Imagine the
Possibilities
Mathematical Biology
University of Utah
Length Regulation of Flagellar Hooks and
Filaments in Salmonella
J. P. Keener
Department of Mathematics
University of Utah
Length Regulation of Flagellar Hooks – p.1/36
Imagine the
Possibilities
Mathematical Biology
University of Utah
Introduction
The first of many slides – p.2/36
Imagine the
Possibilities
Control of Flagellar Growth
Mathematical Biology
University of Utah
The motor is built in a precise step-by-step fashion.
• Step 1: Basal Body
• Step 2: Hook (FlgE secretion)
• Step 3: Filament (FliC secretion)
Basal Body
Length Regulation of Flagellar Hooks – p.3/36
Imagine the
Possibilities
Control of Flagellar Growth
Mathematical Biology
University of Utah
The motor is built in a precise step-by-step fashion.
• Step 1: Basal Body
• Step 2: Hook (FlgE secretion)
FlgE
Hook
• Step 3: Filament (FliC secretion)
Basal Body
Length Regulation of Flagellar Hooks – p.3/36
Imagine the
Possibilities
Control of Flagellar Growth
Mathematical Biology
University of Utah
The motor is built in a precise step-by-step fashion.
FliC
• Step 1: Basal Body
Filament
Hook−filament
junction
• Step 2: Hook (FlgE secretion)
Hook
• Step 3: Filament (FliC secretion)
Basal Body
Length Regulation of Flagellar Hooks – p.3/36
Imagine the
Possibilities
Control of Flagellar Growth
Mathematical Biology
University of Utah
The motor is built in a precise step-by-step fashion.
FliC
• Step 1: Basal Body
Filament
Hook−filament
junction
• Step 2: Hook (FlgE secretion)
Hook
• Step 3: Filament (FliC secretion)
Basal Body
• How are the switches between steps coordinated?
• How is the hook length regulated (55 ±6 nm)?
• How is the length of the filament "measured"?
Length Regulation of Flagellar Hooks – p.3/36
Imagine the
Possibilities
Mathematical Biology
Proteins of Flagellar Assembly
University of Utah
Length Regulation of Flagellar Hooks – p.4/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
• FliK is the "hook length regulatory" protein.
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
• FliK is the "hook length regulatory" protein.
• FliK is secreted only during hook production.
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
• FliK is the "hook length regulatory" protein.
• FliK is secreted only during hook production.
• Mutants of FliK produce long hooks; overproduction of
FliK gives shorter hooks.
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
• FliK is the "hook length regulatory" protein.
• FliK is secreted only during hook production.
• Mutants of FliK produce long hooks; overproduction of
FliK gives shorter hooks.
• Lengthening FliK gives longer hooks.
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Mathematical Biology
Hook Length Regulation
University of Utah
• Hook is built by FlgE secretion.
• FliK is the "hook length regulatory" protein.
• FliK is secreted only during hook production.
• Mutants of FliK produce long hooks; overproduction of
FliK gives shorter hooks.
• Lengthening FliK gives longer hooks.
• 5-10 molecules of FliK are secreted per hook (115-120
molecules of FlgE).
Length Regulation of Flagellar Hooks – p.5/36
Imagine the
Possibilities
Hook Length Data
Mathematical Biology
250
250
200
200
200
150
100
50
0
0
Number of hooks
250
Number of hooks
Number of hooks
University of Utah
150
100
50
20
40
60
(a) L(nm)
80
Wild type
(M = 55nm)
100
0
0
150
100
50
20
40
60
(b) L(nm)
80
Overexpressed
(M = 47nm)
100
0
0
50
100
150
(c) L(nm)
200
250
Underexpressed
(M = 76nm)
Length Regulation of Flagellar Hooks – p.6/36
Imagine the
Possibilities
The Secretion Machinery
Mathematical Biology
University of Utah
• Secreted molecules are
chaperoned to prevent
folding.
MS ring
CM
membrane
components
FlhA
C
ring
FlhB
N
FliJ
FliH
FliI
C
Step 1
Length Regulation of Flagellar Hooks – p.7/36
Imagine the
Possibilities
The Secretion Machinery
Mathematical Biology
University of Utah
• Secreted molecules are
chaperoned to prevent
folding.
• FliI is an ATPase
MS ring
CM
C
ring
membrane
components
FlhA
FlhB
N
FliJ
FliH
FliI
Step 2
C
Length Regulation of Flagellar Hooks – p.7/36
Imagine the
Possibilities
