Figures and data in Balancing stability and flexibility when reshaping archaeal membranes

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27 figures, 5 tables and 1 additional file

Figures

Figure 1 with 4 supplements

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Computational model and phase space of bilayer and bolalipid membranes.

(A) Structure of the diether bilayer lipid archaeol (left) and the tetraether bolalipid caldarchaeol, including four cyclopentane rings (right), both present in the membrane of *Sulfolobus acidocaldarius*, a common archaeal model system that lives at high temperatures and low pH (Rastädter et al., 2020). The hydrophilic head of a bolalipid can be composed of different functional groups represented by R1 and R2 (right). (B) Schematics for a bilayer lipid, a bolalipid and its two in-membrane conformations, membranes made of bilayer molecules only, bolalipid molecules only, and a mixture of the two (left to right). (C) Bilayer lipid (left) is described with one head bead and two tail beads straightened by an angular potential of strength $k_{0}$ . Tail beads of different lipids attract with the strength $ϵ_{p}$ and the range $ω$ . Bolalipids (right) consist of two bilayer lipids connected by a bond and straightened by an angular potential of strength $k_{b o l a}$ . (D) Snapshots of bolalipids self-assembling into a flat membrane ( $k_{b o l a} = 0.3 k_{B} T$ ). (E) Cross-section of self-assembled membrane (right), with bolalipids coloured according to their conformation: straight lipids in crimson and U-shaped lipids in orange. (F) Membrane phase behaviour: liquid, gel and gas regions as a function of the effective temperature $T_{e f f}$ and tail interaction range $ω$ for bilayer (top left) and membranes made of flexible $k_{b o l a} = 0$ (top right) and stiff $k_{b o l a} = 5 k_{B} T$ (bottom left) bolalipid molecules. Overlays of all liquid regions (bottom right) show that stiffer lipids exhibit fluid membrane region at higher temperatures. The dashed line marks $ω = 1.5 σ$ , the value we used in the rest of the work.

Figure 1—video 1

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posterframe for video — Self-assembly of flexible bolalipid molecules into a flat membrane ( $k_{b o l a} = 0$ ).

Figure 1—video 2

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Figure 1—video 3

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Figure 1—video 4

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Figure 2 with 2 supplements

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Mechanics of pure bolalipid membranes.

(A) Liquid region as a function of temperature and bolalipid rigidity for pure bolalipid membranes (grey). The dashed line shows the bolalipid membranes of approximately same fluidities. (B) Fraction of bolalipids in the U-shape conformation (θ=0), fitted to $u_{f} (k_{bola}) = 1 / (1 + \exp (β (- 0.16 + 3 k_{bola})))$ (grey dashed line) according to a two-state model. Insets: simulation snapshots with bolalipids coloured according to their conformations. (C) Bending modulus as a function of bolalipid molecule rigidity $k_{bola}$ . Inset: Tilt persistence length $l_{θ}$ as a function of bolalipid rigidity $k_{b o l a}$ . (D) Snapshots of bolalipid membranes at the range of explored curvatures for $k_{bola} = 1 k_{B} T$ . (E) Fraction of bolalipid molecules in the U-shaped conformation as a function of the mean membrane curvature $H = 1 / (2 R)$ for membranes made of flexible ( $k_{b o l a} = 0$ ) and semi-flexible ( $k_{b o l a} = 1 k_{B} T$ ) bolalipid molecules. (F) Bending modulus as a function of curvature. For the flat membrane (𝐻 ≈ 0), the corresponding bending rigidity from (C) is marked by the vertical line and empty circles.

Figure 2—video 1

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Figure 2—video 2

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Figure 3

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Fluidity and rigidity of mixed bilayer/bolalipid membranes.

(A) Single lipid diffusion constant for each species as a function of bilayer fraction $f^{b i}$ ( $k_{b o l a} = 2 k_{B} T$ , $T_{e f f} = 1.3$ ). For $f^{b i} \geq 0.1$ , the resulting mixture becomes liquid. Top: Diffusion trajectories of a bolalipid (crimson) and a bilayer lipid (blue) in a mixture membrane at $f^{b i} = 0.5$ . (B) Bending rigidity $κ$ and (Inset) tilt persistence length $l_{θ}$ as a function of the fraction of bilayer molecules $f^{b i}$ . Top: Snapshots show bilayer lipids (blue) in mixed membranes at two different values of $f^{b i}$ .

