Designing Convergent Overlapping Genes with Transformer Encoder Models and Lightweight Structural Proxies

7 Overlapping genes allow multiple proteins to be encoded from a single DNA sequence, including 8 convergent (antisense; tail-to-tail) orientations across three reading frames (phases 0, 1, and 2), 9 with phase 1 most frequently observed in nature. Designing such overlaps is challenging due to 10 codon degeneracy, phase-specific biases, and the need to preserve structural integrity for both 11 proteins. Here, a purpose-built transformer encoder is introduced, trained on a balanced synthetic 12 dataset of convergent overlaps spanning diverse prokaryotic genomes and GC contents. 13 Controlled amino acid substitutions were incorporated during training to enhance model 14 generalization, particularly for phase 1 overlaps. At inference, Monte Carlo dropout enabled 15 uncertainty-aware sampling of synonymous codon solutions, which were iteratively refined 16 using a windowed, multi-objective optimization framework. Candidate overlaps were scored 17 using composite weighting across secondary structure preservation, substitution similarity, 18 alignment identity, and ESM-2 contact map similarity, with SSIM applied as a rapid proxy for 19 structural fidelity. This approach generated convergent overlaps across all phases, with phase 1 20 showing the highest success rates. Optimization trajectories revealed distinct dynamics, with 21 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 2 secondary structure preservation steadily increasing despite its lower weight. External validation 22 using SwissProt proteins stratified by AlphaFold2 pLDDT confidence supported generalization 23 to proteins with differing rigidity, yielding high secondary structure preservation in silico. These 24

demonstrate that transformer models trained directly at the nucleotide level, when 25 coupled with uncertainty-aware inference and lightweight structural proxies, can support the 26 computational design of synthetic overlapping genes without requiring full structural prediction. 27 This framework offers a scalable path for phase-specific, codon-aware overlap design under 28 realistic constraints. 29

30 The study of overlapping genes is an important area of interest in prokaryotic genetics, attributed 31 to their ability to encode multiple functional products from a single DNA sequence (1–4). For 32 example, a single gene sequence might code for two distinct proteins, each playing a different 33 role in cellular function. Advancements in identifying and designing these overlapping sequences 34 promise the creation of genetic constructs that are both more efficient and compact, potentially 35 boosting the functionality and stability of engineered organisms (5,6). This phenomenon not only 36 poses challenges but also opens new avenues in genetic engineering and synthetic biology (3). 37 Overlapping genes have been described using different approaches, but can be broadly grouped 38 into unidirectional (sense; encoded on the same strand, → →) or convergent (antisense; encoded 39 on the opposite strand) orientations (Fig 1A); convergent overlaps can be further refined into 40 “head-to-head” (← →) or “tail-to-tail” (→ ←) orientations (7). Convergent overlapping genes 41 occur in the same reading frame (phase 0), or frameshifted by one (phase 1) or two (phase 2) 42 nucleotides. For example, in phase 2, convergent overlaps share their third (degenerate) codon 43 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 3 positions. Research into convergent (tail-to-tail; antisense) overlapping gene sequences in 44 prokaryotes has included analyses on the length distribution among overlapping and near-45 overlapping genes (8), investigations into their prevalence (and misannotations) within 46 prokaryotic genomes (9,10), as well as the characterization of convergent gene pairs (11–18). 47 Notably, both convergent and unidirectional overlaps exhibit a non-uniform distribution with a 48 significant phase bias, predominantly favoring phase 1 overlaps (8,19,20). While the majority of 49 convergent overlaps fall into phase 2, when excluding 4 nucleotide overlaps, nearly half are 50 instead found in phase 1 (8). These studies collectively highlight the phase preference and 51 distribution patterns of naturally occurring gene overlaps. 52 The successful prediction of overlapping nucleotide sequences encoding two amino acid 53 sequences poses unique computational challenges, and has been previously explored (6,21–25); 54 the landscape of available methods has been summarized (22) with additional information 55 pertaining to more recently released programs provided in Supplementary Table 1. An early 56 solution for the design of unidirectional overlapping protein sequences included design of an 57 algorithm to identify the shortest DNA sequence which could encode two amino acid sequences 58 (26). More recently, examples included a dynamic programming algorithm (6) and a subsequent 59

that combines dynamic programming with a hidden Markov model (HMM) (5,27). The 60 strategy was further extended to arbitrary pairs of natural protein domains (28). These 61 approaches have collectively shown that it is possible to design fully embedded overlapping 62 genes that encode functional proteins with high homology to starting sequences, underscoring 63 the potential for engineered overlapping genes in prokaryotic genetics. Another approach, termed 64 overlapping, alternate-frame insertion (OAFI) demonstrated the ability to design unidirectional 65 overlaps through insertion of one gene into an alternate reading frame of another by targeting 66 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 4 sites that may tolerate insertions (29). Recently, Byeon et al. (2025) demonstrated that deep 67 generative models can be used to design synthetic overlapping genes spanning distinct protein 68 families, achieving high in silico and experimental success rates even under the constraints of the 69 standard genetic code (30). Their results indicate that sequence space may contain many viable 70 overlapping solutions, suggesting broader applicability of computational approaches to both 71 unidirectional and convergent overlaps. 72 Transformer models have demonstrated superior performance in various sequence modeling 73 tasks due to their ability to capture long-range dependencies and contextual information (31–33). 74 This study leverages a novel application of the transformer encoder-based model architecture to 75 predict synthetic convergent overlapping genes in prokaryotes. Unlike dynamic programming 76 approaches, transformers utilize self-attention mechanisms to process the entire input sequence 77 simultaneously (32), enabling the model to capture patterns and dependencies that span across 78 long sequences. In this study, Monte Carlo (MC) dropout is applied during inference to enhance 79 prediction performance. Dropout, more generally, is a method of reducing overfitting in deep 80 neural networks during training (34). MC dropout retains dropout activity during inference, 81 effectively sampling from a distribution of possible model predictions (34–36). This provides a 82 practical Bayesian approximation of model uncertainty (37), which is especially advantageous 83 for predicting overlapping genes where the degeneracy of the genetic code permits many distinct 84 but functionally viable nucleotide encodings. By sampling multiple plausible solutions, MC 85 dropout enables exploration of the many synonymous coding possibilities inherent to the genetic 86 code, increasing the likelihood of identifying overlaps that preserve the structural and functional 87 integrity of both proteins. 88 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 5 The objectives of this study are to (i) evaluate the ability of transformer encoder models to 89 recover convergent overlaps across all three phases, (ii) quantify the impact of overlap length 90 and sequence variability on prediction success, and (iii) assess multi-objective optimization as an 91 alternative to full structural prediction in improving overlap fidelity. 92

