J4 (M ·X) = det M and (14), we obtain for arbitrary χ ∈ F1,M·X(QN )

2√

3E1(M ·X, QN ) = 2N2 C

(det M)q + η

= 2N2 C(1

2N2

∑X∈vertexes

in QN

∑4∈faces inQNconnected to X

J4 (χ (X)))q

+ η

.

By convexity, we have∑X∈vertexes

in QN

∑4∈faces inQNconnected to X

C

(J4 (M ·X))q + η

<∑

X∈vertexesin QN

∑4∈faces inQNconnected to X

√3

2C

(J4 (χ (X)))q + η

for any χ ∈ F1,M·X(QN ). The energy of the deformation χ = M ·X is smaller than the energyof any other deformation χ, which shows χmin = M ·X and thus completes the proof. Strictconvexity implies the uniqueness of the minimiser.

So far we have restricted us to two-dimensional deformations on a bounded two-dimensionaldomain. In the fully three dimensional situation, i.e.

χ ∈W 1,p(ΓX ; R3) with det∇Xχ > 0

with ΓX beeing the surface of the RBC in the undeformed state, i.e. reference configuration, wefind

Fin-plane(χ) =∫

ΓX

12

∑i=1...6

VWLC(ξT · FΓT · FΓ · ξ) + 2C

(JΓq + η)dµ, (15)

where FΓ ≡ (∇Xχ) ·PX = P · (∇Xχ) ·PX ∈ R3×3 is the three-dimensional surface deformationtensor, PX = 1 − nX ⊗ nX the surface projection operator with respect to the reference con-figuration (indicated by the subscript X), P = 1 − n ⊗ n the one with respect to the currentconfiguration (nX and n are outer unit normals of the surface), and the surface Jacobian JΓ.The latter is given by JΓ =

√det g with the metric tensor g of the surface, defined as

(g)ij = (FΓ · ti)T · (FΓ · tj) with i, j ∈ 1, 2

where t1 and t2 be two orthogonal vectors in the tangent space of the surface.

3.4 Characterisation of Fin-plane using invariants of FΓ

For simplicity, let us first restrict to Taylor expansions VWLC of VWLC around 1 up to order 2,i.e. p = 2. We thus find the following energy density Ψ:

Ψ =12

∑i=1...6

(VWLC (1) + V ′WLC (1) (ξT · F ΓT · F Γ · ξ − 1)

(16)+

12V ′′WLC (1) (ξT · F ΓT · F Γ · ξ − 1)2

)+ 2

C

JΓq + η

with VWLC as in (5).

12

The energy (16) is isotropic, which is a well known result. (In the general case p > 2 theenergy density (15) is not isotropic and has a hexagonal symmetry!) This allows us to rewritethe energy density in terms of invariants of the deformation tensor, a standard approach innon-linear elasticity. Throughout the literature different invariants are used in the context ofhyperelastic materials. Here, we work with the following invariants [43]:

I1 ≡ JΓ − 1 = λ1λ2 − 1,(17)

I2 ≡ tr(FΓ · (FΓ)T)− 2 = λ21 + λ2

2 − 2,

where λ1 and λ2 are the principal stretches of the surface deformation tensor. Using the relationbetween interaction link length and invariants (c.f Appendix A) as well as dropping constantterms, we find:

Ψ =32V ′WLC(1)I2 +

316V ′′WLC(1)(4I2 − 8I1 + 3I2

2 − 4I21 ) + 2

C

(Iq1 + η).

For the rest of this section let us assume that the membrane skeleton is initially not stressed.This assumption might not hold true in all cases. Pre-stress is an important concept in biology[22, 44]. However, our assumptions allow us the explicit specification of the constant C. Con-sidering deformations consisting only of compression, or alternatively dilatation, i.e. I2 = 2I1,we obtain the following expansion

Ψ = 2C + (3V ′WLC(1)− 2Cq(1 + η)2

)I1 +O(I21 ).

