Molecular noise of capping protein binding induces macroscopic instability in filopodial dynamics.

Abstract

Capping proteins are among the most important regulatory proteins involved in controlling complicated stochastic dynamics of filopodia, which are dynamic finger-like protrusions used by eukaryotic motile cells to probe their environment and help guide cell motility. They attach to the barbed end of a filament and prevent polymerization, leading to effective filament retraction due to retrograde flow. When we simulated filopodial growth in the presence of capping proteins, qualitatively different dynamics emerged, compared with actin-only system. We discovered that molecular noise due to capping protein binding and unbinding leads to macroscopic filopodial length fluctuations, compared with minuscule fluctuations in the actin-only system. Thus, our work shows that molecular noise of signaling proteins may induce micrometer-scale growth-retraction cycles in filopodia. When capped, some filaments eventually retract all the way down to the filopodial base and disappear. This process endows filopodium with a finite lifetime. Additionally, the filopodia transiently grow several times longer than in actin-only system, since less actin transport is required because of bundle thinning. We have also developed an accurate mean-field model that provides qualitative explanations of our numerical simulation results. Our results are broadly consistent with experiments, in terms of predicting filopodial growth retraction cycles and the average filopodial lifetimes.

Full text

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Molecular noise of capping protein binding induces
macroscopic instability in filopodial dynamics
Pavel I. Zhuravlev and Garegin A. Papoian1
Department of Chemistry, University of North Carolina, Chapel Hill, NC 27599-3290;
Edited by Peter G. Wolynes, University of California at San Diego, La Jolla, CA, and approved April 30, 2009 (received for review December 15, 2008)
Capping proteins are among the most important regulatory pro-
teins involved in controlling complicated stochastic dynamics
of filopodia, which are dynamic finger-like protrusions used by
eukaryotic motile cells to probe their environment and help guide
cell motility. They attach to the barbed end of a filament and
prevent polymerization, leading to effective filament retraction
due to retrograde flow. When we simulated filopodial growth in
the presence of capping proteins, qualitatively different dynam-
ics emerged, compared with actin-only system. We discovered
that molecular noise due to capping protein binding and unbind-
ing leads to macroscopic filopodial length fluctuations, compared
with minuscule fluctuations in the actin-only system. Thus, our
work shows that molecular noise of signaling proteins may induce
micrometer-scale growth–retraction cycles in filopodia. When
capped, some filaments eventually retract all the way down to
the filopodial base and disappear. This process endows filopodium
with a finite lifetime. Additionally, the filopodia transiently grow
severaltimeslongerthaninactin-onlysystem,sincelessactintrans-
port is required because of bundle thinning. We have also devel-
opedanaccuratemean-fieldmodelthatprovidesqualitativeexpla-
nations of our numerical simulation results. Our results are broadly
consistent with experiments, in terms of predicting filopodial
growth retraction cycles and the average filopodial lifetimes.
amplification | stochastic chemical kinetics | growth–retraction cycles |
stochastic switch | mechano-chemical sensing
E
motility (1). They play important roles in neuronal growth (2),
woundhealing(3),andcancermetastasis(4).Thefilopodialstruc-
ture consists of parallel actin filaments, cross-linked into bundles
by actin-binding proteins, all of which is enclosed by the cell’s
plasma membrane (1, 5). Despite their importance in eukary-
otic biology and human health, the physical mechanisms behind
filopodial regulation and dynamics are poorly understood, includ-
ing what drives ubiquitous growth–retraction cycles and eventual
filopodial disappearance (1–7). Using stochastic simulations of
filopodial dynamics, we have discovered that molecular noise due
to binding/unbinding of a capping protein results in macroscopic
growth–retraction fluctuations, compared with minuscule fluctu-
ations in the actin-only system. Because of rare fluctuations some
filaments eventually retract all the way down to the filopodial
base and disappear. In contrast to prior computational models
that predicted stable filopodia at steady state, our simulations
show that filopodial lifetimes are finite. We have also developed
an accurate mean-field model that provides insights into filament
disappearance kinetics.
