Difference in polymerization and steady-state dynamics of free and gelsolin-capped filaments formed by alpha- and beta-isoactins

The polymerization of scallop β-like actin is significantly slower than that of skeletal muscle α-actin. To reveal which steps of polymerization contribute to this difference, we estimated the efficiency of nucleation of the two actins, the rates of filament elongation at spontaneous and gelsolin-nucleated polymerization and the turnover rates of the filament subunits at steady-state. Scallop actin nucleated nearly twice less efficient than rabbit actin. In actin filaments with free ends, when dynamics at the barbed ends overrides that at the pointed ends, the relative association rate constants of α- and β-actin were similar, whereas the relative dissociation rate constant of β-ATP-actin subunits was 2- to 3-fold higher than that of α-actin. The 2- to 3-fold faster polymerization of skeletal muscle versus scallop Ca-actin was preserved with gelsolin-capped actin filaments when only polymerization at the pointed end is possible. With gelsolin-induced polymerization, the rate constants of dissociation of ATP-actin subunits from the pointed ends were similar, while the association rate constant of β-actin to the pointed filament ends was twice lower than that of α-actin. This difference may be of physiological relevance for functional intracellular sorting of actin isoforms.     

Mechanism of Cdc42-induced Actin Polymerization in Neutrophil Extracts

Cdc42, activated with GTPγS, induces actin polymerization in supernatants of lysed neutrophils. This polymerization, like that induced by agonists, requires elongation at filament barbed ends. To determine if creation of free barbed ends was sufficient to induce actin polymerization, free barbed ends in the form of spectrin-actin seeds or sheared F-actin filaments were added to cell supernatants. Neither induced polymerization. Furthermore, the presence of spectrin-actin seeds did not increase the rate of Cdc42-induced polymerization, suggesting that the presence of Cdc42 did not facilitate polymerization from spectrin-actin seeds such as might have been the case if Cdc42 inhibited capping or released G-actin from a sequestered pool.Electron microscopy revealed that Cdc42-induced filaments elongated rapidly, achieving a mean length greater than 1 μm in 15 s. The mean length of filaments formed from spectrin-actin seeds was <0.4 μm. Had spectrin-actin seeds elongated at comparable rates before they were capped, they would have induced longer filaments. There was little change in mean length of Cdc42-induced filaments between 15 s and 5 min, suggesting that the increase in F-actin over this time was due to an increase in filament number. These data suggest that Cdc42 induction of actin polymerization requires both creation of free barbed ends and facilitated elongation at these ends.     

The role of actin turnover in retrograde actin network flow in neuronal growth cones

The balance of actin filament polymerization and depolymerization maintains a steady state network treadmill in neuronal growth cones essential for motility and guidance. Here we have investigated the connection between depolymerization and treadmilling dynamics. We show that polymerization-competent barbed ends are concentrated at the leading edge and depolymerization is distributed throughout the peripheral domain. We found a high-to-low G-actin gradient between peripheral and central domains. Inhibiting turnover with jasplakinolide collapsed this gradient and lowered leading edge barbed end density. Ultrastructural analysis showed dramatic reduction of leading edge actin filament density and filament accumulation in central regions. Live cell imaging revealed that the leading edge retracted even as retrograde actin flow rate decreased exponentially. Inhibition of myosin II activity before jasplakinolide treatment lowered baseline retrograde flow rates and prevented leading edge retraction. Myosin II activity preferentially affected filopodial bundle disassembly distinct from the global effects of jasplakinolide on network turnover. We propose that growth cone retraction following turnover inhibition resulted from the persistence of myosin II contractility even as leading edge assembly rates decreased. The buildup of actin filaments in central regions combined with monomer depletion and reduced polymerization from barbed ends suggests a mechanism for the observed exponential decay in actin retrograde flow. Our results show that growth cone motility is critically dependent on continuous disassembly of the peripheral actin network.     

