PhyloSOLID: Robust phylogeny reconstruction from single-cell data despite inherent error and sparsity

15 While lineage tracing based on somatic mutations in single -cell sequencing data offers a 16 powerful approach to reconstructing cellular histories in vivo, its reliability is fundamentally 17 limited by pervasive technical artifacts—specifically, high error rates and data sparsity. These 18 issues introduce false phylogenetic signals that corrupt tree topology and lead to spurious 19 evolutionary conclusions. To overcome these limitations, we present PhyloSOLID, a 20 phylogenetic algorithm designed to be inherently robust to these data imperfections. 21 PhyloSOLID employs a progressive scaffolding strategy that begins with graph -based 22 construction of a low -resolution, high-confidence backbone tree from reliable and uniformly 23 covered mutations. This scaffold is then refined through the iterative integration of remaining 24 data, guided by a Bayesian statistical model that penalizes phylogenetic inconsistencies to 25 effectively separate the true evolutionary signal from technical artifacts. Benchmarking on both 26 simulated datasets and multiple ground-truth datasets demonstrates that PhyloSOLID achieves 27 superior lineage reconstruction accuracy over existing methods, for both single -cell RNA-seq 28 and DNA -seq data. Additionally, a user -friendly web interface enables customized quality 29 assessment, artifact removal, and interpretation of lineage structures. PhyloSOLID provides a 30 powerful solution for decoding cellular evolution in developmental and disease contexts. 31 Main: 32 Somatic mutations serve as natural recorders of cellular lineage, providing a powerful means 33 to reconstruct developmental and evolutionary trajectories in vivo1–3. The advent of single-cell 34 sequencing has brought this promise into sharper focus by enabling the resolution of mutational 35 histories at the cellular level4,5. However, the technical challenges inherent to these methods—36 including significant noise and extreme data sparsity—often obscure true biological signals. In 37 single-cell whole-genome sequencing (scWGS), whole -genome amplification can introduce 38 pervasive amplification artifacts6. In single-cell RNA sequencing (scRNA-seq), technical noise 39 from library preparation and sequencing is compounded by biological processes such as RNA 40 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 2 editing7. A universal challenge is allele dropout (ADO), where the signal from one allele is lost 41 due to stochastic sampling, uneven coverage, or amplification bias. In scRNA -seq, this is 42 further confounded by transcriptional dropout 8, which occurs when a gene or an allele is not 43 expressed in a cell, preventing the detection of its underlying genotype. These technical 44 challenges lead to high error rates in somatic mutation calling and lineage tracing, especially 45 when detecting mosaic mutations in non -cancerous cells, even with the most advanced 46 detection pipelines9. 47 Despite the high error rates in somatic mutation detection, most existing phylogenetic tree 48 construction methods for single -cell data incorporate these noisy sites without proper 49 identification, filtering, or consideration during tree construction 10–13. Additionally, most 50 current methods fail to account for critical issues including allele dropout, extreme data sparsity, 51 and cell -type-specific gene expression biases 14–17. Consequently, the resulting phylogenetic 52 trees are often inaccurate, distorting lineage relationships and complicating the interpretation 53 of cellular evolutionary history (Fig. 1a -b). Given the importance of precise phylogeny 54 reconstruction for understanding developmental processes and disease progression, there is a 55 pressing need for error -tolerant methods that can effectively address the noise and technical 56 artifacts inherent in single-cell sequencing data. 57 To address this critical methodological gap, we developed PhyloSOLID, a robust phylogenetic 58 reconstruction algorithm designed for single -cell somatic mutation data. Unlike conventional 59 methods, PhyloSOLID incorporates several key innovations (Fig. 1c, Supplementary Fig. 1, 60 Methods): (1) It learns the properties of artifact sites, germline variants, and true somatic 61 mutations from multiple cross-validated, real-world data, excluding most germline and artifact 62 sites prior to phylogeny construction (Supplementary Fig. 1, Supplementary Table 1)18–23. (2) 63 It calculates genotyping posterior probabilities while accounting for allele imbalance issues. (3) 64 It identifies correlated clones and mutations using a graph -based method, by explicitly 65 modeling the inherent noise in the data (Supplementary Fig. 1b-e. (4) It first constructs a 66 backbone tree of high-confidence clones supported by mutations from uniformly covered 67 regions and then places remaining mutations onto this scaffold using a Bayesian penalty 68 approach (Methods)24, preventing tree structure distortion from cell -type-specific gene 69 expressions. (5) It refines the final topology by correcting misplaced variants, removing likely 70 false-positive mutations and filtering out suspected doublet cells. (6) It provides an interactive 71 website for user -customized quality control of the tree structure. Together, these features 72 enable PhyloSOLID to overcome the inherent challenges of single -cell data, such as noise, 73 sparsity and allele dropouts. 74 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 3 75 Fig. 1: Development and benchmarking of PhyloSOLID. a-b, Single-cell phylogenetic tree 76 structure can be distorted by several factors: germline variants misclassified as somatic 77 mutations, erroneous mutation calls, allelic/depth dropout, and the presence of doublet cells. c, 78 PhyloSOLID workflow. The PhyloSOLID framework takes a list of candidate somatic 79 mutations and raw sequencing data as input. It then reconstructs an accurate phylogenetic tree 80 by explicitly accounting for the major sources of error that distort tree structure. See Methods 81 for details. d, Benchmarking phylogenetic inference with varying false -positive levels. 82 PhyloSOLID and other tools were evaluated on a scDNA -seq dataset containing a controlled 83 gradient of artifactual and germline variants (false positives). Phylogenetic accuracy was 84 quantified using the normalized Robinson-Foulds (RF) distance. PhISCS execution failed at a 85 90% false positive rate. e, Tanglegram comparison at 50% false-positive rate. Compared with 86 other tools, t he tree inferred by PhyloSOLID remains highly congruent with the reference 87 topology, as shown by the high node correspondence. f, Resolving distinct clones from sparse 88 data. A sparse simulated cell-by-mutation matrix was generated, comprising two major clones 89 (red and blue cells, rows). PhyloSOLID was benchmarked against other tools and correctly 90 partitioned the cells into two independent clones. g, across six single-cell datasets with 91 established ground -truth tree structures, PhyloSOLID consistently achieved superior 92 phylogenetic accuracy. h, Accurate phylogenies from scRNA -seq data. Phylogenetic trees 93 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 4 inferred by PhyloSOLID from somatic mutations in scRNA-seq data are highly congruent with 94 the ground-truth phylogeny defined by CRISPR barcode mutations. In the LUAD-2 sample, no 95 clone-specific mosaic mutations of C1-3 were detected. Instead, we identified a founding clone 96 (Cm) that gave rise to the three descendent clones. “ncells” represents the number of cells. 