protein phosphatase 2A catalytic subunit, alpha isoform (PPP2CA) : Time behavioural study of 3rd order combinations in WNT3A stimulated HEK 293 cells

protein phosphatase 2A catalytic subunit, alpha isoform (PPP2CA) : Time behavioural study of 3rd order combinations in WNT3A stimulated HEK 293 cells shriprakash sinha Independent Researcher; Orcid ID : orcid.org/0000-0001-7027-5788 Address : 104-Madhurisha Heights Phase 1, Risali, Bhilai-490006, India Corresponding author email : [email protected] Abstract PPP2CA encodes the phosphatase 2A catalytic subunit and is one of the four major Serine/threonine-protein phosphatases. It consists of a common heteromeric core enzyme, composed of a catalytic subunit and a constant regulatory subunit, that associates with a variety of regulatory subunits and it is implicated in the negative control of cell growth and division. Gujral and MacBeath [1] provides a quantitative, and dynamic study of WNT3A-mediated stimulation of HEK 293 cells, where they record time based expression profiles of several response genes which correlated significantly with proliferation and migration. By monitoring the dynamics of gene expression using self-organizing maps, they identified clusters of genes that exhibit similar expression dynamics and uncovered previously unrecognized positive and negative feedback loops. However, their study depicts/uses singular measurements of individual gene expression at different time snapshots/points to infer the system wide analysis of the pathway. At any particular time point, it is often the case that genes are working synergistically in combinations, even though their expression measurements are singular in nature. Here, I •enumerate and rank all 2415 PPP2CA related 3rd order combinations in a forest of 71C3combinations using four different sensitivity methods; •show the conserved rankings for PPP2CA-X-X combinations, which point to existence of biological synergy of some of these combinations across the different sensitivity methods; and •study the behaviour of some of these combinations related to WNT3A response genes that are ranked by the machine learning search engine (Sinha [2]) in time. Patterns of combinations emerge, some of which have been tested in wet lab, while others require further wet lab analysis. Keywords: Sensitivity analysis, Support vector ranking, Hilbert Schmidt Independence Criterion indices (HSIC) and Sobol indicies, WNT3A ITime behavioural study of 3-odr PPP2CA comb. in WNT3A stimulated cells 1Aspects of unpublished work were presented in a poster session at Cell Symposia: Technology. Biology. Data Science, 9-11 October 2016, Berkeley, California, USA. Preprint submitted to Preprint March 12, 2025 1. Significance Sinha [2] recently demonstrated the use of machine learning based search engine to rank/reveal gene combinations at 2nd order for the time series data by Gujral and MacBeath [1] and showed how it is possible to locate combinations of priority that might be working synergistically, using sensitivity methods and powerful support vector ranking algorithm. However, the problem explodes combinatorially with even a small set of 71 recorded genes in the study by Gujral and MacBeath [1], when one steps to explore 3rd order combinations. With the total number of 71C3(= 57155) combinations, it becomes nearly impossible for any biologist to study the system wide dynamics of any pathway. Also, the amount of time usually needed to search for and test a combination is far more than the search down by the machine learning based search engine. Here, I extend the research work by Sinha [2] to conduct a behavioral study of 3rd order PPP2CA related combinations using individual gene expressions measured in time, in WNT3A stimulated HEK 293 cells. 2. Introduction The details of the machine learning based search engine has been recently published in Sinha [2] and deployed to explore the 2nd order combinations of genes in the data set provided by Gujral and MacBeath [1]. Nevertheless, here, I point to the fundamentals of the published work for completeness. 