Conditionally Site-Independent Neural Evolution of Antibody Sequences

Stephen Zhewen Lu; Aakarsh Vermani; Kohei Sanno; Jiarui Lu; Frederick A Matsen; Milind Jagota; Yun S. Song

cs.LGFeb 21, 2026

Conditionally Site-Independent Neural Evolution of Antibody Sequences

Authors: Stephen Zhewen Lu, Aakarsh Vermani, Kohei Sanno, Jiarui Lu, Frederick A Matsen, Milind Jagota, Yun S. Song

Organizations: University of California, Berkeley · Mila - Québec AI Institute · Université de Montréal · Fred Hutchinson Cancer Research Center ·

Abstract

Common deep learning approaches for antibody engineering focus on modeling the marginal distribution of sequences. By treating sequences as independent samples, however, these methods overlook affinity maturation as a rich and largely untapped source of information about the evolutionary process by which antibodies explore the underlying fitness landscape. In contrast, classical phylogenetic models explicitly represent evolutionary dynamics but lack the expressivity to capture complex epistatic interactions. We bridge this gap with CoSiNE, a continuous-time Markov chain parameterized by a deep neural network. Mathematically, we prove that CoSiNE provides a first-order approximation to the intractable sequential point mutation process, capturing epistatic effects with an error bound that is quadratic in branch length. Empirically, CoSiNE outperforms state-of-the-art language models in zero-shot variant effect prediction by explicitly disentangling selection from context-dependent somatic hypermutation. Finally, we introduce Guided Gillespie, a classifier-guided sampling scheme that steers CoSiNE at inference time, enabling efficient optimization of antibody binding affinity toward specific antigens.

Figures & tables

Figure 1: Overview of CoSiNE . Given an antibody sequence $x$ , the neural network outputs site-specific rate matrices conditioned on the full sequence. Each matrix is evolved for duration $t$ to yield a per-site transition distribution $p(y_{l}\mid x,t)$ . Assuming conditional independence, we take the product of the per-site transition probabilities to yield the full sequence transition probability $p(y\mid x,t)$ .

Figure 2: Mean per-site likelihood of CoSiNE versus DASM+Thrifty on held out evolutionary transitions from the test set . CoSiNE achieves better model fit, especially on transitions with longer branch lengths ( $t\geq 0.25$ ).

Figure 3: Categorical Jacobian for antibody 47D11 from CoV-AbDab. The heatmap displays the sensitivity of the model’s output predictions (y-axis) to specific mutations in the input sequence ( $x$ -axis). Sensitivity is measured as the Frobenius norm of the change in predicted rate matrix, averaged over all possible mutations. Cyan squares denote the CDR regions for both chains.

Model	Expression			Binding
Model	Koenig (H)	Koenig (L)	Adams	Koenig (H)	Koenig (L)	Shaneh. (119)	Shaneh. (120)
AbLang-2	$\phantom{+}0.096$	$-0.127$	$-0.097$	$-0.090$	$-0.011$	0.253	0.209
DASM	$\phantom{+}\underline{0.596}$	$\phantom{+}0.474$	$\phantom{+}0.270$	$\phantom{+}\underline{0.415}$	$\phantom{+}0.327$	0.450	0.536
PRISM	$\phantom{+}0.069$	$\phantom{+}0.129$	$\phantom{+}\underline{0.297}$	$\phantom{+}0.005$	$\phantom{+}0.061$	0.348	0.286
ESM2-150M	$\phantom{+}0.413$	$\phantom{+}0.485$	$-0.112$	$\phantom{+}0.112$	$\phantom{+}0.266$	0.236	0.205
ESM2-650M	$\phantom{+}0.326$	$\phantom{+}0.429$	$\phantom{+}0.124$	$\phantom{+}0.063$	$\phantom{+}0.265$	0.227	0.360

Table 1: Comparison of deep protein models on VEP benchmarks across expression and binding landscapes as measured by Spearman correlation. The Koenig assays are split depending on which chain is mutated, denoted by (H) and (L) for the heavy and light chains respectively. To maintain the same sequence length across samples, the Shanehsazzadeh (Shaneh.) assay is split into two datasets with heavy chains of lengths 119 and 120. Best performing models are shown in bold ; second-best are underlined .

Figure 4: DMS evaluation results for CoSiNE across expression (green) and binding (purple) assays. A solid color indicates the log-likelihood,

\log p_{\theta}(y\mid x,t)

, and hatching indicates the selection score defined in Equation 5 , which utilizes Thrifty likelihoods to separate selection from neutral mutation.

Figure 5: Guided Gillespie consistently steers the predicted binding affinity of sampled leaf sequences. We plot the change in predicted binding affinity from the naive root sequence used to start sampling. Known SARS-CoV-1 binders are in red.

Figure 6: Antibodies sampled by CoSiNE under guidance (

\gamma=5

) maintain high structural quality (AbodyBuilder3 pLDDT) and humanness (OASis). We compare against unguided leaf samples (black) and CoV-AbDab binders (red).

