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* Department of Bioengineering, Imperial College London, London, United Kingdom; and School of Physics, University of Exeter, Exeter, United Kingdom
Correspondence: Address reprint requests to Dr. Gianluca Marcelli, Dept. of Bioengineering, Chemical Engineering Bldg., Imperial College London, Prince Consort Rd., London SW7 2AZ, UK. E-mail: g.marcelli{at}imperial.ac.uk.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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The RBC at rest assumes a biconcave discoid shape with a diameter of 
8 µm and it is capable of passing through capillaries with less than half the diameter. To allow the RBC to deform inside a capillary, the resistance to deformation of the RBC membrane must not be too large. The resistance, on the other hand, must not be too small, otherwise the cell's integrity would not be preserved during normal flow in the circulatory system. Bending deformability of the RBC membrane is also shown by its mechanical out-of-plane and in-plane fluctuations with amplitudes in the range of 10–400 nm (1
–3).
A long-standing problem in the study of RBC structure is that the observation of thermal fluctuations of nanometer-size seems to be consistent only with a vanishingly small shear modulus (2), whereas static deformation experiments such as micropipette aspiration (4) report a large shear modulus. Strey et al. (2) studied the flickering of RBC to derive the mean-square amplitudes of the first three azimuthal eigenmodes and they obtained a shear modulus, µ, 100-times smaller than the static value measured by micropipette aspiration, µ = 6 x 10–6 N/m (4), with a fluorescent multiparticle tracking system µ = 1–10 x 10–6 N/m (3) or by optical tweezers, µ = 2.5 x 10–6 N/m (5) and µ = 11.1–17.7 x 10–6 N/m (6). They concluded that the shear resistance for small deformations vanishes and therefore it does not contribute to thermal shape fluctuations.
We have chosen to analyze the problem considering both shear and bending moduli in the attempt to reconcile a finite value of the shear modulus with the presence of the nanometer-scale fluctuations observed experimentally. The essential assumptions of the model are that the membrane preserves a constant global area under thermal fluctuations and its mechanical behavior can be described by two material parameters, an out-of-plane bending modulus, B, which describes the resistance to bending, and an in-plane shear modulus, µ, which describes the resistance to shape changes of the membrane surface. It is usually assumed that the bending modulus, B, is dominated by the microscopic properties of the lipid bilayer. Equivalently the shear modulus, µ, is thought to be dominated by the microscopic properties of the cytoskeleton, although protein-protein interactions or domains of lipids in solid-state form can also contribute. Throughout the article, we shall use "membrane" to refer to the composite structure formed by the lipid bilayer, its integral proteins, and associated membrane skeleton.
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MODEL DETAILS |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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). In our model we keep the global surface area constant, so we can assume that KA 
. The two-dimensional network is embedded in a three-dimensional closed surface where it is allowed to fluctuate in all directions (see Fig. 1 a). Equations of motion of the virtual particles are solved in a molecular-dynamics fashion (8,9) to derive the time-evolution of the system and calculate the time-dependent properties.
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 | (1) |
100 nm). This is the minimum length for which it is reasonable to expect the particles to be homogeneous. If a smaller length-scale is chosen (for example, that of membrane thickness) it would be necessary to consider heterogeneous particles to account for inhomogeneity at the molecular scale.
A dihedral angle potential is applied between adjacent triangles. This potential is used in molecular dynamics simulations to describe interactions arising from torsional forces in molecules (8,9). The expression of the dihedral angle potential we used is (10)
 | (2) |
ijml is the dihedral angle between the triangles 
ijm and jml (see Fig. 1 b). This potential has a minimum when two adjacent triangles are co-planar. Intuitively it can be seen as a mechanism for the particles to resist out-of-plane bending. The dihedral angle potential is commonly used as a procedure for curvature discretization in a triangulation (11) and it can be shown that the continuum limit of Eq. 2 is the Kantor and Nelson's bending energy for solid membranes (12). Aronovitz and Lubensky (13) stressed that this bending energy is in general different from the bending energy which Helfrich used to describe a fluid membrane (14). However, for the small displacements that occur in our system, the bending energies for solid and fluid membranes are identical. Furthermore, Gompper and Kroll (15) reported a more general expression for the bending energy that includes the effect of changes in area of triangular elements. They showed that when equilateral triangles are used, the area dependency cancels out. Since we use equilateral triangles as the equilibrium configuration, the effect of dynamic area changes will be negligible on average because of our constant area constraint.
