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asymmetry. This asymmetry is reversed next time the centrosome passes the middle
point (Fig.
25b
). Moreover, the microtubules are engaged with the pulling surface
more to one side of the centrosome than to the other. The other side of the aster
becomes engaged during the reverse swing (Fig.
25b
). Similarly to the Kozlowski
et al. model for the pronucleus oscillations in worm eggs, it can be observed that the
distal ends of microtubules in the Kim-Maly model do not move appreciably during
the oscillation cycle. This should be attributed to the cytoplasm viscosity dampen-
ing propagation of the elastic perturbation along the microtubules from their proxi-
mal parts which may be pulled by the cortex and which are coupled to the moving
centrosome. As a result, when microtubules on one side are pulled and the centro-
some shifts, the proximal parts of microtubules on the opposite side become lifted
off the synaptic surface (Fig.
25
). This makes the tug of war nonlinear: Whenever
one side is winning, this weakens the opposing side. The temporal instability can be
ascribed to this effect. Importantly for the cyclical nature of the movement that
emerges from the instability, its range is limited by the deformation of the microtu-
bules on the winning side. Their distal parts are bent against the side of the cell, and
because of this the zone where they can contact the pulling surface cannot extend to
the edge of the flat synaptic zone. Movement toward the edge therefore diminishes
the pulling force. This gives the elastic relaxation of the trailing microtubules time
to catch up and bring their proximal parts in apposition with the pulling surface.
At this point, the microtubules that trailed are lying relatively flat on the synapse
and are therefore experiencing a greater pulling force than the microtubules that led
and are now contacting the synapse only with their highly curved parts. Movement
in the reverse direction ensues (Fig.
25
).
From the thermodynamic standpoint, the dissipative oscillations are driven by the
non-potential forces of pulling. They arise from the energy-dissipating biochemistry
of the dynein motors, which is coupled to the nonequilibrium metabolism of the liv-
ing cell. In the course of the oscillations these non-potential active forces work against
the similarly non-potential forces of viscous drag. At the same time, the mechanics of
the described hysteretic loop is orchestrated by the potential forces of microtubule
elasticity and by the heritable structural constraints of the cell body and boundary.
Simulations with different pulling force densities showed (Kim and Maly
2009
)
that the basic frequency of the oscillations is fairly insensitive to this parameter. Yet
the overall pattern of oscillations changes abruptly when the pulling force density
crosses a certain value (Fig.
27
). Below approximately 140 pN/μm, the oscillations
appear multiperiodic and continuous (Fig.
27a
). Above approximately 150 pN/μm,
the oscillations are pulse-like (Fig.
27c
). In the relatively narrow range between
approximately 140 and 150 pN/μm, the oscillations are continuous and pure, i.e.,
they exhibit a single frequency and amplitude. Only in this narrow intermediate range
does the distance of the centrosome from the synaptic plane not oscillate (Fig.
27b
).
The experimental estimates of the force exerted by a single cytoplasmic dynein
molecule interacting with a microtubule, on the order of pN (Ashkin et al.
1990
),
limit the range of the pulling force densities that are of interest to 20-200 pN/μm.
Below this range, there would be only a few molecular motors pulling on a given
microtubule, giving rise to stochasticity. Above this range the number of motors
