Pollard and Berro 1 8/25/08

MATHEMATICAL MODELS AND SIMULATIONS OF CELLULAR PROCESSES BASED ON

ACTIN FILAMENTS

Thomas D. Pollard*1 and Julien Berro

1, 2, 3

1Departments of Molecular, Cellular and Developmental Biology, Cell Biology, and Molecular

Biophysics and Biochemistry, Yale University, New Haven, CT 06520-8103 2Institut Camille Jordan, UMR CNRS 5208 and

3Centre de Génétique Moléculaire et Cellulaire, UMR

CNRS 5534, Université Lyon 1, F-69622 Villeurbanne cedex, France

Running head: Modeling of actin structures

Address correspondence to: Thomas D. Pollard, KBT548, Yale University, New Haven, CT 06520-8103;

Fax: 203-432-6161: E-mail: thomas.pollard@yale.edu

Abstract: Actin filaments help to maintain the

physical integrity of cells and participate in

many processes that produce cellular

movements. Studies of the processes that depend

on actin filaments have progressed to the point

where mathematical models and computer

simulations are an essential part of the

experimental toolkit. These quantitative models

integrate knowledge about the structures of the

key proteins, rate and equilibrium constants for

the reactions for comparison with a growing

body of quantitative measurements of dynamic

processes in live cells. Models and simulations

are essential, since it is impossible to appreciate

by intuition alone the properties that emerge

from a network of coupled reactions, particularly

when the system contains many components and

force is one of the parameters.

We use a few examples to illustrate how

mathematical models advance understanding of

the actin system from side chain motions of

proteins to the behavior of whole cells. Readers

will find references to experimental work in the

papers cited. Diverse methods (Supplemental

Box) are required given the range of complexity

(single proteins to cells), dimensions (10-9

to 10-4

m) and time (10-12

to 102 s).

Actin molecule and polymerization

Internal motions of actin monomers -- Actin

consists of four subdomains surrounding a cleft

that binds ATP or ADP (Figure 1). In molecular

dynamics (MD) simulations the DNase loop in

subdomain 2 is the most flexible part of the

protein with a weak tendency to form a !- or "-

helix (1,2), so crystal contacts may stabilize the

helix when it is present. The nucleotide-binding

cleft of actin remains closed in MD simulations

with bound ATP or ADP or without bound

nucleotide (1-3), while the cleft of actin-related

protein, Arp3, tends to open. This difference depends

on a C-terminal extension of Arp3, which fits into the

groove between subdomains 1 and 3 and stabilizes

the open conformation (2). Profilin binding in this

groove also promotes cleft opening and nucleotide

exchange (4).

Actin filament nucleation and elongation -- Pure

actin monomers (Figure 1) spontaneously polymerize

into helical filaments under physiological conditions.

Kinetic simulations of the complete time course of

polymerization of actin monomers showed that

formation of dimers and trimers is extremely

unfavorable (5). Brownian dynamics simulations

showed that the long pitch (end to end) dimer is

favored over the short pitch dimer and that the third

subunit binds laterally to form a trimer nucleus (6).

Brownian dynamics simulations showed that

electrostatic forces favor elongation at the barbed end

over the pointed end as observed (7).

Effect of bound nucleotide on polymerization –

The nucleotide bound to actin influences every aspect

of polymerization. In cells actin monomers are

saturated with ATP. When incorporated into a

filament actin hydrolyzes bound ATP 40,000-fold

faster than monomeric actin (8). In spite of enough

crystal structures and MD simulations to formulate an

hypothesis (2,9,10) for conformational changes

associated with the ATPase cycle, we do not

understand how polymerization stimulates hydrolysis

or how the presence of #-phosphate influences the

affinity actin for profilin, thymosin-ß4 and cofilin.

At the fast-growing barbed end of filaments

ADP-actin binds slower and dissociates faster than

ATP-actin. ADP-Pi-actin associates only slightly

faster than ADP-actin but dissociates much slower

(11). All of these reactions are slower at the pointed

ends of filaments. An enduring mystery has been how

ATP hydrolysis by polymerized actin makes the

critical concentration for elongation about 10 times

http://www.jbc.org/cgi/doi/10.1074/jbc.R800043200The latest version is at JBC Papers in Press. Published on October 20, 2008 as Manuscript R800043200

Copyright 2008 by The American Society for Biochemistry and Molecular Biology, Inc.

