Abstract
Red blood cells
(RBCs) undergo remarkably large deformations when subjected to external forces but return to their
biconcave discoid resting shape as the forces are withdrawn. In many experiments, such as
when RBCs are subjected to a shear
flow and undergo the tank-treading motion, the membrane elements are also
displaced from their original (resting) locations along the cell surface with respect to
the cell axis, in
addition to the cell being deformed. A shape memory is said to exist if after the flow is stopped the RBC
regains its biconcave shape and the membrane elements also return to their original locations.
The shape memory
of RBCs was demonstrated by Fischer [“Shape memory of human red blood cells,” Biophys. J.
86, 3304–3313 (2004)] using shear flow go-and-stop experiments. Optical tweezer and
micropipette based stretch-relaxation experiments do not reveal the complete
shape memory
because while the RBC may be deformed, the membrane elements are not significantly displaced from their original
locations with respect to the cell axis. Here we present the first three-dimensional computational
study predicting the complete shape memory of RBCs using shear flow go-and-stop simulations. The influence of
different parameters, namely, membrane shear elasticity and bending rigidity, membrane
viscosity,
cytoplasmic and suspending fluid
viscosity,
as well as different stress-free states of the RBC is studied. For all cases, the RBCs
always exhibit shape
memory. The complete recovery of the RBC in shear flow go-and-stop
simulations occurs over a time that is orders of magnitude longer than that for
optical tweezer
and micropipette based relaxations. The response is also observed to be more complex and
composed of widely disparate time scales as opposed to only one time scale that
characterizes the optical
tweezer and micropipette based relaxations. We observe that the recovery
occurs in three phases: a rapid compression of the RBC immediately after the
flow is
stopped, followed by a slow recovery to the biconcave shape combined with membrane rotation, and a final
rotational return of the membrane elements back to their original locations. A fast time scale
on the order of a few hundred milliseconds characterizes the initial compression phase
while a slow time scale on the order of tens of seconds is associated with the rotational
phase. We observe that the response is strongly dependent on the stress-free state of the
cells, that is,
the relaxation time decreases significantly and the mode of recovery changes from
rotation-driven to deformation-driven as the stress-free state becomes more non-spherical.
We show that while membrane shear elasticity and non-spherical stress-free shape are
necessary and sufficient for the membrane elements to return to their original locations, bending
rigidity is needed for the “global” recovery of the biconcave shape. We also perform a
novel relaxation simulation in which the cell axis of revolution is not aligned with the shear plane
and show that the shape
memory is exhibited even when the membrane elements are displaced
normal to the imposed flow direction. The results presented here could motivate new
experiments to determine the exact stress-free state of the RBC and also to clearly
identify different tank-treading modes.