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## Nonlinear Differential Equations & Applications

The relevant journal articles are listed at the end, they are useful to read but not essential.
Glycolysis is the metabolic pathway that generates energy from glucose. It involves 10 di erent enzymes and a huge
number of intermediate chemical compounds and chemical reaction steps. Mathematical biologists have searched for
di erent ways to usefully approximate such a complicated system using as few variables and di erential equations as
possible. The extremely simpli ed model investigated in this assignment was proposed by Schnakenberg in 1979 and is
an extension of ideas from Higgins in 1967 and Selkov in 1968. In 2009 Wilhelm wrote a review paper which lists all
the simplest abstract chemical reactions that have interesting mathematical properties (see the Table in his paper).
Let S be the concentration of the starting chemical (in this case glucose) and let P be the concentration of the
most important product chemical. The concentration E of the most important enzyme is assumed to be a constant.
Consider
dP
dt
= + k1ES + k2P2S ? k3P
dS
dt
= k0 ? k1ES ? k2P2S
where the parameters ki are all positive. k0 is the constant rate at which glucose enters the system from outside (e.g.
when you eat something), k1 and k2 are the rates of the two most important intermediate chemical reactions and k3
is the relative rate at which the product is used up inside the body (e.g. when you need energy).
1. Show that the ODEs can be rescaled to give
dx
d
= + by + x2y ? x
dy
d
= a ? by ? x2y
Hint: Use the same scaling factor for P and S, and also choose = k3t.
2. Sketch both nullclines in the rst quadrant and nd their intersection point and add direction arrows on and
between the nullclines.
3. Prove that no trajectories can leave the rst quadrant. In other words, the model has the nice property that
chemical concentration can never become negative.
4. Let a = 1
2 be constant but allow b > 0 to vary as your bifurcation parameter. Evaluate the Jacobian at the
equilibrium point and give a list of all the types of equilibria that occur as b is varied and calculate numerically
where the bifurcations occur. The sign of the determinant and trace are easy enough to study by hand (the
quadratic does NOT factorise nicely). Use a calculator or a spreadsheet to study the discriminant numerically
and nd a numerical approximation to when it changes sign.
5. Challenge: There is an irregular pentagonal region (see image below) with the property that no trajectories
can leave that region. In other words, the model has the nice property that chemical concentration are bounded
and never become TOO large.
The sloping side of the pentagon has slope ?1. Find such a pentagon, and if you can, nd the smallest such
pentagon.
Hint 1: Study each of the 5 sides separately, you have done two of the sides in Q3 already.
Hint 2: Explain why it is useful to know that x_ + y_ < 0 on the sloping side.
x
y
References
Higgins, J. (1967) The theory of oscillating reactions.” Industrial and Engineering Chemistry 59 18{62.
Selkov, E. E. (1968) Self-oscillations in glycolysis. I. A simple kinetic model.” European Journal of Biochemistry 4
79{86.
Schnakenberg, J. (1979) Simple chemical reaction systems with limit cycle behaviour.” Journal of Theoretical Biology
81 389{400.
Wilhelm, T. (2009) The smallest chemical reaction system with bistability.” BMC Systems Biology 3 90-99.
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