INTRODUCTION

TOP ABSTRACT INTRODUCTION DESCRIPTION OF THE MODEL ANALYSIS OF THE MODEL RESULTS BIOLOGICAL IMPLICATIONS OF THE... APPENDIX REFERENCES

Cell motility

Recent advances in cell biology have uncovered molecular mechanisms that control cytoskeletal dynamics underlying cell motion. The significance of such research is clear because the migration of eukaryotic cells plays a fundamental role in morphogenesis, wound healing, immune surveillance, and carcinogenesis (Bray, 1992). The crawling motion of a cell (such as a keratocyte) relies on the extension of its leading edge, the lamellipod, and requires growth of the cytoskeleton; in particular, of the actin network that is its main structural component (Tilney et al., 1991). The fact that motility is based on dynamic changes in the cytoskeleton has been known for well over a decade, but the idea that actin polymerization can, by itself, generate the force of protrusion that pushes the cell front forward (Tilney et al., 1991) was only recently confirmed quantitatively (Peskin et al., 1993; Mogilner and Oster, 1996a; Gerbal et al., 2000). This paper explores key details underlying actin-based lamellipodial protrusion using mathematical modeling. Our main goal is to understand how details of actin polymerization, nucleation, disassembly, and regulation work together in a spatially distributed way to generate and regulate protrusion of the cell front.

Cell motility is a complex, dynamic process in which cytoskeletal assembly, adhesion to extracellular matrix, and contractile forces interact in a spatially heterogeneous, complex geometry. This level of complexity has led some investigators to argue that exclusion of any one of these effects would seriously weaken the validity of a model. Nevertheless, as our main focus is on protrusion, our approach is based on the premise that it is worth investigating and understanding the biochemistry of cytoskeletal assembly as a prelude to more complex and more complete model investigations of cell motion as a whole.

To justify this approach, we temporarily put aside a longer-term goal of understanding the motility of cells such as Dictyostelium, fibroblasts, and leukocytes that undergo dramatic shape changes, transient and erratic locomotion, and complex, heterogeneous dynamic adhesion (Munevar et al., 2001; Beningo et al., 2001). Actin growth at the leading edge does not generally match the rate of migration: these cells have a "slippery clutch" (Theriot and Mitchison, 1992; Cameron et al., 2000). Such examples are, at present, beyond the scope of theoretical modeling as outlined in this paper and we do not attempt to model their motion in terms of actin dynamics alone. For reasons explained further (under "Choice of model system"), our main concern is with keratocyte motion. We first briefly review the relevant biological details required as a background for the model (see also Figs. 1 and 2).

View larger version (20K): [in this window] [in a new window]   FIGURE 1   Bottom: A schematic diagram of a migrating fish keratocyte cell as seen from above (bottom left) and from the side (bottom right) showing typical shape and dimensions. Top: The rectangular portion of the lamellipod indicated in the bottom left view is here magnified, and forms region of interest for the model: its dimensions are length L = 10 µm, width W ~ 1-5 µm, and thickness H ~ 0.1-0.2 µm. Actin filaments are represented schematically by a few diagonal arrows. (The filaments are growing away from their pointed ends and toward the membrane at the top). The edge-density of leading barbed ends, B, is the number of barbed ends at the top surface of the box divided by w.

View larger version (25K): [in this window] [in a new window]   FIGURE 2   The sequence of events associated with lamellipodial protrusion based on the dendritic-nucleation model. (1, 2) Extracellular signals stimulate receptors that activate WASp/Scars; (3) WASp/Scars activate the Arp2/3 complex; (4) Arp2/3 nucleates a new actin filament barbed end by branching from some preexisting filament; (5) barbed ends of the filaments get capped in the cytoplasm; (6) PIP2 inhibits capping at the leading edge; (7) ADF/cofilin accelerates depolymerization of actin filaments at their older, ADP-actin portions; (8) ADF/cofilin, profilin, and thymosin 4 form complexes with G-actin, thymosin sequesters the monomers, while profilin catalyzes exchange of ADP for ATP on the actin monomers; (9) profilin-ATP-G-actin intercalates into the gap between a filament and the membrane, and assembles onto the barbed ends of actin filaments. This pushes the membrane forward.

The lamellipod

The basic engine of motion causing forward protrusion of the cell edge is the lamellipod (Pollard et al., 2000; Abraham et al., 1999; Small et al., 1995; Svitkina et al., 1997; Svitkina and Borisy, 1999). This structure is a broad, flat, sheet-like structure, tens of microns in width, and 0.1-0.2 µm thick (Abraham et al., 1999); see bottom panels in Fig. 1. Lamellipodial actin filaments form a highly ramified, cross-linked, polarized network; fibers subtend a roughly 55° angle with the front edge of the cell in a nearly square-lattice structure (Maly and Borisy, 2001).