97 98 To evaluate performance under noise and data sparsity, we first used a scDNA-seq dataset18 (22 99 cells; 78 somatic mutations) and simulated varying noise levels by introducing artificial false-100 positive sites (Methods, Supplementary Table 2). PhyloSOLID consistently achieved superior 101 accuracy, as measured by the normalized Robinson-Foulds (RF) distance25 to the ground-truth 102 tree (Methods, Fig. 1d). Notably, even at a 90% false-positive rate, PhyloSOLID maintained a 103 normalized RF distance below 0.1, with a representative example illustrated in Fig. 1e. To 104 further test robustness under realistic data sparsity , we used a second simulated dataset 105 comprising 50 mutations across 1,761 cells with a high dropout rate of ~90% (Supplementary 106 Table 3). Here, PhyloSOLID accurately reconstructed the ground-truth monoclonal phylogeny 107 (Fig. 1f, Supplementary Fig. 2), whereas other widely used tools (e.g., HUNTREE, CellPhy, 108 BCITE, ScisTree) erroneously inferred polyclonality. 109 After compiling a benchmark of eight ground truth datasets —comprising four scDNAseq 110 benchmarks post whole -genome amplification 18,19, two single -cell colony benchmarks with 111 published phylogenies20,21, and two scRNAseq benchmarks with knock -in CRISPR barcode 112 mutations26—we further evaluated the performance of PhyloSOLID against eight state-of-the-113 art methods ( e.g., HUNTREE, CellPhy, ScisTree). Across all the scDNA -seq benchmarks, 114 PhyloSOLID consistently and significantly outperformed other methods (Methods, Fig. 1f, 115 Supplementary Fig. 3). 116 Notably, on two scRNA-seq benchmarks comprising over a thousand cells each, PhyloSOLID 117 correctly assigned 94.03% and 96.24% of mutant cells to their original clones, as validated by 118 the CRISPR barcode trees, achieving high phylogenetic accuracy despite the inherent noise 119 and sparsity. In contrast, other tools exhibited high clone misassignment rates, ranging from 120 ~30% to 80% (Fig. 1g, Supplementary Fig. 4-6). Remarkably, using only somatic mutations, 121 PhyloSOLID achieved phylogenetic resolution comparable to CRISPR-based barcoding across 122 the scRNA-seq benchmarks tested. This result demonstrates that our method successfully 123 unlocks the potential of somatic mutations for in vivo lineage tracing. Furthermore, the 124 projection of cell lineages onto the gene expression UMAP revealed distinct transcriptional 125 profiles for each clone ( Supplementary Fig. 6), demonstrating that lineage history is coupled 126 with changes in cell state and fate. 127 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 5 128 Fig. 2: Advanced visualization and user -friendly interface . a-c, A suite of tailored tree 129 layouts designed to intuitively integrate and visualize multiple layers of data —such as per -130 clade mutation burden, cell -type annotations, mutation profiles, and highlighting specific 131 mutations or cells —in a clear and publication -ready format. d, A standard rectangular tree 132 layout is also provided, offering flexibility to accommodate diverse user preferences. e, 133 Interactive quality control and data exploration. A circular phylogenetic layout sorts and 134 displays shared mutations across concentric rings, enabling direct visual comparison of input 135 and output genotypes for each cell. This is integrated into an interactive web interface that 136 allows users to perform point -and-click operations for in -depth analysis, such as inspecting 137 read-level evidence, flagging potential false -positive mutations, and annotating doublet cells. 138 An example of an "orphaned mutation" (black triangle) is illustrated here. This variant shares 139 only one mutant cell with its assigned mother mutation yet causes many false -negative allele 140 flips. This pattern suggests the lack of a true clonal mother mutation, likely resulting from a 141 sequencing error, a doublet cell, or an independent mutation misassigned to this lineage. 142 143 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 6 Phylogenetic trees reconstructed from single-cell data often involve large numbers of cells and 144 mutations, along with additional layers of information such as genotype calls and cell -type 145 annotations. These trees can be highly complex, making well -designed visualization schemes 146 essential for meaningful interpretation. However, existing phylogenetic tools typically offer 147 limited or inflexible visualization options —ill-suited to diverse scenarios such as trees with 148 few mutations across many cells, or vice versa —and often require advanced coding skills for 149 customization27. Furthermore, even robust methods like PhyloSOLID are not immune to 150 limitations; residual false -positive mutations or doublet cells may still be incorporated and 151 distort tree topology. Manual inspection and quality assessment are therefore crucial for 152 accurate biological interpretation. Yet most current approaches deliver only a basic tree 153 structure15,28, leaving users without adequate support for systematic quality evaluation. 154 To meet the demand for advanced visualization, we developed a suite of tailored tree 155 visualization schemes (Fig. 2a –c). Each is designed to present multiple data layers —such as 156 mutation number, cell -type annotations, and mutational profiles—in a clear, efficient, and 157 publication-ready format. A rectangular tree layout is also provided to accommodate different 158 user preferences (Fig. 2d). Beyond these general layouts, we developed specialized 159 visualizations to suit different data structures and biological questions. These include trees that 160 integrate single -cell mutation burdens, shared variants, and cell -type annotations (Fig. 2a); 161 representations optimized for large numbers of mutations across few cells, with optional 162 mutational signature overlay (Fig. 2b); and layouts tailored to large cell sets that emphasize a 163 limited set of key driver mutations (Fig. 2c). Together, these options allow users to adapt tree 164 visualization to their specific dataset and analytical goals. 165 For streamlined phylogeny quality control, we introduced a circular layout in which shared 166 mutations are sorted and displayed across concentric rings, enabling direct comparison of input 167 and output genotypes per cell (Fig. 2 e). We provide an interactive web interface that enables 168 users to perform a range of exploratory and quality-control operations. For example, users can 169 visually inspect read-level evidence for mutations, flag potential false-positive sites, and mark 170 questionable mutations and likely doublet cells —all through an intuitive, point -and-click 171 environment. This interactive system greatly lowers the barrier to rigorous phylogenetic quality 172 assessment, helping ensure robust interpretation of lineage relationships. 173 The PhyloSOLID source code is being prepared for public release. It will be available on 174 GitHub ( https://github.com/douymLab/PhyloSOLID) by March 1, 2026, or upon the 175 manuscript's formal acceptance, whichever comes first. An interactive website is accessible 176 online at https://phylosolid.westlake.edu.cn. 177

