pCoMole: Pareto-Constrained Molecule Editing with Discrete Flows

Tong Chen; Maximilian Holsman; Lin Zhao; Pranam Chatterjee

cs.LGOct 1, 2026

pCoMole: Pareto-Constrained Molecule Editing with Discrete Flows

Authors: Tong Chen, Maximilian Holsman, Lin Zhao, Pranam Chatterjee

Organizations: Department of Computer and Information Science, University of Pennsylvania · Department of Computer Science, Duke University · Department of Bioengineering, University of Pennsylvania

Abstract

Biomolecular therapeutics often start from known sequences and require targeted editing to improve multiple properties while satisfying hard biochemical and manufacturability constraints. However, existing generative methods do not jointly support multi-objective optimization, hard feasibility, and sequence editing in discrete, variable-length biological spaces. In this work, we introduce Pareto-Constrained Molecule Editing (pCoMole), a framework built on discrete flow matching that steers a pre-trained Edit Flow toward user-specified preferences while enforcing terminal feasibility. pCoMole defines a feasibility-gated terminal distribution using an augmented Tchebycheff utility and realizes the resulting preference tilt through a Doob-h transform of the underlying edit process. To make this construction practical, we approximate the required harmonic function using short Monte Carlo rollouts over candidate edits, yielding an efficient guided editor with provable preference consistency. We validate pCoMole by shrinking GFP while retaining fluorescence-related properties, shortening diverse Cas9 orthologs while preserving PAM specificity, and compressing peptide binders into short peptidomimetics that optimize seven drug-related properties under hard constraints. In wet lab testing, two 229-residue pCoMole-designed eGFP variants retained clear green fluorescence in BL21 cells after 10 deletions, with either one or two substitutions. Together, pCoMole enables constraint-aware, Pareto-aligned editing of biomolecular sequences in discrete, variable-length spaces.

Figures & tables

Figure 1: Overview of pCoMole and representative designs. (A) Starting from an intermediate sequence $x_{t}$ , pCoMole samples candidate edits from a pretrained Edit Flows velocity $u_{t}$ and uses short Monte Carlo rollouts to estimate terminal utility and feasibility. The resulting approximate Doob- $h$ guidance reweights transitions into a tilted velocity $\tilde{u}_{t}$ , steering generation toward feasible, high-utility terminal sequences. (B) In GFP miniaturization, pCoMole reduces sequence length while preserving the canonical $\beta$ -barrel and improving predicted optical objectives. (C) In Cas9 shrinkage, pCoMole produces a shortened SpCas9 variant that satisfies the PAM constraint while maintaining high predicted Cas9-likeness and plausible structure. (D) In short peptidomimetic binder design, pCoMole shortens a known 7JVS binder while improving predicted properties. The binder is shown in yellow, the target in light blue, and dark blue highlights target motif regions. Insets zoom into the binding interface (see Figure S6 for enlarged views).

Edit Flow	Train Loss	Test Loss	Diversity	Validity Rate
UniRef S	594.39	602.47	0.68	—
UniRef L	3621.89	3573.07	0.45	—
GFP	147.61	436.65	0.53	0.91
Cas9	2430.42	2498.37	0.25	0.93
Peptidomimetic	36.26	38.40	0.80	0.70

Table 1: Edit Flow training and evaluation metrics across sequence domains.

Method	$\lvert\lambda_{\mathrm{exc}}-488\rvert$ ( $\downarrow$ )	Brightness
RayGun	22.3594	9.4207
SCISOR	20.1279	19.0333
pCoMole (UniRef S Edit Flow)	5.8213	43.5116
pCoMole (GFP Edit Flow)	8.4372	49.0121

Table 2: GFP design comparison between pCoMole, RayGun, and SCISOR under a fixed length constraint. For each method, we generate 100 candidates and report average metrics. All methods enforce a fixed terminal length of 213 residues and achieve a validity rate near 100%.

Task	Method	Low Tier (s)	Medium Tier (s)	High Tier (s)
GFP	RayGun	N.A.	N.A.	N.A.
	SCISOR	14	613	N.A.
	pCoMole	266	333	665
Cas9	RayGun	N.A.	N.A.	N.A.
	SCISOR	6.86	76.61	240.28
	pCoMole	131.69	129.15	138.96

Table 3: Compute-normalized time to first predicted-usable sequence. We report the time required to obtain a sequence satisfying each task-specific quality tier under a 3600-second trial budget. Lower is better. “N.A.” indicates that no valid sequence was found within the budget.

