{"paper_id":"0095f4da-9f6a-4660-b496-fbaf3daeeaf1","body_text":"Deciphering Cell Cycle Dynamics and Cell States in Single-cell RNA-1 \nseq data with SPAE 2 \n 3 \nJiahao Yi (伊嘉豪) 1,# , Jiajia Liu (刘佳佳) 2,#, Peng Guo (郭鹏) 1, Yuan-Nong Ye (叶远浓) 1,*, 4 \nXiaobo Zhou (周小波) 2,3,4,* 5 \n 6 \n1Bioinformatics and Biomedical Big Data Mining Laboratory, Department of Medical Informatics, 7 \nSchool of Biology and Engineering, Guizhou Medical University, Anshun, Guizhou 561100, China 8 \n2Center for Computational Systems Medicine, McWilliams School of Biomedical Informatics, The 9 \nUniversity of Texas Health Science Center at Houston, Houston,TX 77030, USA 10 \n3McGovern Medical School, The University of Texas Health Science Center at Houston, Houston, 11 \nTX 77030, USA 12 \n4School of Dentistry, The University of Texas Health Science Center at Houston, Houston, TX 13 \n77030, USA 14 \n 15 \n# These authors contributed equally to this work 16 \n* Corresponding author. 17 \nE-mail: yyn@gmc.edu.cn (Y uan-Nong Ye), Xiaobo.Zhou@uth.tmc.edu (Xiaobo Zhou). 18 \n19 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nAbstract 20 \nRapid advances in single-cell RNA sequencing (scRNA-seq) technology have enabled 21 \nthe investigation of gene expression changes at the single -cell level, particularly for 22 \nelucidating the heterogeneity among cells and complex biological processes.  This 23 \ntechnique reveals subtle molecular differences within individual cells, thereby offering 24 \na unique viewpoint for the investigation of cell cycle progression, cellular 25 \ndifferentiation, and  disease pathogenesis. However, accurately identifying and 26 \nanalyzing cell cycle dynamics in scRNA-seq data remains challenging due to the 27 \ncomplexity of the data and the subtle differences between cell states. To address this 28 \nchallenge, we developed the integrated Sinusoidal and Piecewise AutoEncoder (SPAE), 29 \nan autoencoder -based piecewise linear model, for characterizing the cell cycle 30 \ndynamics and cell states in scRNA-seq data. Compared with existing methods, SPAE 31 \ndemonstrates substantially improved  accuracy and robustness in cell cycle 32 \ncharacterization. Additionally, SPAE can accurately predict cancer cell cycle transitions 33 \nand effectively facilitate the removal of cell cycle effects  from gene expression data . 34 \nSPAE is available for non-commercial use at https://github.com/YaJahn/SPAE. 35 \n 36 \nKeywords: scRNA-seq; Autoencoder; Cell cycle dynamics; Cell cycle effects  37 \n 38 \nIntroduction 39 \nIn recent years, the field of biomedical research has experienced significant 40 \nadvancements due to the development of single -cell RNA sequencing (scRNA-seq) 41 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ntechnology [1-3]. This technology represents a significant transition from tissue  level 42 \nto single-cell level, providing a unique perspective for a deeper understanding of cell 43 \nheterogeneity [4]. Moreover, scRNA -seq has demonstrated immense potential in 44 \nstudying two fundamental biological processes: the cell cycle and cell differentiation  45 \n[5-7].  46 \nThe cell cycle, a fundamental framework in cellular fate, is essential for 47 \norganismal growth and development. Its stages , G1, S, G2, and M phases , are 48 \ninterconnected by complex molecular events and regulatory networks. Precise 49 \nregulation of these stages is crucial for maintaining tissue homeostasis and enabling 50 \ndevelopmental adaptation [8, 9]. Moreover, it is intricately associated with cell states 51 \nand plays a pivotal role in tumorigenesis.  Consequently, accurate identification and 52 \ncomprehension of cell cycle stages are imperative for the in -depth exploration of 53 \ncellular behaviors [10, 11]. 54 \nscRNA-seq enables precise quantification of gene expression at the individual-cell 55 \nlevel, thereby offering novel and comprehensive insights into the cell cycle  [12, 13]. 56 \nDespite the rich dataset s provided by scRNA -seq for cell cycle analysis, the 57 \ninterpretation of these data presents several challenges. A primary hurdle is the accurate 58 \ninference of specific cell cycle stages from the scRNA -seq data. Traditional 59 \nexperimental methods, though able to identify cell cycle stages, are not only time -60 \nconsuming and labor -intensive but also lack the capability for the quantitative 61 \nmeasurement of cell cycle phase  [14]. Moreover, the technical variability and data 62 \nsparsity inherent to scRNA -seq [15], compounded by the transient and overlapping 63 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nnature of cell cycle stages, significantly complicate the analysis  [16]. This complexity 64 \nunderscores the need for advanced methodologies to accurately interpret cell cycle 65 \ndynamics from scRNA-seq data. 66 \nAddressing the challenges associated with scRNA -seq data interpretation, the 67 \nscientific community has recently introduced several computational methodologies. 68 \nThese encompass supervised machine learning methods, which predominantly utilize 69 \nknown cell cycle genes. For instance, cyclone [5] and the CellCycleScoring function in 70 \nSeurat [17] are notable examples. These methods employ annotated cell cycle genes for 71 \npredicting the cell cycle phases (G1, S, or G2/M) of individual cells. Additionally, 72 \nreCAT [18] is an innovative approach combining the Travelling Salesman Problem and 73 \nHidden Markov Model  to reconstruct cell cycle pseudotime series. However, a 74 \nsignificant limitation of these methods is their dependency on datasets with pre -75 \nannotated cell cycle genes and experimental labels, which constrains their broader 76 \napplication. 77 \nTo circumvent these constraints, unsupervised techniques have been developed.  78 \nCCPE [19] is a representative linear autoencoder-based model for cell cycle analysis. It 79 \nprojects single-cell gene expression profiles into a low -dimensional latent space using 80 \na linear encoder and reconstructs the input with a linear decoder. A helical structure is 81 \nthen fitted in the latent space to capture the continuous cyclic trajectory of the cell cycle. 82 \nWhile CCPE performs well when the data distribution is approximately linear, its linear 83 \nencoder limits its ability to model nonlinear or multi-stage cell cycle transitions, which 84 \nmotivated the development of SPAE. Other unsupervised methods include Cyclum [20] 85 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nand CYCLOPS [21]. Cyclum utilizes an autoencoder model that integrates both linear 86 \nand non-linear elements within the hidden layer, aimed at inferring the pseudotime of 87 \ncellular cyclical processes. Conversely, CYCLOPS employs a linear -projection 88 \nautoencoder, mapping data onto a closed elliptical curve in a low -dimensional space, 89 \noffering a fresh perspective in understanding cell cycle dynamics. However, it ’s 90 \nimportant to note that CYCLOPS, while designed to simulate circadian rhythms, 91 \nincorporates complex operations such as square roots and division in its neural network, 92 \npotentially posing challenges in optimization. Moreover, tools like cyclone and reCAT, 93 \nwhile useful, do not effectively eliminate cell cycle effects from expression data, 94 \nhighlighting the need for continued advancements in this field.  