178 The rapid expansion of single-cell sequencing has established somatic mutation and barcode -179 based lineage tracing as a central technique for decoding cellular fate, development, and disease. 180 However, the intrinsic noisiness and sparsity of single -cell data—including pervasive allele 181 dropout and low mutation -calling accuracy —pose a fundamental challenge to reliable 182 phylogenetic inference. When unaddressed, these technical artifacts can be misinterpreted as 183 biological signal, leading to incorrect reconstructions of cellular relationships and, 184 consequently, erroneous biological conclusions. Here, we introduced PhyloSOLID, a 185 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 7 computational framework designed to be inherently robust to these pervasive data 186 imperfections. 187 The core strength of PhyloSOLID lies in its progressive scaffolding strategy, which prioritizes 188 high-confidence phylogenetic signals before integrating more ambiguous data. By first 189 constructing a robust backbone from reliable mutations and then iteratively refining it with a 190 Bayesian model that penalizes phylogenetic inconsistencies, our method effectively 191 disentangles true evolutionary history from technical noise. Extensive benchmarking on 192 datasets with known ground truth demonstrated that PhyloSOLID consistently outperforms 193 existing methods in reconstruction accuracy across both scRNA -seq and scDNA -seq 194 modalities. Its performance on large -scale scRNA-seq datasets was particularly noteworthy, 195 where phylogenetic trees built solely from somatic mutations recapitulated the clonal structure 196 defined by independent CRISPR barcoding with remarkably high accuracy . This result 197 underscores that somatic mutations, when analyzed with a sufficiently robust phylogenetic 198 framework, can provide resolution for lineage tracing that rivals or even surpasses engineered 199 barcoding approaches. Beyond constructing phylogenies from somatic mutations, 200 PhyloSOLID is a flexible toolkit that can also leverage barcode mutations for phylogenetic 201 inference. 202 Looking forward, the need for reliable single cell phylogenetics will become ever more 203 pressing. PhyloSOLID meets this need by providing a statistically rigorous, error -aware 204 framework for lineage reconstruction. To maximize its accessibility and utility for the broader 205 research community, we have complemented the core algorithm with a user -friendly web 206 interface for quality assessment, artifact removal, and lineage interpretation. We anticipate that 207 PhyloSOLID will become an essential tool, empowering researchers to extract trustworthy 208 evolutionary insights from the complex and noisy landscape of single -cell data and thereby 209 ensuring that the burgeoning field of cellular lineage tracing is built upon a foundation of 210 computational rigor. 211 Data availability 212 All scWGS and scRNA-seq data used in this study are publicly available under the following 213 accession codes: dbGaP: phs001485.v3.p1, SRA: SRA053195, NDA: NDAR#2330, EGA: 214 EGAD00001007032, and GEO: GSE144239, GSE234814 and GSE161363. 215 Code availability: 216 PhyloSOLID is implemented in Python and R and is licensed under the MIT License. The 217 source code, documentation and examples are available on GitHub at 218 https://github.com/douymLab/PhyloSOLID. The PhyloSOLID source code is being prepared 219 for public release. It will be available on GitHub by March 1, 2026, or upon the manuscript's 220 formal acceptance, whichever comes first. 221

This work was supported by the Westlake Laboratory of Life Sciences 222 and Biomedicine (Hangzhou 310024, Zhejiang, China), under the grant "Key R&D Program 223 of Zhejiang" (2024SSYS0032) as well as the National Natural Science Foundation of China 224 (32270682) to Y.D. We thank the High-Performance Computing Center for technical support. 225 We gratefully acknowledge the creators and submitters of the datasets used in this study. Data 226 used in this study under the dbGaP accession phs001485.v3.p1 was generated by Drs. 227 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 8 Christopher A. Walsh and Peter J. Park with funding from NINDS grant R01NS032457. The 228 public datasets were obtained from dbGaP at http://www.ncbi.nlm.nih.gov/gap, EGA at 229 https://ega-archive.org/, NDA at https://nda.nih.gov/ and GEO at 230 https://www.ncbi.nlm.nih.gov/geo/. 231 Author contribution: Y.Dou conceived and supervised the project and secured the funding. 232 N.L. supervised the website construction. Q.Y. and Y.L. both made significant contributions 233 to the project, with Q.Y. developed the software and accomplished bioinformatics analysis, 234 and Y.L. constructed the website with help from J. Y. X.W., Y.X, Z.Y., J.L, Y.Z., J.L., M.Y., 235 Y.Du and H.L. assisted in single cell genotyping , variant detection, and Y. X. assisted in 236 visualization of phylogenies . Y.Dou and Q.Y. wrote the manuscript. All authors carefully 237 reviewed and approved the final manuscript. 238 Inclusion & Ethics statement : This study was conducted in accordance with ethical 239 guidelines and principles. It did not involve human participant or animal subjects. There are no 240 sequencing data generated in this study. The authors ensure that the study adheres to the highest 241 ethical standards in research, data generation, and usage. 242 Competing interests: The authors declare no competing interests. 243 244 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 9

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353 The PhyloSOLID framework 354 1. Pre-phylogeny genotyping and variant filtering 355 We collected eight single-cell benchmarks, comprising six scDNA-seq benchmarks with 356 orthogonally validated ground-truth mutations (Supplementary Table 4) and two scRNA-seq 357 benchmarks with ground-truth CRISPR barcoding trees (Supplementary Table 5-6), from 358 public sources. Mutations were genotyped using MosaicSC24, a Bayesian approach that our lab 359 have developed, accounting for single -cell-specific challenges such as allelic imbalance, 360 sequencing errors, and population allele frequencies. Following genotyping, we applied a 361 logistic regression classifier trained with an " leave-one-donor-out" approach to filter artifacts 362 and germline variants. This model distinguished true mosaic mutations based on 10 (for 363 scDNA-seq) or 14 (for scRNA -seq) features derived from raw sequencing reads, including 364 mismatch frequency, base quality, read alignment position, etc. (Supplementary Table 1). We 365 further excluded mutations with a low mutant-allele fraction (VAF 0.1% in gnomAD. For scRNA-seq-derived mutations, we further excluded thos e 367 mapping to known sites in three classical RNA editing databases29–31 or residing within allele-368 specific expressed (ASE) genes 32, those with significantly imbalanced base‑substitution 369 profiles, and those with low UMI consistency. Of note, when BAM fil es are unavailable, 370 PhyloSOLID offers a practical alternative by constructing phylogenies directly from a cell-by-371 mutation matrix. 372 2. Leveraging clonal architecture to further filter germline variants 373 To distinguish true somatic mutations from germline variants with allele dropout in single-cell 374 DNA sequencing data, we leverage the fundamental principle that genuine somatic mutations 375 exhibit structured patterns of co -occurrence and mutual exclusivity reflective of underlying 376 clonal architecture. Specifically, mutations belonging to the same subclone tend to co -occur 377 within the same group of cells, while those originating from distinct, mutually exclusive 378 subclones are distributed across different cell populations. In contrast, germline variants 379 affected by stochastic allele dropout demonstrate no systematic association with clonal groups; 380 instead, they appear randomly distributed across cells and show no consistent correlation with 381 other mutations, resulting in a nonspecific and biologically incoherent pattern. 382 2.1) Identification of mutations with co-occurrence patterns 383 Let 𝑛 and 𝑚 be the number of cells and mutations retained after initial filtering, respectively. 