Lecture 16: Gene regulation and networks pt. 2
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
A simple model of gene autoregulation
Constitutive expression:
Autoregulation:
Basal transcription rate; depends on promoter binding to RNAP
Autoregulation function; depends on promoter binding to gene protein product
Steady-state RNA approximation
mRNA reaches a constant concentration very fast! You don’t lose much by treating it as constant all the time.
Steady-state RNA approximation for autoregulation
One equation!
Network Motifs
“Out of the many possible patterns that could appear in the network, only a few are found significantly—the network motifs.” (Alon, An Introduction to Systems Biology)
Network 1
Network 2
(“natural” E. coli network)
(random network)
Feed-forward loop
Self-regulation/autoregulation
Motif #1: Autoregulation
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!
Autorepression allows fast response
What’s a less simplified way to model-self regulation mathematically?
For constitutive expression, there’s a tradeoff between how low a concentration you can maintain and how fast you can make protein.
Not so with autorepression!
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!
Motif #2: Feed-forward Loop
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
Et cetera . . .
All feed-forward loops!
Finding motifs by comparing E. coli to random networks
Network 1
Network 2
(“natural” E. coli network)
(random network)
Feed-forward loop
How many FFLs do you expect in a random network?
FFL is a 3-node sub-network
How many FFLs are in E. coli?
Random network:
E. coli:
42!
Like self-regulation, feed-forward loops are very important to gene regulation in E. coli!
(Alon, An Introduction to Systems Biology)
Why so many?
What is the function of the FFL?
Two categories of FFLs
Coherent
Net effect of X/Y/Z interaction matches X/Z
Incoherent
Net effect of X/Y/Z interaction does not match X/Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
Two types of FFL are especially common
X
Y
Z
“Coherent Type-1”
“Incoherent Type-1”
X
Y
Z
What is the function of these motifs?
Quick review before analyzing FFL
Turning on an activator or lifting a repressor causes a slow rise in protein concentration.
Turning off an activator or turning on a repressor causes a slow decay in protein concentration.
Analyzing the C1 FFL
X
Y
Z
“Coherent Type-1”
Many C1 FFLs obey AND transcriptional logic (more on this Friday!), meaning that both X and Y must be present for Z to be expressed:
X
Z
Y
AND
These are generally systems where X and Y are transcription factors that can bind to signal molecules and become active, and then able to activate Z.
Let’s use what we’ve learned about modeling gene expression dynamics to see how this works!
Analyzing the C1 FFL
“Coherent Type-1”
X can change into an active conformation when it binds a signal molecule, which could be a metabolite or signaling molecule.
X
Z
Y
AND
X
Promoter X
Y
Promoter Y
Z
Promoter Z
Inactive X, can’t activate Y/Z
Signal molecule, Sx
Active X, can activate Y/Z
This can happen extremely fast.
Sx
Analyzing the C1 FFL
“Coherent Type-1”
X can change into an active conformation when it binds a signal molecule, which could be a metabolite or signaling molecule.
X
Z
Y
AND
X
Promoter X
Y
Promoter Y
Z
Promoter Z
Inactive X, can’t activate Y/Z
Signal molecule, Sx
Active X, can activate Y/Z
Inactive Y
Signal molecule, Sy
Sx
Sy
Analyzing the C1 FFL
“Coherent Type-1”
X can change into an active conformation when it binds a signal molecule, which could be a metabolite or signaling molecule.
X
Z
Y
AND
X
Promoter X
Y
Promoter Y
Z
Promoter Z
Inactive X, can’t activate Y/Z
Signal molecule, Sx
Active X, can activate Y/Z
Inactive Y
Signal molecule, Sy
Sx
Sy
If Sx goes away, Z will stop being expressed, but will go away slowly due to separation of time scales! Let’s look more closely at the dynamics.
Analyzing the C1 FFL
“Coherent Type-1”
X can change into an active conformation when it binds a signal molecule, which could be a metabolite or signaling molecule.
X
Z
Y
AND
X becomes activated by Sx
X
X*
Analyzing the C1 FFL
“Coherent Type-1”
X
Z
Y
AND
X becomes activated by Sx
Analyzing the C1 FFL
“Coherent Type-1”
X
Z
Y
AND
X becomes activated by Sx
What about when Sx goes away?
Analyzing the C1 FFL
“Coherent Type-1”
X
Z
Y
AND
Sx disappears
Fast!!
Analyzing the C1 FFL
“Coherent Type-1”
X
Z
Y
AND
Sx disappears
Fast!!
Analyzing the C1 FFL
“Coherent Type-1”
X
Z
Y
AND
Sx disappears
Fast!!
Analyzing the C1 FFL
Sx disappears
Fast!!
Sx appears
No delay!
Called a “sign-sensitive delay element”
C1 FFL: a sign-sensitive delay element
“Coherent Type-1”
X
Z
Y
AND
“Sign-sensitive” means the delay is sensitive to whether you’re adding something (positive sign) or removing something (negative sign).
C1 FFL: a sign-sensitive delay element
An example of a sign-sensitive delay element: an elevator door
Sign-sensitive delay in an E. coli circuit
ara system: arabinose metabolism
lac system: lactose metabolism
Create GFP reporters of these genes
Sign-sensitive delay in an E. coli circuit
Add cAMP
Remove cAMP
This motif prevents response to noisy input!
What have we learned?
Autoregulation enables fast gene expression responses
A coherent feed-forward loop can act as a sign-sensitive delay, which filters out noisy input