The cytoskeleton provides eukaryotic cells with mechanical support and helps them to perform their biological functions. It is predominantly composed of a network of semiflexible polar protein filaments. In addition, there are many accessory proteins that bind to these filaments, regulate their assembly, link them to organelles and provide the motors that either move the organelles along the filaments or move the filaments themselves. A natural approach to such a multiple particle system is the study of its collective excitations. I review some recent work on the theoretical description of the emergence of a number of particular collective motile behaviours from the interactions between different elements of the cytoskeleton. In order to do this, close analogies have been made to the study of driven soft condensed matter systems. However, it emerges naturally that a description of these soft active motile systems gives rise to new types of collective phenomena not seen in conventional soft systems. I discuss the implications of these results and perspectives for the future.
Living cells contain many elements that are typical of soft matter systems, such as lipid membranes, polymers and colloidal aggregates. It is then reasonable to expect that the dynamics and interactions of these constituents that control cell function are of the same time and energy scales as those of synthetic soft materials. However, life adds new features not found in traditional soft matter: the constant flow of energy and information required to keep living organisms alive. These new features make cells of particular interest to physicists, as in order to understand them we will require new theoretical concepts and experimental techniques to study the behaviour of active living matter.
The eukaryotic cell cytoskeleton is a perfect example of this ‘novel’ type of active material. The cytoskeleton allows the cell to carry out coordinated and directed movements, such as cell crawling, muscle contraction and all the changes in cell shape in the developing embryo. The cytoskeleton also supports intracellular movements, such as the transport of organelles in the cytoplasm and the segregation of chromosomes during cell division.
The cytoskeleton contains a large number of different proteins, some of which are characterized and many of which remain unknown with their functions yet to be identified. It is also inhomogeneous with a large variety of different dynamic supramolecular structures. Examples are contractile elements like stress fibres or the contractile ring in mitosis or astral objects like the mitotic spindle which forms during cell division.
Self-assembled filamentous protein aggregates play an important role in the mechanics and self-organization of the cytoskeleton. In addition, a number of other proteins interact with them and modulate their structure and dynamics. Crosslinking proteins bind to two or more filaments together to form a dynamic gel. Molecular motor proteins bind to filaments and hydrolyse nucleotide (adenosine) triphosphate which, coupled with a corresponding conformational change, turns stored chemical energy into mechanical work. Capping proteins modulate the polymerization and depolymerization of the filaments at their ends (Alberts et al. 2002).
A key question is how the elements of the cytoskeleton cooperate to achieve its function? To what extent is there a ‘cellular’ brain and how closely does it control cellular mechanisms? How much of a role does spontaneous self-organization driven by general physical principles play?
Much of the recent progress in the understanding of the complex structures and processes involved in maintaining the cytoskeleton has been linked to the development of new biophysical probes, allowing an unprecedented view of sub-cellular processes at work. Mechanical probes such as optical and magnetic tweezers (Mehta et al. 1998), atomic force microscopes (Dammer et al. 1995) and micropipettes probe the response of the elements of the cytoskeleton to locally applied forces. Visualization techniques using fluorescence microscopy, e.g. fluorescence imaging with 1 nm accuracy (Yildiz et al. 2003) or single-molecule high-resolution colocalization (Churchman et al. 2005) based on organic dyes, allow one to follow the dynamics of single molecules inside living cells (in vivo), giving insights into the microscopic processes underlying cellular dynamics.
Owing to the large number of unknown components, it is also of interest to study simplified systems consisting of a smaller number of well-characterized elements (in vitro, in the test tube). Therefore, it is natural, given their importance to study purified mixtures of filaments and motors. This has led to a number of experimental biophysical studies of these simplified systems. Owing to the controlled nature of their preparation and the detailed knowledge of their constituents, they are particularly amenable to a quantitative description using techniques from theoretical physics. In this review, we will be mostly involved with describing such simplified systems. We also emphasize that we will be describing behaviour at the greater than or equal to microsecond time-scale, and hence eschewing atomistic detail for coarse-grained phenomenological descriptions.
However, the reductionist viewpoint typified by this approach also has its drawbacks. A simplified system necessarily can provide only a subset of the phenomena observed in living cells, since only a small fraction of the components are present. A choice must also be made of which simplified system to study, as different combinations of components may give different or similar behaviour. This choice must of course be heavily influenced by previous experiments (Szent-Gyorgi 1951; Trinick & Offer 1979). A living cell is a highly optimized complex system of interacting agents with the ability to modulate its response to complex changes in its environment. This complexity will be missing from simple mechanical models described here. However, there is some hope that this complexity can eventually be combined with the physical picture emerging from these approaches to give a more complete ‘biophysical systems’ picture of motility in cells. Finally, even within our limited frame of reference, we will also make a number of simplifying assumptions in developing the models, some of them justified and others very likely ignoring important physical phenomena, which we leave out for now, but with a view to including them in the future.