95 \nIn this study, we present a novel computational framework, the Integrated 96 \nSinusoidal and Piecewise AutoEncoder (SPAE), designed to concurrently analyze cell 97 \ncycle dynamics and cell states from single-cell RNA-seq data. The motivation behind 98 \nSPAE arises from the need to more accurately capture both the cyclic and piecewise 99 \nlinear characteristics of cellular processes observed in scRNA -seq data. Existing 100 \nmodels such as CCPE  [19] use a purely linear encoder, which limits their ability to 101 \nrepresent complex nonlinear gene expression trajectories, while models like Cyclum 102 \nrely on sinusoidal transformations, they cannot explicitly distinguish multiple cell states 103 \nthat deviate from a single smooth cycle. To overcome these limitations, SPAE 104 \nintegrates two complementary components: a nonlinear component to represent the 105 \nperiodic nature of the cell cycle and a piecewise linear component to model transitions 106 \nbetween distinct cellular states that often follow locally linear patterns in gene 107 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nexpression space. This combination allows SPAE to more faithfully reconstruct cyclic 108 \nand branching trajectories while simultaneously assigning cells to specific states.  We 109 \nrigorously assessed SPAE ’s efficacy in estimating cell cycle pseudotime and 110 \ndetermining cell stage classifications. Our comparative analysis includes established 111 \nmethodologies such as CCPE, cyclone, Seurat, Cyclum, CYCLOPS, and reCAT. 112 \nFurthermore, we demonstrate SPAE’s utility in predicting cancer cell cycle transitions 113 \nand in mitigating the confounding effects of cell cycle variations. 114 \n       115 \nMethod 116 \nDatasets 117 \nWe utilized datasets presented in Table 1 and Table 2 to assess the performance of 118 \nSPAE. The Quartz -Seq dataset of mouse embryonic stem cells  (mESCs) [22], 119 \nsequenced through Quartz -Seq technology, provided cell cycle stage and gene 120 \nexpression data for 33,412 genes. The H1 human embryonic stem cells  (hESCs) [23] 121 \ndataset utilized a fluorescence ubiquitination -based cell cycle indicator to stage 247 122 \ncells. For the E-MTAB-2805 mESCs dataset [6], 288 mouse embryonic stem cells were 123 \nsequenced using the HiSeq 2000 sequencing system, covering 38,293 genes. To assess 124 \nmodel robustness under different dropout rates, we utilized the E-MTAB-2805 mESCs 125 \ndataset, which initially had a dropout rate of 2 4%. We first applied the MAGIC  [24] 126 \nmodel to impute the missing data, and subsequently introduced artificial dropout rates 127 \nof 0%, 20%, 50%, and 70%.  Additionally, the nutlin-treated multiple cancer cell lines 128 \ndataset [25] included single-cell RNA-seq data from 24 cancer cell lines , treated with 129 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nDMSO or nutlin on the 10× Genomics platform, highlighting cell cycle arrest induced 130 \nby nutlin in cells expressing wild -type TP53. The FELINE Breast Cancer Single -Cell 131 \nGenomics dataset  [26] comprised patients with ER+ breast cancer undergoing 132 \nneoadjuvant endocrine therapy (letrozole) with or without a CDK4/6 inhibitor 133 \n(ribociclib), sampled at the start of treatment, after 14 days, and after 180 days of 134 \ntreatment, using 10× technology for single-nucleus RNA sequencing. 135 \nMoreover, to evaluate SPAE ’s performance in removing cell cycle effects, we 136 \nanalyzed datasets from mouse embryonic stem cell , human myoblasts (hMyo), and 137 \nbreast cancer . The mouse embryonic stem cell dataset  [27] utilized droplet 138 \nmicrofluidics for high-throughput barcoding and RNA sequencing of individual cells. 139 \nThis dataset focused on the impact of leukemia inhibitory factor withdrawal. It included 140 \ndata from different stages of mouse embryonic stem cells  at different stages:  141 \nundifferentiated, and 2, 4, and 7 days post LIF withdrawal. The scRNA -seq data of 142 \nhuman myoblasts  [7], developed by Trapnell et al. (2014), included differentiating 143 \nmyoblasts sampled at various time points, namely at 0, 24, 48, and 72 hours. At 0 hours 144 \n(0h), myoblast cells are actively proliferating and remain undifferentiated, with 145 \ntranscriptional profiles dominated by cell cycle and proliferation -associated genes. By 146 \n24 hours (24h), after switching to a differentiation -induction medium, the cells begin 147 \nexiting the cell cycle and initiating early differentiation programs, marked by 148 \ndownregulation of cell cycle genes and upregulation of early myogenic markers like 149 \nMYOD1. At 48 hours (48h), the cells show active differentiation with increased 150 \nexpression of muscle-specific genes such as MYOG (myogenin), reflecting a transition 151 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nfrom proliferative myoblasts to early muscle precursors. By 72 hours (72h), the 152 \ndifferentiation process is largely complete, with transcriptional profiles dominated by 153 \nlate myogenic markers like MYH genes, indicating mature muscle cells with a 154 \nheterogeneous mix of late differentiation states. These time points correspond to critical 155 \nstages in an experimental setup where the growth medium was changed to an induction 156 \nmedium, triggering the transition of proliferating myoblasts into a differentiation 157 \nprogram. This process promotes the differentiation of myoblasts into more specialized 158 \ncell types, allowing the capture of dynamic transcriptional changes associated with cell 159 \nfate transitions.  Breast cancer dataset  [28] involves high throughput sequencing of 160 \nMDA-MB-231 breast tumor cells, exploring the role of CSL (CBF1/RBP -161 \nJkappa/Suppressor of Hairless/LAG-1) in cancer. It includes four cell types: CSLKO1 162 \nand CSLKO2 (CSL gene knockouts) and WT1 and WT2 (wild-type controls). 163 \n 164 \nSPAE model 165 \nSPAE models the distinct cell states using a piecewise linear regression framework .  166 \nPiecewise linear regression is a modeling approach in which the relationship between 167 \nvariables is represented by multiple linear segments with different slopes across distinct 168 \nregimes [29]. The mathematical formulation, including the continuity constraints and 169 \nobjective function, is detailed in Supplementary Note 1 . In SPAE, we employ an  170 \nautoencoder-based piecewise linear model  in which  the encoder consists of  both 171 \nnonlinear and piecewise linear components. In the nonlinear component, we use a 172 \nstandard multi-layer perceptron with hyperbolic tangent activation functions to map the 173 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ntranscriptome profile 𝑋 to 𝑧𝑐 which represents the pseudotime along the cell cycle 174 \nprocess ( Figure 1). The p iecewise linear component assigned cells into different 175 \nclusters. Suppose we have 𝑘 clusters, the gate function in our piecewise linear model 176 \ndetermined which cluster a cell 𝑥𝑛 belongs to, defined as 177 \n𝑔𝑖(𝑥𝑛) = {𝑥𝑛,   𝑐𝑒𝑙𝑙 𝑛 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝑐𝑙𝑢𝑠𝑡𝑒𝑟 𝑖 \n0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒   , (𝑖 = 1, … , 𝑘)      (1)                                       178 \nThe transformations of the encoder can be represented as 179 \n𝑧𝑛 = [𝑧𝑛\n𝑐\n𝑧𝑛\n𝑝] = [𝑤3\n𝑁𝐿tanh (𝑤2\n𝑁𝐿 tanh(𝑤1\n𝑁𝐿𝑥𝑛 + 𝑏1\n𝑁𝐿) + 𝑏2\n𝑁𝐿)  \n∑ (𝑤𝑖\n𝑝𝑥𝑛 + 𝑏𝑖\n𝑝) ∙ 𝑔𝑖(𝑤𝑖\n𝑝𝑥𝑛 + 𝑏𝑖\n𝑝)𝑘\n𝑖=1\n]          (2)                                       180 \nWhere 𝑤𝑁𝐿  (collection of 𝑤1\n𝑁𝐿 , 𝑤2\n𝑁𝐿 , 𝑤3\n𝑁𝐿 ) and 𝑏𝑁𝐿  are the weight and 181 \nbias matrices of the nonline encoder, 𝑤𝑝  represents weight in the piecewise linear 182 \ncomponent. In the decoder, we used 𝑉 as the weight matrix of the decoder and 183 \nperformed linear transformations, as follows 184 \n               𝑥𝑛̂ = 𝑥̂𝑛\n𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 + 𝑥̂𝑛\n𝑝𝑖𝑒𝑐𝑒𝑤𝑖𝑠𝑒 185 \n= [𝑉𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟 𝑉𝑝𝑖𝑒𝑐𝑒𝑤𝑖𝑠𝑒]   [\n𝑠𝑖𝑛𝑧𝑐\n𝑐𝑜𝑠𝑧𝑐\n∑ (𝑤𝑖\n𝑝𝑥𝑛 + 𝑏𝑖\n𝑝) ∙ 𝑔𝑖(𝑤𝑖\n𝑝𝑥𝑛 + 𝑏𝑖\n𝑝)𝑘\n𝑖=1\n]     (3) 186 \n                     = 𝑉𝑧𝑛 187 \nThe optimization problem using the least square error is formulated as 188 \nmin\n𝑤𝑁𝐿,𝑤𝑝,𝑉\n∑ ‖𝑥𝑛 − 𝑥𝑛̂‖2𝑁\n𝑛=1 + 𝜆‖𝑤𝑁𝐿‖2 + ∑ 𝛼𝑖‖𝑤𝑖\n𝑝‖\n2\n+ 𝛽‖𝑉‖2𝑘\n𝑖=1    (4) 189 \nWhere 𝜆, 𝛼𝑖, and 𝛽 are regularization coefficients controlling the complexity of the 190 \nnonlinear encoder, the piecewise linear component, and the decoder, respectively. 