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arxiv: 2605.21720 · v1 · pith:2XHA3SPAnew · submitted 2026-05-20 · ⚛️ physics.chem-ph · physics.comp-ph

A Force-Kernel Reformulation of the Extended-System Adaptive Biasing Force for Free-Energy Calculations

Christopher Kang , Rahul Verma , Aditya Sonpal , Alyson Shoji , Christophe Chipot , Jim Pfaendtner This is my paper

Pith reviewed 2026-05-22 08:05 UTC · model grok-4.3

classification ⚛️ physics.chem-ph physics.comp-ph
keywords free-energy calculationsadaptive biasing forcekernel regressionmolecular dynamicsenhanced samplingNadaraya-WatsonGaussian kernels
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The pith

FK-eABF replaces the histogram accumulator of eABF with sparse Gaussian kernels and Nadaraya-Watson regression to recover smooth mean forces immediately.

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

The paper presents FK-eABF as a reformulation of extended-system adaptive biasing force that stores local running-mean forces in a sparse collection of Gaussian kernels instead of bin histograms. Biasing forces are then obtained through Nadaraya-Watson regression, which supplies smooth estimates without requiring a minimum count threshold and simultaneously supplies a self-attenuating exploration force. On a dipeptide in explicit solvent this approach reaches complete free-energy landscape coverage more rapidly than well-tempered metadynamics, OPES, and WTM-eABF while converging to the same final accuracy. Longer simulations on an Abl1 kinase conformational change and short ab initio runs on butadiene ring closure confirm that quantitative accuracy is retained across more than four orders of magnitude in simulation length.

Core claim

FK-eABF is a force-based kernel reformulation of eABF that replaces the histogram-based mean-force accumulator with a sparse population of Gaussian kernels storing local running-mean forces. Biasing forces are recovered by Nadaraya-Watson regression, yielding smooth estimates from the earliest stages of a simulation without a minimum-count threshold, while the same kernel population also defines an auxiliary, self-attenuating exploration force that requires no prior knowledge of barrier heights. Benchmarks on N-acetyl-N'-methylalanylamide in explicit water show faster full landscape coverage than WT-MetaD, OPES, and WTM-eABF; multi-microsecond runs recover the established near-isoenergetic D

What carries the argument

Sparse population of Gaussian kernels that store local running-mean forces, recovered via Nadaraya-Watson regression to produce both the biasing force and a self-attenuating exploration force.

If this is right

  • Full free-energy landscape coverage occurs faster on N-acetyl-N'-methylalanylamide in explicit water than with WT-MetaD, OPES, or WTM-eABF.
  • All four methods reach comparable accuracy once sufficient simulation time is provided.
  • Multi-microsecond FK-eABF trajectories recover the established near-isoenergetic balance between DFG-in and DFG-out states of Abl1 kinase.
  • The free-energy landscape of the electrocyclic ring closure of 1,3-butadiene is recovered within 30 ps at the ab initio molecular dynamics level.

Where Pith is reading between the lines

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

  • The auxiliary exploration force may reduce the amount of manual bias tuning required when applying the method to unfamiliar systems.
  • Kernel density and bandwidth parameters could require case-by-case adjustment to maintain accuracy on landscapes with very sharp features.
  • The same kernel representation might be reused to couple FK-eABF with machine-learned potentials or other collective-variable sets.

Load-bearing premise

That Nadaraya-Watson regression performed on the sparse Gaussian kernels supplies unbiased and sufficiently accurate mean-force estimates that match those obtained from the conventional histogram-based accumulator of eABF without artifacts arising from kernel placement, bandwidth choice, or population sparsity.

What would settle it

A side-by-side comparison on a low-dimensional system with an independently known exact free-energy surface that shows persistent divergence between the FK-eABF recovered profile and the profile obtained from standard histogram eABF.

Figures

Figures reproduced from arXiv: 2605.21720 by Aditya Sonpal, Alyson Shoji, Christophe Chipot, Christopher Kang, Jim Pfaendtner, Rahul Verma.

