How to model biochemical pathwaysan ODE approach
How do we model activation of an enzyme, e.g:
CaMKII, PP2b by calcium?
We will first show a simple abstract example where a single
molecule of Ca can bind and activate a substrate S
Well mixed system – explain.
T
T
[S]
=
S
−
[S
⋅
Ca]
;
[Ca]
=
Ca
Ca]
 −[S⋅ 

d[S ⋅ Ca]
= k1[S][Ca] − k−1[S ⋅ Ca]
dt
= k1 (S T − [S ⋅ Ca])(CaT − [S ⋅ Ca]) − k−1[S ⋅ Ca]
= k1S T CaT − ( k1(S T + CaT ) + k−1 )[S ⋅ Ca] + k−1[S ⋅ Ca]2
OOPS
€
Can assume: CaT>>ST, so that [Ca]=CaT
This simplifies matters.
Using this we get: (show at home)
This is a first order linear ODE
Which has an exponential solution
With fixed point:
And time constant:
Cooperative activation of a substrate
correct
Assume:
Cooperative activation of a substrate
Assume:
Under this assumption this is a linear equation for
the vector S=(S, S1, S2)T
where
The fixed point is:
Where:
These types of results are often used to make the
following claim: If the relationship has power greater
than 1, the reaction is assumed to be cooperative.
This is usually done by fitting an equation of the form:
T n
(Ca
)
*
[S2] = Smax
;
n
T n
( kd ) + (Ca )
To experimental data
This is called
a hill equation, n is a ‘hill coefficient’.
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An irreversible enzymatic reaction.
Michaelis-Menten approach.
What is the F.P here?
What is the equilibrium here?
=0
Pseudo steady state
hypothesis (an approximation,
not always a good one)
where
Use:
so
where
This is the Michalis-Menten equation
where
Notes:
Lets create now a very simple, ‘toy model’ for
calcium dependent bidirectional synaptic plasticity.
Assume synaptic weight is
proportional to Ap.
Fixed point:
Time constant:
Assume:
dA p
= η(Ca)(Ω(Ca) − A p )
dt
Obtain:
Ω
η
Optional Homework ( I will drop the worst homework
grade, can submit after final)
a. Program the full Michaelis-Menten kinetics.
and show the dynamics at different values of the coefficients
(ki), as well as initial [E] and [S] values.
b. Show the dynamics of the reduced system, in the QSSA
case. Compare to the full dynamics above, and look at the
derivative, of the complex for different parameters. Are these
dynamics consistent with the QSSA? Do the values of the
coefficients influence if the assumptions are reasonable or
not.
c. Code the calcium dependent plasticity model.
It is your job to make assumptions about K(Ca) and P(Ca).
Find K(Ca) and P(Ca) that will generate and LTD/LTP model
such that for Ca<0.3 we get no change, 0.3<Ca<0.8 we
obtain LTD and for Ca>0.8 we get LTP.
d. Compare the rate at which you converge to maximal LTD at
Ca=0.5 and to maximal LTP at Ca=1.2. Is there a difference,
if there is explain why.
Summary