Length Regulation of Flagellar Hooks and Filaments...

transcript

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)

Basal Body

Hook−filamentjunction

Filament

Basal Body

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

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

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

(c) L(nm)

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

The Secretion Machinery

• Secreted molecules arechaperoned to preventfolding.

Step 1

MS ring

CM FlhA FlhB

componentsmembrane

• FliI is an ATPase

Step 2

MS ring

CM FlhA FlhB

membranecomponents

• FlhB is the gatekeeperrecognizing the Nterminus of secretants.

Step 3

MS ring

CM FlhA FlhB

ATP ADP+Pi

membranecomponents

• FlhB is the gatekeeperrecognizing the Nterminus of secretants.

• once inside, molecularmovement is by diffu-sion.

Step 4

MS ring

CM FlhA FlhB

membranecomponents

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

dt= αr(L)Pc(L)(1 − P )

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

dt= βr(L)∆

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

Binding Probability

It follows thatdP

β∆Pc(L)(1 − P )

− ln(1 − P (L)) = κ

Pc(L)dL

Check the Data

0 20 40 60 80 1000

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

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

(c) L(nm)

Wild type Overexpressed Underexpressed

0 10 20 30 40 50 60 70 80 90 1000

Hook Length (nm)

OverexpressedWTUnderexpressed

Check the Data

0 10 20 30 40 50 60 70 80 90 1000

Hook Length (nm)

0 10 20 30 40 50 60 70 80 90 1000

Hook Length (nm)

OverexpressedWTUnderexpressed

κ = 0.9κ = 1κ = 4

− ln(1 − P (L)) = κ

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.

Secretion Model

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

Secretion Model

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

Secretion Model

• For short hooks, folding prevents FlhBcleavage.

Secretion Model

• For short hooks, folding prevents FlhBcleavage.

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

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.

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

∂t= −

∂x(F (x)P ) + D

∂x2− g(x)P,

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

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

Probability of Cleavage

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

Dd2πc

dx2+ F (x)

dx− g(x)πc = 0

subject to π′

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

Results

0 20 40 60 80 1000

(a) L(nm)

0 20 40 60 80 1000

(b) L(nm)

0 50 100 150 200 2500

(c) L(nm)

0 20 40 60 80 1000

(a) L(nm)

0 20 40 60 80 1000

(b) L(nm)

0 50 100 150 200 2500

(c) L(nm)

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.

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,

∂x= 0

J = −D∂p

Remark: Jl

= flux in molecules per unit time.

Rate of Secretion

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

Step 3

MS ring

CM FlhA FlhB

ATP ADP+Pi

membranecomponents

Rate of Secretion

Step 3

MS ring

CM FlhA FlhB

ATP ADP+Pi

membranecomponents

Rate of Secretion

Step 2

MS ring

CM FlhA FlhB

membranecomponents

= Kon(1 − P )

on rate,

Rate of Secretion

Step 4

MS ring

CM FlhA FlhB

membranecomponents

= Kon(1 − P ) − koffP

on rate, off rate,

Rate of Secretion

��

Step 4 Blocked

ATP ADP+Pi

MS ring

CM FlhA FlhB

membrane

components

= 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

MS ring

CM FlhA FlhB

membrane

components

= Kon(1 − P ) − koff (1 − p(0, t))P

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

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

Rate of Polymerization

Stage 3: Polymerization

l= kpp

at the polymerizing end x = L.

��

Then, the growth velocity is

dt= β

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

1 − j− Kb

where j = JlKon

, λ = lLKon

D, Ka = Kon

koff, Kb = Kon

A good approximation J ≈1

for large L

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1flux vs. length

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

Filament Length Control

Introducing FlgM and σ28:

Class 1→ Class 2

→ Class 3

Basal Body

Class 1 → Class 2

→ Class 3

Basal Body

Class 1 → Class 2

→ Class 3

Filament

Basal Body

FlgM-σ28 Chemistry

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ

FlgM-σ28 Chemistry

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ

• FlgM inhibits σ28 activity;

FlgM-σ28 Chemistry

FliCJ(L)

*Eσ∗

FlgM FlgM

FlgM28 σ

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

FlgM-σ28 Chemistry

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.

Basal Body

• FlgM is secreted during filamentgrowth.

Filament

Basal Body

• FlgM is secreted during filamentgrowth.

Filament

Basal Body

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

Tracking Concentrations

FlgM (M ):

dt= rate of production − rate of secretion

Flagellin (FliC) (F ):

dt= rate of production − rate of secretion

Filament Length (L):

dt= β ∗ rate of FliC secretion

Tracking Concentrations

FlgM (M ):

KM + M− α

F + MJ

Flagellin (FliC) (F ):

KM + M− α

F + MJ

Filament Length (L):

dt= β

M + FJ

with J = 1

(which is length dependent!) .

Filament Growth

0 200 400 600 800 1000 1200 14000

25Filament Length vs Time