Velocity oscillations in actin-based motility
Azam Gholami1, Martin Falcke1, Erwin Frey2
1Hahn-Meitner-Institute, Dept. Theoretical Physics, Glienicker Str. 100, 14109 Berlin, Germany
2Arnold Sommerfeld Center for Theoretical Physics and Center of NanoScience,
Ludwig-Maximilians-Universität, Theresienstr. 37, 80333 München, Germany
(Dated: June 28, 2021)
We present a simple and generic theoretical description of actin-based motility, where polymer-
ization of filaments maintains propulsion. The dynamics is driven by polymerization kinetics at the
filaments’ free ends, crosslinking of the actin network, attachment and detachment of filaments to
the obstacle interfaces and entropic forces. We show that spontaneous oscillations in the velocity
emerge in a broad range of parameter values, and compare our findings with experiments.
PACS numbers: 05.20.-y, 36.20.-r, 87.15.-v
Force generation by semiflexible polymers is versatilely
used for cell motility. The leading edge of lamellipodia
of crawling cells [1] is pushed forward by a polymerizing
actin network and bacteria move inside cells by riding on
a comet tail of growing actin filaments [2, 3]. In vivo sys-
tems are complemented by in vitro assays using plastic
beads and lipid vesicles [4, 5, 6]. The defining feature of
semiflexible polymers is the order of magnitude of their
bending energy which is in the range of kBT . They un-
dergo thermal shape fluctuations and the force exerted
by the filaments against an obstacle arises from elastic
and entropic contributions [7, 8].
Mathematical models have quantified the force gener-
ated by actin filaments growing against obstacles [7, 8, 9].
The resisting force depends on the obstacle which is
pushed. In case of pathogens, it has a small com-
ponent from viscous drag of the moving obstacle but
consists mainly of the force exerted by actin filaments
bound to the surface of the bacteria and pulling it back-
wards [10, 11]. The tethered ratchet model [12] is a math-
ematical formulation of these experimental findings in
terms of the dynamics of the number of attached and de-
tached polymers. The starting point of our approach will
be the dynamics of the distributions of the free length of
both polymer populations.
Actin polymerization in living cells and extracts is con-
trolled by a complex molecular network [3]. Nucleation
of new filaments, capping of existing ones, exchange of
ADP for ATP on actin monomers, buffering of monomers
etc. all contribute to that control and have been modeled
[12, 13, 14]. Our goal is not to model the full complex-
ity of that biochemical network. Rather we focus on the
core process of force generation and force balance ensuing
from the interplay between bound pulling filaments and
polymerizing pushing filaments, the transition between
these two groups and the motion of the whole force gen-
erating configuration. This is motivated by recent ob-
servations of complex dynamics in simple reconstituted
systems: the velocity of beads or pathogens propelled
by actin polymerization may oscillate [16, 17, 18]. Our
goal is to describe the dynamics of such biochemically
compressed
under tension
viscous
  force
cross-linked actin network
cross-linkers
obstacle
brush
FIG. 1: (color online) Schematic representation of an ensem-
ble of actin filaments oriented at ϑ = 0 with respect to the
normal n̂ of an obstacle interface, which may either be a cell
membrane or a bacterium. While attached filaments are un-
der tension and pull the interface back, detached filaments
are compressed, elongate by polymerization with rate kon
and push the interface forward. All filaments in the brush
are firmly anchored in a cross-linked network, whose front
advances with velocity vg reducing the free length l of the
filaments. Attached filaments detach with stress dependent
rate kd and detached filaments attach with constant rate ka.
vo is the interface velocity in the extracellular medium, and
x is the distance between the front of the network and the
interface.
simpler systems and find a robust microscopic descrip-
tion for oscillation mechanisms, which may then be con-
trolled by higher order processes. Such a study is meant
to complement investigations based on a continuum ap-
proach [17, 18].
We consider a fixed number N of actin filaments [19]
firmly anchored into a rigid cross-linked network, which
advances with velocity vg; for an illustration see Fig. 1.
Filaments of variable length l are either attached to the
obstacle interface via a protein complex or detached from
it, with time-dependent number distributions denoted by
Na(l, t) and Nd(l, t), respectively. In the detached state,
filaments polymerize at a velocity vp(l, x), which depends
on both the polymer length l and the distance x between
rigid support and obstacle. Transitions between the two
filament populations occur with a constant attachment
rate ka and a stress-dependent detachment rate kd [20].
This results in the evolution equations
vg(l)− vp
Nd = −ka Nd + kd Na , (1a)
vg(l)
Na = ka Nd − kd Na . (1b)
The right hand side of Eq. 1 describes attachment and
detachment process. The second term on the left hand
side accounts for the gain and loss of attached and de-
tached polymers due to the dynamics of the polymer
mesh, growing with velocity vg, and the polymerization
kinetics of the filaments in the brush. The correction
factor l/x in front of vg is due to the fact that for bent
polymers the rigid network swallows by this amount more
in contour length than for straight filaments. This factor
is equal to 1 for l ≤ x.
