MATH3888

Thursday, week 3 (August 18th), 23:59

1 Enzyme kinetics in two dimensions
The irreversible enzyme kinetics scheme,
S + E
k 1
?
k −1
C
k 2
* 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 s 0 = s(0) = 1, c 0 = 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 (s 0 ,c 0 ) 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 (s 0 ,c 0 ) = (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 s 0 = s(0) = 1, c 0 = 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’.