# Compute steady states of gene expression

1. Mar 13, 2019 at 12:18 PM

1. The problem statement, all variables and given/known data
A bacteria that normally divides every 20 minutes express gene X. The production rate of protein X is 5nM/min. The protein is stable and does not degrade.
1. What is the concentration of X in the steady state?

2. The same bacteria enter into a stress state at t=0 for 3 hours (before t=0 the level of protein X is
at the steady state you found in a). In this state of X increases to 20nM/min but the cell division time changes to 60 min. Calculate the dynamic level of protein X starting t=0. What is the steady state in this case?

3. At t=3 hours the bacteria gets a signal that cases the protein X to become unstable with half-life of 10 min. Calculate now the dynamic level of protein X and the steady state. How would the maximal level of X change if the bacteria got this signal after only 1 hour?
2. Relevant equations
$$\beta ... \text{production rate}$$

• Degradation/dilution rate in units of 1/time
$$x\alpha = \alpha_{dil} + \alpha_{deg}$$

• Change in concentration of Y
$$dY/dt = \beta - \alpha Y$$

• Steady state (solving for dY/dt=0)
$$Y_{ST} = \beta / \alpha$$

Further information on "An Introduction to Systems Biology" by Uri Alon.

3. The attempt at a solution

For the first, I have to determine alpha and beta. Beta is obviously 5, and I thought alpha is 1/20, thus 0.05. I am not sure if this is true since the assignment states that the protein does not degrade. Forgive me, I am not that into biology. So the steady state should be 5/0.05 = 100.

I am not sure how to do that with the time shift. I tried to plot it and see how it is looking, so I let MatLab solve the differential equation stated above with initial condition 100. When t is below 180 (3 hours) I set beta to 20 and alpha to 1/60. And when t is higher than 180 then I set the parameters as before. However, that looks like the following:
Unfortunately, I don't know how to calculate the dynamic level of protein X starting with t=0, that's the main problem which I also need for 3.

For 3 I've no idea currently.
I appreciate any help!

2. Mar 14, 2019 at 12:17 PM

### epenguin

I guess this is about some kind of bacteriostatic device like a Chemiostat keeps the bacterial population density constant, hremoving them at the same rate the population increases? More explanation would have been helpful to understand what the problem is.
I agree with your 100 nM
You could at least calculate the nM for the second steady-state and third steady states in the same kind of way independent of your simulations as a check.
What is Y? Is it what you previously called X?
Will try and come back when I when I have more time.

3. Mar 14, 2019 at 12:31 PM

Thanks a lot for your reply. Oh yes, Y is X, sorry. The book denotes it as Y and the assignment as X. The second steady state would then be 1200, however, I'm not sure if my simulation is ok, looks weird to me. To be honest, I am not really able to provide you more background, it is just a "simple transcriptional process". Perhaps you may like to take a look at the book (pdf), ~page 18: https://www.google.com/url?sa=t&rct...cedownload=1&usg=AOvVaw0vBzFeIJ3J9W7Lbe2sU4BK

4. Mar 14, 2019 at 7:41 PM

### epenguin

It will be the weekend before I have the time to come back.
Maybe a more arithmetic rather than formulaic reasoning would tell you if your steady states are right and reasonable.
I see the model is different from what I imagined, It is essential to know what the model is!
Now I know that your second steady state looks right to me.
As well as Matlab, your text at that point tells you the solution of this standard and elementary differential equation, so you can also calculate it that way.

5. Mar 17, 2019 at 8:49 AM

Unfortunately, I am not able to edit my post above, and I've noticed another bug in the degradation/dilution rate formula, I am sorry. Here are the correct equations:

2. Relevant equations
$$\beta ... \text{production rate}$$

• Degradation/dilution rate in units of 1/time
$$\alpha = \alpha_{dil} + \alpha_{deg}$$

• Change in concentration of X
$$dX/dt = \beta - \alpha X$$

• Steady state (solving for dX/dt=0)
$$X_{ST} = \beta / \alpha$$

I think the model is about gene regulation, thus we have a transcription factor that regulates a gene and produces a protein. And since the level of transcription changes with time, we can describe it with differential equations. So, the model can be basically described as $$\frac{dX}{dt} = \beta - \alpha X$$.
I am still confused with 3, however, I will try harder tomorrow. I'd still be happy for any further help.