Processes contributing to the growth of the rigid poly-
mer mesh are entanglement and crosslinking of filaments
in the brush. Both imply a vanishing vg for l → 0, since
short polymers do not entangle and crosslinking proteins
are unlikley to bind to them. At the same time vg can
not grow without bound but must saturate at some value
vmaxg due to rate limitations for crosslinker binding. This
suggests to take the following sigmoidal form
vg(l) = v
g tanh(l/l̄) , (2)
with a characteristic length scale l̄.
The polymerization rate is proportional to the prob-
ability of a gap of sufficient size d (≈ 2.7 nm) between
the polymer tip and the obstacle for insertion of an actin
monomer [7]. This implies an exponential dependence of
vp on the force Fd by which the polymer pushes against
the obstacle,
vp(l, x) = v
p exp [−d · Fd(l, x)/kBT ] . (3)
Here, vmaxp ≈ 500 nm s−1 [7] is the free polymerization
velocity. For the entropic force Fd we use the results ob-
tained in Ref. [8] for D = 2, 3 spatial dimensions, where
we take the accepted value of `p ≈ 15 µm [21] for the
persistence length of F-actin.
The dynamics of the distance x between grafted end
of the filament and the obstacle interface (see Fig. 1) is
given by the difference of the average vg and the velocity
of the obstacle
∂tx = −
dl vg(l) [Na(l, t) +Nd(l, t)] (4)
dl [Na(l, t) Fa(l, x) +Nd(l, t) Fd(l, x)] ,
where ζ is an effective friction coefficient of the obsta-
cle. The force Fa(l, x) acting on the obstacle interface
results from the compliance of the filaments attached
to it by some linker protein complex, which we model
as springs with spring constant kl and zero equilibrium
length. This complex has a nonlinear force-extension re-
lation which we approximate by a piece-wise linear func-
tion; for details see the supplementary material. Let
R‖ ≈ l[1− l(D− 1)/4`p] be the equilibrium length of the
filament. Then, the elastic response of filaments expe-
riencing small compressional forces (x ≤ R‖) is approxi-
mated by a spring constant k‖ = 12kBT `2p/(D−1)l4 [22].
For small pulling forces (x ≥ R‖), the linker-filament
complex acts like a spring with an effective constant
keff = klk‖/(kl + k‖). In the strong force regime, the
force-extension relation of the filament is highly nonlinear
and diverges close to full stretching [23]. Therefore, only
the linker will stretch out. The complete force-extension
relation is captured by
−k‖(x−R‖) , x ≤ R‖ ,
−keff(x−R‖) , R‖ < x < l ,
−kl(x− l)− keff(l −R‖) , x ≥ l .
Finally, we specify the force-dependence of the detach-
ment rate by
kd = k
d exp [−d · Fa(l, x)/kBT ] . (6)
Here, k0d ≈ 0.5 s
−1 [12] is the detachment rate in the
absence of forces and we have followed Ref. [20].
Eq. 1a has a singularity at vp(ls) = vg(ls)ls/x since
the coefficient of the derivative of Nd with respect to
l is zero at ls. To illustrate the key physical features
at that singularity, we start with the simple equation
∂tNd−∂l[vg(l)l/x−vp(l, x)]Nd = 0 with x kept constant.
Then those parts of the distribution of Nd with l < ls will
grow and catch up with ls since vg(l)l/x−vp(l, x) is posi-
tive there, while the parts with l > ls will shorten towards
ls. As a consequence the whole distribution will become
concentrated at ls. To quantify this heuristic argument
we expand vg(l)l/x − vp(l, x) up to linear order around
ls like v1(l− ls) and use the method of characteristics to
solve the equation. Starting initially with a Gaussian dis-
tribution we obtain Nd(l, t) = c(t) exp[−(l− l̄(t))2/w(t)2]
with c(t) = c0 exp(v1t), l̄(t) = ls + (l̄0 − ls) exp(−v1t)
and w(t) = w0 exp(−v1t). This shows that Nd evolves
to a monodisperse distribution which is localized around
ls. Its width decreases exponentially with time while its
height grows exponentially. The time scale for this con-
traction is given by [∂l(vgl/x− vp)]−1.