Actin

Actin, the major component of the lamellipodial cytoskeleton, exists in monomeric (G-actin) and rod-like polymerized filament (F-actin) forms. The actin network is regulated by a host of actin sequestering, capping, severing, nucleating, and depolymerizing proteins (Pantaloni et al., 2001; Ressad et al., 1999; Chen et al., 2000; Southwick, 2000; Machesky, 1997; Machesky and Insall, 1999; Pollard et al., 2000). There are tens to hundreds of proteins involved in actin turnover in motile cells. However, only a small number of those are essential for protrusion. The discovery of this fact (Loisel et al., 1999; Pantaloni et al., 2001) is of fundamental importance for understanding the lamellipodial dynamics. Furthermore, this makes the system amenable for modeling. Actin filaments are not homogeneous along their length. At the newly assembled end, ATP nucleotides are attached to actin; these undergo progressive hydrolysis and subsequent dissociation of the -phosphate over time. Nucleotide hydrolysis has been identified as the main factor determining filament half-life. Factors that accelerate filament disassembly include ADF/cofilin and gelsolin, both displaying higher affinity to the older ADP-actin sites along a filament.

Barbed ends

The ends of an actin filament have distinct polymerization kinetics, with fast-growing barbed (plus) ends directed toward the cell membrane and shrinking pointed (minus) ends directed toward the cell interior. There is evidence that most of the uncapped barbed ends are concentrated close to the cell edge, where they rapidly assemble ATP-G-actin (actin monomers with ATP attached). The leading barbed ends (terminology used in this paper for filament ends pushing the membrane) provide the force for protrusion. In our model we will be primarily concerned with the relationship between the number of leading barbed ends per unit length of membrane and the protrusion velocity of the cell. This velocity depends, among other things, on monomer availability to the growing barbed ends, a factor that must be carefully considered in understanding the mechanism. We will also be concerned with regulation of this edge-density of barbed ends by nucleation and capping.

Capping controls growth of the actin network

If polymerization were unregulated at the front edge, the pool of actin monomers would be depleted in seconds by barbed end growth. Capping of these barbed ends on the time scale of 4 s¹ (Pollard et al., 2000) is likely one of the main (though still not fully understood) regulatory factors (Carlier and Pantaloni, 1997). In the cytosol, uncapping is extremely slow and can be neglected on this time scale. At the leading edge, however, phosphoinositides such as PIP₂ remove barbed end caps, creating a local environment where capping is effectively reduced (Hartwig et al., 1995; Schafer et al., 1996). Barbed ends of nascent filaments close to the edge may further be protected from capping by a Cdc42-dependent mechanism (Huang et al., 1999).

Nucleation controls growth of the actin network

New barbed ends are nucleated along preexisting filaments as branches by a molecular complex, Arp2/3, known to be abundant (Kelleher et al., 1995) and essential (Schwob and Martin, 1992) for cell motility (Ma et al., 1998; Pollard et al., 2000). Under optimal conditions, each activated Arp2/3 complex initiates a new actin filament branch point (Higgs et al., 1999) at an ~70° angle (Mullins et al., 1998); the Arp2/3 becomes integrated into the structure. It is still to be clarified whether the Arp2/3 complex binds at the side (Amann and Pollard, 2001) or at the barbed end of an actin filament (Pantaloni et al., 2000), or possibly both.

An interesting scenario of spatial and temporal self-organization in the lamellipod, called the dendritic-nucleation model, has been proposed (Mullins et al., 1998; Pollard et al., 2000); see Fig. 2. On a time scale of seconds (Gerisch, 1982), external signals such as chemoattractants or growth factors activate cell-surface receptors that signal a family of WASp/Scar proteins; these interact transiently with, and activate Arp2/3 complexes that can nucleate actin branching. (Blanchoin et al. 2000a; Machesky et al., 1999; Higgs and Pollard, 1999). (Indirect evidence suggests that the level of activated WASp/Scar is low relative to Arp2/3, so that activation by WASp/Scar is likely to be a limiting factor (Pollard et al., 2000).)

Signaling pathways are under current intense study (Carlier et al., 1999a, 2000; Egile et al., 1999), but it is as yet unclear whether activation occurs at the plasma membrane or in the cortical region, and exactly where branching dominates. Barbed ends have been observed mainly within 0.1 or 0.2 µm from the cell membrane, while Arp2/3 complexes appear to be more widely distributed (from the membrane up to 1.0-1.5 µm into the cell) (Bailly et al., 1999). Similarly, nucleation sites were observed in a strip <1 µm wide at the extreme leading edge (Svitkina and Borisy, 1999). Such observations suggest that activation, nucleation, and branching all occur in a narrow "activation zone" (a few hundreds of nanometers wide) at the leading edge of the lamellipod (Wear et al., 2000).

