Alex Mogilner, UC Davis

Leah Keshet, UBC, Canada

Actin dynamics and the regulation of cell motility

Brief introduction to the cytoskeleton

PolymerPolymer ActinActin MicrotubuleMicrotubule Intermediate Intermediate filamentfilament

Subunit Actin monomer

Tubulin dimer helical

Elongate by Polymeriz. Polymeriz. ?

Bound nucleotide

ATP GTP None

Treadmilling Very slow Slow No

Track for motors

Yes , 20 myosins

Yes, dynein, kinesin

No

Fluorescence micrograph in cells

TD Pollard (2003) Nature 422: 741

Heath & Holifeld (1993) in: Cell Behaviour, Adhesion, and Motility, Jones, Wigley, Warn, eds Soc Exp Biol Symp 47

Ridley AJ. (2001)J Cell Sci.:114(Pt 15):2713-22

Cell motility is a complex process involving adhesion to the surrounding substrate, forward extension (protrusion), and contraction that retracts the rear portion of the cell. Here we will consider only protrusion. The main component implicated in protrusion is ACTIN.

GOAL:

To relate the protrusion of the cell front to the underlying biochemistry of actin.

To derive some connection between the biochemical parameters* and the cell velocity.

* Binding rates, reaction rate constants, etc

Brief review of properties of actin

cell membrane

Front edge of cell

Actin cytoskeleton

Actin filament “barbed end” “pointed end”

Keratocyte: Top view Side view

1 µm

Svitkina & Borisy (1999) J. Cell Biol., 145(5): 1009-1026.

lamellipod

Actin filament

Pointed end Barbed end

monomers

Polymerization kineticsOn rate: kon a

Off rate: koff

Barbed end grows rapidly

kon a > > koff

Barbed end capped quickly to prohibit explosive growth

cappers

Control of polymerization by capping

Depolymerization dominates at pointed end

Polymerization kinetics cont’d

The actin monomers are modified by attached nucleotides (ATP, ADP)

ADP-actin

ATP-actin

ATP-actin polymerizes fastest at barbed end

As the filament ages, the ATP attached to its monomers is gradually hydrolized

newold

ADP-actin

Cofilin chops and fragments actin filaments, preferentially at older (ADP-actin) parts of an actin filament

There are mechanisms for converting “spent” ADP-actin monomers into their active form

ADP-actin

ATP-actin

Recycling mechanism

ADP-actin

ATP-actin

Profilin facilitates conversion of “spent” ADP-actin to “new” ATP-actin, preparing the monomers for polymerization

Recycling actin monomers

“spent”

“new”

Thymosin “sequesters” actin monomers

I.e. acts as a reservoir for spare actin, to avoid excessive polymerization

A reservoir for actin monomers

Total profilin 10 M

Total thymosin 200 M

Total Cofilin 10 M

Total F-Actin 200 M

Total G-actin ~50 M

Typical concentrations in lamellipod

F-ActinCofilin-actin

Profilin-actin(ADP)

Profilin-actin(ATP)

Thymosin actin

FilamentsSpent monomers

Activated monomers

Reservoir

Filamentss

pa

Rate constants for all these biochemical events (values known from the literature), will enter into the model.

Actin filament dynamics

Actin-binding proteins

•cap barbed end, cut, or degrade a filament

• crosslink filaments into bundles or networks

• nucleate new filaments by branching off a pre-existing filament

Arp2/3 (when activated) forms branch points and nucleates new, growing barbed ends.

T. M. Svitkina and G.G. Borisy, J. Cell Biol., 145(5): 1009-1026, 1999

Arp2/3

actin

We’ll be interested in the number of actin filament barbed ends that are at the cell edge pushing on the membrane.

We’ll take into account the nucleation of new ends by Arp2/3, as well as the (rapid) capping of loose ends inside the lamellipod.

Geometry and spatial aspects of actin dynamics

Keratocyte:

• shape does not change as cell moves

•adhesion/contraction relatively constant as cell moves.

