GO: Biological Process

• Definition: The biological objective to which the gene or gene product contributes.
• Composition: A process is accomplished by one or more assemblies of molecular functions.
• Characteristics: These processes often involve chemical or physical transformations.
• Examples:
  • High Level: “Signal Transduction”, “Cell Growth and Maintenance”
  • Lower Level: “Pyrimidine Metabolism”, “cAMP Biosynthesis”

GO: Molecular Function

• Definition: The biochemical activity of a gene product (the specific “job”).
• The “What”: Describes what is done, rather than why, where, or when.
• Examples:
  • High Level: “Enzyme”, “Transporter”
  • Lower Level: “Adenylate Cyclase”, “Toll Receptor Ligand”

GO: Cellular Component

• Definition: The place in the cell where a gene product is active.
• Specific Locations:
  • Example: “Nuclear membrane”
• Multi-protein Complexes: Can also refer to stable entities made up of multiple gene products.
  • Examples: “Ribosome”, “Proteasome”

GO Terms: The Anatomy of an Entry

[Term]
id: GO:0003678
name: DNA helicase activity
namespace: molecular_function
alt_id: GO:0003679
alt_id: GO:0004003
def: "Unwinding of a DNA helix, driven by ATP hydrolysis." [GOC:jl]
synonym: "ATP-dependent DNA helicase activity" EXACT []
xref: EC:3.6.4.12
xref: Reactome:R-HSA-5693589 "D-loop dissociation and strand annealing"
is_a: GO:0004386 ! helicase activity
is_a: GO:0008094 ! ATP-dependent activity, acting on DNA
property_value: skos:exactMatch EC:3.6.4.12

GO Annotations

GO Annotations

• Gene Association Files (GAFs): Collaborating databases (like FlyBase, MGI, or UniProt) prepare and maintain these files.
• Cardinality (\(0\) to \(n\)):
  • A gene product may have zero entries if its function is entirely unknown.
  • A single gene product often has multiple entries across the three domains (Cellular Component, Molecular Function, and Biological Process).
• Granularity: One protein can be localized to the nucleus and the cytoplasm, while acting as both a transcription factor and a dimerizing agent.

GO Annotations

• Gene Association Files (GAFs): Collaborating databases prepare and maintain these files to link genes to terms.
• Cardinality (\(0\) to \(n\)):
  • A gene product may have zero entries if nothing is known about it.
  • One gene product often has multiple entries covering its Cellular Component, Molecular Function, and Biological Process.

Example: Deoxyribonuclease II

An annotation record typically includes:

GO Evidence Codes

How do we know that a gene product has a certain function?

GO: Overrepresentation Analysis

Counting

• Common things are common
• For instance, extracellular exosome is annotated to 1801 genes in the human genome
• On the other hand, lipoxin B4 metabolic process is annotated to only 2 genes

• But what does more often than we would expect by random chance? mean in this context?

The Binomial Coefficient (review)

Binomial Distribution (review)

• The binomial distribution models the probability of exactly \(k\) successes (e.g., “heads”) in \(n\) trials of a boolean experiment, with the probability of success in a single trial being \(p\).
• The probability of a particular order of trials with \(k\) successes and \(n-k\) failures is obtained by multiplying the probabilities of the individual trials; \[ p^k(1-p)^{n-k} \]
• We have seen in the previous slide that there are \({N\choose k}\) orderings of outcomes (heads and tails) that yield exactly \(k\) successes
• The probability mass function of the binomial is therefore \[ P(X=k) = \underbrace{{N\choose k}}_{\# \text{ways of choosing } k \text{ successes from } n \text{ trials}}\times\underbrace{p^k(1-p)^{n-k}}_{\text{probability of one such trial}} \tag{2}\]

Binomial Test (review)

• If we are interested in testing whether the number of heads is more than we would expect by chance, we can formulate the null and alternative hypetheses as follows
• Let \(p\) be the expected probability of an outcome such as heads
• The null hypothesis (\(H_0\)) is that the true likelihood of heads is not more than \(p\), and the alternative hypothesis is that the true likelihood of heads is more than \(p\) \[ \begin{align} H_0: \pi_{s} &\leq p \\ H_a :\pi_s &> p \end{align} \]

\(n\), \(k\), and \(p\)

Binomial vs hypergeometric distribution

GO Overrepresentation with the Hypergeometric Distribution

The statistical analysis counts up all the ways of getting \(k\) genes annotated to \(t\) and \(n-k\) remaining genes not annotated to \(t\), and compares this number to the number of all possible outcomes.

