In red blood cell (RBC) diseases, the spleen contributes to anemia by clearing the damaged RBCs, but its unique ability to mechanically challenge RBCs also poses the risk of inducing other pathogenic ef- fects. We have analyzed RBCs in hereditary spherocytosis (HS) and hereditary elliptocytosis (HE), two typical examples of blood disor- ders that result in membrane protein defects in RBCs. We employ a two-component protein-scale RBC model to simulate the traversal of the inter-endothelial slit (IES) in the human spleen, a stringentbiome- chanical challenge on healthy and diseased RBCs that cannot be di- rectly observed in vivo. In HS, our results confirm that the RBC loses surface due to weakened cohesion between the lipid bilayer and the cytoskeleton, and reveal that surface loss may result from vesicu- lation of the RBC as it crosses IES. In HE, traversing IES induces sustained elongation of the RBC with impaired elasticity, and frag- mentation in severe disease. Our simulations thus suggest that in inherited RBC disorders, the spleen not only filters out pathological RBCs but also directly contributes to RBC alterations. These results provide a mechanistic rationale for different clinical outcomes docu- mented following splenectomy in HS patients with spectrin-deficient and ankyrin-deficient RBCs, and offer insights into the pathogenic role of human spleen in RBCdiseases.

Spleen | Erythrocytes | Blood disorders | Vesiculation | Cell  fragmenta-  tion

body.

Healthy RBCs have remarkable deformability and stabil-

ity, which enable them to undergo repeated deformation in microcirculation during their lifespan of approximately 120 days. In blood disorders of hereditary spherocytosis (HS) and elliptocytosis (HE), however, defects in RBC membrane pro- teins weaken the cohesion between the lipid bilayer and the cytoskeleton (in HS) or the integrity of the cytoskeleton (in HE), thereby compromising the deformability and stability of RBCs (7). In HS, defects in cytoskeletal proteins or membrane proteins that anchor the cytoskeleton to the lipid bilayer desta- bilize the RBC membrane, leading to membrane surface loss through release of vesicles (8, 9). Loss of membrane surface causes a decrease in the ratio of surface area-to-volume (S/V) of RBCs and promotes the formation of spherical RBCs. Due to their reduced deformability, spherical RBCs are removed prematurely by the spleen, resulting in mild to severe forms of anemia, dependent on the extent of surface area loss (10). In HE, defects in the cytoskeletal proteins impair the elasticity of RBCs by disrupting the integrity of the membrane cytoskele- ton (9, 11). As a result, RBCs in HE undergo irreversible elongation after large deformation and progressively transform into elliptical shape (12). In severe forms of HE, RBCs become mechanically unstable, resulting in cell fragmentation and ly- sis (9, 11). The ensuing ill-shaped RBCs and cell fragments are cleared from circulation by the spleen, causing hemolytic anemia.

The function of spleen in sensing and clearing RBCs with

H.L., L.L., X.L., P.B., M.D., G.E.K and S.S. designed research. H.L.and L.L. performed research. H.L., L.L., X.L, P.B., M.D., G.E.K and S.S. analyzed data.  H.L., L.L., X.L., P.B., M.D., G.E.K and

19. wrote the article.

The authors declare no conflict of interest.

1 H.L. and L.L. contributed equally to this work.

2 To whom correspondence should be addressed. E-mail: george_karniadakis@brown.edu, ssuresh@mit.edu.

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXXPNAS    | April 6, 2018 | vol. XXX | no. XX | 1–6

alterations in their size, shape and deformability, has been elucidated through in vivo (13, 14), ex vivo (3–5) and in  vitro (15–17) experiments, and computational modeling (18– 20). However, the role of IES in filtering RBCs with significant membrane protein defects, as seen in HS and HE, has not been investigated in sufficient detail. To address these questions, we employ a two-component coarse-grained molecular dynamics (CGMD) RBC model (21) to simulate the traversal dynamics of healthy, HS and  HE  RBCs  through  IES.  In  particular, we introduce specific transmembrane protein defects into the RBC model and correlate the molecular basis of the diseases to different clinical expressions of RBCs in HS and HE.

Results

IES model and two-component RBC model. To capture the geometry of IES in the human spleen, we  construct a model that comprises four  solid  elements, as  illustrated  in  Fig.1. The two  vertical bars represent annular fibers with a width of   1 µm, whereas the two horizontal bars represent endothelial cells. The thickness of the slit wall is 1.89 µm and the width and height of the slit is 4 µm and 1.2 µm, respectively, based on the experimentally documented slit geometry (18).

Fig. 1. Simulating RBC passage through IES. The membrane of the RBC is explicitly represented by CG particles. A: actin junctions, B: spectrin particles, C: glycophorin particles, D: band-3 particles, E: lipid particles. The width and height of the simulated slit are 4 µm and 1.2 µm, respectively. The width of the vertical bars is 1 µm and the thickness of slit wall is 1.89 µm.