Assuming that the network is initially unstressed, the energy should be at a minimum and henceall linear terms should vanish. We therefore find

C =3(1 + η)2V ′WLC(1)

2q

and thus are left with one constant less, allowing a better comparison with experiments. A directrelation of the microscopic parameters with macroscopic moduli is possible and is considered inSection 4.3.

Of course the same approach can be applied in the case p > 2. For p > 2 however, thehexagonal symmetry is reflected in (16). Whether a stress tensor with a hexagonal symmetry isbiologically realistic or not is unclear. On one hand it is not clear in which direction the symmetryaxis would be pointing, on the other hand the spectrin network is not perfectly symmetric, it onlyshows largely a hexagonal symmetry (over 80% degree-6 vertexes). Further, it is not clear howa hexagonal symmetry could be identified from an experimental point of view. For simplicity,we average over all possible directions of symmetry axis in the case p > 2: Ψ =

∫ 2π0 Ψdφ, c.f.

Appendix A. The assumption of a hexagonal symmetry is however necessary for the sake of thederivation of a homogenisation formula. Only this homogenisation formula allows the directlinkage between microscopic and macroscopic models and corresponding parameters.

4 A relaxation approach

The models considered above are based on a static description in terms of energy functionals, i.e.shapes of RBCs are determined by energy minimisation. The corresponding energy minima aretypically calculated using the corresponding Euler-Lagrange equations [45]. Physically speaking,the corresponding forces of the energies are determined and one looks for a shape where all forcesequilibrate.

13

force

laser

Figure 3: Illustration of a typical optical tweezers experiment.

Instead of using the Euler-Lagrange equations, we consider a relaxation approach based ona dynamic formulation in the framework of conservation of mass and linear momentum. Suchan approach is of course somewhat more complex, but has several advantages. On one handit can be easily extended to situations, where dynamics play a role, as well as it can be easilycompared with existing dynamics models [46, 20]. On the other hand it can be encoded directlyin the FEM-framework Gascoigne.

We adopt the same assumption as in Section 1.2: the time scale of the experiment (order ofseconds) is fast compared to relaxation, thus that we consider the regime of membrane elasticity.Additionally, we assume that the cytosol is locally incompressible as well as a globally constantsurface area, rather than choosing very large kvolume and kvolume in Model 1. Moreover, weneglect inertial effects of the membrane, since its mass is relatively small.

Model 2 The mechanics of a RBC clamped into an optical tweezers are determined by thefollowing model (for some fixed time T > 0):

du

dt= v in Ωcell(t)× [0, T ),

ρdv

dt= −∇p+∇ · σcytosol in Ωcell(t)× [0, T ),

0 = ∇ · v in Ωcell(t)× [0, T ),

(σcytosol + pout1) · n = Ftweezers −∇Γqsurf. + Fmembrane on ∂Ωcell(t)× [0, T ),

|∂Ωcell(t)| = |∂Ωcell(0)| for [0, T ).

The evolution of Ωcell(t) is given by the speed v on the boundary.

As usual ρ is the local material density, v the local material speed, u is the local materialdisplacement, i.e. the current position of a particle originating position X in the referenceconfiguration is given by x = X + u(X) and p is the local volume pressure, accounting forincompressibility of the cytosol. The global surface pressure qsurf. is constant on the surfacesince the single constraint |∂Ωcell(t)| = |∂Ωcell(0)| requires only one Lagrange multiplier. ∇ = ∇x

is the gradient with respect to the laboratory coordinate system and ∇Γ = P · ∇ is the surfacegradient, where P is the surface projection operator. Here, we have assumed that σcytosol isgiven by the Stokes stress tensor, i.e.

σcytosol = η(∇v + (∇v)T

),

and Ftweezers is the force density due to the optical tweezers, i.e.

Ftweezers = χtweezersftweezers

with χtweezers is the characteristic function of the microbeads and ftweezers the constant force den-sity due to the optical tweezers. The boundary force density Fmembrane describes the resistance to

14

stretching and bending of the membrane and will be discussed in detail below. Relation ∇·v = 0in Ωcell(t)× [0, T ) ensures incompressibility of the volume (cytosol) and |∂Ωcell(t)| = |∂Ωcell(0)|ensures incompressibility of the membrane.

