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REVIEW 3 major objections 5 minor 114 references

Hydrodynamic coupling for particle-based solvent-free membrane models

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that anisotropic Langevin dynamics with Stokes-derived friction and diffusion tensors gives solvent-free particle-based membrane models the same large-scale undulation kinetics as continuum hydrodynamic theory.

desk verdict A useful, well-derived hydrodynamics scheme for solvent-free membranes, with a real but addressable compressibility caveat. read the letter →

arxiv 1909.02722 v3 pith:NBRFLGKH submitted 2019-09-06 physics.comp-ph physics.bio-phphysics.flu-dyn

Mohsen Sadeghi , Frank Noé This is my paper · ORCID
classification physics.comp-phphysics.bio-phphysics.flu-dyn
keywords solvent-freemembranemodelshydrodynamicinteractionsanisotropicLangevindynamicsStokesequationsundulationsdispersionrelationscoarse-grainedsimulationpairwisediffusiontensors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tackles a known weakness of solvent-free coarse-grained membrane simulations: they can reproduce equilibrium membrane shapes but typically have unrealistic kinetics because the surrounding fluid is absent. It proposes to keep the model solvent-free and instead encode the fluid into anisotropic Langevin dynamics, with pairwise friction and diffusion tensors computed from Stokes-flow solutions for idealized membrane geometries. The claim is that this reproduces the wavelength-dependent relaxation rates (dispersion relations) of membrane undulations predicted by continuum elastic-membrane theory, for free-standing patches and for patches near a solid wall. If that is right, particle-based models gain the hydrodynamic kinetics needed for large-scale, millisecond-timescale studies of processes such as membrane remodelling, without the cost of explicit solvent or grid-based flow solvers.

What carries the argument

The load-bearing object is the anisotropic diffusion tensor of Eq. (4) inside the overdamped Langevin update of Eq. (2). Its normal-only pair part means solvent-mediated forces between distinct particles act along membrane normals, which keeps the tensor assembly cheap and makes the divergence terms vanish for flat membranes. The out-of-plane response is computed from Stokes equations with a Gaussian boundary condition of width $\alpha$ (Eq. (9)); Hankel transforms turn that boundary condition into stress or velocity response fields (Eqs. (19) and (20)), and numerical integration over hexagonal-lattice particle patches yields the friction and diffusion coefficients of Eq. (28). The derivation enforces a no-slip, incompressible membrane at the boundary, which separates these kernels from plain point-force (Oseen) tensors and leads to a far-field pair diffusion of roughly $kT/(4\pi\eta r_{ij})$—half the $zz$ component of the Oseen tensor. In-plane mobility is appended separately via the standard diffusion formula for inclusions in a thin fluid sheet.

What would settle it

Repeat the same membrane model with explicit solvent or a grid-based hydrodynamic solver and compare the undulation dispersion relations, especially at short wavelengths and small particle separations; a systematic mismatch there would show where the normal-only pairwise tensor misses in-plane or many-body coupling. A second check is to use a much softer membrane, since the derivation's incompressible-boundary assumption should break down more visibly at high q.

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Extended reading notes

Core claim

The central discovery is that the solvent's effect can be folded into an anisotropic diffusion tensor: a self term $D_{ii} = D^{\parallel}_i I + (D^{\perp}_{ii} - D^{\parallel}_i) n_i n_i$ and a pair term $D_{ij} = D^{\perp}_{ij} n_i n_j$ for $i \neq j$, where $n_i$ is the local membrane normal. The out-of-plane coefficients $D^{\perp}_{ij}$ are obtained by solving the Stokes equations with Gaussian velocity or stress patches applied on the membrane surface, for three geometries—a single planar membrane, parallel planar membranes, and a spherical vesicle—and integrating the resulting response over each particle's area. The in-plane mobility uses the standard formula for diffusion of a cylindrical inclusion in a fluid sheet. In overdamped Langevin simulations of a $0.5\,\mu\mathrm{m}$ membrane patch, the equilibrium undulation spectrum follows the continuum relation $\langle h_q h_q^* \rangle/L^2 = kT/(\kappa (qL)^4)$, and the two relaxation branches—the slow inter-leaflet slipping mode and the fast hydrodynamic mode—match the continuum dispersion relation given by Eq. (34). For a membrane near a wall, inverting the friction matrix from the parallel-membrane Stokes solution reproduces the continuum bound-membrane dispersion relation of Eq. (36), including an additional slow timescale when the wall is close.

Load-bearing premise

The load-bearing premise is that each pair of particles feels an independent solvent push along the local membrane normal, spread over a Gaussian patch, while the membrane itself is treated as a no-slip, incompressible sheet; in-plane and many-body hydrodynamic effects are left out.

