Pulsatile Lipid Vesicles under Osmotic Stress

Abstract

The response of lipid bilayers to osmotic stress is an important part of cellular function. Recent experimental studies showed that when cell-sized giant unilamellar vesicles (GUVs) are exposed to hypotonic media, they respond to the osmotic assault by undergoing a cyclical sequence of swelling and bursting events, coupled to the membrane’s compositional degrees of freedom. Here, we establish a fundamental and quantitative understanding of the essential pulsatile behavior of GUVs under hypotonic conditions by advancing a comprehensive theoretical model of vesicle dynamics. The model quantitatively captures the experimentally measured swell-burst parameters for single-component GUVs, and reveals that thermal fluctuations enable rate-dependent pore nucleation, driving the dynamics of the swell-burst cycles. We further extract constitutional scaling relationships between the pulsatile dynamics and GUV properties over multiple timescales. Our findings provide a fundamental framework that has the potential to guide future investigations on the nonequilibrium dynamics of vesicles under osmotic stress.

Introduction

In their constant struggle with the environment, living cells of contemporary organisms employ a variety of highly sophisticated molecular mechanisms to deal with sudden changes in their surroundings. One often encountered environmental assault on cells is osmotic stress, where the amount of dissolved molecules in the extracellular environment drops suddenly (1, 2). If left unchecked, this perturbation will result in a rapid flow of water into the cell through osmosis, causing it to swell, rupture, and die. To avoid this catastrophic outcome, even bacteria have evolved complex molecular machineries, such as mechanosensitive channel proteins, which allow them to release excess water from their interior (3, 4, 5, 6). This then raises an intriguing question of how might primitive cells, or cell-like artificial constructs, that lack the sophisticated protein machinery for osmosensing and osmoregulation, respond to such environmental insults and preserve their structural integrity.

Using rudimentary cell-sized giant unilamellar vesicles (GUVs) devoid of proteins and consisting of amphiphilic lipids and cholesterol as models for simple protocells, we showed previously that vesicular compartments respond to osmotic assault created by the exposure to hypotonic media by undergoing a cyclical sequence of swelling and poration (7). In each cycle, osmotic influx of water through the semipermeable boundary swells the vesicles and renders the bounding membrane tense, which in turn, opens a microscopic transient pore, releasing some of the internal solutes before resealing. This swell-burst process, depicted in Fig. 1 A, repeats multiple times producing a pulsating pattern in the size of the vesicle undergoing osmotic relaxation. From a dynamical point of view, this autonomous osmotic response results from an initial, far-from-equilibrium, thermodynamically unstable state generated by the sudden application of osmotic stress. The subsequent evolution of the system, characterized by the swell-burst sequences described above, occurs in the presence of a global constraint, namely constant membrane area, during a dissipation-dominated process (8, 9).

Figure 1.

Homogeneous GUVs made of POPC with 1 mol % Rho-DPPE exhibit swell-burst cycles when subject to hypotonic conditions. (A) Schematic of a swell-burst cycle of a homogeneous GUV under hypotonic conditions. Dashed blue arrows represent osmotic influx of solvent through the lipid membranes, while solid blue arrows represent the leak-out of the inner solution through the transient pore. (B) Micrographs of a swollen (left), ruptured (middle), and resealed (right) GUV. Scale bars represent 10 μm. Pictures extracted from Movie S1. (C) Typical evolution of a GUV radius with time during swell-burst cycles in 200 mM sucrose hypotonic conditions. The GUV radius increases continuously during swelling phases, and drops abruptly when bursting events occur. Pore opening events are indicated by red triangles. Dashed line represents the estimated initial radius R₀. See also Movie S2. More GUV radius measurements are shown in Fig. S1. (D) Experimental area strain ϵ_exp as a function of time. To see this figure in color, go online.

The study of osmotic response of lipid vesicles has a rich history in theoretical biophysics, beginning with the pioneering work by Koslov and Markin (10), who provided some of the early theoretical foundations of osmotic swelling of lipid vesicles. In this work, they predicted that the response of a submicrometer-sized vesicles to osmotic stress is likely pulsatile and due to the formation of successive transient pores (see Fig. 9, in (10), for a schematic of the volume change of the vesicle over time). They further approximated the characteristic quantities of swell-burst cycles (e.g., swelling time, critical volumes), based on the probability of the membrane overcoming the nucleation energy barrier to form a pore. Independently, the dynamics of a single transient pore in a tense membrane were first theorized by Litster (11), and later investigated theoretically and experimentally by Brochard-Wyart and co-workers (12, 13). Idiart and Levin (14) combined the osmotic swelling theory and pore dynamics, and calculated the dynamics of a pulsatile behavior assuming a constant lytic tension. These modeling efforts made great strides in our understanding of some of the essential physics underlying vesicle responses to osmotic stress.