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 2 8/25/08

more favorable at barbed ends than pointed

ends. Analytical models and Monte Carlo

simulations consistent with experimental data

show that the difference arises from faster

dissociation of phosphate from ADP-Pi-subunits

near both ends than from the interior subunits

and lower affinity of phosphate for terminal

subunits at pointed ends than barbed ends (11).

At steady state with ATP in the medium these

reactions coupled with random ATP hydrolysis

and Pi release in filaments produce gradients of

subunits containing bound ATP, ADP-Pi and

ADP from the ends toward the interior of

filaments (12), small length fluctuations at

barbed ends (13, 14) and net addition of subunits

at barbed ends balanced by net loss of subunits

at pointed ends (~0.1 s-1

).

Force production by polymerizing actin --

Experiments and theory agree that polymerizing

actin filaments produce a few piconewtons of

force. Single filaments as short as 700 nm long

buckle when they elongate between two

attachment sites (15). Given the stiffness of actin

filaments, the force is >1 pN (16).

Pioneering studies (17, 18) proposed that

elongating filaments move objects by a

Brownian ratchet mechanism. The original

model considered how elongation of rigid

filaments could rectify the thermal motion of a

diffusing object. A later elastic Brownian ratchet

model considered not only the motion of the

object but also the thermal motion of flexible

elastic filaments (Figure 2C). When diffusion

opens a gap between the end of the polymer and

an object, insertion of a subunit prevents the

object from reentering this space. The elongation

rate of a single filament against such a load is

the elongation rate of a free filament times the

probability that a gap exists between the tip and

the load, which is given by a Boltzmann term e-

!E/kT where !E is the energy required to create

the gap, k is Boltzmann’s constant and T is

absolute temperature. Calculation of the

distribution of positions of the object over time

gave a logarithmic dependence of force (in the

pN range) on the rate of elongation (in the range

of 0-110 subunits/second). The velocity depends

on actin monomer concentration, elongation rate

constant, length of the filaments and angle of

incidence between the filament and the barrier,

with an optimum angle near 45°. Remarkably

filament growth and branching at the leading edge of

motile cells selects filaments oriented at angles near

45° relative to the membrane (19). Note that long

filaments buckle, filaments parallel to the barrier

exert no force and bending of filaments normal to the

barrier opens only a small gap for elongation.

Brownian dynamics simulations (20) and Monte

Carlo simulations (21) confirmed the general features

of the elastic Brownian ratchet mechanism.

Physical properties of actin filaments -- Massive

all atom MD simulations and normal mode analysis

of coarse-grained models (3) reproduced the

observed stiffness of ATP- and ADP-actin filaments.

The "-helical DNase loop assumed for ADP-actin

has weaker short pitch interactions and no long pitch

interactions, accounting for the greater flexibility of

ADP-actin filaments. However, we do not understand

how #-phosphate dissociation from filaments alters

their structure and influences subunit reactions at the

ends.

Proteins that regulate actin polymerization

Cells use dozens of proteins to regulate the time

and place of actin polymerization. Other proteins

shape and reinforce structures composed of actin

filaments. Here we use two proteins that direct actin

assembly and one that promotes disassembly to

illustrate how modeling contributes to research on

actin-binding proteins.

Arp2/3 complex -- Arp2/3 complex nucleates

actin filaments as 78° branches on the sides of pre-

existing actin filaments. Five protein subunits hold

the two actin related proteins, Arp2 and Arp3, close

together but separated enough to prevent them from

initiating an actin filament. Kinetic simulations (22)

based on a partial set of rate constants showed that

the favored pathway begins with a nucleation-

promoting factor such as WASp binding actin and

then Arp2/3 complex. This ternary complex has no

nucleation activity until it binds very slowly to the

side of a filament. Then a daughter filament grows at

its free barbed end from the side of the “mother”

filament (Figure 2).

Formins -- Formins are homodimers with

multiple domains, including formin homology-2

(FH2) domains that associate with barbed ends of

actin filaments (reviewed by (23)). Formins stimulate

formation of unbranched actin filaments in cables,

filopodia and cytokinetic contractile rings.

Simulations of experimental data suggest that FH2

domains nucleate filaments by stabilizing actin

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 3 8/25/08

dimers (24) and show that free actin monomers

account for all nucleation in the presence of

profilin (25).

