# From Atoms to Dynamics: Learning the Committor Without Collective Variables

Sergio Contreras Arredondo, Chenyu Tang, Radu A Talmazan, Alberto Megías, Cheng Giuseppe Chen, Christophe Chipot

► To cite this version:

Sergio Contreras Arredondo, Chenyu Tang, Radu A Talmazan, Alberto Megías, Cheng Giuseppe Chen, et al. From Atoms to Dynamics: Learning the Committor Without Collective Variables. Nature Computational Science, 2026, 10.48550/arXiv.2507.17700. [https://doi.org/10.1038/s41540-026-00546-9]

HAL Id: hal-05469657

https://hal.science/hal-05469657v1

Submitted on 21 Jan 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
From Atoms to Dynamics: Learning the
Committor Without Collective Variables

Sergio Contreras Arredondo<sup>1</sup>, Chenyu Tang<sup>1</sup>, Radu A. Talmazan<sup>1</sup>,
Alberto Megías<sup>2</sup>, Cheng Giuseppe Chen<sup>1</sup>, Christophe Chipot<sup>1,3,4*</sup>

¹*Laboratoire International Associé Centre National de la Recherche Scientifique et University of Illinois at Urbana-Champaign, Unité Mixte de Recherche n°7019, Université de Lorraine, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France.

²Complex Systems Group and Department of Applied Mathematics,
Universidad Politécnica de Madrid, Av. Juan de Herrera 6, E-28040 Madrid,
Spain.

³Department of Biochemistry and Molecular Biology, University of Chicago,
Chicago, USA.

⁴Theoretical and Computational Biophysics Group, Beckman Institute, and
Department of Physics, University of Illinois at Urbana-Champaign, Urbana,
USA.

**Abstract**

We introduce a graph-neural-network architecture built on geometric vector perceptrons to predict the committor function directly from atomic coordinates, bypassing the need for hand-crafted collective variables (CVs). The method offers atom-level interpretability, pinpointing the key atomic players in complex transitions without relying on prior assumptions. Applied across diverse molecular systems, the method accurately infers the committor function and highlights the importance of each heavy atom in the transition mechanism. It also yields precise estimates of the rate constants for the underlying processes. The proposed approach assists in understanding and modeling complex dynamics, by enabling CV-free learning and automated identification of physically meaningful reaction coordinates of complex molecular processes.

<footnote>*Corresponding author(s). E-mail(s): chipot@illinois.edu;</footnote>
Introduction

A wide variety of computational methods and strategies have been developed to address rare
transitions in complex molecular systems¹. Many foundational concepts can be illustrated
using a prototypical model featuring two metastable states, *A* and *B*, within the framework
of transition path theory². A particularly important quantity in this context is the committor
function, which quantifies the probability that a trajectory starting from any given configu-
ration will reach state *B* before visiting state *A*³. The committor is often regarded as the
optimal one-dimensional reaction coordinate (RC), capturing the progress of the system dur-
ing the transition between *A* and *B*⁴,⁵. Many approaches have been put forth to estimate the
committor, from shooting strategies⁶⁻⁹, to the application of a variational principle expressed
in terms of its time-correlation function¹⁰,¹¹. More recently, machine learning (ML) methods
have emerged as a promising avenue for discovering complex RCs expressed as nonlinear
functions of the CVs ¹²⁻¹⁴. Furthermore, to enhance sampling of the transition, biasing poten-
tials are applied to a small set of CVs. These CVs are functions of the atomic positions, and
serve as a low-dimensional embedding that preserves the dynamic features of the transition.
Said differently, an effective CV should, in general, discriminate between different confor-
mational states and ideally capture the slow modes of the system ¹⁵,¹⁶. The choice of CVs has
conventionally been guided by physical and chemical intuition, often relying on predefined
descriptors such as interatomic distances, torsional angles, or more complex combinations
thereof. However, this selection based on intuition often lacks quantitative criteria to assess
their suitability to encapsulate the correct dynamics ¹⁷. Unfortunately, low-dimensional mod-
els of the RC are not always sufficient to describe transitions in complex molecular systems
with adequate accuracy¹⁵. To address these limitations, empirical methods have recently been
replaced by advanced ML approaches, including artificial neural networks (ANNs), which
enable a nonlinear combination of large sets of descriptors to define flexible CVs¹⁸. Still,
these methods rely on the selection of appropriate input features that, collectively, contain
the essential information about the transition at hand. Lately, much effort has been invested
in learning meaningful CVs directly from Cartesian coordinates¹⁹,²⁰, yet often resulting in
a substantial increase of the computational cost during molecular simulations. In this work,
we propose to bypass the use of explicit CVs altogether by learning the committor function
directly from Cartesian coordinates, $x = \{x_i\}_{i=1}^N \in \mathbb{R}^{3N}$. Toward this end, we employ graph
neural networks (GNN), which can extract relevant geometric and physical information from
molecular structures in a data-driven fashion²¹. GNNs have become a central tool in ML for
structured data, enabling the modeling of relationships in systems ranging from social net-
works to molecules. Early applications in molecular science have leveraged GNNs primarily
for their relational reasoning capabilities, treating atoms or residues as nodes, and encoding
interactions through edge features, such as distances or chemical bonds²². However, early
GNN architectures lacked a consistent way to incorporate and transform geometric infor-
mation, for instance, orientation and spatial symmetry, essential for the accurate description
of molecular objects. This limitation has been addressed by the development of equivariant
GNNs models that respect symmetry in Euclidean space, recently with the implementation of
the more efficient and scalable geometric vector perceptron (GVP) GNNs (GVP-GNNs)²³.

The GVP-GNN architecture meets the need for simultaneous geometric and relational
reasoning by augmenting message-passing GNNs with GVPs, which operate on both scalar
features, $S \in \mathbb{R}^n$, and vector features, $V \in \mathbb{R}^{\nu \times 3}$. These transformations preserve rotation
and reflection equivariance on the vector features, while keeping the universal approximation of continuous equivariant functions²³. A key innovation in later versions of GVPs lies in the inclusion of vector gating, allowing scalar information to modulate vector channels without breaking equivariance. This makes GVP-GNNs highly expressive, while remaining computationally tractable. This architecture has demonstrated strong empirical performance across diverse structural biology tasks, such as protein-model quality assessment and computational protein design, often outperforming models based on convolutional or purely scalar GNN layers²⁴.

