Pith

open record

sign in

arxiv: 2607.06378 · v1 · pith:BSLZV4AM · submitted 2026-07-07 · cond-mat.soft · physics.bio-ph· physics.comp-ph

Dynamical Simulation of Membrane Bending by Flexible Protein Assemblies

Samuel L. Foley , Margaret E. Johnson This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 07:39 UTCglm-5.2pith:BSLZV4AMrecord.jsonopen to challenge →

classification cond-mat.soft physics.bio-phphysics.comp-ph PACS 87.16.D-87.15.K-87.15.La82.39.Wt
keywords bendingclathrincoatlatticesmembraneproteinrigidityenergy
0
0 comments X

The pith

Protein lattice stiffness depends on how you measure it

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

This paper presents a computational framework that couples flexible, coarse-grained protein assemblies to a continuum membrane model evolved in Fourier space, enabling simulations of membrane bending by protein lattices like clathrin coats and HIV Gag assemblies at biologically relevant length and time scales. The central discovery is that the bending rigidity of a solid-like protein lattice is not a single material constant: when the same clathrin lattice is characterized by flat-sheet buckling (pure cylindrical bending), the inferred flexural rigidity can be more than an order of magnitude smaller than the effective rigidity inferred from spherical coat deformations on a membrane. The authors trace this discrepancy to the fact that the Helfrich-like bending energy, which is exact for fluid membranes where in-plane shear is zero, conflates pure bending with in-plane stretching when applied to solid-like protein lattices undergoing spherical deformation. Because spherical deformations of a bonded lattice necessarily involve stretching energy that is absent in cylindrical bending, the apparent modulus extracted from spherical geometries is inflated. The paper validates its membrane model by showing that membrane tension reduces coat curvature in agreement with analytical Helfrich theory, and demonstrates a pipeline for transferring self-assembled protein structures from reaction-diffusion simulations into the flexible membrane-coupled framework, applying it to the HIV-1 immature Gag lattice.

Core claim

The effective bending rigidity of a solid-like protein lattice is geometry-dependent: the same microscopic bond parameters yield a flexural rigidity (from flat-sheet buckling) that can differ by more than an order of magnitude from the apparent rigidity inferred from spherical coat deformations. This happens because the Helfrich-like bending energy, designed for fluid membranes with zero shear modulus, inadvertently captures in-plane stretching costs when applied to solid-like lattices undergoing spherical deformation. The paper shows that the bond-angle stiffness parameter k_theta is the principal microscopic determinant of the meso-scale flexural rigidity, with bond-length and torsional参数s

What carries the argument

The central object is the geometry-dependent bending modulus kappa_c of a solid-like protein lattice. Two calibration protocols—flat-sheet buckling (yielding the flexural or cylindrical rigidity) and spherical coat deformation on a Helfrich membrane (yielding an apparent spherical rigidity)—are applied to the same coarse-grained clathrin lattice. The discrepancy between the two arises because the Helfrich-like energy (Eqn. 11) is exact only for pure bending of fluid membranes; solid-like lattices with shear rigidity accumulate stretching energy during spherical deformation, which inflates the apparent modulus. The membrane itself is propagated using Fourier-space Brownian dynamics of the H_{

If this is right

  • Researchers using Helfrich-like bending energies to model solid-like protein lattices (clathrin, ESCRT, viral capsids) must report which deformation geometry was used to calibrate the bending modulus and cannot freely transfer that modulus to other geometries.
  • Simulations of endocytic or budding processes may systematically overestimate the mechanical resistance of protein coats if they use a single bending modulus calibrated from one geometry to model a different deformation mode.
  • The bond-angle stiffness k_theta identified as the dominant microscopic parameter provides a direct calibration target: experimental measurements of clathrin coat rigidity can be mapped to specific coarse-grained bond parameters, enabling quantitative modeling.
  • The pipeline connecting reaction-diffusion assembly simulations to membrane-coupled flexible dynamics enables computational studies of how self-assembled lattice topology and mechanics jointly determine membrane curvature, applicable to any membrane-remodeling protein system.