A comprehensive computational model of a filopodium should
contain the following features: mechanical interactions includ-
ing membrane dynamics, protrusion force, and retrograde flow;
chemical interactions, including actin polymerization and depoly-
merization; and biological signaling interactions that control the
filopodiadynamicsturnover.Thefirstmean-fieldmodelforfilopo-
dial growth addressed the interplay between filament growth and
diffusional actin transport (8). A subsequent work highlighted
the importance of the interactions between the membrane and
filament barbed ends (9). Our own previous study treated both
ukaryotic motile cells project finger-like protrusions, called
filopodia, to probe their environment and help guide cell
polymerization and diffusion in a fully stochastic fashion (10).
In that model, the filopodia grow to some steady-state length,
and subsequently exhibit only slight fluctuations (10). Thus, no
essential dynamics occurs after the steady state is reached, a find-
ing similar to those of other prior filopodial simulations (8, 9),
implyinganessentiallyinfinitefilopodiallifetimeandnoturnover.
Although it is not well known whether the turnover is driven
externally or internally, it is plausible that internal biochemi-
cal reaction network dynamics is a significant contributor. For
instance, capping proteins bind to the barbed ends of actin fila-
mentspreventingpolymerization(11).Theireffortsarecountered
by formins (12), anticapping processive motors that attach to the
barbed ends and may effectively increase polymerization rate up
to fivefold (12). In this work, we investigated the influence of
two regulating proteins on the filopodial turnover process. The
mechanochemical model that we use here to describe the filopo-
dial dynamics is fully stochastic, using the Gillespie algorithm
to calculate simulation steps (Fig. 1). It consists of the follow-
ing processes: (i) the diffusion of proteins from the cytosol at the
filopodial base to the tip; (ii) the force applied by the membrane
on individual filaments; (iii) the actin filament polymerization
and depolymerization at the barbed end; (iv) the depolymeriza-
tion at the pointed end and the induced retrograde flow vretr
of the filopodium as a whole; (v) capping of the filaments that
stops polymerization (11); (vi) binding of formin to the barbed
end that increases effective polymerization rate fivefold (12).
The filopodium is split into compartments with diffusion real-
ized as stochastic hops between them. Membrane force is taken
into account by effectively decreasing the polymerization rates
(see Materials and Methods and ref. 10 for more details). The
parameters, such as reaction rates and concentrations, are given
in Table 1.
The addition of capping proteins and formins results in very
different, more complex growth dynamics (Fig. 2), in contrast to
the actin-only model in ref. 10, where the observed length of the
filopodium was stationary. In particular, filopodial length fluctu-
ations become macroscopic, increasing from <100nm in a model
without these proteins to a few micrometers, on the order of the
length of filopodia. In addition, a clearly identifiable retraction
phase appears, with a lifetime of ∼100 s. The mechanism is this:
whenanindividualfilamentiscapped,theretrogradeflowmakesit
retract—the filament will eventually disappear if uncapping does
not occur quickly enough. If the filament number becomes less
than a minimum needed to overcome the membrane force, the
filopodium collapses.
Theselarge-amplitudeoscillationsoffilopodiallengthobserved
in our simulations are the consequence of the amplification of
molecular noise of capping protein. This amplification is possible
because of the timescale separation between fast polymerization
and retrograde flow processes and slow off-rates of formin and
capping protein. Because the regulatory proteins are present in
Author contributions: P.I.Z. and G.A.P. designed research, performed research, analyzed
data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1To whom correspondence should be addressed. E-mail: gpapoian@unc.edu.
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Fig. 1.
this work. Simulation details are elaborated in the text.
