Dividing Liquid Droplets as Protocells

Original paper: Growth and division of active droplets provides a model for protocells


In the beginning there was… what, exactly? Uncovering the origins of life is a notoriously difficult problem. When a researcher looks at a cell today, they see the highly-polished end product of millennia of evolution-driven engineering. While living cells are not made of any element that can’t be found somewhere else on earth, they don’t behave like any other matter that we know of. One major difference is that cells are constantly operating away from equilibrium. To understand equilibrium, consider a glass of ice water. When you put the glass in a warm room, the glass exchanges energy with the room until the ice melts and the entire glass of water warms to the temperature of the room around it. At this point, the water is said to have reached equilibrium with its environment. Despite mostly being made out of water, cells never equilibrate with their environment. Instead, they constantly consume energy to carry out the cyclic processes that keep them alive. As the saying goes, equilibrium is death[1]: the cessation of energy consumption can be thought of as a definition of death. The mystery of how non-equilibrium living matter spontaneously arose from all the equilibrated non-living stuff around it has perplexed scientists and philosophers for the better part of human history[2].

An important job for any early cell is to spatially separate its inner workings from its environment. This allows the specific reactions needed for life, such as replication, to happen reliably. Today, cells have a complicated cell membrane to separate themselves from their environment and regulate what comes in and what goes out. One theory proposes that, rather than waiting for that machinery to create itself, droplets within a “primordial soup” of chemicals found on the early Earth served as the first vessels for the formation of the building blocks of life. This idea was proposed independently by the Soviet biochemist Alexander Oparin in 1924 and the British scientist J.B.S. Haldane in 1929[3]. Oparin argued that droplets were a simple way for early cells to separate themselves from the surrounding environment, preempting the need for the membrane to form first.

In today’s paper, David Zwicker, Rabea Seyboldt, and their colleagues construct a relatively simple theoretical model for how droplets can behave in remarkably life-like ways. The authors consider a four-component fluid with components A, B, C, and C’, as shown in Figure 1. Fluids A and B comprise most of the system, but phase separate from each other such that a droplet composed of mostly fluid B exists in a bath of mostly fluid A. This kind of system, like oil droplets in water, is called an emulsion. Usually, an emulsion droplet lives a very boring life. It either grows until all of the droplet material is used up, or evaporates altogether. However, by introducing chemical reactions between these fluids, the authors are able to give the emulsion droplets in their model unique and exciting properties.

 

modelSchematic_fig1b
Fig. 1: Model schematic. A droplet composed mostly of fluid B (green) within a bath of fluid A (blue). Inside the droplet, B degrades into A. Outside the droplet, fluids C and A react to form fluids B and C’. Adapted from Zwicker and colleagues.

 

The chemical reactions in the model are fairly simple (see figure 1). Fluid B spontaneously degrades into fluid A and diffuses out of the droplet. While fluid A cannot easily turn back into fluid B (since spontaneous degradation implies going from a high energy state to a low one), fluid C can react with A to create fluids B and C’, and this fluid B can diffuse back into the B droplet.

B \to A \quad \text{and} \quad A+C \to B+C'

If C and C’ are constantly resupplied and removed, respectively, they can be kept at fixed concentrations. Without C and C’, the entire droplet would disappear by degrading into fluid A, reaching equilibrium. Here, C and C’ act as fuel that constantly drives the system away from equilibrium, creating what the authors dub an “active” emulsion. Active matter systems like this one have had success in describing living things because they, like all living matter, fulfill the requirement of being out-of-equilibrium.

Because the equations that describe how fluids A and B flow over time are so complicated, the authors solve their model using a computer simulation. When they do, something remarkable happens. Emulsions with no chemical reactions with their surrounding fluids never stop growing as long as there is more of the same material nearby to gobble up. This process is called Ostwald ripening[4]. The authors find that an active emulsion system, due to the fact that material is constantly turning over, suppresses Ostwald ripening and allows the emulsion droplet to maintain a steady size.

