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arxiv: 2606.05541 · v1 · pith:BUVETKWHnew · submitted 2026-06-04 · ⚛️ physics.chem-ph · cond-mat.soft· q-bio.BM

Methods for Inferring Interaction Potentials from Cross-Linking Mass Spectrometry Data

B\"orries von Seggern , Mohsen Sadeghi This is my paper

Pith reviewed 2026-06-27 23:45 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.softq-bio.BM
keywords cross-linking mass spectrometryXL-MSinverse Henderson probleminteraction potential inferencephase separationcoarse-grained modelscoordination numbersoptimization algorithms
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The pith

Three optimization algorithms recover correct interaction parameters from XL-MS observables in a three-phase mixture.

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

The paper develops numerical methods to turn cross-linking mass spectrometry measurements into parameters for coarse-grained particle models of proteins, including cases where the mixture separates into multiple phases. It connects the task to the inverse Henderson problem and adapts existing inversion techniques such as iterative Boltzmann inversion together with new gradient-based and predictor-corrector variants. On ten-component test systems the methods work accurately for uniform fluids; the central demonstration is that Adam, low-density gradient descent, and exact-gradient Newton iteration still recover the input potentials when three distinct phases coexist. This matters because many functional protein assemblies and aggregates involve phase separation, so a direct route from experiment to model would let researchers simulate those assemblies without guessing the effective forces by hand.

Core claim

By framing the inference of pair potentials from XL-MS-derived coordination numbers and similar RDF functionals as an instance of the inverse Henderson problem, the authors show that several adapted optimization algorithms can parameterize multi-component interaction models even when the system undergoes three-phase separation. In homogeneous fluids every tested method converges efficiently; in the three-phase case Adam, gradient descent using the low-density derivative, and Newton iteration with the exact gradient each recover the original parameters to high accuracy.

What carries the argument

The adapted inverse Henderson framework that converts functionals of the radial distribution function (coordination numbers, etc.) into effective pair potentials via iterative inversion and gradient optimization.

If this is right

  • All tested algorithms reach high accuracy and efficiency for homogeneous fluids.
  • Adam, low-density gradient descent, and exact-gradient Newton iteration succeed on three-phase systems.
  • The resulting potentials can be used directly in coarse-grained simulations of phase-separated protein mixtures.
  • The same numerical pipeline applies to any set of RDF functionals obtained from experiment or simulation.

Where Pith is reading between the lines

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

  • The framework could be tested on experimental XL-MS datasets from known biomolecular condensates to check whether the inferred potentials reproduce observed droplet compositions.
  • If the low-density derivative approximation remains accurate at higher densities, it would simplify computations for larger mixtures.
  • Extending the method to include higher-order correlation functions might relax the uniqueness assumption in systems with fewer phases.

Load-bearing premise

XL-MS observables supply enough independent constraints to determine a unique set of interaction potentials despite the general non-uniqueness of inverting RDF functionals.

What would settle it

Apply one of the three successful algorithms to a synthetic three-phase ten-component system whose true potentials and resulting coordination numbers are known in advance; the recovered potentials must reproduce the input observables within statistical error when the model is re-simulated.

Figures

Figures reproduced from arXiv: 2606.05541 by B\"orries von Seggern, Mohsen Sadeghi.

Figure 1
Figure 1. Figure 1: FIG. 1. Representative plot of the potential given by eqn. 18 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Learning curves for the optimization algorithms introduced in Section II B, for values of (a) [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Similar to Fig. 2 for optimization in the multiphase system, employing MD simulations for the evaluation of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Value of [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Local density profiles in the homogeneous reference system. The species type is specified in the top-left corner of each plot, while the [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Local density profiles in the multiphase reference system. The species type is specified in the top-left corner of each plot, while the [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Snapshot from one of the reference simulations of the homogeneous reference system. The simulation box is outlined in blue and [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Snapshot from one of the reference simulations of the multiphase reference system. The simulation box is outlined in blue and [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Distributions of the local density in the homogeneous reference system. The species type is specified in the top-left corner of each [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Distributions of the local density in the multiphase reference system. The species type is specified in the top-left corner of each [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
read the original abstract

Cross-linking mass spectrometry (XL-MS) has emerged as a powerful quantitative technique for probing intra-protein structural information as well as protein-protein interactions at an unprecedented scale. XL-MS data yield information on the pairwise spatial proximity of proteins through inter-molecular linkers. However, systematic methods for adapting such data for coarse-grained interacting particle models remain limited. Predominant focus is put on directly fitting radial distribution functions (RDFs), while numerous observables, e.g. coordination numbers, which are functionals of the RDF, cannot be uniquely inverted. In this work, we develop a framework for parameterizing interaction potentials from such observables in potentially phase-separated mixtures, as encountered in XL-MS results. We establish a connection between this problem and the inverse Henderson problem and adapt algorithms such as Iterative Boltzmann Inversion and Iterative Monte Carlo to its numerical solution. We derive exact and low-density limit gradient approximations and propose two new algorithms based on an adaptation of the predictor-corrector~framework. In total, we evaluate several optimization algorithms on biologically realistic ten-component test systems. We demonstrate that for homogeneous fluids, all methods achieve exceptional efficiency and accuracy. Critically, we further demonstrate successful parametrization in a challenging three-phase system. Here, three algorithms, namely Adam and gradient descent employing the low-density derivative as well as Newton's method with the exact gradient, reliably recover the correct parameters. These results establish a clear pathway from XL-MS experiments to coarse-grained protein models for systems where phase separation governs biological function, potentially enabling new investigations of biomolecular condensates and protein aggregation.

