The (w)hole story

Dissipative structures are governed by non-equilibrium thermodynamics. Here, we describe a size-dependent transition of active droplets into active spherical shells–a dissipative structure that originates from reaction-diffusion gradients.
Published in Chemistry
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The motivations
Broadly speaking, our labs are interested in the spatial regulation of chemical reactions in the cell cytoplasm and at the origin of life. In particular, we investigate mixtures composed of chemically unstable building blocks that require the addition of fuel to persist in solution. We devote special focus on the case where phase separation occurs in such solutions, giving rise to active droplets. While the Weber and Jülicher groups focus on minimal theoretical descriptions of active droplets, using tools from non-equilibrium statistical physics, the BoekhovenLab aims to experimentally achieve active behavior in self-assembling structures and, in particular, active coacervate systems.

Genesis of the project
After carefully examining ancient folders and conversations via email, we could date the project's origin back to the spring of 2021. Giacomo Bartolucci, part of the Weber group, had just uploaded on arXiv a paper discussing the emergence of ring-like patterns in active emulsions confined in two dimensions. This finding triggered the curiosity of Carsten Donau from the Boekhoven Lab, who had just found a similar morphology in active coacervates. In particular, by injecting a sufficient amount of fuel into the system, they observed transient round coacervates that, after running out of fuel, turned into short-lived spherical shells before ultimately dissolving (see Figure 1).

Figure 1. Chemically fueled droplets under batch-fueling. a. Batch-fueling induces the transient formation of active droplets. After 7 min, active droplets became unstable and swelled to form a spherical shell. b. The volume distribution of active droplets (blue) and active liquid shells (red) is shown over time. The solid line represents the median, and the dotted lines represent the upper and lower quartiles. Larger active droplets formed spherical shells earlier than smaller active droplets. * P-values < 0.05 between the shell and droplet population for each time point. c. 3D reconstruction from confocal microscopy data shows the spherical nature of the shell.

It became immediately clear that the collaboration required the expertise of Jonathan Bauermann and Alexander Bergmann. Because of the ongoing pandemic, the four leading authors, together with the supervisors, assembled and got to know each other via online meetings. Nevertheless, the project was quickly grounded, and Jonathan came out with a physical explanation of the phenomenon involving spinodal decomposition. The excitement kept rising, but it soon became clear that, within the initial setup, it was hard to link the observation of spherical shells to the non-equilibrium nature of the chemical reaction.
Indeed, transient spherical shells could, in principle, occur in passive coacervates as well, upon changing the system control parameters.

What we ambitiously wanted to demonstrate, instead, is that spherical shells are a new class of steady states sustained by the presence of the fuel, which keeps the chemical reaction away from equilibrium. For this proof, however, we needed a way to keep the fuel concentration constant in time. 

The turning point
We initially tried to achieve steady states by adding small batches of fuel repeatedly to sustain the spherical shells. However, it became apparent quite quickly that the fuel addition through pipetting ​leads to problematic inhomogeneities, and stirring disturbs the microscopic observation. In a lucky coincidence, we discovered in a control experiment for another project that the addition of excess diisopropylcarbodiimide (DIC) was capable of inducing droplet formation. To our surprise, we observed the formation of spherical shells that were sustained much longer than in previous attempts. However, this approach suffered from reproducibility problems resulting from constant droplet fusion, fuel gradients, and evaporation of DIC. Alexander Bergmann overcame these problems by applying the same principles to droplet-based microfluidics, establishing continuously fueled microreactors (Figure 2a). Quickly, we managed to reproduce spherical shell formation in these microreactors as well as an underlying size-dependence which we postulated from our initial set of experiments (Figures 2b and c). Additionally, the Weber and Jülicher labs managed to find solutions of the Cahn Hilliard equations, which were reminiscent of the experimental patterns.

Figure 2. Spherical shells are a stable, non-equilibrium state. a. Experimental setup to form microreactors that continuously fuel active droplets. b. A macroscopic view of multiple microreactors shows that large reactors formed spherical shells while small microreactors contained droplets. The Z-plane through the center of each individual droplet and shell is shown. The gray circles represent the individual microfluidic reactors. The color scale is given next to the images. c. The volume of the total coacervate material is shown for every individual microreactor that contained an active droplet (blue) or an active shell (red). Above a critical reactor volume, active droplets with a volume bigger than Vunstable transformed into spherical shells.

Getting quantitative
Motivated by these results and the close similarity of the microreactor setup to the setup of the theoretical model, we wanted to get quantitative. However, it turned out that the ideas of the theoreticians were not always compatible with the experimentalists’ ones and vice versa. The following discussions about which parameters are experimentally achievable and which predictions are possible revealed quite quickly that the project was far from over and had just begun. The first big milestone in converging experiments and the model was the fitting of a phase diagram, followed by the development of a simplified theoretical description of spherical shells within the so-called effective droplet model. Combining the two achievements enabled numerical calculations on the spherical shell formation, confirming for the first time that the mechanism was plausible for our set of experimental parameters. After several iterations of further refining the experimental data and the model, we were finally able to quantitatively model the experimental data as well as to predict the behavior of active spherical shells for parameters that were not easily accessible in the experiments (Figure 3). 

 

Figure 3. The mechanism of spherical shell formation. a-b. Theoretically computed concentration profiles of a large chemically fueled droplet with r > runstable (a) before transitioning into a spherical shell and after the transition (b). c. The system’s behavior as a function of steady-state concentration fuel and reactor volume. The shaded areas represent the stable state calculated by the model. The red-blue shaded area represents the coexistence of stable droplets and shells. The markers show the phase-separated state of the experimental data. Overlapping data points are shown with an offset.

Conclusions
For this project, a team composed of experimentalists and theoreticians joined forces to investigate and predict the behavior of active droplets. We designed continuously fueled microreactors that are conceptually similar to commonly used model setups to quantitatively compare experimental data with numerical calculations for active droplets. Here, we observed a thermodynamically unfavored and size-dependent transition of spherical droplets into spherical shells–a dissipative structure that theory and experiment verified to originate from reaction-diffusion gradients.

The details of this work are available from Nature Communications.

https://doi.org/10.1038/s41467-023-42344-w

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