On the contribution of dynein-like activity to twisting in a three-dimensional sliding filament model.

It has been shown (Hines, M., and J. J. Blum, Biophys.J., 1984, 46:559-565) that passive moment-bearing links do not contribute appreciable twist resistance to an axoneme nor do they cause appreciable twisting in response to internal shear forces. We now examine the contribution of active moment-bearing links such as dynein arms to the generation of twist within an axoneme. The dynein model used causes distal sliding of the adjacent doublet by a force dependent on the angle of attachment of the arms. Attachment of the arms occurs at a specified angle relative to the angle of minimum potential energy. The steady state shape consistent with the forces applied by the attached dyneins is computed. It is shown that the twist generated in an active region is counterclockwise as viewed from tip to base and therefore accumulates at the end of the axoneme. For realistic forces and twist resistances, cumulative twist should not exceed a few degrees.     

On the contribution of moment-bearing links to bending and twisting in a three-dimensional sliding filament model

Previously (Hines, M., and J.J. Blum 1983, Biophys. J., 41:67-79), a method was developed that allowed one to compute curvature and twist for a three-dimensional sliding filament model. In that formalism it was difficult to specify the shear and bending moments arising from moment-bearing interfilament links such as fixed 5-6 bridges or dyneins. Euler's equation offers a straightforward method for computing these bending and shear moments when the potential energy stored in the links as a function of axonemal shape is specified. We used this approach to examine the effect of 5-6 bridges on curvature and twist for several distributions of internal shear moments. Twist changes the angle that a link makes with a doublet and thus in some circumstances may reduce the potential energy stored in those links. Twist is a second-order effect proportional to the square of the distance between an outer doublet and the neutral axis. Fixed links will not generate twist if they are symmetrically located around the axoneme.     

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.     

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.     

Effect of anisotropic bending rigidity and finite twisting rigidity on statistical properties of DNA model filaments

The persistence length and effective long-range bending rigidity are derived for a discrete model of an anisotropically bending filament and shown to be independent of the torsional rigidity. The twisting persistence length is found to be independent of the anisotropic bending rigidity. Other statistical properties are briefly discussed, including the dependence of tangent vector projections on contour length. The dependence of a tensor contraction on contour length is derived for an isotropically bending filament with no equilibrium twist.     

A physical model of microtubule sliding in ciliary axonemes.

Ciliary movement is caused by coordinated sliding interactions between the peripheral doublet microtubules of the axoneme. In demembranated organelles treated with trypsin and ATP, this sliding can be visualized during progressive disintegration. In this paper, microtubule sliding behavior resulting from various patterns of dynein arm activity and elastic link breakage is determined using a simplified model of the axoneme. The model consists of a cylindrical array of microtubules joined, initially, by elastic links, with the possibility of dynein arm interaction between microtubules. If no elastic links are broken, sliding can produce stable distortion of the model, which finds application to straight sections of a motile cilium. If some elastic links break, the model predicts a variety of sliding patterns, some of which match, qualitatively, the observed disintegration behavior of real axonemes. Splitting of the axoneme is most likely to occur between two doublets N and N + 1 when either the arms on doublet N + 1 are active and arms on doublet N are inactive or arms on doublet N - 1 are active while arms on doublet N are inactive. The analysis suggests further experimental studies which, in conjunction with the model, will lead to a more detailed understanding of the sliding mechanism, and will allow the mechanical properties of some axonemal components to be evaluated.     

Mechanical Heterogeneity Favors Fragmentation of Strained Actin Filaments

We present a general model of actin filament deformation and fragmentation in response to compressive forces. The elastic free energy density along filaments is determined by their shape and mechanical properties, which were modeled in terms of bending, twisting, and twist-bend coupling elasticities. The elastic energy stored in filament deformation (i.e., strain) tilts the fragmentation-annealing reaction free-energy profile to favor fragmentation. The energy gradient introduces a local shear force that accelerates filament intersubunit bond rupture. The severing protein, cofilin, renders filaments more compliant in bending and twisting. As a result, filaments that are partially decorated with cofilin are mechanically heterogeneous (i.e., nonuniform) and display asymmetric shape deformations and energy profiles distinct from mechanically homogenous (i.e., uniform), bare actin, or saturated cofilactin filaments. The local buckling strain depends on the relative size of the compliant segment as well as the bending and twisting rigidities of flanking regions. Filaments with a single bare/cofilin-decorated boundary localize energy and force adjacent to the boundary, within the compliant cofilactin segment. Filaments with small cofilin clusters were predicted to fragment within the compliant cofilactin rather than at boundaries. Neglecting contributions from twist-bend coupling elasticity underestimates the energy density and gradients along filaments, and thus the net effects of filament strain to fragmentation. Spatial confinement causes compliant cofilactin segments and filaments to adopt higher deformation modes and store more elastic energy, thereby promoting fragmentation. The theory and simulations presented here establish a quantitative relationship between actin filament fragmentation thermodynamics and elasticity, and reveal how local discontinuities in filament mechanical properties introduced by regulatory proteins can modulate both the severing efficiency and location along filaments. The emergent behavior of mechanically heterogeneous filaments, particularly under confinement, emphasizes that severing in cells is likely to be influenced by multiple physical and chemical factors.     

