MoleBridge: Synthetic Space Projecting with Discrete Markov Bridges
Authors: Rongchao Zhang
Yu Huang
Yongzhi Cao
Hanpin Wang
Venue: NeurIPS 2025 poster ()
Keywords: Markov Bridges
Diffusion Models
AIGC
Generation
Primary area: deep_learning

Abstract: Molecular synthetic space projecting is a critical technique in de novo molecular design, which aims to rectify molecules without synthesizability guarantee by converting them into synthetic postfix notations. However, the vast synthesizable chemical space and the discrete data modalities involved pose significant challenges to postfix notation conversion benchmarking. In this paper, we exploit conditional probability transitions in discrete state space and introduce MoleBridge, a deep generative model built on the Markov bridge approach for designing postfix notations of molecular synthesis pathways. MoleBridge consists of two iterative optimizations: i) Autoregressive extending of notation tokens from molecular graphs, and ii) generation of discrete reaction postfix notations through Markov bridge, where noisy token blocks are progressively denoised over multi-step iterations. For the challenging second iteration, which demands sensitivity to incorrect generative probability paths within intricate chemical spaces, we employ a thinking and denoising separation approach to denoise. Empirically, we find that MoleBridge is capable of accurately predicting synthesis pathways while exhibiting excellent performance in a variety of application scenarios.

Mole Synthetic Space Projecting with Discrete Markov Bridges
Rongchao Zhang', Yu Huang"; Yongzhi Cao', Hanpin Wang'
Keey Laboratory of High Confidence Software Technologies (Peking University), Ministry of Education, School of Computer Science, Peking University National Engineering Research Center for Software Engineering, Peking University rczpku@163. com, hy@pku. edu. cn
Abstract
Molecular synthetic space projecting is critical technique in de novo molecular design, which aims to rectify molecules without synthesizability guarantee by converting them into synthetic postfix notations. However, the vast synthesizable chemical space and the discrete data modalities involved pose significant chal- lenges to postfix notation conversion benchmarking. In this paper, exploit conditional probability transitions discrete state space and introduce MoleBridge, deep generative model built on the Markov bridge approach for designing postfix notations of molecular synthesis pathways. MoleBridge consists of two iterative op- timizations: i) Autoregressive extending of notation tokens from molecular graphs, and ii) generation of discrete reaction postfix notations through Markov bridge, where noisy token blocks are progressively denoised over multi-step iterations. For the challenging second iteration, which demands sensitivity to incorrect genera- tive probability paths within intricate chemical spaces, we employ a thinking and denoising separation approach to denoise. Empirically, we find that MoleBridge is capable of accurately predicting synthesis pathways while exhibiting excellent performance in a variety of application scenarios.
Introduction
De novo molecular design has garnered considerable attention across various research domains in life sciences [77 63 2] Among these developments, the majority of cutting-edge breakthroughs
are driven by deep generative approaches 48
42 21
With the promise comes challenge: unlike
traditional combinatorial optimization approaches
constrained by virtual libraries, generative
models typically generate structures that lie outside the synthesizable chemical space [18] Of these, only vanishing small percentage will be experimentally realizable. Recently, the immense potential of projecting synthetic space for the rectification of non-synthesizable molecules has given rise to a milestone paradigm [46), where desired synthesizable molecules BI from structurally similar analogs are made available. This paradigm is centered generating synthetic pathways from purchasable chemical building blocks and deriving designed molecules in postfix notations, which can rival chemically expert- defined rules.
Despite their promising advancements, there are still significant gaps to fill before generative models become practical for synthesizable pathway design. i) First, chemical space theory predicts that the number of compounds synthesizable by humans could reach 1003 10 enormous space [7] For this purpose, it is essential that the model can explore, on sequence-to-sequene basis, synthetic pathways of arbitrary length. ii) Second, unlike natural language, molecular postfix notation sequences
"Corresponding Author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Molecular Geometric Graph Representation
Autoregressive Extension
Markov Bridge Process
Denoised Postfix Notation Sequence
Autoregressive Transformer Po(t"
T=T
M
Structural
Denoised Sequence
RXN
RXN
BB
BB
Ag
Transformer
Graph
BB
...
BB
RXN
BB
BB
RXN
BB
BB
8B
RXN
Embeddings
Discrete Tokoons
Figure 1: Overview of MoleBridge. MoleBridge sequentially extends postfix notation tokens from molecular graphs conditioning on previous blocks. By performing Markov bridge, MoleBridge retains scalable capability for the increasing synthetic space and supports higher-quality generation.
are composed a limited number of building blocks, with minimal redundancy and a lack of semantic coherence. Therefore, the model should pay attention to the deep-level connections between postfix notations. iii) Third, early errors in pathway generation propagate irreversibly. multi- step synthesis, selecting incompatible building block certain step constrains all subsequent reactions, potentially invalidating the entire pathway. This necessitates a generation mechanism that allows progressive refinement of pathway segments.
In this paper, we introduce MoleBridge, a novel Markov bridge generative model for designing postfix notations of molecular synthesis pathways. Markov bridge models exhibit increased flexibility (the noise addition and removal processes resemble powerful data augmentation, compelling the model to form deeper relationships among features). However, constrained by the noise scheduling process, they follow fixed generation length 3 73 52), making difficult to explore synthetic pathways of variable lengths. Thus, instead of adding Gaussian noise to entire fixed-lengto synthesis pathway, MoleBridge introduces probabilistic perturbations to notation token blocks at current autoregressive steps. By approximately referencing the Markov bridge process [7] 29 78). MoleBridge learns to progressively refine perturbed reaction sequences, thereby generating more rational designs. In identify errors introduced
pursuit of stable pathways and semi-autoregressivve error robustness, the sequence steps by thinking at each denoising time step and correct these errors based on the selected positions in the current noisy sequence block. This innovation is more natural for molecular synthesis pathways than for language data. Empirically, we demonstrate that MoleBridge excels in variety of scenarios.
Our contributions can be summarized as follows:
We introduce MoleBridge, a novel generative approach for generating molecular postfix no- tations based the Markov bridge, which expands chemical space via semi-autoregressive process while applying iterative refinement to each block sequentially.
We employ a thinking strategy, which performs denoising when an error occurs at a certain step, ensuring the feasibility of the synthesis pathway.
Through experiments, demonstrate the effectiveness of MoleBridge in various scenarios, such as bottom-up synthesis, structure-based drug design, and target-directed generation.
2
Related Work
Synthesizability of molecules. Synthesizable molecule design aims to generate novel molecular structures that can be realized through practical chemical synthesis pathways. Early approaches 60 24 relied on proposing a large number of potential candidate molecules and screening them using scoring functions to estimate energy and identify stable molecules. Groundbreaking deep

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learning approaches [9 have now been developed to predict reaction outcomes in a "template- free" manner. For instance, MoleculeChef [6] encodes and decodes a set of initial reactants from purchasable building blocks, enumerates possible one-step synthetic paths, and selects the best molecule among the products as the output. It allows chemists to interrogate the properties of the generated molecules. DoGs 7 employs recurrent neural network to generate sequences of actions from latent codes, achieving molecular generation through sampling in the latent space. It also demonstrates strong capabilities to satisfy the feasibility of the synthesis paths. However, empirical validation remains challenging, since they are generally less effective at producing convergent synthesis paths and structurally complex molecules. The synthetic space projecting [46] aims to generate structurally similar and synthesizable analogs by converting them into synthetic postfix notations. It is capable of guaranteeing bottom-up synthesis planning and exploring the locally synthesizable chemical space around hit molecules. Despite this progress, directly applying these approaches to nostaix-nttaiion path generation remains challenging and requires further design.
