Introduction:

Red blood cells are the life cell of the human body. Their job is to carry oxygen from the lungs to the rest of the body. During the transportation it’s necessary for red blood cells (erythrocytes) to squeeze through capillaries and small openings in passageways. In order to do this the cell must be extremely flexible. An erythrocyte’s flexibility is controlled by the cell’s membrane. If the membrane is somehow damaged or its flexibility decreased it follows that the transportation of the cell will be inhibited and its usefulness negated. When this happens it can be crippling to the body as a whole.

Spherocytosis is a genetic disease that affects the membrane of a red blood cell. This disease decreases the ratio of surface area to volume of the erythrocyte translating into a less flexible cell. The greater packed the cell the more susceptible it is to rupture or breakdown of the membrane. When erythrocytes are affected by Spherocytosis they often will rupture prematurely and lead to anemia (shortage of RBCs) which can cause lots of problems for the human body.

As you can see the membrane of the red blood cell is very important not only to its structural performance but also to its ultimate goal of delivering oxygen. Therefore, the first step in understanding ways to protect against such diseases as Spherocytosis is to investigate the form of the membrane and match it with its functions. The goal of the group was to develop an accurate physical model of the RBC membrane and use this to investigate different aspects of how it works.

Membrane Structure:

An RBC membrane has two main components: a lipid bi-layer (similar to those found in most human cells) and a protein skeleton. The skeleton forms an extensive web around the bilayer and is joined to it at what are called junction complexes (JC). The following picture shows the overall structure of the lipid bilayer attached to the protein skeleton. Together they are what form the flexible membrane and will be discussed in more detail in the following paragraphs.

The lipid bilayer consists of two lipid layers stacked tail to tail. A lipid is a cellular unit with a head and a tail. The head is hydrophilic while the tail is hydrophobic, which causes the tails to point toward each other and away from the water on the interior and exterior as shown below. This creates a sort of liquid wall between the inside and outside of the cell. This lipid wall has very little structural stiffness on its own and requires a protein skeleton for support.

The protein skeleton as seen above consists of actin protofilments, ankyrin, band 3, protein 4.2, and spectrin. Actin protofilments provide the anchors for the skeleton. They are embedded in the lipid bilayer and provide limited range of motion. Several spectrin strings are attached or wrap around the actin and spread out like a web. Halfway between actin anchors there are what are known as suspension complexes (SC). These consist of ankyrin, band 3, and protein 4.2. Each suspension complex acts to pin the spectrin down to the bilayer. SCs are able to float around the membrane but are restricted by the length of the spectrin. The spectrin however, have built in buffer zones in cases of high tension. These folded domains consist of coiled areas of the spectrin strings that expand when the spectrin is stretched beyond its natural length as shown in the figure below. Folded domains in the spectrin safeguard against premature breakage of the spectrin skeleton increasing overall malleability of the membrane.

Modeling:

In order to best model the entire red blood cell membrane the model was broken down into three levels: Static, Quasi-static, and Dynamic or levels I, II, and III respectively. Level I handled the spectrin (cable) modeling along with the folded domains. Level II dealt with the junction complex (JC) including the actin protofilments and the suspension complex (SC) pinning sites. Level III modeled the cell membrane as a whole including the interactions between the lipid-bilayer and protein skeleton. These three models are coupled through an information-passing multiscale algorithm, in which predictions of Level I and Level II models are employed as constitutive laws in the Level II and Level III models, respectively.

LEVEL #1: Folded/Unfolded Domains	LEVEL #2: Molecular-Detailed JC	LEVEL #3: Complete-cell

Level I, the spectrin domains, were modeled as linear springs. The force extensibility relationship resulting from the model was then passed into the Level II model. The second level incorporated this relationship into calculating the interactions and forces between the molecular compounds of the junction complex and bilayer. The mathematical characteristics and results were then passed to the third level. This last level used all of the preceding information to develop a model for the interactions of the bilayer and cytoskeleton. It even incorporated a method for predicting sliding between the two induced by the mobility of the pinning points.

Tests & Results:

RBC Traveling Through Small Channel:

RBC travelling through small channel	RBC travelling through small channel

RBC in Stokes Flow:

When testing the tri-level erythrocyte model it was assumed that the cell was filled with interior cytoplasm fluid and surrounded by blood plasma. The viscoelasticity of the membrane was obtained from calculations of the viscosity of the lipid bilayer, cytoskeleton, skeleton/bilayer viscous friction of the moving pins, and the viscosity of the surrounding blood plasma. In order to simulate the cell in a Stokes flow a finite element method (FEM) and a boundary element method (BEM) were coupled using a staggered coupling algorithm. Using this technique, we first simulated RBC dynamics in capillary flow and found that the protein density variation and bilayer- skeleton interaction forces are much lower than those in micropipette aspiration, and the maximum interaction force occurs at the trailing edge. Then we investigated mechanical responses of RBCs in shear flow during tumbling, tank-treading, and swinging motions. The dependencies of tank-treading frequency on the blood plasma viscosity and the membrane viscosity we found match well with benchmark data. The simulation results show that during tank-treading the protein density variation is insignificant for healthy erythrocytes, but significant for cells with a smaller bilayer-skeleton friction coefficient, which may be the case in hereditary Spherocytosis.

Snapshots of a 3D RBC in Simple Shear Flow Over Time