Actin monomer flux

During steady state, constant extension due to the net rate of polymerization at barbed ends of filaments has to be balanced by the net rate of depolymerization at the opposite (pointed) ends. Single filaments have been noted to undergo "treadmilling" in vitro; i.e., apparent translocation by addition of monomers at the barbed end and loss at the pointed end. However, G-actin concentration would have to be two orders of magnitude greater than the in vitro treadmilling concentrations to account for the observed rates of extension (tenths of a micron per second (Pollard et al., 2000)), given the experimentally determined rate constants for actin polymerization (Pollard, 1986). Furthermore, the (slow) rate of depolymerization of the minus ends would have to increase to allow the minus ends of the filaments to keep up with the margin (Coluccio and Tilney, 1983; Wang, 1985). The actual flux of actin monomers across the lamellipod depends on rates of filament disassembly, on diffusion of these monomers in a variety of complexes, and on agents that sequester these monomers in an unusable form. Such effects are incorporated into our model.

Filament disassembly

Although filament disassembly does not appear to contribute directly to protrusion, it plays an important role in the recycling of monomers from rear portions of the lamellipod to the front. ATP nucleotides attached to actin undergo hydrolysis and eventual dissociation of the -phosphate, a process that regulates eventual disassembly of a filament. Conversion from ATP-actin to ADP-actin takes 10-30 s in rapidly migrating cells (Pollard et al., 2000). ADF/cofilin and other fragmenting proteins attach rapidly to ADP-F-actin, catalyzing dissociation of subunits from filament minus ends, or cutting filaments at ADP-actin regions (Korn et al., 1987; Pollard et al., 2000; McGrath et al., 2000). Cutting creates new minus ends and accelerates depolymerization further (Southwick, 2000).

There are several views about depolymerization: one is that each actin subunit dissociates independently from its polymer (Hill and Kirschner, 1982). The opposing vectorial hydrolysis model is that hydrolysis occurs only at the interface between ADP-P_i- and ATP-actin subunits (Carlier et al., 1986) on the filament. Recent studies provide yet a new view (Blanchoin et al., 2000b), namely that some reaction decreases the affinity of Arp2/3 to a pointed end at a branch point. Arp2/3 dissociation would then free a pointed filament end for fast "unraveling," with a cascade of rapid debranching and disassembly. This process is called fiber-by-fiber renewal of the whole population of filaments (Carlier et al., 1999b). This view contrasts with previously held ideas that individual fibers continually grow at their plus ends and shrink at their minus ends.

Monomer sequestering and recycling

The concentration of unpolymerized actin in a lamellipod is estimated to be lower than 100 µM (Pollard et al., 2000), but up to an order of magnitude greater than needed to account for observed rates of protrusion. A large pool of actin monomers is sequestered by actin-binding proteins in a form unavailable for polymerization; for example, in complexes with thymosin 4 and ADF/cofilin. Thymosin 4 is involved in a rapid exchange with profilin, a small protein, which also competes with ADF/cofilin for ADP-actin monomers (see Fig. 3). Profilin facilitates ADP-ATP exchange on the monomers, shifting the equilibria to ATP-G-actin-profilin complexes, which associate to barbed ends exclusively. Thus, profilin serves as a carrier between the sequestered actin monomeric pool (unavailable for polymerization) and the barbed ends of the filaments (Pantaloni and Carlier, 1993).

View larger version (12K): [in this window] [in a new window]   FIGURE 3   Exchange reactions for G-actin complexes of various forms: cofilin-ADP-actin (CAD), profilin-ADP-actin (PAD), profilin-ATP-actin (PAT), thymosin 4-ATP-actin (TAT), and filamentous actin (F-actin). The italic letters adjacent to the boxes are symbols used in the model for the concentrations of these intermediates; values of the reaction rate constants shown beside the arrows are given in Table 1.

It is not currently feasible to experimentally measure the G-actin concentration profile (let alone the relative abundance of G-actin in various complexed forms) across the lamellipod, so there is no direct information about monomer availability at the front edge, where barbed ends are growing. However, from indirect information, including biochemical parameters governing association and dissociation of monomers with ADF/cofilin, thymosin, and profilin, and careful estimates of diffusion, and depolymerization, we can arrive at an estimate of the desired monomer profiles. This analysis forms an important contribution of this paper.

