kringle containing transmembrane protein 1 (KREMEN1) : Time behavioural study of 3rd order combinations in WNT3A stimulated HEK 293 cells

kringle containing transmembrane protein 1 (KREMEN1) : Time behavioural study of 3rd order combinations in WNT3A stimulated HEK 293 cells shriprakash sinha Independent Researcher; Orcid ID : orcid.org/0000-0001-7027-5788 Address : 104-Madhurisha Heights Phase 1, Risali, Bhilai-490006, India Corresponding author email : [email protected] Abstract kringle containing transmembrane protein 1(KREMEN1), encodes a high-affinity dickkopf homolog 1 (DKK1) transmembrane receptor that functionally cooperates with DKK1 to block wingless (WNT)/β-catenin signaling. Gujral and MacBeath [1] provides a quantitative, and dynamic study of WNT3A-mediated stimulation of HEK 293 cells, where they record time based expression profiles of several response genes which correlated significantly with proliferation and migration. By monitoring the dynamics of gene expression using self-organizing maps, they identified clusters of genes that exhibit similar expression dynamics and uncovered previously unrecognized positive and negative feedback loops. However, their study depicts/uses singular measurements of individual gene expression at different time snapshots/points to infer the system wide analysis of the pathway. At any particular time point, it is often the case that genes are working synergistically in combinations, even though their expression measurements are singular in nature. Here, I •enumerate and rank all 2415 KREMEN1 related 3rd order combinations in a forest of 71C3combinations using four different sensitivity methods; •show the conserved rankings for KREMEN1-X-X combinations, which point to existence of biological synergy of some of these combinations across the different sensitivity methods; and •study the behaviour of some of these combinations related to WNT3A response genes that are ranked by the machine learning search engine (Sinha [2]) in time. Patterns of combinations emerge, some of which have been tested in wet lab, while others require further wet lab analysis. Keywords: Sensitivity analysis, Support vector ranking, Hilbert Schmidt Independence Criterion indices (HSIC) and Sobol indicies, WNT3A ITime behavioural study of 3-odr KREMEN1 comb. in WNT3A stimulated cells 1Aspects of unpublished work were presented in a poster session at Cell Symposia: Technology. Biology. Data Science, 9-11 October 2016, Berkeley, California, USA. Preprint submitted to Preprint March 1, 2025 1. Significance Sinha [2] recently demonstrated the use of machine learning based search engine to rank/reveal gene combinations at 2nd order for the time series data by Gujral and MacBeath [1] and showed how it is possible to locate combinations of priority that might be working synergistically, using sensitivity methods and powerful support vector ranking algorithm. However, the problem explodes combinatorially with even a small set of 71 recorded genes in the study by Gujral and MacBeath [1], when one steps to explore 3rd order combinations. With the total number of 71C3(= 57155) combinations, it becomes nearly impossible for any biologist to study the system wide dynamics of any pathway. Also, the amount of time usually needed to search for and test a combination is far more than the search down by the machine learning based search engine. Here, I extend the research work by Sinha [2] to conduct a behavioral study of 3rd order KREMEN1 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, 2 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. kringle containing transmembrane protein 1 (KREMEN1) Prothrombin, plasminogen, urokinaseand tissuetype plasminogen activators contain structures known as kringles. They correspond to autonomous structural and folding domains which mediate the binding of these multidomain proteins to other proteins. Patthy et al. [4] show that the amino acid sequences of the type II structures of the gelatin-binding region of fibronectin are homologous with those of the proteasekringles. Kringle domain, a triple loop, 3-disulfide bridge structure, is conserved in diverse proteins which play important roles in various biological processes. Nakamura et al. [5] cloned KREMEN, a novel member of kringle-containing proteins, which is likely to be a type-I transmembrane protein composed of 473 amino acid residues. KREMEN has a kringle domain, a WSC domain, and CUB domains in the extracellular region, while the intracellular region has no conserved motif involved in signal transduction. Increased KREMEN mRNA levels in mouse embryo and its expression in the adult mouse in a variety of tissues suggested a potential role for KREMEN in the regulation of cellular responses upon extracellular stimulus or cellcell interaction in neuronal and/or muscle cells. Canonical WNT signalling via β-catenin pathway is transduced by two receptor families - Frizzled (FZD) proteins and lipoprotein-receptor-related proteins 5 and 6 (LRP-5/6) which bind WNTs and transmit their signal by stabilizing intracellular βcatenin. However, this pathway is inhibited by the secreted protein Dickkopf1 (DKK1) by binding to and antagonizing LRP-5/6. Mao et al. [6] show that the transmembrane proteins KREMEN-1/2 are high-affinity DKK1 receptors that cooperate with DKK1 to block WNT/β-catenin signalling. Upon binding to DKK1, KREMEN proteins (both KREMEN-1/2) are recruited into a complex with LRP-5/6, which leads to rapid endocytosis and removal of LRP5/6 from the plasma membrane. This formation of a ternary complex composed of KREMEN, DKK, and LRP-5/6 (the coreceptor of WNT) inhibits WNT/β-catenin signalling. The inhibitory function of DKKs depends on the presence of appropriate KREMEN proteins. Nakamura et al. [7] present the identification and structure of KREMEN, KREMEN as a negative regulator in WNT signalling, expression of KREMEN and its role in development and its involvement in WNT signalling and cancer. I present 3rd order combinations of KREMEN1 with other genes, that the machine learning based search engine points to, as possible synergistic combinations that might be working in time. 3 3. Methods Please refer to sections of Sinha [2] for methods, design of study and analysis of data for 2nd order combinations. The same method and design of study is used to generate results for 3rd order combinations presented in this study. 4. Time series data Gujral and MacBeath [1] present a set of 71 WNT-related gene expression values for 6 different times points over a range of 24-hour period using qPCR. The changes represent the fold-change in the expression levels of genes in 200 ng/mL WNT3A-stimulated HEK 293 cells in time relative to their levels in unstimulated, serum-starved cells at 0hour. Gujral and MacBeath [1] state that qPCR data are the means of three biological replicates. Only genes whose mean transcript levels changed by more than two-fold at one or more time points during the 24-hour time course were considered significant. Positive (negative) numbers represent up (down) -regulation. We have already covered the issues related to these data sets in detail in Sinha [8]. 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 [8]. 5. Design of experiment 5.1. Pipeline for time series data For the case of time series data, interactions among the contributing factors are studied by comparing triplets of fold-changes at single time points. The prodecure begins with the generation of distribution around measurements at single time points with added noise is done to estimate the indices. A distribution is generated for the fold changes at single time points. Then for every gene, there is a vector of values representing fold changes as well as deviations in fold changes for different time points and durations between time points, respectively. Next a listing of all Cn kcombinations for knumber of genes from a total of ngenes is generated. kis ≥2 and ≤(n−1). Each of the combination of order krepresents a unique set of interaction between the involved genetic factors. After this, the datasets are combined in a specifed format which go as input as per the requirement of a particular sensitivity analysis method. Thus for each pth combination in Cn kcombinations, the dataset is prepared in the required format from the distributions for two separate cases which have been discussed above. (See .R code in mainScript-1-1.R). After the data has been transformed, vectorized programming is employed for density based sensitivity analysis and looping is employed for variance