Modeling the shapes of actin-based protrusions

Thesis for the degreeMaster of Science

by: Gilad OrlyAdviser: Prof. Nir Gov

Cell grow protrusions:

microvilli

Squamous nasal epithelial cells with microvilli (Dennis Kunkel )

Length (µm): ~ 0.5

Diameter (µm): 0.1-0.2

Life time: 1 min – cell life time

Filopodia

Length (µm): 1-50

Diameter (µm): 0.1-0.2

Life time: 1– 100 min

Stereocilia

Length (µm): 1-50

Diameter (µm): 0.2-1

Life time: Organism life time

Cochlear stereocilia structureVestibular stereocilia structure

Frolenkov 2004

Rzadzinska et al. 2004

Schwander, Kachar, Müller 2010

Furness 2008

Stereocilia formation and structures

Few facts:• The stereocilia grow out of homogeneously distributed

small and thin microvilli into a longer and thicker stereocilia that are well organized.

• The remaining microvilli are then disappear. P0

P0

P0

P2

P2

P4

P20

P20

KALTENBACH et al. (1994)

Stereocilia formation and structures

Few facts:• The first row may be much higher than the rest.• The second row may be the thickest.• The stereocilia at the cochlea’s apex are longer than at

the base

Fettiplace R and Hackney CM (2006)Zampin et al. (2011))Frolenkov et al. (2004)

Stereocilia formation and structures

Few facts:• Change in expression levels of regulating proteins result

in changes in height and possibly width

Rzadzinka et al. (2005)

Zampin et al. (2011)

Research Questions

• What determines the shape and dynamics of the protrusions ?

• How is it possible to get multiple steady-state height in the same cell (stereocilia)?

Actin polymerizationInteractions with Actin: • Myosin as cargo carriers• Myosin as actin

membrane connectors • Actin cross-linkers• Promotor proteins (PP)• Inhibitor proteins (SP)• Severing proteins (SP)• ..

+ end

- end

"skyscraper on quicksand"

• Bending the membrane generates a restoring force

• The cytoskeleton is dynamic and can be regarded as viscous-elastic gel

• The actin bundle can be thought of as a “skyscraper on quicksand”.

• To maintain st.st one most either:– Eliminate bending force– Stiffen the cytoskeleton– Keep on growing to counter the sinking into the

cytoskeleton

The source of protrusion’s growth –Force equations (1/4)

• The Actin’s Pushing force:

• Tail’s treadmilling velocity:

• Tail’s surface area:

• Protrusion’s local radius

• A – polymerization velocity• β – severing velocity• R – actin bundle’s local radius

Sc(t)l(t)Fa

The cell: A viscous gel ()

Z=h

Z=0

Z

R

The source of protrusion’s growth –Force equations (2/4)

The membrane restoring force:• Membrane deformation force (for a

cylindrical protrusion):

• Friction force of a flowing membrane (for a cylindrical protrusion):

Sm(t)h(t)

Fm

The outer cell: A viscous gel ()

~𝜇

The source of protrusion’s growth –Force equations (3/4)

Actin-membrane connectors (myosin) restoring force :• One can speculate that myosin that

connect the actin to the membrane can apply a downward force on the actin as they walk towards the tip:

• depends on the distribution of the myosin connectors. For a uniform distribution

For a cylinder

• Equation of forces:

• Extracting we get:

• Steady-state height (0):

The source of protrusion’s growth –Force equations (4/4)

determines the protrusion’s height

– polymerization velocity – severing velocity– actin- membrane restoring force – restoring force– cell viscosity – tail’s surface area

𝐴

𝛽

𝐹𝑚𝑎

Z=h

Z=0

Z

base

tip

Tail

𝜸𝒄

remainder

Proteins concentration along the protrusion \ at the protrusion’s tip

𝐴 (h )=𝐴 𝑓 +𝐴𝑝𝐾 𝑝𝐶𝑝 (h)

1+𝐾 𝑝𝐶𝑝 (h )+𝐾 𝑖𝐶𝑖(h)

