Length Regulation of Flagellar Hooks and Filaments...

University of UtahMathematical Biology

theImagine Possibilities

Length Regulation of Flagellar Hooks andFilaments in Salmonella

J. P. Keener

Department of Mathematics

University of Utah

Length Regulation of Flagellar Hooks – p.1/36



Introduction

The first of many slides – p.2/36



Control of Flagellar Growth

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








• Step 3: Filament (FliC secretion)

FlgE

Hook

Basal Body









FliC

Hook−filamentjunction

Hook

Filament

Basal Body









FliC


Hook

Filament

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



Proteins of Flagellar Assembly




Hook Length Regulation

• Hook is built by FlgE secretion.






• FliK is the "hook length regulatory" protein.






• FliK is the "hook length regulatory" protein.• FliK is secreted only during hook production.






• 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.







FliK gives shorter hooks.• Lengthening FliK gives longer hooks.







FliK gives shorter hooks.• Lengthening FliK gives longer hooks.• 5-10 molecules of FliK are secreted per hook (115-120

molecules of FlgE).




Hook Length Data

0 20 40 60 80 1000

50

100

150

200

250

Num

ber

of h

ooks

(a) L(nm)0 20 40 60 80 100

0

50

100

150

200

250

Num

ber

of h

ooks

(b) L(nm)0 50 100 150 200 250

0

50

100

150

200

250

Num

ber

of h

ooks

(c) L(nm)

Wild type Overexpressed Underexpressed(M = 55nm) (M = 47nm) (M = 76nm)




The Secretion Machinery

• Secreted molecules arechaperoned to preventfolding.

Step 1

FliI

FliJ

Cring

MS ring

CM FlhA FlhB

C

N

FliH

componentsmembrane






• FliI is an ATPase

FliJ

Step 2

N

C

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliH

FliI







• FlhB is the gatekeeperrecognizing the Nterminus of secretants.

N

C

Step 3

Cring

MS ring

CM FlhA FlhB

FliJ

ATP ADP+Pi

FliH

membranecomponents

FliI







• FlhB is the gatekeeperrecognizing the Nterminus of secretants.

• once inside, molecularmovement is by diffu-sion.

C

Step 4

FliJ

N

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliI

FliH




Secretion Control

Secretion is regulated by FlhB

• During hook formation, only FlgE and FliK can be secreted.

• After hook is complete, FlgE and FliK are no longersecreted, but other molecules can be secreted (thoseneeded for filament growth.)

• The switch occurs when the C-terminus of FlhB is cleavedby FlK.

Question: Why is the switch in FlhB length dependent?




Hypothesis: How Hook Length isdetermined

• The Infrequent Molecular Ruler Mechanism. FliK issecreted once in a while to test the length of the hook.

• The probability of FlhB cleavage is length dependent.




Binding Probability

Suppose the probability of FlhB cleavage by FliK is a function oflength Pc(L). Then, the probability of cleavage at time t, P (t), isdetermined by

dP

dt= αr(L)Pc(L)(1 − P )

where r(L) is the secretion rate, α is the fraction of secretedmolecules that are FliK, and

dL

dt= βr(L)∆

where β = 1 − α fraction of secreted FlgE molecules, ∆ lengthincrement per FlgE molecule.




Binding Probability

It follows thatdP

dL=

α

β∆Pc(L)(1 − P )

or

− ln(1 − P (L)) = κ

∫ L

0

Pc(L)dL




Check the Data

0 20 40 60 80 1000

50

100

150

200

250

Num

ber

of h

ooks

(a) L(nm)0 20 40 60 80 100

0

50

100

150

200

250

Num

ber

of h

ooks

(b) L(nm)0 50 100 150 200 250

0

50

100

150

200

250

Num

ber

of h

ooks

(c) L(nm)

Wild type Overexpressed Underexpressed

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hook Length (nm)

P(L

)

OverexpressedWTUnderexpressed




Check the Data

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

Hook Length (nm)

−ln

(1−

P)

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

Hook Length (nm)

−κ

ln(1

−P

)

OverexpressedWTUnderexpressed

κ = 0.9κ = 1κ = 4

− ln(1 − P (L)) = κ

∫ L

0

Pc(L)dL?Length Regulation of Flagellar Hooks – p.13/36



Hypothesis: How Hook Length isdetermined

• The Infrequent Molecular Ruler Mechanism.

• The probability of FlhB cleavage is length dependent. Whatis the mechanism that determines Pc(L)?




