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Red Blood Cell Membrane Fluctuations and Shape Controlled by ATP-Induced Cytoskeletal Defects

N. S. Gov ^* and S. A. Safran 

^* Department of Materials and Interfaces, and Department of Chemical Physics, The Weizmann Institute of Science, Rehovot, Israel

Correspondence: Address reprint requests to Dr. Nir S. Gov, Weizmann Institute of Science, Dept. of Chemical Physics, P.O.B. 26, Rehovot 76100, Israel. Tel.: 972-8-934-3323; E-mail: nirgov{at}wisemail.weizmann.ac.il.

	ABSTRACT

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
We show theoretically how adenosine 5'-triphosphate (ATP)-induced dynamic dissociations of spectrin filaments (from each other and from the membrane) in the cytoskeleton network of red blood cells (RBC) can explain in a unified manner both the measured fluctuation amplitude as well as the observed shape transformations as a function of intracellular ATP concentration. Static defects can be induced by external stresses such as those present when RBCs pass through small capillaries. We suggest that the partially freed actin at these defect sites may explain the activation of the CFTR membrane-bound protein and the subsequent release of ATP by RBCs subjected to deformations. Our theoretical predictions can be tested by experiments that measure the correlation between variations in the binding of actin to spectrin, the activity of CFTR, and the amount of ATP released.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
It is well established that both the amplitude of membrane fluctuations and overall red blood cell (RBC) shape are influenced by metabolic activity controlled by the intracellular concentration of adenosine 5'-triphosphate (ATP). RBC fluctuations were investigated in a series of experiments (Tuvia et al., 1998; Levin and Korenstein, 1991), indicating that this activity is related to a cytoskeleton component, since the fluctuations of RBC shells (cells where the lipid membrane is removed) behave similarly to intact cells. Various chemical and physical changes that interfere with the cytoskeleton-bilayer coupling were shown to reduce or even eliminate the ATP-induced membrane fluctuations (Tuvia et al., 1998; Levin and Korenstein, 1991). The effect of ATP on RBC membrane fluctuations can be treated theoretically, in terms of an enhanced effective temperature that then enters the usual equilibrium thermodynamic description. The physical origin of this effect and the reason behind the success of the effective temperature model is not yet known. The phenomenon of shape changes of RBC in response to changes in ATP levels is also well documented (Wong, 1999; Lim et al., 2002); here again, the reason for the dependence on ATP is not known (Hoffman, 2001, 2004; Nakao, 2002).

In this article, we show that cytoskeletal defects induced by ATP can account for both the enhanced membrane fluctuations as well as the overall cellular shape changes. By focusing on ATP-induced spectrin-actin dissociations (Sackmann, 1994), we predict the magnitude of the effective temperature and its dependence on ATP concentration, in agreement with published experiment data. These transient dissociations modify the cytoskeleton shear modulus and the effective surface tension of the membrane, which in turn changes the relative area difference between the two leaflets of the lipid bilayer. This can change the shape of the RBC from the normal discocyte to the abnormal echinocyte at low levels of ATP. We also suggest that deformation-induced defects can activate the membrane-bound cystic fibrosis transmembrane conductance regulator CFTR, leading to the release of ATP into the extracellular environment.

We now briefly review the structure of the RBC membrane: The surface of the RBC is a composite material containing an outer lipid bilayer, and an inner, two-dimensional cytoskeleton network (Bennett, 1989) attached in a sparse manner to the lipid bilayer (Gov et al., 2003; Gov and Safran, 2004). The links of the network consist of flexible spectrin molecules (R 80–100 nm), cross-linked at the network nodes by a complex containing a short (30 nm) actin filament, band-4.1, and other proteins (Bennett, 1989; Takakuwa, 2001).

The spectrin cytoskeleton of the RBC is slightly stretched and under tension, even in the un-deformed state of RBC (Zeman et al., 1990; Petrov et al., 1979; Fournier et al., 2004). This is because the spectrin filaments are attached to the bilayer and are therefore constrained from accessing all possible network configurations. This tension is balanced by the curvature energy of the bilayer, which consequentially develops small amplitude undulations with a wavelength of the order of the spectrin filament length (Zeman et al., 1990), and a typical height of 15 nm (Fig. 1). Equating the stretching force of the spectrin (modeled by a spring with spring constant µ and equilibrium length R₀) with the force of curving the membrane (related to its bending modulus, ), we find

(1)

where R 80–100 nm is the average equilibrium length of the spectrin filaments, R₀ 70 nm is an estimate of the radius of gyration the free spectrin filament in solution, and is the number of spectrin filaments per network unit cell (Fig. 1), which is equal to 3 for a defect-free, triangular network. Solving this equality, using R₀ 70 nm, J 10 k_BT (Zilker et al., 1987), and µ 6 x 10^–6 J/m² (Discher et al., 1994; Heinrich et al., 2001; Engelhardt et al., 1984), we find that the average spectrin molecule is stretched by 20%: (Fig. 1).