Since the same kind of singularity also occurs in the
full set of dynamic equations, Eqs. 1, we may readily
infer that Na and Nd evolve into delta-functions with
that dynamics. This is well supported by simulations,
and allows us to continue with the ansatz
Nd(l, t) = nd(t) δ(l − ld(t)) , (7a)
Na(l, t) = na(t) δ(l − la(t)) . (7b)
It defines the dynamic variables nd(t), ld(t), na(t) and
la(t). Upon inserting Eqs. 7 into Eqs. 1 and Eq. 4, we
obtain the following set of ordinary differential equations
∂tld(t) = vp(ld, x)−
vg(ld) + kd
(la − ld) , (8a)
∂tla(t) = −
vg(la) + ka
(ld − la) , (8b)
∂tna(t) = −kd(la, x) na(t) + ka nd(t) , (8c)
∂tx(t) =
[na(t) Fa(la, x) + nd(t) Fd(ld, x)]
[vg(la) na(t) + vg(ld) nd(t)] , (8d)
where nd(t) = N − na(t) since we keep the total number
of filaments fixed.
The values of many parameters in the dynamics can be
estimated using known properties of actin filaments. We
choose the linker spring constant kl ≈ 1 pN nm−1 [12]
and assume N = 200 [12] filaments to be crowded behind
the obstacle. A realistic value of the drag coefficient ζ is
10−3 pN s nm−1 but results did not change qualitatively
for a range from 10−5 pN s nm−1 to 1 pN s nm−1.
We have numerically solved Eqs. 8 in both D = 2
and D = 3 dimensions, and found the dynamic regimes
shown in Fig. 2: stationary states and oscillations. The
existence of an oscillatory regime is very robust against
changes of parameters within reasonable limits including
the spatial dimension. We checked robustness against
changes in the parameter values for the number of poly-
mers N , l̄ (see Eq. 2), kl, vmaxp and k
d, in addition to the
examples shown in Fig. 2. In general, we find that oscil-
lations occur for vmaxg . 500 nm s
−1 and within a range
of values for ka. Note that the oscillatory region in pa-
rameter space depends on the orientation ϑ of filaments
with respect to the obstacle surface, i.e. oscillating and
non-oscillating sub-populations of filaments may coexist
in the same network.
Oscillations appear with finite amplitude and period at
the lower boundary of the oscillatory region; compare the
example shown in Fig. 3a. The stationary state changes
stability slightly inside the oscillatory regime and oscil-
lations set in with a finite period. That is compatible
with oscillations appearing by a saddle node bifurcation
of limit cycles. The upper boundary of the oscillatory
region is determined by a Hopf bifurcation. An exam-
ple of an oscillation close to that bifurcation is shown
in Fig. 3b. More details on the phase diagram will be
published elsewhere [15].
We start with the description of oscillations in the
phase with vg > vp, i.e., decreasing lengths x, la and ld;
see Fig. 3. Then the magnitude of pulling and pushing
forces increases due to their length-dependence. When
the pushing force becomes too strong, an avalanche-like
detachment of attached filaments is triggered and the ob-
stacle jerks forward; compare the steep rise in ld, la and x
FIG. 2: (color online) Phase diagram of Eqs. 8a - 8d outlining
stationary and oscillatory regimes with ζ = 10−3 pN s nm−1
for (a-c) and (a) D = 2, ϑ = 0 , (b) D = 2, ϑ = π/4, (c)
D = 3, ϑ = 0 and (d) D = 3, ϑ = 0, ζ = 10−5 pN s nm−1.
l̄=100 nm, all other parameter values are specified in the text.
 1000
 1500
 2000
 2500
 0  200  400  600  800  1000
time (s)
na(t)
la(t) x(t)ld(t) (a)
 0  10  20  30  40  50  60  70
time (s)
na(t)
la(t) x(t)ld(t) (b)
FIG. 3: (color online) x , la , ld (in nm) and na as a function
of time, as obtained from a numerical solutions of Eqs. 8a
- 8d with vmaxg = 300 nm s
−1 and (a) ka = 0.143 s
−1 (b)
ka = 3.49 s
−1. D = 3, l̄=100 nm in both panels.
shown in Fig. 3. That causes a just as sudden drop of the
pushing force. With low pushing force now, polymeriza-
tion accelerates and increases the length of detached fila-
ments. The restoring force of attached filaments is weak
in this phase due to their small number. Hence, despite
of not so strong pushing forces, the obstacle moves for-
ward. In the meantime, some detached filaments attach
to the surface such that the average length and number of
attached filaments increases as well. When the detached
filaments are long enough to notice the presence of the
obstacle interface, they start to buckle. This, in turn, in-
creases the pushing force and slows down the polymeriza-
tion velocity. Therefore, the graft velocity now exceeds
the polymerization velocity and the average lengths of
attached and detached filaments start to decrease again
and the cycle starts anew. The period of oscillations is
dependent on the parameter values. It reduces from 240 s
in Fig. 3a to 13 s in Fig. 3b as ka increases from 0.143 s−1
to 3.49 s−1 at vmaxg = 300 nm s
The oscillations in x correspond to the saltatory mo-
tion of the obstacle in the lab frame and the oscillations
of its velocity since vg stays essentially constant. An il-
lustration is shown in Fig. 4 for a given set of parameters
which leads to oscillations with periods of the order of
100 s and velocity of the order of 0.7 µm s−1. This is
in good agreement with the results of experiments on os-
cillatory Listeria propulsion [16]. The period of velocity
oscillations with beads propelled by actin polymerization
differs from those of Listeria by one order of magnitude
(8− 15 min [18]). Periods of that length can be obtained
within our model upon using values for ka close to the
lower boundary of the oscillatory regime.
