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Continuous Variational Synthesis

Alan N. Amin, Mattia G. Gollub, Andrei Slabodkin, Elizabeth B. Wood, Eli N. Weinstein

arXiv:2609.35083v1q-bio.BMcs.LG

TL;DR

Discrete synthesis constraints make variational synthesis models difficult to train for pre-training and fine-tuning. The paper trains free models continuously and quantizes them afterward to impose synthesis constraints. Across diverse design challenges, cVS advances the quality-diversity frontier, with strong in silico performance maintained in vitro.

  • Problem

    Chemical constraints can force variational synthesis parameters into discrete choices, making them difficult to search and limiting available training approaches.

    Source: §0.1 · p. 2

  • Method

    cVS trains variational synthesis models with gradient-based optimization and variance reduction, then quantizes parameters to impose synthesis constraints.

    Source: §1 · p. 3

  • Results

    Across design challenges, cVS substantially advances the quality-diversity Pareto frontier, and its strong in silico performance is maintained in vitro.

    Source: §2 · p. 5

  • Takeaways & Limitations

    The approach supports variational synthesis design across enzymes, peptides, antibodies, and regulatory elements, with in silico performance maintained in vitro.

    Source: §3 · p. 10

    Designs beyond single oligos, such as full-length CYP, require assembly or enzymatic synthesis, which can introduce constraints and errors whose impact may vary by reward model.

    Source: §3 · p. 10

Abstract

from arXiv · show

Biological machine learning was long bottlenecked by the ability to synthesize designed DNA. Variational synthesis models control chemical reactions to physically manufacture quadrillions of designed sequences in DNA. However, training these generative models is challenging: constraints on chemical synthesis can force many parameters into a discrete space, limiting the ability to pre-train and fine-tune. In this article we train ``free'' variational synthesis models using stochastic gradient descent in continuous space, and then discretize with post-training quantization to impose hardware and wetware constraints. This enables variational synthesis models to satisfy stringent reward criteria, while still synthesizing diverse designs, achieving a strictly dominating quality-diversity Pareto frontier. We demonstrate by training variational synthesis models of enzymes, peptides, antibody CDRH3s, and regulatory DNA elements. In silico performance is maintained in vitro.

0.1 BACKGROUND: VARIATIONAL SYNTHESIS

Variational synthesis controls stochastic chemical reactions to produce large libraries of designs, but training is difficult when chemistry restricts model parameters to discrete choices. The section describes this synthesis setup and its constraints alongside related probabilistic-modeling approaches.

0.1 BACKGROUND: VARIATIONAL SYNTHESIS

  • Stochastic synthesis samples designs during chemical reactions, rather than sampling computationally and synthesizing each design individually.Learned parameters control the reactions; each molecule is an independent model sample, and synthesis yield can reach trillions or quadrillions.Source: §0.1 · pp. 1–2
  • Chemical constraints can restrict reaction-mixture parameters to a finite catalog, preventing them from being arbitrary vectors in the simplex.One common restriction uses equal-weight nucleotide mixtures; mixtures may also be shared across positions and wells.Source: §0.1 · p. 2
  • Training maximizes data log likelihood, approximately minimizing KL(p∥qθ), while prior EM methods are prone to local maxima.Concurrent PGLD instead optimizes expected reward under qθ(x), handling constraints through a meta-distribution over parameters.Source: §0.1 · p. 2

0.2 RELATED WORK

  • Earlier work controlled stochastic DNA synthesis without optimizing a probabilistic library model, while related methods studied reverse-KL objectives with one or multiple reaction wells.Zhu et al. (2024) studied M = 1; concurrent Sussex et al. (2026) studied complex synthesis models with M > 1.Source: §0.2 · p. 2
  • The authors frame variational synthesis as mixture-model inference used to control experiments, under constraints distinct from posterior approximation.They connect this perspective to approximate Bayesian computation and other work using probabilistic inference to control generative models.Source: §0.2 · p. 2

1 METHOD

The method trains unconstrained variational synthesis models with gradient-based optimization, then quantizes their parameters to meet synthesis constraints. It combines forward-KL pre-training, reverse-KL fine-tuning, and variance-reduction strategies for the model’s mixture structure and autoregressive priors.

