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Persistence of activity in noisy motor-filament assemblies
Raghunath Chelakkot,† Arvind Gopinath,‡ L. Mahadevan †∗
†School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA;
‡Max Planck Institute for Dynamics and Self-Organization, Goettingen 37077, Germany;
∗Department of Physics, Harvard University, Cambridge, MA 02138, USA.
Abstract
Long,elasticfilamentscross-linkedanddeformedbyactivemolecularmotorsoccurinvariousnaturalsettings.The
5
overall macroscopic mechanical response of such a composite network depends on the coupling between the active
1
0 and the passive properties of the underlying constituents and nonlocal interactions between different parts of the
2 composite. In a simple one dimensional system, using a mean field model, it has been shown that the combination
of motor activity and finite filament extensibility yields a persistence length scale over which strain decays. Here
n
a we study a similar system, in the complementary limit of strong noise and moderate extensibility, using Brownian
J multi-particle collision dynamics-based numerical simulations that includes the coupling between motor kinetics
2 andlocalfilamentextensibility.Whilethenumericalmodelshowsdeviationsfromthemeanfieldpredictionsdueto
1 thepresenceofstrongactivenoisecausedbythevariationsinindividualmotoractivity,severalqualitativefeatures
are still retained. Specifically, for fixed motor attachment and detachment rates, the decay is length is set by the
]
h ratio of the passive elasticity to the active shear resistance generated by attached motors. Our study generalizes
p the notion of persistence in passive thermal systems to actively driven systems with testable predictions.
-
o Submitted XXXX, and accepted for publication XXXX
i
b
*Correspondence: [email protected].
.
s Address reprint requests to :
c
i Editor
s
y
h
p Introduction thatthewavelengthsattainedinnatureareself-limiting
[ [2, 4] even as the flagella themselves range from tens
Theactivestraingeneratedbymolecularmotorsmoving
of microns to nearly a centimeter. This strongly sug-
1 on elastic filaments is the principal mechanism of force
v gests that over large lengths, mechanical information
generation and motion in cell biology [1, 2, 4, 5, 6, 7, 9,
1 transmission degrades substantially.
8]. In natural and reconstituted versions of such active
9 Inapassivecontext[6],ithasbeenrecentlydemon-
7 composites, the localized strains due to motors deform
strated that when bundles of filaments are forced to
2 thefilamentsdynamically,duetotheinterplaybetween
bend,theshearingforcesbetweenthemaremediatedby
0 elasticity,geometricconstraints,activemotorforcesand
. extensibility,leadingtoacharacteristicscaleoverwhich
1 noise due to fluctuations in the ambient medium and
mechanicalinformationistransmitted.Inanactivesys-
0 in motor activity. These deformations in turn regu-
tem such as an array of soft filaments driven by molec-
5 late the activity of the motors themselves [3, 13]. A
1 ular motors, a possible consequence of this finite decay
structurally ordered example of such a system is the
: lengthofstrainisthattheweakextensibilityofthefila-
v eukaryotic flagellum that is made of relatively stiff fila-
Xi ments(microtubules),motors(dynein)andpassiveelas- mentscanlimittherangeofmechanicalsignaltransmis-
sion between molecular motors and restrict the length
tic elements (nexins) which together oscillate with well
r scale over which motor coordination can occur. This
a defined wavelengths and frequencies [2, 11, 12, 13, 14].
will significantly affect the motor activity in filaments
While a variety of models of varying degree of com-
oflengthsmuchlargerthanthestraindecaylengthand
plexity [4, 5, 6, 7, 9, 8] are consistent with observations
could bring in spatial inhomogeneities in both strain
ofincreasingwavelengthwithincreasingflagellalength,
and activity. As an example, in Fig. 1(a), we show two
experimental evidences indicate that this relationship
distinct motor patches labeled I and II cross-linking a
is altered for very long lengths [15, 16, 17, 18] and
pair of thin filaments that may bend and shear. For
(cid:13)c 2013TheAuthors
0006-3495/08/09/2624/12 $2.00 doi:10.1529/biophysj.
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complete two way coupling between the motor activity
and filament dynamics that arises as the motor activ-
itycausesfilamentstrain,influencesthecollectivemotor
dynamics.Alltheseeffectsarenotcapturedinthemean
field model.
Here, we present a microscopic numerical model, in
which we incorporate the effect noise arising from vari-
ations in motor kinetics, and implement the two-way
couplingbetweenmotorsandfilamenttostudyitsinflu-
ence on the persistence of active strain by comparing
theresultswiththemeanfieldpredictions. Predictions
of the microscopic model agree qualitatively with the
mean-field results, despite the effect of strong active
noise. However, motor noise is shown to result in richer
dynamical features including modification of the motor
duty ratio and localized regions of coherent oscillations
due to the interplay between the local active shear
resistance and filament elasticity.
