Chapter 22 Force Generation by Cellular Polymers

Chapter 22

Force Generation by Cellular Polymers

George Oster and Alex Mogilner

CONTENTS

I. Introduction 741 II. Polymerization Forces 742 III. Depolymerization Forces 744 IV. Examples of Cellular Polymerization Motors 746 V. Other Mechanisms of Force Generation 747 VI. Discussion 749 References 750

I. INTRODUCTION

Polymers perform many tasks in living cells. We discuss their roles in generating mechanical forces that drive important cellular motions. These can be classified into three general categories:

1. Actin and tubulin polymerization and depolymerization can generate mechanical forces using the free energy of monomer binding and/or nucleotide hydrolysis as their energy source. 2. Polymers can also store elastic energy during their polymerization that can be released later to generate mechanical forces that drive some of the most rapid of cellular motions. 3. Actin and tubulin are tracks for walking motors powered by nucleotide hydrolysis. These motors fall into three classes: myosins, kinesins, and dyneins. The plural applies because, while the force generating principle within each motor type is the same, there is substantial diversity in their dynamical behavior and the cargo they propel. However, the polymer tracks themselves probably play only a passive role as highways for intracellular transport.1

In this chapter we will focus on categories 1 and 2. Polymer motors do not operate in a cyclic fashion like other protein motors that cycle through a number of conformational steps and eventually reset themselves to their initial configuration. Rather they are "one-shot" engines that, after assembly, are disassembled. Nevertheless, these specialized motors play an important role in many cellular processes.

1 There is some evidence that the polymer track may play a more active role in some protein motors [1,2].

741 i 742 SUPRAMOLECULAR POLYMERS

II. POLYMERIZATION FORCES

Polymers can convert the binding free energy of their constituent monomers into mechanical force to push an axial load. Two mechanisms are generally referred to as power strokes and Brownian ratchets [3-5]. In a power stroke, the binding reaction is mechanically coupled to movement and generation of force. For example, if the chemical reaction of a monomer binding to a filament tip triggered a conformational change in the monomer that elongated it, then the conformational change would directly drive an object in front of the polymer tip. In a Brownian ratchet, the role of the monomer binding reaction is to prevent backward fluctuations of the load, rather than to apply a mechanical force directly to it. That is, the load is driven by its own Brownian fluctuations, and the binding energy of the polymer rectifies its diffusive motion [6,7]. To be concrete, suppose an object, say a small sphere with diffusion coefficient D, is aligned ahead of the growing polymer that is anchored at its left end (see Figure 1 (a)). The polymer need not actually "push" the object, but it can "rectify" its Brownian motion. We can view the advancement of the polymerization as moving down on a free energy landscape, AG(x), as shown in Figure l(b) and (c), where the load is moving up the load potential, VL = Fh • x (i.e., the load force, FL = dV^/dx). A ratchet potential energy has a "staircase" profile with step heights much larger than kBT as shown in Figure l(b) (kBT ~ 4.1 pNnm is the "unit" of thermal energy, where kB is Boltzmann's constant, and T the absolute temperature [10]). A power stroke is an inclined path with barriers only a few kBT high, as shown in Figure l(c). Of course, this nomenclature only characterizes the extreme situations; anything in between is possible. The object, D, diffuses uphill against the load force until it reaches the vertical drop that represents the monomer binding step. If the free energy of binding is AGb >> kBT, a backward step is very unlikely, and so work is done against the load force. Thus chemical energy is expended to preferentially select forward steps (or prevent backward steps) and hence to favor forward motion of the load. The difference between ratchets and strokes is only a matter of degree: a power stroke

(a) (b)

Figure 1 Power stroke and ratchet, (a) A helical polymer (e.g., actin) composed of monomers of size 8 polymerizes against an object with diffusion coefficient D. A load force, FL opposes the polymerization, (b) The Brownian ratchet [8]. The polymerization motor can be viewed as a point (black circle) moving on a staircase free energy surface whose step width is 8 and whose height is equal to the free energy of polymerization, that is, the binding free energy of a monomer to the polymer tip,is much larger than thermal energy: AGb » kBT. (c) Power stroke. If, after binding, a monomer develops an internal stress (e.g., via nucleotide hydrolysis) that increases its axial length, then the load will be pushed to the right — corresponding to the inclined free energy segment. A closer inspection of this process would reveal that the power stroke comprises a sequence of thermally activated steps whose magnitude is of the order ~ kBT [9]. FORCE GENERATION BY CELLULAR POLYMERS 743 biases thermal motion by a sequence of small free energy drops while a ratchet rectifies diffusion by a sequence of large free energy changes [5,6,11]. We can make this intuitive picture quantitative as follows. If the polymer assembly is unobstructed, its elongation rate is simply Vp == 8(konM — koff) where 8 [nm] is the size of the monomer (~5.4 nm for actin), M [/xM] the monomer concentration, and fcon(l/[/iMsec]),fcoff [1/sec] are the polymerization and depolymerization rate constants, respectively. In the case of a helical actin polymerization, as shown in Figure 1, the step size is half the monomer diameter ^5.4 = 2.7 nm. First, assume that the polymer is perfectly rigid, and that the diffusion coefficient, D, is large. In order for a monomer to bind to the end of the filament the object must open up a gap of size <5 by diffusing away from the tip, and remaining there for a time ~l/&onM to allow polymerization to take 2 place. In the limiting case when diffusion is much faster than polymerization, that is, konM <$C D/8 , the elongation rate is given by the simple formula vp — 8{konM • p(8,F\J — kos). That is, the polymerization rate is weighted by the probability p(8,F) that the gap size is 8 or larger [12,13]. This probability depends on the load force FL pushing the object to the left; in this simple case p = exp(—Fi^S/k-Ql), where FL<5 is the work required for moving the object by a distance L. So the load-velocity relationship is given by

