1

! PS

#3 d

ue to

day

!(h

ere,

in c

lass

or R

m. 1

3-20

42 b

efor

e 3P

M)

2

222

2

22

xiD

iak

tixa

Da

iak

rta

ii

i

aa

aa

∂∂+

−=

∂∂∂∂

+−

+=

∂∂

γ

γ

Rev

iew

Turin

g-G

iere

r-M

einh

ardt

mod

els

Loca

l exc

itatio

n, g

loba

l inh

ibiti

on

a:co

ncen

tratio

n ac

tivat

ori:

conc

entra

tion

inhi

bito

rt:

time

x:

posi

tion

r a:

basa

l act

ivat

or s

ynth

esis

rate

k a, k

i: ra

te c

onst

ant f

or s

ynth

esis

γ a,γ

i :

deca

y ra

tes

Da,

Di:

diffu

sion

con

stan

ts

varia

bles

cons

tant

s(p

aram

eter

s)

3

222

2

22

xiD

iak

tixa

Da

iak

rta

ii

i

aa

aa

∂∂+

−=

∂∂∂∂

+−

+=

∂∂

γ

γ

()

2

22

2

22

1

sIP

IA

QτI

sAA

IAR

τA

∂∂+

−=

∂∂∂∂

+−

+=

∂∂

choo

sedi

men

sion

less

varia

ble

norm

aliz

e4

varia

bles

2 )1(

1 +=

+=

RI

RA

hom

ogen

eous

solu

tion

0/

/=

∂∂

=∂

∂t

s

only

one

fixe

dpo

int,

sinc

e bo

thA

and

I >0

4

A I

s s

A I

hom

ogen

eous

solu

tion

0/

/=

∂∂

=∂

∂t

s

5

stab

ility

of h

omog

eneo

us s

olut

ion

−+

+−

+−=

−−−

QQ

RRR

RR

QQA

IARIAR

)1(2

)1(

11

2

12

22

2

trace

< 0

det >

0

011 >

<+−

Q

QRR

),

(')

,(

),

(')

,(

ττ

ττ

sI

Is

Is

AA

sA

+=

+=

inho

mog

eneo

usso

lutio

n:

or in

gen

eral

real

par

t of e

igen

valu

es >

0

6

A I

s s

A I

inho

mog

eneo

usso

lutio

n

I’(s,τ)

),

(')

,(

),

(')

,(

ττ

ττ

sI

Is

Is

AA

sA

+=

+=

7

2

2

2

2

2

''

')

1(2

'

''

)1(

'11

'

sIP

QI

AR

QI

sAI

RRA

RRA

∂∂+

−+

=∂∂

∂∂+

+−

+−=

∂∂ ττ

)co

s()

(ˆ)

,('

)co

s()

(ˆ)

,('

ll sI

sI

sA

sA

ττ

ττ

==

),

(')

,(

),

(')

,(

ττ

ττ

sI

Is

Is

AA

sA

+=

+=

trial

sol

utio

n:

8

A I

s s

A II’(

s,τ)

),

(')

,(

),

(')

,(

ττ

ττ

sI

Is

Is

AA

sA

+=

+=

)co

s()

(ˆ)

,('

)co

s()

(ˆ)

,('

ll sI

sI

sA

sA

ττ

ττ

==

9

IP

QA

RQ

dId

IRR

ARR

dAd

ˆˆ )

1(2

ˆ

ˆ)

1(ˆ

111

ˆ

2

22

+−

+=

+−

−+−

=

l

l

ττ

)co

s()

(ˆ)

,('

)co

s()

(ˆ)

,('

ll sI

sI

sA

sA

ττ

ττ

==

01

11

012

111

22

22

<

−+−

−+

>+

+

+

−

+−−

ll

ll RR

PQ

RQR

PQ

RRst

abili

tyin

hom

ogen

eous

solu

tion

11+−

>RR

PQ

10

hom

ogen

eous

sta

bilit

y:

stab

ility

aga

inst

spa

tial d

istru

banc

e:

11+−

>RR

Q

11+−

>RR

PQ

I

s

II’(

s,τ)

if P

< 1

(Di<

Da)

, sys

tem

s is

alw

ays

stab

le, a

gain

st a

nype

rturb

atio

n bo

th s

patia

l and

tem

pora

l

11

I

s

I

hom

ogen

eous

ly s

tabl

e:

I re

laxe

s ba

ck to

prev

ious

val

ue a

fter

smal

l uni

form

dis

turb

ance

I

s

I

stab

le a

gain

st s

patia

ldi

stur

banc

e:

I’ re

laxe

s ba

ck to

afte

r sm

all s

patia

ldi

stur

banc

e

I

12

Topi

c I:

Syst

ems

Cel

l Bio

logy

Spat

ial o

scill

atio

n in

E. c

oli

sim

ilar t

o ge

netic

osc

illat

ors,

but

now

we

cann

ot ig

nore

the

spat

ial d

imen

sion

s

biol

ogic

al fu

nctio

n:

dete

rmin

e th

e ce

nter

of t

he c

ell,

to p

repa

re fo

r pro

per c

ell d

ivis

ion

13

Intro

duci

ng th

e m

olec

ules

:

- Fts

Z fu

nctio

n: A

ssem

bly

of a

pol

ymer

ic ri

ng o

f the

tub

ulin

-like

GTP

ase

FtsZ

(Z ri

ng).

The

Z-rin

g is

loca

lized

to th

e ce

nter

by

the

actio

ns o

fth

e M

inC

, Min

D, a

nd M

inE

prot

eins

.

- Min

C in

hibi

ts th

e in

itiat

ion

of th

e Z

ring.

Min

C c

oloc

aliz

es w

ith M

inD

. I

n w

ild-ty

pe (W

T) c

ells

, Min

C/D

form

s a

pola

r pat

tern

tha

t osc

illat

es b

etw

een

the

pole

s, k

eepi

ng th

e ce

nter

fre

e fo

r ini

tiatio

n of

cel

l div

isio

n.

Thus

, virt

ually

all

of M

inC

/D d

ynam

ical

ly a

ssem

bles

on

the

mem

bran

e in

the

shap

e of

a te

st tu

be c

over

ing

the

mem

bran

efro

m o

ne p

ole

up to

app

roxi

mat

ely

mid

cell.

14

Mos

t of M

inE

accu

mul

ates

at t

he ri

m o

f thi

s tu

be, i

n th

e sh

ape

of a

ring

(the

E ri

ng).

The

rim o

f the

Min

C/D

tube

and

asso

ciat

ed E

ring

mov

e fro

m a

cen

tral p

ositi

on to

the

cell

pole

unt

il bo

th th

e tu

be a

nd ri

ng v

anis

h. M

eanw

hile

, a n

ewM

inC

/D tu

be a

nd a

ssoc

iate

d E

ring

form

in th

e op

posi

te c

ell

half,

and

the

proc

ess

repe

ats,

resu

lting

in a

pol

e-to

-pol

eos

cilla

tion

cycl

e of

the

divi

sion

inhi

bito

r.A

full

cycl

e ta

kes

abou

t 50

s.

min

Em

inC

/D

15gf

p-m

inC

16

GFP

-min

D

17gf

p-m

inE

is lo

caliz

edin

a ri

ng

min

Em

inD

18

gfp-

min

E

19

FtsZ

is n

eces

sary

for f

orm

ing

the

sept

um

In F

tsZ-

cel

ls,

gfp-

Min

D a

lso

osci

llate

s

20

2122

How

doe

s th

is w

ork

?

mod

elin

g ef

forts

:

• Mei

nhar

dt a

nd d

e B

oer,

PN

AS

98,

142

02 (2

001)

;

• How

ard

et a

l., P

hys.

Rev

. Let

. 87,

278

102

(200

1);

• Kru

se, B

ioph

ys. J

. 82,

618

(200

2);

• Hua

ng, M

eir,

and

Win

gree

n, P

NA

S 1

00, 1

2724

(200

3).