191 \nSpecifically, 𝜆 ∥ 𝑤𝑁𝐿 ∥2 ensures the smoothness of the learned latent manifold by 192 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nregularizing the nonlinear weights, while  𝛼𝑖 constrains the slope parameters of each 193 \ncell state.  To train the model, we employ an alternating optimization strategy that 194 \niterates between refining the piecewise thresholds and updating the autoencoder 195 \nweights. Detailed descriptions of the initialization and the two -step optimization 196 \nalgorithm are provided in Supplementary Note 2. 197 \n 198 \nResults 199 \nOverview of SPAE 200 \nAs illustrated in Figure 1, SPAE integrates an autoencoder to analyze single-cell RNA 201 \nsequencing (scRNA-seq) data by capturing both cell cycle dynamics and different cell 202 \ntypes. The model consists of two key components: a nonlinear encoder for cell cycle 203 \nestimation and a piecewise linear regression model for identifying distinct cell types 204 \nbased on their gene expression profiles. In the encoder, a multi -layer perceptron with 205 \nhyperbolic tangent activation functions reduces the data ’ s dimensionality while 206 \nmapping cells along a pseudotime trajectory, capturing their progression through the 207 \ncell cycle. To account for the periodicity of the cell cycle, SPAE employs sine and 208 \ncosine functions in the decoder, allowing for precise estimation of pseudotime and cell 209 \ncycle phases. Once the cyclic behavior is modeled, SPAE integrates a piecewise linear 210 \nregression model, which allows the program to treat inferred cycle processes as 211 \nconfounding factors and, after discounting confounding cell cycle effects, to make 212 \npredictions for multiple cell types. 213 \n 214 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nSPAE accurately infers cell cycle pseudotime  215 \nTo assess the performance of SPAE in predicting cell cycle pseudotime, we compared 216 \nSPAE with Cyclum, CYCLOPS, and reCAT [18, 20, 21], and the widely used trajectory 217 \ninference tool Monocle  using the scRNA -seq data of mESCs. Figure 2A shows the 218 \ndistribution of cell cycle pseudotime estimated by each method.  We calculated 219 \nstatistical measurements using the interquartile distances between boxplots, which 220 \nhelped quantify the separation of pseudotime  of cells in different cell cycle  phases. 221 \nSpecifically, we measured the distance between the lower quartile of the inferred cell 222 \ncycle pseudotime in the S phase  and the upper quartile of the pseudotime in the G1 223 \nphase, as well as the distance between the lower quartile of the pseudotime in the G2/M 224 \nphase and the upper quartile of the pseudotime in the S phase. Both SPAE and Cyclum 225 \nretain the correct cell cycle order, from G1 to S and then to G2/M. CYCLOPS 226 \neffectively distinguishes between S and G2/M phases, while reCAT can distinguish G1 227 \nand S phases but not G2/M. Compared to Cyclum, SPAE exhibits superior performance 228 \nin separating S phase and G2/M phase.  To quantitatively validate these observations, 229 \nwe calculated Spearman’s rank correlation coefficient ( ρ)[30] between each method’s 230 \ninferred pseudotime and the true, biologically ordered cell -cycle stages. Our results 231 \ndemonstrate that SPAE achieves the strongest monotonic correlation with the true cell-232 \ncycle order (ρ = 0.866, P = 1.90×10⁻¹¹), substantially outperforming other methods such 233 \nas Cyclum (ρ = 0.699, P=2.96×10⁻⁶), reCAT (ρ = 0.591, P = 1.87×10⁻⁴), Monocle (ρ 234 \n= 0.468, P=0.004587) and CYCLOPS ( ρ = -0.276, P = 0.1087). Notably, while 235 \nMonocle is effective for branching trajectories, its lower correlation here suggests 236 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nlimitations in capturing the closed-loop topology of the cell cycle compared to SPAE. 237 \nThese statistics confirm that SPAE most accurately reconstructs the biological sequence 238 \nof cell-cycle stages, effectively capturing the continuous transition consistent with the 239 \nvisual patterns in Figure 2A.  We calculated the Pearson correlation between the gene 240 \nexpression and cell cycle pseudotime inferred by SPAE. The genes with the highest 241 \ncorrelation with cell cycle pseudotime were Aurora kinase A ( Aurka), cell division 242 \ncycle associated 2 ( Cdca2) and karyopherin alpha 2 ( Kpna2). The correlation 243 \ncoefficients of Aurka, Cdca2, and Kpna2 are 0.73, 0.73, and 0.71, respectively (Figure 244 \n2B). Aurka is a kinase that plays an important role in cell cycle regulation and control. 245 \nIt affects the cell cycle mainly by regulating chromosome separation in the preparatory 246 \nphase of cell division. Aurka maintains the stability of the cell cycle by interacting with 247 \nother cell cycle proteins [31]. Cdca2 has a crucial role in controlling the G1/S transition, 248 \nwhich is a critical stage in the cell cycle. Cdca2 depletion led to cell cycle arrest at the 249 \nG1 phase, suggesting that Cdca2 is required for proper cell cycle progression [32]. 250 \nKpna2 is also involved in cell cycle regulation and restriction of cell cycle progression 251 \n[33]. To complete the comparison, we extend this analysis to CYCLOPS, Cyclum, 252 \nreCAT and Monocle, showing the Pearson correlation between gene expression and 253 \ninferred cell cycle  pseudotime for the top six genes identified by each method  254 \n(Supplementary Figure S 1-S4). Notably, most of the top genes identified by 255 \nCYCLOPS, Cyclum, and reCAT have little to no direct involvement in the cell cycle, 256 \nwhich suggests a potential limitation in these methods for capturing key cell cycle -257 \nrelated dynamics.  Additionally, we included specific plots comparing the Pearson 258 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ncorrelation for Aurka, Cdca2, and Kpna2 between all methods. While SPAE 259 \nconsistently identified these genes as highly correlated with cell cycle pseudotime (with 260 \ncorrelations of 0.73, 0.73, and 0.71, respectively), their correlations were significantly 261 \nlower in CYCLOPS, Cyclum, and reCAT  (Supplementary Figure S 1B-S3B). For 262 \ninstance, Monocle showed weaker correlations for key regulators like Aurka (R = 0.55) 263 \nand Cdca2 (R = 0.58), although it maintained a comparable correlation for Kpna2 (R = 264 \n0.77) (Supplementary Figure S 4B). This indicates that general-purpose tools like 265 \nMonocle and other specific methods may fail to capture the importance of these well -266 \nknown cell cycle regulators, further highlighting SPAE ’s strength in identifying 267 \nbiologically relevant genes associated with cell cycle progression. Figure 2C displays 268 \na heatmap illustrating several G2/M phase marker genes with relatively high 269 \ncorrelations to cell cycle pseudotime estimated by SPAE, which are all highly expressed 270 \nin the G2/M phase. 