Method	Unique $\uparrow$	$E_{\text{dist}}$	IntDiv $\uparrow$	OASis $\uparrow$	$\Delta\,\text{Bind}_{\text{mean}}$ $\uparrow$	$\Delta\,\text{Bind}_{\text{max}}$ $\uparrow$	$N_{\text{oracle}}$ $\downarrow$
Greedy*	0.996	5.00	6.21	0.807	0.385	0.466	2756
Genetic Algo.	0.991	4.47	7.30	0.789	0.196	0.363	4.81
AbLang-1 -PoE	0.999	4.97	7.71	0.849	0.167	0.390	0.58
ESM2 -PoE	1.000	4.98	8.47	0.802	0.149	0.355	0.58
CoSiNE (ours)	0.988	4.56	6.83	0.822	0.198	0.395	5.00

Table 2: Constrained CDR optimization of a SARS-CoV-1 binder. Each variant is limited to 5 mutations within the CDR regions and is generated using at most 5 oracle calls. Greedy * is exempt from the oracle-call budget and serves as a reference upper bound.

E_{\text{dist}}

reports the mean edit distance from the seed sequence; IntDiv reports the mean pairwise edit distance between generated variants;

N_{\text{oracle}}

reports the average number of oracle calls per generated sequence. Best results are bolded ; second-best underlined .

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9

				Chain Length
Original Source	Assay Type	Dataset Name	Mutated Chain	Heavy	Light	Num. Seqs	Avg. Subs
Koenig et al. (2017)	Expression	Koenig Expression (H)	Heavy	120	108	2261	1
		Koenig Expression (L)	Light	120	108	2014	1
	Binding (VEGF)	Koenig Binding (H)	Heavy	120	108	2261	1
		Koenig Binding (L)	Light	120	108	2014	1
Adams et al. (2016)	Expression	Adams	Heavy	117	112	2803	1.98

Appendix

Table S1: Overview of Deep Mutational Scanning (DMS) Datasets. Avg. Subs denotes the average number of amino acid substitutions per mutant relative to wildtype.

Figure S1 : Clonal Tree used for Guided Affinity Maturation Experiment in Section 5.4

Figure S2 : As

\varepsilon

increases, SNR-weighting (green) reduces the relative Frobenius norm error between the estimated and true rate matrices compared to unweighted MLE (blue), while the Full MLE (yellow) is constant.

Panel of Figure 13 — Figure S3 : Gillespie samples are more consistent with the true transition probability distribution than samples from the factorized CoSiNE likelihood at all levels of epistasis. The difference increases smoothly with the epistasis strength $\varepsilon$ in the underlying rate matrix. Ground truth matrix generation and estimation protocols are in Section B.1 .

Panel of Figure 14 — Figure S4 : Gillespie sampling exhibits lower sampling error and better preserves evolutionary distances on real antibody clonal families. a) Comparison of sampling fidelity: Gillespie samples are closer (in Hamming distance) to the true held-out leaf sequence in 52.0% of cases, compared to 38.8% for the factorized matrix exponential. b, c) Correlation between the root-to-leaf Hamming distances of real versus simulated leaf sequences. Gillespie sampling (b) achieves a higher Pearson correlation ( $r=0.62$ ) with the observed evolutionary distances than the factorized matrix exponential (c) approach ( $r=0.53$ ).

Panel of Figure 15 — Figure S5 : Per-site entropy across increasing branch lengths. Entropy profiles for branch lengths 0.01, 0.05, and 0.1 (top to bottom). Red bars denote CDRs. Entropy scales with branch length, showing higher mutational variability in CDRs.

Branch Length	WAG NLL	CoSiNE NLL
$[0,0.01)$	0.0717	0.0663
$[0.01,0.05)$	0.171	0.147
$[0.05,0.15)$	0.454	0.378
$[0.15,\infty)$	0.836	0.676

Appendix

Table S2 : Comparison of WAG and CoSiNE per-token NLL on test transitions by branch length.

Model	Val NLL	Test NLL
WAG	1.170	1.200
LG+G4	1.139	1.168
CoSiNE	1.005	1.027

Appendix

Table S3 : Comparison of WAG, LG+G4, and CoSiNE per token NLL on held out transitions from TrRosetta.

Figure S6 : Analysis of CoSiNE scoring methods against log Enrichment Ratios for the Koenig DMS assay measuring expression (Light Chain). Left : Log-Likelihood performance (

\rho=0.439

). Right : Selection Score performance (

\rho=0.509

). Points are colored by the edit distance from wildtype codon to mutant codon. We assume the mutant codon is the one which requires the smallest number of nucleotide edits.

Model	Petersen (319-345)	Petersen (222-1C06)
AbLang-2	0.199	$\phantom{+}0.060$
DASM	0.395	$\phantom{+}\underline{0.286}$
PRISM	0.312	$-0.073$
ESM2-150M	0.250	$-0.139$
ESM2-650M	0.278	$\phantom{+}0.013$
ProGen2-Small	0.329	$\phantom{+}0.024$

Appendix

Table S4 : VEP Performance of CoSiNE against baselines on Petersen binding datasets (Spearman).