384 The final matrix used for analysis is 𝐼 = {0, 1, 𝑁𝐴}!×#, where 𝑁𝐴 values are excluded from 385 calculations. This matrix is derived from three input matrices: 386 • 𝑃$,& ∈ [0,1]!×#: the matrix of somatic posterior probabilities, where 𝑃$,& represents the 387 probability that the somatic mutation 𝑗 is present in cell 𝑖 . Somatic mutation 388 likelihoods were calculated based on a beta-binomial sampling model. 389 • 𝑀$,& ∈ [0,1]!×#: the mutant allele frequency matrix, where 𝑀$,& represents the mutant 390 allele frequency of somatic mutation 𝑗 in cell 𝑖. 391 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 14 • 𝐶$,& ∈ ℤ'( !×# : the sequencing co verage matrix, where 𝐶$,& represents the number of 392 reads (coverage depth) at the locus of mutation 𝑗 in cell 𝑖. 393 For each mutation 𝑗 in each cell 𝑖, an entry in the binary matrix is defined as: 394 𝐼$,& = 5 1 𝑖𝑓 𝑀$,& > 0 ⋀ 𝑃$,& > 0.5 0 𝑖𝑓 𝑀$,& = 0 ⋀ 𝑃$,& ≤ 0.5 ⋀ 𝐶$,& ≥ 1 𝑁𝐴 𝑖𝑓 𝐶$,& = 0 (1) 395 For each mutation 𝑗, the set of cells in which it was detected was defined as: 396 𝒮(𝑗) = A𝑖 ∈ {1,2, … , 𝑛}D 𝐼$,& = 1E (2) 397 Mutation processing order: To ensure robust phylogenetic reconstruction, mutations are 398 processed in descending order of their cellular prevalence | 𝒮(𝑗)|, with higher -confidence 399 mutations processed first. This sorting strategy prioritizes mutations with stronger phylogenetic 400 signals for initial tree construction. 401 For any pair of mutations 𝑗) and 𝑗*, the following contingency counts were computed: 402 𝑁)) (𝑗) , 𝑗*) = F 𝟏(𝐼$,&! = 1 ⋀ 𝐼$,&" = 1) ! $+) 403 𝑁)( (𝑗) , 𝑗*) = F 𝟏(𝐼$,&! = 1 ⋀ 𝐼$,&" = 0) ! $+) 404 𝑁() (𝑗) , 𝑗*) = F 𝟏(𝐼$,&! = 0 ⋀ 𝐼$,&" = 1) ! $+) 405 𝑁(( (𝑗) , 𝑗*) = F 𝟏H𝐼$,&! = 0 ⋀ 𝐼$,&" = 0I ! $+) (3) 406 where 𝟏(∙) was the indicator function, returning 1 if its argument is true and 0 otherwise. 407 We applied the Jaccard index to measure the similarity between the sets of cells containing 408 each mutation: 409 𝐽(𝑗) , 𝑗*) = 𝑁)) (𝑗) , 𝑗*) 𝑁)) (𝑗) , 𝑗*) + 𝑁)( (𝑗) , 𝑗*) + 𝑁() (𝑗) , 𝑗*) (4) 410 Jaccard index was selected for scRNAseq data because this metric ignores double -negative 411 cells ( 𝑁(( ), which is suitable for sparse scRNAseq data. However, this measure alone is 412 insufficient for identifying parent -child clonal relationships, as th e Jaccard index between a 413 parent and its subclone may not necessarily be high. To specifically capture such hierarchical 414 relationships, we introduced an additional metric defined as follows: 415 𝑓(𝑗) , 𝑗*) = 𝑁)) (𝑗) , 𝑗*) |𝒮(𝑗) )| (5) 416 𝑓(𝑗*, 𝑗) ) = 𝑁)) (𝑗) , 𝑗*) |𝒮(𝑗*)| (6) 417 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 15 The Two mutations above, 𝑗) and 𝑗*, were considered correlated if they satisfied either of the 418 following criteria: 419 • 𝑁)) (𝑗) , 𝑗*) > 0 ∧ 𝐽(𝑗) , 𝑗*) ≥ 0.08, 420 • 𝑁)) (𝑗) , 𝑗*) > 0 ∧ 0 < 𝐽(𝑗) , 𝑗*) < 0.08 ∧ max (𝑓(𝑗) , 𝑗*), 𝑓(𝑗*, 𝑗) )) ≥ 0.5. 421 2.2) Exclusion of likely germline variants 422 To systematically identify and exclude germline variants misclassified as somatic mutations, 423 we employed an exhaustive search approach that ev aluates the patterns of co -occurrence and 424 mutual exclusivity across all mutations. 425 A “founder mutation” is defined as a genetic alteration that initiates a clonal expansion and is 426 subsequently present in all cells of that clone and all its subclones. To systematically identify 427 such mutations and their associated clonal groups, we proceed as follows: 428 For each mutation 𝑟 ∈ {1, … , 𝑚} with a mutant cell fraction (MCF) >5%, we consider it as a 429 candidate founder mutation. For each such candidate 𝑟, we identify the set of mutations 𝒥, 430 that are significantly correlated with 𝑟, as described in Section 2.1. 431 The underlying rationale is that if 𝑟 is a true founder mutation, the set 𝒥, likely represents 432 subsequent mutations that occurred within the clone founded by 𝑟, and thus should exhibit a 433 strong pattern of co-occurrence with 𝑟. 434 2.2.1) Genotype imputation of the “founder mutation” 435 In theory, these correlated clones descend from a single founder clone. Consequently, all 436 daughter clones are expected to inherit the founder's mutations. Assuming 𝑟 as the founder 437 mutation of these correlated clones, we created a combined muta tion set that includes both 𝑟 438 and all mutations significantly correlated with it: 439 𝒥,- = {𝑟} ∪ 𝒥, (7) 440 For each cell, we then quantified 𝑞$, the number of mutations from the combined set 𝒥,- that 441 are present in that cell: 442 𝑞$ = F 𝐼$,& &∈𝒥#$ (8) 443 Where: 444 • 𝑞$: The number of mutations from the set 𝒥,- that are present in cell 𝑖. 445 Impute founder status: A cell is classified as carrying the founder mutation 𝑟 if it possesses at 446 least two mutations from the set 𝒥,-. This threshold helps mitigate false negatives caused by 447 allele dropout or sequencing errors affecting a single mutation. 448 𝒰, = A𝑖 ∈ {1, … , 𝑛}DH𝐼$,& = 1I ∨ (𝑞$ ≥ 2)E (9) 449 Where: 450 • 𝒰,: The set of cells inferred to belong to the clone founded by mutation 𝑟. 451 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 16 Let 𝒰,____ denote the complement of 𝒰, with respect to covered cells for any mutation 𝑗, i.e., 452 𝒰,____ = {𝑖 ∉ 𝒰,, 𝐶$,& ≥ 1}. 453 For each mutation 𝑗 ∈ 𝒥, -, we calculated the total number of mutant cells outside the infe rred 454 founder clone: 455 𝑁&,, 0123$45 = DA𝑖 ∈ 𝒞&\𝒰,ED (10) 456 Where: 457 • 𝒞6 denotes the set of j-mutant cells. 458 Since all cells carrying mutation 𝑗 ∈ 𝒥, - are descendants of the founder clone, no mutant cells 459 should exist outside of it. We then defined a false -positive penalty score for mutation 𝑗 by 460 normalizing 𝑁&,, 0123$45 by the total number of 𝑗-mutant cells: 461 𝑆&,, 78 = 𝑁&,, 0123$45 D𝒞&D (11) 462 If r and 𝑗 are both real mosaic mutations, the n 𝑆&,, 78 will be low. Conversely, a high score 463 suggests otherwise. 464 2.2.2) Enumeration over all candidate founder mutations and identification of likely 465 germline variations 466 The above procedure (Sections 2.2.1 and 2.2.2) was repeated for each mutation 𝑟 ∈ {1, … , 𝑚}. 467 For each mutation r, we calculate a summary leakage score: (cv\std\mean, fig) 468 𝑆,78eeeeee⃗ = 1 |𝒥, -| F 𝑆&,, 78 &∈𝒥#$ (12) 469 This score represents the average phylogenetic inconsistency of all mutations in the clonal set 470 𝒥,- founder by 𝑟. 471 To identify germline variants, all candidate mutations 𝑟 are sorted in descending order of 𝑆,78. 472 Germline variants form a distinct group at the top of this sorted list. We identify the cutoff 473 point by calculating differences between consecutive sorted scores: 474 ∆9= 𝑠9 − 𝑠9-) (13) 475 where 𝑠9 represents the 𝑙-th highest score in the sorted sequence. Mutations preceding a large 476 drop in scores (∆9> 𝜇∆ + 2𝜎∆, where 𝜇∆ and 𝜎∆ are the mean and standar d deviation of all 477 differences) are classified as germline. 478 These identified germline variants are designated as ℳ;52, placed at the root of the clonal tree, 479 and excluded from subsequent lineage reconstruction. This approach effectively separated 480 germline variants from real mosaic mutations. 481 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 17 3. Scaffold tree construction using uniformly covered mutations 482 Following the exclusion of germline mutations, we proceed to identify high -confidence 483 somatic mutations located in genomic regions with ubiquitous expression and uniform 484 coverage. These mutations serve as backbone features for initial phylogenetic reconstruction. 485 3.1) Initial filtration based on genotype posterior and mutant reads count 486 We begin with the matrices defined in Section 2.1 and apply additional quality filters to ensure 487 reliable detection of ubiquitously expressed mutations. 