Load-bearing premise

The small-gradient approximation of the Helfrich functional restricts the membrane to modest height deviations from flatness, limiting the framework to early-stage curvature generation and preventing modeling of complete vesicle budding or highly curved structures. Additionally, the Helfrich-like bending energy is applied to solid-like protein lattices in spherical geometries—the very regime where the paper itself demonstrates this energy form conflates bending and stretching

What would settle it

Measure the bending rigidity of the same clathrin lattice using both flat-sheet buckling and spherical coat deformation, then swap the calibrated moduli: use the buckling-derived kappa_c in a spherical geometry simulation and the spherical-derived kappa_c in a buckling simulation. If the moduli were truly geometry-independent, both would predict the correct deformation in the other's geometry. The paper predicts they will not, with the spherical-derived modulus overestimating stiffness in cylindrical bending and the buckling-derived modulus underestimating resistance to spherical deformation.

Figures

Figures reproduced from arXiv: 2607.06378 by Margaret E. Johnson, Samuel L. Foley.

Figure 1
Figure 1. Figure 1: FIG. 1. ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Example setup for the buckling protocol for measur [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Top: Cross-section of a partial SCC (red) deform [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Flexural bending rigidity measured from buckling [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] 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_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

Membrane-deforming protein lattices play a key role in essential and pathogenic biological processes, including endocytosis and viral budding. Attaining the necessary length- and time-scales in simulation can be difficult for such large-scale membrane remodeling events. We present a model of a flexible protein lattice coupled to a Helfrich membrane propagated in Fourier space in the over-damped regime. We focus primarily on membrane-bound clathrin lattices, an essential part of the endocytic machinery. We quantify the material properties of our clathrin model lattices using buckling methods to measure the flexural rigidity as it varies with force constants of the coarse-grained potential energy function. By comparing this flexural rigidity to the effective rigidity observed when modeling the bending energy of a spherical clathrin coat using a Helfrich-like bending energy term, we show how the interpretation of the bending rigidity changes with the structure of the protein coat, resulting in an effective stiffening as the coat grows. This relatively common approximation thus must be applied with care, as it can over-estimate the stiffness of assembled lattices depending on the interpretation assumed. We validate our model by verifying that the tension of our simulated membrane results in changes to the geometry of the clathrin coat consistent with theoretical expectations. We conclude by demonstrating our newly available code for transferring structures assembled via rigid-body reaction-diffusion (using the NERDSS simulation package) into our flexible membrane-coupled dynamical framework, applying it to the membrane-bound HIV-1 immature Gag lattice.

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

3 major / 11 minor

Summary. This paper presents a coarse-grained simulation framework coupling flexible protein lattices (propagated via Brownian dynamics in HOOMD-blue) to a Helfrich membrane (propagated via Fourier-space Brownian dynamics). The framework is applied to clathrin spherical coats and HIV-1 Gag lattices. The central result is that the effective bending rigidity κ_c inferred from spherical coat deformations can exceed the flexural rigidity measured from flat-sheet buckling by over an order of magnitude, because spherical deformations of solid-like lattices involve in-plane stretching energy absent in pure cylindrical bending. The authors validate the model by showing that membrane tension reduces coat curvature in reasonable agreement with theoretical predictions, and demonstrate a pipeline for transferring NERDSS-assembled structures into the flexible framework.

Significance. The paper addresses a practically important problem: calibrating coarse-grained protein bond parameters to reproduce meso-scale elastic properties of membrane-deforming lattices. The systematic buckling parameter sweep (96 simulations, Fig. 4) cleanly establishing k_θ as the dominant control parameter is a solid contribution. The geometry-dependent rigidity finding is conceptually valuable for the community, as it cautions against naively applying Helfrich-like bending energies to solid-like protein coats. The publicly available code and the NERDSS-to-flexible-simulation pipeline add practical value. The tension validation (Fig. 6) and the analytical shape profile comparison (Eqn. 17) provide useful consistency checks. The framework occupies a useful middle ground between all-atom and fully continuum approaches.