(1)

The stall load, Fstaii, is reached when the work done in moving the object by a distance 8 is just equal to the free energy of the binding reaction; that is, v = 0 when the load force is Fstaii = (kftT/8) \n(konM/koff), which corresponds to the equilibrium thermodynamic expression. For actin polymerization, M is usually in the micromolar range, the polymerization rate is konM ~ 100/sec, and the depolymerization rate is koff ~ 1/sec, and each monomer added to the polymer tip increases its length by 8 & 2.7 nm. Therefore, without significant load, v ^ 0.1-1 /xm/sec. When stalled, a filament generates a force of ~5-7 pN, similar to that generated by myosin and kinesin [14,15]. These estimates apply in the limit when the object's diffusion is very fast, which is not always the case. However, actin filaments are not rigid, and their thermal bending undulations are very fast (~104/sec). The analysis in this case is similar, and it turns out that the above expression for the stall force of an "elastic ratchet" is still valid [12]. The filament length is an important factor in determining the amplitude of its thermal fluctuations. The effective elastic constant of an actin filament of length L tilted at angle a to an obstacle is k fcs AXk&T/{I? sin2(a)) [12], where A is the persistence length, which is in micron range [16]. This formula indicates that, if the filament is too short (less than ~70 nm), or the angle a is too acute (less than ~30°), the filament is effectively too stiff for the elastic ratchet to work because thermal fluctuations are insufficient to create a gap large enough for monomers to intercalate. On the other hand, if the filament is too long, it becomes too "soft," and it buckles under load forces of less than a picoNewton. For microtubule polymerization, the mathematics is more involved [17-19]. If all 13 microtubule protofilaments are considered as independent force generators, "subsidizing" each other as described in the text, then the theoretically predicted stall force is Fstaii ~ 7 pN (for relevant parameters, see [17,18]). More work is needed on treating the interdependence of the protofilament force generation [19]. Polymerization motors are simple and reliable, and in terms of energy consumption, they are mod- erately efficient. Indeed, the efficiency, rj, can be estimated from the ratio of the work performed, FL • 8, to the monomer binding free energy: AGb = kBT\n(konM/koff) — 4.1[pN nm] • ln(11.6[|uM/sec] • 10|>M]/(l [1/sec]) « 20[pN nm]). Thus r\ = 5[pN] • 2.7[nm]/20[pN nm] *s 0.68. However, there is a large cost associated with controlling when and where polymerization occurs in the cell. This control depends on the enzymatic activity of actin: each monomer of actin binds and hydrolyzes one ATP molecule, whose free energy of hydrolysis is ~80 pN nm. Comparing this to the work 744 SUPRAMOLECULAR POLYMERS

performed in a step gives a "control" efficiency of only ηc ~ 15%. For microtubule, the energy of hydrolysis ~26 pNnm per dimer is used to generate ~7 pN • (8/13) nm of work, so the control

efficiency is again ~15%. When polymerization is fast, ηc drops below 10% far from stall.

III. DEPOLYMERIZATION FORCES

While polymerizing microtubules can generate a pushing force, depolymerizing microtubules can develop pulling forces, although the mechanism is less obvious. For example, depolymerizing microtubules can pull particles at ~l/μm/sec against viscous drag forces of ~10 pN [20]. Also, plastic beads coated with plus-end-directed microtubule motors are carried toward the microtubule minus ends as the microtubule depolymerizes [21]. Depolymerizing microtuble pulling forces may be important in moving chromosomes to the cell pole. Hill proposed the earliest model for depolymerization force generation, Figure 2(a). He assumed that the tip of the depolymerizing microtubule slides through the hole in a sleeve-like docking protein on the kinetochore that allows tubulin dimers to dissociate freely from the microtubule tip [22]. The interior of this sleeve has a high affinity for the microtubule lattice so that when subunits dissociate from the microtubule tip, the binding free energy gradient favors deeper insertion of the microtubule into the sleeve. Thus Brownian motion will drive the docking protein toward the microtubule minus end, producing a pulling force. Movement of the microtubule into the sleeve requires previous interactions to be broken and reformed. This creates a potential energy barrier to the movement of the sleeve that increases the deeper the microtubule penetrates the sleeve and slows further movement of a microtubule. However, subunit loss at the tip will allow the sleeve to follow the tip of the depolymerizing microtubule. According to this model the speed of depolymerization- coupled movement will remain constant over a wide range of load forces because the steady-state force generated by the sleeve adapts to an opposing load by adjusting the average length of the fiber inside the sleeve such that the speed of the load is equal to the depolymerizatio rate [24]. Another mechanism for generating pulling forces is the "conformation wave" model proposed by Mitchison [23] (Figure 2(b)). Depolymerizing microtubule ends consist of two-dimensional sheets that appear frayed and curved [25]. This is thought to arise from GTP binding to tubulin monomers inducing a conformational change that permits polymerization. When GTP is hydrolyzed the monomer tries to relax to its stress free state, but it cannot because it is trapped in the microtubule lattice. Thus the microtubule contains stored elastic energy that can only be released as the microtubule depolymerizes, thus the "banana peel" curvature of the frayed ends [26]. The outward curving protofilaments at the disassembling plus end drive the sliding collar toward the minus end as suggested in Figure 2(b). In this case, the force driving the movement of the sleeve is the release of mechanical strain stored in the lattice during microtubule polymerization. The bending of the protofilament sheets induced upon disassembly is analogous to a power stroke. This model has not been treated quantitatively, but the force it generates can be estimated knowing the strain energy stored in the microtubule lattice from the GTP hydrolysis: AG ~ 26 pN nm/dimer [27]. Dividing by the fiber length increment after one act of unbinding gives 26 pNnm/(8 nm/13) ~ 45 pN. Finally, Peskin et al. modeled experiments [21] in which a bead coated with high affinity tubulin-binding proteins undergoes rotational diffusion along the microtubule polymer lattice (Figure 2(c)) [8,28]. The binding energy gradient prevents the bead from detaching from the plus end of the microtubule. As it rolls, the bead binds to the microtubule via an immobile kinesin con- struct. This weakens the bonds between neighboring tubulin dimers and facilitates depolymerization at the tip. This mechanism is a ratchet: rotational diffusion of the bead is biased by the depolymerizing plus end of the polymer. The ratchet model can be used to estimate the forces generated by the depolymerizing microtubule [8]. Each disassembly event allows the motor to rotate toward the minus end of the FORCE GENERATION BY CELLULAR POLYMERS 745

