Quick grading update
Lecture 16: Gene regulation and networks
Today:�
A simple model of gene regulation
DNA
mRNA
protein
transcription
translation
Rate of change of mRNA conc.
transcription from DNA
Active degradation by RNase
Rate of change of protein conc.
Dilution from cell growth
translation from mRNA
Constitutive expression:
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”
(Started from an unrealistic condition of no mRNA or protein)
Because mRNA reaches a constant level so quickly, we’ll make a simplification:
“steady state” assumption
Constitutive expression
What is the steady-state protein concentration?
Constitutive expression
Stable steady state!
Constitutive expression
What is the steady-state protein concentration?
Directly proportional to translation rate
Directly proportional to transcription rate
To maintain an unregulated protein at a low concentration, you have to synthesize it slowly!
(Will become important when we talk about regulation)
Gene regulation
Gene X
Gene Y
X promoter
X coding sequence
Y promoter
Y coding sequence
Y
repression
This interaction depends critically on the promoter DNA sequence!
Network Motifs
If there are innumerable possibilities for gene regulatory networks, and mutation can constantly change, them, how do you figure which networks are important to understand?
Network motifs:
How does this work?
Network Motifs
We will look at networks as graphs.
These circles represent genes. Called “nodes” in network language.
These lines represent regulatory relationships. Could be repression or activation. Will call the lines “arrows”
Gene X
Gene Y
X represses Y
Gene X
Gene Y
X activates Y
Both of these scenarios would be represented by the graph above.
Network Motifs
We will quantify the structure of the graph with a few metrics
Network 1
Network 2
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(“natural” E. coli network)
(random network)
Network Motifs
How do you generate a set of random graphs to compare to the E. coli network?
Node/gene
Arrow/regulatory interaction
Erdös-Rényi algorithm:
5 nodes
6 arrows
Identify structures or “motifs” and count how many there are of each!
“feed-forward loop”
Self-regulation
Et cetera . . .
Paul Erdös
Eccentric, traveling mathematician from the 20th century
Authored papers with over 500 different collaborators!
Prolific collaboration led to the creation of “Erdös number”, how many degrees one is away from Erdös in terms of co-authoring papers.
Erdös number
Kevin Bacon number
What are the properties of the E. coli network?
This subset of the network contains:
Alon, An Introduction to Systems Biology
What are the properties of the E. coli network?
This subset of the network contains:
This network is actually very “sparse”: it has a low proportion of the possible connections. To see this, again imagine you have a network of 5 nodes/genes. How many possible connections are there?
5 possible connections for the first gene.
Because regulation can go in either direction, every other gene can also have 5 connections. So the maximum possible number of connections is:
# genes
# connections each can make
What motifs dominate the E. coli network?
Alon, An Introduction to Systems Biology
Two network motifs very common compared to random:
How many autoregulatory interactions do we expect to see in a random network?
If we know the mean number of self-regulations, and self-regulation is a rare, independent event, how is the number of self-interactions distributed in the population of all random networks?
For a random network like E. coli:
How many autoregulatory interactions do we expect to see in a random network?
For a random network like E. coli:
How many autoregulatory interactions are actually in E. coli?
40
Self-regulation is a very important component of gene regulation in E. coli. We should understand its dynamics and what its function might be!
Next time we’ll look at the other major motif in E. coli, the feed-forward loop.
Auto-regulation
How do we incorporate auto-regulation into our gene expression model?
DNA
mRNA
protein
Auto-repression, for example:
Auto-regulation
Transcription rate
How do we account for this mathematically?
Let’s use this to see what one very important function of self-repression is.
Self-repression enables faster response
Remember that for unregulated expression, the steady-state protein concentration depended on transcription and translation rate:
Synthesis rates depend on sequence of promoter and ribosome-binding site of mRNA
To keep the concentration low…
…the cell has to synthesize it slowly
How can a cell maintain a low concentration, but be able to synthesize the protein rapidly? Self-repression!
Self-repression enables faster response
Let’s look at self-repression with the simplified step-function regulation:
Time
…synthesis can be very fast and reach a low concentration quickly!
For an unregulated gene, this would need to be very slow!
Self-repression enables faster response
An experimental test of this prediction:
Two strains of E. coli:
GFP
Induced constitutive GFP expression
Induced self-repressed GFP expression
GFP
What do they see?
Self-repression enables fast gene expression response!
What’s a less simplified way to model-self regulation mathematically?
The Hill Function
maximum transcription rate; depends on RNA polymerase binding to promoter.
transition concentration; depends on transcription factor affinity for promoter sequence.
sets how steep the transition is
(n = 5 on the left)
(= 15 here)
The Hill Function
Hill function for activation
One problem here is that with this formula, there is no transcription at all without the activator.
That is not necessarily the case. How do we take that into account?
Hill function for activation
First, some math:
Now introduce one more term to the formula.
Hill function for activation
What about transcriptional repression?
Hill function for repression
What kind of gene expression dynamics does autoregulation predict?
We saw with our simplified, step-function self-repression that there is a stable, steady-state protein concentration. We can see that from the more detailed Hill function too:
Stable steady-state!
The steady-state concentration will depend on synthesis rates (RNAp affinity for promoter), K (repressor affinity for promoter), and cell growth rate!
What about self-activation? Let’s do the same graphical analysis.
Stable steady-state
Self-activation can have bi-stability!!!!
Stable steady-state
Unstable steady-state!!
Stable states of self-activated genes
protein concentration rate of change
Depending on relative synthesis rates, binding affinities, and cell growth rate, self-activation can lead to bistability or one stable steady state.
Steady state over there
bistability
Stable steady state
Bistability not to be common because parameters need to be fine-tuned
Can be realized experimentally though!
Next time: feed-forward loop!