2.1. A combinatorial problem Sensitivity analysis plays a major role in computing the strength of the influence of involved factors in any phenomena under investigation. When applied to expression profiles of various intra/extracellular factors that form an integral part of a signaling pathway, the variance and density based analysis yields a range of sensitivity indices for individual as well as various combinations of factors. These combinations denote the higher order interactions among the involved factors. Computation of higher order interactions is often time consuming but it gives a chance to explore the various combinations that might be of interest in the working mechanism of the pathway. For example, in a range of fourth order combinations among the various factors of the Wnt pathway, it would be easy to assess the influence of the destruction complex formed by APC, AXIN, CSKI and GSK3 interaction. But the effect of these combinations vary over time as measurements of fold changes and deviations in fold changes vary. So it is imperative to know how an interaction or a combination of the involved factors behave in time and Sinha [2] develops a procedure to track the behaviour by exploiting the influences of these involved factors. 2 2.2. A possible solution In this work, after estimating the individual effects of factors for a higher order combination, the individual indices are considered as discriminative features. A combination, then, is a feature set in higher order (≥2 ,i.e multivariate). With an excessively large number of factors involved in the pathway, it is difficult to search for important combinations in a wide search space over different orders. Exploiting the analogy with the issues of prioritizing webpages using ranking algorithms, for a particular order, a full set of combinations of interactions can then be prioritized based on these features using a powerful ranking algorithm via support vectors Joachims [3]. Recording the changing rankings of the combinations over time reveals how higher order interactions behave within the pathway and when an intervention might be necessary to influence the interaction within the pathway. 2.3. protein phosphatase 2A catalytic subunit, alpha isoform (PPP2CA) Protein phosphorylation is a reversible protein posttranslational modification (PTM). Phosphorylation involves the transfer of phosphate groups from ATP to the enzyme, the energy for which comes from hydrolysing ATP into ADP or AMP. Protein kinases (PKs) are the effectors of phosphorylation and catalyse the transfer of a γ-phosphate from ATP to specific amino acids on proteins. Proteins are phosphorylated predominantly on Ser, Thr and Tyr residues. In contrast, protein phosphatases (PPs) are the primary effectors of dephosphorylation and can be grouped into three main classes based on sequence, structure and catalytic function. Dephosphorylation releases phosphates into solution as free ions, because attaching them back to ATP would require energy input. PPP2CA is one of the four major serine/threonine-protein phosphatases. Shi [4] studied the mechanism through structure of the serine/threonine phosphatases. The enzyme protein serine/threonine phosphatase acts upon phosphorylated serine/threonine residues as follows (from Wikipedia contributors [5]) : [protein]−serine/threonine−phosphate+H2O= [protein]−serine/threonine+phosphate (1) Jones et al. [6] localized the the gene for the αisoform of the catalytic subunit of human PPP2CA to chromosome 5 using somatic cell hybrids, and then more finely mapped to chromosome region 5q23−>q31 by in situ hybridization using a tritiated cDNA probe. In their review, Seshacharyulu et al. [7] focus on the structural complexity of serine/threonine phosphatase PP2A and summarize its expression pattern in cancer while discussing about the PP2A interacting and regulatory proteins and substrates. Finally, they also review the mouse models developed to understand the biological role of PP2A subunits in an in vivo model system. Further, Reynhout and Janssens [8] summarize current knowledge on physiologic functions of PP2A in germ cell maturation, tumor suppression, metabolic regulation, embryonic development, and homeostasis of adult brain, liver, heart, immune system, lung, kidney, intestine, skin, eye and bone, all of which were retrieved from in vivo studies using PP2A transgenic, knockout or knockin mice. 3 The phosphatases PP1 and PP2A catalytic subunits exist in cells in form of holoenzymes, which impart substrate specificity and their contribution to the recognition of substrates is unclear. Hoermann et al. [9] develop a phosphopeptide library approach and a phosphoproteomic assay to demonstrate that the specificity of PP1 and PP2A holoenzymes towards pThr and of PP1 for basic motifs adjacent to the phosphorylation site are due to intrinsic properties of the catalytic subunits. PP2A holoenzyme complex comprises a scaffolding (A), regulatory (B), and catalytic (C) subunit, with PPP2CA being the principal catalytic subunit. To define the full scope of PP2A substrates in cells, Brewer et al. [10] employed dTAG proteolysistargeting chimeras to efficiently and selectively degrade dTAG-PPP2CA in homozygous knock-in HEK293 cells. Via unbiased global phospho-proteomics, they identified 2,204 proteins with significantly increased phosphorylation upon dTAG-PPP2CA degradation, implicating them as potential PPP2CA substrates and through bioinformatic analyses, they revealed involvement of the potential PPP2CA substrates in spliceosome function, cell cycle, RNA transport, and ubiquitin-mediated proteolysis. Using quantitative phospho-proteomic analysis, they identified a total of 39,103 phosphopeptides belonging to 5,829 proteins, in DMSOand dTAG-13-treated dTAG/dTAGPPP2CA HEK293 cell extracts. Of these, 2,651 phospho-peptides corresponding to 1,149 proteins showed a significant increase in abundance of >2-fold in dTAG-13-treated cells compared to DMSO-treated controls, while 6,280 phospho-peptides corresponding to 2,204 proteins showed a significant increase in abundance of >1.5-fold. I present 3rd order combinations of PPP2CA with other genes, that the machine learning based search engine points to, as possible synergistic combinations that might be working in time. 3. Methods Please refer to sections of Sinha [2] for methods, design of study and analysis of data for 2nd order combinations. The same method and design of study is used to generate results for 3rd order combinations presented in this study. 4. Time series data Gujral and MacBeath [1] present a set of 71 WNT-related gene expression values for 6 different times points over a range of 24-hour period using qPCR. The changes represent the fold-change in the expression levels of genes in 200 ng/mL WNT3A-stimulated HEK 293 cells in time relative to their levels in unstimulated, serum-starved cells at 0hour. Gujral and MacBeath [1] state that qPCR data are the means of three biological replicates. Only genes whose mean transcript levels changed by more than two-fold at one or more time points during the 24-hour time course were considered significant. Positive (negative) numbers represent up (down) -regulation. We have already covered the issues related to these data sets in detail in Sinha [11]. Readers are requested to go through them in the pointed reference. The tools of study which are used here have been published in another foundational work in Sinha [11]. 4 5. Design of experiment 5.1. Pipeline for time series data For the case of time series data, interactions among the contributing factors are studied by comparing triplets of fold-changes at single time points. The prodecure begins with the generation of distribution around measurements at single time points with added noise is done to estimate the indices. A distribution is generated for the fold changes at single time points. Then for every gene, there is a vector of values representing fold changes as well as deviations in fold changes for different time points and durations between time points, respectively. Next a listing of all Cn kcombinations for knumber of genes from a total of ngenes is generated. kis ≥2 and ≤(n−1). Each of the combination of order krepresents a unique set of interaction between the involved genetic factors. After this, the datasets are combined in a specifed format which go as input as per the requirement of a particular sensitivity analysis method. Thus for each pth combination in Cn kcombinations, the dataset is prepared in the required format from the distributions for two separate cases which have been discussed above. (See .R code in mainScript-1-1.R). After the data has been transformed, vectorized programming is employed for density based sensitivity analysis and looping is employed for variance based sensitivity analysis to compute the required sensitivity indices for each of the pcombinations. This procedure is done for different kinds of sensitivity analysis methods. After the above sensitivity indices have been stored for each of the pth combination, the next step in the design of experiment is conducted. Since there is only one recording of sensitivity index per combination, each combination forms a training example which is alloted a training index and the sensitivity indices of the individual genetic factors form the training example. Thus there are Cn ktraining examples for kth order interaction. Using this training set SVMRank learn Joachims [3] is used to generate a