# 辅导MATH3888、辅导Java，C++程序

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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.

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).

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.

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).