Quasi-steady approximations
Quasi-steady approximations:
k1
k
2
S + E −→
←− C −→ P + E
k3
substrate + enzyme =⇒ complex =⇒ further reaction to product + enzyme
k
1
S + E −→
C
Ċ = k1 SE − k3 C − k2 C
Ṡ = −k1 SE + k3 C
Ė = −k1 SE + k2 C + k3 C
Ṗ
= k2 C
E + C = conserved quantity = E0
Take (C (0) = 0) so we can eliminate E

Ċ = k1 S(E0 − C) − k3 C − k2 C
linear stability shows that
 ⇒ C = 0, S = 0
Ṡ = −k1 S(E0 − C) + k3 C
is stable fixed point.
Ṗ = k2 C
Suppose E0 1 (small amount of enzyme); then also small amount of C.
C = E0 c
⇒ E0 ċ = k1 S(1 − c)E0 − k3 E0 c − k2 E0 c
Ṡ = −k1 S(1 − c)E0 + k3 E0 c
To see leading order behavior, T = E0 t
⇒ E 0 cT
ST
= k1 S1 (1 − c) − k3 c − k2 c
= −k1 S(1 − c) + k3 c
Leading order: set E0 Ċ = 0
⇒c =
k1 S1
k1 S1 + k 2 + k 3
like a steady–state, if
S was a constant
Ṡ = −k1 S(1 − c) + k3 c
More on quasi-steady solutions:
Consider again the Vander Pol oscillator:
dv
dt
dI
L
dt
= I/C
˜
= −V − f(I)
Let
y = −V · C ⇒
1 dy
= −I/C
c dt
L
dI
y
= − f˜(I)
dt
c
˜ = cubic function
f(I)
1
We’ve already studied the weakly nonlinear case, e.g. A, B 1
−(AI − BI 3 )
There we saw that the limit cycle behavior was given by sin, cos oscillations.
In contrast to this case, we can also look at the case LC 1
dy
= −I
dt
,
dI
= y − f (I)
|{z}
dt
C f˜(I)
In this limit we have a quasi-stationary behavior. Consider the phase plane:
y
y=f(I)
I
ẏ > 0 for I < 0
ẏ < 0 for I > 0
Also,
I˙ < 0 for y < f (I)
I˙ > 0 for y > f (I)
Then the vector field looks like
y
I
Note also that I = 0, y = 0 is an unstable fixed point.
In the limit → 0 we can say a bit more using the quasi-steady approximation:
Setting
I˙ = 0
ẏ = −I
⇒
y = f (I)
This suggests that the solution closely follows the curve y = f (I).
This is true, as long as the vector field pushes the solution near the curve. However, for values
of I where f 0 (I) < 0, the vector field pushes the solution away from the curve y = f (I).
In fact, a closer look at the vector field shows that the dynamics are essentially horizontal (I
varies rapidly) when y 6= f (I) and 1.
2
y
I˙ =
I
y − f (I)
1
for y 6= f (I)
1
Then the oscillations proceed as follows: y = f (I) until reaching a “knee” of the curve y = f (I),
then a rapid transition to the other part of the curve y = f (I), where f 0 (I) > 0.
y
y=f(I)
(I)
Then the time behavior is
I
y
t
t
These are called relaxation oscillations: each
oscillation has a “slow” and “fast” part.
This is in contrast to the sinusoidal oscillations that we observed in the weakly nonlinear limit.
Recognizing the slow and fast part of the oscillations allows for an easy computation of the
period of the oscillations.
Since the fast transitions take very little time, the period can be calculated by determining the
time spent near the curve y = f (I).
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y
y1
I
←− time spent here ×2 = period
I1
y2
I2
f (I) = −I + I 3 /3
For
f 0 (I) = 0 ⇒ −1 + I 2 = 0
⇒ transition points at ± 1
⇒ y1 = 1 − 1/3 = 2/3
T
,
y2 = −1 + 1/3 = −2/3
ZI2
Zy2
Zt2
d −I + I 3 /3
dy
=2
= 2 dt = 2
ẏ
−I
y1
t1
I1 = 2
,
I2 = 1
I1
= 2 ln I − 2 · I 2 /2
1
2
= −2 ln 2 + 3
We’ve made a simple approximation to the behavior during the fast transition – essentially we have
taken y = const in this region.
Suppose we want to look closer in this region
δY = (y − 2/3) ⇒
δY
= −I
dt
dI
= δY − f (I) + 2/3
dt
If we let δ −1 t = T (“fast” time T ) δ = , we have
dI
∼ −f (I) + 2/3
dT
dY
= −I
dT
which gives a separable equation for Y (I)
dY
I
=
dI
f (I) − 2/3
This gives the following behavior
y
O()
y =const.
I
dY
I
=
dI
f (I) − 2/3
O()
Notice that as → 0, the approximation of y = ±2/3 in the fast transition improves.
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