model on default value Cvalue of 20. In the current experiment on toy model Cvalue has not been tunned. The training set helps in the generation of the model as the different gene combinations are numbered in order which are used as rank indices. The model is then used to generate score on the observations in the testing set using the SV MRank classi f y Joachims [3]. Note that due to availability of only one example per combination, after the model has been built, the same training data is used as test data to generates the scores. This procedure is executed for each and every sensitivity analysis method. This is followed by sorting of these scores along with the rank indices (i.e the training indices) already assigned to the gene combinations. The end result is a sorted order of the gene combinations based on the ranking score learned by the SV MRank algorithm. Finally, this entire procedure is computed for sensitivity indices generated for each and every fold change at time point and deviations in fold change at different durations. Observing the changing rank of a particular combination at different times and different time periods will reveal how a combination is behaving. Note that the following is the order in which the files should be executed in R, in order, for obtaining the desired results (Note that the code will not be explained here) - • use source(”mainScript-1-1.R”) with arguments for Dynamic data •source(”SVMRankResults-D.R”), to rank the interactions (again this needs to be done separately for 5 different kinds of SA methods), •use source(”Combine-Time-files.R”), if computing indices separately via previous file, •source(”Sort-n-Plot-D.R”) to sort the interactions. Note that the sorting is chages the interaction ranking in time. Thus •use source(”Interaction-Priority-Intime.R”) to find the prioritized ranking of each and every interaction over the different time points and finally •use source(”Print-RankingAND-Interaction-Rank.R”) to print individual ranking of the required input factor with other interaction factors. 6. Results & Discussion 6.1. Time series data by Gujral and MacBeath [1] NOTE - Ranking was assigned on scores that were sorted in DECREASING values. So, 1 was assigned to highest score and vice versa. Results for the 3rd order interactions are presented here. The results first discuss the behaviour of interactions across the snapshots of time using the computed sensitivities on fold change measurements per time snapshot. The analysis was done using 4 different sensitivity indices. Out of the 71C3combinations, I consider/present only those combinations that show a ranking within first 10,000 out of 57,155. This choice is liberal and biologists/oncologists can have a more stricter choice as per need. Two observations are made, •the ranking of a particular combination is conserved (i.e within the 10,000 range) in a particular time point or in the early phase or late phase of WNT3A stimulation, across the majority of the four sensitivity methods, which is a strict criteria of assessment or •the ranking of a particular combination is conserved across time points/phase (i.e they are within the 10,000 range) and the majority of the four sensitivity methods, which is relaxed criteria of assessment. Applying this filter helps reveal important combinations of interest that might be working synergistically at a higher order level in the cell. Regarding technical points of implementation, the rankings were generated without scaling/normalizing the time series data provided by Gujral and MacBeath [1]. For estimating the sensitivity indices, a small gaussian distribution using the function rnorm that generates a vector of normally distributed random variables given a vector length n (here 9, the 10th one is the mean/recorded gene regulation itself), a population mean µand population standard deviation σ. The syntax for using rnorm is as follows: rnorm(n, mean, sd). Further, I use the jitter funtion to add a little bit of noise to the data. This helps to see if the generated rankings are robust or not. 6.2. Enumeration and ranking of 2415 PPP2CA-X-X combinations from Gujral and MacBeath [1] In the supplementary section, I present four files, each containing the rankings of 3rd order combinations, that wary in time (shown for 5 time points). Each file represents the rankings computed using a particular sensitivity method. The changing rankings in time for a particular combination represents the importance of contribution/role that combination plays in the cell stimulated with WNT3A. The sensitivity methods used 6 are Hilbert Schmidt Independence Criterion indices (HSIC) indices (with rbf and linear kernel in Da Veiga [12]) and Sobol indicies (with 2002 implementation in Saltelli [13] and martinez implementation in Martinez [14] and Baudin et al. [15]). 