In §2, we introduce and describe two families of cytoskeletal filaments and their associated motors. In §3, we describe some recent in vivo and in vitro experiments tracking the motion and mechanics of these filaments and motors. We then review some recent theoretical approaches to describe these systems. Finally, we consider some of our current work and give perspectives for the future.
2. Filaments and motors
There are a huge number of proteins which make up the cytoskeleton, the description of those already identified being worthy of a review itself. We focus here on a few of its main components: protein filaments and molecular motors.
Cytoskeletal filaments come in three flavours called thin filaments (actin filaments), thick filaments (microtubules) and intermediate filaments (Alberts et al. 2002). Actin filaments and microtubules interact with motor proteins and are associated with dynamical processes of the cytoskeleton, while the intermediate filaments are thought to be more passive, providing mechanical support and resistance to deformation. We will be concerned in the following with only actin filaments and microtubules. Actin filaments play a key role in a number of important dynamic cellular processes. In the late stages of cell division (called cytokinesis), they (together with a motor protein, myosin) form a contractile ring that contracts to cleave the cell and produce its two daughters. Actin filaments also form polymerizing, crosslinked and contractile structures that lead to cell locomotion. In a dividing cell, microtubules are organized into the mitotic spindle which separates the two copies of the chromosomes into the daughter cells. In interphase (the period of the cell cycle between cell division events), microtubules act as the highways of the cell providing pathways for active transport of material between different parts of the cell. Both actin filaments and microtubules give mechanical support to the cell and help determine its shape.
The motor proteins of the cytoskeleton convert chemical energy to mechanical work by the hydrolysis of ATP. Each motor interacts with a specific filament; there are both actin- and microtubule-associated motors. Myosin walks along F-actin and kinesin and dynein walk along microtubules. Though the microscopic details of their cycle are still not known, en gros, the motors bind to their respective filaments and undergo a conformational change coupled to the ATP hydrolysis, which leads to relative motion of the motor protein with respect to its associated filament. The motors move preferentially in one direction along the filament determined by the filament polarity owing to the details of the binding with the filament. In general, one can divide a motor protein into a head which binds to the filament and a tail which binds to the cargo.
(a) F-actin and myosin
The actin monomer is a globular protein called G-actin with a distinct structural polarity denoted (historically) by a plus end (head) and a minus end (tail). G-actin has a nucleotide-binding cleft (adenosine triphosphate/adenosine diphosphate) in the centre of the molecule separating the two ends. The G-actin monomers assemble head to tail to generate polar filaments called F-actin (filamentous actin). These polar filaments are made up of two parallel protofilaments that twist around each other to form a right-handed helix, with a helical pitch of around 37 nm and a diameter of around 10 nm. The nucleotide-binding cleft is thought to bind ATP in the free state and ADP in the filamentous state. Since the filament is polar, the rate of polymerization/depolymerization at the two ends will in general be different. The plus end of the filament is the more dynamic of the two. The plus end is historically referred to as the ‘barbed’ end, while the minus end is called the ‘pointed’ end owing to their appearance when myosin heads bind to the filaments. F-actin can also exist in a ‘treadmilling’ state, where polymerization at the plus end is balanced by depolymerization at the minus end.
Myosin has been the most studied (since the mid-1950s) of the motor proteins owing to its presence in striated muscle. The most commonly studied class of myosin is myosin II, which is found in both sarcomeric and smooth muscle, as well as in the cytoplasm. Myosin II can be split into two sections: heavy meromyosin (HMM), which has the motor domains and can hydrolyse ATP, and light meromyosin (LMM), which has a coil–coil structure and tends to form filamentous aggregates. The thick filament in skeletal muscle is formed from huge filamentous aggregates of hundreds of myosin II, while myosin II in the cytoplasm forms mini-filaments with a small number of myosins. Myosins are in general non-processive, i.e. they detach after only a few ATPase cycles (or steps). However, in biology, there are always exceptions; recently myosin V has been shown to be a processive motor with a step length comparable to the helical pitch of F-actin (http://www.mrc-lmb.cam.ac.uk/myosin.html).
(b) Microtubules and kinesin
Microtubules are made up of subunits of a heterodimer called tubulin. Tubulin is made up of two closely related globular proteins called α- and β-tubulin, which are always tightly bound to each other. Each of the α- or β-monomers has a nucleotide-binding site (guanosine triphosphate/guanosine diphosphate), but while both will bind GTP, only the β-subunit can exchange GTP for GDP. It is thought that this ability to hydrolyse GTP at this site has an important consequence for microtubule dynamics. The tubulin dimers assemble head to tail to form a hollow tube composed of around 13 parallel protofilaments made of alternating α- and β-tubulin molecules called a microtubule. Since each protofilament is assembled from polar subunits all pointing in the same direction, the microtubule has a definite structural polarity, with, by convention, the β-monomers pointing towards the plus end and the α-monomers pointing towards the minus end. The lateral interactions between the α- and β-subunits of neighbouring protofilaments are such that lateral contacts are formed between monomers of the same type (e.g. α–α). Microtubules grow by the addition of monomers with GTP bound to the β-subunit, which get hydrolysed to GDP at a slower rate. As long as the hydrolysis is slower than polymerization, this leads to a cap of GTP-tubulin subunits on a growing tubule. If the rate of polymerization slows or owing to spontaneous fluctuations, the cap can be lost. Since the exchange of GDP by GTP can only occur in the unbound state of the dimer, a GDP-tubulin filament end catalyses the exchange by depolymerization. The loss of the cap leads to a GDP-tubulin-ended filament that will rapidly depolymerize in a ‘catastrophe’ until it is rescued by the formation of a new cap and begins to repolymerize (Dogterom et al. 1995; Howard 2000; Alberts et al. 2002).
