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How nematode sperm crawl

Dean Bottino^1,*, Alexander Mogilner², Tom Roberts³, Murray Stewart⁴ and George Oster^1,

¹ Department of Molecular and Cellular Biology, University of California, Berkeley, CA 94720-3112, USA
² Department of Mathematics, University of California, Davis, CA 95616, USA
³ Department of Biological Science, Florida State University, Tallahassee, FL 32306-3050, USA
⁴ MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England
^* Present address: Physiome Sciences, 150 College Road West, Princeton, NJ 08540-6608

Author for correspondence (e-mail: goster{at}nature.berkeley.edu)

Accepted October 12, 2001

	SUMMARY

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
Sperm of the nematode, Ascaris suum, crawl using lamellipodial protrusion, adhesion and retraction, a process analogous to the amoeboid motility of other eukaryotic cells. However, rather than employing an actin cytoskeleton to generate locomotion, nematode sperm use the major sperm protein (MSP). Moreover, nematode sperm lack detectable molecular motors or the battery of actin-binding proteins that characterize actin-based motility. The Ascaris system provides a simple ‘stripped down’ version of a crawling cell in which to examine the basic mechanism of cell locomotion independently of other cellular functions that involve the cytoskeleton. Here we present a mechanochemical analysis of crawling in Ascaris sperm. We construct a finite element model wherein (a) localized filament polymerization and bundling generate the force for lamellipodial extension and (b) energy stored in the gel formed from the filament bundles at the leading edge is subsequently used to produce the contraction that pulls the rear of the cell forward. The model reproduces the major features of crawling sperm and provides a framework in which amoeboid cell motility can be analyzed. Although the model refers primarily to the locomotion of nematode sperm, it has important implications for the mechanics of actin-based cell motility.

Movies available on-line.

Key words: Cell motility, Major sperm protein, Nematode sperm cell, Amoeboid movements, Cytoskeleton

	Introduction

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
Crawling cells move by using the actin cytoskeleton to power a simple mechanical cycle whereby the leading edge protrudes and adheres to the substratum. The cell body is then pulled forward in a process generally called retraction (Abercrombie, 1980; Roberts and Stewart, 2000). Delineating the mechanochemical events that drive this cycle has proven elusive because of the large number of proteins involved in cell locomotion and the intricacy of the intracellular control system. Moreover, the involvement of actin in a range of other cellular functions, such as endo- and exocytosis, trafficking and maintenance of cell shape, has frustrated the interpretation of many experiments. Therefore, we have focused on a simple and specialized cell: the sperm of a nematode, Ascaris suum. In these cells, the locomotion machinery is dramatically simplified, thereby providing a unique and powerful perspective for evaluating the molecular mechanism of cell crawling (Italiano et al., 2001; Roberts and Stewart, 2000).

Nematode sperm exhibit the same cycle of protrusion, adhesion and retraction as actin-driven amoeboid cells. This shared motile behavior suggests that both types of cell employ analogous molecular mechanisms to generate locomotion (Roberts and Stewart, 2000). However, nematode sperm lack the actin machinery typically associated with amoeboid cell motility; instead, their movement is powered by a cytoskeleton built from major sperm protein (MSP) filaments^*. MSP is a highly basic 14.5 kDa polypeptide that polymerizes in a hierarchical fashion (Roberts and Stewart, 1995; Roberts and Stewart, 1997). The protein forms symmetrical dimers in solution that polymerize into helical subfilaments, which wind together in pairs to form larger filaments. Because of their unique structure, MSP filaments can spontaneously assemble into higher-order assemblies using the same interaction interfaces employed to assemble subfilaments into filaments (King et al., 1994b; Stewart et al., 1994). Thus, in contrast to actin, MSP polymerization and bundling does not require a broad spectrum of accessory proteins. Moreover, within subfilaments, the polymer has no overall structural polarity (Bullock et al., 1998). This lack of structural polarity implies that motor proteins are unlikely to play a major role in MSP-mediated cell motility, as motors require substrate polarity to define the direction of their movement.

The MSP cytoskeleton of Ascaris sperm is organized into 20-30 branched, densely packed filament meshworks, called fiber complexes, that span the lamellipod from the leading edge to the base where they join the cell body (Fig. 1). Filaments extend out laterally from adjacent fiber complexes and intertwine so that the entire cytoskeleton forms a thixotropic (shear thinning) gel that operates mechanically as a unit^*. Filaments are assembled and bundled into fiber complexes along the leading edge and disassembled at the base of the lamellipod. Thus, as the cell crawls forward, the cytoskeleton treadmills continuously rearward through the lamellipod (Italiano et al., 2001; Roberts and Stewart, 2000). The rate of centripetal cytoskeletal flow matches that of locomotion. Thus, elongation of the fiber complexes is coupled to protrusion of the leading edge, whereas retraction of the cell body is associated with disassembly of the cytoskeleton at the opposite end of the lamellipod. These observations form the basis of a proposed ‘push-pull’ mechanism for crawling movement, whereby protrusion and retraction are linked reciprocally to the assembly status of the MSP machinery (Italiano et al., 1999; Roberts and Stewart, 2000).

View larger version (50K): [in this window] [in a new window]   Fig. 1. (A) Top view of a crawling Ascaris sperm. The lamellipod can be divided into three major regions: (LE), the leading edge where polymerization and gel condensation into macrofibers and ribs takes place. In Ascaris, but not in other nematode sperm, hyper-complexed branched MSP ‘ribs’ are prominent and originate in protuberances called villipodia. (PR) the perinuclear region where the MSP gel solates and generates a contractile stress. (IR) the intermediate region between the LE and PR where the gel density is nearly constant. The proximal-to-distal pH gradient affects the polymerization and depolymerization rates at the LE and in the PR. (B) Schematic diagram showing the Ascaris sperm lamellipod in cross section. The ventral-fiber complexes branch dorsally. The MSP gel forms at the leading edge and is connected mechanically to the substratum through the membrane. As the cell moves forwards, the gel remains stationary with respect to the substratum, eventually entering the perinuclear region where it solates and contracts. (C) The graphs show how the pH, adhesion, gel density and the elastic stress vary with position in the lamellipod.

