pith. sign in

arxiv: 2408.14500 · v1 · submitted 2024-08-23 · ⚛️ physics.chem-ph

Some challenges of diffused interfaces in implicit-solvent models

Mauricio Guerrero-Montero , Michal Bosy , Christopher D. Cooper This is my paper

Pith reviewed 2026-05-23 21:24 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords diffuse interfacePoisson-Boltzmannimplicit solvent modelsolvation energybinding energyfinite elementboundary element
0
0 comments X

The pith

The shape of the diffuse interface transition strongly affects solvation and binding energies

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

The paper investigates the effect of using a smooth, diffuse transition for permittivity and salt concentration at the solute-solvent boundary instead of the conventional sharp jump. It employs a hyperbolic tangent profile controlled by the parameter k_p and a mixed FEM-BEM discretization to compute electrostatic energies. Results on the FreeSolv database indicate that solvation energies match molecular dynamics data best near k_p = 3, but binding energies are highly sensitive to k_p with good performance across 2 to 20. This shows that the precise form of the interface profile is a key modeling choice with direct consequences for accuracy in implicit solvent calculations.

Core claim

Our results suggest that the shape of the function (represented by k_p) has a large impact on solvation and binding energy. An optimal value of k_p for solvation energies was around 3. However, more challenging binding free energy tests make this conclusion more difficult, as binding showed to be very sensitive to small variations of k_p. In that case, optimal values of k_p ranged from 2 to 20.

What carries the argument

Hyperbolic tangent function tanh(k_p x) used to define the smooth variation of permittivity and ionic strength across the diffuse interface, handled by a coupled finite-element and boundary-element numerical scheme.

If this is right

  • Steep interface profiles (high k_p) approach the sharp-interface limit but demand fine meshing near the surface to maintain numerical stability.
  • Solvation free energies obtained with k_p near 3 align most closely with molecular-dynamics benchmarks from the FreeSolv set.
  • Binding free energies vary significantly with small changes in k_p, requiring values between 2 and 20 for reasonable agreement.
  • The conclusions apply to any smooth interface function, not only the tanh form tested here.

Where Pith is reading between the lines

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

  • Binding calculations may require separate calibration of the interface parameter from solvation calculations.
  • Models with diffuse interfaces could benefit from adaptive k_p values that depend on the local molecular geometry or the property being computed.
  • The numerical challenges at high k_p suggest that intermediate diffuse profiles may offer a practical compromise between physical realism and computational cost.

Load-bearing premise

Comparisons to FreeSolv molecular-dynamics data alone can identify an optimal k_p independently of other approximations in the implicit-solvent model.

What would settle it

Computing binding free energies for additional protein-ligand pairs at k_p values of 1, 3, 10, and 30 and checking whether the error relative to experiment or explicit-solvent simulation stays low only inside the 2-20 window.

Figures

Figures reproduced from arXiv: 2408.14500 by Christopher D. Cooper, Mauricio Guerrero-Montero, Michal Bosy.

Figure 1
Figure 1. Figure 1: Representation of a dissolved molecule (Ω [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Visualization of the domains for the two-surface model. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Following Fig. 3, Γ [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Visualization of the different surface definitions. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: behaviour of the hyperbolic tangent for different values of [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of the scheme to calculate [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cross-section of the permittivity in the intermediate domain for cyclohexanol, [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Thermodynamic cycle for binding energy. Top and bottom states are solvated [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cross sectional plane of the three volumetric meshes for a the spherical shell [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Solvation energy of the sphere for different values of [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Solvation energy of 469 molecules from the FreeSolv Database for different values [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