 0  50  100  150  200  250  300  350  400
time (s)
(a)speed
displacement
 0  50  100  150  200  250  300  350  400
time (s)
(b)speed
displacement
FIG. 4: (color online) Velocity and displacement of the obsta-
cle as a function of time with (a) ka = 0.9 s
−1, (b) ka = 1 s
vmaxp = 750 nm s
−1, vmaxg = 75 nm s
−1, k0d = 0.1 s
l̄=100 nm ζ = 10−3 pN s nm−1 and D = 3 in both panels.
We have also studied the system when the network
is oriented at an angle ϑ = π/4. In this case, the spring
constant of the attached filaments parallel to n̂ for D = 2
reads k−1‖ (ϑ) = 4`
+ e−�/2 − 1 + cos 2ϑ( 1
e−2� −
e−�/2) − cos2 ϑ(e−�/2 − 1)2]/kBT , where � = l/lp and
R‖(ϑ) = l(1 − l/4`p) cosϑ [22]. For the pushing force
of a filament grafted at ϑ = π/4, we use the results of
the factorization approximation given in Ref. [8], which is
well valid for a stiff filament like actin. A numerical solu-
tion of Eqs. 8a-8d results in the phase diagram shown in
Fig. 2(b) with the adapted forms of Fd and Fa. The main
effect is that one needs higher values for the attachment
rates and lower values for vg to obtain oscillations.
In summary, we have presented a simple and generic
theoretical description of oscillations arising from the
interplay of polymerization driven pushing forces and
pulling forces due to binding of actin filaments to the
obstacle. The physical mechanism for such oscillations
relies on the load-dependence of the detachment rate and
the polymerization velocity, mechanical restoring forces
and eventually also on the cross-linkage and/or entangle-
ment of the filament network. The oscillations are very
robust with respect to changes in various parameters, i.e.
are generic in this model. Therefore, complex biochem-
ical regulatory systems supplementing the core process
described here may rather stabilize motion and suppress
oscillations than generate them.
Oscillations of the velocity were observed during
propulsion of pathogens by actin polymerization. There,
the core mechanism described here is embedded into a
more complex control of polymerization, which e.g. also
comprises nucleation of new filaments and capping of ex-
isting ones. Hence, the study presented here can not be
expected to fully capture all features of such processes.
Our results still agree well with respect to velocity spike
amplitudes and periods in Listeria experiments reported
in Refs. [16, 17]. The velocity in between spikes appears
to be smaller in experiments than in our simulations.
This may be accounted for in our model by including
capping of filaments upon dissociation from the obsta-
cle. Periods may also become longer when capping and
nucleation were included since it would take longer to
restore the pushing force after the avalanche like rup-
ture of attached filaments. Altogether, qualitative and
quantitative comparison with experiments suggests that
our model may be a promising candidate for a robust
mechanism of velocity oscillations in actin-based bacte-
ria propulsion.
We thank R. Straube and V. Casagrande for inspiring
discussions. E.F. acknowledges financial support of the
German Excellence Initiative via the program ”Nanosys-
tems Initiative Munich (NIM)”. A.G. acknowledges fi-
nancial support of the IRTG ”Genomics and Systems
Biology of Molecular Networks” of the German Research
Foundation.
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[2] J. Plastino and C. Sykes, Curr. Opin. Cell Biol, 17, 62
(2005).
[3] E. Gouin, M.D. Welch, P. Cossart, Curr. Opin. Microbiol,
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[4] T.P. Loisel, R. Boujemaa, D. Pantaloni, M.F. Carlier,
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[5] Y. Marcy, J. Prost, M.F. Carlier, C. Sykes, Proc. Natl.
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[6] S.H. Parekh, O. Chaudhuri, J.A. Theriot, D.A. Fletcher,
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[9] T.L. Hill, Proc. Natl. Acad. Sci. USA, 78, 5613 (1981).
[10] L.A. Cameron et al., Curr. Biol. 11, 130, (2001).
[11] S.C. Kuo and J.L. McGrath, Nature, 407, 1026 (2000).
[12] A. Mogilner and G. Oster, Biophy. J. 84, 1591 (2003).
[13] A.E. Carlsson, Biophys. J. 84, 2907 (2003).
[14] M.E. Gracheva and H.G. Othmer, Bull. Math. Biol. 66,
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[15] A. Gholami, M. Falcke, E. Frey (unpublished).