Prompting protein’s (PP) concentration Inhibiting protein’s (IP) concentration

a

Prot

eins

co

ncen

trati

on

h

c

III

III

z/h

Prot

eins

co

ncen

trati

on

b

Prot

eins

co

ncen

trati

on

z/h0 0 11

a

a

Poly

mer

izatio

n ra

te

h

c

III

III

h

Poly

mer

izatio

n ra

te b

Poly

mer

izatio

n ra

te

h

a

Free diffusion Walking to the base Walking to the tip

Myosin-XVa (a), Myosin-I (d), Myosin-III (e), Myosin-VI (f)concentration profiles in stereociliaSchneider et al, Journal of Neuroscience, 2006 (a-e)Sakaguchi et al, cell motility and the cytoskeleton, 2008 (f)

f

The severing profile (β)

• If freely diffusing the profile should be constant (β = β0)

• If interacting with the Actin \Myosin VI will be restricted to the base ()

Yang C, Czech L, Gerboth S, Kojima S, Scita G, et al. (2007)

Force   balance

The restoring force must increase with h !!Force generated by membrane-actin myosin connectors

Protrusion height

forc

es

Steady state height of cylindrical protrusion as a function of radius\cell viscosity

• In the case of a steep increase of A(h) there can be a bifurcation point in and in

𝜸𝒄

remainder – polymerization velocity – severing velocity– actin- membrane restoring force – restoring force– cell viscosity – tail’s surface area

𝐴

𝛽

𝐹𝑚𝑎

Z=h

Z=0

Z

base

tip

Tail

h

A

Protrusion radius

Prot

rusi

on h

eigh

t

Protrusion radiusCell viscosity

How is the protrusion radius determined?

• Random initial conditions…– Filopodia?

• Proteins with a fixed length form the tip…– ?

• Actively regulated – Experiments show that the radius is sensitive to

expression level of some regulating proteins. – Observations indicates that filaments are only added or

removed to the tip-complex at the rim

The protrusion’s radius• steady-state system has a finite non-zero value

for the radius only if either addition or removal has a dependence on

• We propose a model the rate of tip-complex growth depends on while the rate of removing filaments is constant:

�̇�= 𝑓 (𝑅 )−𝜂𝑛

creation annihilation

The protrusion’s radius• The model:– Adding a new filament to the

bundle requires a nucleator, CL and G-actin at the tip’s radius

– CL reach the tip only through the rim

– CL are deactivated across the tip𝜌 𝑓 (𝑡 ,𝑟 )=𝐷𝛻𝑟

2 𝜌 𝑓 (𝑡 ,𝑟 )−𝑘𝑜𝑛𝜌 𝑓 (𝑡 ,𝑟 )+𝑘𝑜𝑓𝑓 𝜌𝑏(𝑡 ,𝑟 )

𝜌𝑏 (𝑡 ,𝑟 )=𝑘𝑜𝑛𝜌 𝑓 (𝑡 ,𝑟 )−(𝑘¿¿𝑜𝑓𝑓 +𝐴)𝜌𝑏(𝑡 ,𝑟 )¿

𝜌 𝑓 (𝑟 )=𝐶 (h ,𝑅)∙𝐵𝑒𝑠𝑠𝑒𝑙(0 , 𝑖√ 𝑘𝐷𝑟 )≡𝐶 (h ,𝑅) ∙𝑔𝜆(𝑟 ) 𝑘≡ 𝐴

𝑎𝑘𝑜𝑛

𝑘𝑜𝑓𝑓 +𝐴𝑎

Steady-state

Nucleator

Actin

Cross-Linker

R

Dependence of height trough

𝑟 /𝑅𝑡𝑖𝑝

CL

Du

vm

ub

uf

Finding - CL transportation model• The myosin bound to the

crosslinker can either walk on the actin with a velocity , or freely diffuse with diffusion coefficient

• Conservation of current:

,

The protrusion’s radius

• For small there might not be a stable non zero solution for

�̇�=𝑘𝑜𝑛 𝜌 𝑓 (𝑅 )𝑐𝑛𝑢𝑐−𝜂𝑛

creation annihilation

• The steady-state radius:

𝜌 𝑓 (𝑅𝑠𝑠 )=𝜂𝑛

𝑘𝑜𝑛𝑐𝑛𝑢𝑐

Protrusion radius

CL c

once

ntra

tion

(ρf)

𝜂𝑛𝑢𝑐 /𝑘𝑜𝑛𝑐𝑛𝑢𝑐

Combining both models for a constant polymerization rate

a

Protrusion radius

Prot

rusi

on h

eigh

t

• A single stable solution.• Constrain on the minimal possible height.

