Fig 1.
Random graph with specified statistical properties generated on the sphere.
(a) Random graph on a spherical surface generated with a specified node density N/S = 300/μm2 mean node degree μ = 5, and edge length distribution σ(D). Note that, while the density of nodes is consistent with the value for a whole red cell, the surface area of the sphere used in this test is a small fraction of the total surface area of a red cell. (b) and (c) Histograms demonstrating that the generated graph has the specified edge-length distribution σ(D) and Poisson node degree distribution, respectively.
Fig 2.
Comparison between the continuum shear energy and the energy of a random network of worm-like chains.
(a) The triangulation associated with the random polymer network, (b) Relaxation to equilibrium of random network, (c)-(d) Comparison of energies from continuum shear formulation and random spring network on two periodic patches with R = 1 μm and R = 2 μm, respectively. The red curves are the corresponding energies computed from the continuum model using upper and lower values of measured shear moduli E obtained from the literature [9]. Error bars are computed by running simulations with several randomly generated networks.
Fig 3.
Cryoelectron tomography and pattern matching of RBC skeletons.
(a) A virtual slice of a cryoelectron tomogram of a red blood cell skeleton. Putative junctional complexes containing the actin protofilaments are indicated with arrows. Structures inside dashed lines are uncharacterized protein clusters (circles) and lipidic remains (squares), (b) Cross-correlation map resulting from comparing the actin template with the tomogram. The brighter the spot, the higher the probability of a good match, (c) The corresponding cross-correlation coefficient plot with a line indicating the cutoff value used. The 2nd knee of the curve is used to avoid false negatives, (d) The identified actin protofilaments overlaid on the actual densities of the tomogram with obvious false positives discarded (for example, some spots with high cross-correlation values are found at locations of lipid remains and unknown protein clusters).
Fig 4.
(a)-(d) The centers of mass of the actin protofilaments shown in Fig 3(d) are used as nodes for the watershed procedure that segments the cytoskeleton. This segments the cytoskeleton into regions associated with the nodes, as illustrated here with different colors. (e) Histogram of nodes per box compared to the Poisson distribution, (f) Plot of observed pairwise distances compared to the density PDF for uniformly distributed points in a 3D slab.
Fig 5.
Results of the segmentation algorithm.
(a)-(c) Probability density functions of edge-lengths versus distances between nodes, contour lengths, and probability mass function of the number of connections per node computed from tomograms by our segmentation algorithm.
Fig 6.
Red cell ghost in steady shear flow with fixed cytoskeletal topology.
See also S2–S4 Videos. (a) Side view of a red cell tank-treading in shear flow with capillary number G = 0.54 and period T = 0.026 seconds. Based on the results of the section “Elastic response of random graphs”, the capillary number is calculated using the shear modulus E = 6 × 10−3 dyn/cm. A material point is marked in red to illustrate the counterclockwise rotation of the cell membrane, (b) Top view of the same cell, (c) Frequency of tank-treading versus capillary number for several values of G and comparison to previous results [1, 41], (d) The distribution of edge lengths in the cytoskeletal network is observed to oscillate during tank-treading.
Fig 7.
Behavior of a dynamic cytoskeleton under shear flow.
(a) I2/I1 over time for koff = 0, 10, and 100 s−1, (b) The lower effective shear modulus for a dynamic network leads to a higher dimensionless capillary number and greater deformations, as evidenced by the greater maximum value of I1 for the network with the fastest rate of remodeling, (c) The total number of irreversibly broken edges is plotted versus time. The network with the fastest dynamics, i.e. koff = 100s−1, accumulates the most irreversibly broken edges in shear flow. The benefit of having more edges that spontaneously disconnect before breaking is outweighed by the cost of decreased shear resistance and greater extension.
Fig 8.
Mechanical properties of red cell model with static spectrin networks under deformation typical of an optical tweezer experiment.
(a) Initial shape of membrane with overlaid cytoskeleton, (b) Final state (see also S6 Video), (c) Comparison of the simulated force-extension curve to optical tweezer experiments [53], in the axial (upper curve) and transverse (lower curve) directions. Error bars from the simulation were computed by comparing forward and reverse deformations.
Fig 9.
Comparison of static and dynamic spectrin networks under deformation typical of an optical tweezer experiment.
(a) Heat map on the cell surface representing the number of irreversibly broken edges on the static (blue) and dynamic with koff = 10s−1 (red) networks at several instants in time. The static network has an excess of irreversibly broken edges at the cell ends, as illustrated by the deep blue color, (b) Dynamic networks (koff = 10s−1, red symbols and koff = 100s−1, black symbols) accumulate fewer irreversibly broken edges under a prescribed strain than the static network (koff = 0s−1, blue symbols). Error bars were computed by performing each simulation at 64 × 64 × 128 and 128 × 128 × 256 grid resolutions, (c) Final state of stretched membrane with irreversibly broken edges overlaid from dynamic network (red edges) and static network (blue edges) simulations.