On the shape memory of red blood cells

Transcription

1 On the shape memory of red blood cells DANIEL CORDASCO AND PROSENJIT BAGCHI # Mechanical and Aerospace Engineering Department Rutgers, The State University of New Jersey Piscataway, NJ 08854, USA # Corresponding author. pbagchi@jove.rutgers.edu 1

2 Abstract Red blood cells (RBCs) undergo remarkably large deformations when subjected to external forces but return to their biconcave discoid resting shape as the forces are withdrawn. In many experiments, such as when RBCs are subjected to a shear flow and undergo the tank treading motion, the membrane elements are also displaced from their original (resting) locations along the cell surface with respect to the cell axis, in addition to the cell being deformed. A shape memory is said to exist if after the flow is stopped the RBC regains its biconcave shape and the membrane elements also return to their original locations. The shape memory of RBCs was demonstrated by Fischer [T.M. Fischer, Shape memory of human red blood cells, Biophys. J. 86, 3304 (004)] using shear flow go and stop experiments. Optical tweezer and micropipette based stretch relaxation experiments do not reveal the complete shape memory because while the RBC may be deformed, the membrane elements are not significantly displaced from their original locations with respect to the cell axis. Here we present the first threedimensional computational study predicting the complete shape memory of RBCs using shear flow goand stop simulations. The influence of different parameters, namely, membrane shear elasticity and bending rigidity, membrane viscosity, cytoplasmic and suspending fluid viscosity, as well as different stress free states of the RBC is studied. For all cases, the RBCs always exhibit shape memory. The complete recovery of the RBC in shear flow go and stop simulations occurs over a time that is orders of magnitude longer than that for optical tweezer and micropipette based relaxations. The response is also observed to be more complex and composed of widely disparate time scales as opposed to only one time scale that characterizes the optical tweezer and micropipette based relaxations. We observe that the recovery occurs in three phases: a rapid compression of the RBC immediately after the flow is stopped, followed by a slow recovery to the biconcave shape combined with membrane rotation, and a final rotational return of the membrane elements back to their original locations. A fast time scale on the order of a few hundred milliseconds characterizes the initial compression phase while a slow time scale on the order of tens of seconds is associated with the rotational phase. We observe that the response is strongly dependent on the stress free state of the cells, that is, the relaxation time decreases significantly and the mode of recovery changes from rotation driven to deformation driven as the stress free state becomes more non spherical. We show that while membrane shear elasticity and nonspherical stress free shape are necessary and sufficient for the membrane elements to return to their original locations, bending rigidity is needed for the `global recovery of the biconcave shape. We also perform a novel relaxation simulation in which the cell axis of revolution is not aligned with the shear plane, and show that the shape memory is exhibited even when the membrane elements are displaced normal to the imposed flow direction. The results presented here could motivate new experiments to determine the exact stress free state of the RBC and also to clearly identify different tank treading modes.

3 1. Introduction Red blood cells (RBCs), the main cellular component of human blood, experience remarkably large deformations in vivo as they transit through the body s complex vascular network. The RBCs ability to endure such deformations is enabled by their large surface area to volume ratio as well as their structural composition and mechanical properties. The RBC consists of a very thin membrane which encapsulates a viscous cytosol hemoglobin solution. The membrane is composed of a lipid bilayer coupled to an underlying two dimensional cytoskeletal network, and it resists shearing and bending deformations as well as surface area dilation. Under resting conditions, a normal healthy cell acquires a biconcave discoid shape. The overall viscoelastic behavior of the RBC is due to the combined contributions of the composite membrane and internal fluid. The deformation of the RBC is reversible; the cells persistently return to their resting biconcave shape despite continuously undergoing extreme deformations in the vascular systems, e.g., while flowing through narrow capillaries and slits in the spleen (Mebius and Kraal 005). Many in vitro experiments, e.g. stretching and relaxation using micropipettes and optical tweezers, also demonstrate that the cells recover their biconcave shape as the external forces are withdrawn. In a classic experiment, Fischer (004) applied shear flow to deform the RBCs, and studied their recovery to the biconcave shape following flow cessation. The motion of the cell membrane was also tracked by adhering microbeads to their surfaces. Fischer observed that not only had the cells recovered their resting biconcave shape, but that the beads also had returned to the same spots where they were initially located before the application of the flow. This observation is an illustration of the existence of the so called shape memory of the RBCs. Because of the shape memory, the cells not only recover their resting biconcave shape, but the membrane elements also return to their initial locations with respect to the cell axis that they occupied before the external forces were applied. For example, the membrane elements that are located near the dimple or rim before the application of the flow must return to dimple or rim, respectively, after the flow is stopped. The cell is then said to possess a shape memory. The concept is illustrated in Figure 1. If on the other hand the membrane elements can return anywhere after recovery from the deformed shape the cell would lack any shape memory. Thus the shape memory is not just the recovery of the biconcave shape, and may involve the rotation of the membrane as observed by Fischer (004). A full shape memory is not manifested in the stretch relaxation experiments using micropipette or optical tweezers because the membrane is not displaced significantly with respect to the cell axis. The movement of the RBC membrane during the relaxation process was also observed in experiments by Braunmüller et al (01) who studied the relaxation of cells from a deformed parachute shape back to the resting biconcave shape as they exited a narrow constriction into a stagnant reservoir. They observed that the cell s relaxation back to the biconcave resting shape was accompanied by a rotation of the cell membrane in which membrane elements seemingly returned to their original locations suggesting the existence of the shape memory. 3

4 Figure 1. Schematic illustrating the shape memory of the RBC. The RBC is first subjected to a shear flow resulting in a tank treading motion. The major axis in shear plane with respect to the flow is followed with the angle, and a membrane marker point (black dot in the figure) with respect to the cell major axis is tracked with the angle. The flow is then stopped suddenly when the membrane marker has been displaced from its original position by an amount S. In the center figure, the black dot is the instantaneous location of the marker point and the broken dot is its initial location with respect to the major axis. Following the flow cessation, the cell relaxes back to the biconcave shape. If the marker point also returns to its original position ( 0 ), then the cell is said to have a shape memory. 4