6.3. Conserved machine learning rankings for tested PPP2CA-X-X combinations A total of 2415, 3rd order combinations involving PPP2CA were obtained from a full set of 71C3= 57155 combinations. Further, from this selected set, using the above criteria for conserved rankings, I report/tabulate the meaningful combinations that might be working synergistically. Tables 2, 3 and 4 show the rankings for the same combinations as in table 1, but using rbf kernel for HSIC, 2002 implementation for SOBOL and martinez implementation for SOBOL, respectively. As one tallies the rankings of across these tables for a particular combination, one finds that the role of the combination of interest is conserved. This conservation points to the existence of the biological synergy, whether the combination has been tested or unexplored/untested. 6.3.1. Examining the behaviour of AXIN-PPP2CA-X combinations Willert et al. [16] show that AXIN is dephosphorylated in response to WNT signaling and the dephosphorylated AXIN binds β-catenin less efficiently than the phosphorylated form. Thus, WNT signaling lowered AXIN’s affinity for β-catenin, thereby disengaging β-catenin from the degradation machinery. Ikeda et al. [17] show that the heterodimeric form of PP2A directly binds to AXIN, and PP2A complexed with AXIN dephosphorylated APC phosphorylated by GSK3β. Taken together, their results suggest that GSK3β-dependent phosphorylation of APC can be modulated by β-catenin and PP2A complexed with AXIN. Looking at the tables above, one finds the following combinations for AXIN1 along with PPP2CA, to be prominent at 3rd order level - AXIN1-FZD2-PPP2CA, AES-AXIN1-PPP2CA and AXIN1-FOXN1-PPP2CA. All these combinations indicate the existence of a possible synergy when they take a higher rank in the list of combinations. 6.3.2. Examining the behaviour of APC-PPP2CA-X combinations Seeling et al. [18] show that PP2A regulatory subunit, B56, interacted with APC in the yeast two-hybrid system and expression of B56 reduced the abundance of β-catenin and inhibited transcription of β-catenin target genes in mammalian cells and Xenopusembryo explants. Further, the B56-dependent decrease in β-catenin was blocked by oncogenic mutations in β-catenin or APC, and by proteasome inhibitors. B56 may direct PP2A to dephosphorylate specific components of the APC-dependent signaling complex and thereby inhibit WNT signaling. Their study suggested that PP2A heterotrimers containing the B56 regulatory subunit functioned in the WNT signaling complex to down-regulate β-catenin, perhaps through an interaction of B56 and the NH2-terminus of APC, to dephosphorylate specific components of the APC-dependent signaling complex and thereby inhibit WNT signaling. Looking at the tables above, 7 RANKING @tiUSING HSIC - LINEAR 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 CXXC4-FZD7-PPP2CA 90 9819 23105 12397 36836 FZD7-PPP2CA-SFRP4 121 6838 2610 6467 34834 FZD1-FZD7-PPP2CA 188 20213 26368 10322 10221 AES-AXIN1-PPP2CA 213 21529 16225 49328 28118 CSNK2A1-MYC-PPP2CA 271 37740 40109 23353 26864 APC-PITX2-PPP2CA 334 14753 2900 19245 30663 NLK-PORCN-PPP2CA 339 30605 43435 41734 4166 PPP2CA-SFRP4-WNT2B 341 52722 25356 52978 1988 KREMEN1-PPP2CA-SFRP4 362 833 2893 48271 49787 FZD7-NKD1-PPP2CA 369 8802 35791 35537 25608 DKK1-JUN-PPP2CA 408 27394 50340 28816 37806 BCL9-PORCN-PPP2CA 415 15959 34237 38189 34019 DIXDC1-NKD1-PPP2CA 440 13162 40123 35144 6197 CXXC4-PORCN-PPP2CA 481 22027 38709 38555 36190 DVL2-JUN-PPP2CA 504 43997 37157 2445 3613 FZD7-PPP2CA-SENP2 540 11483 354 3539 34512 CCND3-PPP2CA-SFRP4 631 37304 7579 35575 51606 PPP2CA-WNT1-WNT3A 732 6546 10800 53535 9982 DKK1-PPP2CA-SFRP4 815 3000 44035 44920 42238 FBXW11-LRP6-PPP2CA 822 25174 30818 7903 41062 FSHB-FZD2-PPP2CA 922 15170 15248 37534 18116 CSNK1G1-NKD1-PPP2CA 926 17356 35232 35369 12912 DVL1-FOXN1-PPP2CA 977 25908 15392 4810 48741 DAAM1-FOXN1-PPP2CA 1014 50289 35190 8828 44977 CTBP1-GSK3A-PPP2CA 1015 17069 883 43553 18661 FOSL1-FOXN1-PPP2CA 1072 22257 16994 9632 25457 CSNK1G1-FZD2-PPP2CA 1082 23722 6311 56537 35909 FOSL1-FZD7-PPP2CA 1103 37962 25642 24479 50931 FZD8-GSK3A-PPP2CA 1156 55038 1341 50154 22462 CTNNBIP1-JUN-PPP2CA 1160 17287 25595 27276 6859 CSNK1D-FOXN1-PPP2CA 