Symbolic Neural Generation with Applications to Lead Discovery in Drug Design

Abstract We investigate a relatively under-explored class of hybrid neurosymbolic models that integrate symbolic learning with neural reasoning to construct data generators meeting formal correctness criteria. In Symbolic Neural Generators (SNGs), symbolic learners examine logical specifications of feasible data from a small set of instances–sometimes just one. Each specification in turn constrains the conditional information supplied to a neural-based generator, which rejects any instance violating the symbolic specification. Like other neurosymbolic approaches, SNG exploits the complementary strengths of symbolic and neural methods. The outcome of an SNG is a pair (H, X), where H is a symbolic description of feasible instances constructed from data, and X a set of generated new instances that satisfy the description. We introduce a semantics for such systems, based on the construction of appropriate base and fibre partially-ordered sets combined into an overall partial order. We implement an SNG combining a restricted form of Inductive Logic Programming (ILP) with a large language model (LLM) and evaluate it on early-stage drug design. Our main interest is the description and the set of potential inhibitor molecules generated by the SNG. On benchmark problems – where drug targets are well understood – SNG performance is statistically comparable to state-of-the-art methods. On exploratory problems with poorly understood targets, generated molecules exhibit binding affinities on par with leading clinical candidates. Experts further find the symbolic specifications useful as preliminary filters, with several generated molecules identified as viable for synthesis and wet-lab testing. Data Availability The data and code are available at: https://github.com/tirtharajdash/LMLFStar. Notes Biochemical terms used throughout this paper are briefly explained in Appendix C.1. We will sometimes use SNG to stand for the computation process of symbolic neural generation with the context making it clear whether we mean a system or the process. In practice, neither component does just one function or the other. For example, any non-trivial learning (induction) usually requires reasoning (deduction), and any sufficiently complex reasoning would usually be accompanied by some trial-and-error learning. Therefore, this categorisation is necessarily a simplification. Later we suggest alternatives and extensions that apply to wider classes of hybrid neurosymbolic systems. Thus, if the neural component is a stochastic model that defines a probability distribution \(P_N\) over \(\mathcal U\), \(N(H) = \{{u \in \mathcal U}: P_{N(H)}(u) > 0 \}.\) Question (i) is concerned with constructing symbolic hypothesis, given examples and background knowledge. Techniques developed in ILP are especially good at this. Question (iii) is about searching graphs efficiently, for which several alternatives are known. We do not have to explicitly construct \(\operatorname {ext}(H|B)\) for this. Instead, we include x in S iff \(B \wedge H \models \Sigma (x)\). This check can be performed by a model-checker (or in some restricted cases, by an inference engine). Also we do not accept any instance x that vacuously satisfies the requirement that \(B \wedge H \models \Sigma (x)\). The results are with a sample-size of 30 on each iteration. For a uniform sampler, the probability of a positive instance in any single random draw is approximately \(27/27000 \approx 0.001\). The chance of not obtaining at least 1 positive instance in a sample of 30 is \(\approx 1-(0.999)^{30} = 0.03\). It is unclear if this applies directly to language models, which have access to significant amounts of prior chess knowledge. Unlike the chess problem, we no longer have a symbolic hypothesis. We conjecture that the use of the symbolic learner, as is done here, may result in more reliable hypotheses and require fewer experiments to be conducted by the