Donut-shaped FH2 dimers (Figure 3)

encircle an actin filament (26) and remain

associated with a growing barbed end through

thousands of cycles of subunit addition at rates

up to 100 s-1

(27, 28), (25). FH2 domains slow

elongation of barbed ends by 10-99%. Much

work remains to determine how FH2 domains

track reliably on growing barbed ends. One idea

(26) with solid theoretical support (29) is that

the leading FH2 domain steps off the end before

the next actin subunit binds (Figure 3A lower).

Alternatively, the step may occur after each new

actin subunit adds (Figure 3A upper) (25). One

might expect an FH2 dimer to rotate along the

path of the growing actin helix, but this is not

observed if the formin and distal parts of the

filament are both anchored to a surface (15).

One idea is that an FH2 domain tracks with the

growing helix for about six subunits and then

steps around the filament axis in the opposite

direction to relieve any accumulated torsion

(30).

Flexible FH1 domains adjacent to FH2

domains contain multiple poly-proline

sequences that bind complexes of profilin-actin

and transfer actin onto the barbed end of the

filament (Figure 3B) at rates >1000 s-1

(31).

Transfer is more favorable from proximal than

distal polyproline sequences (25).

Theoretical work (32) showed that

elongation in association with a protein like an

FH2 domain can produce more force if subunit

addition is coupled to hydrolysis of ATP bound

to actin. However, formins can use ADP-actin

monomers to elongate filaments (28) and

elongation rates mediated by FH1FH2 constructs

with ATP-actin monomers can exceed the ATP

hydrolysis rates by 300-fold, so any coupling

must be indirect.

Cofilin – ADF/cofilin proteins stimulate

actin filament turnover. ADF/cofilins bind ADP-

actin subunits with higher affinity than ATP- or

ADP-Pi-actin subunits and sever filaments.

Stochastic simulations (33) and mathematical

analysis (34) showed that ATP hydrolysis and

phosphate dissociation by actin subunits leads to

a gradient of ADF/cofilin severing activity from

the oldest to the youngest part of a filament.

This aging process can explain the rapid turnover and

large stochastic fluctuations in the length of growing

filaments observed experimentally.

Models of actin-based cellular motility

Polymerization of branched actin filaments

pushes the plasma membrane forward at the leading

edge of motile cells. Variations of the dendritic

nucleation hypothesis (Fig. 2) are the basis for

models of these processes. Nucleation promoting

factors associated with the inside of the plasma

membrane are proposed to activate Arp2/3 complex

to form many generations of growing branches,

which produce force an elastic Brownian ratchet (17)

(18). Capping proteins terminate branch growth and

all of the proteins recycle back to the cytoplasmic

pool.

Analytical and numerical solutions of a system of

partial differential equations describing the dendritic

nucleation hypothesis operating at steady state

produced several insights (35). All of the filaments

were assumed to share the load equally and actin

subunits diffused after disassembly. The model was

approximately one-dimensional in space. When the

concentration of growing filaments is high,

polymerization consumes actin monomers and

creates a modest sink of monomers at the leading

edge, such that diffusion of actin monomers bound to

thymosin-ß4 and profilin to the leading edge is rate

limiting for movement. The rate of movement

depends on the density of growing filaments reaching

an optimum of about 0.2 "m/s with 20-60 filaments

per "m, depending on the resistance. Resistance

slows polymerization at suboptimal end densities and

monomer depletion slow polymerization at high end

densities.

Stochastic models consider each individual

filament in a heterogeneous population. This

approach allows consideration of how geometry

determines the work (force x distance) performed by

each filament. Carlsson (36) made Monte Carlo

simulations of the growth of networks of rigid,

branched actin filaments against a rigid obstacle,

using the reactions in the dendritic nucleation model

and taking into account the positions of every subunit

in each filament. He assumed a uniform

concentration of reactants, branch formation only

near the obstacle (at rates to give spacing similar to

those in cells) and resistance to polymerization

similar to a Brownian ratchet. Simulations produced

different geometries depending on other assumptions

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 4 8/25/08

such as branching from the sides of filaments or

only at barbed ends. Velocity was remarkably

independent of resistance (as observed in

experiments with Listeria), because resistance

bends the leading filaments, allowing more

filaments to contact the obstacle.

