Wrapping up E. coli Chemotaxis (L7 & L8)
Main points of last 2 lectures:
L7: Biological background
reduction
what is the function of the individual molecules ?
L8: modeling of all possible chemotactic reactions
why doesn’t this model reproduce experimentally
observed perfect adaptation ?
L8-9: strip down full model to essentials based on
assumptions that are experimentally justified
(or sometimes not)
1
Figure 1A in Mittal, N., E. O. Budrene, M. P. Brenner, and A. Van Oudenaarden.
"Motility of Escherichia coli cells in clusters formed by chemotactic aggregation." Proc Natl Acad Sci U S A.
100, no. 23 (Nov 11, 2003): 13259-63.
Copyright (2003) National Academy of Sciences, U. S. A.
Images removed due to copyright considerations.
3
Absence of chemical attractant
Tumble
Run
Image by MIT OCW.
4
Presence of chemical attractant
Attractant
Tumble
Run
Chemical Gradient Sensed in a Temporal Manner
Image by MIT OCW.
5
Figure 1 of Spiro, P. A., J. S. Parkinson, and H. G. Othmer.
"A model of excitation and adaptation in bacterial chemotaxis."
Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8.
Copyright (1997) National Academy of Sciences, U. S. A.
key player: Tar-CheA-CheW complex
6
te
a
i
ed
m
r
e
int
slow
fast
Figure 2 of Spiro, P. A., J. S. Parkinson, and H. G. Othmer.
"A model of excitation and adaptation in bacterial chemotaxis."
Proc Natl Acad Sci U S A 94, no. 14 (Jul 8, 1997): 7263-8.
Copyright (1997) National Academy of Sciences, U. S. A.
First reduction
T2p
LT2p
keff1 (L)
kpt
T2
LT2
[3]
α≡
[2]+[3]
in steady state:
k
keff4 (L)
1-α
keff3 (L)
T3p
LT3p
kpt
keff2 (L)
keff4 (L)
keff3 (L)
T3
LT3
α
=(1 − α)k
(L)+ αk
(L)
phos
eff1
eff2 8
net phosphorylated rate
perfect adaptation for small L
safe zone
perfect adaptation for large L
fine-tune: net phoshorylation rate and keff1 and keff2 so that α falls in safe9zone
net phosphorylated rate
no safe zone
never perfect adaptation
10
Second reduction
T2p
LT2p
keff1(L)
keff3(L)
kpt
T2
LT2
keff4(L)
T3p
LT3p
keff2(L)
keff3(L)
kpt
T3
LT3
additional assumption:
- CheB only demetylates phosphorylated receptors
experimental backup:
- not possible to directly measure if CheB demethylates
only active receptors
- rate of methylation drops immediately after addition
of ligand indicates that CheB works on active receptors
11
Third reduction
rin
keff4
additional assumption:
- [CheR]<< [receptors],
methylation operates at saturation
(rin is independent
of receptor concentrations)
T3p
LT3p
keff2(L)
rin
experimental backup:
- Michaelis constant of CheR binding << [receptors]
so Rtot ~ Rbnd
kpt
T3
LT3
ok
12
Fourth reduction
rin
keff4
additional assumption:
- demethylation is identical for
bound and unbound receptors, so
keff4 is independent of L.
T3p
LT3p
keff2(L)
rin
experimental backup:
- kinetics of demethylation almost independent of
level of methylation and ligand binding.
kpt
T3
LT3
ok
13
rin
keff4
This final module obeys
perfect adaptation for any
value of L.
2r
in
[3 ]=
p k
eff4
T3p
LT3p
keff2(L)
rin
kpt
T3
LT3
14
Coming lectures:
Biological oscillators
Biological relevance:
Cell division cycle
Circadian Rhythms
etc.
What do you need to make an oscillator ?
How is an oscillator different from the systems we
already discussed (switches, chemotactic network) ?
⇒ Graphical way to represent differential equations.
15
Refresh: Autocatalysis (Problem set #1).