Secretion Model

Hypothesis: FliK binds to FlhB during translocation to causeswitching of secretion target by cleaving a recognition sequence.

• FliK molecules move through the growingtube by diffusion.

L

x

0




Secretion Model



• They remain unfolded before and duringsecretion, but begin to fold as they exit thetube.

L

x

0




Secretion Model




• Folding on exit prevents back diffusion,giving a brownian ratchet effect.

L

x

0




Secretion Model





• For short hooks, folding prevents FlhBcleavage.

0

L

x




Secretion Model





• For short hooks, folding prevents FlhBcleavage.

• For long hooks, movement solely by diffu-sion allows more time for cleavage.

0

L

x




Stochastic Model

Follow the position x(t) of the C-terminus us-ing the stochastic langevin differential equa-tion

νdx = F (x)dt +√

2kbTνdW,

where F (x) represents the folding force act-ing on the unfolded FliK molecule, W (t) isbrownian white noise.

L

x

0




Fokker-Planck Equation

Let P (x, t) be the probability density of being at position x attime t with FlhB uncleaved, and Q(t) be the probability of beingcleaved by time t. Then

∂P

∂t= −

∂

∂x(F (x)P ) + D

∂2P

∂x2− g(x)P,

anddQ

dt=

∫ b

a

g(x)P (x, t)dx.

where g(x) is the rate of FlhBcleavage at position x.

g(x)

0 L x




Probability of Cleavage

To determine the probability of cleavage πc(x) starting fromposition x, solve

Dd2πc

dx2+ F (x)

dπc

dx− g(x)πc = 0

subject to π′

b(a) = 0 and πb(b) = 1.Then Pc(L) = πc(a).

Pc(L)

L




Results

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(a) L(nm)

CD

F

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(b) L(nm)

CD

F

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

(c) L(nm)

CD

F

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

(a) L(nm)

PD

F

0 20 40 60 80 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

(b) L(nm)

PD

F

0 50 100 150 200 2500

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

(c) L(nm)

PD

F

Wild type Overexpressed Underexpressed




Difficulties

• There is no direct experimental evidence either for oragainst this proposed length measurement mechanism.




II - Flagellar Length Detection

• Flagella grow at a velocity thatdecreases as they get longer.






• If a flagellum is broken off, it willregrow at the same velocity aswhen it first grew.






• If a flagellum is broken off, it willregrow at the same velocity aswhen it first grew.

Question: How does the bacterium measure flagellar length?




How Do Flagella Grow?

• Step 1: Secretion

• Step 2: Diffusion

• Step 3: Polymerization

��








��








��




Modelling Flagellar Growth

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 attime t. Then,

∂p

∂t+

∂J

∂x= 0

where

J = −D∂p

∂x.

Remark: Jl

= flux in molecules per unit time.




Rate of Secretion

Step 1: SecretionLet P (t) be the probability that ATP-ase is bound

N

C

Step 3

Cring

MS ring

CM FlhA FlhB

FliJ

ATP ADP+Pi

FliH

membranecomponents

FliI




Rate of Secretion


N

C

Step 3

Cring

MS ring

CM FlhA FlhB

FliJ

ATP ADP+Pi

FliH

membranecomponents

FliI

dPdt

=




Rate of Secretion


FliJ

Step 2

N

C

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliH

FliI

dPdt

= Kon(1 − P )

on rate,




Rate of Secretion


C

Step 4

FliJ

N

Cring

MS ring

CM FlhA FlhB

membranecomponents

FliI

FliH

dPdt

= Kon(1 − P ) − koffP

on rate, off rate,




Rate of Secretion


��

��

��

��

Step 4 Blocked

ATP ADP+Pi

N

C

C

ring

MS ring

CM FlhA FlhB

FliJ

membrane

components

FliH

FliI

dPdt

= Kon(1 − P ) −koff (1 − p(0, t))P

on rate, off rate, restricted if blocked by another molecule inthe tube.




Rate of Secretion


��

��

��

��

Step 4 Blocked

ATP ADP+Pi

N

C

C

ring

MS ring

CM FlhA FlhB

FliJ

membrane

components

FliH

FliI

dPdt

= Kon(1 − P ) − koff (1 − p(0, t))P

on rate, off rate, restricted if blocked by another molecule inthe tube. Thus,

Jl

= koff (1 − p(0, t))P at x = 0 (A Robin boundary condition).




Rate of Polymerization

Stage 3: Polymerization

J

l= kpp

at the polymerizing end x = L.