In this manuscript we describe a detailed physical model that accounts for the major aspects of nematode sperm motility and that provides a conceptual framework for evaluating the contribution of different aspects of cytoskeleton dynamics and assembly to locomotion. The layout of the paper is as follows. In the following section we describe a 2D mechanical model for the Ascaris lamellipod. In the third section, we discuss the various possible physical and chemical rationalizations for the assumptions underlying the model and the experimental observations that impact upon each mechanism. In the fourth section, we compare the results of model simulations with the observed motile behavior of sperm and the results of selected experimental manipulations of motility. Finally, in Section 5, we discuss the unified view that the model brings to the integrative aspects of lamellipodial motility.

	Description of the model

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
In this section we describe a computational model developed on the basis of a finite element representation of the MSP cytoskeletal gel system. As the details are quite complicated, mathematical particulars of the model are given in the Appendices. The physical basis for assumptions underlying the model is discussed in the next section.

The model we present is a quantitative biophysical formulation of the ‘push-pull’ hypothesis (Roberts and Stewart, 2000). The physical property of the MSP filaments that underlies the model’s behavior is their propensity to spontaneously associate laterally into higher order filament structures and networks as a result of their unusual macromolecular structure (King et al., 1994b; Stewart et al., 1994). This association produces a fibrous gel that crosslinks via a number of self-association sites on the surface of the MSP filaments. The polymerization and bundling of MSP can be modulated by altering cellular pH (Italiano et al., 2001; Italiano et al., 1999; Roberts et al., 1998). A range of observations lead to the conclusion that assembly generates a protrusive force (Italiano et al., 1996), whereas disassembly generates a contractile force (Italiano et al., 1999), a process suggested by several authors for actin-based systems (Mogilner and Oster, 1996b; Oster, 1988; Oster and Perelson, 1988; Taylor et al., 1979). The model we present here shows how the mechanical balance of forces explains the major features of the protrusion-adhesion-retraction cycle that propels the cell. We also propose a plausible physical basis for each of the component forces.

In Ascaris sperm, cell polarity coincides with a proximal-distal pH gradient of 0.2 pH units (King et al., 1994a). Experimental manipulation of this gradient alters MSP cytoskeletal organization and dynamics, suggesting that intracellular pH contributes to the regulation of the motility machinery (Italiano et al., 1999; King et al., 1992). The origin of the pH gradient is not known but probably results from the mitochondria that are excluded from the MSP gel and so cluster in the cell body at the base of the lamellipod. A range of experimental studies (Roberts and Stewart, 2000) have demonstrated that polymerization and gel formation occurs in the more basic environment at the leading edge of the lamellipod, whereas solation and depolymerization take place predominantly in the more acidic environment of the proximal region near the cell body.

The general idea of the model is as follows. MSP polymerizes at the leading edge to form filaments that spontaneously assemble into bundles that form a fibrous gel. This bundling process pushes the cell front forwards. We propose that gel assembly also stores elastic energy in the form of a tensile stress in the gel. As the cell moves forward, the MSP gel moves proximally towards the cell body where the environment is more acidic until solation of the gel is triggered. The solation process releases the elastic energy in the gel generating a contractile stress. We further propose that adhesion of the lamellipodium to the substratum also decreases in the acidic environment at the rear of the cell. Thus, the gel contraction accompanying solation pulls the cell body forwards rather than pulling the leading edge back.

The remainder of this section is devoted to describing how the finite element model implements these forces. The Appendix explains the model in greater detail and formulates it as mathematical equations. To discuss the various physicochemical processes taking place, we divide the lamellipodium of crawling sperm into three general regions, shown in Fig. 1A: (1) the leading edge (LE); (2) the intermediate region (IR) comprising the bulk of the lamellipodium; and (3) the solation, or perinuclear, region (PR).

The finite element model
The complexity of the interactions and the geometry preclude direct mathematical analysis; therefore, we use a finite element model to investigate the dynamic consequences of the physical forces described above^*. The bulk of the MSP gel is located at the ventral surface of the lamellipod; the dorsal volume does not contribute significantly to locomotion. Moreover, while there is probably significant water flow in the 3D body of the cell, close to the ventral boundary, there is little fluid flow. Therefore, a 2D model is sufficient to represent many aspects of the mechanics of crawling. In a subsequent study, we will extend the model to three dimensions and incorporate fluid flow explicitly into the model.

The basic physics of the MSP gel model can be captured in a simple 1D model that represents an anterior-posterior transect, as shown in Fig. 2A. The 2D version that is the basis for the finite element model is given in the Appendix. Consider a strip of cytogel with unit cross sectional area extending from the leading edge to the cell rear. Denote by u(x) the displacement of a material point from its initial position. For small displacements, the strain is just the gradient in the displacement (x)=u(x)/x [or, in 2D: (x)=½(u + u^T)]. For small strains, the stress in a small element of an elastic body, (x), at position x is related to the strain, (x), by Hooke’s law (Landau and Lifshitz, 1995):

View larger version (44K): [in this window] [in a new window]   Fig. 2. (A) The continuum model of a 1D cytogel strip representing the lamellipod (see Appendix). (B) The finite element model of the lamellipod. The lamellipod is triangulated so that each node represents a mass of cytoskeleton contained in the surrounding (Voronoi) polygon. (C) Detail of a finite element consisting of an elastic element in parallel with a tensile element. The dashpot with damping coefficient µ connected to the substratum accounts for the viscous dissipation associated with making and breaking attachments as well as the dissipation associated with cytoskeleton-fluid friction.

(1)

where Y is the elastic modulus (Young’s modulus) of the material.

We propose that the MSP gel differs from a simple elastic material in several ways, the most important being the ability to store elastic energy as it coalesces into higher-order macromolecular assemblies such as fiber complexes. We shall discuss the physical basis for this ‘bundling stress’ below. The mechanical effect of bundling is to add a tensile stress term to equation (1):

(2)

Here (x) represents the combined dilating effects of the gel osmotic pressure (this includes the gel entropic motions and the counterion pressure) and the ‘bundling’ stress discussed below. Finally, an additional body force must be added to (2) owing to the adhesive forces holding the lamellipod to the substratum. As the cell moves forward, these forces manifest themselves as a frictional drag proportional to the speed of locomotion:

(3)

where µ(x) is the drag coefficient that characterizes the effective resistance to motion. µ includes the making and breaking of adhesions of the lamellipod to the substratum as well as the viscous resistance of the cytoplasm to the forces exerted by the cytoskeleton. Using (3), the force balance on a small element of the cytogel strip is (Landau and Lifshitz, 1995):

(4)

The MSP gel polymerizes at the right hand (leading) boundary of the strip and depolymerizes at the left (rear) boundary. At these boundaries a load-velocity relationship must be specified; the boundary conditions and the solutions to equation (4) are given in the Appendix.