5 The shape memory is physiologically important in terms of RBC health. The return of membrane elements back to their original locations suggests that either the cytoskeleton filaments and cytoskeleton bilayer anchors are not completely dissociated under the action of the external forces, or they reassociate at the same locations where they dissociate. Adenosine triphosphate (ATP) which is released by the RBCs and also by the endothelial cells is known to dissociate the cytoskeleton filaments and increase the cell deformability (Wan et al, 008; Forsyth et al, 011). A total dissociation of the skeleton would result in the formation of a new skeleton, and hence a loss of shape memory. Depletion of ATP, on the other hand, results in a stiffening of the membrane, which in turn could result in capillary blockage by the RBCs, and thereby deregulation of blood and oxygen supply to tissues. Fischer (004) observed that the shape memory persisted even in cells subjected to shear for more than four hours in vitro, and noted that it is an intrinsic property that would be present also in vivo. The shape memory response of an RBC is important also in extracting its mechanical properties, such as the shear elasticity and membrane viscosity. For example, the static stretching data obtained using optical tweezers can be used to extract the shear elasticity, but a dynamic relaxation response would be needed to extract the membrane viscosity (Henon et al 1999, Mills et al 004, Li et al 005). The relaxation of the deformed cell occurs by dissipation of the energy stored in the membrane during deformation, and the response is different depending on the way the cell is deformed. In the dynamic optical tweezing, in which an RBC is allowed to relax after stretching, an exponential decay time constant on the order of 100 to 300 milliseconds was observed (Henon et al 1999, Mills et al 004). Similar time scales were also found for RBC relaxations following micropipette aspirations (Hochmuth, Worthy and Evans 1979, Evans 1989). Bronkhorst et al (1995) used optical tweezers to induce a 3 point bending of the cell and obtained a time scale similar to the extensional relaxation experiments. In contrast, in Fischer s (004) shear flow go and stop experiment in which membrane elements are significantly displaced by tank treading, a time of 5 to 0 seconds was reported for the elements to return to their original locations. This is a significantly longer time than that obtained in optical tweezer and micropipette experiments. Such a slow relaxation was also observed by Braunmüller et al (01) in their experiment as noted above. Furthermore, Braunmüller et al (01) reported two coupled time scales of relaxation: an initial fast response with a time scale of 0.11 s to 0.5 s that is associated with deformation, followed by a slow relaxation with a time scale of 9 s to 49 s associated with membrane rotation. Evidently, a complete shape memory response does not manifest itself unless the membrane elements are displaced from their resting positions, and this may be a reason why it has not been widely studied. More importantly, no fully three dimensional numerical study exists that has addressed the shape memory response, despite a significant progress in recent years in numerical modeling of the large deformation of RBCs under external shear or other forces. It is even unknown whether the existing numerical models could actually predict the shape memory response. To that end, in this article we use a 3D continuum model of the RBC, and numerically replicate the shear flow go and stop experiment of Fischer (004). We find that the continuum model does predict the RBC shape memory. The relaxation dynamics observed in the simulations is quite complex and includes the presence of multiple phases as 5

6 well as multiple and widely disparate time scales. These phases are explained in terms of membrane energy, and the time scales are extracted. We then ask what mechanical properties are responsible for the shape memory. Specifically, the role of shear elasticity, bending stiffness, non sphericity and the stress free shape of the cell is examined. Fischer (004) noted that the presence of the shape memory implies that the stress free state of the cell is not a sphere. However, additional information about the stress free state could not be obtained from the experiment. Here we consider simulations with different RBC stress free states and study their influence on the shape memory recovery. Then the relative importance of shear elasticity and bending rigidity on the shape memory is studied. It is shown that while shear elasticity is sufficient for the membrane elements to return to their original locations, bending rigidity is needed for a global shape recovery. We also consider the influence of membrane and fluid viscosities and study their relative effect on the time scales. Lastly, the shape memory response when the membrane is displaced normal to the plane of shear is considered. In most numerical simulations of RBCs in shear flow, the cell s axis of revolution lies along the shear plane. This is a situation that may be difficult to maintain in an experiment, and most experimental studies including Fischer s (004) only reported on cells with the axis lying along the shear plane. Recently, the full threedimensional dynamics of RBCs with off shear plane orientations has been a topic of interest (Dupire, Socol, and Viallat 01; Omori et al 01; Cordasco and Bagchi 013; Peng and Zhu 013; Cordasco, Yazdani, and Bagchi 014; Peng, Salehyar, and Zhu 015). These studies demonstrated that the membrane could reconfigure as the elements moved normal to the shear plane. Then, a natural question that arises is whether the shape memory exists for an off shear plane membrane displacement also, and, if so, how it compares with the in plane response. To address this, we simulate shear flow goand stop experiments with an off shear plane orientation of the cell axis, and study the shape memory response. The simulation technique is presented in the next section followed by the results. Then, a discussion of the results and a conclusion are presented.. Problem setup and simulation technique Fully three dimensional numerical simulations of deformable RBCs are performed using a continuum modeling of the cells. The numerical methodology and its validation are described in full detail in our previous publications, e.g. Doddi & Bagchi (008), Yazdani & Bagchi (01), Yazdani and Bagchi (013), Cordasco, Yazdani & Bagchi (014), and Cordasco and Bagchi (014, 016). Here a short description is given for the sake of completeness. The cell is modeled as a capsule, that is, a viscous liquid drop enveloped by a zero thickness viscoelastic membrane. The resting shape is taken as the experimentally observed biconcave discocyte (Fung 1993) of 7.8 micron end to end length. The interior and suspending fluids are assumed to be incompressible and Newtonian with viscosities 0 and 0, respectively. The membrane is assumed to possess resistance against shear deformation, area dilatation and bending. The shear deformation and area dilation are modelled following Skalak et al (1973) using an in plane strain energy function, GS WS I1 I1 I CI, (.1) 4 6

7 where GS is the membrane shear elastic modulus, I1 1 and I 1 1are the strain invariants of the Green strain tensor, and 1 and are the principal stretch ratios. The membrane area dilatation is limited by the parameter C. The principal elastic stresses are given by 1 e WS 1 and S 1 1 e 1 W. (.) The viscoelastic behavior of the membrane is resolved using the Kelvin Voigt model in which the total e v membrane stress is the sum of the elastic and viscous components, that is, τ τ τ, where v 1 τ m D tr( D) I S, (.3) is the viscous stress, m is the membrane shear viscosity, D is the strain rate tensor on the membrane, and IS is the surface projection tensor. The effect of membrane dilatational viscosity is neglected since the RBC surface is nearly area incompressible. The surface is discretized using Delaunay triangles. A finite element method is used to compute the total stress tensor τ for each surface element. In the numerical implementation, τ is computed in terms of the strain history using a time convolution integral. Then, the viscoelastic force for each element can be obtained by using the principle of virtual work. The finite element method for an elastic membrane was described in Doddi & Bagchi (008). Details of the implementation of the full viscoelastic membrane model are given in Yazdani & Bagchi (013). The bending resistance is modelled following Helfrich s formulation for bending energy (Zhong Can and Helfrich 1989), EB W B c0 ds, (.4) S where EB is the bending modulus, is the mean curvature, c0 is the spontaneous curvature and S is the surface area. The value chosen for spontaneous curvature depends on the stress free shape of the membrane used and we refer to our earlier work (Cordasco, Yazdani, and Bagchi 014). For numerical implementation, an expression for the bending force density derived from (.4) is used which requires evaluation of the mean and Gaussian curvatures, and the Laplacian on the triangulated surface. Details of the implementation of the bending stiffness and the validation are given in Yazdani & Bagchi (01). The fluid motion interior and exterior to the cell is governed by the Stokes equations. The flow variables are treated in an Eulerian fashion. A combination of spectral and finite difference schemes is used to compute the flow variables. The coupling between the flow and membrane deformation is obtained using a front tracking/immersed boundary method. In this approach the Stokes equations are modified by adding a source term as 7