based sensitivity analysis to compute the required sensitivity indices for each of the pcombinations. This procedure is done for different kinds of sensitivity analysis methods. After the above sensitivity indices have been stored for each of the pth combination, the next step in the design of experiment is conducted. Since there is only one 4 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 SV MRank classi f y Joachims [3]. Note that due to availability of only one example per combination, after the model has been built, the same training data is used as test data to generates the scores. This procedure is executed for each and every sensitivity analysis method. This is followed by sorting of these scores along with the rank indices (i.e the training indices) already assigned to the gene combinations. The end result is a sorted order of the gene combinations based on the ranking score learned by the SV MRank algorithm. Finally, this entire procedure is computed for sensitivity indices generated for each and every fold change at time point and deviations in fold change at different durations. Observing the changing rank of a particular combination at different times and different time periods will reveal how a combination is behaving. Note that the following is the order in which the files should be executed in R, in order, for obtaining the desired results (Note that the code will not be explained here) - • use source(”mainScript-1-1.R”) with arguments for Dynamic data •source(”SVMRankResults-D.R”), to rank the interactions (again this needs to be done separately for different kinds of SA methods), •use source(”Combine-Time-files.R”), if computing indices separately via previous file, •source(”Sort-n-Plot-D.R”) to sort the interactions. Note that the sorting is chages the interaction ranking in time. Thus •use source(”Interaction-Priority-Intime.R”) to find the prioritized ranking of each and every interaction over the different time points and finally •use source(”Print-RankingAND-Interaction-Rank.R”) to print individual ranking of the required input factor with other interaction factors. 6. Results & Discussion 6.1. Time series data by Gujral and MacBeath [1] NOTE - Ranking was assigned on scores that were sorted in DECREASING values. So, 1 was assigned to highest score and vice versa. Results for the 3rd order interactions are presented here. The results first discuss the behaviour of interactions across the snapshots of time using the computed sensitivities on fold change measurements per time snapshot. The analysis was done using 4 different sensitivity indices. Out of the 71C3combinations, I consider/present only those combinations that show a ranking within first 10,000 out of 57,155. This choice is liberal and biologists/oncologists can have a more stricter choice as per need. Two observations are made, •the ranking of a particular combination is conserved (i.e within the 10,000 range) in a particular time point or in the early phase or late phase of WNT3A stimulation, across the majority of the four sensitivity methods, which is a 5 strict criteria of assessment or •the ranking of a particular combination is conserved across time points/phase (i.e they are within the 10,000 range) and the majority of the four sensitivity methods, which is relaxed criteria of assessment. Applying this filter helps reveal important combinations of interest that might be working synergistically at a higher order level in the cell. Regarding technical points of implementation, the rankings were generated without scaling/normalizing the time series data provided by Gujral and MacBeath [1]. For estimating the sensitivity indices, a small gaussian distribution using the function rnorm that generates a vector of normally distributed random variables given a vector length n (here 9, the 10th one is the mean/recorded gene regulation itself), a population mean µand population standard deviation σ. The syntax for using rnorm is as follows: rnorm(n, mean, sd). Further, I use the jitter funtion to add a little bit of noise to the data. This helps to see if the generated rankings are robust or not. 6.2. Enumeration and ranking of 2415 KREMEN1-X-X combinations from Gujral and MacBeath [1] In the supplementary section, I present four files, each containing the rankings of 3rd order combinations, that wary in time (shown for 5 time points). Each file represents the rankings computed using a particular sensitivity method. The changing rankings in time for a particular combination represents the importance of contribution/role that combination plays in the cell stimulated with WNT3A. The sensitivity methods used are Hilbert Schmidt Independence Criterion indices (HSIC) indices (with rbf and linear kernel in Da Veiga [9]) and Sobol indicies (with 2002 implementation in Saltelli [10] and martinez implementation in Martinez [11] and Baudin et al. [12]). 