Figure 2B shows the tessellation of the 2D lamellipod into triangular elements. Each branch of the tessellation represents the mechanical element shown in Figure 2C. It consists of a spring with elastic modulus, , in parallel with an extensional force generator that applies a tension to the element. This represents the tensile stress, , in equation (2). The retarding force attributable to the substratum adhesions is represented by the sliding friction element between the element node and the substratum. There is also internal dissipation in the gel owing to the relative motion between the fibers and the solvent. This is represented in Figure 2C as an additional dashpot, shown by a dashed line, in parallel with the spring and force generator. In our simulations we have incorporated this internal dissipation into the sliding friction dashpot to the substratum. The equations of the finite element model are constructed by collecting together all of the force balances at each node of the tessellation, each equation being the finite version of equation (4); for example, for node i:

(5)

Here n is the number of branches incident on node i, and N is the total number of nodes. Nodes in the intermediate region of the lamellipodium are governed by equations of the form (5). Elements at the leading edge and in the perinuclear region must be handled differently. The Appendix lays out the complete computational algorithm driving the model.

Elements at the leading edge
As the tessellated cell moves forward, new nodes must be introduced at the leading edge to represent polymerization and bundling of the MSP gel. Since actual polymerization takes place within a few tens of nanometers of the leading edge membrane (Italiano et al., 1996), we cannot represent this process explicitly in a model for the whole lamellipod. Therefore, we represent the composite process of protrusion and gel condensation as follows.

As a result of MSP polymerization and bundling, the leading edge advances by the extension of bulbous protrusions called villipodia. These protuberances do not touch the substratum as they expand, but eventually they settle into contact and establish adhesions. Thus the leading node of the model does not have a viscous element (dashpot) connecting to the substratum, and the leading branch consists only of the elastic and tensile elements in parallel. This implies that a leading branch is stress free: the tensile force that represents the (negative) gradient in the free energy of crosslinking balances the elastic force.

Each leading edge branch extends by lengthening at a velocity determined by the rate of MSP polymerization and bundling. These two processes are generated by factors in the leading edge of the lamellipod membrane (Italiano et al., 1996). When a branch reaches twice its original length, a node is inserted at the midpoint along with a viscous element to the substratum so that a new LE and IR branch is formed.

Elements in the solation region
Each node surrogates for a volume of MSP gel represented by the Voronoi polygon surrounding the node (Bottino, 2001)^*. In order to conserve mass, the number of nodes must be conserved. Therefore, as new nodes are introduced at the leading edge, nodes must be removed at the cell rear at the same average rate. As the gel solates in the acidic environment near the cell body, the elastic energy stored in each branch is released to pull the cell forwards. This is accomplished as follows. The most posterior nodes are anchored to the cell body and are permanent. When a node approaches the cell rear closer than a threshold distance, that node is removed along with its dashpot to the substratum. Solation is modeled by removing the tensile elements in parallel with the remaining springs, which have been held in tension. This allows the springs in series to contract to their original rest length. In this way the free energy of crosslinking that was supplied to the system at the front end is released to pull up the cell rear. This algorithm combines both the solation-contraction process and the ultimate depolymerization of the MSP gel to dimers, which are then recycled to the leading edge.

Protrusion and solation rates are modulated by pH
An essential feature of the model is that MSP assembly and disassembly are separated spatially. Although clearly both are influenced by pH, it is likely that protrusion is controlled primarily by factors present in the membrane at the leading edge of the lamellipod provided that the pH is above about 6.8. Under these conditions, a membrane protein (VP) acts in conjunction with at least two soluble cytoplasmic proteins (SF) to facilitate local MSP polymerization (Roberts et al., 1998). This process can be inhibited when the pH is lowered: addition of pH 6.35 – 6.7 external acetate buffer stops MSP polymerization and bundling at the leading edge of the lamellipod (Italiano et al., 1999). However, under these conditions MSP unbundling and depolymerization still continues at the cell body and generates a force that places the cytoskeleton under tension. The pH at the site of cytoskeletal disassembly is lower than that at the leading edge (King et al., 1994a), and so depolymerization and unbundling could be initiated when the pH falls below a critical value near the cell body. Although the precise role played by pH in either assembly or disassembly has yet to be established, the pH is clearly a good marker for these processes. Therefore, we have used pH as a convenient surrogate to model the way in which the balance between MSP assembly and disassembly changes in the lamellipod between its leading edge and the cell body. In our calculations, we compute the proton distribution throughout the lamellipod at each time step. We assume that the proton source is located at the boundary between the lamellipod and the cell body. Protons also leak out from the lamellipod at the boundaries. Since the diffusion rate of protons is very rapid, we can assume that the concentration profile is always at its steady state for a given boundary profile. At each time step, the concentration is updated according to the changed boundary shape, and the polymerization and depolymerization rates are computed accordingly (see Appendix).

	Forces in the lamellipod

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
In this section we discuss the physicochemical basis for the forces introduced in the finite element model and present the relevant experimental data.

Protrusion
Lamellipodial protrusion has been reconstituted in vitro in cell-free extracts of Ascaris sperm. Vesicles derived from the leading-edge membrane induce the localized assembly of MSP filaments that arrange into cylindrical meshworks, called fibers, that push the vesicle forward as they elongate (Italiano et al., 1996). These vesicles contain a phosphorylated form of VP that recruits SF to the membrane where it nucleates polymerization. Thus, MSP dimers are ‘activated’ by VP and SF at the leading edge, whereupon they become polymerization competent and quickly polymerize into filaments. Unlike actin, MSP does not bind to nucleotides directly (Bullock et al., 1996; Italiano et al., 1996). ATP hydrolysis is required for phosphorylation of VP, but the precise role of ATP in MSP polymerization is unclear. The filaments formed in the vicinity of the membrane assemble laterally into higher order filament complexes (Sepsenwol et al., 1989). In Ascaris, fiber complexes are visible by light microscopy as ‘rope-like’ ribs that project from the leading edge to the cell body, forming a branched tree-like pattern. However, these large fiber complexes are prominent in Ascaris compared to sperm from other nematode species (e.g. C. elegans), indicating that their size is probably not crucial to generating locomotion. Therefore, we do not include them as a separate level of gelation in the model.

The concentration of SF and the activation of VP, rather than that of MSP itself, appear to be limiting for polymerization (Italiano et al., 1996; Roberts et al., 1998). The mechanism by which these proteins are localized and controlled is not known; in the Appendix, we present several theoretical possibilities. In the model, polymerization depends only on local pH near the boundary of the lamellipod. The equations governing the polymerization process are given in the Appendix.