��

Then, the growth velocity is

dL

dt= β

J

l≡ V

where β =length of filament per monomer (0.5nm/monomer)

· · · a moving boundary problem.




Diffusion Model

After some work, it can be shown that

λ =1

j−

Ka

1 − j− Kb

where j = JlKon

, λ = lLKon

D, Ka = Kon

koff, Kb = Kon

kp.

A good approximation J ≈1

KJ+LD

≈DL

for large L

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1flux vs. length

j

So, why is growth length dependent? – p.26/36



Filament Length Control

Introducing FlgM and σ28:






Class 1→ Class 2

σ28

FlgE

FlgKL

FlgM

FliK

Eσ28

→ Class 3

FliC

FliD

FlgM

Basal Body






Class 1 → Class 2

σ28

FlgE

FlgKL

FlgM

FliK

Eσ28

→ Class 3

FliC

FliD

FlgM

FlgE

Hook

Basal Body






Class 1 → Class 2

σ28

FlgE

FlgKL

FlgM

FliK

Eσ28

→ Class 3

FliC

FliD

FlgM

FliC


Hook

Filament

Basal Body




FlgM-σ28 Chemistry

σ

σ

E28

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ




FlgM-σ28 Chemistry

σ

σ

E28

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ

• FlgM inhibits σ28 activity;




FlgM-σ28 Chemistry

σ

σ

E28

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ


• Therefore, during stage 3, FlgM inhibits its own production(negative feedback);




FlgM-σ28 Chemistry

σ

σ

E28

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ


• Therefore, during stage 3, FlgM inhibits its own production(negative feedback);

• And, FlgM inhibits the production of Flagellin (FliC).




FlgM-σ28 Secretion Dynamics

• FlgM is not secreted during hookgrowth.

FlgE

Hook

Basal Body






• FlgM is secreted during filamentgrowth.

FliC


Hook

Filament

Basal Body






• FlgM is secreted during filamentgrowth.

FliC


Hook

Filament

Basal Body

So, how fast is FlgM secreted, and why does it matter?




Tracking Concentrations

FlgM (M ):

dM

dt= rate of production − rate of secretion

Flagellin (FliC) (F ):

dF

dt= rate of production − rate of secretion

Filament Length (L):

dL

dt= β ∗ rate of FliC secretion

.




Tracking Concentrations

FlgM (M ):

dM

dt=

K∗

KM + M− α

M

F + MJ

Flagellin (FliC) (F ):

dF

dt=

K∗

KM + M− α

F

F + MJ

Filament Length (L):

dL

dt= β

F

M + FJ

with J = 1

KJ+LD

(which is length dependent!) .




Filament Growth

0 200 400 600 800 1000 1200 14000

5

10

15

20

25Filament Length vs Time

Leng

th (

mic

rons

)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)

Intracellular FlgM and FliC

Num

ber

of M

olec

ules

• Before secretion begins FlgM concentration is large. Whensecretion begins, FlgM concentration drops, producing FliCand more FlgM.




Filament Growth

0 200 400 600 800 1000 1200 14000

5

10

15

20


Leng

th (

mic

rons

)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)


Num

ber

of M

olec

ules


• As the filament grows, secretion slows, FlgM concentrationincreases, shutting off FliC and FlgM production.




Filament Growth

0 200 400 600 800 1000 1200 14000

5

10

15

20


Leng

th (

mic

rons

)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)


Num

ber

of M

olec

ules


• As the filament grows, secretion slows, FlgM concentrationincreases, shutting off FliC and FlgM production.

• If filament is suddenly shortened, secretion suddenlyincreases, reinitiating the growth phase. Length Regulation of Flagellar Hooks – p.31/36



Observations

0 200 400 600 800 1000 1200 14000

5

10

15

20


Leng

th (

mic

rons

)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)


Num

ber

of M

olec

ules

• Because the flux is inversely proportional to length, theamount of FlgM in the cell is a direct measure of the lengthof the filament.




Observations

0 200 400 600 800 1000 1200 14000

5

10

15

20


Leng

th (

mic

rons

)

Time (minutes)0 200 400 600 800 1000 1200 1400

0

500

1000

1500

2000

2500

3000

3500

4000

Time (minutes)


Num

ber

of M

olec

ules

• Because the flux is inversely proportional to length, theamount of FlgM in the cell is a direct measure of the lengthof the filament.

• Because of negative feedback, the cell "knows" to produceFliC only when it is needed.




Acknowledgments

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 theproduction or presentation of this talk.

The End


Date post:	17-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

Length Regulation of Flagellar Hooks and Filaments...

Documents