1252 17427 16752 21509 34632 FBXW11-FOXN1-PPP2CA 1357 8568 16917 13742 17017 FRZB-JUN-PPP2CA 1383 13534 31714 933 1610 CSNK2A1-FOXN1-PPP2CA 1415 18676 25465 10950 9597 PPP2CA-T-WNT3A 1435 46308 13460 20855 21652 FRZB-FZD2-PPP2CA 1437 17024 10096 32430 33434 CSNK1G1-FSHB-PPP2CA 1518 8523 49087 1235 27660 CCND1-JUN-PPP2CA 1525 10554 35624 43398 54711 CSNK2A1-FSHB-PPP2CA 1544 21794 56458 12205 17377 KREMEN1-PPP2CA-WNT2B 1585 14425 197 53141 37098 DKK1-FOXN1-PPP2CA 1604 39842 25588 9674 22711 FRZB-GSK3A-PPP2CA 1624 17689 1801 14917 16232 BTRC-PPP2CA-WNT4 1638 35087 32587 22025 51750 FZD5-CCND3-PPP2CA 1659 15307 15361 43028 36850 PPP2CA-WNT1-WNT4 1668 20515 14866 23920 21265 AES-FOXN1-PPP2CA 1682 1873 24710 15282 25404 FZD8-PORCN-PPP2CA 1713 51044 29439 20913 20485 FZD5-PPP2CA-SFRP4 1743 159 4915 40308 39122 CSNK1A1-LRP6-PPP2CA 1764 14481 17648 2115 28490 FOSL1-NKD1-PPP2CA 1845 14335 40783 24581 23211 FZD7-PPP2CA-WNT2B 1854 16103 178 39687 33132 APC-BTRC-PPP2CA 1858 9257 27876 20401 45249 FOSL1-PORCN-PPP2CA 1917 33808 39279 26918 22451 FRZB-NKD1-PPP2CA 1923 12714 42040 19751 23511 AXIN1-FZD2-PPP2CA 2006 16397 13459 38219 30167 DKK1-PPP2CA-SENP2 2068 7041 27771 41618 51588 CSNK1D-FGF4-PPP2CA 2103 20756 11349 39534 15407 FZD8-JUN-PPP2CA 2138 54751 19454 329 37826 DKK1-NKD1-PPP2CA 2158 21388 52610 49493 9273 CSNK1G1-JUN-PPP2CA 2194 17201 31808 2244 24092 FBXW11-JUN-PPP2CA 2233 16460 49863 18037 53295 FZD6-GSK3A-PPP2CA 2263 8382 9486 47912 53738 FOSL1-GSK3A-PPP2CA 2355 22861 911 45587 39936 DAAM1-LRP6-PPP2CA 2361 55059 33326 5010 40824 CSNK1A1-GSK3B-PPP2CA 2386 9444 19245 4822 2538 BCL9-JUN-PPP2CA 2418 12028 23042 32377 49157 FRZB-PPP2CA-SFRP4 2426 1736 4164 8420 12593 CTBP2-FOXN1-PPP2CA 2521 39676 20597 7309 30523 DKK1-GSK3A-PPP2CA 2529 18354 4501 38825 41214 BCL9-FGF4-PPP2CA 2534 11917 24166 51167 48006 FRAT1-JUN-PPP2CA 2547 32152 28002 1650 1276 APC-PORCN-PPP2CA 2593 14870 33789 20601 17503 CXXC4-FZD2-PPP2CA 2607 29974 9988 18663 50107 FRAT1-GSK3A-PPP2CA 2638 38412 1286 35232 13767 BTRC-PPP2CA-SFRP4 2649 45450 55216 41081 48220 FRZB-PORCN-PPP2CA 2663 20449 38661 30250 22743 EP300-FZD2-PPP2CA 2702 12448 13262 50046 20911 DAAM1-GSK3A-PPP2CA 2745 56040 14776 51085 48063 CCND3-PPP2CA-WNT2 2799 33277 2300 17044 49518 FBXW2-NKD1-PPP2CA 2839 23721 34814 16169 56279 PITX2-PORCN-PPP2CA 2847 24874 33030 43172 21878 CTBP1-FOXN1-PPP2CA 2940 12144 24777 14758 54034 FZD7-PPP2CA-WNT2 2964 18295 2936 723 46075 CCND3-PPP2CA-SENP2 3064 34905 2328 40674 48547 PPP2CA-WNT1-WNT5A 3127 17958 20275 50785 22700 AXIN1-FOXN1-PPP2CA 3158 9114 17327 16618 10071 NKD1-PPP2CA-SFRP4 3169 46409 13326 29656 43339 FGF4-FOSL1-PPP2CA 3175 29507 37541 48577 51007 AES-EP300-PPP2CA 3218 22740 43678 26060 41565 CTBP1-FGF4-PPP2CA 3224 17798 20777 35300 21473 DIXDC1-FOXN1-PPP2CA 3235 4976 13715 8671 2172 APC-FOXN1-PPP2CA 3259 6614 11418 7307 27730 CXXC4-FOXN1-PPP2CA 3277 19301 19575 24116 33763 CCND3-NKD1-PPP2CA 3285 34449 42225 49257 26308 GSK3B-LRP6-PPP2CA 3297 38834 21470 36021 39453 FOSL1-JUN-PPP2CA 3303 25232 24882 2685 4764 FBXW11-FZD2-PPP2CA 3313 19812 19095 30619 15139 FOSL1-PPP2CA-SFRP4 3331 18404 4076 19432 6798 CXXC4-FGF4-PPP2CA 3340 9402 29074 35299 53290 CCND1-NKD1-PPP2CA 3363 11274 37598 54652 44854 PPP2CA-TCF7-WNT3A 3403 35126 15557 53522 39901 FBXW2-PORCN-PPP2CA 3520 27173 22022 46128 27903 AES-DVL1-PPP2CA 3635 6018 44845 39717 52837 CXXC4-JUN-PPP2CA 3661 32213 26859 462 30021 DIXDC1-FZD2-PPP2CA 3662 17928 10686 40372 29250 DAAM1-FGF4-PPP2CA 3665 54152 33362 55780 29602 FZD8-NKD1-PPP2CA 3681 49836 33279 27543 15965 APC-FZD6-PPP2CA 3769 15339 20803 56646 49253 FZD6-PORCN-PPP2CA 3788 4773 43421 3483 7109 BTRC-PPP2CA-T 3795 43011 19216 31904 44253 FBXW11-GSK3A-PPP2CA 3852 26830 8953 53718 5344 FZD5-PORCN-PPP2CA 3939 10638 39482 21947 53216 CSNK2A1-NKD1-PPP2CA 4025 15782 45180 27270 33423 CSNK1D-GSK3A-PPP2CA 4064 22624 794 37163 9946 DIXDC1-PITX2-PPP2CA 4123 13781 1911 14325 45522 DKK1-DVL2-PPP2CA 4138 22771 24516 8573 52303 FZD8-LEF1-PPP2CA 4177 56827 23717 17918 45636 FRAT1-PORCN-PPP2CA 4204 33092 39327 31761 14446 CCND3-PPP2CA-RHOU 4207 37434 5715 53381 50362 CSNK2A1-GSK3A-PPP2CA 4214 31336 13790 24722 34871 PPP2CA-WNT1-WNT2B 4221 27679 5501 15908 49412 FZD1-PORCN-PPP2CA 4258 20050 38961 32098 15000 FZD7-PPP2CA-TLE2 4271 5239 7231 4827 26456 CTNNB1-GSK3A-PPP2CA 4281 19241 845 32420 18538 CCND1-CTNNBIP1-PPP2CA 4294 35830 34555 37986 51377 KREMEN1-PPP2CA-SENP2 4308 6165 521 42127 55415 Table 1: Rankings of PPP2CA-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - linear one finds the following