To model protrusion at the leading edge we must provide a load-velocity relationship that prescribes the force generated at the leading edge by the formation of the MSP gel. The proposal that polymerization drives the extension of lamellipodia in actin-based systems has a long history going back to the classic work of Abercrombie (Abercrombie, 1980). Recently, Mogilner and Oster examined the physics of force generation by a semi-stiff polymerizing actin filament – the elastic polymerization ratchet model (Mogilner and Oster, 1996a; Mogilner and Oster, 1996b). However, MSP filaments appear to be somewhat more flexible than actin, and so the polymerization ratchet mechanism may not be as effective in generating a protrusive force in nematode sperm. However, the process of bundling filaments into higher order macromolecular assemblies can also contribute to the protrusive force; the calculation that supports this assertion is discussed in the Appendix. In the MSP model, we introduce a pressure at the cell boundary that pushes the cell periphery outwards in the direction normal to the local edge tangent. This pressure arises from the assembly and bundling of MSP filaments into a gel as follows.

Newly polymerized MSP filaments are created stress free and are relatively flexible (i.e. they have a short persistence length). Because of the unusual way in which MSP filaments are generated (i.e. by wrapping two helical subfilaments around one another), they have a series of mutual interaction sites arranged on their surface so that they are able to form bundles spontaneously without the specific bundling proteins actin requires (Stewart et al., 1994). This property is seen most dramatically in the macrofibers formed when MSP is assembled in vitro (King et al., 1994b) but is also probably responsible for the various higher order aggregates of MSP filaments observed in vivo. Consequently, the distribution of these mutual interaction sites on the surface of MSP filaments means that filaments that diffuse into contact with one another will adhere and assemble into higher order filament bundles spontaneously^*. This assembly process forces the filaments within a higher-order aggregate to assume an end-to-end distance that is larger than it was in solution. That is, the enthalpic part of the free energy of assembly dominates the entropy loss accompanying lateral association, so that filaments are held in a ‘stretched’ configuration. Thus, bundles of MSP filaments contain the stored elastic energy of their constituent filaments (analogous to a pre-stressed concrete beam) and are stiffer. These bundles of MSP form a thixotropic gel-like cytoskeleton within the lamellipod. The cytoskeletal gel is a fibrous material, so that when filaments bundle laterally they generate a protrusive force longitudinally (Poisson expansion). This may help extrude the leading edge into the characteristic protuberances (villipodia) that characterize the motile sperm (Sepsenwol et al., 1989).

Near the leading edge membrane, a number of associated processes take place. As MSP molecules interact with one another, both during filament polymerization and macrofiber assembly, counterions are released and the local gel osmotic pressure decreases. Moreover, fiber-associated ‘vicinal’ water associated with both filament polymerization and lateral association is released (Pollack, 2001). The sensitivity of the polymerization and bundling of MSP to pressure may be due to this water release and/or to weakening of lateral hydrophobic interactions between filaments (Roberts et al., 1998). It is difficult to assess the quantitative effect of these processes, but they probably also contribute to lamellipodial protrusion and villipodia formation.

Adhesion
To move forward the lamellipod must adhere to the substratum. Examination of crawling sperm by interference reflection microscopy has revealed that the adhesive sites are located primarily in the lamellipod, with few if any in the cell body. In these cells, close contacts form between the lamellipod membrane and the glass substratum. The pattern of these contacts varies; in some cells almost the entire underside of the lamellipod is attached to the glass, whereas others exhibit a series of discrete contact sites. In all cases, the contacts form just behind the leading edge, remain stationary as the cell progresses, and release when the lamellipod-cell body junction passes over the contact site (T. Rodriguez and T. Roberts, unpublished observations). Thus, the pattern of adhesion appears nearly constant from the leading edge to the transition region in the perinuclear region, that is the strength of adhesion appears to be nearly a ‘step’ function. The adhesion gradient determines the direction of crawling by preventing the leading edge from being pulled back by gel contraction; instead the cell rear is pulled forward. For the purposes of the model, we assume that the adhesion strength is piecewise linear: strong at high values of pH, weak at low values of pH, with a linear transition at some intermediate value of pH (Fig. 6). The Appendix discusses the adhesion and release processes in more detail.

View larger version (8K): [in this window] [in a new window]   Fig. 6.

This picture is supported by the following observations on the MSP-associated motion of single vesicles. Italiano et al. (Italiano et al., 1996) demonstrated that membrane vesicles reconstituted from motile sperm can nucleate cylindrical MSP ‘tails’ and propel these vesicles forwards, similar to the actin tails growing behind microspheres coated with ActA (Cameron et al., 1999). When these MSP gel tails are exposed to acidic conditions, they shrink. If one end of the tail is attached to the substratum, the vesicle at the other end is pulled towards the attachment point (L. Miao and T. Roberts, unpublished).

An interfacial tension effect probably also contributes to pulling the rear of the cell forward. The density of the MSP filament gel decreases across the gel-sol transition region, which is typically very narrow. Along phase transition boundaries such as this interface, a tangential stress will develop, similar to the interfacial tension at a liquid-vapor interface. This interfacial tension, combined with the stress in the low density region, pulls the cell body forward. The response of Ascaris sperm to manipulation of intracellular pH supports the idea that interfacial tension is involved in cell body retraction. For example, treatment of the cells with acetate buffer at pH<6 causes the MSP cytoskeleton to disassemble completely. When the acid is washed out, intracellular pH rebounds, and the cytoskeleton is rebuilt by reconstruction of the fiber complexes along the lamellipod membrane. These newly formed complexes lengthen by assembly at their membrane-associated ends. The opposite ends move rearward through the lamellipod, creating an interface between the proximal boundary of the reforming cytoskeleton and the lamellipod cytoplasm. Retraction of the cell rear does not commence until this interface reaches the cell body, implying that depolymerization of the fiber complexes is necessary for retraction. In some cells, this reassembly is asymmetric within the lamellipod; when the reforming fiber complexes contact one side of the cell body before the other, a turning moment develops that moves the cell towards the direction of contact (Italiano et al., 1999).

