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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

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(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

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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

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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)

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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