Ann. Henri Poincaré 4, Suppl. 2 (2003) S671 – S678
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S671-8
DOI 10.1007/s00023-003-0952-8
Annales Henri Poincaré
Active Behaviors in Living Cells
Frank Jülicher
Abstract. We review work on the force and motion generation in cells. Starting from
the physics of individual motor molecules, we discuss how complex dynamic behaviors emerge from the interplay of many active elements. The concepts developed for
simple systems can be used to adress the active mechanical properties of complex
cellular structures.
1 Introduction
One of the most visible feature of living cells is their ability to generate motion
and exhibit active behaviors [1, 2]. Examples are muscle contractions, cell locomotion on substrates and cell division. Cells constantly explore their environment
by generating small protrusions, such as so-called lamellipods, which are sheet-like
structures that emerge from the edges of a cell. They grow and shrink and lead
to constantly readjusted cell shapes. Such structures are built around filaments of
the cytoskeleton which form networks and gels. Their complex dynamic behaviors
are the result of an interplay of a large number of enzymes which interact with
cytoskeletal filaments.
The cytoskeleton is the prototype system for the study of force and motion
generation in cells. The dynamical properties of the cytoskeleton and its associated
proteins are governed by phenomena on different scales. On the molecular level,
highly specialized motor proteins consume chemical energy of ATP and are able
to generate motion and forces [3, 4, 5]. These forces are stochastic in nature. On
larger scales, where the cytoskeleton forms complex structures, dynamical behaviors emerge from an interplay of many active processes on the molecular scale.
Certain types of such behaviors can be described as arising via self-organization
phenomena. This can be illustrated by focusing on simple examples, where complex patterns in space and time can be understood to be generated by molecular
motors interacting with cytoskeletal filaments [6, 7, 8].
A simple situation arises if many aligned filaments form a bundle. Such a
bundle can have active properties if motor molecules or aggregates of motors form
mobile cross linkers [9, 10, 11]. These motors slide filament pairs with respect to
each other which introduces a rich dynamics in the system and active mechanic
properties. With the presence of motors, the filament bundle has been transformed
in a nonlinear dynamic system which undergoes bifurcations and dynamic instabilities. Dynamic systems have interesting properties in the vicinity of dynamic
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(a)
minus
plus
(b)
minus
plus
Figure 1: Schematic representation of molecular motors and track filaments. (a)
Many myosin molecules interacting with an actin filament. (b) Kinesin moving
along a microtubule. Both types of filaments are polar and periodic, their two
different ends are denoted “plus” and “minus”.
instabilities which cells could use to achieve particular goals. An important example are spontaneous oscillations which can be used to detect and amplify periodic
signals [12, 13]. Mechanosensory cells of the ears of vertebrates exhibit complex
cytoskeletal structures which form bundles of rod-like stereocilia [14]. These hair
bundles have been shown to generate spontaneous mechanical oscillations [15, 16].
In the subsequent sections, we discuss physical approaches to understand basic
features of such active phenomena in cells.
2 Motor proteins: single motors and collective effects
A motor protein of the cytoskeleton interacts specifically with a certain type of filament along which it is able to move in presence of Adenosinetriphosphate (ATP)
which is a chemical fuel [3]. The filaments serve as guides or tracks for the motion.
Two types of filaments play this role: microtubules and actin filaments. Both are
formed by a polymerization process from identical monomers (actin and tubulin
monomers, respectively), leading to a regular and periodic structure. An important feature is their polarity: The filaments are asymmetric with respect to their
two ends. This symmetry has its origin in the asymmetry of the monomers which
form a polar filament structure with two different ends which are denoted “plus
end” and “minus end”. This polar symmetry is essential for motor operation as
it defines the direction of motion. A given motor molecules moves in a particular direction along a filament. Motor proteins are classified into several families:
myosins, kinesins and dyneins. Myosins move always along actin filaments while
kinesins and dyneins move along microtubules, see Fig. 1. The energy source for
motion generation is the hydrolysis of ATP. The motor protein (or more precisely,
the head domain containing the ATP binding site) undergoes a chemical cycle.
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It binds ATP, hydrolyzes the bound ATP and releases the products ADP and P
(phosphate). After completion of the cycle the motor is unchanged. The different
conformations which occur during the chemical cycle have different structures and
can in particular have different interaction characteristics with respect to the filament. As a result, the motor protein undergoes chemistry-driven changes between
strongly and more weakly bound states (“attachments” and “detachments”). This
coupling between chemistry and binding permits the creation of motion along a
polar filament [17, 18, 4, 5].
