OTMol: Optimal Transport Molecular Alignment

• OTMol is a molecular alignment method that reformulates atom-to-atom matching as an optimal transport problem, ensuring chemically valid one-to-one correspondences.
• It fuses supervised label matching with Gromov–Wasserstein transport to integrate geometric and topological constraints, preserving bond connectivity and chirality.
• OTMol demonstrates superior scalability and accuracy over heuristic methods, effectively handling challenging cases like water clusters, neon clusters, and complex biomedical molecules.

Searching arXiv for OTMol and closely related optimal-transport molecular alignment work. OTMol is a molecular alignment method that formulates atom-to-atom correspondence and rigid superposition as an optimal transport problem rather than as a heuristic assignment followed by Kabsch alignment. Introduced in “Robust Molecular Structure Comparison via Optimal Transport” (Wei et al., 1 Sep 2025), it is designed for structural comparison tasks in which root-mean-square deviation (RMSD) is sensitive to atom ordering, molecular symmetry, indistinguishable cluster configurations, and chirality inversion. Its central technical claim is that meaningful RMSD requires a chemically valid one-to-one correspondence, and that such a correspondence can be obtained more robustly by fused supervised Gromov–Wasserstein optimal transport, which incorporates atom labels together with intra-molecular geometric and topological relations (Wei et al., 1 Sep 2025).

1. Problem formulation and motivation

RMSD is widely used to assess structural similarity in systems ranging from flexible ligand conformers to complex molecular cluster configurations, but its interpretation depends on the existence of a correct one-to-one atom mapping (Wei et al., 1 Sep 2025). If atom correspondence is incorrect, RMSD may become artificially large or artificially small. OTMol is intended to address precisely this prerequisite step.

The method targets three failure modes identified for conventional RMSD pipelines. First, RMSD depends on atom ordering: the same molecule may be represented with different atom orders by different software, databases, or prediction tools, and a wrong correspondence invalidates the comparison (Wei et al., 1 Sep 2025). Second, symmetric or indistinguishable cluster configurations, such as water clusters or noble gas clusters, admit many nearly equivalent permutations; a valid method must avoid splitting one molecule across several others or assigning atoms from one water molecule into multiple different water molecules (Wei et al., 1 Sep 2025). Third, chirality inversion can arise when reflections are allowed during rigid alignment, yielding a smaller RMSD for a mirrored but chemically invalid structure; the paper identifies this as a major defect of ArbAlign-style procedures (Wei et al., 1 Sep 2025).

These issues motivate OTMol’s reformulation of molecular alignment as an optimal transport problem. Rather than treating atom assignment as an isolated combinatorial step driven by a hand-designed inter-atom cost matrix, OTMol uses the molecule’s own internal geometry and topology to determine correspondence (Wei et al., 1 Sep 2025). This suggests a shift from “atom similarity” in the narrow sense to relation-preserving matching across whole structures.

2. Mathematical framework: from OT to fused supervised Gromov–Wasserstein

OTMol is built on Gromov–Wasserstein (GW) optimal transport. In the paper’s formulation, standard Kantorovich OT uses a cost matrix $C_{ij}$ between source and target atoms, whereas GW matches objects by preserving relations within each object rather than by comparing atoms directly across objects (Wei et al., 1 Sep 2025). For molecules, this is natural because molecular identity is encoded not only by atom labels but by local geometry and topology.

The paper states that GW seeks

$$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$

with

$$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$

Here $\mathbf P_{ij}$ is the probability that atom $i$ in molecule A matches atom $j$ in molecule B, and the objective penalizes mappings that fail to preserve pairwise distances across the two structures (Wei et al., 1 Sep 2025).

OTMol does not rely on GW alone. Its core formulation is fused supervised Gromov–Wasserstein (fsGW), which combines a supervised OT term enforcing label consistency with a GW term preserving structural geometry (Wei et al., 1 Sep 2025):

$$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$

The parameter $\alpha \in [0,1]$ controls the balance between label matching and structure matching. The supervised cost matrix satisfies

$$\mathbf C_{ij}=0 \text{ if labels match, } \infty \text{ otherwise,}$$

so chemically forbidden assignments are excluded (Wei et al., 1 Sep 2025). The distance matrix entering the GW term is defined as

$$\mathbf D = (1-c)\mathbf D_E + c\mathbf D_G, \qquad c\in[0,1],$$

where $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$0 is the Euclidean distance matrix and $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$1 is the graph geodesic distance matrix (Wei et al., 1 Sep 2025). Euclidean distances preserve 3D shape, whereas geodesic distances preserve bond and connectivity structure.