8 B f f ds (.5) S ev b where fev and fb are the membrane viscoelastic and bending force density, is the 3D Dirac delta function that is zero everywhere except at the location of the cell surface, and S represents the cell surface area. Once the fluid velocity is obtained, the membrane is advected as d x m / dt um, where xm is the coordinates of a surface node, and um is its velocity computed by interpolating the surrounding fluid velocity using the delta function. An indicator function is used for identifying the fluids interior and exterior to the cell. The fluid domain is a cubic box of length a0, where a 0 is the radius of a sphere having the same 3 volume as the RBC. The domain is discretized using an Eulerian mesh of10 points, and the cell surface is discretized using 0,480 triangular elements. Two different stress free configurations of the cell are assumed. The biconcave stress free configuration (BCSF) assumes that the membrane of the resting biconcave shape is free of any stress. The resting shape can then be written using an empirical relation derived from experimentally observed RBC shapes (Fung 1993). The nearly spherical stress free configuration (SSF) assumes that the membrane is free from any stress when it is in the shape of an oblate spheroid of reduced volume A / 3 4 3/ V0 Vobl /, where V obl is the spheroid volume and A m is the surface area of a normal RBC. For the SSF case, the biconcave shape has to be generated numerically from the 3 spheroid by deflation until the desired volume 94.1 m of a normal RBC is reached. The resting V cell shape of the SSF cells and BCSF cells are similar and they agree well with the experimental resting shape as given in Fung (1993), and have a reduced volume of 0.644, the same as a normal RBC. A spontaneous curvature value of.09 is used for the BCSF shape and +4.0 for the SSF shape. Additional details on deflation process are given in Cordasco, Yazdani, and Bagchi (014). Additional validations can be found in Cordasco & Bagchi (014) for optical tweezer stretching simulations, and Cordasco & Bagchi (016) for capsules in oscillating shear flow. The go and stop simulations are performed in two parts as shown in figure 1. First, the cells are subjected to simple shear flow u y,0,0, where is the shear rate. The capillary number of the applied shear flow is defined as Ca 0 a0 / GS. After several tank treading cycles, the flow is suddenly stopped, and relaxation of the cell is followed. The major dimensionless parameters are the viscosity ratio, and the ratio of membrane bending to elastic energy EB / a0gs. When membrane viscosity is considered, an additional dimensionless parameter is introduced as m m 0a0. The dimensionless time is taken as t t / s where 6 s 0 a 0 / G S. For a normal, healthy RBC, a 0. 8 µm, GS N/m (Henon et al 1999), 8

9 19 7 EB J (Mohandas & Evans 1984, Scheffer et al 001, Boal 00), and m Ns/m (Tran Son Tay, Sutera & Rao 1984, Hochmuth, Worthy & Evans 1979). Also, the physiological value of 5. Parametric ranges considered in the simulations are , 0.1, and 0 m 100. Most simulations are done with 0. 03so that a numerically stable biconcave shape can be generated using the deflation process noted earlier (Cordasco & Bagchi 014), and 0. 1to ensure large deformation and a tank treading motion during the shear flow, unless stated otherwise. Most simulations also are done with 0 for a faster computation. 3. Results m First, the simulation results from `in plane relaxations are presented in 3.1. We demonstrate that the shape memory exists in the numerical model. Here we present a detailed analysis of the influence of stress free state, membrane viscosity, fluid viscosity, as well as membrane shear elasticity and bending rigidity. Then in 3., the `off plane relaxation is considered In plane relaxation Existence of shape memory The in plane refers to the scenario when the axis of revolution of the RBC remains aligned in the plane of the shear flow (figure 1). The RBC is first subjected to a steady shear flow. The cell deforms in to an ellipsoidal shape and aligns at an angle with respect to the flow direction. The membrane and the interior fluid rotate in a tank treading manner as shown in figure (a). After a few tank treading cycles, the shear flow is stopped and the cell is allowed to relax. The relaxation process is shown in figures (b) and (c). Marker points on the cell surface are tracked throughout the tank treading and relaxation processes. Figure shows the evolution of a marker point that is initially located near the rim of the cell. The instantaneous location of the marker point is identified by an angle with respect to the cell s major axis (see figure 1). The angular displacement of the marker point from its initial location (i.e. before the application of shear flow) 0 is of interest and is denoted by 0. The displacement of the marker at the instant of flow cessation is denoted by S. Small values of S mean the flow is stopped when the instantaneous membrane configuration is close to the original configuration. The limiting case is S 0. Larger values of S mean the flow is stopped when the instantaneous membrane is significantly displaced from its original configuration. The limiting case is then S /. The flow is stopped at different times resulting in different values of S in the range 0 to /. Subsequently, a relaxation simulation is done for each case. Figures (b) and (c) show two such relaxation processes with S / and 0.54, respectively. As evident from both these sequences, the cell regains its biconcave shape and the marker point rotates along the membrane gradually returning back to its initial location corresponding to the undeformed state of t 0. Therefore, the RBC shape memory is clearly observed in our simulations. 9

10 Figure 3(a) shows the results from ten relaxation simulations with varying S. The same marker point that is originally located at the rim of the undeformed cell (as in figure ) is followed. The magnitude of the angular displacement / is plotted over time during the relaxation process. For all values of, the marker point returns to its original location resulting in the final becoming zero, S Figure. Simulation results showing the shape memory of an RBC for in shear plane dynamics. (a) The cell is first subjected to a steady shear flow with Ca=0.. The initial shape ( t 0 ) and the location of the marker point (blue bead) are shown, followed by the tank treading motion of the cell in shear flow. The cell assumes an elongated shape as the membrane and interior fluid rotate in a tank treading manner. After a few tank treading cycles the flow is stopped and the cell is allowed to relax. (b) and (c) show relaxation of the cell by stopping the flow at two different time instants for which the membrane displacements at the instant of flow cessation are S / and 0.54, respectively. The cell returns to the biconcave shape and the marker point returns to its original location in the undeformed state. Examples here show two recovery modes: In (b), the membrane rotates clockwise (CW) while the cell rotates counterclockwise (CCW); in (c), the membrane rotates CCW while the cell rotates CW. Here 0.03, 0. 1, and 0. The stress free state is SSF. m 10