Schaus et al. made a stochastic 2D

simulation of every filament in a dendritic

nucleation model under the plasma membrane

with (19), an extension of Maly and Borisy (37)

without their simplifications. They assumed

actin monomer diffusion, spontaneous formation

of branches at some distance from the previous

branch, a zone with “protection from capping”

within 5 nm of the plasma membrane (essential

but unproven), 70° branches (not critical) and

elastic behavior of both the filaments (assumed

stiffness critical) and the membrane (assumed

stiffness not critical). Starting with randomly

oriented filaments, the mechanism generated a

self-organized network of branched filaments

strongly oriented at ±35° to the plasma

membrane as observed by electron microscopy.

This resulted from capping being faster than

branching for filaments of other orientations.

The model moved at 8 "m/minute and was able

to change direction in about a minute after 15

generations of branches. The maximum velocity

was achieved if the filaments shared the work

equally, but this is impossible with stiff

filaments and hard objects. If elongating

filaments are flexible, they can bend to various

degrees to share the load, push rapidly and

approach perfect thermodynamic efficiency.

Flexibility of the membrane contributes

effectively to load sharing. Tethers between the

load and the filaments reduce performance. The

performance of such a system depends on the

size of the subunits in the polymer and the size

of the actin molecule is nearly ideal for a

Brownian ratchet mechanism driven by

filaments with the physical properties of actin.

Models have also addressed other

remarkable features of the leading edge,

nucleation of most filaments very near the

plasma membrane and growth of filaments in a

plane only 200 nm thick oriented at about ±35°

relative to inside of the membrane. To restrict

nucleation to the front of the cell, Atilgan et al

(38) proposed that nucleation promoting factors

concentrate where the plasma membrane has the

smallest radius of curvature, but the relevant

transmembrane anchors have yet to be identified.

Maly and Borisy (37) proposed that contact of

growing barbed ends with the plasma membrane

inhibits capping. This favors elongation of filaments

growing toward the front and termination of

filaments growing in other directions. Their model

correctly reproduced the distribution of orientations

of filaments relative to the leading edge.

A simple analytical model (39) accounts for

several features of motile keratocytes including

constant surface area, limited variation of shapes,

constant velocity and ability to recover these

characteristics after an insult, which rounds up the

cell. The model assumes protrusion force produced

by actin polymerization against a uniform surface

tension in a fluid but inextensible membrane. A key

feature is a gradient of barbed ends (measured with a

fluorescent natural product) from the middle of the

leading edge to the margins of the cell, where the

force produced by actin polymerization matches the

tension resisting movement. The biochemical origin

of this gradient is not known.

Models of actin-based bacterial motility

Certain intracellular bacteria usurp the cellular

actin system to assemble a comet tail of filaments for

propulsion. For example, ActA on the surface of

Listeria is a nucleation-promoting factor for Arp2/3

complex. ActA attached to plastic beads also

produces actin comet tails in cellular extracts or

mixtures of purified proteins. Tethers to the actin

filament comet tail limit diffusion of the bacterium.

Both deterministic and stochastic models show that

transient tethers are compatible with an elastic

Brownian ratchet (40).

Stochastic object-oriented simulations of

dendritic nucleation by ActA on a bacterium (41)

followed reactions of thousands of molecules in short

time steps. Collisions produced forces, which were

dissipated by movements apart, but the model did not

include force-velocity relationships. Pauses between

intervals of constant velocity emerged in complicated

ways from the ensemble of reactions rather than from

a fundamental step such as subunit addition to barbed

ends.

Macroscopic theories consider the tangled actin

filaments at the rear of Listeria as a continuous

viscoelastic gel. Stress accumulates in the gel as

polymerization takes place at the surface of a

bacterium. Release of this stress can produce

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 5 8/25/08

sustained or intermittent movements as

observed. Mathematical analysis of an

expanding gel model gave a non-linear force-

velocity relationship and simulations reproduced

the hopping movements of bacteria (42). Lipid

vesicles (43) and oil droplets (44) coated with

ActA produce comet tails, which compress the

sides and pull at the rear of these spherical

particles. A model with compression by a

viscoelastic gel accounts for the observed shapes

of these particles (44).

Filopodia

Filopodia (also called microspikes or

microvilli) are slim projections of the plasma

membrane supported by a bundle of actin

filaments, similar to a finger in a glove. In some

cases the filaments turn over by addition of

subunits to the barbed ends of the filaments at

the tip balanced by loss at the base of the bundle.

