MATH2307辅导、R程序设计辅导
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MATH2307/MATH6100
Assignment 4
Instructions: Please submit your line online via the Assignment Submission Tool on the
Wattle course page. Multiple files are allowed to be submitted.Please be sure to add your uni
id in each file’s name. PLEASE WORK ON YOUR OWN, not with other students.
Cover sheet: Remember to include the Assignment Cover Sheet with your assignments. A
pdf of the cover sheet form can be downloaded from this Assignment section of the Wattle
course page.
Due Date: 5pm Friday 28th October, 2022 (end of Week 12)
Total: 15 marks
Late policy: The policy for late assignments is that we will deduct 5% of your mark for each
working day late. Working day means weekends and public holidays are not counted as extra
days, i.e. Friday to Monday counts as 1 day. Please see the Assignment Submission section
on the Wattle course page for more details.
1
Question 1 (3 marks)
In §2.3 of the Biological Modelling notes we derived a formula linking the total rate of elimination
of substrate, V , to the input substrate concentration ci of the steady state version of the liver model.
The idea of this question is to make an R plot to illustrate how the elimination rate is related to its
asymptotic behaviour in the “homogeneous” and “flow-limited” regimes.
(i) For the parameter values Vmax = 1 mmol/min, F = 1 `/min, K = 0.1 mmol/` make a plot
of V (on the vertical axis) against ci (on the horizontal axis) for 0 ≤ ci ≤ 3 mmol/`. Make
sure to label the axes.
(ii) Add to the plot dashed curves for the “homogeneous” and “flow-limited” regime functions.
Also add a vertical dashed line to represent the changeover point c?.
Include the plot in the main pdf file of your assignment. Also upload the source file of your R
code in a file called Asst4-Q1-u1234567.R with “u1234567” replaced by your university
id number. (IMPORTANT: please name the file exactly according to this format - it makes the
taask of marking 50 assignments much easier!)
Marks will be deducted for (1) code that contains no comments, and (2) code in which the values
of the parameters Vmax, F and K are not set at the beginning. It should be possible to run the code
with different parameter values by changing only the lines in which the parameters are set.
2
Question 2 (3 marks)
Consider the steady-state solution to the liver model studied in §2.3 of the notes.
(i) Define the logarithmic mean c? of the input and output substrate concentrations ci and co as
c? =
ci ? co
log ci ? log co . (1)
Show that the total rate of removal of substrate V satisfies the Michaelis-Menton reaction rate law
corresponding to a substrate concentration c?, i.e. show that
V = Vmax
c?
c?+K
. (2)
(ii) Suppose the enzyme distribution is constant along the length of each sinusoid. Show that the
concentration profile c(x) is a decreasing, concave upward function of the distance x from the be-
ginning of the sinusoid.
In your answer you may quote without proof any equation from the notes, but please state the
equation number.
3
Question 3 (6 marks)
Let a distance function d for sequences x1, x2, x3, x4, x5 be given by the following matrix:
Md x
1 x2 x3 x4 x5
x1 0 6 14 8 7
x2 6 0 16 10 9
x3 14 16 0 10 13
x4 8 10 10 0 7
x5 7 9 13 7 0
Prove that d is indeed a distance function. Find the unrooted tree that generates d using the
neighbour-joining method. Does d satisfy the four point condition? (Hints: To check the triangle
inequality, EITHER do it by hand by finding an argument to avoid having to check every possi-
ble combination of three nodes OR write some R code. Similarly, for the the neighbour-joining
algorithm I don’t mind whether you do it by hand or write R code to do some or all of it. If you
choose to use R, write the program yourself. Do NOT just download a package, such as ape, with
a ready-made function. Include a printout of any source-code you do write in the main pdf file of
your assignment, but don’t upload the code as a separate .R file. )
4
Question 4 (3 marks)
Write out the 4×4 matrix corresponding to the distance function d for the following unrooted tree:
1 2
3
4
1 1 2
5
7
Apply the UPGMA algorithm to this distance function to obtain a molecular clock tree. Is the
resulting tree topologically equivalent to the original unrooted tree? Infer whether d is ultrametric.
