In this document margin notes are used for code and comments that are not essential to the discussion, but that can still be interesting.

# Gene expression

A gene expression *experiment* measures messenger RNA
concentrations —or values correlated with RNA concentration, such as
luminescence— for one or more genes under a specific condition. A
*series of experiments* measure gene expression through several
conditions.

Typically the results of a series of experiments is represented as a
matrix, with one row for each gene and one column for each condition. We
can find many experiments on NCBI Gene Expression Omnibus (GEO)
database. In this database each single experiment is called
** sample** and is identified with a code like

`GSM2715492`

. All sample ids begin with prefix
`GSM`

.In the same way, series of experiments are called
** series** and are identified with codes starting
with

`GSE`

, such as `GSE101789`

. We also find
**that describe the experimental method: the microarray slide, the qPCR primers, or equivalent characteristics. The platform can introduce some artifacts, so it is important to only compare samples using the exact same platform. Platforms ids begin with**

*platforms*`GPL`

, such as
`GPL23823`

.For this article we will use the series `GSE101789`

with
the gene expression of *Pseudomonas syringae* pv. tomato exposed
to light of different colors.

As you can guess, there are many ways to access NCBI GEO besides the
web interface. In R we can use the *GEOquery* library^{1}, but it is beyond the scope of this
article. Each series contains several components. We will care about
three parts: the gene ** expression** in different
experimental conditions, the

**— experimental conditions—, and the**

*phenotypes***of each gene.**

*features*To simplify this discussion I separated the components of the NCBI
GEO file in several plain text files. We start with the gene expression data, which
we load in R and we transform into a matrix^{2}
with the following code

```
<- read.delim("GSE101783_expression.txt")
ex <- as.matrix(ex) ex
```

The expression matrix has 15744 rows and 16 columns, so it is not
practical to print it. Instead in Figure 1 we show a *heat map*
of the matrix. Lighter colors represent higher gene expression. The
dendrograms on the left side and above suggest how data can be
clustered.

We can see that each column corresponds to a different sample. We
would like to know the response of each gene to the experimental
** condition**. Thus, we need to know the

*conditions*of each experiment. In a gene expression experiment, the conditions or treatments can be for example:

- growth medium,
- environmental conditions, such as oxygen, light, temperature, or similar,
- time of exposure to the environment,
- healthy versus sick cells, as in the case of cancer,
- wild type cells versus mutant cells.

The file `GSE101783_phenotype.txt`

describes the experimental conditions for each sample in detail. We load
it with the following command, taking care of keeping text as
*strings* so we can handle the labels before creating the
*factor* variables.

`<- read.delim("GSE101783_phenotype.txt", stringsAsFactors=FALSE) phenotype `

geo_accession | treatment.ch1 |
---|---|

GSM2715489 | Darkness |

GSM2715490 | Darkness |

GSM2715491 | Darkness |

GSM2715492 | Darkness |

GSM2715493 | Blue light |

GSM2715494 | Blue light |

GSM2715495 | Blue light |

GSM2715496 | Blue light |

GSM2715497 | Red light |

GSM2715498 | Red light |

GSM2715499 | Red light |

GSM2715500 | Red light |

GSM2715501 | White light |

GSM2715502 | White light |

GSM2715503 | White light |

GSM2715504 | White light |

The phenotype table has 16 rows —one for each experimental condition—
and 39 columns, describing several characteristics. For this article we
will care about one characteristic: *treatment in channel 1*, as
shown in Table 1. More details can be seen at the end of this article,
in the Appendix.

Our goal is to identify the underlying response of each gene to the
experimental *condition*. Thus, we prepare a *factor* to
describe the condition, taking the `treatment.ch1`

description and dropping the suffix `" light"`

.
Alternatively, we can create a factor manually, although it can be hard
to do it for large data sets.

```
<- factor(sub(phenotype$treatment.ch1,
condition pattern = " light", replacement = ""))
```

For the rest of this article we will focus on two arbitrary genes,
chosen only as good examples. They correspond to expression matrix’s
rows 1159 and 3342. To give a name to these genes we need to load the
file `GPL23823_features.txt`

with the gene descriptions. As before, there are several columns, but we
will only show the gene description.

```
<- read.delim("GPL23823_features.txt")
features c(1159, 3342), c("ORF", "DESCRIPTION")] features[
```