Editorial extensions

If this is right

  • Solvent-free particle-based membrane models can run with a $0.5\,\mathrm{ns}$ timestep while preserving realistic undulation kinetics, because the hydrodynamic coupling is only a pair-list update rather than a solvent or grid solve.
  • The continuum free-standing dispersion relation is reproduced without fitting the solvent response; the only adjusted parameter is the inter-leaflet friction, which mostly controls the slow slipping mode.
  • Wall-induced hydrodynamic slowdown emerges from the same tensors: taking the inverse of the friction matrix from the parallel-membrane solution reproduces the bound-membrane dispersion relation, with a clear separation of timescales for wall distances below about $20\,\mathrm{nm}$.
  • The range of hydrodynamic interactions controls non-equilibrium kinetics: longer cut-offs make the largest undulation mode relax faster toward equilibrium, indicating that long-range coupling sets large-scale timescales.
  • Because the spherical-vesicle solutions converge to the planar results even for small vesicle radii, the planar diffusion tensors remain usable for moderately curved membranes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the pair coupling is restricted to normal-normal components, in-plane hydrodynamic interactions—shear through the membrane and through the solvent—remain outside the model; phase separation or protein aggregation dynamics would need a generalized tensor.
  • Editorial inference: the Stokes derivation assumes an incompressible, no-slip membrane while the simulated model is compressible; the close match in the tested wavelength range suggests the correction is small there, but softer or highly compressed membranes could show a deviation.
  • Editorial inference: the length scale $\alpha$ is a free parameter (the effective particle size for the fluid); since results are stable when $\alpha$ changes by a factor of ten, one could calibrate $\alpha$ from a single explicit-solvent simulation and transfer it across resolutions, which the paper does not itself demonstrate.
  • Editorial inference: the same tensors provide a way to test the paper's scaling argument that out-of-plane solvent dissipation grows as $\eta R^3/\kappa$ and eventually dominates budding kinetics; driving a membrane patch with time-dependent curvature forces and comparing relaxation rates would expose that crossover.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a framework for coupling solvent-free particle-based membrane models to solvent hydrodynamics. The authors propose anisotropic Langevin dynamics with pairwise friction/diffusion tensors whose out-of-plane components are obtained from solutions of the Stokes equations under localized Gaussian velocity or stress boundary conditions on the membrane. Closed-form and semi-analytical results are derived for a single planar membrane, a pair of parallel planar membranes (also modeling a membrane near a wall), and a spherical vesicle. The tensors are then used in Langevin dynamics simulations of a previously developed particle-based membrane model, and the dispersion relations of undulation modes are compared with Seifert-type continuum predictions for free-standing membranes and membranes near a wall. The fast hydrodynamic mode agrees well with the continuum model, the equilibrium undulation spectrum is correct, and the near-wall results reproduce the expected wall-distance dependence. The slow slipping mode is matched by choosing the inter-leaflet friction coefficient b as a fitting parameter.

Significance. If the framework holds, it is a practically valuable contribution to multiscale membrane simulation: it offers a way to reintroduce long-range hydrodynamic interactions into solvent-free particle-based models without the cost of explicit solvent or grid-based solvers. The derivation from Stokes equations is transparent, and the agreement of the fast hydrodynamic mode with the Seifert continuum model is a nontrivial validation, since that mode is not tuned by any fitted parameter. The paper also demonstrates correct equilibrium sampling (the undulation power spectrum follows Eq. (33a)) and shows systematic improvement with increased hydrodynamic cutoff radius. These strengths make the method useful for the target community. The main weakness is the mismatch between the incompressible-membrane boundary condition used in deriving the diffusion tensors and the compressible nature of the simulated membrane and of the reference continuum model; this gap needs to be addressed before the central claim is fully supported.

major comments (3)
  1. [Sec. IV (after Eq. (15)) and Sec. VI A (Eq. (34), Table I)] The diffusion tensors used in the Langevin equation are derived under the incompressible membrane boundary condition iq w_parallel(q) = 0 at z = 0 (paragraph after Eq. (15)). However, the simulated membrane has a finite area compressibility modulus K_area = 0.270 N/m (Table I), and the reference continuum model (Eq. (34)) explicitly includes a density field rho_q and a slow slipping mode; the simulated membrane can therefore sustain in-plane velocity and density fluctuations. The agreement with Eq. (34) in Figs. 4 and 5 is thus not a direct validation of the derived mobility for the system being simulated; it may rely on the fitted inter-leaflet friction b and on the insensitivity of the fast mode to the in-plane boundary condition. Please address this by (i) deriving the normal-normal mobility under a compressible boundary condition and showing that the difference is negligible at the probed wavevectors, or (ii) demonstrating numerically that the in-plane membrane velocity in the simulated model is small enough that the incompressible condition is a good approximation for the relevant modes. Without such a check, the claim that the dispersion relations follow from first-principle hydrodynamics is not fully established.
  2. [Sec. VI A, paragraph beginning 'In using Eq. (34)...'] The inter-leaflet friction coefficient b is chosen as b = 10^6 N s/m^3 to give a good match between the theoretical frequencies and simulation results, and the text acknowledges that b mostly affects the slipping mode. Consequently, the filled-symbol data in Fig. 4(a,b) constitute a fit of the slow mode rather than a parameter-free prediction. The conclusion in Sec. VII states that the framework predicts 'realistic timescales' relying solely on properties such as bending modulus and solvent viscosity, 'removing the need for ad hoc corrections after the fact.' This statement is too strong for the slow mode. Please either provide an independent estimate of b (e.g., from the model's inter-leaflet friction or from density relaxation measurements) or explicitly scope the parameter-free claim to the fast hydrodynamic mode.
  3. [Sec. VI C and Fig. 5] For the near-wall case, the authors fit triple exponentials to the undulation autocorrelation and compare the dominant mode with Eq. (36), but the two additional fitted timescales are not validated against any continuum model. Since the membrane is compressible and the two leaflets experience different hydrodynamic environments, these modes likely arise from density and slipping dynamics, but no reference description is provided. Please either supply a continuum counterpart for these modes (for instance, an extension of Eq. (34) with wall boundary conditions) or state explicitly that the additional slow modes are phenomenological and not yet validated. This is important because the presence of a third slow timescale is presented as a qualitative result in the text.
minor comments (5)
  1. [Secs. III and VI] The paper does not state whether the divergence terms ∇_j · D_ij in Eq. (2) (given in Eq. (5)) were computed and included in the reported simulations. For flat membranes they vanish, but for the near-wall simulations, small membrane curvature may make them nonzero. Please clarify whether these terms were included or neglected.
  2. [Sec. VI A, Fig. 4] The dispersion relations obtained from biexponential fits are presented without error bars or any measure of fit uncertainty. Given that the slow mode may have a small amplitude at some wavevectors, please report confidence intervals or at least the number of independent samples used for the fits.
  3. [Eq. (31)] The notation 'lim_{rij→∞, rij ≫ rc}' is ambiguous. Please define r_c and state the precise limiting procedure used to obtain the far-field approximation D⊥_ij ≈ kT/(4πηr_ij).
  4. [References] Several references contain formatting errors, e.g., Ref. [81] 'H IROMI Y AMAKAWA', Ref. [87] 'T ransform. Appl. Handb.', and Ref. [91] 'GNU Scientific Library Reference, Technical Report July'. Please unify and correct the reference list.
  5. [Sec. VII] The statement 'We believe this to be the only viable means to reliably investigate complex, membrane-involved biological processes close to their native time-scales' is too categorical for the evidence presented. Please soften it to reflect that the approach is one promising route.

