Abstract
An attempt was made to discuss and connect various modeling approaches which have been proposed in the literature in order to shed further light on the erythrocyte membrane relaxation under isotonic and hypotonic conditions. Roles of the main membrane constituents: (1) the actin‐spectrin cortex, (2) the lipid bilayer, and (3) the transmembrane protein band 3 and its course‐consequence relations were considered to estimate the membrane relaxation phenomena. Cell response to loading conditions includes the successive sub‐bioprocesses: (1) erythrocyte local or global deformation, (2) the cortex‐bilayer coupling, and (3) the rearrangements of band 3. The results indicate that the membrane structural changes include: (1) the spectrin flexibility distribution and (2) the rate of its changes influenced by the number of band 3 molecules attached to spectrin filaments, and phosphorylation of the actin‐spectrin junctions. Band 3 rearrangement also influences: (1) the effective bending modulus and (2) the band 3‐bilayer interaction energy and on that base the bilayer bending state. The erythrocyte swelling under hypotonic conditions influences the bilayer integrity which leads to the hemolytic hole formation. The hemolytic hole represents the excited cluster of band 3 molecules.
Keywords
- packing state changes of band 3 clusters
- reversible hemolytic hole formation
- the lipid bilayer bending state changes
- the spectrin inter‐ and intrachain interactions
- mathematical modeling
1. Introduction
Erythrocyte mechanics under isotonic and hypotonic conditions has been studied from engineering and biomedical stand points [1–14]. Rheological response of the erythrocyte membrane depends on the main membrane constituent rearrangement: (1) the actin cortex, (2) the lipid bilayer, and (3) the transmembrane protein band 3. The membrane fluctuations under isotonic condition induce alternating expansion and compression of the membrane parts in order to ensure surface and volume conservation. The membrane relaxation occurs within (millisecond order) affine regime and (second order) nonaffine regime. The affine regime corresponds to the spectrin interchain interactions while the nonaffine regime corresponds to the spectrin intrachain interactions. However, the membrane fluctuations under hypotonic condition induce volume increase by ensuring surface conservation. The membrane response under hypotonic condition includes the successive sub‐processes: (1) erythrocyte swelling, (2) lifetime of the lipid structural integrity and the rearrangement of transmembrane protein band 3, and (3) the reversible hemolytic hole formation and hemoglobin (Hb) release to surrounding solution. Duration of the membrane relaxation depends on contributions of three sub‐processes: (1) time for cell swelling
The band 3 rearrangement significantly influences the state of both, the lipid bilayer and the actin‐spectrin cortex. Space distribution of band 3 molecules and their lateral diffusion influence the bending state of the lipid bilayer and its free energy [15–17]. Local changes of the bilayer bending state enhance anomalous sub‐diffusion and eventually lead to hop‐diffusion of lipids. These effects induce anomalous nature of energy dissipation during lipids structural ordering and could be quantified by the effective viscosity [18]. The bilayer structural changes have the feedback effects to band 3 protein‐lipids positive hydrophobic mismatch effects [19–21]. De Meyer et al. [19] pointed to the cholesterol role in lipid mediated protein‐protein interactions. These effects could lead to the protein tilt angle changes and the protein clustering. This clustering can be modulated by homotypic interactions of the protein transmembrane domains and electrostatic protein‐lipid interactions [21]. Tilt angle changes influence packing state of band 3 clusters and their association‐dissociation to spectrin. Band 3 molecules form various complexes with spectrin and influence its conformational changes. In‐homogeneous distribution of band 3 molecules and their ability to clustering influence the rheological response of: (1) the bilayer, (2) the cortex, and (3) the nature of the bilayer‐cortex mechanical coupling. Ehrig et al. [22] pointed that the lipid bilayer phase separation can be strongly affected by interaction with the actin cortex, which, depending on the temperature and membrane composition, can either lead to precipitation of highly dynamic membrane domains (rafts), or prevent large‐scale phase separation.
