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Transcript of variables constants (parameters)web.mit.edu/biophysics/sbio/PDFs/L17_notes.pdf · PDF...
1
! P S
#3 d
ue to
da y
! (h
er e,
in c
la ss
o r R
m . 1
3- 20
42 b
ef or
e 3P
M )
2
22 2
2
2 2
xi D
i a k
ti xa
D a
ia k
r ta
i i
i
a a
a a
∂∂ +
− =
∂∂ ∂∂
+ −
+ =
∂∂
γ
γ
R ev
ie w
Tu rin
g- G
ie re
r- M
ei nh
ar dt
m od
el s
Lo ca
l e xc
ita tio
n, g
lo ba
l i nh
ib iti
on
a: co
nc en
tra tio
n ac
tiv at
or i:
co nc
en tra
tio n
in hi
bi to
r t:
tim e
x:
po si
tio n
r a :
ba sa
l a ct
iv at
or s
yn th
es is
ra te
k a , k
i: ra
te c
on st
an t f
or s
yn th
es is
γ a ,γ
i :
de ca
y ra
te s
D a,
D i:
di ffu
si on
c on
st an
ts
va ria
bl es
co ns
ta nt
s (p
ar am
et er
s)
3
22 2
2
2 2
xi D
i a k
ti xa
D a
ia k
r ta
i i
i
a a
a a
∂∂ +
− =
∂∂ ∂∂
+ −
+ =
∂∂
γ
γ
( )
2
2 2
2
2 2
1
sI P
I A
Q τI
sA A
IA R
τA
∂∂ +
− =
∂∂ ∂∂
+ −
+ =
∂∂
ch oo
se di
m en
si on
le ss
va ria
bl e
no rm
al iz
e 4
va ria
bl es
2 )1 (
1 + =
+ =
R I
R A
ho m
og en
eo us
so lu
tio n
0 /
/ =
∂ ∂
= ∂
∂ t
s
on ly
o ne
fi xe
d po
in t,
si nc
e bo
th A
a nd
I >0
4
A I
s s
A I
ho m
og en
eo us
so lu
tio n
0 /
/ =
∂ ∂
= ∂
∂ t
s
5
st ab
ili ty
o f h
om og
en eo
us s
ol ut
io n
− +
+ −
+− =
−− −
Q Q
R R R
RR
Q Q A
IAR IAR
)1 (2
)1 (
11
2
1 2
2 2
2
tra ce
< 0
de t >
0
011 >
< +−
Q
Q RR
) ,
(' )
, (
) ,
(' )
, (
τ τ
τ τ
s I
I s
I s
A A
s A
+ =
+ =
in ho
m og
en eo
us so
lu tio
n:
or in
g en
er al
re al
p ar
t o f e
ig en
va lu
es >
0
6
A I
s s
A I
in ho
m og
en eo
us so
lu tio
n
I’( s, τ)
) ,
(' )
, (
) ,
(' )
, (
τ τ
τ τ
s I
I s
I s
A A
s A
+ =
+ =
7
2
2
2
2
2
' '
' )
1( 2
'
' '
) 1(
' 11
'
sI P
Q I
A R
Q I
sA I
RR A
RR A
∂∂ +
− +
= ∂∂
∂∂ +
+ −
+− =
∂∂ ττ
) co
s( )
(ˆ )
, ('
) co
s( )
(ˆ )
, ('
ll s I
s I
s A
s A
τ τ
τ τ
==
) ,
(' )
, (
) ,
(' )
, (
τ τ
τ τ
s I
I s
I s
A A
s A
+ =
+ =
tri al
s ol
ut io
n:
8
A I
s s
A I I’(
s, τ)
) ,
(' )
, (
) ,
(' )
, (
τ τ
τ τ
s I
I s
I s
A A
s A
+ =
+ =
) co
s( )
(ˆ )
, ('
) co
s( )
(ˆ )
, ('
ll s I
s I
s A
s A
τ τ
τ τ
==
9
I P
Q A
R Q
dId
I RR
A RR
dAd
ˆ ˆ )
1( 2
ˆ
ˆ )
1( ˆ
1 11
ˆ
2
2 2
+ −
+ =
+ −
− +−
=
l
l
ττ
) co
s( )
(ˆ )
, ('
) co
s( )
(ˆ )
, ('
ll s I
s I
s A
s A
τ τ
τ τ
==
0 1
11
0 12
1 11
2 2
2 2
<
− +−
− +
> +
+
+
−
+− −
l l
l l RR
P Q
R Q R
P Q
RR st
ab ili
ty in
ho m
og en
eo us
so lu
tio n
11 +−
> RR
PQ
10
ho m
og en
eo us
s ta
bi lit
y:
st ab
ili ty
a ga
in st
s pa
tia l d
is tru
ba nc
e:
11 +−
> RR
Q
11 +−
> RR
PQ
I
s
I I’(
s, τ)
if P
< 1
(D i<
D a)
, s ys
te m
s is
a lw
ay s
st ab
le , a
ga in
st a
ny pe
rtu rb
at io
n bo
th s
pa tia
l a nd
te m
po ra
l
11
I
s
I
ho m
og en
eo us
ly s
ta bl
e:
I re
la xe
s ba
ck to
pr ev
io us
v al
ue a
fte r
sm al
l u ni
fo rm
d is
tu rb
an ce
I
s
I
st ab
le a
ga in
st s
pa tia
l di
st ur
ba nc
e:
I’ re
la xe
s ba
ck to
af te
r s m
al l s
pa tia
l di
st ur
ba nc
e
I
12
To pi
c I:
Sy st
em s
C el
l B io
lo gy
Sp at
ia l o
sc ill
at io
n in
E . c
ol i
si m
ila r t
o ge
ne tic
o sc
ill at
or s,
b ut
n ow
w e
ca nn
ot ig
no re
th e
sp at
ia l d
im en
si on
s
bi ol
og ic
al fu
nc tio
n:
de te
rm in
e th
e ce
nt er
o f t
he c
el l,
to p
re pa
re fo
r p ro
pe r c
el l d
iv is
io n
13
In tro
du ci
ng th
e m
ol ec
ul es
:
- F ts
Z fu
nc tio
n: A
ss em
bl y
of a
p ol
ym er
ic ri
ng o
f t he
t ub
ul in
-li ke
G TP
as e
Ft sZ
(Z ri
ng ).
Th e
Z- rin
g is
lo ca
liz ed
to th
e ce
nt er
b y
th e
ac tio
ns o
f th
e M
in C
, M in
D , a
nd M
in E
pr ot
ei ns
.
- M in
C in
hi bi
ts th
e in
iti at
io n
of th
e Z
rin g.
M in
C c
ol oc
al iz
es w
ith M
in D
. I
n w
ild -ty
pe (W
T) c
el ls
, M in
C /D
fo rm
s a
po la
r p at
te rn
t ha
t o sc
ill at
es b
et w
ee n
th e
po le
s, k
ee pi
ng th
e ce
nt er
f re
e fo
r i ni
tia tio
n of
c el
l d iv
is io
n.
Th us
, v irt
ua lly
a ll
of M
in C