19 7
64 are generative models achieving state-of- the-art performance across various domains, including the generation of images [62] 44), video [66], or molecular 770 69). It achieves high-quality and diverse sampling from unknown data distributions by approximating the simple density (i.e., Gaussian density) to the stochastic differential equation of the unknown data density [25). A notable highlight of the success of diffusion models in the field of molecular design is their potential to generate molecules that can serve as the foundation for novel medicinal compounds previously unseen [32. Multiple approaches have been explored to achieve this. For instance, 3D diffusion methods such MDM (28]. DiffLinker [30), and PIDiff [3] can generate candidate drug molecules relevant to chemistry. Most relevant to our work, diffusion models have also achieved promising results in designing discrete DNA sequences [61] and optimizing stable molecules [35] 76,, (5). Recently, some studies [4] 45 propelled the development of molecular design technologies by utilizing more reliable diffusion processes.
Diffusion generative models. Diffusion models 12
38 arises from an intriguing connection between statistical physics and probability theory. Its goal is to find the most probable evolution between a given initial and final distribution relative to a specified reference stochastic process (22). A significant characteristic of the SB problem is the ability to choose any distribution as the initial and terminal distributions [33], which has advanced the resolution of various generative model issues. Building upon the SB framework, diffusion bridge models have shown cutting-edge results in various fields, including imaging [55 36), speech (37, and physical fields [41]. The recently proposed Markov bridge models 778 29 extend these models into the discrete domain,
Schrödinger bridge. The Schrödinger Bridge (SB) problem 53
focusing on environments with categorical distributions. In this work, we apply the Markov bridge for molecular postfix notation path synthesis.
3 Preliminary
3.1 Notations and Problem Formulation
The synthetic space is constructed by recursively applying reaction rules to all possible molecular com- binations, starting from the initial building blocks [46). Mathematically, a synthetic space is repre- sented as the closure of molecules generated by set of n. building blocks B = {bb,, b² ..... 6th} } E S and a set of nt, reaction rules R = (r²2, r...... T" function from the reactant space the product: r := X' x ) + S, (X, )")
}. where each reaction rule r defines a mapping
Z. Here, X, y S represents the sets of molecules applicable to the reaction r", and Z represents the main reaction product. In the synthetic space construction process, synthesis pathways are represented as notation where each token t E BUR indicates pushing a building block onto the stack or caleulating the product and pushing it back onto the stack, with n, indicating the length of the tokens. The goal of the synthetic space projecting problem is to identify a structurally similar and practically synthesizable analog y by projecting the designed molecule M into the synthesizable chemical space S. Thus, an ideal model, parameterized by 9, should be capable of learning the mapping from any molecule M its corresponding postfix notation distribution pa(y MM)
sequences y = [t_ .2..... )
3.2 Autoregressive Models
The success of large models in the natural language processing and computer vision domain demon- strates their scalability and the universality of sequence data modeling [3] 68 65 75). Given
3

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postfix notation sequence [t', .e......tt" t" where the subscript 1 S & S specifies an order, autoregressive models assume that the probability of observing the current token t depends only on its prefix [t', t² ...., -. This unidirectional token dependency assumption allows the likelihood of the sequence to be factorized as:
n2
lo
Σ
log Lo (
where po t(*) is parameterized directly with a neural network. Therefore, autoregressive models can be efficiently trained through "next token prediction" 71 However, due to sequential dependencies, autoregressive models require n, steps to generate n, tokens.
3.3 Diserete Markov Bridge
Markov bridge model fits model (-) (-) to reverse the forward corruption process [7 29 17 This process is pinned to specific data points the beginning and the end, modeling the dependencies between the discrete spaces X and Y. For the sample pair (æ, y) ~ px,y (ar, y) and the time step sequence 7 = 0, 1...... T, it defines the Markov process sequence of random variables (t. T= starting from to = æ, and satisfying the Markov property:
p(tr to, L 1,, y) p(t, 1,,1,)) v).
To ensure the process terminates at the data point by = y. we introduce an additional requirement:
p(t y 1 y) 1.
Suppose the distributions px and py are categorical distributions with a finite sample space 1.... K, and we can represent the data points as K -diiennsional one-hot vectors: x, y. t, I and define the transition probabilities (Eq. 2 as follows:
p(t++1 t,, y) Cat Q+t+):
where Cat(;p) is a categorical distribution with probabilities given by p. and Q is transition matrix parameterized as:
Qy: = a, + +(1
(5)
where I denotes K x K identity matrix, and a, is schedule parameter transitioning from ou = 1 to a- = 0. Then, 4, can be efficiently sampled from p(dr+1 to, br) (4+++++ Qr4n), where Q = Q Qo = I + (1 a. j a cumulative product matrix, and a, = II ag..
During training, the Markov bridge approximates y using the neural network vho ÿ (tr,T).
4 Methods
In this section, we introduce MoleBridge, a Markov bridge model for synthetic space projecting. We first define how interpolate between semi-autoregrrssive and Markov bridge models by defining autoregressive distributions over tokens. Next, we provide an objective for maximum likelihood estimation and efficient training and sampling algorithms. Finally, describe the denoising network architecture employed to approximate the Markov bridges.
4.1 Markoy Bridge for Synthetic Space Projecting
Generally, molecular synthesis paths are highly discrete processes that strictly adhere to chemical rules. common scenario to construct the pathway from simple building blocks to complex target molecules through sequential decisions alone. In this work, we propose to introduce Markov bridges into molecular synthesis pathways, modeling blocks tokens autoregressively. We group the postfix notation tokens into C notation blocks, each of length Åe. where C nu/ (assuming C is integer). We denote each block tych. from token at positions to cń, for blocks c E (1....,C) as to for simplicity.
Interpolation process. As discussed in Section the matrix Q, can efficiently model various transition probabilities in the discrete state space, including masking, random token changes, and

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related word substitutions. When considering the noise process modulated by the masking vector the interpolation process gradually transforms the data point tl into the initial molecular state to ~ po(C), with the transition rate controlled by the masking vector. In this case, the noise interpolation can be represented as:
78 29
g(t 0,, = v + - -u-)y
(6)
where 8,- Bernoulli(rr.)) is masking latent vector and 3, denotes scheduler. At 0, the conditional marginal converges to the initial distribution t.. i.e. 8, = When 7 + T, 3, is set
close to 0 and the distribution closer to the target distribution.
Reverse process. Accordingly, the reverse process in block c can be written as:
(o Vr. t (59) = Σ gt Ur t,, Eppo Uy, t,, (59)
(7)
where the denoising base model predicts clean token t" given the noisy sequence th. Since masked diffusion requires building a complete probability distribution p(t" +t)) over all possible values at each position& e E (1.....I to |} rather than directly identifying corrupted locations {e 1. * t=O,, it is difficult achieve accurate denoising probability estimation [56 47). As evidenced by recent works [ 56 51], the core difficulty stems from denoising probability estimation po being the only key component in the diffusion process that requires neural network approximation. Unfortunately, while uniform diffusion (45) allows token values to be corrected throughout the sampling process, by modeling the posterior q g(t vr. tG, t²) as shown in Eq. 7 its performance does not consistently outperform masked diffusion, particularly in tasks like image or language modeling.