Figure 4 with 4 supplements

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Reshaping of pure bolalipid membranes.

(A) Simulation snapshots of the membrane wrapping a cargo bead adhering to it. Above the onset adhesion energy $ϵ_{m c}^{*}$ , the cargo is fully wrapped by the membrane and buds off the mother membrane. (B) Onset energy $ϵ_{m c}^{*}$ as function of the bolalipid molecule rigidity $k_{b o l a}$ (for the parameters defined by the line in Figure 2A). (C) Bottom: Fraction of bolalipids in the U-shape conformation $u_{f}$ in the outer and inner layers of the membrane bud, and in the flat mother membrane, as a function of the bolalipid molecule rigidity $k_{b o l a}$ . Top: Snapshots and cross-sections of the membrane around the cargo bud. At high bolalipid rigidity, the pores form around the cargo, and are lined with bolalipid molecules lying flat around the pore in a straight conformation, with both heads in the outer layer (coloured in white). The rest of bolalipids was coloured according to their head-to-head angle as before. (D) Bottom: Average diameter of transient pores in the membrane bud and the mother membrane as a function of the bolalipid molecule rigidity $k_{b o l a}$ . Pores are defined as membrane openings through which a sphere of diameter $1 σ$ can cross. Top: Snapshots of the membrane surface with outer and inner leaflet surface coloured in purple and orange, respectively, intersecting at the rim of the pore (grey).

Figure 4—video 1

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Figure 4—video 2

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Figure 4—video 3

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Figure 4—video 4

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Figure 5 with 1 supplement

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Curving of the mixed membranes, made of bilayer and bolalipid molecules.

(A) Onset energy required to form the membrane bud, $ϵ_{m c}^{*}$ , as a function of bilayer head fraction $f_{h}^{b i}$ (for the parameters defined in Figure 3) (**B and C**) Fraction of U-shaped bolalipid molecules $u_{f}$ (B) and bilayer molecules (C) in the outer and inner layers of the membrane bud and in the flat mother membrane as a function of the bilayer head fraction $f_{h}^{b i}$ . Top panels show the respective snapshots of membrane surface around cargo, where bilayer lipids are shown in light blue as in Figure 4. (D) Average diameter of transient pores in the membrane bud and the mother membrane as a function of bilayer head fraction $f_{h}^{b i}$ and respective snapshots of membrane leaflet surfaces surrounding the bud (top panel).

Figure 5—video 1

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Appendix 1—figure 1

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Histogram of the cosine of the angle $θ$ between halves of a bolalipid ( $k_{b o l a} = 0$ ) for the last frame of the simulation of a flat membrane of flexible bolalipids.

Appendix 1—figure 2

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Time series of $u_{f}$ , the fraction of bolalipids in the U-shaped conformation, for the simulations done for flexible bolalipids ( $k_{b o l a} = 0$ ), where the system was pre-assembled with three different initial values $u_{f} |_{t = 0}$ .

The red dashed line marks the typical equilibration time of $10^{4} τ$ .

Appendix 1—figure 3

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Time series of fraction of bolalipids in the U-shaped conformation $u_{f}$ , for simulations of pure bolalipid membranes with different values of the bolalipid rigidity $k_{b o l a}$ .

The dashed line marks the timestamp used for defining the system as equilibrated. All equilibrium measures are taken from frames after this point.

Appendix 1—figure 4

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Diffusion constant for different pure membranes at $w = 1.5 σ$ versus temperature $T_{e f f}$ .

The discontinuity (jump) in $D$ marks the transition from the gel phase to the liquid phase. The dashed black line marks the minimum diffusion constant that is required to classify as a liquid membrane at $D = 5 \times 10^{- 4} σ^{2} / τ$ .

Appendix 1—figure 5

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Path of the complex component of the Fourier transform of membrane height at $n = (1, 0)$ (A) and $n = (2, 0)$ (B) for bolalipid pure membrane at $k_{b o l a} = 0.5 k_{B} T$ .

Left plots show the trajectory in the complex plane, while on the right, we plot their phase and norm versus time. The mode in (A), with an autocorrelation time of roughly $10^{4} τ$ , has only 10 uncorrelated points. On the other hand, the mode in (B), with an autocorrelation time of approximately $10^{3} τ$ , has $\approx 60$ uncorrelated samples and thus crosses the chosen threshold of 20 samples for being considered equilibrated.

Appendix 1—figure 6

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Fluctuation spectra and corresponding fits according to Equation A6 for a model without and with tilt modulus, for a bilayer membrane at $T_{e f f} = 1.2 k_{B} T$ , a flexible bolalipid membrane ( $k_{b o l a} = 0$ ) and a rigid bolalipid membrane ( $k_{b o l a} = 5 k_{B} T$ ).

In vertical dashed lines, we marked the interval of wave numbers that selects the modes used for fitting. The first rigid bolalipid equilibrated mode is to the left of this interval and thus excluded from the corresponding fit.

Appendix 1—figure 7

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Fluctuation spectrum fit comparisons for pure bolalipid membranes (A) and bolalipid/bilayer mixture membranes (B).

In the first row, we show the results of the fit of Equation A5 while in the second row, we used Equation A6.

Appendix 1—figure 8

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Fluctuation spectrum fit comparisons for pure bilayer membranes (A), flexible (B) and rigid pure bolalipid membranes (C).

For each type of membrane, in the first row, we show the results of the fit of Equation A5 while in the second row, we used Equation A6, as a function of $T_{e f f}$ .

Appendix 1—figure 9

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Fluctuation spectrum fit results, with (top) bending and (bottom) tilt modulus, for bilayer, flexible bolalipid and rigid bolalipid membranes at $w = 1.5$ as a function of $T_{e f f}$ .

Appendix 1—figure 10

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Fraction of lipid heads belonging to U-shaped bolalipids, for both outer (upwards triangles) and inner (downwards triangles) leaflets in cylinder membranes versus mean curvature $H$ , for bolalipid membranes at $k_{b o l a} = 0$ and $k_{b o l a} = 1 k_{B} T$ .

Appendix 1—figure 11

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Bending modulus $κ$ versus U-shaped bolalipid fraction, using the same data as Figure 2B and C.

For reference, the bilayer bending modulus at $T_{e f f} = 1.1$ is shown in blue.

Appendix 1—figure 12

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Area per head bead measurements, in the inner and outer layer of the membrane bud, as well as the flat mother membrane, for (A) bolalipid membranes as a function of $k_{b o l a}$ and (B) membranes of a mixture of bilayer lipids and bolalipids as a function of the bilayer lipid fraction $f^{b i}$ .

Appendix 1—figure 13

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Lipid conformation and location in the membrane bud for pure bolalipid membranes, excluding lipids near pores (solid lines) and exclusively considering lipids near pores (dashed lines).

Appendix 1—figure 14

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Lipid conformation and location in the membrane bud for pure bolalipid membranes, before budding, (A) for pure bolalipids at $k_{b o l a} = 1 k_{B} T$ and for (B) a mixture of bolalipids with 30% bilayer, averaged over 100 frames spaced over a $2000 τ$ time interval.

For both cases, a snapshot (left) is shown with the front right half of the membrane cut away, showing the profile shape, matching the (right) time-averaged spatially varying conformation or specie fraction. Visible at $z > 10 σ$ and for small $r_{x y}$ is the inner region of the neck. While for mixture membranes the effect on bilayer fraction is visible using a linear scale, for bolalipid membranes at $k_{b o l a} = 1 k_{B} T$ the effect spans multiple orders of magnitude so a logarithmic scale was used. The flat membrane values, taken as average values of the bins at $r_{x y} > 20 σ$ , are indicated by a black mark on the colour bar.

Appendix 1—figure 15

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Phase diagram for pure bilayer and bolalipid membranes (flexible and rigid), as in Figure 1F, for (A) $T_{e f f}$ and (B) $T$ scaling.

Appendix 1—figure 16

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Liquid region (blue) for bolalipid membranes with potentials setup as in the main text as a function of $k_{b o l a}$ and $T_{e f f}$ (A), and as a function of $k_{b o l a}$ and $T$ (B), with U-shape fraction in colour.

Appendix 1—figure 17

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Comparison between using $T_{e f f}$ for Figure 2 (A), with scaling temperature $T$ for the same parameters (B).

The tilt contribution for $k_{b o l a} < 1 k_{B} T$ is negligible.

