Addressing the needle in a haystack problem in time behavioural study of 3rd order gene combinations in WNT3A stimulated HEK 293 cells

Addressing the needle in a haystack problem in time behavioural study of 3rd order gene 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 Gujral and MacBeath [1] provides a quantitative, and dynamic study of WNT3Amediated 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 WNT 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. 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. However, the problem explodes combinatorially with even a small set of 71 recorded genes in the above study, when one steps to explore 3rd order combinations. With the total number of C71 3(= 57155) combinations, it becomes nearly impossible for any biologist to study the system wide dynamics of any pathway. Here, I •enumerate and rank all C71 3combinations using four different sensitivity methods; •show the conserved rankings for PORCN-WNT-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 the combinations related to WNT3A response genes that are ranked by the search engine in time. This study demonstrates how biologists can use the machine learning based search engine to address the needle in a haystack problem of discovering meaningful combinations of higher order in a vast search forest, which on further wet lab test might assist in intervening the pathway at a IBehavioural study of 3-odr gene 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 February 26, 2025 combinatorial level, in time. Keywords: Sensitivity analysis, Support vector ranking, Hilbert Schmidt Independence Criterion indices (HSIC) and Sobol indicies, WNT3A 1. Integration, Innovation and Insight 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. The problem explodes combinatorially with even a small set of recorded genes, when one steps to explore higher (here 3rd) order combinations. With the huge total number of combinations, it becomes nearly impossible for any biologist to study the system wide dynamics of any pathway as well as locate combinations of genuine interest. This study demonstrates how biologists can use a machine learning based search engine to address the needle in a haystack problem of discovering meaningful combinations of higher order in a vast search forest, while cutting down the time required to search the same. Further wet lab test might assist in intervening the pathway at a combinatorial level, in time. 2. 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 gene combinations using individual gene expressions measured in time, in WNT3A stimulated HEK 293 cells. 2 3. 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 2Aspects of unpublished work presented as poster in the Berkeley Cell Symposyia : Technology, Biology & Data Science, 2016, Berkeley, USA 2 provided by Gujral and MacBeath [1]. Nevertheless, here, I point to the fundamentals of the published work for completeness. 3.1. 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. 3.2. 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. 3.3. Wnt signaling and secretion Sharma [4]’s accidental discovery of the Wingless played a pioneering role in the emergence of a widely expanding research field of the Wnt signaling pathway. A majority of the work has focused on issues related to •the discovery of genetic and epigenetic factors affecting the pathway Thorstensen et al. [5] & Baron and Kneissel [6], •implications of mutations in the pathway and its dominant role on cancer and other diseases Clevers [7], •investigation into the pathway’s contribution towards embryo development Sokol [8], homeostasis Pinto et al. [9] & Zhong et al. [10] and apoptosis Pe´ cina-ˇ Slaus [11] and •safety and feasibility of drug design for the Wnt pathway 3 Kahn [12], Garber [13], Voronkov and Krauss [14], Blagodatski et al. [15] & Curtin and Lorenzi [16]. The Wnt phenomena can be roughly segregated into signaling and secretion part. The Wnt signaling pathway works when the WNT ligand gets attached to the Frizzled(FZD)/LRP coreceptor complex. FZD may interact with the Dishevelled (DVL) causing phosphorylation. It is also thought that Wnts cause phosphorylation of the LRP via casein kinase 1 (CK1) and kinase GSK3. These developments further lead to attraction of Axin which causes inhibition of the formation of the degradation complex. The degradation complex constitutes of AXIN, the β-catenin transportation complex APC, CK1 and GSK3. When the pathway is active the dissolution of the degradation complex leads to stabilization in the concentration of β-catenin in the cytoplasm. As β-catenin enters into the nucleus it displaces the GROUCHO and binds with transcription cell factor TCF thus instigating transcription of Wnt target genes. GROUCHO acts as lock on TCF and prevents the transcription of target genes which may induce cancer. In cases when the Wnt ligands are not captured by the coreceptor at the cell membrane, AXIN helps in formation of the degradation complex. The degradation complex phosphorylates β-catenin which is then recognised by F BOX/WD repeat protein β-TRCP. β-TRCP is a component of ubiquitin ligase complex that helps in ubiquitination of β-catenin thus marking it for degradation via the proteasome. Contrary to the signaling phenomena, the secretion phenomena is about the release and transportation of the WNT protein/ligand in and out of the cell, respectively. Briefly, the WNT proteins that are synthesized with the endoplasmic reticulum (ER), are known to be palmitoyleated via the Porcupine (PORCN) to form the WNT ligand, which is then ready for transportation. It is believed that these ligands are then transported via the EVI/WNTLESS transmembrane complex out of the cell B¨ anziger et al. [17] & Bartscherer et al. [18]. The EVI/WNTLESS themselves are known to reside in the Golgi bodies and interaction with the WNT ligands for the later’s glycosylation Kurayoshi et al. [19] & Gao and Hannoush [20]. Once outside the cell, the WNTs then interact with the cell receptors, as explained in the foregoing paragraph, to induce the Wnt signaling. Of importance is the fact that the EVI/WNTLESS also need a transporter in the from of a complex termed as Retromer. 4. 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. 5. 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 04 hour. 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 [21]. 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 [21]. 6. Design of experiment 6.1. 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 SV MRank 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 SVMRank 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 5 Sensitivity method Time point t=1hr t=3hr t=6hr t=12hr t=24hr HSIC - Linear 01234 HSIClinear−tp−1 −3 −2 −1 0 HSIClinear−tp−2 01234567 HSIClinear−tp−3 0 1 2 3 4 5 HSIClinear−tp−4 −4 −3 −2 −1 0 HSIClinear−tp−5 HSIC - RBF 01234 HSICrbf−tp−1 −7 −6 −5 −4 −3 −2 −1 0 HSICrbf−tp−2 012345 HSICrbf−tp−3 −4 −3 −2 −1 0 HSICrbf−tp−4 −4 −3 −2 −1 0 1 HSICrbf−tp−5 SOBOL - 2002 −1e+05 0e+00 1e+05 SB2002−tp−1 −600 −400 −200 0 200 400 600 SB2002−tp−2 −6e+05 −4e+05 −2e+05 0e+00 2e+05 4e+05 6e+05 SB2002−tp−3 −3e+06 −2e+06 −1e+06 0e+00 1e+06 2e+06 3e+06 SB2002−tp−4 −200 −100 0 100 200 SB2002−tp−5 SOBOL - martinez −4 −3 −2 −1 0 1 2 SBmartinez−tp−1 −2 0 2 4 SBmartinez−tp−2 −3 −2 −1 0 1 2 3 4 SBmartinez−tp−3 −2 −1 0 1 2 3 SBmartinez−tp−4 −150 −100 −50 0 50 SBmartinez−tp−5 Table 1: Rows - Sensitivity methods; Columns - Time points; A graph shows the ranking scores of combinations being arranged in descending order from left to right. order of the gene combinations based on the ranking score learned by the SVMRank 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. Table 1 shows the ranking scores in discending order for 3rd order combinations using four different sensitivity methods (i.e rows) and at five different time points (i.e columns). 7. Results & Discussion 7.1. 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 C71 3combinations, I consider/present only those combinations that show a ranking within first 10,000 out of 57,155. This choice is lib6 eral 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. 7.2. Enumeration and ranking of C71 3=57155 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. A particular row represents a particular rank given by a number. Following this, are combinations taking up that particular rank at different time points. The methods used are Hilbert Schmidt Independence Criterion indices (HSIC) indices (with rbf and linear kernel in Da Veiga [22]) and Sobol indicies (with 2002 implementation in Saltelli [23] and martinez implementation in Martinez [24] and Baudin et al. [25]). Of importance to note is that using these files, one can see a which priority/ranking one wants to investigate and see what combinations are taking up that priority in time. Changing combinations for a particular rank indicate that at a particular point in time, a particular combination might be predominant among others. 7.3. Conserved machine learning rankings for tested PORCN-WNTX combinations 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. [26] 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, 7 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. Liu et al. [27] 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. [28]. Liu et al. [29] 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. [30] developed 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. Based on these experimental tests and documented literature, the synergy of PORCNWNT can be used to see if the above machine learning based engine gives appropriate ranking to 3rd order combinations of PORCN-WNT-X (X, a particular gene/protein). If the rankings are appropriate, then we can infer that the search engine indeed points to combinatorial synergies, whether tested or unexplored, at biological level. Gujral and MacBeath [1] recorded the regulations of PORCN along with WNT1, WNT2B, WNT3A, WNT4 and WNT5A. Here, I present and demonstrate the conservation of rankings of PORCN-WNT-X combinations across different sensitivity methods. Using the linear kernel and HSIC sensitivity analysis method, table 2 shows rankings of combinations within the first 10,000 range (with low numerical value meaning a very high priority/role) mostly during the first phase (or after t = 1 hour of WNT3A stimulation). These point to the possible role of combinations during the early phase of WNT3A stimulation. As time passes, the rankings of these combinations get lower ranks (i.e higer numerical values) pointing to