Active elastic network: Cytoskeleton of the red blood cell
Nir S. Gov
Department of Chemical Physics, The Weizmann Institute of Science, P.O.B. 26, Rehovot, Israel 76100
Received 25 September 2006; published 19 January 2007
In red blood cells there is a cortical cytoskeleton; a two-dimensional elastic network of membrane-attached
proteins. We describe, using a simple model, how the metabolic activity of the cell, through the consumption
ofATP, controls the stiffness of this elastic network. The unusual mechanical property of active strain softening
is described and compared to experimental data. As a by-product of this activity there is also an active
contribution to the amplitude of membrane fluctuations. We model this membrane as a field of independent
“curvature motors,” and calculate the spectrum of active fluctuations. We find that the active cytoskeleton
contributes to the amplitude of the membrane height fluctuations at intermediate wavelengths, as observed
experimentally.
DOI: 10.1103/PhysRevE.75.011921 PACS numbers: 87.68.+z, 83.60.-a, 87.16.Ac, 87.17.-d
I. INTRODUCTION
The red-blood cell RBC has a cortical cytoskeleton,
which is a two-dimensional network of cross-linked elastic
proteins that are attached to and cover the entire inner sur-
face of the cell membrane 1,2. Such a cortical cytoskeleton
gives the RBC the elasticity to survive repeated deformations
3. The outstanding problem of understanding the physical
properties of the RBC cytoskeleton has been to reconcile the
measurements of the elasticity of the cell 3, the observed
shapes of the RBC 4, and the measurements of the dynamic
shape fluctuations of the membrane 5,6. Our model at-
tempts to reconcile these observation by showing that they
follow naturally from our recently proposed mechanism of
active remodeling of the RBC cytoskeleton 7. In this paper
we extend our recent theoretical modeling 7,8 and calculate
several additional features of the active cytoskeleton. i We
calculate the unique effect of strain-softening in this active
network. The active dissociations of the network can account
for the observed strain-softening of the RBC due to the dis-
sociated filaments spending a longer time disconnected when
the network is stretched. ii We then calculate the spectrum
of active fluctuations, where we find that these contribute to
the membrane height fluctuations at intermediate wave-
lengths, as observed experimentally.
The RBC cytoskeleton is a self-assembled network of
flexible spectrin filaments that are end-attached to node com-
plexes Fig. 1. On average approximately six spectrin fila-
ments attach to each actin node, mainly through the protein-
4.1 1,9, with a specific chemical affinity of order 7k
B
T.
The flexible spectrin filaments are much longer 200 nm
than the distance between the nodes that they connect
R 80-100 nm, Fig. 1a, and can be treated as wormlike
chains with an average end-to-end distance of 10:
R
0
50- 60 nm. Each filament behaves for small deforma-
tions as a Hookean entropic spring, with a spring constant of
11: k 20-40k
B
T / R
2
1 10
-5
J/m
2
. At larger extensions
a semiflexible chain behaves nonlinearly, and exhibits strain
stiffening 12,13 and unfolding 14. Note that the spectrin
filaments are additionally connected to the membrane
through the ankyrin-band-3 complex at random locations
along the filaments, in between the actin-band-4.1 complexes
at their ends 1. It is the connections at the actin nodes that
determine the overall connectivity of the network, and there-
fore dominate in controlling its elastic properties. We will
therefore, also for simplicity of modeling, deal here with the
active processes occurring at the filaments ends alone.
The elasticity of such a two-dimensional network, at-
tached to the fluid bilayer, can be modeled as a static elastic
network, with fixed connectivity 4,11,15. Nevertheless, ob-
servations of membrane fluctuations 6 led to the specula-
tion that ATP-induced phosphorylation of cytoskeleton pro-
teins induces an overall softening. More recently, numerical
simulations 4 recover the observed discocyte shape only
FIG. 1. A schematic illustration of our model of ATP-induced
spectrin dissociation. The main parts of the figure show a side view
of the RBC membrane, while the small parts show a schematic top
view of the triangular cytoskeleton network. a The fully connected
network, showing the balance between the stretched spectrin and
curved bilayer. b The dissociated spectrin releases its stored ten-
sion. The phosphorylated node is indicated by the empty circle.
PHYSICAL REVIEW E 75, 011921 2007
1539-3755/2007/751/0119216 ©2007 The American Physical Society 011921-1