Objective Guidance $\lvert\lambda_{\mathrm{exc}}-488\rvert$ Brightness	$\lvert\lambda_{\mathrm{exc}}-488\rvert$ ( $\downarrow$ )	Brightness	Length	Validity Rate
$\checkmark$ $\checkmark$	3.7557	54.0159	214	1.0
$\checkmark$ $\times$	2.9810	31.2600	213	1.0
$\times$ $\checkmark$	12.9500	62.1200	214	1.0
$\times$ $\times$	32.8200	25.0200	213	1.0

Table 4: Objective guidance ablation for pCoMole on GFP design. For each objective guidance setting, we report the average metrics over 100 pCoMole-shrunk GFP sequences. Length reduction guidance is applied in all settings.

Figure 2: Wet-lab fluorescence validation of pCoMole-shortened eGFP designs. 488-nm fluorescence images of the 239-residue in-house eGFP construct, a manually selected 228-residue negative control, and two 229-residue pCoMole designs. pCoMole_eGFP_1 contains 10 deletions and one substitution; pCoMole_eGFP_2 contains 10 deletions and two substitutions.

Input Cas9	Length Change	Cas9 Likelihood	PAM Distr. CE	PAM Match Rate	PAM matched
SpCas9	$1368\rightarrow 1211\;(-157)$	$1.00\rightarrow 0.97$	$0.97\rightarrow 0.90$	1.00	NGG
St1Cas9	$1121\rightarrow 1023\;(-98)$	$1.00\rightarrow 0.98$	$0.91\rightarrow 0.89$	1.00	NNANAA
St3Cas9	$1388\rightarrow 1301\;(-87)$	$1.00\rightarrow 0.95$	$0.94\rightarrow 0.90$	1.00	NGGNG
GeoCas9	$1087\rightarrow 1035\;(-52)$	$0.96\rightarrow 0.95$	$0.93\rightarrow 0.89$	1.00	NNNNCNAA

Table 5: pCoMole shrinks common Cas9 sequences while maintaining PAM specificity. “Cas9 Likelihood” is the score of a pre-trained Cas9 classifier. “PAM Distribution CE” is a cross-entropy between the predicted PAM distribution of the input and that of the generated sequence. “PAM Match Rate” is the fraction of generated sequences whose predicted PAM exactly matches the input. “PAM matched’ is the predicted PAM which pCoMole maintained during shrinkage.

Target	Ligand	Length	Non-Toxicity	Solubility	Permeability	Half-life (h)	Affinity	Motif Score	Specificity
PTH1R	Teriparatide	259	0.8890	0.6905	0.2793	2.8301	0.9148	0.2696	0.9834
PTH1R	pCoMole	217	0.8729	0.7268	0.2959	2.9488	0.9358	0.3593	0.9706
p53	p28	170	0.8533	0.6737	0.2627	2.2015	0.6369	0.5418	0.9873
p53	pCoMole	133	0.8076	0.6309	0.3071	3.8517	0.6885	0.5014	0.9649
GLP-1R	Semaglutide	248	0.8872	0.7656	0.2583	1.8571	0.9467	0.4691	0.9827
	pCoMole	186	0.8122	0.7240	0.2985	4.0277	0.9367	0.5212	0.9778

Table 6: pCoMole designs shorter peptidomimetics with improved predicted properties. For each target, we compare the original peptide drug binder with the average over 100 pCoMole-designed peptidomimetics. Length denotes the number of tokenized SMILES tokens. All pCoMole designs satisfy the specified task constraints. For GLP-1R, motif score and specificity are computed relative to Semaglutide and are therefore reported only for Semaglutide and its pCoMole designs.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

PDB	Constraints	Length	Non-Toxicity	Solubility	Permeability	Half-life (h)	Affinity	Motif Score	Specificity	$R_{\text{PM}}$	$R_{\text{LEN}}$
3AMA	Baseline (Known Binder)	141	0.901	0.854	0.2528	1.4508	0.6731	0.9171	0.9941	—	—
	PM+LEN	129	0.8085	0.4991	0.3049	4.8599	0.6946	0.9798	0.9429	1.00	1.00
	PM only	112	0.7444	0.5769	0.3537	14.4684	0.6666	0.8521	0.9482	1.00	0.27
	LEN only	131	0.8261	0.486	0.3081	4.9182	0.6947	0.9782	0.9443	1.00	1.00
	None	21	0.5104	0.8422	0.6186	68.4154	0.4696	0.4247	0.9613	0.10	0.02
5JHF	Baseline (Known Binder)	95	0.8029	0.8744	0.2431	1.5936	0.5457	0.8484	0.9926	—	—