Figure 2 Depolymerization forces, (a) Hill's model for depolymerization driven transport of a kinetochore along a microtubule [22]. The free energy gradient due to the affinity of the sleeve for the microtubule keeps the microtubule within the sleeve as monomers dissociate from the plus (right) end. (b) The "conformation wave" model of pulling force generation by a depolymerizing microtubule [23]. The elasticity of the protofilaments that curve outward at the disassembling plus end drives a sliding collar on the kinetochore toward the minus end (power stroke), (c) A bead coated with tubulin-binding proteins undergoes rotational diffusion along the microtubule. The binding free energy gradient prevents the bead from detaching from the plus end of the microtubule and, as it rolls, the bead facilitates depolymerization at the plus end. Rotational diffusion of the bead is ratcheted by the depolymerizing plus end [8]. (d) The load-velocity curve passes through a maximum. This is because low load forces increase the residence time of the diffusing object at the microtubule end where it promotes depolymerization. Higher load forces obviate this effect and the curve decreases, w is the work (in units of k^T) to move the load force F\_ by a distance of one subunit S. microtubule with a step size, 8 determined by the thermal energy required to step against the load force, FL :FL • 8 ~ k^T or <5 & k#T/F^. Thus if dimers disassemble from the tip with a rate fcOff, then the velocity of the bead is Vp & koff8 — k^k^T'/FL- Thus, the depolymerization velocity is inversely proportional to the load force. According to this formula the velocity can never reach zero. However, at a load force a few folds greater than k^T/S, depolymerization would slow down significantly. Thus this depolymerization motor can develop force only in the pN range. A peculiarity of the depolymerization motor is that the velocity rises at low loads, passes through a maximum, then decreases monotonically to zero at the stall force, as shown in Figure 2(d) [8]. This apparently paradoxical behavior has a simple explanation. Small pushing loads increase the residence time of the load at the microtubule end where it stimulates monomer dissociation, enhancing motion to the minus end (left in Figure 2(c)). At larger loads, this effect is overridden by the load and the velocity decreases monotonically. 746 SUPRAMOLECULAR POLYMERS

IV. EXAMPLES OF CELLULAR POLYMERIZATION MOTORS

A number of intracellular pathogens propel themselves through their host's cytoplasm by hijacking the cell's actin machinery to its own ends. These organisms are studied as a simplified model system for eukaryotic cell motility. One particular bacterium, Listeria monocytogenes, propels itself by assembling the host cell actin into a comet-like tail of cross-linked filaments, with their polymerizing barbed ends oriented toward the bacterial surface [29]. Listeria moves through the host cytoplasm at velocities of ~0.2 /xm/sec [30]. Only one cell surface protein, Act A, is required for motility [31]. In addition to actin monomers and ActA, only ATP and a handful of cytoplasmic proteins are essential, including the nucleating and branching complex, Arp2/3, capping proteins, and the actin severing and depolymerizing factor, ADF-cofilin (see Figure 3(a)) [32]. The rigid polymerization ratchet theory was originally applied to Listeria propulsion by consid- ering the cell itself as the thermally fluctuating object in front of the filament tips, as in Figure 1 [13].