Schematic representation of the mechanochemical model used in
very low concentrations (<100nM), their noise is highly discrete
(13–21), randomly driving back-and-forth transitions from fast
growth to fast retraction for each filament. Such a highly fluc-
tuating behavior should be advantageous from the point of view
of efficiency of a filopodium as an environmental sensor, in anal-
ogytonear-criticalsystems,wherelargefluctuationsareindicative
of large corresponding susceptibilities. Indeed, we found that the
filopodial length is significantly more sensitive to the change in
membrane force in a system with capping proteins and formins
than in a system with just actin.
To compare our predictions with prior and future experiments,
we computed filopodial lifetime distribution from 2,048 Gillespie
trajectories (Fig. 2 Lower). Experimentally reported lifetimes are
on the order of several minutes, consistent with our results (6, 22–
26). We found that computed filopodial lifetimes strongly depend
on the amplitude of the individual filament length fluctuations
(Fig. 3).
Table 1. Model variables and parameters
Mechanics
Half-actin monomer size
Number of filaments
Membrane force
Diffusion rate
Membrane fluctuation
Retrograde flow speed
(De)Polymerization rates
Free barbed end
Formin-anticapped
Depolymerization
(Anti)Capping rates
Formin on-rate
Formin off-rate
Capping rate
Uncapping rate
Bulk concentrations
Actin
Capping protein
Formin
δ = 2.7 nm
N = 16
f = 10 pN
kD= 5μm2s−1(2,000s−1)∗
σd= 20 nm
vretr= 70 nm/s
k+
k+
k−
A= 11.6μM−1s−1(21.8s−1)
FA= 53.2μM−1s−1(100s−1)
A= 1.4s−1
k+
k−
k+
k−
F= 10μM−1s−1(18.8s−1)
F= 0.0667s−1
C= 3.5μM−1s−1(6.6s−1)
C= 0.04s−1
CA= 10μM
CC= 50 nM
CF= 40 nM; 80 nM
∗Reaction and diffusion rates in parentheses in “seconds” units depend on
the compartment volume. In our computations the compartment volume
was fixed, with compartment length of lD= 50 nm and a filopodial diameter
of 150 nm.
Fig. 2.
(Upper) Sixteen individual trajectories are shown from stochastic simulations
of filopodial growth and retraction with 80 nM formin and 50 nM capping
protein. The average over trajectories is indicated with a thick black line.
Individual trajectories undergo turnover—growth–retraction oscillations on
a micrometer scale—which is induced by molecular noise of regulatory pro-
teins. (Lower) Distribution of the filopodial lifetime calculated from 2,048
trajectories at 40 nM formin and 50 nM capping protein.
Microscopic oscillations endow a filopodium with a finite lifetime.
To gain further insight into the observed noise amplification
phenomenon,weinvestigatedhowthecappingproteinandformin
concentrations influence the magnitude of the filopodial macro-
scopicfluctuations.Tomeasurethem,wecreatedastationarystate
by allowing the tips of disappearing filaments to stay at the filopo-
dial base and eventually uncap. As one might have anticipated,
the fluctuation amplitude grows with increasing capping protein
concentration. However, formin quenches these oscillations such
that the fluctuation amplitude depends mainly on the ratio of the
capping protein to formin concentrations (Fig. 3 Upper). Over-
all, filopodial lifetime strongly depends on the amplitude of the
individual filament length fluctuations (Fig. 3 Lower).
Toshedlightintothekineticsoffilopodialretraction,weaddress
the question of how such long timescale processes (hundreds of
seconds) emerge from the much faster constituent kinetic rates
(filament uncapping rate, k−
model). We elaborate below on a simple mean-field model to esti-
mate the rate for filament disappearance. We consider a filament
in the bundle of N anticapped filaments of stationary length, L,
computed in our prior work (10) (see Table 1 for the definition of
variables):
?CAδ
We keep track of a single filament as various binding/unbinding
eventsoccur.Afterforminunbinds(withoff-rate,k−
C= (25s)−1is the slowest rate in our
L =kDlD
Nvretr
−
?