The Secretion Machinery
Mathematical Biology
University of Utah
• Secreted molecules are
chaperoned to prevent
folding.
• FliI is an ATPase
MS ring
CM
• FlhB is the gatekeeper
recognizing the N
terminus of secretants.
C
ring
membrane
components
N
FlhA FlhB
FliJ
FliH
FliI
ATP
C
ADP+Pi
Step 3
Length Regulation of Flagellar Hooks – p.7/36
Imagine the
Possibilities
The Secretion Machinery
Mathematical Biology
University of Utah
• Secreted molecules are
chaperoned to prevent
folding.
N
• FliI is an ATPase
CM
• FlhB is the gatekeeper
membrane
components
MS ring
FlhA
C
ring
FlhB
C
recognizing the N
terminus of secretants.
• once inside, molecular
movement is by diffusion.
FliH
FliI
FliJ
Step 4
Length Regulation of Flagellar Hooks – p.7/36
Imagine the
Possibilities
Secretion Control
Mathematical Biology
University of Utah
Secretion is regulated by FlhB
• During hook formation, only FlgE and FliK can be secreted.
• After hook is complete, FlgE and FliK are no longer
secreted, but other molecules can be secreted (those
needed for filament growth.)
• The switch occurs when the C-terminus of FlhB is cleaved
by FlK.
Question: Why is the switch in FlhB length dependent?
Length Regulation of Flagellar Hooks – p.8/36
Imagine the
Possibilities
Mathematical Biology
University of Utah
Hypothesis: How Hook Length is
determined
• The Infrequent Molecular Ruler Mechanism. FliK is
secreted once in a while to test the length of the hook.
• The probability of FlhB cleavage is length dependent.
Length Regulation of Flagellar Hooks – p.9/36
Imagine the
Possibilities
Mathematical Biology
Binding Probability
University of Utah
Suppose the probability of FlhB cleavage by FliK is a function of
length Pc (L). Then, the probability of cleavage at time t, P (t), is
determined by
dP
= αr(L)Pc (L)(1 − P )
dt
where r(L) is the secretion rate, α is the fraction of secreted
molecules that are FliK, and
dL
= βr(L)∆
dt
where β = 1 − α fraction of secreted FlgE molecules, ∆ length
increment per FlgE molecule.
Length Regulation of Flagellar Hooks – p.10/36
Imagine the
Possibilities
Binding Probability
Mathematical Biology
University of Utah
It follows that
dP
α
=
Pc (L)(1 − P )
dL
β∆
or
− ln(1 − P (L)) = κ
Z
L
Pc (L)dL
0
Length Regulation of Flagellar Hooks – p.11/36
Imagine the
Possibilities
Check the Data
Mathematical Biology
250
200
200
200
150
100
50
0
0
Number of hooks
250
Number of hooks
250
150
100
50
20
40
60
(a) L(nm)
80
100
150
100
50
0
0
Wild type
20
40
60
(b) L(nm)
80
100
0
0
Overexpressed
50
100
150
(c) L(nm)
200
250
Underexpressed
1
Overexpressed
WT
Underexpressed
0.9
0.8
0.7
0.6
P(L)
Number of hooks
University of Utah
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
Hook Length (nm)
70
80
90
100
Length Regulation of Flagellar Hooks – p.12/36
Imagine the
Possibilities
Check the Data
Mathematical Biology
University of Utah
10
Overexpressed
WT
Underexpressed
−ln(1−P)
8
6
4
2
0
0
10
20
30
40
50
60
70
80
90
100
70
80
90
100
Hook Length (nm)
−κ ln(1−P)
10
κ = 0.9
κ=1
κ=4
8
6
4
2
0
0
10
20
30
40
50
60
Hook Length (nm)
− ln(1 − P (L)) = κ
Z
L
Pc (L)dL?
0
Length Regulation of Flagellar Hooks – p.13/36
Imagine the
Possibilities
Mathematical Biology
University of Utah
Hypothesis: How Hook Length is
determined
• The Infrequent Molecular Ruler Mechanism.
• The probability of FlhB cleavage is length dependent. What
is the mechanism that determines Pc (L)?
Length Regulation of Flagellar Hooks – p.14/36
Imagine the
Possibilities
Secretion Model
Mathematical Biology
University of Utah
Hypothesis: FliK binds to FlhB during translocation to cause
switching of secretion target by cleaving a recognition sequence.
• FliK molecules move through the growing
tube by diffusion.
L
0
x
Length Regulation of Flagellar Hooks – p.15/36
Imagine the
Possibilities
Secretion Model
Mathematical Biology
University of Utah
Hypothesis: FliK binds to FlhB during translocation to cause
switching of secretion target by cleaving a recognition sequence.
• FliK molecules move through the growing
tube by diffusion.
• They remain unfolded before and during
secretion, but begin to fold as they exit the
tube.
