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MATH3888

Semester 2 Interdisciplinary Project (Stream 1) 2022

WEEK 2 HOMEWORK GUIDELINES

Submission:

As outlined in the information sheet of this interdisciplinary project course, you will create reports using

the (maths) editing software LaTeX:

https://en.wikibooks.org/wiki/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:

https://www.overleaf.com

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).

This report is worth 5% of your final mark.

Deadline is Thursday, week 3 (August 18th), 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!

In addition, you need to submit your MATLAB source file (*.m);

Again, additional MATLAB instructions are given within the text.

1 Enzyme kinetics in two dimensions

The irreversible enzyme kinetics scheme,

S + E

k1

k?1

C

k2? P + E,

presented in our lecture defines through mass-action kinetics a corresponding four-dimensional system

of ODEs. This model can be reduced to a two-dimensional model by identifying two conservation laws.

Consider the following dimensionless version of this 2D model:

ds

dt

= αc? s(1? c),

dc

dt

= ?β(αc? s(1? c))? εc,

(1)

with initial conditions s0 = s(0) = 1, c0 = c(0) = 0, and dimensionless parameters ε, α, β ≥ 0 given by

Parameter Value

α 0.6

β 1.25

ε 0.4

1. As a first task in your LaTeX homework file, replicate the above paragraph starting with ’Consider

....’ up to and including the parameter table. Make sure you use the correct LaTeX environments

to create maths formulae and tables.

2. Next, use MATLAB to study the dynamics of system (1).

(a) Implement system (1) in MATLAB and integrate forward the initial condition (s0, c0) for a

sufficiently long time such that you get sufficiently close to the asymptotic end state (but not

too long either). Present:

? a time series plot that includes both variables plotted in different colours; include a legend

in this plot so that the variables can be clearly identified;

? a phase space plot where you include two trajectories: one with the original initial condi-

tions and another with (s0, c0) = (0, 1).

Save both MATLAB plots in, e.g., png format (do NOT create screen shots!). Include these

png figures in your LaTeX homework file by using the LaTeX command \includegraphics.

Set the figure width to 9cm. Include a caption for each plot: ‘Time series s(t) and c(t) with

initial condition s0 = s(0) = 1, c0 = c(0) = 0 integrated for t ∈ [0, T ]’ and ‘System (1) phase

space plot’. Provide your T value in the caption that matches your integration time!

(b) Discuss whether there exists a further one-dimensional approximation to the dynamics of

the substrate concentration s(t) when ε becomes sufficiently small, and argue where such an

approximation would be valid in phase space.

Hint: plot the nullclines. Include a corresponding figure in your LaTeX file and provide your

sufficiently small ε value in the caption.

Please name your MATLAB source file my W2 homework.m where you replace my by ‘your initials’

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