Single filaments cannot support the forces (tens

of pN) required to protrude the membrane, but

packing N filaments into a bundle increases their

stiffness by a factor of N to N2, depending on the

extent of crosslinking and breaks in the

filaments (45). Elongation of many barbed ends

depletes the local pool of monomeric actin,

which is limited by diffusion along the length of

the filopodium and restricted by the close

apposition of the membrane (46). A calculation

made before formins were implicated showed

that 30 filaments are optimal to produce a

process a few "m long (46), similar to numbers

observed in cells. Crosslinking restricts the thermal

motion of the barbed ends, so the ability of the

filaments to grow against the membrane is attributed

to fluctuations of the membrane (47).

Cytokinesis

It has been appreciated for three decades that a

contractile ring of actin filaments and myosin-II is

responsible for cleavage of cells at the end of mitosis,

but progress on mechanisms awaited extensive

inventories of the numerous participating proteins

from genetics in yeast and RNAi experiments in flies

and worms. Both yeast and animal cells depend on

formins associated with the plasma membrane for

assembly of the actin filaments. Myosin-II might

simply capture these filaments and pull them into a

ring, but Monte Carlo simulations of contractile ring

assembly in fission yeast ruled out a simple search

and capture mechanism (48). Further experiments

using fission yeast suggested that connections

between growing actin filaments and clusters of

myosin-II break about every 20 seconds. Simulations

of models including search, capture, traction and

release account for cellular observations (48).

Analytical solutions to partial differential equations

show that force between clusters of myosin around

the mid-section of a cylindrical cell can generate a

contractile ring and cleavage furrow (49). The force

generated by such a bundle of actin filaments and

myosin depends on the lengths of the filaments and

the extent of crosslinking between the filaments (50).