Arp2/3 Complex from Acanthamoeba Binds Profilin and Cross-links Actin Filaments

The Arp2/3 complex was first purified from Acanthamoeba castellanii by profilin affinity chromatography. The mechanism of interaction with profilin was unknown but was hypothesized to be mediated by either Arp2 or Arp3. Here we show that the Arp2 subunit of the complex can be chemically cross-linked to the actin-binding site of profilin. By analytical ultracentrifugation, rhodamine-labeled profilin binds Arp2/3 complex with a K_d of 7 μM, an affinity intermediate between the low affinity of profilin for barbed ends of actin filaments and its high affinity for actin monomers. These data suggest the barbed end of Arp2 is exposed, but Arp2 and Arp3 are not packed together in the complex exactly like two actin monomers in a filament. Arp2/3 complex also cross-links actin filaments into small bundles and isotropic networks, which are mechanically stiffer than solutions of actin filaments alone. Arp2/3 complex is concentrated at the leading edge of motile Acanthamoeba, and its localization is distinct from that of α-actinin, another filament cross-linking protein. Based on localization and actin filament nucleation and cross-linking activities, we propose a role for Arp2/3 in determining the structure of the actin filament network at the leading edge of motile cells.     

Identification of Novel Graded Polarity Actin Filament Bundles in Locomoting Heart Fibroblasts: Implications for the Generation of Motile Force

We have determined the structural organization and dynamic behavior of actin filaments in entire primary locomoting heart fibroblasts by S1 decoration, serial section EM, and photoactivation of fluorescence. As expected, actin filaments in the lamellipodium of these cells have uniform polarity with barbed ends facing forward. In the lamella, cell body, and tail there are two observable types of actin filament organization. A less abundant type is located on the inner surface of the plasma membrane and is composed of short, overlapping actin bundles (0.25–2.5 μm) that repeatedly alternate in polarity from uniform barbed ends forward to uniform pointed ends forward. This type of organization is similar to the organization we show for actin filament bundles (stress fibers) in nonlocomoting cells (PtK2 cells) and to the known organization of muscle sarcomeres. The more abundant type of actin filament organization in locomoting heart fibroblasts is mostly ventrally located and is composed of long, overlapping bundles (average 13 μm, but can reach up to about 30 μm) which span the length of the cell. This more abundant type has a novel graded polarity organization. In each actin bundle, polarity gradually changes along the length of the bundle. Actual actin filament polarity at any given point in the bundle is determined by position in the cell; the closer to the front of the cell the more barbed ends of actin filaments face forward.

By photoactivation marking in locomoting heart fibroblasts, as expected in the lamellipodium, actin filaments flow rearward with respect to substrate. In the lamella, all marked and observed actin filaments remain stationary with respect to substrate as the fibroblast locomotes. In the cell body of locomoting fibroblasts there are two dynamic populations of actin filaments: one remains stationary and the other moves forward with respect to substrate at the rate of the cell body.

This is the first time that the structural organization and dynamics of actin filaments have been determined in an entire locomoting cell. The organization, dynamics, and relative abundance of graded polarity actin filament bundles have important implications for the generation of motile force during primary heart fibroblast locomotion.

     

Actin cable distribution and dynamics arising from cross-linking,motor pulling,and filament turnover

The growth of fission yeast relies on the polymerization of actin filaments nucleated by formin For3p, which localizes at tip cortical sites. These actin filaments bundle to form actin cables that span the cell and guide the movement of vesicles toward the cell tips. A big challenge is to develop a quantitative understanding of these cellular actin structures. We used computer simulations to study the spatial and dynamical properties of actin cables. We simulated individual actin filaments as semiflexible polymers in three dimensions composed of beads connected with springs. Polymerization out of For3p cortical sites, bundling by cross-linkers, pulling by type V myosin, and severing by cofilin are simulated as growth, cross-linking, pulling, and turnover of the semiflexible polymers. With the foregoing mechanisms, the model generates actin cable structures and dynamics similar to those observed in live-cell experiments. Our simulations reproduce the particular actin cable structures in myoVΔ cells and predict the effect of increased myosin V pulling. Increasing cross-linking parameters generates thicker actin cables. It also leads to antiparallel and parallel phases with straight or curved cables, consistent with observations of cells overexpressing α-actinin. Finally, the model predicts that clustering of formins at cell tips promotes actin cable formation.     