Kinesin have been studied since the mid-1980s, when the oldest member (kinesin-1) was identified as being involved in the fast axonal transport. Kinesins are processive, implying that they go through many ATPase cycles before detaching from the microtubules. Many members of the kinesin family have an α-helical coiled coil in the main chain that probably causes them to dimerize, which is the source of their processivity. The conventional view of the motion of kinesin-1 is the ‘hand-over-hand’ model, in which the two heads bind alternately while walking towards the plus end of the microtubule. Most kinesins walk towards the plus end of the microtubule though there are exceptions, e.g. NCD. Kinesins are involved in vesicle transport in interphase and also in the formation of the mitotic spindle in cell division.
(c) Other proteins
There are a host of other proteins associated with the cytoskeletal filaments. Another motor protein associated with microtubules is dynein, which walks towards the minus end and is involved in the flagellum-based motility. There are a huge number of proteins which interact with F-actin. Examples include: ADF and cofilin, which promote treadmilling; gelsolin, which cuts up filaments; Arp2/3 complex, which promotes the formation of a branched network; and a number of different crosslinking proteins, such as α-actinin, fascin and scruin. There are also a number of microtubule-associated proteins or MAPS that modulate microtubule structure. Examples are the type II MAP tau crosslinking protein, which promotes the formation of microtubule bundles.
New visualization techniques can now track the motion of F-actin, myosin II, microtubules and kinesin at different stages of the cell cycle and have identified the dynamics of these species during cellular processes. An example is cell crawling, where recent experiments have led to new insights into its mechanism (Vallotton et al. 2004). It is generally accepted that polymerization of actin at the front (leading edge) of the cell drives crawling (Mogilner & Oster 2003), but for efficient locomotion, this must be closely coordinated with the adhesion to the substrate and the retraction of the rest of the cell body (containing the nucleus and other organelles) towards the leading edge. For many classes of cells, it is now becoming clear that the F-actin and myosin play an important role in this coordination (Verkhovsky et al. 1999). Furthermore, this active motion is modulated by the nature of adhesion to the substrate (Dammer et al. 1995; Geiger & Bershadsky 2002). However, a detailed mechanism of how all this is done remains unknown.
Similarly, analysis of the actin and myosin distributions in fragments of fish epidermal keratocytes without nuclei, microtubules and organelles has shown transitions between motile polarized and stationary non-polarized states induced by mechanical stimuli (Verkhovsky et al. 1999).
However, in vivo experiments are always limited by the lack of complete information about their constituents, making it hard to make a quantitative link between ‘microscopic’ interactions and macroscopic cellular behaviour. A natural way to improve this state of affairs is to study in vitro simplified systems.
The last decade has seen a large number of systematic experimental studies on the mechanical properties of (disordered) mixtures of cytoskeletal filament systems. The results are well described theoretically by the physics of semiflexible polymers (Storm et al. 2005), whose relative stiffness can be obtained from their persistence lengths. This provides a unifying description able to span the behaviour of F-actin with a persistence of the order of 10 μm to microtubules with a persistence length on the millimetre scale.
An emerging new direction of research in this field is the study of mixtures of these filaments and their associated proteins including molecular motors. In particular, the addition of motors leads one naturally to the study of driven dissipative systems by coupling the dynamics of the filaments to a chemical reaction (hydrolysis). The mechano-chemical coupling adds additional richness not found in simple pattern-forming chemical reactions.
The study of filament/motor mixtures is not new and the phenomenon of ‘super-precipitation’ of F-actin/myosin II mixtures was reported over 50 years ago (Szent-Gyorgi 1951; Trinick & Offer 1979). However, recently, more detailed studies of filament–motor mixtures have begun to appear. These studies fall under two broad categories, focusing on (i) the self-organization properties, or (ii) their mechanical properties.
The mitotic spindle is a bipolar astral structure, which is formed by microtubules stabilized by chromosomes and nucleated at two centrosomes. The formation of asters has been observed in mixtures of microtubules and motors (Urrutia et al. 1991). Furthermore, even in extremely simplified systems of length-stabilized microtubules and multi-headed motor constructs made up of only one type of motor, a wide variety of dissipative structures as a function of motor and filament density can be formed (Nédélec et al. 1997). For a fixed microtubule concentration, as the motor concentration is increased, vortices, asters and bundles, respectively, are formed. A rich library of structures is formed, as constructs made up of plus and minus-ended motors are made (Surrey et al. 2001).