We employ the computer code OpenRBC (21) to model healthy and diseased RBCs. As shown in Fig.1, the lipid bilayer and the cytoskeleton as well as transmembrane proteins are explicitly represented in the RBC model. This allows simulations of RBC vesiculation and of cell morphological changes induced by protein defects in blood disorders. Further details of the RBC model can be found in SI.

PressuregradientsandS/Vratiodeterminethepassageof RBCs through IES. Over the lifespan of about 120 days, the cytoskeleton of the RBC stiffens and its membrane compo- nents undergo degradation, resulting in loss of surface area through release of vesicles (22–24). Senescent RBCs with reduced surface area become less deformable and thus are amenable to sequestration and removal in the spleen. In this section, we simulate healthy RBCs with reduced surface area traversing the splenic IES. Pressure gradients of 3, 5, 8, 10,  15 and 20 Pa µm−1 are applied to drive the RBC through IES. These pressure gradients fall within the range found in in vitro experiments (17), i.e., 1–30 Pa µm−1. Such values were sufficient to reproduce the physiological dynamics of RBCs transiting through IES in rat spleen (14).   First,  we  model    a RBC with surface area of 140 µm2 and volume of 90 µm3. Our simulation results show that when driven by a pressure

gradient of 3 Pa µm−1, the RBC is retained by IES (Figs.2A and B). When the pressure gradients  are  equal  to  or  larger than 5 Pa µm−1, RBCs are able to pass through IES. This critical pressure gradient is  on  the  same  order  of  magnitude as  the  pressure gradient of              1 Pa µm−1, which was sufficient to drive RBCs  of  all  sizes  found  in  blood  through  the  slits in microsphere experiments, designed to mimic RBC passage through IES (25). The origins of the difference between the critical pressure gradient inferred from this work and the val-  ues reported from previous studies are discussed in detail in    SI.

Fig.2.(A–B) A RBC with surface area of 140 µm2 and volume of 90 µm3 is retained by IES under a pressure gradient of 3 Pa µm−1 because of an insufficient driving force. (C–D) A RBC with surface area of 110 µm2 and volume of 90 µm3 is retained by IES under a pressure gradient of 8 Pa µm−1 due to reduced surface area. (E–H) Four sequential snapshots of the RBC with surface area of 140 µm2 and volume of 90 µm3 passage through IES under a pressure gradient of 20 Pa µm−1 (see Movie S1). Only one half of the RBC is displayed for clarity.

Figures2(E–H) illustrate a sequence of shape changes of the RBC (surface area of 140 µm2 and volume of 90 µm3) during its deformation though IES.  When  the  RBC  moves  into  the slit (Fig.2F), the portion inside the slit is being squeezed  whereas the rest of the RBC membrane is expanded to accom- modate the excluded volume, thus forming a dumbbell shape. As the RBC moves toward the downstream side of IES, the downstream bulge expands while the upstream bulge shrinks. Subsequently, the cell membrane in the slit infolds toward the cell body and creates a concave region, as shown in  Fig.2G. After crossing the IES, the deformed RBC  gradually  spreads out the inward-folded membrane, turning to a bullet-shape (Fig.2H). The dynamics of the RBC  model  passage  through IES are consistent with the in vivo microscopic observations of the transilluminated rat spleen (14) and the  in vitro microflu- idic study of human RBCs passage through IES-like slits (17). When driven by increased pressure gradients, the RBC dy- namics are similar except that the  higher  pressure  gradients lead to faster RBC traversal.

Next we  show results for cases where the pressure gradi-  ent is 5 Pa µm−1 and the surface area of RBCs is reduced from 140 µm2 to 100 µm2 in decrements of 10 µm2. For the surface areas of 130 µm2 and 120 µm2, RBCs are still able to pass through IES. However, in the case of the surface area of 110 µm2, representing about 21% surface area loss, the simu- lated  RBC  is  retained  by  IES  (Figs.2C  and  D).  This  result is consistent with prior ex vivo experimental observation that RBCs with more than 18% surface area loss were mostly en- trapped in the spleen (5). This finding is also validated by an analytical model (18, 26) that defines the relationship between the critical surface area and volume for healthy RBCs, beyond which traversal through IES is predicated to be compromised. Given the geometry of the slit, the minimum surface area (A), below which the RBCs with fixed volume (V ) are retained by the slit, is given by (18,26),

A= 4{2π[f−1(V)]2 − πf−1(V)g1(V)}

+ 2πL g (V ){ Ds+ Ls− Ls sin[g2(V )] } [1]

s  2 2 2

2g2(V )

where V = 2[ 4 πR3 − 1 πh2(3R − h)] + 2πAcyc. All other sym-

bols are defined in SI. This model predicts a critical surface  area of 113.1 µm2 for the RBC volume of 90 µm3, which vali- dates the critical area obtained from our simulations. Then, we increase the pressure gradients to 8, 10, 15 and 20 Pa µm−1, respectively, and we find  that  RBCs  with  surface  areas  of 110 µm2 and 100 µm2 still cannot pass through IES when driven by these increased pressure gradients.

