Coordination and collective properties of molecular motors: theory

Available online at www.sciencedirect.com

Coordination and collective properties of molecular motors:theoryThomas Guerin1, Jacques Prost1,2, Pascal Martin1 and Jean-FrancoisJoanny1

Many cellular processes require molecular motors to

produce motion and forces. Single molecule experiments

have led to a precise description of how a motor works.

Under most physiological conditions, however, molecular

motors operate in groups. Interactions between motors

yield collective behaviors that cannot be explained only

from single molecule properties. The aim of this paper

is to review the various theoretical descriptions that

explain the emergence of collective effects in molecular

motor assemblies. These include bidirectional motion,

hysteretic behavior, spontaneous oscillations, and

self-organization into dynamical structures. We discuss

motors acting on the cytoskeleton both in a prescribed

geometry such as in muscles or flagella and in the

cytoplasm.

Addresses1 Laboratoire Physico-Chimie Curie, CNRS, Institut Curie, UPMC, 26 rue

d’Ulm, F-75248 Paris Cedex 05, France2 E.S.P.C.I, 10 rue Vauquelin, 75231 Paris Cedex 05, France

Corresponding author: Joanny, Jean-Francois (jean-

[email protected])

Current Opinion in Cell Biology 2010, 22:14–20

This review comes from a themed issue on

Cell structure and dynamics

Edited by Arshad Desai and Marileen Dogterom

Available online 13th January 2010

0955-0674/$ – see front matter

# 2009 Elsevier Ltd. All rights reserved.

DOI 10.1016/j.ceb.2009.12.012

IntroductionMolecular motors are proteins that convert chemical

energy into mechanical work, in general by hydrolyzing

ATP, to move along filaments from the cytoskeleton and

produce forces [1]. Under physiological conditions, these

enzymes are fundamentally out of equilibrium. Motors

have many functions in cells. For example, kinesin and

myosin V transport biological materials along microtu-

bules and actin filaments, respectively. Myosin II is

responsible for muscle contraction and is involved in cell

locomotion. Dynein powers beating of flagella and cilia.

The typical velocity is of the order of 1 mm/s and the

typical opposing force that stalls the motor is of the order

of a few piconewtons.


Molecular motors have been first discussed theoretically

in the context of muscles [2]. In his pioneering work, AF

Huxley introduced the image of a motor with several

conformational states depending on interaction with the

filament. These conformational states have been later

related to the ATP-ase cycle of the motor [3]. A more

recent class of theoretical descriptions (reviewed in [4,5])

highlights general features that do not depend much on

the precise molecular nature of the motors and the tracks.

These models describe how directed motion occurs,

estimate bounds to their efficiency, provide typical

force–velocity relations and investigate the role of fluctu-

ations.

In general, motors do not work in isolation but in groups.

In muscles, the number of myosins can reach a few 1019.

Beating of cilia or flagella involves roughly 104 dyneins.

Even in intracellular transport, about 10 motors coordi-

nate their motion in order to transport vesicles or pull

membrane nanotubes. One of the first approaches to

describe collective behavior in motor assemblies intro-

duced a difference between ‘rowers’ and ‘porters’ [6].

Rowers, such as myosins II, spend most of their time

unbound from their cytoskeletal filament. They cannot

work alone but evoke motion of relatively large velocity

when operating within a large assembly. Conversely,

porters, such as kinesin, have more difficulty to work

collectively than individually, because the presence of

the other motors can impede the motion of a single

molecule within the assembly. Theoretical models have

shown that the interaction between motors can result in

complex phenomena such as bidirectional motion, oscil-

lations, hysteresis and the formation of dynamical struc-

tures. Intracellular transport, cell motility, muscle

oscillation and spindle formation during mitosis are a

few examples that may be described by such theories.

Collective effects are often reminiscent of the apparition

of phase transitions in thermodynamic systems and bifur-

cations in nonlinear physics. They are new features

emerging from the cooperation of a large number of

components.

The aim of this short review is to present the various

theoretical descriptions which have been proposed to

describe the behavior of groups of molecular motors.

We discuss in particular the possible mechanisms of

coordination between molecular motors, the conditions

of existence of spontaneous oscillations and the activity of

the cytoskeleton induced by molecular motors. A review

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http://dx.doi.org/10.1016/j.ceb.2009.12.012

Coordination and collective properties of molecular motors Guerin et al. 15

of the experimental biological and biophysical studies of

coordinated molecular motors is given by Holtzbauer and

Goldman in the same issue [7].