We stress here that the bending and shear moduli are not independent (16), so in general it is not possible to change k (or D) and independently change the value of µ (or B). Details are given below about the derivation of µ and B that we adopted in our work.
In general the free energy of a system is given by two contributions, F = E–TS. In the case of a spring-network, for example, the energetic contribution E is given by the sum of 
over all the particles. The entropic contribution TS depends in general on the number of configurations available to the system at a particular state. Both contributions influence the system elasticity. A clear sign that the elasticity is entropy-dominated is that the shear modulus increases when the temperature increases. It is not clear yet which of the two contributions is the dominant one in determining the RBC membrane elasticity (7). In the present model the elasticity of the system is dominated by the energetic contribution. This should not restrict our analysis since we study membrane fluctuations at a given temperature and with a given shear modulus. Furthermore, the constraint is readily overcome since removing bonds between neighboring particles within our model allows simulation of a system where the entropy can be the dominant contribution to the elasticity (17).
As in any standard molecular dynamics simulation, we define the temperature, T, of the system as
 | (3) |
are the velocity components of particle i; m is the mass of the particle; and 
...
represents the time-average. The temperature is kept constant to the desired value using a Nosé-Hoover thermostat (8,9). The value used for the temperature is T = 309 K (37 C).
Since the lipid bilayer is relatively incompressible, we require that the global surface area of the membrane, defined as the sum of the areas of all triangles by which the surface is tessellated, remains constant. To do this we apply Lagrange's method of undetermined multipliers. We chose the global area, Atot, to be the value given by triangles whose sides are all equal to the equilibrium length, r0:
 | (4) |
) pointed out that the induced flow of the cytoplasm is the most significant among these contributions. Tuvia et al. (19) studied the influence of extracellular fluid macroviscosity (EFM) on RBC membrane fluctuations. They stressed that thermally driven membrane fluctuations cannot be damped by an increase of EFM. They stated that the partial attenuation of the fluctuations they observed with elevation of EFM can be associated with metabolic forces, which are, to some extent, responsible for the fluctuations; however, they did not specify the nature of the metabolic forces.
The system used in the simulations consists of N particles (vertices) embedded in a closed surface. For the initial particle-configuration we followed Discher et al. (20), who used two parallel sheets in the form of a hexagon. Each particle on the perimeter of one sheet is connected to two particles on the perimeter of the other sheet. There are six "corners" on each hexagonal sheet, each of which has only fivefold coordination. The minimum number of fivefold defects required by topology for the triangulation of a spherical surface is 12 and so this configuration represents one possible triangulation of a spherical surface. All particles move in space but the connectivity is fixed, in that each particle has a fixed set of neighbors: 12 particles have five neighbors and all the others have six neighbors.
The molecular dynamics simulations were performed using the DL_POLY simulation package version 2.12 developed in Daresbury Laboratory, Cheshire, UK (10). The equations of motion of the particles were integrated using a Verlet leapfrog integration algorithm in conjunction with a Nosé-Hoover thermostat. We developed a subroutine that applies Lagrange's method of undetermined multipliers to keep the total surface area constant.
Because of computational time-demand considerations we used N = 5582 for number of particles. In most simulations we used an equilibrium length r0 = 100 nm. The area per particle, Aparticle, is given by
 | (5) |
5 x 107 nm2. This value is approximately half the surface area of a real red blood cell (108 nm2). We used two values of the spring constant, k1 = 5 x 10–6 N/m and k2 = 25 x 10–6 N/m, and two values of the dihedral-angle potential constant, D1 = 6 x 10–20 J and D2 = 60 x 10–20 J. We chose these values because the derived shear and bending moduli are in the range of the experimental measurements. We observed that using higher values of the k and D gives an increasingly rigid behavior of the model-network. On the other hand, lower values of k and D generate crumpled configurations, which were not worth studying with the present model that lacks a repulsive interaction between particles and a constraint on the cell volume. Therefore, we performed four main simulations combining the two chosen values of k and D. As we report in the following section, during the simulations we were able to keep track of the thermal fluctuations and to estimate the shear and bending moduli of the membrane.