View larger version (20K): [in this window] [in a new window]   FIGURE 1  (a) Overview of the (almost) regular triangular network of the RBC cytoskeleton. Below we sketch a sideways view of the equilibrium height profile of the RBC membrane: the stretched spectrin filaments (bold jagged lines) are balanced by static height fluctuations of the bilayer. (b) A sketch of an ATP-induced dissociation of a spectrin filament, resulting in two buckled fivefold nodes.

Larger-scale, topological defects, whereby the connectivity of the hexagonal network is modified, are more difficult to create. In these cases the hexagonal symmetry is broken by random defects, with nodes that have either five or seven attached spectrins (Fig. 1). These defects are likely to be induced by a combination of thermal and chemical (ATP) energy. The creation and motion of these defects allows the network to constantly rearrange, in response to external deformations.

We now summarize the main results: We begin by analyzing, in detail, the effects that changes to the cytoskeleton induce on the observed height fluctuations of the membrane in Analysis of the Static Fluctuation Spectrum. We show that as a function of the ATP concentration, a renormalization of both the elastic properties and the effective temperature gives a consistent description of the experimental data. In ATP-Driven Membrane Fluctuations we introduce our model of ATP-induced, transient cytoskeleton dissociations, and show how this mechanism can account for the ATP contribution to the observed height fluctuations of the RBC membrane. The effect of the ATP concentration on the overall elasticity of the cytoskeleton is described in ATP-Driven Steady-State RBC Shape. In particular, we are able to quantitatively relate the ATP concentration to the discocyte-echinocyte shape transition. Topological defects in the cytoskeleton and their role in RBC deformation and ATP-release are discussed in Deformed RBC: Topological Defects, and CFTR Activation and ATP Release, respectively.

	ANALYSIS OF THE STATIC FLUCTUATION SPECTRUM

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
We analyze in this section the experimental measurements of the static fluctuations in RBC, that is, the height fluctuations as a function of the wavevector q (Zilker et al., 1987). We have previously analyzed this data for normal RBC (Gov et al., 2003) in terms of equilibrium thermodynamics, which we shall extend here to include ATP-driven fluctuations as well as for the stomatocyte and echinocyte cells. This analysis is complicated by the mixing of the thermal and nonthermal contributions to the fluctuation amplitude. In the absence of ATP, such as in depleted cells or washed ghosts (Tuvia et al., 1998; Levin and Korenstein, 1991), the root-mean square (RMS) amplitude of the fluctuations is This is therefore a measure of the amplitude of the thermal fluctuations alone. When ATP is added, the fluctuation amplitude rises and saturates at physiological concentrations, giving In our previous phenomenological model (Gov et al., 2003), we have shown that a consistent description of both the static and dynamic data is possible if we treat the ATP effect as introducing a higher effective temperature T_eff/T 3.

We have previously proposed a phenomenological model to describe the fluctuation spectra of the RBC (Gov et al., 2003). This model describes a soft cytoskeleton shell, which is sparsely attached to the bilayer. The average bilayer-shell distance is related to the observed amplitude of thermal fluctuations, and depends on the undulations of both the bilayer and spectrin filaments (Fig. 1). The spectrin filaments form a connected quasi-two-dimensional shell that has a three-dimensional thickness of d 30 nm, due to the conformations of the soft filaments out of the plane of the shell. This can be modeled by a restoring force on the bilayer, which we account for via an harmonic potential. The force acts at all bilayer-spectrin distances due to the relatively delocalized and soft nature of the spectrin network. Additionally, the sparse attachment points of the membrane to this potential induce an effective surface tension in the bilayer (Gov and Safran, 2004). The resulting free energy is therefore given by

(2)

(Gov et al., 2003), where is the effective surface tension and describes the confining harmonic potential. The spectrin filaments behave as entropic springs, where they can have random conformations in the half-space beneath the lipid bilayer, since they are not strongly adsorbed to the bilayer, except at their ends (Everaers et al., 1996). This would predict a fluctuation amplitude given by

(3)

where is the observed surface area of the cell and the bare bending modulus is 2 x 10^–20 J. We can use this to define a renormalized effective bending modulus, _q (see Fig. 3),

(4)

View larger version (19K): [in this window] [in a new window]   FIGURE 3  Calculated normalized effective bending modulus q (Eq. 4) compared with the experimental data (Zilker et al., 1987): normal cells, circles (solid line), stomatocytes, squares (dashed-dot line), and echinocytes, triangles (dashed line). The vertical dotted line shows the crossover wavevector q0. Inset shows the same data in the limit of small q-values, to determine the ratio /.