1 METHOD

  • cVS combines reverse-KL fine-tuning, unified gradient-based optimization with variance reduction, and quantization that imposes synthesis constraints while preserving model quality and diversity.The method extends forward-KL pre-training with reward-model-based fine-tuning.Source: §1 · p. 3

1.1 FINE-TUNING

  • Reverse-KL fine-tuning minimizes divergence to a prior tilted by a sequence reward, prioritizing high-reward samples even if some sequence-space regions are omitted.The reward model scores sequences, and the prior may be a generative model pre-trained on human or evolutionary data.Source: §1.1 · p. 3

1.2 GRADIENT-BASED TRAINING OF FREE VS

  • The authors relax synthesis constraints to train a free model, then use Adam and partially analytic approximations to reduce gradient-estimation variance.For reverse KL, naive REINFORCE estimators have very high variance; the approximations exploit model structure.Source: §1.2 · p. 3

1.2.1 ANALYTIC ENTROPY

  • The entropy decomposition separates within-well diversity from sequence overlap between wells, allowing analytic gradients for its first two terms.Only the final term requires a REINFORCE estimate.Source: §1.2.1 · p. 4

1.2.2 PRIOR RAO-BLACKWELLIZATION

  • For autoregressive priors, the model’s forward pass provides probabilities for all letters at a position, enabling analytic computation of the conditional expectation instead of Monte Carlo.The synthesis model’s conditional sequence probabilities are also tractable, supporting this Rao-Blackwellized estimate.Source: §1.2.2 · p. 4

1.3 QUANTIZATION

  • Post-training quantization maps free parameters to a finite mixture catalog by minimizing distance, rounding them to physically achievable values.For protein designs, nucleotide-mixture constraints are translated into amino-acid mixture constraints before discretization.Source: §1.3 · p. 4

2 EMPIRICAL RESULTS

Across four design challenges, cVS advances the quality–diversity frontier, and its strong in-silico performance is maintained in vitro. The experiments span low-capacity synthesis settings and include enzyme, peptide, antibody, and regulatory-DNA libraries.

2 EMPIRICAL RESULTS

  • Across four design challenges, cVS substantially advances the quality–diversity Pareto frontier, with strong in-silico performance maintained in vitro.The study focuses mainly on low-capacity synthesis settings, where training-algorithm differences can have especially large effects.Source: §2 · pp. 4–5

2.1 DESIGNING FULL LENGTH ENZYMES

  • cVS designs of full-length CYP2C9 were structurally similar to the human enzyme (TM above 0.8) and diverse, while only 2 × 10^-6 EM designs lacked internal stop codons.The cVS designs also had pTM and pLDDT nearly as high as evolutionary sequences.Source: Figure 1 · p. 5, §2.1 · p. 6

2.2 DESIGNING HLA-A*02:01-PRESENTED PEPTIDES

  • cVS substantially advanced the peptide-design frontier, especially at high diversity, with a relatively modest performance gap between free and quantized cVS.The reproduced PGLD frontier closely matched the published one, and learning-rate tuning plus larger NMC improved PGLD.Source: §2.2 · pp. 6–7, Figure 2 · p. 6

2.3 DESIGNING TCR MIMICKING ANTIBODIES

  • For scFv CDRH3 design, cVS Pareto dominated EM VS and PGLD, achieving higher reward and higher diversity.The comparison used forward-KL pre-training followed by reverse-KL fine-tuning; removing either stage harmed performance.Source: §2.3 · pp. 7–8, Figure 3 · p. 7
  • Adding pre-set nucleotide mixtures systematically improved post-quantization antibody-design performance, bringing it closer to free cVS.The authors note that the infinite-mixture limit need not approach free cVS because independent nucleotide draws cannot produce every amino-acid distribution.Source: Figure 3 · p. 7, §2.3 · p. 8
  • Increasing the number of reaction wells improved free-cVS performance smoothly, with the gains carrying over after discretization.As the number of wells increased, the optimal learning rate decreased and performance approached a limiting value.Source: Figure 3 · p. 7, §2.3 · p. 8

2.4 ENGINEERING REGULATORY ELEMENTS

  • For EF1α regulatory DNA, cVS advanced the frontier while keeping designs close to the human-genome prior, and learned tiled motifs unlike PGLD.The quantization penalty was small for equal-nucleotide mixtures.Source: Figure 4 · p. 8, §2.4 · p. 9
  • In-vitro EF1α designs maintained in-silico quality, reward, and prior likelihood, and substantially outperformed even in-silico PGLD across the reported metrics.The synthesis yielded 1.6 nM, or 9.6 × 10^14 molecules; the estimated in-vitro Pareto position uses an optimistic entropy estimate.Source: §2.4 · pp. 9–10, Figure 5 · p. 9

3 DISCUSSION

cVS relaxes chemical-synthesis constraints into continuous space for gradient-based training, then quantizes for synthesis on diverse hardware and wetware. The paper reports that in-silico performance is maintained in vitro, while synthesis beyond single oligos can introduce additional constraints and errors.

