Publications: Physical Model for the Geometry of Actin-Based Cellular Protrusions; Biophys. J. 107:576–587 A biophysical model for the staircase geometry of stereocilia; PLoS ONE. PONE-D-15-00778R1

1. Modeling the shapes of actin-
based protrusions
Thesis for the degree
Master of Science
by: Gilad Orly
Adviser: Prof. Nir Gov

2. Cell grow protrusions:

3. 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

4. Filopodia
Length (µm): 1-50
Diameter (µm): 0.1-0.2
Life time: 1– 100 min

5. Stereocilia
Length (µm): 1-50
Diameter (µm): 0.2-1
Life time: Organism life time
Cochlear stereocilia structure
Vestibular stereocilia
structure
Frolenkov 2004
Rzadzinska et al. 2004
Schwander, Kachar, Müller
2010
Furness 2008

6. 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)

7. 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)

8. 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)

9. 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)?

10. Actin polymerization
Interactions 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

11. "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

12. The source of protrusion’s growth –
Force equations (1/4)
• The Actin’s Pushing force:
𝐹𝑎 𝑡 = 𝛾𝑐 ∙ 𝑆𝑐(𝑡) ∙ 𝑣 𝑎(𝑡)
• Tail’s treadmilling velocity:
𝑣 𝑎 𝑡 = 𝐴 𝑡, ℎ − ℎ(𝑡)
• Tail’s surface area:
𝑆𝑐 𝑡 = 2𝜋
−𝑙(𝑡)
0
𝑅 𝑧, 𝑡 1 + 𝑅′(𝑧, 𝑡)2 𝑑𝑧
• Protrusion’s local radius
𝜕𝑅
𝜕𝑡
= [𝐴 𝑡, ℎ − ℎ] ∙
𝜕𝑅
𝜕𝑧
− 𝛽(𝑧, 𝑡, ℎ)
• A – polymerization velocity
• β – severing velocity
• R – actin bundle’s local radius
Sc(t)l(t)Fa
The cell: A viscous gel
(𝛾𝑐)
Z=h
Z=0
Z
R

13. The source of protrusion’s growth –
Force equations (2/4)
The membrane restoring force:
• Membrane deformation force (for a
cylindrical protrusion):
𝐹 𝑚𝑑 =
𝜎 ∙ 𝑅 + 𝜅 𝑅 ℎ ℎ ≤ ℎ 𝑐
𝜎 ∙ 𝑅 + 𝜅 𝑅 ℎ 𝑐 ℎ ≥ ℎ 𝑐
𝑐 ∙ 𝑒 𝜖𝑆 𝑚 ℎ ≫ ℎ 𝑐
• Friction force of a flowing membrane
(for a cylindrical protrusion):
𝐹𝑚𝑒 = 𝜇 ∙ ℎ(𝑡)
Sm(t)h(t)
Fm
The outer cell:
A viscous gel (𝛾 𝑚)
𝜇

14. 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:
𝐹𝑚𝑎 = 𝛼 ∙ # 𝑜𝑓 𝑚𝑦𝑜𝑠𝑖𝑛 = 𝛼 ∙ 2𝜋𝑅 ∙ 𝑓(ℎ)
• 𝑓(ℎ) depends on the distribution of the
myosin connectors. For a uniform
distribution 𝑓 ℎ ∝ ℎ
For a cylinder