271 \n 272 \nSPAE demonstrates superior accuracy and robustness in cell cycle 273 \ncharacterization 274 \nTo evaluate SPAE ’s performance in predicting cell cycle progression, we follow the 275 \ncomparison strategy in Cyclum  [20]. SPAE was compared with several models, 276 \nincluding CCPE, Cyclum, CYCLOPS, cyclone, Seurat, and reCAT.  The continuous 277 \npseudotime generated by SPAE, CCPE, Cyclum, and CYCLOPS was converted into 278 \ndiscrete cell cycle phases using a three -component Gaussian mixture model. The 279 \nperformance was evaluated using seven classification metrics including Accuracy, 280 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nPrecision, Recall, F -score, Rand Index (RI), Normalized Mutual Information (NMI), 281 \nand Adjusted Rand Index (ARI), across three datasets (mESCs Quartz-Seq, E-MTAB-282 \n2805 mESCs, and H1 hESCs).  To account for the stochasticity of machine learning 283 \nmodels, we evaluated each method ten times on each dataset and then calculated the 284 \naverage values for each performance metric. The radar plots demonstrate SPAE ’s 285 \noutstanding performance in the analysis of H1 hESCs  dataset, with the clustering 286 \nmetrics achieving the highest values among all methods ( Figure 3A, where 0.2, 0.4, 287 \n0.6, and 0.8 represent different thresholds). In the E -MTAB-2805 dataset analysis, 288 \nSPAE led in all individual metrics, showcasing its superior performance compared to 289 \nother models  (Figure 3B) . Furthermore, in the analysis of the m ESCs Quartz Seq 290 \ndataset, SPAE continued to demonstrate strong performance ( Supplementary Figure 291 \nS5), confirming its robustness across different datasets. 292 \nWe further evaluated the performance of SPAE under different sample sizes by 293 \nsubsampling the scRNA-seq data from the H1 hESC dataset with fewer cells or genes. 294 \nWe conducted an analysis on seven sub-datasets with a diverse range of gene numbers, 295 \nextending from 50 to 600, and five sub -datasets with a range of cellular numbers, 296 \nextending from 10 to 100. Our results indicated a gradual increase in the median value 297 \nof clustering metrics for SPAE , CYCLOPS and Cyclum with increasing number of 298 \ngenes ( Figure 3C , Supplementary Figure S 6). The findings of our analysis 299 \ndemonstrated the superiority of SPAE over CYCLOPS and Cyclum in terms of seven 300 \nclustering metrics. In particular, SPAE exhibited a greater accuracy in predicting cell 301 \ncycle stages using fewer genes compared to CYCLOPS and Cyclum. Moreover, SPAE 302 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ndisplayed better performance with smaller numbers of cells compared to CYCLOPS 303 \nand Cyclum. As the number of cells increased, the performance of SPAE declined 304 \ngradually and reached a stable point ( Figure 3D , Supplementary Figure S 7). 305 \nConversely, the performance of Cyclum and CYCLOPS exhibited fluctuations but 306 \nconsistently remained below that of SPAE . However, we did not include cyclone, 307 \nSeurat, and reCAT in Figures 3C -D due to their poor performance in the context of 308 \nsubsampling. When randomly sampling cells, the accuracy of all performance metrics 309 \nfor these methods was extremely low. Additionally, when randomly sampling genes, 310 \nthe number of selected genes was often too small, and many of the genes were not 311 \nmarker genes used by these models, resulting in their failure to produce any meaningful 312 \nresults. Therefore, we believe including these methods in the subsampling analysis 313 \nwould not provide valuable insights, as their performance was not comparable to SPAE, 314 \nCyclum and CYCLOPS under these conditions. Our analysis suggests that SPAE is 315 \nmore robust and exhibits higher prediction accuracy for sub -datasets with smaller 316 \nnumbers of genes or cells. A comprehensive comparison of the algorithmic features and 317 \narchitectures of these methods is provided in Supplementary Table S1. 318 \n 319 \nSPAE is robust to dropout events in the E-MTAB-2805 mESCs dataset 320 \nTo assess SPAE’s robustness against dropout events in scRNA-seq, we utilized the E-321 \nMTAB-2805 mESCs dataset, which initially had a dropout rate of 24%. We first applied 322 \nthe MAGIC  [24] model to impute the missing data, and subsequently introduced 323 \nartificial dropout rates of 0%, 20%, 50%, and 70%. Our analysis indicated that SPAE’s 324 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nperformance was influenced by the dropout rate. However, the evaluation of the 325 \nclustering metrics in Figure 4 revealed that SPAE outperformed Cyclum, and 326 \nCYCLOPS when the dropout rate was below 70%. At a dropout rate of 70% , all 327 \nmethods (SPAE, Cyclum and CYCLOPS) were no longer performant in estimating the 328 \ncell cycle stages based on precision, recall and F1 score. For lower levels of dropout 329 \nevents, however, SPAE ’s clustering metric values remained higher than those of 330 \nCyclum and CYCLOPS. Therefore, our findings suggest that the SPAE is in general 331 \nmore robust to dropout events compared to Cyclum and CYCLOPS, albeit at higher 332 \nlevels of dropout events none of the methods were able to obtain sufficient information 333 \nto deliver performant cell cycle stage estimation. 334 \n 335 \nSPAE identifies differentially expressed genes enriched in key cell cycle pathways 336 \nPerforming differential gene expression analysis based on inferred cell cycle phases 337 \nallows us to uncover variations in gene expression across distinct cell cycle stages.  338 \nUsing the DESeq2 package  [34] within R/Bioconductor, we identified differentially 339 \nexpressed genes (DEGs) from cell cycle stages as inferred by SPAE. Subsequently, we 340 \ncompared these findings to those obtained from Cyclum , CYCLOPS, cyclone, Seurat, 341 \nand reCAT using the E -MTAB-2805 mESCs dataset, applying stringent criteria 342 \n(P.adjusted ≤ 0.05 and |log2FC| ≥ 1).  Our gene set enrichment analysis [35] unveiled 343 \nthat DEGs identified by SPAE are predominantly related to cell cycle pathways  and 344 \nrank highly in enrichment analysis. Furthermore, these DEGs exhibited enrichment in 345 \nbiological processes closely associated with the cell cycle, encompassing pathways 346 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nsuch as the p53 signaling pathway, progesterone -mediated oocyte maturation. 347 \nContrastingly, DEGs identified by Cyclum showed minimal relevance to the cell cycle, 348 \nhighlighting a significant difference  (Figure 5A ). CYCLOPS, cyclone, Seurat, and 349 \nreCAT also exhibit some degree of cell cycle relevance, these methods are less effective 350 \nand exhibit lower correlations compared to SPAE, failing to capture key pathways 351 \nclosely related to the cell cycle with the same level of significance  (Supplementary 352 \nFigure S8). We also explored the expression patterns of four genes enriched in the cell 353 \ncycle pathway, Cdc20, Fzr1, Cdk1 and Ccnb1, which are G2/M phase marker genes  354 \n(Figure 5B). These genes were identified through a rigorous two-step selection process. 355 \nFirst, differential gene expression (DEG) analysis was conducted using the DESeq2 356 \npackage in R/Bioconductor, based on cell cycle stages inferred by SPAE from the E -357 \nMTAB-2805 mESCs dataset. Genes meeting the criteria of an adjusted P -value ≤ 358 \n0.05 and |log2FC| ≥ 1 were designated as DEGs. Second, gene set enrichment 359 \nanalysis of the DEGs revealed significant enrichment of Cdc20, Fzr1, Cdk1, and Ccnb1 360 \nin both the cell cycle pathway and the progesterone -mediated oocyte maturation 361 \npathway. The critical role of these pathways in cell cycle regulation provided a strong 362 \nrationale for their selection for further investigation. 