Previously, we used these ideas to provide a qualitative interpretation of pulsatile behavior of GUVs (see schematics in Fig.7, h and i of (7)). However, a general framework that quantitatively describes the response of pulsatile vesicles to osmotic stress at all relevant timescales is still missing. The success of such a model must rely on 1) the integration of vesicle dynamics, pore dynamics with nucleation, and long-time solute concentration dynamics within a unified framework; and 2) the assessment of the model predictions with respect to experimental measurements, to establish the physical relevance of the essential parameters that govern the system dynamics. Here, we build on the findings and theories reported previously (10, 11, 13, 14, 15, 16) to develop such a quantitative model for the dynamics of swell-burst cycles in giant lipid vesicles subject to osmotic stress.

In analyzing the pulsatile dynamics of GUVs, a number of general questions naturally arise: 1) Is the observed condition for membrane poration deterministic or stochastic? 2) Is poration controlled by a unique value of membrane tension (i.e., lytic tension) introduced by the area-volume changes, which occur during osmotic influx, or does it involve coupling of the membrane response to thermal fluctuations? 3) Does the critical lytic tension depend on the strain rate, and thus the strength of the osmotic gradient? Such questions arise beyond this context of vesicle osmoregulation in other important scenarios where the coupling between the dissipation of osmotic energy and cellular compartmentalization has important biological ramifications (17, 18, 19, 20).

Motivated by these considerations, we carried out a combined theoretical-experimental study integrating membrane elasticity, continuum transport, and statistical thermodynamics. We gathered quantitative experimental data to address the questions above, and developed a general model that recapitulates the essential qualitative features of the experimental observations, emphasizes the importance of dynamics, and places the heretofore neglected contribution of thermal fluctuations in driving osmotic response of stressed vesicular compartments.

Materials and Methods

The detailed materials and methods used in this work are available in Supporting Materials and Methods. The experimental configuration is similar to that already described (7, 21). Briefly, we prepared GUVs consisting essentially of a single amphiphile, namely 1-palmitoyl-2-oleoyl-sn-1-glycero-3-phosphocholine (POPC), doped with a small concentration (1 mol $\text{\%}$) of a fluorescently labeled phospholipid (1,2-dipalmitoyl-sn-glycero-3-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl), Rho-DPPE) using standard electroformation technique (21). The GUVs thus obtained were typically between 7 and 20 μm in radius, encapsulated 200 mM sucrose, and were suspended in the isotonic glucose solution of identical osmolarity. Diluting the extravesicular dispersion medium with deionized water produces a hypotonic bath depleted in osmolytes, subjecting the GUVs to osmotic stress. Shortly (∼1 min delay) after subjecting the GUVs to the osmotic differential, GUVs were monitored using time-lapse epifluorescence microscopy at a rate of one image per 150 ms, and images were analyzed using a customized MATLAB (The MathWorks, Natick, MA) code to extract the evolution of the GUV radii with time, with a precision of ∼0.1 μm.

We developed a mathematical model predicting the pulsatile behavior of GUVs in a hypotonic environment. Essentially, the model couples pore nucleation by thermal fluctuations, osmotic swelling, and solute transport. These aspects are represented by Eqs. 2, 3, and 4, respectively, and discussed below. Details regarding the theory and its numerical implementation are reported in the Supporting Material.

Results

Homogeneous GUVs Display Swell-burst Cycles in Hypotonic Conditions

A selection of snapshots, revealing different morphological states, and a detailed trace showing the time-dependence of the vesicle radius R and corresponding area strain (${\mathit{ϵ}}_{\text{exp}}=\left(R^2-R_0^2\right)/R_0^2$, where and R₀ is the resting initial vesicle radius), are shown in Fig. 1, B–D, for a representative GUV. Swelling phases are characterized by a quasi-linear increase of the GUV radius, whereas pore openings cause a sudden decrease of the vesicle radius.

We outline here three key observations about the dynamics of swell-burst cycles from these experiments:

• 1.The period between two consecutive bursting events increases with each cycle, starting from a few tenths of a second for the early cycles, to several hundreds of seconds after the tenth cycle.
• 2.The maximum radius and therefore the maximum strain at which a pore opens decreases with cycle number, suggesting that lytic tension is a dynamic property of the membrane.
• 3.The observed transient pores are short lived, stay open for about a hundred milliseconds, and reach a maximum radius of up to 60% of the GUV radius.

We seek to explain these observations through a quantitative understanding of the pulsatile GUVs in hypotonic conditions. To do so, we first investigate the mechanics of pore nucleation and its relationship to the GUV swell-burst dynamics.