1 +k−
Aδ
vretr
?
1
k+
FA
efδ/NkBT
?
.
[1]
F),thefilament
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Fig. 3.
centrations of regulating proteins, mainly, on their ratio. These fluctuations,
in turn, largely determine the mean filopodial lifetime.
The amplitude of filament length fluctuations depends on the con-
may either become capped and start retracting or rebind formin,
withthelatterbeingmorelikely.Theratioofprobabilitiesforcap-
ping and anticapping is proportional to the ratio of the on-rates
for capping protein and formin, CCk+
CFare capping protein and formin concentrations, respectively.
The regulatory proteins are not consumed during polymerization
(unlike actin), thus, we can assume bulk concentrations at the tip.
Overall, the average capping time for a filament may be estimated
as ¯ τc= (k−
The computational model for filopodial dynamics is essen-
tially a multidimensional lattice on which a stochastic propagator
enacts a random walk. If the filament is capped, it may either
fully retract and disappear or become uncapped and grow back.
Thus,manypossibletrajectoriesonthelatticemustbeconsidered.
Each distinct trajectory is characterized by a certain probability of
occurrence and the overall time for filament disappearance. The
trajectories may be grouped according to how may times the fil-
ament has been capped and uncapped before it disappears. In
the following analytical estimation, we calculate average time of
disappearance in each group of paths described above by using a
mean-field approximation, instead of carrying out a full path inte-
gral calculation (Fig. 4). Such a group of paths may be thought
of as a particular event scenario. We then average the rates from
each group with statistical weights of the scenarios; the weights
are computed exactly.
In a bundle of N filaments the average time to wait for a cap-
ping event is ¯ τc/N. Once a filament is capped, it starts to retract
under retrograde flow. It may either fully shrink and disappear
or uncap and regrow (first trajectory bifurcation; see Fig. 4). If
uncapping does not occur, it takes τd ≈ L/vretrfor the filament
to fully shrink and disappear. The probability to follow this sce-
nario is therefore p = exp(−k−
filament uncapping and resuming growth after average time ¯ τsof
shrinking—occurswith1−pprobability.Tofindthetypicalshrink-
ing time (the time it takes for a capped filament to uncap), ¯ τs, one
has to average over the exponential distribution for uncapping
up to τd(since for longer times the filament has fully retracted
according to the first scenario),
Cand CFk+
F, where CCand
F)−1(CCk+
C)−1(CFk+
F).
Cτd). The alternative scenario—the
¯ τs=
?τd
0
τP(τ)dτ =
1
k−
C
?1 − e−k−
Cτd?1 + k−
Cτd
??.
[2]
Fig. 4.
which a stochastic propagator enacts a random walk. Eventually, a trajec-
tory arrives to a sublattice where one filament is capped. Mean-field average
time for that is ¯ τc/N. From there some trajectories lead to filament disappear-
ance due to retrograde flow and some to filament uncapping with respective
statistical weights of p and 1 − p and mean-field average times τdand ¯ τs.
After uncapping, all trajectories will pass through the phase area where this
filament is capped again after mean-field average time ¯ τc. Then again, it
may either disappear after time τd or uncap after time ¯ τs with weights p
and 1 − p.
