Model for the curling dynamics
Effect of outer fluid dissipation
Mabrouk et al.1 have reported that curling is driven by a molecularly imposed curvature c0 of the
membrane opposite to and much higher than the initial curvature 1/R of the polymersome of radius
R. According to their model, once a pore nucleates, it grows due to the outer curling of the
membrane, releasing bending energy at a rate Pe. The dynamics of the rim is then controlled solely
by the balance between Pe, and the viscous dissipation Pv due to the movement of the growing rim
in the outer medium. We propose to describe the iRBC bursting with the same approach considering
that, at one point during the parasite development, the iRBC membrane acquires a spontaneous
curvature c0 whose origin is yet to be determined. The time variation of the rim bending energy is
related to the variation in curvature of the spherical membrane elements from a curvature of 1/R to
an opposite higher curvature of 1/L when wrapping on the rim: Pe ≈ 1 κ 12 2πrr˙ , with L ≪ R being
2
L
the radius of the rim. For the viscous dissipation, Mabrouk et al. estimated Pv using the Stokes
€
friction of a cylinder of radius L and length 2πr (perimeter of the rim), moving at a speed r˙ in a
€
fluid of viscosity η: Pv=
4πη
2πrr˙ 2 . The balance of these two terms leads to the following
1/ 2 + ln(2πr / L)
differential equation for r assuming that the slowly varying logarithmic term at the denominator is a
€
constant ln, that we set to 4 considering L ≈ 0.5 µm, r ≈ 3 µm:
r˙ =
D 1
κln 1 1
=
2η 4π L2 2rc c 20 L2
(1)
€
Assuming a compact curling with a membrane thickness e, mass conservation leads to
: 2
1/ 2
1
L = c −1
0 (1+ r / rc ) , hence a polynomial solution of Equation 1 reported in Mabrouk et al. r +2rrc =
Dt, with D=κln/2eη and rc=2π/ec02. For iRBCs however, the pore nucleation is immediately
€
followed by the circular opening up to a radius r0, reached at a time t0. This time delay before
curling
brings
a
correction
to
the
mass
balance
πe(r 2 − r02 ) = 2πr(πL2 − πc −2
0 )
and
to
2
1/ 2
L = c −1
0 (1+ (r − r0 / r) / rc ) . The solution of Equation 1 becomes:
€
r
D(t − t 0 ) = (r − r0 )(r + r0 + 2rc ) − 2r02 ln( )
r0
€
(2)
€
Effect of membrane flow dissipation
In order to calculate the dissipation rate created by the wrapping movement of the membrane during
curling, one has to calculate the flow occurring in the membrane. Such a flow is complex to
estimate. However, one crude approach has been suggested also in the supplementary files of
Mabrouk et al.1. The authors considered that the flow around the curling membrane edge could be
represented by the same term as in the case of a circular pore opening2. In this case, the membrane
r˙
r
internal viscous dissipation can then be written as πr 2 ηS ( ) 2 , where ηS is the membrane viscosity
(N.s/m). Added to the energy rate balance we obtained a new differential equation for r:
r˙ =
€
1 1
D
2 2
η ln
1 + S 2rc c 0 L
8πηr
(3)
The new equation for the evolution of r becomes more complex and is given by the following
€
expression:
r
D(t − t 0 ) = (r − r0 )(r + r0 + 2rc ) − 2r02 ln( ) +
r0
ln ηS
r
((r − r0 ) 2 + rrc ln( ))
4πηr
r0
(5)
€
Moreover, based on our observations of iRBCs, r varies from r0 to R that are of the same order of
magnitude. Thus one good approximation is to consider that D is rescaled by a parameter equal to:
D' =
D
η ln
1+ S
8πηr0
(4)
€
Therefore the possible correction of D that could be attributed to in-plane membrane flow
dissipation will depend on the ratio ηS/ηr0.
REFERENCES
1. Mabrouk E, Cuvelier D, Brochard-Wyart F, Nassoy P, Li MH. Bursting of sensitive
polymersomes induced by curling. Proc. Natl. Acad. Sci. USA. 2009;106:7294-7298.
2. Sandre O, Moreaux L, Brochard-Wyart F. Dynamics of transient pores in stretched vesicles.
Proc. Natl. Acad. Sci USA. 1999;96:10591-10596.
Curling of an elastic stripe in a viscous fluid
We present in this section a preliminary study we performed on the curling in a viscous fluid of a
polypropylene elastic stripe, which displays a spontaneous curvature. As shown and discussed in
supplemental Figure 1A, we observe that viscous friction prevents a compact curling. We checked
that the effect of inertia is negligible on curling compaction by performing experiments for
increasingly viscous fluids where the ratio between the inertial and viscous forces represented by
the Reynolds number of the flow Re = 2Lρr˙/η is decreased, with ρ being the outer fluid density. In
supplemental Figure 1B, we represent the interlayer thickness eN between one layer and the
€
previous one with the number of turns N equal to 0, 1 and 2 as a function of Re for a stripe of fixed
c0, κ and e. We find that eN normalized by c0-1 is a slow varying function of Re and seems to reach a
constant value for Re as low as 10−5. To transpose this result in the case of the curling of an iRBC
membrane and estimate the corresponding eN, we are currently performing experiments at lower Re
(≈ 10−7–10−6 as computed from our data) using different values of c0, κ and e. The full description
of such experiments is out of the scope of this paper. However, supplemental Figure 1B shows that
eN is a fraction of c0-1 close to the Re regime for iRBCs membrane curling.
Figure S1. Curling of an elastic stripe in a viscous fluid. (A) Sequence of images
during curling of a stripe in a referential following the rim: e=90 µm, κ=1.3×10-4 J, c0-1=2.1 mm.
Scale bar = 4.2 mm. (B) eNc0 as a function of the Reynolds number Re = 2Lρr˙/η, where ρ and η are
€
respectively the fluid density and viscosity. Re is the ratio of inertial forces to the viscous forces.
Small Re can be obtained for instance at very high viscosity or when the object size is very small. In
both cases, Re<<1, indicates that viscous forces are dominating the dynamics. Three turns are
represented in the figure: e0, e1 and e2. Re is calculated for four different viscosities: 102, 103, 104
and 105 cSt, for velocities
respectively equal to 25.7, 2.6, 0.3 and 0.02 cm/s. The size of the rim
2L is taken to be the median value between the initial and the final size of the rim after 3 rotations:
0.67-0.86, 0.66-0.96, 0.58-0.73 and 0.575-0.73 cm. To reach Re corresponding to the iRBC scale,
one should have a Re in the range [10-8-10-6] very small compared to 1.