Self-organization in the lamellipod

The synergistic action of the Arp2/3 complex, capping protein, ADF/cofilin, and profilin creates a metastable state: the growing plus ends of actin filaments localize at the extreme leading edge, the disassembling minus ends dominate away from the edge, and sequestering proteins shuttle monomers from the back to the front by simple diffusion. This mechanism replenishes monomers at the front where they are used up. Barbed-end capping produces an excess of free pointed ends. This keeps the supply of monomers plentiful and accelerates growth of any barbed ends that are temporarily uncapped, an effect termed funneling (Dufort and Lumsden, 1996; Carlier and Pantaloni, 1997). A summary of the pertinent phenomena is given in Fig. 7.

Choice of model cell

Some particularly simple systems involving motion based purely on actin polymerization exist biologically. One of these is the intracellular parasite, Listeria. Here, the connection between actin polymerization and speed of motion has been established (Marchand et al., 1995), and preliminary models have been developed for the force of propulsion (Mogilner and Oster, 1996a; Gerbal et al., 2000; Laurent et al., 1999). However, the motility of Listeria is not ideally suited for aspects on which we focus in this paper (though investigating Listeria for its own unique features would be of interest in a future treatment of this type). In Listeria, the role of the actin sequestering cycle is still too poorly understood for modeling to be effective. A major issue is that Listeria swims in a complex, biochemically heterogeneous 3D environment of the host cell. (Neither the geometry of in vitro chambers nor the biochemical milieu of cell extracts used in such chambers make the situation clearcut or simple.) This complicates a geometric treatment of the motion, but more importantly, it dissociates the tight, near-1D spatial coupling between the monomeric state of actin and the assembly of actin, a coupling that we will argue is fundamental to protrusion.

The case of keratocytes appears to be more tractable for the specific purposes we have in mind in this paper: 1) the shape of the cell is almost constant as it moves; 2) the motion is smooth and uniform. Hence the approximation of steady-state motion is quite good; 3) the geometry close to the front edge of its thin lamellipod can be approximated as one-dimensional: biochemical gradients are mainly directed along an axis pointing into the cell. (The "axis" of a Listeria cell is not similarly reducible to 1D, due to diffusion of monomers in its 3D environment); 4) most of the lamellipodial actin network is stationary relative to the substratum, with negligible retrograde flow (Cameron et al., 2000; Theriot and Mitchison, 1991); 5) in the steady-state mode of keratocyte movement, there is a high degree of coordination among protrusion, adhesion, and retraction, compared with other cells. For a given period of observation, spatial and temporal changes in adhesion or contractility are small so that the net effect of these other forces on the process of protrusion is nearly constant (Theriot and Mitchison, 1991; Mitchison and Cramer, 1996). This means that such confounding effects can be factored out of the model.

The above factors make it reasonable to speculate that locomotion of a keratocyte represents protrusion/treadmilling in its purest form, determined predominantly by the dynamics of actin network assembly, but see Lee and Jacobson (1997) and Oliver et al. (1999) for other opinions. It is further reasonable to conclude that the rate of migration of these cells is closely matched with the growth of actin filaments at the front edge. Indeed, as will be shown, predictions of our model for this rate of migration, based on underlying biochemistry, agree with experimentally measured cell velocity.

Goals of this paper

Several fundamental questions arise about actin-based protrusive motion: how is protrusion regulated? How many uncapped barbed ends should be kept available to grow at any given time in the cell? With too few barbed ends, it would be impossible to generate a force sufficient to drive protrusion and motion of the cell. Conversely, if there are too many growing ends, their competition for monomers would quickly deplete the pool, and this would retard growth. The goal of this paper is to understand the quantitative details of this observation within the context of the biochemical and biological parameters whose values are known. To do so, we will find it essential to address some related questions, including how monomers are distributed across the lamellipod. Specifically, we would like to estimate the optimal number of uncapped barbed ends for rapid protrusion. Another goal is to obtain a theoretical estimate of the rate of protrusion of cells under conditions of rapid steady-state motion. An important part of these goals is a comparison of theoretical estimates with experimental observations. Our model relies on the regulation of actin dynamics and treadmilling by a small host of essential proteins (Loisel et al., 1999; Pantaloni et al., 2001).

The model introduced in the next section is an initial attempt to elucidate general principles of spatial and temporal regulation of actin pools in the cell and determine how actin dynamics optimal for protrusion can be achieved. We will describe the dynamics of actin and of its essential associated proteins by a system of reaction-diffusion-advection equations. A sketch of the analysis of our model is provided in the following section (with further details in the Appendix). Complemented with the force-velocity relation for actin filaments, these equations will reveal the way that the protrusive rate of motion depends on key biochemical parameters and on membrane tension (Results). Biological implications of the model will be discussed in the last section.

