$\langle f,g\rangle_{p} = (\sum_{k=1}^{n}\mid f_k.g_k\mid^p)^\frac{1}{p}$
(2)
By this notation, Sobolev inner product, norm and meter of degree $k$ respectively can be defined by:$\langle f,g\rangle_{p,a}^{S} = \langle f,g\rangle_p+\alpha\langle D^kf,D^kg\rangle_p$
(3)
$\mid\mid f\mid\mid_{p,k,\alpha}^S = \sqrt{\langle f,f\rangle_{p,\alpha}^S}$
(4)
$d_{p,k,\alpha}^S(f,g) = \mid\mid f-g\mid\mid_{p,k,\alpha}^S$
(5)
where $D^k$ is the $k$th differential operator. For the special case $p=2$ and $\alpha=1$ an interesting connection to the Fourier-transform of analysis can be made; let $\hat{f}$ be the Fourier-transform $f$$\hat{f}(\omega_k) = \sum_{j=1}^{N-1}g_jexp(-i\frac{2\pi kj}{N})$
(6)
Where $\omega_k=\frac{2\pi k}{N}$ and $i=\sqrt{-1}$. Finally the norm can be written as$\mid\mid f\mid\mid_{2,k,1}^S = \sqrt{\sum_{j=1}^{N-1}(1+\omega_j)^k\mid \hat{f}(\omega_j)\mid^2}$
(7)
In this work metric (5) with norm (7) and $k=1$ was used. #### Fisher metric In this section we use definitions and notations such as in [3]. To define Fisher information metric we first introduce the n-simplex $P_n$ defined by$P_n=\{x\in R^{n+1}:\forall i, x_n \ge0,\sum_{i=1}^{n+1}xi=1\}$
(8)
The coordinates $\{x_i\}$ describe the probability of observing different outcomes in a single experiment (or expression value of a gene in $i$th cell type). The Fisher information metric on $P_n$ can be defined by$Jij = \sum_{k=1}^{n+1}\frac{1}{x_k}\frac{\partial x_k}{\partial x_i}\frac{\partial x_k}{\partial x_j}$
(9)
We now define a well-known representation of the Fisher information as a pull-back metric from the positive n-sphere $S_n^+$$S_n^+=\{x\in R^n;\forall i,x_n\ge0,\sum_{i=1}^{n+1}x^2=1\}$
(10)
The transformation $T: P_n\to S_n^+$ defined by$T(x)=(\sqrt{x_1}, \dots, \sqrt{x_n+1})$
(11)
pulls back the Euclidean metric on the surface of the sphere to the Fisher information on the multinomial simplex. Actually, the geodesic distance for $x,y \in P_n$ under the Fisher information metric may be defined by measuring the length of the great circle on $S_n^+$ between $T(x)$ and $T(y)$$d(x,y) = acos(\sum_{i=1}^{n+1}\sqrt{x_iy_i})$
(12)
**The LNCRNA2GOA method can also be used on other novel genes besides lncRNAs.** ### Example #### Scores + Enrichment The following example is an arbitrary use-case. Meaning that this is just an example and does not (necessarily) imply a certain question/real life use-case. `id_select_vector` represents a vector of gene ids that you'd want to use for annotation enrichment (if left empty, algorithm will use all available gene ids in the ExpressionSet). ```{r lncRNApred} # Example with default dataset, take a look at the data documentation # to fully grasp what's going on with the making of the filter etc. (Biobase # ExpressionSet) # keep everything that is a protein coding gene (for annotation) filter_vector <- fData(GAPGOM::expset)[( fData(GAPGOM::expset)$GeneType=="protein_coding"),]$GeneID # set gid and run. gid <- "ENSG00000228630" result <- GAPGOM::expression_prediction(gid, GAPGOM::expset, "human", "BP", id_translation_df = GAPGOM::id_translation_df, id_select_vector = filter_vector, method = "combine", verbose = TRUE, filter_pvals = TRUE) kable(result) %>% kable_styling() %>% scroll_box(width = "100%", height = "500px") ``` Here we display the results, you can see that it has 6 columns; - `GOID` Describing the significantly similar GO terms - `Ontology` Describing the ontology of the result. - `Pvalue` The P-value/significance of the result. - `FDR` bonferoni normalized P-value - `Term` The description of the GO term. - `used_method` Used scoring method for getting the result. Besides this, there is an optional parameter for different GO labeling/annotation; `id_translation_df`. This dataframe should contain the following; - rownames $\to$ rownames of the expression set - first column $\to$ Gene ID (e.g. EntrezID). Gene should be the same as in the expression dataset. - second colum $\to$ GO IDs. This can also drastically improve calculation time as most of the time is