Figure1:(a)Anactive,motor-filamentcompositeillus-
tratingtwodistinctmotoraggregates,groupsofmotors
(I) and (II). Of interest is the distance between two
Persistence of strain in a passive composite
motor groups beyond which they act as if isolated. (b)
Schematic and geometry of the passive filament com- Tounderstandthecouplingbetweenshear,bendingand
posite with length (cid:96), effective thickness D +2w. The extension in a passive setting, we start with a mini-
arc-lengthismeasuredalongthetangentt.(c)Asketch mal model of two elastic filaments of unstrained length
of the coarse-grained active gel (top) corresponding to (cid:96), lateral width b and thickness w (cid:28) (cid:96) held at a dis-
the mean field approximation of the microscopic fila- tance D (cid:28) (cid:96) apart by a series of passive, linearly
ment / motor system (bottom) illustrating the motors elastic springs with areal density ρN and stiffness kN
(blue) and cross-linking passive linkers (red). (Fig. 1(b)-(c)). The passive springs are compliant in
shear along the axial direction but prevent any change
in the distance between the filaments.
At length scales large compared to the spacing
between these springs, the composite acts as a filament
inextensible filaments, mechanical activity by group (I) of thickness D, with an effective shear modulus that
andshear(sliding)inducedbythispatchistransmitted depends on the spring stiffness, the density of springs,
by attached motors in (II) over arbitrarily large inter- and the inter-spring spacing when b (cid:29) max(w,D). By
aggregatedistances.Forextensiblefilaments,thereisan considering a sliding displacement of the top filament
interplay between shear / slide and filament elongation relative to the bottom due to a shear acting on its sur-
along its contour length and leads to a decrease in the face of area b(cid:96), with a stress σ , the net force is σ b(cid:96).
s s
abilityofgroup(I)tomechanicallylinkwithgroup(II). Further,ifthearealdensityofspringsisρ ,theshearis
N
balanced by ρ b(cid:96) springs, all acting at the same time,
N
Using a mean field approach valid in the noise-less and each contributing to shear resistance with spring
limit [10], we recently showed that the decay length of constant k . Assuming a shear (sliding) displacement
N
localized steady or oscillatory strains is determined by of magnitude ∆, force balance
a combination of filament elasticity, passive shear resis-
G∗∆
tanceimposedbycross-linkersandtheactiveviscoelas- ∆k ρ b(cid:96)∼σ b(cid:96)∼ b(cid:96)
N N s D
tic properties of the motor aggregates. This analysis
was performed by coarse-graining the stochastic nature yields an effective shear modulus from these passive
of motor-filament interactions, and ignoring variations springs given by G∗ ∼ Dk ρ . For a strip of lateral
N N
in motors kinetics due to the finite number of motors. width w, we have G∗w/D ∼ wk ρ - consistent with
N N
However, even if thermal noise can be ignored (such as previousanalysesofapassivecross-linkedrailwaytrack
when motors are rigidly fixed at the base and there- model [7, 5].
foredonotdiffuse)thereisstillaneffectivemotornoise With this estimate in mind, we choose a local coor-
that arises due to discreteness of binding and unbind- dinate system characterized by an arc-length variable s
ingeventsandfluctuationsintheattachmenttimesdue so that the filament is in the range s ∈ (0,(cid:96)). For two
to the finite number of motors. Moreover, there is a points r (on filament 1) and r (on filament 2) (Fig.
1 2
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1(b)) which face each other when the filaments are in Allowing for sliding deformations at s = 0 and
theundeformedstate,wedefinetheslidingdeformation keeping bending, extension as well as sliding terms, we
∆(s) as difference between r and r , relative to its ini- repeat the minimization procedure to obtain the two
1 2
tial difference before deformation. The angle made by coupled equations,
thetangenttothecentreline,θ(s)isthenrelatedtothe
B θ +DK (Dθ −∆ ) = 0 (7)
slide ∆(s) and the displacement u(s) by geometry via pas ss pas ss ss
therelationsdr1/ds=t,dt/ds=Cnanddn/ds=−Ct, Kpas(∆ss−Dθss)−Gpas∆ = 0 (8)
where n and t are the normal and tangent vectors, and
In this general case, the second equation can rewritten
C =dt/ds·n≈θ isthelocalcurvature.Inthedeformed
s as ∆ −(cid:96)−2∆=0 and indicates that variations in the
state, the position r relative to r can be written as, ss ∗
2 1 slidingdisplacement∆areassociatedwithapersistence
(decay) length
r ≈r +Dn+ut. (1)
2 1
(cid:96) (cid:96) w/D
Here we have assumed (D+w) (cid:28) (cid:96). The axial strain (cid:96)∗ ≡ (cid:112) B E ∼ (cid:112) (Kpas/Gpas)21 . (9)
(cid:96)2 +(cid:96)2 1+w2/D2
along the filament is us where d(.)/ds≡(.)s. Using the B E
relations ∆s = D(|(r1)s|−|(r2)s|) and (r1)s = t, we Similarlength scalesappearin avarietyof softsystems
obtain, where bending, shear and extension are coupled, both
in microscopic and macroscopic contexts [19, 20].