L
0
x
Length Regulation of Flagellar Hooks – p.15/36
Imagine the
Possibilities
Secretion Model
Mathematical Biology
University of Utah
Hypothesis: FliK binds to FlhB during translocation to cause
switching of secretion target by cleaving a recognition sequence.
• FliK molecules move through the growing
tube by diffusion.
• They remain unfolded before and during
secretion, but begin to fold as they exit the
tube.
L
• Folding on exit prevents back diffusion,
giving a brownian ratchet effect.
0
x
Length Regulation of Flagellar Hooks – p.15/36
Imagine the
Possibilities
Secretion Model
Mathematical Biology
University of Utah
Hypothesis: FliK binds to FlhB during translocation to cause
switching of secretion target by cleaving a recognition sequence.
• FliK molecules move through the growing
tube by diffusion.
• They remain unfolded before and during
secretion, but begin to fold as they exit the
tube.
• Folding on exit prevents back diffusion,
L
x
giving a brownian ratchet effect.
• For short hooks, folding prevents FlhB
0
cleavage.
Length Regulation of Flagellar Hooks – p.15/36
Imagine the
Possibilities
Secretion Model
Mathematical Biology
University of Utah
Hypothesis: FliK binds to FlhB during translocation to cause
switching of secretion target by cleaving a recognition sequence.
• FliK molecules move through the growing
tube by diffusion.
L
• They remain unfolded before and during
secretion, but begin to fold as they exit the
tube.
• Folding on exit prevents back diffusion,
giving a brownian ratchet effect.
• For short hooks, folding prevents FlhB
0
x
cleavage.
• For long hooks, movement solely by diffu-
sion allows more time for cleavage.
Length Regulation of Flagellar Hooks – p.15/36
Imagine the
Possibilities
Mathematical Biology
University of Utah
Stochastic Model
Follow the position x(t) of the C-terminus using the stochastic langevin differential equation
p
νdx = F (x)dt + 2kb T νdW,
where F (x) represents the folding force acting on the unfolded FliK molecule, W (t) is
brownian white noise.
L
0
x
Length Regulation of Flagellar Hooks – p.16/36
Imagine the
Possibilities
Fokker-Planck Equation
Mathematical Biology
University of Utah
Let P (x, t) be the probability density of being at position x at
time t with FlhB uncleaved, and Q(t) be the probability of being
cleaved by time t. Then
∂P
∂
∂2P
= − (F (x)P ) + D 2 − g(x)P,
∂t
∂x
∂x
and
dQ
=
dt
Z
b
g(x)P (x, t)dx.
a
g(x)
where g(x) is the rate of FlhB
cleavage at position x.
0
L
x
Length Regulation of Flagellar Hooks – p.17/36
Imagine the
Possibilities
Probability of Cleavage
Mathematical Biology
University of Utah
To determine the probability of cleavage πc (x) starting from
position x, solve
dπc
d2 πc
− g(x)πc = 0
D 2 + F (x)
dx
dx
subject to πb′ (a) = 0 and πb (b) = 1.
Then Pc (L) = πc (a).
Pc(L)
L
Length Regulation of Flagellar Hooks – p.18/36
Imagine the
Possibilities
Results
Mathematical Biology
1
1
0.9
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
CDF
1
0.9
CDF
CDF
University of Utah
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
20
40
60
80
0
0
100
20
(a) L(nm)
40
60
80
0
0
100
50
(b) L(nm)
0.1
0.09
0.09
0.08
0.08
0.07
0.07
0.06
0.06
0.05
0.04
0.03
0.03
0.02
0.02
0.01
0.01
200
250
200
250
0.035
0.03
0.025
0.05
0.04
100
150
(c) L(nm)
0.04
PDF
0.1
PDF
PDF
0.5
0.02
0.015
0.01
0
0
20
40
60
(a) L(nm)
Wild type
80
100
0
0
0.005
20
40
60
80
(b) L(nm)
Overexpressed
100
0
0
50
100
150
(c) L(nm)
Underexpressed
Length Regulation of Flagellar Hooks – p.19/36
Imagine the
Possibilities
Mathematical Biology
University of Utah
Difficulties
• There is no direct experimental evidence either for or
against this proposed length measurement mechanism.
Length Regulation of Flagellar Hooks – p.20/36
Imagine the
Possibilities
Mathematical Biology
II - Flagellar Length Detection
University of Utah
• Flagella grow at a velocity that
decreases as they get longer.
Length Regulation of Flagellar Hooks – p.21/36
Imagine the
Possibilities
Mathematical Biology
II - Flagellar Length Detection
University of Utah
• Flagella grow at a velocity that
decreases as they get longer.
• If a flagellum is broken off, it will