Nucleated polymerization of actin from the membrane-associated ends of microvillar filaments in the intestinal brush border

We examined the nucleated polymerization of actin from the two ends of filaments that comprise the microvillus (MV) core in intestinal epithelial cells by electron microscopy. Three different in vitro preparations were used to nucleate the polymerization of muscle G- actin: (a) MV core fragments containing "barbed" and "pointed" filament ends exposed by shear during isolation, (b) isolated, membrane-intact brush borders, and (c) brush borders demembranated with Triton-X 100. It has been demonstrated that MV core fragments nucleate filament growth from both ends with a strong bias for one end. Here we identify the barbed end of the core fragment as the fast growing end by decoration with myosin subfragment one. Both cytochalasin B (CB) and Acanthamoeba capping protein block filament growth from the barbed but not the pointed end of MV core fragments. To examine actin assembly from the naturally occurring, membrane-associated ends of MV core filaments, isolated membrane-intact brush borders were used to nucleate the polymerization of G-actin. Addition of salt (75 mM KCl, 1 mM MgSO4) to brush borders preincubated briefly at low ionic strength with G- actin induced the formation of 0.2-0.4 micron "growth zones" at the tips of microvilli. The dense plaque at the tip of the MV core remains associated with the membrane and the presumed growing ends of the filaments. We also observed filament growth from the pointed ends of core filaments in the terminal web. We did not observe filament growth at the membrane-associated ends of core filaments when the latter were in the presence of 2 microM CB or if the low ionic strength incubation step was omitted. Addition of G-actin to demembranated brush borders, which retain the dense plaque on their MV tips, resulted in filament growth from both ends of the MV core. Again, 2 microM CB blocked filament growth from only the barbed (tip) end of the core. The dense plaque remained associated with the tip-end of the core in the presence of CB but usually was dislodged in control preparations where nucleated polymerization from the tip-end of the core occurred. Our results support the notion that microvillar assembly and changes in microvillar length could occur by actin monomer addition/loss at the barbed, membrane-associated ends of MV core filaments.     

Actin-filament stochastic dynamics mediated by ADF/cofilin

BACKGROUND: The rapid dynamics of actin filaments is a fundamental process that powers a large number of cellular functions. However, the basic mechanisms that control and coordinate such dynamics remain a central question in cell biology. To reach beyond simply defining the inventory of molecules that control actin dynamics and to understand how these proteins act synergistically to modulate filament turnover, we combined evanescent-wave microscopy with a biomimetic system and followed the behavior of single actin filaments in the presence of a physiologically relevant mixture of accessory proteins. This approach allows for the real-time visualization of actin polymerization and age-dependent filament severing. RESULTS: In the presence of actin-depolymerizing factor (ADF)/cofilin and profilin, actin filaments with a processive formin attached at their barbed ends were observed to oscillate between stochastic growth and shrinkage phases. Fragmentation of continuously growing actin filaments by ADF/cofilin is the key mechanism modulating the prominent and frequent shortening events. The net effect of continuous actin polymerization, driven by a processive formin that uses profilin-actin, and of ADF/cofilin-mediating severing that trims the aged ends of the growing filaments is an up to 155-fold increase in the rate of actin-filament turnover in vitro in comparison to that of actin alone. Lateral contact between actin filaments dampens the dynamics and favors actin-cable formation. A kinetic simulation accurately validates these observations. CONCLUSIONS: Our proposed mechanism for the control of actin dynamics is dominated by ADF/cofilin-mediated filament severing that induces a stochastic behavior upon individual actin filaments. When combined with a selection process that stabilizes filaments in bundles, this mechanism could account for the emergence and extension of actin-based structures in cells.     

Electron microscopic determination of the actin filament end at which cytochalasin B blocks monomer addition using the acrosomal actin bundle from horseshoe crab sperm

G-actin freed from exogenous ATP was added to the pieces of isolated acrosomal actin bundles from horseshoe crab sperm to form filaments as reported earlier (Tilney, L.G., Bonder, E.M., & DeRosier, D.J. (1981) J. Cell Biol. 90, 485-494). The growth of a filament was far more rapid at one end (the preferred end) than the other end. These ends were shown to correspond to the barbed and pointed ends, respectively, by decoration of the filaments with myosin subfragment 1. Cytochalasin B inhibited the monomer addition at the preferred end. This technique is useful in determining the ends to which actin filament end-binding proteins from nonmuscle cells bind, which are considered to regulate the actin polymerization in the cells.     