[16] I. Lasa et al., EMBO J. 16, 1531 (1997).
[17] F. Gerbal et al., Biophys J. 79, 2259 (2000).
[18] A. Bernheim-Groswasser, J. Prost, C. Sykes, Biophys J.
89, 1411 (2005).
[19] A constant number is assumed to simplify matters. It has
been shown, however, that a variable number of filaments
is not required for propulsion; see e.g. W.M. Brieher, M.
Coughlin and T.J. Mitchison, J. Cell Biol. 165 233 (2004)
[20] E. Evans and K. Ritchie, Biophys. J. 76, 2439 (1999).
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et al., Phys. Rev. Lett. 89, 258101 (2002).
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	References

  We present a simple and generic theoretical description of actin-based
motility, where polymerization of filaments maintains propulsion. The dynamics
is driven by polymerization kinetics at the filaments' free ends, crosslinking
of the actin network, attachment and detachment of filaments to the obstacle
interfaces and entropic forces. We show that spontaneous oscillations in the
velocity emerge in a broad range of parameter values, and compare our findings
with experiments.

Velocity oscillations in actin-based motility
Azam Gholami1, Martin Falcke1, Erwin Frey2
1Hahn-Meitner-Institute, Dept. Theoretical Physics, Glienicker Str. 100, 14109 Berlin, Germany
2Arnold Sommerfeld Center for Theoretical Physics and Center of NanoScience,
Ludwig-Maximilians-Universität, Theresienstr. 37, 80333 München, Germany
(Dated: June 28, 2021)
We present a simple and generic theoretical description of actin-based motility, where polymer-
ization of filaments maintains propulsion. The dynamics is driven by polymerization kinetics at the
filaments’ free ends, crosslinking of the actin network, attachment and detachment of filaments to
the obstacle interfaces and entropic forces. We show that spontaneous oscillations in the velocity
emerge in a broad range of parameter values, and compare our findings with experiments.
PACS numbers: 05.20.-y, 36.20.-r, 87.15.-v
Force generation by semiflexible polymers is versatilely
used for cell motility. The leading edge of lamellipodia
of crawling cells [1] is pushed forward by a polymerizing
actin network and bacteria move inside cells by riding on
a comet tail of growing actin filaments [2, 3]. In vivo sys-
tems are complemented by in vitro assays using plastic
beads and lipid vesicles [4, 5, 6]. The defining feature of
semiflexible polymers is the order of magnitude of their
bending energy which is in the range of kBT . They un-
dergo thermal shape fluctuations and the force exerted
by the filaments against an obstacle arises from elastic
and entropic contributions [7, 8].
Mathematical models have quantified the force gener-
ated by actin filaments growing against obstacles [7, 8, 9].
The resisting force depends on the obstacle which is
pushed. In case of pathogens, it has a small com-
ponent from viscous drag of the moving obstacle but
consists mainly of the force exerted by actin filaments
bound to the surface of the bacteria and pulling it back-
wards [10, 11]. The tethered ratchet model [12] is a math-
ematical formulation of these experimental findings in
terms of the dynamics of the number of attached and de-
tached polymers. The starting point of our approach will
be the dynamics of the distributions of the free length of
both polymer populations.
Actin polymerization in living cells and extracts is con-
trolled by a complex molecular network [3]. Nucleation
of new filaments, capping of existing ones, exchange of
ADP for ATP on actin monomers, buffering of monomers
etc. all contribute to that control and have been modeled
[12, 13, 14]. Our goal is not to model the full complex-
ity of that biochemical network. Rather we focus on the
core process of force generation and force balance ensuing
from the interplay between bound pulling filaments and
polymerizing pushing filaments, the transition between
these two groups and the motion of the whole force gen-
erating configuration. This is motivated by recent ob-
servations of complex dynamics in simple reconstituted
systems: the velocity of beads or pathogens propelled
by actin polymerization may oscillate [16, 17, 18]. Our
goal is to describe the dynamics of such biochemically
compressed
under tension
viscous
  force
cross-linked actin network
cross-linkers
obstacle
brush
FIG. 1: (color online) Schematic representation of an ensem-
ble of actin filaments oriented at ϑ = 0 with respect to the
normal n̂ of an obstacle interface, which may either be a cell
membrane or a bacterium. While attached filaments are un-
der tension and pull the interface back, detached filaments
are compressed, elongate by polymerization with rate kon
and push the interface forward. All filaments in the brush
are firmly anchored in a cross-linked network, whose front
advances with velocity vg reducing the free length l of the
filaments. Attached filaments detach with stress dependent
rate kd and detached filaments attach with constant rate ka.
vo is the interface velocity in the extracellular medium, and
x is the distance between the front of the network and the
interface.
simpler systems and find a robust microscopic descrip-
tion for oscillation mechanisms, which may then be con-
trolled by higher order processes. Such a study is meant
to complement investigations based on a continuum ap-
proach [17, 18].