The ability of GVP-GNNs to learn functions that are sensitive to spatial structure and symmetry makes them particularly well-suited for computing physically meaningful quantities in molecular systems, such as the committor, which is inherently dependent on the spatial arrangement of atoms. Approaches for estimating the committor that rely on predefined CVs¹⁶,²⁵ may miss relevant geometric correlations, a shortcoming that can be overcome using the equivariant and relational capacity of GVP-GNNs.

In this Brief Communication, we introduce a methodology for learning the committor, $q$, leveraging the power of GNNs. To achieve this goal, we make use of the well-established committor variational principle⁵,¹⁰,¹⁶,²⁵, based on the functional optimization of a time-correlation function as explained in qGNN architecture of Methods. This variational principle, as well as the proper boundary conditions of the committor definition defines the loss function (see equation (2)) that penalizes the learning of a GVP-GNN model that approximates the committor function in the $\mathbb{R}^{3N}$ space, which we will call the committor-GNN (qGNN). A sketch of the qGNN learning is depicted in Fig. 1. This framework paves the way to the automated, data-driven discovery of RCs in complex molecular systems. We illustrate the generality of our approach across a variety of molecular processes of increasing complexity, including the conformational equilibria of two short peptides, namely $N$-acetyl-$N'$-methylalanylamide (NANMA)—also known as dialanine or alanine dipeptide—and a capped trialanine, a Diels—Alder reaction²⁶ and the reversible folding of Trp-cage²⁷,²⁸. For these different molecular systems, in which the qGNN architecture is maintained identical, we obtain accurate committor predictions based on raw atomic coordinates, and provide meaningful atom-level insights into the RC by means of sensitivity analysis (see Sensitivity analysis of Methods) as we discuss in Results and Discussion.

# Results and Discussion

## NANMA isomerization

The first application of our methodology is the all-familiar isomerization of NANMA in vacuum, at 300 K. The transition between the $C_{7\text{eq}}$ and $C_{7\text{ax}}$ conformations has been studied previously¹⁶,²⁵, choosing the backbone dihedral angles, $\phi$ and $\psi$, as the features of the ANNs. In stark contrast, use will be made here of the full set of heavy-atom Cartesian coordinates.

Our results are gathered in Fig. 2, where the learned committor is projected onto the $(\phi, \psi)$-subspace for visualization purposes. As depicted in Fig. 2A, the resulting committor map, learned from a biased simulation (see Simulation details of Methods), agrees with previous findings obtained using feed-forward neural networks¹⁶,²⁵, and with numerical results from shooting strategies²⁹ shown in Fig. 2C and Fig. 2D, respectively. The committor map
exhibits a well-defined separatrix, meaning, the $q = 0.5$ isocommittor hyperplane when projected onto the $(\phi, \psi)$ subspace$^{16}$. Additionally, the node-sensitivity analysis, summarized at Fig. 2B, highlights the key heavy atoms that contribute predominantly to the transition process. As expected, our findings confirmed that the most important atoms for the model are the four (14, 0, 16, and 12) that define the dominant dihedral angle, namely $\phi$. Furthermore, the $\psi$ torsional angle, formed by atoms 14, 0, 16 and 2, is also found to be relevant for the transition, as revealed by our analysis.

## Trialanine conformational equilibrium

A similar application was conducted on trialanine. The dihedral angles, $\phi_1$, $\phi_2$, and $\phi_3$, have been reported to be the important CVs to describe the conformational transitions between metastable states $A$ ($\phi_1 = 60, \phi_2 = -70, \phi_3 = 60$) and $B$ ($\phi_1 = -70, \phi_2 = 60, \phi_3 = -70$)$^{30,31}$.

Our results are presented in Extended Data Figure 1, where the inferred committor is projected onto the $(\phi_1, \phi_2)$-, $(\phi_2, \phi_3)$-, and $(\phi_1, \phi_3)$-subspaces for visualization purposes. The basins associated with both metastable states $A$ and $B$ are clearly identifiable in this representation. The node-sensitivity analysis, shown in Extended Data Figure 1B, highlights the atoms that are most relevant to the transition. Consistent with previous findings$^{30,31}$, the putative three dominant backbone dihedral angles, $\phi_1$ (atoms 6, 8, 10, and 16), $\phi_2$ (atoms 16, 18, 20, and 26), and $\phi_3$ (atoms 26, 29, 30, and 36), play a central role as confirmed by our analysis. The sensitivity profile supports the notion that these torsional angles are the primary degrees of freedom driving the conformational transition in trialanine.

## Diels–Alder reaction

We have applied the methodology to the reaction of ethylene and but-1-en-3-yne (vinyl-acetylene) to form cyclohexa-1,2-diene, which proceeds via a Diels-Alder mechanism, ubiquitous in organic chemistry. This [4+2] cycloaddition reaction necessitates a quantum-chemical treatment to investigate the creation of the two carbon-carbon bonds. To accelerate sampling, the semi-empirical GFN2-xTB level of theory was chosen as the quantum-chemical framework, which has proven accurate in Diels-Alder reaction benchmarks$^{32}$. A more detailed description of the simulations is available in Simulation details of Methods.

The initial sampling was performed by biasing the two distances that represent the newly formed bonds, $d_1 = \|\mathbf{x}_0 - \mathbf{x}_2\|_2$ and $d_2 = \|\mathbf{x}_5 - \mathbf{x}_8\|_2$, where $\|\mathbf{x}_i - \mathbf{x}_j\|_2$ refers to the Euclidean distance between the Cartesian coordinates of atoms i-th and j-th, represented in Extended Data Figure 2D.

The results in Extended Data Figure 2A show the learned committor, with the position of its separatrix in line with earlier theoretical findings$^{26}$, and in agreement with the results obtained with the CV-based ANN reported in Ref. 16, as shown in Extended Data Figure 2C. The node-sensitivity analysis, the results of which can be found in Extended Data Figure 2B, brings to light the four atoms (0, 2, 5, and 8) that participate in the newly created bonds, indicating that the qGNN can discriminate the relevant internal coordinates, namely a pair of distances, amid the Cartesian coordinates. It is noteworthy that atoms 1 and 4 are also found to be substantial, forming with atom 0 a 180° angle in the reactant state, which evolves to 120° in the product state.
Trp-cage reversible folding

As a final application to a realistic biological problem, and to assess the generalizability of our approach, we studied the reversible folding of the 20-residue protein Trp-cage in an explicit aqueous solvent<sup>28,33</sup>. Toward this end, use was made of molecular dynamics (MD) trajectories kindly provided by D.E. Shaw research<sup>28</sup>. We refer the reader to the original work for a full detailed description of the simulations. The trajectory spans 208 µs, and captures multiple folding and unfolding events with atomistic resolution.