robot. Some computations, like \(\textit{Affinity}(\cdot )\), can be quite involved, requiring knowledge of the target-site, and the use of a molecular modelling program for estimating binding affinity of the molecule to this site. w(H, X) is replaced by an indicator function that is 1 if \(w(H,X) > 0\). In effect, this rejects hypotheses for which the neural generator is unable to generate consistent instances. From the rest, the search returns the hypothesis with highest Q-score, and therefore highest posterior probability. Neural Markov Logic Networks (Marra and Kuželka, 2021) are a neurosymbolic relational generative model demonstrated on molecule generation, and would be a natural point of comparison. We attempted this using the authors’ official implementation. The reported NMLN molecule experiments used 1073 molecules restricted to exactly 8 heavy atoms, whereas our molecules are drug-like (13 to 53 heavy atoms, median 31). The NMLN formulation grounds all predicates over a fixed domain of N constants per molecule, and the unary one-of-N constraint requires exactly N heavy atoms per training example; setting \(N=32\) retained only about 350 of about 4000 molecules. Even with reduced potential arity (\(k=3\)), training on a 24 GB GPU completed only \({\sim }3000\) of the default 200, 000 iterations after 48 hours of GPU-saturated runtime, well short of convergence. We therefore do not report a direct comparison: NMLN was developed and evaluated in a small-fragment regime, and scaling it to drug-like sizes lies outside the scope of the current work. Dr. Dey is an expert in structural biology and drug discovery, specialising in the molecular modeling and cryo-EM analysis of membrane proteins to target cardiovascular and infectious diseases. 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The authors thank Professors Suman Kundu and Sumit Biswas for their insightful discussions on DBH. Author information Authors and Affiliations Contributions AS: Formulated the notion of symbolic neural generation (SNG); formulated the semantics of SNG, developed the algorithms, and devised the experiments reported in the paper; TD: Worked with AS in implementing the algorithms, conducted the experiments and analysed the results; AB: Worked with AS in developing the semantics of SNG; MB: Worked with AS in clarifying the notion of SNG and in identifying related work; SD: Structural biology assessment of experimental results of using SNG; MB: Synthetic chemistry assessment of experimental results of using SNG. Corresponding author Additional information Editor: Celine Vens. Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Tirtharaj Dash: Part of this work was carried out while the author was at the University of Cambridge, UK. The major revision of this work was completed at BITS Pilani. Appendices A Additional Conceptual Details for SNG 1.1 A.1 Note on Extending Other Hybrid Systems The general principle of a symbolic base combined with neural-fibres can be applied to characterise systems in each of the cases (A)–(D) in Fig. 2a Example 4 (Hybrid systems with a symbolic base) Let us assume the following: - A fixed universal set \(\mathcal U\) (instances, embeddings, predictions, programs etc). - Background knowledge B; - A symbolic component that can enumerate elements from a set of symbolic hypotheses \(\mathcal H\); and - A neural component that – given any element from \(H \in \mathcal{H}\) – can generate elements from \(\mathcal{U}\) that are consistent with H. Additionally, the set \(\mathcal{H}\) is assumed to a partially ordered set, with ordering \(\ge _\mathcal{H}\). Associated with each element H in the ordering is a fibre-poset F(H) that is obtained from the neural component. Each element of the base set is combined with elements from the fibre-sets to give a partially ordered set \(\mathcal {F}\) of hybrid systems. Example hybrid systems in the categorisation in Fig. 2(a) that can be characterised in this way are: Case | Symbolic | Neural | Base | Fibre | \(\mathcal F\) | |---|---|---|---|---|---| A | Reasoning | Reasoning | Hypotheses ordered by entailment | Neural reasoners consistent with the symbolic entailments | Pairs ordered by stronger symbolic entailment and higher neural agreement | B | Learning | Reasoning | Hypotheses ordered by specialization | Neural approximators of the symbolic extensions | Pairs ordered by more specific hypotheses and more confident neural outputs | C | Reasoning | Learning | Hypotheses ordered by entailment | Neural representations trained to preserve symbolic reasoning patterns | Pairs ordered by stronger entailment and representational inclusion | D | Learning | Learning | Hypotheses ordered by generality | Neural learners minimizing a loss defined by the symbolic hypothesis | Pairs ordered by more general hypotheses and lower training loss | This example shows how the setting of a symbolic base with neural fibres can be used to characterise other hybrid systems where the symbolic component has the primary role. Further variants – of less relevance to this paper – but of wider applicability refer to: (a) Hybrid neurosymbolic systems where the neural component has the primary role (a base of neural states with symbolic fibres); and (b) Hybrid systems where both neural and symbolic components are equally important. In this case the base consists of pairs and each pair is associated with a hybrid instance-set.Footnote 16 1.2 A.2 Correctness of Procedure \(\lambda \) Gen Proposition 2 The set \(M_n\) returned by Procedure 1 is an element of F(H) and \(w_n \in [0,1]\). Proof \(\lambda \) Gen executes the loop (Steps 5–12) n times. We claim the following is loop invariant: At the start of the iteration (\(i = 1\)), \(M_{i-1} = \emptyset \) and the invariant is trivially true. Assume the invariant holds at the start of the \(k^\text {th}\) iteration. That is, \(M_{k-1} \subseteq N(H) \cap \operatorname {ext}(H|B)\) and \(|M_{k-1}| \le s\times (k-1)\). On the \(k^\text {th}\) iteration: (i) \(S_k \subseteq N(H)\); and (ii) in Step 10 \(M_k = Z_k \cup M_{k-1}\), where \(Z_k = \{x: x \in \mathcal{U},~ (true,x) \in D_k\}\). Since \((true,x) \in D_k ~\text {iff}~ x \in S_k \cap \operatorname {ext}(H|B)\). So \(Z_k \subseteq N(H)~ \cap ~ \operatorname {ext}(H|B)\). Therefore \(M_k \subseteq N(H)~ \cap ~ \operatorname {ext}(H|B)\). Also since \(|D_k| \le s\), \(|M_k| \le s + |M_{k-1}|\). Since \(|M_{k-1}| \le s \times (k-1)\), it follows that \(|M_k| \le s \times k\). The loop variable i is incremented to \(k+1\), and at the start of the next iteration, clearly \(M_{i-1}=M_k\) which is a subset of \(\operatorname {ext}(H|B)\) and has at most \(s \times k\) elements. The procedure clearly terminates since i is bounded by n, and the procedure returns the set \(M_{n}\) which is a subset of \(\operatorname {ext}(H|B)\) and \(w_n \in [0,1]\). \(\square \) B Algorithmic Details: Application to Lead Discovery 1.1 B.1 Setting Up the Base Poset We first introduce some definitions that are helpful in clarifying both the set of hypotheses \(\mathcal{H}\) and the ordering over that set. Definition 7 (Interval-Vectors) An n-dimensional interval-vector \(\textbf{v}\) \(=\) \(([a_1,b_1],\ldots ,[a_n,b_n])\) is an element of \((\mathbb {R} \times \mathbb {R})^n\) where \(a_i \le b_i\) for \(i \in \{1,\ldots ,n\}\). We will sometimes denote \((\mathbb {R} \times \mathbb {R})\) as \(\mathcal{I}\) and \((\mathbb {R} \times \mathbb {R})^n\) as \(\mathcal{I}^n\). The set \(\mathcal{I}^n\) is therefore the set of n-dimensional hyper-rectangles, and an interval-vector is a hyper-rectangle. The following definition is useful later. Definition 8 (Interval-Vector Containment) Given interval-vectors \(\textbf{v}_1, \textbf{v}_2 \in \mathcal{I}^n\) If \((\textbf{v}_2[1] \subseteq \textbf{v}_1[1]) \wedge \cdots \wedge (\textbf{v}_2[n] \subseteq \textbf{v}_1[n])\) then we will say \(\textbf{v}_2\) is contained by \(\textbf{v}_1\) (resply. \(\textbf{v}_1\) contains \(\textbf{v}_2\)) We denote this by \(\textbf{v}_2 \sqsubseteq \textbf{v}_1\) (resply. \(\textbf{v}_1 \sqsupseteq \textbf{v}_2\)). If there exists at least one \(j \in \{1,\ldots ,n\}\) s.t. \(\textbf{v}_2[j] \subset \textbf{v}_1[j]\), we will say \(\textbf{v}_2\) is properly contained by \(\textbf{v}_1\) (resply. \(\textbf{v}_1\) properly contains \(\textbf{v}_2\)). We denote this by \(\textbf{v}_2 \sqsubset \textbf{v}_1\) (resply. \(\textbf{v}_1 \sqsupset \textbf{v}_2\)). Clearly, if \(\textbf{v}_1 \sqsubset \textbf{v}_2\) then \(\textbf{v}_1 \sqsubseteq \textbf{v}_2\). Definition 9 (Factors) Let \(\mathcal{U}\) be a set of instances. A factor is a function \(f: \mathcal{U} \rightarrow \mathbb {R}\). Definition 10 (Factor Specification) Let \(F = (f_1,\ldots ,f_n)\) be a sequence of factors. A factor specification is the pair \((F,\mathbf {\Theta })\) and \(\mathbf {\Theta } = ([f_1^-,f_1^+],\ldots ,[f_n^-,f_n^+]) \in \mathcal{I}^n\). For \(i \in \{1,\ldots ,n\}\), \([f_i^-,f_i^+]\) is the range of values for the factor \(f_i\). A factor-specification allows us to define the notion of an experiment: Definition 11 (Experiment) Let \(\mathcal{U}\) be a set of instances. Let \(((f_1,\ldots ,f_n),\mathbf {\Theta })\) be a factor-specification. An experiment \(\textbf{e}\) given the factor-specification, or simply an experiment, is an interval-vector in \(\mathcal{I}^n\) s.t. \(\mathbf {\Theta }\) subsumes \(\textbf{e}\). Experiments specify conjunctive logical constraints on factors. We adopt the terminology in Inductive Logic Programming (ILP) and each experiment will be associated with a hypothesis. Definition 12 (Hypothesis) Let \(\mathcal{U}\) be a set of instances, \((F,\mathbf {\Theta })\) be the factor specification where \(F = (f_1,\ldots ,f_n)\). Let \(\textbf{e}\) be an experiment given \((F,\mathbf {\Theta })\). Then a hypothesis given an experiment is the clause: The hypothesis space \(\mathcal{H}\) is the set of such hypotheses. The ordering on this set will be same as introduced in Defn. 2, namely pointwise inclusion of extensions. 1.2 B.2 Bayesian Scoring of Hypotheses Definition 13 (McCreath’s Q-Heuristic) Let \(\mathcal{U}\) denote the set of all instances. Let h be a hypothesis as defined in Defn. 12. Let \(E^+\) denote a set of positive examples and \(E^-\) denote a set of negative examples s.t. \((|E^+| \ge 1\) and \(|E^-| \ge 0\). Let \(D = E^+ \cup E^-\). Let \(\text {ext}(h|B) = \{x: x \in \mathcal{X}, B \wedge h \models Feasible(x)\}\); and for any \(S \subseteq \mathcal{U}\), \(\theta (S) = \frac{|S|}{|\mathcal{X}|}\). Let \(\epsilon \) be the probability that an instance is randomly assigned to \(E^+\) (resply. \(E^-\)). Let B denote background knowledge, \(TP(H|B,D) = \{e: e \in E^+, e \in \operatorname {ext}(H|B)\}\); \(TN(H|B,D) = \{e: \lnot e \in E^-, B \wedge H \wedge e \not \in \operatorname {ext}(h|B)\}\); and \(FPN(H|B,D) = D \setminus (TP(H|B,D) \cup TN(H|B,D))\). Then, dropping the inclusion of B, D for convenience, the fixed-example model in (Mccreath and Sharma, 1998) defines the quality of a hypothesis as: For the special case of \(\epsilon = 0\), the quality of a hypothesis in the fixed-example setting simplifies to: In (Mccreath and Sharma, 1998) it is shown that maximising Q(H|B, D) maximises the Bayesian posterior \(\text {P}(H|B,D)\), along other theoretical results including a proof of (probabilistic) convergence to a target concept. Assuming the entailment relation \(\models \) can be checked, the practical difficulties in using the Q-heuristic are in obtaining the values for \(\theta (\operatorname {ext}(H))\) and P(H). We note the following: We will need the following to be able to use the Q-heuristic here: - a. In order to obtain the sets TP, TN and FPN we will require B to contain all the definitions needed to evaluate the constraint in the hypothesis (that is, B will need to contain definitions for the \(f_i(\cdot )\)). - b. By definition, \(\text {ext}(h)\) is the set of feasible instances as defined in Defn. 1. We can estimate \(\theta (ext(h))\) on a random sample \(X \subset \mathcal{X}\) as follows: Let \(S = \{x: x \in X, \Phi (x) \text { is } true\}\). Then the (maximum-likelihood) estimate of \(\theta (\text {ext}(h))\) is \(\hat{\theta }(ext(h)) = \frac{|S|}{|X|}\).Footnote 17 Remark 4 These results follow straightforwardly from the definition in Defn. 13: - \(Positive-only data.