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.
- 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
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 tsw ∈ [5, 100 s], (2) membrane lifetime (where is the hemolytic time), and (3) time for Hb release from already formed hemolytic hole during successive open‐closed state changes trelease ∈ [1,5 s] [11, 13].
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 . The bilayer structural changes have the feedback effects to band 3 protein‐lipids positive hydrophobic mismatch effects [19–21]. De Meyer et al.  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 . 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.  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.  and Kodippili et al. . Band 3‐ankyrin complexes are located near the center of spectrin tetramers. These complexes could survive the hypotonic conditions. Golan and Veatch  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  and Kodippili et al. . This subpopulation is located at the spectrin‐actin junction complexes. The junctions of the network link 4–7 spectrin filaments . The third one is freely diffusing subpopulation (∼30%). The part of freely diffusing subpopulation increases under hypotonic conditions. Golen and Veatch  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 . 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 . 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 . Long‐range self‐associative tendency is induced by positive hydrophobic mismatch effects . The band 3 clustering is pronounced under hypotonic conditions . In this case, ∼25% of band 3 molecules make aggregate of ∼5000 mers . 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) . On that base, they influence spectrin conformations . Gov  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 (where is the length of the filament part between two mid-point attachments and is the spectrin persistence length ). The spectrin filament parts are as follows: (1) flexible if , (2) semiflexible if , and (3) rod‐like if . The average length of spectrin parts is equal to (where is the spectrin contour length  and is the average number of attached band 3 molecules per single spectrin filament ). The length of the spectrin parts depends on the rearrangement of band 3 molecules which include their lateral diffusion and self‐associative tendency . Consequently, rheological behavior of the cortex is related to the spectrin flexibility distribution and the rate of its changes [35, 36]. Deeper insight into coarse‐consequence relations between the band 3 rearrangement and the spectrin inter‐ and intrachain interactions as well as the bilayer bending offers a possibility for understanding the complex nature of the membrane relaxation phenomena.
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 . 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 representing the change of cluster part volume with entropy for given number of molecules (i.e., considering it as a canonical ensemble):
The two limits of the compactivity have been introduced: (1) corresponds to the most compact particle rearrangement and (2) corresponds to the least compact particle rearrangement. When , the system picks out one particular configuration as being most likely, and when , the system picks out all configurations as being equally likely. Probability distribution of band 3 cluster states could be expressed as:
where is effective volume of the cluster part (corresponding to the free energy F in classical statistical mechanics) and is the “volume function” corresponding to the Hamiltonian and is constant adjusting dimensions. Then, we can write:
Cluster is considered as canonical ensemble during short‐time rearrangement. The partition function relating with the volume function of a cluster part can be written in the form:
where integration goes over all geometrical degrees of freedom (DOFs) of all molecules in the cluster, is the partition function. The key step is the identification of an exact volume function which makes it possible to pinpoint the configuration phase space and evaluate its dimensionality. We consider weakly interacting systems. To be precise, the interactions must be sufficient to lead to thermodynamical equilibrium, but weak enough that these interactions have negligible effects on the effective volume of individual molecules.
where is the volume function of single molecule. Rearrangement of rigid molecules such as band 3 within the cluster could be described using low degrees of freedom (DOFs). For this case, the DOFs describe molecule's orientation in the cluster part located at and its coordination number.
The band 3 cluster excitation under isotonic condition induces the molecule orientation changes presented in Figure 1. The excitation occurs during alternating expansion and compression of the membrane parts in order to ensure surface and volume conservation.
where is the minimum specific volume of single molecule of band 3 is the minimum of the molecule Stokes radius, is the volume increment caused by the molecule orientation equal to , is the maximum specific volume of single molecule of band 3 , is the maximum of the molecule Stokes radius, accounts the molecule's orientation equal to while accounts various directions, is the volume increment contributions caused by the DOFs changes. For molecule loose packing, we can take and a molecule has high values of the degrees of freedom in such system so values are close to 1.
The band 3 cluster excitation under hypotonic conditions is more intensive than that obtained under isotonic condition due to changes the bilayer bending state during erythrocyte swelling. The excitation could induce changes the packing state from close packing to ring‐like structure which represents the reversible hemolytic hole. These changes are influenced by the positive hydrophobic mismatch effects which could induce protein tilting . Zade‐Oppen  reported that the average hole opening time period is 270 ms, while the average hole closing time period is 260 ms. Seeman et al.  experimentally determined the diameter of the reversible osmotic holes in the range between 10 and 100 nm for human erythrocyte under hypotonic condition at pH = 7. Accordingly, the smaller hemolytic hole corresponds to ∼4 band 3 molecules while the larger one corresponds to ∼40 molecules [18, 30]. The result points out that cluster size is not the main factor for the hole formation. The main factor could be the hydrophobic mismatch effects between band 3 and the surrounding lipid bilayer as was shown in Figure 1. The corresponding volume function is as follows:
where accounts the coordination number, is the volume increment caused by changes the molecule coordination number equal to is the relaxation time, is the thickness of the lipid bilayer for already swollen erythrocyte, is the radius of hemolytic hole, is the number of molecules per cluster located at . DOFs can have values in the range 0 to 1. When and at , the volume function is . When and at , the volume function is equal to . This excited cluster state represents the hemolytic hole.