23

Min

C

FtsZ

Min

D

Min

E

Sum

mar

y of

mai

n fu

nctio

ns o

f pro

tein

s:

poly

mer

izes

in a

con

tract

ile Z

-rin

gth

at in

itiat

es s

eptu

m fo

rmat

ion

inhi

bits

form

atio

n of

Z-r

ing

mem

bran

e as

soci

ated

pro

tein

that

recr

uits

min

C a

nd m

inE

to m

embr

ane

ejec

ts m

inC

/min

D fr

om m

embr

ane

into

cyto

plas

m24

How

ard

et a

l. m

odel

(PR

L)

min

d

min

D

cyto

plas

m

mem

bran

e

e

D ρσρ

σ' 1

1

1+e

dρ

ρσ

2

min

e

min

E

D

e ρσρσ

' 4

4

1+E

Dρ

ρσ

3

in w

ords

:

- firs

t ord

er re

actio

ns f

or o

wn

spec

ies

- e in

hibi

ts m

embr

ane

ass

ocia

tion

of D

(MM

)- e

enh

ance

s m

embr

ane

dis

soci

atio

n of

d (

linea

r)- D

enh

ance

s m

embr

ane

ass

ocia

tion

of E

(re

crui

tmen

t, lin

ear)

- D in

hibi

ts m

embr

ane

dis

soci

atio

n of

E (M

M)

- d a

nd e

do

not d

iffus

e- D

and

E d

iffus

e

25

How

ard

et a

l. m

odel

(PR

L)

min

d

min

D

cyto

plas

m

mem

bran

e

e

D ρσρ

σ' 1

1

1+e

dρ

ρσ

2

min

e

min

E

D

e ρσρσ

' 4

4

1+E

Dρ

ρσ

3

asso

ciat

ion

of c

ytop

lasm

icm

inD

with

mem

bran

e is

inhi

bite

d by

min

e in

mem

bran

eM

M ta

kes

care

of s

ingu

larit

yas

min

E g

oes

to z

ero.

biol

ogic

al in

terp

reta

tion:

min

e in

mem

bran

e sp

atia

llybl

ocks

mem

bran

e fo

r min

Dsi

mila

r to

min

C b

lock

ing

FtZ

asso

ciat

ion

with

mem

bran

e

26

How

ard

et a

l. m

odel

(PR

L)

min

d

min

D

cyto

plas

m

mem

bran

e

e

D ρσρ

σ' 1

1

1+e

dρ

ρσ

2

min

e

min

E

D

e ρσρσ

' 4

4

1+E

Dρ

ρσ

3

diss

ocia

tion

of m

embr

ane

min

d is

stim

ulat

ed b

y m

ine

in m

embr

ane,

afte

r min

d is

ej

ecte

d m

ine

stay

s in

mem

bran

e

biol

ogic

al in

terp

reta

tion:

bind

ing

of m

ine

to m

ind

low

ers

affin

ity o

f min

d w

ith m

embr

ane

but m

embr

ane

affin

ity o

f min

ere

mai

ns u

ncha

nged

27

How

ard

et a

l. m

odel

(PR

L)

min

d

min

D

cyto

plas

m

mem

bran

e

e

D ρσρ

σ' 1

1

1+e

dρ

ρσ

2

min

e

min

E

D

e ρσρσ

' 4

4

1+E

Dρ

ρσ

3

diss

ocia

tion

of m

embr

ane

min

e is

inhi

bite

d by

min

Din

cyt

opla

smM

M ta

kes

care

of s

ingu

larit

y

biol

ogic

al in

terp

reta

tion:

?

28

How

ard

et a

l. m

odel

(PR

L)

min

d

min

D

cyto

plas

m

mem

bran

e

e

D ρσρ

σ' 1

1

1+e

dρ

ρσ

2

min

e

min

E

D

e ρσρσ

' 4

4

1+E

Dρ

ρσ

3

asso

ciat

ion

of c

ytop

lasm

icm

inE

with

mem

bran

e is

stim

ulat

ed b

y m

inD

in c

ytop

lasm

afte

r del

iver

y of

min

E to

the

mem

bran

e, m

inD

div

es b

ack

in th

e cy

topl

asm

biol

ogic

al in

terp

reta

tion:

min

D-m

inE

com

plex

has

hig

haf

finity

to m

embr

ane

sinc

e th

e di

ffusi

on o

f thi

s co

mpl

exdo

esn’

t app

ear i

n th

e m

odel

itsh

ould

be

very

fast

.