Trp-cage folds within approximately 14 µs, making it a challenging system to benchmark
data-driven approaches for folding mechanisms and committor predictions. Just like with the
previous molecular systems, we used the atomic configurations to construct molecular graphs,
but, for simplicity, only considered the Cartesian coordinates of the Cα atoms, thus provid-
ing a coarse-grained, yet informative representation of the protein structure. The qGNN was
fed with this subset of atoms to learn the committor. Moreover, results for the full set of
heavy-atom coordinates, together with a comparison to the committor learned from varia-
tional committor network<sup>16</sup>, are provided in Supplementary Figure 15 in the Supplementary
Information (SI) and show good agreement with the results presented here.

Due to the lack of a reference committor for this system, we initially chose to project the learned committor onto the RMSD and the end-to-end distance, $d_{\text{end}} \equiv \|\mathbf{x}_1 - \mathbf{x}_{261}\|_2$, for ease of visualization. As depicted in Extended Data Figure 3A, the committor projected onto these two coordinates reveals a clear separatrix, as well as the two regions associated with the folded ($q = 1$) and unfolded ($q = 0$) states. In Extended Data Figure 3B, we present the node-sensitivity analysis, alongside a molecular representation of Trp-cage with the top four most influential atoms highlighted. Based on this information, we computed the edge sensitivity—defined in terms of pairwise distances—among these four most important atoms. The rendered image of the protein also indicates the top three distances ranked by edge sensitivity (see Supplementary Figure 7 of the SI). While the differences in importance are not as pronounced as in the other systems examined in this Brief Communication, it is worth noting that the first four atoms are involved in three physically meaningful distances, that is, an end-to-end distance proxy, $d_1 \equiv \|\mathbf{x}_{15} - \mathbf{x}_{261}\|_2$, a distance related to the extension of the helix, $d_2 \equiv \|\mathbf{x}_{15} - \mathbf{x}_{73}\|_2$, and an intermediate distance associated with the separation between the two arms of the protein, $d_3 \equiv \|\mathbf{x}_{73} - \mathbf{x}_{177}\|_2$ (see Extended Data Figure 3C). Based on an edge-sensitivity analysis of the distance features (see Supplementary Figure 11 of the SI), we find that the learned model identifies $d_1$ and $d_2$ as the most informative ones for describing the folding process. Accordingly, in Extended Data Figure 3D, we project the committor map onto the ($d_1, d_2$)-subspace. Additional projections involving $d_3$ can be found in Supplementary Figure 13 of the SI.

While the (d₁, d₂)-subspace is expected to be informative based on the sensitivity anal-
ysis, the qGNN operates over the full Cartesian-coordinate space. To explore the structural
diversity near the transition region, we performed a clustering analysis using the K-means
algorithm<sup>34</sup>, focusing on structures with committor values falling in the (0.45, 0.55) range,
which approximates the separatrix. This analysis, detailed in Supplementary Section 4 of the
SI, revealed four representative structural clusters within the transition ensemble, illustrated
in Extended Data Figure 3E. We observe that all the transition states are characterized by
a loss of secondary structure with respect to the native folded state, notably in the region
encompassing residues 7–16, while the structural features at both ends of the mini-protein are essentially preserved.

In Extended Data Table 1, we gather all computed rate constants for the different systems, and compare them with published values. As there is no available experimental estimate for the Diels–Alder reaction, we used the Eyring equation to compute this quantity from previously reported coupled-cluster calculations²⁶.

Finally, to further demonstrate the scalability of our approach, we applied the qGNN model to the reversible folding of the 35-residue protein villin, using the trajectory reported in Ref. 28, where the graphs are built on all heavy atoms. This system represents a more complex conformational transition compared to Trp-cage, involving 470 heavy atoms. The qGNN model is able to learn the committor directly from the atomic coordinates, rendering an accurate estimate of the transition rate (see Extended Data Table 1) and providing atom-level interpretability through sensitivity analysis. All methodological details and results for this system are provided in Supplementary Section 5 of the SI.

## Discussion

In this Brief Communication, we introduce a GVP-GNN architecture designed to learn the committor function directly from atomic coordinates, emancipated from the need of predefined, human-intuited CVs. The resulting qGNN model captures both spatial and relational information relevant for the dynamics of the molecular system at hand. To evaluate the performance of our approach, we trained our model on MD simulations of four prototypical molecular processes of increasing complexity, namely the conformational equilibria of NANMA and trialanine, the formation of cyclohexa-1,2-diene, and the reversible folding of Trp-cage. In all cases, our model produces an accurate and physically meaningful rendering of the underlying dynamics, with committor-derived rate constants consistent with known experimental or theoretical studies. The CV-independence of the qGNN, combined with a clustering analysis, allows transition-ensemble structures to be studied directly in the full Cartesian-coordinate space, as demonstrated in the case of the Trp-cage reversible folding. This approach avoids loss of information associated with CV-based dimensionality reduction. In addition, the node-sensitivity analysis have proven able to discriminate between the most relevant atoms in the transitions, providing valuable information for the definition of physically interpretable CVs, thereby helping improve our understanding of the underlying molecular mechanisms. Altogether, these results highlight the power and generality of our model to learn complex dynamical features from molecular simulations, offering a scalable and interpretable tool for RC discovery in high-dimensional systems, with an architecture resilient to the complexity of the molecular system. The committor-based framework presented here could, in principle, be generalized to systems with more than two metastable states by learning a set of state-resolved committor functions. These multistate committors would provide a natural foundation for building Markov state models, enabling a systematic characterization of slow dynamical processes, key metastable states, transition hubs, and dominant pathways³⁵. Although qGNN effectively addresses the dependence on pre-defined CV in committor learning, in contrast to previous works¹⁶,²⁵, we acknowledge that the present work still depends on exhaustive sampling, including rare transitions, which is typically obtained using CV-based enhanced-sampling schemes. While our approach does
not directly address the challenge of designing relevant CVs for rare-event sampling, it provides a flexible Cartesian-coordinate-based formalism that renders an accurate reproduction of the committor, which can be leveraged to infer physically interpretable CVs towards enhanced-sampling simulations. Alternatively, the committor itself can be employed readily as a CV to enhance sampling of rare events$^{11,14}$. Other ongoing efforts aim at learning committors and transition pathways directly from Cartesian coordinates, in the spirit of recent ANN-based approaches$^{25}$, thereby enabling future integration with adaptive-sampling and CV-optimization strategies.

