porcupine O-acyltransferase (PORCN) : Time behavioural study of 3rd order combinations in WNT3A stimulated HEK 293 cells

porcupine O-acyltransferase (PORCN) : 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 104-Madhurisha Heights Phase 1, Risali, Bhilai-490006, India Abstract PORCN belongs to the family of membrane bound O-acyl transferase (MBOAT). MBOAT members contain multiple transmembrane domains and carry two conserved residues, a conserved histidine (His) embedded in a hydrophobic stretch of residues and an asparagine (Asn) or histidine within a more hydrophilic region some 30-50 residues upstream. It can add palmitoleate groups to WNT proteins that is necessary for WNT ligand secretion. 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 PORCN related 3rd order combinations in a forest of 71C3combinations using four different sensitivity methods; •show the conserved rankings for PORCN-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 PORCN comb. in WNT3A stimulated cells Email address: [email protected] (shriprakash sinha) 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 June 14, 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 PORCN 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. Porcupine (PORCN) The Drosophila segment polarity gene product Porcupine (Porc) was first identified as being necessary for processing Wingless (Wg), a Drosophila Wnt (Wnt) family member. Tanaka et al. [4] identified Mouse (Mporc) and Xenopus (Xporc) homologs of porc and found that they encode endoplasmic reticulum (ER) proteins with multiple transmembrane domains. Further, Mporc mRNA was differentially expressed during embryogenesis and in various adult tissues, demonstrating that the alternative splicing is regulated to synthesize the specific types of Mporc. In transfected mammalian cells, they found all types of Mporc affected the processing of mouse WNT1, WNT3A, WNT4, WNT6, and WNT7B but not WNT5A. Lastly, they also found that all types of Mporc co-immunoprecipitated with various WNT proteins. Their results suggested that Mporc may function as a chaperone-like molecule for WNT. Caricasole et al. [5] report that the human Porcupine locus (MG61/PORC) spans 15 exons over approximately 12 kb of genomic sequence on Xp11.23. Like its mouse and Xenopus homologues, MG61/PORC encodes four protein isoforms (AD) generated through alternative splicing and expressed in a tissue-specific fashion. They present evidence indicating that MG61/PORC can influence the activity of a human WNT7A expression construct in a T-cell factor-responsive reporter assay. Liu et al. [6] indicate that post-translational modification of WNTs includes lipid modification and glycosylation. The former is performed by PORCN. PORCN is a membrane-bound O-acyltransferase located in the endoplasmic reticulum and can add palmitoleate groups to WNT