Computer simulation of flagellar movement. III. Models incorporating cross-bridge kinetics.

A computer simulation procedure is used to analyze the generation of propagated bending waves by flagellar models in which active sliding is generated by a cycle of cross-bridge activity. Two types of cross-bridge cycle have been examined in detail. In both cycles, cross-bridge attachment is followed immediately by a configurational change in the cross-bridge, which transfers energy to a stretched elastic element and generates a shearing force between the filaments. In the first model, which has cross-bridge behavior close to current ideas about cross-bridge behavior in muscle, cross-bridge attachment is proportional to curvature of the flagellum and detachment is an exponential decay process. The configurational change is equivalent to an angular deviation of pi/5 radians. In the second type of cross-bridge cycle, cross-bridge attachment occurs rapidly when a critical curvature is reached, and detachment occurs when a critical curvature in the opposite direction is reached. With this cycle, an unrealistically large angular deviation of the cross-bridges, equivalent to 3.0 radians, is required to obtain bending waves of normal amplitude. Both models generate bending wave patterns similar to those obtained in earlier work. However, the behavior of the second type of cross-bridge model more closely matches the actual behavior of flagella under experimental conditions: the chemical turnover rate per beat cycle remains constant as the viscosity is increased, and reduction in the number of active cross-bridges can cause a reduction in beat frequency, with little change in amplitude or wavelength.     

Bend propagation in flagella. II. Incorporation of dynein cross-bridge kinetics into the equations of motion.

The cross-bridge formalism of T. Hill has been incorporated into the nonlinear differential equations describing planar flagellar motion in an external viscous medium. A stable numerical procedure for solution of these equations is presented. A self-consistent two-state diagram with curvature-dependent rate functions is sufficient to generate stable propagating waves with frequencies and amplitudes typical of sperm flagella. For a particular choice of attachment and detachment rate functions, reasonable variation of frequency and wave speed with increasing viscosity is also obtained. The method can easily be extended to study more realistic state diagrams.     

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.     

Stretch activation and nonlinear elasticity of muscle cross-bridges.

When active insect fibrillar flight muscle is stretched, its ATPase rate increases and it develops "negative viscosity," which allows it to perform oscillatory work. We use a six-state model for the cross-bridge cycle to show that such "stretch activation" may arise naturally as a nonlinear property of a cross-bridge interacting with a single attachment site on a thin filament. Attachment is treated as a thermally activated process in which elastic energy must be supplied to stretch or compress the cross-bridge spring. We find that stretch activation occurs at filament displacements where, before the power stroke, the spring is initially in compression rather than in tension. In that case, pulling the filaments relieves the initial compression and reduces the elastic energy required for attachment. The result is that the attachment rate is enhanced by stretching. The model also displays the "delayed tension" effect observed in length-step experiments. When the muscle is stretched suddenly, the power stroke responds very quickly, but there is a time lag before dissociation at the end of the cycle catches up with the increased attachment rate. This lag is responsible for the delayed tension and hence also for the negative viscosity.     

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.     

Cross-bridge behavior in rigor muscle

Properties of the rigor state in muscle can be explained by a simple cross-bridge model, of the type which has been suggested for active muscle, in which detachment of cross-bridges by ATP is excluded. Two attached cross-bridge states, with distinct force vs. distortion relationships, are required, in addition to a detached state, but the attached cross-bridge states in rigor muscle appear to differ significantly from the attached cross-bridge states in active muscle. The stability of the rigor force maintained in muscle under isometric conditions does not require exceptional stability of the attached cross-bridges, if the positions in which attachment of cross-bridges is allowed are limited so that the attachment of cross-bridges in positions which have minimum free energy is excluded. This explanation of the stability of the rigor state may also be applicable to the maintenance of stable rigor waves on flagella.     

Changes in force-velocity properties of trachealis due to oscillatory strains.