The standard Poisson-Boltzmann model for molecular electrostatics assumes a sharp variation of the permittivity and salt concentration along the solute-solvent interface. The discontinuous field parameters are not only difficult numerically, but also are not a realistic physical picture, as it forces the dielectric constant and ionic strength of bulk in the near-solute region. An alternative to alleviate some of these issues is to represent the molecular surface as a diffuse interface, however, this also presents challenges. In this work we analysed the impact of the shape of the interfacial variation of the field parameters in solvation and binding energy. However we used a hyperbolic tangent function ($\tanh(k_p x)$) to couple the internal and external regions, our analysis is valid for other definitions. Our methodology was based on a coupled finite element (FEM) and boundary element (BEM) scheme that allowed us to have a special treatment of the permittivity and ionic strength in a bounded FEM region near the interface, while maintaining BEM elsewhere. Our results suggest that the shape of the function (represented by $k_p$) has a large impact on solvation and binding energy. We saw that high values of $k_p$ induce a high gradient on the interface, to the limit of recovering the sharp jump when $k_p\to\infty$, presenting a numerical challenge where careful meshing is key. Using the FreeSolv database to compare with molecular dynamics, our calculations indicate that an optimal value of $k_p$ for solvation energies was around 3. However, more challenging binding free energy tests make this conclusion more difficult, as binding showed to be very sensitive to small variations of $k_p$. In that case, optimal values of $k_p$ ranged from 2 to 20.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The manuscript examines challenges in modeling diffused interfaces in implicit-solvent Poisson-Boltzmann calculations using a hyperbolic tangent transition function controlled by parameter k_p. Through a coupled finite-element/boundary-element numerical scheme, it demonstrates that the interface shape parameter k_p strongly influences computed solvation and binding energies. Optimal k_p values are identified by fitting to FreeSolv molecular dynamics data, yielding k_p ≈ 3 for solvation energies and a broader range 2–20 for binding free energies.

Significance. If the numerical fidelity and isolation of the k_p effect can be established, the results would be significant for the implicit-solvent community by quantifying how interface diffuseness affects energies and highlighting the need for careful parameter tuning in binding calculations. The coupled FEM-BEM approach is a positive technical contribution for treating the near-interface region.

major comments (2)
  1. [Abstract] Abstract (FreeSolv comparison paragraph): The reported optimal k_p values are obtained by matching computed energies to existing FreeSolv MD data. Without a control calculation (e.g., sharp-interface PB with identical atomic radii, charges, and non-polar terms) or error decomposition, it is impossible to determine whether the diffuse-interface correction is the dominant remaining discrepancy or a compensatory adjustment for other model errors.
  2. [Abstract] Abstract (numerical methodology): No error bars, convergence checks, or mesh-refinement details are provided for the reported energies, despite the abstract noting that high k_p values induce high gradients and present a numerical challenge where careful meshing is key. This directly affects confidence in the claimed optimal k_p ranges.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript accordingly to improve clarity on controls and numerical validation while preserving the focus on interface sensitivity.

read point-by-point responses

  1. Referee: [Abstract] Abstract (FreeSolv comparison paragraph): The reported optimal k_p values are obtained by matching computed energies to existing FreeSolv MD data. Without a control calculation (e.g., sharp-interface PB with identical atomic radii, charges, and non-polar terms) or error decomposition, it is impossible to determine whether the diffuse-interface correction is the dominant remaining discrepancy or a compensatory adjustment for other model errors.

    Authors: We agree that a direct comparison to a sharp-interface PB calculation with identical parameters would help isolate the diffuse-interface effect. Our manuscript primarily demonstrates the large sensitivity of solvation and binding energies to k_p (rather than asserting that the diffuse model is the dominant correction to MD discrepancies). To address this point, the revised manuscript will include sharp-interface reference results for the solvation energies using the same radii, charges, and non-polar terms. revision: yes

  2. Referee: [Abstract] Abstract (numerical methodology): No error bars, convergence checks, or mesh-refinement details are provided for the reported energies, despite the abstract noting that high k_p values induce high gradients and present a numerical challenge where careful meshing is key. This directly affects confidence in the claimed optimal k_p ranges.

    Authors: We acknowledge that explicit numerical validation details are needed to support the reported energies and optimal k_p ranges, particularly given the high-gradient issues at large k_p. The revised manuscript will add a methods subsection with mesh-refinement studies, convergence criteria, and error estimates for the FEM-BEM calculations. revision: yes

Circularity Check

0 steps flagged

No circularity; k_p calibration uses independent external benchmark

full rationale

The paper numerically evaluates solvation and binding energies for different values of the interface parameter k_p via a coupled FEM-BEM discretization of the diffuse-interface Poisson-Boltzmann model, then reports an optimal k_p range by direct comparison to the external FreeSolv MD database. This is ordinary parameter tuning against an independent data source and does not reduce any claimed derivation or first-principles result to the fitted values by construction. No self-citations, uniqueness theorems, ansatzes smuggled via prior work, or renamings of known results appear in the load-bearing steps. The derivation chain (discretization of tanh(k_p x) permittivity and ionic-strength profiles, energy evaluation, and external comparison) remains self-contained against the benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim that k_p controls solvation and binding energies rests on one fitted parameter and two standard modeling assumptions; no new entities are postulated.