The main features and principles of this mechanochemical coupling can be
captured by a simple two-state model where we assume that two chemical conformations, an attached state 1 and a weakly bound state 2 are characterized by two
energy landscapes [19, 20, 21, 22]. These energy landscapes are periodic potentials
which have broken symmetry. This broken symmetry reflects the polarity of the
track filaments. Transitions between the states represent conformational changes
driven by the ATP hydrolysis cycle. Since the system operates far from equilibrium, it attains a steady state with nonvanishing average velocity v and mechanical
properties characterized by the relationship between v and an externally applied
force fext . If the periodic potentials are symmetric, no net motion can occur by
symmetry in the absence of applied external forces.
In many interesting situations, a large number of motor molecules operates
collectively. Generalizing the simple two state model to situations where large numbers of motors are coupled rigidly, we have predicted the possibility of dynamic
instabilities in the force-velocity curve [23, 24]. An interesting possibility is that
even if the potentials are symmetric, motion can occur via spontaneous symmetry breaking. In this case, two moving states with opposite velocities coexist. In
practice, this dynamic symmetry breaking transition is concealed by noise. If the
number of collectively operating motors is finite, they generate fluctuations which
induce transitions between the two oppositely moving states. As a consequence,
the system exhibits bidirectional motion where it moves a certain time in one
direction before it switches stochastically to motion in the opposite direction.
Recently, such bidirectional motion has been observed in so-called motility
assays [25]. Motor molecules are attached to a substrate at high density and drive
the motion of microtubules which adhere to the motor-coated substrate. Usually,
the generated motion occurs with one particular end of the microtubule in front.
For a particular mutant of a kinesin motor, the observed behavior was significantly
different. Microtubules switched their direction of motion after times of several
seconds until up to a minute and motion was bidirectional. At the same time,
individual motors of this type were not able to generate motion. This suggests
that the individual motors have lost their directionality because of their mutation.
At the same time, collections of these motors are still able to generate motion via
a symmetry breaking dynamic instability. We have shown that this interpretation
provides a natural explanation of the experimental observations [26]. Interesting
collective effects of many motors can also occur due to crowding of many motors
on a filament along which they advance [27, 28]
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3 Self-organization of motors and filaments
If motors form small aggregates, they can interact with two or more filaments at the
same time. Such aggregates then play the role of cross-linkers of a filament network.
Since a motor can move along a given filament, these cross-links are mobile and
the resulting polymer network intrinsically dynamic [29, 30, 31]. Experimentally,
such filament systems in the presence of mobile cross linkers can be studied using
artificial constructs linking motors together. It has been shown that such systems
self-organize and generate spatiotemporal patterns such as asters and vortices
[7, 8, 32, 33].
A case of particular importance is the situation where filaments are aligned
and form a bundle [9, 10, 11]. The dynamic and mechanic properties of active bundles can be discussed in very simplified physical models. Using a one-dimensional
description of the bundle, we denote by c+ (x) and c− (x) the densities of filaments
pointing their plus ends to the right and left, respectively. Assuming that mobile
cross-links induce interactions predominantly of filament pairs, we can write nonlinear dynamic equations for the filament densities. In this description, filament
currents are resulting from active behaviors of motors, which induce the relative
sliding of filament pairs.
The interaction terms which induce this sliding can be obtained from symmetry arguments. We have to distinguish between two types of interactions. Interactions between filaments of equal orientation, characterized by an interaction
strength α and interactions between filaments of opposite orientation with strength
β. We find that the interaction between equally oriented filaments generates in
general a contractile tension and induces the shortening of a bundle. For sufficiently large α, a homogeneous density profile becomes unstable and contracts to
become a localized distribution. If furthermore the interaction between oppositely
oriented filaments is acting, described by a finite value of β, a dynamic instability
still occurs. However, the homogeneous state now becomes unstable with respect
to propagating modes if periodic boundary conditions are imposed. These solitary
waves can upon further increase of α become again unstable and oscillating waves
appear.
Even though we start from a simplified description based on basic rules of
filament sliding, we obtain a rich scenario of bifurcations and dynamic transitions.
We can incorporate in this description the mechanical properties of a bundle using
momentum balance and balances of internal forces. We show that in a homogeneous
density profile, the mechanical tension
Σ = αη3 ((c+ )2 + (c− )2 )
,
(1)
is generated by the interaction of parallel filaments of strength α. Here, η is a
friction coefficient and the filament length. This tension is partly relaxed when the
density profile becomes unstable and a localized filament distribution is generated.
