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: [email protected]
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.
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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
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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
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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
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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).
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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).
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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: [email protected]
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.
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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
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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:
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