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arxiv: 1907.02992 · v1 · pith:EVVZ562Ynew · submitted 2019-07-05 · ⚛️ physics.bio-ph · cond-mat.stat-mech· q-bio.SC

Towards a theory of assembly of protein complexes: lessons from equilibrium statistical physics

Pablo Sartori , Stanislas Leibler This is my paper

Pith reviewed 2026-05-25 01:26 UTC · model grok-4.3

classification ⚛️ physics.bio-ph cond-mat.stat-mechq-bio.SC
keywords protein complexesself-assemblyequilibrium thermodynamicsmultifarious assemblychimeric assemblyheterogeneous compositionsparse component usecellular noise
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The pith

Equilibrium thermodynamics shows heterogeneous compositions and sparse component use enable reliable assembly of many distinct protein complexes.

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

The paper constructs an equilibrium thermodynamic model of self-assembly to determine how cells can produce multiple different protein complexes without forming incorrect mixed structures amid noise. The model produces four regimes, and only in the multifarious assembly regime do distinct complexes coexist stably. This regime appears when complex compositions are heterogeneous enough and when each protein component participates in only a few complexes. Database checks indicate that actual cellular complexes meet both requirements.

Core claim

Our equilibrium thermodynamic model of self-assembly exhibits four behaviors: diluted solution, liquid mixture, chimeric assembly, and multifarious assembly. In the multifarious regime different protein complexes coexist without forming erroneous chimeric structures. Two conditions must be met: complex compositions must be sufficiently heterogeneous and component usage across complexes must be sparse. Analysis of protein complex databases suggests cellular systems have evolved to satisfy both conditions.

What carries the argument

The multifarious assembly regime in the equilibrium self-assembly model, which permits stable coexistence of distinct complexes when compositions are heterogeneous and component sharing is sparse.

If this is right

  • Distinct protein complexes can form and coexist without producing chimeric errors under the two stated conditions.
  • Heterogeneous composition of complexes is required to reach the reliable regime.
  • Sparse use of each component by only a few complexes is also required.
  • Public databases of protein complexes are consistent with cells having evolved to meet both conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Cells may use active, energy-consuming mechanisms to reinforce the sparsity and heterogeneity that equilibrium already favors.
  • The same two conditions could serve as design rules for engineering synthetic multi-protein systems that avoid cross-talk.
  • If sparsity is strictly necessary, the total number of distinct complexes that can be maintained is bounded by the size of the component pool.
  • The approach may extend to other noisy self-assembly problems such as virus capsid formation or organelle biogenesis.

Load-bearing premise

An equilibrium thermodynamic model captures the essential physical constraints on protein complex formation despite cells operating far from equilibrium.

What would settle it

A survey of protein complexes showing either largely overlapping compositions or dense component sharing across many complexes would indicate cells do not satisfy the conditions for the multifarious regime.

Figures

Figures reproduced from arXiv: 1907.02992 by Pablo Sartori, Stanislas Leibler.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A. The histogram reveals a large abundance of high￾heterogeneity complexes, hc ∈ [0.8, 1.0], which supports the arguments presented in this work. From Fig. 4A it is also apparent that a significant number of complexes have intermediate values of heterogeneity, hc ∈ [0.4, 0.6). Given that our theory applies to structures without any geometrical symmetry (as described in Materials & Methods), we can ask whet… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: We considered each of the c = 1, . . . , K different complexes to be a square arrangement of square components representing proteins with lateral size √ M, so that Mc = M (all complexes have the same size). Each complex contains N different component species, Nc = N ≤ M (all complexes have the same number of species). The total number of component species is Ntot ≥ N, and for each complex its N different c… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 2
Figure 2. Figure 2: In this case the usage of the available component species is “dense”, and complexes are fully heterogeneous. [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16 [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17 [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18 [PITH_FULL_IMAGE:figures/full_fig_p024_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19 [PITH_FULL_IMAGE:figures/full_fig_p026_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20 [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
read the original abstract

Cellular functions are established through biological evolution, but are constrained by the laws of physics. For instance, the physics of protein folding limits the lengths of cellular polypeptide chains. Consequently, many cellular functions are carried out not by long, isolated proteins, but rather by multi-protein complexes. Protein complexes themselves do not escape physical constraints, one of the most important being the difficulty to assemble reliably in the presence of cellular noise. In order to lay the foundation for a theory of reliable protein complex assembly, we study here an equilibrium thermodynamic model of self-assembly that exhibits four distinct assembly behaviors: diluted protein solution, liquid mixture, "chimeric assembly" and "multifarious assembly". In the latter regime, different protein complexes can coexist without forming erroneous chimeric structures. We show that two conditions have to be fulfilled to attain this regime: (i) the composition of the complexes needs to be sufficiently heterogeneous, and (ii) the use of the set of components by the complexes has to be sparse. Our analysis of publicly available databases of protein complexes indicates that cellular protein systems might have indeed evolved so to satisfy both of these conditions.

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

1 major / 1 minor

Summary. The paper develops an equilibrium thermodynamic model of multi-protein self-assembly exhibiting four regimes (diluted solution, liquid mixture, chimeric assembly, multifarious assembly). It derives that the multifarious regime—reliable coexistence of distinct complexes without erroneous chimeras—requires (i) sufficiently heterogeneous complex compositions and (ii) sparse component usage across complexes. Public protein-complex databases are analyzed to argue that cellular systems appear to have evolved to satisfy both conditions.

Significance. If the central derivation holds, the work supplies a concrete statistical-physics criterion for reliable multifarious assembly and supplies an independent empirical check via database statistics. This is a strength: the conditions are falsifiable against composition data and could inform evolutionary hypotheses. The equilibrium framing, however, leaves open whether the identified conditions remain load-bearing once active cellular processes are included.

major comments (1)
  1. [Abstract and biological-implications section] Abstract (final sentence) and the section on biological implications: the inference that 'cellular protein systems might have indeed evolved so to satisfy both of these conditions' treats the equilibrium-model boundaries as directly relevant to vivo assembly. Because cells operate far from equilibrium with continuous energy dissipation, chaperones, and regulated disassembly, the heterogeneity/sparsity conditions may be neither necessary nor sufficient in vivo; a concrete test would be to recompute the regime diagram after adding a minimal non-equilibrium term (e.g., ATP-driven dissociation rate) and to check whether the multifarious window shrinks or disappears.
minor comments (1)
  1. [Abstract] The abstract lists the four regimes but does not explicitly name 'diluted protein solution, liquid mixture, chimeric assembly and multifarious assembly'; adding the names would improve immediate readability.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the constructive report. The central point concerns the scope of our equilibrium model and the phrasing of its biological implications. We address this below and will make targeted revisions to clarify the model's limitations while preserving the core results.

read point-by-point responses

  1. Referee: [Abstract and biological-implications section] Abstract (final sentence) and the section on biological implications: the inference that 'cellular protein systems might have indeed evolved so to satisfy both of these conditions' treats the equilibrium-model boundaries as directly relevant to vivo assembly. Because cells operate far from equilibrium with continuous energy dissipation, chaperones, and regulated disassembly, the heterogeneity/sparsity conditions may be neither necessary nor sufficient in vivo; a concrete test would be to recompute the regime diagram after adding a minimal non-equilibrium term (e.g., ATP-driven dissociation rate) and to check whether the multifarious window shrinks or disappears.

    Authors: We agree that the model is strictly at equilibrium and that in vivo assembly involves active processes. Our statement is intended only as an observation that the two conditions required for the multifarious regime in equilibrium are statistically satisfied by real protein-complex data; we do not claim these conditions are necessary or sufficient once energy dissipation, chaperones, or regulated disassembly are present. We will revise both the abstract and the biological-implications section to (i) state explicitly that the analysis is equilibrium, (ii) describe the database result as an empirical consistency check rather than evidence of evolutionary optimization, and (iii) note that non-equilibrium mechanisms may relax or replace the identified requirements. The suggested numerical test with an ATP-driven term would require an entirely new non-equilibrium formulation and is therefore outside the scope of the present work. revision: partial

standing simulated objections not resolved
  • Recomputing the regime diagram after introducing a minimal non-equilibrium term (e.g., ATP-driven dissociation) would require developing a new dynamical model, which lies beyond the equilibrium framework of the manuscript.

Circularity Check

0 steps flagged

No circularity: model-derived conditions checked against independent database

full rationale

The paper defines an equilibrium thermodynamic model of self-assembly, enumerates four regimes (diluted, liquid mixture, chimeric, multifarious) from its partition function and free-energy analysis, and derives the two conditions (heterogeneous composition, sparse component usage) as the parameter regime boundaries that suppress chimeras. These conditions are outputs of the model's equations rather than inputs or self-definitions. The subsequent analysis of public protein-complex databases constitutes an external empirical test of whether evolved systems occupy that regime; it does not feed back into the derivation or rename fitted parameters as predictions. No self-citations, ansatzes smuggled via prior work, or uniqueness theorems appear in the load-bearing steps. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive enumeration; the model necessarily introduces effective interaction energies or concentrations whose values are not specified here.

free parameters (1)
  • interaction energies or binding affinities
    Thermodynamic models of assembly typically fit or choose component interaction strengths to produce the observed regimes.
axioms (1)
  • domain assumption Protein complex assembly can be usefully approximated by an equilibrium thermodynamic model
    The paper explicitly studies an equilibrium thermodynamic model of self-assembly.

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Reference graph

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    The Glauber transition probability was calculated, W = (1 + exp(∆U))−1

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  54. [54]

    A random number p∈ [0, 1] was generated

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  55. [55]

    peeling-off

    The species change was performed or not, depending on whether p<W or not. Steps 1− 5 constituted one Monte Carlo step. Typical simulations were run for ∼ 106 latt, where 1 latt is a lattice sweep and corresponds to L Monte Carlo steps, with L = √ L× √ L the size of the square lattice. Unless otherwise specified, we used √ L = 40. 12 In order to quantify ou...

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  56. [56]

    random model

    and assigns a p−value for the interactions. Again, we clustered the data using the ClusterOne algorithm with weights given by 1−p (parameters as before). The histograms of qα for datasets IV and V are shown in panels B and C of Fig. 20. Panel B exhibits a very similar trend to panel A, with highly participatory proteins that cannot be explained by our sim...

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