The elastic properties of the combined bilayer-cytoskeleton membrane are bounded by the two limits:

1. The cytoskeleton does not form a shell unless it is adsorbed onto the lipid bilayer. In this case the spectrin network forms an array of polymers that are grafted to the bilayer (Laradji, 2002; Hiergeist and Lipowsky, 1996). The bending modulus of the bare lipid bilayer is then renormalized (typically enhanced by a factor of 1.5–2.5) and an effective surface tension appears.
   
2. On the other hand, if the cytoskeleton does form a rigid shell on its own, the bilayer assumes its bare bending modulus; however, the rigid shell of the cytoskeleton constrains the thermal fluctuations of the bilayer to be smaller in amplitude than the distance to the rigid shell. The actual RBC cytoskeleton is somewhere in between these two limiting cases. In the absence of the bilayer, it does form a rigid shell, but only as a metastable structure, that survives for 20 min (Tuvia et al., 1998; Levin and Korenstein, 1991).
   

We now analyze the static fluctuation spectrum, that is, the height fluctuations as a function of the wavevector q (Zilker et al., 1987). In the limit of short wavelengths (large q) we find that the data for the mean-square amplitude is reasonably described by the expression (Fig. 2)

(5)

At the largest wavevectors the observations are noisy due to experimental limitations. We view this value of as indicative of the bare value of the bilayer, since in this limit, the bilayer is largely free of the sparse cytoskeleton connections. The amplitude of fluctuations is also small compared with the average bilayer-cytoskeleton distance of d 30 nm, so that interaction between the two is small.

View larger version (12K): [in this window] [in a new window]   FIGURE 2  Amplitude of height fluctuations as a function of wavevector q. The calculation for small q-values (Eq. 3) is shown by dashed lines, and for large q-values (Eq. 5) by the solid line. The experimental data (Zilker et al., 1987): normal cells, circles; stomatocyte, squares. The vertical dotted line shows the crossover wavevector q0.

View larger version (16K): [in this window] [in a new window]   FIGURE 4  Calculated ATP-induced RMS height fluctuations as a function of the ATP concentration, using (Eq. 13) (solid line), compared with the experimental data (asterisks, ghost; squares, shell) (Tuvia et al., 1998). Both are normalized by the thermal contribution. In the inset we plot the calculated and measured effective temperature Teff/T.

For comparison, we use the data for the stomatocyte cell (Zilker et al., 1987) (Figs. 2 and 3). These cells are known to have a weakened cytoskeleton and a much expanded bilayer (Lim et al., 2002) (see also ATP-Driven Steady-State RBC Shape, below). The first effect is taken into account by a reduction of the surface tension and by 15%, since both are linearly dependent on the shear modulus of the cytoskeleton µ (Gov and Safran, 2004), which is reduced by this amount (see ATP-Driven Steady-State RBC Shape). The weaker cytoskeleton allows the bilayer to have a shape transition into a stomatocyte, which is characterized by a stretched-out bilayer (Lim et al., 2002). In this case, ATP-induced defects in the cytoskeleton do not translate into normal motion of the bilayer, and the effective temperature is therefore found to be T_eff/T 1.2.

The data for the echinocyte cell (Zilker et al., 1987) (Fig. 3) fits with our expectation of increased surface tension and harmonic potential parameters by a factor of 2.2, compared to the values for the normal RBC (see ATP-Driven Steady-State RBC Shape). We also expect the effective temperature to be essentially given by its bare, thermal value, since ATP activity is absent in the cytoskeleton of the echinocyte (see ATP-Driven Steady-State RBC Shape). Indeed, measurements of the effective absolute fluctuation amplitude are found to be 3.8 times smaller than for the normal RBC.

The model described above, whereby the effect of the ATP activity is taken into account as a modification of the elastic moduli and the effective temperature, will be justified in the remainder of this article. In addition, further predictions of this model for other RBC properties will be discussed. In the next section, we relate ATP activity to local defects in the cytoskeleton (Fig. 1). These local spectrin-actin dissociations may result in a normal motion of the membrane. A locally weakened cytoskeleton was indeed observed to produce normal deformations of the membrane on the lengthscale of single network filaments (Reinhart and Chien, 1987). Such local buckling and dissociation of the cytoskeleton allow the bilayer to have normal motion larger than the bilayer-cytoskeleton separation d (Fig. 1).

We note that our analysis of the dynamic fluctuation data (Gov et al., 2003) is consistent with an effective value of the average bilayer-cytoskeleton distance d, which is determined by the soft potential strength, , deduced from our analysis of the static fluctuation data. The slow water flow appropriate to the measurements of the low frequencies of dynamic bilayer fluctuations is such that the cytoskeleton is nearly impermeable. This is because on the slow timescale of the water flow, the spectrin network fluctuates and effectively covers the entire two-dimensional plane. The transient, ATP-induced defects are much faster than these slow flows, so that the water is bounded on one side by the bilayer and on the other side by the spectrin network (Gov et al., 2003) which, on average, is at a distance d from the bilayer. Since the restoring force is constant, the ATP activity is best taken into account through an increased effective temperature, i.e., increased agitation force.