Figure 3 Models for the molecular propulsion machinery of Listeria [9,33]. (a) Microscopic model [9]. Actin polymerizes at the bacterial surface forming a "comet tail" of short filaments that is anchored to the host cytoskeleton. The posterior surface is coated with ActA that promotes actin polymerization at the surface. ActA in concert with VASP activates the Arp2/3 complex and forms a scaffold for actin assembly. This scaffold adds actin monomers to the growing tips by recruiting profilin-actin complexes. Activated Arp2/3 complex nucleates and branches actin filaments. Actin turnover is regulated by capping proteins that limit filament growth, while ADF-cofilin accelerates actin disassembly further from the bacterial surface. Profilin sequesters actin monomers and restricts polymerization to the region adjacent to the surface. a-Actinin cross-links actin filaments into a denser gel than the surrounding cellular cytoskeleton. The "mother" filaments are tethered transiently to the surface and resist the forward movement of the bacterium. The tethered filaments are in tension and eventually detach to become "working" filaments in compression whose thermal undulations exert the propulsive pressure on the bacterial wall [9]. (b) Continuum model. The actin gel polymerizes at the bacterial surface generating a circumferential stress. This sqeezing force produces an axial thrust that propels the bacterium forward [33]. (c) Vesicle propulsion by actin polymerization [34,35]. An actin tail growing from a lipid vesicle deforms the membrane. Measuring the difference in membrane curvature along the surface gives an estimate of the forces acting on the surface. The propulsive forces from the working filaments exceed the retraction forces from the attached filaments, so the vesicle is propelled forward. FORCE GENERATION BY CELLULAR POLYMERS 747

This model predicted that the bacterial velocity should depend on its diffusion coefficient, and thereby on its size. However, experiments showed that the velocity did not depend on the cell size, so the model was modified to allow thermal fluctuations of the actin filament tips [36]. This resolved the size independence issue but the model ran afoul of another observation: the actin tail appeared to be attached to the surface of the cell [30,37,38]. This problem was resolved by a further gener- alization of the model. The "tethered ratchet" model assumed that the filaments are initiated while attached to the bacterial surface, but subsequently detach and become "working" filaments as in the elastic ratchet model (Figure 3(a), [19]). The attached fibers are in tension and resist the forward progress of the bacterium. At the same time, the dissociated fibers are in compression, and generate the force of propulsion, each filament developing a force of a few pN. The effect of load forces on the velocity of Listeria can be measured by increasing the viscosity of the medium using methylcellulose. These experiments show that loads between 10 [39] and 100 pN [40] are required to slow the bacteria. The discrepancy between these measurements probably arises from differences in the biochemistry and experimental conditions that may alter the number of working filaments. Both are in agreement with the tethered ratchet theory, depending on the number of working/tethered filaments. Finally, there are suggestions that the actin tail generates not only axial propulsive forces, but torques as well (Robbins et al. observed that Listeria rotates about its long axis as it moves through the cytoplasm [41]). This torque could be generated by actin binding proteins that attach laterally and trap torsional fluctuations of the tethered filaments, analogous to the action of scruin in the acrosomal process of Limulus (see below) [42,43]. Other intracellular pathogens, such as Shigella, the spotted fever bacterium Rickettsia, and the Vaccinia virus all utilize the cellular actin assembly machinery for propulsion [44]. Although the molecular details vary, the physical mechanism is the same. For example, Shigella uses the surface protein IcsA instead of ActA to stimulate actin polymerization. Vaccinia utilizes tyrosihe phosphorylation of a unique protein A36R (absent in Listeria and Shigella). Rickettsia does not employ Arp2/3 complex as an actin nucleating/branching center, and thus its tail consists of long actin filaments arranged in a parallel array, rather than short filaments cross-linked at acute angles as in Listeria, Shigella, or Vaccinia. Plastic beads and lipid vesicles coated with either ActA, or WASP proteins, grow actin tails and move in the same way as the pathogens, and also deform the vesicles Figure 3(c) [31,34,35]. Measuring the curvatures of the vesicle gives an estimate of the balance of propulsive and retraction forces exerted on the surface by the working and tethered filaments, respectively. Experiments such as these promise to become useful tools in uncovering the secrets of cell motility.

V. OTHER MECHANISMS OF FORCE GENERATION

Although the ratchet model explains most of the experimental observations, there are alternative proposals for polymerization force generation. Generally, they fall into two categories: hypothetical protein motors and macroscopic phenomenological models. An example of the former is the molecular ratchet motor proposed by Laurent et al. that posits that frequent attachment and detachment of VASP on the cell surface to F-actin allows it to slide along a growing filament, driven by the free energy of monomer addition [45]. Another example of a hypothetical force generator is an affinity- modulated, clamped-filament elongation mechanism that exploits the intrinsic ATPase activity of actin monomers [46]. Kuo and McGrath observed that Listeria appeared to advance in discrete steps of 5.5 nm, similar to the size of an actin monomer [38]. These steps could suggest some intrinsic molecular scale mechanism at the interface between filaments and the surface. Finally, myosin is involved in some way during protrusion of filopodial-like actin bundles at the lamellipodial leading edge that are organized by inhibition of capping and subsequent cross-linking by fascin [47,48]. 748 SUPRAMOLECULAR POLYMERS