proteins that is necessary for WNT ligand secretion, and it is a member of the membrane-bound O-acyltransferases (MBOATs). Lipid modification is necessary for Wnt activity, and the opposite is true for glycosylation as observed by Willert et al. [7]. Liu et al. [8] developed a screen for small molecules that blocked WNT secretion and discovered LGK974, a potent and specific small-molecule PORCN inhibitor. They show that LGK974 inhibits WNT signaling, including reduction of the WNT-dependent LRP6 phosphorylation and the expression of WNT target genes, like as AXIN2. The inhibitor is effective in multiple tumor models at well-tolerated doses. Together, their findings provide a strategy and and a tool for targeting WNT-driven cancers through the inhibition of PORCN. Further down the line, Madan et al. [9] de3 veloped a novel potent, orally available PORCN inhibitor, ETC-1922159 that blocked the secretion and activity of all WNTs. ETC-1922159 is remarkably effective in treating RSPO-translocation bearing colorectal cancer (CRC) patient-derived xenografts. This is the first example of effective targeted therapy for this subset of CRC. By this demonstration they show that inhibition of WNT signaling by PORCN inhibition holds promise as differentiation therapy in genetically defined human cancers. Sutton [10], conver a range of PORCN related developmental disorders. Till now, most research work has focused on the role of PORCN along with WNTs. In this research work, I present 3rd order combinations of PORCN with other genes, apart from the WNTs, 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]. 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 4 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 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. 5 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 PORCN-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 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 6.3. Conserved machine learning rankings for tested PORCN-X-X combinations A total of 2415, 3rd order combinations involving PORCN 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 CXXC4-PORCN-X combinations Ekici et al. [16] identified a homozygous splice site mutation in the CCDC88C gene as a novel cause of a complex hydrocephalic brain malformation, via positional cloning in a consanguineous family with autosomal recessive hydrocephalus. CCDC88C encodes DAPLE (HkRP2), a Hook-related protein with a binding domain for the central WNT signalling pathway protein DVL and DVL is inhibited by CXXC4 and CCDC88C, mediates Wnt signalling via inhibition of GSK3. Looking at the tables above, one finds the following combinations for CXXC4 along with PORCN, to be prominent at 3rd order level - CXXC4-PORCN-SENP2, CXXC4-PORCN-WNT2B, CXXC4-PORCN-TCF7, CXXC4-PORCN-WNT4, CXXC4-PORCN-SFRP4, CXXC4-PORCN-FBXW4, CXXC4PORCN-PPP2R1A, CXXC4-PORCN-PPP2CA and CXXC4-PORCN-WNT5A. 