ORF | DESCRIPTION | |
---|---|---|

1159 | PSPTO_2413 | PSPTO_2413;Product: urease accessory protein UreE |

3342 | PSPTO_5142 | PSPTO_5142;Product: hypothetical protein |

**We will model the gene expression as the sum of two
components: an underlying condition-dependent expression level,
and a noise component independent of the experimental
condition. We want to identify the first component and control the
second one. Moreover, we want to see if the underlying expression level
changes when conditions change.**

# Expression analysis of gene 1159 (PSPTO_2413)

Table 2 and Figure 2 show the ** expression
level** of gene 1159 (PSPTO_2413) and the corresponding

**on each sample.**

*condition*expression | condition | |
---|---|---|

GSM2715489 | 8.65 | Darkness |

GSM2715490 | 8.66 | Darkness |

GSM2715491 | 9.12 | Darkness |

GSM2715492 | 9.19 | Darkness |

GSM2715493 | 9.48 | Blue |

GSM2715494 | 8.48 | Blue |

GSM2715495 | 9.14 | Blue |

GSM2715496 | 8.35 | Blue |

GSM2715497 | 7.58 | Red |

GSM2715498 | 11.27 | Red |

GSM2715499 | 8.39 | Red |

GSM2715500 | 8.27 | Red |

GSM2715501 | 8.77 | White |

GSM2715502 | 9.02 | White |

GSM2715503 | 9.16 | White |

GSM2715504 | 8.76 | White |

## Linear model without intercept

We want to know what is the expression of gene 1159 under each
condition *and compare them*. As discussed in a previous article,
we can find the gene expression for each condition with a linear model
*without intercept*.

```
<- lm(ex[1159,] ~ condition + 0)
model0 summary(model0)
```

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

conditionBlue |
8.863 | 0.439 | 20.19 | 1.25e-10 | * * * |

conditionDarkness |
8.905 | 0.439 | 20.29 | 1.182e-10 | * * * |

conditionRed |
8.878 | 0.439 | 20.22 | 1.225e-10 | * * * |

conditionWhite |
8.928 | 0.439 | 20.34 | 1.147e-10 | * * * |

The fitted model’s coefficients correspond to the mean expression
level for each condition. Each 𝑝-value corresponds to the null
hypothesis “real gene expression is zero”^{3}. As
expected, all 𝑝-values are small, since the gene is expressed in all
conditions.

The stars in the last columns are used to show which 𝑝-values are below the traditional thresholds. Three stars means 𝑝-value below 1%, two stars represent “under 5%”, and so on.

One way to understand the model is the following diagram. Each arrow represents one comparison made when we fit the linear model.

In this basic case, each circle represent the average expression for each condition, and the arrows represent the estimation calculated by fitting the linear model. It is important to know that we can have only four arrows in this diagram, since we have four conditions, and that limits how many combinations we can take.

The expression levels are hard to interpret without context. Are they
high or low? The mean values are different. Are they *significantly
different*? In other words, are the underlying expression levels
really different, or the difference of means corresponds only to the
noise?

## Linear model with intercept

To answer these questions it is more useful to fit a linear model
*with intercept*, that will show the difference between a
reference condition against all other conditions.

```
<- lm(ex[1159,] ~ condition)
model1 summary(model1)
```

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
8.863 | 0.439 | 20.19 | 1.25e-10 | * * * |

conditionDarkness |
0.0425 | 0.6208 | 0.06846 | 0.9465 | |

conditionRed |
0.015 | 0.6208 | 0.02416 | 0.9811 | |

conditionWhite |
0.065 | 0.6208 | 0.1047 | 0.9183 |

The *intercept* shows the expression level of the reference
condition. Again, we can represent this model with a diagram.

In this case the reference is automatically chosen to be “the first
level of the `condition`

factor”, which in this case is
`Blue`

. This is because, unless we say otherwise, the factor
levels are sorted alphabetically.

`levels(condition)`

`[1] "Blue" "Darkness" "Red" "White" `

This does not necessarily corresponds to the most interesting
question^{4}. **We need a way to choose the
reference level.**

## Choosing the reference level

There are several ways to choose the reference level. We will discuss
some of the easy methods. The easiest one is to use the function
`relevel()`

to create a new factor with the same levels in
different order.