Analysis of the Actin–Myosin II System in Fish Epidermal Keratocytes: Mechanism of Cell Body Translocation

While the protrusive event of cell locomotion is thought to be driven by actin polymerization, the mechanism of forward translocation of the cell body is unclear. To elucidate the mechanism of cell body translocation, we analyzed the supramolecular organization of the actin–myosin II system and the dynamics of myosin II in fish epidermal keratocytes. In lamellipodia, long actin filaments formed dense networks with numerous free ends in a brushlike manner near the leading edge. Shorter actin filaments often formed T junctions with longer filaments in the brushlike area, suggesting that new filaments could be nucleated at sides of preexisting filaments or linked to them immediately after nucleation. The polarity of actin filaments was almost uniform, with barbed ends forward throughout most of the lamellipodia but mixed in arc-shaped filament bundles at the lamellipodial/cell body boundary. Myosin II formed discrete clusters of bipolar minifilaments in lamellipodia that increased in size and density towards the cell body boundary and colocalized with actin in boundary bundles. Time-lapse observation demonstrated that myosin clusters appeared in the lamellipodia and remained stationary with respect to the substratum in locomoting cells, but they exhibited retrograde flow in cells tethered in epithelioid colonies. Consequently, both in locomoting and stationary cells, myosin clusters approached the cell body boundary, where they became compressed and aligned, resulting in the formation of boundary bundles. In locomoting cells, the compression was associated with forward displacement of myosin features. These data are not consistent with either sarcomeric or polarized transport mechanisms of cell body translocation. We propose that the forward translocation of the cell body and retrograde flow in the lamellipodia are both driven by contraction of an actin–myosin network in the lamellipodial/cell body transition zone.     

Processive acceleration of actin barbed-end assembly by N-WASP

Neuronal Wiskott–Aldrich syndrome protein (N-WASP)–activated actin polymerization drives extension of invadopodia and podosomes into the basement layer. In addition to activating Arp2/3, N-WASP binds actin-filament barbed ends, and both N-WASP and barbed ends are tightly clustered in these invasive structures. We use nanofibers coated with N-WASP WWCA domains as model cell surfaces and single-actin-filament imaging to determine how clustered N-WASP affects Arp2/3-independent barbed-end assembly. Individual barbed ends captured by WWCA domains grow at or below their diffusion-limited assembly rate. At high filament densities, however, overlapping filaments form buckles between their nanofiber tethers and myosin attachment points. These buckles grew ∼3.4-fold faster than the diffusion-limited rate of unattached barbed ends. N-WASP constructs with and without the native polyproline (PP) region show similar rate enhancements in the absence of profilin, but profilin slows barbed-end acceleration from constructs containing the PP region. Increasing Mg²⁺ to enhance filament bundling increases the frequency of filament buckle formation, consistent with a requirement of accelerated assembly on barbed-end bundling. We propose that this novel N-WASP assembly activity provides an Arp2/3-independent force that drives nascent filament bundles into the basement layer during cell invasion.     

Tropomyosin regulates elongation by formin at the fast-growing end of the actin filament

The balance between dynamic and stable actin filaments is essential for the regulation of cellular functions including the determination of cell shape and polarity, cell migration, and cytokinesis. Proteins that regulate polymerization at the filament ends and filament stability confer specificity to actin filament structure and cellular function. The dynamics of the barbed, fast-growing end of the filament are controlled in space and time by both positive and negative regulators of actin polymerization. Capping proteins inhibit the addition and loss of subunits, whereas other proteins, including formins, bind at the barbed end and allow filament growth. In this work, we show that tropomyosin regulates dynamics at the barbed end. Tropomyosin binds to constructs of FRL1 and mDia2 that contain the FH2 domain and modulates formin-dependent capping of the barbed end by relieving inhibition of elongation by FRL1-FH1FH2, mDia1-FH2, and mDia2-FH2 in an isoform-dependent fashion. In this role, tropomyosin functions as an activator of formin. Tropomyosin also inhibits the binding of FRL1-FH1FH2 to the sides of actin filaments independent of the isoform. In contrast, tropomyosin does not affect the ability of capping protein to block the barbed end. We suggest that tropomyosin and formin act together to ensure the formation of unbranched actin filaments, protected from severing, that could be capped in stable cellular structures. This role, in addition to its cooperative control of myosin function, establishes tropomyosin as a universal regulator of the multifaceted actin cytoskeleton.     