References

1. Zheng, X., Diraviyam, K., and Sept, D. (2007) Biophys J 93(4), 1277-1283

2. Dalhaimer, P., Pollard, T. D., and Nolen, B. (2008) J. Molec. Biol. 376, 166-183

3. Chu, J. W., and Voth, G. A. (2005) Proc Natl Acad Sci U S A 102(37), 13111-13116

4. Minehardt, T. J., Kollman, P. A., Cooke, R., and Pate, E. (2006) Biophys J 90(7), 2445-2449

5. Frieden, C. (1983) Proc. Natl. Acad. Sci. USA 80, 6513-6517

6. Sept, D., and McCammon, J. A. (2001) Biophys. J. 81, 667-674

7. Sept, D., Elcock, A. H., and McCammon, J. A. (1999) J. Molec. Biol. 294(5), 1181-1189

8. Rould, M. A., Wan, Q., Joel, P. B., Lowey, S., and Trybus, K. M. (2006) J Biol Chem 281(42),

31909-31919

9. Wriggers, W., and Schulten, K. (1999) Proteins 35(2), 262-273

10. Nolen, B. J., and Pollard, T. D. (2007) Mol Cell 26(3), 449-457

11. Fujiwara, I., Vavylonis, D., and Pollard, T. D. (2007) Proc Natl Acad Sci U S A 104(21), 8827-

8832

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 6 8/25/08

12. Bindschadler, M., Osborn, E. A., Dewey, C. F., Jr., and McGrath, J. L. (2004) Biophys J 86(5),

2720-2739

13. Vavylonis, D., Yang, Q., and O'Shaughnessy, B. (2005) Proc Natl Acad Sci U S A 102(24), 8543-

8548

14. Stukalin, E. B., and Kolomeisky, A. B. (2006) Biophys. J. 87, 3679-3689

15. Kovar, D. R., and Pollard, T. D. (2004) Proc Natl Acad Sci U S A 101(41), 14725-14730

16. Berro, J., Michelot, A., Blanchoin, L., Kovar, D. R., and Martiel, J. L. (2007) Biophys J 92(7),

2546-2558

17. Peskin, C. S., Odell, G. M., and Oster, G. F. (1993) Biophys. J. 65(1), 316-324

18. Mogilner, A., and Oster, G. (1996) Biophys J 71(6), 3030-3045

19. Schaus, T. E., Taylor, E. W., and Borisy, G. G. (2007) Proc Natl Acad Sci U S A 104(17), 7086-

7091

20. Carlsson, A. E. (2000) Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 62(5 Pt B),

7082-7091

21. Burroughs, N. J., and Marenduzzo, D. (2005) J Chem Phys 123(17), 174908

22. Beltzner, C. C., and Pollard, T. D. (2008) J Biol Chem 283(11), 7135-7144

23. Goode, B. L., and Eck, M. J. (2007) Annu Rev Biochem 76, 593-627

24. Pring, M., Evangelista, M., Boone, C., Yang, C., and Zigmond, S. H. (2003) Biochemistry 42(2),

486-496

25. Paul, A. S., and Pollard, T. D. (2008) Curr Biol 18(1), 9-19

26. Otomo, T., Tomchick, D. R., Otomo, C., Panchal, S. C., Machius, M., and Rosen, M. K. (2005)

Nature 433(7025), 488-494

27. Romero, S., Le Clainche, C., Didry, D., Egile, C., Pantaloni, D., and Carlier, M. F. (2004) Cell

119(3), 419-429

28. Kovar, D. R., Harris, E. S., Mahaffy, R., Higgs, H. N., and Pollard, T. D. (2006) Cell 124(2), 423-

435

29. Shemesh, T., and Kozlov, M. M. (2007) Biophys J 92(5), 1512-1521

30. Shemesh, T., Otomo, T., Rosen, M. K., Bershadsky, A. D., and Kozlov, M. M. (2005) J Cell Biol

170(6), 889-893

31. Vavylonis, D., Kovar, D. R., O'Shaughnessy, B., and Pollard, T. D. (2006) Mol Cell 21(4), 455-

466

32. Dickinson, R. B., and Purich, D. L. (2006) Biophys J 91(4), 1548-1563

33. Michelot, A., Berro, J., Guerin, C., Boujemaa-Paterski, R., Staiger, C. J., Martiel, J. L., and

Blanchoin, L. (2007) Curr Biol 17(10), 825-833

34. Roland, J., Berro, J., Michelot, A., Blanchoin, L., and Martiel, J. L. (2008) Biophys. J. 94, 2082-

2094

35. Mogilner, A., and Edelstein-Keshet, L. (2002) Biophys J 83(3), 1237-1258

36. Carlsson, A. E. (2001) Biophys J 81(4), 1907-1923

37. Maly, I. V., and Borisy, G. G. (2001) Proc Natl Acad Sci U S A 98(20), 11324-11329

38. Atilgan, E., Wirtz, D., and Sun, S. X. (2005) Biophys J 89(5), 3589-3602

39. Keren, K., Pincus, Z., Allen, G. M., Barnhart, E. L., Marriott, G., Mogilner, A., and Theriot, J. A.

(2008) Nature 453, 475-480

40. Mogilner, A., and Oster, G. (2003) Biophys J 84(3), 1591-1605

41. Alberts, J. B., and Odell, G. M. (2004) PLoS Biol 2(12), e412

42. Gerbal, F., Chaikin, P., Rabin, Y., and Prost, J. (2000) Biophys J 79(5), 2259-2275

43. Upadhyaya, A., Chabot, J. R., Andreeva, A., Samadani, A., and van Oudenaarden, A. (2003)

Proc Natl Acad Sci U S A 100(8), 4521-4526

44. Boukellal, H., Campas, O., Joanny, J. F., Prost, J., and Sykes, C. (2004) Phys Rev E Stat Nonlin

Soft Matter Phys 69(6 Pt 1), 061906

45. Bathe, M., Heussinger, C., Claessens, M. M., Bausch, A. R., and Frey, E. (2008) Biophys. J. 94,

2955-2964

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 7 8/25/08

46. Mogilner, A., and Rubinstein, B. (2005) Biophys J 89(2), 782-795

47. Atilgan, E., Wirtz, D., and Sun, S. X. (2006) Biophys J 90(1), 65-76

48. Vavylonis, D., Wu, J.-Q., Hao, S., O'Shaughnessy, B., and Pollard, T. D. (2008) Science 319, 97-

100

49. Shlomovitz, R., and Gov, N. S. (2008) Biophys J 94(4), 1155-1168

50. Carlsson, A. E. (2006) Phys Rev E Stat Nonlin Soft Matter Phys 74(5 Pt 1), 051912

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 8 8/25/08

Figure legends

FIGURE 1. Ribbon diagram of the actin molecule based on pdb file 1ATNM and a space filling model of

an actin filament. Numbers 1-4 indicate the four subdomains. Images from T.D. Pollard and W.C.

