To non-dimensionalize the Michaelis-Menten equations in a way appropriate for the equilibrium
approximation (rather than the quasi-steady-state approximation), we let c = e0 x, and scale s by
s0 , the initial amount of substrate, s = s0 σ. With τ = k2 t, the equations are
dσ
dτ
dx
dτ
where α =
e0
s0 ,
β=
k1
k−1 s0
=
= αx − αβσ(1 − x),
(1)
= βσ(1 − x) − (1 + )x,
(2)
s0
K1 .
The quasi-steady state approximation for these equations is apparent. We set to zero to obtain
x − βσ(1 − x) = 0, or
βσ
x=
.
(3)
1 + βσ
The slow behavior is found by adding together the two equations to find
d
(σ + αx) = −x.
dx
(4)
d
βσ
βσ
(σ + α
)=−
.
dx
1 + βσ
1 + βσ
(5)
Eliminating x, we find
which is the slow-manifold equation.
The problem with this equation is, of course, that it cannot satisfy the correct initial data. Thus,
we need a different scaling of time to study the initial behavior. If, instead, we take τ = k−1 t, we
find
dσ
dτ
dx
dτ
= αx − αβσ(1 − x),
(6)
= βσ(1 − x) − (1 + )x,
(7)
Here, setting to zero, we find that σ + αx = 1 (using initial data σ(0) = 1, x(0) = 0), and then
dσ
= 1 − σ − βσ(α − 1 + σ).
dτ
(8)
Although the analytical solution to this equation can be found, it is sufficiently informative to
observe that the solution initially moves rapidly along the curve σ + αx = 1 until it reaches the
βσ
quasi-equilibrium value at x = 1+βσ
, from where the slow evolution (4) takes over.
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