Depolymerization
Following solation (i.e. disassembly) of the MSP filament bundles and their subsequent entropic contraction, the MSP gel must be depolymerized so that subunits (probably MSP dimers) can be recycled to the leading edge. Since depolymerization creates a proximal-distal subunit gradient, diffusion is sufficient to accomplish this recycling. It is possible that factors other than pH are involved. For example, there is evidence for an ‘MSP depolymerization factor’ that could also be involved in regulating the depolymerization rate (J. Italiano and T. Roberts, unpublished). In the model, we assume that the depolymerization takes place quickly in a narrow region at the rear of the lamellipod, adjacent to the cell body. Figure 1C summarizes how the gradients in adhesion, gel density and elastic stress follow the pH gradient.

	Comparing the model with observations

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
In the Appendix, we consider a ‘minimal’ model consisting of a 1D cytogel strip running along the length of the cell. This model can be explored analytically, and it illustrates how the lamellipod length and migration velocity are regulated to maintain constant values. This regulation comes about from a negative-feedback loop that reduces the rate of protrusion in longer lamellipods and increases the rate of protrusion in shorter ones so that protrusion and retraction are matched, corresponding to the observation of coordinated polymerization and depolymerization in living crawling sperm (Italiano et al., 2001; Roberts and Stewart, 2000).

Of course, the 1D model cannot properly address the issue of lamellipodial shape and area regulation, and so a 2D finite element model is required to reproduce the shapes and rates of locomotion of the Ascaris sperm cell. To our knowledge, this is the first mathematical model that simulates locomotion using simple dynamic principles of coordination of protrusion, graded adhesion and retraction. The model combines the mechanics of protrusion and contraction with regulatory biochemical pathways and shows how their coupling generates stable rapid migration. The dynamic behavior of the model can best be appreciated by viewing the QuickTime^TM movies that can be downloaded from http://www.CNR.Berkeley.EDU/~bottino/research/wormsperm/. Figs 3-5 show frames from these movies (for movie legends, see http://jcs.biologists.org/supplemental).

View larger version (36K): [in this window] [in a new window]   Fig. 3. Frames from the simulation movie showing the progression of the MSP cytoskeleton as the cell moves forward. The same time interval elapses between successive shaded cell ‘shadows’. Bottom, simulation of translocation with ‘normal’ substrate friction. Top, simulation with the cell body friction increased four-fold over the normal run. Note that the cell body moves much more slowly, but the lamellipod shape changes very little.

View larger version (28K): [in this window] [in a new window]   Fig. 5. A study of effects of extracellular pHext on the simulated cell. In all figures, the red filled region is the original outline of the cell. The final position of the cell in all three cases is shown after the same amount of simulation time has elapsed (1 sec of real time). Bottom, pHext=7.6. This is the case of normal motility. At the front, pH reaches the value of >6.15, so that both the storage of elastic energy, cytoskeletal assembly and adhesion are strong at the leading edge. Middle, pHext=6.75. At this pH motility is impaired. At the front, pH drops to less than 6.15, but both the storage of elastic energy and adhesion are still strong at the leading edge. However, the cytoskeletal assembly is attenuated significantly. The cell body moves forward, but the leading edge is nearly stationary. Top, at pHext=6.35 this motility ceases. At the front the pH decreases to less than 6.1, so that all of the protrusion-supporting processes – adhesion, storage of elastic energy and cytoskeletal assembly – are inhibited. The contraction of the lamellipod takes place transiently owing to the elastic energy stored prior to the change in extracellular pH. This contraction moves the cell body forward slightly, at the same time pulling the leading edge backward significantly. The adhesion of the cell body is now greater than the adhesion of the lamellipod.

Velocity and shape
The model reproduces observed properties of cell locomotion: a steady-state velocity with a shape (length to width ratio of 1:1 to 3:1) consistent with those observed in crawling sperm (Royal et al., 1995; Sepsenwol et al., 1989; Sepsenwol and Taft, 1990) (Fig. 3) (see Movie 1 at http://jcs.biologists.org/supplemental).

Shape regulation
To stabilize the length in the 1D model and the area in the 2D model, a pH gradient alone is insufficient. It is necessary to introduce a limiting quantity whose concentration decreases as the lamellipodial size increases. There are several possible candidates, which we discuss below. However, for the model calculations we assumed that the vesicle protein is the limiting quantity. If the amount of vesicle protein is constant, then its concentration in the leading edge dilutes as the lamellipodial area increases. This reduces the MSP polymerization rate. Our simulations show that this is sufficient to regulate the lamellipodial area to a stable average size. Note that the depletion mechanism creates a global negative feedback, whereas pH regulation is local and activating. Thus, a combination of local activation and global inhibition is needed for size regulation.

Persistence
In the absence of external cues, locomotion is persistent: in the computer model, the cell travels many body lengths before it deviates significantly from the initial direction of migration. This is consistent with the observed behavior of crawling cells (Sepsenwol, 1990; Royal et al., 1995).

Robustness
The speed and shape of the lamellipod is not significantly altered by changing the explicit forms of the force-velocity relation at the rear, depolymerization kinetics, pH and density dependencies of the polymerization rate, bundling stress and adhesion strength and elastic stress-strain relation.

Traction forces
We computed the map of traction forces that the lamellipod exerts on the substratum during retrograde flow (Fig. 4). There are significant differences between the pattern of traction forces generated by the model from those measured in fish keratocytes and fibroblasts (Dembo et al., 1996; Jacobson et al., 1996; Oliver et al., 1995; Oliver et al., 1999). Gliding sperm develop much smaller traction forces. Also, the distribution of forces in sperm is much more uniform compared with fibroblasts, where there are alignment and ‘pinching forces’ in the direction of migration in fibroblasts and normal to this direction for keratocytes. This prediction could be evaluated experimentally using elastic films and photobleaching experiments.

View larger version (19K): [in this window] [in a new window]   Fig. 4. A vector field plot of forces at nodes computed after the cell moved two body lengths. The interior node forces (red) are applied to the substrate; the cell body interface node forces (blue) are applied throughout the substrate beneath the cell body. The magnitude of the forces are proportional to the lengths of the arrows. The forces applied to the center of the cell body interface are 100 pN. The total forward translocation force on the cell body is 1000 pN. The traction forces at the rear of the lamellipod are 10 pN per node. Because of the strong substrate adhesion and because there is no gradient in the bundling stress, the traction forces decrease to 1 pN per node at the leading edge. There is also no noticeable anisotropy in the traction forces.