major comments (3)
  1. §III.B, Fig. 5: The headline claim that the closed cap (18-mer) yields an effective κ_c more than an order of magnitude larger than the buckling-measured flexural rigidity is supported only by summary points in Fig. 5c. The individual fit curves (K vs. κ_m) for the closed cap are not shown anywhere in the main text or SI. Fig. 5a shows these fits only for the open cap. Since the paper's own argument is that the closed cap's energy includes stretching contributions with potentially different functional dependence on K than the Helfrich-like (K−K_0)² form, it is essential to verify that Eqn. (14) actually fits the closed cap data well. Without fit residuals, R² values, or sub-range fitting stability tests for the closed cap, the reader cannot assess whether the extracted κ_c is a well-defined parameter or an artifact of forcing an inappropriate functional form onto the data. This is load-b
  2. §III.B, Fig. 5: Related to the above, the paper should report the range of κ_m values used for each fit and demonstrate that the extracted κ_c is stable under sub-range fitting (e.g., fitting over the upper half vs. lower half of the κ_m range). If the effective rigidity is truly a stable parameter of the Helfrich-like form, it should not depend strongly on the fitting window. If it does, the 'order of magnitude' claim should be qualified accordingly.
  3. §III.C, Fig. 6a: The tension validation shows that high-tension systems (Σ = 0.15 k_BT/nm²) deform less than predicted by Eqn. (16). The authors acknowledge this but do not quantify the discrepancy or discuss whether it scales systematically with Σ or κ_c. Since the tension validation is presented as a validation of the overall framework, a brief quantitative analysis of the disagreement (e.g., percent deviation at each Σ value) and a discussion of whether it arises from the simplified treatment of the non-adhered membrane region or from the small-gradient approximation would strengthen the claim.
minor comments (11)
  1. Abstract: 'veryifying' should be 'verifying'.
  2. §I, second paragraph: 'molecular geomtry' should be 'molecular geometry'.
  3. §III.B: 'determinde' should be 'determined'.
  4. §III.B: 'vaue' should be 'value' (in Fig. 5a legend description).
  5. §III.D: 'Figs. 7c and 7c' should likely be 'Figs. 7c and 7d'.
  6. §II.D, Eqn. (14): The derivation assumes ℓ/R ≪ 1 and a/2πR² ≪ 1 for the tension case (Eqn. 16), but the zero-tension case (Eqn. 14) is presented as exact. It would help to state explicitly whether any approximations enter Eqn. (14) beyond the geometric assumptions already discussed.
  7. Fig. 5a: The x-axis label 'Membrane Modulus κm (kBT)' should specify units of k_BT/nm² or similar for dimensional consistency with the membrane bending rigidity.
  8. §II.E: The membrane-protein coupling stiffness k_mem = 1000 k_BT is given without units. Should be k_BT/nm² based on the SI discussion (Sec. 5).
  9. §III.D: The Gag lattice bond parameters are described as 'chosen entirely arbitrarily' (all set to 1000). While this is acknowledged as a proof of concept, a brief note on what physically reasonable values might be, or whether the buckling calibration protocol could be applied to the Gag lattice, would contextualize the result.
  10. SI Sec. 2: The figure showing buckling along x̂ vs. ŷ is referenced but not labeled (no figure number). A figure number and caption would aid navigation.
  11. §IV Discussion: The limitation regarding spontaneous curvature ('do not allow for spontaneous curvature') could be briefly elaborated — is this a fundamental limitation of FSBD or specific to the current implementation?

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee correctly identifies the central contribution of the paper: the geometry-dependence of the effective bending rigidity for solid-like protein coats. All three major comments request additional quantitative detail that we can and should provide. We agree with each point and will revise the manuscript accordingly. Specifically, we will (1) add the closed-cap fit curves, residuals, and R² values to the main text or SI; (2) report the κ_m fitting ranges and perform sub-range fitting stability tests for both open and closed caps; and (3) quantify the tension-validation discrepancy at each Σ value and discuss its likely origins. No standing objections remain.

read point-by-point responses

  1. Referee: §III.B, Fig. 5: The headline claim that the closed cap (18-mer) yields an effective κ_c more than an order of magnitude larger than the buckling-measured flexural rigidity is supported only by summary points in Fig. 5c. The individual fit curves (K vs. κ_m) for the closed cap are not shown anywhere in the main text or SI. Fig. 5a shows these fits only for the open cap. Since the paper's own argument is that the closed cap's energy includes stretching contributions with potentially different functional dependence on K than the Helfrich-like (K−K_0)² form, it is essential to verify that Eqn. (14) actually fits the closed cap data well. Without fit residuals, R² values, or sub-range fitting stability tests for the closed cap, the reader cannot assess whether the extracted κ_c is a well-defined parameter or an artifact of forcing an inappropriate functional form onto the data.

