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arxiv: 2604.01109 · v3 · pith:AUI7UPGGnew · submitted 2026-04-01 · ❄️ cond-mat.soft

High-symmetry ill-fitting subunits in 3D form aggregates of all dimensions

Elena N. Govorun , Martin Lenz This is my paper

Pith reviewed 2026-05-19 17:48 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords self-assemblyprotein aggregatesfilament formationdeformable subunitsill-fitting objectsmorphology predictionthree-dimensional assembly
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0 comments X

The pith

Ill-fitting deformable subunits in three dimensions self-assemble into clusters, filaments, layers or bulks depending on adhesivity and elasticity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines how identical but geometrically ill-fitting deformable subunits, meant to mimic globular proteins, come together in solution. Deformations accumulate with aggregate size and eventually limit further growth. The authors map the mechanics onto two incompatible interconnected networks to analytically predict the stable shapes that minimize energy. They show that zero-dimensional clusters, one-dimensional filaments, two-dimensional layers and three-dimensional bulks all arise as ground states for different values of adhesivity and elasticity. A reader cares because the result supplies a purely mechanical route to filament formation that does not require special molecular recognition.

Core claim

We analytically predict the ground state morphologies of the resulting aggregates as a function of the subunit adhesivity and elasticity by mapping their mechanics onto those of two incompatible, interconnected networks. We find that zero-dimensional clusters, three-dimensional bulks as well as symmetry-broken one-dimensional filaments and two-dimensional layers can all form depending on assembly parameters. Poorly compressible, moderately adhesive subunits favor filaments.

What carries the argument

Mapping subunit mechanics onto two incompatible, interconnected networks that analytically predicts ground-state morphologies from adhesivity and elasticity.

Load-bearing premise

The mechanics of the self-assembling subunits can be mapped onto those of two incompatible, interconnected networks to analytically predict ground state morphologies as a function of adhesivity and elasticity.

What would settle it

Molecular-dynamics runs in which compressibility and adhesivity are varied across the predicted boundary should show a switch from filaments to isotropic clusters; absence of that switch would falsify the mapping.

Figures

Figures reproduced from arXiv: 2604.01109 by Elena N. Govorun, Martin Lenz.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Illustration of a subunit geometry that gives rise to the continuum model studied here. Note that our continuum models is built on the basis of symmetry considerations and is thus not restricted to this specific design. (a) Case of well-adjusted subunits. Here each subunit is a cube with its vertices alternately colored yellow and red. The yellow set of points and the red set of points each form a regular … view at source ↗
Figure 3
Figure 3. Figure 3: Displacement and stress in two-dimensional slabs at mechanical equilibrium. (a) Difference in the displacements of the networks and (b) stress tensor element [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Incompressible subunits are very difficult to deform into a bulk, as indicated by the divergence of and as approaches at fixed and . 5 Asymptotic morphological transitions for near-incompressible subunits To provide more transparent expressions of the implicit aggregate energies given in Sec. 3, here we derive their explicit asymptotic forms in the incompressible limit. As shown in [PITH_FULL_IMAGE:figure… view at source ↗
read the original abstract

Proteins can combine into functional elements in living cells or self-assemble into unwanted structures in a number of diseases. The resulting aggregates often display filamentous morphologies across a large range of protein shapes and molecular interactions. This has led to the suggestion that filament formation could be a generic outcome of the aggregation of geometrically complex, ill-fitting objects, although such a mechanism has not been demonstrated in three dimensions. To address this problem, we theoretically study the self-assembly of three-dimensional identical, ill-fitting deformable subunits mimicking globular proteins in solution. In our model, self-assembling subunits incur deformations that accumulate as the aggregate size increases and can eventually hamper further assembly. We analytically predict the ground state morphologies of the resulting aggregates as a function of the subunit adhesivity and elasticity by mapping their mechanics onto those of two incompatible, interconnected networks. We find that zero-dimensional clusters, three-dimensional bulks as well as symmetry-broken one-dimensional filaments and two-dimensional layers can all form depending on assembly parameters. Poorly compressible, moderately adhesive subunits favor filaments. These findings hint at a generic pathway to control self-assembly in three dimensions and suggests that such mechanisms could be investigated in more realistic protein models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript theoretically studies the self-assembly of three-dimensional identical ill-fitting deformable subunits mimicking globular proteins. By mapping the mechanics of these subunits onto two incompatible interconnected networks, the authors analytically predict ground-state aggregate morphologies (zero-dimensional clusters, one-dimensional filaments, two-dimensional layers, and three-dimensional bulks) as a function of subunit adhesivity and elasticity, finding that poorly compressible, moderately adhesive subunits favor filaments.