Direct measurement of inter-doublet elasticity in flagellar axonemes.

The outer doublet microtubules in ciliary and flagellar axonemes are presumed to be connected with each other by elastic links called the inter-doublet links or the nexin links, but it is not known whether there actually are such elastic links. In this study, to detect the elasticity of the putative inter-doublet links, shear force was applied to Chlamydomonas axonemes with a fine glass needle and the longitudinal elasticity was determined from the deflection of the needle. Wild-type axonemes underwent a high-frequency, nanometer-scale vibration in the presence of ATP. When longitudinal shear force was applied, the average position of the needle tip attached to the axoneme moved linearly with the force applied, yielding an estimate of spring constant of 2.0 (S.D.: 0.8) pN/nm for 1 microm of axoneme. This value did not change in the presence of vanadate, i.e., when dynein does not form strong cross bridges. In contrast, it was at least five times larger when ATP was absent, i.e., when dynein forms strong cross bridges. The measured elasticity did not significantly differ in various mutant axonemes lacking the central-pair microtubules, a subset of inner-arm dynein, outer-arm dynein, or the radial spokes, although it was somewhat smaller in the latter two mutants. It was also observed that the shear displacement in an axoneme in the presence of ATP often took place in a stepwise manner. This suggests that the inter-doublet links can reversibly detach from and reattach to the outer doublets in a cooperative manner. This study thus provides the first direct measure of the elasticity of inter-doublet links and also demonstrates its dynamic nature.     

Molecular Simulation of Mechanical Properties and Membrane Activities of the ESCRT-III Complexes

The endosomal sorting complex required for transport (ESCRT) machinery carries out the membrane scission reactions that are required for many biological processes throughout cells. How ESCRTs bind and deform cellular membranes and ultimately produce vesicles has been a matter of active research in recent years. In this study, we use fully atomistic molecular dynamics simulations to scrutinize the structural details of a filament composed of Vps32 protomers, a major component of ESCRT-III complexes. The simulations show that both hydrophobic and electrostatic interactions between monomers help maintain the structural stability of the filament, which exhibits an intrinsic bend and twist. Our findings suggest that the accumulation of bending and twisting stresses as the filament elongates on the membrane surface likely contributes to the driving force for membrane invagination. The filament exposes a large cationic surface that senses the negatively charged lipids in the membrane, and the N-terminal amphipathic helix of the monomers not only acts as a membrane anchor but also generates significant positive membrane curvature. Taking all results together, we discuss a plausible mechanism for membrane invagination driven by ESCRT-III.     

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.     

Origin of Twist-Bend Coupling in Actin Filaments

Actin filaments are semiflexible polymers that display large-scale conformational twisting and bending motions. Modulation of filament bending and twisting dynamics has been linked to regulatory actin-binding protein function, filament assembly and fragmentation, and overall cell motility. The relationship between actin filament bending and twisting dynamics has not been evaluated. The numerical and analytical experiments presented here reveal that actin filaments have a strong intrinsic twist-bend coupling that obligates the reciprocal interconversion of bending energy and twisting stress. We developed a mesoscopic model of actin filaments that captures key documented features, including the subunit dimensions, interaction energies, helicity, and geometrical constraints coming from the double-stranded structure. The filament bending and torsional rigidities predicted by the model are comparable to experimental values, demonstrating the capacity of the model to assess the mechanical properties of actin filaments, including the coupling between twisting and bending motions. The predicted actin filament twist-bend coupling is strong, with a persistence length of 0.15-0.4 μm depending on the actin-bound nucleotide. Twist-bend coupling is an emergent property that introduces local asymmetry to actin filaments and contributes to their overall elasticity. Up to 60% of the filament subunit elastic free energy originates from twist-bend coupling, with the largest contributions resulting under relatively small deformations. A comparison of filaments with different architectures indicates that twist-bend coupling in actin filaments originates from their double protofilament and helical structure.     

