Title: MoleBridge: Synthetic Space Projecting with Discrete Markov Bridges
Venue: NeurIPS 2025
Accepted: yes
Keywords: Markov Bridges

Diffusion Models

AIGC

Generation
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.

MoleBridge: Synthetic Space Projecting with Discrete Markov Bridges

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.

1 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 a challenge: unlike traditional combinatorial optimization approaches [59, 9] constrained by virtual libraries, generative models typically generate structures that lie outside the synthesizable chemical space [18]. Of these, only a 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 [31] from structurally similar analogs are now made available. This paradigm is centered on 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 10 63 -an enormous space [5, 27]. For this purpose, it is essential that the model can explore, on a sequence-to-sequence basis, synthetic pathways of arbitrary length. ii) Second, unlike natural language, molecular postfix notation sequences

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are composed of 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. In multistep synthesis, selecting an incompatible building block at a 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 a fixed generation length [3, 73, 52], making it difficult to explore synthetic pathways of variable lengths. Thus, instead of adding Gaussian noise to an entire fixed-length synthesis pathway, MoleBridge introduces probabilistic perturbations to notation token blocks at current autoregressive steps. By approximately referencing the Markov bridge process [17, 29, 78], MoleBridge learns to progressively refine perturbed reaction sequences, thereby generating more rational designs. In pursuit of stable pathways and semi-autoregressive error robustness, we identify errors introduced in 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 a variety of scenarios.

Our contributions can be summarized as follows:

We introduce MoleBridge, a novel generative approach for generating molecular postfix notations based on the Markov bridge, which expands chemical space via a 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, we 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

learning approaches [6, 19, 7] have now been developed to predict reaction outcomes in a 'templatefree' 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 a 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 postfix-notation path generation remains challenging and requires further design.

Diffusion generative models. Diffusion models [12, 64] are generative models achieving state-ofthe-art performance across various domains, including the generation of images [62, 44], video [66], or molecular [70, 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 as MDM [28], DiffLinker [30], and PIDiff [13] 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, 55]. Recently, some studies [43, 45] propelled the development of molecular design technologies by utilizing more reliable diffusion processes.

Schrödinger bridge. The Schrödinger Bridge (SB) problem [53, 4, 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 [72]. 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 [15, 36], speech [37], and physical fields [41]. The recently proposed Markov bridge models [78, 29, 17] extend these models into the discrete domain, 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 combinations, starting from the initial building blocks [46]. Mathematically, a synthetic space S is represented as the closure of molecules generated by a set of n b building blocks B = { b 1 , b 2 , . . . , b n b } ∈ S and a set of n r reaction rules R = { r 1 , r 2 , . . . , r n r } , where each reaction rule r i defines a mapping function from the reactant space to the product: r i := X × Y → S , ( X , Y ) ↦→Z . Here, X , Y ∈ S represents the sets of molecules applicable to the reaction r i , and Z represents the main reaction product. In the synthetic space construction process, synthesis pathways are represented as notation sequences y = [ t 1 , t 2 , . . . , t n t ] , where each token t ℓ ∈ B∪R indicates pushing a building block onto the stack or calculating the product and pushing it back onto the stack, with n t 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 ϑ , should be capable of learning the mapping from any molecule M to its corresponding postfix notation distribution p ϑ ( y |M ) .

3.2 Autoregressive Models

The success of large models in the natural language processing and computer vision domain demonstrates their scalability and the universality of sequence data modeling [3, 68, 65, 75]. Given a

postfix notation sequence y = [ t 1 , t 2 , . . . , t n t ] , where the subscript 1 ≤ ℓ ≤ n t specifies an order, autoregressive models assume that the probability of observing the current token t ℓ depends only on its prefix [ t 1 , t 2 , . . . , t ℓ -1 ] . This unidirectional token dependency assumption allows the likelihood of the sequence y to be factorized as:

where p ϑ ( t ℓ | t <ℓ ) is parameterized directly with a neural network. Therefore, autoregressive models can be efficiently trained through 'next token prediction' [54, 71, 74]. However, due to sequential dependencies, autoregressive models require n t steps to generate n t tokens.