regrow at the same velocity as
when it first grew.
Length Regulation of Flagellar Hooks – p.21/36
Imagine the
Possibilities
Mathematical Biology
II - Flagellar Length Detection
University of Utah
• Flagella grow at a velocity that
decreases as they get longer.
• If a flagellum is broken off, it will
regrow at the same velocity as
when it first grew.
Question: How does the bacterium measure flagellar length?
Length Regulation of Flagellar Hooks – p.21/36
Imagine the
Possibilities
How Do Flagella Grow?
Mathematical Biology
University of Utah
111111
000000
• Step 1: Secretion
• Step 2: Diffusion
• Step 3: Polymerization
Length Regulation of Flagellar Hooks – p.22/36
Imagine the
Possibilities
How Do Flagella Grow?
Mathematical Biology
University of Utah
111111
000000
• Step 1: Secretion
• Step 2: Diffusion
• Step 3: Polymerization
Length Regulation of Flagellar Hooks – p.22/36
Imagine the
Possibilities
How Do Flagella Grow?
Mathematical Biology
University of Utah
111111
000000
• Step 1: Secretion
• Step 2: Diffusion
• Step 3: Polymerization
Length Regulation of Flagellar Hooks – p.22/36
Imagine the
Possibilities
Modelling Flagellar Growth
Mathematical Biology
University of Utah
Step 2: Diffusion
Important Fact: Filament is a narrow hollow tube, so movement
(diffusion) is single file.
Let p(x, t) be the probability that a molecule is at position x at
time t. Then,
∂p ∂J
+
=0
∂t
∂x
where
∂p
J = −D .
∂x
Remark:
J
l
= flux in molecules per unit time.
Length Regulation of Flagellar Hooks – p.23/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
MS ring
CM
C
ring
membrane
components
N
FlhA FlhB
FliJ
FliH
FliI
ATP
C
ADP+Pi
Step 3
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
MS ring
CM
C
ring
membrane
components
N
FlhA FlhB
FliJ
FliH
FliI
ATP
C
ADP+Pi
Step 3
dP
dt
=
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
MS ring
CM
C
ring
membrane
components
FlhA
FlhB
N
FliJ
FliH
FliI
Step 2
C
dP
dt
= Kon (1 − P )
on rate,
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
N
membrane
components
MS ring
CM
FlhA
C
ring
FlhB
C
FliH
FliJ
FliI
Step 4
dP
dt
= Kon (1 − P ) − kof f P
on rate, off rate,
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
11
00
00
11
00
11
00
11
00
11
00
11
00
11
CM
membrane
MS ring
C
components
N
FlhB
FlhA
111
000
000
111
000
111
000
111
000
111
000
111
000
111
ring
FliJ
FliH
FliI
ATP
C
ADP+Pi
Step 4 Blocked
dP
dt
= Kon (1 − P ) −kof f (1 − p(0, t))P
on rate, off rate, restricted if blocked by another molecule in
the tube.
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Secretion
Mathematical Biology
University of Utah
Step 1: Secretion
Let P (t) be the probability that ATP-ase is bound
11
00
00
11
00
11
00
11
00
11
00
11
00
11
CM
membrane
MS ring
C
components
N
FlhB
FlhA
111
000
000
111
000
111
000
111
000
111
000
111
000
111
ring
FliJ
FliH
FliI
ATP
C
ADP+Pi
Step 4 Blocked
dP
dt
= Kon (1 − P ) − kof f (1 − p(0, t))P
on rate, off rate, restricted if blocked by another molecule in
the tube. Thus,
J
l
= kof f (1 − p(0, t))P at x = 0 (A Robin boundary condition).
Length Regulation of Flagellar Hooks – p.24/36
Imagine the
Possibilities
Rate of Polymerization
Mathematical Biology
University of Utah
Stage 3: Polymerization
111111
000000
000000
111111
J
= kp p
l
at the polymerizing end x = L.
Then, the growth velocity is
dL
J
=β ≡V
dt
l
where β =length of filament per monomer (0.5nm/monomer)
· · · a moving boundary problem.
Length Regulation of Flagellar Hooks – p.25/36
Imagine the
Possibilities
Diffusion Model
Mathematical Biology
University of Utah
After some work, it can be shown that
1
Ka
λ= −
− Kb
j
1−j
where j =
J
lKon ,
λ=
lLKon
D ,
A good approximation J ≈
Ka =
Kon
kof f ,
1
L
KJ + D
≈
D
L
Kb =
Kon
kp .
for large L
flux vs. length
1
0.9
0.8
0.7
j
0.6
0.5
0.4
0.3
0.2
0.1
So, why is growth length dependent? – p.26/36
0
Imagine the
Possibilities
Filament Length Control
Mathematical Biology
University of Utah
Introducing FlgM and σ 28 :
Length Regulation of Flagellar Hooks – p.27/36
Imagine the
Possibilities
Filament Length Control
Mathematical Biology
University of Utah
28 :
Introducing FlgM and
σ