Indeed, the transition probability can be decomposed into scheduling probability, which assesses whether the data has been corrupted, and a denoising probability that determines the new value [45)
Therefore, we first think the probability of each position in the sequence being corrupted by noise [4). conditioned on the current state t. and time step т, and define:
v, = 0 Pa (z+= ( ((1 B,) Do (t,, B,t+), if 0, = 1
(t U7, t,, t0)
(8)
where p0(2
tt) denotes the probability that each position is corrupted. and 2 {0. 1]
is latent variable to denote if dimension is corrupted. If position is likely to be corrupted (2e = =1 tς) x= 1), the transition probability will lean toward the output of the denoiser. On the other hand, when a position is likely to be clean (po = 1 t s 0), the transition probability will tend to preserve the current state. In mask diffusion case, this can be readily read out from the masked token. But in the uniform diffusion case, we need to compute/approximate this probability instead.
Training, We obtain principled learning objective for the model the (-) by applying the variational bound on negative log-likelihood logpa(y t) to each term, which has the following closed-form expression:
log po()) & (t, == Σ E Egtegles " log to (t.,7))
where y denotes the true tokens in block c. The derivation of (t, e) expresses the training loss as a reweighted standard multiclass cross-entropy loss 78 26). which computed on the labels that have not yet been converted into the base truth y = Ir. Compared to the simpler cross-entropy loss computed over all labels, this new approach assigns greater weight the labels that need refinement. Since the model is conditioned on t<c, the dependency between t t and i is explicated in L,
The training of pe can be simplified to a binary classification task, aiming to estimate the probability that each position is corrupted by noise. Specifically, the training objective is formulated as:
3,
n. BCE (po(25² = 1115),11(()) + ) (10)
0) = Σ (0
B
c=1
where 2. is a variable indicating whether the f-th position in block c is corrupted at time step F, t.f the noisy token at that position, e. e is the true value, T(-) is the indicator function, and BCE(.) represents binary cross-entropy. The final loss function is the sum of the two terms: L= L, + Cp.
5

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Sampling, A postfix notation of synthesis y is sequence that contains four types of tokens: building block tokens 6¹ B, reaction tokens r' R. start token [START], and an end token [END]. Each building block token b is associated with the fingerprint of the corresponding molecule, represented as [BB, [B. where jE (0, } 250 is the Morgan fingerprint of length 256 and radius [46] 49]. A reaction token r², denoted as [RXN, .11, represents the index i of the reaction. During sampling, we generate the pathway token-by-tokens and each token is produced through a block-wise Markov bridge refinement process. The conditional distribution po po(4++1 Vr f,, t<c) is used to sample from the model. Starting from the given ty ~ Po (t"), the process iterates to predict the data point ÿ = (ts 7) and then derive + Po (ti+1 Ur, ty. (5a) while incrementing the time step from 0 to T - 1. Notably, our algorithm allows us to sample sequences of arbitrary lengths, whereas traditional Markov bridge models are limited to fixed-length generation.
In the synthesis process, the postfix stack is initialized as empty and is progressively populated by the generated tokens [46]. When building block token b' is generated, we retrieve the corresponding molecule from the fingerprint and push onto the stack. If reaction token is generated, we first pop the required number of molecules from the stack, then use the reaction template with RDKit g to predict the product, which is subsequently pushed onto the stack. there are insufficient molecules on the stack or if the reaction cannot be applied, the inference process halts. Finally, the process ends when the [END] token is generated, marking the completion of the synthesis. The most recent product molecule is then considered as the input molecule for the synthetic space projecting.
4.2 Architecture Design
Molecular graph representation. Following [46,, we represent molecule M as a graph, where nodes are connected based on chemical bonds. For atoms. we convert them into initial embedding vectors ha Rdd based on their atomic numbers; for chemical bonds, we capture bond type information and incorporate it into the graph structure, represent as h E Rdde
Networks. We use transformer (58) as the backbone network to approximate the final state of the Markov bridge process. Typically, a transformer only takes the synthesis path sequence as input, but our task also requires integrating time steps and structural conditions into the model. The time-step embedding network encodes temporal information using sinusoidal functions, converts into vector e,, E IR and integrates it into the cross-attention mechanisms. We use a L-layer graph transformer [67 to capture the molecular topology and inter-atomic interactions, with the final atomic
representations h serving as structural information. Finally, using the customized transformer decoder architecture, the network takes both sequence embeddings and molecular graph embeddings as input. For the thinking network p= 1 t),, we use multilayer perceptron with gelu(-) activation and sigmoid(-) output layer. The entire architecture trained end-to-end, and the output types include BB, RXN, and END, which together form the complete molecular synthesis pathway.
5 Experiments
5.1 Experimental Setup
Datasets. We use the SynNet reaction template set [19 for reaction templates R, which is based on two publicly available template collections from Hartenfeller et al [24]. and Button et al [10. After removing duplicates and rare reactions, final set of 91 reaction templates obtained. The set includes 13 unimolecular and 78 bimolecular reactions. For building blocks B, we use the Enamine US Stock catalog [ as the data source. Entries containing multiple molecules (c.g., salts or hydrates) filtered by retaining the largest molecule and removing the rest. Any building blocks that fail RDKit sanitization or do not match any reaction templates are excluded, as are duplicates. We use K-means clustering based on Morgan fingerprints to group the blocks into 128 clusters, reserving one structurally distinctive cluster for testing and using the remaining 127 clusters for training. Additionally, include a challenging test set: molecules extracted from the ChEMBL database (00). which have been previously reported as "unreachable" target compounds [19 46).
Implementation details. In our experiments, we use Morgan fingerprints 49 to featurize molecular
structures, with a radius of 2 and a bit length of 256. At the data initialization stage, we randomly initialize the reaction path stack using weighted sampling, assigning an initial weight of 0.90 to

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Table Performance comparison between MoleBridge and baseline methods. The evaluation is performed on both the standard test set and the ChEMBL [20]. Best results are highlighted in bold.
Dataset
Method
Success (t)
Recons. %(()
Sim.(Morgan) (1)
Sim.(Seaffold) (f)
Sim.(Gobbb) (7)
SynNet
0.4205
10.7%
0.4575
0.5109
0.3465
Test Set
ChemProjector
0.4875
28.4%
0.7167
0.7791
0.7273
MoleBridge (Ours)
0.4915
43.5%
0.8455
0.8695
0.8287
SynNet
0.4250
5.4%
0.4270
0.4174
0.2678
ChEMBL ChemProoector
0.4940
13.3%
0.5978
0.5869
0.5570
MoleBridge (Ours)
0.4970
14.6%
0.6159
0.6188
0.5789
Target:
Target:
Sim, [Beddkld:: L,000
(Scatfolly: Sim. (Gabbi)
Sun. (Guble):0.22 0.748
Sim (Morgen): 0.859
Sim, (Morganj:0.90 0.990
Figure 2: Examples of molecules generated by Pocket2Mol and projected by MoleBridge into analogs, demonstrating clear synthetic pathways and the preservation of high structural similarity.
building blocks. The model is trained on NVIDIA 4090 GPUs with batch size of 128 and data loader workers. We use the Adam optimizer 34 with an initial learning rate of 3 10-1, and momentum parameters 31 0.90, 32 = 0.999. platcau-based learning rate scheduler is used, reducing the learning rate by factor of 0.6 when validation performance plateaus, with patience of 5 validation cycles and a minimum learning rate of 1 x 10-5
and
Metrics. We conduct a comprehensive evaluation of the MoleBridge model using multiple quanti- tative metrics: i) Synthesis path success rate: the percentage of valid postfix notations and further multiplied by 1/2. ii) Reconstruction rate: the percentage of proposed synthesis paths that result in same product as the input molecule. iii) In cases of partial success, we evaluate the molecular similarity between the generated and target compounds; similarity score of ( is assigned to failed (invalid) syntheses. The similarity score is calculated using three types fingerprint representations: Morgan fingerprints of length 4096 and radius 2 [49], Murcko scaffold-baad based fingerprints, and Gobbi pharmacophore fingerprints [22). All three similarity scores are normalized to the [0, 1], reflecting chemical similarity in terms of overall structure, scaffold structure, and pharmacophoric properties.
