nr. 477 - 2011 - Institut for Natur, Systemer og Modeller (NSM)

A.4 Michaelis-Menten Kinetics 133
of (0, 0) in R n locally taking orbits of the nonlinear flow generated by the vector field g
to those of the linear flow
As with the implicit function theorem it can be stated for an arbitrary equilibrium,
but through a change of variables we can always move the equilibrium to the origin
(Crawford, 1991, p.993).
A.4 Michaelis-Menten Kinetics
This appendix 2 serves to introduce the idea behind the Michaelis-Menten function which
Marée et al. (2006) uses in their model, to make the cytokine-induced apoptosis more
realistic, i.e. saturated. It all comes from the fact that saturation is a feature inherent
to all biological systems, and therefore it is important to incorporate saturation when
modelling such systems – The Michaelis-Menten saturation function is one way of doing
so.
To arrive at the Michaelis-Menten saturation function let us imagine that we have some
kind of bacteria in a petri dish, that grows when supplied with a nutrient of sorts. In
fact its growth rate depends on the availability of nutrient. How should one go about
modelling such a growth rate? To begin with we could ask ourselves how the bacteria
consumes the nutrient – as it turns out, the nutrient is mediated into the bacterial cell
via receptors. Thus a receptor can either be occupied or it can be available for business.
Sometimes the nutrient is broken down to a product and sometimes it is “set free”. If
we denote an external nutrient molecule with N, an occupied receptor with Ro, an
available receptor with Ra, and the product with P , we can write the reactions as
N + Ra
k1
⇋ Ro
k−1
k2
−→ P + Ra
(A.7)
where k1, k−1, k2 are reaction rates. This equation should be read as: an external nutrient
molecule, N, comes into contact with an available receptor, Ra, thus constituting
the nutrient-receptor complex, Ro. Either the nutrient is relinquished, at rate k−1, or
it is broken down, at rate k2, yielding a reaction product, P , and an available receptor.
Denoting concentrations by appropriate lower-case letters we can transform this equation
into the following system of nonlinear autonomous differential equations
dn
dt = k−1ro − k1nra (A.8)
dra
dt = k−1ro − k1nra + k2ro (A.9)
dro
dt = k1nra − k−1ro − k2ro (A.10)
dp
dt = k2ro (A.11)
With initial conditions: n(0) = n0, ra(0) = ra,0, ro(0) = 0, p(0) = 0. The observant
reader may have noticed that if we add equations A.9 and A.10 a little miracle happens,
2 This appendix is based on Edelstein-Keshet (1988) and Murray (2002).