Significance. If the mapping is rigorously validated, the work identifies a generic, parameter-controlled mechanism for dimension selection in 3D self-assembly of geometrically complex objects. This could explain the prevalence of filamentous protein aggregates across diverse molecular systems and provide a framework for designing or controlling assembly outcomes in soft-matter and biological contexts.

major comments (2)
  1. [Network mapping and energy minimization] The central analytical mapping (described in the section on the network model and used to derive the morphology predictions) must be shown to reproduce the energy minima of the original 3D deformable-subunit Hamiltonian. Specifically, the mapping needs to demonstrate how local ill-fitting accumulates under full 3D geometric closure constraints rather than assuming additive pairwise strains or lower-dimensional networks; without this verification the selection of symmetry-broken filament states for poorly compressible, moderately adhesive subunits risks being an artifact of the reduction.
  2. [Results on morphology dependence] The phase diagram or morphology selection as a function of adhesivity and elasticity (presented in the results) should include an explicit error analysis or sensitivity check on the network incompatibility parameters; the abstract states the mapping is 'analytical' yet the favored filament regime appears to depend on specific ranges of the two free parameters, requiring a clear statement of how ground states are identified without fitting.
minor comments (2)
  1. Notation for adhesivity and elasticity should be defined consistently when first introduced and cross-referenced in any phase diagrams or tables.
  2. The abstract would benefit from a one-sentence statement of the key assumption underlying the two-network mapping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive comments, which help us clarify the foundations of our analytical mapping. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Network mapping and energy minimization] The central analytical mapping (described in the section on the network model and used to derive the morphology predictions) must be shown to reproduce the energy minima of the original 3D deformable-subunit Hamiltonian. Specifically, the mapping needs to demonstrate how local ill-fitting accumulates under full 3D geometric closure constraints rather than assuming additive pairwise strains or lower-dimensional networks; without this verification the selection of symmetry-broken filament states for poorly compressible, moderately adhesive subunits risks being an artifact of the reduction.

    Authors: The mapping is derived by projecting the local 3D geometric mismatch of the deformable subunits onto two orthogonal, interconnected networks whose incompatibility encodes the cumulative strain that arises when multiple subunits close into a 3D structure. Because the network energy is minimized globally subject to the connectivity constraints of the aggregate, the accumulation of ill-fitting is not treated as a simple sum of independent pairwise terms; the 3D closure is enforced through the requirement that the two networks remain compatible at every vertex. We will add an appendix that explicitly derives the network incompatibility parameters from the 3D subunit geometry and shows that the resulting energy expression reproduces the leading-order deformation cost of the original Hamiltonian for both open and closed clusters. This addition will make the non-artifactual character of the filament selection explicit. revision: yes

  2. Referee: [Results on morphology dependence] The phase diagram or morphology selection as a function of adhesivity and elasticity (presented in the results) should include an explicit error analysis or sensitivity check on the network incompatibility parameters; the abstract states the mapping is 'analytical' yet the favored filament regime appears to depend on specific ranges of the two free parameters, requiring a clear statement of how ground states are identified without fitting.

    Authors: Ground-state morphologies are identified by direct analytic minimization of the total network energy (adhesion plus elastic deformation) with respect to the possible aggregate topologies (0D cluster, 1D filament, 2D layer, 3D bulk). The two free parameters are the dimensionless adhesivity and the compressibility modulus; the filament regime is the region in which the energy minimum lies at a linear, symmetry-broken configuration. We will revise the results section to state the minimization procedure explicitly and to include a brief sensitivity analysis demonstrating that the boundaries of the filament regime shift only quantitatively, not qualitatively, under small variations of the incompatibility parameters. revision: yes

Circularity Check

0 steps flagged

Analytical mapping to incompatible networks is self-contained derivation

full rationale

The paper derives morphologies by mapping subunit deformations to two incompatible interconnected networks, then analytically predicting ground states (0D clusters, 1D filaments, 2D layers, 3D bulks) as functions of adhesivity and elasticity. No quoted equation or step reduces a claimed prediction to a fitted parameter or self-citation by construction; the mapping is introduced as an independent analytical step that encodes accumulation of deformations. The derivation remains self-contained against the model Hamiltonian without load-bearing self-citations or renaming of known results. This is the normal case of an honest theoretical construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model rests on deformation accumulation with aggregate size and the network-mapping approximation; adhesivity and elasticity act as tunable inputs rather than derived quantities.

free parameters (2)
  • subunit adhesivity
    Controls binding strength and is varied to determine which morphology is favored.
  • subunit elasticity
    Controls deformability and compressibility; poorly compressible case favors filaments.
axioms (1)
  • domain assumption Self-assembling subunits incur deformations that accumulate as the aggregate size increases and can eventually hamper further assembly.
    Invoked to justify the mapping to incompatible networks and the existence of a ground-state morphology.

pith-pipeline@v0.9.0 · 5737 in / 1299 out tokens · 61110 ms · 2026-05-19T17:48:19.644130+00:00 · methodology

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