Why Do Red Blood Cells Have Asymmetric Shapes Even in a Symmetric Flow?

Understanding why red blood cells (RBCs) move with an asymmetric shape (slipperlike shape) in small blood vessels is a long-standing puzzle in blood circulatory research. By considering a vesicle (a model system for RBCs), we discovered that the slipper shape results from a loss in stability of the symmetric shape. It is shown that the adoption of a slipper shape causes a significant decrease in the velocity difference between the cell and the imposed flow, thus providing higher flow efficiency for RBCs. Higher membrane rigidity leads to a dramatic change in the slipper morphology, thus offering a potential diagnostic tool for cell pathologies.

Figure 1

Top panel: The behavior of the equilibrium lateral position of the center of mass $Y_G$ as a function of $\nu$ for different values of the flow parameter $v_{\max }$. The full lines are fits with $({\nu }_c-\nu {)}^{1/2}$. Middle panel: The slip velocity (normalized by $v_{\max }$) as a function of $\nu$. The slip velocity (or lag) is defined as the difference between the vesicle (when its vertical position has reached a plateau with time) and the corresponding unperturbed velocity at the position of the vesicle center of mass ($Y_G=0$ for the parachute shape, and $Y_G\ne 0$ for the slipper). Bottom panel: The tank-treading velocity. The link with physical units is explained in the discussion.

Figure 2

Phase diagram in the plane of reduced volume and maximum imposed velocity. Evolution of the shapes is shown. Filled squares represent the boundary of the symmetry-breaking bifurcation, and the solid line is a guide for the eyes. The horizontal dashed line represents the boundary below which the parachute shape has a negative curvature at the rear. Here $W/R_0=10$.

Figure 3

Evolution of the morphology of a slipper as a function of membrane rigidity. $\kappa =1$ refers to the typical value ${10}^{-19}\mathrm{J}$ for vesicles and RBCs. Here we have taken reference values (typical in venules), namely $V_{\max }=800\mu \mathrm{m}/\mathrm{s}$ and $W/R_0=10$.