Lecture 3: Models of gene regulation

DNA

Replication

RNA ProteinTranscription Translation

Bacterial growth on sugars

maxmax

Batch reactor:

( ) bacterialgrowth

1substrateconsumption

Substrate metabolised by a Michaelis Menten enzyme:

( ) , enzyme concentrationM

dCS C

dtdS dC

dt Y dt

SS

S K

How does the cell know which enzymes to express?

hours

[conc.]

glucose lactose

Bacterial growth on two substrates:

Two things to notice:

1.Glucose metabolized before lactose

2.Lag between glucose and lactose growth phases

[Monod, Thesis 1942]

• Monod realized this was operating like a genetic “switch”– Genes required for lactose metabolism

turned off in presence of glucose– But turned on in absence of glucose and

presence of lactose

Input/Ouput relation: Lactose

+ -

Glucose+ - -- + -

crp

lacI lacZ lacY lacA

Three main components:

1. Genes: encode protein sequence

2. Promoters: RNAP binding sites

3. Operators: Transcription factor binding sites

lacZ

The lac Operon

The Prokaryotic Promoter

NNNNTTGACANNNNNNNNNNNNNNNNNTATAATNNN

-35 -10

17 bp

• The promoter is a binding site for the protein RNA polymerase, responsible for transcription

DNA

Replication

RNA ProteinTranscription Translation

Transcription factor (protein that repress or activate)Rate usually depends on transcription factor

Gene regulation functions(rate of transcription as a function of factors or regulators)

21

12

2

1

21

(

( )

(

,

)

)n

repression n n

n

n

n

activation n n

nn

n

n

KV R A

K RThe ef

V R R A

fect of two regulators

R

RV R A

K R

K R

K

K R

Dynamics: single regulated gene

deg, 1 2

model of protein conc.:

accumulation=- degradation+ synthesis

( , ,...)pp p

A

dCk C AV R R

dt

Single regulated gene(U= protein conc or expression level)

1

.

Steady state: . ( )( )

Piecewise linear model: ( )

0,Steady state:

/ ,

n

n n

n n

s sn n

s

dU KkU A

dt K R

AK AU U H K R

k K R k

dUkU AH K R

dtR K

UA k R K

Graphical analysis of steady states and stability

sU

, repressor concentrationRR K

U

Mutual inhibition network of transcription factors

11 1 1 1 2

22 2 2 2 1

( )

( )

dRk R AH R

dtdR

k R A H Rdt

R1R2

After eliminating the mRNA variables

Nullclines

1R

2R

2 2/A k

2 1R

2R

1

1 1/A k

1 0dR

dt2 0

dR

dt

Phase plane

2R

2 2/A k

2

1

1 1/A k1R

1 1 2 2 1 23 Steady states: ( / ,0), (0, / ), ( , )A k A k

Domain of bistability

For both genes: maximal expression level should be able to repress the repressor

1 22 1

1 2

, .A A

k k

1 1A k

2 2A k

2

1

HW: Sketch the phase planes for the 4 different regimes in the model

Expression of both

R1 off

Mutual repression circuit

Toggling the switch(transients: IPTG affects R1, temp affects R2)

Parametric dependenceof steady states

(GFP- flourescence protein)

Parametric dependenceof steady statesindividual cells