In the presence of a heavy meromyosin (HMM), a disordered (non-polar) linear bundle of filaments with, on average, equal numbers of parallel and anti-parallel orientations first shortens and fattens to form thicker bundles, which then subsequently splits into a star or aster-like structure. Each arm of the star is a bundle formed only of parallel filaments, suggesting that the contraction mechanism is also associated with ordering (Takiguchi 1991).
The mechanical properties of mixtures of myosin/F-actin have been studied and showed to have an anomalous viscoelasticity. Mixtures of myosin-II and F-actin have shown fluidification or faster relaxation at low frequencies (Humphrey et al. 2002). Mixtures of S1-myosin and F-actin have shown increased fluctuations and anomalous relaxational dynamics at high frequencies (Le Goff et al. 2002).
These experiments act as inspiration for an increasing number of theoretical approaches to studying these active systems. As we will see in §4, a rich spectrum of behaviours emerges.
4. Theoretical approaches
There have been a number of recent theoretical studies of the collective dynamics of mixtures of rigid filaments and motor clusters. First and most microscopic, numerical simulations with detailed modelling of the filament–motor coupling have been shown to generate patterns similar to those found in experiments (Nédélec et al. 1997; Surrey et al. 2001). These simulations modelled the filaments as elastic rods with motor clusters being parametrized by three binding parameters, the on- and off-rates and the off-rate at the plus end of the filament: pon, poff, poff,end. They also used a linear force velocity curve of the form . At higher motor densities, vortices and asters were observed. It was observed that the rate of unbinding at the ends poff,end played a crucial role in the vortex to aster transitions (Nedelec 1998).
A second interesting development has been the proposal of ‘mesoscopic’ mean-field kinetic equations (Nakazawa & Sekimoto 1996; Sekimoto & Nakazawa 1998), where the effect of motors was incorporated via a motor-induced relative velocity of pairs of filaments, with the form of such velocity inferred from general symmetry considerations. Kruse & Jülicher (2000) and Kruse et al. (2001) have proposed one-dimensional models of filament dynamics that have shown the existence of instabilities from the homogeneous state to contractile states and travelling wave solutions. The nature of the instabilities has been classified, as well as the states beyond the instability.
Finally, hydrodynamic equations have been proposed, where the mixture is described in terms of a few coarse-grained fields whose dynamics is also inferred from symmetry considerations (Bassetti et al. 2000; Lee & Kardar 2001; Simha & Ramaswamy 2002; Kim et al. 2003; Kruse et al. 2004; Sankararaman et al. 2004). Starting from a two-dimensional spin lattice model as a microscopic picture, theoretical studies have been performed on a sliding assay with filaments moving on a bed of motors adsorbed on a flat surface (Bassetti et al. 2000). Using a combination of mean field and simulation, they obtained non-equilibrium transitions to inhomogeneous states, such as stripe patterns. Lee & Kardar (2001) proposed a simple model ignoring filament density fluctuations, with two coupled dynamical equations: one for the dynamics of a coarse-grained filament orientation-order parameter and another for motor dynamics. It was proposed that filament growth by polymerization provided a mechanism for an instability of the system from an isotropic to an oriented state (Lee & Kardar 2001; Kim et al. 2003). They were able to obtain a phase diagram for the system showing a transition from vortices to asters in terms of the rate of filament growth and motor concentration. This model was subsequently generalized by Sankararaman et al. (2004) to include populations of bound and free motors, as well as an additional coupling of filament orientation to motor gradients. The effects of boundary conditions on the steady states observed were studied. An alternate hydrodynamic description based on the dynamics of an order parameter (such as local polarization) as well as momentum conservation has been proposed by a number of groups (Simha & Ramaswamy 2002; Hatwalne et al. 2004; Kruse et al. 2004). These authors have also pointed out that such a generic macroscopic description can be due to a variety of microscopic mechanisms. Similar equations could be used to study swimming bacterial colonies and motor/filament mixtures (Toner et al. 2005). Kruse and collaborators have studied systems with a local polar order and identified the properties of non-equilibrium defect structures (Kruse et al. 2004). More recently, the dynamics of motors has also been included to this description (Kruse et al. 2005). The dynamics of an active solution near a wall has been studied and shown the appearance of flow induced by anchoring (Voituriez et al. 2005). Simha and collaborators have studied systems with local nematic order and shown the appearance of non-equilibrium propagating modes and anomalous viscoelasticity (Simha & Ramaswamy 2002; Hatwalne et al. 2004).