Gerbal et al. constructed a continuum model of Listeria propulsion based on the elastic shear stress developed by growth of the actin mesh work at the cell surface [33,49,50]. In this model, the polymerization of actin develops circumferential stresses in the actin meshwork of the tail surrounding the posterior portion of the cell. This developing stress "squeezes" the cell until a yield stress is reached whereupon the cell "squirts" forward, relieving the stress, and the cycle repeats (see Figure 3(b)). This stress-relaxation cycle produces step-like propulsion (with micronsized steps) similar to that observed in the movement of ActA-coated plastic beads [51]. The model predicts that the propulsive force should depend on the surface curvature; this can be tested experimentally. Being macroscopic, this model complements the microscopic elastic ratchet model; indeed, the latter provides a rationale for the polymerization induced stresses that develop in the continuum gel model. Lipid vesicles coated with ActA also grow actin tails and move. The vesicles deform, and the stress distribution exerted by the actin can be computed from their shape [35]. These experiments confirm the existence of large (~nN) "squeezing" stresses on the vesicle, but "squirting" movement cycles were not observed. These experiments also indicate that a spatial separation between tethered and working filaments develops. Tethered filaments are swept to the very rear of the vesicle, while working filaments concentrate at the sides. A simple explanation for this separation phenomenon is that ActA attached to an immobile tethered filament drifts to the rear along the lipid surface as the vesicle is propelled forward by working filaments that keep up with the vesicle's sides. Several mechanisms other than polymerization are probably involved in pushing out the cell's leading edge. In most crawling cells, the force of protrusion is generated locally [52]. Localized protrusive forces can be generated in actin gels because they are highly charged [53]. Because of the counterions to the actin fixed charges, the filaments of a cross-linked polyelectrolyte gel, such as the actin cytoskeletal network, are always in a state of elastic tension. At equilibrium, the elastic tension in the gel filaments is just balanced by the ion osmotic pressure. This is discussed in more detail in [54,55]. In transiently motile cells, the actin gel adjacent to the leading edge membrane may partially solate, for example, by the action of severing proteins, such as gelsolin triggered by calcium influx. This weakens elasticity of the gel so that the local osmotic pressure expands the gel boundary to a new equilibrium. Subsequently, the gel solidifies again stabilizing the protrusion. Some indirect evidence in favor of this scenario is the observations that raising external osmolarity inhibits protrusion, and that prior to protrusion, the lamellipodial leading edge of some cells swells and becomes softer [56]. A very simple and specialized cell, the nematode sperm of Ascaris suum, provides an important example of pushing out the front by a specialized form of gel swelling [57,58]. Nematode sperm lack the actin machinery associated with eukaryotic cell motility; instead, their movement is powered by a cytoskeleton built from major sperm protein (MSP) filaments [59]. This is a positively charged and partially hydrophobic protein that associates into symmetrical dimers that polymerize into helical filaments. Unlike actin, MSP polymerization and bundling does not require a broad spectrum of accessory proteins. The same hydrophobic and electrostatic interaction interfaces allow these filaments wind together in pairs to form larger bundles, and eventually congregate into higher order rope-like arrays [60]. MSP filaments are more flexible than actin, and so the polymerization ratchet mechanism may not be as effective in generating a protrusive force in nematode sperm. However, this assembly process forces the filaments within a higher order aggregate to assume an end- to-end distance that is larger than its persistence length in solution. In this fashion, bundles of MSP filaments are stiffer than, and contain the stored elastic energy of, their constituent filaments. These bundles of MSP form a thixotropic (i.e., shear thining) gel-like cytoskeleton within the lamellipod. This gel is a fibrous material, so that when filaments bundle laterally they generate a protrusive force longitudinally [58]). Finally, osmotic gel swelling appears to propel certain bacterial locomotion,, a phenomenon called "gliding motility" [61,62]. Cyanobacteria and myxobacteria glide on surfaces by hydration and extrusion of a gel from nozzle-like organelles [61,63,64]. FORCE GENERATION BY CELLULAR POLYMERS 749

Figure 4 Energy storage in the acrosomes of Limulus acrosome (redrawn from Mahadevan, L., J. Shin, G. Waller, K. Langsetmo, and P. Matsudaira. J. CellBiol. 2003,162:1183-1188. With permission), (a) The filaments are twisted in the coiled state but straight in the discharge state, (b) During the acrosomal reaction, the actin filaments untwist and unbend going from the coil to the discharge state. The actin bundle uncoils and projects rapidly from the top of the cell.

The smallest free-living organisms are the Mollicutes [65]. Some glide on surfaces, others swim, but all appear to generate the forces for propulsion by cyclically altering elastic properties of cytoplasmic filaments [66-69]. The largest and fastest reversible entropic motor is employed by spasmoneme in Vorticellid ciliates [70]. In this motor, a giant polymer chain is held in a distended configuration by the repulsion of its fixed charges. A rise in cytosolic calcium drastically reduces its rigidity by shielding the polymer-associated charges, triggering an entropic contractile force strong enough to retract at ~8 cm/sec [71]. Another dramatically fast "one-shot" polymerization engine is employed when the sperm of the sea cucumber Thy one encounters the egg jelly coat. An explosive actin polymerization reaction ensures pushing out the acrosomal process and enabling the sperm plasma membrane to penetrate the egg and fuse with the plasma membrane of the egg [72]. In this process, fast and tran- sient actin polymerization is limited by actin delivery to the tip of the process, rather than by force. Water influx coupled with actin polymerization may contribute to force generation by a hydrostatic mechanism without treadmilling and/or nucleotide hydrolysis [73]. A particularly startling polymer motor is the acrosomal process of the Limulus sperm whose acrosomal process consists of a bundle of 30 to 50 actin filaments cross-linked by the protein scruin (Figure 4(a)). During polymerization of the acrosomal process, elastic energy is stored in the filament bundle by using scruin binding to trap torsional thermal fluctuations as elastic strain energy. Later, this strain energy is released to generate the force required to push the actin rod into the egg cortex [43,71]. Shin et al. have modeled this process and found that the protrusion is characterized by a stationary untwisting front that converts the coiled bundle into a straight bundle. Interestingly, while the torsional strain is distributed continuously along the filament bundle, the bending energy is concentrated in a series of "kinks." These kinks rotate sequentially so that the untwisting rotary motion is converted into axial extension, as shown in Figure 4(b) [43,74]. A mathematical model quantifying the mechanism leads to an equation for the balance of torques and an expression for the constant velocity of the front.