6.3. Conserved machine learning rankings for tested KREMEN1X-X combinations A total of 2415, 3rd order combinations involving KREMEN1 were obtained from a full set of 71C3= 57155 combinations. Further, from this selected set, using the above criteria for conserved rankings, I report/tabulate the meaningful combinations that might be working synergistically. Tables 2, 3 and 4 show the rankings for the same combinations as in table 1, but using rbf kernel for HSIC, 2002 implementation for SOBOL and martinez implementation for SOBOL, respectively. As one tallies the rankings of across these tables for a particular combination, one finds that the role of the combination of interest is conserved. This conservation points to the existence of the biological synergy, whether the combination has been tested or unexplored/untested. NOTE - It is important to note that KERMEN1 along with DKK1 and LRP-5/6 were downregulated (−ive value) during the first half of the WNT3A stimulation in Gujral and MacBeath [1]. 6.3.1. Examining the behaviour of DKK1-KREMEN1-X combinations Based on the above literature, KREMEN1 is known to interact with DKK1 to inhibit the WNT signaling pathway. Looking at the tables above, one finds the following com6 RANKING @tiUSING HSIC - LINEAR 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 FZD5-JUN-KREMEN1 96 24278 50801 44288 10781 AES-AXIN1-KREMEN1 127 47825 39191 53628 788 FOXN1-KREMEN1-WNT2B 170 28253 38493 15670 30910 DVL2-JUN-KREMEN1 178 24966 37758 33223 9256 FOXN1-KREMEN1-PPP2R1A 187 38390 45372 22596 7296 FOXN1-KREMEN1-WNT3 217 53012 56130 3088 18960 CCND1-CTBP1-KREMEN1 234 57106 44293 35945 33357 CTNNBIP1-JUN-KREMEN1 241 6571 55388 29592 14471 FOXN1-KREMEN1-LRP5 260 52346 51858 34049 42979 FSHB-FZD2-KREMEN1 286 34277 55693 24595 31187 FBXW11-FOXN1-KREMEN1 302 14589 4660 6330 19835 KREMEN1-PPP2CA-SFRP4 362 833 2893 48271 49787 CSNK1G1-FOXN1-KREMEN1 390 10815 10745 12917 12924 EP300-JUN-KREMEN1 445 18545 22047 46724 18178 DKK1-FOXN1-KREMEN1 480 13941 53110 3917 4043 CTNNB1-FOXN1-KREMEN1 484 1965 14838 7676 28339 FOXN1-KREMEN1-SFRP4 510 51802 54641 8830 39747 FZD5-FOXN1-KREMEN1 511 1786 8577 7989 16170 CSNK1D-FGF4-KREMEN1 529 10037 50110 50880 28493 FZD1-FZD7-KREMEN1 651 16749 42649 57051 9750 KREMEN1-PORCN-WNT4 693 6753 23243 1864 44869 FOSL1-FOXN1-KREMEN1 742 14009 6113 8523 21187 CTNNBIP1-FRAT1-KREMEN1 758 28255 42529 10768 13750 CSNK2A1-FOXN1-KREMEN1 775 7975 18477 6845 18012 FZD5-GSK3A-KREMEN1 798 32935 3687 34447 2546 CCND1-FGF4-KREMEN1 833 56773 18833 37224 15499 FZD8-GSK3A-KREMEN1 914 37178 3268 52107 6645 FSHB-GSK3A-KREMEN1 921 38254 26495 28443 3064 DIXDC1-FOXN1-KREMEN1 939 687 5376 7847 13303 FOXN1-KREMEN1-SFRP1 985 42401 54457 12806 414 DVL2-FGF4-KREMEN1 1025 11180 25345 35577 33719 FZD5-CCND2-KREMEN1 1049 44629 22542 28884 49843 KREMEN1-PORCN-SENP2 1055 20253 16697 487 37387 KREMEN1-PYGO1-WNT3A 1064 30070 1340 20277 52737 CSNK1D-FOXN1-KREMEN1 1122 3656 11331 13933 38262 DKK1-FGF4-KREMEN1 1129 19854 18903 14425 19943 CSNK1A1-CTBP2-KREMEN1 1221 5094 21366 17954 17531 FZD1-FZD2-KREMEN1 1236 35400 37152 3827 42898 CSNK1G1-FGF4-KREMEN1 1239 30116 44329 46230 30098 KREMEN1-WNT1-WNT4 1280 5994 3234 2746 2189 APC-GSK3A-KREMEN1 1292 17817 5446 51090 6601 KREMEN1-PORCN-WNT3A 1377 25809 2588 12830 26801 KREMEN1-PORCN-WNT3A 1377 25809 2588 12830 26801 KREMEN1-NKD1-WNT2B 1400 34496 34759 43275 20181 KREMEN1-PORCN-WNT2B 1401 36444 47192 17886 28451 KREMEN1-NKD1-SENP2 1463 18745 6060 20808 42111 FZD5-FZD2-KREMEN1 1552 30795 21588 5315 28264 CSNK1G1-JUN-KREMEN1 1571 14622 52390 27682 7035 KREMEN1-RHOU-TCF7 1573 55772 27141 223 46704 KREMEN1-PPP2CA-WNT2B 1585 14425 197 53141 37098 FRZB-FZD2-KREMEN1 1603 