The system dynamics is executed on a multidimensional lattice on
After uncapping, the filament starts to regrow until the next cap-
ping.Ifxisthepositionofthebarbedendwithrespecttofilopodial
base, then the regrowth speed is vg(x) = ˙ x = CA(1 − x/L)k+
vretr, where the steady-state G-actin concentration gradient has
beentakenintoaccount(CAisthebulkconcentrationatthefilopo-
dial base). We solve this equation to obtain the filament length
afterregrowingfortime ¯ τc,theaveragetimeuntilthenextcapping
event:
FAδ −
x(¯ τc) =˜L −?˜L − x(0)?exp?−CAk+
where˜L = L(1 −
provided by formin, the exponential factor is essentially zero, so
the new length is x(¯ τc) ≈˜L ≈ L, as vretr ? CAk+
if a capped filament is uncapped, it quickly catches up with the
others at steady-state length. Thus, a filament either retracts after
averagetime ¯ τc/N+τdwithprobabilityp,orreturnstoinitialstate
after time ¯ τc/N + ¯ τswith probability 1 − p. This filament will be
capped again after average time ¯ τc, and there will be second tra-
jectorybifurcation:itwilleitherretractwithprobability(1−p)pat
time ¯ τc/N+¯ τs+¯ τc,orstartgrowingagainwithprobability(1−p)2
attime ¯ τc/N +¯ τs+¯ τc+¯ τs.Eachsubsequenttrajectorybifurcation
leadstoalongertime(lowerrate)forfilamentdisappearance,but
also is less likely to occur. To get the full rate one has to average
the disappearance rates over all scenarios:
FAδ¯ τc/L?,
[3]
vretr
CAk+
FAδ). With a very large polymerization rate
FAδ. Therefore,
λ =
∞
?
n=0
pn
τn
=
∞
?
n=0
p(1 − p)n
a + bn
=p
a
2F1
?
1,a
b;a
b+ 1;1 − p
?
, [4]
where2F1is the hypergeometric function; a = ¯ τc/N + τd; b =
¯ τc+ ¯ τs.
Thus, the average time for filament disappearance is ¯ τ = 1/λ,
where λ is given in Eq. 4. We ran simulations with different num-
bers of filaments at two formin concentrations, looking for an
average time of disappearance of one filament. Because there are
multiple filaments in these simulations, and several of them may
be capped at the same time, we need to take into account a pos-
sibility that the first capped filament is not the first to disappear,
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Fig. 5.
ments at CC= 50 nM and CF= 40 nM (circles, solid lines) and 80 nM (squares,
dashed lines). The symbols show simulation results. Solid and dashed lines are
computed by combining the disappearance for a single filament, given by
Eq. 4, with the scenarios of other filaments disappearing, which had capped
later. We used the following expression to estimate the average time for any
of the bundle filaments to fully retract, τmulti= (1 − p)nc−1λ−1+ (1 − (1 −
p)nc−1)(¯ τcnc/N + τd), where, nc= λ−1/(¯ τc/N), indicates the average number
of simultaneously capped filaments, and the remaining symbols are defined
in text. In this expression, the first term corresponds to the first capped fila-
ment disappearing first, whereas the second term accounts for the possibility
of one of the other nc−1 capped filaments following the scenario of quickly
disappearing without uncapping even once. (Inset) Bare disappearance times
for single filaments, λ−1, computed from Eq. 4, where the effect of other
filament retractions is not taken into account.
Rate of individual filament disappearance from a bundle of N fila-
to directly compare the results from simulations with the ana-
lytical estimates. This possibility might occur if the first capped
filament uncaps and grows back, while another capped filament
quicklyretracts.InFig.5,theaveragefilamentdisappearancetime
is computed by considering not only ¯ τ = 1/λ, but also taking
intoaccountthecontributionsfromthesimultaneousretractionof
otherfilaments.Thecomparisonbetweensimulationsandanalyti-
calresults,wherethelatterdidnotcontainadjustableparameters,
showed good agreement (see Fig. 5).
