Questions
Equations 2-10 in Lecture 5 are used to calculate the length of the spindle at steady-state.
(a) Explain why at steady-state,
V_{poly }= V_{depol}
and use this and Equations 2-8 to derive the spindle length, S, and the length of antiparallel overlap, L (Equations 9 and 10).
(b) Using values for the spindle parameters shown in the table below (Goshima et al) calculate the minimum sliding motor activity (value of α) which gives a stable spindle.
V_{poly} |
0.06 µm s^{-1} |
k_{B}T |
4.2 x 10^{-21} J = 4.2 × 10^{-3} pN µm |
N |
100 |
S_{0} |
10 µm |
V_{sliding,max} |
0.1 µm s^{-1} |
V_{d,0} |
0.01 µm s^{-1} |
V_{dep,max} |
0.06 µm s^{-1} |
α |
100 pN µm^{-1} |
β |
80 pN µm^{-1} |
F_{kt,0} |
100 pN |
δ |
4 nm = 4 × 10^{-3} µm |
2 A 50 nm vesicle is produced outside the nucleus, containing cargo required at the cell periphery, 8 microns away. The cytoplasm of the cell has viscosity 3 × 10^{-2} Pa s, and the cell is maintained at a temperature of 37 °C.
(a) Ignoring the physical constraints of the cell, calculate the expected time required for the vesicle to reach its destination by thermal diffusion. Use the Stokes-Einstein equation for the passive diffusion coefficient:
(b) The vesicle is instead trafficked to the cell periphery by molecular motor complexes along unpolarised arrays of microtubules (so that there is no preferred direction of trafficking, towards and away from cell periphery are equally likely). Estimate the enhancement in the diffusion coefficient, assuming that the motor complexes have an average run duration of 1s and the unbound time is negligible. The motor complexes have processive velocity 0.8 μm ^{-1}.
(c) What advantage does polarised transport have over unpolarised trafficking?
Answers
1 The polymerisation and depolymerisation rates must be equal in the steady-state since the size of the spindle and the amount of incorporated tubulin are constant over time. For this to be true, the rate at which tubulin is added to the spindle, proportional to V_{poly}, must equal the rate of tubulin removal, which is proportional to V_{dep}.
Rearrange Equation 8:
In steady state, Equation 4 is set equal to zero, hence:
F_{tension} = F_{sliding} – F_{kt,0}
Substitute from Equations 6 and 8 into Equation 4:
and substitute from Equation 8 into Equation 5 to obtain Equations 9 and 10 as required:
(b) Spindle length is calculated from Equation 9:
The minimum sliding motor activity is given when L = S; a lower motor activity would require an antiparallel overlap region greater than the length of the spindle, which is not possible.
Set L = 11.1 in Equation 10:
2 (a) The value of the diffusion coefficient is calculated as:
Taking the case of 1D diffusion:
(b) For the random walk, each step is now the run length of the motor complex. Therefore, the step distance is:
d =vt_{i}
N steps are made in time Nt_{i. } The stochasticity of a random walk means that every run is different, however average properties can be calculated. For a random walk, net distance travelled, L, is given by:
Substitute into the 1D diffusion coefficient equation above:
Thus the active diffusion coefficient represents an order of magnitude enhancement over the passive coefficient. It depends on the motor velocity and run duration, rather than the size of the particle being trafficked or the viscosity (though the motor stepping velocity and unbinding probability may be influenced by the viscous drag force created by the particle; the force-dependence of motor transitions is neglected here).
Derivation of random walk equation:
(c) The case of unpolarised trafficking rapidly distributes cargo across the cell but cannot discriminate between regions in the cell. In polarised trafficking the plus- and minus-ends of microtubules distinguish the cell centre and the cell periphery, and so by having certain cargo trafficked specifically by plus-end directed motor complexes and some by minus-end directed complexes, concentration gradients of the cargo molecules can be set up within the cell. These gradients are essential in defining the polarity of cells, for example in tissues such as the epithelium, where cells must establish and maintain a layered geometry for correct function.