Thermal Fluctuations Drive the Dynamics of Pore Nucleation

In the framework of classical nucleation theory (11), the energy potential V(r,ϵ) of a pore of radius r in a lipid membrane under surface tension σ, is the balance of two competitive terms: V_s(ϵ), the strain energy, and V_p(ϵ), the pore energy. The strain energy tends to favor the opening and enlargement of the pore while the pore closure is driven by the pore line tension γ. Accordingly, the energy potential reads

$$\begin{array}{c}V\left(r,\mathit{ϵ}\right)=V_s\left(\mathit{ϵ}\right)+V_p\left(r\right)\\ =\frac{1}{2}{\kappa }_{\text{eff}}{A}_0{\mathit{ϵ}}^2+2\pi r\gamma .\end{array}$$ (1)

The area strain is defined as ϵ = (A – A₀)/A₀, where A = 4πR² – πr² is the surface of the membrane, and $A_0=4\pi R_0^2$ is the resting vesicle area. Here V_s(ϵ) is assumed to have a Hookean form, where κ_eff is the effective stretching modulus, which relates the surface tension to the strain as σ = κ_effϵ (see next section for a discussion on κ_eff). These two energetic terms oppose each other, resulting in an energy barrier that the system has to overcome for a pore to nucleate. The competition between the strain and pore energy is expressed by the ratio r_b = γ/σ, which is the critical radius associated with the crossing of the energy barrier. That is, if a pore in a tensed membrane has a radius r < r_b, the pore energy V_p(r) dominates and the pore closes. On the contrary, for r > r_b, the strain energy V_s(ϵ) prevails and the pore grows. The energy required to open a pore of radius r in a tensed GUV is given by $\Delta V\left(r,\mathrm{ϵ}\right)=V\left(r,\mathit{ϵ}\right)-V\left(0,\mathit{ϵ}\right)$ and is represented in Fig. 2 A. The corresponding critical radius of the energy barrier r_b is shown as a function of the strain $\mathit{ϵ}$ in Fig. 2 B. The height of the energy barrier and its critical radius are both dependent on the membrane strain; the more the membrane is stretched, the lower the energy barrier is, and the smaller the amount of energy required to nucleate a pore.

Figure 2.

Lytic tension is a dynamic quantity governed by thermodynamic fluctuations. (A) The energy required to open a pore of radius r in a GUV without fluctuations, for various membrane tensions. The energetic cost to open a pore in a tense GUV shows a local maximum, which has to be overcome for a pore to open. (B) For a given strain $\mathrm{ϵ}$, the energy barrier is located at a pore radius r_b = γ/σ(ϵ) = γ/(κ_effϵ) (where κ_eff = 2 × 10⁻³ N/m as discussed in the text). (C and E) If the critical strain is fixed at a constant value, ϵ^∗, as in the deterministic approach, then a pore is nucleated whenever the strain reaches ϵ^∗, regardless of the strain rate. Prescribing various linear strain rates ($\mathit{ϵ}=\dot{\mathit{ϵ}}t+{\mathit{ϵ}}_0$, with $\dot{\mathrm{ϵ}}$= 10⁻², 10⁻³, and 10⁻⁴ s⁻¹, ϵ₀ = 0.05) does not alter the strain at which a pore will form (C) and the pore nucleation radius r_b will be constant (E). (D and F) In the stochastic approach, however, the nucleation threshold is replaced by a fluctuating pore (blue line in (F) as computed by Eq. 2), inducing a dependence of the lytic strain on the strain rate (D). This is due to the fact that, for lower strain rates, the probability of a large pore fluctuation to reach r_b is higher (F), producing a lower lytic tension on average. To see this figure in color, go online.

The amplitude of this energy barrier is strictly positive for finite strain values, making pore nucleation impossible without the addition of external energy. This issue has often been resolved by assuming a predetermined and constant lytic strain (ϵ^∗) corresponding to a critical energy barrier under which the pore opens (Fig. 2, C and E). However, this approach is in contradiction with our experimental observations that the lytic strain in the membrane varies with each swell-burst cycle (Fig. 1 D), due to a dependence on the strain rate (15). To account for this variation, we included thermal fluctuations associated with the pore nucleation barrier in our analysis (22, 23). In this scenario, increasing the membrane tension of the vesicle reduces the minimum pore radius r_b at which a pore opens (Fig. 2, A and B), lowering the energy barrier down to the range of thermal fluctuations, eventually letting the free energy of the system overcome the nucleation barrier (Fig. 2, D and F). The stochastic nature of the fluctuations can then explain a distribution of pore opening tensions, eliminating the need to assume constant lytic tension.