Rigidly linked molecular motorsWe first consider the case where the motors are linked to a

common rigid backbone. This geometry is close to that of

muscles or of in vitro motility assays. Cooperativity arises

because any global motion of the filament modifies the

position of the motors, thereby changing both the force

they produce and their probability to bind to or unbind

from the filament.

In the first class of models, the center of mass of each

motor is considered as a particle rigidly linked to the

backbone [8]. Each motor interacts with the filament (e.g.

actin or microtubule) according to a state-dependent

potential. These potentials are not known in detail, but

they must be periodic and asymmetric, to reflect the

periodicity and the polarity of the filament, respectively.

The simplest case is that of a two-state model, in which

the motors have an unbound state with a flat potential (no

interaction with the filament) and a bound state with a

Figure 1

Mechanisms of motor coordination. (a) Stiff motors linked to a rigid backbone

on position x along the filament. The backbone moves at velocity v relative t

(crossbridge model). We show four motors taken at different instants of the

bound to a binding site and has just produced its power stroke. The third m

filament and backbone. (c) Processive motors elastically linked to a rigid back

finite velocity vm which depends on the load k � y. (d) Processive motors pulli

move at a smaller velocity vðFÞ than the velocity v0 of the motors along the

molecular motors pull on a vesicle in two opposite directions. (f) Hydrodyna

velocity field (streamlines are sketched in red), that helps ‘motor 1’ moving


saw-tooth potential (Figure 1a). More complex cases can

be mapped on this simple case [8]. The motors switch

between the two states according to binding and unbind-

ing rates that depend on their positions along the fila-

ment, and on ATP hydrolysis. ATP consumption is

fundamental in maintaining the motors away from ther-

mal equilibrium. At thermal equilibrium, in the case of a

uniform binding rate, the motors would unbind more at

the top of the potential than at the bottom. A localized

unbinding rate near the minimum of the potential con-

stitutes a suitable choice that reflects energy consump-

tion. Equivalently, in agreement with the existence of

binding sites, binding can be localized near the potential

maximum and unbinding can be uniform. For a suffi-

ciently long filament with motors distributed either ran-

domly (such as in motility assays) or with a period

incommensurate with that of the filament (as in muscles),

the system is invariant by translation. If the filament

moves at constant velocity, the force produced by the

motor assembly reaches a steady-state value. One can

then define a ‘force–velocity relation’. Near stall con-

dition, the force produced by the motors varies linearly

with velocity and, remarkably, amounts to negative

. Interaction potentials W1 and W2 and transition rates v1 and v2 depend

o the filament. (b) Elastic motors of stiffness k linked to a rigid backbone

ir cycle: the first and the fourth motors are unbound, the second one is

otor has released its strain y because of the relative sliding between

bone. Contrary to the crossbridge model, each motor moves by itself at a

ng a membrane tube. Owing to the load F, the motors at the leading edge

tube, which leads to cluster formation. (e) Tug-of-war: two groups of

mic coupling: ‘motor 2’, moves to the right at velocity v2 and creates a

to the right.


16 Cell structure and dynamics

Figure 2

Load-dependent unbinding generates nonmonotonic force–velocity

relations (model shown in Figure 1c). (a) If the unbinding rate depends on

strain, the number of bound motors Nbound depends on velocity v:

movement in the negative direction (v< 0) stretches the motors and

promotes unbinding. (b) Velocity v as a function of load f for a single

motor. (c) External force Fext = Nbound � f as a function of velocity v for a

motor assembly (red; product of curves shown in (a) and (b)). Under a

constant load F*, bidirectional motion results from stochastic switching

between velocities vþ and v�. Under elastic loading, stall condition

(v ¼ 0) is unstable and the filament oscillates (trajectory shown in blue).

friction. If the activity of the motors is large enough,

negative friction can overcome passive friction, and the

force–velocity relation is nonmonotonic [8–10] (Figure 2).

Note that the force–velocity relation of a single (proces-

sive) motor is always monotonic. At constant external

load, the velocities in the region of the force–velocity

relation with negative slope are unstable. When operating

in this region, two velocities are allowed, which is the


signature of a discontinuous dynamical phase transition

and implies hysteretic behavior. This dynamical instabil-

ity holds true even for a symmetric potential. In this case,

although a single motor cannot move in a given direction

and diffuses along the filament, a collection of motors

displays directed movements as a result of spontaneous

symmetry breaking. If some motors are pulling in one

direction, they encourage their teammates to join in and

push in the same direction [9]. As a consequence, the

system shows a directed motion in one of the two possible

directions [8]. With a finite number of motors, fluctuations

impose a finite probability for the motors to switch

between the two stable velocities, resulting in a bidirec-

tional motion [11]. In the presence of an electric field,

motility gliding assays with the acto-myosin system

confirm this prediction: close to stall condition, the actin

filaments move in either direction, or even reverse their

motion [12]. With a Ncd mutant, equal probabilities of

jumping forward and backward have been measured in

single molecule experiments, whereas motility assays

reveal bidirectional motion of the filaments [13]. These

observations provide support for the existence of spon-

taneous symmetry breaking in motor assemblies.