93 Synthetic Overlapping Gene Dataset Construction for Model Training 94 While naturally occurring convergent overlaps have been systematically assessed and 95 documented (9), there is no readily available database of convergent overlapping genes with 96 equally distributed overlap lengths. Given this limitation, a database of synthetic convergent 97 overlaps was generated using a diverse set of open reading frames (ORFs) from prokaryotic 98 genomes. Synthetic convergent overlapping genes were created using ORF sequences from 24 99 different genomes from prokaryotes (Supplementary Table 2), using the R (version 4.3.1) and 100 Python programming languages (version 3.8.8). Genomes were selected to have a wide range of 101 genomic GC content (ranging from 23.3% to 74.9%, with a mean of 52.41%), with the aim of 102 controlling for potential amino acid compositional bias driven by GC content (38). The between-103 strain coding GC content (calculated using the set of ORF sequences from each strain) ranged 104 from 20.7% to 72.7%, with a mean of 50.5%. 105 Briefly, for each strain, sequences were downloaded from the NCBI database 106 (https://www.ncbi.nlm.nih.gov/genome/) and each FASTA file was processed to extract ORFs, 107 which were stored in separate list for each strain (Fig 1B). ORF sequences were filtered to 108 remove sequences shorter than 450 nucleotides, to exclude sequences with insufficient 109 opportunity for varied random sampling during overlap construction. A function was defined to 110 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 6 randomly select in-frame segments from the ORF sequence lists, to include a primary and 111 secondary sequence either with or without a convergent overlapping region. First, a contiguous 112 in-frame 312-nucleotide length section from each ORF sequence was extracted and used as the 113 basis for the primary sequence. The primary sequence segments were modified to ensure they 114 contained start and randomly selected stop codons, as these were removed during initial 115 sequence processing. 116 The reverse complement of this synthetic gene was selected to generate the secondary sequence 117 with a defined convergent overlap length. A uniform set of overlap lengths was generated for 118 each length by extracting the reverse complement, adding a stop codon, and removing any in-119 frame stop codons with a random codon. Both the added stop codon and in-frame replacements 120 were selected such that stop codons in the forward direction were not introduced. However, 121 owing to the methodology employed, a portion of generated secondary sequences with 122 overlapping sequences contained more than one stop codon and were subsequently filtered from 123 the dataset to ensure all sequences contained only one stop codon and encoded a single amino 124 acid sequence. 125 The amino acid sequence pairs with known overlaps were modified to incorporate up to 4 of 126 BLOSUM62 weighted amino acid changes. The changes included altering amino acid sequences 127 such that 1) the internal reverse stop codon was removed from both amino acid sequences 128 (primary and secondary), while keeping the known overlap for the original amino acid 129 sequences, or 2) the internal reverse stop codon was removed from both amino acid sequences 130 (primary and secondary), while also introducing additional amino acid modifications per 131 sequence. This yielded modified primary and secondary amino acid sequences, with an 132 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 7 unmodified known convergent overlap nucleotide sequence capable of encoding amino acid 133 sequences with high amino acid sequence identity to the primary and secondary amino acid 134 sequences. 135 The length of the input amino acid sequences was limited to 104 amino acids to manage 136 computational complexity during training and inference phases. Each amino acid was separated 137 by a space and the overall included a terminal asterisk (*) and the primary and secondary 138 sequences were concatenated to generate a single input sequence for training. 139 To balance the representation of different overlap lengths, data were stratified based on the 140 length of the overlap. The overlapping gene training data were balanced, ensuring equal 141 representation for each strain and each length of nucleotide overlap (specifically, convergent 142 overlaps of length 199 to 312 nucleotides). The final dataset was composed of synthetic gene 143 overlaps, each represented by a pair of modified amino acids sequences with a known target 144 convergent overlap sequence. This dataset served as the training set for the transformer encoder-145 based models. 146 Initial Model Description and Training 147 The synthetic overlapping gene dataset was used to train transformer-encoder based models with 148 the aim of generating convergent (tail-to-tail) overlaps for any two specified amino acid 149 sequences. A total of 8,892,000 combined pairs, corresponding to 78,000 pairs per convergent 150 overlap length. 151 Transformers have demonstrated superior performance in various sequence modeling tasks, 152 particularly where understanding long-range dependencies is important (31–33). As such, this 153 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 8 study utilizes a PyTorch based transformer encoder model for overlap prediction (39). 154 Tokenization and text vectorization were performed using the TensorFlow Keras API (40), and 155 data were split into training and validation sets. During training, the models each utilized 156 156 embedding dimensions, 13 attention heads, 192 feed-forward network dimensions, 5 blocks, a 157 dropout rate of 0.1, and a total of 802,983 trainable parameters (Fig. 2D). Models were compiled 158 using the Adam optimizer with cross-entropy as the loss function. 159 Each model was trained for 3 epochs with a batch size of 32 on a single NVIDIA GeForce GTX 160 5070 Ti GPU with 16GB of memory, and reached a validation accuracy of ≥90% by the third 161 and final epoch. 162 Test Dataset Generation and Model Inference 163 Amino acid pairs were generated as described above for preparation of the training dataset, but 164 without the final modification step; however, a different set of strains (n = 12) were selected than 165 those used for training, with a wide range of genomic GC content (ranging from 23.6% to 74.1%, 166 with a mean of 50.2%) (Supplementary Table 3). Given that inference was performed using 167 only coding sequences, the coding sequence GC content was calculated for each strain, and the 168 corresponding mean GC value was used for subsequent analyses (varying from 24.2% to 72.1%, 169 and a mean of 50.0%). 