A direct consequence of the fluctuation-mediated pore nucleation is that the membrane rupture properties become dynamic. Indeed, fluctuations naturally cause the strain at which the membrane ruptures, to be dependent on the strain rate, as illustrated in Fig. 2 D. To understand this dynamic nucleation process, consider stretching the membrane at different strain rates $\dot{\mathit{ϵ}}$. Doing so decreases the radius of the nucleation barrier at corresponding speeds, as shown in Fig. 2 F. For slow strain rates, as r_b tends to zero, it spends more time in the accessible range of the thermal pore fluctuations, increasing the probability that a fluctuation will overcome the energy barrier. On the other hand, at faster strain rates, r_b decreases quickly, reaching small values in less time, lowering the probability for above-average fluctuations to occur during this shorter time.

We use a Langevin equation to capture the stochastic nature of pore nucleation and the subsequent pore dynamics. This equation includes membrane viscous dissipation, a conservative force arising from the membrane potential, friction with water, and thermal fluctuations for pore nucleation (see Supporting Material for detailed derivation). This yields the stochastic differential equation for the pore radius r as follows:

$$\stackrel{\text{viscous}\text{drag}}{\overbrace{\left(h{\eta }_m+C{\eta }_{s}r\right)}}\underset{\text{change}\text{of}\text{pore}\text{radius}}{\underbrace{\frac{d}{dt}\left(2\pi r\right)}}=\stackrel{\text{surface}\text{and}\text{line}\text{tension}}{\overbrace{2\pi \left(\sigma r-\gamma \right)}}+\underset{\text{thermal}\text{pore}\text{fluctuations}}{\underbrace{\xi \left(t\right)}},$$ (2)

where the noise source $\xi \left(t\right)$ has zero mean and satisfies $\langle \xi \left(t\right)\xi \left(t^{\prime }\right)\rangle =2\left(h{\eta }_m+C{\eta }_{s}r\right)k_{\text{B}}T\delta \left(t-t^{\prime }\right)$ according to the fluctuation dissipation theorem (24). Here, η_m and η_s are the membrane and solute viscosities, respectively, h is the membrane thickness, C is a geometric coefficient (16, 25), k_B is the Boltzmann constant, and T is the temperature. We assume here that the pore nucleation probability is independent of the total membrane surface area. The values of the different parameters used in the model are given in Table S1.

Model Captures Experimentally Observed Pulsatile GUV Behavior

In addition to pore dynamics (Eq. 2), we need to consider mass conservation of the solute and the solvent. We assume that the GUV remains spherical at all times and neglect spatial effects. The GUV volume changes because of osmotic influx through the semipermeable membrane and the leakout of the solvent through the pore. The osmotic influx is the result of two competitive pressures—the osmotic pressure driven by the solute differential (Δp_osm = k_BT N_A Δc), and the Laplace pressure, arising from the membrane tension (Δp_L = 2σ/R), resulting in the following equation for the GUV radius R:

$$\underset{\text{change}\text{of}\text{GUV}\text{volume}}{\underbrace{\frac{d}{dt}\left(\frac{4}{3}\pi R^3\right)}}=\stackrel{\begin{array}{c}\text{influx}\text{of}\text{solvent}\\ \text{through}\text{the}\text{membrane}\end{array}}{\overbrace{\frac{P{\nu }_s}{k_{\text{B}}T{N}_{\text{A}}}\left(\Delta p_{\text{osm}}-\Delta p_L\right)A}}-\underset{\text{leakout}\text{through}\text{the}\text{pore}}{\underbrace{v_L\pi r^2}}.$$ (3)

Here A = 4πR² is the membrane area, P is the membrane permeability to the solvent, ν_s is the solvent molar volume, and N_A is the Avogadro number. Assuming low Reynolds number regime, the leakout velocity is given by v_L = Δp_Lr/(3πη_s) (25, 26).

Mass conservation of solute in the GUV is governed by the diffusion of sucrose and convection of the solution through the pore, which gives the governing equation for the solute concentration differential Δc, as follows:

$$\underset{\text{molar}\text{differential}\text{of}\text{solute}}{\underbrace{\frac{d}{dt}\left(\frac{4}{3}\pi R^3\Delta c\right)}}=-\pi r^2\left(\stackrel{\text{diffusion}\text{through}\text{the}\text{pore}}{\overbrace{D\frac{\Delta c}{R}}}+\underset{\text{convection}\text{through}\text{the}\text{pore}}{\underbrace{v_L\Delta c}}\right),$$ (4)

where D is the solute diffusion coefficient. These three coupled equations ((2), (3), (4)) constitute the mathematical model.