3863 36101 9766 48273 EP300-FOXN1-KREMEN1 1620 7884 6802 739 13419 CSNK1G1-CTNNB1-KREMEN1 1675 48319 35467 28250 6432 KREMEN1-NKD1-TCF7L1 1699 28241 2070 41242 30439 DIXDC1-FZD2-KREMEN1 1727 6226 28247 11263 35521 FZD1-GSK3A-KREMEN1 1738 40748 5530 55553 1476 KREMEN1-MYC-SENP2 1739 20338 4422 34221 40748 CTNNBIP1-FZD2-KREMEN1 1837 23176 37249 4014 33708 FBXW11-GSK3A-KREMEN1 1865 56429 31282 41254 1352 DAAM1-JUN-KREMEN1 1890 51281 41166 36350 16680 FOXN1-KREMEN1-WNT2 1894 42716 55280 13922 40195 KREMEN1-PORCN-TCF7L1 1902 38888 15726 2313 30252 FRAT1-JUN-KREMEN1 1937 45988 42307 30915 9629 CXXC4-FGF4-KREMEN1 1948 42829 14565 52294 56300 APC-FOXN1-KREMEN1 2000 4298 6205 6426 23498 DIXDC1-JUN-KREMEN1 2010 2457 38608 53030 25504 FZD5-FGF4-KREMEN1 2013 27018 21260 27088 23655 KREMEN1-WNT1-WNT2B 2036 35623 4394 5538 18503 KREMEN1-FBXW4-WNT3A 2114 13176 100 22815 19325 DAAM1-GSK3A-KREMEN1 2129 51563 36659 50263 21382 FZD6-GSK3A-KREMEN1 2225 27912 27936 52258 25906 FBXW2-FGF4-KREMEN1 2253 32946 50021 47804 26837 CTBP1-GSK3A-KREMEN1 2271 26478 2298 49251 1887 AES-DVL1-KREMEN1 2316 39410 26030 52356 23834 KREMEN1-T-WNT2B 2344 47077 5424 49942 15572 KREMEN1-PORCN-SFRP4 2368 33549 23605 2332 43954 CTBP1-FOXN1-KREMEN1 2451 10517 9466 7533 49640 KREMEN1-MYC-TLE2 2509 13737 10240 55345 25198 FBXW2-GSK3A-KREMEN1 2527 40868 10027 47061 40112 KREMEN1-LRP6-TCF7 2533 55286 41418 32082 2935 DAAM1-FGF4-KREMEN1 2579 47103 51872 50968 4413 KREMEN1-NKD1-TLE2 2619 18016 16139 44657 37610 APC-BTRC-KREMEN1 2657 22730 7427 25126 52825 FZD5-CCND1-KREMEN1 2661 25140 10521 9081 26597 KREMEN1-NKD1-WNT3A 2675 23678 2816 47552 40285 CCND3-FOXN1-KREMEN1 2685 41560 9399 7692 43041 BTRC-FOXN1-KREMEN1 2723 18292 12669 9692 56435 FBXW2-FOXN1-KREMEN1 2728 8404 19667 17910 51326 KREMEN1-WIF1-WNT3A 2762 721 11586 56871 31145 KREMEN1-WNT3-WNT3A 2788 47193 12084 27016 19027 FZD1-JUN-KREMEN1 2822 32928 46013 50199 15289 DKK1-FZD2-KREMEN1 2860 30210 53732 5383 20250 CXXC4-FZD7-KREMEN1 2872 42910 42232 51357 48270 CSNK1G1-CTBP2-KREMEN1 2889 6171 37322 13856 5323 KREMEN1-PYGO1-WNT2B 2904 37451 19332 29917 48436 FZD7-JUN-KREMEN1 2975 27209 49555 48377 22318 AXIN1-DVL1-KREMEN1 3005 6666 41605 52750 7478 BTRC-GSK3A-KREMEN1 3010 54393 56679 45265 52103 FOSL1-GSK3A-KREMEN1 3017 11044 1525 53908 3915 KREMEN1-NKD1-FBXW4 3108 22301 16832 29059 23025 KREMEN1-LRP6-SENP2 3111 32111 14514 33645 6872 FOXN1-KREMEN1-SLC9A3R1 3159 39744 45905 6948 40383 FRAT1-GSK3A-KREMEN1 3217 49686 3676 49905 14302 FBXW11-FGF4-KREMEN1 3258 55985 54859 34318 13486 FZD8-JUN-KREMEN1 3275 44731 52219 20333 8524 FBXW11-JUN-KREMEN1 3339 52289 38426 40741 34907 DIXDC1-GSK3A-KREMEN1 3347 13841 2607 50326 30275 CCND2-FOXN1-KREMEN1 3355 3908 6975 5408 46170 DVL1-EP300-KREMEN1 3366 29392 28100 54892 20990 KREMEN1-MYC-WNT3A 3433 19129 6220 28682 38858 CXXC4-FRAT1-KREMEN1 3542 41044 29584 5685 51990 KREMEN1-SLC9A3R1-WNT3A 3580 17181 25435 54188 25643 KREMEN1-PORCN-FBXW4 3663 26907 15869 622 32460 FBXW2-JUN-KREMEN1 3735 35906 54609 50369 55231 KREMEN1-PORCN-TCF7 3746 56109 39312 3880 34922 CSNK1G1-CTNNBIP1-KREMEN1 3759 25578 23550 7886 28116 KREMEN1-LRP6-FBXW4 3766 33137 25609 45562 1844 AES-EP300-KREMEN1 3796 45394 51834 52960 6499 KREMEN1-MYC-TCF7L1 3819 32963 15399 45595 28737 KREMEN1-NKD1-WNT4 3871 11426 4535 41285 47213 CTNNB1-FGF4-KREMEN1 3963 19788 15003 22803 26132 CTBP2-GSK3A-KREMEN1 3981 56103 1110 40270 1391 KREMEN1-PORCN-TLE2 4003 29172 18083 3949 32896 KREMEN1-T-TCF7L1 4058 45731 11965 33080 25646 KREMEN1-WNT1-WNT5A 4086 1014 15715 19712 13437 KREMEN1-PORCN-TLE1 4111 17653 11096 399 32135 Table 1: Rankings of KREMEN1-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - linear binations for DKK1 along with KREMEN1, to be prominent at 3rd order level - DKK1FOXN1-KREMEN1, DKK1-FGF4-KREMEN1 and DKK1-FZD2-KREMEN1. All these combinations indicate the existence of a possible synergy when they take a higher rank in the list of combinations. 