The physically dynamic environment of the lung constantly modulates the mechanical properties of airway smooth muscle. In vitro experiments have shown that contractility of the muscle is compromised by oscillatory strains, perhaps through disruption of cross-bridge interaction and organization of the contractile filaments. To understand the mechanism by which oscillation affects contractility, functional changes of the muscle in terms of force-velocity relationship were assessed before and after imposition of length oscillation in both relaxed and activated states. The oscillation protocol was designed to reduce isometric force by 15-20%, followed by measurement of force-velocity properties. Maximal velocity and power changed by +8 and -14%, respectively, after oscillation applied in the relaxed state and changed by -15 and -25%, respectively, after oscillation applied during contraction. A simple model of reduced activation could not account for the results; neither could the results be explained satisfactorily by the current cross-bridge theory of contraction. The results, however, could be explained if the possibility of reorganization of the contractile filaments due to oscillatory strains was considered.     

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.     

Force maintenance in smooth muscle: analysis using sinusoidal perturbations

The "latch state" or force maintenance may be due to the emergence of a distinct set of dephosphorylated, slowly cycling "latch" cross-bridges, slowing of the overall cross-bridge cycling rate, or a non-cross-bridge contribution. This was investigated by sinusoidally oscillating strips of intact rabbit portal vein or aorta. Tissue strips were activated with KCl depolarization, resulting in a sustained increase of MLC(20) phosphorylation or 10 microM phenylephrine, resulting in a transient increase in MLC(20) phosphorylation. Stiffness was calculated from the force response to a small, sine-wave oscillation in muscle length (1-100 Hz). The results produced a 3-dimensional plot of stiffness versus the frequency of oscillation (Hz) versus time (s), or stiffness distribution profile. During KCl depolarization, the stiffness distribution profile displayed a shift toward lower frequencies, suggesting a general slowing in the overall cross-bridge cycling rate during force maintenance. On the other hand, phenylephrine stimulation did not display a significant change in the stiffness distribution profile, suggesting that the overall cross-bridge cycling rate did not significantly change during force maintenance.     

Ciliary contractile model applied to sperm flagellar motion

The contractile model developed previously to describe ciliary motions was applied to the case of invertebrate sperm flagella. The model could describe waveforms of these flagella as a function of the flagellar frequency and as a function of the external viscosity. Rapid spontaneous transients from rest to motion were also correctly described by the model. The calculations showed that the time course of the internal, active, moments during a cycle is only weakly displayed in the time course of the transverse motion of the flagella. This is apparently due to nonlinearities which suppress higher harmonics in the frequency spectrum of the active moments. The local curvature of the flagella was found to follow the time course of the internal, active moments directly. It is concluded that to derive the time course of the active contractile moments from the transverse motion, data on the flagellar motion with a position accuracy of 0·1 μm and with a time resolution of 1 msec are required. For curvature data an accuracy of 50 cm^?1 and a time resolution of 2·5 msec would suffice.     

Direct tests of muscle cross-bridge theories: predictions of a Brownian dumbbell model for position-dependent cross-bridge lifetimes and step sizes with an optically trapped actin filament.

Force and displacement events from a single myosin molecule interacting with an actin filament suspended between optically trapped beads (Finer, J. T., R. M. Simmons, and J. A. Spudich. 1994. Nature. 368:113-119) can be interpreted in terms of a generalized cross-bridge model that includes the effects of Brownian forces on the beads. Steady-state distributions of force and displacement can be obtained directly from a generalized Smoluchowski equation for Brownian motion of the actin-bead "dumbbell," and time series from Monte Carlo simulations of the corresponding Langevin equation. When the frequency spectrum of Brownian motion extends beyond cross-bridge transition rates, the inverse mean lifetimes of force/displacement pulses are given by cross-bridge rate constants averaged over a Boltzmann distribution of Brownian noise. These averaged rate constants reflect the strain-dependence of the rate constants for the stationary filament, most faithfully at high trap stiffness. Hence, measurements of the lifetimes and displacements of single events as a function of the resting position of the dumbbell can provide a direct test of different cross-bridge theories of muscle contraction. Quantitative demonstrations are given for Huxley models with 1) faster binding or 2) slower dissociation at positive cross-bridge strain. Predictions for other models can be inferred from the averaging procedure.     