Appendix 1—figure 18

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Height fluctuation spectrum, for a rigid bolalipid membrane at $T_{e f f} = 1.8$ .

Solid lines mark the −2, −4 slopes. The vertical dashed line marks the limit of the data used for fitting. The fits are different dashed lines; the fit that models both tension and tilt overlaps with the fit that only includes tilt.

Appendix 1—figure 19

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Height fluctuation spectrum, for a bilayer membrane at $T_{e f f} = 1.1$ simulated under different integrators.

The resulting fit parameters together with the normalized $χ^{2}$ are shown in Appendix 1—table 5.

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(A) Schematic showing the decay of the power spectrum as a function of the wave number q in the tilt model (top), in the tension model with positive membrane tension (middle), and in the tension model with negative membrane tension (bottom). (B) Fitted power spectrum as a function of q for rigid bolalipid membranes (k_bola=5k_BT). The fit shows that while the model with tension (dashed line) cannot fit the data, the model with tilt nicely fits the spectrum (solid-dashed line). The combined model including both tension and tilt does not fit the spectrum any better (dotted line).

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Height fluctuation spectrum, for a bilayer membrane at T_eff =1.

1, simulated with Langevin dynamics (pink, ‘langevin‘), our setup (purple, ‘nph+langevin‘), and under an isothermal-isobaric ensemble (blue, ‘npt‘); fits are shown as dotted lines.

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Height fluctuation spectrum, for a bilayer membrane at T_eff =1.

1, as simulated in the main text (grey, for 60⇥10³τ), for longer duration (1_.44⇥10⁶τ) (pink), and with the longer duration and halved timestep = 0.005τ_(purple); fits are shown as dotted lines (tension and tilt) or dash-dot lines (tilt only).

Tables

Appendix 1—table 1

Parameters and measurements for bilayer lipid cylinders.

$N$ is number of lipids.

			$H$	$\| \frac{P_{x} + P_{y}}{2 P_{z}} \|$	$κ$
$Δ t$	$N$	$l_{z}$
0.001	19457	120	(3.08+/-0.04)e-02	0.03+/-0.06	8.9+/-2.4
	10000	70	(3.447+/-0.004)e-02	0.03+/-0.05	8.4+/-1.1
	10000	80	(3.923+/-0.008)e-02	0.04+/-0.04	8.1+/-0.6
	5000	40	(3.948+/-0.006)e-02	0.05+/-0.05	8.8+/-1.3
		50	(4.942+/-0.009)e-02	0.011+/-0.021	8.6+/-0.6
		60	(5.857+/-0.005)e-02	0.018+/-0.017	8.22+/-0.33
		70	(6.82+/-0.02)e-02	0.010+/-0.013	8.27+/-0.30
		80	(7.17+/-0.03)e-02	0.009+/-0.011	8.7+/-0.3
0.005	5000	90	(7.86+/-0.03)e-02	0.010+/-0.006	9.16+/-0.21
0.005	5000	96	(8.13+/-0.03)e-02	0.085+/-0.013	8.87+/-0.27

Appendix 1—table 2

Parameters and measurements for flexible bolalipids tethers.

			$H$	$\| \frac{P_{x} + P_{y}}{2 P_{z}} \|$	$κ$	$u_{f}$
$Δ t$	$N$	$l_{z}$
0.001	19457	120	(3.024+/-0.004)e-02	0.06+/-0.08	9.2+/-1.6	0.534+/-0.006
	10000	70	(3.354+/-0.004)e-02	0.03+/-0.08	9.7+/-2.1	0.536+/-0.004
	10000	80	(3.848+/-0.004)e-02	0.04+/-0.05	9.5+/-1.4	0.537+/-0.005
	5000	40	(3.907+/-0.006)e-02	0.04+/-0.07	9.6+/-2.0	0.531+/-0.004
		50	(4.821+/-0.004)e-02	0.03+/-0.04	9.7+/-1.0	0.542+/-0.005
		60	(5.672+/-0.006)e-02	0.021+/-0.024	9.3+/-0.6	0.544+/-0.005
		70	(6.35+/-0.02)e-02	0.020+/-0.018	9.2+/-0.5	0.546+/-0.005
		80	(7.19+/-0.03)e-02	0.008+/-0.012	9.1+/-0.4	0.551+/-0.005
0.005	5000	90	(7.82+/-0.03)e-02	0.095+/-0.007	9.70+/-0.16	0.5583+/-0.0015
0.005	5000	96	(8.13+/-0.03)e-02	0.088+/-0.012	9.35+/-0.28	0.563+/-0.004

Appendix 1—table 3

Parameters and measurements for bolalipid tethers at $k_{b o l a} = 1 k_{B} T$ .