170 It is noted that inference was performed using the trained models along with the same tokenizer 171 to vectorize the input sequence. Using a “greedy decoder” (41), the token with the highest 172 probability at each step of the sequence generation process was selected. 173 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 9 MC dropout was implemented during inference to allow exploration of the uncertainty in model 174 predictions (eg, variation in predicted overlap sequences), leveraging the benefits of Bayesian 175 inference, as previously described (34,36,42). During inference, depending on the objective, 176 feedforward dropout was set to 0.275 for initial predictions. 177 An algorithm was developed to generate overlapping gene sequences from input amino acid 178 sequences using the trained models. The algorithm first back-translated the input amino acid 179 sequences into a nucleotide sequence, leveraging E. coli codon usage frequencies to reflect the 180 probabilistic distribution of codons for each amino acid (43). After back-translation, the models 181 were used to predict potential overlaps within these nucleotide sequences. Predicted overlaps 182 were then integrated into the original nucleotide sequences, resulting in a collection of modified 183 sequences that incorporated the predicted overlapping regions downstream of the remaining 184 back-translated nucleotide sequence. These modified sequences (ie, the back-translated and 185 predicted regions) were then translated into amino acid sequences to assess the accuracy of the 186 predictions. In instances where the correct amino acid sequence was not identified, additional 187 rounds of prediction and back-translation were performed. This iterative process continued until 188 either an overlap yielding amino acid sequences meeting the pre-specified criteria was identified, 189 or the specified number of inference rounds were completed without a predicted overlap (defined 190 as an unsuccessful prediction). 191 An alignment score was calculated for each sequence prediction. The predicted sequence was 192 aligned against the known overlap using the Needleman-Wunsch algorithm (implemented via 193 pairwise2 from Biopython (44)). For sequences with a known overlap and a predicted overlap 194 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 10 sequence, the calculated alignment score was based on the length of the overlap, normalizing 195 scores to a scale of 0 to 1 to facilitate comparison across predictions. 196 Secondary Structure Prediction, Substitution Score, and Structure Prediction 197 The prediction of secondary structures was performed using S4PRED, which enabled predictions 198 based on an input amino acid sequence alone without the need for multiple sequence alignments 199 or known homologous sequences (45). S4PRED outputs were formatted as “ss2” files, from 200 which the predicted secondary structure sequence was extracted for downstream analyses. The 201 model employs a simplified three-state classification—α-helix (H), β-sheet (E), and coil/loop 202 (C), which corresponds to a coarse-grained version of the DSSP (Dictionary of Secondary 203 Structure of Proteins) annotations (46). Minor code modifications were implemented to enable 204 efficient batch processing, substantially reducing runtime. These predicted structures were used 205 to evaluate the structural compatibility of convergent overlapping gene candidates. Substitution 206 matrices utilized were BLOSUM62 (47,48) and ProtSub (49). Protein structures were predicted 207 either with AlphaFold3 or ColabFold (50–52). 208 Optimization of predicted sequences 209 An iterative, windowed optimization procedure was used to convert paired amino-acid inputs 210 into convergent (tail-to-tail) overlapping nucleotide sequences while preserving fold-relevant 211 features of both proteins and maintaining sequence similarity. Input amino-acid sequences were 212 first back-translated to an initial nucleotide scaffold using E. coli codon usage frequencies. The 213 region targeted for overlap design was tiled into partially overlapping amino-acid windows; in 214 the experiments reported here windows were 10 amino acids with an 8-amino-acid stride (10 AA 215 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 11 window, 8 AA stride; model receptive field used = 105 AA). During optimization, only 216 nucleotides mapping to the current window were permitted to change; flanking nucleotides were 217 held fixed except at user-designated residues enforced by token-level biases (for example to 218 preserve active-site residues or motifs). Bracketed constraints on the forward strand were 219 mirrored on the reverse-complement strand so that enforced bases remained consistent with the 220 convergent orientation. 221 Within each window a transformer-encoder was executed in an uncertainty-aware mode: Monte-222 Carlo (MC) dropout was retained at inference to generate diverse synonymous codon proposals. 223 Token-level fixed-logit (base) biases were applied at specified nucleotide positions to enforce 224 required bases during sampling; combined forward/reverse fixed-logit maps ensured constraints 225 were respected on both strands. Each sampled nucleotide pair was translated to amino-acid 226 sequences and duplicate amino-acid solutions were removed prior to scoring. 227 Structural compatibility was assessed using single-sequence secondary structure prediction 228 (S4PRED; three-state H/E/C). Secondary structure preservation was reported as the percentage 229 of residues for which a candidate’s predicted state matched the original sequence prediction. 230 Substitution similarity was computed using BLOSUM62 and normalized to the original 231 sequence’s self-score to yield a percent similarity; global alignment identity was also measured. 232 ESM-2 (53) (model: esm2_t12_35M_UR50D) was used to generate residue-residue contact 233 probability matrices, having demonstrated utility as a fast and effective single-sequence 234 prediction feature (54). Contact map similarity was assessed using the structural similarity index 235 (SSIM) (55), computed between ESM-2-predicted contact maps of candidate overlap sequences 236 and their respective originals. SSIM quantifies local and global structural pattern agreement 237 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 12 (range 0-1, with 1 indicating identical maps). Raw values were linearly scaled to a percentage (0-238 100) to yield the ESM-2 contact map score. This metric served as a rapid, computationally 239 efficient proxy for structural preservation, enabling large-scale overlap design iterations while 240 retaining sensitivity to perturbations in contact map topology. 241 Candidates were ranked by a fixed weighted composite score across all experiments: 0.15 242 (secondary structure preservation), 0.15 (substitution similarity), 0.1 (alignment identity), and 0.6 243 (ESM2 contact map similarity). Although this weighting scheme was not exhaustively 244 optimized, empirical testing with randomly selected SwissProt proteins indicated that it yielded 245 stable rankings across experiments and supported overlap generation while balancing structural 246 fidelity with sequence similarity (Figs. S3, S4, and S5). The top-ranked candidate for the 247 window was integrated, and optimization proceeded to the next window. Multiple full passes 248 across windows were permitted for refinement. After the first structurally successful candidate, 249 global feedforward and attention dropout levels were reduced to bias sampling from exploratory 250 search toward fine-tuning, though the weighting scheme itself remained unchanged. 