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 FOSL1-PORCN-X combinations The ability of the right ventricle (RV) to adapt to an increased pressure afterload determines survival in patients with pulmonary arterial hypertension. The WNT pathway plays an important role in the development of the RV and may also be implicated in adult cardiac remodeling. Nayakanti et al. [17] show that •WNT/β-catenin signaling molecules are upregulated in RV of patients with pulmonary arterial hypertension and animal models of RV overload (pulmonary artery banding-induced and monocrotaline rat models); •Activation of WNT/β-catenin signaling led to RV remodeling via transcriptional activation of FOSL1 and FOSL2; •Further, the mRNA expression profiling of WNT signaling molecules and immunofluorescence staining of aCTNNB1 (active β-catenin) and PORCN in RV myocardial tissues from human donors (control) and patients with idiopathic PAH (IPAH) demonstrated a significant upregulation of WNT/β-catenin signaling and •finally, pharmacological inhibition of WNT signaling using inhibitor of PORCN, LGK-974 attenuated fibrosis and cardiac hypertrophy leading to improvement in RV function in both, pulmonary artery bandingand monocrotaline-induced RV overload and the blockade using LGK-974 in WNT3Astimulated cardiac fibroblasts resulted in significant downregulation of mRNA and pro7 RANKING @tiUSING HSIC - LINEAR 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 CXXC4-PORCN-SENP2 10 33446 17033 17387 32253 CXXC4-PORCN-WNT4 49 6186 19448 20672 51388 CXXC4-PORCN-WNT2B 60 17097 46765 26682 32940 CXXC4-PORCN-SFRP4 80 28448 27819 28835 44315 CCND3-PORCN-WNT4 91 42809 21302 8841 38191 CCND3-PORCN-FBXW4 143 52831 15807 7893 14041 CXXC4-PORCN-TCF7 147 56042 37081 25343 32394 CXXC4-PORCN-FBXW4 168 26465 19028 12201 29004 CSNK1D-PORCN-SENP2 254 26488 43944 21480 22511 PITX2-PORCN-SENP2 270 20159 27718 25665 18827 FOSL1-PORCN-SFRP4 292 44876 11834 15903 36394 APC-PORCN-WNT4 310 33562 14239 15555 28291 FBXW2-PORCN-SFRP4 311 28602 1442 40221 38787 APC-PORCN-TCF7L1 320 33618 6586 15674 22420 FZD7-PORCN-FBXW4 331 24784 14647 7522 19704 NLK-PORCN-PPP2CA 339 30605 43435 41734 4166 FOSL1-PORCN-SENP2 349 31659 11861 10245 21849 NLK-PORCN-SFRP4 368 53128 52278 37793 4619 FZD5-PORCN-SENP2 377 32693 14759 8273 55426 FZD6-PORCN-WNT2B 379 19259 55786 24330 16739 FZD1-PORCN-SENP2 382 25494 17903 13765 16485 FZD6-PORCN-WNT4 394 39861 46523 785 6046 DKK1-PORCN-SENP2 400 29700 53693 2734 52517 CSNK1G1-PORCN-SENP2 410 36322 35831 12245 17913 BCL9-PORCN-PPP2CA 415 15959 34237 38189 34019 CXXC4-PORCN-PPP2R1A 424 27798 37533 50501 37618 FOSL1-PORCN-WNT4 455 37570 20729 10105 38487 PITX2-PORCN-WNT5A 460 36097 17787 53210 23094 APC-PORCN-SENP2 466 12387 7261 13497 19215 FRZB-PORCN-SENP2 467 13159 11733 10122 22264 LRP5-PORCN-SENP2 