	DESCRIPTION OF THE MODEL

TOP ABSTRACT INTRODUCTION DESCRIPTION OF THE MODEL ANALYSIS OF THE MODEL RESULTS BIOLOGICAL IMPLICATIONS OF THE... APPENDIX REFERENCES

The meanings and values for rates and parameters in the model are given in Table 1. A list of the main variables and their definitions is given below.

t	time (s);
x	distance from the leading edge (µm);
B(t)	edge density of the uncapped leading barbed ends (µm1);
s(x, t)	density of ADP-G-actin sequestered by ADF/cofilin (µM);
p(x, t)	density of ADP-G-actin-profilin complexes (µM);
a(x, t)	density of ATP-G-actin-profilin complexes (µM);
(x, t)	density of ATP-G-actin-thymosin 4 complexes (µM);
f(x, t)	length density of F-actin (µm/µm2);
m(x, t)	density of uncapped minus ends (µm2);
mc(x, t)	density of capped minus ends (µm2);
V	protrusion velocity (µm s1).

Geometry of the model

We neglect curvature of the lamellipodial leading edge and any variation in the actin density in a direction parallel to the cell front or across the thickness of the sheet. In the case of the broad fan-like, thin lamellipod of keratocytes, this approximation is an excellent one. This idealization allows us to consider a 1D model, and greatly increases mathematical tractability. We assume very strong adhesion, and, as a consequence, no slippage at the cell front. This assumption is supported by observations of the rapidly moving keratocytes (Theriot and Mitchison, 1991).

We consider a thin strip of lamellipod perpendicular to the cell edge (see rectangular box in Fig. 1). We use a coordinate system moving with the front edge of the cell: our x axis will coincide with the length of the strip with x = 0 at the leading edge, and x representing the distance into the cell. The length of the lamellipod, L, is a model parameter. We define a barbed end "edge density," B(t), as the number of uncapped leading barbed ends at x = 0 per unit edge-length (see Fig. 7). These leading barbed ends do the work of pushing the membrane. Concentrations of actin and actin-associated proteins are in micromoles. All densities and concentrations vary with time t and position x.

As a simplification for modeling purposes, we assume a perfect angular order in the actin network, with each filament oriented at ±35° relative to the direction of motion of the cell. (A 70° angle between branching filaments and a flat edge implies that each filament subtends an acute angle of ±55° at the leading edge. The observed angular distribution of the lamellipodial F-actin supports our approximation (Maly and Borisy, 2001).) When a monomer of size 5.4 nm adds onto the tip of a filament oriented in this way, the tip advances by roughly  = (5.4/2) · cos(35°) 2.2 nm along our chosen x axis. (The factor (1/2) stems from the fact that polymerized actin forms a double helix.)

The differential equations of the model are derived below. Figs. 1, 3, and 7 help capture the geometry, notation, and basic assumptions of the model. Because our coordinate system is moving with the edge of the cell (whose velocity in the absence of slippage is the protrusion rate V), most equations contain terms of the form Vdc/dx, where c is some concentration or density. This is simply a transformation to the moving coordinate system, which keeps the leading edge at the origin.

Barbed ends

The density of the barbed ends is governed by the following equation:

(1)

The first term in the right-hand side of this equation represents a rate of initiation of new barbed ends (branching) by Arp2/3 with the rate n [s¹ µm¹] at the leading edge. Actin branching takes place within the "activation zone." The barbed ends nucleated within this zone grow rapidly toward the cell membrane, where the growth is stalled significantly by membrane resistance. This creates a steep gradient of barbed end density from a high level at the leading edge down to zero at the rear of the activation zone. We assume that the width of this zone is small (much smaller than all other spatial scales inherent to the process). This allows us to treat the uncapped barbed end density as an essentially 1D "edge density," defined as the number of ends per unit length of the leading edge, rather than the number per unit area in the lamellipod.

The last term in Eq. 1 represents loss of barbed ends due to capping at rate . The rates of nucleation and capping in Eq. 1 are assumed to be constant model parameters. This simplification is based on the assumption that the level of activated Arp2/3 is a limiting factor, and that branching sites on preexisting filaments are in abundant supply. Furthermore, we also assume that activation by WASP (rather than the level of Arp2/3) is the rate-limiting step that determines the level of activated Arp2/3 complexes available for branching. (We thus justify the simplification in which full dynamics of Arp2/3 can be omitted.) These assumptions and the constant effective capping rate are discussed in the last section.