11 indicating the presence of the shape memory. The time taken to return to the initial location depends on the value of S. If S 0, the resulting membrane rotation is small, and the marker returns to its original location relatively quickly. For larger values of S, it takes longer for the marker to return to the original location. During the recovery, the membrane may rotate either clockwise (CW) or counterclockwise (CCW) with respect to the cell s major axis in the shear plane as illustrated in figure. For the recovery shown in figure (b) the membrane rotates CW, whereas in figure (c) it rotates CCW. Which direction of membrane rotation is realized depends on the amount that the membrane has been displaced at the time of flow stoppage. Because the top and bottom halves of the cells are symmetric, there are two energetically identical locations that a marker point can return to. For example, a membrane element originally on the top dimple is in an equivalent state when it occupies the bottom dimple. As a consequence of this symmetry, the membrane can return either to its original configuration or to the energetically equivalent configuration. Which one of these two equivalent configurations is acquired depends on how close that configuration is to the deformed configuration of the membrane when the flow is stopped. Accordingly, the membrane rotation during shape recovery can be in either the clockwise or counterclockwise direction. The versus time plots in figure 3(a) generally suggest a complex shape memory response of the cells, particularly for S / In this range, at least three different phases can be identified as marked in the figure. The initial phase is a rapid drop in over a short time immediately after the flow cessation. This is followed by an intermediate phase of a relatively slower and complex decay. Note that the plateau that appears in this stage does not mean a temporary pause in the membrane return. It appears when the marker is near the inflection region of the cell contour. The intermediate phase is followed by a final decay period. Interestingly, the durations and the nature of the curves for the final phases are similar. However, the duration of the intermediate phase varies depending on the value of. S Figure 3(b) shows the time history of the end to end cell length L L / a0 in the shear plane for different values of S. A steep decrease in the cell length is noted immediately after the flow is stopped. In fact, the length drops below that of the undeformed shape implying that the cell undergoes a compression of its major axis in shear plane. The amount of compression increases with increasing S. The compression stage is followed by a slower elongation stage during which the cell length recovers to its undeformed value and the biconcave shape is established. The return of the marker point can be illustrated also using its actual displacement along the arc length of the cell contour. This is shown in the inset of figure 3(a). For larger values of S, these plots show an initial slower membrane rotation followed by a relatively faster rotation. Also notice that the plateau S 11

12 observed in the t plots is not observed here implying that the membrane does not pause during the relaxation. Figure 3. (a) Return of a marker point on the cell surface to its original undeformed location after the flow is stopped. A marker point originally located at the rim of the undeformed cell (as in figure ) is tracked using the angular displacement 0. The figure shows the results from ten relaxation simulations with varying S. For all values of S, the marker point returns to its original location resulting in the final becoming zero. Inset: the same marker point is tracked using arc length displacement. (b) End to end cell length L L / a0, and (c) total membrane strain energy Wtotal versus time during relaxation. Only the relaxation part is shown for all figures. Here 0. 03, 0. 1, and m 0. The stress free state is SSF. 1

13 Some insights into the relaxation process can be gained by considering the decay of the energy stored in the membrane. Figure 3(c) shows the total strain energy of the membrane W total as a function of time. Wtotal is obtained using Eq. (.1) integrated over the entire cell surface. Similar to L and, Wtotal shows a complex decay for larger values of S Wtotal. Immediately after the flow is stopped drops rapidly within a short time. This is followed by a slower decay in the intermediate phase, and then a relatively faster decay in the final phase. The intermediate phase gets extended as approaches / and appears as a plateau. S The distribution of the membrane strain energy W S (Eq..1) over the cell surface at different instants during the recovery process is shown in figure 4 for two cases, S 0 and /. For both cases, the strain energy is mostly stored in the rim of the cell during the shear flow. For the smaller S, the excess membrane energy is quickly dissipated and the ground state is recovered. For the larger S, the relaxation is prolonged, and the three phases of the relaxation as noted earlier can be observed. A large drop of strain energy occurs within a short time after the flow cessation. This is the initial compression phase. Subsequently, regions of local higher energy emerge at the opposite ends of the cell. The appearance of these higher energy regions is due to the compression of the cell. These regions persist for a long time as the cell slowly elongates to return back to the biconcave shape, and result in a plateau in the W total t curves (figure 3c). The high energy regions take a much longer time to dissipate and act to slow down the membrane rotation during the intermediate phase. The maximum value of W S at the instant of flow cessation is higher for larger values of S, resulting in a greater amount of cell recoil, and hence a greater amount of compression. This, in turn, increases the time for the membrane to rotate back to the original configuration. As these regions are dissipated, the final phase of the membrane rotation follows. 13

14 Figure 4. Distribution of membrane strain energy over the RBC surface at different time instances during the recovery for S / 0.03 (a), and (b). Contours range from 5X10 5 (blue) to (red). Maximum values are in (a), and in (b). A locally high energy region near the rim of the cell in (b) is noted to persist for a long time (shown by the arrow). Here 0. 03, 0. 1, and m 0. The stress free state is SSF. Figure 5. The cell axis rotation R during the recovery is plotted as a function of the membrane displacement S at the time of flow cessation. Positive and negative values represent CCW and CW rotations, respectively. The parameters and stress free state are the same as in figure 3. 14