3 DISCUSSION

  • cVS achieves state-of-the-art in-silico performance on forward and reverse KL objectives for DNA and protein design, with performance maintained in vitro.The approach uses continuous relaxation for gradient-based training, followed by post-training quantization.Source: §3 · p. 10
  • Beyond single oligos, designs such as full-length CYP require assembly or enzymatic synthesis, which can introduce additional constraints and errors.Error impacts depend on the reward model: chromatin accessibility showed minimal impact, but other reward models may be more sensitive.Source: §3 · p. 10
  • cVS is presented as a way to train, fine-tune, and deploy variational synthesis models like language models for programmable wet-lab exploration of biological sequence space.The paper frames this as redeploying language-modeling ideas beyond in-silico exploration.Source: §3 · p. 10

A ADDITIONAL TRAINING DETAILS

The training details add variance reduction and a well-resampling strategy to improve optimization. Ablations report a minor performance drop without the control variate and a performance drop without split-merge.

A ADDITIONAL TRAINING DETAILS

  • The authors introduce additional training strategies, including a customized control variate and split-merge well resampling.The passage introduces these strategies before describing their mechanisms and ablations.Source: A ADDITIONAL TRAINING DETAILS · p. 14
  • A customized leave-one-out control variate reduces gradient variance, and its ablation shows a minor performance drop.The method applies REINFORCE with a control variate.Source: A ADDITIONAL TRAINING DETAILS · p. 14
  • Removing split-merge causes a performance drop in ablation, supporting its use to address wells that get stuck in local maxima.Low-weight wells are resampled by jittering parameters from the highest-weight well.Source: A ADDITIONAL TRAINING DETAILS · p. 14

B.1 DETAILS ON ENZYME DESIGN

The enzyme-design setup uses an aligned sequence core, restores 13 fixed insertions, and selects a forward-KL cVS model by held-out loss.

B.1 DETAILS ON ENZYME DESIGN

  • The design models the aligned core positions 23:470, restores 13 wild-type insertion residues as fixed positions, and selects among five forward-KL cVS models using held-out loss.Ten percent of alignment sequences are held out for early stopping.Source: §B.1 · p. 14

B.2 DETAILS ON HLA-A*02:01 EPITOPE REPERTOIRE RESULTS

The HLA-A*02:01 repertoire comparison sweeps learning rates for both cVS and PGLD, while maximizing PGLD's NMC. The peptide reward and diversity measures are also specified for evaluation.

B.2 DETAILS ON HLA-A*02:01 EPITOPE REPERTOIRE RESULTS

  • Sweeping five learning rates per α and maximizing NMC improves PGLD's frontier; all experiments use learning-rate sweeps and maximal NMC.The same learning-rate sweep is used for cVS and PGLD to account for differences across methods and α values.Source: §B.2 · p. 15
  • The reward is log10 of presentation percentile divided by 100, and final performance is measured as one minus the expected percentile divided by 100.The parameterization sharply penalizes low scores.Source: §B.2 · p. 15
  • cVS pre-training optimizes a forward-KL objective using 20 million prior samples reweighted by reward to an effective sample size of one million.PGLD pre-training samples starting templates from the same reward-tilted distribution.Source: §B.2 · p. 15

B.3 DETAILS ON TCRM DESIGN

The TCRM design uses a reward model trained on scFv–pHLA interactions and trains cVS with forward-KL pretraining followed by reverse-KL fine-tuning. Promoter-engineering experiments evaluate designs in EF1α and a selected genomic context, using BPnet and MarinDNA models.