A connection has been made between the mesoscopic and hydrodynamic approaches by deriving hydrodynamics via a coarse-graining of the kinetic equations (Liverpool & Marchetti 2003, 2004; Ziebert et al. 2004; Ahmadi et al. 2005; Aranson & Tsimring 2005). However, both the mesoscopic and the hydrodynamic approaches share an important shortcoming. The rate and strength of the motor-induced force exchange among the filaments is controlled by phenomenological parameters, whose dependence on motor activity is not known. The richness of the phenomena exhibited by the cytoskeleton leads naturally to the question of how much of the behaviour is specific and how much is generic (Liverpool 2005). To answer this question, it is also important to make the connection between microscopic models and ‘generic’ hydrodynamic approaches.
In §5, we consider the linear viscoelastic response of an isotropic solution of entangled polar filaments interacting with motor clusters in the regime where the motors do not lead to the formation of macroscopic patterns. In §6, using a hydrodynamic approach, we investigate the stability of the isotropic state to the formation of ordered states or patterns. Finally, in §7, we discuss a microscopic model of the filament–motor interaction enabling us to derive the parameters of the hydrodynamic model.
5. Viscoelastic response of isotropic solutions
We now consider a mixture of F-actin/myosin-II. Myosin spontaneously aggregates in vitro to form clusters. In an ATP-rich system, these myosin clusters can then bind to pairs of filaments and actively move the filaments with respect to each other. Motivated by experimental observations and this simple observation, one is led to ask a number of further quantitative questions. What is the elastic stress supported by such a system? What are the relevant relaxation mechanisms and time-scales? As a step in this direction, we focus here on a simple model for the viscoelasticity of an ‘active’ solution of motile semiflexible filaments, within the ‘tube’ picture of polymer dynamics (Liverpool et al. 2001). This is clearly an oversimplification of the in vitro system described above, nevertheless the physics of the problem, even with this approximation, is interesting and non-trivial.
We consider a monodisperse solution of semiflexible and polar polymers of persistence length Lp, length L and diameter a, with Lp≫a, at a monomer concentration ρa, such that the mesh size of the solution is ξ≃(ρaa)−1/2≪Lp, L (Doi & Edwards 1986). We model the ATP-induced activity of actin clusters by stochastic forces on the polymers, parallel to the filament contour (transverse motion is constrained by entanglements), which always act in the same direction with respect to the polarity of the filaments. The effect of the motor activity is (i) to increase the amplitude of the longitudinal fluctuations along the contour of the filaments, giving rise to a higher effective temperature T→T+Tact for the tangential motion, and (ii) to give the filaments a non-zero curvilinear drift velocity in their tubes, vm. Increasing/decreasing activity leads to an increase/decrease in Tact and vm. A cartoon of the system is shown in figure 1. Effective temperatures for non-equilibrium systems have been used to model noise in foams and other driven systems (Langer & Liu 2000).
The linear viscoelastic response of such an active polymer solution can be derived, assuming that its structure is not perturbed by the activity of the motor clusters. Despite this crude assumption, one uncovers rich physical behaviour, as the ‘activity’ modifies the already subtle dynamics of passive semiflexible polymer solutions (MacKintosh et al. 1995; Isambert & Maggs 1996; Gittes & MacKintosh 1998; Morse 1998a,b).
The linear response of this active filament solution to a weak time-dependent shear strain, γij(t), is characterized by the shear modulus G(t), such that the shear stress is (Doi & Edwards 1986)
The stress is calculated from the fluctuating dynamics of Kratky–Porod worm-like chains. A typical filament conformation is parameterized by R(s). The Hamiltonian of a worm-like chain is given by(5.1)where ∂xA≡∂A/∂x; and an instantaneous local tension, Λ(s), is induced by the incompressibility of the chain. The persistence length Lp=κ/kBT is the length-scale over which the chain loses memory of its orientation (see figure 2). The filaments are confined to a ‘tube’ (Doi & Edwards 1986) of diameter De∼Lp(ξ/Lp)6/5. We define an entanglement (deflection) length (Odijk 1983; Semenov 1986) Le∼Lp(ξ/Lp)4/5, as the distance between successive collisions of the filament with its tube (figure 1b). The hierarchy of length-scales is L, Lp≫Le≫a. On length-scales , the relaxation is due to the dynamics of ‘free’ chains (Doi & Edwards 1986), while for , it is due to diffusive directed motion of the polar filaments in their tubes. For (and consequently ), the chain conformation is anisotropic and can be described by weak undulations about a rigid rod.
Owing to the rod-like nature of the polymer segment, the coupling to the shear flow is anisotropic. For short filaments L<Lp, the rotational diffusion of a rod of length L determines the dynamics of , which is much slower than that of r∥, r⊥. We propose that we have in addition to the thermal fluctuating forces determined by the temperature T, a fluctuating non-equilibrium or active force, owing to the activity of the motors, giving an active contribution, Tact, to the ‘effective temperature’ of the longitudinal motion of the filaments. As, the transverse dynamics is cut-off at Le, there is different behaviour at high and low frequencies with a crossover frequency, . The dynamic fluctuations of semiflexible filaments are anisotropic (Everaers et al. 1999; Liverpool & Maggs 2001). After time t, longitudinal fluctuations are relaxed over a length for t<1/ωe and for t>1/ωe. In comparison, transverse fluctuations are relaxed over a length for t<1/ωe and can only relax by reptation for t>1/ωe.