• Let \(m\) be the number of genes on the microarray chip.

  • \(m_t\) be the number of these genes that are annotated to GO term \(t\).
  • \(n\) be the number of DE genes (the study set).
  • \(k\) be the number of DE genes that are annotated to \(t\).

• We have: \[ P(k\mid n,m,m_t) = \dfrac{\binom{m_t}{k}\binom{m-m_t}{n-k}}{\binom{m}{n}} \tag{4}\]

• Numerator: \(\binom{m_t}{k}\binom{m-m_t}{n-k}\): The number of ways to choose exactly \(k\) annotated genes AND \(n-k\) non-annotated genes.

• Denominator: \(\binom{m}{n}\): The total number of ways to choose any \(n\) genes from the population \(m\).

GO Overrepresentation with the Fisher’s Exact Test

Equation 4 is known as the hypergeometric distribution.

• Analogously to the situation with the binomial distribution, we need to calculate the probability of observing some number \(k\) or more genes annotated to \(t\) in order to have a statistical test
• The sum over the tail of the hypergeometric distribution is known as the Fisher Exact Test:

\[ P(X \geq k) = \sum_{i=k}^{\min(n,\, m_t)} \dfrac{\binom{m_t}{i}\binom{m-m_t}{n-i}}{\binom{m}{n}} \tag{5}\]

• Here, \(X\) is the (random) number of study-set genes annotated to \(t\); \(n\) is the total count of differentially expressed genes; \(m_t\) is the count of genes in the population annotated to GO term \(t\)

Answer

\[ P(X \geq k) = \sum_{i=k}^{\min(n,\, m_t)} \dfrac{\binom{m_t}{i}\binom{m-m_t}{n-i}}{\binom{m}{n}} \]

• It is impossible to draw more annotated genes than the size of the study set itself (\(n\))
• It is impossible to draw more annotated genes than exist in the population annotated to term \(t\) (\(m_t\))
• So the maximum possible value of \(k\) is whichever of these two is smaller — hence \(\min(n, m_t)\).

Term-for-Term

For each tested GO term, we have:

• Null Hypothesis (\(H_0\)): there is no positive association between genes annotated to the term \(t\) and the study set; that is, there is no overrepresentation of term \(t\).

• Alternative Hypothesis (\(H_1\)): There is overrepresentation of the term.

• The null hypothesis corresponds to the probability of observing a test statistic that is at least as extreme as the one that was observed given that the null hypothesis is true.

• Therefore, the null hypothesis corresponding to a one-sided test is rejected if the probability of observing \(n_t\) genes annotated to term \(t\) in the study set by chance is less than \(\alpha\)

• By convention, \(\alpha\) is usually set to 0.05

  This is given by:

\[ P(X_t\geq n_t|H_0) = \sum_{k=n_t}^{\min(m_t,n)} \frac{\displaystyle{m_t \choose k}\displaystyle{{m - m_t} \choose {n - k}}}{\displaystyle{m \choose n}}. \]

Interpretation of significant GO terms

• if \(P(X_t\geq n_t|H_0) < \alpha\), the null hypothesis is rejected, and we declare the term \(t\) to be in the study set.
• The biological interpretation of significant overrepresentation is usually taken to be that the term \(t\) represents an important biological characteristic of the study set.

Example

• Suppose that there is a population set of \(m=18\) genes, of which \(m_t=4\) genes are annotated to the term \(t\).

• The outcome of an experiment conducted on all 18 genes of the population yields a set of 5 genes. This means that the study set consists of \(n=5\) genes.

• Moreover, we observe that a total of \(n_t=3\) genes from the genes of the study set are annotated to term \(t\)

  We would now like to analyze whether term \(t\) is significantly overrepresented in the study set and thus can be interpreted to represent an important result of the experiment:

\[ P(X_t\geq 3|H_0) = \frac{\displaystyle{5 \choose 3}\displaystyle{{13} \choose {2}}}{\displaystyle{18 \choose 5}} + \frac{\displaystyle{5 \choose 4}\displaystyle{{13} \choose {1}}}{\displaystyle{18 \choose 5}} = 0.044. \] - Since \(P(X_t\geq 3|H_0)=0.044<0.05=\alpha\), the null-hypotheses is rejected and the term may be interpreted as being characteristic of the experiment.