Circularity Check

1 steps flagged · score 4.0 of 10

Slow-mode comparison fits inter-leaflet friction b to simulation; fast-mode and near-wall checks are independent.

  1. fitted input called prediction [Section VI A, paragraph following Eq. (34); Fig. 4]
    "The only remaining parameter is the inter-leaflet friction, b. ... Here, we have found a value of b = 10^6 N s m−3 to give a good match between the theoretical frequencies and simulation results (Fig. 4). However, in our range of inspection, the inter-leaflet friction mostly affects the slipping mode and has little effect on the hydrodynamic mode, which is the focus of our investigation."

    The inter-leaflet friction b is the only free constant in the Seifert reference model used to produce the theoretical dispersion curves omega1 and omega2. The paper states b was chosen ('we have found a value') to make those curves match the simulation results. Hence the claimed agreement of the simulated slow 'slipping' mode with omega1 is not an independent prediction: the reference curve was tuned to the data being compared. The fast hydrodynamic mode is explicitly insensitive to b, and the near-wall comparison (Eq. 36) contains no fitted b, so the framework retains independent validation; the circularity is confined to the slow-mode check.

full rationale

The central derivation of the diffusion/friction tensors is self-contained: D_ij is obtained by solving Stokes equations with Gaussian velocity/stress boundary conditions (Sec. IV) and integrating the resulting fields (Eq. 28); the target dispersion relations are not used as inputs, and the fast-mode dispersion matches Seifert's independent continuum model (Eq. 34) without fitting. The near-wall results also compare against the parameter-free Seifert expression (Eq. 36). Self-citations [25,58] identify the particle membrane model used as a testbed; they do not supply the hydrodynamic result. The one genuinely circular comparison is the slow 'slipping' mode: the inter-leaflet friction b in the reference model is explicitly chosen to match the simulation frequencies, so that mode's agreement is by construction and cannot validate the method. The paper acknowledges this by stating that b mostly affects the slipping mode and that the hydrodynamic mode is the focus. The incompressible boundary condition (v_parallel=0) used in the Stokes solution versus the finite K_area of the simulated membrane is a modeling-assumption concern, not circularity. Overall, partial circularity is confined to a secondary benchmark comparison; the core claim retains independent content.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the Stokes-flow response functions and the anisotropic diffusion tensor ansatz, plus the model parameters alpha and b. alpha is a hand-chosen length-scale; b is fitted to the data. No new physical entities are introduced. The membrane force field parameters (Table II) are carried over from prior work [25] and are treated as given inputs, not fitted here.

free parameters (2)
  • Hydrodynamic width alpha = alpha = 0.01a and 0.1a (and 0.5a for tensor calculations)
    Sets the width of the Gaussian velocity/stress boundary condition used to represent each particle in the fluid domain (Eq. 9). It is a hand-chosen model parameter; the simulated dispersion relations, especially the slow mode, depend on it (Sec. VI A).
  • Inter-leaflet friction coefficient b = 10^6 N s m^-3
    Fitted to match the simulated dispersion relations against the continuum model of Seifert et al. (Eq. 34). The paper states b=10^6 N s m^-3 'gives a good match' (Sec. VI A). This parameter mostly affects the slow slipping mode, not the fast hydrodynamic mode.
assumptions (6)
  • domain assumption Stokes equations (Eq. 7) govern the solvent flow
    Assumes incompressible Newtonian fluid at zero Reynolds number, standard for nanoscale membrane hydrodynamics.
  • ad hoc to paper No-slip and membrane incompressibility at the fluid interface (v_parallel = 0 at z = 0)
    Used to derive the single-membrane response (Eqs. 15-20). The simulated membrane is compressible with finite area compressibility, so this boundary condition is an approximation.
  • ad hoc to paper Pairwise-additive, normal-only hydrodynamic interactions (Eq. 4)
    The diffusion tensor for distinct particles has only n_i n_j components; in-plane hydrodynamics and many-body effects are neglected. Justified for small out-of-plane displacements, but load-bearing for the simulation method.
  • domain assumption Decoupling of in-plane and out-of-plane dynamics
    Cross-terms in the diffusion tensor are zero; the paper states this is justified for small displacements (Sec. III).
  • domain assumption For parallel membranes, the opposing surface is stationary (v_z = v_parallel = 0 at z = h)
    Modeling a rigid wall or held membrane; enters the boundary conditions in Eq. (21).
  • ad hoc to paper For the spherical vesicle, axisymmetric flow and frozen in-plane motion (v_theta = 0)
    Stronger than the planar incompressibility condition; required for the stream-function solution (Sec. IV C).