3.3 Discrete Markov Bridge

Markov bridge model fits a model ψ ϑ ( · ) to reverse the forward corruption process q [78, 29, 17]. This process is pinned to specific data points in the beginning and in the end, modeling the dependencies between the discrete spaces X and Y . For the sample pair ( x , y ) ∼ p X,Y ( x , y ) and the time step sequence τ = 0 , 1 , . . . , T , it defines the Markov process as a sequence of random variables ( t τ ) T τ =0 , starting from t 0 = x , and satisfying the Markov property:

To ensure the process terminates at the data point t T = y , we introduce an additional requirement:

Suppose the distributions p X and p Y are categorical distributions with a finite sample space 1 , . . . , K , and we can represent the data points as K -dimensional one-hot vectors: x , y , t τ ∈ R K , and define the transition probabilities (Eq. (2)) as follows:

where Cat( · ; p ) is a categorical distribution with probabilities given by p , and Q τ is a transition matrix parameterized as:

where I K denotes a K × K identity matrix, and α τ is a schedule parameter transitioning from α 0 = 1 to α T -1 = 0 . Then, t τ can be efficiently sampled from p ( t τ +1 | t 0 , t T ) = Cat( t τ +1 ; ¯ Q τ t 0 ) , where ¯ Q τ = Q τ Q τ -1 . . . Q 0 = ¯ α τ I K +(1 -¯ α τ ) y 1 ⊤ K a cumulative product matrix, and ¯ α τ = ∏ τ s =0 α s . During training, the Markov bridge approximates y using the neural network ψ ϑ : ˆ y = ψ ϑ ( t τ , τ ) .

4 Methods

In this section, we introduce MoleBridge, a Markov bridge model for synthetic space projecting. We first define how to interpolate between semi-autoregressive 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, we describe the denoising network architecture employed to approximate the Markov bridges.

4.1 Markov Bridge for Synthetic Space Projecting

Generally, molecular synthesis paths are highly discrete processes that strictly adhere to chemical rules. A common scenario is 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 of tokens autoregressively. We group the postfix notation tokens into C notation blocks, each of length ˆ n c , where C = n t / ˆ n c (assuming C is an integer). We denote each block t : c ˆ n c from token at positions 0 to c ˆ n c for blocks c ∈ { 1 , . . . , C} as t c for simplicity.

Interpolation process. As discussed in Section 3.3, the matrix Q τ can efficiently model various transition probabilities in the discrete state space, including masking, random token changes, and

related word substitutions. When considering the noise process modulated by the masking vector [78, 29], the interpolation process gradually transforms the data point t 1: T into the initial molecular state t 1: 0 ∼ p 0 ( t ) , with the transition rate controlled by the masking vector. In this case, the noise interpolation can be represented as:

where v τ ∼ Bernoulli( ¯ β τ -1 ) is a masking latent vector and β τ denotes a scheduler. At τ = 0 , the conditional marginal converges to the initial distribution t 1: 0 , i.e. β τ = 1 . When τ → T , β τ is set close to 0 and the distribution is closer to the target distribution.

Reverse process. Accordingly, the reverse process in block c can be written as:

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where the denoising base model predicts clean token t c given the noisy sequence t c τ . Since masked diffusion requires building a complete probability distribution p ( t c | t c τ ) over all possible values at each position ℓ ∈ { 1 , . . . , | t c |} rather than directly identifying corrupted locations { ℓ | t c,ℓ τ = t c,ℓ } , it is difficult to achieve accurate denoising probability estimation [56, 47]. As evidenced by recent works [11, 56, 51], the core difficulty stems from denoising probability estimation p ϑ 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 ( t c τ +1 | v τ , t c τ , t c ) 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 [45], conditioned on the current state t c τ and time step τ , and define:

where p θ ( z c τ = 1 | t c τ ) denotes the probability that each position is corrupted, and z c τ ∈ { 0 , 1 } ˆ n c is a latent variable to denote if a dimension is corrupted. If a position is likely to be corrupted ( p θ ( z c τ = 1 | t c τ ) ≈ 1 ), the transition probability will lean toward the output of the denoiser. On the other hand, when a position is likely to be clean ( p θ ( z c τ = 1 | t c τ ) ≈ 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 a principled learning objective for the model ψ ϑ ( · ) by applying the variational bound on negative log-likelihood log p ϑ ( y | t ) to each term, which has the following closed-form expression:

where y c denotes the true tokens in block c . The derivation of L τ ( t , ϑ ) expresses the training loss as a reweighted standard multiclass cross-entropy loss [78, 26], which is computed on the labels that have not yet been converted into the base truth y = t T . Compared to the simpler cross-entropy loss computed over all labels, this new approach assigns greater weight to the labels that need refinement. Since the model is conditioned on t <c , the dependency between t <c and ϑ is explicated in L τ .

The training of p θ 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:

̸

where z c,ℓ τ is a variable indicating whether the ℓ -th position in block c is corrupted at time step τ , t c,ℓ τ is the noisy token at that position, y c,ℓ is the true value, I ( · ) is the indicator function, and BCE( · ) represents binary cross-entropy. The final loss function is the sum of the two terms: L = L τ + L p .

Sampling. A postfix notation of synthesis y is a sequence that contains four types of tokens: building block tokens b j ∈ B , reaction tokens r i ∈ R , a start token [START] , and an end token [END] . Each building block token b j is associated with the fingerprint of the corresponding molecule, represented as [BB,j] , where j ∈ { 0 , 1 } 256 is the Morgan fingerprint of length 256 and radius 2 [46, 49]. A reaction token r i , denoted as [RXN,i] , represents the index i of the reaction. During sampling, we generate the pathway token-by-token, and each token is produced through a block-wise Markov bridge refinement process. The conditional distribution p ϑ ( t c τ +1 | v τ , t c τ , t <c ) is used to sample from the model. Starting from the given t c 0 ∼ p 0 ( t c ) , the process iterates to predict the data point ˆ y c = ψ ϑ ( t c τ , τ ) and then derive t c τ +1 ∼ p ϑ ( t c τ +1 | v τ , t c τ , t <c ) while incrementing the time step τ 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 a building block token b j is generated, we retrieve the corresponding molecule from the fingerprint and push it onto the stack. If a reaction token r i is generated, we first pop the required number of molecules from the stack, then use the reaction template with RDKit [8] to predict the product, which is subsequently pushed onto the stack. If 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 a molecule M as a graph, where nodes are connected based on chemical bonds. For atoms, we convert them into initial embedding vectors h (0) a ∈ R d based on their atomic numbers; for chemical bonds, we capture bond type information and incorporate it into the graph structure, represent as h (0) e ∈ R de .

Networks. We use a 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 it into vector e τ ∈ R d , 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 ( L ) a 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 θ ( z c τ = 1 | t c τ ) , we use a multilayer perceptron with gelu( · ) activation and a sigmoid( · ) output layer. The entire architecture is 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, a final set of 91 reaction templates is obtained. The set includes 13 unimolecular and 78 bimolecular reactions. For building blocks B , we use the Enamine US Stock catalog [1] as the data source. Entries containing multiple molecules (e.g., salts or hydrates) are 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, we include a challenging test set: molecules extracted from the ChEMBL database [20], 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

Cc1cc(Nc2cc(C(F)(F)F)ccn2)nc(-c2ccnc(F)c2)c1

Sim. (Gobbi): 0.748

Sim. (Morgan): 0.859

Sim. (Gobbi): 0.918

Sim. (Morgan): 0.905

building blocks. The model is trained on 4 NVIDIA 4090 GPUs with a batch size of 128 and 4 data loader workers. We use the Adam optimizer [34] with an initial learning rate of 3 × 10 -4 , and momentum parameters β 1 = 0 . 90 , β 2 = 0 . 999 . A plateau-based learning rate scheduler is used, reducing the learning rate by a factor of 0 . 6 when validation performance plateaus, with patience of 5 validation cycles and a minimum learning rate of 1 × 10 -5 .

