Alex Mogilner, UC Davis Leah Keshet, UBC, Canada Actin dynamics and the regulation of cell motility

Brief introduction to the cytoskeleton

PolymerActinMicrotubule Intermediate filament SubunitActin monomer Tubulin dimer  helical Elongate byPolymeriz. ? Bound nucleotide ATPGTPNone TreadmillingVery slowSlowNo 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):

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): lamellipod

Svitkina & Borisy (1999) J. Cell Biol., 145(5): µm

v v v v v Actin filament Pointed end Barbed end monomers

v v v v v Polymerization kinetics On rate: k on a Off rate: k off Barbed end grows rapidly k on a > > k off

v v v v v Barbed end capped quickly to prohibit explosive growth cappers Control of polymerization by capping

v v v v v Depolymerization dominates at pointed end Polymerization kinetics cont’d

v v v v v The actin monomers are modified by attached nucleotides (ATP, ADP) ADP-actin ATP-actin ATP-actin polymerizes fastest at barbed end

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

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

v v v v v 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 profilin10  M Total thymosin200  M Total Cofilin10  M Total F-Actin200  M Total G-actin~50  M Typical concentrations in lamellipod

F-Actin Cofilin-actin Profilin-actin (ADP) Profilin-actin (ATP) Thymosin actin

Filaments Spent monomers Activated monomers Reservoir

Filaments s 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): , 1999 v v v v v v v v v 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:

Length of actin monomer2.72 nm Abraham et al 1999 length of lamellipod~ 10  Abraham et al 1999 thickness of lamellipod0.1  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: 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 v v v v v

Work done to create gap Thermal energy An important ratio for thermal ratchets:

Probability of a gap forming on rate off rate (very small) Speed of motion of one filament barbed end

Neglecting depolymerization: To know the velocity of barbed end, we need to know the local actin monomer conc. Compare with “free polymerization velocity”:

Force per filament, f protrusion velocity, V Load-Velocity relation for single filament Free polymerization velocity Monomer size Load force Thermal energy v v v v v

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 actin30  2 /sMogilner/LEK 2002 thermal energy kT4.1 pN nmPeskin et al 1993 actin monomer on-rate11.6 /  M /sPollard 1986 capping rate4 /sSchafer et al 1996 Arp2/3 attachment rate1-10 /s speculative Arp2/3 diffusion coef3  2 /scalculated monomers in 1  M actin600/  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=0 motion Arp2/3 nucleation capping nucleation of barbed ends at front edge:

Filaments s pa  actin monomer exchange:

exchange & activation diffusion of monomers across lamellipod s p a  a(x,t) = conc. actin monomers (ATP form) s(x,t) = conc “spent” monomers (ADP form) Source From depolym Similar terms + Etc..

s p a  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 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 p a  For steady-state motion with D>>VL

X=0 front edge x=L rear of lamellipod net polymerization balances total depolymerization da/dx = J p /D at x=0 (BC) For steady-state motion:

 a profiles of s, p nearly constant with x s p

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

Net polymerization equal to depolym flux (J p =JL)

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

Total actin in all forms Amount not available for polymerization Time for monomer to depolymerize, become activated, and diffuse across lamellipod

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) “available actin”

Limiting case 2: Large barbed end density (B large) 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 ~  /s, in good agreement with experimental values Optimal barbed end density is roughly proportional to membrane resistance. For resistance force F~ pN/  optimal barbed end density is B ~ 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

Comet tails assembled by 0.5  bead with 37.5% of its surface area coated with ActA in 40% Xenopus egg extract Cameron,Svitkina,Vignjevic, Theriot, Borisy (2001) Current Biology 11(2),2001, Current Biology 11(2

Time sequence: phase-contrast and fluorescence microscopy images of an actin cloud that spontaneously breaks symmetry (a), and a bead propelled by polymerizing filaments after symmetry-breaking (b). The bead shown in b moves with a constant velocity of 0.12  m s – van Oudenaarden, Theriot (1999) Nat Cell bio 1(8)

Arp2/3 nucleation on and close to bead barbed ends B > B thresh at bead surface leads to motion direction of motion determined by direction of greatest force (from barbed ends) Adriana Dawes and Stan Maree Recent and ongoing work