WNT-1/2B/3A/4/5A : Time Behavioural Study of 3rd Order Combinations in WNT3A Stimulated HEK 293 Cells

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 ⁷¹$C_3$ (= 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 WNT 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. 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. 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 Pećina-Šlaus [11] and • safety and feasibility of drug design for the Wnt pathway 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 $\beta$-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 $\beta$-catenin in the cytoplasm. As $\beta$-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 $\beta$-catenin which is then recognised by F BOX/WD repeat protein $\beta$-TRCP. $\beta$-TRCP is a component of ubiquitin ligase complex that helps in ubiquitination of $\beta$-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 Bänziger 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.

3. Chronological Literature Review of Time Series Analysis of WNT Signaling Pathway

Cyclic activation of the Wnt/$\beta$-catenin signaling pathway controls cell fusion-mediated somatic cell reprogramming. TCFs in complex with $\beta$-catenin, bind and regulate WNT target genes. Aulicino et al. [21] show that WNT/ $\beta$-catenin signaling needs to be off during the early reprogramming phases of mouse embryonic fibroblasts (MEFs) into iPSCs. In MEFs undergoing reprogramming, senescence genes are repressed and mesenchymal-to-epithelial transition is favored. This is correlated with a repressive activity of TCF1, which contributes to the silencing of WNT/$\beta$-catenin signaling at the onset of reprogramming. In contrast, the WNT pathway needs to be active in the late reprogramming phases to achieve successful reprogramming. The authors show that time-dependent activation of WNT/$\beta$-catenin pathway controls reprogramming of four-factor-induced MEFs by presenting experimental scheme of time course analysis of FACS-sorted cells and time-dependent expression of MET and senescence markers. They conclude that continued activation or inhibition of the WNT/$\beta$-catenin signaling pathway is detrimental to the reprogramming of MEFs while, temporal perturbation of the pathway is essential for efficient reprogramming, and the "WNT-off" state can be considered an early reprogramming marker.

Experimental and computational studies have revealed characteristics of the WNT pathway, however a cell-specific spatial perspective have been lacking. Tan et al. [22] provide a novel 3D confocal quantitation protocol to acquire spatial (two cellular compartments: nucleus and cytosol-membrane) and temporal quantitative data on target protein (e.g. $\beta$-catenin and N-cadherin) concentrations in Human Epithelial Kidney cells (HEK293T) during perturbation (with either cycloheximide or WNT3A). Computational models of the Wnt pathway are constructed and interrogated based on this data. Huang et al. [23] showed by transcriptome analysis of cells along human pluripotent stem cells (PSC) differentiation to early and definitive neuroepithelia via time-course data, that downregulation of genes associated with TGF-$\beta$ and canonical WNT/$\beta$-CATENIN signaling, confirmed the roles of classical signaling in human neural induction. Their study revealed known and novel expression patterns of genes that are involved in DNA methylation, histone modi cation, as well as epithelial-mesenchymal transition, which play crucial roles of those genes and signaling pathways during neural differentiation.

To examine signal propogation and its effect on transcription levels in individual cells within a population, Kafri et al. [24] conducted real-time $\beta$-catenin dynamics in a cell population in response to WNT signaling. They observed that $\beta$-catenin levels increased simultaneously at adherens junctions and the centrosome, and a membrane-centrosome transport system was revealed. Correlating $\beta$-catenin nuclear dynamics to CCND1 transcriptional activation showed that the nuclear accumulation rate of change of the signaling factor correlated with the transcriptional output of the pathway.

Madan et al. [25] performed time-course analysis of WNT-dependent signaling pathways in an orthotopic model of WNT-addicted pancreatic cancer, using a PORCN inhibitor and validated key results in additional WNT-addicted models. Their temporal analysis of the drug-perturbed transcriptome demonstrated direct and indirect regulation of more than 3,500 WNT-activated genes. Further, their study identified a central role of WNT/$\beta$-catenin/MYC/ribosome axis, in cancer proliferation.

Using a combination of time-resolved CUT&RUN against $\beta$-catenin, ATAC-seq, and perturbation assays in different cell types, Pagella et al. [26] showed that WNT/$\beta$-catenin physical targets are tissue-specific, $\beta$-catenin "moves" on different loci over time, and its association to DNA accompanies changing chromatin accessibility landscapes that determine cell behavior.

In all the above experimental results till now, we find that time course analysis is a common way to study and decipher the dynamics of the signaling pathway. However, their studies 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, 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 ⁷¹$C_3$ (= 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 time behavioral study of 3rd order WNT-(1,2B,3A,4,5A) related combinations using individual gene expressions measured in time, in WNT3A stimulated HEK 293 cells. Using this method, it is possible to find in time, which combinations at 3rd order are working synergistically. The work can be extended to explore higher order combinations also (i.e ≥ 4), with adequate computational resources.

Finally, from the computational perspective, Ding and Bar-Joseph [27] provide a review of data, methods, and analysis for reconstructing dynamic regulatory network models from high-throughput time-series data sets. They enlist a range of softwares for time-series GRN analysis methods that are basically based on correlation, regression, Granger causality, Probabilistic graph model and Graphical embedding. In comparison with these methods, the current manuscript uses sensitivity analysis based indices (computed via package in R by Pujol et al. [28]) to capture influence of involved factors in the signaling pathway and further use it as feature vectors in advanced support vector ranking algorithm by Joachims [3]. Thus, this is the first machine learning based method that can capture/identify/reveal higher order combinations in time and help biologists study the working of pathway.