This formulation gives OTMol its “chemistry-aware” character in the paper’s sense: element or atom-type consistency can be imposed by $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$2, while internal geometry and bond topology are preserved by the GW component (Wei et al., 1 Sep 2025). A plausible implication is that the method is less dependent on local hand-crafted atom costs than Hungarian-style heuristics.

3. Construction of atom mappings and rigid alignment

The transport plan $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$3 produced by fsGW is a probability matrix rather than a permutation. OTMol therefore includes an explicit projection step that converts transport into a hard assignment. The paper defines

$$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$4

and then checks whether $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$5 is a valid permutation matrix (Wei et al., 1 Sep 2025). This validity test is essential because RMSD requires a one-to-one mapping and because it prevents many-to-one collapse.

Once a valid permutation is obtained, OTMol applies the Kabsch algorithm to compute the optimal rigid-body superposition and then the RMSD (Wei et al., 1 Sep 2025). Chirality preservation is enforced by using Kabsch without reflection, so only proper rotations are allowed. The paper emphasizes that this is crucial for ATP and the FGG peptide, where reflected alignments can reduce RMSD while inverting stereochemistry (Wei et al., 1 Sep 2025).

For single molecules with the same number of atoms, the pipeline is specified as follows (Wei et al., 1 Sep 2025). The inputs are coordinates $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$6, atom labels $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$7, and optional adjacency matrices $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$8. The cost matrix is constructed by label equality, the distance matrices are formed using $$\mathop{\arg \min}_{\mathbf P\in \Gamma} \sum_{i,j,k,l} (\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}$$9, and the method sweeps over

$$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$0

For each $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$1, fsGW is solved, $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$2 is projected to $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$3, invalid assignments are discarded, Kabsch is applied, and the best result is selected by lowest bond connectivity inconsistency (BCI) first and then lowest RMSD (Wei et al., 1 Sep 2025).

The prioritization of BCI reflects a substantive methodological choice. The paper states that the mapping should preserve bonds: a bond in the reference should still be a bond after mapping (Wei et al., 1 Sep 2025). This makes the accepted correspondence close to a graph isomorphism rather than merely an RMSD-minimizing permutation.

4. Specialized variants for clusters and highly symmetric systems

A notable feature of OTMol is that it does not use a single undifferentiated procedure for all molecular systems. The paper distinguishes among single molecules, noble gas clusters, and clusters of the same molecule, especially water clusters (Wei et al., 1 Sep 2025).

For noble gas clusters, label information is absent, so the supervised cost matrix is unnecessary. The method builds the Euclidean distance matrix, optionally raises it elementwise to a Hadamard power $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$4, solves GW, converts the transport plan to a permutation, applies Kabsch, and then refines the result by solving a Kantorovich OT problem on the post-alignment distance matrix:

$$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$5

where

$$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$6

The paper describes this as a two-stage procedure that is particularly useful for highly symmetric systems (Wei et al., 1 Sep 2025).

For clusters of the same molecule, especially water clusters, OTMol introduces a hierarchical alignment strategy because molecule-level integrity must be preserved. The problem is that atoms belonging to one water molecule must not be mapped across several different waters (Wei et al., 1 Sep 2025). The solution is to represent each molecule by a centroid or other representative coordinate, add perturbations to these representative coordinates to generate multiple candidate molecule-level assignments, solve GW or OT at the representative level, and then restrict atom-level matching to the assigned molecular partner by a block cost matrix (Wei et al., 1 Sep 2025). Kabsch is then applied, RMSD is computed, and the process is repeated over many perturbations.

The paper notes that the optimal atom-level correspondence may not arise from the optimal molecule-level correspondence, so multiple suboptimal molecule-level matchings are sampled by adding Gaussian noise before GW (Wei et al., 1 Sep 2025). This is especially important in symmetric systems such as water clusters, where naïve methods can achieve deceptively good RMSD values while producing mismatched water molecules.