28

σ





 FlgE
Class 1→ Class 2
FlgKL



FlgM




FliK












 FliC 
28
Eσ
→ Class 3
FliD







FlgM




Basal Body
Length Regulation of Flagellar Hooks – p.27/36
Imagine the
Possibilities
Filament Length Control
Mathematical Biology
University of Utah
28 :
Introducing FlgM and
σ

28

σ





 FlgE
Class 1 → Class 2
FlgKL



FlgM




FliK












 FliC 
28
Eσ
→ Class 3
FliD







FlgM




FlgE
Hook
Basal Body
Length Regulation of Flagellar Hooks – p.27/36
Imagine the
Possibilities
Filament Length Control
Mathematical Biology
University of Utah
28 :
Introducing FlgM and
σ

28

σ





 FlgE
Class 1 → Class 2
FlgKL



FlgM




FliK











 FliC 

28
Eσ
→ Class 3
FliD







FlgM




FliC
Filament
Hook−filament
junction
Hook
Basal Body
Length Regulation of Flagellar Hooks – p.27/36
Imagine the
Possibilities
FlgM-σ 28 Chemistry
Mathematical Biology
University of Utah
28
σ
28
Eσ
FlgM J(L)
FliC
FlgM
FlgM
σ∗
Eσ*
Length Regulation of Flagellar Hooks – p.28/36
Imagine the
Possibilities
FlgM-σ 28 Chemistry
Mathematical Biology
University of Utah
28
σ
28
Eσ
FlgM J(L)
FliC
FlgM
FlgM
σ∗
Eσ*
• FlgM inhibits σ 28 activity;
Length Regulation of Flagellar Hooks – p.28/36
Imagine the
Possibilities
FlgM-σ 28 Chemistry
Mathematical Biology
University of Utah
28
σ
28
Eσ
FlgM J(L)
FliC
FlgM
FlgM
σ∗
Eσ*
• FlgM inhibits σ 28 activity;
• Therefore, during stage 3, FlgM inhibits its own production
(negative feedback);
Length Regulation of Flagellar Hooks – p.28/36
Imagine the
Possibilities
FlgM-σ 28 Chemistry
Mathematical Biology
University of Utah
28
σ
28
Eσ
FlgM J(L)
FliC
FlgM
FlgM
σ∗
Eσ*
• FlgM inhibits σ 28 activity;
• Therefore, during stage 3, FlgM inhibits its own production
(negative feedback);
• And, FlgM inhibits the production of Flagellin (FliC).
Length Regulation of Flagellar Hooks – p.28/36
Imagine the
Possibilities
FlgM-σ 28 Secretion Dynamics
Mathematical Biology
University of Utah
•
FlgM is not secreted during hook
growth.
FlgE
Hook
Basal Body
Length Regulation of Flagellar Hooks – p.29/36
Imagine the
Possibilities
FlgM-σ 28 Secretion Dynamics
Mathematical Biology
University of Utah
FliC
•
FlgM is not secreted during hook
growth.
• FlgM is secreted during filament
Filament
Hook−filament
junction
Hook
growth.
Basal Body
Length Regulation of Flagellar Hooks – p.29/36
Imagine the
Possibilities
FlgM-σ 28 Secretion Dynamics
Mathematical Biology
University of Utah
FliC
•
FlgM is not secreted during hook
growth.
• FlgM is secreted during filament
Filament
Hook−filament
junction
Hook
growth.
Basal Body
So, how fast is FlgM secreted, and why does it matter?
Length Regulation of Flagellar Hooks – p.29/36
Imagine the
Possibilities
Tracking Concentrations
Mathematical Biology
University of Utah
FlgM (M ):
dM
= rate of production − rate of secretion
dt
Flagellin (FliC) (F ):
dF
= rate of production − rate of secretion
dt
Filament Length (L):
dL
= β ∗ rate of FliC secretion
dt
.
Length Regulation of Flagellar Hooks – p.30/36
Imagine the
Possibilities
Tracking Concentrations
Mathematical Biology
University of Utah
FlgM (M ):
dM
K∗
M
J
=
−α
dt
KM + M
F +M
Flagellin (FliC) (F ):
K∗
F
dF
=
−α
J
dt
KM + M
F +M
Filament Length (L):
dL
F
=β
J
dt
M +F
with J =
1
L
KJ + D
(which is length dependent!) .
Length Regulation of Flagellar Hooks – p.30/36
