The lac Operon, Study Guides, Projects, Research of Topology

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The lac Operon

Section of DNA that acts as an ON/OFF

switch for genes that control the metabolism

of lactose.

Source: http://en.wikipedia.org/wiki/Lac_operon

Promoter: DNA section where RNA polymerase binds to initiate transcription of the lac genes.

The lac Operon

Section of DNA that acts as an ON/OFF

switch for genes that control the metabolism

of lactose.

Source: http://en.wikipedia.org/wiki/Lac_operon

Repressor: protein that shuts off operon. If bound to operator, it prevents RNA polymerase from binding to the promoter.

The lac Operon

Section of DNA that acts as an ON/OFF

switch for genes that control the metabolism

of lactose.

Source: http://en.wikipedia.org/wiki/Lac_operon

The lac Operon

Section of DNA that acts as an ON/OFF

switch for genes that control the metabolism

of lactose.

lacZ: gene encoding for beta-galactosidase, which metabolizes lactose. Source: http://en.wikipedia.org/wiki/Lac_operon

Control Mechanisms in the lac Operon

When lactose is in small concentrations, the repressor is free to bind to the operator. Presence of the repressor blocks RNA polymerase from binding to the promoter. (^) Transcription of the 3 structural genes does not occur. Source: http://ghs.gresham.k12.or.us/science/ps/sci/ibbio/chem/nucleic/chpt16/chpt16.htm

Control Mechanisms in the lac Operon

When lactose is in high concentrations, (an isomer of) lactose binds to the repressor, changing its shape and can’t bind to the operator. Absence of the repressor allows RNA polymerase to bind to the promoter and begin transcription. RNA polymerase transcribes the 3 structural genes, whose mRNA is translated into proteins that aid in lactose metabolism. Source: http://ghs.gresham.k12.or.us/science/ps/sci/ibbio/chem/nucleic/chpt16/chpt16.htm We will show how CoCoA can be used to make a model for the positive feedback control mechanism.

Network Topology

Let f = ( f

1

,…,f

n

) : k

n

 k

n

, k a finite field.

The directed graph G = ( V,E ) is called the wiring diagram for f where

V = { xi | i = 1.. n } E = { ( xi , xj ) | xi  supp( fj ) = variables in fj }

Let x

1

:= M, x

2

:= B, x

3

:= A, x

4

:= L, x

5

:= P.

f

1

= x

3

f

2

= x

1

f

3

= x

3

+ x

2

x

4

+ x

2

x

3

x

4

f

4

= x

5

+ x

4

( x

2

+1) + x

4

x

5

( x

2

f

5

= x

1

Network Dynamics

Let f = ( f

1

,…,f

n

) : k

n

 k

n

, k a finite field.

The directed graph G = ( V,E ) is called the state space for f where

V = k n (the states) E = { ( a, f ( a )) | a  k n } (the transitions)

LS Algorithm

Input : State transitions T = { s 1 ,…, st } in k n ; a term order. Output : Polynomial dynamical system F

1. Find one interpolator f 0 = ( f 1 ,…,fn ) with f 0 ( si ) = si +.
2. Construct ideal of vanishing functions I = < g (x): g ( si ) = 0 >. All PDSs that fit T : f 0 + I := { ( f 1 ,…,fn ) + ( g 1 ,…,gn ) }.
3. Select PDS F = ( F 1 ,…,Fn ) with Fi = fi % GröbnerBasis ( I ). Can construct the wiring diagram and state space from F.