Multiple testing correction

Ontologizer

• HOMEWORK You will process a dataset with the Ontologizer to gain intuition about GO TfT analysis.

Gene set enrichment analysis

• Drawbacks of GO Overrepresentation Analysis (GOOA)
  • After correcting for multiple hypotheses testing, no individual gene may meet the threshold for statistical significance, because the relevant biological differences are modest relative to the noise inherent to the microarray technology.
  • The threshold for signicance is arbitrary. What about genes with p=0.051? Maybe they would become significant if we did more repetitions in our RNA-seq experiment
• Gene Set Enrichment Analysis (GSEA):
  • Do not divide genes into study (differentially expressed) and population (all genes)
  • Instead, rank genes according to degree of expression or differential expression
  • Determine whether members of a gene set S tend to occur toward the top (or bottom) of the list L
  • In this case the gene set is correlated with the phenotypic class distinction.

GSEA: concepts

• GSEA considers experiments with genomewide expression profiles from samples belonging to two classes, labeled 1 or 2.
• Genes are ranked based on the correlation between their expression and the class distinction by using any suitable metric
• Concretely, class 1 can be Annotated to GO term t and class 2 Not annotated to t

GSEA: Enrichment Score

• The enrichment score (ES) that reflects the degree to which a set S is overrepresented at the extremes (top or bottom) of the entire ranked list L.
• The score is calculated by walking down the list L, increasing a running-sum statistic when we encounter a gene in S and decreasing it when we encounter genes not in S.
• The magnitude of the increment depends on the correlation of the gene with the phenotype.
• The enrichment score is the maximum deviation from zero encountered in the random walk; it corresponds to a weighted Kolmogorov–Smirnov-like statistic

GSEA: Enrichment score setup

• Input: A ranked list of N genes, ordered by some gene-level statistic

  • correlation with phenotype
  • differential expression
  • a t statistic

• The most differentially expressed genes are placed at the top of the list, the least differential at the bottom.

The original GSEA implementation had a type of signal to noise ratio (Difference of class means divided by sum of class standard deviations. This is essentially a variance-stabilized fold-change)

\[ r_j = \frac{\mu_{j,A} - \mu_{j,B}}{\sigma_{j,A} + \sigma_{j,B}} \]

• Favors genes with a large, consistent (low-variance) difference between groups over genes with a large but noisy difference.
• Similar in intent to a t-test, which most implementations use now

GSEA: Running Sum

The running-sum construction - Walk down the ranked list from position \(j=1,\ldots,N\) - At each step, maintain two running sums:

\[ P_{hit}(S,i) = \sum_{\substack{j \le i \\ g_j \in S}} \dfrac{|r_j|^p}{N_R} \quad\quad \bullet N_R = \sum_{g_j\in S}|r_j|^p \tag{6}\]

• The hit term of Equation 6 increases whenever it encounters a gene that belongs to \(S\), weighted by the magnitude of its ranking statistic \(r_j\)

and

\[ P_{miss}(S,i) = \sum_{\substack{j \le i \\ g_j \notin S}}\dfrac{1}{N-N_R} \]

• The miss term increases whenever it encounters a gene not in S, with a uniform step size.

• Both are normalized so each is a proper empirical CDF: they start at 0, and \(P_{hit}(S,N) = P_{miss}(S,N) = 1\).

GSEA: Running Sum

• The enrichment score is the maximum deviation from zero encountered in the random walk
• \(D = ES(S) = 0.699\) at rank 64
• Leading-edge subset size: 16 of 20 gene-set members

Kolmogorov–Smirnov

• HOMEWORK We will review the Kolmogorov–Smirnov (KS) test in the homework.
• GSEA is “KS-like” but not exactly a KS-test
• The classical two-sample KS test statistic is

\[ D = \sup_x |CDF_1(x) - CDF_2(x)| \tag{8}\]