488 Cell Filtering: 489 The set of retained cells is defined as those containing sufficient mutation evidence: 490 𝒞 = A𝑖 ∈ {1,2, … , 𝑛}DH∃𝑗 𝑀$,& > 0I ∧ H∃𝑗 𝑃$,& > 0.5IE (14) 491 where 𝑀 and 𝑃 are the mutant allele frequency and somatic posterior probability matrices. 492 Mutation filtration: 493 For each mutation 𝑗, we compute its average mutant allele frequency (MAF) across cells where 494 it is detected. The set of cells where mutation 𝑗 is detected is defined as: 495 𝒞& = A𝑖 ∈ 𝒞D𝑀$,& > 0 ∧ 𝑃$,& > 0.5 E (15) 496 The average MAF for mutation 𝑗 is: 497 𝑀<___ = 1 D𝒞& D F 𝑀$,& $∈𝒞% (16) 498 The set of retained mutations is: 499 ℳ = o𝑗 ∈ {1,2, … , 𝑚}p q𝑚𝑎𝑥 $ 𝑀$,& ≥ 0.3t ∧ H𝑀<___ ≥ 0.1Iu (17) 500 The filtered matrices are then: 501 𝑃ℳ,𝒞 = 𝑃[ℳ, 𝒞] = A𝑃$,&D𝑖 ∈ 𝒞, 𝑗 ∈ ℳE (18) 502 𝑀ℳ,𝒞 = 𝑀[ℳ, 𝒞] = A𝑀$,& D𝑖 ∈ 𝒞, 𝑗 ∈ ℳE (19) 503 504 3.2) Selection of uniformly covered mutations 505 To identify high -confidence mutations within ubiquitously expr essed genomic regions, we 506 applied two complementary coverage-based filters. 507 For mutation 𝑗 in cell type 𝑡, the proportion of cells lacking sequencing coverage is denoted 508 as: 509 𝑝?@ (&,2) = DA𝑖 ∈ 𝒮2|𝐶$,& = 0ED 𝑁2 (20) 510 Where: 511 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 18 • 𝒮2 denotes the set of all cells that belong to cell type 𝑡; 512 • 𝐶$,& denotes the sequencing coverage for mutation 𝑗 in cell 𝑖; 513 • 𝑖 denotes a cell 𝑖; 514 • 𝑁2 denotes the total number of cells of cell type 𝑡. 515 For each mutation 𝑗, we define 𝑁& as the number of cell types in which a sufficient proportion 516 of cells exhibit sequencing coverage: 517 𝑁& = F 1 x𝑝?@ (&,2) ≤ 95%z 2∈𝒮& (21) 518 Where: 519 • 𝑝?@ (&) = D𝑖E𝐶$,& = 0F ? : NA proportion for mutation 𝑗 across all cells. 520 • 𝑝?@ (&,2): NA proportion for mutation 𝑗 in cell type 𝑡; 521 Based on measurements from real-world single cell sequencing data, mutations are retained for 522 subsequent analysis as candidate scaffold mutations if they meet: 523 • Present in >10% cells across all cells when th ere’s only one single dominant cell type, or 524 no cell type annotation information is available. 525 • Present in ≥2 cell types (𝑁& ≥ 2). 526 For each mutation 𝑗, we first required that the median sequencing depth across all cells is at 527 least 1x. 528 We then calculate d the coefficient of variation (CV) to assess coverage uniformity. For 529 mutation 𝑗 in cell 𝑖, we defined a binarized coverage indicator: 530 𝐵$,& = |1, 𝑖𝑓 𝐶$,& ≥ 1 0, 𝑖𝑓 𝐶$,& < 1 (22) 531 Where: 532 • 𝐶$,& indicates the read coverage in cell 𝑖 at the genome location of mutation 𝑗; 533 𝐵$,& is the binary indicator of whether mutation 𝑗 in cell 𝑖 has sufficient read coverage. 534 For each mutation 𝑗, the mean read coverage along with the standard deviation, and coefficient 535 of variation are computed as follows: 536 𝐶𝑉& = 𝜎& 𝐵<~ = 1 𝑁 ∑ H𝐵$,& − 𝐵<~ I * $ 1 𝑁 ∑ 𝐵$,&$ (23) 537 Where: 538 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 19 • 𝐶$,& denotes the sequencing coverage for mutation 𝑗 in cell 𝑖; 539 • 𝑁 is the total number of cells. 540 • 𝐵<~ is the mean of 𝐵$,& across all cells for mutation 𝑗; 541 • 𝜎& is the standard deviation of 𝐵$,& across all cells for mutation 𝑗; 542 • 𝐶𝑉& is the coefficient of variation for the coverage indicator of mutation 𝑗. 543 Based on the measurements in real-world data, only mutations with 𝐶𝑉& < 6 were retained as 544 candidate scaffold mutations. 545 3.3) Noise-robust quantification of pairwise mutation correlations 546 Theoretically, two somatic mutations in an individual can only have one of four phylogenetic 547 relationships: A is ancestral to B, B is ancestral to A, they co-occur on the same node, or they 548 arise on independent branches. Consequently, if two mutations share an ancestral relationship 549 or occur on the same node, they are supposed to co-exist in at least a subset of cells. In contrast, 550 mutations on independent branches are not expected to to -exist in any single cell. Therefore, 551 the co-occurrence patterns of mutations across single cells can be used to identify correlated 552 mutations. 553 However, single cell sequencing data are inherently noisy. Sequencing errors and the presence 554 of doublet cells can obscure the true phylogenetic relationships between mutations. To establish 555 robust co-occurrence patterns while mitigating stochastic effects, we developed a consensus 556 approach based on ensemble clone identification through randomized runs. 557 Beginning from each scaffold mutation 𝑗 ∈ 𝑀G, we sequentially evaluated its correlation with 558 every other scaffold mutation using the method described in Section 2.1. Mutations found to 559 be correlated with 𝑗 were assigned to the same clonal population with 𝑗. This process was 560 repeated iteratively until all mutations were assigned to some clone. To ensure robustness, the 561 scaffold mutation order was randomly shuffled, and the entire procedure was repeated 100 562 times. 563 The occurrence frequency of each distinct clone 𝐶 across 100 runs were then calculated: 564 𝑓& (𝐶) = 1 100 F 𝟏 q𝐶 ∈ 𝐶,1! (&) t )(( $25,+) (24) 565 Only clones with 𝑓&(𝐶) > 0.1 were retained, denote as 𝐶& ∗. The weight for each clone 𝐶 to 566 exist was then computed as: 567 Ω(𝐶) = F 𝑓& (𝐶) ∗ 𝟏H𝐶 ∈ 𝐶& ∗I &∈ℳ' (25) 568 For each clone with |𝐶| > 2, we extracted all H|J| * I unordered mutations pairs. Each pair 569 (𝑗, 𝑗K) inherited weight Ω(𝐶) from its parent clone. The consensus weight for each distinct 570 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 20 mutation pair was obtained by summing contributions from a ll clones containing both 571 mutations: 572 𝜔(𝑗, 𝑗K) = F Ω(𝐶) J∈⋃ J% ∗% &,&)∈J (26) 573 For singleton clones where |𝐶| = 1, although they do not contribute to edge weights, their 574 weights Ω(𝐶) are retained and provide valuable information about mutation independence. 575 Mutations with high singleton weights typically exhibit weak correlations with other mutations 576 and are more likely to form distinct clonal groups in subsequent clustering analysis. 577 We constructed a weighted graph 𝐺M0!35!313 = (𝒱, ℰ, 𝒲) where: 578 • 𝒱 = ℳG: vertices represent mutations. 579 • ℰ = {(𝑗, 𝑗K)|𝜔(𝑗, 𝑗K) > 0}: edges connect co-occurring mutations. 580 • 𝒲(𝑒) = 𝜔(𝑗, 𝑗K): edge weights reflect co-occurrence strength. 581 This graph serves as the input for subsequent community detection. In this way, the relationship 582 between each mutation pair was robustly quantified, making the approach resistant to stochastic 583 noise. 584 3.4) Partitioning of scaffold mutations into clonal groups via weighted graph clustering 585 The weighted consensus correlation graph 𝐺M0!35!313 was partitioned into maximal mutation 586 groups using the Leiden community detection algorithm. This approach identifies 587 phylogenetically coherent groups of mutations that exhibit strong internal correlations and 588 minimal external correlations, with each group defining the mutational signature of a distinct 589 clonal population. 590 The algorithm produces a complete partition of the scaffold mutation set 𝒢 = {𝑔) , 𝑔*, … , 𝑔N }, 591 where ⋃ 𝑔O = ℳG N O+) and 𝑔O ∩ 𝑔O) = ∅ for 𝑘 ≠ 𝑘K. Each mutation group 𝑔O defines the 592 characteristic mutational profile of a putative cellular subpopulation, representing clones that 593 likely evolved through distinct evolutionary trajectories. 594 From each phylogenetically coherent mutation gr oup 𝑔O ∈ 𝒢, we selected a single backbone 595 mutation to serve as the phylogenetic anchor and operational founder of that clonal group. 