## Methods

Here, we introduce the theoretical foundations and technical attributes of the qGNN, as well as the computational details of the MD simulations carried out in this work.

### qGNN architecture

The qGNN learning framework builds upon the well-established committor variational principle$^{5,10,16,25}$, based on the functional optimization of a time-correlation function. This variational principle arises from the net forward scalar flux, $J_{AB}$, from a reactant state A to a product B, which can be written in the context of TPT as$^5$

$$ 
\begin{aligned}
J_{AB}[q; \tau] &= \frac{C[q; \tau]}{\tau}, \\
C[q; \tau] &= \frac{1}{2} \langle (q(\tau) - q(0))^2 \rangle,
\end{aligned} 
\quad (1) $$

where $\tau$ is the time lag and $q$ fulfills the boundary constraints $q(\mathbf{x}) = 0$ if $\mathbf{x} \in A$, and $q(\mathbf{x}) = 1$ if $\mathbf{x} \in B$, consistent with its definition. The committor minimizes $J_{AB}[q; \tau]$ in functional terms satisfying the necessary condition $\delta J_{AB}[q; \tau]/\delta q = 0$, where the choice of $\tau$ guarantees Markovianity of the transition mechanism. Hence, the time-correlation function $C[q; \tau]$ serves as a loss function in the learning process. A more detailed derivation of the variational principle can be found in Ref. 5 and in Supplementary Section 1 of the SI. Hence, given an output of the qGNN, $q_\omega$, we define the following loss function

$$ 
\begin{aligned}
\mathcal{L}[q_\omega; \tau] &= 2C[q_\omega; \tau] + \lambda \mathcal{L}_{AB}[q_\omega], \\
\mathcal{L}_{AB}[q_\omega] &= |q_\omega(\mathbf{x})|_{\mathbf{x} \in A}^2 + |q_\omega(\mathbf{x}) - 1|_{\mathbf{x} \in B}^2,
\end{aligned} 
\quad (2) $$

where $\lambda \in \mathbb{R}^+$ is a positive scalar the value of which is selected beforehand, and $\omega$ represents the set of parameters of the qGNN model. Essentially, this loss penalizes the temporal correlation between committor values along both biased and unbiased trajectories$^{16,25}$, encouraging the model to find functions that minimize this correlation, while imposing the boundary conditions. Said differently, we optimize the qGNN towards outputting the true committor,

$$ q = \arg \min_{q_\omega} \mathcal{L}[q_\omega; \tau]. \quad (3) $$
Owing to the use of biasing forces in the simulations, corrupting the dynamics of the process at hand, reweighting is necessary¹²,¹⁶ to determine the unbiased time-correlation functions,

$$ \langle f(\tau)g(0) \rangle = \frac{\langle e^{\delta W(\tau)/2k_B T} f(\tau) e^{\delta W(0)/2k_B T} g(0) \rangle_b}{\langle e^{\delta W/k_B T} \rangle_b} \quad (4) $$

where the subscript b stands for the average using the biased ensemble, $\delta W(z)$ is the perturbed potential of mean force (PMF), and, $f$ and $g$ are any functions³⁶. To guarantee accurate estimates, we only use at the learning stage the part of the simulation that corresponds to a converged free energy. Moreover, as discussed in previous works¹⁶,²⁵, the validity of the reweighting scheme is limited to small values of the time lag, $\tau$. This limitation was demonstrated in the SI of Ref. 25 for several cases, including a random-walk hopping model on a lattice, the Smoluchowski equation, and first-order perturbative expansions of the propagator in both overdamped and underdamped Langevin dynamics. The present reweighting approach has been validated in systems where the qGNN was trained on biased simulations, such as NANMA isomerization, trialanine conformational equilibria, and the Diels-Alder reaction. In these cases, the computed rate constants agree well with previous estimates (see Extended Data Table 1), and for NANMA, the committor function quantitatively matches unbiased numerical estimates (Fig. 2). Nevertheless, alternative approaches exist in the literature for estimating unbiased time-correlation functions from biased trajectories, such as the Girsanov reweighting formula for path ensembles³⁷,³⁸, but have limitations of their own.

To compute the committor function using our qGNN model, we follow the workflow summarized in Algorithm 2 in the SI, which maps molecular configurations to graph-level predictions. We start by constructing molecular graphs, $\mathcal{G}$, from atomic configurations extracted from MD trajectories. Thus, for every conformation we define $\mathcal{G} = (\mathcal{N}, \mathcal{E}, \mathcal{A})$, where $\mathcal{N}$ and $\mathcal{E}$ denote the sets of node and edge features, respectively, and $\mathcal{A}$ encodes the molecular connectivity. In other words, from the configuration $\{\mathbf{x}_i\}_{i=1}^N$ we construct a graph $\mathcal{G}$ in which both nodes and edges are endowed with scalar and vector features, that is, $\mathcal{N} = (S_n, V_n)$ and $\mathcal{E} = (S_e, V_e)$. Next, node and edge features are propagated into a high-dimensional latent space using GVP layers, represented by the operator $\text{GVP}[\cdot]$, which processes jointly scalar and vector components while respecting geometric equivariance. This operator maps a given pair $(S, V) \in \mathbb{R}^n \times \mathbb{R}^{\nu \times 3}$ into another pair in a higher-dimensional subspace, that is, $(S, V) \in \mathbb{R}^n \times \mathbb{R}^{\nu \times 3} \xrightarrow{\text{GVP}} (S', V') \in \mathbb{R}^m \times \mathbb{R}^{\mu \times 3}$, with $m > n$ and $\mu > \nu$. In the next step, information is propagated through the graph using a message-passing scheme²³, where each node updates its representation based on the features of its neighbors and connecting edges. For the last step, to obtain a graph-level prediction that represents the committor of each configuration, we apply a permutation-invariant pooling operation, which, in our implementation, simply consists of a summation over the updated node embeddings. This pipeline enables the model to learn meaningful global features that are then associated with the committor function.