5. Chemical constraints: chirality, bond consistency, and one-to-one molecular integrity

OTMol’s defining claim is not merely that it lowers RMSD, but that it does so while preserving chemically meaningful structure (Wei et al., 1 Sep 2025). Three constraints are emphasized.

The first is chirality preservation. By using Kabsch without reflection, OTMol prevents mirror inversions that would flip stereocenters (Wei et al., 1 Sep 2025). The paper contrasts this with ArbAlign-style procedures, where reflections among candidate rigid transforms may lead to chemically wrong but numerically favorable alignments.

The second is bond connectivity consistency. The use of graph geodesic distances $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$7 and BCI-based selection is intended to ensure that if two atoms are bonded in the reference, their mapped counterparts remain bonded after alignment (Wei et al., 1 Sep 2025). This is particularly important for peptides, sugars, lipids, DNA, and atmospheric clusters. The paper’s discussion treats BCI as a structural filter rather than as a secondary descriptive statistic.

The third is one-to-one molecular integrity in cluster comparison. For molecular clusters, OTMol enforces one-to-one mappings between entire molecules, thereby avoiding erroneous many-to-one alignments that can arise when equivalent monomers are present (Wei et al., 1 Sep 2025). In water clusters, this prevents a water molecule’s oxygen and hydrogens from being distributed across multiple waters (Wei et al., 1 Sep 2025).

These constraints distinguish OTMol from approaches that optimize only a geometric objective. The paper argues that lower RMSD alone can be misleading if achieved through bond-breaking mappings, high atom mismatch inconsistency (AMI), or chirality inversion (Wei et al., 1 Sep 2025).

6. Empirical evaluation across molecular systems

The paper evaluates OTMol on a broad set of systems that includes biomedical molecules, cyclic peptides, lipids, sugars, DNA, water clusters, atmospheric hydrates, and noble gas clusters (Wei et al., 1 Sep 2025). The listed systems include ATP, Imatinib, cyclic peptides, phospholipid DLP, unsaturated fatty acid EIC, sugar systems BGC and BGC_GLC, DNA (215D), water clusters, FGG tripeptides, S1-MA-W1 atmospheric hydrate, and neon clusters (Wei et al., 1 Sep 2025).

The reported quantitative findings are summarized below.

System category	Reported OTMol result	Reported comparison
Biomedical molecules	$$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$8 BCI on all aligned biomedical examples	ArbAlign often had substantial bond mismatches
ATP	OTMol preserved molecular integrity	ArbAlign produced mappings with mean BCI around $$\Gamma = \left\{\mathbf P \in \mathbb R_+^{n\times n} : \mathbf P\mathbf 1_n = \frac1n\mathbf 1_n,\; \mathbf P^T\mathbf 1_n = \frac1n\mathbf 1_n \right\}.$$9, $\mathbf P_{ij}$0 AMI, and chirality inversion in some cases
Imatinib	OTMol preserved molecular integrity	ArbAlign had $\mathbf P_{ij}$1 BCI and $\mathbf P_{ij}$2 AMI in the reported example
DLP	OTMol preserved molecular integrity	ArbAlign mean BCI around $\mathbf P_{ij}$3
BGC	OTMol preserved molecular integrity	ArbAlign mean BCI around $\mathbf P_{ij}$4
BGC_GLC	OTMol preserved molecular integrity	ArbAlign mean BCI around $\mathbf P_{ij}$5
EIC	OTMol preserved molecular integrity	ArbAlign mean BCI around $\mathbf P_{ij}$6
DNA	OTMol preserved molecular integrity	ArbAlign had $\mathbf P_{ij}$7 BCI in that table
Similar-dihedral cyclic peptides	RMSD below $\mathbf P_{ij}$8 for all pairs	$\mathbf P_{ij}$9 mismatched bond percentage
Water clusters	Preserved one-to-one molecular assignments	ArbAlign could obtain lower RMSD in some cases, but often by producing mismatched water molecules
Neon clusters	Aligned all pairs closely except one 200-atom case	Scaled much better than ArbAlign for larger systems

For biomedical molecules, OTMol achieved $i$0 BCI on all aligned biomedical examples, meaning bond connectivity was fully preserved (Wei et al., 1 Sep 2025). In contrast, ArbAlign often exhibited substantial bond mismatches: ATP around $i$1, Imatinib $i$2, DLP $i$3, BGC $i$4, BGC_GLC $i$5, EIC $i$6, and DNA $i$7 in that table (Wei et al., 1 Sep 2025). In ATP, ArbAlign also produced mappings with $i$8 AMI and chirality inversion in some cases; in Imatinib, the reported example had $i$9 AMI (Wei et al., 1 Sep 2025).