Monomer recycling and exchange

We assume that almost all of the G-actin in the lamellipod occurs in complexes with one of three essential sequestering proteins, ADF/cofilin, profilin, or thymosin 4. Residual free ATP-G-actin certainly occurs, as in its absence, ATP-G-actin-profilin concentration would decay to zero. Actin-based movement of pathogenic bacteria is possible without profilin, through assembly of ATP-G-actin (Loisel et al., 1999). However, we demonstrate in the Appendix that in the presence of large amounts of profilin the steady-state concentration of ATP-G-actin is very small, and ATP-G-actin-profilin is the main species polymerizing at the barbed ends. (This theoretical conclusion has been established experimentally in Pantaloni and Carlier, 1993.) We show that at observed high concentrations of profilin and thymosin (Pollard et al., 2000), ATP-G-actin concentration adjusts rapidly to the level determined by the slowly varying concentrations of G-actin in sequestered forms. Mathematically, this means that on time scales characteristic to the processes described by the model, ATP-G-actin concentration can be expressed as a function of the sequestered actin concentrations.

The following equations account for actin monomers in the forms ADP-G-actin-ADF/cofilin (s), ADP-G-actin-profilin (p), ATP-G-actin-profilin (a), and ATP-G-actin-thymosin 4 () complexes (see Figs. 3 and 7):

(2)

(3)

(4)

(5)

Above, we have captured the dynamics of ADP-G-actin sequestered by cofilin and profilin (Eqs. 2 and 3, respectively), and ATP-G-actin sequestered by thymosin 4 and profilin (Eqs. 4 and 5). Terms of the form Vdc/dx in the equations stem from the moving coordinate system and second derivative terms represent simple molecular diffusion. Sequestering agents are small molecules, and hence their complexes with actin share roughly similar diffusion coefficients, denoted by D: see the Appendix.

In Eq. 2 the term J_d(x) depicts the distribution of sources of ADP-G-actin-cofilin from depolymerization of filaments. All other terms in these equations represent rates of exchange of actin between its sequestering agents and between the ADP-G-actin and ATP-G-actin forms as shown in Fig. 3. For example, the terms ±k₂p describe ADP-ATP exchange on profilin-actin complexes. In the Appendix we provide estimates for the associated reaction rate constants.

We assume that the variables s, p, and , satisfy no-flux boundary conditions at the leading edge, x = 0, and at the base of the lamellipod, x = L. However, because ATP-actin-profilin complexes are used up at the leading edge in the course of the polymerization of the filaments abutting the cell membrane, the appropriate boundary condition at x = 0 for a(x, t) is a given flux of these complexes:

(6)

Here (Da/x(0) + Va(0)) is the sum of diffusive and convective fluxes of ATP-actin-profilin complexes at the leading edge. This flux is directed out of the cytoplasm, in the direction of motion, and therefore carries a negative sign (J_p < 0). The magnitude of this flux, J_p, is given by the rate of monomer addition per filament, V/ ([s¹]), multiplied by the density of leading uncapped barbed ends that are growing and using up monomers at the membrane, B. The factor 1/ converts the flux into suitable dimensions: [µm¹ s¹] into [µM · µm s¹]. (  100 µM¹ µm² is a constant, converting concentrations from µM to number of monomers per µm² in the fixed-thickness lamellipod; see Appendix.) We assume that the variable a satisfies no-flux boundary conditions at the base of the lamellipod, x = L.

Depolymerization

The G-actin distribution depends on a function to be specified, namely the source J_d(x) of ADP-G-actin-ADF/cofilin disassembling from F-actin. We can only speculate about the form of J_d(x) because, as discussed in the Introduction, details of depolymerization are not yet well understood biologically. We will consider a specific source function in the framework of the array treadmilling model (Svitkina and Borisy, 1999).

Debranching (or "pruning") of actin filaments may occur by spontaneous dissociation or by ADF/cofilin-induced dissociation of Arp2/3 from a Y-junction in the actin network. Each such event creates an uncapped minus end. Here, the filament begins to unravel, producing a source of ADP-G-actin-ADF/cofilin until disassembly is complete. The dynamics of the minus ends (density m(x, t)), and those still capped by Arp2/3 (density m_c(x, t)), can be described by the following equations:

(7)

(8)

Spatial derivative terms arise from the moving coordinate system. Terms proportional to m_c describe uncapping of the minus ends. We assume that uncapping is a slow Poisson process characterized by rate 1/t₁ ~ 30 s. A term proportional to m accounts for elimination of uncapped minus ends due to complete disassembly of a filament. The average filament lifetime, t₂, depends on the average filament length, l, and on the rate of depolymerization of the filaments, V_dep. The former can be estimated as the ratio of filament growth rate to filament capping rate: l  V/. (Using the observed V ~ 0.5 µm s¹ and  ~ 4 s¹, we obtain l = V/ ~ 0.1 µm.) The effective rate of depolymerization of ADF/cofilin-F-actin from the minus end is unknown, but a convincing theoretical argument (Carlier et al., 1999b) suggests that its order of magnitude is V_dep ~ 0.1 µm s¹. Combining these estimates, we arrive at the approximation t₂ = l/V_dep ~ 1 s.