The Diaphanous-related formin dDia2 is required for the formation and maintenance of filopodia

Formins have important roles in the nucleation of actin and the formation of linear actin filaments, but their role in filopodium formation has remained elusive. Dictyostelium discoideum Diaphanous-related formin dDia2 is enriched at the tips of filopodia and interacts with profilin II and Rac1. An FH1FH2 fragment of dDia2 nucleated actin polymerization and removed capping protein from capped filament ends. Genetic studies showed that dDia2 is important for cell migration as well as the formation, elongation and maintenance of filopodia. Here we provide evidence that dDia2 specifically controls filopodial dynamics by regulating actin turnover at the barbed ends of actin filaments.     

Actin dynamics at pointed ends regulates thin filament length in striated muscle

Regulation of actin dynamics at filament ends determines the organization and turnover of actin cytoskeletal structures. In striated muscle, it is believed that tight capping of the fast-growing (barbed) ends by CapZ and of the slow-growing (pointed) ends by tropomodulin (Tmod) stabilizes the uniform lengths of actin (thin) filaments in myofibrils. Here we demonstrate for the first time that both CapZ and Tmod are dynamic on the basis of the rapid incorporation of microinjected rhodamine-labelled actin (rho-actin) at both barbed and pointed ends and from the photobleaching of green fluorescent protein (GFP)-labelled Tmod. Unexpectedly, the inhibition of actin dynamics at pointed ends by GFP-Tmod overexpression results in shorter thin filaments, whereas the inhibition of actin dynamics at barbed ends by cytochalasin D has no effect on length. These data demonstrate that the actin filaments in myofibrils are relatively dynamic despite the presence of capping proteins, and that regulated actin assembly at pointed ends determines the length of thin filaments.     

A model for actin-filament length distribution in a lamellipod

A mathematical model is derived to describe the distributions of lengths of cytoskeletal actin filaments, along a 1 D transect of the lamellipod (or along the axis of a filopod) in an animal cell. We use the facts that actin filament barbed ends are aligned towards the cell membrane and that these ends grow rapidly in the presence of actin monomer as long as they are uncapped. Once a barbed end is capped, its filament tends to be degraded by fragmentation or depolymerization. Both the growth (by polymerization) and the fragmentation by actin-cutting agents are depicted in the model, which takes into account the dependence of cutting probability on the position along a filament. It is assumed that barbed ends are capped rapidly away from the cell membrane. The model consists of a system of discrete-integro-PDE's that describe the densities of barbed filament ends as a function of spatial position and length of their actin filament “tails”. The population of capped barbed ends and their trailing filaments is similarly represented. This formulation allows us to investigate hypotheses about the fragmentation and polymerization of filaments in a caricature of the lamellipod and compare theoretical and observed actin density profiles. Received: 19 May 2000 / Revised version: 12 March 2001 / Published online: 19 September 2001     

Actin filament uncapping localizes to ruffling lamellae and rocketing vesicles

Regulated actin filament assembly is critical for eukaryotic cell physiology. Actin filaments are polar structures, and those with free high affinity or barbed ends are crucial for actin dynamics and cell motility. Actin filament barbed-end-capping proteins inhibit filament elongation after binding, and their regulated disassociation is proposed to provide a source of free filament ends to drive processes dependent on actin polymerization. To examine whether dissociation of actin filament capping proteins occurs with the correct spatio-temporal constraints to contribute to regulated actin assembly in live cells, I measured the dissociation of an actin capping protein, gelsolin, from actin in cells using a variation of fluorescence resonance energy transfer (FRET). Uncapping was found to occur in cells at sites of active actin assembly, including protruding lamellae and rocketing vesicles, with the correct spatio-temporal properties to provide sites of actin filament polymerization during protrusion. These observations are consistent with models where uncapping of existing filaments provides sites of actin filament elongation.     