15. • Equation of forces:
0 = 𝑚𝑎 = 𝐹𝑎 𝑡 − 𝐹 𝑚𝑑 𝑡 + 𝐹𝑚𝑎 𝑡 + 𝐹𝑚𝑒(t)
• Extracting ℎ(𝑡) we get:
ℎ =
𝛾𝑐∙𝑆 𝑐 ℎ 𝑡 ∙𝐴 ℎ 𝑡 ,𝑡 − 𝐹 𝑚𝑎 ℎ 𝑡 − 𝐹 𝑚𝑑 ℎ 𝑡
𝛾𝑐∙𝑆 𝑐 ℎ 𝑡 + 𝜇
• Steady-state height (ℎ = 0):
𝛾𝑐 ∙ 𝑆𝑐 ℎ ∙ 𝐴 ℎ = 𝐹𝑚𝑎 ℎ + 𝐹 𝑚𝑑
The source of protrusion’s growth –
Force equations (4/4)
𝑨(𝒉) , 𝜷(𝒉, 𝒛) , 𝑭 𝒎𝒂 (𝒉) determines the protrusion’s height
𝑨 – polymerization
velocity
𝜷 – severing velocity
𝑭 𝒎𝒂– actin- membrane
restoring force
𝑭 𝒎 – restoring force
𝜸 𝒄 – cell viscosity
𝑺 𝒄 – tail’s surface area
𝐴
𝛽
𝐹𝑚𝑎
Z=h
Z=0
Z
base
tip
Tail
𝑆𝑐
𝜸 𝒄
remainder

16. Proteins concentration along the
protrusion at the protrusion’s tip
𝐴 ℎ =
𝐴 𝑓 + 𝐴 𝑝 𝐾 𝑝 𝐶 𝑝(ℎ)
1 + 𝐾 𝑝 𝐶 𝑝 ℎ + 𝐾𝑖 𝐶𝑖(ℎ)
Prompting protein’s (PP)
concentration
Inhibiting protein’s (IP)
concentration
a
Proteinsconcentration
h
c
I
II
III
z/h
Proteinsconcentration
b
Proteinsconcentration
z/h0 0 11
a
a
Polymerizationrate
h
c
I
II
III
h
Polymerizationrate
b
Polymerizationrate
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 stereocilia
Schneider et al, Journal of Neuroscience, 2006 (a-e)
Sakaguchi et al, cell motility and the cytoskeleton, 2008 (f)
f

17. 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 (𝛽 ≈ 0 𝑎𝑡 ℎ > 0)
Yang C, Czech L, Gerboth S, Kojima
S, Scita G, et al. (2007)

18. Force balance
The restoring force must increase with h !!
Force generated by membrane-actin myosin
connectors
Protrusion height
forces

19. Steady state height of cylindrical protrusion as
a function of radiuscell viscosity
• In the case of a steep increase of A(h) there
can be a bifurcation point in ℎ(𝑅) and in ℎ(𝛾𝑐)
𝜸 𝒄
remainder
𝑨 – polymerization
velocity
𝜷 – severing velocity
𝑭 𝒎𝒂– actin- membrane
restoring force
𝑭 𝒎 – restoring force
𝜸 𝒄 – cell viscosity
𝑺 𝒄 – tail’s surface area
𝐴
𝛽
𝐹𝑚𝑎
Z=h
Z=0
Z
base
tip
Tail
𝑆𝑐
h
A
Protrusion radius
Protrusionheight
Protrusion radiusCell viscosity

20. 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

21. 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 𝑅𝑡𝑖𝑝
• We propose a model the rate of tip-complex
growth depends on 𝑅𝑡𝑖𝑝 while the rate of
removing filaments is constant:
𝑅 = 𝑓 𝑅 − 𝜂 𝑛
creation annihilation