From the the sake of argument I will assume that a good reference
level is `"Darkness"`

. Thus, the new factor will be called
`dark_vs_`

. We create and examine it as follows.

```
<- relevel(condition, "Darkness")
dark_vs_ levels(dark_vs_)
```

`[1] "Darkness" "Blue" "Red" "White" `

We see that this new factor `dark_vs_`

contains the same
values as the old factor `condition`

, but the levels are in a
different order. Now we can fit the linear model again.

`<- lm( ex[1159,] ~ dark_vs_ ) model2`

`summary(model2)`

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
8.905 | 0.439 | 20.29 | 1.182e-10 | * * * |

dark_vs_Blue |
-0.0425 | 0.6208 | -0.06846 | 0.9465 | |

dark_vs_Red |
-0.0275 | 0.6208 | -0.0443 | 0.9654 | |

dark_vs_White |
0.0225 | 0.6208 | 0.03624 | 0.9717 |

In the new results the *intercept* shows the expression level
for `"Darkness"`

, and the other coefficients show the
differential expression for `"Blue"`

, `"Red"`

and
`"White"`

light.

Why doing this complicated way? We could just calculate the
difference between the averages? Because of the 𝑝-values. This time the
null hypothesis is “the expression levels are the same”. Small 𝑝-values
show that we do not have enough evidence to reject the null hypothesis.
In other words, **there is no statistically significant difference
between the expression levels**. The gene 1159 does not respond
to the light.

# Expression analysis for gene 3342 (PSPTO_5142)

For comparison, let us see the case of gene 3342, which has the following expression levels.

Using the same modified factor `dark_vs_`

, this time we
get these results

```
<- lm(ex[3342,] ~ dark_vs_)
model3 summary(model3)
```

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
13.74 | 0.146 | 94.14 | 1.38e-18 | * * * |

dark_vs_Blue |
-2.875 | 0.2065 | -13.92 | 9.078e-09 | * * * |

dark_vs_Red |
-2.877 | 0.2065 | -13.94 | 8.989e-09 | * * * |

dark_vs_White |
0.7775 | 0.2065 | 3.765 | 0.002694 | * * |

This time all the differences are statistically significant. Moreover, we know that gene 3342 is repressed under blue or red light, and it is activated under white light. It is easy to find confidence intervals for the change in expression.

`confint(model3, level=0.99)`

0.5 % | 99.5 % | |
---|---|---|

(Intercept) | 13.2990193 | 14.190981 |

dark_vs_Blue | -3.5057119 | -2.244288 |

dark_vs_Red | -3.5082119 | -2.246788 |

dark_vs_White | 0.1467881 | 1.408212 |

## Using *Contrast* matrices

There is another approach to compare gene expressions, more flexible than just reorganizing factor levels. As usual, higher flexibility implies higher complexity.

To make the example more clear, we use the original
`condition`

factor, without reordering.

The function `contr.treatment()`

can be used to create a
*contrasts* matrix, like this.

`<- contr.treatment(levels(condition), base = 2) contrast1 `

` contrast1`

Blue | Red | White | |
---|---|---|---|

Blue | 1 | 0 | 0 |

Darkness | 0 | 0 | 0 |

Red | 0 | 1 | 0 |

White | 0 | 0 | 1 |

This function takes the names of factor’s levels and the number
indicating the *base level* used for reference. The resulting
contrasts matrix has one row and one column for each level of
`condition`

, except that there is no column for the base
level. The row that only has zeros will be the base level.

Now when we fit the linear model we need to include a
`contrasts=`

option. This option takes a *list*, where
each element has the name of a factor. There can be several factors in a
model, each one with its own contrast. In our case we have only one
factor.

`<- lm(ex[3342,] ~ condition, contrasts = list(condition=contrast1)) model3b `

`summary(model3b)`

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
13.74 | 0.146 | 94.14 | 1.38e-18 | * * * |

conditionBlue |
-2.875 | 0.2065 | -13.92 | 9.078e-09 | * * * |

conditionRed |
-2.877 | 0.2065 | -13.94 | 8.989e-09 | * * * |

conditionWhite |
0.7775 | 0.2065 | 3.765 | 0.002694 | * * |

We see that we recovered the same result as `model3`

without re-leveling the factor. It is more complicated, but will have
some advantages.