TraversingIEScausesintra-splenicvesiculationofRBCsin HS. In HS, defects occur in the RBC membrane proteins, such as ankyrin, protein 4.2, band-3 and spectrin (8, 10). These protein deficiencies alter the RBC membrane in two distinct ways (8). In RBCs with deficiencies in the band-3 or protein 4.2, the vertical connections between the spectrin filaments and the lipid bilayer are diminished whereas the spectrin content  is normal (Fig.3B). On the other hand, spectrin-deficient or ankyrin-deficient RBCs are characterized by depleted spectrin content and reduced numbers of actin junction complexes, while the overall structure of the membrane cytoskeleton is preserved (Fig.3C). These alterations in the RBC membrane weaken the vertical linkages between the cytoskeleton and  the lipid bilayer, causing surface area loss through release of vesicles (8, 10). HS RBCs with reduced surface area trans- form progressively from a biconcave shape to a near-spherical shape. The concomitant decrease in cell deformability leads to RBC retention and premature removal by the spleen (5, 27). Although the genetic basis and clinical consequences of HS are known, the mechanics of membrane loss has hitherto not been explored in detail. In this section, we simulate band-3-deficient and spectrin-deficient RBCs traveling through IES. Varying degrees of protein deficiency are examined for these two types of HS RBCs. The deformability of the HS RBCs following reduction in surface area is evaluated by stretching the RBCs, similar to the manner in which RBC deformation was induced in optical tweezers experiments (28, 29). A deformability index (DI) is computed based on the deformation of the stretched RBC. Details of the calculations of DI are given in SI.

We first simulate the band-3-deficient  RBCs  by  reducing the connectivity between the band-3 proteins and spectrin fila- ments (vertical connectivity) in the RBC model. The vertical connectivity is reduced from 100% to 0% in  decrements  of 20%, representing elevated degrees of band-3 deficiency. The surface area and volume of the simulated RBCs are 140 µm2 and 90 µm3, respectively. Figs.3(D–I) show a sequence of shape changes of a HS RBC with a  vertical  connectivity  of 60% as it passes through an IES (top views). Due to weakened cohesion between the lipid bilayer and the  cytoskeleton,  the lipid bilayer detaches from the cytoskeleton when the RBC traverses IES (Fig.3F). One detachment  separates  from  the RBC  and  forms  a  vesicle,   whereas  the  other  develops  into a tubular vesicle and passes through IES behind the RBC (Figs.3G and H). Eventually,  this  tubular  vesicle  detaches  from the RBC and reshapes into a separate spherical vesicle (Fig.3I). These simulations confirm that reduced vertical con- nectivity compromises the cohesion between the cytoskeleton

Fig.3.(A) Cytoskeleton in a healthy RBC model. (B) In the band-3-deficient HS RBC model, the band-3 binding sites are randomly removed (highlighted by red dotted circles) to represent the effect of band-3 deficiency. (C) In the spectrin-deficient HS RBC model, the density of spectrin network is reduced to represent the effect of spectrin-deficiency. (D–I) Six sequential snapshots (top views) of a HS RBC with a vertical connectivity of 60% passage through IES under a pressure gradient of 10 Pa µm−1 (see Movie S2). Reduced vertical connectivity leads to the detachment of the lipid bilayer from the cytoskeleton and subsequent RBC vesiculation. The lipid CG particles (red particles) are plotted at a smaller size to visualize the cytoskeleton below (green particles).

and the lipid bilayer and that surface area loss of RBCs in HS occurs when they traverse the IES, thereby clearly establishing the connection between the spleen and shape alterations in diseased RBCs.

We also examine RBCs with various vertical connectivities. At each vertical connectivity, pressure gradients of 5, 8, 10, 15 and 20 Pa µm−1 are applied. The blue curve in Fig.4A shows that RBCs shed more surface area as the vertical connectivity    is reduced. In other words, the increased degree of band-3 deficiency exacerbates membrane loss of the HS RBCs. No- tably,  the fraction of surface area loss from the HS RBCs with    a vertical connectivity of 0% is larger than the IES retention threshold (the black dashed line in Fig.4A) inferred from ex  vivo experiments (5), implying that these HS RBCs do not successfully pass through IES. The corresponding DI of the band-3-deficient RBCs (the blue curve in Fig.4B) decreases as the degree of band-3 deficiency increases. These results illus- trate the correlation between the degree of protein deficiency and clinical expressions of RBCs in HS.