combinations for AXIN1 along with PPP2CA, to be prominent at 3rd order level - APC-PITX2-PPP2CA, APC-BTRC-PPP2CA, APC-PORCNPPP2CA, APC-FOXN1-PPP2CA and APC-FZD6-PPP2CA . All these combinations indicate the existence of a possible synergy when they take a higher rank in the list of 8 RANKING @tiUSING HSIC - RBF 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 CXXC4-FZD7-PPP2CA 7569 8146 8359 8809 29695 FZD7-PPP2CA-SFRP4 55917 24461 55999 42034 109 FZD1-FZD7-PPP2CA 25993 15034 5386 38356 42649 AES-AXIN1-PPP2CA 35492 17204 7775 30194 15980 CSNK2A1-MYC-PPP2CA 1074 26347 29622 32938 20490 APC-PITX2-PPP2CA 4042 33446 8698 18579 52910 NLK-PORCN-PPP2CA 6640 37879 19229 30844 45007 PPP2CA-SFRP4-WNT2B 55580 46429 27467 46270 1672 KREMEN1-PPP2CA-SFRP4 19893 13395 43502 13961 6353 FZD7-NKD1-PPP2CA 15127 35957 31673 37533 1778 DKK1-JUN-PPP2CA 872 31181 33266 21737 30431 BCL9-PORCN-PPP2CA 19443 12204 3989 2860 44188 DIXDC1-NKD1-PPP2CA 9807 43231 37477 30170 1078 CXXC4-PORCN-PPP2CA 9663 20715 7491 12348 51537 DVL2-JUN-PPP2CA 12969 55374 21200 14940 32734 FZD7-PPP2CA-SENP2 32571 2154 49346 21000 26293 CCND3-PPP2CA-SFRP4 46916 36878 55760 47246 5936 PPP2CA-WNT1-WNT3A 4487 6874 35552 24125 19493 DKK1-PPP2CA-SFRP4 55812 1395 43571 11747 8786 FBXW11-LRP6-PPP2CA 1498 23055 36283 10416 6272 FSHB-FZD2-PPP2CA 3195 15838 9446 3827 55246 CSNK1G1-NKD1-PPP2CA 8799 37693 42013 33688 15900 DVL1-FOXN1-PPP2CA 3346 22402 46063 15109 11834 DAAM1-FOXN1-PPP2CA 609 47634 41711 1552 16742 CTBP1-GSK3A-PPP2CA 841 30822 34980 31747 27039 FOSL1-FOXN1-PPP2CA 2050 20570 56654 7068 31115 CSNK1G1-FZD2-PPP2CA 21192 37740 25191 13344 44898 FOSL1-FZD7-PPP2CA 54451 46826 3194 28518 28447 FZD8-GSK3A-PPP2CA 2976 56637 43724 15698 9354 CTNNBIP1-JUN-PPP2CA 38196 36635 20141 33944 38991 CSNK1D-FOXN1-PPP2CA 677 25144 54449 22625 44186 FBXW11-FOXN1-PPP2CA 551 13483 50027 13549 5121 FRZB-JUN-PPP2CA 5873 49297 3845 20493 26715 CSNK2A1-FOXN1-PPP2CA 278 35287 48412 13705 23060 PPP2CA-T-WNT3A 5197 44818 17755 38317 4985 FRZB-FZD2-PPP2CA 5746 32562 6231 841 53357 CSNK1G1-FSHB-PPP2CA 50070 6934 23453 5427 21042 CCND1-JUN-PPP2CA 40595 14062 3971 9088 1571 CSNK2A1-FSHB-PPP2CA 14473 24470 49869 6189 28547 KREMEN1-PPP2CA-WNT2B 25164 28694 34566 35971 17687 DKK1-FOXN1-PPP2CA 318 42768 43072 5776 28884 FRZB-GSK3A-PPP2CA 1260 45624 46852 17140 25632 BTRC-PPP2CA-WNT4 13782 33555 41012 5416 5243 FZD5-CCND3-PPP2CA 12325 45904 23439 31013 47189 PPP2CA-WNT1-WNT4 1316 18383 42717 1659 16550 AES-FOXN1-PPP2CA 2186 37914 47459 13594 25139 FZD8-PORCN-PPP2CA 4402 55143 2005 12002 37055 FZD5-PPP2CA-SFRP4 47311 664 55943 2216 11593 CSNK1A1-LRP6-PPP2CA 16978 31404 24202 26610 21103 FOSL1-NKD1-PPP2CA 21934 29546 42661 35675 17456 FZD7-PPP2CA-WNT2B 54842 24004 41817 52487 4145 APC-BTRC-PPP2CA 4261 9237 29395 1410 54704 FOSL1-PORCN-PPP2CA 857 51996 6161 7188 48966 FRZB-NKD1-PPP2CA 1942 36043 22753 20309 35265 AXIN1-FZD2-PPP2CA 7416 35245 5191 10583 50255 DKK1-PPP2CA-SENP2 54622 26587 44748 26854 52986 CSNK1D-FGF4-PPP2CA 3187 47005 13018 23779 39413 FZD8-JUN-PPP2CA 6710 57073 4320 19608 19584 DKK1-NKD1-PPP2CA 845 43108 38644 15851 21492 CSNK1G1-JUN-PPP2CA 3740 31530 16470 29436 25461 FBXW11-JUN-PPP2CA 9373 17254 50937 17677 11233 FZD6-GSK3A-PPP2CA 5274 18421 36558 40884 12260 FOSL1-GSK3A-PPP2CA 12185 33599 52257 41997 37270 DAAM1-LRP6-PPP2CA 7255 52778 39048 19324 16287 CSNK1A1-GSK3B-PPP2CA 49785 6326 16521 6418 41571 BCL9-JUN-PPP2CA 32542 20885 2900 20877 26335 FRZB-PPP2CA-SFRP4 52697 31347 51732 14973 22371 CTBP2-FOXN1-PPP2CA 11045 38092 52532 1356 32277 DKK1-GSK3A-PPP2CA 55 13432 43272 11984 17851 BCL9-FGF4-PPP2CA 11758 12541 2912 24152 44954 FRAT1-JUN-PPP2CA 5774 39371 22955 20982 28298 APC-PORCN-PPP2CA 1049 35875 5774 24955 45838 CXXC4-FZD2-PPP2CA 11564 33338 12458 3001 54858 FRAT1-GSK3A-PPP2CA 761 45444 39194 15449 37603 BTRC-PPP2CA-SFRP4 23701 47316 54163 28713 3631 FRZB-PORCN-PPP2CA 4450 31541 6488 11176 47491 EP300-FZD2-PPP2CA 8059 21295 864 5359 44323 DAAM1-GSK3A-PPP2CA 5817 54945 44951 39069 16387 CCND3-PPP2CA-WNT2 36713 24526 54134 13742 8094 FBXW2-NKD1-PPP2CA 15124 44885 17926 12733 2217 PITX2-PORCN-PPP2CA 9603 23189 295 19554 39138 CTBP1-FOXN1-PPP2CA 5258 32520 39917 10063 30654 FZD7-PPP2CA-WNT2 40336 26902 57064 15776 2793 CCND3-PPP2CA-SENP2 42453 28981 45394 1751 13534 PPP2CA-WNT1-WNT5A 1188 16609 32224 18050 32772 AXIN1-FOXN1-PPP2CA 5167 25187 29990 1127 16140 NKD1-PPP2CA-SFRP4 40324 48641 51898 9183 19710 FGF4-FOSL1-PPP2CA 42277 28170 8091 25442 19223 AES-EP300-PPP2CA 40675 14926 3496 32198 22117 CTBP1-FGF4-PPP2CA 3823 27672 4141 