their down play of role when the effect of WNT3A stimulation has subsided in the late phase. These 3rd order synergies indicate the efficacy of the machine learning based search engine in finding meaningful combinations that might be of interest to (developmental)biologists, molecular biologists and oncologists. A total of 2415, 3rd order combinations involving PROCN were obtained from a full set of C71 3= 57155 combinations. Out of these 2415 combinations, those related to PORCN-WNT synergy are selected. Further, from this selected set, using the above criteria for conserved rankings, I report/tabulate the meaningful combinations that might be working synergistically. Tables 3, 4 and 5 show the rankings for the same combinations as in table 2, but using rbf kernel for HSIC, 2002 implementation for SOBOL and 8 RANKING @tiUSING HSIC - LINEAR 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 CXXC4-PORCN-WNT4 49 6186 19448 20672 51388 PITX2-PORCN-WNT4 177 16891 32175 32123 27627 FZD6-PORCN-WNT2B 379 19259 55786 24330 16739 FZD6-PORCN-WNT4 394 39861 46523 785 6046 FOSL1-PORCN-WNT4 455 37570 20729 10105 38487 PITX2-PORCN-WNT2B 630 42884 44910 50269 15186 FZD5-PORCN-WNT4 646 40380 25866 12816 56710 FOSL1-PORCN-WNT2B 670 19667 46545 40946 19709 KREMEN1-PORCN-WNT4 693 6753 23243 1864 44869 FZD7-PORCN-WNT3A 780 35610 948 25632 12174 DKK1-PORCN-WNT2B 1222 26601 56978 25305 51044 KREMEN1-PORCN-WNT3A 1377 25809 2588 12830 26801 BCL9-PORCN-WNT2B 1394 28608 33398 47599 21197 FZD8-PORCN-WNT4 1416 27491 15836 9789 36903 NLK-PORCN-WNT4 1742 49908 50644 30880 6042 GSK3B-PORCN-WNT4 1893 29701 28162 8296 19073 FRAT1-PORCN-WNT4 1985 2835 18017 16053 23730 FZD6-PORCN-WNT2 2024 30877 41431 1558 29140 FZD5-PORCN-WNT2B 2123 44490 45300 33310 53770 FZD6-PORCN-WNT5A 2165 41983 26329 43017 27185 CCND3-PORCN-WNT5A 2270 38617 15634 39601 37699 CXXC4-PORCN-WNT3 2291 2240 24537 6255 30135 FZD6-PORCN-WNT3A 2394 56979 41510 33895 6602 CSNK1D-PORCN-WNT5A 2899 17376 34214 55252 32946 BCL9-PORCN-WNT2 2978 30124 2987 21306 45043 DKK1-PORCN-WNT3A 3027 51411 42480 14175 39772 EP300-PORCN-WNT4 3047 52212 12666 11882 56673 LRP5-PORCN-WNT5A 3051 25077 6874 53519 30711 FZD8-PORCN-WNT2 3084 38943 6461 12608 17348 CSNK1A1-PORCN-WNT4 3257 26979 26665 9094 28864 DIXDC1-PORCN-WNT2B 3266 11238 38406 37800 51881 CSNK1G1-PORCN-WNT2 3410 20366 30313 12508 34531 FOSL1-PORCN-WNT5A 3631 16428 9587 43895 35656 DAAM1-PORCN-WNT4 3680 47587 5186 9818 24761 FZD1-PORCN-WNT2 3737 14464 10387 26137 31141 EP300-PORCN-WNT2B 3826 37202 48507 42391 54306 DIXDC1-PORCN-WNT5A 3885 26867 8379 47688 52084 FOSL1-PORCN-WNT3A 3900 15019 1518 39920 12124 FZD2-PORCN-WNT3 3994 23668 24913 3556 38793 CSNK1A1-PORCN-WNT2B 4109 24272 43637 30113 14274 BCL9-PORCN-WNT3A 4125 17834 1540 51723 23039 FOSL1-PORCN-WNT2 4144 48973 6608 18938 40921 LEF1-PORCN-WNT3 4220 6502 35317 5065 1556 LRP6-PORCN-WNT2B 4288 11513 52986 43798 7888 CTNNB1-PORCN-WNT2B 4381 41494 54608 33159 44891 FZD5-PORCN-WNT3A 4393 44820 4226 33407 54333 GSK3B-PORCN-WNT2B 4525 24848 40685 28860 9721 FZD2-PORCN-WNT4 4576 26380 31174 13954 52829 FZD5-PORCN-WNT2 4674 42137 9005 17361 56340 EP300-PORCN-WNT5A 4719 30282 10559 49731 55789 FRAT1-PORCN-WNT5A 4744 35332 12577 52285 31329 FRAT1-PORCN-WNT3A 4941 28915 3063 37414 10241 KREMEN1-PORCN-WNT2 4959 36712 10209 2355 46390 CTNNBIP1-PORCN-WNT4 5017 38878 13233 4965 42568 FZD7-PORCN-WNT2 5063 48509 3534 18543 37452 FBXW2-PORCN-WNT5A 5093 9753 1927 56299 23944 CSNK1A1-PORCN-WNT5A 5243 23800 15741 37419 31250 FRZB-PORCN-WNT5A 5316 15372 16217 53391 42789 