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Cite this review

Pith. "Pith review of Hydrodynamic coupling for particle-based solvent-free membrane models." pith.science (2026). https://pith.science/paper/NBRFLGKH

@misc{pith2026190902722,
  author       = {Pith},
  title        = {Pith review of: Hydrodynamic coupling for particle-based solvent-free membrane models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBRFLGKH}},
  note         = {Machine review of arXiv:1909.02722}
}
read the original abstract

The great challenge with biological membrane systems is the wide range of scales involved, from nanometers and picoseconds for individual lipids, to the micrometers and beyond millisecond for cellular signalling processes. While solvent-free coarse-grained membrane models are convenient for large-scale simulations, and promising to provide insight into slow processes involving membranes, these models usually have unrealistic kinetics. One major obstacle is the lack of an equally convenient way of introducing hydrodynamic coupling without significantly increasing the computational cost of the model. To address this, we introduce a framework based on anisotropic Langevin dynamics, for which major in-plane and out-of-plane hydrodynamic effects are modeled via friction and diffusion tensors from analytical or semi-analytical solutions to Stokes hydrodynamic equations. Using this framework, we obtain accurate dispersion relations for planar membrane patches, both free-standing and in the vicinity of a wall. We also briefly discuss how non-equilibrium dynamics is affected by hydrodynamic interactions.

Figures

Figures reproduced from arXiv: 1909.02722 by the authors.

Figure 1
Figure 1. Schematic of the introduced framework for hydrodynamic coupling. (a) Components of a comprehensive description of hydrodynamic effects related to the membrane and the surrounding solvent. Distinction is made between the mobility of particles parallel to the membrane (in-plane) and perpendicular to it (out-of-plane), as well as how these mobilities are potentially coupled via hydrodynamic interactions (HI) of solvent… view at source ↗
Figure 2
Figure 2. Stress/velocity distribution on the surface of the membrane in response to a Gaussian velocity/stress boundary conditions (Eq. (9)) for membranes suspended in a solvent with the viscosity η. Results are given for (a) a single planar membrane, (b) a pair of parallel planar membranes with the given separations, h, and (c) spherical vesicles of given radii. For parallel membranes, the boundary conditions are applied on… view at source ↗
Figure 3
Figure 3. Compilation of numerical values of the out-of-plane components of (a, c) friction and (b, d) diffusion tensors, as a function of pairwise particle distances, rij , calculated on a hexagonal assembly of particles with the lattice parameter a. Results are given for a single planar membrane, two sets of parallel membranes with the given separation, and two spherical vesicles with the given radii. The pairwise distances… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Kinetics of a free-standing planar membrane patch of lateral size L = 0.5 µm, suspended in water, modeled with the particle-based membrane model with the lattice parameter of a = 10 nm. (a) Dispersion relations when different cut-off radii are used in the treatment of …
Figure 5
Figure 5. Figure 5: Dispersion relations for planar membrane patches in the vicinity of a wall. The three relaxation frequencies are denoted by empty, color-filled, and gray-filled symbols. Results are shown for different mean distances between the membrane and the wall. Colored dashed li…

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Works this paper leans on

114 extracted references · 80 canonical work pages

  1. [1]

    A LBERTS , A

    B. A LBERTS , A. J OHNSON , J. LEWIS , D. MORGAN , M. R AFF, K. ROBERTS , and P. WALTER, 21 Molecular Biology of the Cell , Garland Science, Taylor & Francis Group, LLC, New Y ork, 6 edition, 2015

    2015

  2. [2]

    K AKSONEN and A

    M. K AKSONEN and A. R OUX, Nat. Rev. Mol. Cell Biol. 19, 313 (2018)

    2018

  3. [3]

    D IMOU , K

    E. D IMOU , K. COSENTINO , E. PLATONOVA, U. ROS, M. SADEGHI , P. KASHYAP , T. KATSINE - LOS, S. W EGEHINGEL , F. N O ´E, A. J. G ARC´IA-S ´AEZ, H. E WERS , and W. N ICKEL , J. Cell Biol. 218, 683 (2019)

    2019

  4. [4]

    B ROCHARD and J

    F. B ROCHARD and J. L ENNON , J. Phys. 36, 1035 (1975)

    1975

  5. [5]

    P ROST , J.-B

    J. P ROST , J.-B. M ANNEVILLE , and R. B RUINSMA , Eur. Phys. J. B 1, 465 (1998)

    1998

  6. [6]