Appendix

Table S1: Constraint ablation for peptidomimetic binder design on 3AMA and 5JHF. For each target, we report the property scores of the known binder (“Baseline”) and the average scores over 100 pCoMole-generated sequences under different constraint settings. Length is the number of tokenized SMILES tokens. PM denotes the hard peptidomimetic constraint and LEN denotes the soft length constraint.

R_{\mathrm{PM}}

and

R_{\mathrm{LEN}}

are the fractions (out of 100) of generated sequences that satisfy the PM and LEN constraints, respectively. Please refer to Section D for details.

Scalarization	Objective Weights Solubility Permeability Affinity	Length	Solubility	Permeability	Affinity	Coverage vs. Reference Front
ATC	1 1 8	38	0.6856	0.3909	0.7488	0.7297
	1 8 1	34	0.7131	0.4724	0.6832
	8 1 1	37	0.8002	0.3851	0.6381
Tchebycheff	1 1 8	34	0.6973	0.4829	0.6748	0.6486
	1 8 1	36	0.7248	0.4002	0.701
	8 1 1	35	0.7197	0.4625	0.6509

Appendix

Table S2: Pareto coverage comparison on PDB 4O56 under different scalarizations and weight vectors. We evaluate three scalarization choices (ATC, Tchebycheff, and linear weighted sum) for multi-objective peptidomimetic binder design optimizing solubility, permeability, and binding affinity. For each method, we run pCoMole with three sets of objective weight vectors and report the average properties of 100 generated sequences. The rightmost column reports each method’s empirical Pareto coverage against a reference front constructed by pooling solutions across all weight settings and methods (higher is better).

Model	Length Diff.	Cas9 Likeli.	PAM Distr. CE	PAM Match %
Base EF ( $s=1$ )	$1121\rightarrow 1119\;$	$1.00\rightarrow 1.00$	$0.91\rightarrow 0.89$	$36$
Base EF ( $s=1000$ )	$1121\rightarrow 1004\;$	$1.00\rightarrow 0.70$	$0.91\rightarrow 0.78$	$0$
pCoMole ( $s=1$ )	$1121\rightarrow 1118\;$	$1.00\rightarrow 1.00$	$0.91\rightarrow 0.94$	$100$
pCoMole ( $s=1000$ )	$1121\rightarrow 1016\;$	$1.00\rightarrow 0.95$	$0.91\rightarrow 0.89$	$100$

Appendix

Table S3: Deletion-rate scaling ablation for pCoMole on St1Cas9 edits. In all cases, the corresponding model was used to generate 50 sequences. “Base EF” denotes sampling from the Edit Flows directly without any guidance. “s” denotes the deletion rate scaling term.

pCoMole Config.	Length Diff.	Cas9 Likeli.	PAM Distr. CE	PAM Match %
pCoMole w/ no ins.	$1121\rightarrow 1021\;$	$1.00\rightarrow 0.98$	$0.91\rightarrow 0.89$	100
pCoMole w/ ins.	$1121\rightarrow 1049\;$	$1.00\rightarrow 0.97$	$0.91\rightarrow 0.90$	100

Appendix

Table S4: Ablation of allowing pCoMole edit steps to make insertions. In both cases, pCoMole was used to generate 50 shrunk variants of St1Cas9.

PDB	#-steps	Length	Non-Toxicity	Solubility	Permeability	Half-life (h)	Affinity	Motif Score	Specificity	Time (s)
2KXQ	0	130	0.5343	0.8607	0.2882	2.2298	0.6373	0.9528	0.8556	—
	10	116	0.8017	0.787	0.3000	3.5262	0.7565	0.9719	0.7786	69.73
	20	111	0.7683	0.7279	0.3113	4.4267	0.8055	0.9690	0.7882	150.94
	30	108	0.7626	0.7036	0.3180	5.3392	0.8106	0.9705	0.7837	257.37
	50	99	0.7541	0.6664	0.3330	6.7532	0.8037	0.9699	0.7928	530.83
8EYA	0	114	0.7220	0.8249	0.2410	4.2260	0.6285	0.9392	0.9967	—

Appendix

Table S5: Sampling-step ablation for peptidomimetic binder design on 2KXQ and 8EYA. For each target, the row with #-steps

=0

reports the known binder (baseline), and the remaining rows report averages over 100 pCoMole-generated sequences under different numbers of sampling steps. Length is the number of tokenized SMILES tokens. We also report the average wall-clock time (seconds) to generate one sequence for each step setting.