• The is, the KS statistic is the largest vertical gap between two empirical cumulative distribution functions.
• GSEA is also the largest gap between two step functions, but there are three key differences
  1. GSEA takes weighted (not uniform) steps. The hit function’s step size at gene \(j\) is proportional to \(∣𝑟_j|^p\) (genes that are more correlated with the phenotype contribute larger jumps when they appear.)
  2. Signed max (GSEA), not absolute max (KS).
  3. Discrete, rank-based domain. The distribution being compared isn’t over a continuous variable (as with KS), it’s over discrete positions in a fixed permutation (the gene rank order), so \(P_{hit}\) and \(P_{miss}\) are step functions over \({1,\ldots, N}\) rather than continuous functions over \(\mathbb{R}\)

Estimation of Significance Level of ES

• The calculation of the p-value in the KS test is .is computed from a known asymptotic distribution (the Kolmogorov distribution), either via an exact formula for small n or asymptotically for large n.

• GSEA has no such closed-form null

• In GSEA, the p-value is estimated empirically using permutations

• permuting phenotype labels changes the ranked gene list itself

• If the actual GSEA statistic is higher than all but 2 of 10,000 permutations, then the p value is calculated as \[ p=\frac{b+1}{m+1} = \frac{2+1}{10000+1} \approx 3\times 10^{-4} \]

• Specifically, we permute the phenotype labels and recompute the ES of the gene set for the permuted data, which generates a null distribution for the ES.

• The empirical, nominal P value of the observed ES is then calculated relative to this null distribution.

• The adjustment for multiple testing is essentially the same as with GO Overrepresentation Analysis

GSEA: Examples

• S1 is significantly enriched and scores highly
• S2 is randomly distributed and scores poorly
• S3 is not enriched at the top of the list but is nonrandom, so it scores well (but somewhat difficult to interpret biologically)

Gene Ontology Causal Activity Modeling (GO-CAM)

• The development of the GO has always been tightly coupled to its use in describing the functions of genes across a wide variety of organisms.
• However, biology is better understood as a collection of coupled pathways rather than a set of individual annotations

GO: Traditional Annotation Modeling

• A standard GO annotation is a gene product associated to a GO term, using an evidence code and a supporting reference (a primary research article, for example).
• These are the annotations we were examining in the previous sections
• each GO annotation is necessarily a partial functional description, and there is no representation of how different annotations for the same gene fit together into a more complete description.

GO-CAM models

- **a)**: set of disconnected GO annotations

Specification of the GO-CAM model

GO-CAMs leverage other ontologies to specify biological activities

• The ontologies used to specificy entities shown in the above Figure include

• For instance, to browse the UBERON ontology, use the OLS site

GO-CAM browser

• negative regulation of canonical Wnt signaling pathway

PAN-GO

• Another fascinating resource is PAN-GO
• explicit evolutionary modelling was applied to all human protein-coding genes
• into models that reconstruct the gain and loss of functional characteristics over evolutionary time
• This allowed additional GO annotations to be inferred, with the resulting set of 68,667 integrated gene functions cover approximately 82% of human protein-coding genes

Conclusion

• A major use case for bioontologies such as GO that are used to annotate biological domains (genes and gene products) is to visualize terms (biological concepts) that are overrepresented in experimental results
• We have examined two major algorithmic approaches:
  • GO Overrepresentation Analysis (Fisher Exact Test)
  • Gene-set Enrichment Analysis
• Both approaches typically require multiple testing correction because they perform hundreds or even thousands of individual tests
• New approaches such as GOCAM and PAN-GO are making GO modelling more sophisticated and providing more comprehensive functional annotations
• A foundation for understanding biology and developing bioinformatics algorithms and analysis pipelines.

Sources

• Bauer S, et al. (2008) Ontologizer 2.0–a multifunctional tool for GO term enrichment analysis and data exploration. Bioinformatics 24:1650-1
• Subramanian A, et al. (2005) Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proc Natl Acad Sci U S A. 102:15545-50
• Mootha VK, et al. (2003) PGC-1alpha-responsive genes involved in oxidative phosphorylation are coordinately downregulated in human diabetes. Nat Genet. 34:267-73
• Thomas PD, et al. (2019) Gene Ontology Causal Activity Modeling (GO-CAM) moves beyond GO annotations to structured descriptions of biological functions and systems. Nat Genet 51:1429-1433
• Feuermann M, et al. (2025) A compendium of human gene functions derived from evolutionary modelling. Nature 640:146-154