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 0-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 [29]. 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 [29].

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 $C_k^n$ combinations for k number of genes from a total of n genes is generated. k is $\ge 2$ and $\le (n-1)$. Each of the combination of order k represents 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 $p^{th}$ combination in $C_k^n$ combinations, 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 p combinations. This procedure is done for different kinds of sensitivity analysis methods.

After the above sensitivity indices have been stored for each of the $p^{th}$ 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 $C_k^n$ training examples for $k^{th}$ order interaction. Using this training set $SV{M}_{learn}^{Rank}$ Joachims [3] is used to generate a model on default value C value of 20. In the current experiment on toy model C value 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 $SV{M}_{classify}^{Rank}$ 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 $SV{M}^{Rank}$ 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("SVMRank-Results-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-Ranking-AND-Interaction-Rank.R") to print individual ranking of the required input factor with other interaction factors.

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 3^rd 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 ⁷¹$C_3$ combinations, 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 $\mu$ and population standard deviation $\sigma$. 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 2415 WNT-X-X Combinations from Gujral and MacBeath [1]

In the supplementary section, I present four files (for each of the WNT member analysed here) 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 [30]) and Sobol indicies (with 2002 implementation in Saltelli [31] and martinez implementation in Martinez [32] and Baudin et al. [33]).

7.3. Conserved Machine Learning Rankings for Tested WNT1-X-X Combinations

A total of 2415, 3rd order combinations involving WNT1 were obtained from a full set of ⁷¹$C_3$ = 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. Table 2, Table 3 and Table 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.

7.3.1. Examining the Behaviour of FZD-WNT1-X Combinations

In their review, Corda and Sala [34] observe that frizzled class receptor 6 (FZD6) binds to WNT1, along with other members of WNT family. Looking at the tables above, one finds the following combinations for members of FZD family along with WNT1, to be prominent at 3rd order level - FZD6-WNT1-WNT2B, FZD1-WNT1-WNT2, FZD6-WNT1-WNT2, FZD5-WNT1-WNT2, FZD6-WNT1-WNT3A, FZD8-WNT1-WNT5A, FZD2-WNT1-WNT3A, FZD2-WNT1-WNT2, FZD2-WNT1-WNT5A, FZD5-WNT1-WNT4, FZD7-WNT1-WNT3A, FZD5-WNT1-WNT3A, FZD6-WNT1-WNT4, and FZD7-WNT1-WNT2B. All these combinations indicate the existence of a synergy when they take a higher rank in the list of combinations. Interestingly, it can also be observed that the WNT1 forms combinations with other WNT members. This phenomena is not seen for combinations of other WNT members as will be discussed later.

7.3.2. Examining the Behaviour of LRP-WNT1-X Combinations

Belenkov et al. [35] showed that data analysis of the group with obstructive coronary artery disease indicated a moderate correlation between WNT1 and low density lipoprotein receptor-related protein 6 (LRP6). Looking at the tables above, one finds the following combinations for members of LRP family along with WNT1, to be prominent at 3rd order level - LRP5-WNT1-WNT2, LRP6-WNT1-WNT2B, LRP6-WNT1-WNT4, LRP5-WNT1-WNT4, LRP5-WNT1-WNT2B, and LRP5-WNT1-WNT5A. All these combinations indicate the existence of a synergy when they take a higher rank in the list of combinations.

7.3.3. Examining the Behaviour of RHOU-WNT1-X Combinations

ras homolog family member U (RHOU) was first identified as a WNT-1-inducible mRNA and subsequently shown to act on the actin cytoskeleton by stimulating filopodia formation and stress fibre dissolution. Schiavone et al. [36] in their study, show that WNT-1-mediated RHOU induction occurs at the transcriptional level. The tables show one combination of RHOU-WNT1-WNT4 to be prominent among the combinations, working in time.

7.3.4. Examining the Behaviour of CTBP-WNT1-X Combinations

Valenta et al. [37] showed that the C-terminal binding protein 1 (CTBP1) associates with the TCF-4 C-terminus and that CtBP1 overexpression in WNT-1-stimulated 293 cells represses transcription from the WNT-responsive reporter and also decreases the activity of the endogenous AXIN2 promoter. Also, Gujral and MacBeath [1] highlighted CTBP1 as a transcription factor that contributes to positive feedback during the early phases of WNT signaling and serves as a novel marker for colorectal cancer progression. Looking at the tables above, one finds the following combinations for members of CTBP family along with WNT1, to be prominent at 3rd order level - CTBP2-WNT1-WNT2B, CTBP2-WNT1-WNT3A, CTBP2-WNT1-WNT2, and CTBP1-WNT1-WNT4. All these combinations indicate the existence of a synergy when they take a higher rank in the list of combinations.