Self-organization of motors and filaments can also be studied in other geometries. Cilia and flagella are hair-like appendages of many cells which contain
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microtubules arranged in a cylindrical geometry together with a large number of
dynein motors [1]. These structures called axonemes are able to generate bending
waves and periodic motion which are used by sperm and some microorganisms to
swim. Such systems can be described as elastic filaments which undergo bending
deformations as a result of internal forces generated by motors. We find that such
systems naturally undergo an oscillating instability or Hopf bifurcation [34, 35]. In
the oscillating state, we can calculate propagating bending waves which can lead
to self-propulsion of the beating filaments.
4 Nonlinear amplification by auditory hair cells
In the previous sections, we have argued that cytoskeletal structures which contain
many filaments and molecular motors exhibit rich dynamic behaviors and can undergo dynamic instabilities. Such phenomena are also relevant for mechanosensory
cells of the inner ears of vertebrates where oscillatory instabilities and spontaneous
oscillations play an important role [36, 12, 13]. These are highly specialized cells
which possess at their surface a bundle of rod-like structures mainly formed by
densely packed actin filaments. This hair bundle is a mechano-electrical transducer,
capable of detecting bundle deflections of a few nm and generating an electrical
membrane potential in response [14].
It has been shown recently that hair bundles in frog ears exhibit spontaneous
oscillatory motion due to an active process in the hair bundle [15, 37, 38, 39]. It has
been suggested that these spontaneous movements are the signature of an active
process which plays the role of an amplifier of mechanical stimuli. Indeed, the
sensitivity of the oscillating hair bundle increases for small stimuli if the stimulus
frequency is in the vicinity of the spontaneous frequency of oscillation of the cell
[37].
This signal amplification can be understood if we assume that the hair bundle
profits from the nonlinear properties of an oscillator in the vicinity of a Hopf
bifurcation [12, 13]. If the system is at the verge of oscillating, it responds to
periodic stimuli of amplitude f with a deformation amplitude x that obeys a
power law
(2)
x ∼ f 1/3
if the stimulus frequency is close to the oscillation frequency. Consequently the
sensitivity x/f ∼ f −2/3 becomes large for small stimulus amplitudes. This nonlinear response is generic for critical oscillators near the oscillation frequency. An
open question is how dynamic oscillators in the ear achieve proximity to the critical point. We have proposed that a general self-regulation mechanism which poses
the system close to the critical point by a feedback control [12].
There exists evidence that all vertebrate ears use active systems for signal
amplification. Most important is the observation of spontaneous sound emissions
which can be recorded from the ears of many different species [40, 41]. How these
oscillations are generated in the ears of different animals and wether different
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species use different physical mechanisms to generate oscillations is still a matter
of debate. The fact the ear uses active amplifiers was indeed foreseen by Thomas
Gold [42] who remarked that the viscous damping in the inner ear would not allow
the exquisite capabilities of the ear if it would rely on passive resonance only.
5 Discussion
The cytoskeleton represents an soft biological material which is inherently dynamic
due to the presence of active proteins such as molecular motors and also due
to the polymerization and depolymerization of filaments. Such active materials
are able to generate motion and forces. General physical principles which govern
such systems can be addressed by studying simplified situations where a small
number of components is taken into account. In general, one finds that inert or
immobile states become unstable and dynamic instabilities occur. In many cases,
spontaneous oscillations are generated and propagating modes appear.
Such simplified situations can in principle be studied experimentally in socalled in vitro systems where several important components are purified and observed under controlled conditions outside the cell. An important example are
motility assays where motors that are attached to a solid substrate set in motion filaments that adhere to the substrate [43, 44, 45]. Such in vitro experiments
together with theoretical studies of simple systems makes it possible to better
understand the interplay and self-organization of several components in the cell.
Addressing the dynamic and mechanic behaviors of complete cells remains a
big challenge. However, the example of hair cell oscillations reveals that general
concepts governing the behaviors of nonlinear and stochastic systems can prove
very useful and provide important insights for the study of cellular dynamics. An
important goal is to understand the dynamic phenomena involved in cell locomotion. Many cells adhere to solid substrates and are able to grow protrusions at an
advancing edge while retracting at the back. This phenomenon involves polymerization and depolymerization of filaments, the action of motor molecules and the
formation and breakage of adhesion bonds with the substrate [46, 47].
I thank M. Badoual, S. Camalet, T. Duke, J.A. Hudspeth, K. Kruse, P.
Martin, A. Parmeggiani, J. Prost for stimulating collaborations.
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Frank Jülicher
Max-Planck-Institut für Physik komplexer Systeme
Nöthnitzer Str. 38
D-01187 Dresden
Germany