Anotherinterestingresultthatweobservedwasthatthelongest
filopodia grow to ≈4.5 times the size of those simulated with-
out formins and capping proteins [in our prior work (10)]. The
explanation for this follows: a key factor limiting the filopodial
length is insufficient actin transport (8, 10). A mean-field esti-
mation for the stationary length (Eq. 1) follows from equating
the actin flux from the established gradient and actin consump-
tion by N polymerizing barbed ends (10). Because the latter is
proportional to N, a smaller number of filaments require less
actin for growth; therefore, a thinner filopodium can grow longer
before the diffusive G-actin transport again becomes a limiting
factor. As filaments cap and disappear, N decreases, hence, the
increase in the length. When too few filaments remain to oppose
the force of the membrane load, the bundle withdraws, result-
ing in the eventual filopodial disappearance. One can speculate
anotherinterestingconsequenceofbundlethinning:itmayinduce
mechanical buckling instability, due to diminution of the mechan-
ical rigidity of the F-actin bundle. In light of the recent suggestion
thatfilopodiawith>10filamentsaremechanicallystable(27),our
current finding of filament bundle thinning may turn out to be an
important mechanism for mechanical collapse of filopodia due to
buckling.
In summary, we have shown in this work that capping proteins
exert a dramatic effect on filopodial dynamics. Introducing them
allowed us to create a computational model that predicts a finite
lifetime of filopodia. The resulting filopodial length dynamics has
a remarkable feature: discrete noise of regulatory proteins that
are in very low concentration becomes greatly amplified. Because
of timescale separation, this slow discrete noise triggers the fast
retrograde flow and polymerization processes resulting in oscil-
lations of filopodial length macroscopic in space and time. This
result suggests that experimentally observed filopodial growth–
retraction turnover dynamics (5–7) may be partially driven by the
internal noise of the filopodial mechanochemical network. Large-
amplitude fluctuations in filopodia may be important with respect
to their sensory role. In particular, a molecular system having
oscillations of this magnitude is expected to be easily perturbed
by small external forces, either mechanical or chemical in origin,
in analogy with large susceptibilities seen in near-critical systems
with large fluctuations. The noise amplification described in this
work, arising from discrete noise in a low copy number of some
of the reaction network species, is related to stochastic switch-
ing in biochemical signal transduction, for example, when a cell
needs to make a binary decision (28). A similar dynamic instabil-
ity is observed in microtubular growth, although in filopodia the
oscillationsareswitchedbythebindingnoise,andinmicrotubular
catastrophesandrescuesitisenzymaticinorigin(29).Inaddition,
by promoting filament disappearance, capping proteins thin the
filopodium,reducingitsmechanicalrigidity.Wespeculatethatthis
may be an important mechanism for creating buckling instabili-
ties. We also investigated in detail the filament retraction kinetics,
and developed a mean-field analytical estimation for the rate of
disappearanceofindividualfilamentsthatmatchesthesimulation
data surprisingly well. We explained the extremely slow timescale
of filament disappearance, compared with bare kinetic rates, by
common occurrence of multiple capping, shrinking, and regrow-
ing events, before the filament fully retracts as a result of a rare
fluctuation.
Materials and Methods
In our stochastic model of filopodial growth (10), represented in Fig. 1, we
assume that the reaction processes are confined to a spatial region having a
linear dimension of ζ [so-called Kuramoto length (30)], such that particles dif-
fuse across the region quickly compared with the typical reaction times. This
allows us to discretize space into compartments and model protein diffusion
along the filopodium as a random walk on a one-dimensional lattice, with
moleculeshoppingbetweenthesecompartmentsatratescalculatedfromdif-
fusion coefficients. At relevant concentrations, ζ is on the order of 100 nm at
the tip of the filopodium. We chose a somewhat more conservative compart-
mentsizeoflD= 50nm(10).Atthefilopodialbasetheproteinconcentrations
are kept at their constant bulk values. Biochemical reactions within each
compartment are simulated by using the Gillespie algorithm (10, 31). Poly-
merization rates at the tip are decreased by the membrane force via the
Brownian ratchet model (8, 10). The experimentally reported uncapping fre-
quency is about once every 30 min, however, it may be greatly increased
through the actions of uncapping proteins, such as PIP2 (11). The variables
and parameters for the model are given in Table 1. Further simulation details
are elaborated below.