596 The founder mutation of a clone is expected to be an early event, present in the vast majority 597 of cells within that clonal lineage. Therefore, for each group 𝑔O, we selected the mutation with 598 the highest Mutant Cell Fraction (MCF), defined as the fraction of cells with adequate 599 sequencing coverage that harbor the mutation: 600 𝑗O ∗ = 𝑎𝑟𝑔 max &∈P* 𝑀𝐶𝐹(𝑗) (27) 601 where 𝑀𝐶𝐹(𝑗) = |𝒮(&)| |J%| . Here, 𝒮(𝑗) = A𝑖 ∈ 𝒞GD𝐼$,& G = 1E is the set of scaffold cells supporting 602 mutation 𝑗, and |𝐶&| denotes the total number of scaffold cells with non -zero sequencing 603 coverage at locus 𝑗. 604 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 21 The stringent, coverage-based filtration applied to define the scaffold set ℳG (Section 3.3) 605 ensures that all candidate mutations have low genome -wide missing data rates ( 𝑝?@ (&) ≤ 90%) 606 or are robustly detected across multiple cell types (𝑁& ≥ 2). This pre-processing step mitigates 607 the risk of selecting mutations with spuzriously high MCF due to technical artifacts, thereby 608 guaranteeing that a high MCF is a reliable indicator of high cellular prevalence. 609 The final set of backbone mutations is defined as the union of all group representatives: 610 ℬ = {𝑗) ∗, 𝑗* ∗, … , 𝑗N ∗ } (28) 611 These backbone mutations provide a robust set of phylogenetically informative markers for the 612 initial scaffold tree construction. This selection provides the foundational framework for the 613 phylogenetic tree. Notably, the designation of an initial founder mutation is operational for the 614 initial tree building and may be subject to re-evaluation during subsequent steps when placing 615 additional mutations, allowing for the integration of more complex phylogenetic signals. 616 Following the selection of backbone mutations as phylogenetic anchors, we define the 617 corresponding backbone clones that constitute the fundamental phylogenetic units in our 618 reconstruction framework. 619 Definition 3.3.4.1 (Backbone Clone). For each backbone mutation 𝑗O ∗ ∈ ℬ representing group 620 𝑔O, the corresponding backbone clone 𝐶O ∗ is defined as the set of scaffold cells exhibiting the 621 characteristic mutation pattern of that phylogenetic lineage: 622 𝐶O ∗ = o𝑖 ∈ 𝐶Gp𝐼$,&* ∗G = 1u 623 To robustly assign cells to backbone clones while accounting for missing data and technical 624 dropouts, we employ a group -wise imputation strategy. For each backbone mutation 𝑗O ∗, an 625 imputed clone vector 𝑉O is constructed by integrating phylogenetic signals from all mutations 626 within group 𝑔O: 627 𝑉O (𝑖) = 1( F 𝐼$,&* ∗G > 0 &∈P* ) 628 where 1(∙) is the indicator function. This approach enhances detection sensitivity by 629 leveraging phylogenetic coherence within each group. 630 Cells may exhi bit mutation patterns consistent with multiple backbone clones due to 631 phylogenetic relationships or technical artifacts. To resolve such ambiguities, a maximum -632 support assignment strategy is applied. Let 𝒜(𝑖) = {𝑘|𝑉O(𝑖) = 1} denote the set of candidate 633 backbone clones for cell 𝑖. The final assignment is determined by: 634 𝑘∗(𝑖) = ⎩⎪⎨ ⎪⎧ ∅ 𝑖𝑓 |𝒜(𝑖)| = 0 𝑘 𝑖𝑓 |𝒜(𝑖)| = 1 𝑎𝑟𝑔 max O∈𝒜($) F 𝐼$,& G &∈P* 𝑖𝑓 |𝒜(𝑖)| > 1 635 In case of ties, the backbone mutation with the highest expression level in the cell is selected: 636 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 22 𝑘∗(𝑖) = 𝑎𝑟𝑔 max O∈𝒜&+,($) 𝐸$,&* ∗ 637 where 𝐸$,& denotes the expression level of mutation 𝑗 in cell 𝑖, and 𝒜2$5 (𝑖) is the set of 638 clones with equal maximal support. 639 The backbone clone matrix 𝐵M90!5 is then constructed as: 640 𝐵M90!5 = o1 𝑖𝑓 𝑘∗(𝑖) = 𝑘 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 641 This matrix provides a robust binary representation of clonal membership, forming the basis 642 for subsequent phylogenetic tree construction. The assignment procedure ensures that each cell 643 is assigned to at most one backbone clone, leveraging phylogenetic coherence to enhance 644 robustness against technical noise while resolving ambiguities using biologically meaningful 645 criteria. 646 647 4. A Bayesian penalty approach to place left mutations onto the scaffold tree 648 The scaffold tree reconstruction proceeds iteratively, starting from the backbone tree 𝑇R 649 containing only the backbone mutations ℬ . Each non -backbone mutation 𝑎 ∈ ℳG ∖ ℬ is 650 sequentially integrated into the current tree 𝑇, which evolves during the reconstruction process. 651 For each candidate mutation 𝑎, we evaluate its phylogenetic relationship to every possible 652 placement position within 𝑇 by computing a binary discordance penalty based on co -653 occurrence patterns and a comprehensive Bayesian penalty that accounts for phylogenetic 654 structure and uncertainty. 655 4.1) Identification of candidate nodes to place the mutation 656 For a given non -backbone mutation 𝑎, we comprehensively evaluate all phylogenetically 657 permissible placement positions within the current tree 𝑇. Candidate positions 𝑏 ∈ 𝑇 includes: 658 (i) any existing node in 𝑇; (ii) a new node introduced along any existing branch, thereby 659 splitting it into two segments; (iii) a new parent node unifying two or more descendants from 660 an existing node; or (iv) a new leaf node attached to any existing node. This exhaustive 661 consideration ensures that all potential evolutionary relationships —including those requiring 662 expansion of the tree topology—are evaluated. 663 To systematically manage these placement options, we employ an anchor-based approach. For 664 each candidate node 𝑏 ∈ 𝑇 (which represents a distinct branch point or phylogenetic position 665 within the tree) and non -backbone mutation 𝑎 ∈ ℳG ∖ ℬ , we compute the standard 666 contingency counts 𝑁)) (𝑏, 𝑎), 𝑁)( (𝑏, 𝑎), 𝑁() (𝑏, 𝑎) and 𝑁(( (𝑏, 𝑎) as defined in Equation 3. 667 To account for uncertainty in mutation calls derived from posterior probabilities 𝑃$,&, we define 668 for each cell in these contingency categories a minimum and maximum possible penalty 669 contribution, based on whether the observed posterior is close to the 0.5 threshold or 670 confidently assigned. 671 The total binary discordance penalty for the pair (𝑎, 𝑏) is defined as: 672 𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U (𝑏, 𝑎) = +𝜔)( ∗ 𝑁)( (𝑏, 𝑎) + 𝜔() ∗ 𝑁() (𝑏, 𝑎) (29) 673 where the weights are assigned as follows: 674 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 23 • 𝜔)( = 1: Full penalty for each cell in 𝑁)( , representing a potential false positive 675 (mutation 𝑎 observed but one expected under position 𝑏; 676 • 𝜔() = 0.1: A reduced penalty for each cell in 𝑁() , representing a potential false 677 negative (mutation 𝑎 not observed but expected under position 𝑏), reflecting the 678 higher empirical rate of false negatives in single-cell data. 679 For each contingency count, we compute the minimum and maximum possible penalty per cell, 680 considering the uncertainty in posterior probability values around the 0.5 threshold: 681 • For a cell in 𝑁)( (𝑏, 𝑎): the penalty contribution ranges from 0.5 (if 𝑃$,& is just 682 above 0.5) to 1 (if 𝑃$,& = 1). 683 • For a cell in 𝑁() (𝑏, 𝑎): the penalty contribution ranges from 0.05 (if 𝑃$,& is just 684 below 0.5) to 0.1 (if 𝑃$,& = 0). 685 Thus, the overall minimum and maximum penalties for the pair (𝑏, 𝑎) are: 686 𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U #$! (𝑏, 𝑎) = 1 ∗ 0.5 ∗ 𝑁)( (𝑏, 𝑎) + 0.1 ∗ 0.5 ∗ 𝑁() (𝑏, 𝑎) 687 𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U #TV (𝑏, 𝑎) = 1 ∗ 1 ∗ 𝑁)( (𝑏, 𝑎) + 0.1 ∗ 1 ∗ 𝑁() (𝑏, 𝑎) (30) 688 To identify candidate anchors, we compare the penalty intervals across all possible 𝑏 ∈ 𝑇. We 689 first identify the mutation 𝑏∗ with the smallest minimum penalty: 690 𝑏∗ = 𝑎𝑟𝑔 min W∈X 𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U #$! (𝑏, 𝑎) (31) 691 The candidate set is then constructed by including all positions 𝑏 for which the minimum 692 penalty is no greater than the maximum penalty of this optimal candidate: 693 ℳT!M;0, = A𝑏 ∈ 𝑇D𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U #$! (𝑏∗, 𝑎) ≤ 𝑃𝑒𝑛𝑎𝑙𝑡𝑦S$!T,U #TV (𝑏∗, 𝑎)E (32) 694 This approach ensures that we consider not only the position with the absolute smallest penalty, 695 but also any other positions whose minimum penalty falls within the uncertainty range of the 696 best candidate. 