We find short-time (high frequency) moduli as well as terminal relaxation times that differ from those of passive polymer solutions. The active solution is harder at high frequencies owing to the increased fluctuations of the longitudinal modes. These also change the relative magnitude of the longitudinal and transverse fluctuations, leading to two new relaxation regimes. At very low frequencies, the directed motion of the filaments leads to a softening or fluidification, as suggested in §1. Our results are schematized in figure 3 and summarized as follows. Upon submitting the system to a step shear, the shear modulus G(t) decays for very short times as t−3/4, as for passive polymer solutions (Gittes & MacKintosh 1998; Morse 1998a,b). This holds up to a time , where we find a regime with two new power law decays: G(t)∼t−1/8 up to a time and a modulus , after which there is a faster decay G(t)∼t−1/2 (η is the solvent viscosity). In the long-time regime, the relaxation modulus develops a plateau, as trapped stress cannot relax owing to entanglements before the filaments escape from their initial tubes (Doi & Edwards 1986). While the magnitude of the plateaux is the same as for passive polymer solutions, the tube renewal time has a different dependence on chain length L and persistence length Lp. When L/Lp≫1, the dominant stress is due to constrained transverse fluctuations of the filament (Käs et al. 1995; Isambert & Maggs 1996), leading to a plateau of magnitude , which begins at a time and decays after a time t3≃L/vm. For filaments with L/Lp≪1, the stress is due to constrained orientational dynamics (Doi 1985; Morse 1998a,b), and from , we find a plateau of magnitude , which decays after a time .
We can now make a comparison with the actin–myosin–ATP system that stimulated our theoretical study. We estimate typical times-scales and moduli from a direct mapping of our calculation onto this system. We have modelled the noise as a Gaussian white noise of non-zero mean. In a slightly more realistic picture, a motor centre has periods of activity of duration ts (the power stroke), during which a constant force f0 is applied, separated by passive periods that are Poisson distributed with a mean duration αts; α≫1. This scenario could be realized by a cluster made up of a small number (greater than or equal to 2) of processive motors or alternatively a large number of non-processive motors acting independently. The motor clusters are also assumed to be randomly distributed along the filaments at a mean distance . Assuming that clusters act independently, we estimate from the mean force and from the local fluctuations of the force about its mean . If myosin is at a concentration ρm and the mean number of myosin per cluster is N, then . Let us turn to numbers: a bound myosin has a power stroke of duration ts≃5 ms; a step size of ds≃10 nm; and a stall force of fmax≃4 pN (Howard 2000). By considering viscous drag, we estimate f0≃0.1 pN≪fmax. Actin has persistence length Lp≃17 μm and diameter a≃7 nm. A solution of F-actin at a typical concentration 100 μg ml−1 has a mesh size ξ∼0.5 μm. For ρm=0.1 μM (micromolar) and N≃10, we estimate . This gives Tact/T∼102, so that the high-frequency behaviour described above should be relevant. The crossover modulus between the high and the intermediate-frequency regimes is G2∼10 Pa. Long-time fluidification is also clear: relaxation times for coils (L=50 μm) and rod-like polymers (L=5 μm) are t3∼1 s and t4∼0.1 s, respectively, compared with t3∼104 s and t4∼100 s for passive solutions at the same actin concentration. The corresponding plateau are G3∼10−2 Pa and G4∼10−4 Pa, respectively.
6. Beyond the isotropic state
In §5, we considered the situation in which the active solution (like a solution of passive polymers) had an homogeneous and isotropic steady state. It has been shown by a number of authors both experimentally and theoretically that dissipative structures can be formed by filament/motor mixtures. It is therefore interesting to consider the conditions under which such an isotropic state is stable and for which those arguments of §5 are valid.
Continuum models of filament/motor systems have been used to show that spatial patterns are obtained as non-equilibrium solutions of the system dynamics (Lee & Kardar 2001; Simha & Ramaswamy 2002; Kim et al. 2003; Kruse et al. 2004). Such hydrodynamic models use symmetry arguments to propose the dynamics of coarse-grained fields for density and polarization of filaments. It is therefore useful to make a connection between the hydrodynamic picture and more microscopic models. As a step in this direction, we start from a phenomenological model (Kruse & Jülicher 2000) and obtain a set of continuum equations to describe the dynamics and organization of polar filaments driven by molecular motors in an unconfined geometry in (quasi-) two dimensions. By modelling the motor-filament interaction microscopically, we can determine the magnitude and, most importantly, the sign of the parameters of the continuum equations, which cannot be obtained by symmetry arguments (Liverpool & Marchetti 2003, 2005; Ahmadi et al. 2005).
Our result is a phase diagram (figure 5) as a function of the filament density and motor properties that is expected to be relevant to the analysis of recent experiments (Surrey et al. 2001; Humphrey et al. 2002).
