Properties of an excitable dynein model for bend propagation in cilia and flagella

Murase & Shimizu (1986, J. theor. Biol. 119, 409) introduced an excitable dynein-microtubule system based on a three-state mechanochemical cycle of dynein to demonstrate bend propagation in the absence of a curvature control mechanism. To examine the essential behavior of this class of models in a viscous fluid, we have represented the force generated by the complex dynein mechanochemistry by a formal model consisting of "force" and "activation" functions vs. sliding distance. Since the model has excitable properties with threshold phenomena and hysteresis switching between two opposed subsystems, it closely resembles the more realistic dynein kinetic scheme in its overall properties but is specified by fewer parameters. This model displays both bend initiation and bend propagation when the filaments at the basal end are either fixed or free to slide. A passive region is necessary at one end of the axoneme in order to obtain stable wave propagation; bends propagate towards the end with the passive region. Stable bend propagation is highly sensitive to small perturbations in external force distribution.     

Thinking about flagellar oscillation

Bending of cilia and flagella results from sliding between the microtubular outer doublets, driven by dynein motor enzymes. This review reminds us that many questions remain to be answered before we can understand how dynein-driven sliding causes the oscillatory bending of cilia and flagella. Does oscillation require switching between two distinct, persistent modes of dynein activity? Only one mode, an active forward mode, has been characterized, but an alternative mode, either inactive or reverse, appears to be required. Does switching between modes use information from curvature, sliding direction, or both? Is there a mechanism for reciprocal inhibition? Can a localized capability for oscillatory sliding become self-organized to produce the metachronal phase differences required for bend propagation? Are interactions between adjacent dyneins important for regulation of oscillation and bend propagation? Cell Motil. Cytoskeleton 2008. (c) 2008 Wiley-Liss, Inc.     

Computer simulation of flagellar movement: VII. Conventional but functionally different cross-bridge models for inner and outer arm dyneins can explain the effects of outer arm dynein removal

Outer arm dynein removal from flagella by genetic or chemical methods causes decreased frequency and power, but little change in bending pattern. These results suggest that outer arm dynein operates within bends to increase the speed of bend propagation, but does not produce forces that alter the bending pattern established by inner arm dyneins. A flagellar model incorporating different cross-bridge models for inner and outer arm dyneins has been examined. The inner arm dynein model has a hyperbolic force-velocity curve, with a maximum average force at 0 sliding velocity of about 14 pN for each 96 nm group of inner arm dyneins. The outer arm dynein model has a very different force-velocity curve, with a maximum force at about 10-15% of V(max). The outer arm dynein model is adjusted so that the unloaded sliding velocity for a realistic mixture of inner and outer arm dyneins is twice the unloaded sliding velocity for the inner arm dynein model alone. With these cross-bridge models, a flagellar model can be obtained that reduces its sliding velocity and frequency by approximately 50% when outer arm dyneins are removed, with little change in bending pattern. The addition of outer arm dyneins, therefore, gives an approximately 4-fold increase in power output against viscous resistances, and outer arm dyneins may generate 90% or more of the power output. Cell Motil.     

Flagellar oscillation: a commentary on proposed mechanisms

Eukaryotic flagella and cilia have a remarkably uniform internal ‘engine’ known as the ‘9+2’ axoneme. With few exceptions, the function of cilia and flagella is to beat rhythmically and set up relative motion between themselves and the liquid that surrounds them. The molecular basis of axonemal movement is understood in considerable detail, with the exception of the mechanism that provides its rhythmical or oscillatory quality. Some kind of repetitive ‘switching’ event is assumed to occur; there are several proposals regarding the nature of the ‘switch’ and how it might operate. Herein I first summarise all the factors known to influence the rate of the oscillation (the beating frequency). Many of these factors exert their effect through modulating the mean sliding velocity between the nine doublet microtubules of the axoneme, this velocity being the determinant of bend growth rate and bend propagation rate. Then I explain six proposed mechanisms for flagellar oscillation and review the evidence on which they are based. Finally, I attempt to derive an economical synthesis, drawing for preference on experimental research that has been minimally disruptive of the intricate structure of the axoneme. The ‘provisional synthesis' is that flagellar oscillation emerges from an effect of passive sliding direction on the dynein arms. Sliding in one direction facilitates force‐generating cycles and dynein‐to‐dynein synchronisation along a doublet; sliding in the other direction is inhibitory. The direction of the initial passive sliding normally oscillates because it is controlled hydrodynamically through the alternating direction of the propulsive thrust. However, in the absence of such regulation, there can be a perpetual, mechanical self‐triggering through a reversal of sliding direction due to the recoil of elastic structures that deform as a response to the prior active sliding. This provisional synthesis may be a useful basis for further examination of the problem.     