5.2 Bottom-Up Synthesis Planning
To validate the effectiveness of MoleBridge, we conduct a comparison with the existing generation methods, SynNet [19] and ChemProjector [46). The results presented in Table[[show that MoleBridge outperforms the baseline methods significantly result is the reconstruction rate on the test set, where MoleBridge achieved 43.5%, significantly outperforming ChemProjector (28.4%) and SynNet (10.7%). Even on the challenging ChEMBL dataset, MoleBridge retains its superior performance.
all evaluation metrics. A particularly notable
5.3 Projecting Molecules Generated by Structure- Based Drug Design Models
Synthetic space projecting has broad application prospects in the field of structure-based novo drug design. Due to limited constraints (23), existing design models often generate chemically invalid structures [46]. To assess the applicability of MoleBridge in drug optimization scenarios, we conduct experiments based on the LIT-PCBA dataset [57], which contains 15 drug targets. Following [46).

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Table 2: Similarity scores between molecules generated by Pocket2Mol and their analogs.
Table The analogs optimized by model
exhibit a significant increase in Vina scores.
Sim.
Sim.
Sim.
Targets
Vina (kcal/mol)
Targets
(Morgan)
(Scaffold)
(Gobbi)
Ref.
Gen.
Analog (4)
ADRB2
0.5149
0.5834
0.4007
ADRB2
-8.70
-8.31
-10.90
-2.59
ALDHI
0.4434
0.3535
0.3304
ALDHI
-5.20
-8.14
-11.00
-2.86
ESR1 ago
0.4641
0.3690
0.2993
ESRI ago
-5.90
-8.23
-9.30
-1.07
ESR1 ant
0,5078
0.4903
0,4229
ESR1 ant
-8.10
-8.77
-11.80
-3.03
FENI 0.4397
0,4308
0.3408
FENI
-580
-6.04
-6.60
-0.56
GBA 0,4267
0.2785
0.2572
GBA
-8.40
-6.86
-6.96
-0.10
IDH1 0.4701
0.3642
0.3224
IDHI
-9.30
-8.62
-9.70
-1.08
KAT2A
0.5123
0.4927
0.4545
KAT2A
-8.00
-7.41
-11.40
-3.99
MAPKΣ
0.4990
0.3955
0.3917
MAPK1
-8.80
-8.20
-9.90
-1.70
MTORCI
0,5351
0,4384
0.3841
MTORCI
8.80
99.32
-12110
-2.78
OPRKI
0,5170
0,5500
0.4560
OPRKI
-9.00
-7.88
10.30
-2.42
PKM2
0.4874
0.4521
0.3927
PKM2
-9.20
-8.25
-1..60
-3.35
PPARG
0.4977
0.4904
0.4616
PPARG
-7.70
-7.60
-9.10
-1.50
TP53
0,5289
0.5595
0,4979
TP53
-6.30
-6.83
-9.80
2,97
VDR
0.5312
0.3990
0.3730
VDR
-8.40
-9.27
-10.30
-1.03
use Pocket2Mol [50] generate candidate molecules for each target and select the top 300 candidates for each target based on QED and SA scores. Subsequently, MoleBridge is applied to design analogs for each candidate molecule, and the optimal analogs selected based Vina
scores [16.. Figure shows examples of molecular synthesis pathways generated by MoleBridge, illustrating the complete synthesis pathway design from Pocket2Mol to the target analog. Table [] presents the structural similarity metrics between the generated analogs and the original molecules. Table 3 displays the estimated binding energies, with optimized analogs demonstrating improved target binding strength.
5.4 Projecting Molecules Explore Local Chemical Space for Hit Expansion
MoleBridge also demonstrates significant appli- cation potential hit expansion. Following the experimental design of Levin et al. [39] and Luo
Malccule JNK3: 0,68
(Scaflold;: 6.867
Sim. (Margan): 0.751
et al. [46], we evaluate the development of c-Jun N-terminal Kinases-3 (JNK3) inhibitors. Using an molecule with a JNK3 score [40] of 0.68 as the starting point, MoleBridge successfully gen- erates several synthesizable structural analogs in Figure 3 Notably, the generated analogs effectively preserve the JNK3 inhibitory activity (0.67, 0.68, and 0.70) while maintaining the in- tegrity of the core scaffold structure. Taking the amino-substituted analog in the lower-left cor- ner as an example, the introduction of an amino functional group not only maintained almost the same JNK3 activity (0.68) but also potentially provided new modification sites for further structural optimization.
Sim. (Gobbije 8.137
JNK 0.67
Sim. (Scaffold): 0.1.000 Sim. (Gobbiy: 0.986 JNKI: 0.68
Sim. (Scaffold): 1,400 (Morgan): 0.326 (Gobbij; JNK3:
Figure 3: Analogs generated from the initial hit expansion compound.
5.5 Projecting Molecules Generated by Target-Directed Generative Models
Target-directed generative models generally suffer from insufficient synthesize-ability with approxi- matcly 70% of generated molecules being labeled non-synthesizable by the ASK-COS [14 [8).

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Table 4: Assessment of property retention after projection of molecules generated by target-directed.
Property
Sim.
Sim.
Sim.
Arg(Objective)
Max(Objectivv)
Morgan
Scaffold
Gobbi
Gen.
Analog
A
Gen.
Analog
Amlodipine MPO
0.55224
0.3734
0.4807
0.84384
0.7600
-0.0838
0.8894
0.8793
-0.0101
Deco Hop
0.52903
0.8650
0.8180
0,9712
0.8633
-0.1788
0.9992
0.9651
+0.0340
Fexofenadine MPO
0.45537
0.487
0.4736
0.94273
0.7813
-0.1613
1.0000
0.8545
+0,1454
Osimertinib MPO
0.43848
0,4830
0.6339
0.9148
0.8484
-0.0663
0.9508
0.9140
-0.0367
Perindopril MPO
0.43919
0.4344
0,5119
0.6832
0.6413
-0.0419
0.7733
0.7374
+0.0358
Ranolazine MPO
0.34952
0.4362
0,4460
0.8868
0.4506
-0.4361
0.9102
0.8310
40.0792
Scaffold Hop
0.3958
0,5879
0,5646
0,9467
0.5401
-0.4065
1.0000
0.8345
+0,1654
Sitagliptin MPO
0,28301
0,2629
0,2825
0,5747
0.0848
-0,4898
0.8315
0.4909
-0,3406
Valsartan SMARTS
0,34513
0,3568
0.3060
0.8331
0.0452
-0.7878
0.9860
0.9283
-0.0577
Zaleplon MPO
0.522717
0.7238
0.62105
0.6384
0.4668
-0.1715
0.7150
0.7071
-0.0079
Next, we evaluate the application potential of MoleBridge this scenario. We generate synthesizable alternative structures for these molecules that were deemed non-synthesizable. As shown in Table 4 the experimental results reveal that MoleBridge not only guarantees the synthesizability of generated molecules but also strikes balance in property degradation. For instance, in the Amlodipine MPO task, the average objective function value decreased by only 0.0838, while the optimal molecular property remained at 0.8793, nearly the same as the original value.
5.6 Ablation Studies
We conduct ablation experiments the test set to validate the key contribu- tions of thinking and cross-attention in MoleBridge's performance, as shown in Table 5 The cross-attention ensures the effective transfer of information across different modalities, while the thinking mechanism optimizes the ex- ecution of the denoising process. To-
Table 5: Ablation results of components. The cross- attention mechanism replaced by fully connected layers when disabled.