\): - Let \(E^+ \ne \emptyset \) and \(E^- = \emptyset \). Let \(\epsilon = 0\). If \(P(H_1) = P(H_2)\), then \((Q(H_1) > Q(H_2))\) iff \((\operatorname {ext}(h_1) < \operatorname {ext}(h_2))\). - \(Negative-only data.\): - Let \(E^- \ne \emptyset \) and \(E^+ = \emptyset \). Let \(\epsilon = 0\). If \(P(H_1) = P(H_2)\), then \((Q(H_1) > Q(H_2))\) iff \((\operatorname {ext}(H_1) > \operatorname {ext}(H_2))\). That is, in the noise-free case, with equal prior probabilities, and positive data only, more specific hypotheses will be preferred; and with negative data only, more general hypotheses will be preferred. 1.3 B.3 Property of the Procedure GenMol Proposition 3 Let \((F,\cdot )\) be a factor-specification, B denote background knowledge. Let \(\textbf{e}_{i}\) (\(1 \le i \le k\)) be an experiment selected by GenMol on the \(i^\text {th}\) iteration s.t. \(\textbf{e}_i\) is contained by \(\textbf{e}_{i-1}\). Let \(H_i = Hypothesis(F,\textbf{e}_i)\), and \(H_{i-1} = Hypothesis(F,\textbf{e}_{i-1})\). Then \(H_{i-1} \models H_i\). Proof First we observe that \(H_{i}(x): (\Sigma (x) \leftarrow (x \in \mathcal{U} \wedge C_i(x))\) and \(H_{i-1}(x): (\Sigma (x) \leftarrow (x \in \mathcal{U}) \wedge C_{i-1}(x))\). It is easy to see that since by construction \(\mathbf {e_{i-1}}\) contains \(\mathbf {e_i}\), \(\forall x (C_{i}(x) \models C_{i-1}(x))\). Now suppose \(H_{i-1} \not \models H_i\). That is, there exists some \(a \in \mathcal{U}\) s.t. \(H_{i-1}(a)\) is true and \(H_i(a)\) is false. Since \(H_{i}(a)\) is false, \(\Sigma (a)\) is false and \(C_{i}(a)\) is true. Since \(C_{i}(x) \models C_{i-1}(x)\) for all x, and \(C_i(a)\) is true, therefore \(C_{i-1}(a)\) is true. Since \(H_{i-1}\) is assumed true, and \(C_{i-1}(a)\) is true, then \(\Sigma (a)\) is true, which is a contradiction. So for all \(a \in \mathcal{U}\), whenever \(H_{i-1}\) is true then \(H_i\) is also true. \(\square \) C Additional Experimental Details: Case Studies 1.1 C.1 Note on Biochemical Terminology A few terms used throughout the paper may be unfamiliar to readers from the ML community. A small molecule is a low-molecular-weight organic compound (typically below 900 Da), often a drug candidate. A target is a biomolecule (usually a protein) whose activity one wishes to modulate to produce a therapeutic effect. An inhibitor is a small molecule that binds the target and reduces its activity, typically by occupying a functional site such as the active site or an allosteric pocket. Binding affinity measures the strength of the molecule–target interaction (commonly reported as \(K_i\), \(K_d\), or \(\textrm{IC}_{50}\)); in this work we estimate it computationally using docking software (GNINA), which predicts a docking score (binding affinity). A hit is a compound showing measurable activity against the target in an initial assay, while a lead is a hit that has been validated and refined to exhibit suitable potency, selectivity, and drug-like properties, serving as a starting point for further medicinal-chemistry optimisation (Bleicher et al., 2003; Hughes et al., 2011). A scaffold is a common structural core shared across a family of molecules, and SMILES is a standard string encoding of molecular structure. Readers interested in a more detailed introduction to small-molecule drug design should see (Di and Kerns, 2015; Doytchinova, 2022; Hughes et al., 2011). Figure 7 illustrates the geometric situation that a docking software attempts to score: a small-molecule inhibitor lodged inside the binding pocket of its protein target. 