free parameters (1)
  • k_p = approximately 3 for solvation; 2-20 for binding
    The steepness parameter in the tanh transition is adjusted until computed solvation energies best match FreeSolv MD data; the same parameter is then tested on binding energies.
axioms (2)
  • domain assumption The hyperbolic tangent function tanh(k_p x) provides a physically adequate representation of the diffuse variation of permittivity and ionic strength across the solute-solvent interface.
    Invoked when the authors state they used tanh(k_p x) to couple internal and external regions.
  • domain assumption The coupled FEM-BEM scheme with a bounded FEM region near the interface accurately captures the physics of the diffuse transition without introducing coupling or discretization artifacts.
    Basis for the claim that numerical results reflect the true effect of k_p.

pith-pipeline@v0.9.0 · 5851 in / 1673 out tokens · 29592 ms · 2026-05-23T21:24:16.980764+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    Shen and F

    J. Shen and F. A. Quiocho, Journal of Computational Chemistry 16, 445 (1995)

    work page 1995

  2. [2]

    N. A. Baker, Methods in Enzymology 383, 94 (2004)

    work page 2004

  3. [3]

    H. Fu, H. Chen, M. Blazhynska, E. Goulard Coderc de Lacam, F. Szczepaniak, A. Pavlova, X. Shao, J. C. Gumbart, F. Dehez, B. Roux, et al., Nature Protocols 17, 1114 (2022)

    work page 2022

  4. [4]

    D. J. Bonthuis, S. Gekle, and R. R. Netz, Physical Review Letters 107, 166102 (2011)

    work page 2011

  5. [5]

    A. H. Boschitsch, M. O. Fenley, and H.-X. Zhou, The Journal of Physical Chemistry B 106, 2741 (2002)

    work page 2002

  6. [6]

    M. D. Altman, J. P. Bardhan, J. K. White, and B. Tidor, Journal of Computational Chemistry 30, 132 (2009)

    work page 2009

  7. [7]

    Xue and S

    C. Xue and S. Deng, Physical Review E 83, 056709 (2011)

    work page 2011

  8. [8]

    P. G. Bolcatto and C. R. Proetto, Journal of Physics: Condensed Matter 13, 319 (2001)

    work page 2001

  9. [9]

    Sihvola and I

    A. Sihvola and I. Lindell, Journal of electromagnetic waves and applications 3, 37 (1989)

    work page 1989

  10. [10]

    Stern, Physical Review B 17, 5009 (1978)

    F. Stern, Physical Review B 17, 5009 (1978)

    work page 1978

  11. [11]

    Movilla and J

    J. Movilla and J. Planelles, Computer physics communications 170, 144 (2005)

    work page 2005

  12. [12]

    Xue and S

    C. Xue and S. Deng, Journal of Computational and Applied Mathematics 326, 296 (2017)

    work page 2017

  13. [13]

    S. Wang, E. Alexov, and S. Zhao, Mathematical biosciences and engineering: MBE 18, 1370 (2021)

    work page 2021

  14. [14]

    S. Wang, Y. Shao, E. Alexov, and S. Zhao, Journal of Computational Physics 464, 111340 (2022)

    work page 2022

  15. [15]

    L. Li, C. Li, Z. Zhang, and E. Alexov, Journal of Chemical Theory and Computation 9, 2126 (2013). 24

    work page 2013

  16. [16]

    Chakravorty, Z

    A. Chakravorty, Z. Jia, L. Li, S. Zhao, and E. Alexov, Journal of Chemical Theory and Computation 14, 1020 (2018)

    work page 2018

  17. [17]

    L. Li, C. Li, and E. Alexov, Journal of Theoretical and Computational Chemistry 13, 1440002 (2014)

    work page 2014

  18. [18]

    L. Wang, L. Li, and E. Alexov, Proteins: Structure, Function, and Bioinformatics 83, 2186 (2015)

    work page 2015

  19. [19]

    S. K. Panday, M. H. Shashikala, A. Chakravorty, S. Zhao, and E. Alexov, Communica- tions in Information and Systems 19 (2019)

    work page 2019

  20. [20]

    Hazra, S

    T. Hazra, S. Ahmed Ullah, S. Wang, E. Alexov, and S. Zhao, Journal of mathematical biology 79, 631 (2019)

    work page 2019

  21. [21]

    Y. Shao, M. McGowan, S. Wang, E. Alexov, and S. Zhao, Applied Mathematics and Computation 436, 127501 (2023)

    work page 2023

  22. [22]

    S. Wang, A. Lee, E. Alexov, and S. Zhao, Applied Mathematics Letters 102, 106144 (2020)

    work page 2020

  23. [23]