So far we have not specified the force densities Fmembrane due to surface mechanics. Using avariational approach we can relate the energies derived in Section 3 with corresponding forces,i.e. forces are given by the steepest decent of the L2-gradient of the free energy. Since thisapproach corresponds to the derivation of the Euler-Lagrange equations, stationary states ofModel 2 are minimisers of the continuum version of Model 1, as derived in Section 3.

In static microscopic (Section 2) and macroscopic (Section 3) descriptions of RBCs in termsof energy functionals, the energies corresponding to bending and in-plane mechanics are notcoupled. Therefore, also the corresponding forces are not coupled and we can split up Fmembrane

into two terms:

Fmembrane = T +N ,

where N is the force due to resistance to bending, and T the force due to in-plane stresses.Both membrane forces N and T are determined by the steepest decent of the L2- gradient ofthe corresponding membrane energies.

4.1 Resistance to in-plane deformations: T

The energy corresponding to the resistance to in-plane deformations is given by (15) with theinitial shape ΓX of the membrane. Let us consider small variations Xε = X + εφ, where φis an arbitrary test function which is infinitely often differentiable, i.e. φ ∈ C∞(Γ; R3). UsingJacobi’s formula, i.e. ∂

∂εJΓ = JΓtr((FΓ)−1 · ∂∂εF

Γ) and ddεF

Γ|ε=0 = ∇ΓX φ =

(∇Γφ

)· FΓ, we find

d

dεFin-plane

∣∣∣∣ε=0

=∫

ΓX

[12

∑i=1...6

V ′WLC(|ξ1|2)ξi · ((∇Γφ) + (∇Γφ)T ) · ξi

− 2CqJΓq−1

(JΓq + η)2JΓtr(∇Γφ)

]dµX,

where ξi = FΓ · ξi is the vector of the ith link after the deformation FΓ. Using∫Γ T · φdµ =

⟨F ′in-plane,φ

⟩= d

dεFin-plane|ε=0 with force density T yields after integration byparts:

T = ∇Γ · τ

with the so-called surface stress tensor

τ =1

2JΓ

∑i=1...6

1Li∂VWLC(Li)

∂Liξi ⊗ ξi − 2q

C

JΓq+1 1. (18)

Boundary terms do not need to be considered since ΓX , and accordingly Γ, is a closed surface.The stress tensor τ defined above generally reflects the hexagonal symmetry of the underlying

network. Considering the case p = 2 or working with averaged energies F =∫ 2π

0 Fdφ withrespect to the symmetry axis (c.f. Section 3.4) the considered energies are isotropic and thus canbe characterised by the invariants of the surface deformation tensor FΓ alone. The variationalapproach yields

τ =2

1 + I1

∂Ψ(I1, I2)∂I2

F Γ · F ΓT +∂Ψ(I1, I2)

∂I1P.

A direct relation of macroscopic moduli with microscopic moduli is given further below inSection 4.3.

15

4.2 Resistance to bending: N

Normal forces N are uniquely due to the resistance of the membrane to bending. Let thebending energy of the membrane be given by the Canham-Helfrich energy. In the following, weconsider small variations Xε = X + εnφ, where n is the outer unit normal of the surface Γ andφ ∈ C∞(Γ; R) is an arbitrary test function. It is sufficient to consider only variations εnφ, sincea variation in the normal direction completely describes the evolution of the interface.

Following [47] (Attention, Willmore uses a different definition of mean curvature, i.e. HWillmore =−H/2), the variation of the mean curvature is given by

d

dεHε

∣∣∣∣ε=0

= −∆Γφ− φ|∇Γn|2

= −∆Γφ− φ(H2 − 2K)

and the variation of the integration measure is given by

d

dεdµε

∣∣∣∣ε=0

= φHdµ.

