Shapes of condensate droplets containing filaments

By combining experiments, simulations, and analytical models, this study reveals how the interplay between surface tension, filament bending energy, and wetting effects governs the shape deformations of biomolecular condensates containing filaments, providing a physical framework for understanding their role in cytoskeletal organization.

The Gist

Imagine a tiny, squishy water balloon floating inside a cell. Now, imagine you drop a long, stiff piece of spaghetti inside that balloon. What happens?

According to this new research, the balloon and the spaghetti don't just sit there; they fight, dance, and reshape each other until they find a comfortable compromise. This paper explores exactly how these "biomolecular condensates" (the squishy balloons) and "cytoskeletal filaments" (the spaghetti) interact to create strange and wonderful shapes.

Here is the story of their relationship, broken down into simple concepts:

1. The Characters: The Balloon and the Spaghetti

• The Balloon (The Condensate): Think of this as a drop of oil in water. It wants to be a perfect sphere because that's the easiest shape to hold together (it minimizes surface tension). It's made of proteins and acts like a liquid droplet inside a cell.
• The Spaghetti (The Filament): This is a long, stiff fiber (like F-actin in our cells). It wants to stay straight because bending it takes energy. It hates being curved.

2. The Great Tug-of-War

When you put the spaghetti inside the balloon, two forces start fighting:

• Force A (The Balloon's Desire): "I want to be a round ball!"
• Force B (The Spaghetti's Desire): "I want to be straight!"

If the balloon is huge compared to the spaghetti, the spaghetti just sits inside, and the balloon stays round. The spaghetti is too short to force the balloon to change shape.

But, if the balloon is small or the spaghetti is very long, the spaghetti gets crowded. It can't fit inside without bending. So, it pushes against the walls of the balloon. The balloon, in turn, squishes and stretches to accommodate the spaghetti.

3. The Shapes They Make

The researchers found that depending on how much "spaghetti" is in the "balloon," they form specific shapes:

• The Sphere: When there's little spaghetti, it's just a round ball.
• The Pancake (Oblate Spheroid): As you add more spaghetti, the balloon gets squashed flat, like a hamburger bun or a pancake. The spaghetti forms a ring around the edge to minimize how much it has to bend.
• The Donut (Torus): If you add even more spaghetti (or shrink the balloon), the center gets pushed out completely! The balloon turns into a donut shape, with the spaghetti forming the ring of the donut.
• The Kettlebell: In computer simulations, they found a weird shape that looks like a kettlebell (a weight with a handle). This happens when the liquid gets trapped in a specific way around the spaghetti ring.
• The Red Blood Cell (Erythrocyte): In extreme cases, the center gets so thin it looks like a human red blood cell (which is a donut with a hole that is just very thin, not empty).

4. The "Wetting" Secret (The Glue)

The paper discovered something crucial: Wetting.
Imagine the spaghetti is covered in a super-sticky glue. The liquid balloon doesn't just touch the spaghetti; it clings to it, forming a thin film around it.

• This "glue" (wetting energy) changes the rules. It means the balloon can hold onto the spaghetti even when the balloon is tiny and the spaghetti is very long—something that shouldn't be possible if you only looked at the bending energy.
• It's like the spaghetti is wearing a wet suit that hugs the liquid, allowing them to stay together in shapes that would otherwise fall apart.

5. Why Does This Matter?

You might ask, "Who cares if a protein drop looks like a donut?"
This is actually a big deal for biology:

• Cell Construction: Cells use these droplets to organize their internal "skeleton." By changing shape, these droplets can help build structures needed for cell movement or division.
• Trapping: Once a filament gets trapped in a droplet, it can get stuck there even if the droplet shrinks. It's like a fly in amber. This could be a way cells store or organize their internal parts.
• Disease: If these shapes go wrong, it might be related to diseases where proteins clump together incorrectly (like in Alzheimer's or Parkinson's). Understanding the shapes helps us understand how these clumps form.

The Big Takeaway

Nature is a master of compromise. When a stiff fiber meets a soft liquid drop, they don't just break; they reshape each other. They trade energy (bending the fiber vs. stretching the surface) to find the most comfortable shape possible. Sometimes that shape is a sphere, sometimes a pancake, and sometimes a donut.

The researchers used real experiments with protein drops, giant water droplets with plastic strings, and super-computer simulations to prove that surface tension (the skin of the drop) and bending energy (the stiffness of the string), combined with a little bit of stickiness, are the architects of these microscopic shapes.

Technical Summary

1. Problem Statement

Biomolecular condensates (liquid-liquid phase-separated droplets) and cytoskeletal filaments (e.g., F-actin, microtubules) frequently coexist within cells. Their interactions lead to mutual deformations that are critical for cellular organization, motility, and mitosis. While previous studies established the concept of the bendocapillary length ($R_{bc}$)—the threshold where adhesion energy overcomes filament bending energy to retain a bent filament—this framework has limitations:

• It assumes a static spherical droplet and does not account for mutual deformations where the droplet shape changes to accommodate the filament.
• It lacks a comprehensive physical model explaining the diverse morphologies observed (e.g., tori, oblate spheroids, erythrocyte-like shapes) when filaments are fully embedded within condensates.
• The interplay between surface tension, filament bending energy, and wetting effects (specifically how liquid wets the filament surface) at the micro-scale of condensates is not fully understood.

2. Methodology

The authors employed a tripartite approach combining wet-lab experiments, analytical modeling, and computational simulations to decouple and analyze the physical forces at play.

