Figure 1.
Schematic of the geometry of the embryo used in the model and the forces involved.
The embryo has the spherical geometry of that of the early Xenopus laevis embyo with the sperm’s pronucleus and its associated microtubule aster positioned close to the cortex. As cargos are transported by minus-end directed motors towards the centrosome they experience an opposing cytosolic drag force (Fdrag, magnified schematic). That force equals the force exerted by the motors on the microtubules and points in opposite directions on the two sides of the centrosome. The net force, Fnet, on the centrosome and associated male pronucleus points towards the far cortical side since the microtubules on that side can grow longer and support more vesicles than those on the other side. The net effect is motion of the growing aster towards the cell center.
Figure 2.
Centrosome dynamics as a function of vesicle velocity.
(A) After an initial ramp up, the centrosome velocity keeps increasing at a much smaller rate. As the difference in number of vesicles moving along microtubules in the far and near cortical sides decreases, the centrosome slows down (arrows). Given that slower vesicles experience a smaller drag force, they lead to a slower centrosome. (B) The corresponding position of the centrosome shows that the small, fast-moving vesicles are sufficient to move the centrosome distances comparable to the motion of the centrosome in fertilized Xenopus laevis embryos. Experiments show that the centrosome moves at least 300 µm in 45 minutes; the dashed lines delineate that region. (calculation parameters: 100 microtubules; 100 nm diameter vesicles; viscosity ratio = 3; 2 vesicles/µm; 250 nm/s MT polymerization rate; vesicle velocity as indicated: 2 µm/s (green), 1 µm/s (red), 0.5 µm/s (black) in that order from top to bottom).
Figure 3.
The number of microtubules comprising the aster does not alter centrosome dynamics.
The randomness in the direction of the microtubules leads to variability in the dynamics. This is shown as error bars representing the standard deviation for multiple runs of the simulation for 100 MTs. The 1000 MT trace lies close to the average and within the error bars indicating similar dynamics. Although the drag arising from having more microtubules (MTs) increases with microtubule number, the number of force generating vesicles increases in the same proportion leading to identical centrosome velocity. (calculation parameters: microtubule numbers as indicated: 1000 (red), 100 (black) in order of increasing duration shown; 100 nm diameter vesicles; viscosity ratio = 3; 250 nm/s MT polymerization rate; 2 vesicles/µm; 2 µm/s vesicle velocity).
Figure 4.
The centrosome takes longer to center for larger effective viscosity ratios.
Since the aster constituents are larger than the vesicles they are also likely to experience a larger effective viscosity arising from the cell’s crowded environment. The effect of viscosity ratio on centering speed is less pronounced (smaller slope) for larger vesicles. The dashed line delineates the centering time experimentally observed. (calculation parameters: 100 microtubules; vesicle diameter: 100 nm (red squares), 200 nm (black circles); 250 nm/s MT polymerization rate; 2 vesicles/µm; 2 µm/s vesicle velocity).
Figure 5.
A larger microtubule polymerization rate leads to limited increase in centrosome speed.
While the centrosome centers within a shorter time for moderate increase in the microtubule (MT) polymerization rate, the centering time saturates for large polymerization rates. This is due to two competing factors: a larger polymerization rate leads to a larger asymmetry in the numbers of force-generating vesicles but also leads to the microtubules touching the far cortical side sooner. Experiments show that centering is completed before the microtubules touch the far cortical side. For all polymerization rates shown except 250 nm/s, the microtubules touch the far cortical side before the centrosome reaches its central position. An average microtubule polymerization rate of 250 nm/s was reported for Xenopus laevis embryos. The dashed line delineates the centering time experimentally observed. (calculation parameters: 100 microtubules; vesicle diameter: 100 nm (red squares), 200 nm (black circles); viscosity ratio = 5; 2 vesicles/µm; 2 µm/s vesicle velocity).
Figure 6.
Using centrosome dynamics to study cytosolic loading of molecular motors.
(A)A schematic sketch of a convex-up force-velocity (F–v) relation for a molecular motor shows that the velocity of the motor decreases only slightly up to an opposing load of about one half its stall force then decreases precipitously. A motor hauling a cargo will experience an opposing load from cytosolic drag that determines its speed at the intersection of the load line (dashed) and the F–v curve. When the aster starts moving towards the center, the minus-end motors on the far cortical side will experience a smaller load and speed up (vvR) while those on the other side will slow down (vvL) (as indicated by the two load lines). For slightly loaded motors, the motor speeds on either side of the centrosome will not differ appreciably, while they will diverge significantly for highly loaded motors. (B) Molecular motors with a concave-up force velocity curve will result in slightly-loaded motors exhibiting a large difference in cargo velocity on either side of the centrosome. Measuring cargo velocities moving along centering aster microtubules can help understand the loading state of the motors if their force-velocity relation is known.