7.3.5. Examining the Behaviour of TLE-WNT1-X Combinations

Gessler et al. [38] indicate that in the WNT OFF state TCF/LEF transcriptional activity is blocked by their association to Groucho (GRO)/Transducin-Like Enhancer of split (TLE) transcriptional repressors. In the WNT ON state nuclear CTNNB1 displaces TLE and facilitates formation of multimeric complexes with transcriptional coactivators which leads to widespread chromatin opening and accessibility. Looking at the tables above, one finds the following combinations for members of TLE family along with WNT1, to be prominent at 3rd order level - TLE2-WNT1-WNT2B, TLE2-WNT1-WNT2, TLE2-WNT1-WNT5A, TLE1-WNT1-WNT3, TLE1-WNT1-WNT2B, TLE1-WNT1-WNT4, TLE1-WNT1-WNT3A, TLE1-WNT1-WNT2, and TLE2-WNT1-WNT3. All these combinations indicate the existence of a synergy when they take a higher rank in the list of combinations.

7.3.6. Examining the Behaviour of TCF7L1-WNT1-X Combinations

Data by Shy et al. [39] indicate that in human mammary tumors, stimulation of WNT/$\beta$-catenin is strongly correlated with decreased nuclear transcription factor 7 like 1 (TCF7L1). Looking at the tables above, one finds the following combinations for TCF7L1 along with WNT1, to be prominent at 3rd order level - TCF7L1-WNT1-WNT2B, TCF7L1-WNT1-WNT4, TCF7L1-WNT1-WNT2, TCF7L1-WNT1-WNT3, TCF7L1-WNT1-WNT5A, and TCF7L1-WNT1-WNT3A. All these combinations indicate the existence of a synergy when they take a higher rank in the list of combinations.

7.4. Conserved Machine Learning Rankings for Tested WNT2B-X-X Combinations

A total of 2415, 3rd order combinations involving WNT2B were obtained from a full set of ⁷¹$C_3$ = 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. Table 6, Table 7 and Table 8 show the rankings for the same combinations as in Table 5, 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.

7.4.1. Examining the Behaviour of PORCN-WNT2B-X 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. [40] 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.

Liu et al. [41] 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. [42]. However, I was unable to find existing literature that document PORCN and post-translational modification of WNT2B via PORCN. Looking at the tables above, one finds the following combinations for PORCN along with WNT2B, to be prominent at 3rd order level - CXXC4-PORCN-WNT2B, CSNK1D-PORCN-WNT2B, PITX2-PORCN-WNT2B, BCL9-PORCN-WNT2B, KREMEN1-PORCN-WNT2B, FRAT1-PORCN-WNT2B, FZD7-PORCN-WNT2B, FZD6-PORCN-WNT2B, FOSL1-PORCN-WNT2B, NLK-PORCN-WNT2B, FBXW11-PORCN-WNT2B, FZD1-PORCN-WNT2B, DKK1-PORCN-WNT2B, and LRP5-PORCN-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.4.2. Examining the Behaviour of FZD-WNT2B-X Combinations

Ortiz-Masia et al. [43] observed that fibrosis and expression of epithelial-mesenchymal transition (EMT) markers were detected in samples from Crohn’s disease (CD) patients irrespective of the clinical behaviour. They show that an increased EMT, associated with increased WNT2B-FZD4 interaction, was detected in intestinal tissue from CD patients with a penetrating behaviour. Normal development of peripheral eye structures is also regulated by Wnt signaling, with WNT2B and FZD4 acting as the primary WNT ligand and receptor. Ohta et al. [44] show that the secreted protein Tsukushi (TSK) functioned as a WNT signaling inhibitor by binding directly to the cysteine-rich domain of FZD4 and competed with WNT2B. Conversely, their targeted inactivation of the TSK gene in mice caused expansion of the ciliary body and up-regulation of WNT2B and FZD4 expression in the developing peripheral eye. Looking at the tables above, one finds the following combinations for members of FZD fmaily along with WNT2B, to be prominent at 3rd order level - FRAT1-FZD1-WNT2B, CXXC4-FZD7-WNT2B, CSNK1G1-FZD2-WNT2B, FZD8-T-WNT2B, FRZB-FZD2-WNT2B, FZD1-WIF1-WNT2B, FZD6-PITX2-WNT2B, FZD1-NLK-WNT2B, FZD7-PORCN-WNT2B, FZD7-NKD1-WNT2B, FZD6-PORCN-WNT2B, AXIN1-FZD2-WNT2B, TCF7L1-WNT1-WNT2B, FZD5-NKD1-WNT2B, FZD8-WNT1-WNT2B, FZD8-JUN-WNT2B, FZD1-T-WNT2B, FZD6-WNT1-WNT2B, FBXW11-FZD2-WNT2B, FZD1-PORCN-WNT2B, FZD1-JUN-WNT2B, FZD8-GSK3A-WNT2B, FZD6-GSK3A-WNT2B, DAAM1-FZD2-WNT2B, FZD7-PPP2CA-WNT2B, FZD8-NKD1-WNT2B and FZD5-DVL2-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.4.3. Examining the Behaviour of CCND-WNT2B-X Combinations