A model of flagellar movement based on cooperative dynamics of dynein-tubulin cross-bridges

A theoretical model based on molecular mechanisms of both dynein cross-bridges and radial spokes is used to study bend propagation by eukaryotic flagella. Though nine outer doublets are arranged within an axoneme, a simplified model with four doublets is constructed on the assumption that cross-bridges between two of the four doublets are opposed to those between the other two, corresponding to the geometric array of cross-bridges on the 6-9 and the 1-4 doublets in the axoneme. We also assume that external viscosity is zero, whereas internal viscosity is non-zero in order to reduce numerical complexity. For demonstrating flagellar movement, computer simulations are available by dividing a long flagellum into many straight segments. Considering the fact that dynein cross-bridge spacing is almost equal to attachment site spacing, we may use a localized cross-bridge distribution along attachment sites in each straight segment. Dynamics of cross-bridges are determined by a three-state model, and effects of radial spokes are represented by a periodic mechanical potential whose periodicity is considered to be a stroke distance of the radial spoke. First of all, we examine the model of a short segment to know basic properties of the system. Changing parameters relating to "activation" of cross-bridges, our model demonstrates various phenomena; for example "excitable properties with threshold phenomena" and "limit cycle oscillation". Here, "activation" and "inactivation" (i.e. switching mechanisms) between a pair of oppositely-directed cross-bridges are essential for generation of excitable or oscillatory properties. Next, the model for a flagellar segment is incorporated into a flagellum with a whole length to show bending movement. When excitable properties of cross-bridges, not oscillatory properties, are provided along the length of the flagellum and elastic links between filaments are presented at the base, then our model can demonstrate self-organization of bending waves as well as wave propagation without special feedback control by the curvature of the flagellum. Here, "cooperative interaction" between adjacent short segments, based on "cooperative dynamics" of cross-bridges, is important for wave propagation.     

Computer simulation of flagellar movement IX. Oscillation and symmetry breaking in a model for short flagella and nodal cilia

A computer model of flagella in which oscillation results from regulation of active sliding force by sliding velocity can simulate the movements of very short flagella and cilia. Of particular interest are the movements of the short (2-3 microm) nodal cilia of the mammalian embryo, which determine the development of the asymmetry of the internal organs. These cilia must generate a counterclockwise (viewed from base to tip) circling motion. A three-dimensional computer model, with active force generated by a simple mathematical formulation and regulated by sliding velocity, can generate this circling motion if a time delay process is included in the control specification. Without the introduction of a symmetry-breaking mechanism, the computer models start randomly in either direction, and maintain either clockwise or counterclockwise circling. Symmetry can be broken by at least two mechanisms: (1) control of dynein activity on one outer doublet by sliding velocity can be influenced by the sliding velocity experienced on an adjacent outer doublet, or (2) a constant twist of the axoneme caused by an off-axis component of dynein force. This second mechanism appears more reasonable, but its effectiveness is highly dependent upon specifications for the elastic resistances of the model. These symmetry-breaking mechanisms need to be present only at the beginning of circling. With these models, once a circling direction is established, it remains stable even if the symmetry-breaking mechanism is removed.     

Computer simulation of flagellar movement. VI. Simple curvature-controlled models are incompletely specified.