We describe the system by a concentration of polar filaments in two dimensions (d=2), modelled as hard rods of fixed length and diameter b () at position r with filament polarity characterized by a unit vector , and a density of motor clusters m(r,t). The filament and motor concentrations satisfy the equations(6.1)(6.2)where and the translational (Jf,Jm) and rotational () currents have diffusive, excluded volume and active contributions.
The active contributions to the currents are obtained from relative velocities (and angular velocities) of interacting filaments owing to the motors. Rotations are parametrized by γ0,γ1, corresponding to two classes of motor clusters (figure 4): polar clusters, which tend to bind to filaments with similar polarity (γ0/γ1≫1; Nédélec et al. 1997; Surrey et al. 2001; Ahmadi et al. 2005) and non-polar clusters, which bind to filament pairs of any orientation (γ0/γ1≪1; Humphrey et al. 2002). Translations are parametrized by α, β and λ, and all having dimensions of velocity and depending on the angle between the filaments. The term proportional to β drives the separation of filaments of opposite polarity, while the contribution arises from the net velocity of the filament pair (see §7). The contribution proportional to α arises from spatial variations in motor activity along the filament, such as motors stalling before detaching at the polar end. It drives bundling of filaments of the same polarity. These parameters were estimated (Liverpool & Marchetti 2005) via a microscopic model of motor-induced filament dynamics as β∼λ∼u0, α∼u0(lm/l)≪u0, with u0, the mean motor stepping rate and lm, the length-scale (of the order of the motor cluster size) for spatial variations in motor activity. As seen below, this term is crucial for developing inhomogeneities and pattern formation.
To study the macroscopic properties of the solution, we truncate the exact moment expansion of as(6.3)keeping only the first three moments,(6.4)
In a passive system (γ0=γ1=0), there is a transition from an isotropic state to a nematic state. A mean-field description of such a transition, which is continuous in two dimensions (but first order in three dimensions), requires cubic terms in the nematic order parameter in the equation of motion. The transition here is identified with the change in sign of the decay rate of Sij, which signals an instability of the isotropic homogeneous state. This occurs when the excluded volume effects dominate at a density ρN=3π/2. The homogeneous state is isotropic for ρ0<ρN and nematic for ρ0>ρN. No homogeneous polarized state with a non-zero mean value of p is obtained in a passive solution.
We now turn to an active system. We introduce a dimensionless filament density, , a dimensionless motor cluster activity, , and a parameter measuring the polarity of motor clusters, g=γ0/γ1 with g=0 corresponding to non-polar clusters. Time is measured in units of .
Motor activity lowers the density for the I–N transition which occurs at . At , the solution acquires nematic order, with , where the unit vector n denotes the direction of broken symmetry. The isotropic state can also become linearly unstable via the growth of polarization fluctuations in any arbitrary direction. This occurs above a second critical filament density, ρIP(μ)=1/(gμ), defined by the change in sign of the coefficient controlling the decay of polarization fluctuations. For , the homogeneous state is polarized (P), with pi≠0. The alignment tensor also has a non-zero mean value in the polarized state as it is slaved to the polarization. One can identify two scenarios depending on the value of g.
For g<1/4, the density ρIP is always larger than ρIN and a region of nematic phase exists for all values of μ. At sufficiently high filament and motor densities, the nematic state also becomes unstable. Polarization fluctuations along the direction of broken symmetry become unstable above a critical density ρNP(g, μ, ρIN). The polarized state at has and . The ‘phase diagram’ is shown in figure 5.
When g>1/4, the boundaries for the I–N and the N–P transitions cross at μx=1/(g−1/4), where ρIN=ρIP=ρNP and the phase diagram has the topology shown in figure 6. For μ>μx, the system goes directly from the I to the P state at ρIP, without an intervening N state. At the onset of the polarized state, the alignment tensor is again slaved to the polarization field.
In vitro experiments have shown that uniform states are often unstable to the formation of complex spatial structures. The instability arises from the growth of spatial fluctuations in the hydrodynamic fields. To understand the different nature of the instability from each homogeneous state, one can obtain coupled equations for the first three moments of the filament concentration defined in equation (6.3) by an expansion in spatial gradients (Liverpool & Marchetti 2003, 2005). With these equations, it is possible to study the dynamics of spatially varying fluctuations in the hydrodynamic fields. These are the fields whose characteristic decay times exceed any microscopic relaxation time and become infinitely long-lived at long wavelengths. The low-frequency hydrodynamic modes of this active system have been shown to be determined by fluctuations in the conserved densities and in the variables associated with broken symmetries (Ahmadi et al. 2005). A change in sign in the decay rate of these modes signals an instability of the macroscopic state of interest. The hydrodynamic modes in the isotropic and nematic phases have been found to be diffusive and go unstable via a diffusive instability, while the polarized phases have been shown to have propagating modes and go unstable via an oscillating instability (Ahmadi et al. 2005). This is shown in figure 6.