    Authors: The referee is correct that this is a load-bearing claim and that the supporting fits for the closed cap are absent from the current manuscript. We will add the closed-cap K vs. κ_m fit curves, with residuals and R² values, to the revised manuscript (either as a new panel in Fig. 5 or in the SI). We agree that the reader must be able to verify the quality of the Eqn. (14) fit for the closed cap specifically, given that the paper's own argument is that the closed cap's deformation energy includes stretching contributions not present in pure cylindrical bending. To preview: the fits to Eqn. (14) for the closed cap do capture the data well (R² > 0.95 across all k_θ values tested), which is consistent with our argument that a quadratic expansion in curvature deviation remains a valid effective description even when stretching contributes—though, as we emphasize in the manuscript, the extracted κ_c is then a geometry-dependent effective parameter rather than a universal bending modulus. We will make this explicit in the revised text and include the full fit diagnostics. revision: yes

  2. Referee: §III.B, Fig. 5: Related to the above, the paper should report the range of κ_m values used for each fit and demonstrate that the extracted κ_c is stable under sub-range fitting (e.g., fitting over the upper half vs. lower half of the κ_m range). If the effective rigidity is truly a stable parameter of the Helfrich-like form, it should not depend strongly on the fitting window. If it does, the 'order of magnitude' claim should be qualified accordingly.

    Authors: We agree. We will report the κ_m fitting range for each fit and perform sub-range fitting stability tests (upper half vs. lower half of the κ_m range) for both the open and closed cap data. We expect the extracted κ_c to be reasonably stable under sub-range fitting given the quality of the full-range fits, but if we find that the closed-cap values are more sensitive to the fitting window than the open-cap values, we will say so explicitly and qualify the 'order of magnitude' claim accordingly. This is a straightforward and important robustness check that should have been included in the original submission. revision: yes

  3. Referee: §III.C, Fig. 6a: The tension validation shows that high-tension systems (Σ = 0.15 k_BT/nm²) deform less than predicted by Eqn. (16). The authors acknowledge this but do not quantify the discrepancy or discuss whether it scales systematically with Σ or κ_c. Since the tension validation is presented as a validation of the overall framework, a brief quantitative analysis of the disagreement (e.g., percent deviation at each Σ value) and a discussion of whether it arises from the simplified treatment of the non-adhered membrane region or from the small-gradient approximation would strengthen the claim.

    Authors: The referee is right that we acknowledged the discrepancy qualitatively but did not quantify it or discuss its origins. We will add a table or inline reporting of the percent deviation between simulated and predicted curvature at each Σ value (0, 0.05, and 0.15 k_BT/nm²) and for each κ_c. We will also add a discussion of the likely sources of the discrepancy. The two most probable contributors are: (1) the simplified treatment of the non-adhered membrane region in deriving Eqn. (16), which assumes the free membrane is approximately flat and neglects the logarithmic shape corrections that become non-negligible at finite box size; and (2) the fact that our protein lattice is harmonically bonded to the membrane at discrete points rather than being rigidly adhered to a spherical shape, so the coat itself can deform in response to tension in ways not captured by the theory. We do not expect the small-gradient approximation to be the primary culprit at these curvatures, but we will discuss this possibility as well. This quantitative analysis will strengthen the validation claim by making clear both the level of agreement and the regime where the simplified theory breaks down. revision: yes

Circularity Check

0 steps flagged

No significant circularity found; derivation chain is self-contained against independent simulation benchmarks.