	ATP-DRIVEN MEMBRANE FLUCTUATIONS

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
The consumption of ATP is known to induce large membrane fluctuations (Gov et al., 2003; Tuvia et al., 1998; Levin and Korenstein, 1991). Since the hydrolysis of ATP releases E_atp 13 k_BT, it provides more than enough energy to dissociate the spectrin from the actin at the nodes of the network; that energy cost is only E_sa 7 k_BT. The rate of energy transfer by this mechanism into shape fluctuations of the membrane is limited by the following quantities: the elastic energy released in the membrane, the concentration of ATP, and the time it takes for the membrane to dissipate this energy and reattach to the cytoskeleton. The ATP-induced dissociations of spectrin are transient, because the dissociated spectrin can re-associate with the actin filament.

The adsorption of ATP to the phosphorylation sites at the spectrin-actin junctions can be described by the equilibrium occupation probability; this assumes that the ATP in the cytoplasm can be treated as a reservoir with concentration n_ATP. Standard thermodynamics (Callen, 1985) gives

(6)

where (in units of k_BT) is the energy to preferentially adsorb at the spectrin-actin sites as opposed to anywhere else on the membrane surface, and is given by the balance

(7)

where S_free-spec is the entropy change due to the release of the spectrin filament. The last term in Eq. 7 represents the mechanical energy released by the spectrin dissociation, which is estimated as due to the large adsorption energy, the ATP occupation of the spectrin-actin sites is saturated at a relatively low ATP concentration, in agreement with experimental data (see Fig. 4).

Next we calculate the height fluctuation of a single network unit-cell (Fig. 1) due to the ATP-induced spectrin dissociation. Each spectrin filament that is dissociated from the actin linkers reduces the local coordination number, N, in Eq. 1 from 3 to 2. In equilibrium this causes the local membrane length, R, to increase by 4%, and changes the local height of the membrane (Fig. 1). On both sides of the dissociated spectrin there are now fivefold nodes that tend to buckle upwards (Seung and Nelson, 1988; Carraro and Nelson, 1993; Lidmar et al., 2003; see also Fig. 1, this article). The bilayer bending modulus resists the cytoskeleton buckling of the fivefold defects, so that the resulting height fluctuation given by balancing the elastic forces is 25 nm. The total bilayer area is of course conserved. We therefore find that the ATP-induced spectrin dissociation converts the stored elastic energy of the filament into normal motion of the membrane.

We show below how this energy is added to the thermal energy k_BT that is dissipated by each membrane mode, giving an enhanced effective temperature of the order of the measured factor of T_eff/T 3 (Gov et al., 2003). Our estimate is valid under the condition that the ATP-induced processes, similar to the thermal motion of the membrane, are spatially incoherent. Note that the use of the effective temperature terminology is a misnomer, since we will show below that the ATP-induced membrane fluctuations have a distinct nonthermal character, such as a dependence on the fluid viscosity. We shall still use this term as an easy way of relating the amplitude of the ATP-induced fluctuations to the amplitude of the thermal component.

We now calculate the local transfer of energy from the ATP to the membrane motion, on the scale of a single network element (Fig. 1). Each unit-cell of the membrane converts the dissociations into local normal motion; the cell motions are independent of each other and of longer wavelength fluctuations. To describe the local dynamics of a single network element (patch), i.e., the spectrin filament and the overlying bilayer, we write the equation of motion for the amplitude h of the membrane fluctuations (in the usual limit of large damping) in one unit cell of the spectrin network as

(8)

where a² is the area of the membrane contained in one network triangle (the mesh size of the spectrin network: a 80 nm), is the viscosity of the surrounding fluid, and (t) is the normalized force, given by (t) = F(t)/(4a²k), where k 1/a and F(t) is the actual force. The first term on the left-hand side of Eq. 8 is the viscous damping of the membrane due to the surrounding fluid, and the second term is the restoring force due to the curvature stiffness of the bilayer. Here, 3_water is a slightly enhanced, effective viscosity due to the cytoskeleton confinement of the membrane fluctuations (Gov et al., 2003; Gov and Safran, 2004).

Note that a full treatment, of the spatial height correlations for all k-modes of the ATP-driven undulations, is beyond the scope of this article. It is not clear, a priori, whether the ATP-induced fluctuations produce a correlation function with a k-dependence that is different from the 1/k⁴ thermal result. For simplicity, we here treat the fluctuations of a single membrane-cytoskeleton patch (unit cell), to calculate the contribution to the overall height fluctuations. The calculated effective temperature will be used for describing the integrated height fluctuations and comparing to the thermal contribution. For a description of the detailed k-dependence it is less useful, although in some cases the active and thermal fluctuations turn out to have the same 1/k⁴ dependence (see summary in Ramaswamy et al., 2000).