It is noteworthy that the architecture of the qGNN proposed herein is identical for all the molecular systems investigated in this Brief Communication. However, the number of nodes of the graph is based on the heavy atoms of the molecular object at hand. In our architecture, we select $n = \nu = 1$ for the initial space of tuples, as we are only considering one scalar
and one vector feature per node and edge, and $m = 2\mu = 16$ for the final space of tuples under the GVP action on the nodes and edges. Additionally, we use tanh as the activation function in the GVP-GNNs, and all the weights in the neural network are initialized according to a Xavier initialization$^{39}$. The hyperparameters of the optimization procedure—through an Adam optimizer$^{40}$, according to the loss function in equation (2), are maintained across the different molecular systems, except for the constant $\lambda$ defined in equation (2). However, even though $\lambda$ is different for each case, a wide range of values of such a parameter leads to comparable results, as explained in Supplementary Section 3 of the SI. For the optimization procedure according to the variational principle, we define the qGNN in a Siamese-like way, as proposed previously in Refs. 16, 25. Further details on GVP-GNN architectures can be found in Supplementary Section 1 of the SI.

## Sensitivity analysis

To identify and interpret the most relevant structural elements for the learned committor, we performed a node-sensitivity analysis on each system. This analysis reflects the importance of a given node—an atom of the molecular object of interest—for the prediction of the committor value. Therefore, given an MD trajectory of $\ell$ time steps, we can define the graph-trajectory $\mathcal{T} = \{\mathcal{G}_j\}_{j=1}^\ell$, where $\mathcal{G}_j \equiv \mathcal{G}(t_j)$ is the graph for the configuration at time $t_j$, $j = 1, \dots, \ell$. Next, the absolute sensitivity of the $i$-th element, either node or edge, of the graph related with the atomic structure, $s_i$, is computed as

$$s_i = \frac{1}{\ell} \sum_{j=1}^{\ell} \left\| \left( \frac{\partial q(\mathcal{G})}{\partial \mathbf{V}_i} \right)_{\mathcal{G}=\mathcal{G}_j} \right\|_2, \quad (5)$$

where $q$ is the committor map obtained from the qGNN, $\mathbf{V}_i$ is the vector component of the $i$-th element for which we are computing the sensitivity, and $\| \cdot \|_2$ is the Euclidean norm. From this absolute sensitivity one can define a relative sensitivity as $s_i/\max_j\{s_j\}$. That is, the subelement of maximum sensitivity will be related to a relative sensitivity of 1, whereas the remaining subelements will take lower values in the interval $[0, 1]$. The relative sensitivity of the nodes will be utilized hereafter to examine the importance of each atom in the molecular process of interest.

## Simulation details

**Molecular dynamics simulations.** In all the studied processes, except for the Trp-cage reversible folding, we performed simulations using the NAMD package$^{41}$ in conjunction with the Colvars library$^{42}$. The enhanced sampling was generated using the well-tempered meta-dynamics extended adaptative biasing force (WTM-eABF) algorithm$^{43}$ on selected CVs with an integration time step of 0.5 fs. Moreover, for all the simulated molecular objects, covalent bonds involving hydrogen atoms were constrained using the RATTLE algorithm$^{44}$.

**NANMA isomerization.** NANMA in vacuum was simulated using the CHARMM22 force field$^{45}$. 100 ns of sampling were generated using as CVs the RMSD with respect to $C_{7\text{eq}}$ and $C_{7\text{ax}}$, limited to the heavy atoms (atoms 0, 1, 2, 4, 8, 12, 13, 14, 16, and 18). The range of the CVs extended from 0.1 Å to 1.8 Å with a bin width of 0.05 Å, and the harmonic walls were set at 1.8 Å with a force constant of 1 kcal/mol Å$^2$. Simulations were performed at a constant
temperature of 300 K, maintained using a Langevin thermostat with a damping coefficient of 10 ps⁻¹. The biasing force was applied once 10,000 samples were accrued in each bin. The parameters of the extended Langevin dynamics were a damping coefficient of 1 ps⁻¹, an extended fluctuation of 0.02 Å and an extended time constant of 100 fs. To avoid sampling of nonphysical regions of large RMSDs, masking of the biasing force was introduced, which was determined beforehand in a 1-ns WTM-eABF simulation. The region was masked for the entire simulation if no sampling was recorded in this prefacing simulation. For the well-tempered metadynamics (WTM) part of the algorithm, the bias temperature was set to 1500 K, with a hill frequency of 1,000, a hill width of [0.1 Å, 0.1 Å], and a height of 0.1 kcal/mol.

**Trialanine conformational equilibrium.** Trialanine in vacuum was simulated using the AMBER ff14SB⁴⁶. 100 ns of sampling were gathered from a biased simulation using as CVs the RMSD with respect to metastable states *A* and *B*, limited to the heavy atoms (atoms 1, 4, 6, 8, 10, 14, 15, 16, 18, 20, 24, 25, 26, 28, 30, 34, 35, 36, and 38). The range of the CVs extended from 0.05 Å to 5.05 Å, with a bin width of 0.1 Å, and the harmonic walls were set at 5.05 Å, with a force constant of 1 kcal/mol Å². The protocol of the simulations was identical to that employed for NANMA isomerization, except for the WTM part of the algorithm, for which the bias temperature was set to 1500 K, with a hill width of [0.2 Å, 0.2 Å], and a height of 0.1 kcal/mol.

**Diels-Alder reaction.** The energies and forces were computed using the external GFN2-xTB quantum-chemical package³². Restraints were applied on the pair of distances corresponding to the newly formed bonds, in order to limit the sampling of irrelevant structures, whereby the two reactants dissociate completely. A harmonic wall was set at 3.5 Å, with a 100 kcal/mol Å² force constant. To prevent contamination of the sampling with equivalent chemical species having different topologies—for instance, typically a flipped ethylene—a restraint was also applied on the dihedral angle between the two reactants (defined by atoms 6,2,0, and 1). The harmonic walls confining the torsion between -80° and +80° were set with a force constant of 10 kcal/mol degree². The reaction was simulated with 32 walkers (only four of them where used in the qGNN learning due to memory requirements), sharing gradient information every 1,000 steps, and 2 ns of simulation time per walker. The temperature was set at 300 K using Langevin dynamics, with a damping constant of 10 ps⁻¹. The parameters of the extended Langevin dynamics were a damping coefficient of 1 ps⁻¹, an extended fluctuation of 0.01 Å and an extended time constant of 200 fs. The sampling along the two distances was partitioned into 73×73 bins, and the biasing force was applied once 5,000 samples were accumulated in each bin. For the WTM part of the algorithm, the metadynamics temperature was set to 2000 K, with a hill frequency of 1,000, a width of [0.028 Å, 0.028 Å], and a height of 0.1 kcal/mol.