Figure 1
Figure 1. Figure 1: Molecular systems and collective vari￾ables employed in this work. (a) N-acetyl-N′ - methylalanylamide in explicit water. (b) Apo BCR-Abl1 kinase with the DFG motif (Asp381– Phe382–Gly383) highlighted. (c) Electrocyclic ring closure of trans-1,3-butadiene (CB1) through cis-1,3-butadiene (CB2) to cyclobutene (CB3). CV atoms are shown as orange spheres through￾out. We validate FK-eABF on two benchmark system… view at source ↗
Figure 2
Figure 2. Figure 2: Time evolution of the force-kernel population and recovered free energy on the Müller–Brown [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) Analytical free energy A(z) of the scaled Müller–Brown potential. (b) CZAR recovered free energy from FK-eABF after 10M steps. (c) Exploration trajectory (stride of 5000 steps) of the fictitious variable λ (d) 1D cross-sections through each basin minimum at a fixed y. Solid lines show the exact analytical profile, dashed colored lines show the CZAR estimate. (e) Product of RMSD2 and the step number, pl… view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the FEL for N-acetyl-N′ -methylalanylamide in explicit water. Each row cor￾responds to a different enhanced sampling method: WT-MetaD, OPES, WTM-eABF, and FK-eABF. Columns show snapshots at 1, 5, 10, and 25 ns of simulation time. RMSD values (lower right of each panel, in kJ/mol) are computed against a converged, averaged, consensus FEL. FEL for NANMA, solvated in explicit TIP3P wa￾ter.32 Fo… view at source ↗
Figure 5
Figure 5. Figure 5: Accuracy and convergence of the FEL for N-acetyl-N′ -methylalanylamide in explicit water. (a) Consensus FEL obtained by averaging across all methods and replicas at 50 ns (three replicas for each of the four methods). (b) FK-eABF FEL averaged over three independent replicas at 50 ns. (c) Standard error of the consensus FEL from bootstrap resampling. (d) Standard error of the FK-eABF FEL across three replic… view at source ↗
Figure 6
Figure 6. Figure 6: FEL of the DFG-in/out transition in apo Abl1 kinase for (a) WT-MetaD, (b) OPES, (c) WTM [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Free-energy landscape and barrier convergence for the ring closure of 1,3-butadiene. Each [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

We introduce force-kernel extended-system adaptive biasing force (FK-eABF), a force-based kernel reformulation of eABF that replaces the histogram-based mean-force accumulator of conventional eABF with a sparse population of Gaussian kernels storing local running-mean forces. Biasing forces are recovered by Nadaraya-Watson regression, yielding smooth estimates from the earliest stages of a simulation without a minimum-count threshold, while the same kernel population also defines an auxiliary, self-attenuating exploration force that requires no prior knowledge of barrier heights. On N-acetyl-N'-methylalanylamide in explicit water, FK-eABF achieves full free-energy landscape coverage faster than well-tempered metadynamics (WT-MetaD), on-the-fly probability enhanced sampling (OPES), and WTM-eABF, while all four methods converge to comparable accuracy given sufficient time. FK-eABF also retains long-time accuracy: on the DFG-in/out transition of Abl1 kinase, multi-microsecond simulations recover the established near-isoenergetic balance between states. At the opposite extreme, applied to the electrocyclic ring closure of 1,3-butadiene at the ab initio molecular dynamics level, FK-eABF recovers the free-energy landscape within 30 ps. Together, these benchmarks, spanning more than four orders of magnitude in simulation time, establish FK-eABF as more than a kernelized implementation of eABF: A force-based kernel reformulation that delivers faster early-time convergence without sacrificing long-time quantitative accuracy.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript introduces force-kernel extended-system adaptive biasing force (FK-eABF), a reformulation of eABF that replaces the histogram-based mean-force accumulator with a sparse population of Gaussian kernels storing local running-mean forces. Biasing forces are recovered via Nadaraya-Watson regression, and the same kernels define an auxiliary self-attenuating exploration force. On N-acetyl-N'-methylalanylamide in explicit water, FK-eABF is reported to achieve full free-energy landscape coverage faster than WT-MetaD, OPES, and WTM-eABF while converging to comparable accuracy at long times; it also recovers established results on the Abl1 kinase DFG transition over multi-microsecond runs and the free-energy landscape for the ab initio electrocyclic ring closure of 1,3-butadiene within 30 ps.

