Exponential cellular growth protein production math problem

tomynator123456

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Hello dear forum, I have been thinking a while over a problem in my studies which is a math problem.

Lets say we have a bacterium B that multiplies itself according to the exponential growth equation: B(t)=B(0)*e^(kt). B(t) is the number of bacteria B at the time point t, B(0) is the number of B at the time point 0, k is the growth rate in 1/h and t the time in h.

The bacterium B is able to produce a protein P at the rate p in 1/s, but when it does so, the growth rate k drops to a lower growth rate h. How can one calculate what the optimal time point d in hours for the start of the protein P production is?

Lets say that:
B(0)=1
k=1/h
t=24h (the total time that is available
P(0)=0
p=1/s=3600/h
h=0.5/h

So the point is to find the optimal trade off between maximum number of production units B that are present, because each B produces one P per second. On the other hand, if we wait longer than there is less time for all the present B to produce P.
And when the B start to produce P they still grow, only slower compared when they produce no P, so if P production starts it will increase over time because the number of producing units B still increases.

I am grateful for every help!

Greetings,
Thomas
 
Dear forum,

I have been thinking a while on a problem regarding my studies where the underlying problem is math. I tried very hard to find a solution but was unable to.

Lets say we have a bacterium that multiplies itself. B is the number of present bacteria. B is growing exponentially according to B(t)=B(0)*e^(kt), where B(t) is the number of B at the time point t, B(0) is the number of B at the time point 0, k is the growth rate in 1/h and t is the time in h.

Each bacterium is able to produce a protein, where P is the number of total present proteins. p is the rate of protein production in 1/s. However, if a bacterium produces the Proteins with the rate p, the growth rate k drops to the lower growth rate f.

If the total time available is tmax, how can one calculate the optimal time point d in hours when the bacteria should start to produce protein so that P is maximized?
The idea is that the longer B is growing with the higher growth rate, the more B will be present to produce P with the fixed rate p. But the longer one waits, the less time is available for all the present B to produce P. Therefore it is desired to calculate the optimum. Also, when the B start to produce P, they still grow and get more, so the production of P is then not static but increases with the growth of B.

I would be interested in a general solution with the stated variables, but for calculation we can make assumptions of values:
B(0)=1
k=1/h
tmax=24h
P(0)=0
p=3600/h
f=0.5/h
d=?


I am very thankful for every response!

Greetings,
Thomas
 
I'm not sure I fully understand; as I read it, there are d hours during which no protein is produced but B increases at exponential rate k, followed by (tmax - d) hours during which protein is produced at a rate proportional to B, which increases at the lower exponential rate f. But then p would not be measured in units per hour, but in units per hour per cell, or something like that. (You appear to be treating the quantities of cells and of protein as dimensionless, which I find confusing.)

To make sure we follow, can you show your calculation of the total amount of protein made given your values, and taking d = 12 hours?
 
This does not sound like homework so I am posting my answer for [imath]0 < h < k[/imath]:
[math]d = T_1 - \frac{1}{h} \ln \left( \frac{k}{k-h}\right)[/math]where [imath]T_1[/imath] is the total time (24 hours). The intermediate steps are slightly hairy so I've left them out for now. I would be interested to hear if someone can think of a way to check this answer without checking the intermediate steps.
BTW, it does not depend on the rate of protein production as long as it stays constant per cell (as Dr.Peterson noticed).
 
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@Dr.Peterson Thank you for your response! I treat the cells and proteins dimensionless because for this model I only want to look at the absolute numbers. What is true that I wrote the protein production rate p wrong, it should not be 1/s but 1/(s*B) so 1 protein per second per cell. I have not yet calculated with the given values, I was thinking generally over this problem and the values that I have written down should just serve as boundaries for calculation. But since math is sometimes black magic for me, I did not knew entirely what I was doing.

@blamocur Thank you very much for your provided solution. It is indeed not a homework, this is a problem I have been thinking on for a while and it frustrated me that I could not find a mathematical answer. At university you get always told that letting the cells grow first and then start production would be beneficial, and I always knew there must be some mathematical explanation for this, but that is not something I was able to learn at university or via google.
 
A question that now also came to my mind is: Is d also independent of the starting value B(0)? So if B(0) is not 1 but 100 or 257, would it still just depend on the two different growth rates and the available time?
 
Good question! And the answer is yes, it is independent on [imath]B_0[/imath]. Consider doubling that amount: would it be different from running too processes independently?
 
Since I am rarely 100% sure about my results I've attached a pdf file with the intermediate steps. Feel free to double check and critique.
 

Attachments

  • x.pdf
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Thank you very much again! I checked your solution and everything should be all right. One last question came to my mind, though.

I don't want to steal your time, you helped me a lot, but if you find it interesting: How would the model change if there is a maximum value that B can reach, Bmax. So that would be the maximum number of cells present because of physical limits.
Lets say according to the equation the optimal time point to start production would be at 22 h, but after 20 h the maximum B is reached in reality. I can think of that then the time point of production start should be earlier than 20 h, because if the bacteria reach the maximum number to soon, they cannot benefit from the lower growth rate anymore, one can say their growth rate is then zero. Then the ones that started before can kinda "catch up", because they can still grow at the lower growthrate of, lets say, 0.5. and they started producing earlier. Do you get what I mean? So when the bacteria are grown until the maximum number of B is reached and then production is started, you do not get the time they would produce AND grow (at a lower rate).

Sry, I would really try ito get to the solution on my own, but I lack such mathematical competences ?
 
I think if the bacteria growth maxes out before the optimal time then you want to start producing protein at exactly the moment when the growth stops, not before that.

My reasoning: when we looked (in the attachment) for the maximum of the function [imath]V(T)[/imath] (proportional to [imath]P(T)[/imath]) we found only one optimal value for [imath]T[/imath] -- let's call it [imath]T_{max}[/imath] here. This means that [imath]P(T)[/imath] keeps increasing when [imath]T < T_{max}[/imath]. Thus the later you start the better, but only as long as the bacteria continues growing at the specified rate.

Does this make sense?
 
Alright, yes it makes sense. Thank you very much again, you have given me a deep understanding of the stated problem, because in university they just tell you that sometimes it is beneficial to start producing at a later time point, but looking at it with a mathematical perspective is very helpful for understanding!

Greetings,
Thomas
 
Alright, yes it makes sense. Thank you very much again, you have given me a deep understanding of the stated problem, because in university they just tell you that sometimes it is beneficial to start producing at a later time point, but looking at it with a mathematical perspective is very helpful for understanding!

Greetings,
Thomas
Glad it helped. But it is worth remembering that this is an idealized mathematical model, which is only an approximation of what happens in real life. It can easily miss some of the factors in real life, so it might be a good idea to verify the results empirically.
 
Yes, you are right. It can serve as a very quick estimation of this behavior. What I find especially beautiful is that it only needs the planned process time T1 and both the growth rates before and after production start. All those informations are really easy to get and one would determine them anyways. So there is no point in not quickly looking at this equation. Probably some scientists have this model and use it, but it it just not findable on the internet or was taught at my university.
 
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