on the distance between the attachments. The growth will occur between the barbed
end and the bound formin, but the detail of how the end of the growing filament was
pushed remains to be clarified. Nevertheless, the magnitude of the forces are consistent with the theoretically predicted values.
7.25 Generation of the Polymerization Force
by Microtubule
Microtubule is much more suitable for the direct measurement of the polymerization
force, because, as described previously, it is far more rigid than an actin filament.
Dogterom and Yurke [211] measured the force exerted by a single polymerizing
microtubule: they allowed microtubule to polymerize in a micro-fabricated narrow
glass chamber (30 μm long and 1 μm wide). They attached polymerization nuclei at
the floor of the chamber to fasten the microtubule and observed the growth by
differential interference contrast microscopy. After two ends of the growing microtubule hit the wall of the chamber, the microtubule started bending, suggesting that
the elongating microtubule pushed against the wall of the chamber and was bent by
the reaction force from the wall. The mechanical properties of microtubule have been
measured [212], and the force exerted on the growing microtubule could be calculated from the arc-shape of the microtubule. They also estimated the velocity of
elongation from the optical micrograph of an individual microtubule and found that
the elongation velocity monotonously decreased with the increase in the force.
To obtain the stall force from the force-velocity curve, Dogterom and Yurke
attempted the curve-fitting for two limiting cases with the equation shown in the last
paragraph in Sect. 7.21.1. In one case, the parameter, t, was set to be zero, which
yielded rather unrealistic k + , k - values and almost linear decrease of velocity with
increasing force was obtained, which could not reproduce the experimental result. In
the other case, the parameter, t, was set to be unity, which resulted in more realistic
the k + value, but the k - value turned out to be negative, and hence, they could not
estimate the stall force; the actual growth velocity rapidly approached zero around
4 pN force [213].
In a subsequent study van Doorn et al. [214] attempted to estimate F stall . From
thermodynamic argument; they derived an expression for F stall : F stall ¼ N(k B T/d)ln
(k + /k - ), where N represents the number of protofilaments in a microtubule (¼ 13),
which bear the load. Physically, this means that the work needed to displace the load,
Fd, is distributed over N protofilaments. With a discrete version of the thermal
ratchet model [208], they obtained the stall force between 9.2 pN and 18.5 pN,
depending to the choice of k + and k - values.
Kolomeisky and Fisher [213] pointed out that the number of protofilament, over
which the load is distributed, may not be 13 and argued that the work of displacing
the load, F, is F, where z represents the shortest distance of the tip from the
obstacle, d, the size of heterodimer and < > denotes the ensemble average. Because
7.25 Generation of the Polymerization Force by Microtubule
151
end and the bound formin, but the detail of how the end of the growing filament was
pushed remains to be clarified. Nevertheless, the magnitude of the forces are consistent with the theoretically predicted values.
7.25 Generation of the Polymerization Force
by Microtubule
Microtubule is much more suitable for the direct measurement of the polymerization
force, because, as described previously, it is far more rigid than an actin filament.
Dogterom and Yurke [211] measured the force exerted by a single polymerizing
microtubule: they allowed microtubule to polymerize in a micro-fabricated narrow
glass chamber (30 μm long and 1 μm wide). They attached polymerization nuclei at
the floor of the chamber to fasten the microtubule and observed the growth by
differential interference contrast microscopy. After two ends of the growing microtubule hit the wall of the chamber, the microtubule started bending, suggesting that
the elongating microtubule pushed against the wall of the chamber and was bent by
the reaction force from the wall. The mechanical properties of microtubule have been
measured [212], and the force exerted on the growing microtubule could be calculated from the arc-shape of the microtubule. They also estimated the velocity of
elongation from the optical micrograph of an individual microtubule and found that
the elongation velocity monotonously decreased with the increase in the force.
To obtain the stall force from the force-velocity curve, Dogterom and Yurke
attempted the curve-fitting for two limiting cases with the equation shown in the last
paragraph in Sect. 7.21.1. In one case, the parameter, t, was set to be zero, which
yielded rather unrealistic k + , k - values and almost linear decrease of velocity with
increasing force was obtained, which could not reproduce the experimental result. In
the other case, the parameter, t, was set to be unity, which resulted in more realistic
the k + value, but the k - value turned out to be negative, and hence, they could not
estimate the stall force; the actual growth velocity rapidly approached zero around
4 pN force [213].
In a subsequent study van Doorn et al. [214] attempted to estimate F stall . From
thermodynamic argument; they derived an expression for F stall : F stall ¼ N(k B T/d)ln
(k + /k - ), where N represents the number of protofilaments in a microtubule (¼ 13),
which bear the load. Physically, this means that the work needed to displace the load,
Fd, is distributed over N protofilaments. With a discrete version of the thermal
ratchet model [208], they obtained the stall force between 9.2 pN and 18.5 pN,
depending to the choice of k + and k - values.
Kolomeisky and Fisher [213] pointed out that the number of protofilament, over
which the load is distributed, may not be 13 and argued that the work of displacing
the load, F, is F
obstacle, d, the size of heterodimer and < > denotes the ensemble average. Because
7.25 Generation of the Polymerization Force by Microtubule
151