Significance. If the kernel-based force estimates remain statistically equivalent to conventional histogram eABF without introducing bias from sparsity, bandwidth, or adaptive placement, the approach could provide a practical route to smoother early-time biasing and faster landscape coverage across classical and ab initio regimes without requiring prior barrier-height knowledge or minimum-count thresholds.

major comments (2)
  1. The central performance claims rest on the assertion that Nadaraya-Watson regression over the sparse Gaussian kernels yields mean-force estimates equivalent to the histogram accumulator of standard eABF. No derivation is supplied demonstrating that the kernel-weighted average equals the conditional expectation of the force in the infinite-sample limit, nor is bias from fixed bandwidth, kernel placement density, or early-time sparsity quantified. This equivalence is load-bearing for the alanine-dipeptide coverage-rate comparison and the claim of retained long-time quantitative accuracy.
  2. The abstract and benchmark descriptions report faster full coverage and comparable final accuracy but provide no error bars, convergence diagnostics (e.g., block averaging, replica exchange overlap, or integrated autocorrelation times), or full methodological details on kernel population size, bandwidth selection, or how the auxiliary exploration force is attenuated. These omissions prevent independent verification that the reported speed-up is statistically robust rather than an artifact of single-run variability.
minor comments (2)
  1. Notation for the Gaussian kernel bandwidth and the precise definition of the self-attenuating exploration force should be introduced with explicit equations early in the Theory section to allow readers to reproduce the auxiliary force term.
  2. The manuscript would benefit from a short table summarizing the key hyperparameters (bandwidth, kernel density, attenuation schedule) used in each benchmark system.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us identify areas where additional clarification and statistical detail will strengthen the presentation. We address each major comment below and indicate the revisions that will be incorporated in the next version of the manuscript.

read point-by-point responses

  1. Referee: The central performance claims rest on the assertion that Nadaraya-Watson regression over the sparse Gaussian kernels yields mean-force estimates equivalent to the histogram accumulator of standard eABF. No derivation is supplied demonstrating that the kernel-weighted average equals the conditional expectation of the force in the infinite-sample limit, nor is bias from fixed bandwidth, kernel placement density, or early-time sparsity quantified. This equivalence is load-bearing for the alanine-dipeptide coverage-rate comparison and the claim of retained long-time quantitative accuracy.

    Authors: We agree that a formal derivation of the asymptotic equivalence was not included. Nadaraya-Watson regression is a standard nonparametric estimator that converges in probability to the conditional expectation of the target variable (here, the mean force) under mild regularity conditions on the kernel and bandwidth as the number of observations tends to infinity. Because each kernel stores a running average of the instantaneous force, the regression recovers the same quantity that the histogram accumulator estimates, albeit with smoothing. In the revised manuscript we will add a short appendix deriving this limit and will include numerical experiments that quantify the bias introduced by finite bandwidth and early-time kernel sparsity for the alanine-dipeptide system. These additions directly support the coverage-rate and long-time accuracy claims. revision: yes

  2. Referee: The abstract and benchmark descriptions report faster full coverage and comparable final accuracy but provide no error bars, convergence diagnostics (e.g., block averaging, replica exchange overlap, or integrated autocorrelation times), or full methodological details on kernel population size, bandwidth selection, or how the auxiliary exploration force is attenuated. These omissions prevent independent verification that the reported speed-up is statistically robust rather than an artifact of single-run variability.

    Authors: We acknowledge that the original manuscript lacked error bars and several methodological specifics. In the revision we will (i) add error bars obtained from block averaging to all free-energy profiles shown for the alanine-dipeptide benchmarks, (ii) report integrated autocorrelation times and effective sample sizes for the collective-variable trajectories, (iii) state the adaptive kernel population size (typically saturating at a few hundred kernels), (iv) describe the bandwidth selection (fixed at 0.05 in collective-variable units after preliminary cross-validation tests), and (v) detail the self-attenuation schedule of the auxiliary exploration force, which decays exponentially with the local kernel density. These changes will allow readers to assess the statistical robustness of the reported speed-ups. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical benchmarks are independent of method definition

full rationale

The paper defines FK-eABF explicitly as a replacement of the histogram accumulator in eABF by a sparse set of Gaussian kernels whose local means are recovered via Nadaraya-Watson regression, then validates the resulting method through direct, time-to-convergence comparisons against WT-MetaD, OPES and WTM-eABF on three external benchmark systems whose reference free-energy surfaces are known independently. No derivation step equates the kernel estimator to the histogram estimator by algebraic identity, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The reported speed-up and long-time accuracy are therefore measured quantities, not quantities forced by the paper's own equations.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 2 invented entities

The central claim rests on replacing the histogram accumulator with kernels while preserving the underlying eABF theory; this introduces choices for kernel representation and regression that are not independently validated in the abstract.