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DEVELOPMENT OF THE MODEL |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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107 time steps), the cell model attains an oblate shape for any of the values of k and D used (see Fig. 2). The oblate shape persists up to the maximum simulation time we reached (4 x 107 time steps). This is also evident in the behavior of the correlation function (reported in Fig. 3). The correlation function is defined as 
where 
is the z-coordinate unit vector, ...t is the time-average, and 
is the instantaneous average of the unit vectors perpendicular to the triangles that fall within r and r+r from the cap-center. In any flat region of the membrane, f(r) has a value close to unity.
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The center of the cell cap is almost flat and parallel to the x-y plane. We exploited this feature and decided to monitor a small portion of the membrane in this region to study the thermal fluctuations. We consider 19 particles located at the center of the cell cap (the lighter patch in Fig. 4), which correspond to an area of 1.6 x 105 nm2. We define the average of the z-displacement of the 19 particles as the out-of-plane instantaneous fluctuations of the patch. We also keep track of the positions of single particles, which correspond to fluctuations of the smallest area accessible to our model (9740 nm2). More specifically, we monitor the particle at the center of the cap and four other particles at orthogonal positions around the cell (see dashed arrows in Fig. 4) and monitor the displacements of the particles in the membrane plane (in-plane fluctuations). Our resolution is comparable with the resolution used in flicker measurements. Tuvia et al. (19), for example, measured membrane fluctuations over a cell patch with an area of 2.5 x 105 nm2 using point-dark-field microscopy.
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),
 | (7) |
represents the maximum wavelength of the fluctuations, and r0 represents the minimum wavelength. To test the validity of Eq. 7 we performed preliminary simulations of a triangular network in two-dimensional space with periodic boundary conditions. We used Eq. 7 to calculate the shear modulus and compared the results with an analytical expression reported by Discher et al. (21) for the shear modulus of a two-dimensional regular triangulation at zero temperature,
 | (8) |
is the density of the patch relative to the stress-free state. We performed the two-dimensional simulations at two different temperatures, T = 309 K (reduced temperature, 
), and T = 10 K (T* 
5 x 10–4), respectively. In both cases Eq. 7 agrees within the variance with the analytical values predicted by Eq. 8. This also suggests that the temperature does not significantly affect the shear modulus for the range of temperatures we used.
To derive the bending modulus we used the formulae reported by Helfrich and Servuss (14),
 | (9) |
 | (10) |
<< 1, where is the tilt angle of the membrane with respect to the x-y plane. In particular Eq. 9 is valid for 
In Eqs. 9 and 10, 
is the lateral tension, 
is the out-of-plane mean-squared displacement of the membrane cap, and Apatch is the difference between the true area of the 19-particle patch and the projected area of the patch on the x-y plane.
From each simulation we can easily calculate and Apatch/Apatch, so we can derive the relative values of B and using Eqs. 9 and 10. In the case in which the lateral tension is zero, Eqs. 9 and 10 simplify to
 | (11) |
 | (12) |
) reported that the lateral tension had no influence on their RBC membrane fluctuation measurements and used Eq. 11 to derive the bending modulus from their measurements obtaining a bending modulus B = 20–70 x 10–20 J. However, this value is larger than most of those reported by other authors: B = 21 x 10–20 J (22); B = 18 x 10–20 J (23); B = 1.3–3.0 x 10–20 J (18); and B = 2.3 x 10–20 J (24). |
RESULTS AND DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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and 
in the x- and y-directions, respectively. 
and 
along the smaller and bigger curvatures, respectively.