VI. DISCUSSION

Force generation by polymerization of single filaments is fairly well understood. However, our knowledge is much less complete concerning how forces are generated during assembly and 750 SUPRAMOLECULAR POLYMERS disassembly of cytoskeletal fibers and networks. Beyond the issue of force generation is the question of how these forces organize themselves, using complementary and antagonistic actions of microtubules and actin, coupled with protein motors, during cell locomotion and division. Moreover, force generation is intimately tied in with the processes of regulation [75], signal transduction [76], adhesion [77], and many other aspects of cell dynamics. Mechanochemical process generally involve many kinds of proteins, and are too complicated to understand without a mathematical model to expose the assumptions and to frame the qualitative picture in quantitative terms. The field needs new models to stimulate new experiments.

REFERENCES

1. Inoue, Y., A. Hikikoshi Iwane, T. Miyai, E. Muto, and T. Yanagida, Motility of single one-headed kinesin molecules along microtubules. Biophys. /., 2001, 81:2838-2850. 2. Ishii, Y. and T. Yanagida, A new view concerning an actomyosin motor. Cell. Mol. Life Sci. (CMLS), 2002, 59:1767-1770. 3. Oster, G., Darwin's motors. Nature, 2002, 417:25. 4. Bustamante, C, D. Keller, and G. Oster, The physics of molecular motors. Ace. Chem. Res., 2001, 34:412-420. 5. Wang, H. and G. Oster, Ratchets, power strokes, and molecular motors. Appl. Phys. A, 2001, 75:315-323. . 6. Oster, G. and H. Wang, Rotary protein motors. Trends Cell Bioi, 2003,13:114-121. 7. Mogilner, A., T. Elston, H.-Y. Wang, and G. Oster, Molecular motors: theory and examples, in Computational Cell Biology, C.R Fall, E. Marland, J. Tyson, and J. Wagner, Eds. 2002, Springer: New York. pp. 321-380. 8. Peskin, C.S. and G.F. Oster, Force production by depolymerizing microtubules: load-velocity curves and run-pause statistics. Biophys. J., 1995, 69:2268-2276. 9. Mogilner, A. and G. Oster, Force generation by actin polymerization II: the elastic ratchet and tethered filaments. Biophys. J., 2003, 84:1591-1605. 10. Howard, J., Mechanics of Motor Proteins and the Cytoskeleton. 2001, Sunderland, MA: Sinauer. 11. Oster, G. and H. Wang, Reverse engineering a protein: the mechanochemistry of ATP synthase. Biochem. Biophys. Acta, 2000, 1458:482-510. 12. Mogilner, A. and G. Oster, The physics of lamellipodial protrusion. Euro. Biophys. J., 1996, 25:47-53. 13. Peskin, C.S., G.M. Odell, and G. Oster, Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. /., 1993, 65:316-324. 14. Visscher, K., MJ. Schnitzer, and S.M. Block, Single kinesin molecules studied with a molecular force clamp. Nature, 1999, 400:184-189. 15. Ruegg, C, C. Veigel, J.E. Molloy, S. Schmitz, J.C. Sparrow, and R.H. Fink, Molecular motors: force and movement generated by single myosin II molecules. News Physiol. Sci., 2002,17:213-218. 16. Janmey, P.A., S. Hvidt, J. Kas, D. Lerche, A. Maggs, E. Sackmann, M. Schliwa, and T.P. Stossel, The mechanical properties of actin gels. Elastic modulus and filament motions. /. Biol. Chem., 1994, 269:32503-32513. 17. Mogilner, A. and G. Oster, The polymerization ratchet model explains the force-velocity relation for growing microtubules. Eur. J. Biophys., 1999, 28:235-242. 18. van Doom, G.S., C. Tanase, B.M. Mulder, and M. Dogterom, On the stall force for growing microtubules. Eur. Biophys. J., 2000, 29:2-6. 19. Stukalin, E. and A. Kolomeisky, Simple growth models of rigid multifilament biopolymers. J. Chem. Phys. 121:1097-1104. 20. Coue, M., V.A. Lombillo, and J.R. Mclntosh, Microtubule depolymerization promotes particle and chromosome movement in vitro. J. Cell Biol, 1991,112:1165-1175. 21. Lombillo, V.A., R.J. Stewart, and J.R. Mclntosh, Minus-end-directed motion of kinesin-coated microspheres driven by microtubule depolymerization. Nature, 1995, 373:161-164. 22. Hill, T.L., Theoretical problems related to the attachment of microtubules to kinetochores. Proc. NatlAcad. Sci. USA, 1985, 82:4404-4408. FORCE GENERATION BY CELLULAR POLYMERS 751