FZD2-PORCN-WNT2 5319 35217 4925 18448 50496 FZD6-PORCN-WNT3 5539 17206 33991 301 6205 DVL2-PORCN-WNT2B 5842 46277 40915 44683 55710 LEF1-PORCN-WNT4 5938 6340 32575 15016 39684 DIXDC1-PORCN-WNT2 6166 9846 4217 17714 53792 DVL1-PORCN-WNT2B 6171 16317 38040 29254 5289 FBXW11-PORCN-WNT2 6311 39888 7951 33180 19070 CTNNB1-PORCN-WNT5A 6382 34261 18812 36691 51406 DVL1-PORCN-WNT4 6631 47068 9083 4176 6383 FRAT1-PORCN-WNT3 6706 18258 19680 8889 11973 DAAM1-PORCN-WNT5A 6758 49371 3391 50318 25339 FBXW11-PORCN-WNT3 6762 31256 9446 8712 8411 FBXW2-PORCN-WNT2 6905 25689 1063 38513 34061 DVL1-PORCN-WNT2 7032 39563 1823 5185 12571 DKK1-PORCN-WNT3 7165 12201 52957 1425 47839 FSHB-PORCN-WNT4 7187 35341 51879 43751 30960 DIXDC1-PORCN-WNT3A 7226 9557 1088 34837 45504 APC-PORCN-WNT3 7285 26324 14843 5204 12325 CSNK2A1-PORCN-WNT2 7317 28374 7626 21080 31795 DVL2-PORCN-WNT5A 7381 46344 27348 46664 56984 NLK-PORCN-WNT3 7444 40963 56870 11439 3041 GSK3B-PORCN-WNT3A 7574 34492 6475 23367 8516 FRZB-PORCN-WNT3A 7687 8222 4498 44883 17262 CTNNBIP1-PORCN-WNT5A 7693 15781 8108 34940 32551 CSNK1G1-PORCN-WNT3A 7824 36551 31135 32075 7370 FZD7-PORCN-WNT3 7919 50229 13866 3977 19719 FBXW2-PORCN-WNT3A 8007 29570 331 44149 23392 CTNNBIP1-PORCN-WNT3 8175 33782 15052 3217 10272 FZD8-PORCN-WNT3 8198 42890 20340 2880 17514 DVL2-PORCN-WNT3A 8315 32904 22980 44056 50976 PORCN-SFRP1-WNT2B 8374 33217 32367 38609 43781 EP300-PORCN-WNT3 8497 44763 16497 4191 45957 PORCN-SFRP1-WNT5A 8531 30769 38133 56398 46745 CSNK1A1-PORCN-WNT3 8682 9887 21818 2357 12384 LRP5-PORCN-WNT3 8787 10685 15101 51335 9714 FOSL1-PORCN-WNT3 8888 41174 18861 4616 20787 AXIN1-PORCN-WNT4 9013 31986 17718 11205 34131 PORCN-WNT4-WNT5A 9093 21416 44185 36030 52780 FRAT1-PORCN-WNT2 9314 30043 8901 28853 31633 CTNNB1-PORCN-WNT3 9325 40433 31660 954 26533 GSK3B-PORCN-WNT2 9371 40431 15296 6282 27880 AES-PORCN-WNT5A 9395 43267 5747 17185 30918 FRZB-PORCN-WNT3 9560 6430 26600 4539 19524 CTBP1-PORCN-WNT3A 9567 4412 1206 29048 13053 MYC-PORCN-WNT2 9597 42141 37704 21380 33950 DAAM1-PORCN-WNT2 9600 49958 2043 12313 45710 CTNNBIP1-PORCN-WNT2 9847 23180 4750 9800 36378 FGF4-PORCN-WNT2B 9991 29929 44511 25174 44265 Table 2: Rankings of PORCN-WNT-X. SA - HSIC; Kernel - linear martinez implementation for SOBOL, respectively. As on 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. At least at the 2nd order, considering the combinations of PORCN-WNT which have already been established in wet lab experiments (in above literature), the tabulated combinations with their appropriate ranks show the promise of the machine learning search engine in effectively locating the PORCN-WNT combinations. Further, the presented rankings point to combinations of PORCN-WNT-X, i.e the 3rd order combinations. So, considering all of the C71 3combinations, the machine learning search engine quickly ranks 9 7.4.6. low density lipoprotein receptor-related protein (LRP) For LRP6, there are 4 combinations involving F-box and WD repeat domain containing 11 (FBXW11) with LRP6. This depicts the possible synergy between LRP6-FBXW11 that need to explored. Wang et al. [42] state that upon activation of the pathway by the binding of Wnt ligand to Frizzled and LRP5LRP6 receptors, the axin complex is inhibited and results in the accumulation of soluble β-catenin that can enter the nucleus, where it interacts with transcription factors of the TCF/LEF1 family to regulate a series of target genes. It is assumed that APC helps phosphorylated β-catenin to dissociate from AXIN, creating a catalytic cycle