Motors can also be described as springs of finite stiffness

([2], reviewed in [1]) (Figure 1b). In this class of models,

often called ‘crossbridge models’, binding to a filament is

accompanied by a rapid conformational change (power

stroke) that sets the spring under tension. The extension

of the spring then varies upon sliding until the motor

unbinds. By specifying binding and unbinding rates, one

can determine a force–velocity relation, which can emu-

late hyperbolic force–velocity relations that have been

measured in contracting muscles [2]. The force–velocity

relation can also be nonmonotonic, which leads to dyna-

mical instabilities similar to those discussed in the

previous section for stiff motors [14]. Within a two-state

model, the force–velocity relation at low velocities can be

computed analytically for arbitrary transition rates [15]. It

was found that a necessary condition for unstable beha-

vior is that the unbinding rate increases upon stretching

the spring. In this case, the unbinding of one motor

increases the external load experienced by the other

motors, and thus their unbinding rates, giving rise to

cooperative unbinding. Conversely, binding of some

motors favors binding of the others. A similar effect has

been invoked to describe collective effects with proces-

sive motors [16] (Figure 1c): although the force–velocity

relation of a single motor is monotonic, that of a collection

can be nonmonotonic (Figure 2). An explicit mechanism

for collective behavior emerges by assuming that the

energy of ATP hydrolysis is not large enough to induce

a power stroke near stall condition [17]. In this case, the

power stroke of one motor is favored by the motion

induced by the others, resulting in instabilities [18]. This

model can explain the tension hysteresis observed at

imposed length in muscle fibers [19�]. Stiff motors and



crossbridge models can be considered as two limits of a

more general description that considers both the periodic

interaction potential between motor and filament and the

finite stiffness of the motors (T Guerin et al., unpub-

lished).

Steric coupling between unlinked molecularmotorsProcessive motors such as kinesin, dynein or myosin V

rarely unbind from their filamentous track. They work

best when acting in small groups or clusters that allow for

the production of the forces needed for example to extract

tubes from the Golgi apparatus. The formation of motor

clusters is not always due to physical linkages between

the motor tails but can also result from steric or short range

interactions. When bound to a liquid membrane, motors

exert forces on their cargo only at the leading edge.

Because they are the only ones to experience a resistance

from the membrane, the motors at the leading edge move

with a slower velocity than the motors that are free to

move along the membrane. This velocity difference leads

to the formation of a cluster (Figure 1d). This mechanism

has been invoked for vesicle transport and tube extraction

from cellular membrane by kinesin [20,21], and for

length-dependent kinetics of microtubule depolymeriza-

tion by protein clusters [22,23]. In all these cases, the size

of the motor cluster results from a balance condition at the

leading edge between the incoming flux of motors and

motor unbinding. In order to determine the properties of

the cluster, one must know the force–velocity relation of

the motors, how the motors share the load, and the force-

dependence of the unbinding rate. A commonly used

approximation, the limits of which have been defined

recently [24,25], is that the motors equally share the force.

Within this approximation, the average cluster size is of

the order of a few tens of motors [21]. Force sharing

reduces the load per motor, and thus increases the

velocity of the cluster. As in crossbridge models, a strong

variation of the motors unbinding rate with the exerted

force leads to instabilities, including oscillations [26]. The

motion of unlinked kinesin motors on a microtubule also

raises interesting questions related to traffic such as the

formation of traffic jams or the propagation of density

waves along the microtubule [27].

Antagonistic motorsVesicles transported inside a cell often interact with two

types of molecular motors that exert forces in opposite

directions (e.g. dynein and kinesin) (Figure 1e), thereby

resulting in a tug-of-war situation [28�,29]. Here also,

load-dependent unbinding rate can result in instability.

If the number of bound motors pulling in one direction

becomes larger, the system moves in this direction. Each

motor pulling in the other direction in turn sustains a

larger force and unbinds more, providing a positive feed-

back that favors the global motion. Doublets of antipar-

allel microtubules that glide on a substrate coated with


kinesin 1 offer a recent experimental realization of tug-of-

war (C Leduc et al., unpublished). An abrupt transition

between slow and fast motion of the doublet is observed

in response to a small change in the length difference

between the two microtubules, which is a signature of

unstable behavior. Bidirectional motion has also been

observed with the acto-myosin system [30].