15 3.1. Timescales The relaxation as shown in figure 3 suggests a complex response that is governed by more than one timescale. For larger values of S approaching /, three phases in relaxation were noted. It was also noted that the duration and shape of the curves in the initial compression phase and in the final rotational phase are similar for different values of S. Therefore, at least two time scales can be extracted from the curves in figure 3. The initial phase that is characterized by the rapid compression of the RBC gives a fast time scale. An exponential curve can be fitted to the initial phases of the W total t plots to obtain a decay constant representing the fast time scale that we denote by t c. Motivation for assuming an exponential decay derives from the optical tweezer stretch relaxation experiments. The simulation results yield the fast time scale 0.1 seconds using an RBC membrane shear elasticity tc 6 6 G s.510 N / m, a m, and It may be noted that the dimensionless times given in the figures can be readily converted to seconds using these parametric values, and the definition of the dimensionless time given in section. For the above parameters, the scaling from dimensionless time to dimensional time (in second) is about 0.1. As the arc length displacement shown in the inset of figure 3(a) suggests, during the initial compression phase the membrane rotation is small. On the other hand, in the final phase, the change in cell length is small, but the membrane rotation is significant. Therefore, the final phase gives a time scale that characterizes the membrane rotation. The rotational time scale denoted by t r can be extracted from the t curves in figure 3(a) assuming an exponential decay. The simulations yield 6 seconds using the aforementioned parameters. tr Remarkably, this is more than an order higher than the compressional time scale t c. It should be noted that the actual duration of the rotation phase is longer than t r, and for the cases shown in figure 3, it is about 30 seconds. For the intermediate phase, the response is too complex to assume an exponential decay, and both membrane rotation and elongational deformation coexist. The duration of this phase depends on the value of S, and is in the same range as the duration of the final phase for larger values of S /. Also note that the rotational time scale tr does not represent the time for a complete return of the membrane to its original configuration as it is based on the final phase of relaxation only. The time for the complete return is longer than t r. As noted above, the complete return time depends on Since approaches zero asymptotically as 15 S. t, we define a return time t95 as the time to reach / S The simulations presented in figure 3(a) give t seconds, which is significantly longer than the typical recovery times observed in optical tweezer and micropipette stretch relaxation experiments Cell axis rotation

16 It was noted in the previous section that the RBC membrane returns to its original configuration by rotating either CW or CCW. The rotation of the membrane must be balanced by a rotation of the entire cell in the opposite direction since there is no external torque present during the recovery. As shown in figure (b), a CW rotation of the membrane (with respect to the cell axis) results in a CCW rotation of the RBC s major axis in the shear plane. In figure (c), a CCW membrane rotation is accompanied by a CW rotation of the RBC. The amount of cell rotation during the shape recovery also depends on S. Figure 5 shows the amount of rotation R of the cell axis as a function of S where R S, and, S and are the orientations of the RBC major axis with respect to the flow direction at the time of flow cessation and at the end of the recovery process, respectively. In the figure, positive values indicate a CCW rotation and negative values indicate a CW rotation. The magnitude of R generally increases with increasing S. The maximum amount of cell rotation takes place when the membrane phase is maximally displaced, that is, S / If the flow is stopped when the membrane is approximately at its original phase, i.e. S / 0, no membrane rotation and cell rotation occur. Therefore, the cell can end up in an orientation with varying inclination between the extremes delimited by stopping the flow when the membrane is displaced by /. The two different modes of recovery, namely, the CW and CCW rotation of the cell, were also observed by Fischer (004) in his experimental study. The varying degree of cell axis rotation during the recovery was also noted in the experiment Effect of RBC stress free state The RBC considered in the previous subsections is assumed to have a nearly spherical stress free (SSF) state. Therefore, the membrane stress and strain energy of the resting biconcave shape of the cell are not zero. We now investigate the role of different stress free states. Specifically, the resting biconcave shape is now taken to be stress free (BCSF case). Go and stop simulations are performed in a similar manner as those described previously for the SSF cells. Figure 6 and an associated multi media view show the recovery process for a BCSF cell by stopping the flow at two different times. Similar to the SSF cells as discussed in 3.1.1, the BCSF cells also exhibit the shape memory. As seen in the figure, the RBC recovers the resting biconcave shape, and the marker points move back to their original locations with respect to the major axis of the cell. Furthermore, both modes of shape return as observed for the SSF cells, namely, the CW membrane rotation accompanied by a CCW cell rotation, and the CCW membrane rotation accompanied by a CW cell rotation, are also observed for the BCSF cells. Note that the SSF and BCSF configurations considered represent the two nearly limiting states of the stress free configuration possible. Thus, the shape memory exists irrespective of the stress free state of the RBC. The recovery process for the BCSF and SSF cells is compared in figure 7 using and L. It is observed to be qualitatively similar for both cases. The three phases noted earlier for the SSF cells are also evident in the figure for the BCSF cells: An initial phase over a short time showing a rapid compression of the RBC 16

17 Figure 6. Existence of shape memory in the BCSF state. The RBC recovers the resting biconcave shape and the marker points return to their original locations after the flow is stopped. Two modes of shape recovery are also observed here: (a) CCW membrane rotation accompanied by CW cell rotation, and (b) CW membrane rotation with CCW cell rotation , 0. 1, and m 0. An animation is provided as multi media file. Figure 7. Effect of stress free states on the RBC relaxation: SSF (continuous lines) and BCSF (dash lines). t and L t are shown in (a) and (b), respectively for different S , 0. 1, and m 0. 17

18 major axis, followed by an intermediate phase characterized by a membrane rotation and an elongation of the cell, and a final phase of membrane rotation as the biconcave shape is fully recovered. As observed before for the SSF cells, the t curves for the BCSF cells also suggest the presence of multiple time scales. The striking difference between the BCSF and SSF cells is that the recovery is faster for the former than the latter. The return time t95 for the BCSF cells is about 3 16 s, as opposed to 8 60 s for the SSF cells as noted in Exponential curves are also fitted to the initial and final phases of the BCSF cell response to extract the fast compressional time scale tc and the slower rotational time scale t r. The simulations yield tc 0. 1 seconds and tr 1. 5 seconds for the BCSF cells. In contrast, for the SSF cells in 3.1., we obtained tc 0. 1 seconds, and tr 6 seconds. Therefore, the fast time scale is independent of the stress free state. In contrast, the rotational time scale is dependent on the stress free state; the BCSF cells have a faster rotational time scale than the SSF cells. The duration of the intermediate phase for the BCSF cells is dependent on S, and is of the order of the final rotational phase, as it was for the SSF cells. Another significant difference between the two stress free states is the greater amount of compression observed in the BCSF cells during the initial phase as evident from figure 7(b). The amount of L min L L / L, where L 0 is the end to end length of the undeformed RBC, is given in figure 8(b) as a function of S. For S 0, two stress free states show almost no compressional overshoot since the membrane configuration at the time of flow stoppage is close to the resting configuration. Differences between the two states become clear as S / as the BCSF cells show a much larger compressional overshoot. compression overshoot defined as 0 0 Interestingly, the amount of cell rotation during the recovery is independent of the stress free state, and depends only on S. This is shown in figure 8(a) where the major axis rotation R is compared for the BCSF and SSF cells. A similar amount of axial rotation is observed for either stress free configuration for a given value of. S Deformation driven recovery Fischer (004) observed the shape recovery by membrane rotation in his experiment and termed it as the tank treading mode of recovery. For the specific examples shown in figure, the marker point returns to its initial undeformed location by rotating about the fluid interior without a significant cell deformation. Therefore, these examples are likely to be the ones that Fischer termed as the tanktreading recovery. He also proposed, but did not observe, that another mode of recovery might exist as the membrane could return to its original configuration by deformation only. In the deformation driven mode the transient RBC shape would deviate significantly from the biconcave shape and a clear 18

19 Figure 8. Comparison of SSF and BCSF cells relaxation. BCSF is represented by red triangles and SSF by blue circles. (a) Amount of cell axis rotation R after flow stoppage as a function of membrane displacement. (b) Normalized compression overshoot L. Parameters are the same as in figure 7. S Figure 9. Deformation dominated recovery is observed for the BCSF cells at reduced values of. (a) 0.01, and (b) Shear flow is applied for 0 t 40. The flow is stopped at t 40, and the RBC is allowed to relax. A significant departure in the transient cell shapes from the biconcave one is noted in (b) during the recovery. The recovery process is also much longer than that in (a). 0. 1, and 0. m 19

20 membrane rotation may not be observable. Fischer (004) further noted that the bifurcating behavior between the CCW and CW membrane rotations as discussed in the previous section might not be present for the deformation driven recovery. In this section we consider the existence of such deformation driven return modes in our simulations. In general, the BCSF cells exhibit some degree of deformation driven recovery. This is apparent when we compare figures and 6 where the SSF and BCSF cases, respectively, are shown. For the SSF cells, no significant deviation from the undeformed biconcave shape is observed during the recovery process. In contrast, a noticeable deviation from the undeformed shape is evident for the BCSF cells. In particular, the BCSF cells develop deeper dimples before recovering to the undeformed biconcave shape. We find that the deformation driven recovery becomes more prominent for the BCSF cells at reduced values of membrane bending to strain energy ratio. Note that the results in figures and 6 are for To study the role of, we consider simulations of the BCSF cells at 0.01 and The results are shown in figure 9. For 0.01, the transient RBC shapes show even more deviation from the biconcave shape compared to what was observed in figure 6. Eventually the cell recovers to the resting biconcave shape. When is further reduced to 0.001, the transient RBC shapes become distinctly different and significantly depart from the biconcave shape with the appearance of multiple folds. Although the RBC eventually recovers the resting biconcave shape without causing any numerical instability, it takes a significantly longer time compared to the cases at 0.01 and higher. Therefore, the deformation driven recovery is observed for the BCSF cells at reduced. In contrast, the tanktreading recovery is observed for the SSF cells, and also for the BCSF cells at higher. For intermediate values of, the BCSF cells exhibit a combination of the two recovery modes. The influence of on the shape recovery is also seen in the versus time plots in figure 10(a). Here results are shown for =0.03 and A longer time is needed for the membrane to return back to its original configuration as is reduced. We extract the membrane return time t95 as 13 and 5 seconds for = 0.03 and 0.001, respectively. However, for the deformation driven recovery, does not give a full description of the shape memory return. This is illustrated in figure 10(b) where shown. For = 0.001, while becomes zero at around t 300, the cell length 0 L versus time is L has yet to reach its undeformed state. Therefore, as becomes smaller the recovery becomes more and more deformation driven, and it takes a much longer time for the deformed shape to return to the biconcave shape than the membrane elements to return back to their initial locations Role of shear elasticity and bending rigidity The observed role of leads us to ask what the fundamental membrane properties are that confer the shape memory of an RBC. To answer this, we consider a recovery simulation where is further reduced to The result from this simulation is shown in figure 11. It is quite clear that the RBC can no longer regain its biconcave shape even after a long time (nearly 16 s). However, the membrane phase indeed returns to its original position as shown in the figure using a marker point. Therefore, when the

21 Figure 10. Influence of on RBC shape recovery. BCSF cells are considered for 0.03 (continuous line), and (dash line). Here S / 0. 43, 0. 1, and 0. m Figure 11. Simulation result for The RBC shapes and the marker points before the flow starts (a) and after a long relaxation period (b) are shown. The flow is stopped at t 39. Two views of the cell are shown in (b) at t 139. The biconcave shape is not recovered even after a long time (about 16 s), although the marker points move back to their original locations. 0. 1, and 0. m Figure 1. Simulation of shape relaxation for initially oblate capsules at = 0.01 (a), and 0 (b). Shear flow is applied until t = 38 and 37, respectively. The capsule is allowed to relax thereafter. While the marker points return to their original locations for both (a) and (b), the cell shape in (b) does not recover to its original shape. 0. 1, m= 0. 1

22 bending rigidity becomes very small, the membrane can return to its original phase, but the overall (global) biconcave shape of the RBC is not recovered anymore. The above observation suggests that in the limit that the bending rigidity is completely absent, the membrane would return to its original phase, but the cell would not recover its undeformed shape. We further confirm this by considering relaxation simulations of initially oblate capsules at = 0.01 and 0. The ratio of the short to long axes of the undeformed capsule is taken to be 0.5. Shear flow go and stop simulations as described before are performed with the oblate capsules. Figure 1 shows the capsule shapes during relaxation. For both values, the marker returns to its original location of t = 0. Thus the rotational shape memory is present for non spherical capsules even in the absence of bending rigidity. However, the overall (global) shape is recovered for = 0.01, but not for = 0. Buckling of the membrane occurs during relaxation for = 0 case. Therefore, the above simulations at varying for both RBC and oblate capsules suggest that only shear elasticity is necessary for the membrane phase to return to its original position, but bending rigidity is necessary for recovery of the global shape Effect of internal and external fluid viscosity The influence of viscosity ratio is considered next. Simulations are performed using the BCSF cells by varying from 0.1 to.0. The recovery process is shown in figure 13 for different values of but with the same membrane displacement S. As seen in the figure, the recovery process slows down with increasing. The qualitative nature of the t and L t curves are, however, similar over the range of considered, and the cells always exhibit the shape memory. As before, the recovery occurs in three phases. The initial response is characterized by a rapid compression of the cell length over a short time. The intermediate phase is longer and comprised of an elongation of the cell length returning back to its undeformed state along with a rotation of the membrane. The final phase shows almost no deformation, but a significant amount of membrane rotation. Multiple time scales are also evident from the response curves. A fast time scale characterizes the compression phase, and the slower time scales characterize the intermediate and final phases. The time scales are extracted from the response curves and plotted in figure 14. As seen here, the time scales are dependent on. As increases from 0.1 to.0, the faster time constant tc increases from 0. s to 0.5 s, and the slower rotational time constant tr increases from about to 5 seconds. The actual duration of the phases also increases with increasing. The durations of the initial compressional phase, the intermediate phase, and the final rotational phase increase from about 1 to 3.5 seconds, 5.5 to 4 seconds, and 8 to 1 seconds, respectively. The 95% return time t95 increases from about 13 to 44 seconds. Notice that the increase in the time scales is not proportional to the increase in ; for a 0 fold increase in, the time scales increase by a factor of about 3 to 4.

23 Figure 13. Influence of viscosity ratio on the RBC shape memory response. (a) Return of a marker point to its original location is shown using the angular displacement. (b) The end to end cell length is shown. Results are shown for recovery part of the simulations. BCSF cells, 0.03, 0. m Figure 14. Influence of viscosity ratio on various time scales of the RBC shape memory response: the fast compressional time t c (red squares), the slower rotational time scale t r (green circles), the recovery time t 95 (blue triangles). BCSF cells, 0.03, m 0. 3

24 3.1.8 Effect of membrane viscosity Next we consider the influence of membrane viscosity on the RBC shape memory response. The physiological value of the dimensionless membrane viscosity m as defined in is in the range We perform simulations using the BCSF cells with varying m from 0 to 100 but keeping and constants. The shape memory response is shown in figure 15 using the phase angle of a marker point, and the end to end cell length. Results are shown for the same value of S. As seen here, the influence of membrane viscosity is to slow down the recovery. The response curves are similar to those observed before. The RBC undergoes a rapid compression of its length immediately after the flow is stopped. The amount of compression decreases and the duration of the compressional phase increases with increasing m. This is followed by an elongation of the cell length nearly to its undeformed state coupled with a slow membrane rotation. The duration of the intermediate phase also increases with increasing m. The final phase is characterized by a significant amount of membrane rotation but nearly no change in the cell shape. Multiple distinct time scales are also evident in the response curves. The relevant time scales, namely the fast compressional scale, the slower rotational time scale, and the 95% recovery time for the membrane phase to return to its original location, are extracted from the response curves and plotted in figure 16 as functions of m. The fast time scale t c is in the range 0.1 to 1 s, the slower scale tr is in the range to 4 s, and t95 is 1 to 5 s. The different time scales are observed to increase with increasing m. However, a nearly 100 fold increase in m results only in a factor of increase in tr and t 95. Furthermore, for the physiological range of, only a marginal increase in the time scales is observed compared their values at m = 0. In our numerical model, the membrane viscosity is assumed to be independent of shear rate. One may ask if the membrane viscosity of an RBC is independent of shear rate, and whether a shear ratedependent viscosity would result in significant changes in the shape memory response. It should be noted that experimental determination of the membrane viscosity is based on the assumption of a constant viscosity. This simplifies the experimental determination as well as the numerical implementation. Further, our predictions with and without membrane viscosity show qualitatively similar response curves, only the response times are changed. Therefore, even if a shear rate dependent viscosity is used, we expect similar qualitative behavior although the response times would be affected. 3. Off plane relaxation In this section we consider the shape recovery of an RBC whose axis of revolution does not initially lie in the plane of shear, and is tilted away as shown in figure 17. The off plane `tilt is defined as the angle between the cell s axis of revolution and its projection on the shear plane, and is denoted by. The m 4

25 Figure 15. Effect of membrane viscosity on the RBC shape memory response: (a) Return of a marker point to its original location is shown using the phase angle for different values of m. (b) End toend cell length. Only the recovery part is shown. BCSF cells, S / 0. 4, 0. 3, Figure 16. Effect of m on recovery time scales: the fast compressional time t c (red squares), the slower rotational time scale t r (green circles), the recovery time t 95 (blue triangles) as extracted from the response curves in figure 15. 5

26 limiting orientations are 0 corresponding to the alignment of the axis in the shear plane, and / corresponding to the alignment of the axis with the direction of the vorticity. The cell with an initial tilt 0 u y,0,0. We consider so that the axis of revolution is initially close to the vorticity axis. First, the off plane dynamics observed in presence of the shear flow is described in figure 18. After the shear flow is started the cell s axis of revolution quickly aligns with the shear plane and the membrane elements are redistributed by moving along the vorticity axis. The initial alignment and redistribution are shown in the top panel of figure 18. The membrane elements initially located on the dimples move out to the rim of the cell and those along the rim move to the mid plane. Two marker points are used in the figure to show the movement of the membrane elements. The green marker point is originally located near the dimple and moves to the rim; the blue marker point is initially located at the rim and moves to the mid plane. Thereafter, a tanktreading motion about the vorticity axis ensues which is shown in the bottom panel in the figure. During the tank treading, the green marker barely moves, while the blue marker traverses the cell surface in the shear plane. is subject to the shear flow After a few tank treading cycles, the flow is stopped, and the cell is allowed to relax. The shape memory response is illustrated in figure 19 and an associated multi media file. The same two marker points shown in figure 18 are followed here. After the flow is stopped, the green marker gradually moves back to the dimple and the blue marker moves back to the rim. Thus the membrane elements return back to their initial positions. At the same time the biconcave shape of the RBC is recovered, and the axis of revolution moves out of the shear plane. Therefore, the shape memory is observed for off plane relaxation as well. Interestingly, the membrane returns to its original configuration by moving normal to the shear plane though the flow had already stopped. The movement of the membrane is accompanied by a rotation of the axis of revolution of the RBC in the opposite direction and away from the shear plane. The rotation of the axis could be either CW or CCW about the x axis. Figure 19 shows an example of a CCW axial rotation and CW membrane rotation when viewed looking down the x axis. A CW axial rotation accompanied by a CCW membrane rotation is also observed in the simulations. Which direction of membrane rotation is realized depends on the membrane configuration at the time the flow is stopped. As noted in 3.1 for the in plane dynamics, due to the symmetry of the biconcave shape the membrane would take the shortest path to return to its resting configuration resulting in either a CW or CCW rotation. The same mechanism also exists for the off plane dynamics. To facilitate tracking of the off plane membrane displacement, we define an off plane phase angle which is the angle between a marker on the cell surface and the axis of revolution, as shown in figure 17. The evolution of for a marker originally located at a dimple and along the axis of revolution is shown in figure 0(a). Also shown in figure 0(b) is the time evolution of the tilt angle. Immediately after the flow starts, the cell axis aligns with the shear plane ( 0), and the marker point moves to the rim ( / ). After the flow is stopped, the membrane rotates off plane and return to either of its 6