spent on querying for this translation. #### Only scores There is also another algorithm that allows you to just calculate scores and skip the enrichment; ```{r lncrnapredscoreonly} # Example with default dataset, take a look at the data documentation # to fully grasp what's going on with making of the filter etc. (Biobase # ExpressionSet) # set an artbitrary gene you want to find similarities for. (5th row in this # case) gid <- "ENSG00000228630" result <- GAPGOM::expression_semantic_scoring(gid, GAPGOM::expset) kable(result[1:100,]) %>% kable_styling() %>% scroll_box(width = "100%", height = "500px") ``` We can see that this function returns a different dataframe; - `original_ids` The identifier of the gene expression row - `score` The similarity score/correlation calculated by one of the methods. - `used_method` The used method used to calculate the score. The rownames also represent the gene expression row. Only the first 100 rows are shown, otherwise the table would be quite big. Enrichment and GO annotation/translation needs to be done manually after this step. However, this should be quite doable with a bit of help from `GOSemSim`. #### Original dataset The original publication used the lncRNA2Function data [4], to test if the results are the same, a small script has been made to reproduce the same results and is located at the package install directory under `scripts`. The `script` folder also contains the two original scripts for the algorithm, but not neccesarily its data. The data (along with the scripts) can be found on the following websites: - [http://tare.medisin.ntnu.no/pred_lncRNA/](http://tare.medisin.ntnu.no/pred_lncRNA/) - [https://tare.medisin.ntnu.no/TopoICSim/](https://tare.medisin.ntnu.no/TopoICSim/) Besides this, the scripts folder also contains a proof-of-concept script for potentially doing the analysis on unannotated transcripts (by finding the closest gene). This is eventually meant as a sort of substitute to the famous [GREAT](http://great.stanford.edu/public/html/) tool. ## TopoICSim ### Background TopoICSim or Topological Information Content Similarity, is a method to measure the similarity between two GO terms given the information content and topology of the GO DAG tree. Unlike other similar measures, it considers both the shortest **and** longest DAG paths between two terms, not just the longest or shortest path(s). The paths along the GO DAG tree get weighted with the information content between two terms. For the information content the following forumla is used;$IC(t) = -log(p(t))$
(1)
Where $t$ is the (GO) term. The IC is calculated by GOSemSim, and based on the frequency of the specific go term ($p(t)$). A GO tree can be described as a triplet $\Lambda=(G,\Sigma,R)$, where $G$ is the set of GO terms, $\Sigma$ is the set of hierarchical relations between GO terms (mostly defined as *is_a* or *part_of*) [5], and $R$ is a triplet $(t_i, t_j, \xi)$, where $t_i,t_j\in G$ and $\xi\in R$ and $t_i\xi t_j$. The $\xi$ relationship is an oriented child-parent relation. Top level node of the GO rDAG is the Root, which is a direct parent of the MF, BP and CC nodes. These nodes are called aspect-specific roots and we refer to them as root in the following. A path $P$ of length $n$ between two terms $t_i,t_j$ can be defined as in (23).$P:G\times G\to G\times G\dots\times G=G^{n+1};\\P(t_i,t_j) = (t_i,t_j+1,\dots,t_j)$
(23)
Here $\forall$ $s$, $i\le s$\mathcal{A}(t_i,t_j)=\underset{P}{\cup}P(t_i,t_j)$
(24)
We use Inverse Information Content (IIC) values to define shortest and longest paths for two given terms $t_i,t_j$ as shown in (25-27).$SP(t_i,t_j) = \underset{P\in A(t_i,t_j)}{argminIIC(P)}$
(25)
$LP(t_i,t_j) = \underset{P\in A(t_i,t_j)}{argmaxIIC(P)}$
(26)
$IIC(P) = \sum_{t\in P}\frac{1}{IC(t)}$
(27)
The standard definition was used to calculate $IC(t)$ as shown in (28)$IC(t) = -log\frac{G_t}{G_\mathrm{Tot}}$
(28)