5
Question 5 (5 marks; for MATH6100 only)
NB: Question 5 is for MATH6100 students and PhB students doing the ASE ONLY. For MATH6100,
the mark out of 5 will be added to the mark for the first 4 questions and the total multiplied by 0.75
to give a mark out of 15.
The methods used to construct the model of a liver described in lectures can also be used to con-
struct a model of a kidney. The kidney differs from the liver in that the flowing solvent passing
through a number of parallel tubules is not blood but water. This water is reabsorbed so that the
flow rate F (x) decreases with distance x through each tubule, and will be assumed here to be time-
independent at any position. The substrate (e.g. glucose) is transported across the tubule surface
by a metabolically driven carrier system capable of saturation and also described by Michaelis-
Menten kinetics. This leads to the following partial differential equation for the substrate concen-
tration c(x, t) at position x and time t in the region 0 ≤ x ≤ L,∞ < t <∞:
. (3)
Here A(x) is the total cross-sectional area of the tubules at position x and ρ(x) and K are parame-
ters analogous to those for the liver describing the Michaelis-Menton reaction kinetics.
(i) Show that this equation can be recast in the form
where
u(x, t) = K log (F (x)c(x, t)) . (5)
(ii) If we assume the substrate concentration to be very small, so that
c(x, t) << K, the c(x, t) in the denominator on the right hand side of Eq.(4) can be set to
zero. Show that if this approximation is made,
u(x, t) = f
is a solution to Eq.(4), where f(·) is any function of a single real argument. (In fact it can be
shown to be the most general solution.)
(iii) Hence show that the input and output substrate concentrations ci(t) = c(0, t) and co(t) =
c(L, t) are related by
co(t) =
Fi
Fo
ci(t? T )e?Φ, (7)
where Fi = F (0) and Fo = F (L) are the input and output flow rates,
T =
∫ L
0
A(ξ)
F (ξ)
dξ (8)
6
is the transit time through the liver and
(iv) For the time dependent case, ci = const. << K, with constant enzyme density ρ = Vmax/L,
and a flow rate that drops linearly with distance from an input rate Fi to and output rate
Fo < Fi, find co in terms of ci and the parameters Fi, Fo, Vmax and K.
7
MATH2307/MATH6100
Assignment 4
Instructions: Please submit your line online via the Assignment Submission Tool on the
Wattle course page. Multiple files are allowed to be submitted.Please be sure to add your uni
id in each file’s name. PLEASE WORK ON YOUR OWN, not with other students.
Cover sheet: Remember to include the Assignment Cover Sheet with your assignments. A
pdf of the cover sheet form can be downloaded from this Assignment section of the Wattle
course page.
Due Date: 5pm Friday 28th October, 2022 (end of Week 12)
Total: 15 marks
Late policy: The policy for late assignments is that we will deduct 5% of your mark for each
working day late. Working day means weekends and public holidays are not counted as extra
days, i.e. Friday to Monday counts as 1 day. Please see the Assignment Submission section
on the Wattle course page for more details.
1
Question 1 (3 marks)
In §2.3 of the Biological Modelling notes we derived a formula linking the total rate of elimination
of substrate, V , to the input substrate concentration ci of the steady state version of the liver model.
The idea of this question is to make an R plot to illustrate how the elimination rate is related to its
asymptotic behaviour in the “homogeneous” and “flow-limited” regimes.
(i) For the parameter values Vmax = 1 mmol/min, F = 1 `/min, K = 0.1 mmol/` make a plot
of V (on the vertical axis) against ci (on the horizontal axis) for 0 ≤ ci ≤ 3 mmol/`. Make
sure to label the axes.
(ii) Add to the plot dashed curves for the “homogeneous” and “flow-limited” regime functions.
Also add a vertical dashed line to represent the changeover point c?.
Include the plot in the main pdf file of your assignment. Also upload the source file of your R
code in a file called Asst4-Q1-u1234567.R with “u1234567” replaced by your university
id number. (IMPORTANT: please name the file exactly according to this format - it makes the
taask of marking 50 assignments much easier!)
Marks will be deducted for (1) code that contains no comments, and (2) code in which the values
of the parameters Vmax, F and K are not set at the beginning. It should be possible to run the code
with different parameter values by changing only the lines in which the parameters are set.