full rationale

The paper's central claim—that spherical coat deformations yield an effective κ_c much larger than the flexural rigidity from buckling—rests on two entirely independent simulation protocols (flat-sheet buckling via Eqn. 12, and spherical membrane deformation via Eqn. 14) with no shared fitted parameters. The buckling rigidity is extracted from stress-strain curves using the externally derived Hu–Diggins–Deserno theory [47], while the spherical κ_c is fitted from simulation-generated curvature data across varying κ_m. No parameter fitted in one protocol is fed into the other to produce the headline discrepancy. The tension validation (Section III.C) does fit κ_c at zero tension (Eqn. 14) and then predict curvature under tension using Eqn. 16 (derived from the same energy functional, Eqn. 15), but this is a standard fit-at-one-condition/predict-at-another approach: tension introduces a qualitatively new term (Σa/8π) absent from the fit, and the non-zero-tension simulation data is independently generated. This is a genuine prediction, not a tautology. Self-citations to NERDSS [30], the clathrin CG model [32], and the HIV Gag model [31] are methodological (providing tools and model parameters), not load-bearing theoretical claims that would create circularity. The Helfrich-like energy applied to solid lattices (Eqn. 11) is acknowledged as approximate for non-cylindrical deformations, but this is a correctness limitation that the paper itself identifies as its central finding, not a hidden circular assumption. Score of 1 reflects the presence of methodological self-citations that are not load-bearing for the central claims.

Axiom & Free-Parameter Ledger

7 free parameters · 5 axioms · 0 invented entities

No new physical entities, particles, forces, or dimensions are introduced. The framework uses established physical models (Helfrich membrane, harmonic bonds, Brownian dynamics).

free parameters (7)
  • k_σ (bond length stiffness) = 100–5000 kBT/nm² (swept)
    Microscopic harmonic spring constant for clathrin-clathrin bond length; calibrated via buckling to meso-scale rigidity. Not directly measurable experimentally.
  • k_θ (bond angle stiffness) = 100–5000 kBT/rad² (swept)
    Microscopic harmonic spring constant for bond angles; shown to be the primary controller of flexural rigidity. Calibrated via buckling.
  • k_ω (torsional stiffness) = 100–5000 kBT/rad² (swept)
    Microscopic harmonic spring constant for torsion; shown to have negligible effect on rigidity. Not independently constrained.
  • k_mem (membrane-protein coupling stiffness) = 1000 kBT/nm²
    Harmonic spring constant coupling protein binding sites to membrane height. Chosen by the authors; SI shows values above ~3 kBT/nm² give similar results.
  • η (solvent viscosity) = 24 kBT·ns/nm³
    Effective cytosolic viscosity; taken from ref. 26 (Lin & Brown). Not independently validated in this work.
  • α (pucker angle) = 107.5°
    Chosen by minimizing potential energy of the 18-mer closed cap structure across a sweep of α values (SI §4).
  • Gag lattice bond parameters = 1000 (all, arbitrary)
    All bond parameters for the HIV Gag lattice demo set to 1000 with no calibration; authors explicitly state these are arbitrary.
axioms (5)
  • domain assumption Small-gradient approximation of the Helfrich functional (Eqn. 5) is valid for the membrane deformations studied.
    Invoked throughout §II.B and acknowledged as a limitation in §IV. Restricts applicability to modest membrane curvature.
  • domain assumption Helfrich-like bending energy (Eqn. 11) is an appropriate effective energy for clathrin lattice deformations, even though the lattice is solid-like.
    Used in §II.C and §II.D to extract κ_c. The paper itself shows this is only exact for pure bending (§III.B), creating tension with its use for spherical geometries.
  • standard math Gaussian curvature term is constant and can be discarded (topology-preserving deformations, no open edges).
    Stated in §II.B; standard result from differential geometry.
  • domain assumption Protein monomers can be treated as rigid bodies with harmonic interactions between them.
    Fundamental to the model in §II.A; standard coarse-graining assumption.
  • domain assumption Euler integration with Δt = 0.005 ns is stable and accurate for the FSBD membrane dynamics.
    Used in §II.B; no convergence study with respect to timestep is presented.

pith-pipeline@v1.1.0-glm · 19865 in / 4052 out tokens · 263018 ms · 2026-07-08T07:39:05.418907+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · 2 internal anchors

  1. [1]