251 Efficiency measures implemented for reproducibility and speed included: (i) caching of 252 secondary structure predictions so identical amino-acid sequences were scored only once, and 253 (ii) caching of ESM-2 embeddings and contact maps. All candidate metadata (attempt, window, 254 per-candidate metrics, and secondary structure predictions) were retained in memory and 255 exported at the end of each input row to permit analysis. 256 Mutual information, PCA, and robustness testing 257 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 13 Terminal regions (104 amino acids) from 250 randomly selected SwissProt-derived proteins 258 were used as seeds to generate ensembles of mutated sequences. For each seed, trajectories were 259 created by stepwise amino acid substitutions sampled from the 20-letter alphabet, introducing 260 one mutation per step for up to 75 iterations. This yielded a progressive set of variant sequences 261 per seed protein. Each sequence in the ensemble was then scored for global alignment identity, 262 ESM-2 contact map similarity, and secondary structure match relative to the reference. These 263 metric vectors formed the input for subsequent mutual information (MI) and principal 264 component analyses. 265 MI was estimated after quantile discretization of continuous scores into 20 bins to evaluate 266 pairwise dependence among objectives. To reduce sensitivity to binning, MI was also estimated 267 using a k-nearest-neighbor approach. Statistical significance of discrete MI values was assessed 268 using permutation testing with 1,000 random shuffles. Principal component analysis (PCA) was 269 applied to z-scored metric vectors (alignment, ESM-2 contact map similarity, secondary 270 structure match) to examine the dimensional structure of variation. Robustness checks included 271 varying the number of bins (5-40) for MI and bootstrap resampling of PCA explained variance. 272 Statistical Analyses and Figure Preparation 273 The R programming language (version 4.3.1) and the ggplot2 package (56,57) were used to 274 perform statistical analyses and generate figures. Prediction success rates are reported as 275 percentages with exact 95% confidence intervals calculated using the Clopper–Pearson method. 276 Correlations between similarity metrics were assessed using Pearson’s correlation coefficient 277 with associated p-values. Relationships between secondary structure composition (coil, helix, 278 sheet) and prediction performance were examined using LOESS smoothing with 95–99% 279 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 14 confidence bands. For external validation experiments, performance metrics were stratified by 280 AlphaFold2 pLDDT confidence brackets and overlap phase. Group differences were evaluated 281 using nonparametric pairwise contrasts with adjusted p-values (Bonferroni correction). Effect 282 sizes are reported as absolute percentage-point differences between groups, alongside confidence 283 intervals. Data are presented as mean values with ranges or confidence intervals where 284 appropriate. Figures were generated with ggplot2, with additional overlays and schematics 285 prepared in Inkscape (version 1.3.2). 286 Declaration of Generative AI and AI-assisted technologies 287 During the preparation of this work, ChatGPT (5.0; OpenAI) was used to assess readability and 288 language, and as a tool to generate code snippets and assist in debugging. After using this 289 tool/service, the author reviewed and edited the content as needed and takes full responsibility 290 for the content of the publication. 291 Code and data availability 292 The code used for synthetic overlap dataset construction, model training, and inference are 293 available at: https://github.com/protosome/convergent_overlaps_aa_change 294

295 Prediction of convergent overlapping genes from amino acid sequence pairs from a 296 dedicated transformer-encoder model set 297 While prior work has established that it is technically feasible to generate overlapping genes 298 from paired amino acid sequences, the performance of dedicated transformer-encoder models for 299 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 15 this task has not been systematically evaluated. In this study, transformer-based models were 300 trained to generate convergent overlapping nucleotide sequences from amino acid pairs, and their 301 ability to recover overlaps across all three phases was assessed. These models were designed to 302 identify convergent overlaps capable of encoding both input proteins, using recent advances in 303 transformer-based sequence modeling to address the large solution space introduced by the 304 degeneracy of the genetic code (31,58). Prediction quality was evaluated using alignment 305 identity scores and secondary structure preservation metrics, allowing comparison across overlap 306 lengths from 199 to 312 nucleotides. 307 Initial filtering was performed using a conservative alignment score cutoff of 34% (28,59). 308 Without Monte Carlo dropout (ie, dropout enabled during inference), successful predictions were 309 substantially reduced, and repeated inference runs converged to identical codon solutions. This 310 indicated that deterministic decoding restricted the search space to narrow, low-diversity 311 outcomes. By contrast, applying Monte Carlo dropout during inference expanded sequence 312 diversity, producing multiple distinct codon assignments that encoded the same amino acid 313 sequences. This stochastic sampling increased the likelihood of recovering overlaps that satisfied 314 both similarity and structural thresholds, underscoring the importance of uncertainty-aware 315 inference in navigating the combinatorial complexity of convergent overlap prediction. 316 To further assess training requirements, the effect of imperfect overlaps and amino acid 317 substitutions in the training data was examined. Models trained only on unaltered overlaps 318 exhibited markedly reduced success rates, particularly for phase 1 (Figs. 3D, 3E). In a shared test 319 set of 5,000 amino acid sequence pairs without a known convergent overlap, training with 320 modified sequences (stop codon removed plus 1–4 substitutions per sequence) substantially 321 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 16 improved prediction rates: phase 0, 95.3% (94.7-95.9%); phase 2, 86.7% (85.7-87.6%); and 322 phase 1, 81.6% (80.5-82.6%). In contrast, models trained on unaltered overlaps performed 323 significantly worse: phase 0, 73.7% (72.4-74.9%); phase 2, 49.1% (47.7-50.5%); and phase 1, 324 9.7% (8.9-10.6%). These findings are consistent with the hypothesis that incorporating 325 controlled sequence variation during training increased model flexibility and suggest improved 326 generalization, particularly when coupled with Monte Carlo dropout during inference. On this 327 basis, models trained with amino acid substitutions were selected for all subsequent analyses. 328 The outputs of these models were further evaluated using BLOSUM62 and ProtSub substitution 329 matrices. For overlap lengths of 310-312 nucleotides, similarity scores were strongly correlated 330 with amino acid identity (r = 0.88; p < 0.0001), and the matrices themselves were highly 331 correlated with each other (r = 0.98; p < 0.0001). 