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 / 1 minor

Summary. The manuscript develops methods to infer coarse-grained interaction potentials from XL-MS observables (e.g., coordination numbers as functionals of the RDF) in multi-component mixtures that may phase-separate. It connects the problem to the inverse Henderson problem, adapts Iterative Boltzmann Inversion and Iterative Monte Carlo, derives exact and low-density gradient approximations, proposes predictor-corrector adaptations, and evaluates multiple optimizers (including Adam, low-density gradient descent, and exact-gradient Newton) on synthetic ten-component test systems, claiming that three of these reliably recover the true parameters even in a challenging three-phase system.

Significance. If the central claim holds after addressing uniqueness, the work would supply a systematic route from XL-MS data to coarse-grained models for phase-separated biomolecular systems, enabling studies of condensates and aggregation where current direct RDF fitting approaches are limited.

major comments (2)
  1. [Abstract] Abstract: The manuscript states that observables such as coordination numbers 'cannot be uniquely inverted,' yet the central claim is that Adam, low-density gradient descent, and exact-gradient Newton 'reliably recover the correct parameters' in the ten-component three-phase system. Recovery of one parameter set does not establish that the chosen XL-MS functionals supply enough independent constraints to eliminate degeneracies; an explicit test ruling out physically distinct potentials that yield identical observables is required to support the claim.
  2. [Abstract] Abstract: The abstract reports successful recovery on synthetic test systems but supplies no quantitative error metrics (e.g., parameter deviations, objective-function values, or cross-validation errors) and no details on how the observables were generated from the target potentials, leaving the strength of the three-phase result difficult to assess.
minor comments (1)
  1. The description of the new predictor-corrector algorithms and their relation to the low-density derivative would benefit from a concise algorithmic pseudocode or flowchart to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We address each of the major comments below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The manuscript states that observables such as coordination numbers 'cannot be uniquely inverted,' yet the central claim is that Adam, low-density gradient descent, and exact-gradient Newton 'reliably recover the correct parameters' in the ten-component three-phase system. Recovery of one parameter set does not establish that the chosen XL-MS functionals supply enough independent constraints to eliminate degeneracies; an explicit test ruling out physically distinct potentials that yield identical observables is required to support the claim.

    Authors: The statement regarding non-unique inversion refers to the general case for functionals of the RDF, as is standard in the inverse Henderson problem. Our central claim is based on empirical recovery in the specific test systems considered, where the optimization algorithms converged to the target potentials. We acknowledge that this does not rigorously prove uniqueness for all possible potentials. In the revised manuscript, we will include an additional analysis testing for potential degeneracies by attempting to recover parameters from observables generated with slightly perturbed potentials and checking if distinct potentials can produce identical observables within numerical tolerance. revision: partial

  2. Referee: [Abstract] Abstract: The abstract reports successful recovery on synthetic test systems but supplies no quantitative error metrics (e.g., parameter deviations, objective-function values, or cross-validation errors) and no details on how the observables were generated from the target potentials, leaving the strength of the three-phase result difficult to assess.

    Authors: We agree that the abstract would be improved by the inclusion of quantitative error metrics to allow readers to better assess the results. In the revised manuscript, we will update the abstract to report key quantitative measures such as the deviations in the recovered interaction parameters and the values of the objective function achieved in the three-phase system. The generation of the synthetic observables is described in detail in the Methods section, where we explain that they are obtained by performing molecular dynamics simulations with the known target potentials and then computing the coordination numbers and other XL-MS functionals from the resulting particle configurations; we will include a brief reference to this in the abstract as well. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation and tests are self-contained

full rationale

The paper adapts known algorithms (IBI, IMC, gradient methods) to infer potentials from XL-MS functionals by connecting to the inverse Henderson problem, derives gradient approximations, and validates via forward simulation of observables from known potentials followed by recovery attempts. This is standard non-circular numerical testing. The abstract's non-uniqueness caveat is stated explicitly and does not enter the method definitions or test constructions as a self-referential loop. No load-bearing step reduces by definition, fitted-input renaming, or self-citation chain to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the numerical solvability of the inverse problem for the chosen observables and on the representativeness of the synthetic test systems; no new physical entities are introduced.

axioms (1)
  • domain assumption XL-MS observables can be treated as functionals of the radial distribution function that are sufficient for numerical inversion via adapted IBI/IMC methods in multi-component phase-separating fluids.
    Invoked when the paper states it develops a framework for parameterizing potentials from such observables.

pith-pipeline@v0.9.1-grok · 5818 in / 1230 out tokens · 21125 ms · 2026-06-27T23:45:13.001910+00:00 · methodology

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Reference graph

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