27 Figure 17. Schematic of an RBC with an off shear plane orientation of the axis of revolution. The shear plane is the (x y) plane. The off plane tilt of the axis of revolution is denoted by. An off plane phase angle is used to define the angular displacement of a marker point normal to the shear plane. The view shown here is looking down the x axis. Figure 18. Off plane dynamics during shear flow. Time lapse images are shown viewed along the vorticity axis. The cell axis is initially placed with an off plane angle 0 5 / 1, or 75 O, from the plane of shear. Two marker points, one (green) initially located along the axis of revolution and another at the rim (blue) are tracked. After the flow starts, the cell axis quickly aligns with the shear plane, and the point initially located near the dimple moves to the side of the RBC, and the one located at the rim moves to the mid plane. The alignment of the axis is shown in the top panel. This is followed by a steady tank treading as shown in bottom panel. SSF cells, 0. 1, 0. 03, 0, Ca=

28 Figure 19. Simulations of off plane shear flow go and stop. The initial RBC orientation and its dynamics during shear flow were shown in figure 18. The flow is stopped at t 39, and only the relaxation part is shown here. The top row shows images looking on the shear plane. The bottom row shows images looking towards the x axis. The same marker points as in figure 18 are followed here. During the relaxation, the green and blue markers move back to their original locations (dimple and rim, respectively). The resting biconcave shape is also recovered. An animation is provided as a multi media view. 8

29 resting configurations 0, or. The rotation of the axis of revolution is also evident in figure 0(b). As shown in the figure, both CW and CCW rotations are observed depending on the when the flow is stopped. The marker either returns to the same dimple ( 0), or to the other ( ), resulting in a CCW or CW rotation of the membrane and a CW or CCW rotation of the cell axis, respectively. The off plane shape memory is observed for both the BCSF and SSF cells. Figure 1(a) compares the responses for the two stress free states. Similar to the in plane relaxation, the off plane relaxation is slower for the SSF cells compared to the BCSF cells. Figure 1(b) compares the off plane and in plane relaxation responses using the respective phase angles and. The qualitative nature of the response curves remains same for the in plane and off plane recovery. Multiple disparate time scales, of the same orders as those exhibited for the in plane relaxation, are also present for the off plane relaxation. For the off plane recovery of the BCSF cells, the rotational relaxation time tr 3 s and the 95% recovery time t 95 is 0 40 s. For the SSF cells, they are about 10 s and s, respectively. 4. Discussion It has been well known that deformed red blood cells regain their resting biconcave shape when the external forces are withdrawn. This type of shape recovery has been observed in several in vitro scenario such as micropipette and optical tweezing stretch relaxation experiments (Evans 1989, Henon et al 1999, Mills et al 004, Li et al 005), and also during passage of RBCs through narrow slits (Mebius and Kraal 005). However, as envisioned by Fischer (004), the presence of shape memory means more than just the recovery of the biconcave shape. The membrane elements, if displaced from their resting locations with respect to the cell axis, must also return to their original locations after withdrawal of the forces. In the aforementioned experiments, the membrane elements do not experience a significant rotational displacement over the cell surface from their original locations. Hence, these experiments do not reveal a complete shape memory. By contrast, in the shear flow go and stop experiment designed by Fischer (004), the membrane elements displaced by the shear flow were observed to return to their original locations after flow cessation indicating the presence of a complete shape memory of RBCs. Apparently, the RBC shape memory in 3D has not been addressed using computational models despite a significant development in this area in recent years. It is however not obvious whether the computational models can predict the shape memory, and it is also unknown what `ingredients are needed in the models for this purpose. Here we employ a continuum description of the RBC neglecting the detailed molecular structure of the cytoskeleton. We mimic Fischer s (004) shear flow go and stop experiment in our simulations, and predict the presence of a complete shape memory of the RBC. The prediction is quite remarkable because the origin of shape memory lies in the molecular structure of the cytoskeleton and cytoskeleton lipid bilayer connections. Although these connections are not explicitly resolved in the continuum model utilized, the model takes into account cytoskeleton elasticity by treating the membrane as a single layer composite in which the cytoskeleton and lipid bilayer are strongly coupled. Recent experiments on the nonlinear relaxation of the cell membrane stress under tension implies that there is some dissociation of the cytoskeletal connections but a long time 9

30 permanent modulus still remains indicating that some connections are still retained (Yoon et al. 008). This notion is supported by the omnipresence of shape memory observed in Fischer s (004) experiments indicating that some cytoskeleton bonds must remain. The simulations presented here represent the first 3D computational study predicting the complete shape memory of RBCs. We perform a novel shear flow go and stop simulation in which the RBC s axis of revolution initially does not align with the plane of shear. Once the flow starts, the membrane elements are displaced normal to the shear plane. After the flow is stopped, the membrane elements are observed to return back to their original locations by moving again normal to the shear plane. The axis of revolution also moves out of the shear plane. This observation is quite remarkable because one might naively think that upon flow cessation the RBC axis could simply remain aligned with the shear plane and the membrane elements could just rotate parallel to the shear plane to return back to geometrically similar locations. But instead, both the axis of revolution and the membrane elements rotate normal to the shear plane to return back to the energetically equivalent configurations. Thus the shape memory is exhibited even when the membrane is displaced normal to imposed shear. The simulations allow a detailed analysis of the shape memory response of the RBCs and the role of different parameters, such as membrane shear elastic modulus and bending rigidity, internal and external fluid viscosity, and stress free states of the cytoskeleton. We observe that when the membrane displacement is significant the RBCs exhibit a complex relaxation response that occurs in three phases: The initial phase which occurs over a short time is characterized by a rapid compression of the cell length below its undeformed state. The compression phase is accompanied by a significant dissipation of the membrane strain energy, but a negligible rotation of the membrane elements. The initial phase is followed by an intermediate phase over a much longer time during which the RBC elongates to regain its biconcave shape, and the membrane elements start to rotate back towards their original locations. The final phase which also occurs over a long time is characterized by mostly the rotational movement of the membrane elements along the cell surface as they return to their original locations. The presence of such multiple phases is not observed in optical tweezer and micropipette based stretch relaxation experiments. 30

31 Figure 0. Off plane relaxation of the RBC. Time evolution of (a) the membrane phase angle and (b) the tilt angle of the axis of revolution are shown throughout the shear flow go and stop simulation. Here 0 5 / 1, and a marker originally located along the axis of revolution ( 0 0) is followed. The thick black line corresponds to the shear flow part, and the colored lines represent the relaxation response observed by stopping the flow at different times. The cell axis and the maker point return to their original locations after relaxation via CW (dash lines) or CCW (continuous lines) rotations. Figure 1. (a) Comparison of off plane relaxation for BCSF (continuous lines) and SSF (dash lines) cells. (b) Comparison of off plane (continuous lines) and in plane (dash lines) relaxation. The respective membrane phase angles and are shown. Only the relaxation part is shown. 31