Bending-wave propagation by microtubules and flagella

The movement of an elastic filament in a viscous medium can be computed from the fourth-order nonlinear partial differential equation obtained by balancing bending moments at all points along the length of the filament. These bending moments result from active forces, elastic resistance to bending, and viscous resistance to movement through the medium. I have studied numerical solutions obtained for two situations of biological interest: For the movement of individual microtubules, the active force is generated by interaction between the microtubule and the substratum over which it is moving, and is directed along the axis of the microtubule. The computations can reproduce the gliding movement of unrestrained microtubules, and also the periodic bending and bend propagation seen when the leading end of the microtubule is restrained. No modulation of active force is required to generate bending waves. For the movement of flagella, the active forces are generated internally as sliding forces between adjacent members of a cylinder of nine microtubular doublets. Without some additional control assumptions, the forces will be balanced and no bending moments will be generated. The problem faced by investigators of flagellar motility is to determine the control mechanisms that operate to make the system asymmetric, so that active bending moments are generated. Computations with models in which the curvature of the flagellum modulates the active-force generators have indicated that this control specification is sufficient to generate oscillation and bend propagation, but is insufficient to completely determine the movement.     

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.     

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.     

Are the local adjustments of the relative spatial frequencies of the dynein arms and the beta-tubulin monomers involved in the regulation of the "9+2" axoneme?

The “9+2” axoneme is a highly specific cylindrical machine whose periodic bending is due to the cumulative shear of its 9 outer doublets of microtubules. Because of the discrete architecture of the tubulin monomers and the active appendices that the outer doublets carry (dynein arms, nexin links and radial spokes), this movement corresponds to the relative shear of these topological verniers, whose characteristics depend on the geometry of the wave train. When an axonemal segment bends, this induces the compressed and dilated conformations of the tubulin monomers and, consequently, the modification of the spatial frequencies of the appendages that the outer doublets carry. From a dynamic point of view, the adjustments of the spatial frequencies of the elements of the two facing verniers that must interact create different longitudinal periodic patterns of distribution of the joint probability of the molecular interaction as a function of the location of the doublet pairs around the axonemal cylinder and their spatial orientation within the axonemal cylinder. During the shear, these patterns move along the outer doublet intervals at a speed that ranges from one to more than a thousand times that of sliding, in two opposite directions along the two opposite halves of the axoneme separated by the bending plane, respecting the polarity of the dynein arms within the axoneme. Consequently, these waves might be involved in the regulation of the alternating activity of the dynein arms along the flagellum, because they induce the necessary intermolecular dialog along the axoneme since they could be an element of the local dynamic stability/instability equilibrium of the axoneme. This complements the geometric clutch model [Lindemann, C., 1994. A “geometric clutch” hypothesis to explain oscillations of the axoneme of cilia and flagella. J. Theor. Biol. 168, 175-189].     

Elastic extension and jump of the flagellar nexin links: a theoretical mechanical cycle.

The functions of the nexin links of a flagellar axoneme have not been clearly demonstrated. Taking into account both the elastic (Hookean) characteristics and the possible jump of the nexin links, we calculated the sliding to bending conversion of a theoretical model in a tip-ward direction step by step, according to the essential principles proposed by the geometric clutch hypothesis [Lindemann, 1994: J Theoret Biol 168:175-189]: the activity of the dynein arms depends on the transverse forces induced by the axonemal curvature. In our calculations, however, the transverse forces that are involved in the regulation of the activities of the dynein arms were due to the extension of the nexin links located upstream of a given abscissa. This allowed us to define a bent segment as the axonemal portion at whose proximal and distal ends the nexin links were relaxed, and as fully extended as possible, respectively. The model creates an apparent disorder in the orientation of the nexin links as already observed [Bozkurt and Wooley, 1993: Cell Motil Cytoskeleton 24:109-118; Wooley, 1997: J Cell Sci 110:85-94]. We propose that the nexin links are involved in a mechanical cycle, whose 3 stages are (1) rapid extension, (2) slow relaxation, and (3) stand-by. The rapid extension is compatible with the mechanical interactions between the nexin links and the inner dynein arms with which they form the dynein regulatory complex. This was correlated with the oscillating properties of the nexin links along the axoneme that allow them to be one of the regulatory elements of the local ATPase activity of the dynein arms.     