WNT2B is a key regulator of downstream target $\beta$-catenin and cyclin D1 (CCND1) in Wnt signaling pathway. Zhu et al. [45] checked the expression of WNT2B, $\beta$-catenin and CCND1, and found that after high glucose treatment, the expression of all three were significantly upregulated (thus enhancing cell proliferation and inflammation, and increase fibrosis-related protein expression), but Hottip silencing could inhibit their expression which could be reversed by miR-455-3p inhibitor. These data indicated that WNT2B was a target of miR-455-3p and Wnt pathway was involved in diabetic nephropathy (DN). Looking at the tables above, one finds the following combinations for members of CCND fmaily along with WNT2B, to be prominent at 3rd order level - CCND2-WNT1-WNT2B, CCND3-NKD1-WNT2B, CCND2-LRP6-WNT2B, CCND3-WNT1-WNT2B, CCND1-WIF1-WNT2B, and CCND1-FGF4-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.4.4. Examining the Behaviour of FBXW11-WNT2B-X Combinations

Nodular and superficial are the most common subtypes of basal cell carcinoma (BCC) and WNT signaling is associated with BCC phenotypic variation. Do Carmo et al. [46] observed that nodular BCC presented up-regulation of CSNK1D, RAB23, PRKACB, PRKX, WNT5A, BMP6, FBXW11, IFT52, and SIAH1; and down-regulation of PRKACA, WIF1, WNT2B, WNT3, WNT10A, WNT10B, WNT16, FKBP8, NPC1, and MAPK1. In conclusion they found that PRKX (up-regulated), WNT3 (down-regulated) and WNT16 (down-regulated) genes, belonging to the WNT signaling pathway, are involved in the tumorigenic process of nodular BCC.

Dalle Carbonare et al. [47] show that FBXW11 is highly modulated during differentiation and that increased levels of FBXW11 cause a decrease in the target $\beta$-catenin levels under normal conditions. Under pathological conditions such as cleidocranial dysplasia (CCD) the relationship between FBXW11 and RUNX2 or $\beta$-catenin changes from normal conditions, suggesting the lack of $\beta$-catenin degradation by FBXW11. It has been reported in Dalle Carbonare et al. [47] that WNT signalling and RUNX2 mutually regulate their expression. In this disease model, FBXW11 levels might be elevated to counteract the dysregulated levels of $\beta$-catenin. Conversely, in osteosarcoma cells, the reduction of FBXW11 protein levels increases $\beta$-catenin. Since $\beta$-catenin promotes tumour formation and progression, reduced FBXW11 levels in osteosarcoma may be considered a negative prognosticfactor.

Looking at the tables above, one finds the following combinations for FBXW11 along with WNT2B, to be prominent at 3rd order level - FBXW11-TCF7-WNT2B, FBXW11-JUN-WNT2B, FBXW11-GSK3A-WNT2B, FBXW11-RHOU-WNT2B, FBXW11-WIF1-WNT2B, FBXW11-NKD1-WNT2B, FBXW11-SLC9A3R1-WNT2B, FBXW11-FGF4-WNT2B, FBXW11-T-WNT2B, FBXW11-GSK3B-WNT2B, FBXW11-FOXN1-WNT2B, FBXW11-PORCN-WNT2B, FBXW11-FZD2-WNT2B and FBXW11-PYGO1-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.4.5. Examining the Behaviour of LRP-WNT2B-X Combinations

To assess the potential role of WNT/TCF signaling in invasive trophoblast differentiation, Pollheimer et al. [48] investigated the role of selected WNT ligands and frizzled receptors. WNT2B, WNT7B, FZD3, FZD6, and LRP6 mRNAs were expressed in all tissues and cell types analyzed, suggesting that functional WNT receptors (FZD/LRP6) could be formed in human trophoblasts. As part of their study, Pollheimer et al. [48] demonstrated mRNA expression of WNT2B, WNT7B, FZD3, FZD6, and LRP6 in all trophoblast cell types strengthening the idea that these and other WNT/FZD factors could play a role in autocrine WNT signaling. Looking at the tables above, one finds the following combinations for members of LRP family along with WNT2B, to be prominent at 3rd order level - LRP6-TCF7-WNT2B, LRP5-SFRP1-WNT2B, LRP5-NKD1-WNT2B, LRP5-NLK-WNT2B, CCND2-LRP6-WNT2B, DAAM1-LRP6-WNT2B, LRP5-PORCN-WNT2B and LRP5-NKD1-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.4.6. Examining the Behaviour of FOSL1-WNT2B-X Combinations

Bonci et al. [49] report downregulation of miRs in B-cell chronic lymphocytic leukemia (CLL) as well as prostate carcinoma where additional WNT-related targets included SMAD7, SOX5, WNT2B and FOSL1. FOSL1 is known as a $\beta$-catenin/WNT signaling target gene, transcribed when WNT signaling is activated. Hachim et al. [50] decided to decipher the role of WNT signaling on bronchial fibroblast biology and their transcriptomics profiling of fibroblasts from asthmatic patients compared to healthy ones confirmed their in silico analysis, as members of canonical WNT signaling (WNT1, FOSL1, WNT2B, LRP5, WNT2, WNT16, and CCND2) were significantly upregulated in fibroblasts from asthmatic patients. Looking at the tables above, one finds the following combinations for FOSL1 along with WNT2B, to be prominent at 3rd order level - FOSL1-SFRP1-WNT2B, FOSL1-PORCN-WNT2B, FOSL1-WIF1-WNT2B, FOSL1-WNT1-WNT2B and FOSL1-NLK-WNT2B. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.5. Conserved Machine Learning Rankings for Tested WNT3A-X-X Combinations

A total of 2415, 3rd order combinations involving WNT3A were obtained from a full set of ⁷¹$C_3$ = 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. Table 10, Table 11 and Table 12 show the rankings for the same combinations as in Table 9, 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.