In addition to limited growth, the authors also find that the droplets undergo a shape instability that leads to spontaneous droplet division (see this movie). This occurs due to the constant fuel supply of C and C’. The chemical reaction A+C ? B+C’ creates a gradient in the concentration of fluids A and B outside the droplet. Just outside the droplet, there is a depletion of B and an abundance of A, while far away from the droplet, A and B reach an equilibrium concentration governed by the rate of their reactions with C and C’. The authors dub this excess concentration of B far away from the droplet the supersaturation. Where there exists a gradient in the concentration of a material, there exists a flow of that material, called a flux. This is the reason a puff of perfume in one corner of a room will eventually be evenly distributed around that room. The size of the droplet is dependent on the flux of fluid B into and out of the droplet.

Two quantities determine the evolution of the droplet. The first is the supersaturation that reaches a steady value once all fluxes stop changing in time, and the second is the rate at which the turnover reaction B?A occurs. For a given supersaturation and turnover rate, the authors can calculate how large the droplet will grow, and they find three distinct regimes. In one regime, the droplet dissolves and disappears as the turnover rate outpaces the flow of fluid B back into the droplet. Another has the droplet grow to a limited size and remain stable, since the turnover and supersaturation balance each other out and give a steady quantity of fluid B. The third and most interesting regime occurs if the droplet grows beyond a certain radius due to the influx of fluid B outpacing its efflux. Here, a spherical shape is unstable and any small perturbation will result in the elongation and eventual division of the droplet (Figure 2).

 

dropletStabilityDiagram_fig2b
Fig. 2: Stability diagram of droplets for normalized turnover rate \nu_-/\nu_0 vs supersaturation \epsilon. For a given value of \epsilon, the diagram shows regions where droplets dissolve and eventually disappear (white), grow to a steady size and remain stable (blue), and grow to a steady size and begin to divide (red). Adapted from Zwicker and colleagues.

 

And that’s it. If you have two materials that phase separate from each other, coupled to a constant fuel source to convert one into the other, controlled growth and division will naturally follow. While these droplets are more sophisticated than regular emulsion droplets, they are still a far cry from even the simplest microorganisms we see today. There is no genetic information being replicated and propagated, nor is there any internal structure to the droplets. Further, the droplets lack the membranes that modern cells use to distinguish themselves from their environments. An open question is whether a synthetic system exists that can test the model proposed by the authors. Nevertheless, these active emulsions provide a mechanism for how life’s complicated processes may have gotten started without modern cells’ complicated infrastructure.

Though many questions still remain, Zwicker and his colleagues have lent considerable credence to an important, simple, and feasible theory about the emergence of life: it all started with a single drop.


[1]: This isn’t exactly true. Some organisms undergo a process called anhydrobiosis, where they purposefully dehydrate and rehydrate themselves to stop and start their own metabolism. Also, some bacteria slow their metabolism to avoid accidentally ingesting antibiotics in a process called “bet-hedging”.

[2]: For example, ancient Greek natural philosophers such as Democritus and Aristotle believed in the theory of spontaneous generation, eventually disproven by Louis Pasteur in the 19th century.

[3]: Oparin, A. I. The Origin of Life. Moscow: Moscow Worker publisher, 1924 (in Russian), Haldane, J. B. S. The origin of life. Rationalist Annual 148, 3–10 (1929).

[4]: Ostwald ripening is a phenomenon observed in emulsions (such as oil droplets in water) and even crystals (such as ice) that describes how the inhomogeneities in the system change over time. In the case of emulsions, it describes how smaller droplets will dissolve in favor of growing larger droplets.

