🧬 biology

Pathway variability, coat stiffening and mechanical adaptation during clathrin-mediated endocytosis

This study introduces a kinetic Monte Carlo simulation framework coupled with an adaptive continuum membrane to demonstrate that clathrin-mediated endocytosis outcomes (flat plaques, stalled invaginations, or closed vesicles) are determined by an adaptive assembly process where coat stiffening and history-dependent curvature emerge dynamically, successfully predicting experimental observations without fitting parameters.

Original authors: Johannes H. H. Dreckhoff (Heidelberg University, Germany), Ulrich S. Schwarz (Heidelberg University, Germany), Leon Lettermann (Heidelberg University, Germany)

Published 2026-06-30

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CC BY 4.0

Original authors: Johannes H. H. Dreckhoff (Heidelberg University, Germany), Ulrich S. Schwarz (Heidelberg University, Germany), Leon Lettermann (Heidelberg University, Germany)

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine a cell trying to swallow a piece of food. To do this, it doesn't just open its mouth; it builds a tiny, temporary "basket" out of a special protein called clathrin to pinch off a bubble of its own skin (the membrane) and pull it inside. This process is called clathrin-mediated endocytosis.

For a long time, scientists were puzzled by a simple question: Why do some of these baskets work perfectly, some get stuck halfway, and others just lie flat and never start?

This paper acts like a detective story, using a super-computer simulation to figure out the rules of the game. Here is the story of what they found, explained simply.

The Problem: The Mystery of the Three Fates

When clathrin proteins start gathering on the cell's surface, they can end up in three different states:

The Flat Pile: They gather but stay flat, like a pancake that never flips.
The Aborted Pit: They start to curve inward like a dimple, but then give up and fall apart before becoming a bubble.
The Closed Vesicle: They successfully curl up into a perfect, closed sphere (a bubble) that gets pulled inside the cell.

The big mystery was: What decides which path a basket takes? Is it the shape of the proteins? The tension of the skin? Or something else?

The Solution: A "Smart" Basket that Learns

The researchers built a computer model where they watched these protein baskets grow, step-by-step. They discovered that the basket isn't just a static structure; it's a living, learning machine that changes its own properties as it grows.

Here are the two main "superpowers" the basket develops:

1. The "Stiffening" Effect (From Rubber Band to Steel)

Think of a single clathrin protein like a flexible rubber band. It's easy to bend. But when you connect thousands of them together into a lattice (a net), something magical happens.

Early on: The net is loose and floppy. It can wiggle easily.
As it grows: Once the net is connected enough, it becomes incredibly stiff. It's like turning that rubber band into a steel cage.
Why? On a flat surface, you can wiggle the net by just twisting the joints. But once the net starts curving into a sphere, you can't just twist anymore; you have to stretch the material. Stretching is much harder than twisting. So, the act of curving itself makes the basket rigid.

2. The "Curvature Memory" (The Basket Remembers Its History)

This is the most fascinating part. Imagine you are building a dome. You lay down the first bricks when the ground is flat. Then you lay down the next layer when the dome is slightly curved. Then the next layer when it's very curved.

The basket remembers the shape it was built in.
The "older" parts of the basket (built early) "want" to be flat. The "newer" parts (built late) "want" to be curved.
The final shape of the basket is a compromise between all these memories. It doesn't just have a fixed shape; it has a preferred shape based on how it grew.

The Two Gates of Destiny

The paper explains that for a basket to become a successful bubble, it has to pass through two specific "gates" or checkpoints:

Gate 1: The Connection Check

The Rule: The proteins must connect to each other tightly enough.
The Result: If the connections are too loose, the basket stays floppy and flat. It can't transmit the force needed to bend the cell's skin. If it passes this gate, it becomes stiff enough to start curving.

Gate 2: The Race Against Time

The Rule: Once the basket starts curving, it enters a race. The basket is trying to close, but the cell's skin is pushing back (like trying to push a beach ball underwater).
The Result:
- If the skin is too tight (high tension), the basket gets stuck halfway. It becomes an aborted pit.
- If the skin is loose enough, the basket wins the race, closes completely, and becomes a vesicle.

The "Cutting" Experiment

To prove their theory, the researchers did a virtual experiment. They took a computer-generated basket that had stopped growing (an "aborted" one) and virtually "cut" a slice out of it, like slicing a piece of a pie.

What happened? The remaining part of the basket immediately relaxed and curved more.
Why? Because cutting it removed the "memory" of the flat parts and the stiffness holding it back. This proved that the basket's shape is indeed a result of its growth history and internal stress.

The Big Picture

The main takeaway is that the fate of these cellular baskets isn't decided by a pre-programmed instruction manual. Instead, the process of building the basket creates the rules.

The act of connecting makes it stiff.
The act of growing writes a memory of the shape.
The environment (how tight the skin is) decides if the race is winnable.

It's like a group of people trying to form a human pyramid. If they don't hold hands tightly (Gate 1), they fall flat. If they hold hands but the floor is too slippery or the top person is too heavy (Gate 2), they might get stuck halfway. But if they coordinate just right, they reach the top. The paper shows us that the "coordination" happens automatically through the physics of how they connect and grow.