Here $G_t$ is the number of genes annotated by the term $t$ and $G_\mathrm{Tot}$ is the total number of genes. The distribution of *IC* is not uniform in the *rDAG*, so it is possible to have two paths with different lengths but with same *IIC*s. To overcome this problem we weight paths by their lengths, so the definitions in (25) and (26) can be updated according to (29) and (30).$wSP(t_i,t_j)=SP(t_i,t_j)\times len(P)$
(29)
$wLP(t_i,t_j)=LP(t_i,t_j)\times len(P)$
(30)
Now let $ComAnc(t_i,t_j)$ be the set of all common ancestors for two given terms $(t_i,t_j)$. First we define the disjuntive common ancestors as a subset of $ComAnc(t_i,t_j)$ as in (31).$DisComAnc(t_i,t_j) = \{x\in ComAnc(t_i,t_j)\mid P(x, root)\cap C(x)=\varnothing\}$
(31)
Here $P(x,root)$ is the path between $x$ and $root$ and $C(x)$ is set of all immediate children for $x$. For each disjuntive common ancestor $x$ in $DisComAnc(t_i,t_j)$, we define the distance between $t_i,t_j$ as the ratio of the weighted shortest path between $t_i,t_j$ which passes from $x$ to the weighted longest path between $x$ and $root$, as in (32-33).$D(t_i,t_j,x) = \frac{wSP(t_i,t_j,x)}{wLP(x,root)}$
(32)
$wSP(t_i,t_j,x) = wSP(t_i,x)+wSP(t_j,x)$
(33)
Now the distance for two terms $t_i,t_j$ can be defined according to (34).$D(t_i,t_j)=\underset{x\in DisComAnc(t_i,t_j)}{min}D(t_i,t_j,x)$
(34)
We convert distance values by the $\frac{Arctan(.)}{\pi/2}$ function, and the measure for two GO terms $t_i$ and $t_j$ can be defined as in (35).$S(t_i,t_j) = 1-\frac{Arcatan(D(t_i,t_j))}{\pi/2}$
(35)
Note that $root$ refers to one of three first levels in the rDAG. So if $DisComAnc(t_i,t_j)=\{root\}$ then $D(t_i,t_j)=\infty$ and $S(t_i,t_j)=0$. Also if $t_i = t_j$ then $D(t_i,t_j)=0$ and $S(t_i,t_j)=1$. Finally let $S=[s_{ij}]_{n\times m}$ be a similarity matrix for two given fenes or gene products $g1, g2$ with GO terms $t_{11},t_{12},\dots,t_{1n}$ and $t_{21},t_{12},\dots,t_{2m}$ where $s_{ij}$ is the similarity between the GO terms $t_{1i}$ and $t_{2j}$. We used a *rcmax* method to calculate similarity between $g1, g2$, as defined in (36).$\begin{aligned}TopoICSim(g_1,g_2)&=rcmax(S)\\&=rcmax\left(\frac{\sum_{i=1}^n\underset{j}{maxs_{ij}}}{n},\frac{\sum_{i=1}^m\underset{i}{maxs_{ij}}}{m}\right)\end{aligned}$
(36)
We also tested other methods on the similarity matrix, in particular average and BMA, but in general $rcmax$ gave the best performance for TopoICSim (data not shown). Besides all this, there's also an algorithm for the geneset level, you can calculate interset/intraset similarities of these with (13) and (14). Or simply use R's `mean` on the resulting matrix.$IntraSetSim(S_k)=\frac{\sum_{i=1}^n\sum_{j=1}^mSim(g_{ki},g_{kj})}{n^2}$
(13)
$InterSetSim(S_k)=\frac{\sum_{i=1}^n\sum_{j=1}^mSim(g_{ki},g_{kj})}{n\times m}$
(14)
(13) is equal to (14) in the specific case of comparing the geneset to itself. All forumulas/explanations are quoted from the paper (references edited) [6]. ### Examples The following example uses the [pfam clan gene set](https://pfam.xfam.org/family/PF00171) to measure intraset similarity. For the single gene, `EntrezID` 218 and 501 are compared. ```{r TopoICSim} result <- GAPGOM::topo_ic_sim_genes("human", "MF", "218", "501", progress_bar = FALSE) kable(result$AllGoPairs) %>% kable_styling() %>% scroll_box(width = "100%", height = "500px") result$GeneSim # genelist mode list1 <- c("126133","221","218","216","8854","220","219","160428","224", "222","8659","501","64577","223","217","4329","10840","7915","5832") # ONLY A PART OF THE GENELIST IS USED BECAUSE OF R CHECK TIME CONTRAINTS result <- GAPGOM::topo_ic_sim_genes("human", "MF", list1[1:3], list1[1:3], progress_bar = FALSE) kable(result$AllGoPairs) %>% kable_styling() %>% scroll_box(width = "100%", height = "500px") kable(result$GeneSim) %>% kable_styling() %>% scroll_box(width = "100%", height = "500px") mean(result$GeneSim) ``` Here we can see the output of TopoICsim, it is a list containg 2 items; - `AllGoPairs` The $n\times m$ matrix of GO terms, including their similarities. Some values might be NA because these were non-occuring pairs. You can add `AllGoPairs` to the