Here, ∆Γ is the Laplace-Beltrami operator (surface Laplace). Hence, we find

d

dε

κ

2

∫Γε

(Hε −H0)2dµε

∣∣∣∣ε=0

= −κ2

∫Γ[2(H −H0)(∆Γφ+ φ(H2 − 2K))− (H −H0)2φH]dµ.

Under the assumption that N is determined by the steepest decent of FCanham-Helfrich, definedin (7), i.e.

∫ΓN · (nφ)dµ = −

⟨F ′bending, φ

⟩= − d

dεκ2

∫Γε

(Hε −H0)2dµε|ε=0, we find

N = κ(∆ΓH + (H −H0)(H2 − 2K)− 12

(H −H0)2H)n,

where we have used the product rule and Green’s theorem for surfaces. Setting the materialspeed v proportional to N and considering the case H0 = 0, the so-called Willmore flow isobtained [47].

4.3 Linear elasticity

The mechanical properties of the membrane within the continuum mechanical Model 2 are de-termined by the underlying continuum energies, which are directly related to the properties thediscrete network. Thus we can relate the modulus of rigidity µ and modulus hydrostatic com-pression K, used in linear elasticity, directly with the properties of the discrete atomistic model.Considering small deformations and restricting ourselves to linear elasticity, i.e. performing aTaylor extension considering only linear terms, we recover from (18)

µ = 3V ′WLC(1) +32V ′′WLC(1),

K =3(1 + η + q − qη)

1 + ηV ′WLC(1) + 3V ′′WLC(1),

16

Table 1: Parameters proposed of the worm-like chain model VWLC [10]

L = 75.00 nm Lmax = 237.75 nmp = 7.50 nm T = 300.00 K

respectively

µ =√

3kBT (−16L3 + 51L2Lmax − 57LL2max + 24L3

max)16(pLmax(−L+ Lmax)3

,

K =3(1 + η + q − qη)

1 + η

√3kBT (4L2 − 9LLmax + 6L2

max

12pLmax(−L+ Lmax)2

+√

3kBTL(−L+ 3Lmax)8p(−L+ Lmax)3

.

Postulated parameters of Li et al. [10] for the worm-like chain model VWLC are summarisedin Table 1. Using further kB = 1.38 · 10−23 J

K , choosing q = 1 and η = 0.005, we find

µ = 8.3 · 10−6N

m,

K = 16.6 · 10−6N

m,

which agrees with experimental results [33, 48].

5 Simulations

In Model 2, the bulk mechanics are given by the incompressible Navier-Stokes equation. Theseare second order equations which are quite well studied from an analytical as well as numeri-cal point of view. The tangential component of the surface mechanics is a typical example ofa hyperelastic material. It is also a second order equation. On the other hand, the normalmechanics of the lipid bilayer involve fourth order derivatives and the structure is nearly iden-tical to the so-called Willmore flow as discussed above. The coupling of the different models,especially the coupling of surface (two-dimensional models on a hypersurface) with bulk mechan-ics (three-dimensional models), makes analytical and computational investigations of Model 2quite challenging. Here, we restrict ourselves to an investigation of the fluid(3D)-structure(2D)-interaction from a numerical point of view.

Since we are only interested in stationary shapes, we simplify Model 2 further by consider-ing stationary Stokes flow instead of the full Navier-Stokes equations. However, an approachconsidering the full Navier-Stokes equations is as well feasible. We would like to work with adiscretisation via the standard Galerkin procedure, i.e. using the same type of ansatz func-tions for all variables. Thus we need to consider a stabilised Stokes approach guaranteeing theBabuska-Brezzi condition [49]:

0 = ∇ · σcytosol in Ωcell(t)× [0, T ),(19)

0 = ∇ · (v + εstab,1∇p) in Ωcell(t)× [0, T )

with an additional natural boundary condition for p, i.e.

n · ∇p = 0 on ∂Ωcell(t)× [0, T ),

17

where εstab,1 is the stabilisation parameter (typically εstab,1 ≈ δx2

η with spatial discretisationsize δx). Typically, the solutions depend only weakly on the exact value of εstab,1. The globalpressure qsurf. is realised via the Chorin-Uzawa scheme [49], i.e. through the introduction of anartificial compressibility:

εstab,2dqsurf

dt= |∂Ωcell(0)| − |∂Ωcell(t)| for [0, T ), (20)

where εstab,2 is again a small constant.