Earnshaw, Cell Biology, second edition, W.B. Saunders, 2007.

FIGURE 2. Biochemical mechanism of actin-based cellular motility. A, Transmission electron

micrograph of the actin fllament network at the leading edge of a keratocyte from the work of Tanya

Svitkina and Gary Borisy. The cell was fixed while moving upward in this orientation. After removal of

the plasma membrane and soluble components, the branched actin filament network was rotary

shadowed. B, drawing of a motile keratocyte. C, dendritic nucleation hypothesis for protrusion of the

leading edge. A nucleation-promoting factor (purple) brings together an ATP-actin monomer and Arp2/3

complex. Binding of this inactive ternary complex to the side of a pre-existing filament activates the

formation of an actin filament branch, which grows from the side of the mother filament at an angle of

78°. Thermal motion of the membrane (1) or the filament tip (2) creates gaps between the barbed end of

the filaments and the membrane, allowing actin subunits bound to profilin to elongate the filaments and

push the membrane by a Brownian ratchet mechanism. Capping protein terminates elongation by

blocking barbed ends. Hydrolysis of ATP bound to ATP-actin subunits (yellow) creates ADP-Pi-actin

subunits (orange), which dissociate phosphate to become ADP-actin subunits (maroon). ADF/cofilin

targets ADP-actin filaments for severing and depolymerization. Profilin catalyzes exchange of ADP for

ATP on dissociated actin monomers, recycling ATP-actin for further rounds of polymerization. Modified

from T.D. Pollard and W.C. Earnshaw, Cell Biology, second edition, W.B. Saunders, 2007.

FIGURE 3. Actin filament elongation mediated by a formin. A, Two pathways of actin subunit addition.

In the upper pathway an actin subunit associates with the barbed end before the formin FH2 domain steps

onto the new subunit. In the lower pathway the formin steps off the end before the new subunits binds. B,

Transfer of actin anchored by profilin on an FH1 domain onto the barbed end of the filament followed by

dissociation of profilin from actin (and from the polyproline sequence of FH1 in this drawing).

NOTE: this figure should be printed in a single column

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 9 8/25/08

NOTE: this figure should be printed in two columns

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 10 8/25/08

NOTE: this figure should be printed in a single column

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 11 8/25/08

Supplemental materials for the mini-review:

MATHEMATICAL MODELS AND SIMULATIONS OF CELLULAR PROCESSES BASED ON

ACTIN FILAMENTS

Thomas D. Pollard*1 and Julien Berro

1, 2, 3

1Departments of Molecular, Cellular and Developmental Biology, Cell Biology, and Molecular

Biophysics and Biochemistry, Yale University, New Haven, CT 06520-8103 2Institut Camille Jordan, UMR CNRS 5208 and

3Centre de Génétique Moléculaire et Cellulaire, UMR

CNRS 5534, Université Lyon 1, F-69622 Villeurbanne cedex, France

Running head: Modeling of actin structures

Address correspondence to: Thomas D. Pollard, KBT548, Yale University, New Haven, CT 06520-8103;

Fax: 203-432-6161: E-mail: thomas.pollard@yale.edu

NOTE: this box will appear in the printed version of Ravi Iyengar’s introduction to the series of mini-

reviews on mathematical modeling.

Descriptions of mathematical methods

Mathematical models and simulations help biologists design experiments, analyze data and test

mechanistic hypotheses about individual components and large systems of proteins in live cells. Modelers

use diverse methods (reviewed in (1,2)), since biological processes occur over a huge range of complexity

(single protein molecules to whole cells), dimensions (Angstroms to tens of micrometers) and time

(picoseconds to minutes).

Modeling biological processes requires equations to describe the properties of the systems. One can

use mathematical analysis to study the properties of the equations, extract relationships between

variables and parameters or to delimit the conditions of validity of a model. Often it is simpler to use

computers to carry out numerical simulations of the equations by calculating changes during a

succession of tiny steps in time or space.

