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University of California, Santa Barbara *
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Course
108C
Subject
Mathematics
Date
Jun 11, 2024
Type
Pages
10
Uploaded by CorporalSquirrelMaster1081
Name: ______________________________ 1/10 Student’s perm #: ________________________ Student’s signature: ________________________ Grade =______________/80 pts. MCDB 108C Spring 2023
MIDTERM EXAM #1 DO NOT OPEN THE EXAM UNTIL YOU ARE INSTRUCTED TO DO SO This exam should have 10 pages. Two blank pages are collated to the exam for you to draft your answers. Please put your name on every page. You are allowed to use one page
of your own hand-written notes. You may not
use a computer or smart phone during the exam, but you shouldn't need them since you can leave your answers in exact form whenever numerical values cannot be provided. Do not spend all your time answering one question, allocate your time among the questions appropriately. For the short-answer questions, you should show how you arrived at your final conclusion(s)
. Each point of this exam represents 1 point of the final grade of the class (total: 500 points). (source: Wikipedia)
Name: ______________________________ 2/10 [ /25 points] Part 1 – Programming Question 1.1
[ /10 pts]: Two vectors are inputted in Matlab by using the following commands: v1 = [0 1 2]; v2 = [2 1 0]; After having run the following for
loop, what are the value(s) of variable w
? W = zeros(3,1); for i
= 1:3 w(i) = v1(i) * v2(i); end Question 1.2
[ /5 pts]: What is the value of variable z
after having run the following commands? mysterious_vector = -3:1:3; z = length(mysterious_vector)
Name: ______________________________ 3/10 Question 1.3
[ /10 points]: Consider scripts 1 and 2. What will be the value of counter_1
and counter_2
after each script has been executed? Will the values of these two counters be the same? Please briefly justify your answer by explaining the algorithmic logic of each script. Script 1: counter_1(1) = 0; for i = 1:9 counter_1(i+1) = counter_1(i) + 1; end disp(counter_1) Script 2: clearvars counter_2(1) = 0; for i = 2:10 counter_2(i) = counter_2(i-1) + 1; end disp(counter_2)
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Name: ______________________________ 4/10 [ /25 points] Part 2 – Derivation and numerical integration Question 2.1
[ /10 points]: The expression of a transcription factor ࠵?
is monitored in a cell over time. The concentration of ࠵?
(mM) is measured at time intervals spaced by 1 second. The graph of the time series of ࠵?(࠵?)
is displayed on the right graph. How will the graph of the rate of change of ࠵?
with respect to time (࠵?࠵?/࠵?࠵?
) look like? Please pick one of the 4 options (A-D) based on your expectations of ࠵?࠵?/࠵?࠵?
and briefly justify your choice. Hint: Provide a reason to rule out the 3 options that do not apply.
Name: ______________________________ 5/10 (continuation of answer to Q2.1) Question 2.2
[ /10 points]: The transcription factor ࠵?
controls the expression of protein Y whose concentration is represented by variable ࠵?(࠵?)
. The ODE system describing the expression of protein Y is: ࠵?࠵?
࠵?࠵?
= ࠵?(࠵?) − ࠵?(࠵?)
where the values of ࠵?(࠵?)
are given in a time series x_t
(values plotted in the graph of question Q2.1). The initial condition is: ࠵?(0) = 0
mM. To compute the time course of ࠵?(࠵?),
you want to numerically integrate the ODE !"
!#
. One line of the script below has a bug and it will not produce the expected result. Please explain what the bug is and how this bug should be corrected based on the principle of Euler’s integration method. Hint: Assume that the loading of file x_t
is successful.
load(
'x_t'
) dt = 1; % second L = length(x_t); y_t = zeros(L,1) for i = 1:(L-1) dydt = x_t(i) - y_t(i); y_t(i+1) = dydt + dt; end
Name: ______________________________ 6/10 (continuation of answer to Q2.2) Question 2.3
[ /5 points]: How should the script presented in question Q2.2 be modified to account for the initial condition is ࠵?(0) = 1
mM? Hint: This change does not require more than one line.
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Name: ______________________________ 7/10 [ /30 points] Part 3 – Modeling drug treatment Question 3.1
[ /15 points]: The viral load of a virus similar to HIV can be modeled by the following ODE: ࠵?࠵?
$
࠵?࠵?
= ࠵?
$
− ࠵?. ࠵?
$
(࠵?)
where ࠵?
represents the viral production and ࠵?
the degradation rate of the virus as a result of the immune response. Two new antiretroviral drugs are tested separately. Drug A completely blocks any new production of the virus. Drug B decreases the production rate of the virus by a factor 2 without abolishing it. The viral load ࠵?
$
is measured at consecutive hours following drug administration. Please associate the expected time course of the viral load following the application of each drug with one of the 3 options presented in the right graph. Please justify your selection with arguments based on the ODE. Hint: You might want to refer to the filter paper analogy.
Name: ______________________________ 8/10 (continuation of answer to Q3.1) Question 3.2
[ /15 points]: What explains that the HIV virus is so prone to develop drug resistance? Why was the mathematical modeling of the viral load observed upon antiretroviral drug treatment instrumental to reach this conclusion? Hint: Your answer can hold in 4-5 sentences. Focus on the broad-picture to answer each question: you don’t need to re-derive any complex mathematical results or provide exact numbers.
Name: ______________________________ 9/10 (continuation of answer to Q3.2)
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Name: ______________________________ 10/10