• actin filaments stationary relative to substrate

• close to front edge, geometry “1D”: i.e., lamellipod is a thin sheet, most gradients into the cell (normal to edge)

x=L x=0

Simplified geometry

X=0 at cell edge; moving coordinate system;

Note direction of positive x into the cell

Abraham, Krishnamurthi, Taylor, Lanni (1999) Biophys J 77: 1721-1732

Length of actin monomer 2.72 nm Abraham et al 1999

length of lamellipod ~ 10 Abraham et al 1999

thickness of lamellipod 0.1 Abraham et al 1999

Typical size scales:

TD Pollard (2003) Nature 422: 741

Spatial features:

• Extracellular signals activate WASp at cell membrane. This then activates Arp2/3

• Arp2/3 nucleates new branches off actin filaments close to the cell membrane, creating new barbed ends that can grow & push.

• uncapping is enhanced, and capping is inhibited close to cell membrane.

• depolymerization dominates at older parts of filament, away from the cell membrane

Thus it is safe to assume that new barbed ends are mainly generated close to the front edge of the cell.

When a filament grows at the front edge, it has to push against a load force (e.g. membrane resistance).

Mogilner & Oster (1996) Biophys J, 71: 3030-3045

We use results of the thermal ratchet model for the extension of an actin filament under a load force.

Thermal fluctuations occasionally create a gap between the cell membrane and the tips of actin filaments. Monomers can fill in this gap to cause the displacement to persist.

Mogilner/Oster Thermal Ratchet Model:

Force per

filament, f

protrusion velocity, V

Thermal Ratchet Model

Mogilner & Oster

Tk

f

B

Work done to create gap

Thermal energy

An important ratio for thermal ratchets:

)( /off

Tkfon keakV B

Probabilityof a gap forming

on rate off rate

(very small)

Speed of motion of one filament barbed end

Tkfon

BeakV /

Neglecting depolymerization:

To know the velocity of barbed end, we need to know the local actin

monomer conc.

akV on0Compare with “free polymerization velocity”:

Force per

filament, f

protrusion velocity, V

)/exp(0 TkfVV B

Load-Velocity relation for single filament

Free polymerization velocity

Monomer size

Load force

Thermal energy

Actin monomers are the fuel that cause extension of the cytoskeleton, and drive the edge of the cell forward.

Our (1D) model aims to determine how much of that fuel is available at the front edge, and how this amount is controlled biochemically.

“ This system has several advantages for modeling: It runs at steady state, the inventory of core proteins is small, the structures and concentrations of these proteins are known, and biophysicists have measured many of the rate and equilibrium constants for the reactions.”

T.D. Pollard (2003) Nature 422, 741-5

Diffusion coeff actin 30 2/s Mogilner/LEK 2002

thermal energy kT 4.1 pN nm Peskin et al 1993

actin monomer on-rate 11.6 /M /s Pollard 1986

capping rate 4 /s Schafer et al 1996

Arp2/3 attachment rate 1-10 /s speculative

Arp2/3 diffusion coef 3 2/s calculated

monomers in 1 M actin 600/3 conversion factor

Examples of representative parameters

Assembling a 1D model for the protrusion of the cell based on actin filaments.

X=0 front edge

x=L rear of lamellipod

direction of motion

L a m e l l i p o d

Ingredients of the model:

• nucleation of barbed ends by Arp2/3 at front edge

• actin monomer exchange between various forms (with realistic values of biochemical rate-constants)

• diffusion of monomers across lamellipod

• assembly at front edge leading to protrusion

How does the protrusion velocity depend on the number of barbed ends ?