Dependence on pH
Finally, we mimicked the experiment wherein crawling sperm were treated with weak acid at pH 6.35 and pH 6.75, respectively (Italiano et al., 1999) (Fig. 5; Movie 3; Movie 4.) In both the real cell and the model, at pH 6.75, the assembly stops, thus arresting protrusion, whereas adhesion and contraction continue, leading to temporary forward translocation of the cell body. At the same time, the lamellipod begins to shrink. At pH 6.35, also in both the actual cell and the model, both assembly and adhesion are disrupted, whereas contraction continues. The MSP cytoskeleton detaches from the leading edge, flows rearward and is disassembled in the proximal region.

	Discussion

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
We have developed a finite element model of the MSP gel system that generates locomotion in nematode sperm. The model accounts for amoeboid motility by providing a mechanical basis for the processes of lamellipodial protrusion, substrate adhesion and cell body retraction. A central feature of the model is the way in which energy stored in the cytoskeleton gel during protrusion is subsequently released to generate a pulling force on the cell body. Thus, although protrusion and retraction can be separated experimentally (Italiano et al., 1999), in vivo, they rely respectively on the assembly and disassembly dynamics of the MSP cytoskeleton. In addition to providing a physical basis to account for the observed amoeboid motility of nematode sperm, our model also has implications for actin-based cell locomotion.

Nematode sperm motility
Previous studies have demonstrated that the amoeboid motility of nematode sperm closely resembles that seen in many actin-based systems (Roberts and Stewart, 2000). Nematode sperm have a cytoskeleton derived from MSP and lack actin, myosin and tubulin. Nevertheless, their motility shows the same lamellipodial protrusion and cell body retraction seen in actin-based systems and their locomotion also relies on adhesion to the substrate to generate forward motion. A range of experiments have demonstrated the crucial role played by the vectorial assembly of MSP filaments and their bundling into large aggregates (Roberts and Stewart, 2000). Direct observation of MSP fiber complexes by light microscopy shows that they treadmill, with material being added continuously at the leading edge of the lamellipod and removed near the cell body. In vitro, MSP polymerization and bundling can move membrane vesicles (Italiano et al., 1996), and both this reconstituted motility and cell locomotion show a remarkable sensitivity to pressure (Roberts et al., 1998). The MSP polymerization that takes place at the leading edge of the cell requires both membrane-bound and soluble factors (Roberts et al., 1998). It has been possible to decouple lamellipodial protrusion, membrane-cytoskeletal attachment and cell body retraction by manipulating pH with acetate buffer (Italiano et al., 1999). At pH 6.75 lamellipodial protrusion is inhibited but retraction continues, whereas at pH 6.35 the adhesion of the cytoskeleton to the membrane is broken and the MSP filament system then moves rearward.

Our finite element model for the MSP cytoskeleton gel system reproduces these features of nematode sperm locomotion and gives an unanticipated insight into how retraction is mechanically related to protrusion. At the leading edge of the lamellipod, MSP is polymerized initially to form filaments that bundle to form large fiber complexes that attach to the membrane and, through it, to the substratum. The filaments condense to form the fiber-complex gel by forming a large number of weak crosslinks between filaments. In this configuration, they are held in a more extended conformation than they are in free solution. That is, their persistence length in these aggregates is greater than that in free solution with a consequent loss of entropy, possibly associated with a compensating release of bound water and ions. In this extended configuration, the fiber complexes contain stored elastic energy that, upon release, will provide the contractile stress to pull the cell body forwards. The fiber complexes maintain their shape as they treadmill, suggesting that there is little remodeling of the constituent filaments and filament bundles. Thus, the stored energy is not released until the filaments unbundle and depolymerize at the base of the lamellipod, whereupon the filaments seek to contract to their equilibrium length. Thus elastic energy stored during bundle formation generates tension in the cytoskeleton to pull the cell body forward when the gel solates. Attachment to the substrate is required for both traction and to mechanically separate the forces of protrusion and retraction that are generated at opposite ends of cytoskeleton gel. For this, the lamellipod must adhere more strongly than the cell body, lest the tension generated in the cytoskeleton generated by MSP depolymerization and solation move the lamellipod rearwards rather than the cell body forward.

The ability of our model to simulate both sperm movement and shape indicates that the forces used are sufficient to account for sperm locomotion. The model can also account for the dynamics of cell shape. To our knowledge, there have been no studies that quantitatively address the dynamics of lamellipodial shape. The ‘graded radial extension’ (GRE) model of Lee et al. sheds light on the kinematic principles underlying lamellipodial shape in fish keratocytes (Lee et al., 1993a; Lee et al., 1993b). This model demonstrated that if extension is locally normal to the cell boundary, and if the rate of extension decreases from the center to the sides of the cell, then the 2D steady-state shape of the traveling lamellipod evolves. The model we present here identifies the dynamic principles underlying self-organization of the lamellipod of the nematode sperm and provides a dynamic mechanism for the GRE model. For example, it is likely that lamellipod size regulation is based on a negative-feedback loop involving a limiting factor rather than on the cellular pH gradient alone. Thus, long lamellipods would grow more slowly than short ones and so converge to a roughly constant length. There are a number of plausible molecular mechanisms by which this feedback could be generated. For example, if material such as MSP dimers or SF is being consumed when the fibers form at the leading edge and subsequently liberated near the cell body when MSP depolymerizes, then the concentration of these factors at the leading edge would depend on diffusion and so would be lower the greater the distance of the leading edge from the cell body. Also, the membrane area of the cell would be greater with longer lamellipodia; increasing the membrane area could reduce the concentration of VP per unit area at the leading edge, thus decreasing the supply of polymerization competent subunits and slowing the rate of protrusion. Finally, increasing lamellipodial size may increase the membrane tension, which could decrease the rate of exocytosis (Raucher and Sheetz, 1999a; Raucher and Sheetz, 1999b; Raucher and Sheetz, 2000). If VP is supplied by exocytosis, this would again decrease the polymerization rate.

The model captures the cellular polarization observed in Ascaris sperm: construction of the MSP cytoskeleton gel and adhesion to the substratum occur at or near the leading edge, whereas gel solation and de-adhesion take place at the base of the lamellipod. We have discussed the possible molecular basis for these events; however, since the molecular details remain uncertain, we have used the pH gradient present within the cell as a surrogate to model the effect of these processes on motility. It is possible – albeit unlikely – that the pH gradient alone generates cellular polarity directly. For example, the high pH present at the leading edge of the lamellipod could exceed a threshold for operation of the VP-SF nucleation complex that generates MSP assembly at the leading edge, whereas the more acidic environment at the cell body could trigger dissociation of the MSP filament bundles and depolymerize individual filaments. However, it is equally likely that the pH gradient is associated with intervening regulatory proteins that perform the actual work of gelation and solation.