Ordinary Differential Equations (ODE) involve variables and their successive derivatives with

respect to only one variable, usually time or one of the space coordinates. Partial Differential Equations

(PDE) consider more than one variable, such as time and several space coordinates. These are

deterministic methods, which employ continuous variables, so ODEs and PDEs are usually used for

modeling macroscopic average variables where stochastic effects are negligible. For example, ODEs are

used to model classical kinetics of bulk samples (3). PDEs are used if space and time matter, as in

diffusion and transport (4). PDEs are also used to model mechanical properties of bulk materials such as

actin gels (5). ODEs and PDEs can also be used to describe probability distributions, such as the

nucleotide states of subunits in filaments (6).

Stochastic simulations (also called Monte-Carlo simulations) take into account the variability and

the randomness of the fate of ensembles of molecules or when their precise spatial distribution is crucial

for the process, such as modeling the position and fate of each actin subunit involved with moving a

bacterium (7). Such simulations where large numbers of particles interact randomly to give rise to

macroscopic behavior are also called Multi-Agent Systems (MAS). Stochastic simulations require more

computer power than ODEs and PDEs, they are thus often used for systems with modest numbers of

molecules (thousands), in a small volume and for short times.

These two approaches are complementary. Differential equations are often used in the mathematical

analysis of stochastic processes. For example, Peskin et al. (8) used a mathematical analysis to calculate

the force produced by polymerization of single actin filaments, whereas Carlsson (9) used a Brownian

dynamics simulation to arrive at similar conclusions.

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Pollard and Berro 12 8/25/08

Two main methods are used to simulate conformational changes within macromolecules or

interactions between molecules. Molecular Dynamics (MD) simulations are based on Newton’s law of

motion taking into account mechanical, electrostatic and van der Waals force fields between the atoms

within a molecule and with the solvent. Equations of motion are integrated in femtosecond steps. Because

of the complexity of a system of many atoms, such as a macromolecule, MD is usually used to simulate

small conformation changes on a time scale of picoseconds to nanoseconds, beginning from a known

atomic structure, such as domain motions of proteins (10-12). Brownian Dynamics (BD) is a stochastic

simulation method based on the Brownian motions of atoms, which are considered to be over-damped

and without inertia. These simplifications allow simulations on longer time scales from the milliseconds

to the seconds. BD is used to study the kinetics of protein interactions (13).

References

1. Zheng, X., and Sept, D. (2007) Methods in Cell Biology 84, 893-910

2. Carlsson, A. E., and Sept, D. (2007) Methods in Cell Biology 84, 911-937

3. Frieden, C. (1983) Proc. Natl. Acad. Sci. USA 80, 6513-6517

4. Mogilner, A., and Edelstein-Keshet, L. (2002) Biophys J 83(3), 1237-1258

5. Bernheim-Groswasser, A., Wiesner, S., Golsteyn, R. M., Carlier, M. F., and Sykes, C. (2002)

Nature 417(6886), 308-311

6. Roland, J., Berro, J., Michelot, A., Blanchoin, L., and Martiel, J. L. (2008) Biophys. J. 94, 2082-

2094

7. Alberts, J. B., and Odell, G. M. (2004) PLoS Biol 2(12), e412

8. Peskin, C. S., Odell, G. M., and Oster, G. F. (1993) Biophys. J. 65(1), 316-324

9. Carlsson, A. E. (2000) Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 62(5 Pt B),

7082-7091

10. Chu, J. W., and Voth, G. A. (2005) Proc Natl Acad Sci U S A 102(37), 13111-13116

11. Zheng, X., Diraviyam, K., and Sept, D. (2007) Biophys J 93(4), 1277-1283

12. Dalhaimer, P., Pollard, T. D., and Nolen, B. (2008) J. Molec. Biol. 376, 166-183

13. Sept, D., Elcock, A. H., and McCammon, J. A. (1999) J. Molec. Biol. 294(5), 1181-1189

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from

Thomas D. Pollard and Julien BerroMathematical models and simulations of cellular processes based on actin filaments

published online October 20, 2008J. Biol. Chem. 

  10.1074/jbc.R800043200Access the most updated version of this article at doi:

 Alerts:

  When a correction for this article is posted• 

When this article is cited• 

to choose from all of JBC's e-mail alertsClick here

Supplemental material:

  http://www.jbc.org/content/suppl/2008/10/21/R800043200.DC1

by guest on February 18, 2020http://w

ww

.jbc.org/D

ownloaded from