Computer simulation is used to examine a simple flagellar model that will initiate and propagate bending waves in the absence of viscous resistances. The model contains only an elastic bending resistance and an active sliding mechanism that generates reduced active shear moment with increasing sliding velocity. Oscillation results from a distributed control mechanism that reverses the direction of operation of the active sliding mechanism when the curvature reaches critical magnitudes in either direction. Bend propagation by curvature-controlled flagellar models therefore does not require interaction with the viscous resistance of an external fluid. An analytical examination of moment balance during bend propagation by this model yields a solution curve giving values of frequency and wavelength that satisfy the moment balance equation and give uniform bend propagation, suggesting that the model is underdetermined. At 0 viscosity, the boundary condition of 0 shear rate at the basal end of the flagellum during the development of new bends selects the particular solution that is obtained by computer simulations. Therefore, the details of the pattern of bend initiation at the basal end of a flagellum can be of major significance in determining the properties of propagated bending waves in the distal portion of a flagellum. At high values of external viscosity, the model oscillates at frequencies and wavelengths that give approximately integral numbers of waves on the flagellum. These operating points are selected because they facilitate the balance of bending moments at the ends of the model, where the external viscous moment approaches 0. These mode preferences can be overridden by forcing the model to operate at a predetermined frequency. The strong mode preferences shown by curvature-controlled flagellar models, in contrast to the weak or absent mode preferences shown by real flagella, therefore do not demonstrate the inapplicability of the moment-balance approach to real flagella. Instead, they indicate a need to specify additional properties of real flagella that are responsible for selecting particular operating points.     

Direction of force generated by the inner row of dynein arms on flagellar microtubules

Our goal was to determine the direction of force generation of the inner dynein arms in flagellar axonemes. We developed an efficient means of extracting the outer row of dynein arms in demembranated sperm tail axonemes, leaving the inner row of dynein arms structurally and functionally intact. Sperm tail axonemes depleted of outer arms beat at half the beat frequency of sperm tails with intact arms over a wide range of ATP concentrations. The isolated, outer arm-depleted axonemes were induced to undergo microtubule sliding in the presence of ATP and trypsin. Electron microscopic analysis of the relative direction of microtubule sliding (see Sale, W. S. and P. Satir, 1977, Proc. Natl. Acad. Sci. USA, 74:2045-2049) revealed that the doublet microtubule with the row of inner dynein arms, doublet N, always moved by sliding toward the proximal end of the axoneme relative to doublet N + 1. Therefore, the inner arms generate force such that doublet N pushes doublet N + 1 tipward. This is the same direction of microtubule sliding induced by ATP and trypsin in axonemes having both inner and outer dynein arms. The implications of this result for the mechanism of ciliary bending and utility in functional definition of cytoplasmic dyneins are discussed.     

Bend propagation in flagella. I. Derivation of equations of motion and their simulation.

A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.     

Structural conformation of ciliary dynein arms and the generation of sliding forces in Tetrahymena cilia

The sliding tubule model of ciliary motion requires that active sliding of microtubules occur by cyclic cross-bridging of the dynein arms. When isolated, demembranated Tetrahymena cilia are allowed to spontaneously disintegrate in the presence of ATP, the structural conformation of the dynein arms can be clearly resolved by negative contrast electron microscopy. The arms consist of three structural subunits that occur in two basic conformations with respect to the adjacent B subfiber. The inactive conformation occurs in the absence of ATP and is characterized by a uniform, 32 degrees base-directed polarity of the arms. Inactive arms are not attached to the B subfiber of adjacent doublets. The bridged conformation occurs strictly in the presence of ATP and is characterized by arms having the same polarity as inactive arms, but the terminal subunit of the arms has become attached to the B subfiber. In most instances the bridged conformation is accompanied by substantial tip-directed sliding displacement of the bridged doublets. Because the base-directed polarity of the bridged arms is opposite to the direction required for force generation in these cilia and because the bridges occur in the presence of ATP, it is suggested that the bridged conformation may represent the initial attachment phase of the dynein cross-bridge cycle. The force-generating phase of the cycle would then require a tip-directed deflection of the arm subunit attached to the B subfiber.     

Induction of beating by imposed bending or mechanical pulse in demembranated, motionless sea urchin sperm flagella at very low ATP concentrations

A basic feature of the movement of eukaryotic flagella is oscillation. Although flagellar oscillation is thought to be regulated by a self-regulatory feedback system including the mechanical signal of bending itself, the mechanism regulating the dynein motile activity to produce oscillation is not well understood. To elucidate the mechanism, we developed a new experimental system which allowed us to analyze the conditions necessary for the induction of oscillation. When a mechanical signal of bending or a pulse was applied by micromanipulation to a demembranated motionless sea urchin sperm flagellar axoneme at very low ATP concentrations (1-3 microM), a localized pair of bends was induced. The bend formation was often followed by further responses including propagation of the distal bend of paired bends, growth and propagation of the paired bends, and cyclical beating. The beating was induced at 2.0 microM or higher concentrations of ATP, but appeared even at 1.5 microM ATP if a few muM of ADP was also present. When the proximal half of a flagellum was attached to a microneedle, beating could not be induced in the distal free region at 2 microM ATP. These results suggest that mechanical signal is involved in the mechanism regulating the motile activity of dynein to produce oscillation. Our results also showed that the presence of a small amount of ADP and the axial difference along the flagellum are factors essential for the induction of flagellar oscillation.     

