- In the last lab you learned how to group genes into clusters based on similar expression patterns.
- In this lab we extend this concept to build gene networks
- Gene networks are graphs that show connections between genes with similar expression.
Clustering vs Networks
Co-expression Network
- The goal is to connect genes with the most similar expression
- One simple way to do this is to use correlation as a measure of expression similarity
- Why might we want to do this?
Correlation Matrix
First calculate the correlation between each gene’s expression across samples
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | 1.00 | 0.12 | 0.75 | 0.86 | 0.49 | 0.32 |
| GeneB | 0.12 | 1.00 | 0.92 | 0.08 | 0.88 | 0.08 |
| GeneC | 0.75 | 0.92 | 1.00 | 0.81 | 0.78 | 0.02 |
| GeneD | 0.86 | 0.08 | 0.81 | 1.00 | 0.28 | 0.59 |
| GeneE | 0.49 | 0.88 | 0.78 | 0.28 | 1.00 | 0.78 |
| GeneF | 0.32 | 0.08 | 0.02 | 0.59 | 0.78 | 1.00 |
Adjacency Matrix
Then create an adjacency matrix with “1” indicating genes that are correlated above a threshold, and “0” indicating below threshold.
Connect genes with a “1”
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | 0 | 0 | 1 | 1 | 0 | 0 |
| GeneB | 0 | 0 | 1 | 0 | 1 | 0 |
| GeneC | 1 | 1 | 0 | 1 | 1 | 0 |
| GeneD | 1 | 0 | 1 | 0 | 0 | 0 |
| GeneE | 0 | 1 | 1 | 0 | 0 | 1 |
| GeneF | 0 | 0 | 0 | 0 | 1 | 0 |
Terminology
- Genes are nodes
- Connections between genes are edges
Mutual Rank Networks
One problem with correlation networks is that it is hard to know what threshold to pick. Further, correlation values can be affected by “noise” in the experiment, that may not be relevant.
An alternative (and in my hands better) approach is to use Mutual Ranks. We can connect genes with the the highest correlations, regardless of their precise value.
Here, we:
- Create a pairwise correlation matrix, as above
- Rank the correlations from strongest to weakest
- Compute the pairwise geometric mean ranks
- Choose a rank-based cutoff to create the adjacency matrix
Correlation Matrix
First calculate the correlation between each gene’s expression across samples
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | 1.00 | 0.12 | 0.75 | 0.86 | 0.49 | 0.32 |
| GeneB | 0.12 | 1.00 | 0.92 | 0.08 | 0.88 | 0.08 |
| GeneC | 0.75 | 0.92 | 1.00 | 0.81 | 0.78 | 0.02 |
| GeneD | 0.86 | 0.08 | 0.81 | 1.00 | 0.28 | 0.59 |
| GeneE | 0.49 | 0.88 | 0.78 | 0.28 | 1.00 | 0.78 |
| GeneF | 0.32 | 0.08 | 0.02 | 0.59 | 0.78 | 1.00 |
Rank Matrix
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | NA | 3 | 4 | 1 | 4 | 3 |
| GeneB | 5 | NA | 1 | 5 | 1 | 4 |
| GeneC | 2 | 1 | NA | 2 | 2 | 5 |
| GeneD | 1 | 4 | 2 | NA | 5 | 2 |
| GeneE | 3 | 2 | 3 | 4 | NA | 1 |
| GeneF | 4 | 4 | 5 | 3 | 2 | NA |
- Rankings in columns
- Note that these are not necessarily symmetrical.
- A-D and D-A are symmetrical.
- E-B and B-E are not symmetrical.
- Use the (geometric) mean of the ranking
Geometric mean
Geometric average of \(x\) and \(y\): \(\sqrt{x*y}\)
| x | y | arith_mean | geom_mean |
|---|---|---|---|
| 1 | 1 | 1.0 | 1.00 |
| 1 | 10 | 5.5 | 3.16 |
| 3 | 2 | 2.5 | 2.45 |
| 3 | 10 | 6.5 | 5.48 |
| 5 | 3 | 4.0 | 3.87 |
| 5 | 20 | 12.5 | 10.00 |
| 100 | 1 | 50.5 | 10.00 |
| 100 | 2 | 51.0 | 14.14 |
| 100 | 20 | 60.0 | 44.72 |
When \(x\) and \(y\) are different, the geometric mean weights the smaller numbers more heavily.
Average Ranks
Geometric average of \(x\) and \(y\)
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | NA | 3.87 | 2.83 | 1.00 | 3.46 | 3.46 |
| GeneB | 3.87 | NA | 1.00 | 4.74 | 1.41 | 4.24 |
| GeneC | 2.83 | 1.00 | NA | 2.00 | 2.74 | 5.00 |
| GeneD | 1.00 | 4.74 | 2.00 | NA | 4.47 | 2.45 |
| GeneE | 3.46 | 1.41 | 2.74 | 4.47 | NA | 1.58 |
| GeneF | 3.46 | 4.24 | 5.00 | 2.45 | 1.58 | NA |
Adjacency Matrix
Mutual Rank <= 3
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | 0 | 0 | 1 | 1 | 0 | 0 |
| GeneB | 0 | 0 | 1 | 0 | 1 | 0 |
| GeneC | 1 | 1 | 0 | 1 | 1 | 0 |
| GeneD | 1 | 0 | 1 | 0 | 0 | 1 |
| GeneE | 0 | 1 | 1 | 0 | 0 | 1 |
| GeneF | 0 | 0 | 0 | 1 | 1 | 0 |
Network
| GeneA | GeneB | GeneC | GeneD | GeneE | GeneF | |
|---|---|---|---|---|---|---|
| GeneA | 0 | 0 | 1 | 1 | 0 | 0 |
| GeneB | 0 | 0 | 1 | 0 | 1 | 0 |
| GeneC | 1 | 1 | 0 | 1 | 1 | 0 |
| GeneD | 1 | 0 | 1 | 0 | 0 | 1 |
| GeneE | 0 | 1 | 1 | 0 | 0 | 1 |
| GeneF | 0 | 0 | 0 | 1 | 1 | 0 |
Network Measures
- Density: how highly connected are the nodes?
- Total edges in the network / All possible edges
- What are the most “important” or “central” nodes?
- Degree centrality: which node has the most number of connections?
- Betweeness centrality: which node has the most number of shortest paths going through it?
Limitations
Correlation and mutual rank networks easy to make and easy to understand but have some limitations
- Is a hard threshold proper?
- What is the right threshold?
- Are correlations even the right measure?
- Directionality?
Additional method:
- Weighted Gene Correlation Networks. Uses a “soft” threshold. (WGCNA) (nice tutorials also YouTube videos)