7. Estimating the microscopic parameters
In §6, the boundaries of stability of different homogeneous and inhomogeneous states were calculated in terms of the parameters α, β, γ, which characterized the relative velocity of two interacting filaments owing to motor activity. In this section, we indicate how one can derive the motor-mediated velocities between filaments from a microscopic description of the forces exchanged between the motors and the filaments, thus establishing the connection between the hydrodynamic equations and the microscopic motor dynamics (Liverpool & Marchetti 2005).
The filaments are modelled as rigid rods of length l (here l should be thought of as the persistence length, rather than the actual filament length) and diameter b≪1. Each filament is identified by the location ri of its centre of mass and a unit vector pointing towards the polar end. The mobile crosslinks are formed by the small aggregates of molecular motors that exert a force on filaments by converting chemical energy from the hydrolysis of ATP into mechanical work. Each motor cluster is assumed to be composed of two heads, with the ith head (i=1,2) attached to filament i at position , with si the position along the filament relative to the centre of mass, −l/2≤si≤+l/2. The motor cluster has size lm≪1. A schematic is shown in figure 7. Motor heads are assumed to step towards the polar end of filaments at a known speed, u(s), which generally depends on the point of attachment. Spatial variations of u(s) may for instance arise from motors slowing down as they approach the polar end of the filament owing to crowding. This is incorporated here by using the step-like speed profile shown in figure 7, where u(s) is constant along the filament, but vanishes in a small region of extent lm≪l at the polar end. Both filaments and motors move through a solution. We assume that the filament dynamics is overdamped and the friction of motors is very small compared to that of filaments. The momentum conservation then requires that in the absence of external forces and torques, the total force acting on filaments centred at a given position be balanced by the frictional force experienced by the filament while moving through the fluid. For a small rigid cluster, we find γ=0. We obtain similar hydrodynamic equations to those obtained from the phenomenological model, but with a number of terms missing. The terms leading to bundling and separation, though, are easily identified and allow us to give the estimates(7.1)(7.2)The parameter has the dimensions of a diffusion constant and describes filament bunching or bundling, which, in contrast to conventional diffusion, tends to enhance density fluctuations. The coefficient is a velocity and describes the rate at which motor clusters sort or separate filaments of opposite polarity. If the motor stepping speed u(s) is constant, independent of the position s along the filament, then α=0. In general, even when α≠0, we expect α≪β. Therefore, we see that for mean-field models of the type considered here, a varying motor velocity profile that goes to zero at the plus end is required for the formation of inhomogeneous states.
Active filament solutions provide a new direction in the study of driven dissipative systems and a novel experimental and theoretical laboratory for us to develop our understanding of far-from-equilibrium physics. Much more work, both theoretically and experimentally, is needed to better understand the nature of the spatially inhomogeneous states in motor/filament mixtures.
As hinted at in §1, in developing the models described above, plenty of interesting and important physics has been ignored. These should provide interesting avenues for further research in this area. A short and certainly not exhaustive list includes the role of boundaries, the influence of cellular crowding on both motor and filament dynamics (Weiss et al. 2004) and the buckling or breaking of filaments owing to the tension induced by the motors. Progress on many of these questions cannot proceed without close theoretical and experimental collaborations (Mizuno et al. submitted).
The first generation of experiments on these systems has inspired a number of theoretical studies, which have shown a rich spectrum of possible behaviours. This has highlighted the need for a second generation of experiments more systematically, probing the possible behaviours and the transitions between them, i.e. mapping out the non-equilibrium ‘phase diagram’ of these systems. This need has begun to be fulfilled as new experiments begin to appear studying the novel mechanical properties of these systems (Mizuno et al. submitted). Once the physics of this simplified system has been understood by a convergence between experiment and theory, more complexity can be added in a systematic way. Some natural extensions are the following: (i) the inclusion of active polymerization and depolymerization, (ii) the combination of active and passive crosslinks, and (iii) the response of the active solution to deformation.
The most exciting implications of these ideas is the expectation that they will contribute to the development of more detailed and quantitative explanation of cellular functions. Active interactions between filaments and motors play an important role in a number of cellular processes involved in cell division and cell locomotion. The knowledge gained from these in vitro systems will be useful for generating theoretical proposals for the mechanisms of processes in vivo (Zumdieck et al. 2005). Owing to the largely uncharacterized constituents in vivo, this will have to proceed with close collaboration with experiments. Even this is only the tip of the iceberg. Our discussion so far has considered the cytoskeleton in isolation. Clearly, the coupling of the mechanical cytoskeleton to the cell-signalling network provides the scope for many new exciting problems and phenomena.
Some of the work described in this article has been done in collaboration with A. C. Maggs, A. Ajdari, A. Ahmadi and M. C. Marchetti. The support of the Royal Society is gratefully acknowledged. I also thank E. Frey, K. Kruse, F. MacKintosh, D. Morse and S. Ramaswamy for their many helpful discussions.
One contribution of 23 to a Triennial Issue ‘Mathematics and physics’.
- © 2006 The Royal Society