7.5.1. Examining the Behaviour of JUN-WNT3A-X Combinations

Sato et al. [51] found that the expression levels of AXIN2 and c-JUN activated by WNT3A stimulation in HEK293T cells were also inhibited in the presence of STOCKS2S-26016 (inhibitors of the WNK signalling pathway as reported in Mori et al. [52]). Looking at the tables above, one finds the following combinations for JUN along with WNT3A, to be prominent at 3rd order level - DKK1-JUN-WNT3A, DVL2-JUN-WNT3A, FOSL1-JUN-WNT3A, CSNK1D-JUN-WNT3A, DIXDC1-JUN-WNT3A, CSNK1G1-JUN-WNT3A, JUN-PYGO1-WNT3A, CTNNBIP1-JUN-WNT3A, FZD7-JUN-WNT3A, BCL9-JUN-WNT3A and FBXW11-JUN-WNT3A. Additionally, one also finds the following combinations for AXIN1 along with WNT3A, to be prominent at 3rd order level - AXIN1-DVL1-WNT3A, AES-AXIN1-WNT3A and AXIN1-MYC-WNT3A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.5.2. Examining the Behaviour of DVL-WNT3A-X Combinations

Zhan et al. [53] state that the WNT canonical pathway is activated upon binding of secreted WNT ligands (for example, WNT3A and WNT1) to FZD receptors and LRP co-receptors. LRP receptors are then phosphorylated by CK1$\alpha$ and GSK3$\beta$, which recruits Dishevelled (DVL) proteins to the plasma membrane where they polymerize and are activated as observed in Metcalfe et al. [54]. Looking at the tables above, one finds the following combinations for members of DVL family along with WNT3A, to be prominent at 3rd order level - DVL2-JUN-WNT3A, AES-DVL1-WNT3A, AXIN1-DVL1-WNT3A, DVL2-PYGO1-WNT3A, CXXC4-DVL2-WNT3A, DVL2-FGF4-WNT3A and DVL1-FOXN1-WNT3A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.5.3. Examining the Behaviour of TCF-WNT3A-X Combinations

The TCF/Lef family of transcription factors utilise $\beta$-catenin as a transcriptional co-activator, and instigate target gene expression in WNT signaling pathway. In order to assess whether a list of 90 differentially expressed genes (DEGs) had been previously identified as direct targets of $\beta$-catenin/TCF-/Lef in other cell types, Szemes et al. [55] compared the gene list with published target genes of canonical WNT signalling, based on using ChIPseq in U87 glioma cells in Zhang et al. [56], HEK293, and the colorectal cancer cell line LS174T in Schuijers et al. [57]. 53 out of the 90 DEGs were identified as having $\beta$-catenin and/or TCF7L2/TCF4 bound to their promoters in one or more of these three data sets. Looking at the tables above, one finds the following combinations for members of TCF family along with WNT3A, to be prominent at 3rd order level - TCF7L1-WNT1-WNT3A, TCF7L1-WIF1-WNT3A and PPP2CA-TCF7-WNT3A . All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.5.4. Examining the Behaviour of FZD / LRP/ CCND / FOSL1 / PITX2 -WNT3A-X Combinations

Amyotrophic lateral sclerosis (ALS) is a chronic neurodegenerative disease characterized by progressive degeneration of the motor neurons in the cortex, brainstem, and spinal cord. Yu et al. [58] identified differentially regulated WNT signaling molecules by comparing the transcription profiles of spinal cord from ALS and wild-type mice at different stages. Among the different WNT molecules, WNT3A was found to be up-regulated at 95 days. Frizzled receptors including FZD1 to FZD8 and coreceptor LRP5 were all up-regulated at 122 days. In this study, they found that CCND-1/2/3, FOSL1 and PITX2, target genes of canonical WNT signaling pathway, were upregulated in the spinal cord of ALS mice at the end stage. Looking at the tables above, one finds the following combinations for members of • FZD family along with WNT3A, to be prominent at 3rd order level - FZD8-FBXW4-WNT3A, FZD7-WNT1-WNT3A, FZD5-FGF4-WNT3A, FZD6-WNT1-WNT3A, FZD7-T-WNT3A, CCND3-FZD2-WNT3A, FZD7-WIF1-WNT3A, FZD7-PYGO1-WNT3A, FZD7-PORCN-WNT3A, FZD7-JUN-WNT3A, FZD5-WNT1-WNT3A, FZD7-NKD1-WNT3A, FZD7-NLK-WNT3A, FSHB-FZD2-WNT3A, FZD6-PORCN-WNT3A, FZD5-CCND2-WNT3A, FZD5-GSK3A-WNT3A and FZD6-GSK3A-WNT3A;

• LRP family along with WNT3A, to be prominent at 3rd order level - FBXW11-LRP6-WNT3A, LRP5-WNT1-WNT3A, CCND2-LRP6-WNT3A, CCND3-LRP6-WNT3A and LRP5-WNT3-WNT3A;

• CCND family along with WNT3A, to be prominent at 3rd order level - CCND3-FOXN1-WNT3A, CCND3-FGF4-WNT3A, CCND3-NKD1-WNT3A, CCND1-WIF1-WNT3A, CCND3-FZD2-WNT3A, CCND3-GSK3A-WNT3A, CCND3-PYGO1-WNT3A, CCND3-PITX2-WNT3A, CCND1-FGF4-WNT3A, CCND1-PYGO1-WNT3A, CCND2-WNT1-WNT3A, CCND3-SFRP1-WNT3A, CCND2-LRP6-WNT3A, CCND3-LRP6-WNT3A, CCND3-CSNK1A1-WNT3A, FZD5-CCND2-WNT3A, CCND3-T-WNT3A and CCND3-MYC-WNT3A;

• FOSL1 along with WNT3A, to be prominent at 3rd order level - FOSL1-JUN-WNT3A, FOSL1-WNT1-WNT3A, FOSL1-PORCN-WNT3A, FOSL1-NLK-WNT3A, FOSL1-FBXW4-WNT3A and FOSL1-FRAT1-WNT3A;

• PITX2 along with WNT3A, to be prominent at 3rd order level - CCND3-PITX2-WNT3A and PITX2-WNT3-WNT3A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.6. Conserved Machine Learning Rankings for Tested WNT4-X-X Combinations

A total of 2415, 3rd order combinations involving WNT4 were obtained from a full set of ⁷¹$C_3$ = 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. Table 14, Table 15 and Table 16 show the rankings for the same combinations as in Table 13, 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.

7.6.1. Examining the Behaviour of PORCN-WNT4-X Combinations

Zhang et al. [59] observe that palmitoylation of WNT4 is required for its binding to FZD1, but this is not essential for its secretion. Rao et al. [60] found that WNT4 secretion was PORCN-independent, and the secreted WNT4 did not exhibit paracrine activity but had a cell-autonomous activity, which indirectly proves the necessity of WNT4 palmitoylation. Zhu et al. [61] showed that low level of a natural health supplement, 3,3’-diindolylmethane (DIM) promoted gastric cancer progression by inducing the PORCN-dependent secretion of WNT4 and the activation of $\beta$-catenin signaling. Looking at the tables above, one finds the following combinations for PORCN along with WNT4, to be prominent at 3rd order level - CXXC4-PORCN-WNT4, CCND3-PORCN-WNT4, APC-PORCN-WNT4, FZD6-PORCN-WNT4, FOSL1-PORCN-WNT4, FBXW2-PORCN-WNT4, FZD5-PORCN-WNT4, KREMEN1-PORCN-WNT4, FBXW11-PORCN-WNT4, NLK-PORCN-WNT4, FRZB-PORCN-WNT4, FRAT1-PORCN-WNT4, PITX2-PORCN-WNT4, FZD7-PORCN-WNT4, DIXDC1-PORCN-WNT4, DKK1-PORCN-WNT4, FZD8-PORCN-WNT4 and GSK3B-PORCN-WNT4. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.6.2. Examining the Behaviour of FZD-WNT4-X Combinations

Ring et al. [62] observe that WNT4/FZD10 and WNT5a/FZD4 in combination with LRP6 show strong specific signal. Lyons et al. [63] found that Madin–Darby Canine Kidney (MDCK) cells express the FZD6 receptor and that WNT4 forms a biochemical complex with the FZD6 CRD. Acute myocardial infarction (AMI)-induced excessive myocardial fibrosis exaggerates cardiac dysfunction. Yin et al. [64] showed that in vitro, hypoxia enhanced the secretion and expression of WNT-2/4 in neonatal rat cardiac myocytes (NRCMs) or fibroblasts (NRCFs). In cultured NRCFs, FZD-2/4 proteins or mRNAs were increased by WNT-2/4 treatment and WNT-2/4 activated NF-$\kappa$B to promote cardiac fibrosis by FZD-4/2 receptor signaling. Looking at the tables above, one finds the following combinations for member of FZD family along with WNT4, to be prominent at 3rd order level - FSHB-FZD2-WNT4, FZD6-PORCN-WNT4, FZD5-PORCN-WNT4, FZD5-WNT1-WNT4, DIXDC1-FZD2-WNT4, FZD5-FOXN1-WNT4, FZD8-FBXW4-WNT4, CSNK1G1-FZD2-WNT4, DAAM1-FZD8-WNT4, APC-FZD2-WNT4, FZD5-PYGO1-WNT4, FZD5-JUN-WNT4, FZD7-PORCN-WNT4, FZD8-PORCN-WNT4, FZD6-WNT1-WNT4, FZD5-CTNNBIP1-WNT4, APC-FZD6-WNT4, DVL2-FZD1-WNT4 and FZD5-NKD1-WNT4. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.6.3. Examining the Behaviour of FOSL1-WNT4-X Combinations

Yu et al. [58] observed upregulation of WNT4 and FOSL1 in spinal cords of neurodegenerative disease amyotrophic lateral sclerosis (ALS) transgenic ${\mathrm{SOD1}}^{G93A}$ mice. Looking at the tables above, one finds the following combinations for FOSL1 along with WNT4, to be prominent at 3rd order level - FOSL1-FRAT1-WNT4, FOSL1-PORCN-WNT4, FOSL1-FOXN1-WNT4, FOSL1-NLK-WNT4 and FOSL1-WNT1-WNT4. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.6.4. Examining the Behaviour of NLK-WNT4-X Combinations