1.2 C.2 Method The following additional details are relevant to the experimental method: - In the controlled (Validate) experiments, we are only attempting to optimise one factor, namely: docking score, which is indicative of binding affinity. This is in line with what was done in (Brahmavar et al., 2024). For the open-ended (Explore) experiments, we will extend this to include: number of synthesis steps, and estimated yield per step. - The factor-set specification also requires identifying minimum and maximum for the initial search space. For all experiments, we use: \(\{\textit{affinity}: [3,10], \textit{molwt}: [200, 700], \textit{SAS}: [0, 7.0]\}\), where \(\textit{affinity}\) is the predicted affinity from GNINA software, \(\textit{molwt}\) is the molecular weight, and \(\textit{SAS}\) denotes synthesis accessibility score. - For the controlled experiments, we use the known inhibitors and non-inhibitors as the dataset D. For the open-ended experiments we use 5 known inhibitors of DBH and 5 randomly sampled molecules from ChEMBL as non-inhibitors. Estimating the Q-heuristic requires a sample of unlabelled molecules. For this we use a randomly drawn set of 1000 molecules from the ChEMBL database. - The description of SearchHyp does not specify a sampling method for obtaining subsumed hyper-rectangles. We use an approach based on Latin Squares Hyper-Rectangle Sampling (LHRS: (McKay et al., 2000)). Some additional prior information may be available that may be used to modify the basic LHRS approach: (a) If we know beforehand that a factor is to maximised (for example, binding affinity), then we do not sample points from the upper-end of the range for the factor. Thus, subsuming rectangles are obtaining by only random placements of the lower-end; (b) Similarly, if we know beforehand that we want to minimise a factor, then only the upper-threshold is sampled. If nothing is known then a standard LHRS approach is adopted. - For all experiments, we use a value of \(s=10\); and \(n=10\) for GenMol and we sample 100 molecules when it returns the optimal hypothesis. The set of feasible molecules during search and in the final generation are considered for evaluation. - For molecule generation from LLM, we use model’s chat-completion API with a maximum output length of \(128\times \mathtt {max\_samples}\) tokens, a temperature of 0.7 to balance diversity and determinism. The prompt content was provided via the \(\texttt{messages}\) parameter (see Appendix C for details). - We assess the results of controlled experiments by examining the range and median docking scores of the molecules generated, and compare those to those generated by the LMLF procedure in (Brahmavar et al., 2024). For the open-ended experiments, we obtain the statistics on docking scores, and compare them to those of the latest generation of molecules used for DBH inhibition (see Sec. 4.2). In addition, we also provide an assessment of the molecules by an expert synthetic chemist. - For each target problem, we assess the novelty of the generated molecules by using the average Tanimoto (or Jaccard) coefficient to the database of known inhibitors. 1.3 C.3 Results The following details refer to the questions in Sec. 4.3. The choice of LLM: Table 6 compares the performance of GenMol when implemented with two different general-purpose LLMs: OpenAI’s GPT-4o and Anthropic’s Claude 3.5 Sonnet (Anthropic, 2024). Both these GenMol variants are evaluated under identical parameter settings and prompts to ensure a fair comparison. Results are shown for both the zero early context (\(|C_0|=0\)) and early context (\(|C_0|=5\)) settings across the three benchmark protein targets. Overall, the two models yield comparable mean binding affinities, with neither showing a clear advantage from the inclusion of early context. This similarity in performance between the two general-purpose LLMs raises the question whether a domain-specific LLM might offer a potential advantage. We examine this next. We implement a GenMol variant using 87 M parameter variant of GPT-2 that has been trained on a large corpus of about 480 million molecules (as SMILES strings) from