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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
−+
+−
+−=
−−−
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