363 \nSPAE accurately detects Nutlin-induced G1 arrest in TP53 wild-type cancer cells 364 \nTo validate SPAE’s efficacy in cell cycle prediction, we analyzed a dataset of cancer 365 \ncells treated with nutlin and DMSO. Nutlin, an antagonist of the MDM2-p53 pathway 366 \n[36], is known to induce cell cycle arrest [37]. Nutlin promotes the stability and activity 367 \nof p53 by inhibiting the interaction between MDM2 and p53, thereby inducing G1 368 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nphase arrest, particularly in TP53 wild-type (WT) cells. As a key tumor suppressor 369 \nprotein, p53 initiates cell cycle checkpoints, especially the G1/S checkpoint, in response 370 \nto cellular stress, preventing damaged cells from entering the S phase. Therefore, it is 371 \nexpected that Nutlin treatment in TP53 WT cells will significantly increase the 372 \nproportion of G1 phase cells and reduce the number of cells in the S and G2/M phases. 373 \nWe utilized two groups of data: one group consisted of cells treated with DMSO (the 374 \ncontrol group), and the other group comprised the same type of cells treated with nutlin 375 \n[25]. The experimental samples comprised 7 TP53 wild-type (WT).  As shown in 376 \nFigure 6A, SPAE’s predictions revealed a significant rise in G1 phase cells in TP53 377 \nWT samples treated with nutlin, compared to the control group.  Further analysis 378 \nspecifically targeting TP53 WT cell lines reinforced this finding, showing a significant 379 \nincrease in the proportion of G1 phase cells, indicating that Nutlin induced G1 phase 380 \narrest in these cells (Figure 6B). Additionally, we identified DEGs for each cell cycle 381 \nstage using D ESeq2 ( Figure 6C), revealing multiple pathways related to cell cycle 382 \nregulation, particularly the regulation pathway of the G2/M checkpoint. In Nutlin -383 \ntreated TP53 WT cells, the G2/M checkpoint process ranks third among the DEGs. 384 \nThese findings not only confirm the high accuracy of SPAE in predicting cell cycle 385 \nstages but also demonstrate its effectiveness in detecting G1 arrest induced by Nutlin 386 \nin TP53 WT cells. 387 \n 388 \nSPAE disentangles cell cycle effects from intrinsic cell states 389 \nIn cellular biology, removing cell cycle effects is critical for accurately identifying cell 390 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ntypes and understanding cellular differentiation [38]. Cells in different stages of the cell 391 \ncycle often exhibit large differences in gene expression, which can obscure true 392 \nbiological signals and hinder functional analysis  [39]. In this study, we evaluated the 393 \nperformance of SPAE in removing cell cycle effects across three datasets: mouse 394 \nembryonic stem cells (mESCs), human myoblasts (hMyo), and breast cancer cells. The 395 \nmESCs dataset includes data from four different withdrawal intervals of leukemia 396 \ninhibitory factor (LIF) at 0, 2, 4, and 7 days. The hMyo dataset comprises scRNA-seq 397 \ndata from human myoblasts collected at various time points, namely 0, 24, 48, and 72 398 \nhours. The breast cancer dataset encompasses cell types with CSL gene knockout 399 \n(CSLKO1 and CSLKO2) and wild-type controls (WT1 and WT2). To comprehensively 400 \nassess SPAE’s performance in removing cell cycle effects, we compared it with four 401 \nother methods: Cyclum, CCPE, ccRemover, and Seurat. We performed dimensionality 402 \nreduction analyses using UMAP [40] on both raw data and the output of each method 403 \nafter cell cycle correction. The data were visualized with UMAP plots labeled by cell 404 \ncycle stages (upper panels in Figure 7A-C) and by cell types, states or time points 405 \n(lower panels in Figure 7A -C). The distribution of cells across different cell cycle 406 \nstages was examined to evaluate the effectiveness of each method in mitigating cell 407 \ncycle-driven clustering. Before the removal of the cell cycle effect, the distribution of 408 \ncells in all three datasets was predominantly influenced by their cell cycle phases, 409 \nobscuring the differences between cell types and hindering the clear separation of 410 \nsimilar cell populations. However, after applying SPAE, cells of the same type, state or 411 \ntime point were more accurately clustered together, no longer being dispersed based on 412 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\ntheir cell cycle stages (Figure 7A-C). Specifically, in mouse embryonic stem cells, the 413 \nraw data can distinguish cells from four different withdrawal intervals of leukemia 414 \ninhibitory factor (LIF) at undifferentiated, 2, 4, and 7 days  (Figure 7A). However, at 415 \neach interval, cells in the three cell cycle stages are not well mixed, indicating that cell 416 \nstate classification is influenced by cell cycle effects. For instance, most ES7d cells are 417 \nin the G1 phase. After applying SPAE to remove cell cycle effects, the distribution of 418 \ncells across the three cell cycle stages in ES7d becomes more uniform. In contrast, 419 \nCCPE effectively mixed cells from different cell cycle stages but failed to distinguish 420 \nthe four LIF withdrawal intervals. ccRemover distinguished the four intervals but did 421 \nnot evenly mix the three cell cycle stages, indicating residual cell cycle influence. 422 \nCyclum neither distinguished the LIF withdrawal intervals nor evenly mixed the three 423 \ncell cycle stages. Seurat was able to distinguish between different LIF discontinuation 424 \nintervals to some extent but failed to effectively and evenly mix cells in the three cell 425 \ncycle stages (Figure 7A).  Similar trends were observed in the hMyo dataset (Figure 426 \n7B) and the breast cancer dataset (Figure 7C). SPAE consistently outperformed other 427 \nmethods, effectively mixing cells across different cell cycle phases while maintaining 428 \ndistinctions between different time points, cell states, or cell types. This improved 429 \nclustering suggests that SPAE successfully removed confounding cell cycle effects, 430 \nallowing the data to reflect true biological differences. In contrast, other methods, 431 \nincluding Cyclum, CCPE, Seurat, and ccRemover, did not achieve comparable results. 432 \nEven after applying these methods, cell cycle -driven clustering remained evident 433 \n(Figure 7A-C). SPAE ranked highest in both mixing cells from different cell cycle 434 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nphases and maintaining clear separation of cells collected from different time points or 435 \nbiological states. These findings highlight SPAE’s robustness in removing cell cycle 436 \neffects, making it a valuable tool for accurately analyzing single -cell transcriptomic 437 \ndata and uncovering true cellular identities. 