Fading memory model for airway smooth muscle dynamic response

This work presents the application of a fading memory model to describe the behavior of contracted airway smooth muscle (ASM) for two biophysical cases: finite duration length steps and longitudinal sinusoidal oscillations. The model parameters were initially determined from literature data on transient step length change response and subsequently the model was applied to the two cases. Results were compared with previously published experimental data on ASM oscillations. The model confirms a trend observed in the experimental data which shows that: (i) the value of tissue length change is the most important factor to determine the degree of cross-bridge detachment and (ii) a strong correlation exists between increasing frequency and declining stiffness until a certain frequency (∼25 Hz) beyond which frequency dependence is negligible. Although the model was not intended to simulate biophysical events individually, the data could be explained by cross-bridge cycling rates. As the frequency increases, cross-bridge reattachment becomes less likely, until no further cross-bridge attachment is possible.     

The effect of cross-bridge clustering and head-head competition on the mechanical response of skeletal muscle under equilibrium conditions.

The effect of cross-bridge clustering and head-head competition on the mechanical response of skeletal muscle under equilibrium conditions is considered. For this purpose, the recent multiple site equilibrium cross-bridge model of Schoenberg (Schoenberg, M., 1985, Biophys. J., 48:467-475) is extended in accordance with the formalism of T.L. Hill (1974, Prog. Biophys, Mol. Biol., 28:267-340) to consider the case where groups of independent cross-bridge heads compete with each other for binding to multiple actin sites. Cooperative behavior between heads is not allowed. Computations indicate that for the double-headed cross-bridge with two independent equivalent heads, the time course of force decay after a stretch is similar to that for the single-headed cross-bridge; that is, the rate constant for force decay is approximately equal to the cross-bridge head detachment rate constant. The results also show that the force decay after a stretch becomes slower than the detachment rate constant of a single head when cross-bridge heads bind adjacently in clusters so that competition between heads for binding to the available actin sites increases. However, if one assumes that the detachment rate constant of an unstrained head in a fiber is comparable to that of an S1 molecule in solution, this effect is not large enough to explain why some of the rate constants for force decay after a stretch in rigor, or in the presence of ATP analogues such as adenyl-5'-yl imidodiphosphate, appear to be significantly slower than the detachment rate constant of S1 from actin in solution.     

Effect of viscosity on mechanics of single, skinned fibers from rabbit psoas muscle.

Muscle contraction is highly dynamic and thus may be influenced by viscosity of the medium surrounding the myofilaments. Single, skinned fibers from rabbit psoas muscle were used to test this hypothesis. Viscosity within the myofilament lattice was increased by adding to solutions low molecular weight sugars (disaccharides sucrose or maltose or monosaccharides glucose or fructose). At maximal Ca2+ activation, isometric force (Fi) was inhibited at the highest solute concentrations studied, but this inhibition was not directly related to viscosity. Solutes readily permeated the filament lattice, as fiber diameter was unaffected by added solutes (except for an increased diameter with Fi < 30% of control). In contrast, there was a linear dependence upon 1/viscosity for both unloaded shortening velocity and also the kinetics of isometric tension redevelopment; these effects were unrelated to either variation in solution osmolarity or inhibition of force. All effects of added solute were reversible. Inhibition of both isometric as well as isotonic kinetics demonstrates that viscous resistance to filament sliding was not the predominant factor affected by viscosity. This was corroborated by measurements in relaxed fibers, which showed no significant change in the strain-rate dependence of elastic modulus when viscosity was increased more than twofold. Our results implicate cross-bridge diffusion as a significant limiting factor in cross-bridge kinetics and, more generally, demonstrate that viscosity is a useful probe of actomyosin dynamics.     

Effects of antibodies against dynein and tubulin on the stiffness of flagellar axonemes

Antidynein antibodies, previously shown to inhibit flagellar oscillation and active sliding of axonemal microtubules, increase the bending resistance of axonemes measured under relaxing conditions, but not the bending resistance of axonemes measured under rigor conditions. These observations suggest that antidynein antibodies can stabilize rigor cross-bridges between outer-doublet microtubules, by interfering with ATP-induced cross-bridge detachment. Stabilization of a small number of cross-bridge appears to be sufficient to cause substantial inhibition of the frequency of flagellar oscillation. Antitubulin antibodies, previously shown to inhibit flagellar oscillation without inhibiting active sliding of axonemal microtubules, do not increase the static bending resistance of axonemes. However, we observed a viscoelastic effect, corresponding to a large increase in the immediate bending resistance. This immediate bending resistance increase may be sufficient to explain inhibition of flagellar oscillation; but several alternative explanations cannot yet be excluded.     