			$H$	$\| \frac{P_{x} + P_{y}}{2 P_{z}} \|$	$κ$	$u_{f}$
$Δ t$	$N$	$l_{z}$
0.001	19457	120	(2.921+/-0.006)e-02	0.020+/-0.032	23.3+/-2.1	0.07401+/-0.00033
	10000	70	(3.260+/-0.002)e-02	0.043+/-0.028	23.3+/-1.7	0.0806+/-0.0009
	10000	80	(3.743+/-0.002)e-02	0.021+/-0.022	22.0+/-1.5	0.0898+/-0.0009
	5000	50	(4.526+/-0.003)e-02	0.009+/-0.022	19.5+/-1.0	0.1053+/-0.0010
	5000	60	(5.375+/-0.004)e-02	0.010+/-0.015	17.8+/-0.5	0.1225+/-0.0008

Appendix 1—table 4

The membrane type, $T_{e f f}$ , the area per lipid, the patch radius $R$ , the folding parameter $ξ$ , the membrane line tension $γ$ , the bending rigidity $κ$ , the Gaussian bending rigidity $\bar{κ}$ , and the ratio of $- \bar{κ} / κ$ .

Memb	$T_{e f f}$	A/lipid [ $σ^{2}$ ]	$R$ [ $σ$ ]	$ξ$	$γ$ [ $k_{B} T / σ$ ]	$κ$ [ $k_{B} T$ ]	$\bar{κ}$ [ $k_{B} T$ ]	$- \bar{κ} / κ$
Bila	1.2	1.3587	7.238±0.046	1.679±0.010	1.237±0.035	4.82±0.08	−4.30±0.22	0.89±0.04
Bola	1.2	1.2902	7.165±0.0000	1.309±0.009	1.957±0.043	7.88±0.14	−5.04±0.37	0.64±0.04
Bila	1.3	1.4238	7.1999±0.122	1.660±0.008	0.917±0.031	3.36±0.05	−2.74±0.18	0.81±0.05
Bola	1.3	1.3475	7.322±0.000	1.3466±0.009	1.381±0.038	4.79±0.14	−2.17±0.35	0.45±0.07

Appendix 1—table 5

Normalized $χ^{2}$ and parameters for fits in Appendix 1—figure 19.

	$χ^{2} / N$	$κ / k_{B} T$	$Σ / (k_{B} T / σ^{2})$	$l_{θ} / σ$
langevin	0.72	8.51±0.12	−0.039±0.0004	0.85±0.05
nph +langevin	0.28	8.71±0.11	−0.050±0.004	0.92±0.04
npt	0.25	8.35±0.09	−0.0492±0.0026	0.86±0.04
nph +langevin (short)	0.17	7.88±0.25	N/A	0.60±0.16
nph +langevin (timestep halved)	0.31	9.13±0.12	N/A	0.97±0.05

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Miguel Amaral
Felix Frey
Xiuyun Jiang
Buzz Baum
Anđela Šarić

(2025)

Balancing stability and flexibility when reshaping archaeal membranes

eLife 14:RP105432.

https://doi.org/10.7554/eLife.105432.3

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Computational model and phase space of bilayer and bolalipid membranes.

Self-assembly of flexible bolalipid molecules into a flat membrane (kbola=0).

Close-up view of a flat membrane, with tail beads shown like sticks.

Close-up view of a flat membrane, with tail beads shown like sticks.

Close-up view of a flat membrane, with tail beads shown like sticks.

Mechanics of pure bolalipid membranes.

Fluctuating flat membrane patch of bolalipid molecules at kbola=0 used for height fluctuation spectrum measurements.

Cylindrical membranes after equilibration, made of bolalipid molecules of intermediate stiffness (kbola=1kBT) at the largest simulated radius R≈17σ.

Fluidity and rigidity of mixed bilayer/bolalipid membranes.

Reshaping of pure bolalipid membranes.

Successful budding, showing only a cargo-centred cross-section.

Successful budding, showing only a cargo-centred cross-section.