We consider a fixed number N of actin filaments [19]
firmly anchored into a rigid cross-linked network, which
advances with velocity vg; for an illustration see Fig. 1.
Filaments of variable length l are either attached to the
obstacle interface via a protein complex or detached from
it, with time-dependent number distributions denoted by
Na(l, t) and Nd(l, t), respectively. In the detached state,
filaments polymerize at a velocity vp(l, x), which depends
on both the polymer length l and the distance x between
rigid support and obstacle. Transitions between the two
filament populations occur with a constant attachment
rate ka and a stress-dependent detachment rate kd [20].
This results in the evolution equations
vg(l)− vp
Nd = −ka Nd + kd Na , (1a)
vg(l)
Na = ka Nd − kd Na . (1b)
The right hand side of Eq. 1 describes attachment and
detachment process. The second term on the left hand
side accounts for the gain and loss of attached and de-
tached polymers due to the dynamics of the polymer
mesh, growing with velocity vg, and the polymerization
kinetics of the filaments in the brush. The correction
factor l/x in front of vg is due to the fact that for bent
polymers the rigid network swallows by this amount more
in contour length than for straight filaments. This factor
is equal to 1 for l ≤ x.
Processes contributing to the growth of the rigid poly-
mer mesh are entanglement and crosslinking of filaments
in the brush. Both imply a vanishing vg for l → 0, since
short polymers do not entangle and crosslinking proteins
are unlikley to bind to them. At the same time vg can
not grow without bound but must saturate at some value
vmaxg due to rate limitations for crosslinker binding. This
suggests to take the following sigmoidal form
vg(l) = v
g tanh(l/l̄) , (2)
with a characteristic length scale l̄.
The polymerization rate is proportional to the prob-
ability of a gap of sufficient size d (≈ 2.7 nm) between
the polymer tip and the obstacle for insertion of an actin
monomer [7]. This implies an exponential dependence of
vp on the force Fd by which the polymer pushes against
the obstacle,
vp(l, x) = v
p exp [−d · Fd(l, x)/kBT ] . (3)
Here, vmaxp ≈ 500 nm s−1 [7] is the free polymerization
velocity. For the entropic force Fd we use the results ob-
tained in Ref. [8] for D = 2, 3 spatial dimensions, where
we take the accepted value of `p ≈ 15 µm [21] for the
persistence length of F-actin.
The dynamics of the distance x between grafted end
of the filament and the obstacle interface (see Fig. 1) is
given by the difference of the average vg and the velocity
of the obstacle
∂tx = −
dl vg(l) [Na(l, t) +Nd(l, t)] (4)
dl [Na(l, t) Fa(l, x) +Nd(l, t) Fd(l, x)] ,
where ζ is an effective friction coefficient of the obsta-
cle. The force Fa(l, x) acting on the obstacle interface
results from the compliance of the filaments attached
to it by some linker protein complex, which we model
as springs with spring constant kl and zero equilibrium
length. This complex has a nonlinear force-extension re-
lation which we approximate by a piece-wise linear func-
tion; for details see the supplementary material. Let
R‖ ≈ l[1− l(D− 1)/4`p] be the equilibrium length of the
filament. Then, the elastic response of filaments expe-
riencing small compressional forces (x ≤ R‖) is approxi-
mated by a spring constant k‖ = 12kBT `2p/(D−1)l4 [22].
For small pulling forces (x ≥ R‖), the linker-filament
complex acts like a spring with an effective constant
keff = klk‖/(kl + k‖). In the strong force regime, the
force-extension relation of the filament is highly nonlinear
and diverges close to full stretching [23]. Therefore, only
the linker will stretch out. The complete force-extension
relation is captured by
−k‖(x−R‖) , x ≤ R‖ ,
−keff(x−R‖) , R‖ < x < l ,
−kl(x− l)− keff(l −R‖) , x ≥ l .
Finally, we specify the force-dependence of the detach-
ment rate by
kd = k
d exp [−d · Fa(l, x)/kBT ] . (6)
Here, k0d ≈ 0.5 s
−1 [12] is the detachment rate in the
absence of forces and we have followed Ref. [20].
Eq. 1a has a singularity at vp(ls) = vg(ls)ls/x since
the coefficient of the derivative of Nd with respect to
l is zero at ls. To illustrate the key physical features
at that singularity, we start with the simple equation
∂tNd−∂l[vg(l)l/x−vp(l, x)]Nd = 0 with x kept constant.