the ZINC database (entropy, 2023), which provides extensive coverage of chemical space and encodes general chemical grammar and structure–property relationships. We refer this model here as Molecule-GPT2. This model is expected to generate syntactically valid and chemically diverse molecules due to its broad exposure to molecular representations. However, because the training data does not contain information specific to the protein targets considered here, the model may be lacking the ability to exploit target-specific binding patterns that are otherwise known to the general-purpose LLMs through their enormous training corpus of research articles and large-scale studies. Nevertheless, Molecule-GPT2 produces competitive predicted affinities for JAK2 and DRD2, which are probably very highly studied proteins and for these substantial domain-knowledge is available from the literature. Interestingly, it outperforms general-purpose LLMs for DBH, a relatively lesser studied target and additional target-specific knowledge has to be inferred from known inhibitor molecules. Although these results indicate that a general-purpose LLM is more suited for target-specific lead generation in the manner done in this study, a more thorough investigation is warranted. Furthermore, while prompting was used directly to generate molecules from the general-purpose LLMs, the same approach was not straightforward for the domain-specific LLM (Molecule-GPT2), primarily because it was not trained in the same manner as GPT-4o or Claude 3.5 Sonnet. For each protein target, we first had to construct a small set of structural scaffolds by examining known inhibitor molecules, and then use these scaffolds as prefixes for GPT2-based molecule generation (see Appendix C for more details on this). This additional preprocessing step may have contributed to the lower performance observed for Molecule-GPT2 relative to the general-purpose LLMs (Table 7). The effect of parameters: We consider 3 parameters: (a) Initial context provided to the LLM; (b) sample-sizes used by the symbolic learner; and (c) LLM temperature. Table 8 shows the changes in mean binding affinity with the initial context (“few-shots”) provided to the LLM. The results suggest that there is little difference in predicted affinity, although there is a large change in the Tanimoto coefficient for the DBH data. Figs. 8–9 shows the effect of varying (in turn): sample-size in GenMol; and LLM “temperature”. GenMol with GPT-4o shows statistically significant difference in predicted binding affinities, with two different sample sizes (\(s=10, 20\)) for the JAK2 and DBH proteins. Although having a high sample size (\(s=20\)) may result in better and more consistent binding affinities, this was not observed for DRD2 protein. Figure 9 shows the effect of sampling temperature on predicted binding affinities for molecules generated by GenMol using GPT-4o and Claude 3.5 Sonnet. For GPT-4o, temperature changes produce statistically significant differences in mean affinity scores, whereas Molecule-GPT2 exhibits minimal variation across temperatures and statistically significant difference in number of molecules generated by GenMol and the mean affinity. We performed a similar investigation using Claude 3.5 Sonnet and observe that it remains largely stable across temperatures, with no statistically significant variation in mean affinity. 1.4 C.4 Prompts We distinguish between 2 types of prompts for the API calls to the LLM (in this paper, GPT): - System prompt: We use this prompt to guide the model’s behaviour and responses. It sets the overall instructions for the model, such as defining its role and the syntactic format in which it should respond. In this work, we use this as: “You are a scientist specialising in chemistry and drug design. Your task is to generate valid SMILES strings as a comma-separated list inside square brackets. Return the response as plain text without any formatting, backticks, or explanations. The response must be formatted exactly as follows: [SMILES1, SMILES2, ...]