476 41616 8750 51278 21981 CXXC4-PORCN-PPP2CA 481 22027 38709 38555 36190 FZD6-PORCN-TLE2 482 39422 46346 2340 25087 FZD6-PORCN-SENP2 497 45212 45263 687 14394 BCL9-PORCN-SFRP4 498 29468 3887 31671 36411 PITX2-PORCN-FBXW4 499 12554 20396 26439 17842 FBXW2-PORCN-WNT4 531 21565 4445 36629 31422 CXXC4-PORCN-WNT5A 535 15961 16253 55679 46631 FZD8-PORCN-SFRP1 556 26745 7918 11285 32679 PITX2-PORCN-TCF7L1 572 42893 22702 25919 13429 CSNK1D-PORCN-WNT2B 573 51683 44617 47981 16208 FZD6-PORCN-SFRP1 615 42793 48666 4863 20350 PITX2-PORCN-WNT2B 630 42884 44910 50269 15186 FZD8-PORCN-SENP2 639 19701 11231 9109 12830 FOSL1-PORCN-WNT2B 670 19667 46545 40946 19709 KREMEN1-PORCN-WNT4 693 6753 23243 1864 44869 DKK1-PORCN-WNT4 724 36114 54617 3866 55268 NLK-PORCN-SENP2 751 51440 56864 21216 6122 FZD7-PORCN-WNT3A 780 35610 948 25632 12174 FZD8-PORCN-PPP2R1A 807 27524 31540 26322 40650 PITX2-PORCN-TLE1 813 43605 21513 22845 15786 FRAT1-PORCN-SFRP4 816 23261 18955 23358 33807 DKK1-PORCN-SFRP4 824 35293 52036 5440 53719 LRP5-PORCN-SFRP4 872 1618 3850 57012 38203 FRAT1-PORCN-SENP2 915 19198 9986 14436 18176 APC-PORCN-SFRP1 973 22402 11999 21396 30025 LRP5-PORCN-WNT2 975 5747 5037 56461 37999 CCND3-PORCN-SFRP1 984 46966 24689 11920 26237 CCND3-PORCN-TLE2 1002 48876 25152 13843 25724 PITX2-PORCN-PPP2R1A 1003 8071 39783 44256 5481 NLK-PORCN-WNT2B 1006 26349 37659 35498 4512 FBXW11-PORCN-WNT2B 1048 54134 36167 45459 15114 CSNK1D-PORCN-TLE2 1050 41629 52347 39317 20436 KREMEN1-PORCN-SENP2 1055 20253 16697 487 37387 FRAT1-PORCN-RHOU 1077 31453 10861 21530 17749 FZD1-PORCN-RHOU 1095 21176 12741 16941 18130 NLK-PORCN-SLC9A3R1 1108 35858 56545 26507 3768 FZD7-PORCN-TLE2 1111 50028 20809 22737 16490 CSNK1D-PORCN-TCF7 1115 16160 38935 28820 29405 LEF1-PORCN-PPP2R1A 1139 34004 43790 26178 5620 NLK-PORCN-TCF7 1154 48490 41769 34836 12221 FZD6-PORCN-TCF7 1180 33602 45874 5530 14682 FZD1-PORCN-WNT2B 1214 23191 49164 44140 14132 PITX2-PORCN-RHOU 1220 38816 23756 29958 13324 DKK1-PORCN-WNT2B 1222 26601 56978 25305 51044 FZD1-PORCN-SFRP4 1272 20986 27308 23520 32421 BCL9-PORCN-RHOU 1284 46597 6968 29490 21809 CSNK1G1-PORCN-TLE2 1291 31155 47884 19907 37757 CSNK1G1-PORCN-RHOU 1299 27191 33972 15284 20376 FOSL1-PORCN-RHOU 1333 44749 10281 23020 23848 APC-PORCN-WNT2B 1336 19495 29454 32718 22662 KREMEN1-PORCN-WNT3A 1377 25809 2588 12830 26801 BCL9-PORCN-WNT2B 1394 28608 33398 47599 21197 KREMEN1-PORCN-WNT2B 1401 36444 47192 17886 28451 CCND3-PORCN-TCF7L1 1408 36413 14249 8914 16004 FZD8-PORCN-WNT4 1416 27491 15836 9789 36903 FBXW11-PORCN-WNT4 1417 38911 18306 23726 35310 FZD8-PORCN-TCF7 1423 29081 35097 18363 11878 FBXW11-PORCN-SFRP4 1424 40575 3082 26157 24346 BCL9-PORCN-TCF7L1 1434 34720 6524 28798 16832 FOSL1-PORCN-TLE2 1446 44769 21226 13796 30066 DKK1-PORCN-TLE2 1461 36221 53569 8435 54639 FZD6-PORCN-SFRP4 1468 43415 46173 1426 23003 FZD7-PORCN-SFRP1 