22. 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 tip
𝜌 𝑓(𝑡, 𝑟) = 𝐷𝛻𝑟
2
𝜌 𝑓(𝑡, 𝑟) − 𝑘 𝑜𝑛 𝜌 𝑓(𝑡, 𝑟) + 𝑘 𝑜𝑓𝑓 𝜌 𝑏(𝑡, 𝑟)
𝜌 𝑏 𝑡, 𝑟 = 𝑘 𝑜𝑛 𝜌 𝑓 𝑡, 𝑟 − (𝑘 𝑜𝑓𝑓+𝐴)𝜌 𝑏(𝑡, 𝑟)
𝜌 𝑓 𝑟 = 𝐶(ℎ, 𝑅) ∙ 𝐵𝑒𝑠𝑠𝑒𝑙 0, 𝑖
𝑘
𝐷
𝑟 ≡ 𝐶(ℎ, 𝑅) ∙ 𝑔 𝜆(𝑟)
𝑘 ≡
𝐴
𝑎
𝑘 𝑜𝑛
𝑘 𝑜𝑓𝑓 +
𝐴
𝑎
Steady-state
Nucleator
Actin
Cross-Linker
R
Dependence of height
trough 𝐴(ℎ)
𝑟/𝑅𝑡𝑖𝑝
CL

23. Du
vm
ub
uf
Finding 𝐶 ℎ, 𝑅 - CL transportation
model
• The myosin bound to the
crosslinker can either walk on
the actin with a velocity 𝑣 𝑚, or
freely diffuse with diffusion
coefficient 𝐷 𝑢
• Conservation of current:
,

24. The protrusion’s radius
• For small ℎ there might not be a stable non zero solution
for 𝑅𝑡𝑖𝑝
𝑅 = 𝑘 𝑜𝑛 𝜌 𝑓 𝑅 𝑐 𝑛𝑢𝑐 − 𝜂 𝑛
creation annihilation
𝑐 𝑛𝑢𝑐 − 𝑛𝑢𝑐𝑙𝑒𝑎𝑡𝑜𝑟 𝑐𝑜𝑛𝑐𝑒𝑛𝑡𝑟𝑎𝑡𝑖𝑜𝑛
𝜂 𝑛 − 𝑎𝑛𝑛𝑖ℎ𝑖𝑙𝑎𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑒
• The steady-state radius:
𝜌 𝑓 𝑅 𝑠𝑠 =
𝜂 𝑛
𝑘 𝑜𝑛 𝑐 𝑛𝑢𝑐
Protrusion radius
CLconcentration(ρf)
𝜂 𝑛𝑢𝑐/𝑘 𝑜𝑛 𝑐 𝑛𝑢𝑐

25. Combining both models for a constant
polymerization ratea
Protrusion radius
Protrusionheight
• A single stable solution.
• Constrain on the minimal possible height.

26. The protrusion’s radius for an increase
in polymerization rate
a) Microvilli (control) b) overexpretion of Eps8 (a capper) reducing A
Zwaenepoel et al. (2012)
a b
a b
Polymerization rate
Protrusionradius
Polymerization rate
Protrusionradius
𝐴 =
𝐴0 + 𝐴 𝑛𝑢𝑐 𝑐 𝑛𝑢𝑐
1 + 𝑐 𝑛𝑢𝑐

27. 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 𝐴 on 𝑐 𝑛𝑢𝑐.
h
A
Protrusion radius
Protrusionheight
𝐴(ℎ) 𝐴(ℎ, 𝑐 𝑛𝑢𝑐)

28. 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

29. Watanabe et al. (2010)
Dynamics
const. polymerization rate
Protrusionheight
Time
Long initial
tail effect
forces
h
Protrusionheight
Time
Analytical - A ≫ ℎ
Analytical - A~ℎ
Numerical
a
forces
h
𝜏ℎ = ℎ 𝑠𝑡/𝐴
No initial tail Long initial tail
Gorelik et al. (2003)

30. h
forces
Protrusionheight
Time
Bifurcation
point effect
Dynamics
increasing polymerization rate

31. Protrusionheight
Time
Analytical - ℎ ≫ 𝛽
Analytical - ℎ ≪ 𝛽
Numerical
d
Dynamics
No polymerization
Rzadinska et al (2004)
Gorelik et al. (2003)

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

33. 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?
Forces
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)

34. 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 γc2γc3γc4…

35. 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
lowhigh
γc1γc2γc3γc4…

36. 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))

37. 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

38. 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.

39. Thank you!