Next, we reduce the spectrin density in the cytoskeleton  by decreasing the number of actin junction complexes in the RBC model (Fig.3C), to mimic spectrin-deficient RBCs. To capture the wide spectrum of clinical severity observed in HS, the spectrin density is decreased from 100% (healthy) to 40% in decrements of 20%. When the spectrin-deficient RBCs pass through IES, the lipid bilayer in the spectrin-depleted area  can bud off and develop into vesicles (8). Fig.4A (red curve) illustrates that the fractional surface area loss of spectrin- deficient RBCs increases with decreased spectrin density, in agreement with clinical evidence (8). Fig.4B shows that

Fig.4.(A) Fractional surface area loss of HS RBCs after passage through IES. For the band-3-deficient RBCs, the surface area loss is increased with the decreased vertical connectivity. For the spectrin-deficient RBCs, the surface area loss is increased with the decreased spectrin density. The error bars are computed based on pressure gradient values of 5, 8, 10, 15 and 20 Pa µm−1 . The black dashed line highlights the critical fraction of surface area loss that determines the retention of RBCs reported by exvivoexperiment (5). (B) DI of the band-3-deficient and the spectrin-deficient RBCs at different levels of HS-related protein deficiency. The brown bar graph shows DI of spectrin-deficient RBCs measured by osmotic gradient ektacytometry at a fixed osmolality of 300 mOsmol/kg (30).

reduced spectrin density leads to decreased DI of the spectrin- deficient RBCs (red curve). In particular, the values  of DI  are consistent with the values experimentally measured by ektacytometry at a fixed osmolality of 300 mOsmol/kg (30, 31), indicating that the trends predicted by the present results are validated by clinical observations.

The surface area and volume of HS RBCs following the formation of vesicles are  plotted  in  Fig.5.  The  blue  and red symbols in Fig.5show that as the HS-related protein defects increase, the S/V ratios of HS RBCs gradually decrease toward the retention threshold predicted by Eq. (1) (black solid line) as well as the retention threshold reported by ex vivo experiments (black dashed line) (5). This plot implies that HS RBCs with low degrees of protein deficiency could pass IES in spite of reduced surface area. Therefore, these RBCs undergo a gradual decrease in their lifespans, leading to mild anemia. Conversely, RBCs with high degrees of protein deficiency undergo rapid drops of S/V ratio after crossing IES. As a result, the expected lifespans of these RBCs are shortened markedly,  leading to severe anemia. These findings provide  a mechanistic rationale for the connections among molecular defects, RBC alterations and the broad spectrum of clinical

severity found in HS.

Fig.5.Prediction of splenic IES retention for healthy RBCs (no protein deficiency) and HS RBCs. Healthy RBCs (green color symbols) with surface area of 140(◆), 130(V), 120( ), 110(K), 100 µm2 (□) and a fixed volume of 90 µm3 are examined, respectively. The surface area and volume of HS RBCs after their passage through IES are plotted. Blue color symbols denote the band-3-deficient RBCs with vertical connectivities of 80%(V), 60%( ), 40%(×), 20%(K) and 0%(□). Red color symbols denote the spectrin-deficient RBCs with spectrin density of 80%(V), 60%( ) and 40%( ). The error bars are computed based on pressure gradient values of 5, 8, 10, 15 and 20 Pa µm−1 . The black solid and dashed lines highlight the RBC retention threshold predicted by an analytical model given by Eq.(1) (18, 26) and the threshold reported by exvivoexperiment (5), respectively. RBCs with surface area and volume above these thresholds are able to cross IES, otherwise RBCs are retained by IES.

Fig.4A shows that as the level of HS-related protein defi- ciency increases, the spectrin-deficient RBCs with dispersed cytoskeleton shed more membrane surface than the band-3- deficient RBCs whose cytoskeleton is dense. Fig.5also shows that as the spectrin density decreases, the spectrin-deficient RBCs (red symbols) undergo a more rapid drop of S/V ratio than the band-3-deficient RBCs (blue symbols) for the same decrease in volume. These results suggest that the spectrin- deficient RBCs are more amenable to vesiculation than the band-3-deficient RBCs when traversing IES. This finding pro- vides a compelling mechanistic rationale for the clinically doc- umented observation that splenectomy prolongs the survival of spectrin/ankyrin-deficient RBCs, but not band-3-deficient RBCs (32). The distinct clinical manifestations between these two types of HS RBCs are likely attributed to the fact that the band-3-deficient RBCs in the spleen shed surface area to the same extent as they do through other mechanisms, such as by antibody triggered vesiculation or trogocytosis (32), while the spectrin-deficient RBCs lose membrane surface predominantly in the spleen.

TraversingIEScausesshapetransitionandfragmentationof RBCs in HE. In HE, defects occur in cytoskeletal proteins of RBCs, such as α-spectrin, β-spectrin and protein 4.1R (11). The α and β-spectrin deficiencies, which account for nearly 95% of HE cases, disrupt the self-association of spectrin dimers, whereas the protein 4.1R deficiency, responsible for 5% HE cases, alters the cohesion of the spectrin–actin–protein 4.1R junctional complexes (11, 33).  The characteristics of RBCs  in HE are increased cell fragility and shape transition from  the biconcave to the elliptical shape (9, 11). In severe forms  of HE, cell fragments have been detected in patients’ blood smear (9). The prevailing hypothesis is that RBCs in HE

become damaged during their transit through narrow pathways in microcirculation (33, 34), although little clinical evidence is available to support this hypothesis. Here, we examine RBCs displaying different degrees of HE-related protein deficiency as they travel through IES and explore how the traversal through IES contributes to the pathological alterations of RBCs in HE. We break the spectrin filaments in the RBC model (inset in Fig.6A), mimicking the disrupted spectrin tetramers. As the severity of HE varies according to the degree of the impaired cytoskeleton (33), we reduce the percentage of the intact spectrin filaments (horizontal connectivity) from 100% to 20% in decrements of 10%. At each horizontal connectivity, pressure gradients of 5, 8, 10, 15 and 20 Pa µm−1 are examined.

Fig.6.(A) Aspect ratios of HE RBCs after passage through IES. When the horizontal connectivity is reduced to 40% or less, HE RBCs break into fragments due to reduced cytoskeleton integrity. The error bars are computed based on pressure gradient values of 5, 8, 10, 15 and 20 Pa µm−1 . Inset shows that in the spectrin-deficient HE RBC model, spectrin filaments are randomly disassociated (highlighted by red dotted circles), mimicking the disrupted spectrin tetramers. (B-D) Three sequential snapshots (top views) of a HE RBC with a horizontal connectivity of 50% crossing IES (see Movie S3). (E-G) Three sequential snapshots (top views) of a HE RBC with a horizontal connectivity of 20% breaking into fragments after crossing IES (see Movie S4).

A notable feature of HE RBCs is the elongation of RBCs af- ter passage through IES (Figs.6(B-D)). Although vesiculation of HE RBCs is also observed, it is less pronounced compared to that of HS RBCs.  Here, we  quantify the shape transition  of HE RBCs by computing the aspect ratio of cells after their egress from IES. The aspect ratio is defined as L_A/L_T, where L_A  is the length of the RBC along its moving direction and  L_T is the length of the RBC perpendicular to its moving di- rection. As shown in Fig.6A, RBCs with decreased horizontal connectivity undergo further elongation during their passage through IES. These elongated HE RBCs cannot fully recover their original biconcave shape due to impaired cell elasticity, leading to the progressive shape transition to elliptical shape.

When the horizontal connectivity is 40% or less (30% and 20% in our simulations), the model predicts that the HE RBCs break into cell fragments. Figs.6(E-G) illustrate three se- quential snapshots of a RBC with a horizontal connectivity of 20% passage through IES. It is noted that the portions of RBCs protruding into the luminal sides of IES are elongated and subsequently break apart from the RBC, forming two cell fragments. These simulations provide a mechanistic rationale for RBC fragmentation in HE and for the presence of cell fragments in the blood smear of HE patients (9,11).

Discussion

Circulating healthy human RBCs have a life span of approxi- mately 120 days, during which they experience gradual degra- dation of membrane components and progressive loss of surface area through release of vesicles (22, 23, 35). Loss of surface area leads to alterations of the shape, size and deformability of RBCs. Removal of aged RBCs from circulation mainly occurs in the spleen where the less deformable RBCs are retained   by IES and subsequently phagocytosed by macrophages (27). Our present simulation results show that S/V ratio of RBCs   is an essential determinant of their passage through IES, con- firming the function of IES in filtering senescent RBCs (1, 3). In particular, our simulations illustrate that the critical sur- face area that allows RBCs with a fixed volume of 90 µm3 to successfully cross IES is between 110 µm2 and 120 µm2. This finding is consistent with the value reported from ex  vivo experiments (5) and with the prediction of an analytical model (18, 26). These results demonstrate the capability of our model in quantitatively assessing RBC filtration function in the spleen.

Altered RBCs in HS are characterized by spherical shape and impaired deformability, which results from the reduced surface area (8–10). Our simulations illustrate that HS RBCs lose surface area through shedding vesicles when traversing IES because of the weakened cohesion between cytoskeleton and lipid bilayer. Loss of surface area from HS RBCs becomes more pronounced as the degree of inherited protein deficiency increases. Our simulation results also demonstrate that the reduction in surface area compromises the deformability of RBCs, potentially leading to their splenic retention by IES.  As shown in Fig.5, the S/V ratios of HS RBCs after losing surface area decrease toward the retention thresholds (black solid and dashed lines). HS RBCs with low levels of protein deficiency undergo a small drop of S/V ratio after passage through IES, indicating a gradual decrease in their lifespans, whereas HS RBCs with higher protein deficiency undergo a larger drop of S/V ratio, implying a marked decrease in their lifespans. These findings correlate well with the severity of anemia and earlier occurrence of chronic complications induced by hemolysis in patients with severe forms of HS. In addition, our simulation results suggest that the spectrin-deficient RBCs are more amenable to vesiculation than the band-3-deficient RBCs when traversing IES. This result provides a possible explanation of the distinct clinical outcomes of splenectomy in HS patients with different pathogenesis (32).

The present simulations do not account for the effect of a putatively impaired splenic RBC filtration function in HS patients (25, 36). Therefore, the prediction of RBC retention in Fig.4A and Fig.5may overestimate the function of IES in retaining the altered RBCs in HS. In addition, the reduced

surface area in HS RBCs can provoke a transient dehydration, leading to a decrease in cell volume and thus an attendant  increase in the S/V ratio (5, 37). As a result, a greater than expected proportion of HS RBCs may escape splenic retention and return to circulation. Although our present study strongly suggests that the surface area loss from HS RBCs occurs when they cross IES in the spleen, this pathogenic process may begin even earlier when reticulocytes (young RBCs) exit from the bone marrow, a process not devoid of mechanical constraints (37). These two mechanisms of surface area loss are not mutually exclusive, as reticulocytes exit the bone marrow only once but cross IES in the spleen several times during their life-span.

The major molecular basis of HE is the α or β-spectrin  deficiency, which disrupts the self-association of spectrin dimer into tetramers or oligomers, leading to weak and fragile mem- brane cytoskeleton (9, 11). Elongated cell shape and decreased membrane stability are the clinical features of RBCs in HE (12). Our simulations demonstrate that due to impaired cytoskele- ton, HE RBCs are elongated after traversing IES, contributing to their shape transition to ellipsoids. When the severity of HE-related protein deficiency is high, RBCs break into frag- ments during their passage through IES. These findings clarify the relationship among the genetics, degree of protein defi- ciency in cytoskeleton, and clinical expressions of RBCs in HE.

Taken together, our simulations provide unique insights into the dual pathophysiological function of the spleen. The spleen not only assesses the mechanical fitness of the healthy and diseased RBCs in circulation, but also contributes directly to the alterations of diseased RBCs. Our results also elucidate the connections among the molecular basis of inherited RBC disorders, the different clinical manifestations of RBCs, the resulting symptoms and the clinical outcomes of splenectomy. The computational framework presented in the current work thus offers new possibilities to study the complex role of the spleen in the pathogenesis of RBC disorders.

Mechanics of diseased red blood cells in human spleen and consequences for hereditary blood disorders

Li et al. 10.1073/pnas.XXXXXXXXXX

Supporting Information (SI)

Two-component RBC Model and Validation

Two-componentRBCmodel.In the coarse-grained molecular dynamics (CGMD) RBC model, the major two components  of the RBC membrane, namely the cytoskeleton and the lipid bilayer, are represented explicitly by coarse grained (CG) par- ticles. As shown in Fig. 1, the cytoskeleton of the membrane consists of spectrin filaments connected at the actin junctional complexes, forming a hexagonal network. The actin junctional complexes, represented by blue particles, are connected to the lipid bilayer via glycophorin proteins (yellow particles). Spec- trin is a protein tetramer formed by two identical heterodimers. Each heterodimer is comprised of an α-chain with 22 triple- helical segments and a β-chain with 17 triple-helical segments. Thus, each spectrin filament is simulated by 39 spectrin parti- cles (green particles). These spectrin particles are connected with unbreakable springs uspectrin= kspectrin(ddeq)2, where kspectrinis the spring constant, dand deqare distance and equi- librium distance between two spectrin CG particles. The lipid bilayer and transmembrane proteins of the RBC membrane are represented by three types of CG particles (Fig. 1). The red CG particles denote aggregates of lipid molecules. The yellow particles signify glycophorin proteins which are connected to the blue particles by unbreakable springs. The black particles represent band-3 proteins that tether spectrin filaments to the lipid bilayer.  These three types of CG particles interact via    a pairwise potential similar to the Lennard-Jones potential. However, the employed interacting potential depends not only on the translational degrees of freedom of two interacting CG particles  di and  dj,  but  also  on  their  rotational  degrees  of freedom ni and nj.  The potential is given by

Detailed information about this RBC  model  can  be found in the authors’ former work in Li et al. (38) and Tang et al. (21). This RBC model can simulate an entire RBC by using                            4 million CG particles using a single shared memory commodity workstation, but it is computationally expensive when simu- lating a long-time dynamic process such as a RBC passage through IES. In order to achieve higher computational effi- ciency, we apply a coarse model by using a fewer number of actin junctions in the RBC model. While the RBC membrane structure is preserved, we model 500 actin junctions in a single RBC (see Fig. 1), instead of a physiological value of              23867. Following the method applied in (39), the parameters in the coarse RBC model are  recalibrated  against experimental data to ensure the mechanical properties of the RBC are preserved. The coarse RBC model consists  of 198965 CG  particles and the length unit  of the  model  is σ = 35  nm.  The energy unit is g = kB T /0.22, where kB is the Boltzmann constant and the temperature of the system T is 300 K. The parameter that determines the bending stiffness of the RBC membrane, α, is

selected to be 2.1.

The mechanical properties of mature RBCs mainly result from their cell membrane as they lack of nucleus and most organelles. The lipid bilayer of the RBC membrane behaves like 2D-fluid and thus the elasticity of RBCs arises primarily from cytoskeleton. In the following sections, we will illustrate that the elastic properties and bending stiffness of the coarse RBC model are consistent with the previously reported exper- imental results by performing optical tweezer simulations and by measuring the thermal fluctuations of the RBC membrane.

ElasticpropertiesoftheRBCmodel.Optical tweezers have been  successfully  implemented  to  study  the  elasticity and

u   (n,n ,x   ) = u   (d) + A(α,a(n,n ,xˆ   ))u   (d),    [1]

deformability of RBCs (28,29). A number of numerical sim-

ij i j ij R

i j ij A

 dc − d8

ulations have mimicked this experimental setup by applying stretching forces on the opposite sides of RBCs to validate

uR(d) = 1.4g(

dc

− deq

) , [2]

the RBC models and to probe the biomechanical behavior

 dc − d4

of diseased RBCs (40, 41). Following previous studies, we

uA(d) = −2.8g( d

− deq

) , [3]

put the RBC model under stretch, analogous to the optical tweezer experiments, to examine the elasticity of the healthy

A(α,a(ni,nj,xˆij)) = 1 + α(a(ni,nj,xˆij) − 1), [4]

a(ni,nj,xˆij) = (ni × xˆij) · (nj × xˆij)

and diseased RBCs. The total stretching force, Fs, is applied at the two ends of the RBCs in diametrically opposite direc-

= ni · nj − (ni · xˆij)(nj · xˆij),    [5]

tions, as shown in Fig.S1A. The stretching force is varied from 0 pN to 200 pN in increments of 20 pN. The stretch re-

where xij = dj      di, d=  xij    and xˆij = xij/d.  αis a parame- ter that tunes the bending stiffness of the RBC membrane. dc is the cutoff distance of the potential and it is selected to be 2.6σ, where σis the length unit of the system. gis the energy unit. Actin and spectrin filaments interact with lipid bilayer and transmembrane proteins via a Lennard-Jones potential,

sponse of the RBC is measured by recording the axial (DA) and transverse (DT ) diameters of the stretched RBC. Our simulations show that as the Fs increases, DA of the RBC model increases whereas DT decreases. Fig.S1B shows a RBC under a stretching force of 200 pN. As plotted in Fig.S2, the overall stretching response measured from the RBC model (blue curve) under various values of Fs is consistent with pre-

U = 4gΣ( σ)12 − ( σ)6Σ d <d   . [6]

vious optical tweezer measurements of healthy RBCs (red

the RBC model.

Fig. S1. Modeling the healthy and HS RBCs under stretch. (A) A stretching force, Fs, is applied at the two ends of a RBC in diametrically opposite directions. The axial (D_A) and transverse (D_T) diameters of the stretched RBC are recorded at a variety of Fs. (B) A healthy RBC under a stretching force of 200 pN. (C) A spectrin-deficient

RBC with spectrin density of 40% under a stretching force of 200 pN. (D) A band-

3-deficient RBC with vertical connectivity of 0% under a stretching force of 200 pN. In the above figures, the lipid particles (red particles) are plotted in a smaller size in order to more clearly visualize cytoskeleton (green particles).

BendingstiffnessoftheRBCmodel.In this section, we vali- date the bending stiffness of the RBC model based on Brownian flicker analysis of membrane thermal fluctuations (42). We perform simulations following the protocol of a prior diffraction phase microscopy experiment (42), where the instantaneous heights of the cell membrane along the upper rim of a RBC were monitored to calculate the fluctuations of a RBC. In our simulation, the bottom of the RBC model is fixed, mimick- ing adhesion of the RBC to a solid substrate. The thermal fluctuation of the membrane is measured on the upper side of the RBC, through which the root-mean-square displacement (rmsd) of membrane fluctuations is calculated. Fig.S3illus- trates that the membrane fluctuation distribution of the RBC model overlaps with the measurements from the healthy RBCs in (42). The rmsd of the membrane fluctuations is computed to be about 90 nm. Based on the analytical expression derived in (43), the bending stiffness of the RBC, kc, can be estimated by  kc= AkB T /(8π3rmsd2),  where A is the surface area of  a RBC. Following this expression, kcof the RBC model is

Fig. S2. The stretching response of a healthy RBC model under stretching forces ranging from 0 pN to 200 pN measured from our simulation and from optical tweezer experiments performed by Suresh et al. (29).

Fig. S3. Membrane fluctuation distributions measured at the upper rim of a healthy RBC measured from simulation (circles) and from experiment performed by Park et al. (42) (solid lines).

calculated to be 2.89 10−19 J, which falls within the range of 2 10−19 J to 7 10−19 J reported in prior experimental studies (44).

Causeofincreasedcriticalpressuregradient.The discrep- ancy between the critical pressure gradient of 5 Pa µm−1 obtained in current work and the value of 1 Pa µm−1 reported in previous studies is mainly attributed to the fact that the  implemented CGMD RBC model is an implicit-solvent model. Implicit representation of solvent particles in a pressure-driven flow could underestimate the effect of the driven pressure.   To confirm this hypothesis, we perform DPD-based simula- tions, as reported in (18), and apply the pressure gradients on different components of blood: (a) RBC membrane only (an analogous simulation setup to our current study), and (b) both RBC membrane and solvent particles (the same simulation setup as in (18)). Indeed, we find that when driving the RBC through IES, the equivalentpressure gradient applied in case