5.1 Weak formulation and discretisation

With respect to the finite element method appropriate weak formulations need to be derived.Apart from the bending forces N of the membrane mechanics the derivation is straight forward.A derivation of an appropriate weak formulation for the Willmore flow can be found in [50]. Slightmodifications of the approach yield a weak formulation of the membrane mechanics consideredhere [21].

The weak formulation of model (2) with stabilisations (19) and (20) is given by:∫Ω(t)

φdu

dtdµ =

∫Ω(t)

φvdµ,

0 =∫

Ω(t)σcytosol · ∇φdµ,

0 =∫

Ω(t)φ∇ · v − εstab,1∇p · ∇φdµ,

and ∫∂Ω(t)

φΓ(σcytosol + pout1) · ndµ = −∫∂Ω(t)

τ · (∇ΓφΓ)dµ

−κ∫∂Ω(t)

12|Y |2(∇Γx) · (∇ΓφΓ)− (∇ΓY ) · (∇ΓφΓ)

+2n⊗ n · (∇ΓY ) · (∇ΓφΓ)dµ

+∫∂Ω(t)

φΓFtweezersdµ,

εstab,2dqsurf

dt= |∂Ωcell(0)| − |∂Ωcell(t)|,∫

∂Ω(t)φΓY dµ =

∫∂Ω(t)

(∇Γx) · (∇ΓφΓ)− φH0ndµ,

where φ ∈ C∞0 (Ω(t); R) and φΓ ∈ C∞(∂Ω(t); R) are arbitrary test functions.The strong and weak formulations of the surface evolution given above are all based on the

Eulerian description. A Lagrangian description is, however, more appropriate with respect toan implementation using standard finite element packages. The weak formulation of Model 2given above has been transformed accordingly to a Lagrangian (fixed) coordinate system (formore details see [21]) and implemented using the finite element library Gascoigne [6].

The software package Gascoigne does not allow the differentiation between surface and bulkvariables using one implicitly time-stepping scheme for membrane and bulk processes. (Since

18

A B

Figure 4: Typical discretisations: (A) initial discretisation; (B) discretisation after grid refine-ment (2048 surface quadrilaterals)

membrane mechanics involve spatial derivatives of fourth order, explicit time stepping schemeswould imply very small time steps [51].) Hence, variables defined only on the boundary haveto be extended appropriately. Here, we have chosen an extension by Laplace’s equation. Thesurface speed vΓ and the bulk speed v are considered as one variable, which allows a directimplicit coupling. Since the weak formulation of Model 2 involves only first order derivatives,we use tri-linear finite elements in the bulk part and accordingly bi-linear finite elements on thesurface. The evolution in time is discretised with a Fractional-Theta scheme. Using Newton’smethod, the resulting non-linear systems are solved. Corresponding linearisations are solvedwith a GMRES preconditioner and a multigrid method. The scheme has been thoroughly testedand shows a reasonable order of convergence larger than 1.5, which is in good agreement withthe results for Willmore flows reported by [51].

Taking advantage of reflection symmetries with respect to the coordinate planes, we canreduce the computational effort by considering only one eighth of the object. Moreover, thefinite element toolkit Gascoigne offers the possibility of local mesh refinement, which allowsa further reduction of computational effort. Since the dynamics of Model 2 are driven bymembrane mechanics, we have chosen an adaptive discretisation refined in the vicinity of thesurface (see Fig. 4).

5.2 Initial shapes of Red Blood Cells

A quantitative characterisation of the rest shape of RBCs has been given by [7], c.f. formula(1) and could be used as an initial condition. On the other hand it has been postulated by [10]that the rest shape is an energy minimiser of the Canham-Helfrich energy FCanham-Helfrich.