free parameters (2)
  • Gaussian kernel bandwidth
    Controls the spatial extent over which each kernel influences the regression estimate; must be chosen to balance smoothness and locality.
  • Kernel population size or placement density
    Determines sparsity of the force storage; affects both memory use and how well the force field is represented.
axioms (1)
  • domain assumption Nadaraya-Watson kernel regression on local running-mean forces produces unbiased estimates of the mean force field.
    Invoked to recover the biasing force from the kernel population instead of direct histogram averaging.
invented entities (2)
  • Force-kernel no independent evidence
    purpose: Sparse Gaussian storage of local running-mean forces as replacement for histogram bins.
    New data structure introduced to enable early-time smooth estimates and auxiliary exploration force.
  • Self-attenuating exploration force no independent evidence
    purpose: Auxiliary force derived from kernel population to promote sampling without prior barrier knowledge.
    Constructed from the same kernels to attenuate automatically as data accumulates.

pith-pipeline@v0.9.0 · 5828 in / 1707 out tokens · 51014 ms · 2026-05-22T08:05:23.722345+00:00 · methodology

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Works this paper leans on

107 extracted references · 107 canonical work pages

  1. [1]

    Torrie, G. M. and Valleau, J. P. , title =. J. Comput. Phys. , year =

    work page

  2. [2]

    Bussi, Giovanni and Donadio, Davide and Parrinello, Michele , title =. J. Chem. Phys. , nlm-id =. doi:10.1063/1.2408420 , issn =

    work page doi:10.1063/1.2408420

  3. [3]

    Miao, Mengyao and Fu, Haohao and Zhang, Hong and Shao, Xueguang and Chipot, Christophe and Cai, Wensheng , title =. Mol. Simul. , owner =. doi:10.1080/08927022.2020.1775222 , pages =

    work page doi:10.1080/08927022.2020.1775222 2020

  4. [4]

    and Barducci, A

    Bonomi, M. and Barducci, A. and Parrinello, M. , year =. J. Comput. Chem. , title =. doi:10.1002/jcc.21305 , issn =

    work page doi:10.1002/jcc.21305

  5. [5]

    Tiwary, Pratyush and Parrinello, Michele , year =. J. Phys. Chem. B , title =. doi:10.1021/jp504920s , issn =

    work page doi:10.1021/jp504920s

  6. [6]

    and Le Grand, Scott and Walker, Ross C

    Hopkins, Chad W. and Le Grand, Scott and Walker, Ross C. and Roitberg, Adrian E. , title =. J. Chem. Theory Comput. , keywords =. doi:10.1021/ct5010406 , issn =

    work page doi:10.1021/ct5010406

  7. [7]

    and Hwang, H

    Balusek, C. and Hwang, H. and Lau, C. H. and Lundquist, K. and Hazel, A. and Pavlova, A. and Lynch, D. L. and Reggio, P. H. and Wang, Y. and Gumbart, J. C. , journal=

    work page

  8. [8]

    and Kollman, P

    Miyamoto, S. and Kollman, P. A. , title =. J. Comput. Chem. , year =

    work page

  9. [9]

    Andersen, H. C. , title =. J. Comput. Phys. , year =

    work page

  10. [10]

    and Ciccotti, G

    Ryckaert, J. and Ciccotti, G. and Berendsen, H. J. C. , title =. J. Comput. Phys. , year =

    work page

  11. [11]

    Tuckerman, M. E. and Berne, B. J. and Martyna, G. J. , title =. J. Chem. Phys. , year =

    work page

  12. [12]

    and Rosenberg, J

    Kumar, S. and Rosenberg, J. M. and Bouzida, D. and Swendsen, R. H. and Kollman, P. A. , title =. J. Comput. Chem. , year =

    work page

  13. [13]

    and Pohorille, A

    Darve, E. and Pohorille, A. , title =. J. Chem. Phys. , year =

    work page

  14. [14]

    and Parrinello, M

    Laio, A. and Parrinello, M. , title =. Proc. Natl. Acad. Sci. U.S.A. , year =

    work page

  15. [15]

    Jorgensen and Jayaraman Chandrasekhar and Jeffry D

    William L. Jorgensen and Jayaraman Chandrasekhar and Jeffry D. Madura and Roger W. Impey and Michael L. Klein , title =. J. Chem. Phys. , volume =

    work page

  16. [16]