    S EIFERT , Adv

    U. S EIFERT , Adv. Phys. 46, 13 (1997)

    1997

  7. [7]

    J. C. S HILLCOCK and R. L IPOWSKY , J. Phys. Condens. Matter 18, S1191 (2006)

    2006

  8. [8]

    S. J. M ARRINK , A. H. DE VRIES , and D. P. TIELEMAN , Biochim. Biophys. Acta - Biomembr. 1788, 149 (2009)

    2009

Show all 114 references
  1. [9]

    D ESERNO , Macromol

    M. D ESERNO , Macromol. Rapid Commun. 30, 752 (2009)

    2009

  2. [10]

    N OGUCHI , J

    H. N OGUCHI , J. Phys. Soc. Japan 78, 041007 (2009)

    2009

  3. [11]

    L IPOWSKY , Understanding Membranes and Vesicles: A Personal Recollection of the Last Two Decades, in Phys

    R. L IPOWSKY , Understanding Membranes and Vesicles: A Personal Recollection of the Last Two Decades, in Phys. Biol. Membr. , pp. 3–44, Springer International Publishing, Cham, 2018

    2018

  4. [12]

    S. J. M ARRINK , V. CORRADI , P. C. SOUZA , H. I. I NG ´OLFSSON , D. P. TIELEMAN , and M. S. SANSOM , Chem. Rev. 119, 6184 (2019)

    2019

  5. [13]

    F RIEDMAN , S

    R. F RIEDMAN , S. K HALID , C. A PONTE -SANTAMAR ´IA, E. A RUTYUNOVA , M. B ECKER , K. J. BOYD, M. CHRISTENSEN , J. T. COIMBRA , S. CONCILIO , C. DADAY, F. J. VAN EERDEN , P. A. FERNANDES , F. G R ¨ATER , D. H AKOBYAN , A. H EUER , K. K ARATHANOU , F. K ELLER , M. J. LEMIEUX ,...

    2018

  6. [14]

    O. H. O LLILA and G. PABST , Biochim. Biophys. Acta - Biomembr. 1858, 2512 (2016)

    2016

  7. [15]

    P OGER , B

    D. P OGER , B. C ARON , and A. E. M ARK, Biochim. Biophys. Acta - Biomembr. 1858, 1556 (2016)

    2016

  8. [16]

    K. A. M CKIERNAN , L.-P. W ANG, and V. S. P ANDE , J. Chem. Theory Comput. 12, 5960 (2016). 22

    2016

  9. [17]

    S. J. M ARRINK , H. J. R ISSELADA , S. Y EFIMOV , D. P. T IELEMAN , and A. H. D E VRIES , J. Phys. Chem. B 111, 7812 (2007)

    2007

  10. [18]

    S. J. M ARRINK and D. P. TIELEMAN , Chem. Soc. Rev. 42, 6801 (2013)

    2013

  11. [19]

    D ESERNO , K

    M. D ESERNO , K. K REMER , H. P AULSEN , C. P ETER , and F. SCHMID , Computational Stud- ies of Biomembrane Systems: Theoretical Considerations, Simulation Models, and Appli- cations, pp. 237–283, Springer International Publishing, Cham, 2014

    2014

  12. [20]

    A RNAREZ , J

    C. A RNAREZ , J. J. U USITALO , M. F. M ASMAN , H. I. I NG ´OLFSSON , D. H. D E JONG, M. N. MELO, X. P ERIOLE , A. H. D E VRIES , and S. J. M ARRINK , J. Chem. Theory Comput. 11, 260 (2015)

    2015

  13. [21]

    L ARADJI , P

    M. L ARADJI , P. B. S. K UMAR , and E. J. S PANGLER , J. Phys. D. Appl. Phys. 49, 293001 (2016)

    2016

  14. [22]

    G. S. A YTON , E. LYMAN , V. KRISHNA , R. D. S WENSON , C. M IM, V. M. U NGER , and G. A. VOTH, Biophys. J. 97, 1616 (2009)

    2009

  15. [23]

    D AVTYAN, M

    A. D AVTYAN, M. S IMUNOVIC , and G. A. V OTH, J. Chem. Phys. 147, 044101 (2017)

    2017

  16. [24]

    F ENG, Y

    S. F ENG, Y. HU, and H. L IANG , J. Chem. Phys. 148, 164705 (2018)

    2018

  17. [25]

    S ADEGHI , T

    M. S ADEGHI , T. R. W EIKL , and F. NO ´E, J. Chem. Phys. 148, 044901 (2018)

    2018

  18. [26]

    H AUCKE , E

    V. H AUCKE , E. N EHER , and S. J. S IGRIST , Nat. Rev. Neurosci. 12, 127 (2011)

    2011

  19. [27]

    J. K. N OEL, F. NO ´E, O. DAUMKE , and A. S. M IKHAILOV , Biophys. J. 117, 1870 (2019)

    2019

  20. [28]

    D AUMKE and G

    O. D AUMKE and G. J. P RAEFCKE , Biopolymers 105, 580 (2016)

    2016

  21. [29]

    A NTONNY , C

    B. A NTONNY , C. BURD, P. DE CAMILLI , E. CHEN, O. DAUMKE , K. FAELBER , M. FORD, V. A. FROLOV , A. F ROST , J. E. H INSHAW , M. M. K OZLOV , M. L ENZ, H. H. L OW, H. M CMAHON , C. M ERRIFIELD , T. D. POLLARD , and P. J. ROBINSON , EMBO J. 35, 2270 (2016)