29

D

eE

De

D

eE

DE

EE

de

e

Dd

de

e

DD

DD t

xD

tt

xD

t

ρσρσ

ρρ

σρ

ρσρσ

ρρ

σρ

ρ

ρρ

σρ

σρσ

ρ

ρρ

σρ

σρσ

ρρ

' 4

43

' 4

43

2

2

2' 1

1

2' 1

12

2

1

1

1

1 +−

=∂∂

++

−∂∂

=∂∂

−+

=∂∂

++

−∂∂

=∂∂

syst

em o

f equ

atio

ns:

30

stab

ility

ana

lysi

s

1. fi

nd fi

xed

poin

t

(e.g

. num

eric

ally

:

how

_hom

og.m

) di

ffere

nt ra

ndom

initi

al c

ondi

tions

rela

x to

sam

e fix

ed p

oint re

sult:

one

fixe

d po

int:

d =

1383

e =

82D

= 1

17E

= 3

00

=∂∂

=∂∂ xt

31

2. fi

nd s

tabi

lity

mat

rix (J

acob

ian)

+−

++

+

+−

−+

−

−+

−−

+

++

+−

=

DD

ED

eD

DE

De

de

De

e

de

De

e

A

' 443

32

' 4

' 44

' 443

32

' 4

' 44

22

' 1

' 11

2' 1

1

22

' 1

' 11

2' 11

10

)1(

10

)1(

)1(

01

)1(

01

σσσ

σσσ

σσσ

σσ

σσ

σ

σσσ

σσ

σσ

σσσ

σσ

σσ

32

3. te

st s

tabi

lity

of fl

uctu

atio

ns a

roun

d ho

mog

eneo

us s

olut

ion

)co

s()

(ˆ)

,(

)co

s()

(ˆ)

,(

)co

s()

(ˆ)

,(

)co

s()

(ˆ)

,(

qxt

dt

xd

qxt

Dt

xD

qxt

et

xe

qxt

Et

xE

====

δδδδ

D

x

δD(x

,t)

33

+−

++

+

+−

−−

+−

−+

−−

+

++

−+−

=

DD

ED

eD

qD

DE

De

de

De

e

de

De

qD

e

AE

D

' 443

32

' 4

' 44

' 442

33

2' 4

' 44

22

' 1

' 11

2' 1

1

22

' 1

' 11

22

' 11

10

)1(

10

)1(

)1(

01

)1(

01

ˆ

σσσ

σσσ

σσσ

σσ

σσ

σ

σσσ

σσ

σσ

σσσ

σσ

σσ

3. te

st s

tabi

lity

of fl

uctu

atio

ns a

roun

d ho

mog

eneo

us s

olut

ion

34

4. -

dete

rmin

e ei

genv

alue

s of

sta

bilit

y m

atrix

,

- fin

d re

al p

art o

f eig

enva

lues

,

- pl

ot th

e la

rges

t as

a fu

nctio

n of

q.

(e.g

. how

_eig

.m)

q =

1.5

(µm

)-1λ

= 2π

/q =

4.2

µm

q =

2.3

(µm

)-1λ

= 2π

/q =

2.7

µm

q

Max(Real(Eigenvalues)) 1/s

35

How

ard

et a

l.: R

esul

ts

min

E

t

min

D x36

Hua

ng, M

eir,

and

Win

gree

n, P

NA

S 1

00, 1

2724

(200

3).

mai

n di

ffere

nces

:

- ATP

cyc

le- 1

D v

ersu

s 3D

(pro

ject

ed o

n 2D

)

37

ρ d:

mem

bran

e bo

und

min

D:A

TP c

ompl

exes

ρ de:

mem

bran

e bo

und

min

D:m

inE

:ATP

com

plex

esρ D

:AD

P:co

ncen

tratio

n cy

topl

asm

ic m

inD

bou

nd to

AD

Pρ D

:ATP

:con

cent

ratio

n cy

topl

asm

ic m

inD

bou

nd to

ATP

ρ E :

conc

entra

tion

cyto

plas

mic

min

E

only

min

D-A

TP c

an a

ssoc

iate

with

mem

bran

em

inE

onl

y bi

nds

min

D-A

TP o

ligom

ers

in m

embr

ane

only

min

D-m

inE

-ATP

com

plex

can

dis

soci

ate

from

mem

bran

e38

Ed

Ee

deE

EE

ρρ

σρ

σdxρ

dD

dtdρ−

+=

2

2

dede

D:ADP

ATP

ADP

DADP

DD

D:ADP

ρσ

ρσ

dxρd

Ddt

dρ+

−=

→2:

2

()

[]

ATP

Dde

ddD

DD:ADP

ATP

ADP

DATP

DD

D:ATP

ρρ

σdxρ

dD

dtdρ

:2:

2

ρρ

σσ

++

−+

=→ (

)[

]ATP

Dde

ddD

DE

dE

dρ

dtdρ:

ρρ

σσ

ρρ

σ+

++

−=

Ed

Ede

dede

ρdtdρ

ρσ

ρσ

+−

=

Rea

ctio

n 1:

min

D-A

TP b

inds

bot

h lin

early

and

auto

cata

lytic

ally

to m

inD

-ATP

in m

embr

ane

min

D fo

rms

poly

mer

s in

mem

bran

e

39

Ed

Ee

deE

EE

ρρ

σρ

σdxρ

dD

dtdρ−

+=

2

2

dede

D:ADP

ATP

ADP

DADP

DD

D:ADP

ρσ

ρσ

dxρd

Ddt

dρ+

−=

→2:

2

()

[]

ATP

Dde

ddD

DD:ADP

ATP

ADP

DATP

DD

D:ATP

ρρ

σdxρ

dD

dtdρ

:2:

2

ρρ

σσ

++

−+

=→ (

)[

]ATP

Dde

ddD

DE

dE

dρ

dtdρ:

ρρ

σσ

ρρ

σ+

++

−=

Ed

Ede

dede

ρdtdρ

ρσ

ρσ

+−

=

Rea

ctio

n 2:

min

E b

inds

min

D-A

TP in

mem

bran

e~

[min

E]*

[min

d]

40

Ed

Ede

deE

EE

ρρ

σρ

σdxρ

dD

dtdρ−

+=

2

2

dede

D:ADP

ATP

ADP

DADP

DD

D:ADP

ρσ

ρσ

dxρd

Ddt

dρ+

−=

→2:

2

()

[]

ATP

Dde

ddD

DD:ADP

ATP

ADP

DATP

DD

D:ATP

ρρ

σdxρ

dD

dtdρ

:2:

2

ρρ

σσ

++

−+

=→ (

)[

]ATP

Dde

ddD

DE

dE

dρ

dtdρ:

ρρ

σσ

ρρ

σ+

++

−=

Ed

Ede

dede

ρdtdρ

ρσ

ρσ

+−

=

Rea

ctio

n 3:

min

D-m

inE

-ATP

com

plex

dis

asso

ciat

esfro

m m

embr

ane

hydr

olyz

ing

ATP

~ [m

ine]

41

Ed

Ede

deE

EE

ρρ

σρ

σdxρ

dD

dtdρ−

+=

2

2

dede

D:ADP

ATP

ADP

DADP

DD

D:ADP

ρσ

ρσ

dxρd

Ddt

dρ+

−=

→2:

2

()

[]

ATP

Dde

ddD

DD:ADP

ATP

ADP

DATP

DD

D:ATP

ρρ

σdxρ

dD

dtdρ

:2:

2

ρρ

σσ

++

−+

=→ (

)[

]ATP

Dde

ddD

DE

dE

dρ

dtdρ:

ρρ

σσ

ρρ

σ+

++

−=

Ed

Ede

dede

ρdtdρ

ρσ

ρσ

+−

=

Rea

ctio

n 4:

char

ging

of m

inD

in c

ytop

lasm

from

AD

P to

ATP

bou

nd

42

()

ee

ATP

Dd

dD

ds

dtdρ

σρ

ρσ

ρ−

+=

:

()

DA

ATP

Dd

dD

ATP

DD

ATP

Ds

dxd

dtd

ρσ

ρρ

σρ

ρ+

+−

=:

2:2

:D (

)e

eE

ed

dEe

dtdρ

σρ

ρρ

σρ

−−

=

()

ee

Ee

ddE

EE

E

dxddtd

ρσ

ρρ

ρσ

ρρ

+−

−=

2

2

D

ee

ADP

DP

ADP

DD

ADP

D

dxd

dtd

ρσ

ρσ

ρρ

+−

=:

2:2

:D

ADP

DP

DA

DD

D

dxddtd

:2

2

ρσ

ρσ

ρρ

+−

=D