332 In addition to substitution metrics, alignment scores and secondary structure scores were 333 compared across phases. Phase 1 overlaps exhibited the highest average alignment and 334 secondary structure scores, phase 0 displayed intermediate values, and phase 2 was generally the 335 lowest. However, score distributions overlapped considerably, suggesting that phase-dependent 336 differences were not absolute. The near-identical secondary structure score distributions for 337 phase 0 and phase 2 further indicated that structural outcomes may be more sensitive to 338 sequence-specific context than to phase alone. 339 To broaden the analysis, overlap lengths from 199 to 312 nucleotides were examined across a 340 larger set of amino acid pairs. Phase-dependent patterns persisted: phase 1 overlaps were 341 increasingly recoverable at shorter lengths, and the relative rank order of secondary structure and 342 alignment scores across phases remained consistent. These findings suggest that phase 1 overlaps 343 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 17 may be predicted to be structurally and codon-compatible favorable, and that performance of the 344 transformer encoder model set is influenced by overlap length and the diversity of sequences 345 available for training. 346 High Secondary Structure Scores Are Linked to Coil-Dominated Precursor Sequences 347 To assess whether the secondary structure composition of precursor amino acid sequences 348 influences the predicted quality of convergent overlaps, the proportion of coil (C), β-sheet (E), 349 and α-helix (H) structures in each input was compared against the model’s predicted secondary 350 structure score. For each amino acid pair, structural proportions were calculated separately and 351 then averaged across both sequences. This approach reflects the biological premise that overlap 352 feasibility arises not from a single sequence in isolation, but from the combined structural 353 constraints and flexibilities of the two proteins that must be co-encoded (60,61). 354 Across all three overlap phases, sequences with higher coil content exhibited elevated secondary 355 structure scores (Fig. 3C). This trend was most apparent in phase 0 and phase 1, where LOESS-356 smoothed curves showed a modest increase in secondary structure score with increasing coil 357 content, followed by a sharper rise beyond approximately 70% coil. In phase 2, a similar increase 358 was observed at high coil levels, but the trend was less consistent, with greater variability at 359 intermediate values. By contrast, α-helix content was inversely associated with secondary 360 structure score. The most pronounced effect occurred in phase 0, where increasing helix content 361 corresponded to a marked decline in secondary structure scores. A similar, though less steep, 362 trend was observed in the other phases. For β-sheet content, no consistent relationship with 363 secondary structure score was observed; across all three phases, smoothed curves were largely 364 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 18 flat, indicating that sheet content neither strongly promoted nor hindered predicted structural 365 quality. 366 To evaluate whether these relationships extended to sequence-level similarity, mean alignment 367 scores were also plotted against precursor structural compositions. No meaningful trends were 368 observed for any secondary structure class (Fig. S1). Alignment scores remained relatively 369 constant regardless of coil, β-sheet, or α-helix content, suggesting that precursor structure did not 370 systematically influence the model's ability to recover the reference overlap sequence. 371 Multi-objective optimization and computational design of EGFP/AmpR convergent 372 overlaps 373 Although the transformer-encoder models were capable of generating convergent overlaps for 374 paired amino acid sequences, the secondary structure preservation achieved in initial predictions 375 was variable and generally lower than values reported in prior work (60). To address this 376 limitation, an iterative sequence optimization algorithm was implemented to refine model 377 outputs using a multi-objective framework that did not rely on MSA availability (Fig. 4A). 378 The optimization procedure employed a windowed feedforward Monte Carlo (MC) dropout 379 strategy to generate multiple candidate solutions for each sequence window (Fig. 4B). Candidate 380 overlaps were then evaluated against two complementary structural criteria: 1) predicted 381 secondary structure states derived from S4PRED (62), providing local fold-relevant constraints, 382 and 2) ESM-2 contact map similarity (63), which encodes sequence-level context that may 383 reflect long-range residue dependencies. A key feature of the framework was the ability to 384 enforce selective preservation of amino acid residues when required. This was achieved by 385 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 19 applying a feedforward mask to specific codon positions, constraining the model to maintain 386 defined residues (for example, active-site motifs) while permitting synonymous variation 387 elsewhere. In practice, this allowed critical sequence features to be retained while enabling 388 exploration of codon-level degeneracy to optimize overlap formation. 389 Relationships among the chosen objectives were next quantified using mutual information (MI) 390 and principal component analysis (PCA). Both approaches indicated that alignment identity, 391 ESM-2 contact map similarity, and predicted secondary structure capture overlapping but distinct 392 constraints. Discrete MI (20 quantile bins) showed that alignment carried moderate information 393 about both ESM-2 contact map similarity (MI = 0.393) and secondary structure match (MI = 394 0.381), whereas the MI between ESM-2 contact map similarity and secondary structure 395 similarity was lower (MI = 0.240), consistent with partial independence. Continuous MI 396 estimates (kNN regression) produced similar results (0.509, 0.508, and 0.340, respectively). All 397 associations were highly significant in permutation tests (p = 0.001). PCA of z-scored metrics 398 supported this interpretation (Fig. S2). A single axis (PC1) explained 72.4% of variance (95% 399 CI: 71.8-73.0) and loaded positively on all three objectives, reflecting a general similarity 400 dimension. PC2 (18.4%; 95% CI: 18.0-18.9) contrasted ESM-2 contact map similarity with 401 secondary structure similarity, while PC3 (9.2%) primarily captured sequence-only variance. 402 Thus, although alignment dominates the shared signal, secondary structure preservation and 403 ESM-2 contact map-based similarity provide orthogonal information that cannot be reduced to 404 alignment alone. This justifies their inclusion as separate objectives during overlap optimization. 405 Metric behavior under progressive sequence divergence was also assessed by subjecting 250 406 randomly sampled SwissProt proteins to iterative single-residue substitution trajectories (75 steps 407 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 20 per sequence). Across trajectories, all four similarity measures declined with increasing 408 mutational distance, but the rate and pattern of decay differed by metric (Fig. S3). Alignment 409 identity and BLOSUM62 similarity decreased rapidly and near-linearly, consistent with their 410 direct dependence on residue-level identity. By contrast, ESM-2 contact map similarity and 411 secondary structure similarity showed more gradual declines. These distinct decay profiles 412 support the interpretation that the objectives capture complementary aspects of overlap fidelity. 