• Experimental Approach:

  • System: In vitro formation of biomolecular condensates using PLPPR3 ICD (a transmembrane protein domain) with polymerizing F-actin.
  • Conditions: Two regimes were tested: (1) Standard conditions (20 µM PLPPR3, 1.24 µM G-actin) and (2) High-actin/Low-condensate conditions (10 µM PLPPR3, 4 µM G-actin) to induce extreme deformations.
  • Validation: A macroscopic control experiment using water droplets on super-hydrophobic taro leaves containing a polylactic acid (PLA) filament was used to validate the analytical model at a larger scale.
  • Imaging: Confocal microscopy was used to visualize 3D shapes and calculate aspect ratios (AR).

• Analytical Modeling:

  • Developed a free-energy minimization model ($E_{total} = E_{bend} + E_{surf}$).
  • Assumptions: The model treats the filament as a dimensionless curve initially, balancing bending stiffness ($\kappa$) and surface tension ($\gamma$).
  • Geometries: Modeled shapes as solids of rotation: spheres, oblate spheroids (lens shapes), and tori.
  • Refinement: Later iterations incorporated a "filament bundle" concept with a finite volume fraction ($\epsilon$) and wetting films to account for realistic packing.

• Computational Simulations:

  • Tool: Coarse-grained Molecular Dynamics (MD) using LAMMPS.
  • Setup: Simulated fluid particles (phase-separating via Lennard-Jones potential) and filament beads (modeled as a chain with harmonic springs for bending and stretching).
  • Regimes:
    1. Non-self-avoiding filaments: Filament loops could overlap (zero volume), isolating surface/bending energy effects.
    2. Self-avoiding filaments: Filaments had finite volume and repulsion, introducing realistic wetting and packing constraints.
  • Analysis: Used Support Vector Machines (SVM) to classify shape phases (spheroid, kettlebell, torus) based on volume ($V$) and filament length ($L$).

3. Key Contributions

• Discovery of the "Kettlebell" Morphology: Identified a novel intermediate shape (kettlebell) that arises during the transition between spheroids and tori, driven by wetting films and volume constraints.
• Refinement of the Bendocapillary Concept: Demonstrated that mutual deformations allow condensates to retain filaments even when the droplet radius is below the classical bendocapillary length, effectively expanding the regime of accessible stable shapes.
• Quantification of Wetting Effects: Showed that as the filament-to-volume ratio increases, wetting interactions (liquid trapping around the filament) become the dominant constraint, dictating shape transitions and volume loss.
• Unified Framework: Successfully integrated experimental data, a fit-free analytical model, and MD simulations to create a predictive phase diagram for droplet-filament shapes.

4. Key Results

• Experimental Observations:

  • Volume Dependence: Smaller condensates exhibit higher aspect ratios (more oblate) as filaments must bend more sharply, increasing bending energy.
  • Morphological Diversity: Depending on the actin-to-condensate ratio, shapes ranged from spherical to oblate spheroids, erythrocyte-like (thin central films), and tori.
  • Macroscopic Validation: The water-PLA system confirmed that the analytical model could predict bending stiffness ($\kappa$) from shape changes, validating the physics of surface tension vs. bending energy.

• Analytical Model Findings:

  • The model accurately predicts a transition from spheroid to torus as volume decreases or filament length increases.
  • However, the initial dimensionless filament model failed to capture the "kettlebell" shape and misaligned with simulation data regarding transition volumes.
  • Correction: Introducing a finite filament bundle volume and wetting film constraints aligned the analytical predictions with simulations, revealing that wetting sets physical boundaries for shape stability.

• Simulation Insights:

  • Shape Phases: Simulations revealed three distinct topological states:
    1. Spheroid: High volume, low curvature.
    2. Kettlebell: Intermediate state where excess liquid forms a bulge due to wetting film saturation.
    3. Torus: Low volume, where the liquid forms a ring around the filament.
  • Volume Loss & Laplace Pressure: Transitions from spheroid to torus involve a significant loss of liquid volume. This is attributed to increased Laplace pressure in the torus geometry, driving liquid out of the droplet (Ostwald ripening) unless stabilized by wetting interactions.
  • Hysteresis: The system exhibits hysteresis; a droplet formed by polymerizing a filament inside may retain a deformed shape (e.g., torus) even if the volume shrinks below the classical stability limit, whereas a simple contact between a small droplet and filament would not retain the filament.

5. Significance

• Biological Implications: The findings suggest that biomolecular condensates are not passive containers but active agents that can restructure the cytoskeleton. The ability to trap filaments in deformed shapes (even below the bendocapillary length) implies a mechanism for organizing cellular architecture, such as filopodia formation or mitotic spindle positioning.
• Pathology: The framework offers a physical explanation for how amyloid fibrils (rigid filaments) might be retained within condensates, potentially serving as early indicators of protein aggregation diseases.
• Prebiotic Chemistry: The shape changes and surface area increases observed could facilitate metabolite uptake in protocells, suggesting a role for filament-induced deformations in the origin of life.
• Physical Framework: The study provides a robust physical framework (combining surface tension, bending rigidity, and wetting) to predict the morphology of soft matter systems containing elastic inclusions, applicable from cellular biology to materials science.

In conclusion, the paper establishes that the shape of condensate droplets containing filaments is a result of a complex competition between surface tension, filament bending energy, and wetting interactions. Mutual deformations expand the stability regime of these systems, allowing for diverse morphologies that are crucial for biological function.