of binding and release of the substrate. Others have suggested that APC acts either upstream of the phosphorylation reactions, by transporting β-catenin to the complex or downstream of the phosphorylation reactions, by recruiting the ubiquitin ligase bTrCP (FBXW11) to the complex. Holt et al. [43] observe that FBXW11 targets include β-catenin, key mediator of WNT signaling, critical to digital, neurological, and eye development. There might be an indirect synergy between LRP6-FBXW11. Gujral and MacBeath [1], LRP6 was found to be up regulated (+ive numbers), along with FBXW11. The search engine confirms the possible existence of the biological synergy between the above components by pointing out 3rd order combinations involving LRP6-FBXW11 interaction. 7.4.7. C-terminal binding protein (CTBP) For CTBP1, there are 6 combinations involving fibroblast growth factor 4 (FGF4) with CTBP1. For CTBP2, there are 5 combinations involving catenin β1 (CTNNB1) with CTBP2. This depicts the possible synergy between CTBP1-FGF4 and CTBP2CTNNB1 that need to explored. Wang et al. [42] observe that during the epithelialmesenchymal transition (EMT) process, TGFβinduced isoform switching of FGF receptors, causing the cells to become sensitive to FGF2. Addition of FGF2 to TGFβ-treated cells perturbed EMyoT by reactivating the MEK-Erk pathway and subsequently enhanced EMT through the formation of MEK-Erk-dependent complexes of the transcription factor δEF1/ZEB1 with the transcriptional corepressor CTBP1. Kim et al. [44] demonstrate that CTBP2 associates with major components of the β-catenin (CTNNB1) destruction complex and limits the accessibility of β-catenin to core transcription factors in undifferentiated embryonic stem cells (ESCs). Thus the synergies between the components have been established. Gujral and MacBeath [1], CTBP-1/2 and CTNNB1 were found to be up regulated (+ive numbers), while FGF4 was down regulated for a major period of time (apart from being upregulated). The search engine confirms the possible existence of the biological synergy between the above components by pointing out 3rd order combinations involving CTBP1-FGF4 and CTBP2-CTNNB1 interactions. 7.4.8. cyclin D (CCND) For CCND1, there are 7 combinations involving fibroblast growth factor 4 (FGF4) with CCND1. For CCND-2/3, there are 4 combinations each involving frizzled class receptor 5 (FZD5) with CCND-1/2. This depicts the possible synergy between CCND1FGF4 and CCND-1/2-FZD5 that need to explored. Bao et al. [45] show that CCND1 16 co-localizes with FGF3, FGF4, and FGF19 at chromosome location 11q13. Brandt et al. [46] show that expression levels of previously described endothelial target genes of β-catenin were studied using qPCR, but no differences was observed in the expression of CCND1 after knockdown of FZD5. But this might not be the case with CCND2/3. Gujral and MacBeath [1], CCND-1/2/3 was found to be up regulated (+ive numbers) along with FGF4, while FZD5 was found to be down regulated (−ive numbers). The search engine confirms the possible existence of the biological synergy between the above components by pointing out 3rd order combinations involving CCND1-FGF4 and CCND-2/3-FZD5 interactions. 7.4.9. Wnt family member (WNT) One peculiarity that we can find in the table under the title of WNT1 is that the machine points to synergistic combinations of WNT1 with other families of WNT. This pattern emerged in all the top 10 ranked combinations. It might be of interest to investigate whether WNT1 works in tandem with other WNT family members, as if one observes in the other columns such behaviour is not observed. 