Lecture 8: Gene expression analysis

class: center, middle, inverse, title-slide

.title[
# Lecture 8: Gene expression analysis
]
.subtitle[
## BE_22 Bioinformatics SS 21
]
.author[
### January Weiner
]
.date[
### 2024-06-10
]

---

## Plan for today

* Gene expression analysis – as an in-depth example of high-throughput
   analyses

---

## Transcriptomic methods

* QPCR: precise, low-throughput
 * Nanostring: precise, mid-throughput (~ 500-1000 genes)
 * Microarray: less exact, high-throughput, pre-defined genes
 * RNA-Seq: very flexible, less exact, high-throughput

---

## QC – quality control

* Method-dependent quality measures
   * aligned reads
   * duplications
   * intron / exon binding
   * Y chromosome binding
   * adapter content
 * Bias? Clustering?

---

## (Demo) Example RNA-Seq QC

Document derived using the `multiQC` package can be retrieved
[here](multiqc.all_samples.all_mates.qc_report.html)

---

## The GSE156063 data set

---

## The GSE156063 data set

<table>
 <thead>
 <tr>
 <th style="text-align:left;"> Col </th>
 <th style="text-align:left;"> Class </th>
 <th style="text-align:right;"> NAs </th>
 <th style="text-align:right;"> unique </th>
 <th style="text-align:left;"> Summary </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> label </td>
 <td style="text-align:left;"> &lt;chr&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> All values unique </td>
 </tr>
 <tr>
 <td style="text-align:left;"> filename </td>
 <td style="text-align:left;"> &lt;chr&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> All values unique </td>
 </tr>
 <tr>
 <td style="text-align:left;"> md5 </td>
 <td style="text-align:left;"> &lt;chr&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> All values unique </td>
 </tr>
 <tr>
 <td style="text-align:left;"> replicate </td>
 <td style="text-align:left;"> &lt;int&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;"> 1 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> title </td>
 <td style="text-align:left;"> &lt;chr&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> All values unique </td>
 </tr>
 <tr>
 <td style="text-align:left;"> geo_accession </td>
 <td style="text-align:left;"> &lt;chr&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> All values unique </td>
 </tr>
 <tr>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> age </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> &lt;dbl&gt; </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 0 </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 68 </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 20.0 [40.0 &lt;50.5&gt; 65.0] 89.0 </td>
 </tr>
 <tr>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> disease_state </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> &lt;chr&gt; </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 0 </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 3 </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> no virus: 100, SC2: 93, other virus: 41 </td>
 </tr>
 <tr>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> gender </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> &lt;chr&gt; </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 0 </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 2 </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> F: 124, M: 110 </td>
 </tr>
 <tr>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> group </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> &lt;chr&gt; </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 0 </td>
 <td style="text-align:right;font-weight: bold;color: white !important;background-color: #D7261E !important;"> 3 </td>
 <td style="text-align:left;font-weight: bold;color: white !important;background-color: #D7261E !important;"> no: 100, SC2: 93, other: 41 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> sizeFactor </td>
 <td style="text-align:left;"> &lt;dbl&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 234 </td>
 <td style="text-align:left;"> 0.0754 [0.5049 &lt;1.1083&gt; 2.0840] 7.4839 </td>
 </tr>
 <tr>
 <td style="text-align:left;"> replaceable </td>
 <td style="text-align:left;"> &lt;lgl&gt; </td>
 <td style="text-align:right;"> 0 </td>
 <td style="text-align:right;"> 1 </td>
 <td style="text-align:left;"> TRUE: 234 FALSE: 0 </td>
 </tr>
</tbody>
</table>

---

How do the count data look like?

---

# QC using PCA

Dimension reduction with PCA allows us to see batch effects and also have a
first preliminary look at the data.

---

.pull-left[

### Before PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-7-1.png)

Correlation between x and y = 0.66

]

.pull-right[

### After PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-8-1.png)

Correlation between PC1 and PC2 = -0.00
]

---

.pull-left[

### Before PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-9-1.png)

Correlation between x and y = 0.84

]

.pull-right[

### After PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-10-1.png)

Correlation between PC1 and PC2 = 0.00
]

---

.pull-left[

### Before PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-11-1.png)

]

.pull-right[

### After PCA:

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-12-1.png)
]

---

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---

## PCA "scree plot"

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-13-1.png)

---

## Heatmaps of the most variable genes

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-14-1.png)

---

## Differential gene expression analysis

* Finding differences between groups
 * Models may vary

---

## GLMs – generalized linear models
 
 * OLS – ordinary least squares regression
 * Fit data (independent + dependent variables) to a model – a bit like in
   OLS
 * Differences to OLS
   * **link function** allows models in which variables have
     non-normal distributions
   * maximum likelihood estimation (rather than least squares)
 * Count data: negative binomial distribution of the dependent variables