Successful budding, showing only a cargo-centred cross-section.

Rotating snapshot of a bud formed from a membrane made out of stiff rigid bolalipid molecules (kbola=5kBT), showing several pores (ϵmc=3.25kBT).

Curving of the mixed membranes, made of bilayer and bolalipid molecules.

Successful budding, showing only a cargo-centred cross-section.

Histogram of the cosine of the angle θ between halves of a bolalipid (kbola=0) for the last frame of the simulation of a flat membrane of flexible bolalipids.

Time series of uf, the fraction of bolalipids in the U-shaped conformation, for the simulations done for flexible bolalipids (kbola=0), where the system was pre-assembled with three different initial values uf|t=0.

Time series of fraction of bolalipids in the U-shaped conformation uf, for simulations of pure bolalipid membranes with different values of the bolalipid rigidity kbola.

Diffusion constant for different pure membranes at w=1.5σ versus temperature Teff.

Path of the complex component of the Fourier transform of membrane height at n=(1,0) (A) and n=(2,0) (B) for bolalipid pure membrane at kbola=0.5kBT.

Fluctuation spectra and corresponding fits according to Equation A6 for a model without and with tilt modulus, for a bilayer membrane at Teff=1.2kBT, a flexible bolalipid membrane (kbola=0) and a rigid bolalipid membrane (kbola=5kBT).

Fluctuation spectrum fit comparisons for pure bolalipid membranes (A) and bolalipid/bilayer mixture membranes (B).

Fluctuation spectrum fit comparisons for pure bilayer membranes (A), flexible (B) and rigid pure bolalipid membranes (C).

Fluctuation spectrum fit results, with (top) bending and (bottom) tilt modulus, for bilayer, flexible bolalipid and rigid bolalipid membranes at w=1.5 as a function of Teff.

Fraction of lipid heads belonging to U-shaped bolalipids, for both outer (upwards triangles) and inner (downwards triangles) leaflets in cylinder membranes versus mean curvature H, for bolalipid membranes at kbola=0 and kbola=1kBT.

Bending modulus κ versus U-shaped bolalipid fraction, using the same data as Figure 2B and C.

Area per head bead measurements, in the inner and outer layer of the membrane bud, as well as the flat mother membrane, for (A) bolalipid membranes as a function of kbola and (B) membranes of a mixture of bilayer lipids and bolalipids as a function of the bilayer lipid fraction fbi.

Lipid conformation and location in the membrane bud for pure bolalipid membranes, excluding lipids near pores (solid lines) and exclusively considering lipids near pores (dashed lines).

Lipid conformation and location in the membrane bud for pure bolalipid membranes, before budding, (A) for pure bolalipids at kbola=1kBT and for (B) a mixture of bolalipids with 30% bilayer, averaged over 100 frames spaced over a 2000τ time interval.

Phase diagram for pure bilayer and bolalipid membranes (flexible and rigid), as in Figure 1F, for (A) Teff and (B) T scaling.

Liquid region (blue) for bolalipid membranes with potentials setup as in the main text as a function of kbola and Teff (A), and as a function of kbola and T (B), with U-shape fraction in colour.

Comparison between using Teff for Figure 2 (A), with scaling temperature T for the same parameters (B).

Height fluctuation spectrum, for a rigid bolalipid membrane at Teff=1.8.

Height fluctuation spectrum, for a bilayer membrane at Teff=1.1 simulated under different integrators.

Height fluctuation spectrum, for a bilayer membrane at Teff =1.

Height fluctuation spectrum, for a bilayer membrane at Teff =1.

Parameters and measurements for bilayer lipid cylinders.

Parameters and measurements for flexible bolalipids tethers.

Parameters and measurements for bolalipid tethers at kbola=1kBT.

The membrane type, Teff, the area per lipid, the patch radius R, the folding parameter ξ, the membrane line tension γ, the bending rigidity κ, the Gaussian bending rigidity κ¯, and the ratio of −κ¯/κ.

Normalized χ2 and parameters for fits in Appendix 1—figure 19.

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Self-assembly of flexible bolalipid molecules into a flat membrane ( $k_{b o l a} = 0$ ).

Fluctuating flat membrane patch of bolalipid molecules at $k_{b o l a} = 0$ used for height fluctuation spectrum measurements.