For understanding the influence of band 3 rearrangement on complex nature of the membrane rheological response, it is necessary to consider three subpopulations of band 3 molecules under isotonic and hypotonic conditions. The first subpopulation (20–40%) as tetramers forms high affinity complexes with ankyrin (quantified by the dissociation constant ∼5 nM) as reported by Tomishige et al. [23] and Kodippili et al. [24]. Band 3‐ankyrin complexes are located near the center of spectrin tetramers. These complexes could survive the hypotonic conditions. Golan and Veatch [25] reported that 25% of band 3 population remains attached to the cortex under ionic strength 26 mM NaPO4 solution at 37°C. The second subpopulation (∼30%) as dimers forms lower affinity complexes with the adducin (the dissociation constant is ∼100 nM) as reported by Franco and Low [26] and Kodippili et al. [24]. This subpopulation is located at the spectrin‐actin junction complexes. The junctions of the network link 4–7 spectrin filaments [27]. The third one is freely diffusing subpopulation (∼30%). The part of freely diffusing subpopulation increases under hypotonic conditions. Golen and Veatch [25] determined that mobile fraction of band 3 molecules under the external solution tonicity 46.0 mM NaPO4 at 21°C was 11 ± 9%. Under tonicity of 5.2 mM NaPO4 at 21°C, the mobile fraction of band 3 molecules was 72 ± 7%. Band 3 molecules are anion at pH = 7. The value of Stokes radius of band 3 dimmer is approximately 7.6 nm while for tetramer is 11 nm at room temperature and pH = 7.2 [28]. The total number of band 3 per single erythrocyte is ∼1 × 106. Thermal fluctuations of the erythrocyte membrane could induce conformational changes of cytoplasmic domain of transmembrane protein band 3 [29]. Such conformational changes result in electrostatic interactions between highly anionic N‐terminal domains of band 3 molecules and on that base intensified their short‐range self‐associative tendency [28]. Long‐range self‐associative tendency is induced by positive hydrophobic mismatch effects [21]. The band 3 clustering is pronounced under hypotonic conditions [30]. In this case, ∼25% of band 3 molecules make aggregate of ∼5000 mers [31]. All band 3 subpopulations through these complexes contribute to the spectrin conformational changes by reducing its mobility and influence the cortex stiffening.
The band 3 molecules within the freely diffusing subpopulation could form low affinity complexes to spectrin (the dissociation constant is ∼1–10 μM) [25]. On that base, they influence spectrin conformations [32]. Gov [32] reported that the parts of the spectrin filament between two mid‐point attachments behave as independent blobs. The main factor which influences the spectrin flexibility and conformations is
2. Rearrangement of band 3 molecules‐cluster packing state changes
Packing state of band 3 clusters and its changes during short‐time lateral motion under isotonic and hypotonic conditions could be estimated by applying Edwards’ statistics [30, 35, 37, 38]. Short‐time motion of band 3 molecules includes: (1) Brownian diffusion within the mesh compartment of the spectrin‐actin cortex and (2) hop diffusion between two compartments. Hop is observed at every 350 ms [23]. Band 3 molecules form clusters caused by positive hydrophobic mismatch effects during their lateral diffusion. This statistical approach is suitable for describing the cluster packing state changes under vibration field. Edwards introduced a new and very significant parameter for description of particle clusters named compactivity of the cluster part
The two limits of the compactivity have been introduced: (1)
where
Cluster is considered as canonical ensemble during short‐time rearrangement. The partition function relating
where integration goes over all geometrical degrees of freedom (DOFs)
where
The corresponding volume function could be expressed as [30, 35]:
where
where
Band 3 molecules change their states during migration. Consequently, for estimating the DOFs temporal changes it is necessary to consider the temporal changes of single molecule velocity. Temporal changes of DOFs under vibration field could be described in the form of Langevin‐type equations [30, 37]:
where the stochastic random force
where
3. Band 3 rearrangement influences the spectrin inter‐ and intrachain interactions and the bilayer bending
The spectrin interchain interactions depend on the number of band 3 molecules attached per single spectrin filament as was shown in Figure 2.
The spectrin filaments have been treated as flexible (
Where
where
Where
where
The presence of the cortex micro domains is related to in‐homogeneous distribution of: (1) band 3 molecules, (2) spectrin flexibility, and (3) the presence of the cortex defects as was shown in Figure 3. Cumulative effects as: (1) the spectrin intrachain interactions which lead to formation of the cortex micro domains, (2) longtime diffusion of band 3 molecules, and (3) longtime bending relaxation of the lipid bilayer are at the order of seconds [18].
These local in‐homogeneities of the cortex are caused by alternating expansion and compression of the membrane and the cortex‐bilayer coupling. The bilayer bending is influenced by conformational changes of the two types of spectrin filaments [44]: (1) type 1—corresponds to the filaments grafted at one end or at both ends but not connected to the stretched cortex and (2) type 2—corresponds to the filaments grafted at both ends and on that base represents a part of the connected stretched cortex. Filaments within the type 1 induce a concave curvature of radius
Where
where
Consequently, the model Eqs. (15) and (16) could be combined to describe the influence of: (1) spectrin conformationals and (2) band 3 molecules migration on the lipid bilayer bending state expressed in the form:
where
where
4. Conclusion
Rheological behavior of the cortex depends on the spectrin flexibility distribution and the rate of its changes [35, 36]. The spectrin flexibility primarily depends on the number of band 3 molecules attached per single spectrin filaments. Rearrangement of the band 3 molecules and their lateral diffusion also influence the bending modulus of the lipid bilayer and the band 3‐bilayer interaction energy. Consequently, the band 3 rearrangement influences the cortex‐bilayer coupling and on that base influences the membrane rheological behavior as a whole. The membrane structural changes induce anomalous nature of energy dissipation caused by these complex multi scale molecular dynamics.
Acknowledgments
This research was funded by grant III46001 from the Ministry of Science and Environmental Protection, Republic of Serbia.
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