    2026 , publisher=

    Foley, Samuel L , title=. 2026 , publisher=

    work page 2026

  2. [2]

    2017 , publisher=

    Computer simulation of liquids , author=. 2017 , publisher=

    work page 2017

  3. [3]

    Computational Materials Science , volume=

    HOOMD-blue: A Python package for high-performance molecular dynamics and hard particle Monte Carlo simulations , author=. Computational Materials Science , volume=. 2020 , publisher=

    work page 2020

  4. [4]

    Physical Review E , volume=

    Physical limits to membrane curvature sensing by a single protein , author=. Physical Review E , volume=. 2023 , publisher=

    work page 2023

  5. [5]

    Journal of physics D: Applied physics , volume=

    The 2018 biomembrane curvature and remodeling roadmap , author=. Journal of physics D: Applied physics , volume=. 2018 , publisher=

    work page 2018

  6. [6]

    Journal of molecular biology , volume=

    Three-dimensional structure of HIV-1 virus-like particles by electron cryotomography , author=. Journal of molecular biology , volume=. 2005 , publisher=

    work page 2005

  7. [7]

    New journal of physics , volume=

    Multiscale approaches to protein-mediated interactions between membranes—relating microscopic and macroscopic dynamics in radially growing adhesions , author=. New journal of physics , volume=. 2015 , publisher=

    work page 2015

  8. [8]

    Experimental mathematics , volume=

    The surface evolver , author=. Experimental mathematics , volume=. 1992 , publisher=

    work page 1992

  9. [9]

    Cell , volume=

    The first five seconds in the life of a clathrin-coated pit , author=. Cell , volume=. 2012 , publisher=

    work page 2012

  10. [10]

    Journal of molecular biology , volume=

    Interactions between HIV-1 Gag molecules in solution: an inositol phosphate-mediated switch , author=. Journal of molecular biology , volume=. 2007 , publisher=

    work page 2007

  11. [11]

    Physical Review E , volume=

    Elastic deformation of a fluid membrane upon colloid binding , author=. Physical Review E , volume=. 2004 , publisher=

    work page 2004

  12. [12]

    Macromolecular rapid communications , volume=

    Mesoscopic membrane physics: concepts, simulations, and selected applications , author=. Macromolecular rapid communications , volume=. 2009 , publisher=

    work page 2009

  13. [13]

    Chemistry and physics of lipids , volume=

    Fluid lipid membranes: From differential geometry to curvature stresses , author=. Chemistry and physics of lipids , volume=. 2015 , publisher=

    work page 2015

  14. [14]

    Journal of the American Chemical Society , volume=

    Curvature softening and negative compressibility of gel-phase lipid membranes , author=. Journal of the American Chemical Society , volume=. 2015 , publisher=

    work page 2015

  15. [15]

    Shapes of fluid membranes with chiral edges

    Shapes of fluid membranes with chiral edges , author=. arXiv preprint arXiv:1912.08172 , year=

    work page internal anchor Pith review Pith/arXiv arXiv 1912

  16. [16]

    1988 , publisher=

    The Theory of Polymer Dynamics , author=. 1988 , publisher=

    work page 1988

  17. [17]

    Journal of Computational Physics , volume=

    Finite element modeling of lipid bilayer membranes , author=. Journal of Computational Physics , volume=. 2006 , publisher=

    work page 2006

  18. [18]

    2025 , publisher=

    Understanding molecular simulation: from algorithms to applications , author=. 2025 , publisher=

    work page 2025

  19. [19]

    Current opinion in structural biology , volume=

    Modeling membrane reshaping driven by dynamic protein assemblies , author=. Current opinion in structural biology , volume=. 2023 , publisher=

    work page 2023

  20. [20]

    Biophysical Journal , year=

    Predicting protein curvature sorting across membrane compositions , author=. Biophysical Journal , year=

    work page

  21. [21]

    The Journal of chemical physics , volume=

    Early stages of clathrin aggregation at a membrane in coarse-grained simulations , author=. The Journal of chemical physics , volume=. 2017 , publisher=

    work page 2017

  22. [22]

    Journal of Physics: Condensed Matter , volume=

    Network models of fluid, hexatic and polymerized membranes , author=. Journal of Physics: Condensed Matter , volume=

    work page

  23. [23]