parameters next time you run TopoICSim, to possibly speed up computations (they will be used as pre-calculated scores to fill in occurring pairs). - `GeneSim` The gene/geneset similarities depending on your input. Can be 1 number, or a matrix displaying all possible combinations. $\to$ The mean of the geneset matrix shows the intraset/interset similarity. #### Custom genes Besides this, you can also define custom genes for TopoICSim. This consists of an arbitrary amount of GO terms. The custom genes have to individually defined in a named list. ```{r} custom <- list(cus1=c("GO:0016787", "GO:0042802", "GO:0005524")) result <- GAPGOM::topo_ic_sim_genes("human", "MF", "218", "501", custom_genes1 = custom, drop = NULL, verbose = TRUE, progress_bar = FALSE) result ``` Here we define a custom gene named "cus1" with the GO terms; "GO:0016787", "GO:0042802", "GO:0005524", it will be added to the first gene vector (218). If you want to **only** have a custom gene, you can define an empty vector with `c()` for the respective vector. #### Other notes. With TopoICSim there is a pre-calculated score matrix available, this can be turned on/off. The scores might become deprecated however, as soon as one of the `org.DB` packages gets an update. For this reason, we advise mostly to keep this option off. You can also pre-calculate some GO's yourself using the custom genes. `all_go_pairs` can then be used as a precalculated score matrix, only intersecting/present GO terms will be used. ## Parrallel processing and big data As of right now, parallel processing is not supported. The other algorithms aren't parallelized yet, because of implementation difficulties regarding the amount and type of dependencies that the algorithms rely on. Possibly, in the future, parallel processing will be supported. It might however be possible to divide the work in TopoICSim (on gene pair level). It runs for each unique pair of genes, which you may be able to generate using the hidden `GAPGOM:::.unique_combos` function. The `All_go_pairs` object in the result can then be combined together with other results. No support is offered however in regards to trying to make this work. Tip: The `all_go_pairs` argument in `topo_ic_sim_genes()` doesn't automatically create a new, bigger matrix, it only uses the overlapping or present GO terms of the analysis based on the input genes. ## Performance and benchmarks The performance of this package is well tested in the [Benchmarks vignette](benchmarks.html), in which benchmarks are also prepared. ## Contact and support For questions, contact, or support use the [(Bioconductor) git repository](https://github.com/Berghopper/GAPGOM/) or contact us via the [Bioconductor forums](). ## SessionInfo ```{r} sessionInfo() ``` ## References - [1] Ehsani R, Drablos F: **Measures of co-expression for improved function prediction of long non-coding RNAs**. *BMC Bioinformatics* 2018. Accepted. - [2] Villmann T: **Sobolev metrics for learning of functional data - mathematical and theoretical aspects**. In: *Machine Learning Reports.* Edited by Villmann T, Schleif F-M, vol. 1. Leipzig, Germany: Medical Department, University of Leipzig; 2007: 1-13. - [3] Lebanon G: **Learning riemannian metrics.** In: *Proceedings of the Nineteenth conference on Uncertainty in Artificial Intelligence; Acapulco, Mexico.* Morgan Kaufmann Publishers Inc. 2003: 362-369 - [4] Jian Q: **LncRNA2Function: a comprehensive resource for functional investigation of human lncRNAs based on RNA-seq data.** In: *BMC Genomics* 2015. DOI: [10.1186/1471-2164-16-S3-S2](https://doi.org/10.1186/1471-2164-16-S3-S2) - [5] Benabderrahmane S, Smail-Tabbone, Poch O, Napoli A, Devignes MD. **IntelliGO: a new vector-based semantic similarity measure including annotation origin.** In: *BMC Bioinformatics*. 2010;11:588. - [6] Ehsani R, Drablos F: **TopoICSim: a new semantic similarity measure based on gene ontology.** In: *BMC Bioinformatics* 2016, **17**(1):296. DOI: [10.1186/s12859-016-1160-0](https://doi.org/10.1186/s12859-016-1160-0)