PDB	#-rollouts	Length	Non-Toxicity	Solubility	Permeability	Half-life (h)	Affinity	Motif Score	Specificity	Time (s)
2QSC	0	111	0.8861	0.7156	0.2874	1.7210	0.6775	0.9323	0.9621	—
	5	87	0.7084	0.6840	0.2940	5.9379	0.7745	0.9371	0.9470	78.79
	10	86	0.7172	0.6858	0.2974	6.2827	0.7809	0.9287	0.9477	124.57
	20	87	0.7447	0.6938	0.3063	6.7057	0.7826	0.9335	0.9464	216.92
	30	84	0.7175	0.6967	0.3042	7.1078	0.7756	0.9294	0.9462	306.83
7ZPY	0	84	0.4621	0.7744	0.2787	2.5357	0.5218	0.3083	0.9954	—

Appendix

Table S6: Rollout ablation for peptidomimetic binder design on 2QSC and 7ZPY. For each target, the row with #-rollouts

=0

reports the known binder (baseline), and the remaining rows report averages over 100 pCoMole-generated sequences under different numbers of rollouts. Length is the number of tokenized SMILES tokens. We also report the average wall-clock time (seconds) to generate one sequence for each rollout setting.

PDB	#-candidates	Length	Non-Toxicity	Solubility	Permeability	Half-life (h)	Affinity	Motif Score	Specificity	Time (s)
2O9V	0	65	0.0826	0.8305	0.2977	2.7590	0.4128	0.9671	0.8806	—
	10	47	0.6998	0.7226	0.4626	16.5440	0.6143	0.9692	0.8327	41.42
	20	45	0.7330	0.7345	0.4633	17.5760	0.6241	0.9646	0.8309	76.76
	30	46	0.7511	0.7323	0.4574	19.6885	0.6278	0.9658	0.8339	58.32
	50	47	0.7532	0.7379	0.4541	20.1591	0.6242	0.9683	0.8312	163.55
8PFT	0	143	0.6156	0.7538	0.185	6.1210	0.5750	0.6579	0.9502	—

Appendix

Table S7: Number of candidates ablation for peptidomimetic binder design on 2O9V and 8PFT. For each target, the row with #-candidates

=0

reports the known binder (baseline), and the remaining rows report averages over 100 pCoMole-generated sequences under different numbers of candidates. Length is the number of tokenized SMILES tokens. We also report the average wall-clock time (seconds) to generate one sequence for each candidate-count setting.

Ortholog	PAM guidance	Protein2PAM exact match	Protein2PAM PWM sim.	CICERO exact match	CICERO PWM sim.
St3Cas9	Enabled	100 [87, 100]	0.896	88 [70, 96]	0.837
St3Cas9	Disabled	16 [6, 35]	0.634	4 [1, 20]	0.758
GeoCas9	Enabled	100 [87, 100]	0.792	72 [52, 86]	0.766
GeoCas9	Disabled	88 [70, 96]	0.392	20 [9, 39]	0.619

Appendix

Table S8: Held-out CICERO evaluation of pCoMole-generated Cas9 designs. We evaluated 25 designs per ortholog and condition. The guidance-disabled control removes the PAM objective and hard PAM constraint while keeping the remaining pCoMole settings fixed. Exact-match values are percentages with Wilson 95% confidence intervals; PWM similarity compares the full predicted PAM distributions.

Method	Length Diff.	Avg. Cas9 Likelihood	PAM Match %
pCoMole w/ Cas9 Edit Flows	984 $\longrightarrow$ 934	0.95	100
pCoMole w/ UniRef Edit Flows	984 $\longrightarrow$ 934	0.98	100
SCISOR	984 $\longrightarrow$ 934	0.82	35
RayGun	984 $\longrightarrow$ 934	0.0	0

Appendix

Table S9: Comparison of pCoMole to SCISOR and RayGun at shrinking CjCas9. All three methods were used to shrink WT CjCas9 to a sequence length of 934.

