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MATH3888 HOMEWORK

创建日期：2024-05-13 17:55:26

所属分类：数学mathematics

标签： MATH3888

HOMEWORK

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MATH3888
WEEK 7 HOMEWORK GUIDELINES
Submission:
As outlined in the information sheet of this interdisciplinary project course, you will create reports using
the (maths) editing software LaTeX:

You are encouraged to use Overleaf to create your LaTeX report which you can access via your browser
through your University of Sydney account:

Use the following basic setup for your LaTex file:
\documentclass[11pt]{article}
\usepackage{fullpage,amsmath,graphicx}
. . .
\begin{document}
. . .
\end{document}
Submission of the corresponding pdf file is via Canvas/turnitin (where it will be checked for plagariasm).
As outlined in the course info sheet, this report is worth 5% of your final mark.
Deadline is Thursday, week 8 (September 22nd), 23:59. No late submission will be accepted!
Constraints:
The ‘fontsize’ is strictly 11 points and the margins of the document are automatically set by the ‘fullpage’
package (as instructed above).
The package ‘amsmath’ might be needed for the mathematical editing, and I let you figure out what the
‘graphicx’ package is needed for. Add any other packages, if needed.
Additional LaTeX instructions are given within the text. Please follow them to avoid losing marks!
Please do not provide any screenshots! Otherwise, you will lose marks!
Oscillations in a calcium flux model
Consider the following calcium flux model (‘Friel model’) introduced in weeks 6 & 7:
c′ = JL1 − JP1 + JL2 − JP2,
c′e =
1
γ
(JP2 − JL2) ,
(1)
where c measures the Ca2+ concentrations in the cytosol and ce measures the Ca
2+ concentration in an
internal ‘compartment’ (ER/SR). The parameter γ is a volume ratio measure between the ER/SR and
the cytosol.
Each term in (1) corresponds to a flux Jx into or out of the cytosol: index ‘1’ indicates fluxes between the
extracellular space and the cytosol, while index ‘2’ indicates fluxes between the ER/SR and the cytosol:
JL1 = kL1(co − c)
JP1 = kP1c
JP2 = kP2c
JL2 = kL2(c)(ce − c)
where co denotes the (fixed) calcium concentration in the extracellular medium and
kL2(c) = kL20 + kL21
 1
1 +
(
Kd
c
)n
 . (2)
Set parameter values to
co = 1000µM, kL1 = 2 · 10−5s−1, kP1 = 0.13s−1, kL20 = 0.013s−1,
kL21 = 0.58s
−1, kP2 = 0.9s−1, Kd = 0.1µM, n = 3, γ = 0.24.
Friel’s model is inspired by experiments of caffeine-induced calcium oscillations in bullfrog sympathetic
neurons. Your goal is to understand the onset and properties of these oscillations, by analysing how
caffeine affects the individual flux components. Here are experimental data:
Figure 1: Experimental data adapted from D. Friel, Biophysical Journal 68 (1995), 1752-1766; cytosolic
calcium response to caffeine exposure.
In this model, the release of Ca2+ from the endoplasmic reticulum (which is modelled by the flux term
JL2 in the equations) is controlled by ryanodine receptors (RyR). The function kL2(c) models the corre-
sponding (cytosolic calcium-dependent) RyR conductivity.
1. Plot the graph of kL2(c) vs. c, for various values of Kd ∈ (0, 1).
(Create a single plot that includes at least 3 representative graphs; use an appropriate figure
environment in LaTeX including a figure caption with sufficient information).
What is the qualitative effect of varying Kd on the affinity of the RyR?
2. Use MatCont to create a bifurcation diagram in the (Kd, c)−plane.
(Provide the plot in an appropriate figure environment in LaTeX including a figure caption with
sufficient information).
For which values of Kd ∈ [0, 1] does the cytosolic calcium concentration oscillate? Give the (nu-
merical) eigenvalues at each Andronov-Hopf bifurcation as provided by MatCont. Based on these
(numerical) eigenvalues, calculate the oscillation period at onset of these Andronov-Hopf bifurca-
tions.
Determine the criticality of the corresponding Andronov-Hopf bifurcations responsible for the onset
and termination of these oscillations by providing the (numerical) 1st Lyapunov coefficient given
by MatCont.
3. Use MatCont to plot the corresponding period of these oscillations as a function of Kd.
(Again, provide the plot in an appropriate figure environment in LaTeX including a figure caption
with sufficient information).
What is the effect of increasing/decreasing Kd in system (1)?
4. Given your numerical work in parts (1)–(3), relate the action of caffeine—which increases RyR
response/affinity—to the variation of a parameter in the Friel model. In particular, interpret your
results directly to the experimental data shown in Figure 1.
Provide representative plots of cytosolic calcium time traces from your model ‘to back up your
story’.
(Again, provide these plots in an appropriate figure environment in LaTeX including a figure caption
with sufficient information).

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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