7 RANKING @tiUSING HSIC - RBF 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 FZD5-JUN-KREMEN1 14265 7847 49848 50707 31671 AES-AXIN1-KREMEN1 37703 35313 40516 18682 15851 FOXN1-KREMEN1-WNT2B 20404 27973 36396 884 30778 DVL2-JUN-KREMEN1 13103 30709 32743 48017 27397 FOXN1-KREMEN1-PPP2R1A 21547 15130 581 7154 55001 FOXN1-KREMEN1-WNT3 13156 54946 25548 2636 21279 CCND1-CTBP1-KREMEN1 28942 57047 8671 40620 6175 CTNNBIP1-JUN-KREMEN1 49360 1571 43983 55548 30690 FOXN1-KREMEN1-LRP5 11126 50667 3123 17750 44144 FSHB-FZD2-KREMEN1 5444 42081 8565 25988 38407 FBXW11-FOXN1-KREMEN1 332 11091 39492 5306 3719 KREMEN1-PPP2CA-SFRP4 19893 13395 43502 13961 6353 CSNK1G1-FOXN1-KREMEN1 7103 1710 43024 36832 31611 EP300-JUN-KREMEN1 4639 142 27356 54948 19911 DKK1-FOXN1-KREMEN1 881 21837 24344 36049 9457 CTNNB1-FOXN1-KREMEN1 3133 5700 50426 39968 6394 FOXN1-KREMEN1-SFRP4 21541 47835 13452 25040 38289 FZD5-FOXN1-KREMEN1 2473 6345 54537 47495 8093 CSNK1D-FGF4-KREMEN1 2147 22653 25065 18882 43497 FZD1-FZD7-KREMEN1 40446 14016 820 47659 32313 KREMEN1-PORCN-WNT4 4572 591 18551 4045 46258 FOSL1-FOXN1-KREMEN1 2901 4767 41791 45598 13163 CTNNBIP1-FRAT1-KREMEN1 23718 42906 43026 55168 47523 CSNK2A1-FOXN1-KREMEN1 393 11453 41930 49554 21564 FZD5-GSK3A-KREMEN1 3906 30061 53828 42745 53934 CCND1-FGF4-KREMEN1 22424 56825 8471 20861 15165 FZD8-GSK3A-KREMEN1 1348 36579 51609 22398 40701 FSHB-GSK3A-KREMEN1 9218 45608 28535 37479 51895 DIXDC1-FOXN1-KREMEN1 3184 5794 52693 37317 689 FOXN1-KREMEN1-SFRP1 27973 46263 7809 24618 51572 DVL2-FGF4-KREMEN1 6004 6605 12570 48822 49957 FZD5-CCND2-KREMEN1 8697 44124 49389 52728 52529 KREMEN1-PORCN-SENP2 2897 40815 14540 5841 26554 KREMEN1-PYGO1-WNT3A 7864 26771 50128 43544 15239 CSNK1D-FOXN1-KREMEN1 3060 1922 41865 30003 8137 DKK1-FGF4-KREMEN1 1554 26908 3678 42614 56156 CSNK1A1-CTBP2-KREMEN1 10123 4733 30223 55392 43989 FZD1-FZD2-KREMEN1 10931 27613 9671 42642 45433 CSNK1G1-FGF4-KREMEN1 12312 33122 8653 27029 51700 KREMEN1-WNT1-WNT4 2412 6480 36977 30144 18633 APC-GSK3A-KREMEN1 127 26706 52155 45367 53973 KREMEN1-PORCN-WNT3A 8946 29482 33980 6745 37128 KREMEN1-PORCN-WNT3A 8946 29482 33980 6745 37128 KREMEN1-NKD1-WNT2B 10715 45017 15095 25349 17172 KREMEN1-PORCN-WNT2B 4060 34624 13586 33545 46659 KREMEN1-NKD1-SENP2 730 3287 51067 20876 22424 FZD5-FZD2-KREMEN1 17381 29667 40888 35883 46791 CSNK1G1-JUN-KREMEN1 6491 2240 25073 41585 18357 KREMEN1-RHOU-TCF7 13632 56061 32578 2640 36177 KREMEN1-PPP2CA-WNT2B 25164 28694 34566 35971 17687 FRZB-FZD2-KREMEN1 9733 3875 18132 24675 50396 EP300-FOXN1-KREMEN1 2486 110 54841 38583 4454 CSNK1G1-CTNNB1-KREMEN1 12259 40883 18405 45871 37363 KREMEN1-NKD1-TCF7L1 6544 33167 55781 7544 9258 DIXDC1-FZD2-KREMEN1 14683 17223 32337 21120 1723 FZD1-GSK3A-KREMEN1 1383 26847 53461 44646 50340 KREMEN1-MYC-SENP2 7375 12701 49420 6987 13204 CTNNBIP1-FZD2-KREMEN1 10840 35951 31211 40651 41730 FBXW11-GSK3A-KREMEN1 2846 54984 50753 24803 36268 DAAM1-JUN-KREMEN1 12541 51560 9266 55905 3551 FOXN1-KREMEN1-WNT2 14202 42807 982 1807 50203 KREMEN1-PORCN-TCF7L1 5105 42564 13754 28956 27851 FRAT1-JUN-KREMEN1 5995 44498 42143 51675 21496 CXXC4-FGF4-KREMEN1 1514 40573 41207 46205 49688 APC-FOXN1-KREMEN1 1997 6943 51297 49623 30810 DIXDC1-JUN-KREMEN1 28857 3 41815 56574 3054 FZD5-FGF4-KREMEN1 1921 28429 50401 52865 52795 KREMEN1-WNT1-WNT2B 7206 28407 30386 2208 20264 KREMEN1-FBXW4-WNT3A 49105 7228 49337 28152 8690 DAAM1-GSK3A-KREMEN1 2003 53652 54181 54441 24386 FZD6-GSK3A-KREMEN1 2997 31223 53428 45556 33165 FBXW2-FGF4-KREMEN1 2821 26088 13154 23517 16921 CTBP1-GSK3A-KREMEN1 1319 10146 55450 49855 53098 AES-DVL1-KREMEN1 23959 30765 28010 19926 10756 KREMEN1-T-WNT2B 5079 45857 6720 48287 28414 KREMEN1-PORCN-SFRP4 1803 22519 801 35021 39778 CTBP1-FOXN1-KREMEN1 5743 17110 49946 41629 26263 KREMEN1-MYC-TLE2 15234 1306 52625 36242 22622 FBXW2-GSK3A-KREMEN1 852 35836 33828 33146 22194 KREMEN1-LRP6-TCF7 3639 53069 30894 20591 19675 DAAM1-FGF4-KREMEN1 870 48494 7124 51889 10006 KREMEN1-NKD1-TLE2 2919 14915 47965 33551 33612 APC-BTRC-KREMEN1 8265 27604 37492 52851 54717 FZD5-CCND1-KREMEN1 9857 36236 47155 44679 56118 KREMEN1-NKD1-WNT3A 9700 7837 52991 45907 10853 CCND3-FOXN1-KREMEN1 1171 34063 53094 50272 4611 BTRC-FOXN1-KREMEN1 2135 23342 46688 49929 2379 FBXW2-FOXN1-KREMEN1 1595 364 52793 37705 895 KREMEN1-WIF1-WNT3A 38760 4243 33739 39807 7207 KREMEN1-WNT3-WNT3A 13113 51425 51324 32502 6247 FZD1-JUN-KREMEN1 15931 13787 9034 41845 30091 DKK1-FZD2-KREMEN1 6261 35724 3154 16366 45481 CXXC4-FZD7-KREMEN1 3201 51583 32100 52166 52493 CSNK1G1-CTBP2-KREMEN1 16581 39296 9396 40247 34672 KREMEN1-PYGO1-WNT2B 2480 27771 5674 47549 30280 FZD7-JUN-KREMEN1 6606 12101 29205 53157 12973 AXIN1-DVL1-KREMEN1 