(r ) ≈t−DCt+u t+uCn. (2)
2 s s Toputtheselengthscalesinperspective,welookat
typical parameters for a flagellar axoneme. We take w
For small deformations we can ignore quadratic/cross
and D as the radius and the spacing of microtubules,
terms in the deformation to get
E as the Young’s modulus of microtubules, k as the
N
stiffness of passive (nexin) links and ρ as the density
|(r ) |=1, |(r ) |≈1−Dθ +u (3) N
1 s 2 s s s
oftheselinks.Usingw ∼20nm,D ∼40nm[3],E ∼1.2
Substituting (3) into (2) we get GPa [3, 4], kN ∼ 16−100 pN µm−1 [23] and wρN ∼
105 − 107 m−1 [17, 23], we estimate (cid:96) ∼ 200 − 500
E
∆ =Dθ −u . (4) µm while (cid:96) ∼ 80−200 µm. This estimate is however
s s s ∗
with just the passive cross linkers contributing to the
Integrating (4) from s = 0 to s using the boundary shearstiffnessandthereforevaluesshouldbetreatedas
condition θ(0)=0 yields ∆(s)=Dθ(s)−u(s). an upper limit. Motor activity will naturally influence
The shape of the filament is then obtained by mini- this scale as attached motors contribute to the instan-
mizing the energy ET of the composite due to bending, taneous shear resistance. Assuming that all motors are
stretching and shear, with bound, with linear density O(108) m−1 and effective
spring stiffness of 10−3 N/m, we obtain (cid:96) ∼ 5−10
(cid:90) (cid:96)(cid:20)(cid:18)B θ2+K u2+G ∆2(cid:19)(cid:21) E
E =b pas s pas s pas ds. (5) µm. Since sperm flagella are often much longer than
T 0 2 this scale [4], the role played by activity in modulating
persistence lengths can be important.
Here B ∼ Ew3, K ∼ Ew and G ∼ ρ k are
pas pas pas N N
the passive bending, stretching and shear moduli, with
E is the Young’s modulus of the material. The Euler- Mean-field model for an active composite
Lagrange equations obtained by minimizing the func-
For the case we treat here, corresponding to the case
tional (5) are given by δE /δ∆ = 0 and δE /δθ = 0
T T
without bending wherein the slide ∆(s) = −u(s)
and lead to equilibrium static solutions.
according to (4), motor activity modifies the decay
We consider solutions to (4) in two complementary
length (cid:96) in the noiseless limit by changing the effec-
limits. When the bending stiffness B →∞, the com- E
pas
tive shear resistance in a manner controlled by motor
posite deforms due to shear and extension alone and
kinetics. We being by summarizing the results of the
∆=−u, so that the minimization of (5) provides
meanfieldanalysis[10],validinthenoise-lesslimitwith
∆ −(G /K )∆=0, (6) the focus on systems with purely extensional modes of
ss pas pas
deformation. While the original analysis [10] allows for
which yields the relaxation length scale for pure exten- amoregeneralformforthemotor-filamentinteractions,
sion (cid:96)2 ≡ K /G ∼ Ew/ρ k . For inextensible fil- hereweconsideraspecialcaseofthecontinuumtheory
E pas pas N N
aments, K →∞, and ∆=Dθ and the minimization consistent with our simulations as shown in Fig.1(b).