The Stochastic Model of Filopodial Growth. A mature filopodium is a bundle
of a few dozen actin filaments enclosed by the cell membrane with a protein
complex at the tip (32). The filopodial base is usually located within a three-
dimensional actin mesh below, which forms the basis for the lamellipodium
(32). In addition, bundling proteins cross-link the filaments in the bundle, but
this was not included in our current model.
Typical growth and retraction speeds are within 0.1–0.2 μm/s, with filopo-
dial lengths reaching 1–2 μm (33). In special cases, the filopodial length may
reach nearly 100 μm (34). Since short filopodia are mostly straight, for sim-
plicity we assume growth of straight filaments. Longer filopodia might tilt
and bend. The possibilities of bending of the filaments and buckling of a
filopodium should be addressed in future work.
Depending on the number of actin filaments, the filopodial diameter typ-
ically varies between d = 100 nm and 300 nm. In our simulations we used
d = 150 nm, a number that was derived from minimizing the membrane free
energy (9) and is reasonable if the number of filaments is not too large.
Our physicochemical model of filopodial growth consists of the following
processes: (i) diffusion of proteins along the filopodial tube, providing pas-
sive transport from cell body to the filopodial tip; (ii) mechanical interactions
between the membrane and individual filaments; (iii) actin filament poly-
merization and depolymerization processes at the barbed end; (iv) depoly-
merization at the pointed end and the induced retrograde flow vretrof the
bundleasawhole;(v)cappingofthefilamentsthatstopspolymerization(11);
and (vi) binding of formin to the barbed end which increases the effective
polymerization rate fivefold (12).
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Polymerization, Depolymerization. Actin filaments (F-actin) are asymmetric,
with one end called “the barbed end” and the other end called “the pointed
end.” Because of ATP binding and subsequent hydrolysis, chemical affinities
for monomers (G-actin) are different at the two ends, such that the polymer-
ization rate at the barbed end is much higher. This leads to the motion of
the filament as a whole in the direction of the barbed end (whereas indi-
vidual monomer units migrate from the barbed end to the pointed end).
This process is called “treadmilling” and is the biochemical basis for the
cytoskeletal dynamics (35, 36), including filopodial protrusion. An actin fil-
ament consists of two protofilaments, wound up in a right-handed helix,
with a pitch distance of 37 nm. Diameter of one globular actin monomer is
5.4 nm. In our simulations we increase the filament length by δ = 2.7nm
on one polymerization event, since there are two protofilaments, and two
monomers are needed to increase length by 5.4 nm.
The resulting double-helical filament is mechanically robust, with a persis-
tent length, Lp∼ 10μm. The buckling length, or the critical length at which
one filament would buckle if subject to a force F, is
?
where kBis the Boltzmann constant and kBT = 4.1pNnm at room temper-
ature (8). The equation gives Lb≈100nm for a force F = 10pN for a single
filament. For weakly cross-linked bundles of N actin filaments, the buckling
length of the bundle is
dles (8). A recent work suggested that the membrane enclosure significantly
increases these estimates (27).
In our model, cytosolic molecules such as G-actin, formin, or capping pro-
tein, which are in the same compartment as the filopodial tip, can attach to
one of the filament ends with a probability, given by the rates k+
These monomers at filament ends can also stochastically dissociate with the
corresponding rates k−
theactinpolymerizationrateratherstronglydependsonthemembraneforce
(37). Theoretical explanations were also provided (8, 38, 39). For instance, the
Brownian ratchet model considers membrane fluctuations at the tip of the
filopodium (38, 40). That the membrane fluctuations are sufficient to allow
a G-actin monomer to fit sterically atop the filament allows a polymerization
with the rate k+
ment equals the “bare” rate kA,0times the probability of the gap opening at
the tip of the nth filament. A convenient relation between the loading force
fnand the polymerization rate was derived earlier (40),
Lb≈π
2
kBTLp
F
, [5]
√NLb, whereas it is NLb/√2 for tightly linked bun-
A, k+
FA, or k+
C.
