CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector 3030 Biophysical Journal Volume 71 December 1996 3030-3045 Cell Motility Driven by Actin Polymerization Alexander Mogilner* and George Oster# *Department of Mathematics, University of California, Davis, California 95616, and #Department of Molecular and Cellular Biology, University of California, Berkeley, California 94720-3112 USA ABSTRACT Certain kinds of cellular movements are apparently driven by actin polymerization. Examples include the lamellipodia of spreading and migrating embryonic cells, and the bacterium Listeria monocytogenes, that propels itself through its host's cytoplasm by constructing behind it a polymerized tail of cross-linked actin filaments. Peskin et al. (1993) formulated a model to explain how a polymerizing filament could rectify the Brownian motion of an object so as to produce unidirectional force (Peskin, C., G. Odell, and G. Oster. 1993. Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys. J. 65:316-324). Their "Brownian ratchet" model assumed that the filament was stiff and that thermal fluctuations affected only the "load," i.e., the object being pushed. However, under many conditions of biological interest, the thermal fluctuations of the load are insufficient to produce the observed motions. Here we shall show that the thermal motions of the polymerizing filaments can produce a directed force. This "elastic Brownian ratchet" can explain quantitatively the propulsion of Listeria and the protrusive mechanics of lamellipodia. The model also explains how the polymerization process nucleates the orthogonal structure of the actin network in lamellipodia. INTRODUCTION Many cell movements appear to be driven by the polymer- in front of it. This process produced an axial force by ization of actin. The most conspicuous example is the la- employing the free energy of polymerization to render uni- mellipodia of crawling cells. Certain gram-negative patho- directional the otherwise random thermal fluctuations of the genic bacteria, such as Listeria, Shigella, and Rickettsia, load. Their model assumed that the polymer was infinitely move intracellularly by polymerizing a "comet tail" of stiff, and so the Brownian motion of the load alone created cross-linked actin filaments that propel them through their a gap sufficient for monomers to intercalate between the tip host's cytoplasm (Sanger et al., 1992; Southwick and and the load. Consequently, this model predicts that velocity Purich, 1994; Marchand et al., 1995). will depend on the size of the load through its diffusion Several lines of evidence suggest that these motions may coefficient. However, recent experiments have cast doubt be a physical consequence of polymerization itself (Stossel, on this mechanism of propulsion: 1) Listeria and Shigella 1993, 1994; Condeelis J, 1992); (Condeelis, 1993; Cramer move at the same speed despite their very different sizes et al., 1994). For example, actin polymerization can drive (Goldberg and Theriot, 1995). 2) The actin network at the polycationic beads placed on the dorsal surface of lamelli- leading edge of lamellipodia is organized into an approxi- podia (Forscher et al., 1992). Moreover, the sperm cells of mately orthogonal network (Small et al., 1995). An approx- the nematode Ascaris crawl via a lamellipodium that ap- imately orthogonal network is also observed in platelet pears identical to that of mammalian cells; however, the cytoskeleton (Hartwig, 1992). This is unexplained by the polymer driving this motile appendage is "major sperm Brownian ratchet model, which treats only collinear fila- protein" (MSP), a protein unrelated to actin (Roberts and ment growth. Stewart, 1995). Vesicles derived from sperm membrane will To remove these limitations, we have generalized the also grow a tail of polymerized MSP and move in a Listeria- Brownian ratchet model to include the elasticity of the like fashion. This suggests that the propulsive force gener- polymer and to relax the collinear structure of growing tips. ated by polymerizing actin filaments has more to do with The principal result of this paper will be an expression for the physics of polymerization than to any property peculiar the effective polymerization velocity of a growing filament to actin. as a function of the load it is working against and its angle Recently, Peskin et al. (1993) formulated a theory for to the load. We use this expression to describe the propul- how a growing polymer could exert an axial force. They sion of Listeria and the protrusion of lamellipodia, and showed that by adding monomers to its growing tip, a discuss the agreement of our estimates with experimental polymer could rectify the free diffusive motions of an object