413 Weighting experiments further clarified the contributions of each objective. When the relative 414 weights assigned to secondary structure similarity, ESM-2 contact map similarity, alignment 415 identity, and BLOSUM62 substitution similarity were varied across randomly selected SwissProt 416 pairs with very high AlphaFold2 pLDDT values (≥90), distinct trade-off dynamics were 417 observed (Fig. S4). Increased weighting on secondary structure similarity consistently improved 418 secondary structure preservation but reduced alignment and substitution similarity, whereas 419 embedding-focused weightings enhanced ESM-2 contact map similarity at the expense of 420 secondary structure similarity. Intermediate combinations (eg, 0.15-0.4 secondary structure with 421 0.6 ESM-2 contact map similarity) yielded more balanced outcomes across the four metrics. 422 Variance patterns indicated that secondary structure weighting stabilized optimization 423 trajectories, while ESM-2 contact map similarity remained comparatively variable across 424 candidate sequences. 425 A convergent overlap between enhanced green fluorescence protein (EGFP) and a TEM-1 β-426 lactamase (ampicillin resistance marker from pCVD004; hereafter AmpR) was designed and 427 refined under the same objectives as a biologically relevant test case (64,65). The combination of 428 MC dropout sampling, secondary structure prediction, and embedding-based scoring consistently 429 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 21 improved fold preservation of computationally designed EGFP/AmpR 311 nucleotide (phase 1) 430 overlaps (Fig. 4C; Fig. 5). Compared to unoptimized outputs (ie, sequences generated from the 431 first window), optimized sequences displayed higher secondary structure similarity rates and 432 increased ESM-2 contact map similarity, demonstrating that overlap designs can be improved 433 using lightweight, MSA-independent objectives. During windowed optimization, alignment and 434 BLOSUM62 metrics remained largely constant or decreased across windows, reflecting their 435 role in constraining candidate sequences to a biologically plausible subset of sequence space. In 436 contrast, secondary structure similarity generally increased throughout optimization, despite 437 being weighted at only 0.15 in the composite score. ESM-2 contact map similarity increased 438 modestly in the early windows before plateauing. These findings suggest that fold-relevant 439 secondary structure metrics may act as the principal discriminative driver of optimization 440 progress, with alignment- and embedding-based metrics providing a stabilizing baseline. 441 Following optimization, a pair of amino acids with the highest combined score was selected for 442 structural comparison using predictions from AlphaFold3 (50), while selecting amino acid 443 sequences in both pairs for preservation based on predicted importance (eg, active site). The 444 AlphaFold3-predicted models of designed overlaps were highly similar to reference predictions 445 (Fig. 4C), with TM-scores of 0.98 (EGFP) and 0.90 (AmpR), the latter with deviations generally 446 localized to a disordered coil not located in the overlapping terminal region (66). 447 Performance of this algorithm in predicting overlapping genes using pLDDT as an 448 orthogonal proxy for intrinsic structure rigidity 449 Analyses on prokaryote-derived amino acid sequence pairs indicated that secondary structure 450 preservation increased with coil content and decreased with helix content, with sheet content 451 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 22 showing no consistent effect. To determine whether this pattern reflected a generalizable 452 property of overlap design for convergent overlap derived from terminal amino acid sequences, 453 an external validation set was assembled from SwissProt proteins stratified by AlphaFold2 (AF2) 454 per-residue confidence (pLDDT), an orthogonal proxy for intrinsic structural rigidity (52,67–69). 455 SwissProt sequences were filtered to 105–200 amino acids to align with the 105-AA receptive 456 field and to bound computational cost, then binned into Low-intermediate (50-69), Confident 457 (70-89), and Very high (90-100) pLDDT groups. For each bracket, 100 non-redundant amino-458 acid pairs were formed, and convergent overlaps were designed across phases 0, 1, and 2 at 310, 459 311, and 312 nt using the same transformer-encoder, windowed Monte-Carlo (MC) dropout 460 inference, objective weights, and constraint handling described previously. 461 Quantitative analysis of secondary structure composition (determined using S4PRED) across 462 pLDDT confidence bins revealed clear trends. Looking specifically at the terminal AA regions, 463 as pLDDT confidence increased from the Low-intermediate to the Very high bins, coil content 464 declined substantially (from ~64% to ~41%), while both helix and sheet content rose (helix from 465 ~29% to ~39%; sheet from ~6% to ~20%) (Figs. 6A, 6B). These shifts suggest that AlphaFold’s 466 higher-confidence regions correspond generally to more ordered structural states, and reinforce 467 the interpretation of pLDDT as an approximate proxy for structural order and rigidity (50,70,71). 468 These AF2-stratified findings were consistent with the earlier composition-based analysis using 469 prokaryotic ORFs in this study. Sequence pairs with lower pLDDT (coil-enriched) were 470 associated with slightly higher secondary structure preservation, whereas higher-confidence 471 inputs tended to show reduced variation (Fig. 6C). Nevertheless, the overall high secondary 472 structure scores across all three brackets emphasize that the algorithm was able to recover 473 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 23 structurally faithful overlaps even for proteins predicted by AlphaFold2 to adopt stable, well-474 defined folds. Alignment identity and substitution similarity remained generally stable across 475 brackets, indicating that secondary structure preservation was the primary dimension along 476 which flexibility exerted its effect. 477 Bracket- and phase-specific effects were evident across all three metrics, consistent with the 478 secondary structure composition of the pLDDT bins. Across overlap phases, sequences in the 479 Low-intermediate (50-69) bracket generally outperformed those in the Confident (70-89) and 480 Very high (90-100) brackets, with the gap being most pronounced in phase 2 (Fig. 6D). In phase 481 1, the bracket effect was clear for secondary structure preservation and for the combined score 482 relative to Very high (90-100), whereas pairwise differences were not evident for ESM-2 contact 483 map similarity; Low-intermediate (50-69) also did not differ from Confident (70-89) for the 484 combined score. Overall, phase 1 maintained the highest preservation, while phase 2 exhibited 485 the largest bracket separations. Although group differences were statistically significant, effect 486 sizes varied by metric: within a phase, combined and ESM-2 contact map similarity differences 487 were typically ~1-4 percentage points, whereas secondary structure differences reached ~5-8 488 points in some phase 2 contrasts. All groups maintained mean secondary structure preservation 489 above 85%. Full pairwise estimates with confidence intervals and adjusted p-values are provided 490 in Supplementary Tables 4 and 5. 491 While statistically significant differences were observed between pLDDT brackets, these 492 differences were relatively small, and overall secondary structure preservation remained 493 consistently high across all conditions (Fig. 6E). These results reinforce the generalizability of 494 the earlier coil-associated trend, but also highlight the robustness of the multi-objective, 495 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 24 uncertainty-aware optimization framework in achieving high predicted structural preservation 496 regardless of precursor rigidity. 497