Equations 7 and 8 must be supplemented with appropriate boundary conditions. We use the fact that Arp2/3-capped minus ends are nucleated at the front simultaneously with barbed ends, i.e., at the same rate, n. Thus, the density of the Arp2/3-capped minus ends at the leading edge, m_c(0), is equal to the nucleation rate divided by the speed of the lamellipodial front. Assuming all minus ends at the leading edge are capped then leads to the boundary conditions:

(9)

The G-actin source J_d(x, t) is proportional to the density of uncapped minus ends:

(10)

Factor 1/ converts the dimension of the source term into [µM s¹].

Clearly, this model of depolymerization is grossly simplified. There may be many other mechanisms acting to liberate actin monomers. In the Appendix, we discuss the feasibility of the alternative tread-severing model (Dufort and Lumsden, 1996). Also, at the rear of the lamellipod, myosin-generated contraction (Svitkina et al., 1997) physically breaks actin fibers, creating many minus ends and massive depolymerization. Recent information points to the involvement of tropomyosin and other actin-associated proteins in regulating F-actin stabilization. Furthermore, there is a pronounced difference between the highly branched actin meshwork seen at the front and the smoother network further into the lamellipod (Blanchoin et al., 2001). This may indicate that the rates of debranching and/or severing vary significantly from one region in the lamellipod to another.

Protrusion velocity

If not for the membrane resistance and depolymerization, the barbed ends at the leading edge would grow with the free polymerization velocity:

(11)

Here, k_on [s¹ µM¹] is the rate constant for monomer assembly, a(0) [µM] is the concentration of ATP-G-actin-profilin complexes at the front, and k_ona(0) is the local rate of assembly of monomers per unit time at the barbed end of a filament. Multiplied by , the length increment due to the addition of a single monomer, this rate becomes the free polymerization velocity.

The cell membrane associated with the actin cortex imposes a resistance to the propulsive motion. Because of this resistance, the velocity of protrusion, V, is smaller than the free polymerization velocity, V₀. We require a relationship between the resistance force per unit length of the leading edge, F [pN/µm], and the protrusion velocity V = V(V₀, F), to close the system of equations forming our model. However, as no measurements are currently available for this force-velocity relation in actin-based lamellipodial protrusions, here we must rely on theoretical arguments for the desired formula. Peskin et al. (1993) and Mogilner and Oster (1996a) derived expressions for the force-velocity relation for a single actin filament growing against a given load force, f. In the limiting case when bending undulations of the filaments and the cell membrane are much faster than polymerization kinetics, and when the average amplitude of such undulations is greater than the size of an actin monomer, the relationship has the form:
where k_on and k_off are the polymerization and depolymerization rate constants, respectively, T is absolute temperature, and k_B is Boltzmann's constant.

Although this is a limiting case, it adequately describes most biological situations based on physiological values of the parameters associated with filament and membrane mechanics and actin concentrations in the cell (Mogilner and Oster, 1996a; Mogilner and Oster, 1999). For fast-moving cells, the rate of depolymerization of actin from barbed ends, k_off, is negligible, so that the force-velocity relation is well approximated by
That is, the free polymerization rate, V₀, is weighted by a Boltzmann factor, where the exponent, f/k_BT, is the work (in units of thermal energy, k_BT  4.1 pN · nm) done against the load by the assembly of one monomer. Note that near stall, when resistance is high, viscous dissipation can be neglected, the work of polymerization is almost reversible, and the above force-velocity relation follows from general thermodynamical arguments that do not depend on a detailed microscopic model (Hill, 1987).

To now apply this relation to a population of filaments, B(t) pushing against the membrane load, we take the simplest and most easily justifiable assumption; namely, that the load is equally divided among the filaments, each bearing a share f = F/B (Mogilner et al., 2001; van Doorn et al., 2000). The resulting form of the load-velocity relation for the lamellipodial front is:

(12)

We will assume this relationship henceforth.

In Eq. 12, values of the constants , k_B, and T are known, but we require estimates for the resistance force, F. Two factors contribute to this force. The first is membrane surface tension with bending modulus determined by the splay of the outer membrane leaflet and compression of the inner leaflet (Evans and Skalak, 1980). The second is binding energy dissipation when the links between the actin cortex and membrane are broken as the membrane is pushed forward. The value of the total resistance force can be estimated to be F ~ 50-500 pN/µm (Dai et al., 1998; Dai and Sheetz, 1999; Raucher and Sheetz, 1999; Erickson, 1980; Petersen et al., 1982). In this paper we will use the value F = 100 pN/µm for the estimates. The velocity dependence of the resistance force is very weak (Hochmuth et al., 1996).