When Bending Comes at A Cost: Vesicle Formation and Size Distribution

Original paper: Formation and size distribution of self-assembled vesicles


I’m going to start this post with an experiment. Find a piece of smooth and unwrinkled A4 or paper of a similar size, and hold it by gripping an edge between your thumb and forefinger. Due to the gravitational force, the paper is pulled down and is bent. Now crumple the same paper, then unfold and hold it by the edge again. What happened? The paper can now resist gravity! This wrinkling strategy is a simple trick to improve the mechanical response of a thin 2D sheet. Astonishingly in biology, by such simple ways, cells tune the mechanics of their thin membrane to form tiny capsules called vesicles in order to uptake nutrients, to dump waste, and to communicate. But how such a thin sheet can address all these needs? What are the mechanisms behind these tunings? Are there consequences other than mechanical improvements? In today’s paper, Changjin Huang and colleagues investigate the critical parameters governing the vesicle formation process (or vesiculation) and the size distribution of vesicles.

The Vesiculation Process

A class of molecules called amphiphiles contain two parts: a water-loving (hydrophilic) head and a water-fearing (hydrophobic) tail. When amphiphiles are dispersed in water, the hydrophobic tails are frustrated and get together (self-assemble) to stay away from water. Based on the geometry of these two parts, different structures emerge (see note [1]). One such structure is the bilayer structure (Fig. 1A).

Slide1
Fig. 1. A through D is the evolution of a vesicle, starting from a membrane patch (A) bending to (B) and closing at (C ) to form vesicles. The spontaneous decrease in systems’ energy by closing the patch is opposed by the energy required to bend the patch. The competition between these two energies is determined by factors such as patch size(l), membrane thickness (d), curvature (1/R), and bending stiffness (k_b). Combination of any of these factors can result in either or combination of above morphologies.

A bilayer structure composed of two layers of molecules with the hydrophobic tails turned inward (Fig 1A). This bilayer arrangement still is not the favored structure, since the water-fearing tails are exposed to water on the edges of the bilayer. An energy is imposed on the system by such exposure. This energy is called the interface energy and usually is shown by ?. This interfacial energy is the only driving force for the bilayer to bend in order to minimize the system’s energy. Thereby, the bilayer attempts to bend into spherical structures (Fig 1B & C).  But bending comes at a cost! The system needs to exert force to bend the bilayer. In other words, energy is required to curve the bilayer. In this work, Huang and colleagues model this process with an energy-minimization approach to realize the critical parameters that determine the fate of this competition.

Parameters Affecting the Vesiculation

The quick paper experiment highlighted the essential role of local curvature in sheet’s rigidity, but that’s not all. The authors of this study theoretically demonstrate that besides local curvature, membrane thickness, membrane bending resistance (bending stiffness) and the membrane patch size (size of the paper sheet) all play a crucial role in the vesiculation process. When the authors considered the role of membrane thickness, they could predict morphologies other than vesicles such as disks and cups which we observe in real-life experiments.

Many models have been developed in recent decades to explain the vesiculation process, and none were able to predict the intermediate morphologies. In all of these models, the membrane is treated as a 2D sheet with no thickness such that when it is bent, only undergoes linear elastic deformation. Before we proceed, let’s briefly elaborate on ” linear elasticity”.

Imagine a spring that is being pulled by a force that you apply. The magnitude of extension is proportional to the force exerted. This example corresponds to a linear response. However, there is a threshold force after which the extension magnitude is not proportional to the force, and to predict the behavior of spring, you may need to consider non-linear terms in the model. The same consideration applies to the vesiculation process. To model the energy required to bend a membrane patch we need to consider non-linear terms since our material is a very, very thin 2D sheet undergoing an enormous deformation when bent. So, for small bendings, the small value of h in Fig 1B, the linear term will suffice. But if membrane bends to final stages of closing itself, larger h, then we need to consider the non-linear term as well.

With this combination of linear and non-linear terms, an energy minimization model is proposed by the authors upon which a critical membrane bending length is obtained. At lengths, smaller than the critical length, the bending energy barrier increases dramatically, making it hard for the membrane to bend. At lengths larger than the critical length, bending energy barrier tends to zero and the membrane can readily bend (see note [2]). Now if we know the parameters to change this critical length, then we would be able to alter the vesicle size or to understand the mechanics of different vesicles produced by both healthy and diseased cells.