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 is formulated as white noise with correlation function , and is the analog of frictional resistance. System structural changes could induce anomalous nature of energy dissipation derivative during molecules migration within the cluster. If the molecule migration causes damping effects as described by Tomishige et al.  (subdiffusion phenomenon), the derivative could be replaced by the fractional derivative (where is the Caputo's fractional derivative, is the order of the fractional derivative such that ). Caputo's definition of the fractional derivative of some function is given as follows : (where is the gamma function). Average equilibrium volume of single molecule is obtained as:
where is the partition function, and is the cluster volume.
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 ()  and semiflexible ()  (where is the spectrin contour length equal to , is the length of i‐th filament part between two mid-point attachments of band 3). The flexibility depends on the number of band 3 molecules attached per single spectrin filament. Spectrin filament without band 3‐spectrin complexes behaves as flexible. Its conformations have been described as :
Where is the number of spectrin filaments per network units, is the surface shear modulus of the cortex equal to , is Boltzmann constant and is temperature, is the end‐to‐end distance of spectrin filaments in the cortex, is the average filaments radius of gyration. If the band 3‐spectrin low affinity complexes exist, the parts of the spectrin filament between the complexes behave as independent blobs . The conformation changes within the blobs are the milliseconds order . When the filament parts are flexible, but if they become semiflexible, while if they behave as rod‐like polymers (where is the average length of the filament part) . Li et al.  treated the whole spectrin filaments as a semiflexible and proposed worm‐like force for modeling of the spectrin conformations:
where is the stretch ratio. The worm‐like force corresponds to the condition . It is in accordance with the fact that the spectrin‐band 3 complexes lead to decrease in the spectrin flexibility. It could be quantified by apparent increase in the spectrin persistence length (where is the effective spectrin persistence length). The concept of effective persistence length has been introduced for describing the nature of interchain structural changes for worm‐like chains such as proteins under stretching [42, 43]. On that base, the effective persistence length in our case could be expressed as :
Where is the effective persistence length of spectrin filament, is the spectrin persistence length for the filaments without mid‐point attachments at the same temperature conditions and is the contribution to the persistence length caused by the band 3 midpoint attachments. The collective phenomena among variously flexible spectrin filaments induce generation of the cortex in‐homogeneities . The in‐homogeneities in the context of the cortex micro domains influence the cortex relaxation. The cortex relaxation modulus could be expressed as [18, 36]:
where is the cortex relaxation modulus within the domain and is the spectrin flexibility distribution caused by the band 3 rearrangement.
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 .
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 : (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 , while the type 2 induce a concave curvature of radius such that . The bilayer‐cortex coupling has been expressed by Helfrich‐type bending‐free energy functional :
Where is the coordinate along the contour, is the bending modulus of the bilayer, is the bilayer width, and are the relative densities of the types 1 and 2 of spectrin filaments, and the corresponding local mean curvature are and . The overall curvature by spectrin filaments is expressed as . Band 3 molecules influence the lipid bilayer bending modulus. Shlomovitz and Gov  formulated the apparent bending modulus equal to:
where is the apparent bending modulus, is the bending modulus of the bilayer without band 3 molecules, is the contribution of band 3 molecules to the bending modulus, is the local surface fraction of the band 3 molecules. Shlomovitz and Gov  expressed the influence of inclusions (band 3 molecules) on the lipid bilayer bending by formulating the free energy functional as:
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: . The collective phenomena related to the spectrin filaments migration is expressed by spatial‐temporal changes of the conservative variable  as:
where is the spectrin collective diffusion coefficient which accounted for the intrachain interactions, is the filaments mobility parameter, is the maximal packing density of the filaments, and is the derivative along the contour . Spectrin filament mobility depends on the number of attached band 3 molecules. Consequently, the effective diffusivity could be formulated as and introduced in eq. 17. Shlomovitz and Gov  modeled the collective migration of band 3 molecules as:
where is the derivative along the space, is the band 3 lateral mobility and is the band 3 diffusion coefficient equal to . Lateral motion of the band 3 molecules induces the anomalous nature of energy dissipation which includes damping effects . These damping effects are induced from band 3 association‐dissociation to spectrin filaments. Pajic‐Lijakovic [35, 36] proposed fractional Langevin equation for describing the lateral diffusion by applying the fractional derivatives . Consequently, the time derivatives from Eqs. (17) to (18) could be replaced by the fractional derivative .
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.
This research was funded by grant III46001 from the Ministry of Science and Environmental Protection, Republic of Serbia.