    2016

  22. [30]

    M. G. S AUNDERS and G. A. V OTH, Annu. Rev. Biophys. 42, 73 (2013)

    2013

  23. [31]

    H OFFMANN , C

    M. H OFFMANN , C. F R ¨OHNER , and F. NO ´E, PLoS Comput. Biol. 15, e1006830 (2019)

    2019

  24. [32]

    S CH ¨ONEBERG , M

    J. S CH ¨ONEBERG , M. L EHMANN , A. U LLRICH , Y. P OSOR , W.-T. L O, G. L ICHTNER , J. S CHMORANZER , V. HAUCKE , and F. NO ´E, Nat. Commun. (2017)

    2017

  25. [33]

    F R ¨OHNER and F

    C. F R ¨OHNER and F. NO ´E, J. Phys. Chem. B , acs.jpcb.8b06981 (2018)

    2018

  26. [34]

    B IEDERMANN , A

    J. B IEDERMANN , A. U LLRICH , J. S CH ¨ONEBERG , and F. NO ´E, Biophys. J. 108, 457 (2015)

    2015

  27. [35]

    V IJAYKUMAR , P

    A. V IJAYKUMAR , P. G. BOLHUIS , P. REIN, A. V IJAYKUMAR , P. G. BOLHUIS , and P. REIN, J. Chem. Phys. 214102, 0 (2015)

    2015

  28. [36]

    G UNKEL , J

    M. G UNKEL , J. S CH ¨ONEBERG , W. A LKHALDI , S. I RSEN , F. NO ´E, U. B. K AUPP, and A. A L- 23 AMOUDI , Structure 23, 628 (2015)

    2015

  29. [37]

    S CH ¨ONEBERG , A

    J. S CH ¨ONEBERG , A. U LLRICH , and F. NO ´E, BMC Biophys. 7, 11 (2014)

    2014

  30. [38]

    S CH ¨ONEBERG , M

    J. S CH ¨ONEBERG , M. H ECK, K. P. HOFMANN , and F. NO ´E, Biophys. J. 107, 1042 (2014)

    2014

  31. [39]

    U LLRICH , M

    A. U LLRICH , M. A. B ¨OHME , J. S CH ¨ONEBERG , H. D EPNER , S. J. S IGRIST , and F. N O ´E, PLoS Comput. Biol. 11, e1004407 (2015)

    2015

  32. [40]

    T URLIER and T

    H. T URLIER and T. BETZ, Fluctuations in Active Membranes, in Phys. Biol. Membr., edited by P. BASSEREAU and P. SENS, pp. 581–619, Springer, Cham, 2018

    2018

  33. [41]

    B ETZ, M

    T. B ETZ, M. L ENZ, J.-F. J OANNY , and C. SYKES , Proc. Natl. Acad. Sci. 106, 15320 (2009)

    2009

  34. [42]

    Q IAN, Annu

    H. Q IAN, Annu. Rev. Phys. Chem. 58, 113 (2007)

    2007

  35. [43]

    R. B. G ENNIS , Biomembranes, Springer Advanced Texts in Chemistry, Springer New Y ork, New Y ork, NY , 1989

    1989

  36. [44]

    F. L. B ROWN, Q. Rev. Biophys. 44, 391 (2011)

    2011

  37. [45]

    J. M. D ROUFFE , A. C. M AGGS , and S. L EIBLER , Science 254, 1353 (1991)

    1991

  38. [46]

    I. R. C OOKE , K. K REMER , and M. D ESERNO , Phys. Rev. E 72, 011506 (2005)

    2005

  39. [47]

    Z. J. W ANG and D. FRENKEL , J. Chem. Phys. 122, 234711 (2005)

    2005

  40. [48]

    F RITZ , K

    D. F RITZ , K. KOSCHKE , V. A. HARMANDARIS , N. F. VAN DER VEGT, and K. KREMER , Phys. Chem. Chem. Phys. 13, 10412 (2011)

    2011

  41. [49]

    G. S. A YTON , J. L. M CWHIRTER , and G. A. V OTH, J. Chem. Phys. 124, 64906 (2006)

    2006

  42. [50]

    S. A. S HKULIPA , W. K. DEN OTTER , and W. J. BRIELS , Phys. Rev. Lett. 96, 178302 (2006)

    2006

  43. [51]

    M. J. H UANG , R. K APRAL , A. S. M IKHAILOV , and H. Y. CHEN, J. Chem. Phys. 137, 055101 (2012)

    2012

  44. [52]

    Z GORSKI and E

    A. Z GORSKI and E. LYMAN , Biophys. J. 111, 2689 (2016)

    2016

  45. [53]

    S TREET and U

    H. S TREET and U. K INGDOM , arXiv:cond-mat , 0607382 (2006)

    2006

  46. [54]

    B OTAN, V

    V. B OTAN, V. D. U STACH, K. L EONHARD , and R. F ALLER , J. Phys. Chem. B 121, 10394 (2017)

    2017

  47. [55]

    P. J. A TZBERGER , P. R. KRAMER , and C. S. P ESKIN , J. Comput. Phys. 224, 1255 (2007)

    2007

  48. [56]