. Avoid any extra text or explanations.” - User prompt: This is the input provided by the user, containing the actual query constructed in manner described below. The LLM generates responses based on this input while considering the instructions set by the system prompt. There are two kinds of user prompts based on whether a set of inhibitors are shown to the LLM during search and generation or not. For revealing known inhibitors, we use: “Generate up to s novel valid molecules similar to the following positive molecules: [...]”. Otherwise, the prompt is simply “Generate up to s novel valid molecules”. We also allow feasible molecules generated in GenMol to be used as “context”. In this case, we use it as a part of the user prompt as: “Additionally, consider these previously generated feasible molecules: [...].” 1.5 C.5 Domain-Specific LLM: Molecule-GPT2 We implemented a variant of GenMol using the publicly available GPT2 model hosted on Hugging Face (entropy, 2023). This model, which we call Molecule-GPT2, is based on the GPT2 architecture and has approximately 87 million parameters and trained with about 480 million molecules (in SMILES representation) from the ZINC database (Irwin et al., 2020). Unlike general-purpose LLMs such as GPT-4o and Claude 3.5 Sonnet, Molecule-GPT2 is not instruction-tuned and does not natively support natural language prompting for molecule generation. Therefore, we employed a scaffold-based prefixing strategy to condition the model’s outputs on specific protein targets. For each target protein, we first identified a small set of representative molecular scaffolds by extracting common substructures from known inhibitors. In all our experiments, we restrict to top 5 scaffolds, with each scaffold of maximum length 8. These scaffolds, encoded in SMILES format, were then used as fixed prefixes during autoregressive generation with the GPT2 model. We performed molecule generation using HuggingFace’s \(\mathtt {model.generate()}\) API with a maximum output length of 120 tokens, nucleus sampling with \(p=0.9\), temperature \(=0.8\), and stochastic sampling enabled. For each scaffold, we generated multiple candidate molecules for downstream binding affinity evaluation for GenMol. Although this approach may not be the optimal one, it does ensure that the generated molecules were chemically valid and retained structural motifs relevant to the target protein. 1.6 C.6 Reproducibility and API Cost A general concern with LLM-based systems is the reproducibility of results obtained via closed, API-only commercial models. We therefore note the approximate API costs incurred in this work. The entire study, conducted over several months, incurred modest API costs: approximately USD 10 each on OpenAI GPT-4o and Anthropic Claude 3.5 Sonnet, across roughly 400 requests and \(\sim \)100,000 generated tokens in total, for each model. The domain-specific Molecule-GPT2 model (entropy, 2023) was run locally and incurred no API cost. We note that nothing in the SNG framework is specific to any particular LLM: the \(\lambda \) Gen procedure (Procedure 1) is agnostic to the choice of generator, requiring only that it can be conditioned on a hypothesis description. To strengthen reproducibility, we have included results with a free, domain-specific Molecule-GPT2 model (Appendix C.3); a fully open general-purpose instruction-tuned LLM (e.g. a Llama-3 or Mistral variant) could be dropped into the same interface. We identify this as a concrete extension. All code and data are available at the repository referenced in Sec. 5. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. About this article Cite this article Srinivasan, A., Dash, T., Baskar, A. et al. Symbolic Neural Generation with Applications to Lead Discovery in Drug Design. Mach Learn 115, 210 (2026). https://doi.org/10.1007/s10994-026-07136-5 Received: Revised: Accepted: Published: Version of record: DOI: https://doi.org/10.1007/s10994-026-07136-5