pas
procedureprovidestherelationθ −(cid:0)D2G /B (cid:1)θ = Here, two cross-linked filaments are held together
ss pas pas
0, thus yielding the persistence scale for pure bending by a combination of active elastic motors and by pas-
(cid:96)2 ≡B /D2G ∼Ew3/D2k ρ . sive elastic spring-like linkers. The active motors act as
B pas pas N N
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linear elastic springs of stiffness k , and can either be the second term gives the motor velocity relative to the
m
attached to or detached from the filament. When the filament, and the third term corresponds to the rate
inter-link spacing and the inter-motor spacing is much at which the mean strain changes due to the kinetics
smaller than the filament length, the motor-filament of motor attachment and arises from the difference in
assembly can be represented by a continuum descrip- extension of attaching and detaching motors. The two
tion. In this case, for an inertialess filament, the sum of dimensionless parameters in (12) A ≡ v k /(øoF )
1 0 m s
forces on the filament arising from filament extensibil- and A ≡ F /(k δ ) relate the microscopic motor
2 s m m
ity, springs and motors must vanish. If F is the active kinetics to the population averaged dynamics. Once we
m
force due to a single attached motor and ρ the motor have N and Y, the motor force F in (10) is written as
m m
density, this yields
F =k δ NY. (13)
m m m
(K u ) −G u+ρ F =0, (10) Equations(10)-(13)providesaclosedsystemforthethe
pas s s pas m m
dynamics of filament displacement u, attached motor
which differs from previous system described by (6) density N and motor extension Y in terms of dimen-
because of the active force density term ρmFm. sionless parameters A1, A2 and Ψ = ø/ωon. These
To estimate the active force density Fm that con- equationshaveahomogeneousstationarysolutiongiven
tributes to an active shear resistance, we consider a byN =(1+ΨF )−1 andY =A A (A +F )−1,and
0 0 0 1 2 1 0
minimal model for the motor-filament interaction. In anaturalquestionisthatoftheirstabilitytovariations
this, active motors attach to the filament with zero in the parameters.
extension (no pre-strain) and once attached walk rel- Before we describe the global distribution of the
ative to the filament with a velocity vm. The motor strainfieldandthepersistenceofactivity,wefirstexam-
velocity vm is linearly related to strain through a linear ine the dynamics of a small fragment of the filament-
relation, vm = yt = ut +v0(1−kmy/Fs), where ut is motor composite of length (cid:96)s (cid:28) (Kpas/Gpas)21. In
the velocity of the filament relative to the motor, Fs is this limit, we can ignore filament extensibility and the
the motor stall force and v0 is the zero load velocity of motion of this segment relative to its neighbors can
the motor. Since one end of the motor is permanently be mapped to that of a homogeneous population of
attached to a rigid substrate, motor movement results motors acting on a rigid segment and working against
in a motor extension y and thereby causes a resisting an external spring with effective stiffness K ∝ G .1
s pas
force kmy. For such a fragment, when Ks > 0, a linear stability
Motorkineticsmaybedescribedbyasetofpopula- analysisofequations(11)-(13)revealsthatthestation-
tion balances relating the attached and detached prob- ary states N and Y become linearly unstable and a
0 0
ability densities to the corresponding fluxes. Ensemble stable oscillatory state emanate via supercritical Hopf-
averaging the terms in these relations provide dynami- Poincarebifurcations[10].Inthestableoscillatorystate,
calequationsforthefractionofattachedmotors,N(s,t) the power input into the system due to activity bal-
and the mean motor extension, Y(s,t) (see [10] for ances the energy dissipated by motor viscosity. The
detailed derivation). The evolution equation for N is effective motor friction characterizing this process is
then given by η ∼ρ N k (d(cid:104)y(cid:105)/du ) where ρ N is the total
act m 0 m t ut=0 m 0
number of attached motors. As Ψ increases, the term
N =ωo (1−N)−ω N, (11)
t on off (d(cid:104)y(cid:105)/dut)ut=0 typicallydecreases,asdoesN0 thuspre-
dicting an increase in the frequency of the emergent
where ωo is the mean attachment rate of motors, and
on oscillations ω .
ø is the load dependent detachment rate, defined as c
To connect these localized solutions to the global
ø = øoF(E,Y), where E ≡ k δ2 /k T with δ being
m m B m extensional field in order to determine decay lengths of
the extension corresponding to motor detachment, and
imposedextensions,weconsideranactivecompositefil-
the scaled mean extension of the attached motors, Y ≡
ament held fixed at one end (s = 0) while free at the
(cid:104)y(cid:105)/δ . Since the motor detachment is strain depen-
m other (s = (cid:96)) which is forced harmonically. We assume
dent, the evolution of N is non-linearly coupled to the
that u(0,t) = 0 and u((cid:96),t) = u ((cid:96)) + (cid:15)uˆReal[eiωt],
mean motor stretch Y, as implied in (11). Population 0
where (cid:15)uˆ/u ((cid:96)) (cid:28) 1 characterizes the deviation from
balance relations then lead to an analogous evolution 0
thestationaryextensionalstateu (s)whichsatisfiesthe
equation for Y that is given by 0
equation
(cid:18) (cid:19)
Y = ut +A ωo (A −Y)+ωo Y 1−N . (12) Kpas(u0)ss−Gpasu0+GactC =0, (14)
t δ 1 off 2 on N
m 1Strictlyspeaking,thereisalsoanactivecontributiontothis
effective spring constant that comes from attached motors in
In (12), the first term on the right hand side is the
neighboringfilaments-butthisjustrescalesKsandthemapping
stretchduetothepassivemotionoftheattachedmotor, stillholds.IfρN=0thenthepassivepartofKs∝Kpas.