8. Conclusion This study demonstrates how biologists can use the machine learning based search engine to address the needle in a haystack problem of discovering meaningful combinations of higher order in a vast search forest, which on further wet lab test might assist in intervening the pathway at a combinatorial level, in time. The problem explodes combinatorially with even a small set of recorded genes in the above study, when one steps to explore 3rd order combinations. With the total number of C71 3(= 57155) combinations in this study, it becomes nearly impossible for any biologist to study the system wide dynamics of any pathway. The manuscript addresses these issues by enumerating and ranking a huge list of 3rd order combinations, demonstrating conserved machine learning rankings for wet lab established combinations across the different sensitivity methods used and presenting some of the patterns in the behaviour of some of the established combinations related to WNT3A response genes. In summary, the work presents a solution to the fundamental needle in a haystack problem of locating higher order gene combinations in a vast search forest, via use of powerful machine learning based search engine. Use of this engine is bound to assist many biologists/oncologists in search for meaningful higher order gene combinations that work in cell biology and make potential discoveries necessary for advancement in the study of cell/developmental biology as well as developement of therapeutics in diseased cells. Competing interests No competing interest is declared. 17 WNT3 stimulated response genes Response gene family Gene family member 3rd order combinations adenomatosis polyposis coli APC-PITX2-SFRP4 APC-FZD6-SENP2 APC-PITX2-SENP2 APC-DIXDC1-WNT2B APC-PITX2-TCF7 APC-FZD6-TLE2 (APC) regulator of WNT signaling pathway APC-FZD2-TCF7L1 APC-PITX2-PPP2CA APC-PORCN-SENP2 APC-PITX2-WNT4 v-myc avian myelocytomatosis CSNK2A1-MYC-SENP2 CSNK2A1-MYC-SFRP4 AES-AXIN1-MYC CSNK2A1-MYC-TCF7L1 FRZB-MYC-SENP2 CSNK2A1-MYC-PPP2CA viral oncogene homolog (MYC) DKK1-MYC-SENP2 FZD5-MYC-SENP2 CSNK2A1-MYC-WNT4 DVL1-MYC-SENP2 frizzled class receptor (FZD) FZD1 FZD2 FZD5 FZD6 FZD7 FZD8 FZD1-NLK-SENP2 FSHB-FZD2-SENP2 FZD5-JUN-WNT2 APC-FZD6-FZD8 FZD7-NKD1-SENP2 FZD8-FBXW4-WNT3A AES-AXIN1-FZD1 FSHB-FZD2-WNT4 FZD5-JUN-FBXW4 APC-FZD6-TLE2 CXXC4-FZD7-PPP2CA AES-AXIN1-FZD8 DVL1-FRZB-FZD1 FSHB-FZD2-FZD7 FZD5-CCND2-FBXW11 AES-AXIN1-FZD6 FZD7-PPP2CA-SFRP4 FZD8-LRP6-RHOU FZD1-FZD7-PPP2CA FSHB-FZD2-KREMEN1 FZD5-CCND2-SENP2 FZD6-PORCN-WNT2B CXXC4-FZD7-SFRP4 FZD8-PORCN-SFRP1 FZD1-FZD7-SFRP4 DKK1-FZD2-LRP5 FZD5-CCND2-FRZB FZD6-PORCN-WNT4 FZD1-FZD7-PPP2CA DIXDC1-FOXN1-FZD8 AES-EP300-FZD1 APC-FZD2-TCF7L1 FZD5-CCND3-SENP2 FZD6-GSK3A-WNT2 FZD1-FZD7-SFRP4 FZD8-PORCN-SENP2 DKK1-DVL2-FZD1 FSHB-FZD2-TLE2 FZD5-JUN-WNT5A CCND1-CTBP1-FZD6 FZD7-PORCN-SENP2 CXXC4-FRAT1-FZD8 CXXC4-FRAT1-FZD1 FSHB-FZD2-LRP5 FZD5-PITX2-SENP2 FZD6-PORCN-TLE2 CSNK1D-FGF4-FZD7 BTRC-FOXN1-FZD8 DVL1-EP300-FZD1 FRZB-FZD2-SENP2 FZD5-CCND2-DKK1 FZD6-PORCN-SENP2 FSHB-FZD2-FZD7 FZD8-LRP6-TCF7 FRAT1-FZD1-SFRP4 FSHB-FZD2-SFRP4 FZD5-MYC-SENP2 FZD6-PORCN-SFRP1 FZD7-PORCN-FBXW4 FZD8-GSK3A-KREMEN1 glycogen synthase kinase 3 (GSK3) GSK3A GSK3B FRZB-GSK3A-SENP2 CSNK1D-FGF4-GSK3B FRZB-GSK3A-PPP2R1A FBXW11-GSK3B-WNT2B DKK1-GSK3A-LRP5 FOSL1-FOXN1-GSK3B FZD6-GSK3A-WNT2 GSK3B-LRP6-SLC9A3R1 BTRC-GSK3A-T CTBP2-GSK3B-RHOU FRZB-GSK3A-LRP5 CSNK1G1-GSK3B-SLC9A3R1 FZD5-GSK3A-SENP2 GSK3B-RHOU-SENP2 