Effect of Curvature

The proposed model in the original study reveals that by introducing wrinkles, we can modify the critical length, however, the model also shows that decrease or increase of the critical length by wrinkles (or membrane spontaneous curvature-see note [3]) depends on curvature direction. Under negative spontaneous curvature, the membrane is curved in the opposite direction of bending (Fig 2A). Under this condition, the model shows that the critical length is larger than when the spontaneous curvature is positive. Note how in Fig 2B, for a negative curvature the bending energy barrier diminishes only at larger critical length. So, if a given membrane bilayer has specific molecules mostly inducing negative curvature, the critical length for that membrane will be larger, meaning that the patch needs to grow more to reach the critical length after which there would be no barrier for bending. Under such condition, larger vesicles will form in contrast to the membrane with positive spontaneous curvature, which can bend itself at lower critical length, thus, forming small vesicles.

 

Fig2
Fig 2. Effect of spontaneous curvature on membrane critical length. (A) schematics showing two types of curvatures; positive (left) and negative (right) both under same bending direction. (B) Total bending energy is calculated with respect to the spontaneous curvature, c_0 and the critical length, a. The heat map shows the barrier energy for bending. Amphiphilic molecules shown with darker tails were aimed to induce curvature based on their geometries.

Effect of Membrane Bending Stiffness

Bending stiffness, shown by k_b is the bending resistance of the membrane and thus it is a membrane property. Sometimes cells recruit molecules such as cholesterol to their membrane to increase the membrane bending stiffness. On the other hand, viruses are known to decrease the membrane stiffness so that they can readily bend the host’s cell membrane. In regard to vesicle size distribution analysis, the proposed model showed that the critical length is proportional to bending stiffness. In other words, for the stiffer membrane, the critical length is larger and these membranes tend to form larger vesicles.

Effect of Membrane Thickness

So far, for our analysis of the membrane (or sheet for our analogy) thickness was fixed. To consider the membrane thickness, the authors adopt a simple approximation. They first argue that membrane stiffness varies as a function of membrane thickness squared (k_b \propto d^2). Then, assuming that membrane is free to bend (its size is larger than the critical length), they obtain the minimum diameter of the vesicle formed from this membrane size as D_{min}=(critical\ length) + (membrane\ thickness). But critical\ length \propto k_b. Therefore, from their argument we can write:

D_{min}=d^{2} + d

With this approach, membrane thickness is considered as a non-linear concept. The proposed model reveals that for thicker membrane the critical length is larger, and thus these membranes will more likely form larger vesicles. In contrast, for the thinner membranes, the critical length is shorter and these membranes are prone to form small-sized vesicles.

Conclusion

The vesiculation model developed by Huang and his colleagues has contributed to our understanding of how vesicles form. Understanding the parameters that govern vesicle formation is critical for the design of vesicles for applications such as drug delivery, where nanoscale vesicles are needed to move drugs into a cell. In addition, the identified vesiculation parameters could be used as diagnostic measures, as it has been shown that the vesicles produced by cancer cells or by cells infected with viruses have mechanical properties different from healthy cells.


[1]  Known as Israelachvili’s packing parameter, the volume of the hydrophobic part divided by the product of effective hydrophilic area and the length of the hydrophobic part,p=\dfrac{v}{l*a}, defines the favored morphology.  when p < \frac{1}{3} spherical micelles, \frac{1}{3} < p < \frac{1}{2} cylindrical micelles, p > \frac{1}{2} bilayer structures are expected to form.

[2] Cut an unwrinkled A4 paper in half and see the bending response. If you continue cutting you will notice that after a certain length the paper doesn’t bend. That length is the critical length.

[3] Spontaneous curvature is the natural curvature of the membrane because of asymmetries between two monolayers of the bilayer. These asymmetries can be due to the presence of proteins or geometrical difference of different amphiphilic molecules making the membrane.