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where G = ρ k , the active analogue of passive betweenfilamentextensibilityandthemeanattachment
act m m
shear modulus and C ≡δmN0Y0 = 1+δΨmF0AA11+AF20. With time of motors - as evidenced by the fact that N0 and
boundary conditions u (0) = 0 and (u ) ((cid:96)) = 0, at Y are independent of u . We now extend this minimal
0 0 s 0 0
leading order, i.e. O((cid:15)0) (14) provides, theory and investigate the role of increasing noise and
fluctuationsonthedecayofbothsteadyandoscillatory
(cid:16) (cid:17)
u0(s)=β 1−αe(cid:96)sE −(1−α) e−(cid:96)sE . (15) localized extensions.
In our model, the continuous filament is made up of
Here,wenotethatthedecaylength(cid:96) =(cid:112)K /G , N discretesphericalmonomersofdiameterσ locatedat
E pas pas
α = e−2(cid:96)(cid:96)E(1+e−2(cid:96)(cid:96)E)−1 and β ≡ Gpas(GAa1ct+AF10A)(21δ+mΨF0) ri, (i=1,...,N), connected by an elastic potential
does not depend on the active shear resistance because N−1
motorssenseonlystrainratesandnottheactualstrain. u(cid:96) = κ2l (cid:88) (|ri+1−ri|−b)2. (18)
At the next order, i.e. O((cid:15)) we find the relations i=1
that determine the linearized response of the system to
Bendingstiffnessisimplementedviaathree-bodybend-
externally imposed small amplitude perturbations. The
ing potential
magnitude ofthe timedependent partof theextension,
uˆ(s) due to the active force from the motors ρ F can N−1
be found by substituting ρmFm =−Gactχuˆ inm(10m) and ub = κ2 (cid:88) (ti+1−ti)2 (19)
yields i=2
K uˆ −(G +G χ)uˆ=0. (16)
pas ss pas act where t = (r −r )/|r −r |. Here the effective
i i i−1 i i−1
where χ(ω) = −δ (Y T + N T ) characterizes bending rigidity κ penalizes angular changes from a
m 0 2 0 1
the complex linearized response of the system local straight geometry. We fix the filament length to
(cid:16) (cid:17)
with T ≡ − iω (iω+ωo ΨF +ωo A ) − beaconstant(cid:96)=80σ,andalsofixthespringconstants
1 δm on 0 off 1 κ = κ = 2×104(cid:0)k T/σ2(cid:1) in order to get a large fil-
(cid:16)−δi(cid:16)mωi(cid:17)ω+(cid:16)ωωωoooofofonFn+0(cid:48)(cid:16)ωNooNY0ff002F(cid:17)0(cid:17)(cid:16)Tiω1+[ω1ωoo0oof]fn.F+0(cid:48)ωNSoo0effeFk0i(cid:17)n(cid:17)g−1soluatniodns Tt2o th≡is emaxmletneentntsmiopaneyarslibssetteifftnrnceeeastlseeBndagratehslaa(cid:96)rpgr(cid:39)eigied2n5oi0nu(cid:96)eg.xhtTeshnoesitbbhleaentdsteihngegmseaenngtd-.
equation of the form uˆ ∼ es/λ yields the following For simplicity and to focus on the role of activity and
expression for the effective persistence length, noise, we did not incorporate permanent passive cross
links in the model. With this simplification, shearing is
√ (cid:18)(cid:113) (cid:19)−12 resisted because of the temporary crosslinks (bridges)
λ ≡Real[λ]= 2 ϕ2+ϕ2+ϕ (17)
E 1 2 1 formed by attached motors. These bridges disappear as
motors detach at sufficiently high loads.
