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 even- tually 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, respec- tively. 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 micro- tubule 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 param- eters, 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.