    R. M. V ENABLE , H. I. I NG ´OLFSSON , M. G. L ERNER , B. S. P ERRIN , B. A. C AMLEY , S. J. MARRINK , F. L. B ROWN, and R. W. PASTOR , J. Phys. Chem. B 121, 3443 (2017)

    2017

  49. [57]

    V ¨OGELE , J

    M. V ¨OGELE , J. K ¨OFINGER , and G. H UMMER , Phys. Rev. Lett. 120, 268104 (2018)

    2018

  50. [58]

    S ADEGHI and F

    M. S ADEGHI and F. NO ´E, Nat. Commun. 11, 2951 (2020)

    2020

  51. [59]

    Z. Y UE, C. L I, G. A. V OTH, and J. M. S WANSON , J. Am. Chem. Soc. 141, 13421 (2019). 24

    2019

  52. [60]

    S CHLAICH , J

    A. S CHLAICH , J. K APPLER , and R. R. N ETZ, Nano Lett. 17, 5969 (2017)

    2017

  53. [61]

    H. L I, H. Y. CHANG , J. YANG, L. L U, Y. H. TANG, and G. LYKOTRAFITIS , Appl. Math. Mech. (English Ed. 39, 3 (2018)

    2018

  54. [62]

    N OGUCHI and G

    H. N OGUCHI and G. GOMPPER , Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 72 (2005)

    2005

  55. [63]

    P. G. S AFFMAN , M. D ELBRUCK , and M. D ELBR ¨UCK, Proc Natl Acad Sci USA 72, 3111 (1975)

    1975

  56. [64]

    P. G. S AFFMAN , J. Fluid Mech. 73, 593 (1976)

    1976

  57. [65]

    P ANZUELA and R

    S. P ANZUELA and R. D ELGADO -BUSCALIONI , Phys. Rev. Lett. 121, 048101 (2018)

    2018

  58. [66]

    B. A. C AMLEY , C. E SPOSITO , T. B AUMGART , and F. L. H. B ROWN, Biophys. J. 99, L44 (2010)

    2010

  59. [67]

    S ORKIN and H

    B. S ORKIN and H. D IAMANT , Persistent collective motion of a dispersing membrane do- main, 2020

    2020

  60. [68]

    B. A. C AMLEY and F. L. B ROWN, J. Chem. Phys. 151, 124104 (2019)

    2019

  61. [69]

    O PPENHEIMER and H

    N. O PPENHEIMER and H. D IAMANT , Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 82 (2010)

    2010

  62. [70]

    O PPENHEIMER and H

    N. O PPENHEIMER and H. D IAMANT , Phys. Rev. Lett. 107 (2011)

    2011

  63. [71]

    A RROYO, A

    M. A RROYO, A. D ESIMONE , and L. H ELTAI, arXiv (2010)

    2010

  64. [72]

    K RAMER , J

    L. K RAMER , J. Chem. Phys. 55, 2097 (1971)

    1971

  65. [73]

    S EIFERT , Phys

    U. S EIFERT , Phys. Rev. E 49, 3124 (1994)

    1994

  66. [74]

    S EIFERT and S

    U. S EIFERT and S. A. L ANGER , Biophys. Chem. 49, 13 (1994)

    1994

  67. [75]

    P FEIFFER , S

    W. P FEIFFER , S. K ¨ONIG , J. F. L EGRAND , T. B AYERL , D. R ICHTER , and E. S ACKMANN , Europhys. Lett. 23, 457 (1993)

    1993

  68. [76]

    K AIZUKA and J

    Y. K AIZUKA and J. T. G ROVES , Phys. Rev. Lett. 96 (2006)

    2006

  69. [77]

    P EUKES and T

    J. P EUKES and T. BETZ, Biophys. J. 107, 1810 (2014)

    2014

  70. [78]

    D. L. E RMAK and J. A. M CCAMMON , J. Chem. Phys. 69, 1352 (1978)

    1978

  71. [79]

    G. G. S TOKES , T rans. Cambridge Philos. Soc. 9, 8 (1851)

  72. [80]

    E INSTEIN , Ann

    A. E INSTEIN , Ann. d. Phys. 322, 549 (1905)

    1905

  73. [81]

    H IROMI YAMAKAWA , Modern Theory of Polymer Solutions , Harper & Row Publishers, New Y ork, 1971

    1971

  74. [82]

    R OTNE and S

    J. R OTNE and S. P RAGER , J. Chem. Phys. 50, 4831 (1969)

    1969

  75. [83]

    Y AMAKAWA , J

    H. Y AMAKAWA , J. Chem. Phys. 53, 436 (1970). 25

    1970

  76. [84]

    N OGUCHI and G

    H. N OGUCHI and G. G OMPPER , Proc. Natl. Acad. Sci. U. S. A. 102, 14159 (2005)

    2005

  77. [85]

    A. H. B AHRAMI and G. H UMMER , ACS Nano 11, 9558 (2017)

    2017

  78. [86]

    G RANEK , J

    R. G RANEK , J. Phys. II 7, 1761 (1997)

    1997

  79. [87]

    P IESSENS , The Hankel Transform, in T ransform

    R. P IESSENS , The Hankel Transform, in T ransform. Appl. Handb., edited by E D. A LEXAN - DER and D. P OULARIKAS , chapter 9, p. 30 pages, CRC Press LLC, Boca Raton, 2 edition, 2000

