Lecture 14: Gene Expression Pt. 2
Today:
Our starting point: the central dogma of biology
DNA
gene
DNA
RNA
protein
transcription
translation
Gene expression in bacteria
promoter
protein coding sequence
gene
RNA polym.
repressor
activator
some
ribo
protein
mRNAs can be destroyed
proteins can be destroyed
Transcription can be repressed by a protein
Transcription can be activated by a protein
Can respond to environment.
mRNA
Goal: use this framework to build toward equations that will let us:
Our basic framework
DNA
mRNA
protein
For a given gene:
Let [m] by the concentration of the mRNA for the gene and [P] be the concentration of the protein.
How to we describe the dynamics of these molecules within the cell?
transcription of DNA into RNA
active degradation of mRNA, dilution due to growth
translation of mRNA into protein
active degradation of protein, dilution due to growth
Can be influenced by the level of another gene or itself through gene regulation!
[m]
[P]
What do we need to build and test a model?
We need to know several rates to set parameter values in our equations
Then we need to know
How do we measure gene expression in cells to compare to models?
Last time!
Today!
Next week!
How is mRNA “lost”? What are the rates of loss?
Ribonuclease (RNase) enzymes actively degrade RNAs.
mRNA
RNase
+
How quickly does this happen to an mRNA in a cell?
What are the half lives of mRNAs?
Very short! Most genes degraded to half their concentration in less than 6 minutes!
Interesting note from these authors: no correlation observed between mRNA half life and 1) abundance, 2) secondary structure, or 3) cell growth rate.
Note: these half lives reflect much faster degradation than would be accounted for by dilution due to cell growth.
Active degradation is the dominant factor for mRNA loss!
How is protein “lost”? What are the rates of loss?
Measured rate of radioactivity of perfusate
Fast component of protein degradation
Slow component of protein degradation
How is protein “lost”? What are the rates of loss?
fast
slow
They are able to estimate two things:
Findings:
Conclusion: active, rapid degradation is not a major component of protein loss in E. coli.
To summarize
DNA
mRNA
protein
Similar synthesis rates
Loss is fast; mainly due to active degradation by RNases
Loss is slow; mainly due to dilution from growth
What we found last lecture
DNA
mRNA
protein
active degradation
transcription
translation
dilution due to cell growth
active degradation
dilution due to cell growth
Thinking about concentrations in the cell:
similar rates
dominant factor
dominant factor
What will these measurements tell us?
DNA
mRNA
protein
transcription
translation
Depending on regulation, this will predict:
In order to test these predictions, we have to measure gene expression. How do we do that?
Measuring gene expression
DNA
mRNA
protein
transcription
translation
vs.
Measuring bacterial gene expression with fluorescent proteins
gene of interest: tapA
tasA
protein coding sequence
PtapA
6. Method 1: transcriptional reporters, transcription-level
promoter
gfp
PtapA
Create a mutant with the following sequence elsewhere in the genome:
Same regulatory sequence. gfp is regulated the same way as tasA in this cell. If you seen no GFP signal in a cell, it’s probably not making a lot of tasA. If you see a lot of signal, it’s probably making a lot.
Measuring bacterial gene expression with fluorescent proteins
1. Method 1: transcriptional reporters
PcitZ-YFP
(B. subtilis)
Measuring bacterial gene expression with fluorescent proteins
gene of interest: tasA
gfp
tasA
protein coding sequence
PtasA
6. Method 2: fluorescent protein fusion, protein-level
Create a mutant with:
The protein is made with a fluorescent protein physically connected.
Only used when spatial protein localization is strictly needed because attaching GFP changes a protein’s behavior in unknown ways!
Method | Molecule(s) measured | Pros | Cons |
Western blot | Protein |
|
|
RNA sequencing | RNA |
|
|
Proteomics w/mass spec | Protein |
|
|
RNA FISH | RNA |
|
|
Transcriptional fluorescent protein reporters | RNA-ish |
|
|
Fusion fluorescent protein reporters | Protein |
|
|
Now back to our model
DNA
mRNA
protein
active degradation
dilution due to cell growth
active degradation
dilution due to cell growth
How do we turn this
into terms for these equations?