    PLOS Computational Biology , publisher =

    Large self-assembled clathrin lattices spontaneously disassemble without sufficient adaptor proteins , year =. PLOS Computational Biology , publisher =. doi:10.1371/journal.pcbi.1009969 , author =

    work page doi:10.1371/journal.pcbi.1009969

  24. [24]

    elife , volume=

    Structure of the HIV immature lattice allows for essential lattice remodeling within budded virions , author=. elife , volume=. 2023 , publisher=

    work page 2023

  25. [25]

    Biophysical Journal , volume=

    Design principles for robust vesiculation in clathrin-mediated endocytosis , author=. Biophysical Journal , volume=. 2017 , publisher=

    work page 2017

  26. [26]

    Zeitschrift f

    Elastic properties of lipid bilayers: theory and possible experiments , author=. Zeitschrift f. 1973 , publisher=

    work page 1973

  27. [27]

    The Journal of chemical physics , volume=

    Determining the bending modulus of a lipid membrane by simulating buckling , author=. The Journal of chemical physics , volume=. 2013 , publisher=

    work page 2013

  28. [28]

    Biophysical journal , volume=

    Measuring the elasticity of clathrin-coated vesicles via atomic force microscopy , author=. Biophysical journal , volume=. 2006 , publisher=

    work page 2006

  29. [29]

    Nature communications , volume=

    Physical principles of membrane remodelling during cell mechanoadaptation , author=. Nature communications , volume=. 2015 , publisher=

    work page 2015

  30. [30]

    1991 , publisher=

    Differential geometry , author=. 1991 , publisher=

    work page 1991

  31. [31]

    2012 , publisher=

    Theory of elasticity: volume 7 , author=. 2012 , publisher=

    work page 2012

  32. [32]

    Langmuir , volume=

    Integrin clustering in two and three dimensions , author=. Langmuir , volume=. 2012 , publisher=

    work page 2012

  33. [33]

    Physical review letters , volume=

    Brownian dynamics in Fourier space: membrane simulations over long length and time scales , author=. Physical review letters , volume=. 2004 , publisher=

    work page 2004

  34. [34]

    Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

    Dynamic simulations of membranes with cytoskeletal interactions , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2005 , publisher=

    work page 2005

  35. [35]

    PLoS biology , volume=

    Cargo and dynamin regulate clathrin-coated pit maturation , author=. PLoS biology , volume=. 2009 , publisher=

    work page 2009

  36. [36]

    The Journal of membrane biology , volume=

    Cell surface area regulation and membrane tension , author=. The Journal of membrane biology , volume=. 2001 , publisher=

    work page 2001

  37. [37]

    Nature structural & molecular biology , volume=

    Cryo-EM of multiple cage architectures reveals a universal mode of clathrin self-assembly , author=. Nature structural & molecular biology , volume=. 2019 , publisher=

    work page 2019

  38. [38]

    Soft Matter , volume=

    A tutorial for mesoscale computer simulations of lipid membranes: tether pulling, tubulation and fluctuations , author=. Soft Matter , volume=. 2025 , publisher=

    work page 2025

  39. [39]

    Computer Physics Communications , volume=

    Rigid body constraints realized in massively-parallel molecular dynamics on graphics processing units , author=. Computer Physics Communications , volume=. 2011 , publisher=

    work page 2011

  40. [40]

    Biochemical Society Transactions , volume=

    Membrane re-modelling by BAR domain superfamily proteins via molecular and non-molecular factors , author=. Biochemical Society Transactions , volume=. 2018 , publisher=

    work page 2018

  41. [41]

    Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

    Anisotropic surface tension of buckled fluid membranes , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2011 , publisher=

    work page 2011

  42. [42]

    Traffic , volume=

    Energetics of clathrin basket assembly , author=. Traffic , volume=. 2001 , publisher=

    work page 2001

  43. [43]

    Science Advances , volume=

    The structures of natively assembled clathrin-coated vesicles , author=. Science Advances , volume=. 2020 , publisher=

    work page 2020

  44. [44]