## Codes and software

The GNN committor model designed has been built using the PyTorch Geometric library⁴⁷ and the Geometric Vector Perceptron library²³, a rotation-equivariant GNN for learning from biomolecular structure and the programming language Python⁴⁸. Trajectories are generated by the popular MD package NAMD⁴¹ together with the use of the Colvars module⁴².
**Supplementary information**

The article has accompanying supplementary information.

**Data availability**

Due to the size of the dataset, all the data in this work are provided upon request. Nevertheless, the necessary configuration files for data reproducibility are provided in https://doi.org/10.5281/zenodo.18259668⁵⁰.

**Code availability**

All the codes generated in this work can be found in https://doi.org/10.5281/zenodo.18259668⁵⁰.

**Acknowledgements**

C.C. acknowledges the European Research Council (project 101097272 "MilliInMicro"), the Université de Lorraine through its Lorraine Université d'Excellence initiative, and the Région Grand-Est (project "Respire"). C.C. also acknowledges the Agence Nationale de la Recherche under France 2030 (contract ANR-22-PEBB-0009) for support in the context of the MAMABIO project (B-BEST PEPR). A.M. is sincerely thankful for the support provided by Universidad Politécnica de Madrid through the "Programa Propio de Investigación" grant EST-PDI-25-C0ON11-36-6WBGZ8, as well as the hospitality of Université de Lorraine during his stay.

**Author Contribution Statement**

C.C. conceived the idea, and designed and supervised the research. S.C.A. designed the neural network, developed the corresponding code, and carried out the qGNN learnings. C.T., R.T. and C.G.C. performed the MD simulations. A.M. performed the shooting analysis of NANMA and worked on the theoretical framework. All the authors contributed in the writing of the manuscript and provided helpful discussions.

**Competing Interests Statement**

The authors declare no competing interests.
References

[1] Peters, B.: Reaction coordinates and mechanistic hypothesis tests. Annu. Rev. Phys. Chem. **67**, 669–690 (2016) https://doi.org/10.1146/annurev-physchem-040215-112215

[2] E, W., Vanden-Eijnden, E.: Transition-path theory and path-finding algorithms for the study of rare events. Annu. Rev. Phys. Chem. **61**, 391–420 (2010) https://doi.org/10.1146/annurev.physchem.040808.090412

[3] Onsager, L.: Initial recombination of ions. Phys. Rev. **54**(8), 554–557 (1938) https://doi.org/10.1103/PhysRev.54.55

[4] Berezhkovskii, A., Szabo, A.: One-dimensional reaction coordinates for diffusive activated rate processes in many dimensions. J. Chem. Phys. **122**(1), 014503 (2005) https://doi.org/10.1063/1.1818091

[5] Roux, B.: Transition rate theory, spectral analysis, and reactive paths. J. Chem. Phys. **156**(13), 134111 (2022) https://doi.org/10.1063/5.0084209

[6] van Erp, T.S., Moroni, D., Bolhuis, P.G.: A novel path sampling method for the calculation of rate constants. J. Chem. Phys. **118**(17), 7762–7774 (2003) https://doi.org/10.1063/1.1562614

[7] Best, R.B., Hummer, G.: Reaction coordinates and rates from transition paths. Proc. Natl. Acad. Sci. U. S. A. **102**(19), 6732–6737 (2005) https://doi.org/10.1073/pnas.0408098102

[8] Peters, B., Trout, B.L.: Obtaining reaction coordinates by likelihood maximization. J. Chem. Phys. **125**(5), 054108 (2006) https://doi.org/10.1063/1.2234477

[9] Ma, A., Dinner, A.R.: Automatic method for identifying reaction coordinates in complex systems. J. Phys. Chem. B **109**(14), 6769–6779 (2005) https://doi.org/10.1021/jp045546c

[10] Krivov, S.V.: On reaction coordinate optimality. J. Chem. Theory Comput. **9**(1), 135–146 (2013) https://doi.org/10.1021/ct3008292

[11] Rotskoff, G.M., Mitchell, A.R., Vanden-Eijnden, E.: Active importance sampling for variational objectives dominated by rare events: Consequences for optimization and generalization. In Mathematical and Scientific Machine Learning (PMLR), vol. 145, pp. 757–780 (2022). https://proceedings.mlr.press/v145/rotskoff22a.html

[12] Ribeiro, J.M.L., Bravo, P., Wang, Y., Tiwary, P.: Reweighted autoencoded variational bayes for enhanced sampling (rave). J. Chem. Phys. **149**, 072301 (2018) https://doi.org/10.1063/1.5025487

[13] Khoo, Y., Lu, J., Ying, L.: Solving for high-dimensional committor functions using artificial neural networks. Res. Math. Sci. **6**, 1–13 (2019) https://doi.org/10.1007/
[14] Lin, B., Ren, W.: A deep learning method for computing committor functions with adaptive sampling. SIAM J. Sci. Comput. 47(5), 1091–1114 (2025) https://doi.org/10.1137/24M1654750

[15] Prinz, J.-H., Wu, H., Sarich, M., Keller, B., Senne, M., Held, M., Chodera, J.D., Schütte, C., Noé, F.: Markov models of molecular kinetics: Generation and validation. J. Chem. Phys. 134(17), 174105 (2011) https://doi.org/10.1063/1.3565032

[16] Chen, H., Roux, B., Chipot, C.: Discovering reaction pathways, slow variables, and committor probabilities with machine learning. J. Chem. Theory Comput. 19(14), 4414–4426 (2023) https://doi.org/10.1021/acs.jctc.3c00028

[17] Ansari, N., Rizzi, V., Parrinello, M.: Water regulates the residence time of benzamidine in trypsin. Nat. Commun. 13 (2022) https://doi.org/10.1038/s41467-022-33104-3

[18] Bonati, L., Trizio, E., Rizzi, A., Parrinello, M.: A unified framework for machine learning collective variables for enhanced sampling simulations: mlcolvar. J. Chem. Phys. 159(1), 014801 (2023) https://doi.org/10.1063/5.0156343

[19] Zhang, J., Bonati, L., Trizio, E., Zhang, O., Kang, Y., Hou, T., Parrinello, M.: Descriptor-free collective variables from geometric graph neural networks. J. Chem. Theory Comput. 20(24), 10787–10797 (2024) https://doi.org/10.1021/acs.jctc.4c01197

[20] Pengmei, Z., Lorpaiboon, C., Guo, S.C., Weare, J., Dinner, A.R.: Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics. J. Chem. Phys. 162(4) (2025) https://doi.org/10.1063/5.0244453

[21] Eismann, S., Townshend, R.J.L., Thomas, N., Jagota, M., Jing, B., Dror, R.O.: Hierarchical, rotation-equivariant neural networks to select structural models of protein complexes. Proteins: Structure, Function, and Bioinformatics 89(5), 493–501 (2020) https://doi.org/10.1002/prot.26033

[22] Battaglia, P.W., Hamrick, J.B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V.F., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., Gülçehre, Ç., Song, H.F., Ballard, A.J., Gilmer, J., Dahl, G.E., Vaswani, A., Allen, K.R., Nash, C., Langston, V., Dyer, C., Heess, N., Wierstra, D., Kohli, P., Botvinick, M.M., Vinyals, O., Li, Y., Pascanu, R.: Relational inductive biases, deep learning, and graph networks. CoRR abs/1806.01261 (2018) https://doi.org/10.48550/arXiv.1806.01261

[23] Jing, B., Eismann, S., Soni, P.N., Dror, R.O.: Equivariant graph neural networks for 3d macromolecular structure. In: Proceedings of the 38 Th International Conference on Machine Learning (PMLR) (2021). https://doi.org/10.48550/arXiv.2106.03843

[24] Townshend, R.J.L., Vögele, M., Suriana, P.A., Derry, A., Powers, A., Laloudakis,
Y., Balachandar, S., Jing, B., Anderson, B.M., Eismann, S., Kondor, R., Altman, R., Dror, R.O.: Atom3d: Tasks on molecules in three dimensions. In: Vanschoren, J., Yeung, S. (eds.) Thirty-fifth Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 1), vol. 1 (2021). https://openreview.net/forum?id=FkDZLpK1MI2

[25] Megías, A., Contreras Arredondo, S., Chen, C.G., Tang, C., Roux, B., Chipot, C.: Iterative variational learning of committor-consistent transition pathways using artificial neural networks. Nat. Comput. Sci. (2025) https://doi.org/10.1038/s43588-025-00828-3

[26] Ajaz, A., Bradley, A.Z., Burrell, R.C., Li, W.H.H., Daoust, K.J., Bovee, L.B., DiRico, K.J., Johnson, R.P.: Concerted vs stepwise mechanisms in dehydro-diels–alder reactions. J. Org. Chem. 76(22), 9320–9328 (2011) https://doi.org/10.1021/jo201567d

[27] Qiu, L., Pabit, S.A., Roitberg, A.E., Hagen, S.J.: Smaller and faster: The 20-residue trp-cage protein folds in 4 µs. J. Am. Chem. Soc. 124(44), 12952–12953 (2002) https://doi.org/10.1021/ja0279141

[28] Lindorff-Larsen, K., Piana, S., Dror, R.O., Shaw, D.E.: How Fast-Folding Proteins Fold. Science 334(6055), 517–520 (2011) https://doi.org/10.1126/science.1208351

[29] Swenson, D.W.H., Prinz, J.-H., Noé, F., Chodera, J.D., Bolhuis, P.G.: OpenPathSampling: A Python framework for path sampling simulations. 1. Basics. J. Chem. Theory Comput. 15(2), 813–836 (2019) https://doi.org/10.1021/acs.jctc.8b00626

[30] Chen, H., Ogden, D., Pant, S., Cai, W., Tajkhorshid, E., Moradi, M., Roux, B., Chipot, C.: A companion guide to the string method with swarms of trajectories: Characterization, performance, and pitfalls. J. Chem. Theory Comput. 18(3), 1406–1422 (2022) https://doi.org/10.1021/acs.jctc.1c01049

[31] Tiwary, P., Berne, B.J.: Spectral gap optimization of order parameters for sampling complex molecular systems. Proc. Natl. Acad. Sci. U. S. A. 113(11), 2839–2844 (2016) https://doi.org/10.1073/pnas.1600917113

[32] Bannwarth, C., Caldeweyher, E., Ehlert, S., Hansen, A., Pracht, P., Seibert, J., Spicher, S., Grimme, S.: Extended tight-binding quantum chemistry methods. Wiley Interdiscip. Rev. Comput. Mol. Sci. 11(2), 1493 (2021) https://doi.org/10.1002/wcms.1493

[33] Meuzelaar, H., Marino, K.A., Huerta-Viga, A., Panman, M.R., Smeenk, L.E.J., Ketelarij, A.J., Maarseveen, J.H., Timmerman, P., Bolhuis, P.G., Woutersen, S.: Folding dynamics of the trp-cage miniprotein: Evidence for a native-like intermediate from combined time-resolved vibrational spectroscopy and molecular dynamics simulations. J. Phys. Chem. B 117(39), 11490–11501 (2013) https://doi.org/10.1021/jp404714c

[34] Lloyd, S.: Least squares quantization in pcm. IEEE Trans. Inf. Theory 28(2), 129–137 (1982) https://doi.org/10.1109/TIT.1982.1056489
[35] Schütte, C., Noé, F., Lu, J., Sarich, M., Vanden-Eijnden, E.: Markov state models based on milestoning. J. Chem. Phys **134**(20), 204105 (2011) https://doi.org/10.1063/1.3590108

[36] Bicout, D.J., Szabo, A.: Electron transfer reaction dynamics in non-debye solvents. J. Chem. Phys. **109**(6), 2325-2338 (1998) https://doi.org/10.1063/1.476800

[37] Donati, L., Hartmann, C., Keller, B.G.: Girsanov reweighting for path ensembles and Markov state models. J. Chem. Phys. **146**(24), 244112 (2017) https://doi.org/10.1063/1.4989474

[38] Donati, L., Weber, M., Keller, B.G.: A review of Girsanov reweighting and of square root approximation for building molecular markov state models. J. Math. Phys. **63**, 123306 (2022) https://doi.org/10.1063/5.0127227

[39] Glorot, X., Bengio, Y.: Understanding the difficulty of training deep feedforward neural networks. In: Teh, Y.W., Titterington, M. (eds.) Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol. 9, pp. 249–256. PMLR, Chia Laguna Resort, Sardinia, Italy (2010). https://proceedings.mlr.press/v9/glorot10a.html

[40] Kingma, D.P., Ba, J.: Adam: A method for stochastic optimization, vol. abs/1412.6980 (2014). https://api.semanticscholar.org/CorpusID:6628106

[41] Phillips, J.C., Hardy, D.J., Maia, J.D.C., Stone, J.E., Ribeiro, J.V., Bernardi, R.C., Buch, R., Fiorin, G., Hénin, J., Jiang, W., McGreevy, R., Melo, M.C.R., Radak, B.K., Skeel, R.D., Singharoy, A., Wang, Y., Roux, B., Aksimentiev, A., Luthey-Schulten, Z., Kalé, L.V., Schulten, K., Chipot, C., Tajkhorshid, E.: Scalable molecular dynamics on cpu and gpu architectures with NAMD. J. Chem. Phys. **153**, 044130 (2020) https://doi.org/10.1063/5.0014475

[42] Fiorin, G., Marinelli, F., Forrest, L.R., Chen, H., Chipot, C., Kohlmeyer, A., Santuz, H., Hénin, J.: Expanded functionality and portability for the colvars library. J. Phys. Chem. B **128**(45), 11108–11123 (2024) https://doi.org/10.1021/acs.jpcb.4c05604

[43] Fu, H., Shao, X., Cai, W., Chipot, C.: Taming rugged free energy landscapes using an average force. Acc. Chem. Res. **52**(11), 3254–3264 (2019) https://doi.org/10.1021/acs.accounts.9b00473

[44] Andersen, H.C.: RATTLE: A “velocity” version of the shake algorithm for molecular dynamics calculations. J. Comput. Phys. **52**, 24–34 (1983) https://doi.org/10.1016/0021-9991(83)90014-1

[45] MacKerell, A.D.J., Bashford, D., Bellott, M., Dunbrack, R.L.J., Evanseck, J.D., Field, M.J., Fischer, S., Gao, J., Guo, H., Ha, S., Joseph-McCarthy, D., Kuchnir, L., Kuczera, K., Lau, F.T.K., Mattos, C., Michnick, S., Ngo, T., Nguyen, D.T., Prodhom, B., Reher, W.E., Roux, B., Schlenkrich, M., Smith, J.C., Stote, R., Straub, J., Watanabe, M.,
Wiórkiewicz-Kuczera, J., Yin, D., Karplus, M.: All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B **102**(18), 3586–3616 (1998) https://doi.org/10.1021/jp973084f

[46] Maier, J.A., Martinez, C., Kasavajhala, K., Wickstrom, L., Hauser, K.E., Simmerling, C.: ff14sb: Improving the accuracy of protein side chain and backbone parameters from ff99sb. J. Chem. Theory Comput. **11**(8), 3696–3713 (2015) https://doi.org/10.1021/acs.jctc.5b00255

[47] Fey, M., Lenssen, J.E.: Fast graph representation learning with pytorch geometric. In: ICLR Workshop on Representation Learning on Graphs and Manifolds, vol. abs/1903.02428 (2019). https://api.semanticscholar.org/CorpusID:70349949

[48] Van Rossum, G., Drake, F.L.: Python 3 Reference Manual. CreateSpace, Scotts Valley, CA (2009)

[49] Juraszek, J., Bolhuis, P.G.: Rate constant and reaction coordinate of trp-cage folding in explicit water. Biophys. J. **95**(9), 4246–4257 (2008) https://doi.org/10.1529/biophysj.108.136267

[50] GNN committor learning repository. GitHub (2025) https://doi.org/10.5281/zenodo.18259668
**Fig. 1: qGNN.** Sketch of the learning process for the qGNN. First, we generate a molecular dynamics trajectory—represented by the molecule between brackets—for the studied system and we use this data for training the model. The qGNN model will return a scalar function according to Algorithm 2 in the SI. Based on the committor variational principle, we optimize the qGNN scalar output, $q_\omega$, to obtain the committor function, $q$.

**Fig. 2: NANMA isomerization.** Learned committor $q$, projected on $(\phi, \psi)$ space, from atomic coordinates of heavy atoms from a biased trajectory of NANMA system. The contour lines represent the free-energy surface (A). Relative node sensitivity, $s_i/\max_j\{s_j\}$, computed for a short biased simulation of 5ns for the same system. Sketch of NANMA and the dihedral angles $\phi, \psi, \theta$ and $\omega$, where carbon atoms are represented in cyan, oxygen in red, nitrogen in blue and hydrogen in white. (B). Committor computed from shooting unbiased trajectories for NANMA according to the OpenPathSampling package$^{29}$ (C) and learned committor from the variational committor network$^{16}$ from a biased trajectory of NANMA sampling the $(\phi, \psi)$ subspace (D).
<table><thead><tr><th rowspan="2">System</th><th colspan="2">Transition rate, k<sub>AB</sub> (ps<sup>-1</sup>)</th></tr><tr><th>GNN</th><th>Reference</th></tr></thead><tbody><tr><td>NANMA</td><td>3.92 × 10<sup>-5</sup></td><td>1.34 × 10<sup>-5</sup> 16</td></tr><tr><td>Trialanine</td><td>5.60 × 10<sup>-5</sup></td><td>3.62 × 10<sup>-5</sup> 30<sup>a</sup></td></tr><tr><td>Diels-Alder</td><td>1.33 × 10<sup>-3</sup></td><td>1.03 × 10<sup>-3</sup> 26</td></tr><tr><td>Trp-cage</td><td>4.67 × 10<sup>-7</sup></td><td>3.2 × 10<sup>-7</sup><br>2.5 × 10<sup>-7</sup><br>8.3 × 10<sup>-7</sup></td></tr><tr><td>Villin</td><td>5.88 × 10<sup>-7</sup></td><td>3.57 × 10<sup>-7</sup> 28</td></tr></tbody></table>

<footnote><sup>a</sup> The reported reference value corresponds to the dominant pathway.</footnote>