    M. K. Gilson, A. Rashin, R. Fine, and B. Honig, Journal of Molecular Biology 184, 503 (1985)

    work page 1985

  24. [24]

    N. A. Baker, D. Sept, M. J. Holst, and J. A. McCammon, Proceedings of the National Academy of Sciences of the USA 98, 10037 (2001)

    work page 2001

  25. [25]

    Rocchia, E

    W. Rocchia, E. Alexov, and B. Honig, The Journal of Physical Chemistry B 105, 6507 (2001)

    work page 2001

  26. [26]

    A. H. Boschitsch and M. O. Fenley, Journal of Chemical Theory and Computation 7, 1524 (2011)

    work page 2011

  27. [27]

    Jurrus, D

    E. Jurrus, D. Engel, K. Star, K. Monson, J. Brandi, L. E. Felberg, D. H. Brookes, L. Wilson, J. Chen, K. Liles, et al., Protein Science 27, 112 (2018)

    work page 2018

  28. [28]

    C. M. Cortis and R. A. Friesner, Journal of Computational Chemistry 18, 1591 (1997). 25

    work page 1997

  29. [29]

    L. Chen, M. J. Holst, and J. Xu, SIAM journal on numerical analysis 45, 2298 (2007)

    work page 2007

  30. [30]

    Xie and S

    D. Xie and S. Zhou, BIT Numerical Mathematics 47, 853 (2007)

    work page 2007

  31. [31]

    S. D. Bond, J. H. Chaudhry, E. C. Cyr, and L. N. Olson, Journal of Computational Chemistry 31, 1625 (2010)

    work page 2010

  32. [32]

    Shaw, Physical Review A 32, 2476 (1985)

    P. Shaw, Physical Review A 32, 2476 (1985)

    work page 1985

  33. [33]

    B. J. Yoon and A. Lenhoff, Journal of Computational Chemistry 11, 1080 (1990)

    work page 1990

  34. [34]

    Juffer, E

    A. Juffer, E. F. Botta, B. A. van Keulen, A. van der Ploeg, and H. J. Berendsen, Journal of Computational Physics 97, 144 (1991)

    work page 1991

  35. [35]

    B. Lu, X. Cheng, J. Huang, and J. A. McCammon, Proceedings of the National Academy of Sciences of the USA 103, 19314 (2006)

    work page 2006

  36. [36]

    Geng and R

    W. Geng and R. Krasny, Journal of Computational Physics 247, 62 (2013)

    work page 2013

  37. [37]

    C. D. Cooper, J. P. Bardhan, and L. A. Barba, Computer physics communications 185, 720 (2014)

    work page 2014

  38. [38]

    S. D. Search, C. D. Cooper, and E. Van’t Wout, Journal of Computational Chemistry 43, 674 (2022)

    work page 2022

  39. [39]

    L. E. Felberg, D. H. Brookes, E.-H. Yap, E. Jurrus, N. A. Baker, and T. Head-Gordon, Journal of computational chemistry 38, 1275 (2017)

    work page 2017

  40. [40]

    S. V. Siryk and W. Rocchia, The Journal of Physical Chemistry B 126, 10400 (2022)

    work page 2022

  41. [41]

    A. Jha, M. Nottoli, A. Mikhalev, C. Quan, and B. Stamm, The Journal of Chemical Physics 158 (2023)

    work page 2023

  42. [42]

    Jha and B

    A. Jha and B. Stamm, arXiv preprint arXiv:2309.06862 (2023)

    work page arXiv 2023

  43. [43]

    A. H. Boschitsch and M. O. Fenley, Journal of Computational Chemistry 25, 935 (2004)

    work page 2004

  44. [44]

    Ying and D

    J. Ying and D. Xie, Applied Mathematical Modelling 58, 166 (2018). 26

    work page 2018

  45. [45]

    M. Bosy, M. W. Scroggs, T. Betcke, E. Burman, and C. D. Cooper, Journal of Compu- tational Chemistry 45, 787 (2024)

    work page 2024

  46. [46]

    S. Pal, S. Balasubramanian, and B. Bagchi, The Journal of chemical physics 120, 1912 (2004)

    work page 1912

  47. [47]

    Sterpone, G

    F. Sterpone, G. Stirnemann, and D. Laage, Journal of the American Chemical Society 134, 4116 (2012)

    work page 2012

  48. [48]