1470 54865 4185 15633 25384 DIXDC1-PORCN-SENP2 1529 21963 6981 10680 50015 FZD7-PORCN-WNT5A 1546 23763 6363 31855 40603 NLK-PORCN-WNT2 1557 46116 57071 36752 14418 FZD1-PORCN-TCF7 1572 38315 38185 25247 19741 CCND3-PORCN-TLE1 1580 36600 16976 3464 18298 CSNK1D-PORCN-SLC9A3R1 1596 41508 51420 34746 16992 PITX2-PORCN-SFRP4 1607 32189 32121 34063 21972 APC-PORCN-TCF7 1608 44043 39040 24555 20881 NLK-PORCN-TLE2 1657 49377 56092 39434 12430 FZD1-PORCN-SLC9A3R1 1698 18363 16380 21532 14659 FZD8-PORCN-PPP2CA 1713 51044 29439 20913 20485 FZD1-PORCN-FBXW4 1723 25108 19068 11515 19611 NLK-PORCN-WNT4 1742 49908 50644 30880 6042 FOSL1-PORCN-SFRP1 1757 36317 8057 17969 48972 FRZB-PORCN-SFRP4 1758 2975 26008 22234 42068 FRZB-PORCN-WNT4 1770 28522 21808 15299 35164 FRAT1-PORCN-WNT2B 1796 26246 45830 36444 11708 CSNK1G1-PORCN-SLC9A3R1 1815 10503 40046 19851 14743 FRZB-PORCN-PPP2R1A 1881 14863 41848 44601 12655 FZD7-PORCN-WNT2B 1892 48697 42677 33490 16364 GSK3B-PORCN-WNT4 1893 29701 28162 8296 19073 FOSL1-PORCN-FBXW4 1896 35711 18446 8569 19117 KREMEN1-PORCN-TCF7L1 1902 38888 15726 2313 30252 PITX2-PORCN-SFRP1 1905 39236 31192 37387 9110 CSNK1D-PORCN-SFRP1 1945 41776 54231 29005 26519 FZD1-PORCN-TLE2 1962 28422 26300 32913 23948 FZD5-PORCN-SFRP1 1980 55503 9426 16493 55566 FRAT1-PORCN-WNT4 1985 2835 18017 16053 23730 DKK1-PORCN-TCF7L1 1996 8327 51626 5019 50393 FRZB-PORCN-WNT2 2018 14494 10314 22043 37904 FZD6-PORCN-WNT2 2024 30877 41431 1558 29140 LRP5-PORCN-RHOU 2053 8242 9590 48746 20172 Table 1: Rankings of PORCN-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - linear tein expression of Wnt/β-catenin signaling and its target genes (β-catenin, PORCN, MYC), transcription factors (FOSL1, FOSL2), proliferation marker (CCND1), and fibroblast-to-myofibroblasts transdifferentiation markers (POSTN, CTGF). Looking at the tables above, one finds the following combinations for FOSL1 along with PORCN, 8 RANKING @tiUSING HSIC - RBF 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 CXXC4-PORCN-SENP2 9869 47028 41747 45009 41195 CXXC4-PORCN-WNT4 12492 2278 24920 46755 55842 CXXC4-PORCN-WNT2B 11272 21120 2288 41634 56160 CXXC4-PORCN-SFRP4 9369 30775 45417 32220 54090 CCND3-PORCN-WNT4 32692 49112 4162 52578 21791 CCND3-PORCN-FBXW4 14645 48562 10056 3289 18987 CXXC4-PORCN-TCF7 8174 50291 19771 4132 42869 CXXC4-PORCN-FBXW4 19853 43651 1014 10933 51164 CSNK1D-PORCN-SENP2 5664 26748 24402 26200 37885 PITX2-PORCN-SENP2 9499 29209 4986 29421 35908 FOSL1-PORCN-SFRP4 1126 41952 37755 38422 51768 APC-PORCN-WNT4 3592 12533 35169 40131 56018 FBXW2-PORCN-SFRP4 6999 27618 7405 7672 31324 APC-PORCN-TCF7L1 2882 26995 28827 39506 53073 FZD7-PORCN-FBXW4 8121 22895 5486 11733 22661 NLK-PORCN-PPP2CA 6640 37879 19229 30844 45007 FOSL1-PORCN-SENP2 314 29835 38646 3075 34319 NLK-PORCN-SFRP4 26548 54453 3595 31340 34837 FZD5-PORCN-SENP2 13359 46673 16639 12543 32648 