Imagine the
Possibilities
Filament Growth
Mathematical Biology
University of Utah
Filament Length vs Time
Intracellular FlgM and FliC
25
4000
3500
Number of Molecules
Length (microns)
20
15
10
3000
2500
2000
1500
1000
5
500
0
0
200
400
600
800
Time (minutes)
1000
1200
1400
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
• Before secretion begins FlgM concentration is large. When
secretion begins, FlgM concentration drops, producing FliC
and more FlgM.
Length Regulation of Flagellar Hooks – p.31/36
Imagine the
Possibilities
Filament Growth
Mathematical Biology
University of Utah
Filament Length vs Time
Intracellular FlgM and FliC
25
4000
3500
Number of Molecules
Length (microns)
20
15
10
3000
2500
2000
1500
1000
5
500
0
0
200
400
600
800
Time (minutes)
1000
1200
1400
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
• Before secretion begins FlgM concentration is large. When
secretion begins, FlgM concentration drops, producing FliC
and more FlgM.
• As the filament grows, secretion slows, FlgM concentration
increases, shutting off FliC and FlgM production.
Length Regulation of Flagellar Hooks – p.31/36
Imagine the
Possibilities
Filament Growth
Mathematical Biology
University of Utah
Filament Length vs Time
Intracellular FlgM and FliC
25
4000
3500
Number of Molecules
Length (microns)
20
15
10
3000
2500
2000
1500
1000
5
500
0
0
200
400
600
800
Time (minutes)
1000
1200
1400
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
• Before secretion begins FlgM concentration is large. When
secretion begins, FlgM concentration drops, producing FliC
and more FlgM.
• As the filament grows, secretion slows, FlgM concentration
increases, shutting off FliC and FlgM production.
• If filament is suddenly shortened, secretion suddenly
increases, reinitiating the growth phase.
Length Regulation of Flagellar Hooks – p.31/36
Imagine the
Possibilities
Observations
Mathematical Biology
University of Utah
Filament Length vs Time
Intracellular FlgM and FliC
25
4000
3500
Number of Molecules
Length (microns)
20
15
10
3000
2500
2000
1500
1000
5
500
0
0
200
400
600
800
Time (minutes)
1000
1200
1400
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
• Because the flux is inversely proportional to length, the
amount of FlgM in the cell is a direct measure of the length
of the filament.
Length Regulation of Flagellar Hooks – p.32/36
Imagine the
Possibilities
Observations
Mathematical Biology
University of Utah
Filament Length vs Time
Intracellular FlgM and FliC
25
4000
3500
Number of Molecules
Length (microns)
20
15
10
3000
2500
2000
1500
1000
5
500
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
0
0
200
400
600
800
1000
1200
1400
Time (minutes)
• Because the flux is inversely proportional to length, the
amount of FlgM in the cell is a direct measure of the length
of the filament.
• Because of negative feedback, the cell "knows" to produce
FliC only when it is needed.
Length Regulation of Flagellar Hooks – p.32/36
Imagine the
Possibilities
Acknowledgments
Mathematical Biology
University of Utah
Help came from
• Kelly Hughes, U of Washington
• Bob Guy, U of Utah
• Tom Robbins, U of Utah
No computers were harmed by Microsoft products during the
production or presentation of this talk.
The End
Length Regulation of Flagellar Hooks – p.33/36