30441 47610 DIXDC1-FOXN1-PPP2CA 1796 26077 33448 2622 1308 APC-FOXN1-PPP2CA 1206 4752 36187 915 15627 CXXC4-FOXN1-PPP2CA 1922 14941 49768 109 32177 CCND3-NKD1-PPP2CA 19772 35706 47585 36014 3731 GSK3B-LRP6-PPP2CA 3750 50098 25527 8818 33308 FOSL1-JUN-PPP2CA 21063 49029 6511 48244 44672 FBXW11-FZD2-PPP2CA 3595 19254 48774 3116 27274 FOSL1-PPP2CA-SFRP4 31966 44588 53340 39850 33169 CXXC4-FGF4-PPP2CA 896 24972 12137 33837 43059 CCND1-NKD1-PPP2CA 31094 31919 45839 10804 5523 PPP2CA-TCF7-WNT3A 36467 12019 36560 49458 39632 FBXW2-PORCN-PPP2CA 13731 19973 12814 22383 21062 AES-DVL1-PPP2CA 19128 12832 8371 36154 55869 CXXC4-JUN-PPP2CA 2697 40480 12704 20114 45550 DIXDC1-FZD2-PPP2CA 8902 30326 5894 8118 11004 DAAM1-FGF4-PPP2CA 1734 53041 30385 36043 10689 FZD8-NKD1-PPP2CA 31727 55399 17443 4434 10950 APC-FZD6-PPP2CA 9255 28099 28345 32780 55276 FZD6-PORCN-PPP2CA 15 24063 29679 24492 40813 BTRC-PPP2CA-T 42753 54274 45558 42965 26218 FBXW11-GSK3A-PPP2CA 5045 19409 45810 19725 16426 FZD5-PORCN-PPP2CA 23568 21263 11481 8780 45819 CSNK2A1-NKD1-PPP2CA 6422 22743 30647 42195 19402 CSNK1D-GSK3A-PPP2CA 1046 40897 47094 14850 34061 DIXDC1-PITX2-PPP2CA 5166 50281 13595 5982 18374 DKK1-DVL2-PPP2CA 2732 42296 39816 1658 9495 FZD8-LEF1-PPP2CA 4960 57112 5320 15616 6681 FRAT1-PORCN-PPP2CA 2795 32747 21472 22562 46364 CCND3-PPP2CA-RHOU 24349 40806 49185 54625 8752 CSNK2A1-GSK3A-PPP2CA 8360 44268 37323 48745 27502 PPP2CA-WNT1-WNT2B 5623 15143 40493 2300 24871 FZD1-PORCN-PPP2CA 4367 38671 4743 23669 51214 FZD7-PPP2CA-TLE2 56558 28223 50599 11481 3392 CTNNB1-GSK3A-PPP2CA 2288 19240 46473 33276 26230 CCND1-CTNNBIP1-PPP2CA 9077 28086 5722 5046 6609 KREMEN1-PPP2CA-SENP2 3092 4451 26948 15879 51552 Table 2: Rankings of PPP2CA-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - rbf combinations. 9 6.3.11. Examining the behaviour of EP300-PPP2CA-X combinations Transcriptional coactivator p300 (or EP300) is required for embryonic development and cell proliferation. Valproic acid, a histone deacetylase inhibitor, is widely used in the therapy of epilepsy and bipolar disorder. Chen et al. [31] report that valproic acid stimulates proteasome-dependent p300 degradation through augmentation of gene expression of the B56γregulatory subunits of protein phosphatase 2A. The B56γ3 regulatory and catalytic subunits of protein phosphatase 2A interact with p300. Overexpression of the B56γ3 subunit led to proteasome-mediated p300 degradation and repressed p300-dependent transcriptional activation, which required the B56γ3 interaction domain of p300. Conversely, silencing of the B56γsubunit expression by RNA interference increased the stability and transcriptional activity of p300. Via quantitative phospho-proteomic analysis, Brewer et al. [10] identified EP300 to be substrate of PPP2CA. Looking at the tables above, one finds the following combinations for Ep300 along with PPP2CA, to be prominent at 3rd order level - EP300-FZD2-PPP2CA and AESEP300-PPP2CA. All these combinations indicate the existence of a possible synergy when they take a higher rank in the list of combinations. 7. Conclusion This manuscript studies the time behaviour of 3rd order combinations of PPP2CA in WNT3A stimulated HEK 293 cells. Based on the extablished 2nd order combinations of the PPP2CA, 3rd order combinations emerge using the machine learning based search engine. These 3rd order combinations might be of interest for further wet lab investigations. Competing interests No competing interest is declared. Author contributions statement SS conceived and designed the experiments; wrote the code; performed the experiments; analyzed the data; wrote the manuscript. Availability of code Code for time series data available at CERN based Zenodo on https://zenodo.org/ records/14637456. 16 Acknowledgments Special thanks to Mrs. Rita Sinha and late Mr. Prabhat Sinha for supporting the author financially, without which this work could not have been made possible. Supplementary The following files (ending with .txt and can be opened in R or in simple text processing program) with these names are made available with this manuscript. For PPP2CA, (1) -3-odr-TP-ranking-linear.txt, (2) -3-odr-TP-ranking-rbf.txt, (3) -3odr-TP-ranking-2002.txt, and (4) -3-odr-TP-ranking-martinez.txt, contain rankings for 3rd order combinations across each time point for, HSIC (linear kernel), HSIC (rbf kernel), SOBOL (2002 implementation) and SOBOL (martinez implementation), respectively. References [1] T. S. Gujral, G. 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