where ϕ1 = (Gpas/Kpas) + (Gact/Kpas)R and ϕ2 = The active motors were modeled as linear springs
(Gact/Kpas)I with R and I being the real and imagi- withstiffnesskmandequilibriumlength(cid:96)m.Themotors
narypartofχrespectively.Wenotethatintheabsence attach with zero mean strain and walk along the fila-
of activity, i.e. Gact = 0, we recover the expression ment with discrete step size dm. As the motor heads
λ=(cid:96)E. move, the accompanying extension of the motor length
In summary, the mean-field analysis predicts that leads to a force F = −κ ((cid:96) − (cid:96) ) (cid:96)ˆ that acts on
m m m
the relative importance of the active (motor) to passive the attached filament. Thus, motor activity results in
(linker) elasticity is controlled by Gact/Gpas, whereas localized active strain on the section of the filament
in the absence of passive linkers the persistence length to which they are attached. Consistent with previous
is controlled solely by Gact/Kpas. Further, (17) implies experimental studies the load dependent stepping rate,
thatthepersistencelengthisinfluencedbytheimposed i.e. the stepping velocity of the motor head - is chosen
frequency ω, though the frequency dependent complex to have the simple form v (1−|F |/f ). However,
m m max
responseterm,χthatcontrolsthecompetitionbetween unlike in mean-field models where the attachment and
intrinsic motor duty ratios and the extrinsic imposed detachment processes are coarse-grained into rates, in
frequency ω in determining the filament extension. the simulations we prescribe microscopic probabilities:
the load independent attachment probabilityp , is the
on
probability of a detached motor to attach to the fila-
Numerical model for the motor-filament composite
ment, while the motor detachment probability p˜ is
off
However, the mean field theory does not take into defined by the piecewise function
account the effect of local motor noise due to dis-
(cid:40)
crete attachment and detachment events and the finite p˜ = poff, if |Fm|≤fcut
off
number of motors. Also, there is no intrinsic coupling 1, if |F |>f .
m cut
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Figure 2: (a) A schematic diagram of the filament-motor system for a small fragment. The filament moves due to
active forces imposed by attached motors - this motion is resisted by a anchoring spring of stiffness K . (b) The
s
time dependent displacement, ∆ (t)/d for a fixed value for the ratio of connecting spring to motor stiffnesses,
a m
K /k = 0.2, and for zero load motor detachment/attachment probabilities, p /p = 1.0 (top), 2 (middle)
s m off on
and 10 (bottom), illustrating various dynamical regimes observed in simulations. (c) Phase diagram indicating the
dynamical regimes as a function of the ratio K /k , and the ratio of zero load detachment-attachment probabili-
s m
ties (p /p ).The observed dynamical regimes are, steady extension (◦), intermittent oscillations ((cid:7)), and steady
off on
oscillations ((cid:4)). Dashed curves are a guide to the eye. (d) The oscillation frequency, ω of a periodically oscillating
c
filament with K /k =0.5, as a function of p /p .
s m off on
This load dependence of the detachment probability of particle with a momentum taken from the Maxwell-
individual motors allows for the crucial two-way cou- Boltzmann distribution with variance ρk T, where ρ is
b
plingbetweenfilamentelasticityandthemotorresponse the fluid density.
and also sets the critical motor extension at which Using the numerical model, we first analyze the
detachment occurs. We take f = f in our sim- effect of noise in dynamical regimes of a short inexten-
cut max
ulations, although in general they are not equal. Fur- sible fragment - this corresponds to the dynamics of
thermore, the constant load independent detachment filaments with length (cid:96)s (cid:28) (Kpas/Gpas)21. This allows
probability p in conjunction with p yields a well us to identify the way noise modified the predictions of
off on
defined equilibrium attachment of motors at zero load, the mean field model, which predicts a sharp separa-
such that, p (1−N)+p N = 1. However, we note tion between two states - a stable, stationary state and
on off
that despite the added noise, our numerical model is astable,oscillatorystate.Weexpectnoisetomaketrue
an approximate one in that excluded volume interac- boundary between these two regimes fuzzy.
tions between the motors are not taken into account; In Fig. 2(a), we show a rigid filament anchored to
theseinteractionswillbesignificantonlyatlargevalues a wall via a linear elastic spring of stiffness K that
s
of p when almost all motors are always attached, a moves under the action of 800 motors. Monitoring the
on
situation that we do not consider here. long time displacement of the free end as a function
Thepositionandvelocityofthemonomerbeadsare of the spring constant K and the probability ratio
s
updated using velocity-Verlet algorithm, in which the p /p yieldsresultssummarizedinFig.2(b)-(d).The
off on
forceactingonthemonomeriscalculatedateverytime interplay between the softness of the filament compos-
step.Theentiresystemisimmersedinaviscousmedium iteK /k andtheactivityp /p yieldsthreedistinct
s m off on
usingtheBrownianmultipleparticlecollisiondynamics dynamicalregimesillustratedinFig.2(b).ForfixedK
s
(MPC)schemewhichneglectstheeffectsofinertia[24]. when p /p (cid:28) 1, there is a large force on the fila-
off on
Accordingtothisscheme,eachmonomerindependently ment, and the free end of the filament attains a steady
performs a stochastic collision with a phantom fluid displacement with a well defined mean attached motor
density, with fluctuations due to the noise imposed by
discrete nature of motor binding and unbinding. For
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Figure 3: (a) Schematic sketch of a composite filament made of 50 rigid segments, with s as the coarse-grained
position measured along the fiament. (b)The mean steady state extension of material points as a function of their
positionfromtheclampedend(s=0).Theverticalbarscorrespondtotherootmeansquaredeviationandindicate
the role of noise. The extension is less correlated for soft assemblies, when K /k =0.01(left), while it is highly
pas m
correlated when K /k = 10. (c) Time averaged steady state extension of the segment at the free end, u ((cid:96)),
pas m 0
as a function of zero load motor attachment/detachment probability, p /p , for various values of passive spring
on off
stiffness, K /k . The extension increases with p /p , and decreases with K /k , consistent with analytical
pas m on off pas m
predictions. For large p /p the extension saturates at a maximum value, set by the finite length of the filament-
on off
motorcomposite.(d)Thefractionofmotorsattachedonacompositefilamentconsistsof50segments,asafunction
of their position from the clamped end for K /k = 10. The mean motor attachment is maximum close to the
pas m
clamped end, where the local strain is relatively small.