Technical Summary: Pathway Variability, Coat Stiffening, and Mechanical Adaptation in Clathrin-Mediated Endocytosis

Problem Statement
Clathrin-mediated endocytosis (CME) is the primary mechanism for eukaryotic cells to internalize receptors and nutrients. During this process, clathrin triskelia assemble into a polygonal lattice that remodels the membrane into a coated pit and eventually a vesicle. However, experimental observations reveal significant pathway variability: assemblies can persist as flat plaques, abort after partial invagination, or successfully close into vesicles. The determinants of these distinct fates remain unresolved. Existing models often treat coat stiffness and preferred curvature as fixed material parameters, failing to account for how these properties might emerge dynamically from the assembly process itself. Furthermore, the relationship between microscopic lattice assembly, topological defects (pentagons/heptagons), and macroscopic membrane curvature generation is unclear.

Methodology
The authors developed a hybrid kinetic Monte Carlo (KMC) simulation framework that couples discrete clathrin agents to an adaptive continuum membrane.

Discrete Component: Individual clathrin triskelia are modeled as mobile nodes on a continuous membrane surface. Each triskelion possesses an elastic Hamiltonian with costs for deviations from a preferred leg length ( $L_0 = 18.4$ nm), in-plane angle ( $\phi_0$ ), and pucker angle ( $\chi_0$ ). Nodes diffuse, bind/unbind, and form or dissolve bonds with stochastic, energy-dependent rates.
Continuum Component: The membrane is described by a spherical-cap geometry defined by area ( $A$ ) and mean curvature ( $H$ ). The membrane curvature adapts adiabatically to the mechanical state of the coat between stochastic assembly events.
Key Innovation: Unlike previous models that prescribed curvature or stiffness, this framework allows the effective coat bending rigidity ( $\kappa_{cl}^{eff}$ ) and the preferred coat curvature ( $H_{min}^{eff}$ ) to emerge naturally during growth.
Validation: The model was validated against experimental data, including high-speed atomic force microscopy (AFM) reconstructions of coat geometries and in silico nanodissection experiments mimicking HS-AFM cutting studies.

Key Contributions and Results

Dynamic Stiffening and Curvature Memory:
Simulations of coats growing on membranes with fixed curvature revealed that the effective bending rigidity increases by nearly two orders of magnitude (from $\sim 20 k_B T$ to $\sim 10^3 k_B T$ ) as the coat matures. This stiffening arises from a geometric transition: in flat or loosely connected lattices, curvature changes are accommodated by soft angular rearrangements. In connected curved lattices, changing curvature induces residual area strain, forcing the system to resist via much stiffer stretching modes.
Simultaneously, the coat develops "curvature memory." The effective energy-minimizing curvature ( $H_{min}^{eff}$ ) shifts from the microscopic preference of individual triskelia toward the curvature at which the material was assembled. This is modeled as a weighted average of growth curvatures, where each incorporated triskelion writes a local reference geometry into the lattice.
Two Sequential Fate Gates:
The model identifies two sequential gates that determine the assembly fate (flat, aborted, or closed) based on the interplay between discrete connectivity and continuum mechanics:
- Gate 1 (Connectivity): A coat must achieve sufficient lattice connectivity to transmit coherent bending stresses. Coats with low connectivity remain floppy and grow as flat plaques. Sufficient connectivity triggers the bending-to-stretching transition, leading to stiffening and curvature generation.
- Gate 2 (Energy Landscape): Once stiffening occurs, the fate is determined by the competition between the moving energy minimum of the system and the closure boundary ( $\phi = 1$ ). The position of this minimum depends on membrane tension, polymerization energy, and the growth history (curvature memory). If the minimum moves beyond the closure boundary before the system reaches it, the coat closes; if the system catches the minimum while it is still within the accessible region, the pit stalls (aborted).
Role of Topological Defects:
The simulations demonstrate that topological defects (pentagons and heptagons) are not the primary trigger for curvature generation. Curvature can initiate before visible non-hexagonal defects are counted. Instead, defects follow and stabilize the curvature generated by a sufficiently connected, stiffened coat, satisfying the topological requirements for a closed cage only after curvature has been established.
Non-Euclidean Elasticity Theory:
The authors derived an analytical theory based on non-Euclidean elasticity that predicts the observed growth laws. The theory posits that the effective stiffness and preferred curvature are functions of a geometric weight ( $w \propto A^2 H^2$ ) and a history-dependent curvature variable. This theory successfully collapses simulation data from various growth curvatures and membrane tensions onto universal trends without fitting parameters.
Validation via In Silico Cutting:
To test the concept of prestress and memory, the authors simulated cutting a connected coat (mimicking nanodissection). The results showed that cutting reduces the effective bending rigidity and increases the effective energy-minimizing curvature. This confirms that the coat stores growth history in its connected lattice; removing low-curvature material from early growth stages shifts the relaxed state of the remaining coat toward higher curvature.

Significance
The paper claims to provide a unifying physical framework that explains the variability of CME outcomes without invoking separate mechanisms for each fate. By demonstrating that coat stiffness and preferred curvature are emergent, history-dependent properties generated by the assembly process itself, the authors bridge the gap between discrete molecular assembly and continuum membrane mechanics.

The study redefines the inputs and outputs of CME modeling: rather than prescribing material parameters, the model shows how the environment (e.g., membrane tension) and the assembly history jointly determine the effective mechanical state of the coat. This suggests that pathway variability is an intrinsic feature of the self-assembly process, governed by mechanical feedback loops where the coat's maturation reshapes the energy landscape it inhabits. The findings reconcile experimental observations of flat, stalled, and closed coats under a single set of physical rules, highlighting that the "fate" of a clathrin coat is a reflection of the mechanical environment in which it grew.