697 For each candidate anchor 𝑏 ∈ ℳT!M;0,, we consider two primary phylogenetic hypotheses: 698 • Placement on position 𝑏 : implying an identical phylogenetic origin ( 𝑎 = 𝑏 ), 699 suggesting that observed differences (𝑁)( , 𝑁() ) are due to sequencing errors. 700 • Placement as a direct descendant of position 𝑏 : implying a parent -child 701 relationship (𝑎 ⊂ 𝑏), where 𝑎 is a subclone of 𝑏. 702 Additionally, placements that introduce new nodes along the branch leading to 𝑏, or that unify 703 𝑏 with other siblings under a new parent node, are evaluated through an extension of this 704 anchor-based framework, leveraging the same penalty structure to ensure phylogenetic 705 consistency. 706 4.2) A Bayesian approach to penalization 707 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 24 We applied a Bayesian approach to place new mutations onto the scaffold phylogenetic tree. 708 The optimal placement was identified by maximizing the posterior probability of the 709 mutation’s location, conditional on the scaffold tree structure and the observed data. This is 710 formally equivalent to minimizing a penalty function defined as the negative log -posterior 711 probability. The Bayesian function is as follows: 712 713 𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇, 𝐷) = 𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇) ∗ 𝑃(𝐷|𝑇, 𝑁𝑜𝑑𝑒 = 𝑏) 𝑃(𝐷|𝑇) (33) 714 Where: 715 • 𝑁𝑜𝑑𝑒: a candidate phylogenetic position in tree 𝑇 where mutation might be 716 placed. 717 • 𝐷 = {𝑃) ,T, 𝑃*,T, … , 𝑃!,T}: the somatic posterior probabilities vector for mutation 718 across all cells. 719 • 𝑃$,T: probability that mutation 𝑎 is present in cell 𝑖. 720 Taking the negative logarithm yields the penalty function: 721 𝑃𝑒𝑛𝑎𝑙𝑡𝑦(𝑎, 𝑏) = −log𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇, 𝐷) (34) 722 Expanding this expression: 723 𝑃𝑒𝑛𝑎𝑙𝑡𝑦(𝑎, 𝑏) = −log𝑃(𝐷|𝑇, 𝑁𝑜𝑑𝑒 = 𝑏) − log𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇) + log(𝐷|𝑇) (35) 724 Since 𝑃(𝐷|𝑇) is constant across candidate placements for the current tree: 725 𝑃𝑒𝑛𝑎𝑙𝑡𝑦(𝑎, 𝑏) ∝ −log𝑃(𝐷|𝑇, 𝑁𝑜𝑑𝑒 = 𝑏) − log𝑃H(𝑁𝑜𝑑𝑒 = 𝑏|𝑇)I (36) 726 Assuming a uniform prior over 𝑁!0453 possible placement nodes in the current tree 𝑇: 727 𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇) = 1 𝑁!0453 (37) 728 −log𝑃(𝑁𝑜𝑑𝑒 = 𝑏|𝑇) = log𝑁!0453 (38) 729 The data likelihood factors over cells based on the current tree structure: 730 𝑃(𝐷|𝑇, 𝑁𝑜𝑑𝑒 = 𝑏) = ¨ 𝑃H𝐷$,TD𝐺$,T I ! $+) (39) 731 − log 𝑃(𝐷|𝑇, 𝑁𝑜𝑑𝑒 = 𝑏) = F log 𝑃H𝐷$,TD𝐺$,TI ! $+) (40) 732 Where: 733 • 𝐺$,T: expected genotype (0 or 1) given placement at 𝑁𝑜𝑑𝑒S; 734 • 𝐷$,T: observed data (posterior probability 𝑃$,T). 735 4.3) Mutation placement by minimizing the penalty function 736 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 25 The theoretical framework translates to the practical penalty function: 737 𝑃𝑒𝑛𝑎𝑙𝑡𝑦(𝑎, 𝑏) = Fx𝜔7? ∗ 𝑁( →) ($) + 𝜔78 ∗ 𝑁) →( ($) + 𝜔?@→) (T) ∗ 𝑁?@→) ($) + 𝜔?@→( (T) ∗ 𝑁?@→( ($) z ? $+) + log 𝑁!0453 (41) 738 Binary discordance indicators (0 or 1): 739 𝑁( →) ($) = 𝟏(𝐺$,T = 1 ⋀ 𝐼$,T = 0) 740 𝑁) →( ($) = 𝟏(𝐺$,T = 0 ⋀ 𝐼$,T = 1) 741 𝑁?@→) ($) = 𝟏(𝐺$,T = 1 ⋀ 𝐶$,T = 0) 742 𝑁?@→( ($) = 𝟏H𝐺$,T = 0 ⋀ 𝐶$,T = 0I (42) 743 Where: 744 • Each indicator function 𝟏(∙) returns 1 if the specified condition is true, 0 745 otherwise. 746 • 𝑁( →) ($) : indicates if cell 𝑖 has a false negative (expected mutant but observed 747 reference). 748 • 𝑁) →( ($) : indicates if cell 𝑖 has a false positive (expected reference but observed 749 mutant). 750 • 𝑁?@→) ($) : indicates if cell 𝑖 has missing data imputed as mutant. 751 • 𝑁?@→( ($) : indicates if cell 𝑖 has missing data imputed as reference. 752 Weight computation from posterior matrix 𝑃: 753 • 𝜔78 = − log(𝑃$,T): penalty for false positives. 754 • 𝜔7? = 0.1 ∗ (− logH1 − 𝑃$,TI): penalty for false negatives, scaled by empirical 755 error ratio. 756 • 𝜔?@→) (T) = − log(𝜔?@ ∗ !! (.) !0 (.)-!! (.)): penalty for imputing NA as mutant. 757 • 𝜔?@→( (T) = − log(𝜔?@ ∗ !0 (.) !0 (.)-!! (.)): penalty for imputing NA as reference. 758 Empirical error ratio incorporation: 759 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 26 Based on extensive analysis of single -cell phylogenetic trees from real datasets, we observed 760 that false negative errors occur approximately 10 times more frequently than false positive 761 errors. This empirical observation ( 𝑓𝑛𝑓𝑝_𝑟𝑎𝑡𝑖𝑜 ≈ 0.1) is incorporated by scaling the false 762 negative penalty, ensuring the penalty function reflects actual error patterns in single cell 763 sequencing data. Both penalties maintain their log-probability interpretation while accounting 764 for real-world error distributions. 765 Where: 766 • 𝑛) (T) = |{𝑖: 𝐼$,T = 1}|: number of cells with observed mutant allele. 767 • 𝑛( (T) = |{𝑖: 𝐼$,T = 0}|: number of cells with observed reference allele. 768 • 𝜔?@ = 0.001: global dropout weight for typical datasets. 769 Additionally, we incorporate a complexity penalty based on the Bayesian Information Criterion 770 (BIC) to avoid overfitting: 771 𝐵𝐼𝐶85!T92U (𝑏, 𝑎) = 𝜑 ∙ log 𝑛 (43) 772 𝑃𝑒𝑛𝑎𝑙𝑡𝑦202T9 (𝑏, 𝑎) = 𝑃𝑒𝑛𝑎𝑙𝑡𝑦(𝑏, 𝑎) + 𝐵𝐼𝐶85!T92U (𝑏, 𝑎) (44) 773 Where: 774 • 𝜑: increase in number of parameters when adding 𝑎; 775 • 𝜆: regularization parameter for complexity control. 776 777 The mutation 𝑎 is placed at the candidate node 𝑏∗ that minimizes the total penalty: 778 𝑏∗ = 𝑎𝑟𝑔 min S∈ℳ.1234# 𝑃𝑒𝑛𝑎𝑙𝑡𝑦202T9 (𝑏, 𝑎) (45) 779 If ℳT!M;0, = ∅, the mutation is placed at the root of the tree, subject to a root-specific penalty 780 calculation. 781 After processing all non -backbone mutations through this iterative placement procedure, the 782 resulting tree structure represents the complete scaffold tree 𝑇G , containing all scaffold 783 mutations at their phylogenetically consistent positions. 784 The scaffold tree 𝑇G provides a robust evolutionary framework grounded in high -confidence 785 mutations from ubiquitously expressed regions, serving as the foundation for subsequent 786 phylogenetic analysis and placement of additional mutations. 787 5. Prune & replace likely artifacts and reconstruct the tree 788 Following initial tree construction, each mutation was evaluated using five quality metrics to 789 identify potential misplacements: 790 a) The false-positive flip count of the mutation itself, 791 b) The false-positive flip count of its correlated mutations, 792 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 27 c) The false-negative flip number of the mutation itself, 793 d) The false-negative flip number of its mother mutations in its assigned subclone, 794 e) The number of mutant cells carrying the mutation withing its subclone. 795 Based on these metrics, mutations were refined as follows: 796 Misplaced clusters: If a mutation, along with its correlated mutations, exhibited an excessive 797 number of false-positive flips (a + b > 0.2), the entire group was considered misplaced. In this 798 case, the cluster was pruned and then re -integrated into the phylogeny at a more plausible 799 location using the Bayesian penalty approach mentioned. 800 Relocation of orphaned mutations : If a mutation displayed a high false -negative flip count 801 (c > 2) concurrent with a low mutant cell count in its subclone (e < 2), it was deemed to lack a 802 valid phylogenetic parent. Such mutations were relocated to an independent clonal branch. 