B.3 DETAILS ON TCRM DESIGN

  • The scFv–pHLA reward model is a 7M-parameter encoder-only transformer trained to predict binding counts from CDRH3 sequence and pHLA.Its training data came from scFv-CAR constructs tested against 100 fluorescently labeled, DNA-barcoded pHLA targets.Source: §B.3 · p. 15
  • cVS pretraining reweights 20 million prior samples to 5% effective sample size, then reverse-KL fine-tunes for 5 million reward evaluations.Pretraining runs to convergence; fine-tuning uses batch size 1024, a five-rate sweep, cosine annealing, and an α annealing schedule.Source: §B.3 · p. 15

B.4 DETAILS ON PROMOTER ENGINEERING

  • Promoter designs target the first 129 bp of EF1α or the center 150 bp of a 2-kb genomic region near accessibility peaks in all six experiments.BPnet and MarinDNA were trained or fine-tuned on the accessibility peaks, with chromosomes 9–11 held out for early stopping.Source: §B.4 · p. 15
  • Promoter variational-synthesis models use the Section B.2 procedure, with pretraining tilted to an effective sample size of 0.5%.The genomic-context designs use chr10:63519858-63521858.Source: §B.4 · p. 15

C In vitro EVALUATION

The in-vitro evaluation sequences synthesized DNA libraries and compares model samples with the target distribution using KSD-B. It also visualizes sample relationships through Hamming-distance-based UMAP embeddings.

C In vitro EVALUATION

  • Sequencing yielded 24,471,996 full-length reads from the synthesized library.The library was prepared from 250 ng of sampled DNA and sequenced in a 2×150 bp configuration.Source: C In vitro EVALUATION · p. 16
  • KSD-B measures how close synthesis models are to the reward-tilted target distribution using an IMQ Hamming kernel and repeated independent draws.The evaluation uses 1,000 samples per model or sequencing data, redrawn five times to report mean ± SEM.Source: C In vitro EVALUATION · p. 16
  • Low-dimensional sample representations are produced by transforming pairwise Hamming distances with an IMQ Hamming kernel, then applying UMAP.The procedure follows Weinstein et al. (2026a).Source: C In vitro EVALUATION · p. 16

D SUPPLEMENTAL RESULTS

Supplemental analyses examine training curves, hyperparameter sensitivity, scaling, diversity, and ablations across peptide, scFv CDRH3, and regulatory-DNA designs. They report narrower diversity for PGLD at low kernel bandwidths and a performance decrease when split-merge resampling is removed.

D SUPPLEMENTAL RESULTS

  • Peptide and scFv CDRH3 training-loss plots compare PGLD with free cVS across reward evaluations, including learning-rate zoom-ins.Figures 6 and 9 include overview curves alongside the results plotted in Figures 2 and 3a, respectively.Source: Figure 6 · p. 16, Figure 9 · p. 18
  • Peptide and scFv CDRH3 sensitivity analyses scan learning rate, Adam settings, baseline-control-variate moving-average rate, and batch size.Each scan is relative to the setting used in the corresponding main figure.Source: Figure 7 · p. 17, Figure 8 · p. 17
  • The scFv CDRH3 analyses vary well count, compare pretraining conditions, and hold total reward evaluations fixed in a from-scratch comparison.The fixed-budget comparison uses 5 million reward and prior evaluations.Source: Figure 10 · p. 18, Figure 11 · p. 18, Figure 12 · p. 19
  • A scFv CDRH3 Pareto frontier measures diversity as the exponential of Shannon entropy under a uniform prior rather than a human prior.The stated diversity measure is effective library size.Source: Figure 13 · p. 19
  • At low RBF kernel bandwidths, cVS achieves much higher diversity than PGLD at the same constraints and average reward, consistent with PGLD finding spread-out but narrow modes.The comparison also reports similar diversity at high bandwidths and similar mean Hamming distance.Source: Figure 14 · p. 19
  • Removing split-merge resampling of wells decreases scFv CDRH3 performance.Removing prior Rao-Blackwellization or the REINFORCE control variate reduces gradient variance but changes final performance only slightly.Source: Figure 16 · p. 20, Figure 15 · p. 20
  • Regulatory-DNA designs in random genomic context are evaluated by cell-type accessibility difference versus KL to the human-genome prior.The figure also depicts predicted accessibility and nucleotide mixtures across well positions.Source: Figure 17 · p. 21
  • EF1α-library KSD-B is also computed with exponential Hamming kernels using scanned bandwidths and, separately, fixed σ = 3 across five redraws.The fixed-bandwidth panel reports SEM across redraws.Source: Figure 18 · p. 21
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