Mechanism
Network
All
Sim
thinking
w/ cress-atization
Success
Recons.
Morgan
ScalTold
Gobbi
0.4840
40.1%
0.8266
0.8470
0.8068
41.1%
gether, they collaborate to achieve op- timal generation performance.
v
0.4915
43.5%
0.8455
0.8695
0.8287
Conclusion and Limitations
In this paper, we introduce MoleBridge, a novel semi-autoregrrssive Markov bridge process for synthetic space projecting. Experimental results demonstrate that MoleBridge excels in various scenarios, such as bottom-up synthesis, structure-based drug design, target-directed generation, and hit expansion. Despite significant progress, MoleBridge still has some limitations. First, the optimization space is limited for molecular structures that contradict existing synthesis logic. Second, for some synthetic paths with specific stereoselective requirements, the model's 's control ability needs improvement. Future work will expand the reaction template library to cover more types of chemical transformations.
7 Broader impacts
Synthetic space projecting technology has a profound impact on the field of novo molecular design by converting molecules that lack synthetic pathways into structurally similar and synthesizable analogs. This approach can significantly accelerate the process from virtual screening clinical candidates, reducing the high failure rate in traditional drug development due to synthesis barriers.

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8 Acknowledgments
This paper was supported by National Key R&D Program of China (No. 2023YFC3502902 2021YFF1201 100), National Natural Science Foundation of China under Grants (62436006). Sanya Science and Technology Special Fund (No. 2024KFJX04), Beijing Natural Science Foundation (No. L257018) and Beijing Nova Program.
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Reviews:
[{"review_id": "pe8NAaLDO4", "reviewer": "Reviewer_WBuU", "summary": "This paper introduces a novel generative model, MoleBridge, for molecular rectification. It semi-autoregressive extends reaction tokens and generates postfix notations based on Markov bridges. The author validated the effectiveness of MoleBridge through a wide range of experiments.", "questions": "1. In the synthesis process, how accurate is the RDKit prediction of the reaction product?\n2. In section 5.1, the author defines synthesis path success rate as the proportion of paths that complete at least 50% of the synthesis steps. Could the author elaborate more on why completing at least 50% of the synthesis steps is considered a success?\n3. How is the number of notation blocks determined? Does the choice of the number of notation blocks affect the performance?", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. Molecule rectification is important in machine learning for molecular design. Many de novo molecule generative model generates non-synthesizable molecules. This paper uses novel methods of Markov Bridges to generate synthesis pathways. \n2. The thinking and denoising separation approach in the paper successfully addresses error accumulation in chemical space projection, offering new insights into machine learning algorithms for chemical reactions. \n\n**Weaknesses**\n1. The MoleBridge generates synthetic postfix notations in the reaction pathways. The author could also discuss some earlier methods, such as JT-VAE, that also generate molecule graphs by sub-parts. \n2. Could the author provide more analysis and discussion on the generated synthetic process compared to the pathway used in the real-world chemical reactions or molecular rectification?", "was_revised": true, "final_justification": "The author has addressed all my questions. I believe the work meets the standard for a NeuIPS acceptance. However, I am still not completely satisfied with the significance of the work and design of the evaluation. Therefore, I will retain my score as it is.", "rebuttal": "Dear reviewer,\n\nWe are grateful for your valuable suggestions! Our detailed responses to the concerns are provided below.\n\n---\n\n>**W1:** The author could also discuss some earlier methods, such as JT-VAE, that also generate molecule graphs by sub-parts.\n\nThank you for sharing this paper. JT-VAE primarily involves continuous embedding and generation of molecular graphs, recursively constructing them through chemical substructures, with the aim of automating molecular design based on specific chemical properties [1]. It serves as a foundational work in the field of molecular research. Specifically, their junction tree variational autoencoder generates molecular graphs in two phases: first by creating a tree-structured scaffold over chemical substructures, and then combining them into a molecule using a graph message passing network. This is an excellent method for graph generation. We recognize the important contributions of methods such as JT-VAE in molecular structure generation.\n\nWe clarify that it can generate molecular structures with specified properties, but whether these molecules can actually be synthesized through existing chemical methods remains an open question. Hence, our work tackles a **distinct yet complementary** problem, addressing downstream challenges related to its application. Thank you for your valuable suggestion. We will cite this work and discuss it. Specifically, we will further discuss the development of this field in the introduction of the revised version, providing the necessary academic discussion and recognition for important works such as JT-VAE.\n\n>**W2:** Could the author provide more analysis and discussion on the generated synthetic process compared to the pathway used in the real-world chemical reactions or molecular rectification?\n\nThank you for this suggestion. Here, we provide additional discussions compared to the pathway used in the real-world chemical reactions from three aspects:\n\n- **First**, real-world chemical reactions typically depend on specific experimental conditions and additional reagents. Similar to most related computational methods, our model does not directly predict these conditions nor provide explicit information about the required reaction types. However, our Markov Bridge framework offers good scalability, allowing for the integration of additional constraints and prior knowledge, which offers flexibility in modeling conditional dependencies. Thus, it can accommodate the extended needs of such a setup.\n\n- **Second**, the chemical validity of our generated results primarily depends on the RDKit product prediction in the inference process. We acknowledge that this validation approach lacks direct evaluation from chemistry experts. To address this issue, we have further evaluated the accuracy of RDKit predictions (please refer to our response in **Q1**). Our analysis indicates that the prediction accuracy of RDKit is acceptable.\n\n- **Third**, we acknowledge that the reaction templates currently used are primarily derived from combinatorial chemistry libraries and may not fully represent the diversity of actual synthesis. These templates are based on validated actual reactions, ensuring chemical feasibility, and cover the most commonly used reaction types in chemistry, fulfilling the needs of most application scenarios. Additionally, future extensions and updates of the templates are also feasible.\n\nFinally, through this work, we hope to improve the design of synthesizable molecules in molecular design. As experiments in real-world scenarios are an extremely time-consuming and costly process. We believe that our contribution will provide a more reliable and practical computational foundation for chemical space projection. In the future, we hope that more excellent works in various fields will benefit from our technical ideas!\n\n>**Q1:** How accurate is the RDKit prediction of the reaction product?\n\nTo evaluate the accuracy of RDKit's prediction of the reaction product, we filter all 11,818 single-step synthetic reactions generated. We evaluate the chemical validity rate (CVR) and prediction coverage (PC). CVR denotes the proportion of chemically valid predicted products, while PC indicates the proportion of samples with successfully generated predictions. As presented in the table below:\n\n| Metric | Value |\n|--------|-------|\n| CVR    | 100%  |\n| PC     | 100%  |\n\nThe results indicate that the RDKit predictions are reliable.\n\n>**Q2:** Could the author elaborate more on why completing at least 50% of the synthesis steps is considered a success?\n\nYour feedback is very important to us. We have realized that there is a problem with our expression. It is computationally equivalent to the **percentage of valid postfix notations**, and further multiplied by 1/2. We acknowledge that our explanation is unclear and caused confusion, so we will restore the definition (\"the percentage of valid postfix notations\") and values in the revised version. We sincerely apologize and hope that this explanation clarifies the issue.\n\n>**Q3:** How is the number of notation blocks determined? Does the choice of the number of notation blocks affect the performance?