To completely define the system, we need to specify the relationship between the membrane surface tension σ and the area strain of the GUV. We note that the GUV has irregular contours during the pore opening event and for a short time afterwards, when nodules are observed at the opposite end from the pore, indicating accumulation of excess membrane generated by pore formation (Fig. 1 B, middle and right panels). In the low tension regime, GUVs swell by unfolding these membrane nodules, and the stretching is controlled by the membrane bending modulus κ_b and thermal energy, yielding an effective unfolding modulus of ${\kappa }_u=48\pi {\kappa }_b^2/\left(R_0^2{k}_{\text{B}}T\right)$ of the order of 10⁻⁵ N/m (27). In contrast, in the high tension regime, elastic stretching is dominant, and the elastic area expansion modulus κ_e is roughly equal to 0.2 N/m (28). Because the maximum area strain plotted in Fig. 1 D is ∼15%, significantly larger than the expected 4% for a purely elastic membrane deformation, the experimental data suggests the occurrence of two stretching regimes: an unfolding-driven stretching, and an elasticity-driven stretching (29, 30, 31). Therefore, for simplicity, we assume an effective stretching modulus κ_eff, which takes into account both unfolding and elastic regimes (28, 32) through a linear dependence between the membrane tension and the strain (σ = κ_effϵ). Note that κ_eff is the only adjustable parameter of the model.

We solved the three coupled equations ((2), (3), (4)) for an initial inner solute concentration of c₀ = 200 mM, and different GUV radii of R₀ = 8, 14, and 20 μm. All the results presented here are obtained for κ_eff = 2 × 10⁻³ N/m, the value that best fits the experimental observations (see Fig. S5 for the effect of this parameter on the GUV dynamics). Dynamics of the GUV radius and the pore radius are shown in Fig. 3 for a typical simulation with R₀ = 14 μm (see Fig. S3 for simulations with different values of R₀). Our model qualitatively reproduces the dynamics of the GUV radius during the swell-burst cycle (compare Figs. 1 C and 3 A). Importantly, we recover the key features of the swell-burst cycle—namely an increase of the cycle period with each bursting event (point 1), and a decrease of the maximum radius with time (point 2). The stochastic nature of the thermodynamic fluctuations leads to variations and irregularities in the pore opening events, and therefore, the cycle period and maximum strain. The dynamics of a single cycle is shown in Fig. 3, C and D. Our numerical results show an abrupt drop in the GUV radius, followed by a slower decrease, suggesting a sequence of two leakout regimes: a fast burst releasing most of the membrane tension, and a low tension leakout. This two-step tension release is confirmed by the pore radius dynamics, which after suddenly opening (release of membrane tension), reseals quasi-linearly due to the dominance of line tension compared to membrane tension in Eq. 2. Furthermore, the computed pore amplitude and lifetime are in agreement with experimental observations (point 3). Overall, our model is able to reproduce the quantitative features of GUV response to hypotonic stress over multiple timescales.

Figure 3.

Dynamics of swell-burst cycles from the model for a GUV of radius 14 μm in 200 mM hypotonic stress. (A and C) GUV radius and (B and D) pore radius as a function of time. The model captures the dynamics of multiple swell-burst cycles, in particular the decrease of maximum GUV radius and increase of cycle period with cycle number (A). Looking closely at a single pore opening event corresponding to the gray region, the model predicts a three-stage pore dynamics (C and D), namely opening, closing, and resealing, with a characteristic time of a few hundred milliseconds. Numerical reconstruction of the GUV is shown in Movies S3 and S4. Results for R₀ = 8 and 20 μm are shown in Fig. S3. To see this figure in color, go online.

If thermal fluctuations are ignored, the strain to rupture needs to be adjusted to roughly 15% to match the range of maximum GUV radius observed experimentally (Fig. S4). However, such a deterministic model does not capture the pulsatile dynamics quite as well as the stochastic model in terms of cycle period and strain rate (Fig. S11), and fails to reproduce a strain rate-dependent maximum stress (Fig. S4).

Solute Diffusion is Dominant during the Low Tension Regime of Pore Resealing

The concentration differential of sucrose decreases exponentially and drops from 200 to ∼10 mM in ∼1000 s (Fig. S3). Even after 2000 s when the concentration differential is as low as 10 mM, the osmotic influx is still large enough to maintain the dynamics of swell-burst cycles (Figs. 1 C and S3). We further observe that every pore opening event produces a sudden drop in inner solute concentration (Fig. 4 A, blue line). This suggests that diffusion of sucrose plays an important role in governing the dynamics of solute. In the absence of diffusive effects, the model does not show the abrupt drops in concentration but, instead, a rather smooth exponential decay (Fig. 4 A, gray line).

Figure 4.