Cylindrical membranes after equilibration, made of bolalipid molecules of intermediate stiffness ( $k_{b o l a} = 1 k_{B} T$ ) at the largest simulated radius $R \approx 17 σ$ .

Rotating snapshot of a bud formed from a membrane made out of stiff rigid bolalipid molecules ( $k_{b o l a} = 5 k_{B} T$ ), showing several pores ( $ϵ_{m c} = 3.25 k_{B} T$ ).

Histogram of the cosine of the angle $θ$ between halves of a bolalipid ( $k_{b o l a} = 0$ ) for the last frame of the simulation of a flat membrane of flexible bolalipids.

Time series of $u_{f}$ , the fraction of bolalipids in the U-shaped conformation, for the simulations done for flexible bolalipids ( $k_{b o l a} = 0$ ), where the system was pre-assembled with three different initial values $u_{f} |_{t = 0}$ .

Time series of fraction of bolalipids in the U-shaped conformation $u_{f}$ , for simulations of pure bolalipid membranes with different values of the bolalipid rigidity $k_{b o l a}$ .

Diffusion constant for different pure membranes at $w = 1.5 σ$ versus temperature $T_{e f f}$ .

Path of the complex component of the Fourier transform of membrane height at $n = (1, 0)$ (A) and $n = (2, 0)$ (B) for bolalipid pure membrane at $k_{b o l a} = 0.5 k_{B} T$ .

Fluctuation spectra and corresponding fits according to Equation A6 for a model without and with tilt modulus, for a bilayer membrane at $T_{e f f} = 1.2 k_{B} T$ , a flexible bolalipid membrane ( $k_{b o l a} = 0$ ) and a rigid bolalipid membrane ( $k_{b o l a} = 5 k_{B} T$ ).

Fluctuation spectrum fit results, with (top) bending and (bottom) tilt modulus, for bilayer, flexible bolalipid and rigid bolalipid membranes at $w = 1.5$ as a function of $T_{e f f}$ .

Fraction of lipid heads belonging to U-shaped bolalipids, for both outer (upwards triangles) and inner (downwards triangles) leaflets in cylinder membranes versus mean curvature $H$ , for bolalipid membranes at $k_{b o l a} = 0$ and $k_{b o l a} = 1 k_{B} T$ .

Bending modulus $κ$ versus U-shaped bolalipid fraction, using the same data as Figure 2B and C.

Area per head bead measurements, in the inner and outer layer of the membrane bud, as well as the flat mother membrane, for (A) bolalipid membranes as a function of $k_{b o l a}$ and (B) membranes of a mixture of bilayer lipids and bolalipids as a function of the bilayer lipid fraction $f^{b i}$ .

Lipid conformation and location in the membrane bud for pure bolalipid membranes, before budding, (A) for pure bolalipids at $k_{b o l a} = 1 k_{B} T$ and for (B) a mixture of bolalipids with 30% bilayer, averaged over 100 frames spaced over a $2000 τ$ time interval.

Phase diagram for pure bilayer and bolalipid membranes (flexible and rigid), as in Figure 1F, for (A) $T_{e f f}$ and (B) $T$ scaling.

Liquid region (blue) for bolalipid membranes with potentials setup as in the main text as a function of $k_{b o l a}$ and $T_{e f f}$ (A), and as a function of $k_{b o l a}$ and $T$ (B), with U-shape fraction in colour.

Comparison between using $T_{e f f}$ for Figure 2 (A), with scaling temperature $T$ for the same parameters (B).

Height fluctuation spectrum, for a rigid bolalipid membrane at $T_{e f f} = 1.8$ .

Height fluctuation spectrum, for a bilayer membrane at $T_{e f f} = 1.1$ simulated under different integrators.

Height fluctuation spectrum, for a bilayer membrane at T_eff =1.

Height fluctuation spectrum, for a bilayer membrane at T_eff =1.

Parameters and measurements for bolalipid tethers at $k_{b o l a} = 1 k_{B} T$ .

The membrane type, $T_{e f f}$ , the area per lipid, the patch radius $R$ , the folding parameter $ξ$ , the membrane line tension $γ$ , the bending rigidity $κ$ , the Gaussian bending rigidity $\bar{κ}$ , and the ratio of $- \bar{κ} / κ$ .

Normalized $χ^{2}$ and parameters for fits in Appendix 1—figure 19.