PDB	Length Change	$\Delta$ Non-Toxicity	$\Delta$ Solubility	$\Delta$ Permeability	$\Delta$ Half-life (h)	$\Delta$ Affinity	$\Delta$ Motif Score	$\Delta$ Specificity
1AYC	59 $\rightarrow$ 38	-0.1288	+0.0163	+0.1436	+12.8558	+0.1001	+0.0268	-0.0386
1B8Q	51 $\rightarrow$ 33	-0.2324	+0.0456	+0.1512	+33.3370	-0.0643	+0.0024	-0.0389
1DDV	41 $\rightarrow$ 28	+0.3973	+0.0293	+0.1945	+31.1299	+0.2025	-0.0214	+0.0098
1E6I	43 $\rightarrow$ 29	-0.1412	+0.1025	+0.1748	+23.5607	+0.1515	+0.0547	-0.0537
2LTV	85 $\rightarrow$ 64	+0.3128	-0.0909	+0.0584	+9.6710	+0.1178	+0.1034	-0.0978
2Q8Y	72 $\rightarrow$ 49	-0.0639	-0.0591	+0.0971	+18.0023	+0.2046	+0.0122	-0.0437

Appendix

Table S10: pCoMole designs short peptidomimetic binders for 15 PDB targets with known peptide binders, achieving substantial length reduction and improved predicted properties. For each target, we report the average scores over 100 designed peptidomimetics and the property score change (

\Delta

) relative to the pre-existing binder. Length is the number of SMILES tokens after tokenization. All designed sequences satisfy the specified constraints, and positive

\Delta

indicates improvement.

Figure S1: Distribution of average pLDDT scores across a set of 1000 sequences generated by the UniRef S Edit Flow, compared to the 1000 true Uniref sequences used as inputs. 1000 UniRef sequences with length $\leq 350$ were sampled and used as $x_{0}$ for these generations.

Figure S2: Objective guidance controls optical trade-offs while preserving GFP structure under shrinkage. Representative pCoMole-designed GFP variants generated under four objective-guidance settings: (A) brightness, excitation alignment, and length reduction (B) brightness and length reduction (C) excitation alignment and length reduction (D) length reduction only. We present their secondary structures together with AlphaFold3-predicted pTM scores, sequence length, predicted brightness, and excitation alignment error.

Panel of Figure 21 — Figure S3: Complex structures of known peptide binders and pCoMole-designed peptidomimetic binders with three target PDB proteins (A) 1AYC (B) 1B8Q (C) 1DDV. Binders are shown in yellow, target proteins in light blue, and the dark-blue surface highlights the target motifs used by pCoMole to enforce motif-specific binding during design. Insets zoom into the binding interface to illustrate key contacts. Predicted property scores for each binder are reported.

Panel of Figure 22 — Figure S4: Complex structures of known peptide binders and pCoMole-designed peptidomimetic binders with three target PDB proteins (D) 1E6I (E) 2LTV (F) 2Q8Y. Binders are shown in yellow, target proteins in light blue, and the dark-blue surface highlights the target motifs used by pCoMole to enforce motif-specific binding during design. Insets zoom into the binding interface to illustrate key contacts. Predicted property scores for each binder are reported.

Panel of Figure 23 — Figure S5: Complex structures of known peptide binders and pCoMole-designed peptidomimetic binders with three target PDB proteins (G) 4GNE (H) 5AZ8 (I) 6MLC. Binders are shown in yellow, target proteins in light blue, and the dark-blue surface highlights the target motifs used by pCoMole to enforce motif-specific binding during design. Insets zoom into the binding interface to illustrate key contacts. Predicted property scores for each binder are reported.

Panel of Figure 24 — Figure S6: Complex structures of known peptide binders and pCoMole-designed peptidomimetic binders with three target PDB proteins (J) 7JVS (K) 7LUL (L) 8CN1. Binders are shown in yellow, target proteins in light blue, and the dark-blue surface highlights the target motifs used by pCoMole to enforce motif-specific binding during design. Insets zoom into the binding interface to illustrate key contacts. Predicted property scores for each binder are reported.

Figure S7: pCoMole effectively shrinks a diverse set of Cas9 orthologs. Representative pCoMole-designed Cas9 sequences generated by shrinking four well-characterized Cas9 orthologs: (A) SpCas9, (B) St1Cas9 (C) St3Cas9 (D) GeoCas9. We present their secondary structures together with AlphaFold3-predicted pTM scores, sequence length, Cas9-likelihood score, and predicted PAM.

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