Then those parts of the distribution of Nd with l < ls will
grow and catch up with ls since vg(l)l/x−vp(l, x) is posi-
tive there, while the parts with l > ls will shorten towards
ls. As a consequence the whole distribution will become
concentrated at ls. To quantify this heuristic argument
we expand vg(l)l/x − vp(l, x) up to linear order around
ls like v1(l− ls) and use the method of characteristics to
solve the equation. Starting initially with a Gaussian dis-
tribution we obtain Nd(l, t) = c(t) exp[−(l− l̄(t))2/w(t)2]
with c(t) = c0 exp(v1t), l̄(t) = ls + (l̄0 − ls) exp(−v1t)
and w(t) = w0 exp(−v1t). This shows that Nd evolves
to a monodisperse distribution which is localized around
ls. Its width decreases exponentially with time while its
height grows exponentially. The time scale for this con-
traction is given by [∂l(vgl/x− vp)]−1.
Since the same kind of singularity also occurs in the
full set of dynamic equations, Eqs. 1, we may readily
infer that Na and Nd evolve into delta-functions with
that dynamics. This is well supported by simulations,
and allows us to continue with the ansatz
Nd(l, t) = nd(t) δ(l − ld(t)) , (7a)
Na(l, t) = na(t) δ(l − la(t)) . (7b)
It defines the dynamic variables nd(t), ld(t), na(t) and
la(t). Upon inserting Eqs. 7 into Eqs. 1 and Eq. 4, we
obtain the following set of ordinary differential equations
∂tld(t) = vp(ld, x)−
vg(ld) + kd
(la − ld) , (8a)
∂tla(t) = −
vg(la) + ka
(ld − la) , (8b)
∂tna(t) = −kd(la, x) na(t) + ka nd(t) , (8c)
∂tx(t) =
[na(t) Fa(la, x) + nd(t) Fd(ld, x)]
[vg(la) na(t) + vg(ld) nd(t)] , (8d)
where nd(t) = N − na(t) since we keep the total number
of filaments fixed.
The values of many parameters in the dynamics can be
estimated using known properties of actin filaments. We
choose the linker spring constant kl ≈ 1 pN nm−1 [12]
and assume N = 200 [12] filaments to be crowded behind
the obstacle. A realistic value of the drag coefficient ζ is
10−3 pN s nm−1 but results did not change qualitatively
for a range from 10−5 pN s nm−1 to 1 pN s nm−1.
We have numerically solved Eqs. 8 in both D = 2
and D = 3 dimensions, and found the dynamic regimes
shown in Fig. 2: stationary states and oscillations. The
existence of an oscillatory regime is very robust against
changes of parameters within reasonable limits including
the spatial dimension. We checked robustness against
changes in the parameter values for the number of poly-
mers N , l̄ (see Eq. 2), kl, vmaxp and k
d, in addition to the
examples shown in Fig. 2. In general, we find that oscil-
lations occur for vmaxg . 500 nm s
−1 and within a range
of values for ka. Note that the oscillatory region in pa-
rameter space depends on the orientation ϑ of filaments
with respect to the obstacle surface, i.e. oscillating and
non-oscillating sub-populations of filaments may coexist
in the same network.
Oscillations appear with finite amplitude and period at
the lower boundary of the oscillatory region; compare the
example shown in Fig. 3a. The stationary state changes
stability slightly inside the oscillatory regime and oscil-
lations set in with a finite period. That is compatible
with oscillations appearing by a saddle node bifurcation
of limit cycles. The upper boundary of the oscillatory
region is determined by a Hopf bifurcation. An exam-
ple of an oscillation close to that bifurcation is shown
in Fig. 3b. More details on the phase diagram will be
published elsewhere [15].
We start with the description of oscillations in the
phase with vg > vp, i.e., decreasing lengths x, la and ld;
see Fig. 3. Then the magnitude of pulling and pushing
forces increases due to their length-dependence. When
the pushing force becomes too strong, an avalanche-like
detachment of attached filaments is triggered and the ob-
stacle jerks forward; compare the steep rise in ld, la and x
FIG. 2: (color online) Phase diagram of Eqs. 8a - 8d outlining
stationary and oscillatory regimes with ζ = 10−3 pN s nm−1
for (a-c) and (a) D = 2, ϑ = 0 , (b) D = 2, ϑ = π/4, (c)
D = 3, ϑ = 0 and (d) D = 3, ϑ = 0, ζ = 10−5 pN s nm−1.
l̄=100 nm, all other parameter values are specified in the text.