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20–60 nm) as observed in experimental measurements (1,2). This is significant because these fluctuations are observed in conjunction with values of the shear and bending moduli that are of the same order of magnitude of those measured experimentally. Case LS in Table 1 is particularly interesting: 
is 55 nm; the calculated bending modulus, B = 3.7 x 10–20 J, agrees well with experimental measurements, 1.3–3 x 10–20 J (18), and 2.3 x 10–20 J (24), as does the calculated shear modulus, µ = 6.7 x 10–6 N/m, with the relative experimental values, 6 x 10–6 N/m (4), 1–10 x 10–6 N/m (3), and 2.5 x 10–6 N/m (5). Moreover, and 
calculated with our simulation (14 and 15 nm), are quite close to the experimental values, 17–19 nm (3). Fig. 5 shows our results for µ and B together with an indication of the experimental measurements obtained with different techniques.
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and coincide within the variance. This means that the flat portion of the membranes behaves like an isotropic two-dimensional network. Furthermore, in cases LL, SS, and SL, and coincide within the variance, showing an isotropic behavior. In case LS, on the other hand, where the differences between the two principal curvatures are more pronounced (see Fig. 2), and do not coincide. The fluctuations in that region are not isotropic. Lee and Discher (3) studied the thermal fluctuation of the red blood cell by measuring the displacement of a fluorescent nanoparticle attached to the membrane. They found isotropic fluctuations when the particle was centrally attached in a spherical contour of the cell, but anisotropic fluctuations when the particle was located in a highly strained region where the principal curvatures were different. We observed the same phenomenon with the out-of-plane fluctuations. The different values of and 
reported in Table 1 are also due to the different curvature in the two regions. In Fig. 6 we show the histogram of the out-of-plane fluctuation amplitude for the case LS (k = 25 x 10–6 N/m, D = 6 x 10–20 J) and for the case LL (k = 25 x 10–6 N/m, D = 6 x 10–19 J). Thermal fluctuations follow a Gaussian distribution. We observe in both cases shown in Fig. 6 that the amplitude can be approximated by a Gaussian distribution and the amplitude for the case LL is narrower due to the higher value of the dihedral constant D. This is expected since a higher value of D gives a higher value of the bending modulus, which induces a lower RMS amplitude in the out-of-plane fluctuations. Equivalently, the RMS amplitude of the in-plane fluctuations is lowered by higher values of the shear modulus, which are obtained using a higher value of the spring constant k.
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The lateral tension reported in Table 1 ranges between 0.5 and 2.6 x 10–6 N/m. These values are three or four orders-of-magnitude smaller than the isotropic lysis tension, 3–12 x 10–3 N/m (25). We stress that the lateral tension proscribes the use of Eqs. 11 and 12, which are valid only for vanishingly small values of the lateral tension.
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CONCLUSIONS |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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Apatch/Apatch (or related quantities) when studying membrane fluctuations experimentally and of properly choosing the minimum wavelength appearing in Eqs. 9 and 10. |
APPENDIX |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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,26). These authors used renormalization group techniques to study the crumpling transition and the fluctuations around a flat phase of flexible D-dimensional crystalline membranes embedded in d-dimensional space. They reported that for fluctuations of wavelength, 
, greater than the nonlinear length, 
nl, the shear modulus is renormalized and behaves as µ() = (1/)
, the nonlinear length being defined as 
for D = 2, where 
and 
are the bare elastic moduli divided by the temperature. Using T = 309 K and the values of D and µT=0 (Eq. 8) to derive nl, the following values are obtained: 
and 
For the shear modulus to be renormalized and then to undergo softening, the critical exponent must be positive and the fluctuation wavelengths of the system must be larger than nl. In cases LL and SL the nonlinear length is comparable with the size of the system, and for this reason the renormalization of the shear modulus is less significant. In cases LS and SS, instead, nl is much smaller than the size of the system and the conditions for the softening of the shear modulus are met. To corroborate further the conclusion that the softening of the shear modulus can be ascribed to renormalization, we performed simulations on a hexagonal sheet (open surface) with the same characteristics as the LS case. We used hexagonal sheets with three different diameters, namely, 40r0, 60r0, and 100r0. We obtained the following values for the shear modulus: µ(40r0) = 9.9(0.1), µ(60r0) = 9.0(0.3), and µ(100r0) = 8.2(0.3) 10–6 N/m, which show that the shear modulus diminishes as the size of the system increases. These are preliminary results and more systematic work should be done, but they seem to show an interesting behavior. Considering that our system is realistic in terms of size and range of elastic moduli, we can infer that the renormalization group may play an important role in the understanding of red-blood cell mechanical properties.