6881 8114 22985 56273 14578 BTRC-GSK3A-KREMEN1 5140 53781 54895 53606 19024 FOSL1-GSK3A-KREMEN1 5655 3612 51883 54394 46466 KREMEN1-NKD1-FBXW4 16975 10848 31634 30290 26950 KREMEN1-LRP6-SENP2 2434 29765 40575 33866 40506 FOXN1-KREMEN1-SLC9A3R1 14501 25331 8315 26659 34832 FRAT1-GSK3A-KREMEN1 1014 49731 56438 48125 38195 FBXW11-FGF4-KREMEN1 3698 53770 4230 517 36721 FZD8-JUN-KREMEN1 8991 45182 43032 29768 21965 FBXW11-JUN-KREMEN1 8267 48029 13598 24626 10882 DIXDC1-GSK3A-KREMEN1 962 5165 52201 49330 13895 CCND2-FOXN1-KREMEN1 8379 10719 51570 46577 14647 DVL1-EP300-KREMEN1 54034 12200 13668 57031 15029 KREMEN1-MYC-WNT3A 10637 5567 56637 35572 1429 CXXC4-FRAT1-KREMEN1 9306 45373 50696 51816 46911 KREMEN1-SLC9A3R1-WNT3A 5894 4234 35325 31271 14229 KREMEN1-PORCN-FBXW4 2024 24361 7927 6273 32503 FBXW2-JUN-KREMEN1 45460 39448 22654 30792 5213 KREMEN1-PORCN-TCF7 4135 52520 18227 7012 25650 CSNK1G1-CTNNBIP1-KREMEN1 44820 20207 8775 44692 55891 KREMEN1-LRP6-FBXW4 4459 21512 25267 25628 35821 AES-EP300-KREMEN1 46085 42522 27876 8199 46619 KREMEN1-MYC-TCF7L1 11844 48709 41046 17332 2258 KREMEN1-NKD1-WNT4 1146 12492 44039 43786 11253 CTNNB1-FGF4-KREMEN1 1779 13542 31545 43294 49835 CTBP2-GSK3A-KREMEN1 2929 53064 51746 47070 37417 KREMEN1-PORCN-TLE2 18984 25116 9384 24961 48839 KREMEN1-T-TCF7L1 1217 40415 35163 33451 5680 KREMEN1-WNT1-WNT5A 4362 35794 22223 40251 36925 KREMEN1-PORCN-TLE1 12519 13926 4026 25562 26223 Table 2: Rankings of KREMEN1-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - HSIC; Kernel - rbf 6.3.2. Examining the behaviour of LRP-5/6-KREMEN1-X combinations Since KREMEN proteins combine with DKK1 along with LRP-5/6, looking at the tables above, one finds the following combinations for members of LRP family along with KREMEN1, to be prominent at 3rd order level - FOXN1-KREMEN1-LRP5, KREMEN1-LRP6-SENP2, KREMEN1-LRP6-FBXW4 and KREMEN1-LRP6-TCF7. All these combinations indicate the existence of a possible synergy when they take a 8 RANKING @tiUSING SOBOL - 2002 3rd order comb. t1t3t6t12 t24 3rd order comb. t1t3t6t12 t24 FZD5-JUN-KREMEN1 40950 45002 28772 48247 9854 AES-AXIN1-KREMEN1 15129 20699 60 19492 50134 FOXN1-KREMEN1-WNT2B 48078 14775 45903 47094 15768 DVL2-JUN-KREMEN1 30808 34898 33501 52840 6443 FOXN1-KREMEN1-PPP2R1A 42868 43393 35692 37283 25597 FOXN1-KREMEN1-WNT3 22912 27457 7524 8198 32025 CCND1-CTBP1-KREMEN1 2059 53588 1909 4100 53006 CTNNBIP1-JUN-KREMEN1 42377 46499 47863 48423 9908 FOXN1-KREMEN1-LRP5 48531 10599 34009 31625 7563 FSHB-FZD2-KREMEN1 36466 29558 45780 49639 2209 FBXW11-FOXN1-KREMEN1 36681 54148 38407 41684 18205 KREMEN1-PPP2CA-SFRP4 47389 38291 46157 33295 21478 CSNK1G1-FOXN1-KREMEN1 12568 9848 1206 5693 56612 EP300-JUN-KREMEN1 11605 22398 6349 16123 55238 DKK1-FOXN1-KREMEN1 49863 2268 35179 34070 52801 CTNNB1-FOXN1-KREMEN1 28350 28602 22304 4596 30984 FOXN1-KREMEN1-SFRP4 7110 32958 11953 11793 25861 FZD5-FOXN1-KREMEN1 40550 9655 41021 31346 25618 CSNK1D-FGF4-KREMEN1 21369 16424 25622 19556 47905 FZD1-FZD7-KREMEN1 51823 19533 33033 29848 22214 KREMEN1-PORCN-WNT4 2994 1634 5445 11732 52786 FOSL1-FOXN1-KREMEN1 5566 2122 4725 4323 49299 CTNNBIP1-FRAT1-KREMEN1 2163 4891 11842 2940 40557 CSNK2A1-FOXN1-KREMEN1 57030 55282 44745 35369 8120 FZD5-GSK3A-KREMEN1 16434 54131 7576 208 46476 CCND1-FGF4-KREMEN1 6300 54164 19681 18160 46828 FZD8-GSK3A-KREMEN1 1673 51151 25679 27789 20543 FSHB-GSK3A-KREMEN1 49626 22553 37390 51741 6459 DIXDC1-FOXN1-KREMEN1 21482 50446 17603 1664 40258 FOXN1-KREMEN1-SFRP1 46908 27207 43387 43980 36486 DVL2-FGF4-KREMEN1 28296 6083 20156 19941 55458 FZD5-CCND2-KREMEN1 48984 49394 53174 47456 2995 KREMEN1-PORCN-SENP2 1460 11304 1074 20122 52679 KREMEN1-PYGO1-WNT3A 52932 56977 53771 37472 14726 CSNK1D-FOXN1-KREMEN1 31821 16389 28671 30403 11306 DKK1-FGF4-KREMEN1 3925 2235 12144 2997 40109 CSNK1A1-CTBP2-KREMEN1 22944 41262 26386 13278 43474 FZD1-FZD2-KREMEN1 15110 16794 24118 25653 14942 CSNK1G1-FGF4-KREMEN1 49711 51438 57107 44900 43052 KREMEN1-WNT1-WNT4 15955 22656 18139 572 9093 APC-GSK3A-KREMEN1 22161 34336 12787 23677 7226 KREMEN1-PORCN-WNT3A 36088 42706 41171 37942 19003 KREMEN1-PORCN-WNT3A 36088 42706 41171 37942 19003 KREMEN1-NKD1-WNT2B 3720 31186 24394 8640 52443 KREMEN1-PORCN-WNT2B 52764 52181 54179 35890 32504 KREMEN1-NKD1-SENP2 56836 50251 32521 51554 1744 FZD5-FZD2-KREMEN1 