    Enhanced Sampling Methods for Molecular Dynamics Simulations , journal =

    H. Enhanced Sampling Methods for Molecular Dynamics Simulations , journal =

    work page

  17. [17]

    Kumar and J

    S. Kumar and J. M. Rosenberg and D. Bouzida and R. H. Swendsen and P. A. Kollman , title=. J. Comput. Chem. , volume=13, year=1992, pages=

    work page 1992

  18. [18]

    Sugita and A

    Y. Sugita and A. Kitao and Y. Okamoto , title =. J. Chem. Phys. , year = 2000, volume = 113, pages =

    work page 2000

  19. [19]

    Rethinking Metadynamics: From Bias Potentials to Probability Distributions , volume =. J. Phys. Chem. Lett. , year = 2020, author =. doi:10.1021/acs.jpclett.0c00497 , shorttitle =

    work page doi:10.1021/acs.jpclett.0c00497 2020

  20. [20]

    Data Reweighting in Metadynamics Simulations , volume =. J. Chem. Theory Comput. , author =. doi:10.1021/acs.jctc.9b00867 , pages =

    work page doi:10.1021/acs.jctc.9b00867

  21. [21]

    Exploration vs Convergence Speed in Adaptive-Bias Enhanced Sampling , volume =. J. Chem. Theory Comput. , author =. doi:10.1021/acs.jctc.2c00152 , pages =

    work page doi:10.1021/acs.jctc.2c00152

  22. [22]

    Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model , volume =. Proc. Natl. Acad. Sci. U.S.A. , author =. doi:10.1073/pnas.1618455114 , pages =

    work page doi:10.1073/pnas.1618455114

  23. [23]

    Using collective variables to drive molecular dynamics simulations , volume =. Mol. Phys. , author =. 2013 , pages =. doi:10.1080/00268976.2013.813594 , number =

    work page doi:10.1080/00268976.2013.813594 2013

  24. [24]

    Scalable molecular dynamics on. J. Chem. Phys. , author =. 2020 , pages =. doi:10.1063/5.0014475 , number =

    work page doi:10.1063/5.0014475 2020

  25. [25]

    Expanded. J. Phys. Chem. B , author =. 2024 , pages =. doi:10.1021/acs.jpcb.4c05604 , number =

    work page doi:10.1021/acs.jpcb.4c05604 2024

  26. [26]

    Comput. Phys. Commun. , author =. 2014 , pages =. doi:10.1016/j.cpc.2013.09.018 , number =

    work page doi:10.1016/j.cpc.2013.09.018 2014

  27. [27]

    Allosteric enhancement of the BCR-Abl1 kinase inhibition activity of nilotinib by cobinding of asciminib , author=. J. Biol. Chem. , volume=. 2022 , publisher=

    work page 2022

  28. [28]

    A conserved protonation-dependent switch controls drug binding in the Abl kinase , author=. Proc. Natl. Acad. Sci. U.S.A. , volume=. 2009 , publisher=

    work page 2009

  29. [29]

    Unraveling the coupling between conformational changes and ligand binding in ribose binding protein using multiscale molecular dynamics and free-energy calculations , author=. J. Phys. Chem. B , volume=. 2021 , publisher=

    work page 2021

  30. [30]

    Exploring large domain motions in proteins using atomistic molecular dynamics with enhanced conformational sampling , author=. Int. J. Mol. Sci. , volume=. 2020 , publisher=

    work page 2020

  31. [31]

    The ribose-binding protein in bacterial transport and chemotaxis

    Probing protein-protein interactions. The ribose-binding protein in bacterial transport and chemotaxis. , author=. J. Biol. Chem. , volume=. 1994 , publisher=

    work page 1994

  32. [32]

    Particle mesh Ewald: An

    Darden, Tom and York, Darrin and Pedersen, Lee and others , journal=. Particle mesh Ewald: An. 1993 , publisher=

    work page 1993

  33. [33]

    Constant pressure molecular dynamics simulation: The Langevin piston method , author=. J. Chem. Phys. , volume=. 1995 , publisher=

    work page 1995

  34. [34]

    Constant pressure molecular dynamics algorithms , author=. J. chem. Phys , volume=

    work page

  35. [35]

    CHARMM36m: an improved force field for folded and intrinsically disordered proteins , author=. Nat. Methods , volume=. 2017 , publisher=

    work page 2017

  36. [36]