438 \n 439 \nPrediction of cell cycle transitions in breast cancer treatment using SPAE 440 \nCell cycle dysregulation manifests as changes in the distribution of cells across various 441 \nstages and alterations in the expression of cell cycle regulatory genes [41]. To assess 442 \nSPAE’s effectiveness and discern changes in single-cell data, we analyzed scRNA-seq 443 \ntumor data from 176,644 cells. These were divided into three treatment groups: 444 \nendocrine therapy alone (letrozole plus placebo), intermittent high -dose combination 445 \ntherapy (letrozole plus ribociclib [600 mg/day, 3 weeks on/off]), and continuous low -446 \ndose combination therapy (letrozole plus ribociclib [400 mg/day]) [26]. Patients 447 \nunderwent six cycles of treatment, with biopsies collected at baseline (day 0), the start 448 \nof treatment (day 14), and at the end of treatment (around day 180 at surgery). SPAE 449 \nwas employed to predict transitions in the cancer cell cycle. Additionally, a cyclic 450 \ngeneralized additive model was used to describe the dynamics of gene expression at 451 \nvarious cell cycle stages. The SPAE-inferred cell cycle stages were used to color cells 452 \n(Figure 8A). Reconstructing the cell cycle based on scRNA -seq data ( Figure 8A), 453 \nSPAE recovered the expected cell cycle stages, including the G1 checkpoint transition, 454 \nwhere cyclin D initially rises, followed by CDK6 expression. Additionally, we observed 455 \na decrease in RB1 expression (a key G1 checkpoint protein) and an increase in E2F3, a 456 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nproliferation gene. Moreover, based on SPAE-inferred cell cycle staging, we calculated 457 \nthe proportion of mitotic (S/G2 phase) cancer cells in each patient’s biopsy (Figure 8B). 458 \nDuring combination therapy, an increase in the proportion of mitotic and proliferating 459 \n(S/G2 phase) cancer cells was observed in each patient. Persistent tumors exhibited an 460 \nincreased frequency of proliferating cells, especially those undergoing high -dose 461 \ncombination therapy. In contrast, patients receiving only endocrine therapy showed 462 \nfewer proliferating cells. These results suggest that in surviving subclonal populations, 463 \nthe ribociclib-enhanced G1/S checkpoint can be effectively bypassed. Subsequently, 464 \napplying the cyclic generalized additive model revealed fluctuations in ESR1 and FOS 465 \ngene expression throughout the cell cycle. By applying this approach to cells sampled 466 \nat different time points and patients undergoing different treatments, we differentiated 467 \nwhether treatment altered expression at specific cell cycle stages, or if gene expression 468 \nwas independent of cell cycle dysregulation. ESR1 showed consistent expression levels 469 \nthroughout the cell cycle ( Figure 8C). However, a decline in ESR1 expression over 470 \ntime, coupled with an increase in FOS expression, was observed across the entire cohort 471 \nreceiving combination therapy (Figure 8C). Additionally, during combination therapy, 472 \na reduction in CDK inhibitor 2A (CDKN2A encoding p14 and p16) and an increase in 473 \nCDK6 expression from G1 to S/G2 phase were observed ( Figure 8C). In summary, 474 \nSPAE accurately estimates cancer cell cycle transitions and can be used to explore the 475 \nphenotypic evolution of cancer cells under endocrine therapy and CDK4/6 inhibitor 476 \ntreatment, as well as the relationship of these phenotypic changes to genomic variations. 477 \nThis information helps to reveal resistance mechanisms in early estrogen receptor -478 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\npositive breast cancer and identify potential therapeutic targets. 479 \n 480 \nSPAE identifies key transcription factors driving cell cycle transitions 481 \nThe SPAE also enables us to identify potential transcription factors (TFs) responsible 482 \nfor the dynamics of gene expression along the cell cycle process. Transcription factors 483 \nbind to specific DNA sequences (binding motifs) and activate the transcription of their 484 \ntarget genes. They encode cellular programs for many functions required by the cell. 485 \nWe used SCENIC [42] to infer TF activity during the cell cycle . SCENIC is a method 486 \nthat computes gene regulatory networks in single -cell transcriptomic data through co -487 \nexpression and motif analysis. Based on the cell cycle predicted by SPAE, we analyzed 488 \ntranscription factors using the mESCs Quartz -Seq dataset and the hESCs scRNA -seq 489 \ndataset, with motif analysis predictions. In the mESCs Quartz-Seq dataset, observed TF 490 \nactivities suggested that the E2f family appears to be a group of key regulators, known 491 \nto act at the onset of the cell cycle, especially during the G1/S transition [43, 44]. E2f1 492 \nand E2f2 peaked between the G1 and S phases, potentially activating genes required 493 \nfor the transition  [45] (Supplementary Figure S 9A). Specificity factor 1 (Sp1) and 494 \nnuclear respiratory factor 1 (Nrf1) were both active in early G1 (Supplementary 495 \nFigure S 9B). For factors emerging from the hESCs scRNA -seq dataset, MYB is 496 \ninvolved in the G2/M transition, functioning during the G2/M transition[46]. Kruppel-497 \nLike Factor 6 (KLF6), a transcription factor active in the G1 phase, can induce cell 498 \ncycle arrest and reduce the rate of cell proliferation  [47]. These results reveal the 499 \ndynamic changes in transcription factor activity across different cell cycle stages, 500 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nfurther supporting the critical role of transcription factors in regulating the cell cycle 501 \nprocess. The observed TF activity patterns reflect the specific needs of cells at each 502 \nstage for transcriptional regulation and offer a new perspective for understanding the 503 \ncell cycle regulation of stem cells. These findings reinforce the central role of 504 \ntranscription factors in cell fate decisions and could provide potential targets for the 505 \ntreatment of cell cycle-related diseases. 506 \n 507 \nDiscussion 508 \nIn this work, we present SPAE, a computational framework that integrates a sinusoidal 509 \nautoencoder with piecewise linear regression to decouple cell cycle dynamics from cell 510 \nstates. By utilizing an alternating optimization strategy, SPAE simultaneously learns 511 \ncontinuous pseudotime and discrete cell clusters. While we employed a Gaussian 512 \nMixture Model (GMM)  [48] to map pseudotime to discrete cell cycle phases (G1, S, 513 \nG2/M), we acknowledge that standard GMMs may not fully capture the cyclic 514 \ncontinuity and non -Gaussian patterns of biological processes. Nevertheless, this 515 \napproach provides a practical approximation for delineating clinical stages from 516 \ncontinuous trajectories. 517 \nComprehensive benchmarking against established methods (including CCPE, 518 \nCyclum, CYCLOPS, Seurat  and Monocle ) across diverse scRNA -seq datasets 519 \ndemonstrated SPAE’s superior performance. Unlike linear models (e.g., CCPE) that 520 \nstruggle with complex nonlinear topologies, SPAE’s piecewise architecture effectively 521 \ncharacterizes multi -stage transitions and nonlinear gene expression patterns. 522 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nConsequently, SPAE exhibited greater accuracy and robustness in pseudotime 523 \ninference, stability against data sparsity (dropout) and subsampling, and the 524 \nidentification of key cell cycle -related transcription factors. Critically, SPAE proved 525 \neffective in removing cell cycle confounding effects, thereby revealing true cell type 526 \nidentities that were otherwise obscured. Biological validation on Nutlin-treated cancer 527 \ncells further confirmed SPAE’s sensitivity in detecting specific G1 arrest and predicting 528 \ntherapy-resistant states, highlighting its translational potential. 529 \nDespite these advancements, limitations remain. Currently, SPAE lacks explicit 530 \nmodeling for complex technical variations, such as batch effects or tissue -specific 531 \nbiases, which may constrain its application in large -scale integrative studies. Future 532 \nwork will focus on incorporating mechanisms, such as adversarial domain adaptation, 533 \nto mitigate these confounding factors and extend SPAE to complex disease models. In 534 \nconclusion, SPAE provides a robust and flexible tool for dissecting the interplay 535 \nbetween cell cycle dynamics and cell fate, offering new insights into biological 536 \nheterogeneity and potential therapeutic targets. 