For cyclic peptides, OTMol gave RMSD below $j$0 for all pairs in the similar-dihedral set and preserved all bonds, corresponding to $j$1 mismatched bond percentage (Wei et al., 1 Sep 2025). For water clusters, the paper reports that OTMol generally outperformed ArbAlign in preserving one-to-one molecular assignments, even though ArbAlign could sometimes obtain a lower RMSD by producing mismatched water molecules (Wei et al., 1 Sep 2025). For neon clusters, OTMol aligned all pairs closely except one 200-atom case and scaled much better than ArbAlign for larger systems (Wei et al., 1 Sep 2025).

The paper also compares against ArbAlign EMD, where the Hungarian solver is replaced by Earth Mover’s Distance from POT. This variant is still generally worse than OTMol, which the authors interpret as evidence that changing the solver does not address the deeper limitation imposed by heuristic atom-cost design (Wei et al., 1 Sep 2025).

7. Computational properties, sensitivity analyses, and limitations

All three OTMol variants are stated to have theoretical time complexity $j$2 (Wei et al., 1 Sep 2025). The paper reports that OTMol is much faster than ArbAlign for large systems: ArbAlign runtime increases steeply beyond 100 atoms, whereas OTMol remains more scalable (Wei et al., 1 Sep 2025). This combination of cubic worst-case complexity and better empirical scalability is important to the method’s positioning as a generalizable structural comparison framework.

The paper includes several sensitivity studies. OTMol sweeps over $j$3, and for some systems different $j$4 values trade off label fidelity and geometry fidelity; in FGG, a higher-quality alignment was obtained at $j$5 than at $j$6 (Wei et al., 1 Sep 2025). The parameter $j$7, which mixes Euclidean and geodesic distances, is often set to $j$8 for strong bond/connectivity preservation in biomolecules and $j$9 for cyclic peptides and FGG (Wei et al., 1 Sep 2025). For noble gas clusters, multiple Hadamard powers $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$0 are tested, and RMSD does not always improve monotonically with increasing $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$1 (Wei et al., 1 Sep 2025). For water clusters, increasing the perturbation count $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$2 helps find better molecule-level mappings; for 10-PP1 and 10-PP2, RMSD decreased as $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$3 increased (Wei et al., 1 Sep 2025).

The method also has explicit assumptions and limitations. It assumes that the two systems are comparable under a one-to-one matching model at the relevant level, often assumes equal-size structures for the main single-molecule formulation, and depends on meaningful atom labels or graph structure when fsGW is used (Wei et al., 1 Sep 2025). The permutation extraction step is heuristic, based on row-wise argmax followed by a permutation check (Wei et al., 1 Sep 2025). The transport solution is not always exactly a permutation, so projection is necessary; the hyperparameters $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$4, $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$5, $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$6, and perturbation count $$\mathop{\arg \min}_{\mathbf P\in \Gamma}\ (1-\alpha)\langle \mathbf C, \mathbf P\rangle_F + \alpha \sum_{i,j,k,l}(\mathbf D^{\mathrm{A}}_{ik} - \mathbf D^{\mathrm{B}}_{jl})^2 \mathbf P_{ij}\mathbf P_{kl}.$$7 affect performance; and for highly symmetric clusters, multiple near-equivalent solutions may exist, making the final result sampling-dependent (Wei et al., 1 Sep 2025). The paper further notes that while OTMol scales better for large systems and is more chemically reliable, it can be more computationally intensive than a naïve single Hungarian solve for tiny systems (Wei et al., 1 Sep 2025).

Within molecular modeling and cheminformatics, OTMol is therefore best understood as a method for robust structural correspondence under geometric, topological, and chemical constraints. Its contribution is not the replacement of RMSD, but the replacement of brittle atom matching by a fused optimal transport formulation that is independent of atom ordering, robust to symmetry and indistinguishable cluster configurations, and chirality-preserving (Wei et al., 1 Sep 2025).