23. Mitchison, T.J., Microtubule dynamics and kinetochore function in mitosis. Annu. Rev. Cell Biol., 1988, 4:527-549. 24. Joglekar, A.P. and A.J. Hunt, A simple, mechanistic model for directional instability during mitotic chromosome movements. Biophys. J., 2002, 83:42-58. 25. Arnal, I., E. Karsenti, and A.A. Hyman, Structural transitions at microtubule ends correlate with their dynamic properties in Xenopus egg extracts. J. Cell Biol, 2000,149:767-774. 26. Hyman, A.A., D. Chretien, I. Arnal, and R.H. Wade, Structural changes accompanying GTP hydrolysis in microtubules: information from a slowly hydrolyzable analogue guanylyl-(alpha,beta)-methylene- diphosphonate. / Cell Biol, 1995,128:117-125. 27. Inoue, S. and E.D. Salmon, Force generation by microtubule assembly/disassembly in mitosis and related movements. Mol. Biol. Cell, 1995, 6:1619-1640. 28. Tao, Y.C. and C.S. Peskin, Simulating the role of microtubules in depolymerization-driven transport: a Monte Carlo approach. Biophys. J., 1998, 75:1529-1540. 29. Tilney, L.G. and D.A. Portnoy, Actin filaments and the growth, movement, and spread of the intracellular bacterial parasite, Listeria monocy to genes. J. Cell Biol, 1989,109:1597-1608. 30. Cameron, L. A., T.M. Svitkina, D. Vignjevic, J.A. Theriot, and G.G. Borisy, Dendritic organization of actin comet tails. Curr. Biol, 2001,11:130-135. 31. Cameron, L.A., M.J. Footer, A. van Oudenaarden, and J.A. Theriot, Motility of ActA protein-coated microspheres driven by actin polymerization. Proc. Natl. Acad. Sci. USA, 1999, 96:4908-4913. 32. Loisel, T.P., R. Boujemaa, D. Pantaloni, and M.F. Carlier, Reconstitution of actin-based motility of Listeria and Shigella using pure proteins. Nature, 1999, 401:613-616. 33. Gerbal, E, P. Chaikin, Y. Rabin, and J. Prost, On the theory of the Listeria propulsion. Biophys. J., 2000, 79:2259-2275. 34. Upadhyaya, A., J. Chabot, A. Andreeva, A. Samadani, and A.V. Oudenaarden, Probing polymerization forces by using actin-propelled lipid vesicles. Proc. NatlAcad. Sci. USA, 2003,100:4521-4526. 35. Giardini, P.A., D.A. Fletcher, and J.A. Theriot, Compression forces generated by actin comet tails on lipid vesicles. Proc. NatlAcad. Sci. USA, 2003,100:6493-6498. 36. Mogilner, A. and G. Oster, Cell motility driven by actin polymerization. Biophys. J., 1996, 71:3030-3045. 37. Noireaux, V., R.M. Golsteyn, E. Friederich, J. Prost, C. Antony, D. Louvard, and C. Sykes, Growing an actin gel on spherical surfaces. Biophys. J., 2000, 78:1643-1654. 38. Kuo, S.C. and J.L. McGrath, Steps and fluctuations of Listeria monocytogenes during actin-based motility. Nature, 2000,407:1026-1029. 39. McGrath, J.L., N.J. Eungdamrong, C.I. Fisher, F. Peng, L. Mahadevan, T.J. Mitchison, and S.C. Kuo, The force-velocity relationship for the actin-based motility of Listeria monocytogenes. Curr. Biol, 2003, 13:329-332. 40. Wiesner, S., E. Heifer, D. Didry, G. Ducouret, F. Lafuma, M.F. Carlier, and D. Pantaloni, A biomimetic motility assay provides insight into the mechanism of actin-based motility. J. Cell Biol, 2003,160:387-398. 41. Robbins, JR. and J.A. Theriot, Listeria monocytogenes rotates around its long axis during actin-based motility. Curr. Biol, 2003,13:R754-R756. 42. Mahadevan, L. and P. Matsudaira, Motility powered by supramolecular springs and ratchets. Science, 2000, 288:95-100. 43. Mahadevan, L., J. Shin, G. Waller, K. Langsetmo, and P. Matsudaira, Stored elastic energy powers the 60um extension of the Limulus polyphemus sperm actin bundle. J. Cell Biol, 2003, 162:1183-1188. 44. Goldberg, M.B., Actin-based motility of intracellular microbial pathogens. Microbiol. Mol. Biol. Rev., 2001,65:595-626. 45. Laurent, V., T.P. Loisel, B. Harbeck, A. Wehman, L. Grobe, B.M. Jockusch, J. Wehland, F.B. Gertler, and M.F. Carlier, Role of proteins of the Ena/VASP family in actin-based motility of Listeria monocytogenes. J. Cell Biol, 1999,144:1245-1258. 46. Dickinson, R.B. and D.L. Purich, Clamped-filament elongation model for actin-based motors. Biophys. J., 2002, 82:605-617. 47. Vignjevic, D., D. Yarar, M.D. Welch, J. Peloquin, T. Svitkina, and G.G. Borisy, Formation of filopodia-like bundles in vitro from a dendritic network. J. Cell Biol, 2003,160:951-962. 48. Sheetz, M.P., D.B. Wayne, and A.L. Pearlman, Extension of filopodia by motordependent actin assembly. CellMotil Cytoskeleton, 1992, 22:160-169. 752 SUPRAMOLECULAR POLYMERS