The ALP-Enigma Protein ALP-1 Functions in Actin Filament Organization to Promote Muscle Structural Integrity in Caenorhabditis elegans

Mutations that affect the Z-disk–associated ALP-Enigma proteins have been linked to human muscular and cardiac diseases. Despite their clear physiological significance for human health, the mechanism of action of ALP-Enigma proteins is largely unknown. In Caenorhabditis elegans, the ALP-Enigma protein family is encoded by a single gene, alp-1; thus C. elegans provides an excellent model to study ALP-Enigma function. Here we present a molecular and genetic analysis of ALP-Enigma function in C. elegans. We show that ALP-1 and α-actinin colocalize at dense bodies where actin filaments are anchored and that the proper localization of ALP-1 at dense bodies is dependent on α-actinin. Our analysis of alp-1 mutants demonstrates that ALP-1 functions to maintain actin filament organization and participates in muscle stabilization during contraction. Reducing α-actinin activity enhances the actin filament phenotype of the alp-1 mutants, suggesting that ALP-1 and α-actinin function in the same cellular process. Like α-actinin, alp-1 also interacts genetically with a connectin/titin family member, ketn-1, to provide mechanical stability for supporting body wall muscle contraction. Taken together, our data demonstrate that ALP-1 and α-actinin function together to stabilize actin filaments and promote muscle structural integrity.     

α-Actinin and Filamin Cooperatively Enhance the Stiffness of Actin Filament Networks

Background

The close subcellular proximity of different actin filament crosslinking proteins suggests that these proteins may cooperate to organize F-actin structures to drive complex cellular functions during cell adhesion, motility and division. Here we hypothesize that α-actinin and filamin, two major F-actin crosslinking proteins that are both present in the lamella of adherent cells, display synergistic mechanical functions.

Methodology/Principal Findings

Using quantitative rheology, we find that combining α-actinin and filamin is much more effective at producing elastic, solid-like actin filament networks than α-actinin and filamin separately. Moreover, F-actin networks assembled in the presence of α-actinin and filamin strain-harden more readily than networks in the presence of either α-actinin or filamin.

Significance

These results suggest that cells combine auxiliary proteins with similar ability to crosslink filaments to generate stiff cytoskeletal structures, which are required for the production of internal propulsive forces for cell migration, and that these proteins do not have redundant mechanical functions.     

Regulation of actin dynamics in rapidly moving cells: a quantitative analysis

We develop a mathematical model that describes key details of actin dynamics in protrusion associated with cell motility. The model is based on the dendritic-nucleation hypothesis for lamellipodial protrusion in nonmuscle cells such as keratocytes. We consider a set of partial differential equations for diffusion and reactions of sequestered actin complexes, nucleation, and growth by polymerization of barbed ends of actin filaments, as well as capping and depolymerization of the filaments. The mechanical aspect of protrusion is based on an elastic polymerization ratchet mechanism. An output of the model is a relationship between the protrusion velocity and the number of filament barbed ends pushing the membrane. Significantly, this relationship has a local maximum: too many barbed ends deplete the available monomer pool, too few are insufficient to generate protrusive force, so motility is stalled at either extreme. Our results suggest that to achieve rapid motility, some tuning of parameters affecting actin dynamics must be operating in the cell.     

The biochemical composition of the actomyosin network sets the magnitude of cellular traction forces

The regulation of cellular force production relies on the complex interplay between a well-conserved set of proteins of the cytoskeleton: actin, myosin, and α-actinin. Despite our deep knowledge of the role of these proteins in force production at the molecular scale, our understanding of the biochemical regulation of the magnitude of traction forces generated at the entire-cell level has been limited, notably by the technical challenge of measuring traction forces and the endogenous biochemical composition in the same cell. In this study, we developed an alternative Traction-Force Microscopy (TFM) assay, which used a combination of hydrogel micropatterning to define cell adhesion and shape and an intermediate fixation/immunolabeling step to characterize strain energies and the endogenous protein contents in single epithelial cells. Our results demonstrated that both the signal intensity and the area of the Focal Adhesion (FA)–associated protein vinculin showed a strong positive correlation with strain energy in mature FAs. Individual contents from actin filament and phospho-myosin displayed broader deviation in their linear relationship to strain energies. Instead, our quantitative analyzes demonstrated that their relative amount exhibited an optimum ratio of phospho-myosin to actin, allowing maximum force production by cells. By contrast, although no correlation was identified between individual α-actinin content and strain energy, the ratio of α-actinin to actin filaments was inversely related to strain energy. Hence, our results suggest that, in the cellular model studied, traction-force magnitude is dictated by the relative numbers of molecular motors and cross-linkers per actin filament, rather than the amounts of an individual component in the cytoskeletal network. This assay offers new perspectives to study in more detail the complex interplay between the endogenous biochemical composition of individual cells and the force they produce.