2
Question 2 (3 marks)
Consider the steady-state solution to the liver model studied in §2.3 of the notes.
(i) Define the logarithmic mean c? of the input and output substrate concentrations ci and co as
c? =
ci ? co
log ci ? log co . (1)
Show that the total rate of removal of substrate V satisfies the Michaelis-Menton reaction rate law
corresponding to a substrate concentration c?, i.e. show that
V = Vmax
c?
c?+K
. (2)
(ii) Suppose the enzyme distribution is constant along the length of each sinusoid. Show that the
concentration profile c(x) is a decreasing, concave upward function of the distance x from the be-
ginning of the sinusoid.
In your answer you may quote without proof any equation from the notes, but please state the
equation number.
3
Question 3 (6 marks)
Let a distance function d for sequences x1, x2, x3, x4, x5 be given by the following matrix:
Md x
1 x2 x3 x4 x5
x1 0 6 14 8 7
x2 6 0 16 10 9
x3 14 16 0 10 13
x4 8 10 10 0 7
x5 7 9 13 7 0
Prove that d is indeed a distance function. Find the unrooted tree that generates d using the
neighbour-joining method. Does d satisfy the four point condition? (Hints: To check the triangle
inequality, EITHER do it by hand by finding an argument to avoid having to check every possi-
ble combination of three nodes OR write some R code. Similarly, for the the neighbour-joining
algorithm I don’t mind whether you do it by hand or write R code to do some or all of it. If you
choose to use R, write the program yourself. Do NOT just download a package, such as ape, with
a ready-made function. Include a printout of any source-code you do write in the main pdf file of
your assignment, but don’t upload the code as a separate .R file. )
4
Question 4 (3 marks)
Write out the 4×4 matrix corresponding to the distance function d for the following unrooted tree:
1 2
3
4
1 1 2
5
7
Apply the UPGMA algorithm to this distance function to obtain a molecular clock tree. Is the
resulting tree topologically equivalent to the original unrooted tree? Infer whether d is ultrametric.
5
Question 5 (5 marks; for MATH6100 only)
NB: Question 5 is for MATH6100 students and PhB students doing the ASE ONLY. For MATH6100,
the mark out of 5 will be added to the mark for the first 4 questions and the total multiplied by 0.75
to give a mark out of 15.
The methods used to construct the model of a liver described in lectures can also be used to con-
struct a model of a kidney. The kidney differs from the liver in that the flowing solvent passing
through a number of parallel tubules is not blood but water. This water is reabsorbed so that the
flow rate F (x) decreases with distance x through each tubule, and will be assumed here to be time-
independent at any position. The substrate (e.g. glucose) is transported across the tubule surface
by a metabolically driven carrier system capable of saturation and also described by Michaelis-
Menten kinetics. This leads to the following partial differential equation for the substrate concen-
tration c(x, t) at position x and time t in the region 0 ≤ x ≤ L,∞ < t <∞:
. (3)
Here A(x) is the total cross-sectional area of the tubules at position x and ρ(x) and K are parame-
ters analogous to those for the liver describing the Michaelis-Menton reaction kinetics.
(i) Show that this equation can be recast in the form
where
u(x, t) = K log (F (x)c(x, t)) . (5)
(ii) If we assume the substrate concentration to be very small, so that
c(x, t) << K, the c(x, t) in the denominator on the right hand side of Eq.(4) can be set to
zero. Show that if this approximation is made,
u(x, t) = f
is a solution to Eq.(4), where f(·) is any function of a single real argument. (In fact it can be
shown to be the most general solution.)
(iii) Hence show that the input and output substrate concentrations ci(t) = c(0, t) and co(t) =
c(L, t) are related by
co(t) =
Fi
Fo
ci(t? T )e?Φ, (7)
where Fi = F (0) and Fo = F (L) are the input and output flow rates,
T =
∫ L
0
A(ξ)
F (ξ)
dξ (8)
6
is the transit time through the liver and
(iv) For the time dependent case, ci = const. << K, with constant enzyme density ρ = Vmax/L,
and a flow rate that drops linearly with distance from an input rate Fi to and output rate
Fo < Fi, find co in terms of ci and the parameters Fi, Fo, Vmax and K.
7