Diffusion of sucrose through a transient pore produces a stepwise decrease of the inner solute concentration. (A) In hypotonic conditions, the model predicts a stepwise decrease of solute concentration differential with time (blue line), which is solely due to diffusion of solute through the transient pore. In comparison, when diffusion is neglected in the model (gray line), the solute concentration differential decreases smoothly (also see Fig. S6 for further analysis on the effect of diffusion). In isotonic conditions (dashed line), the solute concentration differential is constant with time (here t₀ = 40 s.) (B) Time evolution of the normalized fluorescence intensity of a GUV in hypotonic condition, encapsulating fluorescent glucose analog. ΔI is the difference in mean intensity between the inside of the GUV and the background. In hypotonic conditions (solid lines) the normalized intensity decreases with time due to the constant influx of water through the membrane, and shows sudden drops in intensity at each pore opening (indicated by arrows), due to diffusion of sucrose through the pore (see Movie S5). In comparison, GUVs in an isotonic environment (dashed lines) exhibit a rather constant fluorescence intensity (see Movie S6). (C) Micrographs of a GUV in hypotonic condition, encapsulating fluorescent glucose analog, just before bursting (left panel), with an open pore (middle panel), and just after pore resealing (right panel). The leakout of fluorescent dye is observed in the middle frame, coinciding with a drop of the GUV radius. Frames were extracted from Movie S7. (D) Same as (C), with the images processed to increase contrast and attenuate noise. The blue, red, and white lines are the isocontours of the 90, 75, and 60 grayscale values, respectively, highlighting the leakout of fluorescent dye. To see this figure in color, go online.

To experimentally verify the model predictions of sucrose dynamics, we quantified the evolution of fluorescence intensity in GUVs encapsulating 200 mM sucrose plus 58.4 μM 2-NBDG, a fluorescent glucose analog (see Supporting Material and Methods). Fig. 4 B presents the evolution of fluorescent intensity of sucrose in time. GUVs in isotonic conditions (dashed lines) do not show a significant change in fluorescence intensity. GUVs in hypotonic conditions (solid lines) exhibit an overall decrease of intensity due to permeation of water through the membrane. Strikingly, consecutive drops of fluorescence intensity are observed coinciding with the pore opening events (Fig. 4, C and D, middle panels), and highlight the importance of sucrose diffusion through the pore. While the quantitative dynamics of sucrose depends on the value of the diffusion constant (Fig. S6), the qualitative effect of diffusion on the dynamics remains unchanged. On the other hand, leakout-induced convection does not influence the inner concentration of sucrose, as both solvent and solute are convected, conserving their relative amounts. These observations are in agreement with the existence of the low tension pore closure regime discussed above, where Laplace pressure produces negligible convective transport compared to solute diffusion though the pore.

Cycle Period and Strain Rate are Explicit Functions of the Cycle Number and GUV Properties

Given that lytic tension is a dynamic quantity, we asked how do the cycle period and strain rate evolve along with the cycles. We analyzed the simulated dynamics of GUVs with resting radii of 8, 14, and 20 μm, each data point representing the mean and the standard deviation of 10 simulations with identical parameters (the variations are due to the stochastic nature of the model). The details of this burst cycle analysis is reported in Supporting Material. Cycle periods and strain rates show a dependence on the GUV radius, as depicted in Fig. 5 where larger GUVs have slower dynamics, resulting in smaller strain rates and longer cycle periods (Fig. S3). To verify this experimentally, a total of eight GUVs were similarly analyzed with resting radii ranging from 7.02 to 18.76 μm (Fig. S1). The measured cycle period and strain rate as a function of the cycle number (corrected for the lag between the application of the hypotonic stress and the beginning of the observations) are shown in Fig. 5, A and B, respectively. Experimental and model results quantitatively agree, and show an exponential dependence of the cycle period and strain rate on cycle number (Fig. 5, A and B, insets).

Figure 5.

Cycle period and strain rate are exponential functions of cycle number, and power-law functions of solute concentration. (A and C) Cycle period and (B and D) strain rates as functions of cycle number (n) (A and B) and solute concentration (C and D). Insets show the same data in log scale. Each model point is the mean of 10 numerical experiments; error bars represent mean ± SD. The analytical expressions for the cycle period and strain rate (Eq. 5) with ϵ^∗ = 0.15, are plotted in (C and D) for comparison. To see this figure in color, go online.

Two further questions arise: How can we relate the cycle number to the driving force of the process, namely the osmotic differential? And, is there a scaling law that governs the GUV swell-burst dynamics? To answer these questions we computed the cycle solute concentration (defined as the solute concentration at the beginning of each cycle) as a function of the cycle number (Fig. S2). We found that the solute concentration follows an exponential decay function of the cycle number, and is independent of the GUV radius. Additionally, plotting the cycle period and strain rate against the cycle solute concentration (Fig. 5, C and D), we observe that the cycle period increases as Δc decreases, while the strain rate is a linear function of Δc. The data presented in Fig. 5 suggest that the dynamics of GUVs swell-burst cycle can be scaled to their size. From the nondimensional form of Eq. 3, we extracted a characteristic time associated with swelling, defined by τ = R₀/(Pν_sc₀), and scaled the cycle period and strain rates with this quantity. As shown in Fig. 6, all the scaled experimental and model data collapse onto the same curve, within the range of the standard deviations. The scaled relationships can be justified analytically, by estimating the cycle period and strain rates as follows:

$$\frac{T_n}{\tau }\simeq \frac{{\mathit{ϵ}}^{\text{∗}}}{\left(2\sqrt{{\mathit{ϵ}}^{\text{∗}}+1}\Delta c/c_0\right)}\text{and}\tau \dot{\mathit{ϵ}}\simeq \frac{2\sqrt{{\mathit{ϵ}}^{\text{∗}}+1}\Delta c}{c_0},$$ (5)

respectively (see Supporting Material for complete derivation). These analytical expressions are plotted in Figs. 5, C and D and 6, C and D for a characteristic lytic strain of ϵ^∗ = 0.15, and show a good agreement with the numerical data. Taken together, these results suggest that the GUV pulsatile dynamics is governed by the radius, the membrane permeability, the solute concentration, and—importantly—the stochastic pore nucleation mechanism, which determines the strain to rupture.

Figure 6.

The pulsatile dynamics is characterized by the characteristic timescale τ. (A and B) The data from Fig. 5 are scaled by the characteristic time associated with swelling defined as τ. Insets show the same data in log scale. The nondimensionalization by τ allows cycle periods and the strain rates to collapse onto a single curve. (C and D) The analytical expressions for the cycle period and strain rate (Eq. 5) with ϵ^∗ = 0.15, are plotted for comparison. To see this figure in color, go online.

Discussion

Explaining how membrane-enclosed compartments regulate osmotic stress is a first step toward understanding how cells control volume homeostasis in response to environmental stressors. In this work, we have used a combination of theory, computation, and experiments in a simple model system to study how swell-burst cycles control the dynamics of GUV response to osmotic stress. Using this system, we show that the pulsatile dynamics of GUVs under osmotic stress is controlled through thermal fluctuations that govern pore nucleation and lytic tension.

The central feature of a GUV’s osmotic response is the nucleation of a pore. Even though Evans and co-workers (15, 33) identified that rupture tension was not governed by an intrinsic critical stress, but rather by the load rate, the idea of a constant lytic tension has persisted in the literature (8, 14, 34). By coupling fluctuations to pore energy, we have now reconciled the dynamics of the GUV over several swell-burst cycles with pore nucleation and dependence on strain rate. Our model is not only able to capture the experimentally observed pulsatile dynamics of GUV radius and solute concentration (Figs. 3 and 4), but also predicts pore-formation events and pore dynamics (Fig. 3, B and D). We also found that during the pore opening event, a low-tension regime enables a diffusion-dominated transport of solute through the pore (Fig. 4), a feature that, until now, has been neglected in the literature.

Specifically, we have identified a scaling relationship between 1) the cycle period and cycle number and 2) the strain rate and the cycle number, highlighting that swell-burst cycles of the GUVs in response to hypotonic stress is a dynamic response (Fig. 6). One of the key features of the model is that we relate the cycle number, an experimentally observable quantity, to the concentration difference of the solute, a quantity that is hard to measure in experiments (Fig. S2). This allows us to interpret the scaling relationships described above in terms of solute concentration differential. The cycle period increases as the solute concentration difference decreases, while the strain rate is a linear function of the concentration difference. Both relationships are derived theoretically in the Supporting Material. These features indicate long timescale relationships of pulsatile vesicles in osmotic stress.

Thermal fluctuations and stochasticity are known to play diverse roles in cell biology. Well-recognized examples include Brownian motors and pumps (35, 36), noisy gene expression (37), and red blood cell flickering (38). The pulsatile vesicles presented here provide yet another example of how fluctuations can be utilized by simple systems to produce dynamical adaptive behavior. Given the universality of fluctuations in biological processes, it appears entirely reasonable that simple mechanisms similar to these pulsatile vesicles may have been exploited by early cells, conferring them with a thermodynamic advantage against environmental osmotic assaults. On the other hand, if such swell-burst mechanisms were at play, the chronic leakout of inner content could have led protocells to evolve active transport mechanisms to compensate for volume loss, and endure osmotic stress without a high energetic cost.

In this study, we experimentally measure the dynamics of swell-burst cycles in GUVs, and provide, for the first time to our knowledge, a model that captures quantitatively the pulsatile behavior of GUVs under hypotonic conditions for long timescales. To do so, we developed a general framework that integrated parts of existing models (10, 13, 16), with, to our knowledge, novel key elements: 1) the explicit inclusion of thermal pore fluctuations, which enables dynamic pore nucleation; 2) the definition of an effective stretching modulus, which combines membrane unfolding and elastic stretching; and 3) the incorporation of solute diffusion through the pore, which results in a nontrivial contribution to the evolution of the osmotic differential. The coupling of these key features results in a unified model that is valid in all regimes of the vesicle, pore, and solute dynamics.

Although we have been able to explain many fundamental features of the pulsatile GUVs in response to osmotic stress, our approach has some limitations and there is a need for further experiments. We have assumed a linear relationship between stress and strain. Although this assumption is reasonable and appears to work well for these experimental conditions, a more general expression should be considered to include both membrane (un)folding and elastic deformation (39). Another important aspect of biological relevance is membrane composition, where the abundance of proteins and heterogeneous composition leading to in-plane order and asymmetry across leaflets influences the membrane mechanics (40, 41). We have previously found experimentally that the dynamics of swell-burst cycles is related to the compositional degrees of freedom of the membrane (7). Future efforts will be oriented toward the development of a theoretical framework and quantitative experimental measures that provide insight into how the membrane’s compositional degrees of freedom influence the pulsatile dynamics of cell-size vesicles. In addition to osmotic response and membrane composition, we will focus on how membrane components such as aquaporins and ion channels may couple thermal fluctuations with membrane tension to regulate their functions. Additionally, we are also investigating how the properties of the encapsulated bulk fluid phase may affect the response of the GUV in response to osmotic shock. To our knowledge, this work is a first and critical step in these directions.

Author Contributions

J.C.S.H. and A.N.P. designed the experiments. J.C.S.H. performed the experiments. M.C. and P.R. derived the model. M.C. performed the simulations. M.C. and J.C.S.H. analyzed the data. All authors discussed and interpreted results. All authors wrote and agreed on the manuscript.

Editor: Ana-Suncana Smith.

Supporting Citations

References (42, 43, 44, 45, 46, 47, 48, 49, 50, 51) appear in the Supporting Material.

Supporting Material

Movie S1. Membrane Nodule Appearance after Membrane Pore Reseals

Movie assembled from time-lapse fluorescence microscopy images (frame rate, 2 fps; total duration, 17 s; image size, 82.43 μm → 82.43 μm; scale bar, 10 μm) obtained for a population of electroformed GUVs consisting of POPC doped with 1% Rho-DPPE membrane in a hypotonic solution (osmotic differential of 200 mM).

Movie S2. Multiple Swell-burst Cycles of GUVs Subject to Hypotonic Stress

Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 77 s; image size, 82.43 μm → 82.43 μm; scale bar, 10 μm) obtained for a population of electroformed GUVs consisting of POPC doped with 1% Rho-DPPE membrane in a hypotonic solution (osmotic differential of 200 mM).

Movie S3. Model Results Showing Multiple Swell-burst Cycles of a GUV Subject to Hypotonic Stress

GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. (Right panel) Representation of the numerical GUV in time, where the shaded intensity is proportional to the inner sucrose concentration. GUV initial radius is R₀ = 14 μm, initial solute concentration is c₀ = 200 mM. All parameters are shown in Table S1.

Movie S4. Model Results Showing Single Pore-opening Dynamics of a GUV Subject to Hypotonic Stress

GUV radius (top-left panel), pore radius (middle-left panel), and solute differential (bottom-left panel) as a function of time. (Right panel) Representation of the numerical GUV in time, where the shaded intensity is proportional to the inner sucrose concentration. GUV initial radius is R₀ = 14 μm, initial solute concentration is c₀ = 200 mM. All parameters are shown in Table S1.

Movie S5. Solute Leakage of a GUV in Multiple Swell-burst Cycles under a Hypotonic Condition

Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 11 s; image size, 119.14 μm → 125.58 μm; scale bar, 20 μm) obtained for a population of electroformed GUVs consisting of POPC doped with 1% Rho-DPPE membrane in a hypotonic solution (osmotic differential of 200 mM).

Movie S6. GUV under an Isotonic Condition

Movie assembled from time-lapse fluorescence microscopy images (frame rate, 24 fps; total duration, 8 s; image size, 101.11 μm → 101.11 μm; scalebar, 10 μm) obtained for a population of electroformed GUVs consisting of POPC doped with 1% Rho-DPPE membrane in a isotonic solution (no osmotic differential).

Movie S7. Solute Efflux from GUV during One Swell-burst Cycle

Movie assembled from time-lapse fluorescence microscopy images (frame rate, 12 fps; total duration, 8 s; image size, 164.86 μm → 164.86 μm; scale bar, 20 μm) obtained for a population of electroformed GUVs consisting of POPC doped with 1% Rho-DPPE membrane in a hypotonic solution (osmotic differential of 200 mM).