803 Mutations that failed to meet the quality thresholds after one iteration of refinement were 804 discarded from the final phylogeny. This process was repeated iteratively 1 -5 times until the 805 tree topology stabilized. 806 Removal of likely doublet cells: Finally, likely doublet cells contain too many false -positive 807 flips ratio (> 0.2) were excluded from the final tree. 808 Benchmarking of PhyloSOLID 809 For the benchmark, we used real-world single-cell whole-genome sequencing (WGS) data and 810 WGS data from single-cell colonies18–21. The ground-truth mutation lists and phylogenies for 811 these datasets were sourced from their original publications. Germline variants and artifact 812 mutations were identified by randomly sampling reads from the raw BAM files 813 using mpileup (Supplementary Table 4). The second simulated dataset used in Fig. 1f was 814 generated by modeling the allele dropout ratios observed in a realistic scRNA-seq data. 815 For the real -world single-cell RNA-seq data, candidate mosaic mutations were called using 816 MosaicSC24, and the ground -truth CRISPR -barcode phylogenies were obtained from the 817 corresponding original papers (Supplementary Table 5-6). 818 Phylogenetic trees were constructed from both simulated and real -world single -cell data 819 using eight software tools, including HUNTRESS, CellPhy, ScisTree, ScisTree2, BSCITE, 820 SiFit, PhISCS and MPBoot, and compared against PhyloSOLID. Default parameters were used 821 for all tools. The tanglegrams and comparison plots between the barcode trees and mutation 822 trees generated by each method were produced using R packages dendextend33 (v1.17.1) and 823 phangorn34 (v2.11.1). 824 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 28 825 Supplementary Fig. 1: Framework of PhyloSOLID . a, Overall pipeline. See Methods for 826 details. b-c, Principal component analysis (PCA) of extracted sequence features distinguishes 827 somatic mutations from germline variants and technical artifacts. In the resulting PCA space, 828 true somatic mutations form a distinct cluster, separable from both germline variants and 829 artifacts. d-e, Application of a logistic regression classifier to discriminate somatic mutations. 830 Receiver operating characteristic (ROC) curves demonstrate high classification accuracy on 831 both scDNA -seq and scRNA -seq benchmark datasets . f-g, Identification of correlated 832 backbone mutations via a leaden graph. Representative leiden graphs derived from scDNA-seq 833 (f) and scRNA -seq ( g) data are shown. Benchmark dataset details: UMB1465, UMB4638, 834 UMB4643: scDNA-seq data after whole -genome amplification (sample IDs from Lodato et 835 al., Science, 2018). DB6, DB9, DB10: Single-cell colony data (sample IDs from Bae et 836 al., Science, 2018, and Park et al., Nature, 2021, respectively). LUAD-1 and LUAD -2 837 correspond to the original sample IDs 100k and 10k, respectively (Quinn et al., Science, 2021). 838 cSCC_1 (original sample ID: P6_cSCC) and normal_skin_1 (original sample ID: P6_normal) 839 are from Andrew et al. , Cell, 2020 . normal_breast_1 (original sample ID: hbca_n03) and 840 normal_breast_2 (original sample ID: hbca_c11) are from Tapsi et al., Nature, 2023. 841 842 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 29 843 844 Supplementary Fig. 2: Benchmarking PhyloSOLID on simulated data with high sparsity. 845 The ability of each method to recover the true monoclonal structure under high dropout 846 conditions is assessed. For each panel, phylogeny generated by each software was shown on 847 the left and simulated mutation matrix was shown on the right . Rows correspond to cells and 848 columns to simulated mutations. Mutant cells are shown in red, reference-homozygous (refhom) 849 sites in light blue, and sites with no coverage in white. Execution of PhiSCS, CellPhy, 850 ScisTree2 and MPBoot failed on this dataset. 851 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 30 852 Supplementary Fig. 3: Benchmarking PhyloSOLID on scDNA -seq data . Ground-truth 853 phylogeny (leftmost tree, from published data) and the corresponding phylogenies inferred by 854 each software. Each row represents a cell. Tanglegrams were shown to visualize their 855 agreement with the ground-truth tree (the left-most tree). Benchmark dataset details: UMB1465, 856 UMB4638, UMB4643: scDNA-seq data after whole-genome amplification (sample IDs from 857 Lodato et al., Science, 2018). TNBC16: scDNA-seq data after whole -genome amplification 858 (sample ID from Wang et al., Nature, 2014). S316, DB10: Single-cell colony data (sample IDs 859 from Bae et al., Science, 2018, and Park et al., Nature, 2021, respectively). 860 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 31 861 Supplementary Fig. 4: Benchmarking phylogeny reconstruction tools on scRNA-seq data 862 (LUAD-1) with CRISPR barcode ground truth . For each evaluated tool, the output is 863 organized as follows: (Left) Phylogeny inferred from mosaic mutations called from 864 scRNA-seq data. (Middle left) Cell-by-mutation matrix. Rows: cells; columns: mutations. 865 Colors denote mutant cells (red), reference -homozygous sites (light blue), and sites with no 866 coverage (white). (Middle right) Barcode-by-cell matrix. Execution of PhiSCS failed on this 867 dataset. Each column represents one CRISPR barcode mutation. (Right) Published 868 ground-truth clonal structure based on CRISPR barcodes. Benchmark dataset details: the 869 original sample ID of LUAD-1 is “100k” (Quinn et al., Science, 2021). 870 871 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 32 872 Supplementary Fig. 5: Benchmarking phylogeny reconstruction tools on scRNA-seq data 873 with CRISPR barcode ground truth (LUAD-2). For each evaluated tool, the output is 874 organized as follows: (Left) Phylogeny inferred from mosaic mutations called from 875 scRNA-seq data. (Middle left) Cell-by-mutation matrix. Rows: cells; columns: mutations. 876 Colors denote mutant cells (red), reference -homozygous sites (light blue), and sites with no 877 coverage (white). (Middle right) Barcode-by-cell matrix. Execution of PhiSCS failed on this 878 dataset. Each column represents one CRISPR barcode mutation. (Right) Published 879 ground-truth clonal structure based on CRISPR barcodes. Benchmark dataset details: the 880 original sample ID of LUAD-2 is “10k” (Quinn et al., Science, 2021). 881 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint 33 882 Supplementary Fig. 6: Benchmarking PhyloSOLID on scRNA -seq data with CRISPR 883 barcode ground truth. a, b, Analysis of two lung cancer scRNA-seq datasets. For each dataset: 884 (Left) Phylogeny reconstructed from mosaic mutations identified in the scRNA -seq data. 885 (Middle left) Cell-by-mutation matrix. Rows represent cells, columns represent mutations. 886 Mutant cells are shown in red, reference -homozygous (refhom) sites in light blue, and sites 887 with no coverage in white. (Middle right) Barcode-by-cell matrix. Each column corresponds 888 to one CRISPR barcode mutation. (Right) Ground-truth clonal structure derived from the 889 published CRISPR barcode data. c, d, UMAP visualization of cells colored by their assigned 890 clones. Benchmark dataset details: the original sample IDs of LUAD-1 and LUAD-2 are “100k” 891 and “10k” (Quinn et al., Science, 2021). 892 reuse, remix, or adapt this material for any purpose without crediting the original authors. share,this preprint (which was not certified by peer review) in the Public Domain. It is no longer restricted by copyright. Anyone can legally The copyright holder has placedthis version posted February 6, 2026. ; https://doi.org/10.64898/2026.02.04.703905doi: bioRxiv preprint