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REVIEW 3 major objections 5 minor 85 references

From populations to absolute binding affinities in molecular simulations: exact volumetric terms and practical estimators

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper derives an exact relation between binding constants and population weights, Eq.

desk verdict Solid methods paper with a useful warning about single-bin estimators, but the 'exact' framing and the DgoT volume need qualifying. read the letter →

arxiv 2608.04834 v1 pith:G5SPIHMH submitted 2026-08-05 physics.comp-ph physics.app-phphysics.bio-phphysics.chem-phphysics.data-an

Davide Mandelli , Emiliano Ippoliti , Charles Plate This is my paper · ORCID
classification physics.comp-phphysics.app-phphysics.bio-phphysics.chem-phphysics.data-an
keywords bindingfreeenergyconstantstatisticalweightvolumetriccorrectionreactioncoordinatehistogramwell-temperedmetadynamicscucurbit[7]urilligand-protein
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's central claim is that the equilibrium binding constant in the dilute limit is exactly $$K = \frac{P_b}{P_u/V},$$ where $P_b$ and $P_u$ are the statistical weights of the bound and unbound states defined from the canonical density, and $V$ is the accessible volume. The authors derive this from the partition function in the explicit-solvent setting, which brings the unbound-state volumetric term out of the code and shows exactly how a volume restraint propagates into the affinity. From this single relation they build practical estimators that work off histograms of any reaction coordinate, in unrestrained or restrained simulations, with closed-form corrections for spherical, circular, and cylindrical geometries. Applied to the CB7/1-adamantanol host–guest complex and the galactonate–DgoT transporter, the theoretically grounded estimators agree with each other while the commonly used single-bin estimator deviates by about 1 kcal/mol. A sympathetic reader would take the paper to establish that the definition of the bound state and the unbound volume are controlled inputs, not hidden assumptions.

What carries the argument

The load-bearing object is the statistical weight $P_s = \int \rho(q,p)I_s(q,p)\,dq\,dp$, the integral of the canonical phase-space density over the region defining a thermodynamic state. The identity $K = P_b/(P_u/V)$ carries the argument: it turns a binding constant into a ratio of populations, so every volumetric effect reduces to the accessible volume of the unbound state, and an imposed restraint enters simply as $V \to V_R$. The practical estimators then run on histograms of a reaction coordinate $z$ or $r$, kernel density estimates, and pair correlation functions $g_3(r)$ or $g_2(r)$, which supply the convergence diagnostics (a flat plateau in $\bar\rho(z)$ or $g_3(r)\to 1$) that certify the homogeneous-unbound-state assumption.

What would settle it

Take any restrained binding simulation with a known experimental affinity and check whether the unbound density becomes exactly flat over the full interval used for the volume correction, and whether moving the unbound window within that plateau changes the final free energy by more than the statistical uncertainty; if it does, the homogeneous-unbound assumption is the cause. Alternatively, re-bin a fixed trajectory at two different bin widths and verify that the hybrid and single-bin estimators shift by exactly the predicted $-k_BT\log(\Delta z)$ term, while the extended-region estimator does not.

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Extended reading notes

Core claim

The central discovery is that Eq. (10), $K = P_b/(P_u/V)$, is exact under the stated dilute-limit assumptions, and that all practical binding-affinity estimators follow from it. The proof starts from the canonical partition function for $N_A$ ligands, $N_B$ receptors, and solvent, factorizes it in the dilute limit, and identifies the statistically dominant term; the equilibrium condition then yields the ratio of statistical weights. The volume dependence lives entirely in the unbound state, so the standard correction $-k_BT\log(VC_0)$ is not an appended term but a direct consequence of the normalization of $P_u$. For restrained simulations, the same relation becomes $K = P_b^R/(P_u^R/V_R)$ under the physically mild assumption that the unbound density is homogeneous over the restrained region, and the paper derives explicit $V_R$ formulas for spherical, circular, and cylindrical restraints. The applications show that integrated bound-state estimators (extended-region and hybrid) give mutually consistent free energies across restraint sizes, while the single-bin estimator consistently departs by about 1 kcal/mol, and that in a structured bound state the placement of the state boundary can shift the affinity by about $-0.6$ kcal/mol.

Load-bearing premise

The derivation's load-bearing premise is that the unbound state is homogeneous, so the probability density is constant over the entire restricted region; if residual correlations remain in that region, replacing the unbound integral with a constant density times the volume is inexact.

