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.     

Bend propagation drives central pair rotation in Chlamydomonas reinhardtii flagella

Regulation of motile 9+2 cilia and flagella depends on interactions between radial spokes and a central pair apparatus. Although the central pair rotates during bend propagation in flagella of many organisms and rotation correlates with a twisted central pair structure, propulsive forces for central pair rotation and twist are unknown. Here we compared central pair conformation in straight, quiescent flagella to that in actively beating flagella using wild-type Chlamydomonas reinhardtii and mutants that lack radial spoke heads. Twists occur in quiescent flagella in both the presence and absence of spoke heads, indicating that spoke--central pair interactions are not needed to generate torque for twisting. Central pair orientation in propagating bends was also similar in wild type and spoke head mutant strains, thus orientation is a passive response to bend formation. These results indicate that bend propagation drives central pair rotation and suggest that dynein regulation by central pair--radial spoke interactions involves passive central pair reorientation to changes in bend plane.     

The equation of motion for sperm flagella.

The equation of motion for sperm flagella, in which the elastic bending moment and the active contractile moment are balanced by the moment from the viscous resistance of the surrounding fluid, is solved for a wave solution that superimposes partial solutions. Substitution of the expression for the wave solution into the equation leads to an expression for the active contractile moment. This active moment can be decomposed into two parts. The first part describes an active moment that travels over the flagellum with the mechanical flagellar wave, the second part represents a moment in phase over the entire length of the flagellum, which decreases linearly towards the distal tip. The linear synchronous moment, to which an amount of traveling moment has been added as a perturbation, leads to wave solutions that closely resemble flagellar waves. Properties such as wavelength and wave amplitudes and also the shape of the waves in sea urchin sperm flagella at different frequencies are accurately described by the theory. The change in wave shape in sea urchin sperm flagella at raised viscosity is predicted well by the theory. The different wave properties caused in bull sperm flagella by different boundary conditions at the proximal junction are explained. When only a traveling active moment is present in a flagellum, the wave solutions describe waves of a small wave length in a long flagellum. Some properties of the wave motion of sperm flagella are derived from the theory and verified experimentally.     

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.     

Brownian motion of helical flagella

We develops a theory of the Brownian motion of a rigid helical object such as bacterial flagella. The statistical properties of the random forces acting on the helical object are discussed and the coefficients of the correlations of the random forces are determined. The averages , and are also calculated where z and theta are the position along and angle around the helix axis respectively. Although the theory is limited to short time interval, direct comparison with experiment is possible by using the recently developed cinematography technique.     

Derivation of unstirred-layer transport number equations from the Nernst-Planck flux equations.

Since the late 1960s it has been known that the passage of current across a membrane can give rise to local changes in salt concentration in unstirred layers or regions adjacent to that membrane, which in turn give rise to the development of slow transient diffusion potentials and osmotic flows across those membranes. These effects have been successfully explained in terms of transport number discontinuities at the membrane-solution interface, the transport number of an ion reflecting the proportion of current carried by that ion. Using the standard definitions for transport numbers and the regular diffusion equations, these polarization or transport number effects have been analyzed and modeled in a number of papers. Recently, the validity of these equations has been questioned. This paper has demonstrated that, by going back to the Nernst-Planck flux equations, exactly the same resultant equations can be derived and therefore that the equations derived directly from the transport number definitions and standard diffusion equations are indeed valid.     

A field theory of neural nets: I: Derivation of field equations

A model is described in which neural activity is represented by a field quantity ϕ, with the neurons as the sources of ϕ. It is shown that, with certain physically realistic assumptions, ϕ satisfies a moderately nonlinear differential equation. It is also found that this equation is isotropic and of second order if and only if the neuronal connectivity has a dependence on distance,p, of the formp ⁻¹ e ^−1/2βp .     

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.     

Derivation, propagation and differentiation of human embryonic stem cells

Embryonic stem (ES) cells are in vitro cultivated pluripotent cells derived from the inner cell mass (ICM) of the embryonic blastocyst. Attesting to their pluripotency, ES cells can be differentiated into representative derivatives of all three embryonic germ layers (endoderm, ectoderm and mesoderm) both in vitro and in vivo. Although mouse ES cells have been studied for many years, human ES cells have only more recently been derived and successfully propagated. Many biochemical differences and culture requirements between mouse and human ES cells have been described, yet despite these differences the study of murine ES cells has provided important insights into methodologies aimed at generating a greater and more in depth understanding of human ES cell biology. One common feature of both mouse and human ES cells is their capacity to undergo controlled differentiation into spheroid structures termed embryoid bodies (EBs). EBs recapitulate several aspects of early development, displaying regional-specific differentiation programs into derivatives of all three embryonic germ layers. For this reason, EB formation has been utilised as an initial step in a wide range of studies aimed at differentiating both mouse and human ES cells into a specific and desired cell type. Recent reports utilising specific growth factor combinations and cell-cell induction systems have provided alternative strategies for the directed differentiation of cells into a desired lineage. According to each one of these strategies, however, a relatively high cell lineage heterogeneity remains, necessitating subsequent purification steps including mechanical dissection, selective media or fluorescent or magnetic activated cell sorting (FACS and MACS, respectively). In the future, the ability to specifically direct differentiation of human ES cells at 100% efficiency into a desired lineage will allow us to fully explore the potential of these cells in the analysis of early human development, drug discovery, drug testing and repair of damaged or diseased tissues via transplantation.     

Computer Simulation of Flagellar Movement: I. Demonstration of Stable Bend Propagation and Bend Initiation by the Sliding Filament Model

A program has been developed for digital computer simulation of the movement of a flagellar model consisting of straight segments connected by joints at which bending occurs. The program finds values for the rate of bending at each joint by solving equations which balance active, viscous, and elastic bending moments at each joint. These bending rates are then used to compute the next position of the model. Stable swimming movements, similar to real flagellar movements, can be generated routinely with a 25-segment model using 16 time steps/beat cycle. These results depend on four assumptions about internal flagellar mechanisms: (a) Bending is generated by a sliding filament process. (b) The active process is controlled locally by the curvature of the flagellum. (c) Nonlinear elastic resistances stabilize the amplitude of the movement. (d) Internal viscous resistances stabilize the wavelength of the movement and explain the relatively low sensitivity of flagellar movement to changes in external viscosity.     

Hydrodynamic calculations on the movements of cilia and flagella I. Paramecium

Calculations of the velocity profile, force, moment and bending moment using a theoretical model are carried out for the three-dimensional “conical-helical” beat of a cilium of Paramecium multimicro-nucleatum. The mean velocity profile obtained by numerical computation is found to be twisted in form: it is inclined at a slight angle to the effective stroke at the top of the cilia sublayer, but twists around with the recovery stroke in the lower part of the sublayer. The force and moment are larger during the fast effective stroke, but over a complete cycle their contributions are approximately zero. Calculations on the bending moments reveals that they are larger during the recovery stage of the beating cycle.     

Cell envelopes of chemotaxis mutants of Escherichia coli rotate their flagella counterclockwise.

Flagella rotated exclusively counterclockwise in Escherichia coli cell envelopes prepared from wild-type cells, whose flagella rotated both clockwise and counterclockwise, from mutants rotating their flagella counterclockwise only, and even from mutants rotating their flagella primarily clockwise. Some factor needed for clockwise flagellar rotation appeared to be missing or defective in the cell envelopes.