Hydrodynamic coupling between molecularmotorsA moving motor drags the cytosolic fluid, creating a flow

that may influence the motion of other motors (Figure 1f).

It has been shown theoretically that hydrodynamic inter-

actions can increase the motor velocities [31�]. Hydro-

dynamic coupling could be important for the creation of

cytoplasmic streaming in plant cells [32] and ooplasmic

streaming in Drosophila embryos [33]. Hydrodynamic

interactions can also coordinate assemblies of molecular

motors in arrays of beating cilia. There, oscillatory beating

is mediated by assemblies of dynein arms connecting the

microtubule doublets in the axoneme. Hydrodynamic

feedback on the oscillation mechanism can enforce motor

synchronization [34]. Synchronization leads to the appari-

tion of metachronal waves similar to those observed in

many arrays of cilia, such as at the surfaces of organisms

like Paramecium or Volvox [35].

Oscillations of coupled molecular motorsMechanical oscillations occur in a large variety of bio-

logical systems [36], some of which include assemblies of

molecular motors. Insect fibrillar flight muscles develop

oscillatory tension with a rhythm that is asynchronous to

activating nervous impulses [37]. Skinned skeletal and

cardiac muscle fibers can also exhibit spontaneous oscil-

latory contractions (SPOC) in vitro [38]. Beating of the

flagella in sperm cells and bronchi, oscillations of the

mitotic spindle during asymmetric cell division [39], of

some insects’ antennal hearing organs [40], and of

mechanosensory hair bundles in hair cells from the

vertebrate ear [41] offer illustrative examples in nonmus-

cular systems. In the latter case, it has been shown that

mechanical oscillations provide amplification of weak

stimuli and frequency tuning [42]. Finally, oscillations

have also been observed in vitro with a minimal acto-

myosin system under elastic loading [43] which suggests

that oscillatory instabilities may emerge generically from

the dynamical properties of motor assemblies.

A motor system endowed with a nonmonotonic force–velocity relation, as discussed above, generically produces

oscillations when subjected to an elastic load. If the stall

force for the motors lies within the negative-slope region

of the force–velocity relation, a weakly loaded system

cannot reach this dynamically unstable condition.

Because an elastic restoring force perpetually attempts

to halt the motion of the system, the motors are forced to

oscillate (Figure 2). This type of model has been used to



describe the beating of cilia [44], oscillations of the

spindle during mitosis [16], oscillations of the hair bundle

in the inner ear [45], muscle spontaneous contraction

oscillations and waves [46�] and oscillations in in vitrogliding assays under load [43]. In the absence of elastic

loading, oscillations can also emerge with spatial gradients

of motor density [47], or a spatial reorganization of motors

[48�].

Other oscillation mechanisms have been proposed, in

which the force–velocity relation remains monotonic

(for a synthetic classification see [49��]). In hair cells,

spontaneous hair bundle oscillations have been inter-

preted as an interplay between the activity of a myo-

sin-based adaptation motor, negative stiffness provided

by mechanosensitive ion channels and calcium feedback

on the motor force [41]. Similarly, the negative stiffness of

a half-sarcomere in muscle might arise from mechano-

sensitivity of individual acto-myosin crossbridges [18],

resulting in oscillations when combined to load-depend-

ent unbinding of the motors. Other proposed mechanisms

of oscillations, which might be relevant for insect flight

muscle, rely on inertial loading of the contractile appar-

atus and delayed stretch activation [50]. Stretch activation

and effective negative stiffness could emerge if geometri-

cal constraints imposed variations of the available pool of

myosins to actin as a function of the external force [38,51].

Cytoskeleton and motorsUp till now, we have considered the properties of motors,

assuming that the cytoskeleton has a prescribed geometry

and thus that its properties are independent of motor

activity. These conditions are obviously met in bead

assays and in muscles for instance. In the cytoplasm,

however, motor activity changes the cytokeleton organ-

ization and properties. Motors can either quantitatively

modify visco-elastic properties of the filament network in

a nontrivial way, or generate new active behavior. The

introduction of myosins in a crosslinked actin network has

two main consequences on its visco-elastic properties: a

network stiffening by several orders of magnitudes at

frequencies higher than a fraction of 1 Hz [52], and a

nonclassical power law dependence of the elastic

modulus with frequency [53]. These effects are associ-

ated with rearrangements of the network, that depend on

motor unbinding that is likely to be a function of force, as

discussed above. In the presence of active motors, net-

works of filaments acquire dynamical properties, which

are highly relevant to cell biology. Cell shape, cell oscil-

lations, cell motility, blebs, cytokinesis and cell wound

healing all involve the active actin–myosin system as a

key player [54�]. A unified description of these diverse

phenomena can be obtained either from conservation

laws and symmetry arguments [55,56], or from molecular

modeling [57–59]. Whatever the approach, the mechan-

ical activity of the network produces an important novel

contribution to stress, named contractility [60��]. A


detailed review of the rheology and the active behavior

of the cytoskeleton is given by Mackintosh and Schmidt

[61].