Editorial extensions

If this is right

  • Absolute binding affinities can be computed from existing PMF or histogram data by reweighting the bound and unbound populations, without additional simulations.
  • Single-bin estimators should be avoided whenever the bound state has internal structure; in the two molecular systems tested they are off by approximately 1 kcal/mol.
  • The volume in the standard-state correction is the unbound-state accessible volume, so restraint geometries must be designed so that $V_R$ (or $\Sigma\ell$) is well defined and the restraint is inactive in the bound state.
  • Comparing computed affinities to bulk equilibrium experiments requires an inclusive bound state; site-resolved experiments may validly target a single sub-basin, in which case a restricted definition is the correct target.

Reading between the lines

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

  • The construction is not tied to two-body complexes: the same population-ratio logic should yield exact volumetric terms for dimerization equilibria or multi-site binding, where the unbound state has a geometric volume (or area) of its own.
  • The size of the single-bin error is tied to how much bound-state weight a single histogram bin captures; for flexible ligands with multiple poses the deviation could plausibly exceed the 1 kcal/mol seen here.
  • A practical by-product is a consistency test for existing PMF-based pipelines: any implementation whose unbound reference is not the true accessible unbound volume should show a residual dependence of $\Delta F_0$ on the restraint radius, which this formalism can detect.
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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 derives the relation K = Pb/(Pu/V) from the canonical partition function in the dilute limit and uses it to construct practical estimators for binding free energies from histograms of a reaction coordinate. The estimators cover unrestrained and restrained simulations in spherical, circular, and cylindrical geometries, with analytical volumetric corrections. The framework is applied to a Lennard-Jones dimer, the CB7/1-adamantanol host–guest complex, and the galactonate–DgoT transporter system. The central numerical finding is that single-bin estimators deviate from integrated extended-region or hybrid estimators by about 1 kcal/mol in both molecular complexes. The paper also discusses how the bound-state definition should be matched to experimental resolution and introduces a distinction between full-access and partial-access restraining settings.

Significance. If the derivation is accepted, the paper offers a transparent, unified route from population histograms to absolute binding free energies, making the volumetric term explicit and the bound-state definition a controlled input. The Lennard-Jones test with three restraint radii is a clean verification that the volumetric correction cancels the restraint dependence exactly. The demonstration that single-bin estimators can be off by about 1 kcal/mol is practically important for the many methods that use such estimators. The discussion of full-access versus partial-access settings, and the resulting conditional nature of some predictions, is honest and useful. The main weaknesses are a rigor gap in the derivation of Eq. (10) and an untested geometric-volume assumption in the DgoT application; both affect load-bearing claims rather than presentation.

major comments (3)
  1. [SI S1.1, Eqs (S46)–(S51)] The derivation of Eq. (10) passes from the solvent average in Eq. (S46) to the pointwise identity in Eq. (S49) by asserting that the integrand is constant over the support of the solvent distribution. This replaces ⟨log f⟩ with log⟨f⟩ without a rigorous justification: the single-molecule partition functions z_A(S), z_B(S), and z_AB(S) are explicitly S-dependent, and the support of ρ(S) contains many solvent configurations. The final relation is standard and likely correct, but as written the central 'exact' claim rests on an unjustified step. Please either provide a thermodynamic-limit argument showing that fluctuations of the integrand vanish, or derive Eq. (10) directly from the ratio of the averaged partition functions, or qualify the exactness claim accordingly.
  2. [Galactonate–DgoT; SI S1.2.2, Eq (S92)] The DgoT estimator uses the geometric accessible volume V_R = πR²ℓ with R = 0.1 nm, which is exact only if the smooth confining potential is flat inside the cylinder and the unbound density is uniform over the whole cross-section. Unlike the CB7 system, no multi-radius test is reported for DgoT, and the functional form of the smooth potential is not given. The SI footnote in S1.2 explicitly acknowledges that a harmonic soft wall is only an approximation to the ideal reflective boundary and must be flat throughout the accessible region. The reported ΔF0 ≈ −1.1 kcal/mol therefore inherits an unquantified systematic error from this assumption. I request either a direct computation of the effective volume ∫e^{−βV_rest} d³r, a second restraint radius, or an explicit limitation statement with an estimated bound on the induced error.
  3. [Table 2] The extended-region estimates for CB7 are −17.639(4), −17.67(2), and −17.49(2) kcal/mol for R = 0.1, 0.2, and 0.3 nm. The spread of about 0.18 kcal/mol is several times the reported statistical uncertainties (0.01–0.04 kcal/mol). The text describes these values as consistent, but the residual scatter suggests a small systematic component not captured by the geometric volume correction, possibly from soft-wall penetration or an imperfect plateau. Please discuss this residual explicitly and quantify its implications for the claim that the volumetric correction is exact.
minor comments (5)
  1. [Eq (14)] The symbol F̃(x) is introduced for −kT log ρ(x) and then not used again; consider removing it or using it consistently when discussing the dimensional ambiguity of the profile.
  2. [Computational Details, galactonate–DgoT] The 'smooth confining potential' taken from Ref. [32] is not described in this paper; please provide its functional form or cite the specific section of Ref. [32] where it is defined, so that the geometric volume V_R = πR²ℓ can be checked against the actual potential.
  3. [Hybrid estimator, Eqs (31)–(32)] The choice of the single-bin unbound reference z_u* should be justified as lying in a genuinely flat region of the plateau; for DgoT, z_u* = 4.0 nm lies at the upper edge of the stated unbound window [3.8, 4.2] nm, so a brief sensitivity check would strengthen the result.
  4. [SI S2.2] There is a typo: 'implememnted' should be 'implemented' in the description of the Particle Mesh Ewald scheme.
  5. [Discussion, full-access vs partial-access] The last row of Table 3 shows a −0.6 kcal/mol shift when the bound-state definition is changed from the primary pose to the complementary region; this is discussed clearly, but it would help to state explicitly that this shift is of the same order as the single-bin discrepancy and therefore the absolute value of the DgoT affinity remains conditional on the experimental definition of the bound state.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. (10) and the practical estimators follow from the partition function via algebra and stated physical assumptions, with internal restraint-radius checks; the DgoT data reuse is a published external input, not a fitted prediction.