Model	Expression			Binding
Model	Koenig (H)	Koenig (L)	Adams	Koenig (H)	Koenig (L)	Shaneh. (119)	Shaneh. (120)
CoSiNE-Paired	0.613	0.508	0.464	0.456	0.371	0.498	0.536
CoSiNE-Single	0.545	0.509	0.234	0.390	0.375	0.504	0.549

Appendix

Table S5 : Comparison of CoSiNE on zero-shot VEP with both the heavy and light chains provided as context (Paired) versus just the chain with the mutations (Single). Blue indicates datasets with mutations on the heavy chain, and red indicates mutations on the light chain.

Model	Expression			Binding
Model	Koenig (H)	Koenig (L)	Adams	Koenig (H)	Koenig (L)	Shaneh. (119)	Shaneh. (120)
AbLang-2	$\phantom{+}0.153$	$-0.109$	$-0.096$	$-0.114$	$-0.108$	0.263	0.166
DASM	0.688	$\phantom{+}\underline{0.674}$	$\phantom{+}0.221$	$\phantom{+}\underline{0.335}$	$\phantom{+}\underline{0.316}$	0.458	0.518
PRISM	$\phantom{+}0.055$	$\phantom{+}0.145$	$\phantom{+}\underline{0.243}$	$-0.001$	$\phantom{+}0.000$	0.346	0.251
ESM2-150M	$\phantom{+}0.476$	$\phantom{+}0.539$	$-0.119$	$\phantom{+}0.044$	$\phantom{+}0.266$	0.215	0.197
ESM2-650M	$\phantom{+}0.384$	$\phantom{+}0.416$	$\phantom{+}0.097$	$\phantom{+}0.009$	$\phantom{+}0.243$	0.191	0.308

Appendix

Table S6 : Comparison of deep protein models on VEP benchmarks across expression and binding landscapes as measured by Pearson correlation.

Figure S7 : Spearman correlation $\rho$ between the CoSiNE selection score and experimental fitness evaluated over branch lengths $\bm{t\in\{0.01,0.1,0.2,0.3,0.4,10\}}$ . Expression assays are shown in green and binding assays are shown in purple.

Model	Expression			Binding
Model	Koenig (H)	Koenig (L)	Adams	Koenig (H)	Koenig (L)	Shaneh. (119)	Shaneh. (120)
CoSiNE ( $t=0.01$ )	0.370	0.349	0.284	0.240	0.217	0.486	0.537
CoSiNE ( $t=0.1$ )	0.584	0.487	0.441	0.427	0.352	0.505	0.549
CoSiNE ( $t=0.2$ )	0.613	0.508	0.464	0.456	0.371	0.498	0.536
CoSiNE ( $t=0.3$ )	0.624	0.500	0.463	0.471	0.375	0.503	0.533
CoSiNE ( $t=0.4$ )	0.605	0.509	0.425	0.472	0.378	0.505	0.530

Appendix

Table S7 : Underlying Spearman correlation values for Figure S7 . CoSiNE results are bolded where they tie or exceed all non- CoSiNE baselines ( Table 1 ).

Model	Expression			Binding
Model	Koenig (H)	Koenig (L)	Adams	Koenig (H)	Koenig (L)	Shaneh. (119)	Shaneh. (120)
DASM	0.596	0.474	0.270	0.415	0.327	0.450	0.536
ESM2-650M	0.326	0.429	0.124	0.063	0.265	0.227	0.360
ProGen2-Small	0.407	0.513	-0.024	0.098	0.332	0.119	0.070
CoSiNE-ESM2	0.613	0.508	0.464	0.456	0.371	0.498	0.536
CoSiNE-8M	0.503	0.474	0.447	0.363	0.330	0.508	0.531

Appendix

Table S8 : Ablation of the pretrained ESM2 backbone on VEP benchmarks. Spearman correlation (

\rho

) is reported across expression and binding datasets. Best performing models are shown in bold ; second-best are underlined .

Panel of Figure 25 — Figure S8 : TAG guidance matches exact guidance performance with 500–900× speedup . (Left) Fitness improvements and (right) runtime comparison across branch lengths. No significant fitness differences ( $p>0.05$ ), but TAG is orders of magnitude faster.

Sequence Length ( $L$ )	Guided Runtime (s)	Unguided Runtime (s)
$50$	$5.16$	$3.54$
$100$	$5.94$	$3.51$
$250$	$8.53$	$3.51$
$500$	$12.20$	$3.65$

Appendix

Table S9 : Runtime analysis of Guided vs. Unguided Gillespie sampling. Left: Runtime scaling by sequence length

L

with a fixed

N=50

sampling steps. Right: Runtime scaling by sampling steps

N

with a fixed sequence length of

L=100

Panel of Figure 27 — Figure S9 : Guided affinity maturation results for additional naive antibody sequences using CoV-1 (top) and CoV-2 (bottom) oracles. We randomly selected naive IgM heavy sequences from the OAS database and recursively sampled down the tree in Figure S1 .

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