Independed of the chosen initial shape we follow [10] and assume that the attachment of themicrobead to the RBC does not alter the initial shape of the RBC. This is of course a quite crudeassumption, but it should not influence the results quantitatively on the order of experimentalaccuracy. Let the contact area of the microbead with respect to the reference configuration begiven by

∂Ωtweezers,0 = x = (x, y, z) ∈ ∂Ωcell,0 : x2 + z2 ≤ R2tweezers,

with Rtweezers = 1.3µm. This corresponds to an area of contact of approximately 6.7µm2 perbead, i.e. the force due to the two optical tweezers is applied to 10% of the surface area asproposed by [10].

19

Table 2: Experimentally determined parameters

Bending elasticity κ 2 · 10−19J see [37], [33]Cytosol viscosity η 6.0cp = 6.0 · 10−3Pa s see [20]Spont. membrane curvature H0 0− 0.35 · 106m−1 see text

B C

A

Figure 5: Relaxation of the initial shape of a red blood cell given by [7], i.e. formula (1),with R = 3.91µm. Here, we have used an implicit Euler time stepping scheme with time stepsize δt = 0.0003s and parameters given in table 2. Figure (A) shows the initial shape given byformula (1). Figure (B-C) show the relaxed shape after 3s. In (B) the deformation in the radialdirection and in (C) the deformation in the z-direction is shown.

5.3 Simulation results

Here we use the same parameters of the microscopic model [10] (summarised in Table 1) to whichwe would like to compare our continuum approach. Other parameters of Model 2 according to theliterature are summarised in Table 2. The spontaneous membrane curvature can be “guessed”only roughly: According to Evans [7] the volume of the RBC equals V = 1.57R3

0 = 94.10µm3,which corresponds to a ball of radius R = 2.82µm. We therefore expect the spontaneousmembrane curvature in the range from H0 = 0 to H0 = 1

2.82 · 106m−1.

5.3.1 Relaxation experiments

Li and co-workers [10] have proposed that rest shapes of RBCs minimise the Canham-Helfrichenergy (7), since the membrane-bound cytoskeleton is constantly rearranging and thus allowsrelaxation of any stresses on long time scales. The minimal energy configuration of the Canham-Helfrich energy depends solely on the volume/area ratio and the spontaneous curvature H0 [28].The rest shape given by formula (1) is based on an experimental characterisation of RBCs by[7]. Hence, it might not necessarily be a minimal energy configuration with respect to theCanham-Helfrich energy.

Let us neglect the mechanics of the membrane-associated cytoskeleton for the moment. Sim-ulations with an initial shape given by formula (1) with R = 3.91µm are shown in Fig. 5. Theshape has been allowed to relax over 3s (104 time steps), after which the velocity is virtually zero.Since the shape has relaxed only slightly, we can conclude that the minimal energy configurationof the Canham-Helfrich energy with H0 = 0.1 · 10−6m−1 is indeed close to the experimentally

20

t = 1.2 s

t = 0.0 s t = 0.3 s

t = 0.9 st = 0.6 s

t = 10.0 s

Figure 6: Evolution of a red blood cell (Model 2). Parameters of the model are summarised inTable 1 and 2. The results have been obtained using an implicit Euler scheme with time stepsize δt = 1.0 · 10−3s.

observed rest shapes. A variation of H0 within the range 0− 0.2 · 10−6m−1 does not change theresults qualitatively.

5.3.2 Force experiments

Typical experiments consider microbeads subject to a constant force. The objective of such forceexperiments is to measure the longitudinal and transversal radii as a function of the appliedstretching force [14, 10]. Here, we consider the same experiments in silico using Model 2 withthe initial shape determined above (Section 5.3.1). The evolution of a RBC subject to a constantforce applied via optical tweezers is shown in Fig. 6. After some time the RBC converges to afixed shape. The elastic energy stored in the membrane is relaxed and has reached an energyminimum. The relaxed shape (energy minimum) agrees qualitatively with the discrete model of[10] as well as with the shapes observed by [14].

