One method to experimentally determine the stiffness of membranes consists of indent-ing them with the tip of an atomic force microscope (AFM) a lipid vesicle [Li et al.
(2011)] or a lipid [Kocunet al. (2011)] or polymeric [Kocunet al. (2010)] membrane.
modeling of an experiment
CHAPTER 4. MECHANICAL AND PHASE PROPERTIES OF PLANAR MEMBRANES
Figure 4.11: lhs) Sketch of the indentation of a vesicle by an elliptical tip. rhs)
experimentally measured curves of the indentation of liposomes for different vesicle diameters,dves[Liet al.
(2011)].
Since in such experiments the results depend both on the intrinsic membrane prop-erties and the experimental boundary conditions, we run a sequence of simulations to control for the influence of the following properties that might be difficult to control in experiments:
• size of the tip
• speed of the tip
• contact angle between the supporting wall and the vesicle
• size of the vesicle
We perform the simulations using a vesicle sitting on a surface attractive towards the lipid head-groups by an attractiveLennard-Jones potential (wall = 1[kBT]). We de-scribe the tip as an ellipsoidal repulsiveLennard-Jonespotential (tip= 0.0004[kBT]) defined by a normalrn and two equal lateralrlaxis:
rnnormal axis of the ellipsoid,rl
lateral axis of the ellipsoid,di
distance between the ellipsoid
surface and the beadi. Utip(di) := tip
rl
di 12
−tip
rl
di 6
(4.34) d2i := (xnp−xi)2+ (ynp−yi)2+
rl
rn 2
(znp−zi)2 (4.35) We slowly decrease the position of the tip, ztip, and calculate the response forceFz which is the normal component of the total force exerted on the tip.
The stiffness,kstif f is given by the relation [Liet al.(2011)]:
Fz=kstif f∆ztip (4.36)
The interaction between the vesicle and the plane can be mapped to the experimental data via the ratio: vesicle height/vesicle diameter. We have mapped the force of the system comparing the line tension of the pore (see seventh chapter). The abscissa in the graph (see fig:4.12) refers to the distance between the center of the tip and the plane.
In the experimental set-up [Li et al. (2011)] the tip size is rl = rn ' 20[nm] and the tip speed is 3000[nm/s]. This tip speed is too slow to be able to run comparable simulations. We use a tip speed of 0.0625[nm/ns] and we compare the results of the descending tip with a sequence of simulations where the tip remains at a fixed position.
To perform the first calculation we average over five different indentation simulations with different starting conditions (fast motion).
modeling of an experiment
CHAPTER 4. MECHANICAL AND PHASE PROPERTIES OF PLANAR MEMBRANES
Figure 4.12: Indentation versus response for a 20[nm]
radius liposome. The red curve refers to the average force between the tip and the liposome at fix tip height (stop motion), the blue curve refers to the fast speed motion. We have fitted both curves in the first, I, and second, II, indentation region (see text).
To perform the second calculation we have to equilibrate the system for each tip height for around 500[ts] and then average the normal component of the force (stop motion). We use 12 different tip heights to reconstruct the indentation line. The ratio between the time of the two different calculations is estimated to be 10 times. The stop motion simulations are much more time demanding but the fast motion ones present an irregularity in the profile that is visible in the decrease of force with indentation (see fig:4.12). This fall is due to a residual momentum imposed on the vesicle in the transition where the membrane in contact with the tip is flat. The stop motion does not show thickness irregularity and provides a more realistic description of the real experiment as shown in (fig:4.11). The stiffness measured from the experimental
indentation (see fig:4.11) has to be corrected by: kindslope of the indentation curve,
ktip= 0.0390±0.005[nN/nm]
stiffness of the tip of the AFM.
1
kstif f = 1 kind − 1
ktip (4.37)
The non equilibrium free-energy of the vesicle deformation can be calculated using the Jarzynskiequality [Jarzynski (1997)]:
∆Find= ln e−W
(4.38) WhereW is the work done on the vesicle: W =Fz∆ztip.
Both in experiments and in simulations we identify two regions in the indentation curve: the first region corresponds to small indentations (until 35% of the total height), the second corresponds to large indentations (until twice the membrane thickness when the two bilayers start to fuse) (see fig:4.12). The ratio between the slope in the two regimes is around ten times both in experiments and simulations.
The value of the vesicle stiffness is increased by the effect of the fast motion of the tip as observed in experiment [Li et al. (2011)]. The stiffness calculated by the stop motion indentation is 30% smaller than the experimental value and this is probably due to the softness of the model (a similar deviation is observed comparing the area compressibility measured in this chapter (see table:4.3) with the experimental data on synthetic lipid bilayers [Kocunet al. (2011)]).
boundary condition contributions
An interesting physical question is whether the stiffness is dependent on the size of the vesicle.
modeling of an experiment
CHAPTER 4. MECHANICAL AND PHASE PROPERTIES OF PLANAR MEMBRANES
Figure 4.13: lhs) Stiffness of the vesicle as a function of the curvature, the experimental data refer to the indentation of DMPC liposomes [Liet al.
(2011)], the simulation data refer to fast motion
simulations. rhs)
characteristics of the different indentation simulations. αcont
is contact angle between the vesicle and the substrate.
rves rn−rl kstif f αcont
15 6-8 31 36o
15 20-20 49.0 36o
19.2 20-20 23.2 55o
19.2 20-20 23.7 30o
25 10-10 17.1
30 10-10 11.7
[nm] [nm] [pN/nm]
Lipid membranes are much more complex than elastic sheets, and at high curvature the compression of lipids, the thickness variation, and the induced curvature around the tip should influence the response force.
The influence of the tip size on the stiffness measurement is done by simulating the same indentation using different tip radii: rn = 6[nm] andrl= 8[nm] andrn =rl= 20[nm].
From the table (see fig:4.13) we observe a change of 30% in the stiffness.
To analyze the influence of the speed we have compared a single indentation simulation (tip speed=0.0625[nm/ns]) with a sequence of simulations where the tip was at a fixed constant height. In the graph (4.12) we can observe the difference in the indentation curve depending on the speed of the tip.
We observe that the contact angle between the vesicle and the substrateαcont does not influence significantly the indentation curve (see table:4.13).
Figure 4.14: Thickness profile of the vesicle obtained from isolines of the density of the lipid tails correspondent to three different indentation.
A significant change in the stiffness is caused by the radius of the vesicle. We can see that the resistance of the vesicle to indentation decreases of 60% from doubling the vesicle radius from 15 to 30[nm]. With respect to the experiments we calculate smaller values of the vesicle stiffness but we observe a similar behaviour in the dependence between the vesicle radius and the stiffness.
With respect to the thin shell finite element calculation [Li et al. (2011)] we can extract more precise information about the shape deformation of the vesicles. We calculate the density plot of an indented vesicle on a slab passing through the center
conclusion
CHAPTER 4. MECHANICAL AND PHASE PROPERTIES OF PLANAR MEMBRANES of the vesicle and with a width of 4[nm].
We have decided not to calculate the radial density profile because the fluctuation in shape distorts the thickness profile at large radial distances. From density profiles we extract the isolines that give precise information about the local curvature and thinning of the membrane (see fig:4.14), both information are not available neither in experiments nor in finite elements calculations.
This small set of simulations shows clearly that the deformation rate, the size of the indenting ellipsoid and the radius of the vesicle all have a huge effect on the simulated vesicle stiffness. A more detailed analysis of the parameter space will be necessary to map the exact relations and to check if the continuum mechanics approach followed by [Liet al. (2011)] ignores important physical aspects of the experiments.