Metrics. We conduct a comprehensive evaluation of the MoleBridge model using multiple quantitative 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 the same product as the input molecule. iii) In cases of partial success, we evaluate the molecular similarity between the generated and target compounds; a similarity score of 0 is assigned to failed (invalid) syntheses. The similarity score is calculated using three types of fingerprint representations: Morgan fingerprints of length 4096 and radius 2 [49], Murcko scaffold-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 1 show that MoleBridge outperforms the baseline methods significantly on all evaluation metrics. A particularly notable 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.

5.3 Projecting Molecules Generated by Structure-Based Drug Design Models

Synthetic space projecting has broad application prospects in the field of structure-based de 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],

we use Pocket 2 Mol [50] to 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 5 analogs for each candidate molecule, and the optimal analogs are selected based on Vina scores [16]. Figure 2 shows examples of molecular synthesis pathways generated by MoleBridge, illustrating the complete synthesis pathway design from Pocket 2 Mol to the target analog. Table 2 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 to Explore Local Chemical Space for Hit Expansion

MoleBridge also demonstrates significant application potential in hit expansion. Following the experimental design of Levin et al. [39] and Luo et al. [46], we evaluate the development of c-Jun N-terminal Kinases3 (JNK 3 ) inhibitors. Using an molecule with a JNK 3 score [40] of 0 . 68 as the starting point, MoleBridge successfully generates several synthesizable structural analogs as in Figure 3. Notably, the generated analogs effectively preserve the JNK 3 inhibitory activity ( 0 . 67 , 0 . 68 , and 0 . 70 ) while maintaining the integrity of the core scaffold structure. Taking the amino-substituted analog in the lower-left corner as an example, the introduction of an amino functional group not only maintained almost the

Hit Molecule

JNK3: 0.68

Sim. (Morgan): 0.751

Sim. (Gobbi): 0.937

JNK3: 0.67

Sim. (Morgan): 0.535

Sim. (Gobbi): 0.986

JNK3: 0.68

Sim. (Morgan): 0.826

Sim. (Gobbi): 0.935

JNK3: 0.70

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same JNK 3 activity ( 0 . 68 ) but also potentially provided new modification sites for further structural optimization.

5.5 Projecting Molecules Generated by Target-Directed Generative Models

Target-directed generative models generally suffer from insufficient synthesize-ability, with approximately 70% of generated molecules being labeled as non-synthesizable by the ASK-COS [14, 18].

Next, we evaluate the application potential of MoleBridge in 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 a 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 on the test set to validate the key contributions 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 execution of the denoising process. Together, they collaborate to achieve optimal generation performance.

6 Conclusion and Limitations

In this paper, we introduce MoleBridge, a novel semi-autoregressive 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 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 de 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 to clinical candidates, reducing the high failure rate in traditional drug development due to synthesis barriers.

8 Acknowledgments

This paper was supported by National Key R&D Program of China (No. 2023YFC3502902, 2021YFF1201100), 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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{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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).\n\nQUESTIONS:\n- 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.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\n**[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.\n\nQUESTIONS:\n- 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?\n\nLIMITATIONS:\nSee question and weaknesses sections", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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\n\nQUESTIONS:\nIn 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\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

comments: 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.

STRENGTHS AND WEAKNESSES:
**Strengths**
1. 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. 
2. 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. 

**Weaknesses**
1. 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. 
2. 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?

QUESTIONS:
1. In the synthesis process, how accurate is the RDKit prediction of the reaction product?
2. 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?
3. How is the number of notation blocks determined? Does the choice of the number of notation blocks affect the performance?

LIMITATIONS:
Yes.
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03