 1000
 1500
 2000
 2500
 0  200  400  600  800  1000
time (s)
na(t)
la(t) x(t)ld(t) (a)
 0  10  20  30  40  50  60  70
time (s)
na(t)
la(t) x(t)ld(t) (b)
FIG. 3: (color online) x , la , ld (in nm) and na as a function
of time, as obtained from a numerical solutions of Eqs. 8a
- 8d with vmaxg = 300 nm s
−1 and (a) ka = 0.143 s
−1 (b)
ka = 3.49 s
−1. D = 3, l̄=100 nm in both panels.
shown in Fig. 3. That causes a just as sudden drop of the
pushing force. With low pushing force now, polymeriza-
tion accelerates and increases the length of detached fila-
ments. The restoring force of attached filaments is weak
in this phase due to their small number. Hence, despite
of not so strong pushing forces, the obstacle moves for-
ward. In the meantime, some detached filaments attach
to the surface such that the average length and number of
attached filaments increases as well. When the detached
filaments are long enough to notice the presence of the
obstacle interface, they start to buckle. This, in turn, in-
creases the pushing force and slows down the polymeriza-
tion velocity. Therefore, the graft velocity now exceeds
the polymerization velocity and the average lengths of
attached and detached filaments start to decrease again
and the cycle starts anew. The period of oscillations is
dependent on the parameter values. It reduces from 240 s
in Fig. 3a to 13 s in Fig. 3b as ka increases from 0.143 s−1
to 3.49 s−1 at vmaxg = 300 nm s
The oscillations in x correspond to the saltatory mo-
tion of the obstacle in the lab frame and the oscillations
of its velocity since vg stays essentially constant. An il-
lustration is shown in Fig. 4 for a given set of parameters
which leads to oscillations with periods of the order of
100 s and velocity of the order of 0.7 µm s−1. This is
in good agreement with the results of experiments on os-
cillatory Listeria propulsion [16]. The period of velocity
oscillations with beads propelled by actin polymerization
differs from those of Listeria by one order of magnitude
(8− 15 min [18]). Periods of that length can be obtained
within our model upon using values for ka close to the
lower boundary of the oscillatory regime.
 0  50  100  150  200  250  300  350  400
time (s)
(a)speed
displacement
 0  50  100  150  200  250  300  350  400
time (s)
(b)speed
displacement
FIG. 4: (color online) Velocity and displacement of the obsta-
cle as a function of time with (a) ka = 0.9 s
−1, (b) ka = 1 s
vmaxp = 750 nm s
−1, vmaxg = 75 nm s
−1, k0d = 0.1 s
l̄=100 nm ζ = 10−3 pN s nm−1 and D = 3 in both panels.
We have also studied the system when the network
is oriented at an angle ϑ = π/4. In this case, the spring
constant of the attached filaments parallel to n̂ for D = 2
reads k−1‖ (ϑ) = 4`
+ e−�/2 − 1 + cos 2ϑ( 1
e−2� −
e−�/2) − cos2 ϑ(e−�/2 − 1)2]/kBT , where � = l/lp and
R‖(ϑ) = l(1 − l/4`p) cosϑ [22]. For the pushing force
of a filament grafted at ϑ = π/4, we use the results of
the factorization approximation given in Ref. [8], which is
well valid for a stiff filament like actin. A numerical solu-
tion of Eqs. 8a-8d results in the phase diagram shown in
Fig. 2(b) with the adapted forms of Fd and Fa. The main
effect is that one needs higher values for the attachment
rates and lower values for vg to obtain oscillations.
In summary, we have presented a simple and generic
theoretical description of oscillations arising from the
interplay of polymerization driven pushing forces and
pulling forces due to binding of actin filaments to the
obstacle. The physical mechanism for such oscillations
relies on the load-dependence of the detachment rate and
the polymerization velocity, mechanical restoring forces
and eventually also on the cross-linkage and/or entangle-
ment of the filament network. The oscillations are very
robust with respect to changes in various parameters, i.e.
are generic in this model. Therefore, complex biochem-
ical regulatory systems supplementing the core process
described here may rather stabilize motion and suppress
oscillations than generate them.
Oscillations of the velocity were observed during
propulsion of pathogens by actin polymerization. There,
the core mechanism described here is embedded into a
more complex control of polymerization, which e.g. also
comprises nucleation of new filaments and capping of ex-
isting ones. Hence, the study presented here can not be
expected to fully capture all features of such processes.
Our results still agree well with respect to velocity spike
amplitudes and periods in Listeria experiments reported
in Refs. [16, 17]. The velocity in between spikes appears
to be smaller in experiments than in our simulations.
This may be accounted for in our model by including
capping of filaments upon dissociation from the obsta-
cle. Periods may also become longer when capping and
nucleation were included since it would take longer to
restore the pushing force after the avalanche like rup-
ture of attached filaments. Altogether, qualitative and
quantitative comparison with experiments suggests that
our model may be a promising candidate for a robust
mechanism of velocity oscillations in actin-based bacte-
ria propulsion.
We thank R. Straube and V. Casagrande for inspiring
discussions. E.F. acknowledges financial support of the
German Excellence Initiative via the program ”Nanosys-
tems Initiative Munich (NIM)”. A.G. acknowledges fi-
nancial support of the IRTG ”Genomics and Systems
Biology of Molecular Networks” of the German Research
Foundation.
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