The covalent oscillator: A paradigm accounting for the sliding/bending mechanism and wave propagation in cilia and flagella

Summary— In most models of wave propagation in eucaryotic flagella and cilia, a clear distinction is made between the dynein dependent microtubule sliding which represents the oscillatory motor and the bending mechanism which regulates wave propagation. Little is known about the physical elements regulating the latter: in the present model, the bending propagation is postulated to be supported by an open/close cyclic mechanism protease/ligase dependent, which involves transient covalent links between adjacent microtubular doublets; this open/close cycle propagates in register with the powering action of the dynein motor along the exoneme. The implications of the model are discussed in relation to previous data which involve protease/ligase in the axonemal function as well as other data which can be integrated by the proposed model.     

A novel motility pattern in quail spermatozoa with implications for the mechanism of flagellar beating

Background information. The spermatozoon of the quail (Coturnix coturnix L., var japonica) has a ‘9+2’ flagellum that is unusually long. When it moves in a viscous medium, near to the coverslip, it develops a meander waveform. Because of the high viscosity, the meander bends are static in relation to the field of view; bend propagation is therefore manifest as the forward movement of the flagellum through the meander shape. At the same time, the origin of the oscillation typically shifts proximally in a stepwise fashion. These movements have been analysed in the hope of contributing to the resolution of problems in flagellar mechanics. Results. (1) Meander waves originate from spontaneous sigmoid bend complexes. (2) On a given flagellum, fully developed meander bends are uniform in their large angle, curvature and propagation speed; interbends can vary in length and shape. (3) No intra‐axonemal sliding is transmitted through formed bends; sliding related to new bends is accommodated proximally. (4) Sliding reversal is initiated at a threshold shear angle of approx. 1 rad. (5) The arc wavespeed is the product of the arc wavelength and the beat frequency. (6) Physical obstruction to bend development causes a pause in the oscillation. (7) New bend initiation can thus be dissociated from bend propagation on the distal flagellum. (8) The steps in the forward advance of the oscillation site occur during the early phase of bend growth. Conclusions. (1) The main conclusion is that, in meander waves, the mechanical basis of the oscillation appears to be that the propulsive thrust arising from bend propagation acts as a bending stress to trigger sliding reversal, thus perpetuating the rhythmic beating. (2) Oscillations can originate at any position, provided the position is distal to a location where doublet sliding is restrained. (3) Meander waves are an example of new bend development without ‘paradoxical’ classes of sliding.     

A Model Describing Bending in Flagella

In Part I of this paper, we present a modelto account for the force generationproducing bending, and the formation of awaveform in sperm flagella. The model isbased on the observation that dimers, andhence microtubules, possess dipole moments.The electric field these dipoles produce isthe source for storing mechanical work indynein arms. The mechanical work is thenreleased and act on the doublets to producea distally directed force with the resultthat bending occurs. The model described isconsistent with experimental observationsreported in the literature. The flexuralrigidity of a dynein arm is alsocalculated. In Part II of this paper, theconsequences of the bending mechanism arediscussed. It is shown that the sum offorces from dynein arms acting distallyalong doublet microtubules in a flagellumis essentially zero when all dyneins areattached thus resulting in the rigor state.The waveform in a flagellum occurs if oneof the sets of bending moments is zero,that is, a row of dyneins are detached oversome distance along the flagellum. Thedirection of the bend in the waveform isdetermined by which set of dynein arms aredetached with respect to the verticalmedian plane of the flagellum. Thepropagation of a bending wave is the resultof a moving region in which alternate sidesfrom the vertical median plane haveinactive dynein arms. The processes bywhich this moving region occurs and therelationship of the above results to thepropulsion of the flagellum are notconsidered.     