\n\nWe sincerely appreciate your insightful suggestion. We conduct a sensitivity experiment on the block size, keeping other experimental conditions constant while changing only the size. The results are shown in the table below. We can observe that when the size is 1, its performance is not as good as when the size is 4. When the block size is increased to 6, the performance slightly decreases again. This indicates that too large a block size reduces the ability to model semantic dependencies, leading to limited gains. Therefore, we recommend setting the optimal size to 4.\n\n|  | Success | Recons. | Sim. (Morgan) | Sim. (Scaffold) | Sim. (Gobbi) |\n|--------|---------|---------|---------------|-----------------|--------------|\n| 1      | 0.4810  | 34.2%   | 0.7852        | 0.8139          | 0.7311       |\n| 4      | 0.4915  | 43.5%   | 0.8455        | 0.8695          | 0.8287       |\n| 6      | 0.4780  | 41.9%   | 0.8124        | 0.8313          | 0.7721       |\n\n---\n\nThank you again for your efforts and time. We sincerely appreciate the feedback, which helped improve our work!\n\nBest wishes,\n\nAll authors of Paper 1834.\n\n[1] Wengong Jin, Regina Barzilay, Tommi S. Jaakkola: Junction Tree Variational Autoencoder for Molecular Graph Generation. ICML 2018"}, {"review_id": "LgB7GwsSlt", "reviewer": "Reviewer_Nwry", "summary": "The authors present MoleBridge, a method utilizing Markov bridges to semi-autogressively decode postfix notations for synthesizable chemical space projection. They compare their method to recent work in chemical space projection, particularly SynNet and ChemProjector, and find better empirical performance on a variety of tasks, including reconstruction rate, Morgan fingerprint similarity, and SBDD applications.", "questions": "In addition to addressing the above concerns, could the authors answer:\n- Can the authors compare the empirical results with those of SynFormer (current SOTA on chemical space projection)?\n- What are the computational costs associated with the Markov bridge-based method compared to the more simple autoregressive method?\n- Could Figure 1 be made more clear to illustrate how the algorithm works? I find it hard to follow compared to figures demonstrating the algorithm for ChemProjector [1] and RetroBridge [2], which are both similar methods.\n- I'm not sure I follow the definition of \"success rate\" in the evaluation. \"The proportion of paths that complete at least 50% of the synthesis steps\" is not clear to me. ChemProjector also uses a \"success rate\" but this seems to be defined differently, as the numbers are very different.\n\n[1] Luo et al. https://arxiv.org/abs/2406.04628\n[2] Igashov et al. https://arxiv.org/abs/2308.16212", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- Provides an interesting and novel method to the synthesizable molecule projection task, which itself is important\n- The authors demonstrate empirical benefit over ChemProjector, the most similar method explored, in many tasks such as reconstruction rate for bottom-up synthesis planning and analog generation.\n- Overall, the manuscript is well-written and free of glaring errors.\n\nWeaknesses\n- The empirical benefits over ChemProjector do not seem extremely significant in many evaluations (other than reconstruction rate on the less challenging \"test set\"). I am curious why this evaluation does significantly better compared to other evaluations; there is something to say about the fact that this test set is more artificial and less of an extrapolative task.\n- Additionally, the more recent SynFormer method [1] is not compared against, which greatly improves empirical performance compared to ChemProjector.\n- The evaluations are mainly just the same ones done in ChemProjector with slightly better performance. Since the work is proposing a novel modification to the algorithm, I would be more interested in demonstrating the value behind this algorithmic change. Can the authors find ways more specific ways to show that by adding the Markov bridge / semi-autoregressive formulation, there are error modes in ChemProjector that MoleBridge addresses? For example, something like case studies where the autoregressive formulation leads to incoherent, invalid, or generally \"bad\" sequences that MoleBridge otherwise can correct.\n\n[1] Gao et al. https://arxiv.org/abs/2410.03494", "was_revised": true, "final_justification": "The authors develop a novel and interesting methodology for chemical space projection, and demonstrate strong empirical results on certain benchmarks. I retain my score as though the authors addressed my questions, I am not convinced that the autoregressive formulation of prior papers is a significant issue.", "rebuttal": "Dear reviewer,\n\nWe deeply appreciate your thoughtful review and your recognition of our contributions. Below, we provide point-by-point responses to your concerns and suggestions.\n\n---\n\n>**W1:** Empirical advantage.\n\nWe clarify that the test set comes from the largest and most comprehensive source of medchem building blocks in the world, and it is a standard data source. The reconstruction rate is the percentage of proposed synthesis pathways that lead to products identical to the input molecules. MoleBridge improves the error accumulation issue at the algorithmic level, leading to a higher reconstruction rate. An example can be seen in **W3**. Additionally, given the inherent difficulty of the tasks, our improvements in other evaluations have been relatively significant. In drug discovery, due to the time-consuming and labor-intensive nature of the process with low success rates [1], even a small improvement of a few percentage points could translate into millions of dollars in value. Hence, we hope that in the future, we can better demonstrate the practical application value of the algorithm. Furthermore, we hope that other excellent works in various domains will benefit from our technical ideas!\n\n>**W2 and Q1:** Can the authors compare the empirical results with those of SynFormer?\n\nThank you for providing this paper. We observe that SynFormer is currently only published on arXiv and has not yet been peer-reviewed. Despite this, we take the suggestion seriously and have included a comparison with SynFormer.\n\n|      | Success | Recons. | Sim. (Morgan) | Sim. (Scaffold) | Sim. (Gobbi) |\n|-------------|---------|---------|---------------|-----------------|--------------|\n| SynFormer   | 0.4980  | 26.4%   | 0.7517        | 0.6827          | 0.6878       |\n| MoleBridge  | 0.4915  | 43.5%   | 0.8455        | 0.8695          | 0.8287       |\n\nAlthough SynFormer is an excellent parallel work, MoleBridge has advantages on most metrics. While SynFormer has a slight edge on success, we clarify that in practical studies, the primary focus is on finding synthetically viable analogs similar to the target molecule. Despite this, the comparison reveals interesting research directions, and we are exploring how to combine SynFormer’s strengths to enhance success. We will further discuss this work in the revised version.\n\n> **W3:** Case studies where the autoregressive formulation leads to incoherent, invalid sequences that MoleBridge otherwise can correct.\n\nWe believe the suggestion you mentioned has unique value. To address this, we have further analyzed 7314 cases, where sequence error accumulation occurred in 5441 cases in the autoregressive formulation. We present a typical example of an early error for the target molecule, \"NCCCC(=O)N(CC(=O)NN)C(N)=O\". The autoregressive model generates \"CC(=O)Cl\" in an inference step, which is an invalid pathway, leading to the errors of subsequent steps. This is because this step cannot lead to the target structure. MoleBridge corrects the step to \"NCCCC(=O)NC(N)=O\", generating a complete pathway and successfully reconstructing the target molecule.\n\n>**Q2:** What are the computational costs associated with the Markov bridge-based method compared to the simpler autoregressive method?\n\nThank you for your suggestion. With your suggestion, we have discovered an additional advantage. We conduct a computational cost comparison on the NVIDIA 4090:\n\n| Time            | Min | Average | Max   |\n|-------------------|------------|--------------|------------|\n| Autoregressive method    | 0.19s      | 10.84s       | 191.40s    |\n| MoleBridge        | 0.36s      | 9.71s        | 101.30s    |\n\nOur method is slower in inference for short sequences due to the additional overhead from the Markov Bridge. However, this slight difference in inference time is minimal, with a difference of only 0.17s, which is within an acceptable range. In contrast, for longer synthesis pathway sequences, it is faster, owing to the parallel inference of tokens within blocks.\n\n>**Q3:** Could Figure 1 be made clearer to illustrate how the algorithm works?\n\nThank you for your revision suggestion. Since the conference reply format limits us to text only, we will follow the excellent presentation formats of ChemProjector and RetroBridge that you suggested, further simplify the process, and provide clearer and more intuitive module divisions and labeling improvements, which will be reflected in the revised version.\n\n>**Q4:** The definition of \"success rate\" in the evaluation.\n\nYour feedback is very important to us. We clarify that the definition of the success rate we use is the same as the one in ChemProjector, and it is computationally equivalent to the percentage of valid postfix notations, but further multiplied by 1/2, which has caused some confusion. We acknowledge that our explanation was unclear, and we will restore the original definition in the revised version.