Nemo-like kinase (NLK) is an evolutionarily conserved protein kinase involved in WNT/$\beta$-catenin signalling. Ota et al. [65] demonstrate that NLK functions downstream of Dishevelled (DVL) in the WNT/$\beta$-catenin signalling pathway. Swann et al. [66] observe that NLK has previously been demonstrated to either enhance or suppress Wnt signaling in a context-dependent manner; however, the defective migration of the thymic lobes, parathyroids or thyroid was preserved in NLK-deficient, WNT4-transgenic embryos, indicating that NLK is not required to mediate the WNT4-induced signaling pathway that disrupts organ morphogenesis. Yu et al. [67] observed that recombinant WNT4 proteins (rWNT4) stimulation induced the formation of the TAK1-TAB2-NLK complex. Looking at the tables above, one finds the following combinations for NLK along with WNT4, to be prominent at 3rd order level - LRP5-NLK-WNT4, CSNK1G1-NLK-WNT4, FRAT1-NLK-WNT4, NLK-PORCN-WNT4, FGF4-NLK-WNT4, FOSL1-NLK-WNT4, CXXC4-NLK-WNT4, DIXDC1-NLK-WNT4, FRZB-NLK-WNT4 and CSNK1D-NLK-WNT4. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.6.5. Examining the Behaviour of DVL-WNT4-X Combinations

Sikora et al. [68] observed that WNT4 did not activate $\beta$-catenin-dependent transcription in HT1080 cells despite inducing phosphorylation of the DVL-2/3 WNT signaling molecules in HT1080. Emodin, an anthraquinone with multiple physiological activities, has been reported to inhibit proliferation of vascular smooth muscle cells (VSMCs) that might cause intimal arterial thickening. Emodin relieved injury-induced artery intimal thickness. To assess the downstream signal of WNT4 that mediates emodin-induced prevention of artery stenosis, Hua et al. [69] examined levels of DVL-1 and $\beta$-catenin in injured arteries and found that expression of both proteins was activated by vascular balloon injury, but was inhibited in rats administered emodin. Looking at the tables above, one finds the following combinations for members of DVL family along with WNT4, to be prominent at 3rd order level - DVL2-JUN-WNT4, DVL1-EP300-WNT4, DVL1-WNT1-WNT4, DVL1-JUN-WNT4, DVL1-MYC-WNT4, DVL2-FZD1-WNT4 and DVL2-FGF4-WNT4. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.7. Conserved Machine Learning Rankings for Tested WNT5A-X-X Combinations

A total of 2415, 3rd order combinations involving WNT5A were obtained from a full set of ⁷¹$C_3$ = 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. Table 18, Table 19 and Table 20 show the rankings for the same combinations as in Table 17, 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.

7.7.1. Examining the Behaviour of WIF1-WNT5A-X Combinations

Surmann-Schmitt et al. [70] report that WIF1 is highly expressed at cartilage-mesenchyme interfaces of the early developing skeleton andin fetal and postnatal skeletal development, WIF1 is expressed in a sharply restricted zone in the upper hyaline layer of epiphyseal and articular cartilage and in trabecular bone. They found that recombinant WIF1 and WNT factors show specific binding of the former to WNT3A, WNT4, WNT5A, WNT7A, WNT9A and WNT11. Chondrosarcoma (CS) is a rare cancer and Liu et al. [71] in their study, found promoter methylated WIF1 in both CS cell lines (CS-1 and SW1353) and tumor tissues. They confirmed loss of WIF1 expression and activation of Wnt pathway proteins (WNT5-A/B, LRP6, and DVL2) and an inverse correlation between the expression of WNT5-A/B and WIF1. Taken together, their results support the hypothesis that WIF1 restoration could attenuate the activity of WNT pathway and impedes the transmission of WNT signal. Skaria et al. [72] investigated the antagonistic effect of the RYK specific secreted WIF1 on WNT5A-mediated activation/inactivation of LIMK/CFL, and adherens junction disruption in human vascular endothelial cells (VECs). They found that in human coronary artery endothelial cells (HCAEC), treatment with WNT5A enhanced the phosphorylation of LIMK and CFL that was significantly prevented by WIF1. The presence of WIF1 suppressed WNT5A-mediated disruption of $\beta$-catenin and VE-cadherin adherens junctions in HCAEC, thereby preventing barrier dysfunction caused by WNT5A. Vassallo et al. [73] show that loss of WIF1 enhanced the migratory potential of glioblastoma through WNT5A that activated the WNT/Ca²⁺ pathway and metastasis-associated lung adenocarcinoma transcript 1 (MALAT1). Gujral and MacBeath [1] record the activity of WIF1 and WNT5A and it shows that in time, the expression levels of WIF1 are much higher than WNT5A, indicating that WIF1 might be attenuating WNT5A expression. Looking at the tables above, one finds the following combinations for WIF1 along with WNT5A, to be prominent at 3rd order level - CTNNBIP1-WIF1-WNT5A, TLE1-WIF1-WNT5A, JUN-WIF1-WNT5A, CCND1-WIF1-WNT5A, TCF7L1-WIF1-WNT5A and PYGO1-WIF1-WNT5A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.7.2. Examining the Behaviour of FZD-WNT5A-X Combinations