537 \n 538 \nCode availability 539 \nSPAE developed for this study is implemented in Python  3.9 and is available for 540 \ndownload on GitHub  (https://github.com/YaJahn/SPAE). The code has also been 541 \nsubmitted to BioCode at the National Genomics Data Center (NGDC), China National 542 \nCenter for Bioinformation (CNCB) (BioCode: BT008079), which is publicly accessible 543 \nat https://ngdc.cncb.ac.cn/biocode/tool/ BT008079. 544 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nCRediT author statement 545 \nJiahao Yi: Methodology, Software, Formal analysis, Investigation, Writing – original 546 \ndraft. Jiajia Liu:  Methodology, Software, Formal analysis. Peng Guo:  Writing – 547 \noriginal draft. Yuan-Nong Ye: Conceptualization, Supervision, Writing – review & 548 \nediting. Xiaobo Zhou: Conceptualization, Supervision, Writing – review & editing. 549 \nAll authors read and approved the final manuscript. 550 \n 551 \nCompeting interests 552 \nThe authors declare no competing interests. 553 \n 554 \nAcknowledgements 555 \nThis work was  supported by the National Institutes of Health [ R01LM014156, 556 \nR01GM153822 and R01CA241930  to X.Z .], the National Science Foundation 557 \n[2217515, 2326879 to X.Z.] and the National Natural Science Foundation of China 558 \n[32160151 to Y .N.Y .]. 559 \n 560 \nSupplementary material 561 \nSupplementary material is available at Genomics, Proteomics & Bioinformatics online. 562 \n 563 \nORCID 564 \n0009-0006-0791-6005 (Jiahao Yi) 565 \n0000-0002-0038-9592 (Jiajia Liu) 566 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. 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E2f1, E2f2, and E2f3 control E2F target expression and 667 \ncellular proliferation via a p53-dependent negative feedback loop, Mol Cell Biol 2007;27:65-78. 668 \n46. Nakata Y, Shetzline S, Sakashita C et al. c -Myb contributes to G2/M cell cycle transition in 669 \nhuman hematopoietic cells by direct regulation of cyclin B1 expression, Mol Cell Biol 670 \n2007;27:2048-2058. 671 \n47. Trucco LD, Andreoli V, Nunez NG et al. Kruppel -like factor 6 interferes with cellular 672 \ntransformation induced by the H-ras oncogene, FASEB J 2014;28:5262-5276. 673 \n48. Wan C, Chang W, Zhang Y et al. LTMG: a novel statistical modeling of transcriptional 674 \nexpression states in single-cell RNA-Seq data, Nucleic Acids Res 2019;47:e111. 675 \n 676 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 677 \nFigure 1. Overview of the SPAE framework.  678 \nSPAE primarily consists of two components: a nonlinear component and a piecewise 679 \nlinear component. The nonlinear component is employed for the estimation of cell cycle 680 \npseudotime, while the piecewise linear component is dedicated to predicting various 681 \ncell types. Six downstream analyses and applications of SPAE have been employed to 682 \nevaluate its performance. 683 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 684 \nFigure 2. Cell cycle pseudotime analysis of mESCs Quartz-Seq data. 685 \n(A) Boxplots show the distribution of cell cycle pseudotimes inferred by five different 686 \nmethods (SPAE, Cyclum, CYCLOPS, reCAT, and Monocle). The boxplots are colored 687 \naccording to the three stages of the cell cycle (G1, S, G2/M). (B) Correlation between 688 \nthe expression of three cell cycle marker genes ( Aurka, Cdca2, Kpna2) and the cell 689 \ncycle pseudotime estimated by SPAE. The top-left corner of each plot is marked with 690 \nthe correlation coefficient (R) and P-value, demonstrating a strong correlation between 691 \nexpression levels and pseudotime. (C) A heatmap displays several G2/M phase marker 692 \ngenes correlated with cell cycle pseudotime inferred by SPAE. Each column represents 693 \na cell. The color changes in the heatmap represent variations in gene expression levels, 694 \nranging from low (blue) to high (red). 695 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 696 \nFigure 3. Inferring cell cycle stages from real datasets.  697 \n(A) Radar chart showing seven multi -class classification metrics (Fscore, Recall, 698 \nPrecision, Accuracy, NMI, ARI, RI) used to evaluate the cell cycle classification 699 \naccuracy of SPAE, CCPE, cyclone, Seurat, reCAT, Cyclum, and CYCLOPS on H1 700 \nhESCs data. (B) Radar chart displaying the same seven multi -class classification 701 \nmetrics for assessing the performance of SPAE and the six benchmark methods on E -702 \nMTAB-2805 mESCs data. (C) Robustness analysis under gene subsampling. The 703 \nboxplots illustrate the Accuracy, Adjusted Rand Index (ARI), and Normalized Mutual 704 \nInformation (NMI) of SPAE, CYCLOPS, and Cyclum on datasets with varying 705 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nnumbers of genes (ranging from 50 to 600). (D) Robustness analysis under cell 706 \nsubsampling. The boxplots illustrate the performance of SPAE, CYCLOPS, and 707 \nCyclum in terms of Accuracy, ARI, and NMI metrics on datasets with varying cell 708 \ncounts (ranging from 10 to 100). 709 \n 710 \n 711 \n 712 \n 713 \n 714 \n 715 \n 716 \n 717 \n 718 \n 719 \n 720 \n 721 \n 722 \n 723 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 724 \nFigure 4. Robustness of SPAE in handling dropout events in the E -MTAB-2805 725 \nmESCs dataset. 726 \nEvaluation of the impact of different dropout rates (ranging from 0% to 70%) on the 727 \nperformance of SPAE, Cyclum, and CYCLOPS. The comparison utilizes seven multi-728 \nclass classification metrics: RI, ARI, NMI, Accuracy, Precision, Recall, and Fscore. 729 \n 730 \n 731 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 732 \nFigure 5. Differential gene expression analysis of E-MTAB-2805 mESCs Data.  733 \n(A) Comparison of biological pathway enrichment. The left panel (red) shows the top 734 \nten biological processes enriched in DEGs identified by SPAE -inferred stages, 735 \nincluding relevant terms like cell cycle, oocyte meiosis, and the p53 signaling pathway. 736 \nThe right panel (blue) shows processes associated with DEGs identified by Cyclum, 737 \nwhich include unrelated pathways such as T-cell receptor signaling and non-small cell 738 \nlung cancer. (B) Expression dynamics of G2/M phase marker genes. Scatter plots 739 \nshowing the expression variation of Cdc20, Fzr1, Cdk1, and Ccnb1 along the cell cycle 740 \npseudotime inferred by SPAE. Each point represents a single cell, and the red curve 741 \nindicates the smoothed gene expression trend along the inferred pseudotime. 742 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 743 \nFigure 6. Validation of SPAE predictions using scRNA-seq datasets of cells treated 744 \nwith cell cycle perturbants.  745 \n(A) UMAP visualization of 7 cancer cell lines treated with DMSO (control) or Nutlin. 746 \nCells are colored according to cell cycle phases (G1, S, G2/M) inferred by SPAE. (B) 747 \nQuantitative analysis of G1 arrest. Bar chart showing the proportion of cells in different 748 \ncell cycle stages for TP53 wild-type (WT) cell lines in DMSO versus Nutlin treatment 749 \ngroups. SPAE accurately detects the accumulation of cells in the G1 phase upon Nutlin 750 \ntreatment. (C) Functional enrichment analysis. Bar charts showing the top enriched 751 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nbiological processes for Differentially Expressed Genes (DEGs) in the DMSO group 752 \n(top) and the Nutlin group (bottom), highlighting pathways related to cell cycle 753 \nregulation and p53 signaling. 