1

A corrected Problem Set 1has been posted (wrong units , question 3a)

PS1 is due 10/03/06 before 3PM(in class or Rm. 13-2042)

Tomorrow’s recitation topic:‘PS #1 Support’

W 4-5PM Rm. 26-204

2

Review L5: Multistability & introduction λ phage modelλ phage model (Hasty et al.) as example for applyingmass action law.

OR2 OR3

λ λ

λ λ

λλ

OR2 OR3λ λ

OR2 OR3λ λ λ λ

K1

K2

biology

AdkX

nXP2DXtkP2DX

2DX2DX4K2X2DX

*2DX3K

2XD

2DX2K2XD

2X1K2X

→

++→+

→←+

→←+

→←+

→←

math

most importantstep in

modeling !!

most importantstep in

modeling !!

fast

slow

3

AdkX

nXP2DXtkP2DX

2DX2DX4K2X2DX

*2DX3K

2XD

2DX2K2XD

2X1K2X

→

++→+

→←+

→←+

→←+

→←

2K1Krdkγ

rTd0ptnk

α

2K4K

2σ

2K3K

1σ

1γx4x2σ2)x1σ(11

2αxdtdx

=

=

=

=

+−+++

=

mass action

choose elegant (dimensionless, relative) variables !

relative bindingconstants

~ synthesis/basalrate

~ degradation/basalrate

4

graphical stability analysis

5

How to experimentally verify these ideas ?

Synthetic Biology

Build your own designed network ‘from scratch’and test your model

Gardner et al. Construction of a genetic toggleswitch in Escherichia coli. Nature 403, 399 (2000)

Isaacs et al. Prediction and measurement of an autoregulatorygenetic module. PNAS 100, 7714 (2003)

Examples for synthetic genetic switches:

6

Toolbox of the genetic engineer:

1. Restriction enzymes

2. Plasmids

3. PCR (Polymerase Chain Reaction)

4. Fluorescent proteins

First a (very) short intro on ‘Genetic Engineering’

7

1. Restriction Enzymes

8

9

2. Plasmids

10

3. PCR

11

4. Fluorescent proteins

12

13

Isaacs et al. Prediction and measurement of an autoregulatorygenetic module. PNAS 100, 7714 (2003) 14

15 16

17 18

A simple mechanism for persistence: positive feedback

genepromoter

yx

yx

x y

x

yt

x y

genepromoter

yx

x y+

x

yt

19

lacZ lacY lacAPlac

gfpPlac

GFP

LacI

LacYTMG

TMG

Positive feedback in a bacterial regulatory network

x

β

R

y

nT xxR

R)/(1

1

0+=

yRRdt

dyy −

+=

0/11ατ

xydtdx

x −= βτ

yyy

dtdy

n

n

−++

=)()(1

βρβα

maximal activation: α

extracellular TMG: β

repression factor: ρ =(1+RT/R0)-1

)()( ygyfdtdy

−=

20

0.0 0.5 1.0 1.5 2.0

φ (y

)

y

The lac system is bistable

0.0 0.5 1.0 1.5 2.0

-0.2

0.0

0.2

0.4

dy/d

t

y

n = 1

∫ −−=y

dygfy0

')()(φ

gfdtdy

−= yyy

dtdy

n

n

−++

=)()(1

βρβα

21

The lac system is bistable

0.0 0.5 1.0 1.5 2.0

-0.1

0.0

0.1

0.2

dy/d

t

y

0.0 0.5 1.0 1.5 2.0

φ (y

)

y

∫ −−=y

dygfy0

')()(φ

gfdtdy

−=

n = 2

yyy

dtdy

n

n

−++

=)()(1

βρβα

Phase diagram

αβ/ρ

22

0.05 0.10 0.15 0.20

0.2

0.4

0.6

0.0

αβ /

ρ

1/ρ

TMG

~ β TM

G ~ β

Decrease repression

factor

Discontinuous

Network response can be either discontinuous or continuous

Continuous

23

Per

mea

seco

ncen

tratio

n (y

)

Extracellular TMG

Bistability allows memory storage

decrease TMG

increase TMG24

lacZ lacY lacA

gfp

HcRed

HcRed

LacI

Plac

Plac

Pgat

CRP

cAMP TMG

TMGGlu

LacY

LacZ

lactose metabolism

GFP

Experimental protocol:Measure single cell fluorescence histograms (both GFP and HcRed) in ‘steady-state’ as afunction of:

(i) external TMG concentration (continuous variable)(ii) external glucose concentration (continuous variable)(iii) initial condition (binary variable: fully induced versus not induced)

Plac-GFP is integrated in the chromosome; Pgat-HcRed is on a low copy plasmid

25

split singlecolony

> 12 hours liquid culture0 mM TMG(all cells OFF)

> 12 hours liquid culture1 mM TMG(all cells ON)

> 24 hoursliquid culture(~ 24 generations)

0 µM TMG

1 µM TMG2 µM TMG

1 mM TMG

...

0 µM TMG

1 µM TMG2 µM TMG

1 mM TMG

...

Induction protocol, history dependent experiments

Novick and Weiner, PNAS 43, 553 (1957); Cohn and Horibata, J. Bacteriol. 78, 601 (1959)26

2 4 6 8 10 20 40

1

10

100

Gre

en fl

uore

scen

ce

Extracellular TMG (µM)

1

10

100

initial LOW state

initial HIGH state

27

bistable monostablemonostable

2 4 6 8 10 20 40

1

10

100

Gre

en fl

uore

scen

ce

Extracellular TMG (µM)

1

10

100

initial LOW state

initial HIGH state

28

Mapping the bistable region as a function of TMG and glucose concentration

29

0.05 0.10 0.15 0.20

0.2

0.4

0.6

0.0

αβ /

ρ

1/ρ

TMG

TMG

Decrease repression

factor

Discontinuous

Network response can be either discontinuous or continuous

Continuous

30

Turning a binary response intoa graded response

WT +7 lacO +15 lacO

0 100 200 300 4000

5

10

15

20

Gre

en

fluor

esce

nce

Extracellular TMG (µM)

1/ρ = 0.005

1/ρ = 0.16

αβ/ρno hysteresis

31

How to experimentally verify these ideas ?

Synthetic Biology

Build your own designed network ‘from scratch’and test your model

Gardner et al. Construction of a genetic toggleswitch in Escherichia coli. Nature 403, 399 (2000)

Isaacs et al. Prediction and measurement of an autoregulatorygenetic module. PNAS 100, 7714 (2003)

Examples for synthetic genetic switches:

32

Gardner et al. Construction of a genetic toggleswitch in Escherichia coli. Nature 403, 399 (2000)

33

IPTG

atc

34

doesn’t decrease evenafter removal of IPTG

doesn’t increase evenafter removal of atc

35 36

37 38

doesn’t decrease evenafter removal of IPTG

doesn’t increase evenafter removal of atc

39

another example, engineered bistable switch:

vγu12α

dtdv

uβv11α

dtdu

−+

=

−+

=

40

γu12αv

βv11αu

+=

+=

nullclines:

vγu12α

dtdv

uβv11α

dtdu

−+

=

−+

=

41

γu12αv

βv11αu

+=

+=

nullclines:

vγu12α

dtdv

uβv11α

dtdu

−+

=

−+

=

42

γu12αv

βv11αu

+=

+=

nullclines:

vγu12α

dtdv

uβv11α

dtdu

−+

=

−+

=

43

How to obtain this stability diagram ?

Goal of today: - general mathematical frameworkfor stability analysis