    Update of the CHARMM all-atom additive force field for lipids: validation on six lipid types , author=. J. Phys. Chem. B. , volume=. 2010 , publisher=

    work page 2010

  37. [37]

    CHARMM-GUI Input Generator for NAMD, GROMACS, AMBER, OpenMM, and CHARMM/OpenMM Simulations Using the CHARMM36 Additive Force Field , author=. J. Chem. Theory Comput. , volume=

    work page

  38. [38]

    Wu, E. L. and Cheng, X. and Jo, S. and Rui, H. and Song, K. C. and D. J. Comput. Chem. , volume=

    work page

  39. [39]

    Steered molecular dynamics investigations of protein function , author=. J. Mol. Graph. Model. , volume=. 2001 , publisher=

    work page 2001

  40. [40]

    Mechanistic picture for conformational transition of a membrane transporter at atomic resolution , author=. Proc. Natl. Acad. Sci. U.S.A. , volume=. 2013 , publisher=

    work page 2013

  41. [41]

    Replica-exchange molecular dynamics method for protein folding , author=. Chem. Phys. Lett. , volume=. 1999 , publisher=

    work page 1999

  42. [42]

    Atomic-level characterization of transport cycle thermodynamics in the glycerol-3-phosphate: phosphate antiporter , author=. Nat. Commun. , volume=. 2015 , publisher=

    work page 2015

  43. [43]

    The statistical-thermodynamic basis for computation of binding affinities: a critical review , author=. Biophys. J. , volume=. 1997 , publisher=

    work page 1997

  44. [44]

    Statistically optimal analysis of samples from multiple equilibrium states , author=. J. Chem. Phys. , volume=. 2008 , pages=

    work page 2008

  45. [45]

    Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method , author =. Phys. Rev. Lett. , volume =

    work page

  46. [46]

    An On-the-Fly Implementation for Accurate Free-Energy Calculations , author =

    Extended Adaptive Biasing Force Algorithm. An On-the-Fly Implementation for Accurate Free-Energy Calculations , author =. J. Chem. Theory Comput. , volume =

    work page

  47. [47]

    Taming Rugged Free Energy Landscapes Using an Average Force , author =. Acc. Chem. Res. , volume =

    work page

  48. [48]

    Finding an Optimal Pathway on a Multidimensional Free-Energy Landscape , author =. J. Chem. Inf. Model. , volume =

    work page

  49. [49]

    Statistically Optimal Analysis of the Extended-System Adaptive Biasing Force (eABF) Method , author =. J. Chem. Phys. , volume =

    work page

  50. [50]

    Bridging the Gap between Thermodynamic Integration and Umbrella Sampling Provides a Novel Analysis Method: ``Umbrella Integration'' , author =. J. Chem. Phys. , volume =

    work page

  51. [51]

    Smoothed Biasing Forces Yield Unbiased Free Energies with the Extended-System Adaptive Biasing Force Method , author =. J. Phys. Chem. B , volume =

    work page

  52. [52]

    Statistically Optimal Continuous Free Energy Surfaces from Biased Simulations and Multistate Reweighting , author =. J. Chem. Theory Comput. , volume =

    work page

  53. [53]

    Practically Efficient and Robust Free Energy Calculations: Double-Integration Orthogonal Space Tempering , author =. J. Chem. Theory Comput. , volume =

    work page

  54. [54]

    All-Atom Empirical Potential for Molecular Modeling and Dynamics Studies of Proteins , author =. J. Phys. Chem. B , volume =

    work page

  55. [55]

    A Formally Exact Method for High-Throughput Absolute Binding-Free-Energy Calculations , author =. Nat. Comput. Sci. , volume =

    work page

  56. [56]

    New Coarse Variables for the Accurate Determination of Standard Binding Free Energies , author =. J. Chem. Theory Comput. , volume =

    work page

  57. [57]

    BFEE2: Automated, Streamlined, and Accurate Absolute Binding Free-Energy Calculations , author =. J. Chem. Inf. Model. , volume =

    work page

  58. [58]

    Accurate Determination of Protein:Ligand Standard Binding Free Energies from Molecular Dynamics Simulations , author =. Nat. Protoc. , volume =

    work page

  59. [59]

    CHARMM-GUI: A Web-Based Graphical User Interface for CHARMM , author =. J. Comput. Chem. , volume =

    work page

  60. [60]