\n\n---\nThank you once again for your valuable and insightful review!\n\nBest wishes,\n\nAll authors of Paper 1834.\n\n[1] Kang Zhang, Xin Yang, Yifei Wang, Yunfang Yu, Niu Huang, Gen Li, Xiaokun Li, Joseph C. Wu & Shengyong Yang: Artificial intelligence in drug development. Nature Medicine 2025"}, {"review_id": "drOnLdpujw", "reviewer": "Reviewer_xhsG", "summary": "This work is a contribution to chemical space projection, which translates unsynthesizable molecular graph into the postfix notation-based synthesizable molecular representation. This work identified one of the key weaknesses of chemprojector: autoregressive generation is prone to error accumulation, so some error correction mechanisms are needed. Therefore, this work proposes a block-wise autoregressive generation approach. Within each block, a diffusion-like method, specifically Discrete Markov Bridge, is used to generate tokens (only reaction tokens in this work). Improved performance is demonstrated.", "questions": "- Only reaction tokens are modeled in the Markov Bridge. My understanding is that building block tokens have an extremely large vocabulary size (approx. 200k) and it is predicted as fingerprints, so Markov Bridge cannot be applied to fingerprints? What do the authors think the contributions of building blocks token to the error accumulation? Are errors in reactions token more detrimental? For future work, what do the authors think about the potential solution to implementing error correction on building blocks?", "limitations": "See question and weaknesses sections", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**[Strengths]**\n\n- Chemical space projection is an important problem in molecular design because it improves the feasibility of molecules proposed by generative models. This work is a notable contribution to this field.\n- The motivation is clear and sufficient. This work identified the error accumulation issue in previous work and proposed a mixed approach (specifically, block-wise autoregressive and Markov Bridge) to tackle the challenge. The mixed approach combines the strengths of both techniques. Specifically, block-wise autoregressive generation allows indefinite length sequence but lacks error correction, and Markov Bridge allows error correction but requires fixed length.\n- Bottom-Up Synthesis Planning bencnmarks confirm the improvement of the proposed method.\n\n**[Weaknesses]**\n\n- No comparison to previous methods in other experimental settings (SBDD and Target-directed). This evaluation would be more sound if comparison of **similarity** to previous can be included in other experimental settings. It would be even better if improvement in docking/oracle scores can be shown. However, it is still okay if no significant improvement in docking/oracle is achieved because it is understandable that projection focues only on structural similarities but similarities not necessarily correlate to those scores.", "was_revised": true, "final_justification": "This paper is a notable technical contribution to the field and the authors mostly resolved my questions. I will keep my score.", "rebuttal": "Dear reviewer,\n\nWe sincerely appreciate your thoughtful and constructive feedback. Thank you for your positive assessment of our work! We now address the raised questions as follows.\n\n---\n\n>**W:** No comparison to previous methods in other experimental settings (SBDD and Target-directed).\n\nWe sincerely appreciate this valuable suggestion. However, we would like to clarify that we actually compared previous methods in the SBDD and target-directed settings, and the results included in the supplementary materials. This may have been overlooked because we did not highlight the results sufficiently in the main text, and we apologize for that. We appreciate your insightful comments. Thank you once again for your valuable time!\n\n>**Q1:** Building block tokens have an extremely large vocabulary size and it is predicted as fingerprints, so Markov Bridge cannot be applied to fingerprints? What do the authors think the contributions of building block tokens to the error accumulation? Are errors in reaction tokens more detrimental?\n\nThank you for raising this question. We clarify that the Markov Bridge can be applied to building blocks, although there are still further improvements required in our current implementation. However, we agree that the larger vocabulary size of building block tokens indeed presents technical challenges for the direct application of the Markov Bridge. In contrast, the Markov Bridge performs better in lower-dimensional discrete spaces. For instance, it has been successfully applied in protein modeling with only 20 amino acids [1]. Of course, considering that there is still room for improvement in the discrete Markov Bridge, we also hope to further design more advanced modeling techniques. We will further consider the contributions to this specific modeling aspect. \n\nWe have analyzed 7314 cases and found that 1178 cases of pathway errors were caused by building blocks, which is indeed a significant proportion. Therefore, it is also essential to address this issue. However, given the recent significant progress in guided model learning, we acknowledge that there are also better potential solutions to this issue. We will further discuss this in response to your next question (**Q2**). Thank you again for your invaluable suggestions.\n\n| | |\n|-------------|------|\n| Total cases | 7314 |\n| Errors are caused by building blocks | 1178 |\n\n>**Q2:** For future work, what do the authors think about the potential solution to implementing error correction on building blocks?\n\nTo further optimize the issue of error correction in building blocks, we propose three better potential solutions. \n\n- **First**, fingerprints are essentially \"encodings,\" which we believe are not a complete representation of the molecule but a form of information compression. If their length is too large, they even tend to cause more information loss. The Markov Bridge faces the issue of large fingerprint dimensions in its modeling process. We can draw on advanced theories from information compression to optimize its encoding process, thus generating more informative yet easier-to-handle low-dimensional representations.\n\n- **Second**, inspired by guided training methods in LLMs, we could introduce erroneous blocks as random noise in the training data, which might help the model learn erroneous patterns, potentially improving robustness to building block errors.\n\n- **Third**, akin to the second approach, during inference, we could design highly reliable reference templates for comparison to evaluate the model's performance, helping it identify modeling flaws and potentially serving as a correction strategy.\n\nOf course, these are just our preliminary technical ideas. Thank you again for your valuable professional feedback!\n\n---\n\nThank you once again for your thoughtful and positive review!\n\nBest wishes,\n\nAll authors of Paper 1834.\n\n\n[1] Yiheng Zhu, Jialu Wu, Qiuyi Li, Jiahuan Yan, Mingze Yin, Wei Wu, Mingyang Li, Jieping Ye, Zheng Wang, Jian Wu: Bridge-IF: Learning Inverse Protein Folding with Markov Bridges. NeurIPS 2024"}, {"review_id": "NjF8RRmzPu", "reviewer": "Reviewer_bSFf", "summary": "This paper proposes MoleBridge, a technique for predicting a sequence of postfix notations for synthesizing molecules from a database of available building blocks and reactions in the context of chemical space projection. To enable inference of variable length sequences, the authors propose modelling groups of tokens autoregressively, where each group of tokens undergoes a discrete denoising diffusion process by first predicting which position is to be denoised at each iteration, and then denoising that position. The performance of the resulting model is good, beating existing state-of-the-art on a test set (obtained from Enamine US Stock catalog after clustering) and ChEMBL.", "questions": "- I understand the grouping of tokens (equation 6) as necessary from an efficiency/variable length prediction perspective. How does varying the group size affect the performance of your model? My understanding is that with group size = 1, you should recover the fully autoregressive case. Do you have results for that?\n- Line 178: the notation ($t^{1:}_T$) is a bit unclear (the “${1:}$” part). Could you explain that better in the text? Is it because you start from a given molecule which corresponds to $t^{0}$?\n- Equation 7: second $v_\\tau$ should be bold (I think)\n- Equation 9: should $\\beta_t $ be $\\beta_\\tau$ instead?\n- Equation 9: the statement just before this equation states “define the transition probability as follows:” but Equation 9 is not the actual transition probability (but as you point out, it is related to the transition probability via Equation 8. It would be very helpful to discuss how the second term (with $v_\\tau = 1$) is derived.