Gerbal et al. (2000) developed a different, mesoscopic model relying on the elastic shear stress generation due to the growth of the actin gel. They demonstrated that the rate of growth of the actin meshwork is decreased (in comparison with the velocity given by formula (12) due to elastic recoil under load by a factor on the order of (1 + (F/YH)), where Y is the Young modulus of the lamellipodial cytoskeleton and H is the thickness of the lamellipod. In the physiological range of the resistance load, this effect does not introduce a significant correction to Eq. 12 because the lamellipodial network is very stiff: Y ~ 10⁴ Pa (Rotsch et al., 1998), and F/YH < 0.2.

	ANALYSIS OF THE MODEL

TOP ABSTRACT INTRODUCTION DESCRIPTION OF THE MODEL ANALYSIS OF THE MODEL RESULTS BIOLOGICAL IMPLICATIONS OF THE... APPENDIX REFERENCES

Spatial distribution of uncapped barbed ends

The stationary solution for the density of uncapped barbed ends at the leading edge can be found from Eq. 1:

(13)

The distribution of actin monomers

A first important observation concerns the relative magnitudes of the diffusion and drift terms in Eqs. 2-5. In the Appendix we demonstrate that diffusion of the G-actin complexes is much faster than drift on a spatial scale relevant for the lamellipod. This justifies neglecting the drift terms in Eqs. 2-5 for our purposes.

These approximations lead us to the following simplified system for the stationary distribution of sequestered monomers:

(14)

(15)

(16)

(17)

We now comment on the behavior of the solution to this system, below (see Fig. 4 for a plot of the results).

View larger version (15K): [in this window] [in a new window]   FIGURE 4   Stationary concentrations of sequestered G-actin complexes are plotted (using formulae (22, 33, 34) for values of the model parameters listed in Table 1) as functions of distance from the front edge of the cell. The concentrations of ADP-G-actin bound to profilin (PAD) and ADF/cofilin (CAD), 0.35-5 µM, respectively, are constant over the lamellipod. Concentrations of ATP-G-actin complexes with profilin (PAT) or thymosin (TAT) decrease from the rear to the front of the lamellipod.

The depolymerization source

(18)

(19)

View larger version (24K): [in this window] [in a new window]   FIGURE 5   (A) Distribution of the source of cofilin-ADP-actin as given by treadmilling model (dashed line) and corresponding distribution of the source of profilin-ADP-actin obtained numerically (dot-dashed line). The solid line is an approximate constant source of profilin-ADP-actin. (B) Concentrations of cofilin- and profilin-ADP-actin. The dashed curves represent the numerically computed concentrations. The solid lines are approximate constant concentrations.

(20)

ADP-G-actin

(21)

1

1

(22)

ATP-G-actin

(23)

(24)

3

3

1

Protrusion velocity and leading barbed ends

Equation 23 linking actin monomer availability at the leading edge, a(0), to polymerization flux, J_p, leads to our key result, the dependence of the rate of protrusion on biochemical parameters and resistance to motion. Indeed, recalling that the free polymerization velocity is V₀ = k_ona(0), and that the protrusion velocity is V = V₀exp(w/B) by the force-velocity relation, we get
Using expression (6) for the polymerization flux
substituting the expressions for a(0) and J_p into (23) and solving the resulting linear algebraic equation for V leads to the form of the protrusion velocity:

(25)

where

(26)

The algebraic equation (25) expressing the protrusion velocity, V, as a function of the edge-density of leading barbed ends, B, is the main output of the model. We will be interested in what this equations implies about cell motion. A detailed interpretation of its form, and of the parameters in (26), is given in the following section. Subsequently, we draw conclusions about the way that the intricate machinery of the cell regulates rapid locomotion.

	RESULTS

TOP ABSTRACT INTRODUCTION DESCRIPTION OF THE MODEL ANALYSIS OF THE MODEL RESULTS BIOLOGICAL IMPLICATIONS OF THE... APPENDIX REFERENCES

G-actin distribution

The model predicts stationary spatial distribution of sequestered G-actin complexes shown in Fig. 4 (see also Fig. 7). The concentrations of ADP-G-actin are constant and small over the lamellipod. Concentrations of ATP-G-actin complexes with profilin and thymosin are similar. They decrease from the rear to the front of the lamellipod over a range of 20-12 µM. This concentration gradi