The membrane (in one unit cell of the spectrin network) kicked by the dissociation has a characteristic fluctuation frequency of _m /4a³ 10³ Hz. This should also include the reassociation time, _re, which is the Zimm time (Doi, 1996) that it takes for the detached end of the filament to thermally diffuse back to the attachment site, and reattach. Since the detached spectrin is confined by the bilayer and neighboring spectrin filaments, only a short segment of the spectrin molecule (Lenormand et al., 2003) can actually diffuse freely. This results in _re 10^–7 s, which can therefore be neglected.

Note that this short reassociation time makes it highly unlikely that the spectrin filament will reassociate at a node different from the one it was dissociated from. Furthermore, for the dissociated end of the spectrin filament to connect to a different node, it needs to surmount an energy barrier. This barrier, likely of an order of a few k_BT, is due to the steric repulsion from neighboring filaments, which oppose the large-scale rearrangement of the filaments. The ATP-induced dissociations shown in Fig. 1, therefore, mostly produce a transient fivefold node. More extensive defects, involving five- and sevenfold nodes, can be thermally excited in addition to the ATP-induced dissociation stage. These topological defects involve large-scale rearrangement of the filaments, due to thermal excitation or applied stress. The ATP-induced transient dissociations, therefore, do not directly give rise to changes in the overall topology of the cytoskeleton network, as is further discussed in Deformed RBC: Topological Effects, below.

In Fig. 4 the data from an RBC ghost and shell is shown (Tuvia et al., 1998; Levin and Korenstein, 1991). The natural frequency of the fluctuations of the ghost is that of the intact membrane _m. For the exposed cytoskeleton (shell) the dissociated spectrin is no longer confined by the membrane, and the reassociation time for the whole length of the spectrin determines the natural frequency of the cytoskeleton as _re 10 ms _m 10² Hz. At the same time the shell has a lower overall tension in the spectrin filaments, so that the force F due to the ATP-induced disconnections is lower. Together these conflicting trends result in a slightly larger T_eff/T (Eqs. 13 and 16) in the shell compared with the ghost (Fig. 4).

Fourier-transforming Eq. 8 in time, we get

(9)

We model the force correlation function as an exponential in time

(10)

(Liverpool, 2003; Granek, 1999), where n_d is given by Eq. 6 and 1 ms is the estimated average time for the ATP-induced spectrin-actin dissociation. This is the timescale at which spectrin is dissociated from the weakened binding site, due to the ATP (Tuvia et al., 1998, 1992; Levin and Korenstein, 1991; Herscher et al., 1994). The release of a spectrin filament gives a driving force of magnitude of F µ(R – R₀) 0.2 pN. Upon Fourier-transforming in time, we get

(11)

Plugging this into Eq. 9, we find

(12)

Integrating over all frequencies, we find that the equal time (static) correlation function for height fluctuations is

(13)

For the RBC case, where _m ^–1 (in normal conditions), we can write

(14)

The maximal RMS height fluctuation due to ATP is Note that the ATP-induced amplitude is proportional to the ratio of the released tension in the cytoskeleton and the curvature modulus of the membrane. At physiological concentrations of ATP (i.e., in normal RBC), our model predicts a finite-amplitude kick due to ATP, which is in agreement with the time-series analysis of the fluctuation signal (Bitler and Korenstein, 2004). This means that the motion on the lengthscale of the cytoskeleton network ( is not thermal and does not satisfy the fluctuation-dissipation theorem (Liverpool, 2003). However, at larger scales an effective thermal equilibrium description can be used since nonlinear couplings mix the energy of the discrete motion into membranes modes of larger wavelength. Thus, at lengthscales and timescales that are much longer than the scale at which the ATP-induced motion occurs (q << q₀), the continuum effective-temperature treatment is reasonable.

Assuming that the ATP-induced fluctuations are incoherent with respect to the thermal contribution, we predict an additive contribution to the total mean-square fluctuation amplitude (Main,1984), |h|²_total = |h|²_ATP + |h|²_thermal. In Fig. 4 we compare our calculated ATP-driven height fluctuations (Eq. 13) with the available data (Tuvia et al., 1998; Levin and Korenstein, 1991), in a plot of the measured RMS height, normalized by the thermal contribution (corresponding to zero ATP), For the calculated fluctuations we normalize by the calculated thermal contribution (Gov et al., 2003), where 7.7 x 10⁷ J/m⁴ describes the confining effect of the cytoskeleton. Finally we can define an effective temperature for height fluctuations, by comparing with the thermal contribution T_eff/T = 1 + |h|²_ATP/|h|²_thermal (Fig. 4, inset). The agreement of our model with the data is rather good, except for low ATP concentrations, where there seems to be a threshold behavior in the experiments. This could arise from competing ATP-adsorption sites other than the spectrin-actin nodes, which effectively lower the available ATP concentration.