498 Overlapping gene architectures have long been recognized as both a source of evolutionary 499 constraint and an avenue for novel functionality (72–80). In addition to their evolutionary 500 importance, overlapping structures can be purposefully engineered under realistic codon and 501 structural constraints. The present study shows that phase-specific convergent overlaps can be 502 computationally generated and optimized using a transformer encoder-based multi-objective 503 framework without requiring full 3D prediction at the optimization stage; targeted 3D checks 504 were applied post hoc. This framework offers a complementary tool that could expand the range 505 of strategies available for synthetic overlap design, particularly where phase, length, and 506 nucleotide-level control are critical. 507 Recent work has demonstrated that deep generative protein models can discover viable 508 overlapping solutions under genetic code constraints. Byeon et al. (2025) showed that pretrained 509 amino acid-space generative models can design synthetic overlapping genes across multiple 510 reading frames, validating expression experimentally using Gibbs sampling with codon-511 compatibility constraints (30). Xu et al. (2025) further demonstrated that pretrained generative 512 protein language models (ESM-3) can yield viable entangled protein pairs through structure-513 conditioned inverse folding and CAMEOS-based entanglements, filtered by cross-entropy and 514 Potts energy scores (81). Importantly, Xu et al. established that pretrained generative models are 515 capable of identifying functional overlap-compatible solutions, albeit in a limited experimental 516 context (the InfA/AroB system) and without explicit codon-level optimization. 517 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 25 The framework presented here builds on this momentum but takes a different route. Rather than 518 relying on pretrained amino acid-space sampling, it employs a purpose-built, phase-generalizable 519 transformer encoder model set trained on a balanced synthetic dataset of convergent overlaps 520 spanning diverse GC contexts and overlap lengths. Multi-objective optimization is integrated 521 directly into the inference loop, with candidate overlaps iteratively refined through a windowed, 522 uncertainty-aware (MC dropout) search. Evaluation criteria span predicted secondary structure 523 preservation (S4PRED), amino acid substitution similarity (BLOSUM62), pairwise alignment 524 identity, and ESM-2 contact map similarity. This approach emphasizes codon-level control while 525 maintaining structural fidelity, enabling overlap recovery without requiring full 3D structural 526 prediction at each iteration. 527 MC dropout was retained during inference to enable stochastic sampling of synonymous codon 528 solutions, thereby broadening the effective search space (35). Candidate overlaps were ranked 529 using a fixed composite score that balanced structural preservation, substitution similarity, 530 alignment identity, and embedding-based metrics. This weighting scheme served as a practical 531 approach for navigating competing design constraints without requiring full structural prediction 532 at each iteration, and was most closely aligned with prior work arguing that synthetic overlap 533 design requires explicit balancing of competing biological constraints and objectives (5,60). 534 While other multi-objective optimization strategies could be considered in future work (such as 535 ε-constraint methods (82,83), adaptive scalarization (84,85), or evolutionary multi-objective 536 optimizers (86)), the fixed weighted framework proved sufficient to achieve high overlap fidelity 537 under the conditions evaluated. 538 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 26 The synthetic convergent overlapping gene dataset construction method developed as part of this 539 study provides a structured framework for exploring overlapping gene design in silico, as it 540 permits generation of diverse and uniformly distributed overlaps. However, one potential 541

of this approach is the reliance on synthetic, rather than natural, overlapping genes for 542 model training, which were generated using ORFs from a diverse set of prokaryotic strains. This 543 may introduce a bias affecting the predictive ability and variability in sequence output. Despite 544 this concern, the method employed to generate synthetic overlapping genes is in principle 545 analogous to the formation of new overlaps through the process of gene extension following the 546 loss of a stop codon (8). This results in synthetic overlapping genes where the second (extended) 547 strand's encoded amino acid composition is derived from the reverse complement of the first 548 strand, without undergoing further selection. As recently reported, there is extensive evidence 549 suggesting functionality in prokaryotic convergent (antisense) proteins, indicating that non-550 coding RNA regions could also encode functional proteins (87). For same-strand overlaps, a 551 significant difference in overall composition compared to non-overlapping genes has been 552 previously observed (88), along with a bias towards disorder-promoting amino acids (89). The 553 extent to which this affects convergent overlapping genes remains unclear. Future work will 554 therefore aim to (i) characterize compositional differences between natural and synthetic 555 overlaps, (ii) extend design to longer proteins beyond the 199-312 nucleotide range considered 556 here, and (iii) experimentally validate whether in silico-designed overlaps from this framework 557 maintain expression, folding, and function in vivo. 558 In summary, this study establishes that phase-aware transformer models trained directly at the 559 nucleotide level, when coupled with uncertainty-aware inference and multi-objective 560 optimization, can recover computationally designed convergent overlapping genes under realistic 561 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 27 genomic constraints. By integrating codon-level sampling with structural and substitution-based 562 objectives, this framework complements recent amino acid-space generative approaches while 563 offering length- and phase-specific control. These findings provide a computational foundation 564 for experimentally testable designs and point toward a scalable strategy for compact, synthetic 565 gene architectures. 566 567 568 569 Fig 1. Depiction of convergent overlapping genes in three reading frames and the process 570 developed to computationally design synthetic convergent overlaps from coding sequences. 