The relation between applied forces and axial as well as transversal diameters obtained fromseveral simulations (Fig. 7) agrees within experimental accuracies quantitatively well with ex-perimental results reported in the literature [14, 10]. Especially, considering experiments upto forces of 200 pN Model 2 based on a Taylor expansion of the potential VWLC up to secondorder, i.e. p = 2, approximates the proposed microscopic model [10] extremely well (with signif-

21

StretchingFore (pN)

Dia

met

er (

µm)

0 50 100 150 2000

5

10

15

20Microstructure ModelExperimentsContinuum Model with p=2Continuum Model with p=3Continuum Model with p=4

Axial Diameter

Transverse Diameter

Figure 7: Diameters of stretched red blood cell for different prescribed forces, which have beendetermined using finite element simulations of Model 2. Parameters of the model are summarisedin Table 1 and 2. Here we have used an implicit Euler scheme with time step size δt = 0.06s.For p > 2, the Model 2 is based on the averaging scheme outlined in Appendix A, i.e. themembrane is isotropic. The results are compared with experimental results and simulations ofthe microscopic model according to [10] ( c©2005 Biophysical Society).

icantly less computational effort). Only for forces larger than 200 pN the full non-linear wouldbe necessary.

6 Discussion

In this article we have shown how concepts from Γ-convergence, introduced for the derivation ofcontinuum macroscopic models from atomistic interactions in crystal lattices [4], can be gener-alised to biomechanical problems. As a test case we have considered RBCs. From a well studieddiscrete microscopic model in terms of energy functionals we have first derived a correspond-ing continuum energy functional for the membrane bound cytoskeleton. Corresponding energyminimising shapes (observed shapes) have been determined using a relaxation approach. At thesame time an easily extendable continuum mechanical model for the RBC has been derived.

Our multiscale model considers the membrane as a two dimensional hypersurface, whosemechanical properties are determined by the lipid bilayer and membrane bound spectrin network.The latter explicitly includes molecular details inherited by multiscale analysis. Only very fewmodels consider the mechanics of RBCs in a similar realistic setting. Often the membrane isassumed to be a three dimensional solid with small but finite thickness [19, 15] or the bending-resistance of the lipid bilayer is given by quite heuristic models [20]. The multiscale nature of theconsidered continuum model allows to benefit from advantages of continuum models like efficientnumerical schemes based on adaptive discretisation techniques (the discretisation can be chosenindependent of the microscopic geometry) without neglecting important microscopic details. Allmicroscopic parameters can be directly related to macroscopic parameters in a quantitative way.

Our approach required the assumption of a hexagonal symmetry for the sake of a homogeni-sation formula. This implies that in the general case the hexagonal symmetry is reflected in thecontinuum mechanical equations, which might not be realistic from a biological point of view.The spectrin network shows only roughly hexagonal symmetry (over 80% degree-6 vertexes).Further, it is not clear in which direction the symmetry axis would be pointing as well how thehexagonal symmetry could be investigated form an experimental point of view. Therefore, we

22

consider averaged energies and stress tensors obliterating the hexagonal symmetry. In the futureless heurisic averaging approaches, e.g. including stochasticity, should be considered.

Further, we had to introduce a small regularisation parameter η to guarantee boundness ofsteric interaction energies, as well consider Taylor-expansions of pair-interaction energies, whichotherwise could “explode” at finite interaction lengths. These regularisations are necessary tofollow the rigorous approach of Alicandro and Cicalese [4] proving convergence of the microscopicmodel to the macroscopic model. Disregarding these regularisation the approach is only a formalone. It yields a macroscopic model based on a microscopic model, but it cannot be guaranteedthat minimisers of the microscopic model indeed converge to minimisers of the macroscopicmodel, which is non-trivial [52]. Our approach should be generalised in the future, such thatthese regularisations could be lifted.

