ABSTRACT
Elsewhere in this volume some of the biological and biochemical challenges offered by ciliary motion have been discussed. Here I propose to address certain physical aspects of the movement, with a view to examining the constraints placed on the system in respect of the microtubule displacements produced, the forces developed, and the energy expended during ciliary bending. A major objective is to use the observed motion of a cilium to deduce how the assemblies of dynein motor molecules behave within the axoneme, relating in particular the motor activity to microtubule sliding, ciliary bend shape, and beat frequency. After a brief discussion of the relevant biophysical background, the numbers of active arms required to generate different physical characteristics of the motion will be estimated. It will emerge that the numbers required to satisfy the geometry of the movement are different from those required to provide the force and energy, and the reconciliation of this difference provides information about the behavior of the arms within the system. 19
A cilium undergoes cyclic activity, usually with a frequency in the range of 5 - 50 Hz, in which a power, or effective, stroke generates force to induce relative motion between the cell and its surrounding fluid; examples include motion of the mucous layer in the human lung and free-swimming protozoa. The effective stroke is followed by a recovery stroke, generally not the reverse of, and usually dissipating less power than, the effective stroke, to return the cilium to the beginning of its cycle. The power for the motion is provided by the dynein motors, or arms, which utilize the chemical fuel ATP as a source of energy. The arms also have a cyclic action, with a force-generating, or duty, phase as one of the components; in other phases of the cycle, the arm undergoes mech-anochemical changes to prepare it for the next duty phase. A 5-pm-long cilium contains about 4000 inner and outer arms. There is evidence (e.g., Ref. 1) that the inner and outer arms perform different functions in the axoneme, with the inner arms considered to be important in controlling bend shape, while the outer arms control beat frequency. The inner arms, with three distinct types repeating every 96 nm along a microtubule, are more complex in structure and arrangement than the outer arms, which are all of the same type and spaced at 24 nm intervals (2). For the purposes of the current discussion, it will be assumed that the outer arms dominate the bending process and that the contribution of the inner arms to this process is not significant. This is clearly an oversimplification, but it affects neither the main thrust of the argument nor the conclusions drawn.The forces exerted by the motors produce the relative microtubule sliding between the peripheral doublets of the axoneme (3), which leads to ciliary bending. In a number of cells, e.g., Paramecium, an alteration in the speed or direction of locomotion in response to an external stimulus is achieved through a changed pattern of ciliary activity; studies of the relationships between the changes indicate that ciliary motion is controlled by the cell. An appropriate physical model for the cilium is therefore a controlled, active, mechanochemical oscillator. Such models are not, of course, novel. For example, in 1958, Machin (4) represented the flagellum as a vibrating elastic beam and demonstrated that energy must be introduced along the length of an organelle to produce the observed wave patterns with sustained, and sometimes increasing, amplitude, and thereby anticipated the discovery of dynein. Other workers, e.g., Brokaw (5) and Holwill and Miles (6), introduced parameters into the modeling to reflect the mechanochemistry of the system.The available evidence (e.g., Ref. 7) suggests that dynein generates force in one direction only, with the arms on one microtubule (number N) in the axoneme pushing the neighboring microtubule (N + 1) tipwards. One consequence of this is that if a ciliary beat is planar, motor activity alternates between the two halves
of the axoneme separated by the beat plane. Transition between the effective and recovery strokes is achieved by a switching mechanism (e.g., Ref. 8), which will have a frequency equal to that of the cilium. The dynein arm action is therefore subject to considerable control, the mechanism of which is essential to understand in a comprehensive model of ciliary behavior. MICROTUBULE SLIDING DISPLACEMENT
At the end of a typical effective stroke, the angle between the tangents to the ends of the major bend on the cilium is about 100°. From this, it is straightforward to calculate that the maximum distance one microtubule has slid relative to its neighbor is about 100 nm, assuming that the cilium was straight at the beginning of the stroke. The amount of relative sliding is, of course, different for each pair of neighboring microtubules, being greatest for the pair lying most nearly parallel to the beat plane of the cilium. Estimates of the distance, or step size, through which an individual outer dynein arm can push a microtubule in one arm cycle vary from 4 nm (9) to 40 nm (10), with values of 8 nm or 16 nm currently being favored. Given a value of 8 nm, a sliding displacement of 100 nm could be achieved by the action of about 12 dynein arms. This is the minimum number of arms that need to be active on each doublet. Because of the unidirectional character of the dynein motors, a minimum number of 48 arms (12 on each of 4 microtubules) will be active within the whole cilium to produce the complete effective stroke. During the effective stroke, a maximum of 1000 outer arms could be active, so that, based on the geometry of the system, only about 5% need be active to produce the observed sliding. FORCE GENERATION
To produce ciliary beating, the shearing forces generated by the dynein motors must overcome both external and internal resistances to motion. External resistances arise from the interaction between the cilium and its liquid environment and, since the system operates in the low Reynolds number regime of hydrodynamics, are dominated by viscosity. Several authors (e.g., Refs. 11-14) have investigated the propulsive thrust developed by cilia, and the interaction between the organelles and the surrounding medium is well understood. If the precise beat form of a cilium is known, the forces generated and energy dissipated at any stage of its beat can be calculated, but since the patterns of ciliary beating are generally nonuniform, it is more useful to estimate mean values of force and energy production. Using the expressions derived by Sleigh and Holwill (15) and the ciliary force coefficients derived by Lighthill (16), it can be shown that, at every instant during its cycle, a force of the order of 10 pN is exerted by a beating cilium on the liquid environment.
