Modeling Biochemistry

7/27/2019 Modeling Biochemistry

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How to model biochemical pathways-

an ODE approach


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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 ofCa can bind and activate a substrate S

Well mixed system explain.


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[S] = ST [SCa]

; [Ca] = CaT [SCa]

d[SCa]

dt= k

1[S][Ca] k

1[SCa]

= k1S

T [SCa]( ) Ca

T [SCa]( ) k1[SCa]

= k1STCa

T k1(S

T+ Ca

T) + k

1( )[SCa]+ k1[SCa]2

OOPS

Can assume: CaT>>ST, so that[Ca]=CaT

This simplifies matters.


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Using this we get: (show at home)

This is afirst order linear ODE

Which has an exponential solution

With fixed point:

And time constant:


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Cooperative activation of a substrate

Assume:

correct


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Cooperative activation of a substrate

Assume:

Under this assumption this is a linear equation for

the vectorS=(S, S1, S2)T

where


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The fixed point is:

Where:

These types of results are often used to make the

following claim: If the relationship has power greaterthan 1, the reaction is assumed to be cooperative.


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[S2]*= S

max

(CaT)n

kd( )

n

+ (CaT)n;

This is usually done by fitting an equation of the form:

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?


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=0

Pseudo steady statehypothesis (an approximation,

not always a good one)


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Use:

where

so

where


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where

This is the Michalis-Menten equation

Notes:


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Lets create now a very simple, toy model for

calcium dependent bidirectional synaptic plasticity.

Assume synaptic weight is

proportional toAp.


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Fixed point:

Time constant:


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Obtain:

dAp

dt=(Ca) (Ca) A

p( )

Assume:


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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 thederivative, 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.


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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


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Summary

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Modeling Biochemistry