Computer simulation of flagellar movement VIII: coordination of dynein by local curvature control can generate helical bending waves

Computer simulations have been carried out with a model flagellum that can bend in three dimensions. A pattern of dynein activation in which regions of dynein activity propagate along each doublet, with a phase shift of approximately 1/9 wavelength between adjacent doublets, will produce a helical bending wave. This pattern can be termed "doublet metachronism." The simulations show that doublet metachronism can arise spontaneously in a model axoneme in which activation of dyneins is controlled locally by the curvature of each outer doublet microtubule. In this model, dyneins operate both as sensors of curvature and as motors. Doublet metachronism and the chirality of the resulting helical bending pattern are regulated by the angular difference between the direction of the moment and sliding produced by dyneins on a doublet and the direction of the controlling curvature for that doublet. A flagellum that is generating a helical bending wave experiences twisting moments when it moves against external viscous resistance. At high viscosities, helical bending will be significantly modified by twist unless the twist resistance is greater than previously estimated. Spontaneous doublet metachronism must be modified or overridden in order for a flagellum to generate the planar bending waves that are required for efficient propulsion of spermatozoa. Planar bending can be achieved with the three-dimensional flagellar model by appropriate specification of the direction of the controlling curvature for each doublet. However, experimental observations indicate that this "hard-wired" solution is not appropriate for real flagella.     

Testing the geometric clutch hypothesis

The Geometric Clutch hypothesis is based on the premise that transverse forces (t-forces) acting on the outer doublets of the eukaryotic axoneme coordinate the action of the dynein motors to produce flagellar and ciliary beating. T-forces result from tension and compression on the outer doublets when a bend is present on the flagellum or cilium. The t-force acts to pry the doublets apart in an active bend, and push the doublets together when the flagellum is passively bent and thus could engage and disengage the dynein motors. Computed simulations of this working mechanism have reproduced the beating pattern of simple cilia and flagella, and of mammalian sperm. Cilia-like beating, with a clearly defined effective and recovery stroke, can be generated using one uniformly applied switching algorithm. When the mechanical properties and dimensions appropriate to a specific flagellum are incorporated into the model the same algorithm can simulate a sea urchin or bull sperm-like beat. The computed model reproduces many of the observed behaviors of real flagella and cilia. The model can duplicate the results of outer arm extraction experiments in cilia and predicted two types of arrest behavior that were verified experimentally in bull sperm. It also successfully predicted the experimentally determined nexin elasticity. Calculations based on live and reactivated sea urchin and bull sperm yielded a value of 0.5 nN/microm for the t-force at the switch-point. This is a force sufficient to overcome the shearing force generated by all the dyneins on one micron of outer doublet. A t-force of this magnitude should produce substantial distortion of the axoneme at the switch-point, especially in spoke or spoke-head deficient motile flagella. This concrete and verifiable prediction is within the grasp of recent advances in imaging technology, specifically cryoelectron microscopy and atomic force microscopy.     

Does axonemal dynein push, pull, or oscillate?

Dynein is the molecular motor that provides motive force in cilia and flagella. Dynein is anchored to the A-subtubule of the outer doublets by a club-shaped extension called the stem, which supports the large globular head of the molecule. Dynein forms an attachment or cross-bridge to the B-subtubule of the adjacent outer doublet through a slender appendage extending from the head that is called the stalk or alternately the B-link. It is generally thought that the B-link mediates the interdoublet transfer of force that bends the flagellum. This requires that energy released at the site of ATP hydrolysis, located in the globular head, be transferred as mechanical work to the microtubule binding site at the tip of the B-link. It has been proposed that this is accomplished by a sideways or rotational translocation of the B-link caused by a rotation of the globular head. An estimate of the stiffness of the B-link and stem derived from the recently published data of Burgess et al. [2003: Nature 421:715-718] yields a maximum stiffness of 0.47 pN/nm for the B-link and 0.1 pN/nm for the stem. The B-link stiffness would allow transfer of 3.8 pN of force in response to an 8-nm displacement of the B-link tip. However, if as proposed the globular head of the dynein heavy chain is supported by the stem, the B-link and stem elasticity are in series. Thus, the flexibility of the stem would limit the force that can be transferred laterally by the entire dynein heavy chain to 0.6 pN at 8 nm displacement. This force is insufficient to support flagellar motility. So, if the stem were the only support for the globular head, then force would have to be transmitted linearly along the axis defined by the stem and B-link. Because this configuration is never observed, the hypothesis that dynein generates force by lateral displacement of the B-link is more attractive, but requires that the globular head of the dynein is stabilized by an additional means of support during the power stroke. We propose that the microtubule affinity of the tip of the B-link is independent of the ATP-dependent powerstroke, and that detachment from the B-subtubule is regulated by tension. A dynein cross-bridge cycle that incorporates an anchored head, together with a ratchet-like mechanism for microtubule translocation by the B-link, would have distinct advantages. This mechanism may reconcile dynein oscillation and interdoublet sliding within one cross-bridge mechanism.     

Functional state of the axonemal dyneins during flagellar bend propagation

When mouse spermatozoa swim in media of high viscosity, additional waves of bending are superimposed on the primary traveling wave. The additional (secondary) waves are relatively small in scale and high in frequency. They originate in the proximal part of the interbend regions. The initiation of secondary bending happens only in distal parts of the flagellum. The secondary waves propagate along the interbends and then tend to die out as they encounter the next-most-distal bend of the primary wave, if that bend exceeds a certain angle. The principal bends of the primary wave, being of greater angle than the reverse bends, strongly resist invasion by the secondary waves; when a principal bend of the primary wave propagates off the flagellar tip, the secondary wave behind it suddenly increases in amplitude. We claim that the functional state of the dynein motors in relation to the primary wave can be deduced from their availability for recruitment into secondary wave activity. Therefore, only the dyneins in bends are committed functionally to the maintenance and propagation of the flagellar wave; dyneins in interbend regions are not functionally committed in this way. We equate functional commitment with tension-generating activity, although we argue that the regions of dynein thus engaged nevertheless permit sliding displacements between the doublets.     

Protein friction exerted by motor enzymes through a weak-binding interaction.

Recently Vale et al. (1989, Cell 59, 915-925.) reported an observation of the one-dimensional Brownian movement of microtubules bound to flagellar dynein through a weak-binding interaction. In this study, we propose a theoretical model of this phenomenon. Our model consists of a rigid microtubule associated with a number of elastic dynein heads through a weak-binding interaction at equilibrium. The model implies that (1) the Brownian motion of the microtubule is not directly driven by the atomic collision of the solvent particles, but is driven by the thermally-generated structural fluctuations of the dynein heads which interact with the microtubule; (2) dynein heads through a weak-binding interaction exert a frictional drag force on the sliding motion of the microtubule and the drag force is proportional to the sliding velocity the same as in hydrodynamic viscous friction. This protein friction, with such viscous-like characteristics, may well play a role as a velocity-limiting factor in the normal ATP-induced sliding movement of motile proteins.     

The counterbend phenomenon in dynein-disabled rat sperm flagella and what it reveals about the interdoublet elasticity

Rat sperm that have been rendered passive by disabling the dynein motors with 50 muM sodium metavanadate and 0.1 mM ATP exhibit an interesting response to imposed bending. When the proximal flagellum is bent with a microprobe, the portion of the flagellum distal to the probe contact point develops a bend in the direction opposite the imposed bend. This "counterbend" is not compatible with a simple elastic beam. It can be satisfactorily explained by the sliding tubule model of flagellar structure but only if there are permanent elastic connections between the outer doublets of the axoneme. The elastic component that contributes the bending torque for the counterbend does not reset to a new equilibrium position after an imposed bend but returns the flagellum to a nearly straight or slightly curved final position after release from the probe. This suggests it is based on fixed, rather than mobile, attachments. It is also disrupted by elastase or trypsin digestion, confirming that it is dependent on a protein linkage. Adopting the assumption that the elasticity is attributed to the nexin links that repeat at 96 nm intervals, we find an apparent elasticity for each link that ranges from 1.6 to 10 x 10(-5) N/m. However, the elasticity is nonlinear and does not follow Hooke's law but appears to decrease with increased stretch. In addition, the responsible elastic elements must be able to stretch to more than 10 times their resting length without breakage to account for the observed counterbend formation. Elasticity created by some type of protein unfolding may be the only viable explanation consistent with both the extreme capacity for extension and the nonlinear character of the restoring force that is observed.