Concluding remarksWe have reviewed here theoretical descriptions that can

account for a variety of collective effects. Dynamical

phase transitions emerge naturally in the limit of an

infinite number of motors for which fluctuations average

out. Biological systems, however, comprise a finite num-

ber of motors, typically between ten and a few hundreds,

and are thus inherently stochastic. Some effects, such as

bidirectional motion, can be described only by taking

noise into account. Collective effects are concealed if

fluctuations are too intense. More precise theories con-

sidering a finite number of motors are clearly needed to

bridge the gap between single molecule descriptions and

the limit of large motor collections. In this respect,

experiments at the single molecule level provide useful

constraints on parameter values that can be used in

collective motor theories.

Cooperative behaviours are not an exclusive property

feature of molecular motors but can also emerge in other

classes of interacting proteins. Propagation of local con-

formational changes has for instance been described

quantitatively in protein arrays with allosteric states

and can also lead to nonequilibrium phase transitions

[62]. In this example, proteins interact locally with their

nearest neighbors. Because molecular motors can be

mechanically linked to a common rigid backbone, as

for instance in muscle, one motor molecule can be influ-

enced by all the others resulting in global interactions.

Theories describing respectively the collective behavior

of molecular motors and solid friction display deep ana-

logies. In both cases, particles hop between consecutive

potential wells and dissipate elastic energy during attach-

ment-detachment cycles. Saw-tooth oscillations of mol-

ecular motors assemblies could then be related to stick-

slip phenomena. It has also been recently suggested that

crossbridge theories could be modified to describe a class

of friction models [63].

As a final remark, it is widely demonstrated in physics that

‘more is different’ [64]. This short review shows that this

statement holds true for biology. Molecular motors can

interact in many different ways, and most often the inter-

actions lead to nontrivial collective behaviors. Collective

effects cannot be explained by only considering the beha-

vior of an individual motor, but require a theoretical

description at the level of many motor molecules. These

effects include oscillations, bidirectional motion, hystere-

tic behavior and thus formation of dynamical patterns.

They explain complex phenomena such as sound detec-

tion by hair cells, muscle and flagella oscillations, cell

oscillations, cytokinesis and cell wound healing.



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45. Camalet S, Duke T, Julicher F, Prost J: Auditory sensitivityprovided by self-tuned critical oscillations of hair cells. ProcNatl Acad Sci U S A 2000, 97:3183-3188.

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Gunther S, Kruse K: Spontaneous waves in muscle fibres. New JPhys 2007, 9:417.

A simple microscopic model of sarcomere dynamics explains howspontaneous oscillations of individual sarcomeres can lead to the pro-pagation of contraction waves along a muscle fiber.

47. Campas O, Sens P: Chromosome oscillations in mitosis. PhysRev Lett 2006, 97:128102.

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Vogel SK, Pavin N, Maghelli N, Julicher F, Tolic-Norrelykke IM:Self-organization of dynein motors generates meiotic nuclearoscillations. PLoS Biol 2009, 7:e1000087.

A theoretical and experimental study of meiotic oscillations in yeast,which are due to the instabilities associated with molecular motors.

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Vilfan A, Frey E: Oscillations in molecular motor assemblies. JPhys: Condens Matter 2005, 17:S3901-S3911.

A synthetic classification of existing models that lead to oscillations inmolecular assemblies. The authors identify two categories: systems witha delayed force activation and nonmonotonic force–velocity relations.

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Julicher F, Kruse K, Prost J, Joanny JF: Active behavior of thecytoskeleton. Phys Rep 2007, 449:3-28.

This paper is an extensive theoretical review about the properties ofactive gels and their application to the cytoskeleton.

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Bendix PM, Koenderink GH, Cuvelier D, Dogic Z, Koeleman BN,Brieher WM, Field CM, Mahadevan L, Weitz DA: A quantitativeanalysis of contractility in active cytoskeletal proteinnetworks. Biophys J 2008, 94:3126-3136.

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Coordination and collective properties of molecular motors: theory