Submitted on November 11, 2004; accepted for publication June 20, 2005.
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REFERENCES |
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TOPABSTRACTINTRODUCTIONMODEL DETAILSDEVELOPMENT OF THE MODELRESULTS AND DISCUSSIONCONCLUSIONSAPPENDIXREFERENCES |
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2. Strey, H., M. Peterson, and E. Sackmann. 1995. Measurement of erythrocyte membrane elasticity by flicker eigenmode decomposition. Biophys. J. 69:478–488.
3. Lee, J. C. M., and D. E. Discher. 2001. Deformation-enhanced fluctuations in the red cell skeleton with theoretical relations to elasticity, connectivity, and spectrin unfolding. Biophys. J. 81:3178–3192.
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9. Sadus, R. J. 1999. Molecular Simulation of Fluids: Theory, Algorithms and Object-Orientation. Elsevier Press, Amsterdam, The Netherlands.
10. Smith, W., and T. R. Forester. 1999. The DL_POLY_2 Molecular Simulation Package. http://www.cse.clrc.ac.uk/msi/software/DL_POLY.
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15. Gompper, G., and D. M. Kroll. 1996. Random surface discretizations and the renormalization of the bending rigidity. J. Phys. (France). 6:1305–1320.[CrossRef]
16. Nelson, D. R., and L. Peliti. 1987. Fluctuations in membranes with crystalline and hexatic order. J. Phys. E. 48:1085–1092.
17. Plischke, M., D. C. Vernon, B. Joós, and Z. Zhou. 1999. Entropic rigidity of randomly diluted two- and three-dimensional networks. Phys. Rev. E. 60:3129–3135.[CrossRef]
18. Brochard, F., and J. F. Lennon. 1975. Frequency spectrum of the flicker phenomenon in erythrocytes. J. Phys. E. 36:1035–1047.
19. Tuvia, S., A. Almagor, A. Bitler, S. Levin, R. Korenstein, and S. Yedgar. 1997. Cell membrane fluctuations are regulated by medium macroviscosity: evidence for a metabolic driving force. Proc. Natl. Acad. Sci. USA. 94:5045–5049.
20. Discher, D. E., D. H. Boal, and S. K. Boey. 1998. Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys. J. 75:1584–1597.
21. Discher, D. E., D. H. Boal, and S. K. Boey. 1997. Phase transitions and anisotropic responses of planar triangular nets under large deformation. Phys. Rev. E. 55:4762–4772.[CrossRef]
22. Scheffer, L., A. Bitler, E. Ben-Jacob, and R. Korenstein. 2001. Atomic force pulling: probing the local elasticity of the cell membrane. Eur. Biophys. J. 30:83–90.[CrossRef][Medline]
23. Evans, E. A. 1983. Bending elastic modulus of red blood cell membrane derived from buckling instability in micropipette aspiration tests. Biophys. J. 43:27–30.
24. Zilker, A., M. Ziegler, and E. Sackmann. 1992. Spectral analysis of erythrocyte flickering in the 0.3–4-µm–1 regime by microinterferometry combined with fast image processing. Phys. Rev. A. 46:7998–8001.[CrossRef][Medline]
25. Evans, E. A., R. Waugh, and L. Melnik. 1976. Elastic area compressibility modulus of red cell membrane. Biophys. J. 16:585–595.
26. Aronovits, J., L. Golubovic, and T. C. Lubensky. 1989. Fluctuations and lower critical dimensions of crystalline membranes. J. Phys. [E]. 50:609–631.
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versus r/r0. Case LS (
), k = 25 x 10–6 N/m, B = 6 x 10–20 J; Case LL (
), k = 25 x 10–6 N/m, B = 60 x 10–20 J; Case SS (
), k = 5 x 10–6 N/m, B = 6 x 10–20 J; and Case SL (
), k = 5 x 10–6 N/m, B = 60 x 10–20 J.