FZD6-PORCN-WNT2B 47 9685 16997 36922 48852 FZD1-PORCN-SENP2 5513 34056 9744 10153 31014 FZD6-PORCN-WNT4 49 30454 13795 53236 50505 DKK1-PORCN-SENP2 2598 41263 6134 19723 13188 CSNK1G1-PORCN-SENP2 13585 41454 10606 1967 17443 BCL9-PORCN-PPP2CA 19443 12204 3989 2860 44188 CXXC4-PORCN-PPP2R1A 20998 30567 5511 22186 56640 FOSL1-PORCN-WNT4 1429 25705 30547 51837 52319 PITX2-PORCN-WNT5A 11335 32674 6548 15458 54809 APC-PORCN-SENP2 311 33317 27533 12463 51663 FRZB-PORCN-SENP2 2427 25944 18926 35699 48606 LRP5-PORCN-SENP2 11545 45801 9754 1937 37320 CXXC4-PORCN-PPP2CA 9663 20715 7491 12348 51537 FZD6-PORCN-TLE2 1065 45384 18117 3169 43333 FZD6-PORCN-SENP2 3 53444 8379 12707 32509 BCL9-PORCN-SFRP4 18390 13860 4099 36393 45375 PITX2-PORCN-FBXW4 2782 19946 19676 35972 50951 FBXW2-PORCN-WNT4 26557 1122 13989 31473 19951 CXXC4-PORCN-WNT5A 24704 5412 10695 26556 50085 FZD8-PORCN-SFRP1 11923 6846 34413 24802 45677 PITX2-PORCN-TCF7L1 13222 48332 9570 21227 51753 CSNK1D-PORCN-WNT2B 8595 53305 9816 26691 42121 FZD6-PORCN-SFRP1 1350 53618 15134 15153 53828 PITX2-PORCN-WNT2B 6802 25276 16938 13140 54357 FZD8-PORCN-SENP2 1042 32328 3606 28410 19273 FOSL1-PORCN-WNT2B 4296 5489 1279 29512 55035 KREMEN1-PORCN-WNT4 4572 591 18551 4045 46258 DKK1-PORCN-WNT4 9761 29610 8470 23618 51513 NLK-PORCN-SENP2 25839 52725 11882 9881 37272 FZD7-PORCN-WNT3A 7046 46922 42358 23913 36667 FZD8-PORCN-PPP2R1A 3136 21194 9602 10973 55135 PITX2-PORCN-TLE1 13872 39019 23792 7146 28106 FRAT1-PORCN-SFRP4 3261 12036 26076 48387 48335 DKK1-PORCN-SFRP4 10967 45207 3433 23697 54991 LRP5-PORCN-SFRP4 7636 690 3149 40164 40704 FRAT1-PORCN-SENP2 1688 40659 28876 5431 52097 APC-PORCN-SFRP1 7111 14469 39969 4242 53927 LRP5-PORCN-WNT2 19897 6038 17781 5354 56782 CCND3-PORCN-SFRP1 33184 51464 6743 7129 29071 CCND3-PORCN-TLE2 27878 51557 10879 4451 14924 PITX2-PORCN-PPP2R1A 11355 50277 26063 28070 55628 NLK-PORCN-WNT2B 30300 3466 8133 31316 44880 FBXW11-PORCN-WNT2B 4588 54697 1382 3672 49591 CSNK1D-PORCN-TLE2 12595 39793 26325 7235 43651 KREMEN1-PORCN-SENP2 2897 40815 14540 5841 26554 FRAT1-PORCN-RHOU 3460 15685 47256 50127 56007 FZD1-PORCN-RHOU 5740 25675 39166 40916 55717 NLK-PORCN-SLC9A3R1 39803 20122 9428 9998 42527 FZD7-PORCN-TLE2 38186 53192 36658 6999 30131 CSNK1D-PORCN-TCF7 38186 53192 36658 6999 30131 LEF1-PORCN-PPP2R1A 9650 37275 29706 14200 48587 NLK-PORCN-TCF7 14350 41604 26400 15610 37827 FZD6-PORCN-TCF7 9 22463 16490 29529 32076 FZD1-PORCN-WNT2B 12628 31292 8901 34846 55210 PITX2-PORCN-RHOU 11795 30595 6426 8297 54722 DKK1-PORCN-WNT2B 6263 7902 41853 840 49894 FZD1-PORCN-SFRP4 11010 1938 293 36645 52104 BCL9-PORCN-RHOU 13417 51514 7531 35538 56157 CSNK1G1-PORCN-TLE2 25416 24850 21439 2281 53031 CSNK1G1-PORCN-RHOU 18456 21677 26712 34529 54262 FOSL1-PORCN-RHOU 1550 39845 36998 42942 56120 APC-PORCN-WNT2B 3093 4702 5177 37025 54684 KREMEN1-PORCN-WNT3A 8946 29482 33980 