49. Gerbal, R, P. Chaikin, Y. Rabin, and J. Prost, An elastic analysis of Listeria monocytogen.es propulsion. Biophys. J., 2000, 79:2259-2275. 50. Gerbal, R, V. Laurent, A. Ott, M.-R Carlier, P. Chaikin, and J. Prost, Elastic forces propel Listeria monocytogenes. Euro. Biophys. J., 2000, 29:134-140. 51. Bernheim-Groswasser, A., S. Wiesner, R.M. Golsteyn, M.R Carlier, and C. Sykes, The dynamics of actin-based motility depend on surface parameters. Nature, 2002, 417:308-311. 52. Grebecki, A., Membrane and cytoskeleton flow in motile cells with emphasis on the contribution of free-living amoebae. Int. Rev. Cytol, 1994,148:37-80. 53. Oster, G.R, On the crawling of cells. J. Embryol. Exp. Morphol, 1984, 83 (Suppl.):329-364. 54. Mogilner, A. and G. Oster, Shrinking gels pull cells. Science, 2003, 302:1340-1341. 55. Mogilner, A. and G. Oster, Polymer motors: pushing out the front and pulling out the back. Curr. Biol, 2003,13:R721-R733. 56. Rotsch, C, K. Jacobson, J. Condeelis, and M. Radmacher, EGF-stimulated lamellipod extension in adenocarcinoma cells. Ultramicroscopy, 2001, 86:97-106. 57. Wolgemuth, C, A. Mogilner, and G. Oster, The hydration dynamics of polyelectrolyte gels with applications to cell motility and drug delivery. Eur. Biophys. J., 2003, 33:146-158. 58. Bottino, D., A. Mogilner, T. Roberts, M. Stewart, and G. Oster, How nematode sperm crawl. / Cell Sci., 2002,115:367-384. 59. Italiano, J.E., Jr., M. Stewart, and T.M. Roberts, How the assembly dynamics of the nematode major sperm protein generate amoeboid cell motility. Int. Rev. Cytol., 2001, 202:1-34. 60. Roberts, T.M., E.D. Salmon, and M. Stewart, Hydrostatic pressure shows that lamellipodial motility in Ascaris sperm requires membrane-associated major sperm protein filament nucleation and elongation. J. CellBiol., 1998, 140:367-375. 61. McBride, M., Bacterial gliding motility: mechanisms and mysteries. Am. Soc. Microbiol. News, 2000, 66:203210. 62. Hoiczyk, E., Gliding motility in cyanobacteria: observations and possible explanations. Arch. Micro., 2000, 174:11-17. 63. Hoiczyk, E. and W. Baumeister, The junctional pore complex, a prokaryotic secretion organelle, is the molecular motor underlying gliding motility in cyanobacteria. Curr. Biol., 1998, 8:1161-1168. 64. Wolgemuth, C, E. Hoiczyk, D. Kaiser, and G. Oster, How myxobacteria glide. Curr. Biol., 2002,12:369- 377. 65. Trachtenberg, S., Mollicutes-Wall-less bacteria with internal cytoskeletons. /. Struct. Biol., 1998,124:244- 256. 66. Wolgemuth, C, O. Igoshin, and G. Oster, The motility of Mollicutes. Biophys. J., 2003, 85:828-842. 67. Trachtenberg, S., R. Gilad, and N. Geffen, The bacterial linear motor of Spiroplasma melliferum BC3: from single molecules to swimming cells. Mol. Microbiol., 2003, 47:671-697. 68. Trachtenberg, S. and R. Gilad, A bacterial linear motor: cellular and molecular organization of the contractile cytoskeleton of the helical bacterium Spiroplasma melliferum BC3. Mol. Microbiol., 2001, 41:827-848. 69. Miyata, M. and A. Uenoyama, Movement on the cell surface of the gliding bacterium, mycoplasma mobile, is limited to its head-like structure. FEMS Microbiol. Lett, 2002, 215:285-289. 70. Moriyama, Y., H. Okamoto, and H. Asai, Rubber-like elasticity and volume changes in the isolated spasmoneme of giant Zoothamnium sp. under Ca2+-induced contraction. Biophys. J., 1999, 76:993-1000. 71. Mahadevan, L. and P. Matsudaira, Motility powered by supramolecular springs and ratchets. Science, 2000, 288:95-99. 72. Tilney, L.G. and S. Inoue, Acrosomal reaction of Thy one sperm. II. The kinetics and possible mechanism of acrosomal process elongation. J. Cell Biol, 1982, 93:820-827. 73. Oster, G., A. Perelson, and L. Tilney, A mechanical model for acrosomal extension in Thyone. J. Math. Biol, 1982,15:259-265. 74. Winder, S.J., Structural insights into actin-binding, branching and bundling proteins. Curr. Opin. CellBiol, 2003,15:14-22. 75. Pollard, T.D. and G.G. Borisy, Cellular motility driven by assembly and disassembly of actin filaments. Cell, 2003,112:453-465. 76. Bourne, H. and O. Weiner, A chemical compass. Nature, 2002, 419:21. 77. Webb, D.J., J.T. Parsons, and A.F. Horwitz, Adhesion assembly, disassembly and turnover in migrating cells— over and over and over again. Nat. Cell Biol, 2002, 4:E97-E100.