Distributed Representations for Actin-Myosin Interaction in the Oscillatory Contraction of Muscle

In this paper we suggest and test a specific hypothesis relating the attachment-detachment cycle of cross bridges between actin (I) and myosin (A) filaments to the measured length-tension dynamics of active insect fibrillar flight muscle. It is first shown that if local A-filament strain perturbs the rate constants in the cross-bridge cycle appropriately, then exponentially delayed tension changes can follow imposed changes of length; the latter phenomenon is sufficient for the work-producing property of fibrillar muscle, as measured with small-signal forcing of length and at low Ca²⁺ concentration, and possibly for related effects described recently in frog striated muscle. It is not clear a priori that the above explanation of work production by fibrillar muscle will remain tenable when the viscoelastic complexity of the heterogeneous sarcomere is taken into account. However, White's (1967) recent mechanical and electron microscope study of the passive dynamics of glycerinated fibrillar muscle has produced a model of the distributed viscoeleastic structure sufficiently explicit that alternative schemes for cross-bridge force generation in this muscle can now be tested more critically than previously. Therefore, we derive and solve third-order partial-differential equations which relate local interfilament shear forces associated with the perturbed cross-bridge cycles to the over-all length-tension dynamics of an idealized sarcomere. We then show (a) that the starting hypothesis can account approximately for the small-signal dynamics of glycerinated muscle in the work-producing state over two decades of frequency and (b) that the rate constants for cross-bridge formation and breakage, restricted solely by fitting of the model to the mechanical data, determine a cycling rate of cross bridges in the model compatible with recent measurements of ATP hydrolysis rate vs. stretch in this muscle. Finally, the formulation is extended tentatively to the large-signal nonlinear case, and shown to compare favorably with previous suggestions for the origin of the work-producing dynamics of fibrillar flight muscle.     

Effects of viscosity and ATP concentration on the movement of reactivated sea-urchin sperm flagella.

Spermatozoa from the sea urchin, Lytechinus pictus, can be demembranated with solutions containing Triton X-100 and 5mM-CaCl2 and reactivated in ATP solutions containing low concentrations (10(-9)M) Of free Ca2+ ion to give symmetrical bending wave movements, even at very low ATP concentrations. At ATP concentrations of 0.01-0.02 mM the reactivated spermatozoa have beat frequencies near 1 Hz, nearly normal bend angles, and wave-lengths about 50% longer than normal. 2. The effects of increased viscosity, obtained by addition of methyl cellulose to the reactivation solutions, on bend angle and beat frequency decrease with decreasing ATP concentration, and become almost undetectable at 0.01 mM ATP. On the other hand, the effect of increased viscosity on wavelength shows relatively little change with ATP concentration, although it is noticeably reduced at 0.01 mM ATP. 3. These observations suggest that the beat frequency at low ATP concentrations is determined by an intrinsic rate-limiting process, in contrast to the viscocity-limited behaviour at high beat frequencies. Quantitative agreement with the experimental results is obtained with a model in which ATP concentration and viscosity each determine the rates of one step in a two-step reaction cycle which determines the beat frequency. 4. Two other models which can qualitatively explain the effects of ATP concentration and viscosity on flagellar beat frequency fail to show quantitative agreement with the experimental results. In one of these models, ATP concentration determines the maximum rate of shear between filaments. In the other, ATP concentration determines a time delay which is required to bring the active moment into phase with the elastic moments which would be expected to dominate the bending resistance at low beat frequencies.     

Stiffness and force in activated frog skeletal muscle fibers.