Kurayoshi et al. [74] show that the binding of WNT5A to the extracellular domain of FZD5 required palmitoylation of WNT5A but not required for its secretion. Kumawat et al. [75] show that silencing of the noncanonical WNT5A receptors FZD8 and RYK attenuated TGF-$\beta$-induced extracellular matrix (ECM) expression. Traumatic brain injury (TBI) can dramatically increase levels of intracellular calcium (Ca²⁺). In mammalian nerve cells, Niu et al. [76] study the association between WNT5A-FZD2 signaling and Ca²⁺ cellular homeostasis and demonstrated that the WNT5A-FZD2 signaling pathway contributed to increasing intracellular Ca²⁺ in nerve cells under physiological and pathological conditions. Chen et al. [77] observe that endocytosis of FZD4 in human embryonic kidney 293 cells was dependent on added WNT5A protein and was accomplished by the multifunctional adaptor protein beta-arrestin 2 (betaarr2), which was recruited to FZD4 by binding to phosphorylated DVL2. Kawasaki et al. [78] concluded that WNT5A is a natural ligand of FZD3 that triggers the PI3K/AKT signal and promotes adhesion of human dermal fibroblasts. About half of all melanomas harbor a mutation that results in a constitutively active BRAF kinase mutant (${\mathrm{BRAF}}^{V600E/K}$) that can be selectively inhibited by targeted BRAF inhibitors (BRAFis). Anastas et al. [79] suggest that chronic BRAF inhibition elevates WNT5A expression, which promotes AKT signaling through FZD7 and RYK, leading to increased growth and therapeutic resistance. Looking at the tables above, one finds the following combinations for members of FZD family along with WNT5A, to be prominent at 3rd order level - FZD5-JUN-WNT5A, APC-FZD6-WNT5A, FZD6-PORCN-WNT5A, FZD5-PPP2R1A-WNT5A, CSNK1G1-FZD2-WNT5A, APC-FZD2-WNT5A, FRZB-FZD2-WNT5A, FZD1-SFRP1-WNT5A, FZD2-WNT1-WNT5A, FZD5-FGF4-WNT5A, FSHB-FZD2-WNT5A, FZD5-FOXN1-WNT5A, FZD6-WNT1-WNT5A, FZD6-PORCN-WNT5A, FZD5-SFRP1-WNT5A, FZD8-WNT1-WNT5A, FZD5-NKD1-WNT5A, FZD1-NLK-WNT5A, FZD7-NKD1-WNT5A, FZD5-FZD2-WNT5A, FZD5-CSNK1D-WNT5A, FZD8-GSK3A-WNT5A, FZD1-T-WNT5A, AXIN1-FZD2-WNT5A and EP300-FZD2-WNT5A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.7.3. Examining the Behaviour of FGF4-WNT5A-X Combinations

The apical ectodermal ridge (AER) has a vital role in directing embryonic limb development. Allen [80] demonstrate that FGF4 from the AER activates a distal gradient of WNT5A expression in the limb. They observe that when the FGF4 bead is implanted closer to the limb bud, endogenous WNT5A is expressed more strongly, indicating that elevated levels of WNT5A are expressed with increasing FGF4 concentration. Looking at the tables above, one finds the following combinations for FGF4 along with WNT5A, to be prominent at 3rd order level - CSNK1D-FGF4-WNT5A, DIXDC1-FGF4-WNT5A, CCND1-FGF4-WNT5A, DKK1-FGF4-WNT5A, DVL2-FGF4-WNT5A, CTBP1-FGF4-WNT5A, CXXC4-FGF4-WNT5A, FZD5-FGF4-WNT5A, FBXW2-FGF4-WNT5A, CSNK1A1-FGF4-WNT5A, FBXW11-FGF4-WNT5A and CTNNBIP1-FGF4-WNT5A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

7.7.4. Examining the Behaviour of FOXN1-WNT5A-X Combinations

Epithelial–mesenchymal interactions are key to skin morphogenesis and homeostasis. Through functional rescue experiments Hu et al. [81] establish WNT5A as an essential downstream mediator of Notch–CSL signaling, impinging on expression in the keratinocyte compartment of FOXN1, a gene with a key hair follicle regulatory function. Their immunofluorescence analysis confirmed that WNT5A treatment caused a significant increase of FOXN1 expression in hair follicles of cultured (ColI-Cre * ${\mathrm{RBP-J}}_K^{loxP/loxP}$) skin. Looking at the tables above, one finds the following combinations for FOXN1 along with WNT5A, to be prominent at 3rd order level - DVL1-FOXN1-WNT5A, DIXDC1-FOXN1-WNT5A, DKK1-FOXN1-WNT5A, FZD5-FOXN1-WNT5A, FOSL1-FOXN1-WNT5A, FBXW11-FOXN1-WNT5A, CXXC4-FOXN1-WNT5A, BTRC-FOXN1-WNT5A, CSNK1G1-FOXN1-WNT5A, APC-FOXN1-WNT5A, and FBXW2-FOXN1-WNT5A. All these combinations indicate possible existence of a synergy when they take a higher rank in the list of combinations.

8. Conclusion

This manuscript studies the time behaviour of 3rd order combinations of WNT family members in WNT3A stimulated HEK 293 cells. Based on the extablished 2nd order combinations of the WNT members, 3rd order combinations emerge using the machine learning based search engine. These 3rd order combinations might be of interest for further wet lab investigations.