571 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 28 (A) Sequences are represented for convergent (tail-to-tail) and unidirectional (same-strand) 572 overlapping genes in each of the three reading frames (phase 0, phase 1, and phase 2). Note, 573 phase 0 unidirectional overlaps are not represented as these are considered as an alternative start 574 site for the same gene (90). In each phase, the reading frame was shifted by one nucleotide. Start 575 and stop codons are indicated with green and red shading, respectively. (B) The general process 576 for creating synthetic convergent overlapping genes is presented, starting with a pool of ORF 577 sequences filtered by length (≥450 base pair [bp]). The primary sequence and in-frame 105-bp 578 fragment were randomly selected. A randomly selected stop codon from the reverse complement 579 (rc), and the fragment reverse complement was extended to that stop codon to serve as the 580 downstream sequence of the secondary sequence. Given the downstream secondary sequence 581 length, another sequence was randomly selected from the same gene list. A size appropriate 582 fragment was selected from that gene with length equal to the difference between the desired 583 gene length and the upstream overlap fragment (ie, the reverse complement of the primary 584 sequence to the selected stop codon). The upstream fragment was then combined with the 585 downstream fragment to generate the complete secondary sequence. The resultant primary and 586 secondary nucleotide sequences, sharing a known convergent overlapping region, were then 587 translated to amino acid (aa) sequences (either modified or unmodified) for subsequent use in 588 model training or inference. 589 590 591 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 29 Fig 2. Illustration of the overlap transformer-encoder architecture and general structure of 592 the data flow 593 (A) Depiction of the general structure of the data flow, for two input concatenated amino acid 594 sequences (primary and secondary) used during inference. The primary and secondary amino 595 acid sequences are represented with green and blue text, respectively, and the flow proceeds 596 from top to bottom. (B) Illustration of the transformer encoder model architecture. The model 597 input is tokenized concatenated paired amino acid sequences, and the model outputs raw logits. 598 The output logits are processed using a greedy decoder to generate a predicted overlap sequence. 599 600 601 Fig 3. Training data evaluation of secondary structure preservation and convergent 602 overlap-phase performance. 603 (A, D) Results for unmodified amino acid sequences. (A) Scatter density plot of secondary 604 structure (SS) score versus alignment score, with contours and marginal density plots showing 605 the distribution by overlap phase (0 = red, 1 = green, 2 = blue). (D) Fraction of successful 606 predictions across overlap lengths (310, 311, and 312 nucleotides), stratified by phase. Success 607 rates are reported with exact 95% Clopper–Pearson confidence intervals. (B, E) Results for 608 modified amino acid sequences. (B) Scatter density plot of SS score versus alignment score with 609 the same phase-coloring scheme. (E) Fraction of successful predictions across overlap lengths 610 for modified sequences, stratified by phase. Success rates are reported with exact 95% Clopper–611 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 30 Pearson confidence intervals. (C) Relationship between sequence-level SS composition (coil [C], 612 sheet [E], helix [H]) and mean SS score across phases 0, 1, and 2. Dashed lines indicate LOESS 613 regression fits with 99% confidence bands. 614 615 616 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 31 617 Fig 4. In silico design of convergent overlap with EGFP and AmpR 618 Overview of the multi-objective optimization process for EGFP and AmpR proteins using the 619 transformer encoder framework. (A) Schematic of the transformer encoder–based workflow. The 620 terminal 104 amino acids of EGFP and AmpR (across all tested overlap lengths) are 621 concatenated and passed through the model. Outputs are scored by a composite metric 622 integrating alignment, substitution, secondary structure preservation, and long-range interactions. 623 Bracketed amino acids from the input are preserved to enforce positional constraints. The final 624 sequence pair is selected from the output table by the highest average combined score. (B) 625 Multiple-sequence alignment of all output pairs (n=3,624) for EGFP and AmpR across both 626 optimization passes and windows (top to bottom). Diagonal dropout patterns correspond to 627 localized MC dropout applied to the overlap region. Variability reflects bracket-constrained 628 codon degeneracy introduced on the non-fixed partner sequence. (C) Predicted tertiary structures 629 generated with AlphaFold3 for EGFP (left) and AmpR (right). Native and overlapping terminal 630 regions relevant to the 311-nt overlap are highlighted (native in red, overlapped in blue); non-631 overlapping regions are shown in gray. 632 633 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 32 634 Fig 5. Trajectories of composite and component metrics during computational optimization 635 of EGFP/AmpR convergent overlaps 636 Optimization performance for EGFP and AmpR across two passes, each consisting of 13 637 windows. Metrics shown include the combined score (top left), which integrates secondary 638 structure (SS), ESM-2 contact map similarity, and alignment components, as well as the 639 individual SS (top right), ESM-2 (bottom left), and amino acid alignment (bottom right) scores. 640 Solid red (EGFP) and blue (AmpR) lines denote mean values, and shaded regions represent the 641 minimum-maximum range across sequences within each window. 642 643 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 33 644 Fig 6. Comparison of computational overlap predictions across pLDDT brackets, with 645 structural composition and optimization outcomes 646 647 Analyses are based on random sampling of SwissProt protein pairs (n=100 for each bracket) 648 from the AlphaFold2 Protein Structure Database (AF2). (A) Distribution of average pLDDT 649 values for the terminal 104 residues across sampled sequences, with red dashed lines marking the 650 confidence thresholds. (B) Counts of sequences assigned to pLDDT confidence brackets, shown 651 for both total and terminal regions (internal white bars denote terminal regions). (C) Ternary 652 plots of amino acid composition (helix [H], sheet [E], and coil [C]) for the primary and 653 secondary sequence sets, colored by pLDDT bracket (low-intermediate, confident, and very 654 high). Circles with black centers denote the centroid (mean position of all sequences within a 655 .CC-BY 4.0 International licenseavailable under a (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made The copyright holder for this preprintthis version posted November 10, 2025. ; https://doi.org/10.1101/2025.11.07.687268doi: bioRxiv preprint 34 given pLDDT bracket) in ternary space. (D) Optimization performance across convergent 656 overlap phases (0, 1, 2) for Pass 1 (P1) and Pass 2 (P2). Plots show average secondary structure 657 (SS Avg) and ESM-2 contact map similarity (ESM Avg) scores by optimization window, with 658 shaded regions representing the range of values stratified by pLDDT bracket. (E) Density 659 distributions of individual metric scores (SS, ESM, and Alignment) for the highest combined 660 score sequences (primary and secondary) across phases 0, 1, and 2. 661 662

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