Main question addressed:

• B(t) = density barbed ends at edge

x=0motion

Bndt

dB

Arp2/3 nucleation

capping

nucleation of barbed ends at front edge:

Filamentss

pa

actin monomer exchange:

2

2

x

aD

t

a exchange& activation

diffusion of monomers across lamellipod

s

pa

• a(x,t) = conc. actin monomers (ATP form)

• s(x,t) = conc “spent” monomers (ADP form)

t

s Source From

depolym

Similar terms +

Etc..

s

pa

coordinate system moving with cell edge

Monomer exchange

Depolym. source

Boundary conditions cont’d:

No flux of monomers at rear:

da/dx =0, d/dx =0 etc at x=L

No flux of some forms of monomers at front:

d/dx =0 etc at x=0

assembly at front leading to protrusion (BC):

Flux of actin monomers arriving at edge

= Extension of barbed ends

VB

Vax

aD x

0)(

Conversion factor: monomer conc to filament length

Analysis of the model to investigate steady state propulsion, using realistic biochemical parameters

Over relevant spatial scale of lamellipod, diffusion dominates over the apparent convective flux:

D > V L

30 > 0.3 (10 ) 2/s

In the analytical realm, can neglect the first derivative terms for approx solns.

s

pa

For steady-state motion with D>>VL

X=0 front edge

x=L rear of lamellipod

net polymerization balances total depolymerization

da/dx = Jp /D at x=0 (BC)

For steady-state motion:

a

profiles of s, p nearly constant with x

sp

Analysis of the model for steady-state motion:

• for given biochemical parameters, profiles of s, p nearly constant. Model reduces to 2 eqns

Explicit analytical solutions found, and monomer concentration at edge, a(0) determined

0

0

332

2

332

2

akkdx

dD

Jakkdx

adD

0

0

332

2

332

2

akkdx

dD

Jakkdx

adD

0

0,

,0

0

Lx

Lxx

dx

ddx

da

D

JL

dx

da

Net polymerization equal to depolym

flux (Jp=JL)

From the model and biological parameter values, we determine the actin monomer concentration at membrane available to drive protrusion.

recyclingcofilindepolym

JAkk

ka

)()0(33

3

Total actin in all forms

Amount not available for

polymerization

Time for monomer to depolymerize, become activated, and diffuse across lamellipod

BBw

VV

)/exp(

Tk

fwLk

k

kAkV

Bonon

,/,1,3

3

Result:

Protrusion velocity depends on kinetic rate constants and on the number of barbed ends (B) pushing the membrane

F=100 pN/m

F=300 pN/ m

B, # / m

The model predicts:

• There is an optimal barbed end density

• The protrusion drops very rapidly for barbed ends below their optimal density, but drops more gradually above the optimum

There is an optimal density of barbed ends at the leading edge:

• too few: force to drive protrusion insufficient.

• too many: competition for monomers depletes monomer pool too quickly, slowing growth.

Limiting case 1: Small barbed end density (B small)

)/exp( Bw

VV

)/exp( BwA

kV on

“available actin”

Limiting case 2: Large barbed end density (B large)

B

VV

Velocity inversely proportional to B

What did we learn from the model?

BIOLOGICAL IMPLICATIONS

The model predicts:

• optimal barbed end density: suggests that careful regulation of the barbed ends is needed in the cell.

• means that there are optimal nucleation and capping rates

• For biological parameter values, V ~ 0.1- 0.3 /s, in good agreement with experimental values

• Optimal barbed end density is roughly proportional to membrane resistance. For resistance force

F~50-500 pN/

optimal barbed end density is B ~ 25-250 per

• at this optimum density, V is roughly inversely proportional to membrane resistance.

• A greater amount of total actin and a faster rate of actin turnover correlate positively with the rate of locomotion.(Evidence from McGrath et al. and by Loisel et al. who showed that there is a concentration of ADF/cofilin that is optimal for enhancing the rate of depolymerization.)

• Increasing the amount of thymosin slows down locomotion

Further model predictions:

“ These models identify the variables that limit the rate of movement, such as the concentration of actin bound to profilin.

In fact, when the concentration of unpolymerized actin is lowered by releasing an actin monomer sequestering protein in the cytoplasm, that part of a cell stops moving.”

TD Pollard (2003) Nature 422: 741-5

“ The models raise a number of questions that can be answered by further experimentation.”

• profilin-actin really limiting?

• do interactions of filaments with membrane inhibit capping, biasing forward motion?

• how are filaments reshaped at rear?

TD Pollard (2003) Nature 422: 741-5