Single fibers, isolated intact from frog skeletal muscles, were held firmly very near to each end by stiff metal clasps fastened to the tendons. The fibers were then placed horizontally between two steel hooks inserted in eyelets of the tendon clasps. One hook was attached to a capacitance gauge force transducer (resonance frequency up to approximately 50 kHz) and the other was attached to a moving-coil length changer. This allowed us to impose small, rapid releases (complete in less than 0.15 ms) and high frequency oscillations (up to 13 kHz) to one end of a resting or contracting fiber and measure the consequences at the other end with fast time resolution at 4 to 6 degrees C. The stiffness of short fibers (1.8-2.6 mm) was determined directly from the ratio of force to length variations produced by the length changer. The resonance frequency of short fibers was so high (approximately 40 kHz) that intrinsic oscillations were not detectably excited. The stiffness of long fibers, on the other hand, was calculated from measurement of the mechanical resonance frequency of a fiber. Using both short and long fibers, we measured the sinusoids of force at one end of a contracting fiber that were produced by relatively small sinusoidal length changes at the other end. The amplitudes of the sinusoidal length changes were small compared with the size of step changes that produce nonlinear force-extension relations. The sinusoids of force from long fibers changed amplitude and shifted phase with changes in oscillation frequency in a manner expected of a transmission line composed of mass, compliance, and viscosity, similar to that modelled by (Ford, L. E., A. F. Huxley, and R. M. Simmons, 1981, J. Physiol. (Lond.), 311:219-249). A rapid release during the plateau of tetanic tension in short fibers caused a fall in force and stiffness, a relative change in stiffness that putatively was much smaller than that of force. Our results are, for the most part, consistent with the cross-bridge model of force generation proposed by Huxley, A. F., and R. M. Simmons (1971, Nature (Lond.), 213:533-538). However, stiffness in short fibers developed markedly faster than force during the tetanus rise. Thus our findings show the presence of one or more noteworthy cross-bridge states at the onset and during the rise of active tension towards a plateau in that attachment apparently is followed by a relatively long delay before force generation occurs.(ABSTRACT TRUNCATED AT 400 WORDS)     

Intraflagellar transport particle size scales inversely with flagellar length: revisiting the balance-point length control model

The assembly and maintenance of eukaryotic flagella are regulated by intraflagellar transport (IFT), the bidirectional traffic of IFT particles (recently renamed IFT trains) within the flagellum. We previously proposed the balance-point length control model, which predicted that the frequency of train transport should decrease as a function of flagellar length, thus modulating the length-dependent flagellar assembly rate. However, this model was challenged by the differential interference contrast microscopy observation that IFT frequency is length independent. Using total internal reflection fluorescence microscopy to quantify protein traffic during the regeneration of Chlamydomonas reinhardtii flagella, we determined that anterograde IFT trains in short flagella are composed of more kinesin-associated protein and IFT27 proteins than trains in long flagella. This length-dependent remodeling of train size is consistent with the kinetics of flagellar regeneration and supports a revised balance-point model of flagellar length control in which the size of anterograde IFT trains tunes the rate of flagellar assembly.     

Axial and radial forces of cross-bridges depend on lattice spacing

Nearly all mechanochemical models of the cross-bridge treat myosin as a simple linear spring arranged parallel to the contractile filaments. These single-spring models cannot account for the radial force that muscle generates (orthogonal to the long axis of the myofilaments) or the effects of changes in filament lattice spacing. We describe a more complex myosin cross-bridge model that uses multiple springs to replicate myosin's force-generating power stroke and account for the effects of lattice spacing and radial force. The four springs which comprise this model (the 4sXB) correspond to the mechanically relevant portions of myosin's structure. As occurs in vivo, the 4sXB's state-transition kinetics and force-production dynamics vary with lattice spacing. Additionally, we describe a simpler two-spring cross-bridge (2sXB) model which produces results similar to those of the 4sXB model. Unlike the 4sXB model, the 2sXB model requires no iterative techniques, making it more computationally efficient. The rate at which both multi-spring cross-bridges bind and generate force decreases as lattice spacing grows. The axial force generated by each cross-bridge as it undergoes a power stroke increases as lattice spacing grows. The radial force that a cross-bridge produces as it undergoes a power stroke varies from expansive to compressive as lattice spacing increases. Importantly, these results mirror those for intact, contracting muscle force production.