A + X ←⎯⎯→ 2 X
k −1, k +1
x& = k1ax − k −1 x
This equation is of the type:
2
x=[X]
a=[A] = constant
(e.g. enormous surplus)
x& = f (x)
first order differential equation
Analysis recipe:
1. determine fixed points
2. stability analysis
16
x1* = 0
Fixed points: f(x*) = 0:
k1a
x =
k −1
*
2
x& > 0
x& < 0
2 fixed points: one stable and one unstable
17
[X]
Guestimate of dynamics for
different initial conditions
time
18
more quantitative stability analysis:
small perturbation from fixed point:
η (t ) = x(t ) − x*
f ( x* + η ) = f ( x* ) + ηf ' ( x* ) + O(η 2 ) ≈ ηf ' ( x* )
η& ≈ ηf ' ( x* )
η (t ) ~ exp( f ' ( x* )t )
f ' ( x* ) > 0
unstable fixed point
f ' ( x* ) < 0
stable fixed point
19
‘unstable’
f '(x ) > 0
*
‘stable’
f ' ( x* ) < 0
20
Other example:
rin
C& * = (− k pt − keff 4 )C * + keff 2C + rin
*
&
C = k pt C − keff 2C + rin
x& = ax + by + rin
y& = cx + dy + rin
x ≡ C*
y≡C
keff4
T3p
LT3p
keff2(L)
rin
T3
LT3
C*
kpt
C
a ≡ (−k pt − keff 4 )
b ≡ keff 2
c ≡ k pt
d ≡ −b
21
Nullclines:
− rin a
− rin k pt + keff 4
− x=
+
x
y=
keff 2
keff 2
b
b
x& = 0
y& = 0
k pt
− rin c
rin
− x=
+
x
y=
keff 2 keff 2
d
d
y& = 0
y
fixed point (stable or unstable ?)
x& = 0
x
22
x& < 0, y& < 0
y
x& > 0, y& < 0
y& = 0
*
*
(x , y )
x& > 0, y& > 0
x& < 0, y& > 0
x& = 0
x
x& = −(k pt + keff 4 ) x + keff 2 y + rin
y& = k pt x − keff 2 y + rin
23
y
y& = 0
⎛ 2rin rin keff 4 + 2rin k pt
(x , y ) = ⎜
,
⎜k
⎝ eff 4 keff 4 keff 2 ( L)
*
x& = 0
*
x
x& = −(k pt + keff 4 ) x + keff 2 y + rin
y& = k pt x − keff 2 y + rin
24
⎞
⎟
⎟
⎠
increased ligand concentration
y
CTOT
y& = 0
x + y = CTOT
CTOT
x& = 0
x
⎛ 2rin rin keff 4 + 2rin k pt
(x , y ) = ⎜
,
⎜k
⎝ eff 4 keff 4 keff 2 ( L)25
*
*
⎞
⎟
⎟
⎠
Guestimated response
C*
‘activity’
(phosphorylation
level)
time
CTOT
(methylation level)
time
26
Oscillators ?
x& = − x + ay + x 2 y
y& = b − ay − x y
2
nullclines:
model for glycolysis
x
y=
a + x2
b
y=
a + x2
27
y& = 0
y
x& > 0
y& > 0
x& < 0
y& > 0
x& > 0
y& < 0
x& = 0
x& < 0
y& < 0
x
28
y& = 0
y
x& > 0
y& > 0
x& < 0
y& > 0
x& > 0
y& < 0
x& = 0
x& < 0
y& < 0
x
29
limitcycle
y
x
x
time
30
Dynamical response of switches, chemotactic network and oscillators
‘switch’
adaptation
(differentiator,
at least for small
frequencies)
oscillator
31
Dynamical response of switches,
chemotactic network and oscillators
two stable
fixed points
one stable
fixed point
unstable
fixed point
32
nullclines:
α
u= 1
1 + vβ
α
2
v=
1 + uγ
Image removed due to copyright considerations.
α
du
1 −u
=
dt 1 + vβ
α
dv
2 −v
=
dt 1 + uγ
33
Adaptation (one stable fixed point)
y& = 0
y
sfp
⎛ 2rin rin keff 4 + 2rin k pt
(x , y ) = ⎜
,
⎜k
⎝ eff 4 keff 4 keff 2 ( L)
*
x& = 0
*
x
x& = −(k pt + keff 4 ) x + keff 2 y + rin
y& = k pt x − keff 2 y + rin
34
⎞
⎟
⎟
⎠
increased ligand concentration
y
CTOT
y& = 0
x + y = CTOT
CTOT
x& = 0
x
⎛ 2rin rin keff 4 + 2rin k pt
(x , y ) = ⎜
,
⎜k
⎝ eff 4 keff 4 keff 2 ( L)35
*
*
⎞
⎟
⎟
⎠
Oscillator (unstable fixed point)
y& = 0
y
x& > 0
y& < 0
x& > 0
y& > 0
x& = 0
ufp
x& < 0
y& > 0
x& < 0
y& < 0
x
36
y& = 0
y
x& > 0
y& > 0
x& < 0
y& > 0
x& > 0
y& < 0
x& = 0
x& < 0
y& < 0
x
37
38
Image removed due to copyright considerations. See Figures 1, 2, 3 in Elowitz, M. B., and S. Leibler.
"A synthetic oscillatory network of transcriptional regulators." Nature 403, no. 6767 (Jan 20, 2000): 335-8.