    2000

  80. [88]

    D UFFY, T ransform Methods for Solving Partial Differential Equations , Chapman & Hall/CRC, 2 edition, 1994

    D. D UFFY, T ransform Methods for Solving Partial Differential Equations , Chapman & Hall/CRC, 2 edition, 1994

    1994

  81. [89]

    H. F. J OHNSON , Comput. Phys. Commun. 43, 181 (1987)

    1987

  82. [90]

    L EMOINE , J

    D. L EMOINE , J. Chem. Phys. 101, 3936 (1994)

    1994

  83. [91]

    G ALASSI , J

    M. G ALASSI , J. D AVIES , B. G OUGH , G. J UNGMAN , P. A LKEN , and R. U LERICH , GNU Scientific Library Reference, Technical Report July, 2013

    2013

  84. [92]

    N. G OV, A. G. Z ILMAN , and S. S AFRAN , Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys. 70 (2004)

    2004

  85. [93]

    B RENN , Analytical Solutions for T ransport Processes , Mathematical Engineering, Springer Berlin Heidelberg, Berlin, Heidelberg, 2017

    G. B RENN , Analytical Solutions for T ransport Processes , Mathematical Engineering, Springer Berlin Heidelberg, Berlin, Heidelberg, 2017

    2017

  86. [94]

    K IM and M

    K. K IM and M. S ONG, Korean J. Comput. Appl. Math. 4, 179 (1997)

    1997

  87. [95]

    F IXMAN , Macromolecules 19, 1204 (1986)

    M. F IXMAN , Macromolecules 19, 1204 (1986)

    1986

  88. [96]

    G EYER and U

    T. G EYER and U. W INTER , J. Chem. Phys. 130, 114905 (2009)

    2009

  89. [97]

    A. J. B ANCHIO and J. F. BRADY, J. Chem. Phys. 118, 10323 (2003)

    2003

  90. [98]

    S. E. F ELLER , Y. Z HANG , R. W. P ASTOR , and B. R. B ROOKS , J. Chem. Phys. 103, 4613 (1995)

    1995

  91. [99]

    M ARSH , Chem

    D. M ARSH , Chem. Phys. Lipids 144, 146 (2006)

    2006

  92. [100]

    M. H U, J. J. B RIGUGLIO , and M. D ESERNO , Biophys. J. 102, 1403 (2012)

    2012

  93. [101]

    J. F. N AGLE , Faraday Discuss. 161, 11 (2013)

    2013

  94. [102]

    D IMOVA, Adv

    R. D IMOVA, Adv. Colloid Interface Sci. 208, 225 (2014)

    2014

  95. [103]

    J ANOSI and A

    L. J ANOSI and A. A. G ORFE , J. Chem. Theory Comput. 6, 3267 (2010)

    2010

  96. [104]

    J. B. K LAUDA , R. M. V ENABLE , J. A. F REITES , J. W. O’C ONNOR , D. J. T OBIAS , C. MONDRAGON -RAMIREZ , I. V OROBYOV , A. D. M ACKERELL , and R. W. PASTOR , J. Phys. Chem. B 114, 7830 (2010)

    2010

  97. [105]

    R AGHUNATHAN , Y

    M. R AGHUNATHAN , Y. Z UBOVSKI , R. M. V ENABLE , R. W. P ASTOR , J. F. N AGLE , and 26 S. T RISTRAM -NAGLE , J. Phys. Chem. B 116, 3918 (2012)

    2012

  98. [106]

    A. R. B RAUN, J. N. S ACHS , and J. F. NAGLE , J. Phys. Chem. B 117, 5065 (2013)

    2013

  99. [107]

    A NDO, E

    T. A NDO, E. C HOW, and J. S KOLNICK , J. Chem. Phys. 139, 121922 (2013)

    2013

  100. [108]

    S EIFERT and S

    U. S EIFERT and S. K. L ANGER , Euro. Phys. Lett. 23, 71 (1993)

    1993

  101. [109]

    E VANS and A

    E. E VANS and A. Y EUNG , Chem. Phys. Lipids 73, 39 (1994)

    1994

  102. [110]

    Y. A. C HIZMADZHEV , D. A. K UMENKO , P. I. K UZMIN , L. V. C HERNOMORDIK , J. Z IMMER - BERG , and F. S. COHEN , Biophys. J. 76, 2951 (1999)

    1999

  103. [111]

    S. A. S HKULIPA , W. K. D EN OTTER , and W. J. B RIELS , Biophys. J. 89, 823 (2005)

    2005

  104. [112]

    C ICCOTTI , M

    G. C ICCOTTI , M. F ERRARIO , and C. S CH ¨UTTE , Entropy 20, 348 (2018)

    2018

  105. [113]

    O NSAGER , Phys

    L. O NSAGER , Phys. Rev. 38, 2265 (1931)

    1931

  106. [114]

    R AMAKRISHNAN , P

    N. R AMAKRISHNAN , P. B. S UNIL KUMAR , and R. R ADHAKRISHNAN , Phys. Rep. 543, 1 (2014). 27 (a) in-plane mobility solvent-mediated HI membrane-mediated HI (i) (vi) out-of-plane mobility (ii) (iv) (v) (iii) (v) (b) (c) (i) (ii) (iii) Figure 1: Schematic of the introduced frame...

    2014

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