DKK1-GSK3A-WNT2B FSHB-FZD2-GSK3B BTRC-GSK3A-NLK GSK3B-JUN-TLE1 CSNK1D-GSK3A-LEF1 CTBP1-FGF4-GSK3B dishevelled segment polarity protein (DVL) DVL1 DVL2 DVL1-EP300-FRZB DKK1-DVL2-SENP2 AXIN1-DVL1-FBXW2 DKK1-DVL2-FRZB AXIN1-DVL1-FBXW11 DVL2-JUN-FBXW4 DVL1-FRZB-FZD1 DKK1-DVL2-FZD1 DVL1-EP300-GSK3B DVL2-JUN-WNT3A DVL1-EP300-WNT2B DVL2-JUN-WNT2B DVL1-FBXW11-SLC9A3R1 CXXC4-DVL2-FRZB DVL1-EP300-WNT4 DKK1-DVL2-FBXW11 AXIN1-DVL1-RHOU DVL2-JUN-TCF7 DVL1-EP300-FZD1 FZD5-DVL2-FRZB low density lipoprotein receptor related protein (LRP) LRP5 LRP6 DVL1-EP300-LRP5 FBXW11-LRP6-SENP2 DKK1-JUN-LRP5 FBXW11-LRP6-TCF7 LRP5-NLK-WNT4 FBXW11-LRP6-SLC9A3R1 FOXN1-KREMEN1-LRP5 LRP6-TCF7-WNT2B DKK1-FZD2-LRP5 FBXW11-LRP6-SFRP4 FZD5-FOXN1-LRP5 DKK1-LRP6-SENP2 FSHB-FZD2-LRP5 DAAM1-LRP6-SENP2 DKK1-GSK3A-LRP5 CCND2-LRP6-RHOU LRP5-SFRP1-WNT2B DAAM1-LRP6-SLC9A3R1 FRZB-GSK3A-LRP5 CCND2-LRP6-TCF7 C-terminal binding protein (CTBP) CTBP1 CTBP2 CCND1-CTBP1-KREMEN1 CTBP2-CTNNB1-FOSL1 CSNK2A1-CTBP1-PPP2R1A CTBP2-CTNNB1-WNT4 CCND1-CTBP1-FZD6 CTBP2-T-TLE2 CTBP1-FGF4-FBXW4 CTBP2-FOXN1-SENP2 CTBP1-FGF4-FZD1 CTBP2-CTNNB1-FBXW4 CTBP1-FGF4-WNT5A CTBP2-CTNNB1-TLE1 CTBP1-GSK3A-PPP2CA CTBP2-CTNNB1-WNT2B CTBP1-FGF4-TCF7L1 CTBP2-T-WNT2B CTBP1-FGF4-FRZB CTBP2-GSK3B-RHOU CTBP1-FGF4-RHOU CSNK1A1-CTBP2-KREMEN1 cyclin D (CCND) CCND1 CCND2 CCND3 CCND1-FGF4-GSK3B CCND2-LRP6-SENP2 CCND3-PORCN-WNT4 CCND1-FGF4-RHOU FZD5-CCND2-FBXW11 CCND3-PORCN-FBXW4 CCND1-WIF1-WNT2B FZD5-CCND2-SENP2 FZD5-CCND3-SENP2 CCND1-FGF4-SFRP1 FZD5-CCND2-FRZB FZD5-CCND3-FRZB CCND1-FGF4-WNT5A CCND2-LRP6-RHOU AES-AXIN1-CCND3 CCND1-FGF4-FOSL1 CCND2-FBXW11-SLC9A3R1 CCND3-WNT1-WNT4 CCND1-PYGO1-WNT2 CCND2-LRP6-TCF7 CCND3-PORCN-SFRP1 CCND1-FGF4-FBXW4 FZD5-CCND2-DKK1 CCND3-PORCN-TLE2 CCND1-FGF4-PPP2R1A CCND2-LRP6-FBXW4 FZD5-CCND3-CSNK1D CCND1-CTBP1-KREMEN1 CCND2-WNT1-WNT4 FZD5-CCND3-SFRP4 Wnt family member (WNT) WNT1 WNT2B WNT3A WNT4 WNT5A TLE2-WNT1-WNT2B CTNNBIP1-WIF1-WNT2B CSNK1D-FGF4-WNT3A CXXC4-PORCN-WNT4 CSNK1D-FGF4-WNT5A AXIN1-WNT1-WNT4 AES-AXIN1-WNT2B FZD8-FBXW4-WNT3A CSNK1D-FGF4-WNT4 FSHB-T-WNT5A AXIN1-WNT1-WNT2 LRP6-TCF7-WNT2B DKK1-JUN-WNT3A LRP5-NLK-WNT4 CTNNBIP1-WIF1-WNT5A CCND2-WNT1-WNT4 CXXC4-PORCN-WNT2B AES-AXIN1-WNT3A PITX2-PORCN-WNT4 CCND1-FGF4-WNT5A TLE1-WNT1-WNT2B TLE1-WIF1-WNT2B DVL2-JUN-WNT3A FSHB-FZD2-WNT4 FZD5-JUN-WNT5A TCF7L1-WNT1-WNT2B CCND1-WIF1-WNT2B CCND1-FGF4-WNT3A AXIN1-WNT1-WNT4 CCND2-LRP6-WNT5A CCND2-WNT1-WNT5A APC-DIXDC1-WNT2B FBXW2-WNT3A-WNT4 DVL1-EP300-WNT4 PITX2-PORCN-WNT5A TCF7L1-WNT1-WNT4 FBXW11-LRP6-WNT2B JUN-PYGO1-WNT3A FSHB-T-WNT4 CXXC4-PORCN-WNT5A TLE1-WNT1-WNT4 FZD7-NKD1-WNT2B CTNNBIP1-WIF1-WNT3A APC-PORCN-WNT4 LEF1-T-WNT5A AXIN1-WNT1-WNT3 TLE2-WIF1-WNT2B PPP2CA-WNT1-WNT3A DKK1-JUN-WNT4 CXXC4-PORCN-WNT5A Table 6: Top 10 3rd order combinations with conserved machine learning rankings of WNT3 stimulated response genes, across different sensitivity indices. Author contributions statement SS conceived and designed the experiments; wrote the code; performed the experiments; analyzed the data; wrote the manuscript. 18 Availability of code Code for time series data available at CERN based Zenodo on https://zenodo.org/ records/14637456. 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 .R and can be opened in R or in simple text processing program) with these names are made available with this manuscript. (1) HSIClinearTP-Choose-3-NSc-D.R, (2) HSICrbf-TP-Choose-3-NSc-D.R, (3) SB2002-TP-Choose3-NSc-D.R, and (4) SBmartinez-TP-Choose-3-NSc-D.R, 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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