The derived model is challenged by a quantitative comparison with a typical biophysicalexperiment: an optical tweezers experiment. To perform an optical tweezers experiment insilico, we have developed an appropriate finite element scheme, which can handle mechanics on ahypersurface and bulk mechanics including their coupling within one scheme. Simulation resultsshow a good qualitative and quantitative agreement with experiments [14] and simulations ofthe microscopic model [10] (see Fig. 7). The developed numerical approach based on the toolboxGascoigne is much more flexible than many schemes available, e.g. boundary element methods[20]. It can be easily extended to more complicated models, e.g. models considering interactionsbetween mechanics and biochemistry in the cytosol and membrane at the same time, making ita powerful tool for the theoretical investigation of the mechanobiology of single cells.

Since we have found a good qualitative and quantitative agreement between microscopic simu-lations and the derived macroscopic continuum model, our approach could be used for parameterestimation in the near future. Parameter estimation together with the multiscale approach, al-lowing a significant speedup of simulations, would enable us to estimate microscopic mechanicalparameters through upscaling. Thus the microscopic model of spectrin relying on the worm-likechain model originally introduced for DNA could be improved. Hence the experimental re-sults could be better approximated by the microscopic model and therefore also by the derivedmacroscopic continuum model, which is approximating the microscopic one.

The extension of mathematical multiscale techniques and the development of a highly flexiblecomputational framework for single cell mechanics offers the possibility to tackle many unsolvedquestions in the field of mechanobiology in the near future. We believe that theses techniqueswill have a significant impact on modelling in mechanobiology, as underlined by the test casesof RBC mechanics.

A Interaction lengths and invariants

The tensor (FΓ)T ·FΓ considered in Section 3.4 is symmetric and positive definite, hence it canbe diagonalised:

(FΓ)T · FΓ = ΦT ·Λ ·Φ

with

Λ = diag(1 + I2+√I22+4I2−4I21−8I1

2 , 1 + I2−√I22+4I2−4I21−8I1

2 , 0)

and an appropriate transformation Φ, i.e. a rotation with an angle φ. I1 and I2 are the invariantsgiven in (17). Hence, we obtain

ξTi · F T · F · ξi − 1 = (Φ · ξi)T · (Λ− 1) · (Φ · ξi),

23

and thus, c.f. Taylor expansion (16):∑i=1...6

(ξTi · F T · F · ξi − 1

)= 3I2,

∑i=1...6

(ξTi · F T · F · ξi − 1

)2=

34

(4I2 − 8I1 + 3I22 − 4I2

1 ),

∑i=1...6

(ξTi · F T · F · ξi − 1

)3=

38I2(12I2 + 5I2

2 − 12I21 − 24I1)

+316

(I22 + 4I2 − 4I2

1 − 8I1)3/2 cos(6φ),

i.e. the hexagonal structure of the network reflected in the terms (ξTi · F T · F · ξi − 1)j withj <= 2, since these terms depend only on the invariants I1 and I2.

Since the direction of a symmetry axis is not distinguished, we consider the following averagedand thus isotropic terms for p > 2:∫ 2π

0

∑i=1...6

(ξTi · F T · F · ξi − 1

)3dφ =

38I2(12I2 + 5I2

2 − 12I21 − 24I1)∫ 2π

0

∑i=1...6

(ξTi · F T · F · ξi − 1

)4dφ =

364

(192I31 + 48I4

1 − 240I22I1 − 120I2

2I21 + 35I4

2

+192I21 + 48I2

2 − 192I2I1 − 96I2I21 + 120I3

2 ).

Acknowledgements: Dirk Hartmann was supported by the German Science Foundationthrough the International Graduate College 710: “Complex processes: Modeling, Simulation andOptimization”. This work was part of a Doctoral Thesis [21] supervised by Prof. Willi Jager andProf. Marek Niezgodka. The author would like to thank them both for their support. Further,the author thanks Prof. Andrey Piatnitski and Dr. Mariya Pytashnik for very stimulatingdiscussions, as well as Dr. Dominik Meidner and Dr. Thomas Richter for their outstandingGascoigne support.

Visualisations in this article are based on the visualisation toolkits VisuSimple(http://www.visusimple.uni-hd.de) and ParaView (http://www.paraview.org).

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