---

## GLMs – generalized linear models

For a given gene, we assume that its counts in a sample `$i$` have negative
binomial distribution:

`$$\text{counts} \sim f_{\text{binom}}(\mu_i, \alpha)$$`

(Negative binomial distribution is a generalization of a Poisson
distribution that you might know)

Statistical test:

**Null hypothesis `$H_0$`:** For each sample `$i$`, `$\mu_i = \mu_0$`

**Alternative hypothesis `$H_1$`:**

`$$\log{\mu_i} = \beta_0 + x_i \times \beta_G$$`

Where

`$$\beta_G = \left\{\begin{array}{ll}0 & x_i \text{is control} \\ 1 & x_i \text{is treatment}\end{array}\right.$$`

This model is then tested against the `$H_0$` using a Wald test.

---

## Correction for multiple testing

.pull-left[

Statistical testing is like a russian roulette: each time you test, you
have a chance of drawing a False Positive (FP). If you draw many times, you
are bound to have "significant" FP's.

* Family-wise error correction (e.g. Bonferroni): guarantees that the
   chance of having 1 or more FP's in the results significant at `$\alpha$`
   is `$\alpha$`.
 * False-discovery rate (FDR; e.g. Benjamini-Hochberg): guarantees that the
   *fraction* of FP's in the results significant at `$\alpha$` is no more
   than `$\alpha$`.

]

.pull-right[

**Remember:** any results not corrected for multiple testing are *not
 trustworthy!*

]

---

## R equation language

R uses a particular representation of linear model equations:

e.g.

`$$y = \beta_0 + \beta_1 \times y$$`

is represented as

`y ~ x`

Whereas

`$$y = \beta_1 \times y$$`

(no intercept, i.e. intercept `$= 0$`)

becomes

`y ~ 0 + x`

---

<pre class="r-output"><code>## ENSEMBL SYMBOL ENTREZID REFSEQ
## 1 ENSG00000126709 IFI6 2537 NM_002038
## 2 ENSG00000185745 IFIT1 3434 NM_001270927
## 3 ENSG00000137959 IFI44L 10964 NM_006820
## 4 ENSG00000169248 CXCL11 6373 NM_001302123
## 5 ENSG00000134326 CMPK2 129607 NM_001256477
## 6 ENSG00000187608 ISG15 9636 NM_005101
## GENENAME baseMean
## 1 interferon alpha inducible protein 6 603.9613
## 2 interferon induced protein with tetratricopeptide repeats 1 950.0070
## 3 interferon induced protein 44 like 1197.3869
## 4 C-X-C motif chemokine ligand 11 136.6495
## 5 cytidine/uridine monophosphate kinase 2 338.7044
## 6 ISG15 ubiquitin like modifier 755.3958
## log2FoldChange lfcSE stat pvalue padj symbol entrez
## 1 3.405602 0.1994352 17.07623 2.230680e-65 3.564403e-61 IFI6 2537
## 2 3.334884 0.2138449 15.59487 7.887986e-55 6.302107e-51 IFIT1 3434
## 3 3.082872 0.2008791 15.34691 3.714854e-53 1.978655e-49 IFI44L 10964
## 4 5.959395 0.3940944 15.12174 1.164190e-51 3.720517e-48 CXCL11 6373
## 5 2.758852 0.1936366 14.24758 4.641192e-46 8.240179e-43 CMPK2 129607
## 6 3.766232 0.2743033 13.73017 6.697215e-43 1.070148e-39 ISG15 9636
</code></pre>

---

## Heatmap of the DE genes

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-16-1.png)

---

## Plotting individual genes

.pull-left[

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-17-1.png)
]

.pull-right[

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-18-1.png)

]

---

## What next?

* What is our research question?
 * How do we get from genes to functions or pathways?
 * How do we validate our data?

---

# Clustering

* "unbiased" machine learning
 * group data points (samples, genes, ...) by similarity
 * applications in many areas of bioinformatics

---

.pull-left[
## Why cluster the genes?

* Genes which are co-regulated have similar function `$\rightarrow$`
   identification of "transcriptional modules"
 * Analysis of the clusters may give us clues about what is happening in
   the biological system
 * Better visualization if genes are clustered

]

.pull-right[

<img src="weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-19-1.png" width="60%" />
]

---
class:empty-slide,mywhite
background-image:url(images/network_1.png)

---
class:empty-slide,mywhite
background-image:url(images/network_2.png)

---

### From correlations to distances

`$d_{i,j}$` – Distance between genes `$i$` and `$j$`

`$\rho(x_i, x_j)$` – correlation coefficient between expression of `$i$` `$x_i$` and `$j$` `$x_j$`

Hard threshold:

`$$d_{i,j} \equiv \left\{\begin{array}{ll}0 & |\rho(x_i, x_j)| > \tau\\1 & |\rho(x_i, x_j)| \leq \tau\end{array}\right.$$`

Continuouos in `$(0, 1)$`

`$$d_{i,j} \equiv 1 - |\rho(x_i, x_j)|$$`

---
class:empty-slide,mywhite
background-image:url(images/correlation_modules.png)

---
class:empty-slide,mywhite
background-image:url(images/correlation_dependence.png)

---

## Many alternatives

Other measures exist:

* mutual information
   * different types of correlation (Spearman `$\rho$`, Kendall `$\tau$`)
   * distance correlation (Székely 2007)

Bottom line: we need to get at distances

---
class:empty-slide,myinverse
background-image:url(images/desert2.jpg)

.mytop[
The number of grains of sand in this picture accurately represents the
number of clustering algorithms.
]

--
.mytop[
 
 
 
 
 
Each tree in the picture corresponds to a method of clustering which is
robust, reliable and provides automatically high quality clusters. 
]

---
class:empty-slide,myinverse
background-image:url(images/desert2.jpg)

.pull-left[
.transpblock[
**PAM** Partitioning Around Medoids. Similar to k-means;
predefined number of clusters
]

.transpblock[
**SOMs**, Self-organizing maps:
Train a neural network to recognize clusters in the data
(also SOTA: self-organizing trees)
]

.transpblock[
**SVC** – support vector clustering;
based on SVM’s (support vector machines)
]

.transpblock[
**Fuzzy C-means clustering**:
Each sample gets assigned a probability of
belonging to each of the clusters.
Algorithm similar to k-means.
Also similar – “soft k-means”
]

]

.pull-right[

.transpblock[
**Hierarchical clustering:** UPGMA (group-average linkage), Ward's method, neighbor joining, single
linkage, complete linkage...

]

.transpblock[
**Mclust** – model based clustering
The data is fit to a statistical model consisting of K normal distributions.
]

.transpblock[
**Density based clustering:**
identify clusters by local density profiles
]

]

---

### Hierarchical clustering

* Start with each element in its own cluster: `$n_{\text{clust}} = N$`
 * Identify two elements with the smallest distance
 * Recalculate distances using a *linkage* function

E.g. complete linkage:

`$$D(C_i, C_j) = \max_{e_i \in C_i, e_j \in C_j} D(e_i, e_j)$$`

Basically: join the clusters with the smalles distance between the
*furthest* elements of the clusters

E.g. single linkage:

`$$D(C_i, C_j) = \min_{e_i \in C_i, e_j \in C_j} D(e_i, e_j)$$`

Basically: join the clusters with the smalles distance between the
*closest* elements of the clusters

---
class:empty-slide,mywhite
background-image:url(images/hclust.png)

---

## k-means

Naive k-means:

* Assign the elements to random clusters
 * Repeat:
   * Calculate the cluster centroids (midpoints) 
   * Assign the elements to the cluster with the closest centroid
   * Abort if the asignments no longer change

.center[
![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-21-1.png)
]

---

.center[
![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-22-.gif)
]

---

## Measuring clustering performance

.pull-left[

* How many clusters?

* Which clustering method?

* What parameters?
]

.pull-right[

Clustering performance measures:

* Internal: information scientific

* Stability measures: cross-validation, resampling (like bootstrapping in
   phylogenies!)

* External: use external information
   * a priori information
   * biological information (how well the clusters correspond to
      underlying biology)

]

---
class:empty-slide,myinverse
background-image:url(images/desert2.jpg)

.mytop[
The number of grains of sand in this picture accurately represents the
number of available measures of clustering performance.
]

---

## The elbow method

.pull-left[
There is a total amount of variance in the clusters.

We can split it into *between cluster* variance and *within cluster*
variance.

Instead of variance, we use Sum of Squares (SS), because `$Var(x)\equiv \frac{SS}{n-1}$`.

`$$SS_{\text{tot}} = SS_{\text{within}} + SS_{\text{between}}$$`

* If all elements are in one large cluster, `$SS_{\text{between}}= 0$`.
 * If all elements are each in a separate cluster, `$SS_{\text{within}}= 0$`.
]

.pull-right[

![](images/elbow.png)

]

---

## Silhouette plots

.pull-left[

For each element, we plot the difference between distance to the center of
the cluster where that element is and the distance to the closest *other*
cluster center.

]

.pull-right[

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-24-1.png)

]

---

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-25-1.png)

---

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-26-1.png)

---

![](weiner_BE_22_lecture_08_files/figure-html/unnamed-chunk-27-1.png)

---

Another example: clustering of single cell data

![](images/cell_1.png)

---

Another example: clustering of single cell data

![](images/cell_2.png)

---

Another example: clustering of single cell data

![](images/cell_3.png)