measurements, as well as predictions of the model. Received for publication 28 May 1996 and in final form 17 September THE FORCE EXERTED BY A SINGLE 1996. POLYMERIZING FILAMENT Address reprint requests to Dr. George Oster, Dept. of ESPN, University of California, 201 Wellman Hall, Berkeley, CA 94720-3112. Tel.: 510-642- In this section we describe the physical model for the 5277; Fax: 510-642-5277; E-mail: [email protected]. thermal fluctuations of a free end of an actin filament. We C) 1996 by the Biophysical Society explicitly take into account only the entropic forces of the 0006-3495/96/12/3030/16 $2.00 filaments, and ignore the cytoplasmic fluid flow (Grebecki, Mogilner and Oster Cell Motility Driven by Actin Polymerization 3031 1994). We model a polymerizing actin filament as an elastic TABLE 2 Other notation rod whose length grows by addition of monomers at the tip Symbol Meaning at a rate konM [s- 1] and shortens by losing subunits at a rate B modulus of actin filament = where is the Bending A kBT (pN-nm2) k0ff [s-1], klc, [s-1jiM-1] polymerization rate Db Diffusion coefficient of bacterium (,um2/s) and M[,uM] the local molar concentration of monomers near Df Effective diffusion coefficient of filament (Am2/s) the growing tip. The values of all the parameters we use are f Load force (pN) gathered in Tables 1 and 2. fA Stall force (pN) An actin filament can be characterized by its persistence = w/2s = fl/O8 dimensionless load force kBT Unit of thermal energy = 4.1 X 10-'4 dyne-cm = length, A[,um] (Janmey et al., 1994). The theoretical elastic 4.1 pN-nm bending modulus, B, of a filament is related to its persis- N Number of filaments tence length by B = AkBT (Doi and Edwards, 1986). How- P(o, f) Probability of 8-sized gap as a function of the load, f, ever, the experimental persistence length, Aobs, is generally and angle, 0 determined by fitting the observed shape of filaments with i(O, Yo) Probability of 8-sized gap as a function of the angle, 0, and the equilibrium position of the filament tip, a Fourier series, and so depends on the actual length of the Yo observed filaments: B = AobS(f)kBT = O(l)AkBT. For our q = V/(8kn*M) dimensionless polymerization velocity purposes here we shall neglect this distinction. The data on s Ratio of depolymerization and polymerization rates the numerical value of A varies between 0.5 p.m (Kas et al., t Time (s) V of filament 1993, 1996; Gotter et al., 1996) to 15 p.m (Isambert et al., Velocity tip (,Am/s) V* = maximum polymerization velocity (,um/s) 1995) depending on the experimental conditions. We feel V(0,) Vr = 2D/8 ideal ratchet velocity (,um/s) that the lower measurements are more realistic for filaments Vp Free polymerization velocity (,um/s) under cellular conditions, and so we shall use the value A x Position of filament tip (nm) 1 p.m. We focus our attention on the actin filaments that Yo Equilibrium distance of filament tip measured from constitute the "free ends" at the growing surface of a cross- the membrane (nm) linked A = &cos(O) = size of sufficient gap to permit actin gel. To render the model tractable, we shall intercalation of monomer make the following simplifying assumptions: 1) The ther- = K082/2kBT = dimensionless bending energy mal fluctuations of the filaments are planar. 2) All filaments K Elastic constant of an actin filament (pN/nm) impinge on the load at the same angle, 0, and they poly- CO = f 8/kBT dimensionless work to move the load ahead merize with the same angle-dependent rate, V. 3) The free by one monomer ends of each filament are the same length, i?. That is, the growing region is of constant width, behind which the filaments become cross-linked into a gel. 4) We consider sure. However, to add a monomer to the tip of a free only one fluctuation mode, neglecting collective modes of filament end a thermal fluctuation must create a gap suffi- the whole actin network; i.e., we treat the body of the cient to permit intercalation. For a filament approaching the network as a rigid anchor. The assumed spatio-angular load perpendicularly, a gap half the size of an actin mono- structure of the actin network is shown in Fig. 1. mer is necessary to enable a monomer to intercalate be- As the filaments polymerize, their Brownian motions tween the tip and the membrane (the actin filament is a impinge on the load (e.g., the bacterial wall, or the cyto- double helix, so a gap of only - 2.7 nm is required). For plasmic surface of the plasma membrane) exerting a pres- a filament approaching at an angle 0 to the load, the required TABLE I Parameter values Notation Meaning Value Source e Length of free filament end 30-150 nm (Marchand et al., 1995; Small et al., 1995; Tilney et al., 1992a; Tilney et al., 1992b) Stall