The protrusion’s radius for an increase in polymerization rate

a) Microvilli (control) b) overexpretion of Eps8 (a capper) reducing AZwaenepoel et al. (2012)

a b

a b

Polymerization rate

Prot

rusi

on ra

dius

Polymerization rate

Prot

rusi

on ra

dius

𝐴=𝐴0+𝐴𝑛𝑢𝑐𝑐𝑛𝑢𝑐

1+𝑐𝑛𝑢𝑐

Combining both models for an increase in polymerization rate

• There may be two stable solutions.• The radius may increase or decrease with the increase

of height, depending on the dependence of on .

h

A

Protrusion radius

Prot

rusi

on h

eigh

t𝐴(h) 𝐴(h ,𝑐𝑛𝑢𝑐)

dynamics

• Full numerical solutions and approximated analytical solutions for the dynamic growth for: – Constant polymerization rate with no initial tail – constant polymerization rate with a long initial tail – polymerization rate that increases with height – collapse following the termination of the

polymerization

Watanabe et al. (2010)

Dynamicsconst. polymerization rate

Prot

rusi

on h

eigh

t

Time

Long initial tail effect fo

rces

h

Prot

rusi

on h

eigh

t

Time

Analytical -

Analytical -

Numerical

a

forc

es

h

𝜏h=h𝑠𝑡/ 𝐴No initial tail Long initial tail

Gorelik et al. (2003)

h

forc

esProt

rusi

on h

eigh

t

Time

Bifurcation point effect

Dynamicsincreasing polymerization rate

Prot

rusi

on h

eigh

t

Time

Analytical - Analytical -

Numericald

DynamicsNo polymerization

Rzadinska et al (2004)Gorelik et al. (2003)

How can we explain the stereocilia formation and structure using our models?

Possible mechanisms for stereocilia multi heights

• Multi bifurcation. Few promoting proteins each with a different concentration profile

unlikely:– More rows requires more PP– Very sensitive to noise

• Interactions between the stereocilia (auto-oscillations, base angle, Ca+,..). – Different height exists even when the TL are KO– Can a feedback mechanism stabilize the heights?

Forc

es

h

• Base viscosity gradient – probably exists– How is the gradient chosen and maintain?

(a pre-existing state or a regulated one?)Furness et al. (2008)

A gradient in the viscosity – constant polymerization rate

• Multiple steps• No microvilli below some height can exist• The shortest row can be shorter

γc1

γc2 γc3

γc4 γc5

𝛾𝑐

lowhigh γc2 γc3 γc4 …

A gradient in the viscosity – increasing polymerization rate

• A jump of the height of the first row.

Fettiplace R and Hackney CM (2006)

h

A

𝛾𝑐 lowhighγc1 γc2 γc3 γc4 …

A reduction of the polymerization rate due to Eps8 KO

• A reduction in the heights, increase of radius and no jump.

Zampin et al. (2011))

Another example – microvilli

• Overexpression of the Eps81L capper results in a decrease of length of microvilli and an increase of the radius.

• In our model this can be explained by the reduction of polymerization rate.

Reduction of A

Zwaenepoel et al. (2012)

WT Eps8L1 overexpression

summary

• The shape of actin protrusion can be understood in terms of coupling between the biochemistry and physical forces in these systems

• Together with explanation of existing data our model predicts: – Effect of the cytoskeleton viscosity on the height.– The existence of a gradient in cuticular plate.– Reduction of height with increase in myosin I concentration.

• Our model provides a tool to analyze the roles of proteins in protrusions, based on the protrusion shape, instead of the very difficult direct measurements.

Thank you!