18555 33011 7751 196 40221 CSNK1G1-JUN-KREMEN1 6923 4884 26325 13483 55790 KREMEN1-RHOU-TCF7 4926 18206 668 7937 52639 KREMEN1-PPP2CA-WNT2B 2599 25978 4343 22180 25119 FRZB-FZD2-KREMEN1 15195 18589 16392 489 7409 EP300-FOXN1-KREMEN1 23162 5450 14335 12286 51032 CSNK1G1-CTNNB1-KREMEN1 8399 6622 8923 3958 38591 KREMEN1-NKD1-TCF7L1 584 45730 97 3743 56440 DIXDC1-FZD2-KREMEN1 37975 14894 44042 43100 25518 FZD1-GSK3A-KREMEN1 7660 43808 24295 26443 39968 KREMEN1-MYC-SENP2 17023 9287 11185 185 42065 CTNNBIP1-FZD2-KREMEN1 10794 9878 11545 460 10405 FBXW11-GSK3A-KREMEN1 21461 3002 26958 4138 47620 DAAM1-JUN-KREMEN1 52625 28542 37711 31692 4937 FOXN1-KREMEN1-WNT2 9044 42513 11318 10102 41349 KREMEN1-PORCN-TCF7L1 55719 32088 53306 30161 3431 FRAT1-JUN-KREMEN1 40594 22021 35719 33231 5409 CXXC4-FGF4-KREMEN1 9212 1819 7168 7601 47095 APC-FOXN1-KREMEN1 29547 40708 55148 45215 3732 DIXDC1-JUN-KREMEN1 460 53163 11120 19183 46244 FZD5-FGF4-KREMEN1 6216 5651 2403 7647 52951 KREMEN1-WNT1-WNT2B 36576 35166 39887 54348 48410 KREMEN1-FBXW4-WNT3A 45167 47280 56590 48968 2663 DAAM1-GSK3A-KREMEN1 13384 15668 12158 498 56140 FZD6-GSK3A-KREMEN1 12895 32903 20439 2759 51190 FBXW2-FGF4-KREMEN1 22248 48690 8637 9586 46423 CTBP1-GSK3A-KREMEN1 26256 33539 19902 1842 38069 AES-DVL1-KREMEN1 22925 17355 8573 17311 25079 KREMEN1-T-WNT2B 7236 31182 12044 18634 52339 KREMEN1-PORCN-SFRP4 1575 45337 8949 23208 52407 CTBP1-FOXN1-KREMEN1 39964 20046 56465 43731 25806 KREMEN1-MYC-TLE2 45670 47473 48642 49391 21316 FBXW2-GSK3A-KREMEN1 25123 9865 17408 393 41917 KREMEN1-LRP6-TCF7 1727 7976 1357 1445 52140 DAAM1-FGF4-KREMEN1 7127 14101 26191 16548 35458 KREMEN1-NKD1-TLE2 15875 10503 2532 217 43597 APC-BTRC-KREMEN1 37985 37331 55282 41112 5158 FZD5-CCND1-KREMEN1 20691 21099 3090 417 50939 KREMEN1-NKD1-WNT3A 3850 55089 612 11917 55918 CCND3-FOXN1-KREMEN1 51128 56410 37630 41173 22159 BTRC-FOXN1-KREMEN1 50834 8273 50274 53098 8493 FBXW2-FOXN1-KREMEN1 31050 48418 48879 52531 24819 KREMEN1-WIF1-WNT3A 36608 31863 33558 56397 51574 KREMEN1-WNT3-WNT3A 46697 24784 33813 55605 41537 FZD1-JUN-KREMEN1 50807 8217 31011 33476 7864 DKK1-FZD2-KREMEN1 8878 48703 16689 25640 46207 CXXC4-FZD7-KREMEN1 38479 27988 31220 33378 4766 CSNK1G1-CTBP2-KREMEN1 54343 4395 40167 51788 18755 KREMEN1-PYGO1-WNT2B 49601 24278 44028 43212 12272 FZD7-JUN-KREMEN1 13407 6514 26939 4591 30820 AXIN1-DVL1-KREMEN1 14308 33186 25541 10671 49471 BTRC-GSK3A-KREMEN1 8824 50530 21375 2882 33323 FOSL1-GSK3A-KREMEN1 52307 56531 47402 55645 3684 KREMEN1-NKD1-FBXW4 15922 39691 1142 12162 54069 KREMEN1-LRP6-SENP2 3201 51293 12356 14720 31932 FOXN1-KREMEN1-SLC9A3R1 8920 18978 27908 10801 36087 FRAT1-GSK3A-KREMEN1 7872 42308 28346 8689 37391 FBXW11-FGF4-KREMEN1 23854 8666 12120 5305 31743 FZD8-JUN-KREMEN1 44256 39609 38215 56844 3545 FBXW11-JUN-KREMEN1 28638 40636 44640 48839 7440 DIXDC1-GSK3A-KREMEN1 56097 50759 42458 41643 27236 CCND2-FOXN1-KREMEN1 7182 3689 8117 4641 42356 DVL1-EP300-KREMEN1 53244 4343 43692 35471 20160 KREMEN1-MYC-WNT3A 53432 44377 45039 45412 2524 CXXC4-FRAT1-KREMEN1 17526 25193 23946 23550 12450 KREMEN1-SLC9A3R1-WNT3A 7391 2325 5102 8894 56781 KREMEN1-PORCN-FBXW4 55579 11951 48161 33924 4762 FBXW2-JUN-KREMEN1 40033 20545 34038 56164 16905 KREMEN1-PORCN-TCF7 1434 24959 3863 27001 53818 CSNK1G1-CTNNBIP1-KREMEN1 28185 25608 21278 2355 52283 KREMEN1-LRP6-FBXW4 54867 1776 44391 39618 1479 AES-EP300-KREMEN1 29288 36103 39298 31122 2254 KREMEN1-MYC-TCF7L1 56751 50531 56726 45121 4763 KREMEN1-NKD1-WNT4 53306 2099 56540 45244 1241 CTNNB1-FGF4-KREMEN1 52691 24612 41182 40890 28618 CTBP2-GSK3A-KREMEN1 22993 39974 5113 13082 37125 KREMEN1-PORCN-TLE2 54746 5958 46990 32840 4508 KREMEN1-T-TCF7L1 18200 10728 3144 4198 52427 KREMEN1-WNT1-WNT5A 41161 34326 39043 56585 47900 KREMEN1-PORCN-TLE1 2413 51146 10205 24334 52670 Table 3: Rankings of KREMEN1-X-X. A list of approximately first 125 combinations with rankings below 10,000 out of 57,155. SA - SOBOL; Implementation - 2002 higher rank in the list of combinations. 6.3.3. Examining the behaviour of WNT-KREMEN1-X combinations Since there is inhibition of the WNT singaling pathway due to presence of KREMEN activity (and vice versa), combinations of WNTs that might be getting inhibited via KREMEN1 could tell which WNT molecules are getting affected. Looking at the ta9