754 \n 755 \n 756 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 757 \nFigure 7. Comparison of different methods for removing cell cycle effects. 758 \n(A) UMAP of mouse ES cells before and after removing cell cycle effects by SPAE, 759 \nCyclum, CCPE, ccRemover, Seurat . (B) UMAP of human myoblast cells before and 760 \nafter removing cell cycle effects by SPAE, Cyclum, CCPE, ccRemover, Seurat . (C) 761 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nUMAP of breast cancer cells before and after removing cell cycle effects by SPAE, 762 \nCyclum, CCPE, ccRemover, Seurat . Raw represents data before cell cycle effect 763 \nremoval. Upper panel is labeled by cell cycle stages and lower panel is labeled by cell 764 \ntypes, states or time points. 765 \n 766 \n 767 \n 768 \n 769 \n 770 \n 771 \n 772 \n 773 \n 774 \n 775 \n 776 \n 777 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\n 778 \nFigure 8. Analysis of cell cycle dysregulation based on SPAE.  779 \n(A) Cell cycle phases colored based on predictions by SPAE, depicting expression 780 \nchanges of cell cycle regulatory genes CDK6, Cyclin D, RB1, and E2F3 using a cyclic 781 \ngeneralized additive model. The distance from the point to the origin indicates the 782 \nexpression of the cell at that phase.  (B) Changes in the proportion of cells at different 783 \nstages of the cell cycle (S/G2 phase) in patient tumor samples treated with Letrozole 784 \nalone, intermittent high -dose Ribociclib, and continuous low -dose Ribociclib, as 785 \ninferred by SPAE. Blue represents patients sensitive to treatment, and red represents 786 \npatients resistant to treatment, with changes in cell proportions over time. (C) Changes 787 \nin the expression of ESR1, CDK6, CDKN2A, and FOS before, during, and after the cell 788 \ncycle and treatment (columns). Colored lines show the expected gene expression in 789 \ncells throughout the cell cycle before (blue), during (orange), and after (red) treatment. 790 \nThe distance from the center of the circle indicates gene expression at a particular point 791 \nin the cell cycle. 792 \nTable 1. Cell-cycle scRNA-seq datasets. 793 \nTable 2. scRNA-seq datasets of cell cycle effect removal 794 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nSupplementary Figure Legends 795 \nSupplementary Figure S1. Performance of CYCLOPS on pseudotime inference.  796 \n(A) Scatter plots showing the Pearson correlation between gene expression and 797 \npseudotime for the top six genes identified by CYCLOPS. (B) Pearson correlation 798 \nbetween gene expression and pseudotime for known cell cycle markers Aurka, Cdca2, 799 \nand Kpna2 as inferred by CYCLOPS. 800 \n 801 \nSupplementary Figure S2. Performance of Cyclum on pseudotime inference.  802 \n(A) Scatter plots showing the Pearson correlation between gene expression and 803 \npseudotime for the top six genes identified by Cyclum. (B) Pearson correlation between 804 \ngene expression and pseudotime for known cell cycle markers Aurka, Cdca2, and 805 \nKpna2 as inferred by Cyclum. 806 \n 807 \nSupplementary Figure S3. Performance of reCAT on pseudotime inference.  (A) 808 \nScatter plots showing the Pearson correlation between gene expression and pseudotime 809 \nfor the top six genes identified by reCAT. (B) Pearson correlation between gene 810 \nexpression and pseudotime for known cell cycle markers Aurka, Cdca2, and Kpna2 as 811 \ninferred by reCAT. 812 \n 813 \nSupplementary Figure S4. Performance of Monocle on pseudotime inference. (A) 814 \nScatter plots showing the Pearson correlation between gene expression and pseudotime 815 \nfor the top six genes identified by Monocle. (B) Pearson correlation between gene 816 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nexpression and pseudotime for known cell cycle markers Aurka, Cdca2, and Kpna2 as 817 \ninferred by Monocle. 818 \n 819 \nSupplementary Figure S5. Benchmarking classification metrics on mESCs 820 \nQuartz-Seq data. 821 \n Radar chart displaying seven multi -class classification metrics (Fscore, Recall, 822 \nPrecision, Accuracy, NMI, ARI, RI) evaluated on the mESCs Quartz-Seq dataset. The 823 \nperformance of SPAE is compared against Cyclum, CYCLOPS, Seurat, Cyclone, and 824 \nreCAT. 825 \n 826 \nSupplementary Figure S6. Robustness analysis under gene subsampling.  827 \nBoxplots of Fscore, Precision, Recall, and RI metrics indicating the performance of 828 \nSPAE, CYCLOPS, and Cyclum on subsampled datasets with varying numbers of genes 829 \n(ranging from 50 to 600). 830 \n 831 \nSupplementary Figure S7. Robustness analysis under cell subsampling.  832 \nBoxplots of Fscore, Precision, Recall, and RI metrics indicating the performance of 833 \nSPAE, CYCLOPS, and Cyclum on subsampled datasets with varying numbers of cells 834 \n(ranging from 10 to 100). 835 \n 836 \nSupplementary Figure S8. Functional enrichment analysis of benchmark methods. 837 \nBar charts showing the top ten enriched biological processes associated with 838 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nDifferentially Expressed Genes (DEGs) identified by cell cycle stages inferred through 839 \n(A) Seurat, (B) reCAT, (C) Cyclone, and (D) CYCLOPS. 840 \n 841 \nSupplementary Figure S9. SPAE predictions of core transcription factor activity. 842 \nHeatmaps visualizing the motif activity of transcription factors during the cell cycle. 843 \n(A) Transcription factor activity in the mESCs Quartz -Seq dataset. (B) Transcription 844 \nfactor activity in the H1 hESCs scRNA -seq dataset. The color gradient represents the 845 \nvariation in activity levels, ranging from low (blue) to high (red). 846 \n 847 \nSupplementary Notes 848 \nSupplementary Note 1: Piecewise linear regression model formulation. 849 \nSupplementary Note 2: Optimization of the autoencoder-based piecewise model.  850 \nSupplementary Table S1. Summary of computational methods for single -cell cell 851 \ncycle analysis compared in this study. 852 \n 853 \n 854 \n 855 \n 856 \n 857 \n 858 \n 859 \n 860 \n 861 \n 862 \n 863 \n 864 \n 865 \n 866 \n 867 \n 868 \n 869 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nTable 1. Cell-cycle scRNA-seq datasets. 870 \nDataset Tissue \nCell cycle \nstage \nlabeled \nTotal cell count G0/G1  S G2/M Gene \ncount \nmESCs Quartz -\nSeq  mESC Yes 35 20 7 8 33,412 \nH1 hESCs \nscRNA-seq hESC Yes 247 91 80 76 19,084 \nE-MTAB-2805 \nmESCs  mESC Yes 279 95 88 96 38,293 \nNutlin-treated \nMultiple Cancer \nCell Lines \nCancer cell \nline No \n3,097 ( nutlin-treated \ngroup) and 2,381 \n(control group) \n   32,738 \nFELINE Breast \nCancer Single -\nCell Genomics  \nBreast \nTumor No \n46,986 (letrozole \nalone) \n27,790 (high dose \nribociclib)  \n34,543 (low dose \nribociclib) \n   21,279 \n 871 \n 872 \n 873 \n 874 \n 875 \n 876 \n 877 \n 878 \n 879 \n 880 \n 881 \n 882 \n 883 \n 884 \n 885 \n 886 \n 887 \n 888 \n 889 \n 890 \n 891 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint \n\nTable 2. scRNA-seq datasets of cell cycle effect removal 892 \nDataset Tissue Cell type Total cell count Gene count \nMouse ES cells Mouse embryonic stem cellss 4 2717 24,046 \nhMyo Differentiating myoblasts (0 \nh, 24 h, 48 h, 72 h) 4 372 47,192 \nBreast cancer Human Breast Cancer (MDA-\nMB-231) 4 376 16,383 \n 893 \n.CC-BY 4.0 International licensemade available under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is \nThe copyright holder for this preprintthis version posted March 8, 2026. ; https://doi.org/10.64898/2026.03.05.709782doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}