Note that from Eq. 14 we get that T_eff/T E, in the limit where T_eff/T >> 1. This means that the ATP-induced mechanical energy that is given to the membrane is linearly proportional to the effective thermal energy of the membrane fluctuations. This is similar to a generalized equipartition theorem that was suggested for describing the effective temperature in sheared glassy materials (Berthier and Barrat, 2002).

Our model predicts (Eq. 13) that the effective temperature depends on the viscosity as

(15)

since only the membrane frequency depends on the viscosity, whereas the chemical timescale for the spectrin-actin dissociation, , is independent on the viscosity. The first limit pertains to the RBC case, where a 1/ dependence has indeed been measured (Tuvia et al., 1997; Fricke and Sackmann, 1984). It is interesting to note that similar results were observed (Manneville et al., 2001) in the case of membrane fluctuations due to light-activated ion channels. Note that the ATP-driven activity that we propose is not due to molecular motors such as myosin.

We therefore conclude that the ATP-induced membrane height fluctuations induced by spectrin dissociations can quantitatively explain the observations of the concentration and viscosity dependence of this effect (Gov et al., 2003; Tuvia et al., 1998; Levin and Korenstein, 1991).

	ATP-DRIVEN STEADY-STATE RBC SHAPE

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
In addition to enhancing the membrane fluctuations, ATP-induced transient defects of the spectrin network are also important in determining the average shape of RBC (Hoffman, 2001, 2004; Nakao, 2002). The intimate experimental correlation between the amplitude of membrane fluctuations and overall cell shape was demonstrated in Fricke et al. (1986). ATP is clearly important for maintaining the discocyte shape of a free RBC, since it has been observed that depletion of ATP causes a change from the discocyte to the echinocyte shape (Lim et al., 2002). This change of shape suggests that the loss of ATP results in a stiffer cytoskeleton that pulls the bilayer over a smaller cytoskeleton-projected area. The need to accommodate the bilayer area over a smaller projected area results in the appearance of the spicules of the echinocyte shape (Lim et al., 2002). In our model, the physical origin of this effect is simply related to the change in the number of released spectrin filaments, when the number of defects is reduced as ATP is reduced. We estimate this effect by relating it to the pulling force that the cytoskeleton exerts on the bilayer, and predict its dependence on the ATP concentration.

We begin by counting the different numbers of ATP-induced defects in the cytoskeleton. The proportion of dissociated ends of spectrin filaments at any given time, n_dis, is given by the proportion of ATP bound to the activation sites (Eq. 6), times the proportion of the association-reassociation cycle that the spectrin is dissociated: We do not know this last ratio very precisely, but since the RBC is normally in the limit of (Eq. 16), we expect it to be close to one. Considering that there are two dissociation sites per spectrin filament, the proportion of associated spectrin filaments is given by n_a = (1 – n_dis)², whereas the proportion of filaments that are cut at both ends is In Fig. 5 we plot the dependence of these quantities on ATP concentration, using the ratio _m/( + _m) = 0.8. We also plot the average value of the shear modulus of an intact RBC (at physiological levels of ATP n_{ATP, 0} 1.5 mM) normalized by the value for a cytoskeleton shell (no ATP) (Lenormand et al., 2003, 2001; Sleep et al., 1999; Hénon et al., 1999). Note that .

View larger version (24K): [in this window] [in a new window]   FIGURE 5  Calculated proportion of doubly-connected (dotted line) and associated (solid line) spectrin filaments, as a function of the ATP concentration, compared with the critical concentration for two-dimensional bond percolation on a triangular lattice (dashed line). Squares mark the measured normalized shear elasticity of a shell (nATP 0) and intact RBC (Lenormand et al., 2003; Sleep et al., 1999). In the inset we show the path of dissociated filaments in both the triangular and complementary honeycomb lattices.

The effect of the ATP-induced dissociations determines the overall equilibrium shape of free RBC by determining the number of defects in the spectrin cytoskeleton, which, in turn, determines the pulling tension that the cytoskeleton exerts on the bilayer. The increase in the tension µ acting on the inner leaflet of the bilayer is balanced by a variation in the areas of the two leaflets. This balance can be written as

(16)

(Lim et al., 2002), where D 5 nm is the bilayer thickness, a is the relative area difference of the two bilayer leaflets, and a₀ 0 is the value for the discocyte shape (Lim et al., 2002).