A, k−
FA, or k−
C. In prior experiments, it was shown that
A. Thus, the effective polymerization rate kA,non the nth fila-
k+
n= k+exp
?
−fnδ
kBT
?
. [6]
According to Eq. 6 the on-rates for G-actin monomer addition (k+
barbed end; k+
tein (k+
assumed to be independent of the membrane fluctuations due to diminished
steric constraints (12). All of the off-rates do not depend on the membrane
position.
For these effective on-rates, we needed to estimate the membrane force
on each individual filament, which is discussed next.
Afor free
FAfor the barbed end anticapped by formin) and capping pro-
C) are modified after each timestep. The on-rate for formin, k+
F, was
Membrane Force. According to prior studies fluctuations of a membrane
sheet below micrometer scale relax on the microsecond to millisecond
timescale(41–44).Chemicalreactionsoccuronamuchslowertimescale(10−2–
101s), thus, due to this timescale separation, the membrane fluctuations may
be assumed to be equilibrated at the time of each given chemical event. Each
filament experiences an individual membrane force, fn, that depends on the
proximity of the filament tip to the average location of the membrane. Thus,
the total membrane force f is distributed among the individual filaments,
f =?
proportional to the membrane-filament contact dwelling probability, which
is the probability that the membrane touches that filament. That, in turn,
depends on the amplitude of the membrane fluctuation near the filament
and the filament length. Longer filaments are more likely to be in contact
with the membrane, and feel a stronger membrane force. If the membrane
fluctuations are assumed to be described by a Gaussian distribution around
the membrane average position, the dwelling probability pnfor the nth fila-
ment to be in contact with the membrane is proportional to the probability
that the membrane height is found below the filament end
?∞
where σdis the average membrane fluctuation amplitude (discussed next).
Once pnis obtained, the force fnon each filament may be computed,
fn=pn
where p =?
RetrogradeFlow. Thefilopodialdynamicsarestronglycontrolledbythepoly-
merization at its tip (45, 46). As was mentioned earlier, the “treadmilling”
contributes to a steady backward motion of the whole actin filament bun-
dle,calledtheretrogradeflow.Insomecells,itisbelievedthatspecificmyosin
motors participate in creating the retrograde flow (47). All of this is subject
to the regulation by the signaling proteins, but in our current computations,
we neglect these subtleties of the retrograde flow process, and assume a con-
stant average retrograde flow speed, vretr. The retrograde flow speed vretris
≈20–200 nm/s (46, 48–51) and we take vretr= 70 nm/s as the default value in
our simulations.
nfn.
To calculate fn, we assume that on average the force on a filament is
pn∼
h−hn
exp?− z2/σ2
d
?dz,[7]
pf,[8]
npnis a normalization factor. Then we use Eq. 6 to modify the
polymerization rates.
Numerical Scheme. Our computations are based on the Gillespie algorithm
(31). In particular, at each simulation step two independent random num-
bers are chosen, where the first one determines the time of the next event
t + δt (t is the time of last event) based on the reaction rates present in
the system, and the second one determines which event occurs, based on
the magnitude of individual event rates. The event is chosen among the fol-
lowing possibilities: (i) G-actin monomer, capping protein or formin hopping
between filopodial compartments, (ii) individual filament polymerization or
capping events, (iii) depolymerization or uncapping events, and (iv) binding
or unbinding of formin. Then, we update current time, adding δt, the num-
bers of proteins in compartments, filament length, and which protein is on its
end according to the event that happened. We then incorporate retrograde
flow and update the length of filaments again. Finally, the membrane load is
partitioned among the filaments, as described above, which results in recom-
puting individual filament polymerization rates, according to the Brownian
ratchet model.
ACKNOWLEDGMENTS. This work was supported by National Science Foun-
dation Grant CHE-715225. All calculations were carried out using the UNC
Topsail supercomputer.
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