p /p (cid:29)1,theaveragenumberofmotorsattachedto oscillations. While the regular oscillations are consis-
off on
the filament is close to zero, and there is effectively no tent with the mean field predictions, the intermittent
active force. However there is an intermediate range in oscillating state - where the filament-motor composite
p /p , where we observe first intermittent (Fig.2(b)- randomly switches from a stationary to an oscillatory
off on
middle) and then regular (Fig.2(b)-bottom) filament state and back, is a new dynamical state observed only
inthepresenceofnoise.Asweepoftheparameterspace
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Figure 4: (a)-(c)The time averaged amplitude of imposed oscillations of the filaments as a function of scaled posi-
tions, for various elasticity contrast (a) K /k = 10−3, (b) 10−2 and (c) 10−1. An exponential decay length is
pas m
indicated whose value depends on both motor kinetics as well as the elasticity contrast. (d) The decay lengths
obtained by analyzing the amplitude of oscillations as a function of K /k for various p /p .
pas m on off
yields the phase plot in Fig. 2(c). The parameter range frequency (ω (cid:28) d /v ), and small amplitude (∆ (cid:28)
m m 1
over which steady oscillations are seen is qualitatively ∆ ). For sufficiently low frequencies, the results are
a
consistent with theoretical predictions [10]. However, qualitatively independent of ω and hence we will focus
due to discrete motor noise, the boundary separating on results obtained for a frequency ω = 10−3(d /v ).
m m
the steady and oscillatory states is no longer sharp - Quantifying the mean extension of the filament as a
as expected, we observe a region of intermittent oscil- functionofarclengthsoftheextensiblefilamentmotor
lations. Within the oscillatory regime, the frequency compositeshowninFig.3(b),weseethatthemaximum
increases with increasing p /p (Fig. 2(d)), which at the free end is influenced strongly by both p /p ,
off on off on
also agrees with the trend seen from the mean field andK .However,thediscretenatureofmotoractivity
pas
predictions of the theoretical model. causes large fluctuations in mean extension, especially
Next we study the role of motor noise on the exten- forverysoftfilamentswithK /k (cid:28)1(Fig.3b-left).
pas m
sibility of a composite filament that is made up of 50 Motornoisealsoresultsinuncorrelatedspatialdomains
equally sized rigid segments, each linked to adjacent in extension for K /k =0.01 while a greater degree
pas m
neighbors by linear springs of stiffness K (Fig. 3(a)) of correlation is observed for K /k = 10 (Fig.3(b)-
s pas m
that account for the weak extensibility of the compos- right).Concomitantly,variationsinthemeanextension
ite. Since no permanent cross links are present, ρ =0, (shown as vertical bars) are also more correlated for
N
andtheeffectivepassivestiffnessK ∝K .Wechoose stiffer filaments than for softer filaments. The noise
s pas
parameters for the filament elasticity and attachment decreases with increase in p , as shown in Fig. 3(b).
on
and detachment probabilities such that the base state Fig.3(c)showsthatthesteadystateextensionofthe
in the absence of imposed oscillations is steady (non- freeendu ((cid:96))/σincreaseswithp /p .Tocomparethe
0 on off
oscillatory) and stable and oscillate the free end at low numericalpredictionsforsteadystateextensionu (s)to
0
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our mean field theory [10], we simplify (15) by setting An exponential fit to the amplitude along the com-
ρ =0, yielding the solution posite for different parameter values allows us to esti-
N
mate the decay length λ as a function of s (Fig.4(d)).
E
(cid:18)G (cid:19) A A (cid:18) δ (cid:19) This indicates that for a fixed pon/poff, λ is scales as
u0(s)= Kact A 1+F2 1+mΨF s(2(cid:96)−s) (20) (Kpas/km)21. However, the decay length exhibits only
pas 1 0 0 weak dependence on motor kinetics for the range of
attachmentprobabilitiesinvestigated.Wealsofindthat
from which one deduces that there is no decay length the value saturates to a constant in simulations with
scale in the absence of permanent passive linkers. We large K /k , which we ascribe to finite system size
pas m
see from the right panel in Fig. 3(b) that this is needed effects.
the case. While the simple theory predicts a quadratic To compare with theoretically predicted decay
form for the extension consistent with Fig. 3(b), we length to our simulation results, we set ρ = 0 in (17)
N
see that increasing the attachment probability yields a to obtain the decay length
sharper gradient close to the clamped end and a flatter
(cid:18) (cid:19)1 (cid:34) (cid:35)
profile near the free end. Setting Gact =ρmkm, we find λ = 2 Kpas 2 1 (21)
from (20) that the extension is an increasing function E kmρm (cid:112)|χ|+Real(χ)
of k /K and is maximum at the free end, which are
m pas
both consistent with our numerical results (Fig. 3(c)). Since χ is a function of imposed frequency and motor
The dependence on the activity Ψ is also qualitatively kinetics,andistoleadingorderindependentoffilament
captured as the slope increases with the ratio p /p . stiffness, the term in the square brackets is a constant
on off
The extension at the free end u0((cid:96)) obtained from for a given ω and pon/poff. As shown in Fig.4(d), this
(20) indicates that u ((cid:96)) decreases with K /k while prediction agrees very well with our simulations.
0 pas m
it increases with (p /p ) - both these trends are
on off
observed in the numerical simulations(Fig.3(d)).
Conclusions
Sincethemotordetachmentisstraindependent,the
motor imposed extensions in turn regulate the motor The scale over which active deformations persist in a
kinetics - and specifically modify the mean attached fluctuatingenvironmentarebasicquestionsinthestudy
time. In Fig. 3(d) we plot the mean fraction of motors of living matter. Here, we address both these questions
attached, (cid:104)N (cid:105)/N , as a function of the distance in the context of an ordered composite structure con-
attached 0
from the anchored end, s for K /k = 10. When sisting of elastic elements that can be stretched and/or
pas m
p /p (cid:46) 0.1 the motor imposed extension of the fila- sheared by motors that can bind to and unbind from
on off
mentisweakduetolessnumberofmotorsattached,and them, taking to account the individual motor kinetics.
the distribution of attached motors along the compos- Even though extensibility is negligible locally, it affects
itefilamentisapproximatelyuniform.Howeverforlarge the scale over which deformations persist owing to the
values of p /p (p /p (cid:38) 1) the enhanced motor competition between extensibility and shear. In partic-
on off on off
attachment imposes significant extension to the fila- ular, for active systems, we show that the emergence
ment, which is maximum at the free end. The nonuni- of an oscillatory instability in a short segment leads to
form extension of the composite leads to a nonuniform deformations that do not persist forever and instead
distribution of attached motors (Fig.3(d)), where the decayoverlongerlengthscales.Further,thefinitenum-
meanmotorattachmentismaximumneartheanchored ber of motors attached causes an intermediate regime
end (s = 0) and decreases with the distance from that of intermittent oscillations, which was not predicted by
point, as the filament strain increases. This leads to a modelsthatignoreactivenoise.Ouranalysisshowsthat
diminished motor activity near the free end. These fea- the feedback due to the ability of attached motors to
tures are not seen in the mean field predictions and are senselocalextensionratesinfluencesthemotoractivity
thusadirectconsequenceofthenoiseduetothediscrete andmodifiestheeffectivelengthscaleoverwhichstrain
nature of motor binding and unbinding events. decays.
We finally calculate the length-scale corresponding Our simulations suggest that deviations from the
to the decay of the amplitude of the imposed oscil- mean field theory - even for weak extensibility - can
latory strain (uˆ) along the composite, for a range of arise in the limit of very small number of attached
K /k and p /p . For all values of K /k and motors(orsmallattachmentrates)andverylargenum-
pas m on off pas m
p /p , our analysis show that uˆ decays exponen- ber of attached motors (noise acting cumulatively at
on off
tially with distance s from the free end, as shown largeattachmentrates).Wefindthat,foragivenmotor
in Fig. 4(a)-(c). Further, the decay length of the activity, the decay length of strain predicted by the
exponential increases when Kpas/km is varied from mean field theory, λE ∼ 2(Kpas/Gpas)12, is valid even
10−3(Fig.4(a)) to 10−2(Fig.4(b)) and 10−1(Fig.4(c)). in the presence of noise. This confirms a finite range of
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[20] K.Hatch,C.Danilowicz,V.Colijee,M.Prentiss(2008)
Demonstrationthattheshearforcerequiredtoseparate
short double-stranded DNA does not increase signifi-
cantly with sequence length for sequences longer than
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