In Fig. 6 we plot both µ and a as a function of the ATP concentration, where 100% corresponds to physiological levels of ATP. The calculations of Lim et al. (2002) predict that an increase in the relative area difference of a 1.6% from the discocyte shape induces a shape transition to an echinocyte. We find that the critical ATP concentration at which this transition should therefore occur is 10%, which indeed correlates with a sharp increase in the measured probability (Féo and Leblond, 1974) for the echinocyte shape to appear (Fig. 6). We conclude that our model gives a quantitative explanation of the ATP-induced shape transformations of the RBC.

View larger version (16K): [in this window] [in a new window]   FIGURE 6  Measured percentage of echinocytes (solid line and squares) (Féo and Leblond, 1974) as a function of the relative ATP concentration, compared with the relative shear modulus of Fig. 3 (dotted line). The calculated change in the relative area difference between the bilayer leaflets a (Eq. 16) is shown (dash-dot line). The vertical dashed line shows the critical ATP concentration where a difference of a 1.6% occurs relative to the discocyte value, and signals the discocyte-echinocyte transition (Lim et al., 2002).

View larger version (12K): [in this window] [in a new window]   FIGURE 7  The self-regulation of the ATP levels through the cytoskeleton tension. Calculated reduction in shear modulus due to ATP concentration (solid line) (Fig. 5). Derived dependence of the ATP concentration on the membrane shear stress (dashed line) (Johnson, 1994), through its effect on the membrane ion permeability (Ataullakhanov and Vitvitsky, 2002). The equilibrium concentration is found at the crossing point of the graphs, close to the physiological concentration nATP,0 1.5 mM.

On the other hand, increased levels of ATP are associated with hereditary stomatocytosis (Ataullakhanov and Vitvitsky, 2002), where a leaky membrane promotes higher metabolic activity of ion pumps and a concurrent rise in ATP levels, by up to a factor of 5. At these ATP levels we predict the number of doubly-connected spectrins and shear strength to decrease by 12% compared to the normal discocyte. This calculated reduction in cytoskeleton tension is translated into a change of 0.5% in the relative area difference a (Eq. 16), which indeed induces the stomatocyte shape transition (Lim et al., 2002). The saturation shown in Fig. 4 breaks down when the ATP levels increase beyond the transition from discocyte to stomatocyte. In this case the bilayer is fully expanded and the ATP-induced spectrin dissociations do not translate into normal bilayer motion, as in the normal cell (Fig. 1). The effective temperature therefore goes down to the bare thermal value in these abnormal cells (see Analysis of the Static Fluctuation Spectrum, above).

In severe spherocytosis the cytoskeleton is effectively destroyed, and there is therefore no tension in the spectrin filaments, which results in much reduced membrane fluctuations (Zilker et al., 1987) and stronger cell-substrate adhesion properties (Wandersee et al., 2004). Note that in these cells, as the ATP content is reduced with age, so does the adhesion strength reduce (Wandersee et al., 2004)—indicating a certain recovery of cytoskeletal strength, in accordance with our model.

Lastly, agents that increase the binding of the spectrin-actin-protein 4.1 complex, such as Ca²⁺ ions (Nakao, 2002; Wallis et al., 1993; Fricke et al., 1986; Vest et al., 2004), are found to induce cytoskeleton shrinkage and increased connectivity (A. Ostafin, personal communication, 2004), which can be reversed by ATP perfusion (Nakao, 2002), as follows from our model. ATP loss from electroporated RBC explains, according to our model, their shape transformation to echinocytes (Schwarz et al., 1999). Addition of vanadate, which was found to inhibit membrane fluctuations (Tuvia et al., 1998; Levin and Korenstein, 1991), was also found to induce the echinocyte shape (Schwarz et al., 1999), as expected from our model. Antibodies for protein-4.1 block its phosphorylation and therefore lead to reduced dissociations and echinocyte shape, whereas spectrin antibodies prevent reassociations of spectrin filaments and stomatocyte shape (Pestonjamasp and Mehta, 1995).

	DEFORMED RBC: TOPOLOGICAL DEFECTS

TOP ABSTRACT INTRODUCTION ANALYSIS OF THE STATIC... ATP-DRIVEN MEMBRANE FLUCTUATIONS ATP-DRIVEN STEADY-STATE RBC... DEFORMED RBC: TOPOLOGICAL... CFTR ACTIVATION AND ATP... CONCLUSION APPENDIX: MALARIA PARASITE AND... ACKNOWLEDGEMENTS REFERENCES

 
In addition to the transient defects discussed so far, there is likely to be a finite population of nontransient, topological defects. The creation of such defects involves, as a first stage, the dissociation of a spectrin filament, and is thereby related to the ATP-induced dissociations we discussed above. These topological defects will be shown to proliferate when the cell is subject to strong static deformations. These circumstances arise when the RBC is pulled into a thin capillary, stretched by optical tweezers, or flows through a porous filter (Evans, 1983, 1989; Reinhart et al., 1991; Dao et al., 2003; Sprague et al., 1998, 2003). We show that when the deformation of the RBC membrane is large enough, the cytoskeleton buckles and folds through the appearance of long-lived defects. Topology requires that there are 12 fivefold defects in a closed triangulated vesicle (Bowick et al., 2000). We will focus here on the cases when far more numerous defects appear, due to deformations.