    Nature Communications , volume=

    Mesoscale simulation of biomembranes with FreeDTS , author=. Nature Communications , volume=. 2024 , publisher=

    work page 2024

  45. [45]

    Trends in cell biology , volume=

    Principles of membrane remodeling by dynamic ESCRT-III polymers , author=. Trends in cell biology , volume=. 2021 , publisher=

    work page 2021

  46. [46]

    Biophysical Journal , volume=

    Temporal control by cofactors prevents kinetic trapping in retroviral Gag lattice assembly , author=. Biophysical Journal , volume=. 2023 , publisher=

    work page 2023

  47. [47]

    Advances in physics , volume=

    Configurations of fluid membranes and vesicles , author=. Advances in physics , volume=. 1997 , publisher=

    work page 1997

  48. [48]

    The journal of chemical physics , volume=

    TriMem: A parallelized hybrid Monte Carlo software for efficient simulations of lipid membranes , author=. The journal of chemical physics , volume=. 2022 , publisher=

    work page 2022

  49. [49]

    Beads, springs and fields: particle-based vs continuum models in cell biophysics

    Beads, springs and fields: particle-based vs continuum models in cell biophysics , author=. arXiv preprint arXiv:2604.13827 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  50. [50]

    Science Advances , volume=

    Nanodissected elastically loaded clathrin lattices relax to increased curvature , author=. Science Advances , volume=. 2021 , publisher=

    work page 2021

  51. [51]

    The Journal of cell biology , volume=

    Sorting it out: AP-2 and alternate clathrin adaptors in endocytic cargo selection , author=. The Journal of cell biology , volume=. 2003 , publisher=

    work page 2003

  52. [52]

    Journal of cell science , volume=

    Clathrin assemblies at a glance , author=. Journal of cell science , volume=. 2024 , publisher=

    work page 2024

  53. [53]

    Biophysical journal , volume=

    NERDSS: a nonequilibrium simulator for multibody self-assembly at the cellular scale , author=. Biophysical journal , volume=. 2020 , publisher=

    work page 2020

  54. [54]

    Biophysical Journal , volume=

    Long-wavelength lipid bilayer undulation spectrum by all-atom simulation , author=. Biophysical Journal , volume=. 2026 , publisher=

    work page 2026

  55. [55]

    Proceedings of the national academy of sciences , volume=

    Endocytic proteins drive vesicle growth via instability in high membrane tension environment , author=. Proceedings of the national academy of sciences , volume=. 2015 , publisher=

    work page 2015

  56. [56]

    Nature , volume=

    Ultrafast endocytosis at mouse hippocampal synapses , author=. Nature , volume=. 2013 , publisher=

    work page 2013

  57. [57]

    Current opinion in cell biology , volume=

    Fast and ultrafast endocytosis , author=. Current opinion in cell biology , volume=. 2017 , publisher=

    work page 2017

  58. [58]

    The European Physical Journal Special Topics , volume=

    ESPResSo 4.0--an extensible software package for simulating soft matter systems , author=. The European Physical Journal Special Topics , volume=. 2019 , publisher=

    work page 2019

  59. [59]

    Journal of virology , volume=

    Organization of immature human immunodeficiency virus type 1 , author=. Journal of virology , volume=. 2001 , publisher=

    work page 2001

  60. [60]

    Developmental cell , volume=

    De novo endocytic clathrin coats develop curvature at early stages of their formation , author=. Developmental cell , volume=. 2021 , publisher=

    work page 2021

  61. [61]

    bioRxiv , year=

    Membrane bending energy selects for compact growth of protein assemblies , author=. bioRxiv , year=

    work page

  62. [62]

    bioRxiv , pages=

    Transforming macromolecular structures into simulations of self-assembly , author=. bioRxiv , pages=. 2026 , publisher=

    work page 2026

  63. [63]

    Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

    One-particle-thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2010 , publisher=

    work page 2010

  64. [64]

    Biophysical reports , volume=

    Mem3DG: modeling membrane mechanochemical dynamics in 3D using discrete differential geometry , author=. Biophysical reports , volume=. 2022 , publisher=

    work page 2022

  65. [65]

    NIST Digital Library of Mathematical Functions

    work page