Biol 381 Molecular GeneticsProblem Set #7
Due: 10 am, Wednesday, April 6, 2022
This problem uses an Excel spreadsheet model of some of the gene regulation systems we are
discussing in class in order to drive home some of the features of such idealized systems. The model
itself (that is, the spreadsheet) is already prepared for you and is available on the Moodle site
(Simple/NAR Regulation Model). You should be able to download this and then use it to see how
changing parameters of the model affects the outcomes.
When you open the spreadsheet, you will find that there are two pages. Make sure you are on the
first page (Simple Regulation tab at the bottom). This first page, devoted to the Simple regulation
model, is in three sections. The first, found in the upper left and colored light yellow, contains the
System Parameters: time, a, and ß. You should not vary the time (the model is not designed to
explore different time scales), but you can and should change both a and ß. You can enter values
directly in the boxes, or you can use the scroll boxes under each to change them. Note that in order to
get reasonable results for this time scale, ⍺ should be between 0.01 and 1, and ß should be between 1
and 100, as indicated below each scroll bar.
The second section is the prominently displayed graph, which represents the following equation (this
is the solution of the differential equation that describes the simple model):
[𝑌] =
!
”
(1 − 𝑒 #”$ )
As a reminder, the only two constants are a, which describes the rate of removal of [Y]:
rate of removal of [Y]= – a[Y])
and ß, which describes the constant rate of production of [Y]:
rate of production of [Y] = ß
Below the graph are two important measurements: the steady state value (which represents the
asymptote of the curve, or the maximal level of [Y] that is achieved after a long period of time), and
the T1/2 or time to half-maximum, also called the half-life. The graph, the steady state value, and the
T1/2 value, all change as you change a and ß.
The third section, in the lower left labeled Data Points, are the actual calculated data points used to
construct the graph. You will not need to use these numbers at all.
To begin, I suggest you simply play around with values of ⍺ and ß to see how changing them changes
the shape of the graph. Don’t worry about anything except learning how to change parameters in the
model.
Once you are comfy with using the model, here are the specific things I would like you to do:
1.
Set ⍺=0.1 and ß=10. The steady state value (box below graph) with these parameters
should be 100 units, and the T1/2 (other box) should be 6.9 seconds. The steady state
value represents how much [Y] will be present in a cell after transcription has been
turned on and allowed to proceed long enough to get to steady-state. You should be able
to see on the graph as steady state is reached. We will use the T1/2 as a measure of the
“speed of response:” lower values of T1/2 describe a system that can reach steady state
more rapidly, in other words, respond more quickly.
2.
3.
a.
From these starting conditions, describe what happens to the steady state value
and the T1/2 as you increase ⍺ but keep ß at 10.
b.
Similarly, describe what happens to the steady state value and the T1/2 as you
alter ß but keep ⍺ at 0.1.
c.
Can you make any generalizations about what you find, given what you know ⍺
and ß to represent? In other words, does what you find agree with what you
would have predicted, given what ⍺ and ß represent?
Let’s think about the steady state value for a bit:
a.
Does the steady state value depend upon ⍺ only? ß only? Or does it depend
upon both ⍺ and ß? If you don’t have enough data to figure this out, perform
more “experiments” with your model until you can answer this.
b.
Can you find other values of ⍺ and/or ß (besides 0.1 and 10) that result in the
same steady state value? Don’t worry about whether T1/2 is the same for this
question.
c.
Can you deduce any mathematical relationship between the steady state value
and the values of ⍺ and/or ß?
Now let’s think about the T1/2 value for a bit:
a.
Does the T1/2 value depend upon ⍺ only? ß only? Or does it depend upon both ⍺
and ß? Again, if you don’t have enough data to figure this out, perform more
“experiments” with your model until you can answer this.
b.
Can you find other values of ⍺ and/or ß (besides 0.1 and 10) that result in the
same T1/2 value? Don’t worry about whether the steady state value is the same
for this question.
c.
Can you deduce any mathematical relationship between the T1/2 value and the
values of ⍺ and/or ß?
Now let’s move to the second page (Negative Autoregulation tab at the bottom). The set up is similar:
you will find the same section involving the basic parameters in yellow, but there is now an
additional parameter (n) added in a blue section below this. This parameter represents a measure of
the cooperativity, and as is usual in biochemistry, we consider only values of n between 0 and 4,
where 0 represents no cooperativity, and 4 indicates maximum cooperativity.
Start with ⍺ and ß at 0.1, and 10 (the same starting values used in our simple regulation model). Also
start with n=0 (no cooperativity in the system). You should see that the NAR system under these
conditions behaves exactly like the simple system—the graphs coincide, and the steady state and T1/2
values are the same. This should not be a surprise if there is no cooperativity in the system—it is a
simple system.
Now change n, first to 1, then 2, etc., up to 4. You should see the graph reflect differences, as well as
different steady state and T1/2 values.
4.
Let’s compare the response of the NAR system with cooperativity to the simple system.
5.
a.
Does the steady state value change as n is changed (keeping all other
parameters constant)? If so, describe the relationship. If not, why not?
b.
Does the T1/2 value change as n is changed? (keeping all other parameters
constant)? If so, describe the relationship. If not, why not?
Describe as best you can based on these models what adding cooperativity through
negative autoregulation accomplishes within a cell, and why this arrangement is found
so frequently in living cells.
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