First, we’ll look at the case of unregulated or “constitutive” gene expression: mRNA is always being made.
How do we write the terms of our equation?
First, note that we’ll measure concentration in units of molecules per cell.
Remember from the first lecture, 1 molecule per cell is ~1 nM.
What is the rate of mRNA production?
Units of
concentration
time
molecule/cell
time
=
mRNA production
From last lecture:
gene
~1000 nt
RNAp
mRNA
~16.17 sec
So a single RNA polymerase produces ~1 mRNA in 16.17 seconds, or a rate of
But a single gene can fit ~18 polymerases (55 nt footprint), so the maximum rate is then
How do we write the terms of our equation?
What is the rate of mRNA loss?
We saw last lecture that active mRNA degradation results in a half-life of ~5 minutes. What does this mean?
How to write out the mRNA degradation term
We saw last lecture that in a variety of conditions, mRNAs are degraded with a half-life of ~6 minutes in the absence of transcription.
This corresponds to our equation with a transcription rate of 0:
0
A 6-minutes half life means that every 6 minutes, the concentration of mRNA is divided by 2.
The mRNAs are decaying exponentially! What’s the equation for that?
Exponential decay of mRNAs without transcription
0
We now have an equation for the concentration of mRNA in time.
Equation for constitutive mRNA production
We considered the situation where the mRNA had the maximum number of polymerases on it. In reality no promoter binds polymerase well enough for this to be true. A real value will be much lower.
What about protein??
Protein production for this gene
What is the rate of protein production?
Units of
concentration
time
molecule/cell
time
=
Protein production
From last lecture:
mRNA
~1000 nt
ribosome
protein
~16 amino acid / sec
333 codons per mRNA
A single ribosome produces ~1 protein in 21 seconds from an mRNA, or a rate of
But a single mRNA can fit ~28 ribosomes (35 nt footprint), so the maximum rate is then
But there are also [m] mRNAs in the cell and this is per mRNA! The production term is:
What about protein loss?
We saw that for 95% of proteins, active degradation is extremely slow. The major contributor to loss of protein concentration is dilution due to cell growth.
In that case, the half-life for a protein is just the cell doubling time. If the cell volume doubles in one doubling time, then the protein concentration will be reduced by a factor of 1/2 because concentration is inversely proportional to volume. In that case the loss term is:
Our protein equation:
Translation from mRNA
Dilution from cell growth
Equations for the production of a constitutively expressed gene
We can now solve this system of equations in python to look at the dynamics of mRNA and proteins in the cell. What does this look like for a 60-minute doubling time?
Let’s start Jupyter
Our solution:
Due to fast degradation, mRNA reaches a maximum level very quickly!
mRNA production perfectly canceled by mRNA degradation
Due to much slower degradation, protein takes a long time to reach a maximum level!
Concept of “separation of time scales”
Notice both of these y-axes is larger than the values we saw measured in lecture 2.
This is because we assumed the maximum possible transcription rate. Not true for any real gene!
(Started from an unrealistic condition of no mRNA or protein)
Hypothetical scenario
We saw in our previous example that once an mRNA started being produced, the protein for the gene took ~10 hours to reach a steady-state level.
What if at 10 hours a repressor starts being produced that prevents transcription?
protein coding sequence
promoter
DNA
What happens to the protein and mRNA levels?
repressor
How do we use our model for this hypothetical?
The initial condition was:
Our equations from the central dogma were:
We solved them to get the concentration of mRNA and protein after 10 hours.
Now a repressor binds and transcription goes to 0!:
0
Now solve these equations with the initial condition being our first solution at 10 hours:
Repression after 10 hours
The cell senses something and makes a repressor. What happens?
Happily making the gene
???
Repression after 10 hours
The cell senses something and makes a repressor. What happens?
Happily making the gene
mRNA decays very quickly!
Protein sticks around for a while
What have we learned?
Next: