Page 1

Requirements for disordered actomyosin bundle contractility

Martin Lenz∗

James Franck Institute, University of Chicago, Chicago IL 60637, USA

Aaron R. Dinner

James Franck Institute, University of Chicago, Chicago IL 60637, USA

Department of Chemistry, University of Chicago, Chicago IL 60637, USA

Actomyosin contractility is essential for biological force generation, and is well understood in

highly ordered structures such as striated muscle.

sarcomeric bundles comprised only of F-actin and myosin thick filaments can also display contractile

behavior, which cannot be described by standard muscle models. Here we investigate the microscopic

symmetries underlying this process in large non-sarcomeric bundles with long actin filaments. We

prove that contractile behavior requires non-identical motors that generate large enough forces to

probe the nonlinear elastic behavior of F-actin. A simple disordered bundle model demonstrates

a contraction mechanism based on these assumptions and predicts realistic bundle deformations.

Recent experimental observations of F-actin buckling in in vitro contractile bundles support our

model.

In vitro experiments have shown that non-

I.INTRODUCTION

Many living organisms heavily rely on large-scale

movements to sustain themselves, from crawling amoeba

following a chemical cue to animals escaping predators.

All such motions originate on the much smaller scale of

motor proteins, and thus depend on the ability of living

cells to harness the power of their microscopic compo-

nents to generate mesoscopic to macroscopic displace-

ments. The best known example of this capacity is prob-

ably striated muscle, the type of muscle implicated in

virtually all our voluntary movements. Its main force-

generating elements are clusters—also known as “thick

filaments”—of the molecular motor myosin. Thick fila-

ments are able to slide directionally on polar actin fila-

ments (F-actin) by hydrolyzing adenosine triphosphate

(ATP), thus generating the microscopic motion at the

heart of muscle contraction.

transmission of this motion to larger scales critically

depends on the characteristic periodic structure illus-

trated in Fig. 1(a). As ATP is hydrolyzed each period—

or “sarcomere”—contracts, and macroscopic contraction

arises from the sum of the action of many sarcomeres

arranged in series [1].

Despite its familiarity, the sarcomere-like organization

of striated muscle is far from a universal feature of ac-

tomyosin assemblies found in cells. In some instances,

partially periodic arrangement reminiscent of striated

muscle is observed, as in subcellular contractile bundles

known as stress fibers and transverse arcs [2–4], or in

smooth muscle [5]. However, in many other cases no such

organization is known to exist: examples include graded

polarity bundles—a type of actomyosin bundle similar to

a stress fiber [6], as well as multidimensional contractile

structures such as the cell cortex [7] and lamellar net-

In striated muscle, the

∗martinlenz@uchicago.edu

works [8]. Sarcomere-like contraction is unlikely to apply

to these systems, and there is no consensus regarding

their actual deformation mechanism.

In vitro experiments using purified proteins are use-

ful for understanding how these structures convert mi-

croscopic forces into large-scale motion and identifying

the minimum requirements of actomyosin contractility.

Early attempts using dilute actomyosin gels were not able

to induce contractility in the presence of actin and myosin

alone [9–12], although adding the actin cross-linker α-

actinin did produce observable contraction.

a more recent study using denser actomyosin bundles

shows that F-actin and myosin can induce contractil-

ity on their own [13]. The bundles used in these ex-

periments are likely disordered, in that their actin fila-

ments and myosins are not aligned in register and they

lack actin polarity ordering.

contractility cannot be described by a sarcomere-like

model. Nevertheless, they are able to contract at veloci-

ties ≈ 20-600nm·s−1and to generate tensions ≈ 500nN.

These effects are observed in bundles as long as 100µm,

whereas the individual F-actin and thick filaments used

have lengths of order 5µm and 300nm respectively. This

demonstrates that F-actin and myosin are sufficient to

convert small-scale sliding motion into large-scale con-

traction. In these experiments actin polymerization-

depolymerization dynamics is inhibited by phalloidin,

and thus is not necessary for contractility.

Disordered actomyosin systems have attracted theo-

retical attention for several years. A significant fraction

of the literature consists of continuum models, which fo-

cus on cytoskeletal behavior on scales much larger than

the length of an actin filament [14–17]. In these elegant

descriptions, contractility is introduced phenomenologi-

cally, which avoids the question of its emergence from

the microscopic interaction between myosin and F-actin.

Several other studies do however investigate this connec-

tion. In the absence of sarcomeric organization and cross-

linkers, thick filaments are expected to simply translocate

However,

As a consequence, their

arXiv:1101.1058v1 [physics.bio-ph] 5 Jan 2011

Page 2

2

(a)

(b)

actin filament

barbed end

pointed end

passive cross-linker

myosin thick

filament (motor)

contractioncontraction

contraction

extension

Figure 1. Mechanism of muscle contraction, and the prob-

lem of disordered actomyosin systems. (a) In striated mus-

cle, the polar actin filaments are rigidly held at their barbed

ends by passive cross-linkers. The pointed ends of the fila-

ments face each other and are bound to myosin thick fila-

ments. Upon ATP hydrolysis, the thick filaments slide on the

F-actin towards the barbed ends, thus inducing a contraction

of the whole structure. (b) A thick filament localized near

the pointed ends of two F-actin (left) represents a geome-

try similar to the striated muscle case, and leads to contrac-

tion. If located near their barbed ends (right), the tendency

of the myosin to slide towards the F-actin barbed end pushes

the actin out, which constitutes extension. In a large class

of bundles the two effects balance each other and no overall

contraction or extension occurs, as discussed in Sec. III.

F-actin without inducing any contraction [18, 19]. In-

stead, they act as a conveyor belt and sort the F-actin

according to polarity, as observed in some experiments

[20]. This conceptual difficulty can however be overcome

by assuming that thick filaments have a tendency to dwell

at the barbed end of the F-actin after sliding over their

whole length [21–24], or more generally that their veloc-

ity depends on their position relative to the filament [25].

As a consequence, F-actin would tend to have immobi-

lized motors that transiently act as passive cross-linkers

at their barbed ends. This essentially introduces a small

amount of sarcomere-like organization and results in con-

tractility. In the limit where the ratio of F-actin length

to thick filament size (? 20 in Ref. [13]) is much larger

than one, however, the portion of the motor population

bound to actin barbed ends is small and this effect be-

comes negligible, although it could be enhanced by the

presence of myosin traffic jams on F-actin [26]. Also, no

direct experimental evidence of thick filaments dwelling

at the barbed end of F-actin at the end of their runs is

available.

In this paper we undertake a systematic study of the

possible mechanisms leading to the emergence of con-

tractility in bundles of polar filaments and sliding mo-

tors. We focus on mechanisms appropriate for structures

without sarcomere-like organization. In Sec. II, we de-

velop a model of bundles inspired by the experiments of

Ref. [13]. As the internal geometry of the bundles used

in this study is unknown, we keep our discussion at the

level of a bundle with arbitrary connections between mo-

tors and polar filaments, neither assuming nor excluding

ordering at this stage. Sec. III then discusses the general-

ity of the situations described in the previous paragraph

and illustrated in Fig. 1(b), where random assemblies

of motors and filaments fail to induce contractility. We

find that the symmetry properties of the actomyosin bun-

dles impose restrictions on the mechanisms susceptible to

macroscopic contraction. We determine that the the non-

linear elastic behavior of F-actin is most likely to account

for the observations of Ref. [13]. This effect was shown

to induce contractility in a simple bundle geometry in

Ref. [27]. Sec. IV discusses this effect in a more general

disordered bundle model. Finally, we discuss our results

and their relation to experimental situations in Sec. V.

II.BUNDLE MODEL

The constituents of our model are polar filaments, rep-

resenting the F-actin, linkers, representing either molec-

ular motors or passive actin cross-linking proteins, and

“junctions”, representing the interface between filament

and linker. Passive cross-linkers are usually able to bind

actin through two binding sites [1], whereas myosin thick

filaments have numerous (≈ 100) motor heads and are

thus able to bind more than two filaments in principle.

The bundles studied in Ref. [13] have a diameter smaller

than the diffraction limit, indicating that their 5µm-long

actin filaments are essentially parallel to the bundle axis.

Accordingly, we study bundles within which the filaments

all lie along the same direction z, and ask whether these

bundles have a tendency to contract, extend, or neither.

Most previous models of actomyosin represent the

filament-motor interactions through some form of mean-

field approximation, whereby motors couple to some lo-

cal average filament velocity. Such a simplifying assump-

tion enables powerful analytical formulations, but entails

some loss of information about the bundle’s geometry.

Here, we are interested in the role of the bundle struc-

ture and symmetries in contractility, and must therefore

avoid such simplifications if we are to make general state-

ments. Relatedly, in this section we do not impose a

specific topology of connections between linkers and fil-

aments. This approach is required for the discussion of

Sec. III.

In Sec. IIA we introduce the components of our bun-

dle, then describe the topology of their connections in

Sec. IIB. Kinematic and mechanical relationships per-

taining to these connections are discussed in Sec. IIC,

followed by mechanical descriptions of the filaments and

junctions in Secs. IID and IIE, respectively. Finally, we

Page 3

3

filament 1

linker 2linker 1

filament 2

filament 3

junction 2

filament 4

junction 1

junction 3junction 4

filament 5

filament 6filament 7

z

Figure 2.

actin, one myosin thick filament bound in three different sites

and one bound only once. The division into units is indi-

cated, and the bundle geometry in this example is described

by Eqs. (1), (3) and (8).

Schematic of a model bundle comprising three F-

simplify the equations derived in this section in Sec. IIF.

A. Linkers, filaments and junctions

The geometry of the model bundle studied here is il-

lustrated in Fig. 2. To describe it, we divide the bundle

into “units” of three types: filament units, junctions and

linkers.

We define a filament unit as the stretch of F-actin be-

tween two successive locations where an F-actin binds

a thick filament or passive cross-linker, or between such

a location and a free F-actin end. We do not consider

freely floating F-actin, i.e., filament units with two free

ends. We denote by n the number of filament units. We

define Π the n × n diagonal matrix of filament unit po-

larities such that Πii = +1 (Πii = −1) if the pointed

end of filament unit i is oriented in the direction of pos-

itive (negative) z. For example, the bundle represented

in Fig. 2 has

0 00

Π =

1 0

0 1

0 0 −1

0 0

0 0

0 0

0

0

0

0

0

0 0 0

0 0 0

0 0 0

0

0

0

−1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

. (1)

In the following we refer to the positive z direction as the

“right” for convenience, and to the negative z direction

as the “left”. Here we consider a situation similar to that

of Ref. [13], and assume that no actin polymerization or

depolymerization takes place within the bundle.

Linkers represent motors or passive cross-linkers. We

denote their number by n??. Linkers may bind F-actin

through a junction (Fig. 2). One linker can have several

junctions to F-actin. Linkers are assumed to be rigid

objects, and the distance between junctions pertaining

to the same linker is therefore constant. We denote by

n?the number of junctions. Since junctions involve a

portion of F-actin, each of them has a polarity described

by a n?× n?diagonal matrix Π?. Junctions reaching the

very end of an F-actin immediately detach in our model.

Thus a junction is always flanked by exactly two filament

units of the same polarity, and Π?can be expressed as a

function of Π as discussed below [Eq. (4)]. In our model,

each linker is connected to at least one junction, each

junction has exactly two filament unit neighbors, and

each filament unit is connected to one or two junctions.

As a consequence, n??? n?? n.

In the following we do not consider linker attachment

or detachment from the bundle. This is partly motivated

by the fact that these phenomena do not seem to partic-

ipate in the contraction dynamics in Ref. [13]. Indeed,

the number of thick filaments in the bundles studied there

does not change over time, except for a short transient

regime where loosely associated myosins are washed away

by ATP injection. A more general motivation for this

assumption is discussed in Sec. IIIC. Finally, no interac-

tions between linkers and filaments are taken into account

besides those occurring at junctions.

B.Bundle geometry and topology

In the previous section we introduced the filament

units, junctions and linkers forming our bundle. To char-

acterize their interactions and eventually the overall bun-

dle behavior, we need to describe the connections be-

tween them, which we do here.

We first describe the relationship between junctions

and filament units. As stated above, every junction is

flanked by two filaments but the reverse is not true. As

a consequence, n?? n, and 2(n − n?) is the number of

filament dangling ends. We define the n?× n matrices ρ

and λ by

∀i?∈ {1..n?}, i ∈ {1..n}

?

0otherwise

?

0 otherwise.

ρi?i=

1 if i is the right-hand neighbor of i?

(2a)

λi?i=

1if i is the left-hand neighbor of i?

(2b)

For instance, the bundle represented in Fig. 2 is described

by

λ =

1 0 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

, ρ =

0 1 0 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

.

(3)

Consider a vector x = (x1,··· ,xn), where xi is some

quantity associated with filament unit i (e.g., a force,

length, or velocity). The product ρx is a vector of

length n?. The i?th entry of this vector, denoted (ρx)i?,

is equal to the quantity xi associated with the right-

hand neighbor of junction i?. Likewise, we may define

x?= (x?

junction i?. Then ρTx?, where the index T represents the

1,··· ,x?

n?) with x?

i? some quantity associated with

Page 4

4

matrix transpose, is a length n vector. If filament unit

i has a left-hand junction neighbor, then the ith entry

(ρTx?)iof this vector is the quantity x?

this neighbor. If it does not, then (ρTx?)i= 0. Similar

statements hold for matrix λ. These matrices further-

more allow us to express the fact that a junction is al-

ways flanked by exactly two filament units of the same

polarity as

i? associated with

Π?= ρΠρT= λΠλT.(4)

Since each junction has exactly one right-hand (left-

hand) neighbor, there is exactly one entry equal to 1 in

each row of ρ (λ). As each filament unit has at most

one left-hand (right-hand) neighbor, there is zero or one

entry equal to 1 in each column of ρ (λ). This implies

ρρT= λλT= 1 l,(5)

where 1 l denotes the n?×n?identity matrix. Another con-

sequence is that the matrices ρTρ and λTλ are projectors

onto the subspace of filament units that have a left-hand

and a right-hand neighbor, respectively. Equivalently, we

can write

?

0 if it does not.

(ρTρx)i=

xi

if i has a left-hand junction neighbor

(6)

A similar statement holds for λTλ.

We now turn to linkers and define the n??× n?matrix

γ by

∀i??∈ {1..n??}, i?∈ {1..n?}

?

0 otherwise.

γi??i? =

1 if junction i?is connected to linker i??

(7)

For instance, Fig. 2 has

γ =

?1 1 1 0

0 0 0 1

?

.(8)

Note that the description of the bundle topology given

here does not make any assumptions about the dimension

of the transverse direction of the bundle. In particular,

we do not require that the bundle can be represented in

a two-dimensional picture as in Fig. 2: it can a priori be

three-dimensional, or otherwise.

C.Forces and velocities

Having described the bundle’s geometry, we now turn

to its mechanical characterization.

discuss the forces and velocities of its components, and

implement simple force balance and velocity continuity

constraints pertaining to the connections between them.

Due to the one-dimensional nature of the bundle, we only

consider forces and velocities along the z axis.

In this section we

filament i

junction i' linker i'' (has velocity )

connection to junction j':

force

left:

force

velocity

right:

force

velocity

arclength

left:

force

velocity

right:

force

velocity

connection to linker:

force

velocity

end-to-end length

connection to junction k':

force

Figure 3. Summary of the model features defined in the text.

A filament unit, a junction and a linker are represented, along

with the velocities and lengths characterizing their state, as

well as the forces they are subjected to.

Let f be the length n vector of filament unit tensions.

More specifically, fiis the force applied to the right-hand

side of filament unit i (fi< 0 for a compressive force—

see Fig. 3 for a summary of notations). If filament unit

i has a dangling right end, then fi= 0. Filament unit i

is also subjected to two other types of forces: the force

applied to its left-hand side, and viscous friction forces.

Let us estimate the amplitude of viscous forces in the

situation of Ref. [13]. We estimate that actin filaments

in that study are approximately d ≈ 50nm apart in the

directions perpendicular to z, a distance equal to the di-

ameter of a thick filament [28]. Typical filament lengths

and velocities are L = 5µm and v ≈ 10−6m·s−1, respec-

tively, and the viscosity of water is η ? 10−3Pa·s. In the

situation of two neighboring filaments moving at different

speeds, velocity gradients ≈ v/d are generated, leading

to shear stresses ≈ ηv/d exerted over a surface ≈ Ld, and

therefore viscous forces are of order ηvL ≈ 10−15N. For

comparison, thick filaments generate forces ≈ 10−11N,

and we may thus neglect viscous effects. Combining this

observation with force balance on a filament unit, we find

that the force exerted on the left-hand side of filament

unit i is −fi. Thus a filament unit with a dangling left

end also has fi= 0.

The force exerted on the left of junction i?is opposite to

that exerted on the right of its left filament unit neighbor,

and thus equal to −(λf)i?. Likewise, the force exerted

on its right is (ρf)i?. Force balance thus implies that the

linker bound to junction i?exerts a force [(λ − ρ)f]i? on

this junction (see Fig. 3).

The only forces applied to linker i??are those exerted

by junctions, and force balance implies that they have to

sum to zero:

?

or, in matrix form:

i?connected to i??

[(ρ − λ)f]i? = 0,(9)

γ(ρ − λ)f = 0.(10)

Page 5

5

Although this condition comprises n??scalar equations,

those equations are not all independent. This can be seen

by considering the mechanical subsystem formed by all

filaments and junctions (but not including the linkers).

Force balance for this subsystem reads:

n?

?

i?=1

[(λ − ρ)f]i? = −

n??

?

i??=1

[γ(ρ − λ)f]i?? = 0, (11)

where the first equality follows from the fact that γ has

exactly one element equal to 1 per column and zeros ev-

erywhere else. Therefore Eq. (10) expresses only n??− 1

linearly independent scalar conditions.

Let vrbe the length n vector of actin velocities at the

right-hand side of the filament units. In other words,

vr

iis the instantaneous velocity of the rightmost actin

monomer of filament unit i in the laboratory reference

frame. Note that the identity of this monomer may

change as the motor bound to the right-hand side of fil-

ament unit i slides relative to the actin. We define vl

similarly for the filament unit’s left-hand side. Thus the

actin velocity at the left of junction i?is (λvr)i?, and that

at its right is (ρvl)i?. Since the junction is very small, the

flux of actin into and out of it have to be equal, and those

two velocities are identical. This is expressed by

λvr= ρvl. (12)

Finally, we define v??the length n??vector of linker ve-

locities. We choose to work in the reference frame where

the center-of-mass of all linkers is motionless, which reads

n??

?

i??=1

v??

i?? = 0. (13)

Since junctions pertaining to the same linker are held at

a fixed distance, the connection of junction i?to its linker

(see Fig. 3) has a velocity (γTv??)i?.

D.Filament lengths and force-extension relations

We now discuss the relationship between the forces and

velocities introduced in the last section and the filament

units’ lengths. Because the filament units’ lengths mea-

sure the distances between junctions, they are obviously

related to the total bundle length. We thus study their

rates of change to determine whether the bundle con-

tracts or not.

Actin filaments in actomyosin bundles can undergo sig-

nificant thermal fluctuations, or be under compression

due to linker-induced forces. To account for these pos-

sibilities, we define two independent lengths pertaining

to filament unit i: its end-to-end distance ?iand its ar-

clength Li(Fig. 3). These quantities are related to the

force exerted on the filament units through their force-

extension relationships:

∀i ∈ {1..n}

fi= F(?i,Li), (14)

or, in vector notation

f = F(?,L), (15)

where ? and L are length n vectors with elements ?iand

Li, respectively.

Although the reasoning of Sec. III is largely indepen-

dent of the exact form of function F, we may calculate

a reasonable expression for it by modeling the filament

units as worm-like chains with a persistence length ?p.

For small transverse fluctuations, this implies describing

filament unit i with the following free energy functional:

?kBT?p

+fi

2

?Li

0

2

?(∂2

zx)2+ (∂2

?(∂zx)2+ (∂zy)2??

zy)2?

dz,(16)

where x(z) and y(z) are the transverse coordinates of the

filament displacement, and kBT is the thermal energy.

Imposing ∂zx(z = 0) = ∂zy(z = 0) = ∂zx(z = Li) =

∂zy(z = Li) = 0 and averaging over the (fast) thermal

fluctuations, we find

??i− Li? =L2

i

2?p

1 −

?

˜ficoth

˜fi

?

˜fi

, (17)

where˜fi = fiL2

only the averages over thermal fluctuations and thus

drop the bracket for brevity. In approximation of small

transverse fluctuations used here, the end-to-end length

?i of filament unit i diverges for a compressive force

fi → −fb(Li) = −π2kBT?p/L2

buckling force for filament unit i.

Returning to the general argument, the arclength Li

of filament unit i changes as the junctions neighboring

i slide relative to the F-actin.

junction on its right-hand side, the velocity of the linker

there is (λTγTv??)i, and its sliding velocity relative to

the actin is (λTγTv??−vr)i. We may rewrite this sliding

velocity as (λTγTv??− λTλvr)i, which is equal to the

previous expression if i has a right-hand neighbor, and

to zero if it does not. Similar expressions can be written

for the left-hand side, which yields

i/kBT?p. In the following we consider

i, whereby fb(Li) is the

If i has a neighboring

dL

dt

= (λTγTv??− λTλvr) − (ρTγTv??− ρTρvl).(18)

The first (second) term on the right-hand side of this

equation accounts for actin-linker sliding on the right-

hand (left-hand) side of the filaments units, and vanishes

for the filaments units that do not have a right-hand (left-

hand) neighbor junction.

Let us consider a filament unit i that has a right-hand

junction neighbor, and assume for argument’s sake that

its left end is fixed. This filament unit’s end-to-end dis-

tance ?iis defined as the distance between its left end and

its right-hand junction neighbor. It therefore changes at

Page 6

6

a rate (λTγTv??)i, the velocity of the junction. A sim-

ilar statement holds for filament units with a left-hand

junction neighbor. Therefore, the end-to-end length of a

filament unit with two junction neighbors changes at a

rate

d?i

dt

=?(λT− ρT)γTv???

i. (19)

Let us now consider filament units that have one free

end, and thus have a vanishing tension. Differentiating

Eq. (14) with respect to time, this implies

∂F

∂?

d?i

dt+∂F

∂L

dLi

dt

= 0, (20)

which gives the rate of change of ?i for these filament

units. We define the diagonal matrices

?∂F

?∂F

where δij is the Kronecker delta, and note that the el-

ements of ∂F/∂? are always strictly negative when the

expression Eq. (17) is used, which implies that this ma-

trix is invertible. Making use of the fact that matrices

ρTρ and λTλ are projectors (see Sec. IIB), we write an

equation that describes filament units whether they have

one or two neighboring junctions:

∂L

?

?

ij

=∂fi

∂Lj

= δij∂F

∂L(?i,Li) (21a)

∂?

ij

=∂fi

∂?j

= δij∂F

∂?(?i,Li), (21b)

d?

dt= (ρTρλT− λTλρT)γTv??

−(1 l − ρTρλTλ)

?∂F

∂?

?−1∂F

∂L

dL

dt,

(22)

where 1 l denotes the n × n identity matrix. Note that

the first (second) term of this equation vanishes for all

filaments units with one (two) junction neighbors due to

the projector properties of ρTρ and λTλ.

E.Junction force-velocity relationships

The previous section describes the internal mechanics

of the filaments units, while that of the rigid linkers is

quite simple and is captured by Eq. (10). In this sec-

tion we turn to the last type of unit in our system—the

junctions—and characterize the relative motion of actin

and linkers within them.

In the case of a junction between F-actin and a pas-

sive cross-linker, the binding of the cross-linker imposes

that the filament moves at the same velocity as the linker

(no-slip condition). A myosin thick filament, on the other

hand, tends to slide along F-actin in discrete steps when

in the presence of ATP. Here we consider the observa-

tional time scales ? 1s of Ref. [13], much longer than the

typical duration ≈ 10ms of the mechanochemical cycle of

a smooth muscle myosin IIb under saturating ATP condi-

tions [29]. We thus average over a large number of steps,

and consider that the actin-myosin sliding occurs at a

well-defined velocity. It is well-known that sliding veloc-

ities in molecular motors depend on the mechanical load

applied to the motor-filament junction [30]. Considering

some junction i?, this can be expressed as a functional

relationship between the velocity of the linker relative to

the F-actin and the force applied to i?by the linker to

which it is connected :

∀i?∈ {1..n?}

where the function˜V?

Π?

by noting that the linker force-velocity relationship must

not depend on our arbitrary choice of the direction of

positive z. Reversing this choice is equivalent to reversing

the sign of all velocities, forces and polarities. The only

way for Eq. (23) to be invariant under this transformation

is to write

(ρvl−γTv??)i? =˜V?

i? {[(λ − ρ)f]i?}, (23)

i? a priori depends on the polarity

i?i? of the junction. This dependence can be determined

∀i?∈ {1..n?}

where function V?

tor i?, an a priori nonlinear function independent of Π?.

In vector notation,

(ρvl−γTv??)i? = Π?

i?i?V?

i? {Π?

i?i?[(λ − ρ)f]i?},

(24)

i? is the force-velocity relationship of mo-

ρvl− γTv??= Π?V?[Π?(λ − ρ)f].

Note that Eq. (25) may describe a passive cross-linker

if Vi?(f) = 0, ensuring that the linker does not move

relative to the F-actin whatever the force applied to the

junction. Equation (25) does not include the possibility

that motors slow down or stop when reaching the barbed

end of F-actin, consistent with the discussion of Sec. I.

(25)

F.Simplified system of equations

We now have a complete description of the bundle, but

not a very compact one. Here we simplify the equations

given above into a more concise system of 2n nonlinear

first-order ODEs. We use Eqs. (12), (15) and (25) to

eliminate f, ρvland λvrin Eqs. (18) and (22). This

yields

dL

dt

d?

dt= −

= (ρT− λT)Π?V?[Π?(λ − ρ)F(?,L)]

?∂F

(26a)

∂?

?−1∂F

∂L(ρTρ − λTλ)(ρT+ λT)Π?V?[Π?(λ − ρ)F(?,L)] + (ρTρλT− λTλρT)γTv??

s(?,L),(26b)

Page 7

7

where the nonlinear vector function v??

and the following vector equation, obtained by combining Eqs. (10) and (15), then differentiating with respect to time

and inserting Eqs. (26) into the resulting equation:

s(?,L) is the solution of the linear (in v??) system of equations formed by Eq. (13)

γ(ρ − λ)∂F

∂?(ρTρλT− λTλρT)γTv??= γ(ρ − λ)∂F

∂L(ρTρλT− λTλρT)Π?V?[Π?(λ − ρ)F(?,L)]. (27)

Equation (27), just like Eq. (10), has only n??− 1 in-

dependent scalar equations, and supplementing it with

Eq. (13) thus results in a complete set of equations for

v??

and exhaustive description of the dynamics of a bundle

of arbitrary geometry.

s. Thus Eqs. (13), (26) and (27) constitute a compact

III.SITUATIONS WITHOUT TELESCOPIC

DEFORMATION

We now ask whether our model allows for bundle con-

traction or extension. More specifically, we focus on

“telescopic deformation”, whereby the end-to-end veloc-

ity of a bundle under negligible external load is propor-

tional to its length, as observed in Ref. [13].

characteristic of systems formed of a serial arrangement

of many independently deforming elements, the size of

which was estimated to be 5-10µm in Ref. [13].

This is

Here we demonstrate two requirements for telescopic

deformation. In Sec. IIIA we show that it cannot arise

in a bundle with only F-actin and motors having identi-

cal unloaded velocities. Sec. IIIB tackles the case where

motors with different unloaded velocities are present. We

furthermore focus on bundles assembled in a polarity-

independent way, which is likely the case in Ref. [13].

We prove that telescopic deformation cannot arise if the

filaments always stay in their linear elastic regime, e.g., if

they are very stiff. These two properties imply that tele-

scopic deformation in bundles assembled in a polarity-

independent way requires linkers that (i) have a variety of

motor unloaded velocities and (ii) generate large enough

forces to probe the nonlinear elastic behavior of the fil-

aments. Finally, in Sec. IIIC we examine the possible

effects of changes in the bundle topology on the proper-

ties discussed here.

In the following we consider initially relaxed bundles,

i.e., bundles with initial arclengths L(t = 0) and end-to-

end lengths ?(t = 0) such that

f(t = 0) = F[?(t = 0),L(t = 0)] = 0.(28)

This allows us to focus on the active stresses generated

by the motors without having to worry about the pre-

stress of the bundle, which is not essential to the active

contractility studied here.

A.Motors with identical unloaded velocities

We first consider the case where the force-velocity re-

lationships of all junctions present in the bundle have an

identical unloaded velocity v?

junction i?is its sliding velocity in the absence of exter-

nally applied force. Defining v?

with all its components equal to v?

loaded velocity are equal reads

u. The unloaded velocity of

uas the length n?vector

u, the fact that all un-

V?(0) = v?

u. (29)

One special case where this is true is when all motors

in the bundle are identical and no passive cross-linkers

are present, i.e., V?

f. Conversely, junctions satisfying the condition Eq. (29)

but with V?

induce contractility, a point discussed in Sec. IVB.

The solution of Eqs. (26) with an initial condition sat-

isfying Eq. (28) is given by

i?(f) = V?

j?(f) for all i?, j?and for all

i?(f) ?= V?

j?(f) for f ?= 0 are not sufficient to

L0(t) = L(t = 0) + t(ρT− λT)Π?v?

and by choosing ?0(t) as the solution of F[?0(t),L0(t)] =

0. Substituting this solution into Eqs. (25) and (27), we

find that v??= 0 and ρvl= λvr= Π?v?

This means that in the linker center-of-mass reference

frame all linkers are immobile and all right-pointing (left-

pointing) filaments move relative to them with a constant

velocity v?

veyor belt for the actin.Therefore the maximum ve-

locity difference between two actin filament units is 2v?

whatever the topology of the bundle. This constitutes

an upper bound for the overall contraction velocity of

the bundle. Since this number does not scale linearly

with the length of the bundle, we conclude that tele-

scopic contraction cannot take place in bundles in which

motors have identical unloaded velocities. Instead, the

bundle described here sorts the filaments by polarity, as

observed in Ref. [20]. An alternative argument for this

behavior is given in Ref. [19].

It can easily be seen that the conveyor belt regime

described here is not a solution of the equations of mo-

tion when one of the junctions has an unloaded velocity

different from v?

sliding velocity ±v?

regime only when under force, which implies that the

neighboring filament units have to be under tension or

compression. Thus the function F(?,L) does not van-

ish, and using Eq. (27) this implies v???= 0, which is not

consistent with a conveyor belt behavior.

u

(30)

ufor all times.

u(−v?

u): linkers collectively act as a static con-

u

u. Indeed, this junction can adopt the

uconsistent with the conveyor belt

Page 8

8

B.Disordered bundles with linearly elastic

filaments

We now turn to a case where non-identical linkers are

present. Single bundles with non-identical linkers gener-

ically exhibit some amount of contractility or extensility.

We therefore ask whether these bundles have an aver-

age tendency to contract or extend. Since the bundles of

Ref. [13] are assembled through an aggregation process

that is not sensitive to polarity, we assume that for every

bundle topology that can form experimentally, a bundle

with identical topology but opposite polarities can form

with the same probability. The ensemble of all bundle

that can form experimentally weighted by their proba-

bility of occurence is thus polarity-reversal invariant.

Let us consider a bundle B with a given topology char-

acterized by ρ, λ, γ, Π and with given initial conditions

L(t = 0), ?(t = 0) satisfying Eq. (28). The subsequent

evolution of the bundle is described by the solution of

Eqs. (26), which we denote as

L(t) = L(t = 0) + ∆L(t)

?(t) = ?(t = 0) + ∆?(t).

(31a)

(31b)

In this section, we focus on cases where the force-

extension relations of the filament units can be expanded

as

F(?,L) ?∂F

∂?∆? +∂F

∂L∆L,(32)

where ∂F/∂? and ∂F/∂L are constants, i.e.

?∂F

This constitutes a more restrictive definition of the fila-

ment “linear elastic behavior” than that used in Ref. [27],

where the full nonlinear dependence in L is taken into ac-

count. The description Eqs. (32) and (33), applies to two

situations:

d

dt∂?

?

=d

dt

?∂F

∂L

?

= 0.(33)

• Infinitely stiff filaments. For very stiff filaments,

?p and fb are much larger than the filament unit

lengths and the forces exerted by the motors, re-

spectively. In that case, the filament unit force-

extension relationship reads [using Eq. (17)]

F(?i,Li) =90kBT?2

p

L4

i

(?i− Li),(34)

and thus ∂F/∂? and ∂F/∂L are not constants as

they depend on Li, which can change over time.

In the ?p→ +∞ limit, however, Eq. (34) imposes

? = L throughout the dynamics, and the filament

unit tensions can be seen as Lagrange multipliers

enforcing this condition. The force-extension rela-

tions can thus be replaced by the K → +∞ limit

of

F(?i,Li) = K(?i− Li).(35)

i=14

i=11

i=7i=6

i=13i=12

i=17

i=8

i=16

i=3

i=5i=4

i=2i=1

i=10i=9

i=20

i=18

i=15

i=19

Figure 4. (Color online) Example of a path (red dotted line),

as discussed in Sec. IIIB. In this example ?2 = ?4 = ?14 =

?20 = −?7 = 1 and all other ?is are equal to zero.

consequence, Eq. (36) reads dL/dt = d(?4+ ?2− ?7+ ?14+

?20)/dt.

As a

This yields an equivalent description of the bundle,

and in this description ∂F/∂? and ∂F/∂L are con-

stants. Thus the conclusions derived in the remain-

der of this section under the assumption Eq. (33)

apply to infinitely stiff filaments.

• Small strains. Under some conditions, the linear

elastic description of Eqs. (32) and (33) can apply

to filaments with moderate fb and ?p. These in-

clude the short-time dynamics of the bundle, where

∆L = O(t) and ∆? = O(t) have not had the time

to become large yet. Other cases include special ge-

ometries where filament unit lengths undergo small

changes over time (such a geometry is considered in

the example studied in Sec. IVA). If such changes

are small enough for a first-order Taylor expansion

of the force-velocity relationship to be a good de-

scription of the system, then Eqs. (32) and (33)

apply. Note that unlike the infinitely stiff filament

situation discussed above, these cases can involve

finite elastic constants ∂F/∂? and ∂F/∂L.

To determine whether bundle B is contractile, we de-

fine its overall length L as the sum of the lengths of the

filament unit and linker lengths lying along a path going

from the leftmost filament end to the rightmost one, as

pictured in Fig. 4. Since the length of the linkers is as-

sumed to be fixed here, the time derivative of L can be

expressed as the sum of the derivatives of the filament

units’ end-to-end lengths:

dL

dt=

n

?

i=1

?id∆?i

dt

, (36)

where ?iis equal to 1 if the path considered crosses fila-

ment unit i from left to right, to −1 if it crosses it from

right to left, and to 0 otherwise (see Fig. 4).

We now consider the polarity-reversed image˜B of bun-

dle B, with identical initial conditions. The geometry of

˜B is thus characterized by ˜ ρ = ρ,˜λ = λ, ˜ γ = γ,˜Π = −Π.

Substituting Eq. (32) into Eqs. (26) and (27), we find

that the solution of the equations for the dynamics of

˜B have a simple expression as a function of ∆?(t) and

Page 9

9

∆L(t), namely

˜L(t) = L(t = 0) + ∆˜L(t) = L(t = 0) − ∆L(t)(37a)

˜?(t) = ?(t = 0) + ∆˜?(t) = ?(t = 0) − ∆?(t). (37b)

The time derivative of˜B’s length can be assessed by cal-

culating

d˜L

dt=

n

?

i=1

?id∆˜?i

dt

,(38)

where the ?is are identical to the ones used in Eq. (36).

Using Eqs. (37), we thus find that

d˜L

dt= −dL

dt.

(39)

Averaging over the polarity-invariant ensemble of bun-

dles described above, we thus find

?dL

Therefore for every contractile bundle observed in an ex-

periment, a bundle extending with the same velocity is

also observed.

Viewing a long bundle as a serial arrangement of

smaller, independent bundles as in Ref. [13], we thus ex-

pect that the overall deformation velocity of such a bun-

dle is the sum of random, uncorrelated contractile and

extensile velocities with a zero average. Thus the typi-

cal bundle end-to-end velocity scales as

not constitute a telescopic deformation as defined above

and observed in Ref. [13]. This no-telescopic contractility

property holds for disordered bundles with small strains

[obeying the linear elastic regime Eq. (32)] and/or rigid

filaments. Mean-field modeling of dilute actomyosin gels

with force-independent motor velocities and rigid fila-

ments suggest that this symmetry-based reasoning could

have a three-dimensional counterpart [25].

Although the results of this section are dependent on a

linearization of the filament unit force-extension relation-

ship, we note that they are valid whatever the nonlinear

form of the motor force-velocity relationships V?

over, this symmetry argument can easily be extended to

the case of a bundle attached to fixed external walls, im-

posing vl

i= 0 at the ends of the wall-bound

filaments instead of the fi = 0 dangling end condition

used here. In that case, it is found that the force exerted

on the walls is zero on average, similar to the contraction

velocity of Eq. (40). This symmetry also holds in the

presence of friction forces between filaments or between

the filaments and an external medium as long as those

forces are linear in the filament velocities.

Finally, we note that the property described here

does not concern skeletal muscle. Indeed, even though

the actin filaments in sarcomeres are essentially straight

and may therefore be regarded as stiff in the sense of

Eq. (35), the overall structure of the sarcomere is clearly

dt

?

= 0.(40)

√L, which does

i?. More-

i= 0 or vr

not polarity-reversal invariant. Indeed, exchanging the

barbed and pointed ends of filaments in Fig. 1(a) re-

sults in an extensile, not contractile, structure. From

this point of view, the crucial factor favoring contraction

over extension in skeletal muscle is the fact that static

cross-linkers are strongly localized at the barbed ends of

their F-actin, while active, mobile motors are found at

the pointed ends. We do not expect such correlations to

develop in the frame of the assembly process of Ref. [13].

C.Changes in bundle topology

Although the results presented here do not rely on any

specific choice of bundle topology, our model assumes

that this topology does not change over time, and there-

fore does not describe linker attachment or detachment.

As mentioned in Sec. IIA, such events are likely not es-

sential for the bundle of Ref. [13]. Moreover, even if we do

assume that linker attachment/detachment events occur,

the topology-independent reasoning of Sec. IIIA applies

to the new bundles resulting from such events just as well

as to the original bundles. This is true of the property

discussed in Sec. IIIB to a lesser extent. Indeed, for this

property to hold it is required that the linker attach-

ment/detachment events preserve the polarity-reversal

invariance of the ensemble of bundles considered.

One important effect that could violate this symmetry

is known as the “Fenn effect”. This term describes the

fact that a myosin motor detaches preferentially when the

force applied to it points in the direction of the F-actin

barbed end (force assisting the myosin sliding) compared

to the case where the force is directed to the pointed

end (force opposing the myosin sliding) [31]. To visualize

the implications of this effect for the bundle as a whole,

we consider a simple contractile bundle such as the one

depicted in Fig. 5. As tension develops in the contract-

ing bundle, linkers at the F-actin barbed ends are put

under increasing assisting forces, and thus tend to de-

tach. Reciprocally, linkers at the pointed ends tend to

hold on to the F-actin. This generates a situation where

barbed ends are under weaker cross-linking than pointed

ends, a situation exactly opposite to the one that is re-

quired for muscle function (see discussion at the end of

Sec. IIIB). The Fenn effect thus tends to impede con-

tractility in contractile bundles, and similarly it lessens

extensility in extensile bundles. This effect is therefore

unlikely to account for the telescopic behavior observed in

Ref. [13] even though it does break the polarity-reversal

symmetry assumed in the last section.

IV.NONLINEAR ELASTICITY-INDUCED

CONTRACTILITY

Having shown that telescopic deformation of disor-

dered actomyosin bundles requires both non-identical

motors and nonlinear elastic behavior, we argue in this

Page 10

10

(a)

(b)

tension

strong

contraction

weak

expansion

weaker

contraction

stronger

expansion

tension

Figure 5.

fect impede active bundle deformation. (a) Simple bundle

with faster (lighter) motors on the left and slower (darker)

motors on the right, resulting in overall contractility. (b) If

the bundle is attached to some external object, contractility

results in tension. Tension assists the motion of the motors

on the right, which thus tend to detach due to the Fenn effect.

Conversely, it opposes the motors on the left, which tend to

hold on the the actin and might be joined by other motors

from the surrounding solution. As a result of their reduced

number, the motors on the right tend to yield to the applied

tension, which strengthens their extensile contribution. On

the other hand, the more numerous motors on the left are

better able to withstand the tension and thus contract with

a smaller velocity than they would have in the absence of the

Fenn effect. The opposite behavior is observed for an initially

extensile bundle; the Fenn effect thus hinders both types of

active bundle deformation.

Linker attachment/detachment and the Fenn ef-

section that these are sufficient conditions, and that they

generically lead to contractility rather than extensility, as

observed. Roughly, one can understand this tendency by

considering a bundle cross-section. The motor-induced

forces impose both compressive and extensive stresses on

the filaments within this cross-section. As F-actin nonlin-

ear elasticity implies that filaments are more compliant

under compression than extension [see Eq. (17)], com-

pressed filaments strongly deform while stretched ones do

not extend much. Overall, the bundle thus experiences

compression, which is equivalent to contractile behavior.

In contrast to the mechanism underlying skeletal mus-

cle contraction, the effect discussed here does not rely on

the relative motion of F-actin ends [as in, e.g., Fig. 1(b)],

but involves contraction throughout the length of the fil-

aments. In this section, we illustrate this effect by con-

sidering a simple bundle involving only two antiparal-

lel F-actin with periodic boundary conditions, whereby

no filament dangling ends are present. Because of the

ring topology resulting from these boundary conditions,

this bundle can be thought of as a model for the acto-

myosin contractile ring involved in cytokinesis [1] devoid

of the simple filament sorting effects which were shown

in Sec. IIIA to not contribute to overall contractility.

In Sec. IVA, we study a two-filament bundle with

arbitrary random motors in the case where the motor-

induced forces are much smaller than the filament buck-

ling force. We illustrate the point made in Sec. III

that contractility requires both heterogeneous motors

and nonlinear elastic behavior of the filaments by showing

that the amount by which this bundle contracts is pro-

portional to two quantities describing those two charac-

teristics. We then consider a specific motor distribution

and show that our predictions are in order-of-magnitude

agreement with experimental results in Sec. IVB. Finally,

we show in Sec. IVC that very long bundles are not well

described by the small-force regime, and discuss contrac-

tility in the resulting large-force regime.

A.Bundle with two F-actin and random motors

Our model, pictured in Fig. 6, represents a bundle re-

sulting from the assembly of actin with a heterogeneous

population of linkers. We thus take the junctions’ force-

velocity relationships to be arbitrary nonlinear functions

drawn from an as yet unspecified distribution.

functions V?

are independent and identically distributed. We assume

that the thick filaments are symmetric, i.e., junctions

i?and i?+ n?/2 have the same force-velocity relation-

ship. As a result, the bundle is polarity-reversal invari-

ant, and therefore does not contract when actin elasticity

is taken into account only to linear order, as discussed in

Sec. IIIB. We furthermore assume that all filament units

have the same initial length and arclength, and consider

periodic boundary conditions along the z axis. This bun-

dle geometry is described by

?1 l

where n = n?= 2n??and the n/2×n/2 matrix ˜ ρ is defined

as

The

i?(f) chosen at different sites i?∈ {1..n?/2}

Π =

0

0 −1 l

?

, γ =?1 l 1 l?, λ = 1 l, ρ =

?

˜ ρ 0

0 ˜ ρ

?

(41)

,

˜ ρ =

01

1

...

...

...

10

(42)

As a consequence, λTλ = ρTρ = 1 l. Equations (41) and

(42) imply several symmetries between the lengths and

forces pertaining to the top (indexed i = 1,...,n/2) and

bottom (indexed i = n/2 + 1,...,n) filaments units rep-

resented in Fig. 6. Specifically, combining Eqs. (41) and

(42) with Eq. (26b) yields

∀i ∈ {1..n/2}

d?i

dt

= V??

i− V??

i−1=d?i+n/2

dt

. (43)

Force balance moreover imposes

∀i ∈ {1..n/2}

fi= −fi+n/2.(44)

Using this relation and Eq. (15) in Eq. (26a) yields

∀i ∈ {1..n/2}

dLi

dt

= V?

i−1[fi−2− fi−1] − V?

= −dLi+n/2

dt

i[fi−1− fi]

. (45)

Page 11

11

i=1

i'=n/2+1

i=

i=n/2+2

i=2i=n/2i=1

i'=1i'=2i'=n/2-1i'=n/2

...

i=n

i'=n/2+2 i'=n-1 i'=n

i''=1 i''=2i''=n/2i''=n/2-1...

...

n/2+1

i=

n/2+1

Figure 6.

units all have identical initial length and arclength. Linkers

are symmetric (i.e., each have two junctions with identical

force-velocity relationships—represented by identical shades

of gray), but are not identical to one another. The bundle

has periodic boundary conditions.

Bundle geometry studied in Sec. IV. Filament

Finally, using the notation of Eqs. (31), Eqs. (43) and

(45) imply

∀i ∈ {1..n/2}

∆?i(t) = ∆?i+n/2(t)

∆Li(t) = −∆Li+n/2(t).

(46a)

(46b)

In other words, the end-to-end lengths of the top and

bottom filament units change by identical amounts as

imposed by the fact that they are connected to the same

linkers. Because of their opposite polarities, the junctions

connecting them to these linkers have opposite velocities

and their arclengths thus change by opposite amounts.

We first consider a weakly deformed bundle, where ∆?

and ∆L are small, and expand Eq. (14) to second order

in these deformations:

fi=∂F

∂?∆?i+∂F

+1

2

∂L∆Li

i+∂2F

(47a)

∂2F

∂?2∆?2

∂?∂L∆?i∆Li+1

2

∂2F

∂L2∆L2

i

−fi= fi+n/2

=∂F

∂?∆?i−∂F

+1

2

∂L∆Li

i−∂2F

(47b)

∂2F

∂?2∆?2

∂?∂L∆?i∆Li+1

2

∂2F

∂L2∆L2

i,

where the derivatives are taken in the initial state. Com-

bining these equations, we find that fi and ∆?i are of

order ∆Liand ∆L2

?∂F

∆?i= −1

2 ∂?∂L2

i, respectively, and that

?−1

?∂F

Note that the right-hand side of Eq. (48b) involves a

second derivative of F, meaning that ?i (and therefore

the overall bundle length) does not change when only

linear elasticity is considered.

Eq. (17), ∆?i is moreover found to always be negative,

meaning that each section of the bundle always contracts

but never expands. The conservation of the F-actin ar-

clength imposes?n/2

∆Li=

∂L

fi+ O(f3

?−1∂2F

i)

?∂F

(48a)

∂L

?−2

f2

i+ O(f3

i).(48b)

Using the example of

i=1Li = constant. Combining this

with Eq. (48a), we find that to second order in fi

n/2

?

i=1

fi= 0. (49)

In the following we use Eqs. (48) and (49) to calculate

the bundle deformation ∆L =?n/2

focus on the steady state length of the bundle. Imposing

dL/dt = 0 in Eq. (26a), we find

i=1∆?i. We are thus

i∝ −∆L. We moreoverinterested in the quantity?n/2

i=1f2

∀i ∈ {1..n/2}

where the convention f0= fn/2, f−1= fn/2−1represent-

ing the periodic boundary conditions is implied. This

implies that at steady state, all junctions belonging to

the top F-actin have the same velocity v. By symmetry,

the junctions of the bottom F-actin have an equal and

opposite velocity and the thick filaments are immobile.

This implies

V?

i(fi−fi−1) = V?

i−1(fi−1−fi−2), (50)

∀i ∈ {1..n/2}

i)−1is the inverse of function V?

stands for the motor force exerted at junction i?when

sliding at a velocity v. Summing Eq. (51) over i with

f0= fn/2as above, we find

fi− fi−1= (V?

i)−1(v),(51)

where (V?

i, and (V?

i?)−1(v)

n/2

?

i=1

(V?

i)−1(v) = 0, (52)

which must be solved to determine v. The solution to

the recursion Eq. (51) under this constraint reads

fi= −2

n

n/2−1

?

j=0

j(V?

i+j)−1(v),(53)

from which we can calculate the quantity of interest

n/2

?

i=1

f2

i=(n − 2)(n − 1)

6n

n/2−1

?

i=0

?(V?

i)−1(v)?2

+

n/2−1

?

j=1

(n − 2j − 2)(n − 2j)(n + j − 1)

3n2

×

n/2−1

?

i=0

(V?

i)−1(v)(V?

i+j)−1(v). (54)

We average this expression over the distribution of mo-

tor force-velocity relationships. Since the force-velocity

relationships are independent and identically distributed

random functions, we have

?(V?

By squaring and averaging the constraint Eq. (52), we

moreover find

i)−1(v)(V?

j?=i)−1(v)? = ?(V?

i)−1(v)(V?

i+1)−1(v)?. (55)

?(V?

i)−1(v)(V?

j?=i)−1(v)? = −

2

n − 2??(V?

i)−1(v)?2?. (56)

Page 12

12

Combining these two relations with Eqs. (48b) and (54),

we find the change in bundle length in the steady-state

to lowest order in motor-induced force:

∆L =

n/2

?

i=1

∆?i(t = +∞)(57)

= −n(n + 2)

96

?∂F

∂?

?−1∂2F

∂L2

?∂F

∂L

?−2

??(V?

i)−1(v)?2?,

where v is determined by solving Eq. (52). This simple re-

sult is an interesting illustration of the main point of this

article, as here contraction, represented by ∆L, is the

product of the quantity ∂2F/∂L2, which characterizes

the actin nonlinear elastic behavior, and ??(V?

tics. If either of those vanish, no contraction is observed.

Whereas contraction in striated muscle [Fig. 1(a)] relies

purely on filament sliding, in the polarity-reversal sym-

metric bundle considered here sliding alone does not in-

duce contraction. This can be seen in the special case

where all motors are identical, which implies that fila-

ments slide at a non-vanishing velocity ±v = ±V?(0) but

∆L remains zero.

This example also enables us to consider the case where

an external force fextis applied to the bundle. For small

fext, ∆L generically couples to fextlinearly and thus

?∂F

+O

i)−1(v)?2?,

which characterizes the dispersion in motor characteris-

∆L =

∂?

?−1nfext

?

4

+ ∆L(fext= 0)

?(V?)−1?2?

(58)

fext

+ O(f2

ext) + O

??(V?)−1?3?

,

where the proportionality constant in front of fext is

easily determined from the zero motor activity case

(V?)−1= 0, ∆L(fext= 0) is given by Eq. (57) and terms

of order fext× (V?)−1are forbidden by polarity-reversal

symmetry. Therefore a bundle tethered so that its length

is imposed exerts a contractile force

fcont=n + 2

24

∂2F

∂L2

?∂F

∂L

?−2

??(V?

i)−1(v)?2?

(59)

on its surroundings. This remains true if the imposed

bundle length results in a stretching of the filaments,

showing that filament compression is not required for

contractility.

B. A simple motor distribution

To illustrate the results derived above, we consider the

simple case where each motor is of type a with probability

p and of type b with probability 1 − p, where motors of

type a and b are characterized by the linear force-velocity

relationships

V?

V?

a(f) = V?0

b(f) = V?0

a− af

b − bf,

(60a)

(60b)

respectively. The sliding velocity v that is selected by the

bundle depends on the number of motors of each type in

the bundle. Denoting by naand n − nathe number of a

and b motors respectively, we find that Eq. (52) imposes

v =naV?0

b+ na(bV?0

na + na(b − a)

a− aV?0

b)

. (61)

As a consequence, the forces exerted by motors of type a

and b respectively read

Ga(na) =ab(V?

a)−1(v)

V?0

b)−1(v)

V?0

a− V?0

b

=

n − na

n − na+ nab/a

= −

(62a)

Gb(na) =ab(V?

a− V?0

b

na

n − na+ nab/a, (62b)

where the dimensionless motor forces Gaand Gbare de-

fined. Since we consider symmetric linkers with two mo-

tors each, the probability for having namotors of type a

among n reads:

?n/2

To calculate the variance of the force associated with

motor i as required by Eq. (57), we consider two disjoint

cases: either linker i is of type a (probability p), or of

type b (probability 1−p). In each of these cases, each of

the remaining linkers can be of type a or b, and summing

over all of these different subcases yields:

Pn(na) =

na/2

?

pna/2(1 − p)(n−na)/2.(63)

?G2

i? = p

n/2

?

na/2=1

Pn/2−1(na/2 − 1)G2

a(na)

+(1 − p)

n/2−1

?

n/2

na/2 − 1

×pna/2(1 − p)(n−na)/2.

na/2=0

Pn/2−1(na/2)G2

b(na).

=

n/2

?

na/2=1

??(n + 2 − na)(n − na)

(n − na+ nab/a)

(64)

We use Stirling’s approximation and convert this sum to

an integral. In the limit of large n, it is dominated by a

saddle point, which yields the following result to leading

order in n:

?G2

i? =

p(1 − p)

[1 + (b/a − 1)p]2.(65)

Let us denote by ?0 = ?i(t = 0) the initial length of

the filament units and by L0= n?0/2 the initial length

of the bundle. Here we consider only long bundles, i.e.,

bundles with ?0? L0. Moreover, the approximation of

weak transverse fluctuations leading up to Eq. (17) and

the fact that we work with small forces imply that our

approach is valid only for filament units much shorter

than their persistence length: ?0? ?p. Putting the result

of Eq. (65) back in a dimensional form and calculating the

Page 13

13

derivatives of F(?,L) from Eq. (17) yields the following

result to leading order in ?0/L0and ?0/?p:

∆L

L0

where the large numerical factor in the denominator re-

sults from the combinatorics associated with the reason-

ing leading up to Eq. (57) as well as the successive differ-

entiations of function F(?,L). Consistent with Sec. IIIA,

we see that motors with identical unloaded velocities

(V?0

velocity relationships are otherwise different (for instance

a ?= b). Note that the case of motors and passive cross-

linkers can easily be derived from this result by taking

V?0

We evaluate the magnitude of the contractility ex-

pected from our calculation by using the numerical values

?p? 10µm, ?0? 360nm (the average thick filament size

in Ref. [13]), L0 ? 30µm and a typical motor-induced

force

?

[pb + (1 − p)a]2

Strictly speaking, these realistic values are too large for

the expansion Eqs. (47) to hold. We may however use

them in Eq. (66) to evaluate the order of magnitude of

the expected deformation. This yields ∆L/L0≈ −28%,

in agreement with the observations of Ref. [13].

An interesting aspect of Eq. (66) is the predicted de-

pendence of the contractility on ?0, which results from the

competition between two effects occurring as the motor

density is decreased (?0is increased). First, less force is

generated within the bundle; second, the filament units

become longer, and deforming the bundle thus becomes

easier. The latter effect turns out to be dominant, as

evidenced by the increase of ∆L/L0with increasing ?0,

and thus with decreasing myosin density. This effect has

not yet been observed experimentally, but could be tested

through a systematic investigation of the effect of myosin

concentration on the contractility of in vitro bundles.

= −

?4

0L0

11340(kBT)2?3

p

p(1 − p)(V?0

[pb + (1 − p)a]2

a− V?0

b)2

,(66)

a = V?0

b) do not yield contraction even if their force-

b= 0, b = 0.

p(1 − p)(V?0

a− V?0

b)2

≈ 10pN.(67)

C.Long bundles and large forces

Another surprising feature of Eq. (66) is the fact that

the relative contraction ∆L/L0 increases with increas-

ing L0, whereas for simple telescopic contraction ∆L

is expected to be proportional to L0. In other words,

the expansion Eqs. (47) yields a contraction that scales

faster than telescopic with L0. This can be understood

from the recursion Eq. (51), which shows that the force

in i is equal to the force in i − 1 plus some random

“kick” (V?

erage kick is zero, fi follows a random walk as a func-

tion of i, whereby the typical filament force is of the

order of√n × the typical kick ∝

i)−1(v). Since Eq. (52) ensures that the av-

?

?[(V?

i)−1(v)]2?L0/L0.

As for low forces the bundle deformation is proportional

to f2

i(the coupling to fi being forbidden by polarity-

reversal symmetry), this rationalizes the L0-dependence

in Eq. (66).

To understand why telescopic behavior is observed in

Ref. [13], we note that this discussion implies that for ar-

bitrarily long bundles, the forces on individual filament

units become arbitrarily large. This in turn entails a de-

parture from the low-force regime described by Eqs. (47).

Typically, the random walk regime described here is valid

over regions of the bundle of size nrwsuch that the force

in these regions is of order fb:

?

nrw?[(V?

i)−1(v)]2? ≈ fb⇒ nrw≈

f2

b

?[(V?

i)−1(v)]2?.

(68)

Two consecutive such regions are separated by a site

where one of the two filaments is buckled. The tension

of this buckled filament unit vanishes, and force balance

implies that the tension of the filament unit facing it is

also zero. In this regime of strong deformation, a fraction

1/nrw of the filament units are collapsed, and thus the

overall bundle deformation scales like

≈??(V?

b

∆L

L0

i)−1(v)?2?

f2

≈

?4

0

π4(kBT?p)2??(V?

i)−1(v)?2?.

(69)

Thus both the ?0dependence presented in Eq. (66) and

the linear dependence in variance of the motor force are

robust features of the mechanism discussed here. The

surprising proportionality of ∆L/L0to L0, on the other

hand, is not present in the limit of very long bundles.

V.DISCUSSION

In this article we investigate the mechanisms underly-

ing the telescopic contraction of disordered actomyosin

bundles. More specifically, we focus on bundles of long

filaments assembled in a polarity-independent way, which

implies that linkers of different types do not bind pref-

erentially to either end of the F-actin, and therefore for-

bids sarcomere-like organization. We recognize that the

tendency of myosin motors to slide along F-actin in the

presence of ATP does not in itself favor contraction or ex-

tension. Although the influence of actin nonlinear elastic-

ity in bundle contraction has been mentioned in previous

works [27], we conduct the first systematic study of the

possible role of other less obvious effects, including com-

plicated bundle topologies and their interplay with the

linkers’ non-linear force-velocity relationships. We show

that these effects cannot generate contraction by them-

selves, and that there are two necessary steps for con-

tractility in such bundles: 1) linkers with non-identical

unloaded velocities generate stresses inside the bundle;

2) The actin nonlinear elastic behavior allows the com-

pressive stresses to collapse the bundle, while resisting

stretching by extensile stresses.We further illustrate

Page 14

14

this mechanism by studying a simple example of a bun-

dle where motors are randomly distributed, and explic-

itly show that actin worm-like-chain nonlinear elasticity

generically favors contraction over extension. This sim-

ple bundle is moreover expected to be a useful represen-

tation of the local structure of more complicated bundles

on scales smaller than the length of an F-actin.

Our assumption that the F-actin length is much larger

than that of myosin thick filaments is motivated by the

experimental situation encountered in Ref. [13]. In other

contexts, effects pertaining to the F-actin ends might

have a significant influence on the bundle behavior. One

such proposed effect is the dwelling of a thick filament on

an F-actin barbed end following its sliding. Assessing the

importance of this effect would require a measurement

of the hypothetical dwell time τ, which is not currently

available. We can however evaluate the minimum dwell

time required to yield a contraction velocity compara-

ble to the ones observed experimentally. Denoting by

v ≈ 200nm·s−1the typical myosin velocity, the barbed

end of a filament is expected to be occupied by a dwelling

motor a fraction θ = vτ/?0of the time, or all the time

(θ = 1) if vτ > ?0. This yields an overall contraction

velocity per unit length L−1

LF ? 5µm is the F-actin length in Ref. [13]. The con-

traction observed in Ref. [13] has L−1

and therefore dwelling motors can only play a significant

role in it if θv/LF ? 0.1s. This requires θ ≈ 1, and

thus τ ≈ 1s or larger, which should be observable ex-

perimentally. Another F-actin end effect that could play

a role in in vivo contraction is actin polymerization and

depolymerization. For actively treadmilling F-actin, we

expect that immobile, passive cross-linkers tend to ac-

cumulate near the actin pointed (depolymerizing) ends,

while myosins migrate to the barbed ends. Interestingly,

this generates the opposite of a sarcomere-like organiza-

tion [Fig. 1(a)], and therefore actin treadmilling is not

expected to contribute to contractility through the sim-

ple mechanism present in striated muscle.

As discussed in Sec. IIIB, the polarity-reversal sym-

metry considerations presented here strongly constrain

the motion of networks of stiff filaments, which are likely

to be a good model of the microtubule cytoskeleton and

might be relevant for some dense actomyosin structures.

These constraints have not been taken into account in

previous symmetry-based active hydrodynamics models

of the cytoskeleton [14–16], and could help reduce the

large number of phenomenological parameters these for-

malisms usually involve.Relatedly, microscopic mod-

eling of dilute three-dimensional actomyosin gels shows

that no contractility occurs when rigid actin filaments

are coupled by motors with position-independent sliding

velocities [25]. Enforcing our more general microscopic

symmetry considerations at the continuum level is how-

ever not straightforward. An interesting first step would

be to recognize a continuum version of the observation

0d∆L/dt ≈ θv/LF, where

0d∆L/dt ≈ 0.1s−1,

that myosin thick filaments do not in themselves promote

contraction or extension, but rather just translocate fila-

ments. This could be done by modeling thick filaments in

a three-dimensional actin gel not as force dipoles [17], but

rather as generating local torques. By analogy with the

work presented here, we expect that this model would

not display contractility within a linear elastic descrip-

tion, but that taking into account the nonlinear behavior

of the continuum medium could have interesting results.

Evidence of the applicability of the mechanism dis-

cussed here is presented in Ref. [19], where actin buck-

ling in in vitro reconstituted bundles was directly ob-

served under conditions similar to those reported in

Ref. [13]. Although these bundles are comprised of actin

and myosin only, it is likely that their motors have a dis-

tribution of force-velocity relationships because of vari-

ability in thick filament lengths and orientation relative

to the actin, the stochastic nature of the myosin slid-

ing process, and the possible presence of a small frac-

tion of non-ATP-cycling myosin heads. Further experi-

ments could include a systematic study of the influence

of myosin concentration on contractility as suggested

in Sec. IVB, whereby our prediction that contractility

should decrease with increasing myosin density in a cer-

tain range of myosin concentrations could be checked. It

might also be possible to confirm the role of the filament

units’ elasticity by verifying that contractility decreases

for filaments with a larger persistence length.

The question of whether our mechanism describes con-

tractile bundles in vivo also remains open. One often

cited argument against the prevalence of actin buckling

by the cell regards its energetic cost, which would make

it wasteful for the cell to rely on this mode of contraction.

We however argue that this process is relatively cheap; in-

deed, the work required to buckle a 360nm-long section of

actin filament is of order fbL ≈ kBTπ2?p/L ≈ 300kBT,

which corresponds to the free energy liberated by the

hydrolysis of only a dozen ATP molecules, whereas we

estimate that each thick filament hydrolyses ≈ 100 ATP

molecules per second [29]. It is also interesting to note

that although the in vitro bundles discussed here stall

after contracting only by a few tens of percents, many in

vivo structures such as the contractile ring can shrink to

a small fraction of their initial length. If such structures

indeed rely on filament buckling for contraction, an im-

portant open question is whether the buckled filaments

are disposed of and the bundle reorganized to allow fur-

ther contraction.

ACKNOWLEDGMENTS

We thank Margaret Gardel, Yitzhak Rabin, Todd

Thoresen and Tom Witten for inspiring discussions.

Page 15

15

[1] B. Alberts, D. Bray, A. Johnson, J. Lewis, M. Raff,

K. Roberts, and P. Walter, Essential Cell Biology (Gar-

land, New-York, 1998).

[2] J. P. Heath, J. Cell Sci. 60, 331 (March 1983).

[3] J. W. Sanger, J. M. Sanger, and B. M. Jockusch, J. Cell

Biol. 96, 961 (April 1983).

[4] L. J. Peterson, Z. Rajfur, A. S. Maddox, C. D. Freel,

Y. Chen, M. Edlund, C. Otey, and K. Burridge, Mol.

Biol. Cell 15, 3497 (July 2004).

[5] F. S. Fay, K. Fujiwara, D. D. Rees, and K. E. Fogarty, J.

Cell Biol. 96, 783 (March 1983).

[6] L. P. Cramer, M. Siebert, and T. J. Mitchison, J. Cell

Biol. 136, 1287 (March 1997).

[7] O. Medalia, I. Weber, A. S. Frangakis, D. Nicastro,

G. Gerisch, and W. Baumeister, Science 298, 1209

(November 2002).

[8] A. B. Verkhovsky, T. M. Svitkina, and G. G. Borisy, J.

Cell Biol. 131, 989 (November 1995).

[9] L. W. Janson, J. Kolega, and D. L. Taylor, J. Cell Biol.

114, 1005 (September 1991).

[10] D. Mizuno, C. Tardin, C. F. Schmidt, and F. C. Mack-

intosh, Science 315, 370 (January 2007).

[11] P. M. Bendix, G. H. Koenderink, D. Cuvelier, Z. Dogic,

B. N. Koeleman, W. M. Brieher, C. M. Field, L. Ma-

hadevan, and D. A. Weitz, Biophys. J. 94, 3126 (April

2008).

[12] G. H. Koenderink, Z. Dogic, F. Nakamura, P. M. Bendix,

F. C. MacKintosh, J. H. Hartwig, T. P. Stossel, and D. A.

Weitz, Proc. Natl. Acad. Sci. U.S.A. 106, 15192 (Septem-

ber 2009).

[13] T. Thoresen, M. Lenz, and M. L. Gardel, “Contractile

behavior of reconstituted actomyosin bundles,” (2010),

to be published.

[14] K. Kruse, A. Zumdieck, and F. J¨ ulicher, Europhys. Lett.

64, 716 (December 2003).

[15] K. Kruse, J. F. Joanny, F. J¨ ulicher, J. Prost, and K. Seki-

moto, Phys. Rev. Lett. 92, 078101 (January 2004).

[16] K. Kruse, J. F. Joanny, F. J¨ ulicher, J. Prost, and K. Seki-

moto, Eur. Phys. J. E 16, 5 (January 2005).

[17] F. C. MacKintosh and A. J. Levine, Phys. Rev. Lett.

100, 018104 (January 2008).

[18] A. Zemel and A. Mogilner, Phys. Chem. Chem. Phys. 11,

4821 (June 2009).

[19] M. Lenz, T. Thoresen, M. L. Gardel, and A. R. Dinner,

“F-actin buckling underlies the contraction of disordered

actomyosin bundles,” (2011), to be published.

[20] Y. Tanaka-Takiguchi, T. Kakei, A. Tanimura, A. Takagi,

M. Honda, H. Hotani, and K. Takiguchi, J. Mol. Biol.

341, 467 (August 2004).

[21] K. Kruse and F. J¨ ulicher, Phys. Rev. Lett. 85, 1778 (Au-

gust 2000).

[22] K. Kruse and F. J¨ ulicher, Phys. Rev. E 67, 051913 (May

2003).

[23] T. B. Liverpool and M. C. Marchetti, Phys. Rev. Lett.

90, 138102 (April 2003).

[24] F. Ziebert and W. Zimmermann, Eur. Phys. J. E 18, 41

(September 2005).

[25] T. B. Liverpool and M. C. Marchetti, Europhys. Lett.

69, 846 (March 2005).

[26] K. Kruse and K. Sekimoto, Phys. Rev. E 66, 031904

(September 2002).

[27] T. B. Liverpool, M. C. Marchetti, J.-F. Joanny, and

J. Prost, Europhys. Lett. 85, 18007 (January 2009).

[28] R. Craig and J. Megerman, J. Cell Biol. 75, 990 (Decem-

ber 1977).

[29] S. S. Rosenfeld, J. Xing, L.-Q. Chen, and H. L. Sweeney,

J. Biol. Chem. 278, 27449 (July 2003).

[30] K. Svoboda and S. M. Block, Cell 77, 773 (June 1994).

[31] C. Veigel, J. E. Molloy, S. Schmitz, and J. Kendrick-

Jones, Nat. Cell Biol. 5, 980 (November 2003).