    Fumagalli, A

    L. Fumagalli, A. Esfandiar, R. Fabregas, S. Hu, P. Ares, A. Janardanan, Q. Yang, B. Radha, T. Taniguchi, K. Watanabe, et al., Science 360, 1339 (2018)

    work page 2018

  49. [49]

    M. L. Connolly, Journal of applied crystallography 16, 548 (1983)

    work page 1983

  50. [50]

    Lee and F

    B. Lee and F. M. Richards, Journal of molecular biology 55, 379 (1971)

    work page 1971

  51. [51]

    R. M. Howard, Mathematical and Computational Applications 27 (2022), ISSN 2297- 8747

    work page 2022

  52. [52]

    Johnson, J

    C. Johnson, J. N´ ed´ elec, et al., Mathematics of Computation35, 1063 (1980)

    work page 1980

  53. [53]

    O. Steinbach, Numerical approximation methods for elliptic boundary value problems (Springer, New York, 2008), ISBN 978-0-387-31312-2, finite and boundary elements, Translated from the 2003 German original

    work page 2008

  54. [54]

    D. L. Mobley and J. P. Guthrie, Journal of computer-aided molecular design 28, 711 (2014)

    work page 2014

  55. [55]

    Betcke and M

    T. Betcke and M. Scroggs, Journal of Open Source Software 6, 2879 (2021)

    work page 2021

  56. [56]

    Logg and G

    A. Logg and G. N. Wells, ACM Transactions on Mathematical Software (TOMS) 37, 1 (2010)

    work page 2010

  57. [57]

    Dawson-Haggerty et al., trimesh, URL https://trimsh.org/

    work page

  58. [58]

    M. F. Sanner, A. J. Olson, and J.-C. Spehner, Biopolymers 38, 305 (1996)

    work page 1996

  59. [59]

    Decherchi and W

    S. Decherchi and W. Rocchia, PloS one 8, e59744 (2013). 27

    work page 2013

  60. [60]

    C. T. Lee, J. G. Laughlin, J. B. Moody, R. E. Amaro, J. A. McCammon, M. Holst, and P. Rangamani, Biophysical Journal 118, 1003 (2020)

    work page 2020

  61. [61]

    R. C. Harris, A. H. Boschitsch, and M. O. Fenley, Journal of Chemical Theory and Computation 9, 3677 (2013)

    work page 2013

  62. [62]

    R. C. Harris, T. Mackoy, and M. O. Fenley, Journal of chemical theory and computation 11, 705 (2015)

    work page 2015

  63. [63]

    D. D. Nguyen, B. Wang, and G.-W. Wei, Journal of Computational Chemistry 38, 941 (2017)

    work page 2017

  64. [64]

    Izadi, R

    S. Izadi, R. C. Harris, M. O. Fenley, and A. V. Onufriev, Journal of chemical theory and computation 14, 1656 (2018)

    work page 2018

  65. [65]

    A. M. Buckle, G. Schreiber, and A. R. Fersht, Biochemistry 33, 8878 (1994)

    work page 1994

  66. [66]

    U. C. K¨ uhlmann, A. J. Pommer, G. R. Moore, R. James, and C. Kleanthous, Journal of molecular biology 301, 1163 (2000)

    work page 2000

  67. [67]

    D. Lim, H. U. Park, L. De Castro, S. G. Kang, H. S. Lee, S. Jensen, K. J. Lee, and N. C. Strynadka, Nature structural biology 8, 848 (2001)

    work page 2001

  68. [68]

    Brownlow, J

    S. Brownlow, J. H. M. Cabral, R. Cooper, D. R. Flower, S. J. Yewdall, I. Polikarpov, A. C. North, and L. Sawyer, Structure 5, 481 (1997)

    work page 1997

  69. [69]

    J. R. Tame and B. Vallone, Acta Crystallographica Section D: Biological Crystallography 56, 805 (2000)

    work page 2000

  70. [70]

    M. T. Pisabarro, L. Serrano, and M. Wilmanns, Journal of molecular biology 281, 513 (1998)

    work page 1998

  71. [71]

    Michalska, K

    K. Michalska, K. Brzezinski, and M. Jaskolski, Journal of Biological Chemistry 280, 28484 (2005)

    work page 2005

  72. [72]

    J. W. Ponder and D. A. Case, Advances in protein chemistry 66, 27 (2003). 28

    work page 2003

  73. [73]

    T. J. Dolinsky, J. E. Nielsen, J. A. McCammon, and N. A. Baker, Nucleic acids research 32, W665 (2004). 29

    work page 2004