Cargo Transport by Cytoplasmic Dynein Can Center Embryonic Centrosomes

To complete meiosis II in animal cells, the male DNA material needs to meet the female DNA material contained in the female pronucleus at the egg center, but it is not known how the male pronucleus, deposited by the sperm at the periphery of the cell, finds the cell center in large eggs. Pronucleus centering is an active process that appears to involve microtubules and molecular motors. For small and medium-sized cells, the force required to move the centrosome can arise from either microtubule pushing on the cortex, or cortically-attached dynein pulling on microtubules. However, in large cells, such as the fertilized Xenopus laevis embryo, where microtubules are too long to support pushing forces or they do not reach all boundaries before centrosome centering begins, a different force generating mechanism must exist. Here, we present a centrosome positioning model in which the cytosolic drag experienced by cargoes hauled by cytoplasmic dynein on the sperm aster microtubules can move the centrosome towards the cell’s center. We find that small, fast cargoes (diameter ∼100 nm, cargo velocity ∼2 µm/s) are sufficient to move the centrosome in the geometry of the Xenopus laevis embryo within the experimentally observed length and time scales.


Force-generating equations
There are two ways to find the force exerted on a MT by molecular motors: 1) directly use the force-velocity curve for each motor to determine the force corresponding to its velocity; and 2) indirectly calculating the force by finding the drag force the cargo experiences as it is hauled through the cytoplasm; this force will be equal to the force the motors apply to move it (see the main text for more detail). Previous attempts at modeling centrosome centering via cytoplasmically moving cargoes have used the force-velocity curve for a single motor to calculate the force on a given MT (1, 3). This assumption is incorrect since it is well known that individual cargoes in vivo are typically hauled by multiple motors (4,5). As discussed further below, assuming single motors move cargoes led to the inconsistent conclusion that very slow and large cargoes are needed. In the work of Kimura and Onami, the physical equations used to account for the force-velocity curve lead to physically unreasonable motor behavior. In the following we discuss their model and its implications.
In order to calculate the force exerted by each cargo, the authors defined the following system of reference: A microtubule (MT) was defined by a direction vector û that points from the minusend towards the plus-end. Thus, MTs pointing towards the near cortical side will have at least one component of their direction vector pointing in opposite direction to those MTs pointing towards the far cortical side. The motor speed was calculated by the dot product between the direction vector of the MT in which the motor is moving and the velocity vector of the pronucleus. Finally, the force exerted by the motor on the MT was calculated by choosing between 3 possible states depending on the value of the motor speed: The first force value states that when . In C. elegans, the male pronucleus velocity is about 250nm/s. Thus, the model requires that cargoes move at velocities smaller than 250 nm/s to generate the centering forces.
Cargo velocities have been measured in a plethora of systems and in a large number of these (including C. elegans), cargo velocities exceed 1µm/s. Indeed, in experiments performed by the same authors, centering was attributed to forces mediated by fast cargoes in the C. elegans embryo (2). To our knowledge, our work presented in the main text is the only one that demonstrates that fast moving cargoes (> ~1µm/s) are required to generate sufficiently large centering forces to position the centrosome within the experimentally observed length and time scales. This was only possible using a different approach than previously attempted: considering the drag forces on the cargoes rather than the force-velocity curve of the cargoes.

The shape of the force-velocity curve and the number of active motors
Force-velocity curves for molecular motors have been reported to have various shapes (concave up, concave down, linear, etc. (6)(7)(8)(9)). Previous works have assumed a linear F-v curve, under the argument that this shape is representative enough of the behavior of a motor (mainly that it slows down with increasing opposing force). Furthermore, they use this F-v curve to model the velocity and/or the force a motor transmits to the microtubule. Although using the F-v curve to model motor behavior is in principle correct, it can lead to underestimation of the velocity at which a given cargo moves at if used incorrectly. For example, it has been shown that many intracellular cargoes are hauled by multiple copies of molecular motors, and the load the cargo faces is distributed over all the active motors at any given time (5,10). In this case, Fv curves would need to be scaled up or down according to the number of motors active on each cargo. Previous works did not account for this, and in essence are one-motor models thus leading to the underestimation of the velocity the cargoes move at (1, 3). Our work circumvents this pitfall by focusing instead on the behavior of the cargo directly, and not that of the motors.
Regardless of the number of active motors on a given cargo, the drag force experienced by the latter is proportional to its size, velocity and cytoplasmic viscosity. Since this force is provided by all the motors active on the cargo, the force transmitted to the microtubule is identical to the drag force on the cargo. This approach does not require making assumptions about the properties of the motors, and instead allows us to use experimentally observed values for cargo velocities to test whether small, fast moving cargoes can generate sufficiently large forces to center the pronucleus.