## Making the contrast matrix manually

Once we know how the contrast matrix looks, we can make it manually.
In R we use `matrix(data, nrow, byrow)`

to create a new
matrix from a vector `data`

. If we do not include
`byrow=TRUE`

then the matrix will be filled by columns. Try
it and see the difference.

The code to make a matrix like `contrast1`

is this

```
<- matrix(
contrast2 c(1,0,0,
0,0,0,
0,1,0,
0,0,1), nrow=4, byrow=TRUE)
colnames(contrast2) <- c("B_D","R_D","W_D")
rownames(contrast2) <- levels(condition)
```

` contrast2`

B_D | R_D | W_D | |
---|---|---|---|

Blue | 1 | 0 | 0 |

Darkness | 0 | 0 | 0 |

Red | 0 | 1 | 0 |

White | 0 | 0 | 1 |

Obviously, it produces the same model and the same coefficients as before.

## Other combinations

We can make other contrast matrices, but there are technical
constraints on what can be done. One way to easy avoid these contrast is
to build a *single* comparison. For example, an important
biological issue is to compare the gene expression under blue light
versus under red light, and find if the difference is statistically
significant.

To answer this question we build a one-column matrix with a code like this

```
<- matrix(c( 1/2, 0,-1/2, 0), nrow=4)
contrast3 colnames(contrast3) <- c("B-R")
rownames(contrast3) <- levels(condition)
```

which produces the following matrix

` contrast3`

B-R | |
---|---|

Blue | 0.5 |

Darkness | 0.0 |

Red | -0.5 |

White | 0.0 |

Not the fitted model has only two coefficients

```
<- lm(ex[3342,] ~ condition,
model5 contrasts=list(condition=contrast3))
summary(model5)
```

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
12.5 | 0.4476 | 27.93 | 1.117e-13 | * * * |

conditionB-R |
0.0025 | 1.266 | 0.001975 | 0.9985 |

Now it will be useful to keep in mind the mean expression for each condition.

`tapply(ex[3342,], condition, mean)`

```
Blue Darkness Red White
10.8700 13.7450 10.8675 14.5225
```

Close inspection shows that coefficient `conditionB-R`

is
the difference between mean expression in `"Blue"`

and
`"Red"`

conditions. More importantly, we get a 𝑝-value for
the difference. We conclude that, although `"Blue"`

and
`"Red"`

are different from `"Darkness"`

.

We also observe that the *intercept* is the mean of all
expressions.

`mean(ex[3342,])`

`[1] 12.50125`

Unfortunately it is not easy to do this comparison for several conditions at the same time. For this article we will limit us to only one comparison per linear model.

## All versus the average

We have the mean expression for each condition and the global mean of all expressions. It is easy to calculate the difference between the global mean and the mean of each condition.

`tapply(ex[3342,], condition, mean) - mean(ex[3342,])`

```
Blue Darkness Red White
-1.63125 1.24375 -1.63375 2.02125
```

It is not so easy to see if this difference is significant. We can
answer this question using contrast matrices created with the
`contr.sum()`

function.

```
<- contr.sum(levels(condition))
contrast4 colnames(contrast4) <- c("B","D","R")
```

The `contr.sum()`

does not assign names to the columns, so
we assign them manually. The resulting matrix is this

` contrast4`

B | D | R | |
---|---|---|---|

Blue | 1 | 0 | 0 |

Darkness | 0 | 1 | 0 |

Red | 0 | 0 | 1 |

White | -1 | -1 | -1 |

We see that the last row is different from the rest, and indeed it is not included in the output. This is always the case when we do not have enough degrees of freedom. Since we are comparing to the average, we lose one degree of freedom. We can represent this model with the following diagram.

As before, we can have up to four arrows —that is, contrasts— given that we are limited by the number of conditions. To include the mean we must “sacrifice” one of the conditions to make space for the average. The “lost” row is replaced with a row of -1.

The fitted model has the following coefficients

```
<- lm(ex[3342,] ~ condition,
model6 contrasts=list(condition=contrast4))
summary(model6)
```

Estimate | Std. Error | t value | Pr(>|t|) | ||
---|---|---|---|---|---|

(Intercept) |
12.5 | 0.073 | 171.2 | 1.057e-21 | * * * |

conditionB |
-1.631 | 0.1264 | -12.9 | 2.15e-08 | * * * |

conditionD |
1.244 | 0.1264 | 9.836 | 4.279e-07 | * * * |

conditionR |
-1.634 | 0.1264 | -12.92 | 2.113e-08 | * * * |

We observe that the *intercept* is the global average, and the
rest of the coefficients are the difference between the average and the
average expression under each condition. Also, we have 𝑝-values to
evaluate the statistical significance of each difference.

How would you make a

contrastmatrix to compare the global average to“Blue”,“Red”and“White”, omitting“Darkness”?

# Appendix: Materials & Methods

Just to make my biologist friend happy, we can see more details about
the experimental protocol as described in the first row of
*phenotype* file.

`1, 3:31] phenotype[`

status | submission_date | last_update_date | type |
---|---|---|---|

Public on Oct 01 2018 | Jul 23 2017 | Oct 01 2018 | RNA |

channel_count | source_name_ch1 | organism_ch1 |
---|---|---|

1 | Wild type 10 min Dark | Pseudomonas syringae pv. tomato str. DC3000 |

characteristics_ch1 | characteristics_ch1.1 | characteristics_ch1.2 |
---|---|---|

genotype/variation: Wild type | culture type: Liquid | treatment: Darkness |

characteristics_ch1.3 | growth_protocol_ch1 | molecule_ch1 |
---|---|---|

treatment duration: 10 min | Transcriptomic analyses were done with cultures grown in liquid KB medium. Total RNA was extracted from cell cultures collected at O.D.=0.5 after a 10 min treatment with either white, blue or red light. Cultures maintained in the darkness were used as controls. | total RNA |

extract_protocol_ch1 | label_ch1 | label_protocol_ch1 |
---|---|---|

Total RNA from four biological replicates from each treatment was extracted with TRI Reagent (Ambion) as recommended by the manufacturer. RNA was pre-treated with RNase-free DNase I (Roche) plus RNaseOUT (Invitrogen), followed by purification with RNeasy columns (Qiagen). | Cy3 | cDNA was labeled with Cy3 by using the SuperScript III indirect cDNA labeling system (Invitrogen) as described in the instruction manual (One-Color Microarray Based Gene Expression Analysis Manual). |

taxid_ch1 | hyb_protocol | scan_protocol |
---|---|---|

223283 | 600 ng of Cy3 probes were mixed and added to 5 μl of 10x Blocking Agent and Nuclease-free water in a 25 μl reaction. Then, we added 25 μl from 2x GExHybridization buffer and mixed carefully. The samples were placed on ice and quickly loaded onto arrays, hybridized at 65 ºC for 17 h and then washed once in GE wash buffer 1 at room temperature (1 min) and once in GE Wash Buffer 2 at 37 ºC (1 min). | Images from Cy3 channel were equilibrated and captured with a High resolution Scanner (Agilent) and spots quantified using Feature Extraction software (Agilent). |

description | description.1 | data_processing |
---|---|---|

G0 | Gene expression after 10 min darkness treatment. | For local background correction and normalization, the methods normexp and loess in LIMMA were used, respectively. Log-ratio values were scaled using as scale estimator the median-absolute-value. Linear model methods were used for determining differentially expressed genes. Each probe was tested for changes in expression over replicates by using an empirical Bayes moderated t-statistic. To control the false discovery rate, p-values were corrected by using the method of Benjamani and Hochberg, 1995. The expected false discovery rate (FDR) was controlled to be less than 1%. |

platform_id | contact_name | contact_email |
---|---|---|

GPL23823 | Emilia,,López Solanilla | emilia.lopez@upm.es |

contact_institute | contact_address |
---|---|

CBGP-Universidad Politécnica de Madrid (UPM) | Parque Científico y Tecnológico UPM, Campus de Montegancedo, Ctra. M-40, km 38 |

contact_city | contact_state |
---|---|

Pozuelo de Alarcon | Madrid |