It is admittedly an approximation to use the pH gradient to represent both the spatial separation of MSP assembly from disassembly and adhesion from de-adhesion. However, the finite element model simulates motile behavior of nematode sperm in remarkable detail. This simple model can account for persistence in the direction of locomotion, the maintenance of cell shape, the continuation of cytoskeletal flow when the lamellipod is not in contact with substratum and the behavior of tethered cells. Thus the model encapsulates the primary mechanochemistry of the motile process and can serve as a conceptual framework in which the amoeboid motility of nematode sperm can be understood. The model provides a physical realization of the push-pull model of nematode sperm motility and describes how MSP filament assembly and bundling can generate the forces required for cell locomotion. In a subsequent study, we shall refine the model to include the fluid and solid phases of the MSP gel, which will enable us to match a large number of additional observations on cytoplasmic flow (manuscript in preparation).

Implications for actin-based cell motility
The general principles of lamellipodial-driven cell locomotion have been established for some time: a cycle of protrusion, graded adhesion and retraction drives translocation (Abercrombie, 1980). However, many details remain elusive. Broadly speaking, there are two fundamental questions that need to be resolved: (1) What is the physical nature and the molecular basis of protrusion, retraction, and adhesion? (2) How are the three processes coordinated to achieve the observed shapes and rates of migrating cells?

Our simulations of the nematode sperm have implications for the mechanism of locomotion of actin-based cells. Although the mechanical principles of motility seem to be remarkably similar for nematode sperm and many actin-based cells, the force-generating mechanisms, biochemical components and regulatory pathways employed to modulate cytoskeletal polymerization and organization are different. The concept that lamellipodial protrusion is driven primarily by localized filament polymerization and self-organization into the network, which was established in nematode sperm (Italiano et al., 1996), is now generally accepted as the basis for protrusion in actin-based systems as well (Borisy and Svitkina, 2000; Theriot, 2000). The molecular mechanisms, and the relative contributions of polymerization and network organization, differ in the actin and MSP machinery. For example, the polymerization ratchet model, proposed to account for protrusion in actin systems, places the physical basis for force generation on subunit addition at the ends of filaments that bend away from the membrane and then spring back (Mogilner and Oster, 1996a; Mogilner and Oster, 1996b). This mechanism also requires that the filaments be arranged into a branching meshwork by the action of nucleating and minus-end-capping proteins such as Arp2/3 (Blanchoin et al., 2000; Pantaloni et al., 2001; Pollard et al., 2000; Pollard et al., 2001). In the MSP system, in which filaments spontaneously aggregate to form higher order arrays, bundling may be the dominate force generating mechanism, although polymerization is also required.

Another difference between actin- and MSP-driven systems is that the force for retraction in sperm involves depolymerization and unbundling, whereas in many actin-based systems retraction appears to be driven by myosin (Lin et al., 1996; Verkhovsky et al., 1999; Verkhovsky et al., 1995). For example, the dynamic contraction model of Verkhovsky et al. suggests that disruption of the actin cytoskeletal gel by depolymerization and dissociation of Arp2/3 complexes weakens actin in the posterior region (Verkhovsky et al., 1999). This allows collapse of the largely isotropic actin network into bi-polar actin-myosin bundles and subsequent sliding contraction. In contrast, the present model demonstrates that solation and depolymerization of the MSP cytoskeleton can, in principle, use energy stored during the formation of the cytoplasmic gel during protrusion to generate a tension that can be used to pull the cell body forward. Thus, the results obtained using nematode sperm suggest that motor proteins may not be the whole story (Bullock et al., 1998; Italiano et al., 1999). It is plausible that, in actin systems, the energy released when filament networks are taken apart could be used to contribute to the forward motion of the cell body, in addition to the actin-myosin sliding mechanism (Mogilner and Oster, 1996b).

In summary, we have constructed a physical model that accounts for the major features of nematode sperm amoeboid motility and which provides a mechanochemical basis for the ‘push-pull’ theory of locomotion. The model provides a mechanism for how energy stored during lamellipod protrusion can be subsequently used to generate cell body retraction, thus providing a mechanistic link between the MSP assembly dynamics at either end of the cell. In addition to providing a detailed explanation for nematode sperm locomotion, it is likely that many of the concepts explored here are also important in actin-based amoeboid motility. If so, our model could provide a conceptual framework for evaluating many general aspects of cell locomotion.

	Appendix

Top SUMMARY Introduction Description of the model Forces in the lamellipod Comparing the model with... Discussion Appendix REFERENCES

 
Discussion of the basic physical forces
Protrusion and size regulation
Previously, we demonstrated that actin polymerization in the leading edge of the lamellipod of actin based cells is able to generate both the force to overcome the cell membrane tension and the observed rates of protrusion (Mogilner and Oster, 1996b). The mechanism is the polymerization ratchet: thermal undulations of both filaments and the membrane create a gap between the membrane and the filament tip into which monomers can intercalate. Assembly of monomers onto the polymer tips generates force and unidirectional movement by rectifying Brownian motion. MSP protofilaments appear to be more flexible than actin fibers, suggesting that the polymerization ratchet mechanism may not be sufficient to drive protrusion. However, the process of filament ‘bundling’ could generate the necessary force and protrusion rates. Here we illustrate this notion using order-of-magnitude estimates.

Consider a protofilament being incorporated into a higher-order MSP fiber at the leading edge that is growing at a rate V_p. This bundling process proceeds by successive adhesion of hydrophobic and electrostatic patches located periodically along the protofilament. Thus, the extension of the bundle results from the binding of the growing, and thermally undulating, tip of the protofilament to other filaments in the bundle. According to the general theory of the polymerization ratchet, this process can generate a force of the order of k_BT/, where is the distance between the adhesive patches along the protofilaments (Peskin et al., 1993) (A.M. and L. Edelstein-Keshet, unpublished). If 10-40 nm, then, one protofilament generates a protrusion force of the order of few tenths of a piconewton. EM images of the leading edge MSP cytoskeleton show that there is roughly one protofilament per 100 nm² of the cell’s leading edge. Near the stall force, the forces generated by each proto-filament are additive (Van Doorn et al., 2000). The dimensions of the leading edge are 1 µmx10 µm, so the total protrusion force from the bundling ratchet is (10 µm²/0.0001 µm²)x0.1 pN 10⁴ pN. There are no data about the sperm cell membrane tension. Using the scarce data related to some actin-based cells (tens to hundreds of piconewtons per micron of the leading edge), we can estimate the total membrane resistance as 10²–10³ pN. Thus, the force of the bundling ratchet would be sufficient to drive protrusion. In the model, we assume that the membrane resistance force is much less than the maximal stall force of the bundling ratchet, and that MSP cytoskeletal growth at the leading edge is load-free.