full rationale

The central relation, Eq. (10), is derived in SI S1.1 from the canonical partition function by isolating the dominant term in the NAB sum, applying the dilute-limit decoupling, and imposing the equilibrium condition; the paper explicitly notes its equivalence to the independent Woo-Roux result, Eq. (2) of Ref. [19]. The practical estimators (Eqs. 27-34) are obtained by substituting the restricted volume VR under the explicitly stated homogeneous-unbound assumption; no adjustable parameter is fitted to the quantity being predicted. The ~1 kcal/mol discrepancy between the single-bin and the extended-region/hybrid estimators is a formal consequence of whether one integrates the bound-state weight or evaluates a single histogram bin, not a back-fit of data. The CB7 and LJ systems are checked for restraint-radius independence (Tables 1 and 2), providing internal validation that the volumetric correction accounts for the restraint geometry. The DgoT application reuses a trajectory from Ref. [32], which includes overlapping authors, and the flat-restraint/homogeneous-unbound premise for that system is not independently re-tested here; the manuscript itself flags the soft-wall approximation in the SI S1.2 footnote ('the restraining potential must be flat (zero force) throughout the accessible region of interest'). This is a missing-support/correctness caveat rather than circularity, because the cited trajectory is a published external input and the theory's equations do not presuppose the target absolute affinities or the estimated single-bin shift. No uniqueness theorem, ansatz, or renamed empirical pattern is imported from the authors' prior work, and no fitted parameter is relabeled as a prediction. Therefore no circular step can be exhibited, and the derivation is self-contained under its stated assumptions.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

Central claim rests on the standard dilute-limit statistical mechanics of binding, plus the homogeneity assumption for the unbound state. No free parameters are fitted; the bound-state boundary and unbound reference position are user-chosen inputs, which the paper makes explicit. No invented entities.