6745 37128 BCL9-PORCN-WNT2B 7648 28196 20793 10962 34371 KREMEN1-PORCN-WNT2B 4060 34624 13586 33545 46659 CCND3-PORCN-TCF7L1 23178 26953 5435 27259 35443 FZD8-PORCN-WNT4 3177 7307 28634 31520 31714 FBXW11-PORCN-WNT4 6702 42970 13553 51464 33227 FZD8-PORCN-TCF7 3913 45328 27883 32520 22942 FBXW11-PORCN-SFRP4 4481 17010 1946 40382 32523 BCL9-PORCN-TCF7L1 23603 38078 31638 19229 41402 FOSL1-PORCN-TLE2 11375 42064 44489 3406 53873 DKK1-PORCN-TLE2 15980 45554 11547 10317 52314 FZD6-PORCN-SFRP4 58 47582 15962 31907 46706 FZD7-PORCN-SFRP1 46709 56734 39833 5383 34239 DIXDC1-PORCN-SENP2 16346 39480 19973 18502 14705 FZD7-PORCN-WNT5A 20174 37185 18191 15667 34821 NLK-PORCN-WNT2 44692 38504 8829 1995 53685 FZD1-PORCN-TCF7 2001 52600 33800 5805 49403 CCND3-PORCN-TLE1 13616 41666 12307 4061 29372 CSNK1D-PORCN-SLC9A3R1 26693 34397 22670 453 31614 PITX2-PORCN-SFRP4 7652 26224 19994 9630 41059 APC-PORCN-TCF7 671 37810 35108 9656 51194 NLK-PORCN-TLE2 41691 54802 745 25237 46608 FZD1-PORCN-SLC9A3R1 18304 25091 21402 3214 52196 FZD8-PORCN-PPP2CA 4402 55143 2005 12002 37055 FZD1-PORCN-FBXW4 15874 28222 17086 4292 47953 NLK-PORCN-WNT4 33383 45823 7542 34942 49961 FOSL1-PORCN-SFRP1 10522 38362 29766 6077 55724 FRZB-PORCN-SFRP4 6537 11064 22288 34580 50227 FRZB-PORCN-WNT4 9103 50017 36210 40277 54561 FRAT1-PORCN-WNT2B 5506 22998 20346 48317 52835 CSNK1G1-PORCN-SLC9A3R1 33423 29599 9114 6260 50642 FRZB-PORCN-PPP2R1A1 3914 23845 13959 7815 56301 FZD7-PORCN-WNT2B 35742 42851 5084 32980 11716 GSK3B-PORCN-WNT4 7755 14164 10378 27077 53726 FOSL1-PORCN-FBXW4 996 45964 6696 11811 50330 KREMEN1-PORCN-TCF7L1 5105 42564 13754 28956 27851 PITX2-PORCN-SFRP1 20244 35446 37738 24843 55456 CSNK1D-PORCN-SFRP1 36889 38723 2444 15209 50430 FZD1-PORCN-TLE2 29353 42139 4457 3385 52832 FZD5-PORCN-SFRP1 19216 56452 45353 3271 51709 FRAT1-PORCN-WNT4 6345 412 40450 45568 49626 DKK1-PORCN-TCF7L1 10025 8033 19471 39600 52807 FRZB-PORCN-WNT2 7253 15397 42587 3591 56438 FZD6-PORCN-WNT2 586 40987 4888 4733 51712 LRP5-PORCN-RHOU 12060 10648 7628 45521 55586 Table 2: Rankings of PORCN-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - rbf to be prominent at 3rd order level - FOSL1-PORCN-SFRP4, FOSL1-PORCN-SENP2, FOSL1-PORCN-WNT4, FOSL1-PORCN-WNT2B, FOSL1-PORCN-TLE2, FOSL1PORCN-RHOU, FOSL1-PORCN-SFRP1 and FOSL1-PORCN-FBXW4. All these combinations indicate the existence of a possible synergy when they take a higher rank in the list of combinations. 9 [25] Z. Zhong, D. M. Virshup, Recurrent mutations in tumor suppressor fbxw7 bypass wnt/β-catenin addiction in cancer, Science Advances 10 (2024) eadk1031. [26] R. A. Cooney, M. L. Saal, K. P. Geraci, C. Maynard, O. Cleaver, O. N. Hoang, T. T. Moore, R. F. Hwang, J. D. Axelrod, E. K. Vladar, A wnt4-and dkk3-driven canonical to noncanonical wnt signaling switch controls multiciliogenesis, Journal of cell science 136 (2023). 16