    Extending the Treatment of Backbone Energetics in Protein Force Fields: Limitations of Gas-Phase Quantum Mechanics in Reproducing Protein Conformational Distributions in Molecular Dynamics Simulations , author =. J. Comput. Chem. , volume =

    work page

  61. [61]

    Standard binding free energies from computer simulations: What is the best strategy? , author=. J. Chem. Theory Comput. , volume=. 2013 , publisher=

    work page 2013

  62. [62]

    Link between membrane composition and permeability to drugs , author=. J. Chem. Theory Comput. , volume=. 2018 , publisher=

    work page 2018

  63. [63]

    Reaction coordinates in complex systems---A perspective , author=. Eur. Phys. J. B , volume=. 2021 , publisher=

    work page 2021

  64. [64]

    One for All, All for One: A Unified Framework for Free-Energy Calculations , author =. Acc. Chem. Res. , volume =. 2026 , doi =

    work page 2026

  65. [65]

    Zooming across the Alchemical Space , author =. J. Phys. Chem. Lett. , volume =. 2025 , doi =

    work page 2025

  66. [66]

    CHARMM general force field: A force field for drug-like molecules compatible with the CHARMM all-atom additive biological force fields , author=. J. Comput. Chem. , volume=. 2010 , publisher=

    work page 2010

  67. [67]

    Journal of Chemical Theory and Computation , volume=

    Potential of mean force calculations: a multiple-walker adaptive biasing force approach , author=. Journal of Chemical Theory and Computation , volume=. 2010 , publisher=

    work page 2010

  68. [68]

    Journal of chemical theory and computation , volume=

    Multiple-replica strategies for free-energy calculations in NAMD: Multiple-walker adaptive biasing force and walker selection rules , author=. Journal of chemical theory and computation , volume=. 2014 , publisher=

    work page 2014

  69. [69]

    2018 , publisher=

    Density estimation for statistics and data analysis , author=. 2018 , publisher=

    work page 2018

  70. [70]

    Journal of Chemical Information and Modeling , volume=

    Impact of the unstirred water layer on the permeation of small-molecule drugs , author=. Journal of Chemical Information and Modeling , volume=. 2024 , publisher=

    work page 2024

  71. [71]

    Nature Reviews Physics , volume=

    Using metadynamics to explore complex free-energy landscapes , author=. Nature Reviews Physics , volume=. 2020 , publisher=

    work page 2020

  72. [72]

    Physical review letters , volume=

    Well-tempered metadynamics converges asymptotically , author=. Physical review letters , volume=. 2014 , publisher=

    work page 2014

  73. [73]

    Journal of chemical theory and computation , volume=

    Metadynamics with adaptive Gaussians , author=. Journal of chemical theory and computation , volume=. 2012 , publisher=

    work page 2012

  74. [74]

    Journal of chemical theory and computation , volume=

    Efficient sampling of high-dimensional free-energy landscapes with parallel bias metadynamics , author=. Journal of chemical theory and computation , volume=. 2015 , publisher=

    work page 2015

  75. [75]

    Journal of Chemical Theory and Computation , volume=

    Combining Fast Exploration with Accurate Reweighting in the OPES-eABF Hybrid Sampling Method , author=. Journal of Chemical Theory and Computation , volume=. 2025 , publisher=

    work page 2025

  76. [76]

    Accounts of Chemical Research , volume=

    One for All, All for One: A Unified Framework for Free-Energy Calculations , author=. Accounts of Chemical Research , volume=. 2025 , publisher=

    work page 2025

  77. [77]

    Smooth regression analysis , author=. Sankhy. 1964 , publisher=

    work page 1964

  78. [78]

    Theory of Probability & Its Applications , volume=

    On estimating regression , author=. Theory of Probability & Its Applications , volume=. 1964 , publisher=

    work page 1964

  79. [79]

    COMPSTAT 1982 5th symposium held at Toulouse 1982: Part I: Proceedings in Computational Statistics , pages=

    Updating formulae and a pairwise algorithm for computing sample variances , author=. COMPSTAT 1982 5th symposium held at Toulouse 1982: Part I: Proceedings in Computational Statistics , pages=. 1982 , organization=

    work page 1982

  80. [80]

    The American Statistician , volume=

    Algorithms for computing the sample variance: Analysis and recommendations , author=. The American Statistician , volume=. 1983 , publisher=

    work page 1983

Showing first 80 references.