In addition to the membrane tension, two other factors determine the velocity of protrusion: the intrinsic rate of the bundling ratchet and the free elongation rate of the filaments. The order of magnitude of the former can be estimated as D/, where D 10⁷ nm²/sec is the effective diffusion coefficient describing thermal writhing of a filament tip 10–40 nm long. The corresponding velocity, D/ 10³ µm/sec, clearly cannot be the limiting factor of the observed protrusion rate of 1 µm/sec. Thus, in the model we assume that the load-free elongation rate of MSP filaments, V_p, is the rate of advancement of the leading edge. Below, we show that the slippage of the cytoskeleton relative to the substratum can be neglected at the leading edge. Therefore, the velocity of lamellipodial advancement is locally normal to the leading edge boundary, and its magnitude is equal to V_p.

Owing to the extremely high concentration of MSP in the sperm cell cytoskeleton (4 mM), the diffusion of MSP dimers from the rear to the front of the cell is unlikely to limit the rate of assembly. We assume that two factors regulate the net assembly rate. First, polymerization is catalyzed in regions of the leading edge membrane where vesicle protein (VP) and soluble factor (SF) aggregate. The combination of the two correspond to an effective membrane-bound ‘enzyme’ that catalyzes the activation of MSP monomers from a polymerization-incompetent form into a polymerization-competent configuration. (Either MSP/SF complex can be activated by VP or SF is activated before binding to MSP.) Second, phosphorylation of the VP protein controls its nucleation activity. Another factor controls the location along the margin where this protein is phosphorylated. We use the local pH as the marker for the VP/SF activity and thus assume that V_p is a function of the local pH.

Numerical experiments allowed us to determine the character of the pH dependence of the rate of protrusion. When the assembly rate decreased with pH, as in the 1D model (compare with below), then the sides of the lamellipod gradually collapsed towards the center. On the other hand, if the polymerization rate increased with pH, the lamellipod expanded without limit. If the rate increased at smaller pH values and decreased at greater values, unstable ‘mushroom-like’ lamellipod shapes ensued. We observed that the following two simple assumptions were sufficient to produce stable lamellipodial size and shape in the simulated cell: (1) cytoskeletal assembly rate increases monotonically with intracellular pH, and (2) there is a depletable factor which limits the size. There are several likely candidates for this additional factor. First, there is evidence for a cytoskeletal component (P25 filament stabilization factor) that dissociates from the fiber complexes just before they are disassembled. If this factor is in limited supply, then an expansion of the lamellipod can lead to its depletion. Second, the vesicle protein itself can be the limiting quantity. If the amount of vesicle protein in the leading edge membrane is constant, then its concentration in the leading edge is diluted as the lamellipodial area increases, reducing the MSP polymerization rate. Our simulations show that this is sufficient to regulate the lamellipodial area to a stable average size. The reason for this behavior is that regions of the cell frontier that are closer to the center are at the same time farther from the cell body. Consequently, pH values there are greater, and the protrusion rates are greater. This leads to a stable steady shape of the lamellipod in accordance with the GRE model. Meanwhile, the lamellipodial area cannot grow without limit because of depletion of the limiting factor and slowing down of the leading edge extension.

In the simulations we use the formula:

(A1)

where V₀=1 µm/sec is the magnitude of the protrusion velocity. The function f(pH) is a linearly increasing function (Fig. 6), such that f(pH=6.05)=0.1, and f(pH=6.15)=1. A is the area of the lamellipod. A_max is the area at which the limiting factor controlling the protrusion is completely depleted. In the simulations we used A_max=200 µm².

Contractile forces
Individual MSP filaments polymerizing near the leading edge membrane are initially stress free and relatively flexible. There is a distribution of filament lengths, but it is not clear what factors determine this distribution. Therefore, we assume an equilibrium configuration characterized by an end-to-end distance, L₀, that is considerably shorter than the fully extended contour length, L. Soon after formation, filaments begin to associate laterally into higher-order filament bundles. Bundling is due to a combination of hydrophobic and electrostatic interactions. The latter arises because, at high pH, the basic groups release their protons and the resulting negative charges keep the filaments extended beyond their neutral end-to-end length. Thus, newly polymerized MSP filaments find themselves distended beyond their equilibrium length if they were neutral, and hydrophobic bundling locks them into this configuration. This amounts to storing tensile elastic energy that can later be released when the filament unbundle.

We assume that most of the filaments have been extended from their equilibrium length to their contour length: L₀L. A corresponding bundling stress can be estimated as c k_BT((L-L₀)/L₀), where c is the volumetric concentration of crosslinks/entanglements. Individual filaments appear to be fairly flexible; there are no direct measures, but we can estimate Lo to be 10 nm by looking at electron micrographs of negatively stained filaments. The strands (fiber complexes) appear to be very stiff, and the average contour length L can be estimated roughly to be of the order of 100 nm. The reasonable estimate of the average distance between neighboring crosslinks/entanglements is 30 nm. Then, the parameter c can be estimated as (1/30 nm)3 3x10⁴ /µm³. The strain ((L-L₀)/L₀) is of order unity, and the order of magnitude of the bundling stress can be estimated as 100 pN/µm². In the model, we use the value _max=100 pN/µm² as the maximal value of the isotropic bundling stress developed at high values of pH.

After formation, the lamellipodial gel remains nearly static (relative to the substratum) because of the high effective friction (see the next section) and the absence of the bundling stress gradient. Eventually, the gel reaches the low pH environment in the perinuclear region. Under the influence of the acidic milieu, the bundling forces weaken. Indeed, the gel is basic so the protonated sites have a low pK_a (i.e. the potential well for proton binding is not too deep: V/k_BT2.3·pK_a). Therefore, at low proton concentrations the negative binding sites are unoccupied. As the proton concentration increases, the sites become neutralized, thus freeing their counterions to diffuse out of the gel region. This locally lowers the ionic strength, which decreases the difference between the chemical potential of the vapor and the liquid, (µ_v - µ_l). According to the current theory of the hydrophobic effect, this weakens the hydrophobic forces between the gel filaments allowing the gel t