Axonemal activity relative to the 2D/3D-waveform conversion of the flagellum

The waveform of the flagellum of the sea urchin spermatozoon is mainly planar, but its 3D-properties were evoked for dynamic reasons and described as helical. In 1975, the apparent twisting pattern of the sea urchin axoneme was described [Gibbons I. 1975. The molecular basis of flagellar motility in sea urchin spermatozoa. In: Inoué S, Stephens R, editors. Molecular and cellular movement. New York: Raven Press, p. 207-232.] and was considered to be one of the main elements involved in axonemal behaviour. Recently, planar, quasi-planar, and helical waveforms were observed when the flagellum of sea urchin sperm cells was submitted to an increase in viscosity. The quasi-planar conformation seemed to be due to the alternating torsion of the inter-bend segments [Woolley D, Vernon G. 2001. A study of helical and planar waves on sea urchin sperm flagella, with a theory of how they are generated. J. Exp. Biol. 204:1333-1345]. These three waveforms, which are due to a change in axonemal activity, are possibly used by the sperm cells to adapt their movement to variations in the physico-chemical characteristics of the medium (seawater) in which the cells normally swim. We constructed a simple model to describe qualitatively the central shear (between the axonemal doublets and the central pair) and the tangential shear (between the doublets themselves). In this model, the 3D-bending is resolved into components in two perpendicular planes and each of the nine planes of inter-doublet interaction defines a potential bending plane that is independently regulated. These shears were calculated for the three waveforms and their inter-conversion. This allowed us to propose that axoneme is resolved in successive modules delineated by abscissas where the sliding is always nil. We discuss these data concerning the axonemal machinery, and especially the alternating activity of opposite sides of (two) neutral surface(s) that seem(s) to be responsible for this inter-conversion, and for the possible twist of the axoneme during the beating.     

Propagation of force and the induced bending displacement along eukaryotic flagellar axoneme.

The mechanical forces responsible for inducing the bending movement of eukaryotic flagellar axonemes have components that propagate at velocities different from those for the displacement of the medium. These forces are subject to the Third Law of Mechanics which states the null-conservation of acting and reacting forces. Experimental demonstration of the propagating internal tensile force along the axoneme, when not accompanied by simultaneous bending displacement, demonstrates that the active force for the bending of the flagellar axoneme is part of the process of counterbalancing the corresponding reactive force to the ATP hydrolysis underlying the force generation.     

Theoretical analysis of twist/bend ratio and mechanical moduli of bacterial flagellar hook and filament

Certain motile bacteria employ rotating flagella for propulsion. The relative flexibility of two key components of the flagellum, filament and hook, is partially responsible for the mechanistic workings of this motor. A new computational method, the quantized elastic deformational model, was employed in this article to calculate the dimensionless twist/bend ratio (EI/GJ) of the filament and hook, providing a quantitative means to compare their relative stiffness. Both ratios were much <1.0, an average of 0.0440 for the filament and 0.0512 for the hook, indicating that within each structure bending is favored over twisting. These two ratios, along with previous experimental measurements, allowed us to propose a theoretical Young's modulus (E) between 10(6) and 10(7) dyn/cm(2) for the hook. This value is orders of magnitude smaller than experimentally determined Young's moduli of the filament, hence in agreement with empirical evidence linking compliance in the flagellum mainly to the hook.     

One of the Nine Doublet Microtubules of Eukaryotic Flagella Exhibits Unique and Partially Conserved Structures

The axonemal core of motile cilia and flagella consists of nine doublet microtubules surrounding two central single microtubules. Attached to the doublets are thousands of dynein motors that produce sliding between neighboring doublets, which in turn causes flagellar bending. Although many structural features of the axoneme have been described, structures that are unique to specific doublets remain largely uncharacterized. These doublet-specific structures introduce asymmetry into the axoneme and are likely important for the spatial control of local microtubule sliding. Here, we used cryo-electron tomography and doublet-specific averaging to determine the 3D structures of individual doublets in the flagella of two evolutionarily distant organisms, the protist Chlamydomonas and the sea urchin Strongylocentrotus. We demonstrate that, in both organisms, one of the nine doublets exhibits unique structural features. Some of these features are highly conserved, such as the inter-doublet link i-SUB5-6, which connects this doublet to its neighbor with a periodicity of 96 nm. We also show that the previously described inter-doublet links attached to this doublet, the o-SUB5-6 in Strongylocentrotus and the proximal 1–2 bridge in Chlamydomonas, are likely not homologous features. The presence of inter-doublet links and reduction of dynein arms indicate that inter-doublet sliding of this unique doublet against its neighbor is limited, providing a rigid plane perpendicular to the flagellar bending plane. These doublet-specific features and the non-sliding nature of these connected doublets suggest a structural basis for the asymmetric distribution of dynein activity and inter-doublet sliding, resulting in quasi-planar waveforms typical of 9+2 cilia and flagella.