free parameters (3)
  • Bound-state boundary (z range) = CB7: [-0.1, 0.1] nm; DgoT: [-0.2, 0.2] nm
    The DgoT affinity shifts by about -0.6 kcal/mol when the boundary is moved to z in [-0.2, 3.8) nm. The framework makes the choice explicit but it remains user-defined.
  • Unbound reference position z_u* = 1.57 nm (CB7), 4.0 nm (DgoT)
    Should be insensitive to choice if the unbound plateau is flat; the identity of the plateau region is user-assigned.
  • Histogram bin width Delta z = 0.023 nm (CB7), 0.031 nm (DgoT)
    Affects the hybrid and single-bin estimators; the final corrected affinity is formally independent of Delta z, but the intermediate Delta F_{u->b} and the volumetric term depend on it.
assumptions (5)
  • domain assumption Dilute-limit neglect of solute-solute interactions beyond the bound complex
    First invoked in SI S1.1 after Eq (S5): 'in the limit of ideal dilution ... we can set h_AA = h_BB about 0'. This underlies the factorization leading to Eq (S52) and hence Eq (10).
  • domain assumption Homogeneous unbound state with constant pair density rho_bar(r)=rho_bar in the unbound region
    SI S1.2.2 Eq (S69): 'Assuming that in the unbound state there are no interactions between the two molecules, one has rho_bar(r)=rho_bar=constant'. All volumetric corrections (Eqs 29, 32, 34) depend on this.
  • domain assumption Restraint is inactive in the bound state (full-access setting)
    SI S1.2 opening and Discussion: 'a crucial requirement for the validity of the corrected estimators ... is that the restraining potential must be inactive whenever the ligand occupies the bound state'. For DgoT this fails partially and shifts the affinity by -0.6 kcal/mol.
  • domain assumption Harmonic restraining potential can be treated as an ideal reflective wall
    SI S1.2: 'such a soft wall is itself an approximation to the idealized reflective boundary assumed here in the formal derivation'. The estimators rely on ideal-wall volume substitution.
  • standard math Stirling approximation and thermodynamic-limit dominance of one term in the partition-function sum
    SI S1.1 Eqs (S15)-(S20); standard textbook manipulations.

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Pith. "Pith review of From populations to absolute binding affinities in molecular simulations: exact volumetric terms and practical estimators." pith.science (2026). https://pith.science/paper/G5SPIHMH

@misc{pith2026260804834,
  author       = {Pith},
  title        = {Pith review of: From populations to absolute binding affinities in molecular simulations: exact volumetric terms and practical estimators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5SPIHMH}},
  note         = {Machine review of arXiv:2608.04834}
}
abstract

We present a statistical-mechanics framework for computing equilibrium binding constants $K$ in the dilute limit. From first principles, we derive a general expression relating $K$ to the relative populations of the bound and unbound states. Its transparency has twofold advantage: it makes the origin of the unbound-state volumetric term explicit, and it allows one to track exactly how an imposed volume restraint propagates through the expression. This makes $K$ directly computable, as restrained simulations can account for the volumetric contribution exactly, under the physically mild assumption of a homogeneous unbound state. The resulting estimators are computable from histograms of any suitably defined reaction coordinate, and determine unambiguously how the boundaries of the thermodynamic states of interest must be defined. We apply our framework to the cucurbit[7]uril/1-adamantanol host--guest complex and the galactonate--DgoT ligand--protein complex. Our results show that commonly used single-bin estimators depart from the theoretically correct one by $\approx 1$~kcal/mol in both systems. This shift originates in the definition of the bound state: by anchoring that definition to what state-of-the-art experiments resolve, the theory turns it from a hidden assumption into a controlled input, and provides a principled route to absolute binding affinities from molecular simulations.

Figures

Figures reproduced from arXiv: 2608.04834 by the authors.

Figure 1
Figure 1. (a) A spherically symmetric case: a small-molecule ligand binds to a soluble protein in aqueous solution in the absence of a well-defined binding pocket. (b) A circularly symmetric case: two membrane-embedded proteins dimerize within a lipid bilayer. (c) A cylindrically symmetric case: a ligand binds to the buried pocket of a membrane-embedded protein from the aqueous phase. In all panels, restraints are shown as tr… view at source ↗
Figure 2
Figure 2. (Left) View of the system used in simulations. The dimer (magenta) interparticle distance r is restrained within a sphere of radius R (green). (Right) (a) Probability histograms p(r). Dashed lines are quadratic fit of p(r) ∝ r 2 in the region r > 1.8σ. (b) Pair correlation function g3(r) on a logarithmic scale. The dashed horizontal line marks the asymptotic value of g3 = 1. (c) Free energy profile F(r) = −kBT log p… view at source ↗
Figure 3
Figure 3. (Left) Atomistic model of 1-adamantanol (top) and cucurbit[7]uril (bottom) in explicit water (omitted for clarity) used in simulations. Atoms are colored by element: carbon (cyan), oxygen (red), nitrogen (blue), and hydrogen (white). In the unbound state, the center-of-mass of 1-adamantanol is confined to remain within a cylinder of radius R (green). The arrow indicates the z reaction coordinate. (Right) (a) Probabi… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: (Left) A representative snapshot of the simulation system, showing the ligand GAL (in ball-and-stick representation, with atoms colored by element: carbon cyan, oxygen red and hydrogen white) inside the smooth confining potential (black lines) positioned along the bind…

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.