(a) is at least twice as much as that in case (b). In addition, our implicit-solvent model did not consider the lubrication effect induced by the fluid flow around the RBC, which facilitates the RBC traversal process. Moreover, the interaction between the RBC and wall of IES is described by a hard core repul- sive potential (Lennard-Jones potential) in our present model, whereas a soft-core repulsive potential was implemented in the DPD model. Implementation of a hard core repulsive potential also contributes to the increased critical pressure in

2 of3 Li et al.10.1073/pnas.XXXXXXXXXX

the current model. It is noted that in spite of the discrepancy in the critical pressure gradient, we show that the  critical ratio of S/V measured from our simulation, which determines the passage of RBCs through IES, is in agreement with the critical ratio of S/V obtained from the analytical model and DPD-based simulations in (18,26). This finding suggests that application of a pressure gradient of 5 Pa µm−1 in the present model is equivalent to application of a pressure gradient of    1 Pa µm−1 in the DPD-based RBC model.

Analytical framework. Pivkin et.al (18,26) developed a simpli- fied axisymmetric model to elucidate the effects of geometric constraints on RBC–IES interactions. In their work, the slit cross-section in the y–z plane is assumed to be circular (Fig.S4) instead of rectangular, but with the same cross-sectional area such that               

Ds  = 2    HsWs/π. [7]

Thus, the IES is approximated by the surface of a torus. As shown in Fig.S4, the traversing RBC at the critical condition is comprised of a central torus connected with two spheres. The surface area and volume of the RBC, therefore, can be computed by

A = 4 .2πR² − πRhΣ + 2πL θ . Ds+ Ls− Ls sin θΣ [8]

Fig. S4. Schematic of the slit geometry considered in the analytical model. The traversing RBC is signified by the red line. The surface of the RBC is tangential to the surfaces of the endothelial cells at point p. The black line circle on the right represents the cross-section A-A. The rectangular slit is approximated by the circular slit with the same cross-sectional area. Adapted from figure 2 in (18).

Deformability index of healthy and diseased RBCs. To quan-

and

s 2 2 2θ

tify the deformability of the healthy RBC and  HS  RBCs after passage through IES, we subject these RBCs under stretch to compute their deformability index based on the

V=2 Σ 4 πR³ − 1 πh² (3R− h)Σ

deformations of RBCs. The deformability index is defined as

3

+ πLs.

3

+ LsΣ cos (

sin 2 )

DI = (DA − DT )/(DA + DT ), where DA and DT are the axial

2

+ π.

R 2 θR

3 3

L³ sin θ − s 3

θ − Lsθ

[9]

and transverse diameters of the stretched RBCs, as shown in Fig.S1A. The stretch force Fs is selected to be 200 pN such that DI of the healthy RBC model is calculated to be 0.596, consistent with the DI of ∼0.61 measured by ektacytometry

respectively, where h is the height of spherical cap cut by the

y–z plane at intersecting point p (Fig.S4) and

for healthy RBCs at a fixed osmolality of 300 mOsmol/kg (31). The average values of surface area loss reported in Fig. 4A are

used to reconstruct the HS RBCs with various vertical connec-

.‚ . _D + L Σ₂

tivities and spectrin densities for DI measurements. Figs.S1C

and

.R + LsΣ₂

θ= arccos Σ. Ds+ LsΣ /.R+ LsΣΣ [11]

with spectrin density of 40%, and a band-3-deficient RBC with zero vertical connectivity under stretching forces of 200 pN. Due to the decreased S/V ratio, the deformations of these two RBCs are suppressed, compared to the healthy RBC plotted

is an angle of the torus part as shown in Fig.S4. After rewriting the Eq. (9) as V = f (R) and defining its inverse function as R = f ⁻¹(V ), Eq. (10) and Eq. (11) can be

expressed as h= h(R) = h(f−1(V)) = g1(V) and θ= θ(R) = θ(f−1(V)) = g2(V). Based on the axisymmetric theory, the minimum surface area (A), below which the RBCs with fixed volume (V) are retained by the slit, is given by

A= 4 ,2πΣf−1(V)Σ2  − πf−1(V)g1(V),

+ 2πL g (V ) . Ds+ Ls− Ls sin [g2(V )] Σ . [12]

molecular defects in HS is summarized in Fig. 4B.

SI Movies. Movie S1 A healthy RBC passing through IES under a pressure gradient of 20 Pa µm−1. Only one half of the RBC is plotted for clarity.

Movie S2 A HS RBC (top view) with a vertical connec- tivity of 60% passing through IES under a pressure gradient  of 10 Pa µm−1. The lipid CG particles (red particles) are plot- ted at a smaller size to visualize the underneath cytoskeleton (green particles).

Movie S3 A HE RBC (top view) with a horizontal connec-

s  2 2 2

2g2(V)

tivity of 50% passing through IES under a pressure gradient of 10 Pa µm−1.

where Ds is the radius of the slit opening and Ls is the thickness of the sinus wall.

MovieS4A HE RBC (top view) with a horizontal connec- tivity of 20% passing through IES under a pressure gradientof 10 Pa µm−1.