The hexagonal cytoskeleton network in the discocyte RBC is relatively flat on the scale of the spectrin filaments (Sackmann, 1995). The simplest topological defect that can be excited in this hexagonal network involves the dissociation of a spectrin filament at both ends, its rotation by 90°, and its reattachment to neighboring nodes (Fig. 8). We assume that it is much less likely to have two spectrin filaments connecting the same nodes due to their steric repulsion. We therefore have defects with quadrupole symmetry, which contains two sevenfold and fivefold pairs (Magnasco, 1994). A line of adjacent defects results in a separation of the quadrupole into two separated pairs of defective dipoles (Fig. 8). Note that since two ends of a spectrin filament must be simultaneously cut for these excitations, it is highly unlikely to occur due to thermal fluctuation. This initial stage of dissociation of the spectrin filament at both ends is therefore dependent on the ATP-induced process we discussed in the previous sections. The ATP-induced dissociations (Fig. 1) will cause doubly-cut spectrin filaments to exist (Fig. 5), but since they reassociate at the same nodes from which they were dissociated, no topological defects arise directly. For the topological defects to be created, the doubly-cut spectrin filament needs to be rotated by a large angle, so that it finds new nodes to reassociate with. This rotation requires thermal or elastic energy, since there is an energy barrier to overcome. This barrier results from the steric repulsion of neighboring filaments, and is likely to be of the order of a few k_BT. This additional energy comes from thermal excitation or an external stress, and results in cytoskeleton dynamics which are much slower than the timescale of ATP-induced membrane fluctuations (Fischer, 2004). Note that the rearrangement of the filaments away from the hexagonal symmetry is limited by the ability of the short actin filaments at the nodes to cross-link only a finite number of spectrin filaments, probably not more than eight per node (Liu et al., 1987). Large-scale deviations from the hexagonal order are therefore mostly limited to five- to sevenfold defects, and no structural instability occurs.

View larger version (42K): [in this window] [in a new window]   FIGURE 8  Schematic picture of the simplest quadrupole defect in the spectrin network. The defect arises from the rotation of the dashed filament in a, to produce two pairs of fivefold (dashed rings) and sevenfold defects (solid rings) (b). A linear arrangement of such defects produces two separated dipoles (c and d). (e) The shape of a buckled fivefold defect in the presence of the bilayer.

With this, we estimate the critical radius of the RBC membrane below which the cytoskeleton buckles and forms fold-lines. We consider a simplified model, where we compare the elastic energy of a continuously bent membrane in the shape of a cylinder of radius R, with the energy of a membrane of equal area that is folded by linear fold-lines connected by flat faces (Fig. 9),

(17)

where the conserved membrane area is S = 2RL, and the effective length when folded is L' = S/(n2R sin(/2)), with where E₅ 0.7 k_BT is the elastic bending energy of a single fivefold defect (core energy). In this model we assume that the cylindrical bending of the membrane involves mainly curvature energy with very little static shear of the cytoskeleton. If in addition we would include the shear energy, this would only renormalize the prefactor of the bending energy, because the shear term has the same 1/R² dependence as the curvature energy. The turning angle at the fivefold defect (Fig. 8) is 40°, so that we find n = 2/ 5. Plotting these energies in Fig. 9 we find that for radius R < 0.4 µm the membrane prefers to form buckled fold-lines. We emphasize that the above calculation is only semiquantitative. Each fold-line, when it buckles, will have 100 defects along its length.

View larger version (23K): [in this window] [in a new window]   FIGURE 9  Schematic picture showing the buckling transition of a cylindrical membrane from round (a) to polygonal cross-section (b). The latter has fold-lines along its length. Below that we plot the calculated membrane energies for these geometries, given by Eq. 17, as a function of radius of the capillary. Solid line is for smooth bending (a), whereas the dashed line is for fold-lines (b). For radii less than the critical value of Rc 0.4 µm, the RBC membrane forms fold-lines.

We conclude that network defects that contain fivefold nodes induce a spontaneous local curvature. They are therefore likely to spontaneously appear as fold-lines in the membrane of highly deformed RBCs. This prediction must be experimentally tested. The observed shape memory of the RBC can be related to accumulation of static (or long-lived) defects at regions of higher curvature, such as the rim of the discocyte or the spicules of the echinocyte (Gass et al., 1991; Fischer, 2004). The role of the ATP-driven dynamic dissociations described in the previous sections is important not only for the creation of the topological defects (as described in the beginning of this section), but also to allow these structural defects to move (flow) around o