# 辅导MATH3888、辅导Java，C++程序

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Semester 2 Interdisciplinary Project (Stream 1) 2022
WEEK 7 HOMEWORK GUIDELINES
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.
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).