\n- “De novo molecular design has garnered considerable attention across various research domains in life sciences ... Among these developments, the majority of cutting-edge breakthroughs are driven by deep generative approaches ..., exemplified by Proteína ...” The callout to Proteína seems quite unnecessary, especially because it is a protein generative model, not a small molecule generative model. For proteins, the question of synthesizability is not as important as for small molecules because many proteins can be expressed in E. coli/yeast/rabbit/human cells. In the context of this paper which is focused on synthesizability, the authors should emphasize this point better.\n- I feel like the first part of the title “Semi-autoregressive Your Bridge” is a little ill-formed and unclear. I would recommend the authors to modify the title, if possible.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is quite well-written, and the presentation is quite nice. Great job on the diagrams! The modelling makes a lot of sense, and the experiments seem reasonable to me. \n\nI would have liked to see some more discussion on the chemistry aspects of MCSP; in particular, many synthesis pathways produce > 1 reaction products (eg. enantiomers), which seems hard to integrate into the current framework where the reaction output is only a single product. (One would have to represent a distribution over reaction products at each step). Another aspect is tautomerization, where the reaction product essentially exists in > 1 distinct states. Further, is there a way to add in temperature/pH dependence into your reaction templates? These could go a long way into making the approach more practically useful. For example, can your molecular fingerprints capture chirality which could help distinguish between enantiomers to choose for the next reaction?\n\nI would have also liked to see a few more experiments varying the group size of the tokens (ie, measuring the improvement obtained by discrete diffusion vs autoregression).", "was_revised": true, "final_justification": "Experiments with token group size were provided during the rebuttal. I was not extremely convinced because the performance goes down with increasing group size, but I think overall the methodology of the paper is well-founded and their results are strong.", "rebuttal": "Dear reviewer,\n\nWe sincerely appreciate the time you have taken to review our work and your positive comments about our paper! We aim to address your questions below.\n\n---\n\n>**W1:** Some more discussion on the chemistry aspects of MCSP. Is there a way to add temperature/pH dependence into your reaction templates?\n\nThank you for the insightful suggestions! We completely agree that many chemical reactions are often dependent on specific experimental conditions and additional reagents. Different experimental conditions, such as the temperature and pH you mentioned, can indeed lead to enantiomers. However, our Markov Bridge framework is highly scalable, allowing for the integration of additional constraints and prior knowledge, offering flexibility for modeling conditional dependencies, and thus can accommodate the expansion requirements for such settings. For instance, after obtaining a dataset containing condition information such as temperature and pH, the most direct approach would be to **integrate these conditions as additional context into the neural network inference process**. We believe this is a technically feasible extension. The primary limitation at present is the absence of high-quality datasets containing this kind of data. We look forward to future contributions from the community in providing data resources that include detailed reaction condition information.\n\nWe believe that drawing from recent advancements in **contrastive learning techniques could also be a promising solution**. Contrastive learning works by minimizing the distance between positive sample pairs and maximizing the distance between negative pairs, allowing for the learning of more discriminative molecular embeddings, which is useful for distinguishing enantiomers that have similar structures but different chemical properties. \n\nAdditionally, your observation about the influence of molecular fingerprint choice on the results is highly insightful. We believe that molecular fingerprints are essentially vector hash representations of atomic connections, local environments, paths, and other features. Regardless of the method employed, molecular fingerprints serve as encoding systems that compress a subset of the overall molecular information. We acknowledge that any molecular fingerprint systems inherently introduce some information loss. Our molecular fingerprint method iteratively updates the encoding for each atom, combining its characteristics with those of neighboring atoms, thus integrating the local environmental context of the atoms. This design allows for capturing specific substructures and local chemical environments in the molecule [1], providing molecules with similar structures whenever possible, which alleviates the issue of enantiomer differentiation to some extent. However, we also acknowledge that the current methods still require further development and optimization. Ideally, we hope that in the future, a universal encoding system can be developed that both encompasses complete molecular information and adapts to various application scenarios, including deep learning.\n\n>**W2 and Q1:** How does varying the group size affect the performance of your model?\n\nWe conduct a sensitivity experiment on the group size, keeping other experimental conditions constant while changing only the group size. The results are shown in the table below. When the size is 1, the model generates tokens one by one, which is formally equivalent to an autoregressive model. However, it shows that the performance is worse compared to when the block size is 4. With a size of 6, performance slightly decreases again. This indicates that too large a size reduces the ability to model semantic dependencies, leading to limited gains. Therefore, we recommend setting the optimal size to 4.\n\n|  | Success | Recons. | Sim. (Morgan) | Sim. (Scaffold) | Sim. (Gobbi) |\n|--------|---------|---------|---------------|-----------------|--------------|\n| 1      | 0.4810  | 34.2%   | 0.7852        | 0.8139          | 0.7311       |\n| 4      | 0.4915  | 43.5%   | 0.8455        | 0.8695          | 0.8287       |\n| 6      | 0.4780  | 41.9%   | 0.8124        | 0.8313          | 0.7721       |\n\n> **Q2, Q3 and Q4:** Notations.\n\nWe sincerely appreciate your thorough review of our paper and your careful examination of the mathematical details! We deeply apologize for the notation representation errors that appeared in the paper. Your understanding of the meaning of these notations is correct. We will correct all the notation issues you pointed out in the revised version. Thank you once again for bringing these issues to our attention!\n\n>**Q5:** Equation 9 is not the actual transition probability.\n\nWe appreciate you pointing out this issue! Equation 9 actually defines the approximation process of the Markov Bridge. When $v_\\tau$ = 1, in the Markov Bridge, the forward transfer process can be equivalently expressed as $(1-\\beta_\\tau)y^c+\\beta_\\tau t_\\tau^c$. This approximation process is its reverse operation. To clarify, we will revise this claim.\n\n>**Q6:** The callout to Proteína seems quite unnecessary, especially because it is a protein generative model, not a small molecule generative model.\n\nThank you for your valuable insight. We find your perspective very reasonable. After your suggestion, we realized that citing the protein generation model here is indeed inappropriate. We will follow your advice and remove the inappropriate reference to Proteína in the revised version, refocusing on the challenges of synthesizability in small molecule design. We will highlight the enormous waste of development resources caused by designing non-synthesizable molecules.\n\n>**Q7:** I feel like the first part of the title “Semi-autoregressive Your Bridge” is a little ill-formed and unclear.\n\nThank you for your valuable feedback. We acknowledge that there is a lack of clarity in the title. If possible, we will remove \"Semi-autoregressive Your Bridge\" and revise the title according to your feedback. Once again, we sincerely thank you for your highly valuable feedback!\n\n---\n\nThank you again for your constructive feedback! Your feedback has been very helpful in improving our paper, and I believe you are a very responsible reviewer.\n\nBest wishes,\n\nAll authors of Paper 1834.\n\n[1] Jiatong Li, Yunqing Liu, Wenqi Fan, Xiao-Yong Wei, Hui Liu, Jiliang Tang, Qing Li: Empowering Molecule Discovery for Molecule-Caption Translation With Large Language Models: A ChatGPT Perspective. IEEE Trans. Knowl. Data Eng 2024"}]

Author rebuttal:

Meta-review: The paper proposes a Markov Bridge generative model for postfix notations of molecular synthesis pathways. Reviewers found the work well-written, well-motivated, methodologically sound, and supported by strong empirical results. However, some expressed reservations about the significance of the contribution and the scope of evaluations (including missing baselines). These concerns were largely addressed during the rebuttal, and with all reviewers recommending acceptance, I find the contribution notable enough to merit acceptance.
Decision: Accept (poster)